robench-2024b
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<|MaskedSetence|> The commutation relations established in Eqs. <|MaskedSetence|> The internal time and energy do not form a conjugate pair. <|MaskedSetence|> The internal time and temporal oscillation can be treated as self-adjoint operators without contradicting Pauli’s theorem.
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**A**: Therefore, there is no restriction on the internal time spectrum, albeit the system’s Hamiltonian is bounded from below.
**B**: The introduction of the internal time and temporal oscillation operators in Section 3.4 have no conflict with Pauli’s theorem.
**C**: (39) and (42) do not involve energy.
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In view of recent experimental progress with ultra-cold atoms forming a Bose-Einstein condensate, double-well systems are one of the most commonly exploited schemes studied Andrews ; Smerzi ; Milburn ; Menotti ; Meier ; Shin ; Albiez ; Levy ; Salgueiro ; Simon ; Liu . Typically, in this context one assumes that weakly interacting bosons occupying different wells can be described with two independent single-particle orbitals and that the dynamics is governed by two mechanisms: contact two-body interactions acting locally and single-particle tunneling between wells. Then, in the mean-field limit, a corresponding Gross-Pitaevskii equation is introduced and numerically solved for different initial conditions Raghavan ; Ostrovskaya ; Ananikin . <|MaskedSetence|> Although the validity of these simplified two-mode models was confirmed experimentally for weak interactions between particles, they were extended beyond the range of their applicability and adopted for strongly interacting systems, i.e. <|MaskedSetence|> For example, it was shown that for initially imbalanced occupations the dynamics is heavily affected by strong interactions DuttaS . <|MaskedSetence|>
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**A**: in situations when the local interaction energy is much larger than the single-particle tunneling energy.
**B**: Unfortunately, the validity of the model used was not discussed and its predictions were not compared with the exact dynamics governed by a general model.
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**C**: Generalized two-mode models, taking into account additional terms originating from long-range interactions or occupation-dependent tunnelings, are also considered in the literature and relevant corrections to the dynamics are studied Lahaye ; Adhikari ; Bruno .
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7. Realizing Harmonic Oscillators With Coupled Supersymmetry
As previously noted, the quantum mechanical harmonic oscillator is a specific instance of a coupled supersymmetry, albeit a somewhat trivial case in which the two coupled SUSY equations are identical. <|MaskedSetence|> <|MaskedSetence|> they satisfy the same Lie algebra and by virtue of Stone-von Neumann, may be realized in some way as harmonic oscillators. <|MaskedSetence|>
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**A**: If one takes γ=−δ𝛾𝛿\gamma=-\deltaitalic_γ = - italic_δ, then the coupled SUSY equations become
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**B**: This is not the only manner in which the two are connected.
**C**: Indeed, a special class of coupled SUSYs may be realized as harmonic oscillator-like systems, i.e.
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(7). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Then by (7) it is
unstable again but, quite surprisingly R→−Gm2/E→𝑅𝐺superscript𝑚2𝐸R\rightarrow-Gm^{2}/Eitalic_R → - italic_G italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_E i.e. it
would undergo a finite expansion such that the ball reaches a finite.
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**A**: The fourth possibility is that the ball
is gravitationally bounded i.e.
**B**: −Gm2/2E<Rinitial<−Gm2/E𝐺superscript𝑚22𝐸subscript𝑅initial𝐺superscript𝑚2𝐸-Gm^{2}/2E<R_{\rm initial}<-Gm^{2}/E- italic_G italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_E < italic_R start_POSTSUBSCRIPT roman_initial end_POSTSUBSCRIPT < - italic_G italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_E.
**C**: E<0𝐸0E<0italic_E < 0 but is not small
i.e.
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<|MaskedSetence|> However, it is still possible to reconstruct a boundary theory by carefully restricting the doubled coordinates in the bulk region close to the boundary. There are several options by which such a restriction can be performed [43], [44], [45], each coming with advantages and disadvantages. The most important aspect is to keep the desirable effects of T-duality in the limit where the doubled coordinates become irrelevant. It is currently not clear what precisely the boundary encoding map associated to the mixing terms found in the bulk due to the extended gauge symmetry. <|MaskedSetence|> <|MaskedSetence|>
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**A**: Also, effects associated to the extended nature of the strings, from where double field theory extracts its stranger features, seem to be related to mixing of operators in the boundary and to a left-right symmetry which should not otherwise be present [46].
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**B**: Heuristically speaking, it is possible to imagine that stringy modes encoded in the bulk double field theory may have the effect of violating associativity of the operators in the boundary.
**C**:
fails, in the sense that there can be no such simple decomposition as the group associated to the bulk space will include also a de-Sitter component.
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We construct the exterior spacetime in a way that allows for the treatment of the phenomenon in a clear conceptual and technical setting. As explained in Section III (see also Appendix B) the new construction of the exterior spacetime given here is adapted to the needs of the calculation and allows for a more clear conceptual setup. The relation to older constructions is explained in Appendices C, D,
E.
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**A**: In Section III we review and improve the setup for the exterior spacetime, which describes the part of the spacetime well approximated by classical general relativity.
**B**:
The paper is organized as follows.
**C**: Before discussing the black hole case, in Section II we review the case of a particle tunneling through a potential wall in non relativistic quantum mechanics and discuss the timescales involved in this process.
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<|MaskedSetence|> Another important ingredient is the Uniformization Theorem for the stack of 𝒢𝒢\mathscr{G}script_G-torsors on the parahoric Bruhat-Tits group scheme 𝒢𝒢\mathscr{G}script_G due to Heinloth [He]; in fact, its parabolic analogue (cf. <|MaskedSetence|> Finally, yet another ingredient is the splitting of the central extension of the twisted loop group G(𝔻q×)Γq𝐺superscriptsuperscriptsubscript𝔻𝑞subscriptΓ𝑞G(\mathbb{D}_{q}^{\times})^{\Gamma_{q}}italic_G ( blackboard_D start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT over
Ξ=MorΓ(Σ\Γ⋅q,G)ΞsubscriptMorΓ⋅\ΣΓ𝑞𝐺\Xi={\rm Mor}_{\Gamma}(\Sigma\backslash\Gamma\cdot q,G)roman_Ξ = roman_Mor start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT ( roman_Σ \ roman_Γ ⋅ italic_q , italic_G ) and the reducedness and the irreducibility of ΞΞ\Xiroman_Ξ (cf. <|MaskedSetence|> 𝔻qsubscript𝔻𝑞\mathbb{D}_{q}blackboard_D start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT) is the punctured formal disc (resp. formal disc) around q𝑞qitalic_q in ΣΣ\Sigmaroman_Σ..
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**A**: Theorem 10.7 and Corollary 11.5), where q𝑞qitalic_q is a point in ΣΣ\Sigmaroman_Σ and 𝔻q×superscriptsubscript𝔻𝑞\mathbb{D}_{q}^{\times}blackboard_D start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT (resp.
**B**: Theorem 11.3).
**C**:
One of the main ingredients in the proof of this theorem is the connectedness of the ind-group MorΓ(Σ∗,G)subscriptMorΓsuperscriptΣ𝐺{\rm Mor}_{\Gamma}(\Sigma^{*},G)roman_Mor start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT ( roman_Σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_G ) consisting of ΓΓ\Gammaroman_Γ-equivariant morphisms from Σ∗superscriptΣ\Sigma^{*}roman_Σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT to G𝐺Gitalic_G (cf. Theorem 9.5), where Σ∗superscriptΣ\Sigma^{*}roman_Σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is a ΓΓ\Gammaroman_Γ-stable affine open subset of ΣΣ\Sigmaroman_Σ.
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The second question regards how a subsystem of anyons in a many-anyon system may be characterized by certain good quantum numbers. This is an important question because often (e.g., in topological quantum computing) we may be concerned about only the entanglement between such subsystems and ignore what is inside each subsystem. Recall that the Hilbert space of a fermionic or bosonic system is taken as a Fock space, which is the tensor product of the local Hilbert spaces of single-particle states. <|MaskedSetence|> If we can find the complete set of observables (giving rise to good quantum numbers) of the system, we may take their eigenvectors to form the basis of the Hilbert space. As to be seen in this paper, the clarification of the physical meaning of pseudo-species turns out to be the answer to this question too. <|MaskedSetence|> The dependency coefficients are called the statistics parameters. In terms of these extended statistics parameters, we extend Wu’s formula for statistical weight, which lays the foundation of the statistical mechanics of anyons on systems with gapped boundaries. Clearly, such exclusion statistics put the boundary components and bulk anyons on an equal footing. <|MaskedSetence|>
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**A**: We shall dub this kind of exclusion statistics by extended anyonic exclusion statistics..
**B**: To study the Hilbert space structure of a system of non-Abelian anyons, however, there demands a new formulation of the basis, as many-anyon states do not have an obvious Fock space analogy formed by the tensor product of local single-anyon Hilbert spaces.
**C**: More precisely, the good quantum numbers of a subsystem are the eigenvalues of the observables of the relevant pseudo-species.
In this paper, we extend Haldane’s generalized exclusion principle to systems with gapped boundaries, by proposing that the number of available single particle states for additional particles linearly and mutually depends on the number of (every species of) existing anyons and the number of (every boundary type of) existing gapped boundaries.
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<|MaskedSetence|> The paper is organized as follows. <|MaskedSetence|> In Section 3, we give a concise example for the MSR of the mixed-spin (1/2,1/2)1212(1/2,1/2)( 1 / 2 , 1 / 2 ) systems. <|MaskedSetence|> A brief discussion and summary are given in Section 5.
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**A**:
Our study provides an intuitive perspective of a two-spin (s𝑠sitalic_s and 1/2121/21 / 2) system and unveils the intrinsic property of the two-spin system on a Bloch sphere, which shall deepen our comprehension of the spin-(s,1/2)𝑠12(s,1/2)( italic_s , 1 / 2 ) system.
**B**: In Section 2, we study the fundamental theory of the MSR to describe an arbitrary pure state of spin-(s,1/2)𝑠12(s,1/2)( italic_s , 1 / 2 ), through coupling bases.
**C**: In Section 4, we show more applications of our method in the mixed-spin (s,1/2)𝑠12(s,1/2)( italic_s , 1 / 2 ) systems.
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point in the domain (i.e.,formulae-sequence𝑖𝑒i.e.,italic_i . italic_e . <|MaskedSetence|> This property also hold for the
pyramid side functions, i.e.,formulae-sequence𝑖𝑒i.e.,italic_i . <|MaskedSetence|> <|MaskedSetence|>
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**A**: italic_e .
**B**: , Σn=1Nnϕn(𝐫)=Σn=1Nnψn(𝐫)=1subscript𝑁𝑛𝑛1Σsubscriptitalic-ϕ𝑛𝐫subscript𝑁𝑛𝑛1Σsubscript𝜓𝑛𝐫1\overset{N_{n}}{\underset{n=1}{\Sigma}}\phi_{n}(\textbf{$\mathbf{r}$})=%
\overset{N_{n}}{\underset{n=1}{\Sigma}}\psi_{n}(\textbf{$\mathbf{r}$})=1start_OVERACCENT italic_N start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_OVERACCENT start_ARG start_UNDERACCENT italic_n = 1 end_UNDERACCENT start_ARG roman_Σ end_ARG end_ARG italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_r ) = start_OVERACCENT italic_N start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_OVERACCENT start_ARG start_UNDERACCENT italic_n = 1 end_UNDERACCENT start_ARG roman_Σ end_ARG end_ARG italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( bold_r ) = 1.
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**C**: , at non-nodal locations, as well as at
nodal locations), is equal to one.
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Selection 4
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Since our identification will use a certain set of (framed) wild harmonic bundles, we remark that this set does not match any of the usual wild moduli spaces ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Furthermore, we will have the additional data of a “framing”. Hence our moduli space must a priori be different from the usual moduli spaces of wild harmonic bundles.
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**A**: In the usual story of moduli spaces of wild harmonic bundles over a punctured compact Riemann surface, one fixes the singular part of the Higgs field and the parabolic structure at the punctures.
**B**: On the other hand, in our set of wild harmonic bundles we will allow the simple pole of the Higgs field and the parabolic structure to vary.
**C**: Under certain stability conditions, one obtains moduli spaces of these objects, with the natural hyperkähler metric [BB04].
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<|MaskedSetence|> <|MaskedSetence|> Next, all peaks in each power spectra were identified using first and second derivatives (in the frequency dimension) of the power spectral data. Power spectra where the largest peak was less than 8 dB above the noise were excluded from consideration. Power spectra where the largest peak was less than 0.5 fcesubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT or more than 1.1 fcesubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT were also excluded from consideration. Finally, a peak was identified as a potential near-fcesubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT wave when the amplitude of the peak was at least 8 dB larger than the power of the first spectral point on either side of the peak where the derivative of the power spectra (in the frequency dimension) changed sign. This algorithm was applied to the V12subscript𝑉12V_{12}italic_V start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT onboard AC spectral data only, as those data are available for both the first and second perihelion passes. In all, 6.7 hours (∼similar-to\sim∼27,000 individual wave spectra) of data were found to contain near-fcesubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT waves during solar encounter 1 and 9.12 hours (∼similar-to\sim∼37,000 individual wave spectra) during solar encounter 2.
For each spectra identified as containing near-fcesubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT waves, the angle between the solar wind magnetic field vector and the radial direction (θBrsubscript𝜃𝐵𝑟\theta_{Br}italic_θ start_POSTSUBSCRIPT italic_B italic_r end_POSTSUBSCRIPT) was calculated as the mean of the angle between the solar wind magnetic field vector and the radial direction over the NYs corresponding to each power spectral measurement. <|MaskedSetence|>
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**A**: The sense of the radial field (sunward or anti-sunward) was not retained, such that the range of θBrsubscript𝜃𝐵𝑟\theta_{Br}italic_θ start_POSTSUBSCRIPT italic_B italic_r end_POSTSUBSCRIPT is 0∘<θBr<90∘superscript0subscript𝜃𝐵𝑟superscript900^{\circ}<\theta_{Br}<90^{\circ}0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT < italic_θ start_POSTSUBSCRIPT italic_B italic_r end_POSTSUBSCRIPT < 90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT.
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**B**: To examine the degree to which near-fcesubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT waves are observed in association with near-radial solar wind magnetic field, an automated detection algorithm was applied to solar encounter 1 and 2 data to identify wave intervals.
**C**: First, each power spectra was converted to dB relative to background, where the background for each spectrum is defined as the median wave power at each frequency during the day when the observation was made.
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<|MaskedSetence|> only zero in this case) for a single choice of
parameters. <|MaskedSetence|> Alternatively, in the case s2=−12subscript𝑠212s_{2}=-\tfrac{1}{2}italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG 2 end_ARG from
equation −4V22−(1−2V3)≥04superscriptsubscript𝑉2212subscript𝑉30-4V_{2}^{2}-(1-2V_{3})\geq 0- 4 italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 - 2 italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ≥ 0 follows V2=0,V3=1/2formulae-sequencesubscript𝑉20subscript𝑉312V_{2}=0,V_{3}=1/2italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 , italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 1 / 2. The both cases yield an isolated root ±12(1−𝐞3+𝐞12+𝐞123)plus-or-minus121subscript𝐞3subscript𝐞12subscript𝐞123\pm\frac{1}{2}(1-\mathbf{e}_{3}+\mathbf{e}_{12}+\mathbf{e}_{123})± divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 - bold_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT 123 end_POSTSUBSCRIPT ). <|MaskedSetence|>
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**A**: made non-negative (i.e.
**B**: In particular, in the case s1=12subscript𝑠112s_{1}=\tfrac{1}{2}italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG, the requirement −4V22−(1+2V3)2≥04superscriptsubscript𝑉22superscript12subscript𝑉320-4V_{2}^{2}-(1+2V_{3})^{2}\geq 0- 4 italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 + 2 italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 0 yields V2=0,V3=−1/2formulae-sequencesubscript𝑉20subscript𝑉312V_{2}=0,V_{3}=-1/2italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 , italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = - 1 / 2.
**C**: Therefore, in this
algebra in fact there exist only isolated real square root of
𝖡=−𝐞3+𝐞12𝖡subscript𝐞3subscript𝐞12\mathsf{B}=-\mathbf{e}_{3}+\mathbf{e}_{12}sansserif_B = - bold_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT..
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<|MaskedSetence|> For each batch 1111 through 10101010, a feedforward model was trained on that batch. Training of a new model was repeated 30 times on each batch. <|MaskedSetence|> Networks trained on batches 2, 4, 6, and 8 were plotted (Fig. 1B). The accuracy data was reformulated as a function of the distance between the train batch and the test batch (Fig. <|MaskedSetence|> As expected, the accuracy decreases with the time gap between training and testing, demonstrating that, indeed, sensor drift progresses over time..
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**A**: 1C).
**B**: The accuracy of all classifiers were evaluated on every batch.
**C**:
IV-A Drift demonstration
First, the effect of sensor drift on classification accuracy is demonstrated using classifiers trained on a single batch.
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It is well known that there are close relations between entanglement and spin squeezing, and a lot of effort has been devoted to unveiling it Ulam-Orgikh and Kitagawa (2001); Jin and Kim (2007); Jafarpour and Akhound (2008); Messikh et al. (2003); Wang (2004); Yin et al. (2010).
Because spin squeezing is relatively easy to be generated and measured experimentally Genes et al. <|MaskedSetence|> (2008); Takano et al. <|MaskedSetence|> (1992, 1994); Cronin et al. <|MaskedSetence|> (1996); Döring et al. (2010), as well as in making high-precision atomic clocks Sørensen and Mølmer (1999); André et al. (2004); Meiser et al. (2008) and gravitational-wave interferometers Walls and Zoller (1981); Goda et al. (2008), etc.
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**A**: (2009); Bollinger et al.
**B**: (2009), spin-squeezing parameters are promising candidates as measures of many-body correlations.
Improving the precision of measurements is another important application of spin squeezing.
For example, spin squeezing plays an important role in Ramsey spectroscopy Wineland et al.
**C**: (2003); Fernholz et al.
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<|MaskedSetence|> Acknowledgements
During the writing of the paper, I was supported by the starter grant “Categorified Donaldson–Thomas theory” No. <|MaskedSetence|> I was also supported by a Royal Society university research fellowship. <|MaskedSetence|> Finally, I offer my heartfelt gratitude to Paul, Sophia, Sacha, Kristin and Nina, for their help and support throughout the writing of this paper.
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**A**: 759967 of the European Research Council.
**B**: I would like to thank Andrei Okounkov and Olivier Schiffmann for helpful conversations, Tristan Bozec for patiently explaining his work on crystals to me, Lucien Hennecart and Shivang Jindal for helpful comments regarding an earlier version of the paper, and an anonymous referee for a careful reading of the paper and many helpful suggestions.
**C**: 1.7.
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<|MaskedSetence|> <|MaskedSetence|> (2010); Bolognesi et al. <|MaskedSetence|> (2016); Berge et al. (2015).
We use this decay to validate the methodology and as an example of application of statistical methods to distinguish between different spin-parity hypotheses.
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**A**: The third study-case is the decay of Higgs to two Z𝑍Zitalic_Z bosons and their subsequent
decay into four leptons.
This final state has been extensively studied in the past as the golden decay mode of the Higgs boson and in searches for physics beyond the Standard Model Keung et al.
**B**: (2008); Gao et al.
**C**: (2012); Modak et al.
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To our knowledge, Temperley-Lieb algebras have not been studied in the representation stability literature, or within the broader context of representation stability and FIFI\operatorname{FI}roman_FI-modules. It appears that much of the work in representation stability has focussed on algebraic objects which are either close to symmetric groups [5] [14] [8] (Wilson, Putman, Sam, Gunturkun, Snowden ) or are close to Lie groups [14] [17] (Sam, Snowden, Putman). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Thus, representation stability with respect to a chain of diagrammatically defined algebras is not considered in [1] (Barter, Entova-Aizenbud, Heidersdorf).
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**A**: Diagrammatically defined chains of algebras appear to have not been considered as objects whose representation category can be studied through the lens of representation stability.
**B**: The chain with respect to which one is considering representation stability there is of course still the chain of symmetric groups.
**C**: Diagrammatics and representation stability have, however, been uttered in the same breadth, but in a different sense: in [1] (Barter, Entova-Aizenbud, Heidersdorf) the authors produce a functor from the category of FIFI\operatorname{FI}roman_FI-modules modulo finite length FIFI\operatorname{FI}roman_FI-modules to the abelian envelope of the Deligne category.
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Lemaître redshift formula, viz. <|MaskedSetence|> We believe that
this is an oversight, however. <|MaskedSetence|> One such type of VSL exists in the form of
the velocity of light being dependent on the scale factor, e.g. <|MaskedSetence|>
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**A**: 1+z=a−11𝑧superscript𝑎11+z=a^{-1}1 + italic_z = italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT.
**B**: There are certain types of VSL in which
the classic Lemaître redshift formula is no longer applicable,
warranting a revision.
**C**: c∝a−ζproportional-to𝑐superscript𝑎𝜁c\propto a^{-\zeta}italic_c ∝ italic_a start_POSTSUPERSCRIPT - italic_ζ end_POSTSUPERSCRIPT,.
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For N𝑁Nitalic_N particles in each shot, the SQL, also known as the shot-noise limit, can be surpassed by using quantum effects such as entanglement Bollinger et al. (1996); Monz et al. (2011) and squeezing Muessel et al. (2015), reaching the so-called Heisenberg limit (HL), in which the sensitivity exceeds the SQL by 1/N1𝑁1/\sqrt{N}1 / square-root start_ARG italic_N end_ARG Bollinger et al. (1996); Holland and Burnett (1993); Munro et al. (2002).
Many schemes have been proposed to achieve the SQL, such as quantum state transfer from light to atoms Agarwal and Puri (1990); Kuzmich et al. (1997); Moore et al. (1999), quantum nondemolition measurement Appel et al. (2009); Kuzmich et al. <|MaskedSetence|> (2010); Hammerer et al. (2010), one-axis twisting Schleier-Smith et al. (2010); Kitagawa and Ueda (1993); Sørensen and Mølmer (2001); Haine et al. (2014), two-axis countertwisting Kitagawa and Ueda (1993); Ma and Wang (2009), twist-and-turn squeezing Muessel et al. (2015); Law et al. (2001), spin changing collisions Lücke et al. (2011); Duan et al. (2000); Pu and Meystre (2000); Nolan et al. <|MaskedSetence|> <|MaskedSetence|> (2012).
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**A**: (1998); Louchet-Chauvet et al.
**B**: (2018).
Furthermore, the concept of interaction-based readout resolves the dilemma that the states prepared via these schemes require the low-noise detection in order to see significant quantum enhancement Haine (2018); Demkowicz-Dobrzański et al.
**C**: (2016), and adiabatically scanning through a quantum phase transition Lee (2006); Huang et al.
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Here we develop a unified formalism for quantum computation with unitary oracles, which captures all of the above generalizations simultaneously. Thus, we are able to derive the strongest-yet no-go theorem for the if-clause task [36]; the task to implement controlled U𝑈Uitalic_U given an oracle U𝑈Uitalic_U, the quantum version of the conditional statement essential to classical computation. We show that an if clause of any query complexity is impossible, thus losing what could be a fundamental building block in quantum circuits. <|MaskedSetence|> <|MaskedSetence|> To further demonstrate its applicability, we prove limitations of other previously studied unitary-oracle tasks: the neutralization, fractional power, inverse and transpose tasks. The limitations hold for quantum circuits (even with postselection and relaxed causality), but not for linear optics. <|MaskedSetence|>
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**A**: Our method exploits features unique to unitary oracles; its novel topological approach adds to the few previously known lower-bound methods.
**B**: This, for example, limits the flexibility of phase estimation, which assumes the controlled U𝑈Uitalic_U building blocks.
**C**: This motivates developing implementation-dependent algorithms for oracle problems.
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<|MaskedSetence|> <|MaskedSetence|> The custom featurization procedure is laid out in the results and discussion section. Surfaces that had identical features to any other surface in the dataset (and a difference in work function between the two surfaces of <0.1absent0.1<0.1< 0.1 eV) were removed as duplicates from the dataset before training. For benchmarking purposes we used the automatminer testing suite[87] (200 features) and a conventional Coulomb matrix (topmost 5 surface atoms as input, matrix sorted by ℓ2subscriptℓ2\ell_{2}roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-norm of columns – the flattened matrix yields 25 features that are used in a random forest model).[88] For automatminer we use the “express" setting and for comparison we used the bulk unit cell and the topmost 5 atomic layers of the surface slabs as inputs. <|MaskedSetence|>
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**A**: As a baseline model we predict the work function to be the average work function regardless of the surface.
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**B**: II.3 Machine Learning Model Training
The dataset is randomly split into training and test sets (90/10 split) and the hyperparameters are optimized with a grid-search implementing 10-fold cross-validation on the training set.
**C**: Multivariate linear regression, random forest, and neural network models are set up with the scikit-learn package in Python.
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We measured the magnetic response of a small piece of the as-grown sample using an AC Magnetic Susceptometer set-up and found a diamagnetic signal below ∼6similar-toabsent6\sim 6∼ 6 K, as shown with red dots in Fig. 6. This value of the critical temperature is consistent with other Sr doped samples with x∼0.06similar-to𝑥0.06x\sim 0.06italic_x ∼ 0.06 Fujita et al. (2002); Hawthorn et al. <|MaskedSetence|> (2006). <|MaskedSetence|> A small piece of oxygenated sample was measured with a VSM in an applied field of 20 Oe. <|MaskedSetence|>
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**A**: (2003).
The sample was oxygenated at the University of Connecticut through a wet-chemical technique for several months Mohottala et al.
**B**: The rod was cut into pieces to fit it in the cryo-magnets used in the neutron experiments.
**C**: It reveals a single transition for our La1.94Sr0.06CuO4+y sample, shown in Fig. 6 in blue dots.
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<|MaskedSetence|> In a companion analysis [16], some of the present authors examined several algorithmic improvements, showing that we can in fact construct analysis-length time-domain EMRI waveforms in the Schwarzschild limit in under a second. There are two ways that the results of the present work can be incorporated into the framework of [16]. The first is to extend to Kerr inspirals. The main challenges here are due to the higher dimensional parameter space, and the need to sum over more modes, since waveforms depend on an additional harmonic frequency. <|MaskedSetence|> The framework in [16] is designed to overcome these challenges. <|MaskedSetence|>
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The framework and techniques we describe here have not been optimized, so there is much scope for efficiency gains.
**B**: Its neural network interpolation is a promising technique for dealing with the increased dimensionality, and its use of GPU-based hardware acceleration alleviates the computational burden of summing over thousands of additional (l,m,k,n)𝑙𝑚𝑘𝑛(l,m,k,n)( italic_l , italic_m , italic_k , italic_n )-modes.
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**C**: Inspirals also extend deeper into the strong field, so more modes are likely to make strong contributions to the waveform.
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Another technical route is to reform the laser-electron interaction mechanism to relax the power requirement of a high-repetition-rate seeded FEL. Several methods have been developed, such as the optical resonator [43, 44, 45, 46, 47, 48], angular dispersion enabled microbunching (ADM) [49, 50], self-modulation [51, 52] and optical-klystron HGHG [53]. In the optical resonator, the electron beam is stored in an oscillator before it is directed to generate radiation. The repetitive laser-electron interaction is able to amplify the laser pulses and achieve considerable energy modulation on the electron beam. With combination of frequency up-conversion mechanism, e.g., HGHG or EEHG, this method could further extend the radiation to shorter wavelength. <|MaskedSetence|> <|MaskedSetence|> Also, the increased thermal issues in the cavity need to be studied to secure a stable output. ADM takes use of multi-dimensional modulation of the electron beam to relax the requirement on the laser-induced energy modulation. The need for small angular divergence of the electron beam, however, makes this method more suitable for use in storage rings. <|MaskedSetence|> These schemes still need a laser-induced energy modulation at least half of the initial energy spread. And the correspondingly required dispersion to form the density modulation is much larger than that in nominal HGHG, which may cause the spectrum broadening induced by the energy chirps in the electron beam (Appendix).
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**A**: Self modulation or optical-klystron HGHG is based on pre-density modulation method.
**B**: However, the repetition rate of the radiator is non-adjustable at most time.
**C**: 1 MHz repetition rate indicates 300 m long resonator, and 0.5 MHz means 600 m long resonator.
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A little bit more than twenty years after such studies in Refs. <|MaskedSetence|> Witten:1998zw , by using the new-found AdS/CFT correspondence, as proposed in Ref. <|MaskedSetence|> Soon after Witten’s work the authors in Refs. <|MaskedSetence|> It is worthwhile to mention that within holography the thermodynamics quantities are derived from the holographic renormalization of the on-shell euclidian action or the thermodynamics potentials.
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**A**: Bekenstein:1973ur ; Hawking:1974sw the iconic work done by Witten, in Ref.
**B**: Chamblin:1999tk ; Chamblin:1999hg studied the thermodynamic associated to a charged AdS black holes in the holographic context, and then opening up a multitude of possibilities to connect string and gauge theories through various types of black hole and its thermodynamics.
**C**: Maldacena:1997re , relates the Hawking temperature achieved in a curved high-dimensional spacetime to the temperature of a super conformal Yang-Mills theory in a flat four-dimensional spacetime.
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V Discussion and Conclusion
Quantum non-locality demonstrated in EPR steering is strictly weaker than that in the Bell test. We may wonder if the above discussion of Maxwell demon applies to the case of the Bell test. The answer, unfortunately, is negative. To make it explicit, let’s consider a Bell game in which a third party, Charlie, who chooses collaboration with the demons, tries to simultaneously deceive Alice and Bob, who are separated by distant space, into believing that their received qubits are entangled. In order to achieve this goal, two demons have to be sent secretly to Alice and Bob, respectively. The operations of demons are almost the same in the case of EPR steering, i.e., they have access to measurement basis information and rotate the qubits before they are measured. The main issue, however, is how to rotate the qubits based on the basis information. In EPR steering, the demon can randomly rotate the qubit into one eigenstate of measurement basis due to the classical communication between Alice and Bob, and the demon could inform Alice of its operation. <|MaskedSetence|> One might imagine that two demons could share a table list from the beginning, guiding them on how to operate based on the obtained basis information. That would require a table list long enough to cover all the runs, which is physically impossible. <|MaskedSetence|> This can indeed be possible using the quantum circuit model. However, it is a logical circular argument because it uses the non-locality correlation of demons to replace the non-locality correlation between Alice and Bob in the Bell test. <|MaskedSetence|>
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**A**: This is not possible in the Bell test, in which classical communication is prohibited.
**B**: One may also imagine that the random generators of two demons are entangled, such that their operations are correlated.
**C**: We thus conclude that there is no Maxwell demon loophole in the Bell test.
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Acknowledgements
The authors thank Andreas Osterloh, Karol Życzkowski, Rui Perdigão and Yasser Omar for fruitful discussions and correspondence. The authors are especially grateful to Jens Siewert, Andreas Osterloh, Barbara Kraus, and Karol Życzkowski for stimulating conversations which led to formal proof of Proposition 2 while staying at Centro de Ciencias de Benasque Pedro Pascual. A.B acknowledges support from the National Science Center under grant number DEC-2015/18/A ST2/00274 and by NWO Vidi grant (Project No. <|MaskedSetence|> <|MaskedSetence|> thanks the support from Fundação para a Ciência e a Tecnologia (Portugal), namely through projects CEECIND/02474/2018 and project UIDB/50008/2020 and IT project QuantSat-PT. R.A. <|MaskedSetence|>
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**A**: acknowledges support from the Doctoral Programme in the Physics and Mathematics of Information (DP-PMI) and the Fundação para a Ciência e Tecnologia (FCT) through Grant No. PD/BD/135011/2017..
**B**: G.Q.
**C**: VI.Vidi.192.109).
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<|MaskedSetence|> It is an open theoretical question whether the same phenomenology applies and what quantitative changes are necessary in the absence of this fixed initial state, e.g., for an infection spreading into a population which itself is growing. This question also has practical relevance. In human diseases there is often a separation of time scales between the host and viral reproduction rate so that the total population can be assumed constant [10], but this is not always the case, e.g., for chronic diseases like HIV or for countries with high birth rates [7]. <|MaskedSetence|> <|MaskedSetence|> However, these studies have typically not been concerned with the impact of bacterial growth on the infection dynamics
[31]..
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**A**: Similarly, no separation of time scales applies to the inter-microbe interactions that play an essential role in our global biochemical and geochemical cycles [25], e.g., bacteria and the viruses that infect them (bacteriophages, also known as phages) typically have similar growth rates [20].
Extensive experimental and theoretical work has been conducted into the spread of bacteriophage infections through bacterial populations, typically focussed on phage ‘plaques,’ i.e., the clearings formed by bacteriophages in bacterial lawns on semi-solid media such as agar [26, 27, 28, 29, 30].
**B**: These studies have produced quantitative predictions of wave speed [26], and have highlighted the impact on the infection dynamics of effects such as the distribution of phage lysis times [28] or bacterial crowding [29].
**C**: Analysis of FKPP-like equations relies on perturbation around the fixed, unstable initial state into which the front propagates [9].
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In our work, we showed that even though all non-Gaussian states can furnish a low enough Chernoff bound, only the photon-subtracted state and the idler-photon-added state can outperform the coherent state having the same signal strength. We believe that since the performance of the coherent state heavily depends on the signal photon number [44], comparison based on the signal strength is the way to properly access the quantum advantage in the illumination protocol. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Considering local noise modeled by Gaussian distributions, we found that, unlike a noiseless scenario, if the signal transmission line equally affects both the non-Gaussian and coherent states having equal signal strength, all non-Gaussian states give a quantum advantage. Specifically, in the presence of certain critical noise values, benefits via non-Gaussian states increase with the increase of noise..
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**A**: The non-Gaussian probes cannot provide an advantage over the Gaussian TMSV state, even though they can easily outperform the coherent state probe having the same signal strength.
**B**: Thus, we have focused only on the performance of non-Gaussian states and compared their performance with respect to the coherent state probe.
**C**: We also illustrated how converting a large number of copies of the Gaussian probes into a smaller number of non-Gaussian ones can help to obtain a better quantum advantage.
In any experimental implementation, noise is inevitable, and in our work, the effects of different noisy probe states generated via different imperfections on the illumination procedure are investigated.
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It is important to note that the DM capture process in NSs differs significantly from that in the Sun, because of the extreme conditions present. Therefore, Gould’s original formalism requires substantial modification to properly account for the physics of this extreme environment. <|MaskedSetence|> Ref. [53] presented a new formalism that consistently incorporated many of the relevant physical effects. <|MaskedSetence|> The two main physical effects are (i) baryons experience strong interactions in the NS interior and (ii) the momentum transfer in DM-baryon collisions is sufficiently large that baryons cannot be treated as point particles. <|MaskedSetence|> We generalize that analysis here to consider scattering from other baryonic species in the neutron star, namely protons and hyperons. We also extend this treatment to the multiple scattering regime relevant for heavy DM, and project sensitivity limits across a wide DM mass range..
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**A**: These include: the NS internal structure [49, 53, 54], through solving the Tolman–Oppenheimer–Volkoff (TOV) equations [69, 70] coupled to an appropriate equation of state,
correct treatment of Pauli blocking for degenerate targets [49, 53, 54], relativistic kinematics [51, 52, 53, 54], gravitational focusing [39, 60, 53, 54], and multiple scattering effects for heavy DM [71, 52, 53, 37, 54].
In this work, we build on these improvements by focusing on the physics pertinent to baryonic targets in the extremely dense NS interior.
**B**: These features were first introduced in ref. [55], in the context of scattering from neutrons, where we demonstrated an impact on the capture rate of up to three orders of magnitude for heavy NSs.
**C**: Such corrections have gradually been introduced.
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<|MaskedSetence|> <|MaskedSetence|> Bradley et al., however, showed that this obstruction is removed for a limiting form of the Wahlquist solution [16], and also for every D-type metric [17], if the requirement of an asymptotically flatness is removed.
More recent work in this field exists, ”where the existence of a rank-2 generalized closed conformal Killing-Yano tensor with a skew-symmetric torsion ” [18], and ”the separability of the Maxwell equation on the Wahlquist spacetime” are shown [19]. A special feature of this metric is that in the wave equation for a massless scalar particle in the background of this metric gives trivial exponential solutions for two of the coordinates, the differential equation for the angular and the radial coordinates may be arranged to give exactly the same equation yielding general Heun solutions. Then the equation is reduced to the Mathieu equation by putting many of the constants in the original equation to zero. <|MaskedSetence|>
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**A**: Thus, ” the Wahlquist metric can not describe an isolated rotating body”.
**B**:
A negative point about this metric is the paper, [15], where it was shown that ”the Wahlquist perfect fluid space-time can not be smoothly joined to an exterior asymptotically flat vacuum region since the conditions for matching the induced metrics and extrinsic curvatures are mutually contradictory”.
**C**: This makes it possible to calculate the two point function in one less dimension [6]..
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<|MaskedSetence|> By initializing the learning process with a uniform random expander we bias the optimized solution towards expanders that distribute energy throughout the eyebox, in contrast to a quadratic phase profiles[28] that concentrate the energy at fixed points. <|MaskedSetence|> We incorporate pupil-aware optimization[37] to preserve the perceived hologram quality at different eye pupil locations. See Supplementary Note 5 for findings.
Finally, we also investigate 3D étendue expanded holograms. <|MaskedSetence|> We note that existing methods on étendue expanded holography has focused on monochromatic 3D holograms[7, 28, 29]. Photon sieves[21] only achieves 3D color holography for sparse points. See Supplementary Note 4 for a discussion of these findings.
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**A**:
In addition to field-of-view, we also investigate the eyebox that is produced with neural étendue expansion.
**B**: Thus, the viewer’s eye pupil can freely move within the eyebox and observe the wide field-of-view hologram at any location.
**C**: We find that neural étendue expansion also enables higher fidelity étendue expanded 3D color holograms.
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In this tutorial review, we presented theory for reverse osmosis (RO) and electrodialysis (ED), explaining how both technologies are based on the same fundamental transport theory. This is the solution-friction (SF) theory, and for ED we solved it in the absence of convection, thus we did not discuss pressures. We used SF theory for RO but then only described neutral solutes. Finally we solved SF theory for the osmosis experiment based on ions and a charged membrane, and we compared with experimental data. For ED we also developed new equations for Donnan equilibrium that extend the standard ideal expression. We present analytical equations for current efficiency, showing that this is a process parameter, not a membrane material property. For RO we summarized the literature for SF theory for neutral solutes including also the effect of concentration polarization. The general derivation we provided of SF theory also results in the twice-extended Nernst-Planck equation which is generally applicable in describing ion flow in reverse osmosis and nanofiltration of salt solutions.
Topics that we did not address in this tutorial review are first of all that both for RO and NF we must implement the Nernst-Planck equation for ions and a charged membrane in a full module calculation, and beyond that extend the theory from simple 1:1 salt solutions to multi-ionic solutions, also for electrodialysis. Even the addition of one extra type of anion or cation can significantly change the entire modeling framework. In addition, in real water sources also the protonation degree of ions must be considered, which depends on local pH. <|MaskedSetence|> These effects are relevant to study because for instance an ammonium ion is acted on by the electrical field, but the neutral ammonia species is not. Thus rejection of these ions is strongly pH-dependent. <|MaskedSetence|> Ions with a higher charge will be hydrated better, and are expected to be slower. <|MaskedSetence|> Another important assumption is local electroneutrality in channels and in membranes. Especially in reverse osmosis with membranes as thin as 100 nm, it is important to know if possibly Poisson’s equation must be used to replace the assumption of local electroneutrality in the membrane..
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**A**: At high concentrations, ions also associate in ion pairs.
**B**: For very tiny pores, a related topic is the effective size of ions, that has an impact on their partitioning and their mobility within the membranes.
**C**: State-of-the-art theory for simultaneous transport and reaction of ions (such as acid-base reactions between ions) assumes that these reactions are very fast, but it is interesting to investigate whether that is a correct assumption.
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Visualization of QNN extracted features. MNIST-2 classification result is determined by which feature is larger between the two: feature one is the sum of measurement outcomes of qubit 0 and 1; feature 2 is that of qubit 2 and 3. <|MaskedSetence|> <|MaskedSetence|> The circles/stars are samples of digit ‘3’ and ‘6’. All the baseline points (yellow) huddled together, and all digit ‘3’ samples are misclassified. With normalization (green), the distribution is significantly expanded, and the majority of ‘3’ is correctly classified. Finally, after noise injection (red), the margin between the two classes is further enlarged, and the samples are farther away from the classification boundary, thus becoming more robust.
Breakdown of accuracy gain.
Figure 9 shows the performance of only applying noise-injection, only applying quantization, and both. Using two techniques individually can both improve accuracy by 9%. Combining two techniques delivers better performance with a 17% accuracy gain. <|MaskedSetence|>
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**A**: We visualize the two features obtained from experiments on Belem in a 2-D plane as in Figure 8 right.
**B**: The blue dash line is the classification boundary.
**C**: This indicates the benefits of synergistically applying three techniques in QuantumNAT..
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Figure 2: Parameters inferred from a Gaussian rise plus exponential decay model applied to the ZTF light curves. The flux increase is measured relative to the ZTF reference image. <|MaskedSetence|> This requirement selects nuclear supernovae plus all spectroscopically-confirmed ZTF TDEs. <|MaskedSetence|> <|MaskedSetence|>
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**A**: The label ‘TDE?’ indicates accretion flares that occurred in active galaxies (i.e., sources with evidence for accretion prior to the main flare).
**B**: The dashed lines indicates the box that is used to separate accretion flares from regular AGN variability.
**C**: The three events coincident with a high-energy neutrino are indicated with solid symbols..
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Acknowledgements.
L. L., L. S., and R. F. <|MaskedSetence|> L. P., T. P., A. L., and O. C., acknowledge the
support by Grant No. GA19-14988S from the Czech Science Foundation and the project CZ.02.1.01/0.0/0.0/16 026/0008460 of MEYS CR. T. <|MaskedSetence|> L. <|MaskedSetence|> C. acknowledge the project EMPIR 20FUN01 TSCAC, which received funding from the EMPIR programme co-financed by the participating States and from European Union’s Horizon 2020 research and innovation programme. The research leading to these results has received funding from the H2020 European Programme under Grant Agreement No. 951737 NONGAUSS. L. P. acknowledges the internal project of Palacky University IGA-PrF-2021-006.
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**A**: P., A.
**B**: and O.
**C**: are grateful to the support from the Czech Science Foundation under the project GA21-13265X.
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(b) To fit DCDFM, an efficient spectral clustering algorithm called nDFA is designed. <|MaskedSetence|> Benefited from the distribution-free property of DCDFM, our theoretical results under DCDFM are general. Especially, when DCDFM reduces to DFM, our theoretical results are consistent with those under DFM. When DCDFM degenerates to DCSBM, our results also match classical results under DCSBM. Numerical results of both simulated and real-world networks show the advantage of introducing node heterogeneity to model weighted networks.
(c) To measure performances of different methods on real-world weighted network with unknown information on nodes labels, we propose a general modularity as an extension of classical Newman’s modularity [23]. For weighted network in which all edge weights are nonnegative, the general modularity is exactly the Newman’s modularity. <|MaskedSetence|> <|MaskedSetence|> By using two community-oriented topological measures introduced in [24], we find that the modularity is effective and our nDFA returns reasonable community partition for real-world weighted networks with unknown ground-truth nodes labels.
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**A**: We build theoretical framework on consistent estimation for the proposed algorithm under DCDFM.
**B**: Numerical results on simulated network generated under DCDFM for different distributions, and empirical un-weighted and weighted networks with known ground-truth nodes labels support the effectiveness of the general modularity.
**C**: For weighted network in which some edge weights are negative, the general modularity considers negative edge weights.
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At the heart of every quantum resource theory is the definition of free states and operations, corresponding to quantum states and quantum transformations which are easy to establish or implement. One of the basic questions in every quantum resource theory concerns quantum state conversion: is it possible to transform a quantum state into another one by using a free operation? This question is essential for entanglement theory, as many quantum information processing tasks require singlets for achieving optimal performance. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> If a conversion is not possible between two states, they still admit a probabilistic or approximate transformation, and results in this direction have been reported for specific setups [28, 29, 30, 19, 20, 31, 32, 33].
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**A**: It is thus crucial to develop optimal methods to convert a less useful state into another state which is potentially more useful for the specific task.
**B**: A complete solution to this problem has been given in terms of the Schmidt coefficients of the corresponding quantum states [27].
**C**: First results addressing this question concern transformations between pure entangled states.
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<|MaskedSetence|> (5).
The closer the final state is to the initial state, the weaker the perturbation of the electron cloud, which results in a lower probability of shaking.
The physical process at the origin of the vacancy has thus a major impact on the shaking probabilities. They are related to the capture probabilities via Eq. <|MaskedSetence|> Consequently, the atomic model employed to describe the initial and final states is critical to any realistic theoretical prediction. <|MaskedSetence|>
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**A**: It is clear that the BetaShape model fails to provide accurate shaking probabilities, especially for the innermost subshells..
**B**: (1).
**C**: This difference is due to the overlap between the initial and final atomic states involved, as can be seen from Eq.
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However, the above models are built for unweighted networks and they can not model weighted networks, where an edge weight can represent the strength of the connection between nodes. Weighted networks are ubiquitous in our daily life. For example, in the neural network of the Caenorhabditis elegans worm [23], a link joins
two neurons if they are connected by either a synapse or a gap junction and the weight of a link is the number of synapses
and gap junctions [24], i.e., edge weights for the neural network are nonnegative integers; in the network of the 500 busiest commercial airports in the United States [25, 26], two airports are linked if a flight was scheduled between them in 2002 and the weight of a link is the number of available seats on the scheduled flights [24], i.e., edge weights for the US 500 airport network are nonnegative integers; in co-authorship networks [27], two authors are connected if they have co-authored at least one paper and the weight of a link is the number of papers they co-authored, i.e., edge weights for co-authorship network are nonnegative integers; in signed networks like the Gahuku-Gama subtribes network [28, 29, 30, 31], edge weights range in {1,−1}11\{1,-1\}{ 1 , - 1 }; in the Wikipedia conflict network, [32, 33], a link represents a conflict between two users, link sign denotes positive and negative interactions, and the link weight denotes how strong the interaction is. <|MaskedSetence|> To model non-overlapping weighted networks in which a node only belongs to one community, some models which can be viewed as SBM’s extensions are proposed [34, 35, 36, 37, 38, 39] in recent years. <|MaskedSetence|> Though the multi-way blockmodels proposed in [40] can model mixed membership weighted networks (we also use mixed membership to denote overlapping occasionally), it has a strong requirement such that edge weights must be random variables generated from Normal distribution or Bernoulli distribution. This requirement makes the multi-way blockmodels fail to model the aforementioned weighted networks with nonnegative edge weights and signed networks. <|MaskedSetence|>
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**A**: In this paper, we aim at closing this gap by building a general model for overlapping weighted networks..
**B**: For the Wikipedia conflict network, edge weights are real values.
**C**: However, these models can not model overlapping weighted networks in which nodes may belong to multiple communities.
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Acknowledgements
We extend our deepest appreciation to the innumerable colleagues, staff, and loved ones at each of our institutes and in each others’ lives, especially during the COVID-19 pandemic for their continued encouragement and necessary assistance in the research and writing of this paper.
We thank the anonymous reviewer for their insightful comments, which significantly improved the quality and clarity of this paper.
We acknowledge PRACE for awarding us access to MareNostrum at the Barcelona Supercomputing Center (BSC), Spain. <|MaskedSetence|> Frontera is made possible by National Science Foundation award OAC-1818253. <|MaskedSetence|> We acknowledge access to Piz Daint at the Swiss National Supercomputing Centre, Switzerland under the University of Zurich’s share with the project ID uzh18. <|MaskedSetence|>
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**A**: This work made use of infrastructure services provided by S33{}^{3}start_FLOATSUPERSCRIPT 3 end_FLOATSUPERSCRIPTIT (www.s3it.uzh.ch), the Service and Support for Science IT team at the University of Zürich..
**B**: This work was supported in part by a grant from the Swiss National Supercomputing Centre (CSCS) under project IDs s697 and s698.
**C**: This research was partly carried out via the Frontera computing project at the Texas Advanced Computing Center.
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<|MaskedSetence|> On a more theoretical side, because of a wide range of scales and physical processes involved, beta transitions are a perfect laboratory to research and develop the concepts of Effective Field Theory (EFT).
The main ingredients of the theory of beta transitions were worked out by the end of 1950s, see in particular Refs. Lee:1956qn ; Jackson:1957zz ; Weinberg:1958ut .
The flip side is that the habitual language in the literature may sometimes be unfamiliar to contemporary QFT practitioners. <|MaskedSetence|> The advantage, apart from the conceptual side, is that the theory can be smoothly incorporated into the ladder of EFTs spanning various energy scales, from the TeV scale down to MeV. In particular, the EFT for beta transitions can be matched to the so-called WEFT (the general EFT of SM degrees of freedom below the electroweak scale), and via this intermediary to the SMEFT (the general EFT of SM degrees of freedom above the electroweak scale). <|MaskedSetence|>
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**A**: On the other hand, they yield important constraints on physics beyond the SM (BSM), such as leptoquarks and other hypothetical particles contributing to scalar and tensor currents in weak interactions Herczeg:2001vk ; Gonzalez-Alonso:2018omy ; Falkowski:2020pma .
**B**: One of the goals of this paper is to reformulate beta transitions in the modern EFT language.
**C**: This way, the general effects of heavy non-SM particles can be naturally incorporated, along with the more studied SM effects, into the low-energy effective theory of beta decay..
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IV.1.4 Extrapolation of Chain Energies to the Thermodynamic Limit
Given the sub-milliHartree accuracy of these predictions, we now turn to analyzing the performance of our GPR predictions for extrapolating the energies of very long, yet finite chains that approach the thermodynamic limit. In previous studies,Motta et al. (2017) thermodynamic limit predictions were made by assuming the chain energies varied polynomially with N−1superscript𝑁1N^{-1}italic_N start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, with orders ranging from 1 to 3 depending upon the convergence speed exhibited by the data.Motta et al. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|>
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**A**: We contrasted the extrapolations produced by this polynomial fit with GPR results trained once across different bond lengths on chains of 10, 20, and 30 atoms..
**B**: (2017) To make use of such scaling laws, a polynomial must be fit to a large enough number of different chain sizes to capture the correct scaling behavior.
**C**: To compare the performance of our GP regressions against this more conventional fitting procedure, we fit the energies of chains containing 10, 30, and 50 atoms, as was done in Reference Motta et al., 2017.
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<|MaskedSetence|> Many notions of network community structure arise from the analysis of random walks [12, 25], and we expect different types of random walks to yield different community structures [17, 18]. A “community” in a network is a tightly knit set of nodes that is connected sparsely to other tightly knit sets of nodes [12, 33]. Communities are a common feature of many real-world networks, and community structure influences dynamical processes such as the spread of infectious diseases [42] and online content [15, 45]. <|MaskedSetence|> There is intense interest in understanding how community structure and node characteristics combine to influence contagions on networks [24, 34, 37].
We develop community-detection algorithms that account for node-absorption rates. We adapt the widely-used community-detection algorithm InfoMap [35, 36, 41] to absorbing random walks and thereby account for heterogeneous node-absorption rates in the detected communities. In our adaptation, we apply InfoMap to absorption-scaled graphs, which account for absorption by scaling the edge weights of a network [16]. <|MaskedSetence|> We use absorption inverses and results from [16] to study the absorption-scaled graphs that are associated with our adaptations of InfoMap..
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**A**: These absorption-scaled graphs are related to their associated absorbing random walks by a generalized inverse (the so-called “absorption inverse”) and a fundamental matrix [16].
**B**: For example, community structure can affect the size and duration of a disease outbreak [38].
**C**:
In the present paper, we examine absorbing random walks on graphs in which different nodes can have different absorption rates, inducing an “effective” network structure that is reflected only partially by the edge weights of a network.
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<|MaskedSetence|> Moreover, existing methods for nonstationary processes rely only on nonstationary measurements, thus imposing a relatively high demand for nonstationary data. <|MaskedSetence|> Our approach combines the nonstationary measurement of a few observables with the tilted equilibrium measurement of the same set of observables. <|MaskedSetence|> Moreover, if the system satisfies a condition called realizability condition, which says that the nonstationary distribution is exactly realized as a tilted equilibrium, our method provides us with additional information about the process: the exact value of the true EP, the instantaneous EP rate, the nonstationary thermodynamic forces, and a constraint on relaxation trajectories.
Our method applies to arbitrary classical stochastic systems relaxing to equilibrium, including overdamped and underdamped systems, that may have continuous or discrete state spaces..
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**A**: However, most of these studies are concerned with steady states, and only a few of them develop thermodynamic inference methods for nonstationary processes [24, 39, 40, 41].
**B**: From these data, our method allows us to compute the exact value of the minimum EP compatible with the nonstationary data, which constitutes a tight lower bound on the true EP over the relaxation from any intermediate distribution to the final equilibrium.
**C**: It is natural to ask whether we can reduce the required nonstationary data by using additional data from other types of measurements.
In this paper, we propose a method of thermodynamic inference for relaxation processes that uses measurements in tilted equilibrium, i.e., the equilibrium under the application of external fields to the system.
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<|MaskedSetence|> Consequently, it becomes crucial to investigate the impact of random disturbances on epidemic models. Disease spread is inherently stochastic, and the introduction of stochastic noise can notably influence the likelihood of disease extinction during the early stages of an outbreak. While ordinary differential equation (ODE) models provide specific sample solutions, employing a stochastic differential equation (SDE) model allows for the exploration of the stochastic distribution of disease dynamics.
In the current landscape, various stochastic epidemic models have been extensively explored. Notably, articles such as those in the [9, 10, 11, 12, 13, 14, 15, 16, 17] series have delved into stochastic epidemic models influenced by Lévy noise. Motivated by this body of research, we introduce the assumption that the contact rate is perturbed by Lévy noise, emphasizing the need to employ Lévy processes for disease protection and control. The resulting model is detailed in (4), as elaborated in Section 4. Our objective is to derive the basic reproduction number, a critical determinant governing the extinction or persistence of the disease (infection). The reproduction number represents the average number of secondary infections produced by a single infected individual in a susceptible population, making it an essential metric for understanding disease spread dynamics. <|MaskedSetence|> <|MaskedSetence|> This research contributes to the broader understanding of stochastic epidemic modeling and its implications for disease dynamics..
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**A**:
Environmental fluctuations have emerged as a significant factor in the study of diseases, particularly in the context of the coronavirus.
**B**: If the reproduction number exceeds 1, it indicates that each existing infection leads to more than one new infection, resulting in exponential growth of the disease within the population.
**C**: Conversely, if the reproduction number is less than 1, the disease is likely to diminish and eventually disappear, as each infected person infects, on average, fewer than one new individual.
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<|MaskedSetence|> As a consequence, a minimal model to describe these compounds must include strong (long-range) Coulomb interactions [2, 3, *ips-rahul, *ips-rahul-errata].
It is by now well-established that the interactions mediated by the Coulomb forces in Luttinger semimetals stabilize a new non-Fermi liquid (NFL) state – the so-called Luttinger-Abrikosov-Beneslavskii (LAB) phase [27, 28]. The effective field theory for this phase was first studied by Abrikosov and Beneslavskii in the 1970s in a controlled approximation, by using a large-N𝑁Nitalic_N expansion [27, 28]. <|MaskedSetence|> <|MaskedSetence|> From recent analytical works, we do have some other examples of nodal NFLs [50, 51] as well..
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**A**: This NFL phase was later revisited and further reformulated using dimensional regularization and renormalization group (RG) techniques by Moon et al.
**B**:
Since the Luttinger semimetal harbors an isolated Fermi node at the Brillouin zone center, the electron-electron interactions are not effectively screened in these systems.
**C**: [2], with many interesting new predictions.
An important distinction of the LAB phase from other well-known NFLs arising for critical Fermi surfaces [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49] is that the former represents an NFL fixed point at a Fermi node, rather than for a Fermi surface.
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<|MaskedSetence|> It has low-weight stabilizer checks that can be read out using a nearest-neighbour qubit layout Chamberland et al. (2020); Beverland et al. (2021) and it has a fault-tolerant implementation of logical Clifford gates by transversal rotations on its physical qubits. In addition to these inherently fault-tolerant transversal gates, the color code also admits a multitude of fault-tolerant measurement-based code deformations Raussendorf and Harrington (2007); Bombín and Martin-Delgado (2009); Fowler (2011); Horsman et al. (2012); Brown et al. <|MaskedSetence|> The wide variety of available fault-tolerant logic gates can be attributed to the rich structure of its underlying anyon model when viewed as a topological phase Yoshida (2015); Bridgeman et al. (2017); Kesselring et al. <|MaskedSetence|>
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**A**: (2018).
In spite of its practicality, and its favourable properties for performing logical operations, little work has been conducted to determine the resource cost of performing large quantum algorithms with the color code realised on a two-dimensional qubit array.
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**B**: (2017).
**C**: The color code Bombin and Martin-Delgado (2006) is particularly well suited for fault-tolerant quantum computing using a two-dimensional qubit array.
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The science driver for massive galaxy spectroscopic surveys is to extract cosmological information from the clustering of galaxies in the past lightcone. <|MaskedSetence|> The increasing size of galaxy redshift surveys over the decade 2000200020002000-2010201020102010 led ultimately to a 5σ5𝜎5\sigma5 italic_σ detection of BAO by the Baryon Oscillation Spectroscopic Survey (BOSS)[4]. This enabled the BAO to be used as an accurate standard ruler to measure the geometry of the Universe and constrain the cosmic expansion history. <|MaskedSetence|> <|MaskedSetence|>
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**A**: A compilation of results from the Sloan Digital Sky Survey (SDSS) galaxy survey recently demonstrated the power of this technique[5].
The BAO feature is generally blurred by the nonlinear evolution of the Universe reducing its strength as a standard ruler, and various reconstruction methods have been developed to sharpen the BAO peak by undoing the nonlinear evolution of the density field.
**B**: Baryon acoustic oscillations (BAO)[1], formed in the early Universe due to interactions between photons and baryons under pressure and gravity, yield a special clustering pattern of galaxies around a characteristic comoving scale around 150150150150 Mpc, which is one of key probes for dark energy[2, 3].
**C**: The commonly used Lagrangian reconstruction, for example, linearises the density field by shifting the galaxies using the displacement field [6, 7, 8], while for the Eulerian reconstruction, manipulation is performed at the field level without moving the galaxies[9].
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At c=1/2𝑐12c=1/2italic_c = 1 / 2, all configurations occur with equal probability, cf. Eq. (7). In terms of the RW representation of configurations this corresponds to the set of RW excursions. <|MaskedSetence|> In turn, the average height field takes a semi-circular form, Fig. 2(d) (blue squares). <|MaskedSetence|> <|MaskedSetence|>
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**A**: Note that this a phase of large fluctuations and this average height field is not representative of typical sample profiles.
**B**: The average occupation, Fig. 2(c) (blue squares) interpolates between 1 and 0,
and in the thermodynamic limit, N→∞→𝑁N\to\inftyitalic_N → ∞, the average occupation density in the bulk is 1/2121/21 / 2 [11].
**C**: This is in contrast to the other two phases which are exponentially dominated by extremal area configurations, cf. Eq. (7).
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<|MaskedSetence|> <|MaskedSetence|> It would also be beneficial to test the methods as they would be used, i.e. <|MaskedSetence|> Matérn) and combinations,
such as more complex compositional kernels to model correlations at different scales. The benefits of modelling seasonal and weekly patterns should also be explored..
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**A**: In particular, due to data constraints, London was used both for developing the prior and for evaluation.
To test whether the prior can be constructed from other cities additional data needs to be used.
**B**: While the results show that the approach has promise, more evaluation is needed.
**C**: iteratively placing sensors, but this would be much more resource-intensive.
There might be easy performance gains available by experimenting with
kernel families (e.g.
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<|MaskedSetence|> By comparing simulations with observations of surface radio arrays, we know that the in-air radio emission of air showers is well understood [10, 11, 12]. It is however a non-trivial task to extend these simulations to include in-ice radio arrays, a process where both the signals as well as the cascade move between different media.
While the in-air propagation of a cosmic-ray air shower and its radio emission has been studied in detail, only a few studies exist that investigate its propagation into ice [13, 14, 15, 16, 17, 18] or other media [19]. It is the in-ice propagation of the cascade which is the focus of this work. We first illustrate the general properties of high-energy cosmic ray air showers at high altitudes typical for radio neutrino detection sites. Next we describe the simulation setup that was used to simulate the propagation of cosmic ray air shower cores through high-altitude ice layers, and summarize the simulation results. We will discuss the energy deposited in the ice, the longitudinal profile of the in-ice particle shower and the lateral charge distribution in the ice. We will show that both the longitudinal profile and the lateral charge distribution in the ice can be parameterized in a universal fashion. <|MaskedSetence|> <|MaskedSetence|> Furthermore, we show that the ionization trail left in the wake of the cascade should have properties favorable for radar detection [25]..
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**A**: Finally, we briefly discuss the application of our simulations to radio neutrino observatories located at high-altitude ice-sheets.
**B**: In this work we show that indeed the cosmic-ray shower core propagating from air into ice should be detectable by currently existing and future Askaryan radio detectors [20, 21, 22, 23, 24].
**C**:
Therefore, a dedicated study is required to investigate the radio emission from air showers as seen by in-ice radio detectors.
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IV.2 Propagation of non-analyticities
Figure 4: Propagation of non-analyticities in the coupled Yang-Mills system with bare vertices. <|MaskedSetence|> Complex poles in the gluon propagator cause additional branch cuts off the real axis in the ghost propagator, as shown in the plot on the right (see Figure 11 for full size). <|MaskedSetence|> These additional branch cuts generate corresponding additional branch cuts also in the gluon propagator via the ghost loop, demonstrated in the bottom left figure (see Figure 13 for full size). <|MaskedSetence|> In consequence, a single pair of complex poles cannot feature alone in consistent solutions in our truncation. The explicit analytic computation is presented in Appendix D..
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**A**: The displayed calculation is fully analytic.
**B**: This violates the initial assumption of just a single pair of complex poles in the gluon propagator.
**C**: Hence, the Källén-Lehmann representation of the ghost is violated.
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Machine learning tools for classification and regression are entering in the literature of fluid flows. <|MaskedSetence|> Regression problem are arguably more common, encompassing surrogate model-based optimization (Kim and Boukouvala,, 2019), turbulence modeling (Duraisamy et al.,, 2019), non-intrusive Reduced Order Modeling (Daniel et al.,, 2020; Hesthaven and Ubbiali,, 2018; Renganathan et al.,, 2020), aeroacoustic noise prediction (Dominique et al.,, 2021), surrogate modeling (Calado et al.,, 2023; Gkimisis et al.,, 2023) and system identification for prediction and control (Pan and Duraisamy,, 2018; Brunton et al.,, 2016; Huang and Kim,, 2008). <|MaskedSetence|> Both aspects are briefly illustrated in Section 2.
Regardless of the application, the final outcome is a “surrogate” model that can make predictions. These models comes in various shapes and sizes: examples considered in this work are linear combination of radial basis functions (RBFS) (Buhmann,, 2003), Artificial Neural Networks (Goodfellow et al.,, 2016) or recursive expression trees as in Genetic Programming (Banzhaf et al.,, 1997). These models provide analytical representations, hence amenable to analytic differentiation. <|MaskedSetence|>
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**A**: Examples of automatic data-driven classification of flow regimes are provided in Majors, (2018); Hobold and da Silva, (2018); Kang et al., (2020).
**B**: This enables meshless integration of Partial Differential Equations, discussed in Section 3.
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**C**: The main challenge in setting these problems is the choice of the hypotheses set balancing model complexity versus available data, and the implementation of physical constraints in the learning.
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In conclusion, this work proposes using a topological fluid dynamics action as an alternative method for studying Laughlin states. <|MaskedSetence|> <|MaskedSetence|> The no-slip boundary condition is excluded due to the absence of chiral dynamics along the edge caused by the anomaly. In contrast, the no-stress boundary condition leads to chiral dynamics at the boundary, regulated by the compressible boundary layer mechanism. <|MaskedSetence|>
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**A**: This no-stress boundary condition can be derived from an edge action involving an additional auxiliary chiral boson field coupled with the matter density at the boundary..
**B**: Additionally, we demonstrate that the topological fluid action can be mapped onto a Chern-Simons-Ginzburg-Landau theory through a duality transformation.
As with any theory, the requirement for a well-defined variational formulation dictates the permissible set of boundary conditions.
**C**: We show that the anomaly inflow mechanism replaces the no-penetration boundary condition in the presence of an external electric field tangential to the boundary.
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Figure 5: Sketch of the E-336 experimental setup at FACET-II. <|MaskedSetence|> <|MaskedSetence|> It is thereafter imaged with a quadrupole triplet onto a scintillating screen. <|MaskedSetence|> The X-rays produced in the beam-target interaction are measured with profile monitors and spectrometers (not shown).
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**A**: The electron bunch is focused on a target with a hole array.
**B**: A dispersive dipole magnet allows the reconstruction of the energy spectrum.
**C**: The beam after the interaction passes through an optional profile monitor which allows to measure the transverse momentum spread and deflection.
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<|MaskedSetence|> Trace element partitioning during calcite growth has been studied extensively in both natural and laboratory settings (e.g., Lorens, 1981; Carpenter and Lohmann, 1992; Paquette and Reeder, 1995; Nehrke et al., 2007; Tang et al., 2008b; Gabitov and Watson, 2006; Gabitov et al., 2014), and several theoretical models have been developed to explain these observations (DePaolo, 2011; Nielsen et al., 2012, 2013; Jia et al., 2022). Advances in stable isotope analyses of carbonate-incorporated major and trace metals (e.g., Ca, Li, Mg, Sr, Ba) (e.g., Tang et al., 2008a; Böhm et al., 2012; Mavromatis et al., 2013, 2020; Zhang and DePaolo, 2020; Füger et al., 2022; AlKhatib and Eisenhauer, 2017) could provide new insights to calcite growth kinetics and crystallization pathways. While most existing models are based on the classical crystallization pathway (DePaolo, 2011; Nielsen et al., 2012, 2013; Jia et al., 2022), these new observations highlight a more complex and diverse range of carbonate precipitation processes and thus offer additional constraints on previous precipitation models. In this study, with the paired observations of Ca and Sr isotope fractionations (e.g., Tang et al., 2008a, b; Böhm et al., 2012; Wang et al., 2021), we demonstrate the inadequacies of previous models to account for the full range of the observed calcite precipitation processes, and then provide new additions to incorporate the non-classical crystallization mechanisms. <|MaskedSetence|> <|MaskedSetence|>
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The partitioning of trace elements and the fractionation of stable isotopes during calcite precipitation are strongly affected by crystallization kinetics (e.g., Watkins et al., 2017).
**B**: Applying this new framework, we quantify the roles of classical and non-classical crystallization mechanisms at different precipitation rates and supersaturation levels.
**C**: This model framework can also be applied to other crystal systems and tested with other paired element and isotope measurements.
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This provides a strong step forward towards correctly identifying the imprint of IDE. <|MaskedSetence|> <|MaskedSetence|> Although a lot of work has already been done, more needs to be done. <|MaskedSetence|> Moreover, current literature show that an in-depth qualitative analysis of the redshift space distortions (RSDs) with respect to IDE is still necessary, as previous works (e.g. Refs. \refciteFerreira:2014jhn–\refciteBorges:2023xwx) focus on the estimation of cosmological parameters.
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**A**: Several other analyses have used various observations to test IDE models, in most of which cases the IDE models turn out to be compatible with the observations.
**B**: The observational analyses need to go beyond background observables, to the perturbations.
**C**: Despite all these efforts, there is still no definitive answer from observational analysis or fundamental theory as to the true form of IDE.
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With ZTF, we aim to observe all accessible high-quality neutrino alerts from IceCube. We define high-quality alerts as those with a high probability to be of astrophysical origin (‘signalness’ > 50%), or those which are well-localised (a 90% localisation area <<< 10 sq. deg.). <|MaskedSetence|> <|MaskedSetence|> IC201130A). We therefore ignore the labelling of these streams, and select exclusively based on the signalness and localisation.
We have followed up 24 neutrinos in the period from survey start on 2018 March 20 to 2021 December 31, out of a total of 79 neutrino alerts published by IceCube during that time. Table 1 summarises each neutrino alert observed by ZTF. <|MaskedSetence|> In addition to 1 of the 12 alerts under the old selection, ZTF followed up 23 of the 67 alerts published under the V2 selection. Midway through the ZTF program, an additional cut on neutrino alert galactic latitude (|b| > 10 degdegree\degroman_deg) was introduced to avoid crowded fields with many stars..
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**A**: IC211208A) and Gold alerts have been reported with signalness values less than 15% (e.g.
**B**: Though IceCube labels alerts as Gold or Bronze based on average quality, individual Bronze alerts have been reported with signalness values greater than 50% (e.g.
**C**: From 2019 June 17, IceCube published neutrino alerts with improved selection criteria (V2) to provide an elevated alert rate (Blaufuss et al., 2019).
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<|MaskedSetence|> For example, one may “compatify” some (even all) spatial directions, i.e. replace ΛΛ\Lambdaroman_Λ by a quotient group. <|MaskedSetence|> <|MaskedSetence|> In this section we discuss stacking and coarse-graining (in particular how they affect invariants of a code), but compactifications are postponed to future work. Moreover, we explain that the choice of n𝑛nitalic_n (which has to be a common multiple of qubit dimensions) does not matter and that the whole theory reduces to the case when n𝑛nitalic_n is a prime power.
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**A**: 4 Operations on Pauli stabilizer codes
One Pauli stabilizer code may give rise to various other codes.
**B**: Another possibility is stacking of infinitely many copies of a certain code to create a code with higher dimension.
**C**: Finally, one has coarse-graining, which does not change the code, but forgets about some of its translation symmetry.
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<|MaskedSetence|> Except for the success in the industrial applications, such as the image and speech recognitions [8], the autonomous driving, and the game of Go [9], neural networks have been widely adopted to study a broad spectrum of areas in physics, ranging from statistical and quantum physics to high energy and cosmology [10, 11, 12, 13, 14].
Among these successful applications in physical sciences, the more challenging task is to use neural networks to study nonequilibrium problems. Recently, an algorithm of artificial neural networks was proposed to solve the unitary time evolutions in a quantum many-body system [15]. Later developments in this direction can be found in [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].
In nonequilibrium dynamics, a common issue is the emergence of universalities during the critical dynamics of a phase transition. One of the most well-known challenges in this regard is the formation of topological defects. <|MaskedSetence|> <|MaskedSetence|>
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**A**: Number density of topological defects was found to satisfy a universal power-law with respect to the quench rate..
**B**: The recent state-of-the-art neural networks have been shown to provide high efficient representations of such complex states, making the overwhelming complexity computationally tractable [6, 7].
**C**: It is stated that topological defects will arise in the course of a phase transition with symmetry breaking of the system due to the celebrated Kibble-Zurek mechanism (KZM) [27, 28].
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The goal of this paper is to construct the neural network model that can preserve the fluids helicity. Unlike the standard finite element methods based on the weak formulation of the PDE models, Physics-informed neural networks (PINN) model [27] is based on the strong PDE and thus, conservation can be shown to be made much easier without having to introduce a number of auxiliary variables. <|MaskedSetence|> Later [19], [21] made great progress on deep learning solving PDEs and [13] discussed the theory of it. [16] proposed seq2seq strategy which is essential for timing problems. As is well-known the weak formulation seeks the pair of finite elements that is stable as well as is compatible such that the pressure space contains divergence of velocity so that the strong divergence is attained. However, such a construction is extremely difficult in general. Oftentimes, it relies on a discrete differential form point of view such as the finite element exterior calculus, see [9, 10] on structure-preserving discretization for the fluid mechanics with H(div)𝐻divH(\operatorname{div})italic_H ( roman_div )-conforming velocity. <|MaskedSetence|> The flow equation is similar to similar discretization for the Navier-Stokes equations based on the Nédélec edge element can be found in [11]. <|MaskedSetence|>
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**A**: This is really the case for conserving the incompressibility.
**B**: However, helicity-preservation was not addressed there.
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**C**: Recently, helicity has been discussed for the magnetic hydrodynamics equation.
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This approach, whilst providing information on smaller vesicles compared to holography, requires that all parameters other than the refractive index of the vesicle’s contents n𝑛nitalic_n be known and constrained. The most typical method to control for vesicle size involves extruding vesicles through pores, and generates nanoscale vesicles. <|MaskedSetence|> However, the refractive index and thickness for most lipid bilayer compositions is unknown. The presence of bilamellar vesicles is also expected to impact the scattering significantly, given the large surface area to volume ratio of these scatterers [21]. <|MaskedSetence|> Turbidometry must therefore be approached with caution, with complementary methods such as cryogenic electron microscopy to constrain the lamellarity, dynamic light scattering to measure the size, and a good estimate for the refractive index and thickness of the lipid, before the turbidity data can be used to extract the refractive index of the vesicles’ contents.
Figure 7: Experimental absorbance spectra (red) of POPC vesicles diluted into isotonic or hypotonic buffers. The modelled absorbance (gray) has no fitting parameters and was determined using the expected sucrose concentration difference (ΔcΔ𝑐\Delta{c}roman_Δ italic_c) between the interior and exterior of the vesicles. <|MaskedSetence|>
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**A**: At these smaller length scales, the exact refractive index and thickness of the membrane all play a large role in the vesicle’s scattering, relative to the aqueous core [21].
**B**: Vesicles prepared via slightly different methods have slightly different distributions in lamellarity, leading to different amounts of sample scattering (Fig. S8).
**C**: Taking the vesicle size distribution measured using dynamic light scattering (black, dotted) into account makes little difference to the modelled absorbance.
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<|MaskedSetence|> From one perspective, there are essentially no such solutions, because pure gravity in 2+1212+12 + 1 dimensions has no local propagating gravitational degrees of freedom Deser:1983nh . However, the theory can contain matter concentrated on scales much smaller than the AdSAdS\mathrm{AdS}roman_AdS radius without producing a black hole. <|MaskedSetence|> When the scales are very different, we can take the source to be pointlike. <|MaskedSetence|> In higher-dimensional setups, such configurations would necessarily form black holes..
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**A**:
Gravity with defects.
Without knowing the details, we can ask to what extent purely gravitational solutions might give us a guide to that physics.
**B**: Such solutions can be purely gravitational away from the matter: this is allowed in 3 dimensions, where gravitational fields are essentially localized at their sources.
**C**: (For example, in string theories one can have D-branes or other heavy objects whose size is the string scale.) As long as the mass of the matter source is below the BTZ threshold, no horizon will form.
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<|MaskedSetence|> This framework draws upon recent advancements in two key quantum information science areas. First, it utilizes high-dimensional quantum information and to extract underlying structures of classical data, capitalizing on its efficiency in identifying atypical data patterns and surpassing classical machine learning speeds [10]. <|MaskedSetence|> Thus, rather than merely transmitting raw data in compressed quantum states, the QSC framework intelligently extracts semantic information from the data. It then transmits only the essential quantum semantic representations over quantum channels, thereby resulting in more resource-efficient QCNs while maintaining high accuracy. Specifically, we provide a systematic approach for assessing and optimizing the minimality of quantum communication resources needed (e.g., entangled quantum states), and the accuracy of those resources in terms of quantum communication and semantic fidelity, showcasing the tradeoffs that exist. Simulation results validates that the QSC framework results in minimal quantum communication resources, saving 50-75% of the resources compared to semantic-agnostic QCNs, while achieving higher quantum semantic fidelity. The proposed framework provides a promising direction for researchers and engineers to explore the potential of QML and in reducing the resources required in QCNs. <|MaskedSetence|> Subsequent quantum clustering extracts useful semantic concepts, which are then mapped into efficient semantic representations for transmission via entangled qudits over quantum channels. At the receiver end, the fidelity and accuracy of the quantum mapping are verified, followed by the reconstruction of semantic representations and derivation of semantic concepts through quantum measurements.
Figure 1: Illustrative figure showcasing the proposed QSC framework..
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**A**: Figure 1 illustrates the proposed QSC framework, which starts at the transmitter, where raw data is embedded into qudits in high-dimensional Hilbert spaces.
**B**:
Towards this goal, the main contribution of this letter is a novel resource-efficient QCN framework, dubbed the framework.
**C**: Second, the QSC framework delves into quantum semantic representations, highlighting quantum mechanics’ fundamental role in vector modeling and linear algebraic semantics [11, 12].
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<|MaskedSetence|> However, the remaining deviation from ideal behavior is caused by the equivalent series inductance that typically produces self-resonance frequencies above 100100100100 MHz. solutions (2019) The non ideal behavior illustrated here occurs at frequencies smaller than this by at least two and at most eight orders magnitude.
To accurately model decay in a vacuum capacitor one must turn to Maxwell’s equations. <|MaskedSetence|> <|MaskedSetence|> Schade et al. (2019) Applying Maxwell’s equations numerically to the relaxation of an RC circuit yields an initial non-exponential decay due to transit time effects. Preyer (2002) However, this calculation does not predict the other decays that are documented below..
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**A**: However, only a handful of solutions that involve steady state behavior have been addressed analytically.
**B**: Muller (2012) These require a surface charge on the circuit wires to maintain the current as described theoretically Sommerfeld (1952); Heald (1984); Chabay and Sherwood (2019) and utilized experimentally (where the surface charge functions as a Hall probe).
**C**:
These effects are mitigated when using a vacuum capacitor.
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The structure of the paper is as follows. <|MaskedSetence|> In Section 3 we show how the Euclidean space emerges out of 𝔻𝔻\mathbb{D}blackboard_D-module geometry, in particular, Theorem 3 is proved. In Section 4 we obtain several results of screw theory by using the formalism of 𝔻𝔻\mathbb{D}blackboard_D-module geometry. In Section 5 we obtain results that, a posteriori, can be interpreted as a manifestation of the transference principle. Again, they are obtained by using the formalism of 𝔻𝔻\mathbb{D}blackboard_D-module geometry. <|MaskedSetence|> <|MaskedSetence|>
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**A**: We end the paper with some conclusions.
As for notation and terminologies, throughout the work we use, unless otherwise stated, the Einstein summation convention.
**B**: In Section 2 we recall some basic definitions about rings and modules, and prove that the notion of rank is well defined.
**C**: Vectors.
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This nascent appreciation that environments and their (possibly non-Markovian) dynamics could be active and potentially programmable components of future quantum devices raises exciting possibilities that could be accessible with current nanofabrication techniques and experimental probes [26, 27, 28, 29, 30]. In this work we address a relatively unexplored aspect of ‘environment-assisted’ phenomena that is of relevance for all of the examples and topics given above. Most theoretical descriptions of interacting arrays of open quantum systems utilize models in which each component (qubits, chromophores, quantum dots, etc.) interact with ‘local’, independent environments, i.e. while the components may interact with each other, their dissipative environments do not. <|MaskedSetence|> However, as the density of components increases – as required for more sophisticated quantum devices – the independence of these local environments becomes harder to justify [31]: propagating perturbations (excitations) of their common medium at one location become able to affect the dynamics of spatially remote systems, and maybe even do so on timescales that could be comparable to the intrinsic inter-system dynamics (see Fig. 1(a)).
Moreover, due to the retarded nature of these environmental ‘signals’, the subsequent dynamics of each component depends on the whole history of previous system-environment interactions of every component. <|MaskedSetence|> When systems are packed into nanoscale regions, a significant fraction of environmental excitations will encounter neighbouring systems and influence their dynamics (green arrows), even if the systems are uncoupled (yellow arrow indicates coherent coupling). <|MaskedSetence|> depend on the speed of signal propagation and the separation of the systems, providing new time and length scales to their now cooperative dissipative dynamics. Example in 1D1𝐷1D1 italic_D: (b) in photosynthetic reaction centers, pigments are held by a protein scaffold that can dissipatively mediate vibrations and structural reorganisation to coordinate exciton (eh pair) splitting, electron transfer and hole refilling in different locations (separated by 4−5454-54 - 5 nm) on timescales from the fs to the µs [32]. (c) The charge separation process decribed in (b) can be abstracted into an acceptor, bound exciton and charge separated states both coupled to the vibrational modes of their protein scaffold. This is the type of model-system studied in this paper..
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**A**: The resulting complex, multiscale decoherence phenomena present a new challenge for controlling or mitigating the effects of quantum noise, however such spatiotemporal effects may also present a mechanism for collective, co-operative and non-linear feedforward/feedback responses to external perturbations, and this emerging paradigm is the focus of this article.
Figure 1: (a) Spatially distributed systems dissipate energy into their environment and/or cause local deformations that typically propagate and are lost in the bulk medium (red arrows).
**B**: Energy and/or information dissipated into these local environments is forever lost to the global multicomponent system.
**C**: These interactions are retarded, i.e.
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<|MaskedSetence|> <|MaskedSetence|> Thus, one ends up with even loops and isolated vertices. What makes the monopole-dimer model less physical is that configurations have a signed weight and they cannot be interpreted as energies anymore. <|MaskedSetence|> Moreover, it is a perfect square for a 2m×2n2𝑚2𝑛2m\times 2n2 italic_m × 2 italic_n grid graph. A combinatorial interpretation of the square root is given in [Ayy20].
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**A**:
In another direction, a signed version of the monomer-dimer model called the monopole-dimer model has been
introduced [Ayy15] for planar graphs.
**B**: On the other hand, the partition function here can be expressed as a determinant.
**C**: Configurations of the monopole-dimer model can be thought of as superpositions of two monomer-dimer configurations having monomers (called monopoles there) at the same locations.
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Author Biography
{biography}
Massimiliano Lupo Pasini. <|MaskedSetence|> The focus of his undergraduate and master studies was statistics and discretization techniques and reduction order models for partial differential equations. He obtained his PhD in Applied Mathematics at Emory University in Atlanta (GA) in May 2018. <|MaskedSetence|> Upon graduation, Max joined the Oak Ridge National Laboratory (ORNL) as a Postdoctoral Researcher Associate in the Scientific Computing Group at the National Center for Computational Sciences (NCCS). Since 2020 Max has been a Data Scientist in the Scalable Algorithms and Coupled Physics Group in the Advanced Computing Methods for Engineered Systems Section of the Computational Sciences and Engineering Division at ORNL. Max’s research focuses on the development of surrogate models for material sciences, scalable hyper parameter optimization techniques for deep learning models, and acceleration of computational methods for physics applications. <|MaskedSetence|>
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**A**: Massimiliano (Max) Lupo Pasini obtained his Bachelor of Science and Master of Science in Mathematical Engineering at the Politecnico di Milano in Milan, Italy.
**B**: The main topic of his doctorate work was the development of efficient and resilient linear solvers for upcoming computing architectures moving towards exascale.
**C**: He is currently the lead of the Artificial Intelligence for Scientific Discovery thrust of the ORNL Artificial Intelligence Initiative..
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Certain evolution or dynamical maps which although are non-Markovian, do not attribute information backflow. <|MaskedSetence|> We investigate that information backflow can never be activated for those channels even when many copies of such channels are used in series or in parallel combination. However, exploiting two such channels in superposition of different orders, we find that information backflow can be restored. We then find out actual cause of this phenomenon of activation of information backflow by looking at the reduced dynamics of the effective switched channel. We show that though the original invertible dynamics, is not CP-divisible, but it always preserve P-divisibility and hence is unable to generate information backflow. However, for the switched evolution, along with its invertibility, both CP-divisibility and P-divisibility of the dynamics also breaks down. Moreover, the presence of the switching action also leads to the activation of information backflow.
Before concluding, a few remarks are in order. In this work, we are neither demanding that the way of activating information backflow presented in this paper is the only procedure to activate such things nor activation of information backflow is impossible by any higher order quantum process Chiribella et al. (2008). <|MaskedSetence|> (2008) are in order. On the other hand, finding out the fundamental cause behind the activation of quantum channel exploiting indefinite causal order is one of the most important open problems which are yet to be addressed. Although in this paper, we address the question partially, the question remains open for the advantages of indefinite causal order in other tasks especially for quantum communication. Despite the recent attempts to establish the connection between non-Markovianity and causal nonseparability Milz et al. (2018); Utagi (2021), this work explores the dynamical perspectives of indefinite causal order from the backdrop of Lindblad type evolution. <|MaskedSetence|>
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**A**: Therefore, further study regarding the activation of information backflow using causally ordered quantum comb Chiribella et al.
**B**: For such dynamics, the non-Markovian feature is not captured by the usual measure of non-Markovianity based on information backflow.
**C**: Since both indefinite causal order and non-Markovianity are fundamentally related to the memory of quantum systems, further investigations are in order, to decipher the connections between these two novel phenomena..
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Our standard cell approach is useful for reducing the computational complexity related to the placement and routing of quantum circuits. The latter problem is related to quantum circuit design automation, and heuristics have been investigated for the past decades [11, 12] and more recently, for example, by [13, 1]. <|MaskedSetence|> <|MaskedSetence|> Those approaches also use cells, but those are configurable, programmable elements laid out in such a way that arbitrary computations can be implemented; similar to conventional FPGAs, such FPGA-like approaches sacrifice efficiency (speed, area, and energy) for configurability and programmability. In contrast, our cells are non-configurable, pre-programmed elements with significant efficiency advantages. <|MaskedSetence|> Besides classical VLSI and quantum circuits, standard cells have also been proposed for other emerging computing platforms, such as quantum-dot cellular automata [21, 22], where tiles are placed and routes are computed between the tiles..
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**A**: Exact methods for qubit placement and routing have also been investigated, e.g. [14].
**B**: We do not have a cell for routing, like in [19], because we are extracting SWAP schedules after the circuit has been tiled.
Our approach is the opposite of the one presented in [20], in which gates are scheduled neglecting connectivity considerations, while routing operations are added at a later step.
**C**: FPGA-like approaches to quantum circuit compilation have been proposed, for example, in [15, 16, 17, 18].
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<|MaskedSetence|> (2015). <|MaskedSetence|> (2015), in this work, we will just point out some of the main steps to get the observed DM abundance and the maximum size of these DN. In the SM, the main bottleneck that prohibits the synthesis of nuclei is the substantial binding energy per nucleon of helium-4, ∼7similar-toabsent7\sim 7∼ 7 MeV, relative to the following smaller nuclei.131313During SM BBN, almost all the nucleons present wind up in hydrogen and helium-4 while a small fraction leads to the synthesis 4<A<84𝐴84<A<84 < italic_A < 8. There are subsequent bottlenecks post helium-4, such as 12C, where the binding energy per nucleon exceeds helium-4. However, there are no bottlenecks in the formation of large dark nuclei in the exotic sectors. <|MaskedSetence|> DM will thus be produced at low temperatures given that, in this case, the energy term dominates over the binding energy, which favors the generation of bound states, and it will be built up by aggregation, as fusion processes dominate over dissociations and fissions in the low temperature regime Hardy et al. (2015).
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**A**: As the detailed mechanism is explained in Hardy et al.
**B**:
In this section, we argue that composite DM, made of stable dark nuclei (DN) of large dark nucleon number, can evade the constraints from the Bullet Cluster on the DM self-interactions and provide a viable DM candidate Krnjaic and Sigurdson (2015); Hardy et al.
**C**:
As the Coulomb repulsion term is absent in this scenario, unlike in the SM, there can be stable DN up to a large dark nucleon number.
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The data that support the findings of this study are openly available. <|MaskedSetence|> <|MaskedSetence|> cel, in accordance with the Terms of Service of the respective web resources. <|MaskedSetence|> IV.2 and the trajectory is available at figshare.com/articles/dataset/Black-box_models_for_TERP_interpretation/24475003. Source data is provided with this paper as additional material.
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**A**: The AG’s news corpus dataset was obtained from Ref.
**B**: AGw, , and CelebA dataset from Ref.
**C**: Data generation details for the molecular dynamics of alanine dipeptide are provided in Sec.
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<|MaskedSetence|> Hence, the logic behind the index is that its value increases monotonically with the extent to which the feature is expressed. For entities that do not hold the feature at all, they should have the same and the lowest value. If an entity that does not hold the feature has a higher index value than that does hold the feature, the index must be incorrectly designed. <|MaskedSetence|> In the main text, we consider the lowest value to be zero. Indeed, all 21 indexes in this study assign the value 0 to entities that do not hold the feature they are designed to quantify. There could be exceptions. For example, one can design an index equal to the CN value plus 0.5. This is a valid index. <|MaskedSetence|> Indeed, the validity of based on the fact that entities that do not hold the feature are assigned the same and the lowest value, which is guaranteed by the design of the index..
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**A**: Likewise, if two entities that do not hold the feature have different index values, the index is also incorrectly designed.
Currently, there is no unified rule on what value should be the lowest.
**B**:
An index for a topological feature is designed to quantify the expression of this feature.
**C**: However, it is easy to see that our theoretical framework still holds when the lowest value is not zero.
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<|MaskedSetence|> 5 of the main text. Here, κ𝜅\kappaitalic_κ is the decay rate of the resonator, consisting of the intrinsic part κinsubscript𝜅𝑖𝑛\kappa_{in}italic_κ start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT and variable coupling part κcsubscript𝜅𝑐\kappa_{c}italic_κ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT with respect to the transmission line (TL). γmsubscript𝛾𝑚\gamma_{m}italic_γ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT and γmϕsubscript𝛾𝑚italic-ϕ\gamma_{m\phi}italic_γ start_POSTSUBSCRIPT italic_m italic_ϕ end_POSTSUBSCRIPT are the energy relaxation rate and the pure dephasing rate of the m𝑚mitalic_m-th qubit, respectively. <|MaskedSetence|> <|MaskedSetence|> |W3⟩ketsubscript𝑊3|W_{3}\rangle| italic_W start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⟩ will be stored in the resonator till.
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**A**: Initially, the dissipation rate is very small, so that |W3⟩=1/14(|100⟩+2|010⟩+3|001⟩)ketsubscript𝑊3114ket1002ket0103ket001|W_{3}\rangle=1/\sqrt{14}(|100\rangle+2|010\rangle+3|001\rangle)| italic_W start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⟩ = 1 / square-root start_ARG 14 end_ARG ( | 100 ⟩ + 2 | 010 ⟩ + 3 | 001 ⟩ ) can be generated through adiabatic evolution along |ψ2s+⟩ketsubscript𝜓limit-from2𝑠|\psi_{2s+}\rangle| italic_ψ start_POSTSUBSCRIPT 2 italic_s + end_POSTSUBSCRIPT ⟩ in 1.86×2πω−11.862𝜋superscript𝜔11.86\times 2\pi\omega^{-1}1.86 × 2 italic_π italic_ω start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT by choosing g1:g2:g3=1:2:3:subscript𝑔1subscript𝑔2:subscript𝑔31:2:3g_{1}:g_{2}:g_{3}=1:2:3italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : italic_g start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 1 : 2 : 3.
**B**: We choose κin=10−4ωsubscript𝜅𝑖𝑛superscript104𝜔\kappa_{in}=10^{-4}\omegaitalic_κ start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT italic_ω, κc=0.1ωsubscript𝜅𝑐0.1𝜔\kappa_{c}=0.1\omegaitalic_κ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 0.1 italic_ω, γm=10−5ωsubscript𝛾𝑚superscript105𝜔\gamma_{m}=10^{-5}\omegaitalic_γ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT italic_ω, γmϕ=2×10−5ωsubscript𝛾𝑚italic-ϕ2superscript105𝜔\gamma_{m\phi}=2\times 10^{-5}\omegaitalic_γ start_POSTSUBSCRIPT italic_m italic_ϕ end_POSTSUBSCRIPT = 2 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT italic_ω, where ω𝜔\omegaitalic_ω is the resonator frequency.
**C**:
to study the effect of the environment and catch and release of the W𝑊Witalic_W states, as shown in Fig.
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The rest of the paper is organized as follows: the next section discusses thermoelectric transport in mesoscopic systems with two terminals. We use the Onsager matrix that relates heat and charge current to the thermodynamic forces (voltage and temperature bias) and derive the expressions for different thermoelectric coefficients. Next, we elaborate on the proposed model used as a probe for MBS and derive the scattering amplitudes and the transmission probability that will be used to calculate the thermoelectric coefficients. Then we show the variation of thermoelectric coefficients such as Seebeck, Peltier, and the thermal conductance versus the Aharonov-Bohm flux and Fermi energy, both in the absence and presence of MBS. When an Aharonov-Bohm flux is introduced, the thermoelectric coefficients are asymmetric to Fermi energy only when MBS are present and coupled. <|MaskedSetence|> We also plot the thermoelectric coefficients versus the coupling energy of the MBS. We see that in the absence of magnetic flux, the thermoelectric coefficients are symmetric with respect to the coupling between the MBS, however, in the presence of a magnetic flux, the thermoelectric coefficients become asymmetric. Next, we study violation of the Wiedemann-Franz law. We show that WF law is only violated when MBS are present, regardless of the coupling. <|MaskedSetence|> Further, we see that the WF ratio (i.e., the ratio of electrical conductance to thermal conductance) behaves similarly to other thermoelectric coefficients in terms of symmetry versus Aharonov-Bohm flux, as well as Fermi energy and thus, can also be used to probe the existence and nature of MBS. We summarize the outcomes of our study for investigating MBS in Table 1. <|MaskedSetence|>
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**A**: When MBS are absent, there is no violation of WF law.
**B**: In the absence of MBS, the Seebeck and Peltier coefficients vanish, while the thermal conductance remains constant.
**C**: We end with a conclusion, summarizing our results.
2 Theory of Thermoelectric transport in mesoscopic systems.
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Right: Mass vs. coupling relationship that matches the observed abundance. <|MaskedSetence|> <|MaskedSetence|> However, when the exponential suppression is weaker (b<1𝑏1{b<1}italic_b < 1), instantaneous freezeout may no longer be a good approximation for estimating the relic abundance, as shown in Ref. Kim:2019udq ; Kramer:2020sbb . <|MaskedSetence|>
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**A**: The definitions of ΔΔ\Deltaroman_Δ for co-scattering, zombie, and forbidden annihilations are given in Table 1.
Instantaneous freezeout is a good approximation in many scenarios, because the annihilation rate is dropping exponentially with time (or temperature).
**B**: The solid line shows the results from the numerical simulation, while the dashed line shows the analytical solution in Eq. (16).
**C**: For b≪1much-less-than𝑏1b\ll 1italic_b ≪ 1, corresponding to an annihilation with a much lighter particle, the correction can be large.
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We have previously reported the growth of high quality optimally doped a𝑎aitalic_a-axis oriented YBa2Cu3O7-δ (YBCO) thin films [48].
In this work we present the properties of a𝑎aitalic_a-axis oriented YBCO thin films, at various doping levels from the optimally doped down to the insulating state, obtained by a careful oxygen annealing procedure. <|MaskedSetence|> <|MaskedSetence|> We find that the unit cell differs from the bulk even at dimensions of several hundreds of nanometers. <|MaskedSetence|> We observe a strong in-plane anisotropy of the resistance, comparable, also for rather thick films, to the best results achieved on detwinned single crystals. This opens up the possibility of studying potential modifications of the ground state induced by confinement in films with only a few unit cells in thickness.
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**A**: By varying the film thickness up to 800 nm, we have verified the dominant role of strain in this peculiar system.
**B**: The resistance vs temperature measurements are characterized by sharp superconducting transitions, showing that the superconducting properties of the films are rather homogeneous.
**C**: X-ray diffraction (XRD) analysis shows that the films are fully detwinned, implying the presence of CuO chains, aligned throughout the sample along the in-plane b𝑏bitalic_b direction.
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<|MaskedSetence|> We do expect that any measurement induced two-qubit error will result in a faster decay of the control qubit for mcm-rb compared to delay-rb. <|MaskedSetence|> However, from the simultaneous RB protocol [26] we know that in the limit of small two-qubit error the decay of a single-subsystem Clifford twirl will still be approximately exponential, such that we can again quantify the added error on the control using an IRB procedure comparing mcm-rb and delay-rb.
The signature of the ancilla decay curves is not consistent across the various error models that fall under measurement-induced two-qubit error. <|MaskedSetence|> On the other hand, a double excitation error (i.e. an XX𝑋𝑋XXitalic_X italic_X-gate) induced by measurement would result in finite ancilla EPM for both mcm-rb and mcm-rep, while a correlated phase error (i.e. a ZZ𝑍𝑍ZZitalic_Z italic_Z-gate) would not impact the ancilla state.
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**A**: For example, if the error is a coherent excitation exchange between control and ancilla, then we would expect to see decay of the ancilla ground state probability only for the mcm-rb protocol but not the mcm-rep protocol, since the ancilla and control are both initialized in the ground state.
**B**: Unfortunately, the control mcm-rb decay is not guaranteed to be exponential as only the control-qubit is twirled.
**C**: Unlike the error signatures we have thus far considered, there are sufficiently diverse measurement induced two-qubit errors that they will not all result in the same error signature.
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Applications of catalysis in quantum thermodynamics have also been reported by Lipka-Bartosik et al. <|MaskedSetence|> They showed that for every quantum state there exist two effective temperatures – hot and cold temperatures – quantifying the ability of a quantum system to cool down or heat up via thermal operations. <|MaskedSetence|> [136] showed that, if the temperature of the bath is less than the cold temperature, then the system can never gain energy via any thermal operation (and vice versa). <|MaskedSetence|> [136] showed that approximate catalysis leads to much colder or hotter effective temperatures.
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**A**: Precisely, Lipka-Bartosik et al.
**B**: [136], who proposed a notion of temperature of a non-equilibrium state, inspired by the “zeroth law of thermodynamics”.
**C**: Interestingly, Lipka-Bartosik et al.
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In this work, we presented a (magneto)hydrodynamical model for laminar, inviscid flow ahead of and around a solid obstacle of paraboloid form whose axis of symmetry coincides with that of the incident flow. For the case of the fluid being both incompressible and irrotational, we derived compact formulas for the scalar flow potential and the stream function, and hence for the velocity field. This velocity field was then used as an input to the induction equation of ideal magnetohydrodynamics, which we solved for the magnetic field subject to a homogeneous upstream boundary condition of arbitrary inclination. <|MaskedSetence|> <|MaskedSetence|> As a further result of this renormalization procedure, a simple one-to-one mapping between distorted and undistorted coordinate space (where in the latter the obstacle located at downstream infinity) is established that not only eases visualization, but also allows to describe the profound deformations experienced by arbitrary scalar or vector fields through the presence of the obstacle around which they are advected. In one direction, this mapping is available as an explicit formula, while in the opposite direction, recourse to a (quickly converging) numerical algorithm becomes mandatory. <|MaskedSetence|>
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**A**: When interpreting the obstacle’s surface as the magnetopause of a planet in a subsonic stellar wind, typical astrophysical applications include the modeling of transients, such as magnetic clouds or local density enhancements, embedded in the flow that is incident on the magnetosphere.
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**B**: This model constitutes the first (and therefore so far also the only) known nontrivial axially symmetric magnetohydrodynamical solution for obstacle-modulated flow with this property.
Moreover, we used a renormalization procedure to address the problem of the infinite travel time spent by a fluid element moving from upstream infinity to the vicinity of the obstacle.
**C**: The resulting field components can be written in compact and explicit form, involving neither integrals nor special functions.
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We employ the PRFT to analyze multimode driving, and discover that this leads to macroscopic light-matter entanglement at long times due to the matter-system-controlled transport of photon between distinct modes. This light-matter entanglement causes a complete decoherence of the matter system in the basis of the Floquet states. <|MaskedSetence|> The PRFT thus provides a quantum-optical interpretation of Floquet states as the decohering basis for the matter system. As the standard Floquet theory is unable to describe this fundamental decoherence effect, it will incorrectly predict the dynamics of the matter system in general. The PRFT thus demonstrates that even in the semiclassical regime, fundamental physical implications of the standard Floquet theory are not understood and require further investigation. In this context, it is worth noting that light-matter entanglement can arise even in single-mode models due to the photon shot noise [97, 98, 99, 100, 101]. <|MaskedSetence|> A detailed analysis of these issues is presented in Sec. III.1.
Furthermore, the transport-entanglement-related decoherence effect has far-reaching experimental consequences. In particular, we demonstrate that the quantum-optical coherence time is reasonably short (a few ms) for typical optical fields used in experiments, but it can be very long for radio-frequency driving. This implies that the radio-frequency regime is optimal for realizing quantum memories and quantum operations. Furthermore, we argue that quantum time crystals provide a powerful platform for realizing quantum memories irrespective of the driving frequency. Intriguingly, we demonstrate that the light-matter entanglement described by the PRFT can be deployed in a quantum-communication protocol that is intrinsically robust against photon loss. <|MaskedSetence|> Our analysis thus demonstrates that the PRFT can play a pivotal role in the development of future quantum technologies.
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**A**: This effect is depicted for a pardigmatic two-mode Rabi model in Fig. 1(b).
**B**: However, this shot-noise induced entanglement has a much smaller impact on the photonic dynamics compared to the transport-related entanglement revealed by the PRFT.
**C**: In particular, we demonstrate that using coherent light, it is possible for the quantum state transfer rate to reach the 0.1KHz0.1KHz0.1\,\text{KHz}0.1 KHz regime over 500km500km500\,\text{km}500 km, thereby far exceeding the Hz regime that is predicted in current theoretical protocols [102].
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<|MaskedSetence|> There are well-motivated reasons to consider modifications to GR, from anomaly cancellation to explaining observations such as leptogenesis and parity violation. <|MaskedSetence|> In particular, dynamical Chern-Simons gravity (dCS) is a strong candidate to provide a modification to GR, a theory which has been studied and developed extensively [5]. It has been theorized that chiral gravitational waves could emerge from dCS, where a change in a left-handed wave sources the right-handed wave and vice versa. <|MaskedSetence|> We find a new exact solution to the Hamiltonian constraint such that the dCS modification allows for the time variation of the cosmological constant ΛΛ\Lambdaroman_Λ.
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**A**: Chiral gravitational waves could be a way to resolve the argument that the Kodama state is unphysical due to negative helicity states in the expansion of the Kodama state that result in negative energies; a state with negative energy could be recast as a positive-energy state with opposite helicity [6, 7].
In this note, we revisit the Kodama state by adding dCS into the gravitational theory and therefore into the Wheeler-DeWitt equation.
**B**: Nevertheless, the Ashtekar variables have faciliated formulating non-perturbative quantizations in extensions of GR as well.
**C**: Recent work has parametrized gravitational wave propagation in extensions of GR, which allows one to map theory parameters to observables [4].
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<|MaskedSetence|> It is also known that RBCs travel in the body as an oxygen carrier, and easily alter their shapes to pass through 3333-to-4444 μ𝜇\muitalic_μm capillaries [13], where the upper limit of shear stress in the human circulatory system has been estimated to be 15151515 Pa [14]. <|MaskedSetence|> <|MaskedSetence|> Using a capsule model consisting of a Newtonian fluid enclosed by a thin elastic membrane, Matsunaga et al. [19] demonstrated that frequency-dependent deformations of a single spherical capsule become evident for high shear rates and large values of the viscosity contrast between the internal and external fluids. However, it is still unknown whether this knowledge can be used for suspensions of RBCs. The first question in this study is, therefore, whether the cell deformation is reduced or enhanced by varying the oscillatory frequency. Although the recovery of RBCs under oscillatory flow has been investigated in the past via experimental observations [20, 21, 22], and model analysis [23, 24, 25, 26], much is still unknown, especially in relation to the bulk suspension rheology and to the dynamical viscoelasticity of suspensions of RBCs under oscillatory shear flow. Hence, our second question is how the viscoelastic character of the suspension of RBCs differs in an oscillatory shear flow with respect to the steady case.
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**A**: The rheological description of blood under oscillatory flow is thus of fundamental importance not only for a physiological understanding but also in the design of novel artificial blood pumps that minimize mechanical stimuli that may cause the rupture of RBCs, the so-called hemolysis.
**B**:
In real human blood, RBCs are constantly under mechanical stimulation from the plasma flow due to the heart beat (≈\approx≈ 1 Hz) and from the vessel walls in various organs.
**C**: However, when they travel through artificial blood pumps, the cells may experience much higher shear stresses, up to 1000100010001000 Pa [15].
From a physiological viewpoint, the relationship between the deformation, as a mechanical response to oscillatory loading, and the oxygen transport, as a biological function, is therefore of great interest, and several studies have shed some light on this [16, 17, 18].
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II.1 Theoretical model
Quark recombination or coalescence models were first proposed to explain the baryon-over-meson enhancement and valence quark number scaling in RHIC Au+Au collisions Greco et al. <|MaskedSetence|> (2003a, b). In these models, valence quarks are assumed to be abundant in the phase space and recombine into hadrons through quark recombination. <|MaskedSetence|> <|MaskedSetence|> (2003b, a, 2008), the momentum distribution of baryons is given by
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**A**: (2003a, b); Fries et al.
**B**: The hadron formation process is usually assumed to be instantaneous and takes a very thin hypersurface (Δτ≈0Δ𝜏0\Delta\tau\approx 0roman_Δ italic_τ ≈ 0).
**C**: Following Refs. Fries et al.
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On the other hand, the multi-shell model in Ref.[25] (and Appendix A) considers both the time evolution of continuously many null shells and the backreaction of the evaporation in a common time coordinate and examines whether an apparent horizon is formed, without assuming anything about the magnitude of the energy-momentum tensor a priori. <|MaskedSetence|> Thus, the 4D dynamics of (1.1) including the tangential direction makes the difference.
As a quantum-gravity approach, Ref.[44] considers a spherically-symmetric quantum-gravity model and analyzes the time evolution of a collapsing matter to provide a bounce scenario in which a horizon is formed but no singularity appears due to a repulsive effect induced non-perturbatively. <|MaskedSetence|> In our scenario, as explained just above, the large tangential pressure plays a key role in supporting the dense configuration without horizons or singularities, and it is a non-perturbative effect coming from vacuum fluctuations of all modes with an angular momentum [30] (see Sec.6 again). <|MaskedSetence|>
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**A**: It would be interesting to use an idea of quantum-gravity dynamics as in Ref.[44] to describe the regions beyond the semi-classical approximation (see also below).
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**B**: Note here that the 4D dynamics of quantum matter fields is not considered in the analysis.
**C**: Solving (1.1) self-consistently, each shell will never cross its shrinking Schwarzschild radius, and a surface pressure will occur on each shell [25], which becomes the large tangential pressure (2.15) in a continuum limit, leading to the dense configuration without forming horizons.
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where we omit the normalization constant for brevity. <|MaskedSetence|> <|MaskedSetence|> As this procedure is mostly technical, for details about estimating bandwidths and constructing the Gaussian mixture, we refer to Appendix C.
Second, let us consider stke. Suppose high-dimensional samples are resampled so that each sample keeps a certain distance away from the others. <|MaskedSetence|> Then, w(𝐱)𝑤𝐱w(\mathbf{x})italic_w ( bold_x ) can be replaced by the unbiased probability density estimator ρ(𝐱)𝜌𝐱\rho(\mathbf{x})italic_ρ ( bold_x ) in Eq. (29). Thus, the reweighting factor is given by:.
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**A**: Note that many methods can be used for this purpose; however, to facilitate analysis, we use a method from Ref. 24.
**B**: The sum in Eq. (30) is over bandwidths that are automatically estimated and selected to fit that data.
**C**: In that case, the distribution of samples can be viewed as approximately uniform.
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In any theory of quantum gravity, which could break isotropy, the leading order term has to be the Einstein–Hilbert action. <|MaskedSetence|> Motivated by the extra dimensions that arise in string theory, brane world models that produce either a Yukawa [10] or power-law [11] correction to Newtonian gravity have been constructed [12, 13]. <|MaskedSetence|> We also note that most spin-independent corrections to Newtownian gravity result in a linear combination of Yukawa-type and power-law-type corrections [18, 19, 20, 21, 22, 23]. <|MaskedSetence|> If the actual cosmology is described by anisotropic branes [14, 15, 16, 17], there could exist detectable anisotropic corrections to the gravitational field at short distances.
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**A**: Because the short-distance approximation to the Einstein–Hilbert action leads to Newtonian gravity, which preserves isotropy, we expect any anisotropic terms to emerge from corrections to Newtonian gravity.
**B**: Even though most brane world models are isotropic, it has also been possible to construct anisotropic models that produce anisotropic corrections to Newtonian gravity [14, 15, 16, 17].
**C**: An advantage of using brane world models is that, in such models, it is possible for the effective Planck scale to be much larger than the actual Planck scale, and this can lead to corrections to the gravitational field at sub-millimeter and micrometer length scales [24, 25].
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<|MaskedSetence|> Extended periods of idle thrusters are observed during the orbital phase of the mission. The ΔVΔ𝑉\Delta Vroman_Δ italic_V budget is 76.77 m/s, with the Monte Carlo-Lambert guidance accounting for most of it. Despite operating with significant uncertainties about the spacecraft’s state and environment, the 51.08 m/s spent by the Monte Carlo-Lambert guidance before the orbital insertion burn is reasonable. <|MaskedSetence|> <|MaskedSetence|> Compare this to NEAR-Shoemaker’s 31.57 m/s for orbital insertion (OIM) and orbital operations (OCM-1 to OCM-25) 212121https://near.jhuapl.edu/NewMissionDesign/prpevent422.html.
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**A**: The 25.69 m/s ΔVΔ𝑉\Delta Vroman_Δ italic_V in orbital insertion and maintenance is noteworthy.
**B**:
Figure 3f illustrates the control commands during the 10-day operation in close-proximity to Eros.
**C**: To provide context, NEAR-Shoemaker spent 50.38 m/s in the one year after the rendezvous burn (TCM-17), from TCM-18 to TCM-23 202020https://near.jhuapl.edu/NewMissionDesign/prpevent422.html.
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5.3 Metals are masked by unknown stellar opacity
This hypothesis states that the heavy element pollution in DQ white dwarfs is present, and similar to that observed e.g. in DZ stars, but is masked by an unknown stellar opacity source. <|MaskedSetence|> The derived masses of metals in the fully mixed outer layers of the star, as well as the inferred accretion rates, are tied to line strengths. Then, if metal lines were reduced in strength within DQ stars and not in DZ stars, owing to a previously unaccounted for opacity source, there would be an expected trend with effective temperature or carbon abundance (or both). <|MaskedSetence|> <|MaskedSetence|>
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**A**: This hypothesis fails to address most of the other outstanding characteristics of the DQ stars, and would not, for example, prevent the detection of circumstellar material in the infrared, nor account for their distinctive stellar properties..
**B**: This possibility can in principle account for the fact that DQ stars rarely exhibit detectable metal lines, but does not address the bulk of relevant issues discussed in the previous section.
**C**: Neither the deep UVES observations, nor the hundreds of stars observed with the SDSS, show any trends in detected Ca ii line strengths with the DQ stellar parameters.
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A particularly interesting facet of active suspensions is the so-called active or swim pressure produced by the constituent self-propelled particles [35, 39, 40, 37, 38, 18, 36, 42, 34, 41]. In equilibrium systems, pressure can be calculated using thermodynamic, mechanical, and hydrodynamical approaches, leading to the same result. This result follows a state equation and thus varies only with bulk properties such as temperature and density. In active systems, a state equation may not generally exist [40]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Active pressure has been investigated on boundaries with different geometries such as flat walls [37, 36], curved surfaces [35, 34, 41, 36], corners [35], and sinusoidal and flexible interfaces [44, 43]. In addition to the geometry of boundaries, active pressure can vary depending on intrinsic features of active particles such as chirality [41], interparticle interactions, and local concentration [34, 38, 18].
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**A**: Active elongated particles and rods present a different situation where the pressure becomes dependent on particle-wall interactions [40].
**B**: More specifically, in the case of self-propelled spheres next to flat walls, the pressure can be described as a state function using activity-dependent effective temperature and bulk number density of spheres [38, 37].
**C**: Therefore, the pressure is mainly defined via mechanical and hydrodynamical approaches [39, 40, 38].
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<|MaskedSetence|> One can use any standard method for numerically solving ODEs to approximate ϕAB(xi),subscriptitalic-ϕ𝐴𝐵subscript𝑥𝑖\phi_{AB}(x_{i}),italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , i=1,…,N.𝑖1…𝑁i=1,\ldots,N.italic_i = 1 , … , italic_N . <|MaskedSetence|> Indeed, let k𝑘kitalic_k be the smallest integer such that kν∈ℤn𝑘𝜈superscriptℤ𝑛k\nu\in\operatorname{\mathbb{Z}}^{n}italic_k italic_ν ∈ blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and ks∈ℤℓ.𝑘𝑠superscriptℤℓks\in\operatorname{\mathbb{Z}}^{\ell}.italic_k italic_s ∈ blackboard_Z start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT . We have F(x,ϕAB(x))=ϕAB(x)k−f(x)ksxkν=0.𝐹𝑥subscriptitalic-ϕ𝐴𝐵𝑥subscriptitalic-ϕ𝐴𝐵superscript𝑥𝑘𝑓superscript𝑥𝑘𝑠superscript𝑥𝑘𝜈0F(x,\phi_{AB}(x))=\phi_{AB}(x)^{k}-f(x)^{ks}x^{k\nu}=0.italic_F ( italic_x , italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_x ) ) = italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_x ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT - italic_f ( italic_x ) start_POSTSUPERSCRIPT italic_k italic_s end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_k italic_ν end_POSTSUPERSCRIPT = 0 . <|MaskedSetence|> There is a degree k𝑘kitalic_k covering.
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**A**: Consider the algebraic curve 𝒞={(x,y)∈(ℂ*)2|yk−f(x)ksxkν=0}𝒞conditional-set𝑥𝑦superscriptsuperscriptℂ2superscript𝑦𝑘𝑓superscript𝑥𝑘𝑠superscript𝑥𝑘𝜈0{\cal C}=\{(x,y)\in(\operatorname{\mathbb{C}}^{*})^{2}\,|\,y^{k}-f(x)^{ks}x^{k%
\nu}=0\}caligraphic_C = { ( italic_x , italic_y ) ∈ ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_y start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT - italic_f ( italic_x ) start_POSTSUPERSCRIPT italic_k italic_s end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_k italic_ν end_POSTSUPERSCRIPT = 0 } with marked points Z={(x,0)|f(x)=0}.𝑍conditional-set𝑥0𝑓𝑥0Z=\{(x,0)\,|\,f(x)=0\}.italic_Z = { ( italic_x , 0 ) | italic_f ( italic_x ) = 0 } .
**B**: Furthermore, ϕAB(xN)=ϕAB(B)subscriptitalic-ϕ𝐴𝐵subscript𝑥𝑁subscriptitalic-ϕ𝐴𝐵𝐵\phi_{AB}(x_{N})=\phi_{AB}(B)italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_B ) can be used as the initial condition for the next integral over the line segment BC.𝐵𝐶BC.italic_B italic_C .
When the parameters s,𝑠s,italic_s , ν𝜈\nuitalic_ν are rational numbers, we can make use of the fact that the graph (x,ϕAB(x)),𝑥subscriptitalic-ϕ𝐴𝐵𝑥(x,\phi_{AB}(x)),( italic_x , italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_x ) ) , x∈AB,𝑥𝐴𝐵x\in AB,italic_x ∈ italic_A italic_B , satisfies an algebraic equation F(x,y)=0.𝐹𝑥𝑦0F(x,y)=0.italic_F ( italic_x , italic_y ) = 0 .
**C**: with initial condition specified by ϕAB(x1)=ϕAB(A).subscriptitalic-ϕ𝐴𝐵subscript𝑥1subscriptitalic-ϕ𝐴𝐵𝐴\phi_{AB}(x_{1})=\phi_{AB}(A).italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_A ) .
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The reaction participates in the oxidation, i.e., combustion pathway of propane, as the reactant forms through the free radical addition reaction between triplet oxygen and the 1-propyl radical. <|MaskedSetence|> For the simplicity of this tutorial, all the quantum calculations are done using Gaussian09 at the DFT/B3LYP/6-31G*(d,p) level of theory 27. It is clearly worth noticing that more modern theories, including DFT-D and Coupled Cluster theory, can give more accurate results28, 29, 30, 31. <|MaskedSetence|> All the structures are checked using frequency analysis, since the transition state should only contain one imaginative frequency. <|MaskedSetence|>
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**A**: It should be noted that this reaction has been comprehensively researched in a multitude of papers21, 22, 23, 24, 25, 26, and the goal of this tutorial is to provide a simple yet effective approach to building a Master Equation model to study it.
The barrier height E0subscript𝐸0E_{0}italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is essential in using RRKM to model chemical reactions, as it will be used in the RRKM expression (1).
**B**: The value of the barrier height and the value of the imaginative frequency are used in the RRKM subroutine and the tunnelling subroutine.
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**C**: The transition state is located using the QST2 algorithm, and the barrier height is determined by finding the energy difference between the reactant (product) and the transition state.
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2 Evidence the source crystal is Bernal trilayer graphene
Prior to device fabrication, the thickness of the source graphene crystal was determined using its optical contrast against the SiO2 exfoliation substrate in the standard way (Supplementary Fig. <|MaskedSetence|> We checked that the source crystal was Bernal rather than rhombohedral by performing Raman spectroscopy on several areas. The Raman 2D-mode of few layer graphene exhibits distinctive features depending on the stacking configuration, with rhombohedral graphene showing a pronounced asymmetric line shape [47, 50]. <|MaskedSetence|> <|MaskedSetence|> We further confirmed the thickness of the source crystal as trilayer by measuring STM topography across the step leading to the twisted region of the device (Supplementary Fig. 1b). Spectroscopy measured below the step provides additional evidence that the source crystal was Bernal stacked, as described in the main text (Fig. 2h).
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**A**: In our Raman measurements we observe no features indicative of rhombohedral graphene, confirming that our source crystal was stacked in the Bernal configuration (Supplementary Fig.
**B**: 1a, inset).
**C**: 1a).
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<|MaskedSetence|> Ref. [7] discusses exponential concentration in the context of a projected quantum kernel for a specific example embedding.
On the other hand, Refs. [16, 8] provide a rigorous study of the number of measurement shots required to successfully train the fidelity kernel but do not address the issue of exponential concentration. <|MaskedSetence|> We intend our results to be viewed as a guideline to the types of kernels and embeddings to be avoided for successful training. <|MaskedSetence|> For a more detailed survey of how our results fit in the context of prior work see Appendix A.
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**A**: Moreover, our results on noise-induced kernel concentration serve as a warning against using deep encoding schemes in the near-term.
**B**: Here we provide a systematic treatment of the causes and effects of exponential concentration in the presence of shot noise.
**C**: The problem of exponential concentration for the fidelity quantum kernel was first observed in Ref. [6] and later analyzed in Ref. [7, 44, 45] in the context of generalization.
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MPS obtained in [36] is integrable in the sense of [60]. <|MaskedSetence|> In section 2, we first review some results in [36], including the algebraic Bethe ansatz method and the selection rules. <|MaskedSetence|> <|MaskedSetence|> In section 5, we prove the giant graviton [36] is integrable for the cases where ω=0𝜔0\omega=0italic_ω = 0 or 1. Finally, we conclude in section 6.
2 Selection Rules.
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**A**: The root pairing proposed in [36] can be derived from the integrable condition and we generalize the result to a more general matrix product state.
**B**: In section 3, we give the integrable condition for SU(4)𝑆𝑈4SU(4)italic_S italic_U ( 4 ) alternating spin chain and propose a method to prove it.
**C**: In section 4, we find the previous method can be generalized to a more general state.
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<|MaskedSetence|> We use a hybrid trap Lin et al. (2009), combining a magnetic quadrupole field and a red-detuned ‘dimple’ laser beam, focused below the zero of the quadrupole, approximately at the position where the atoms will be finally confined in the shell trap, see Fig. 1. <|MaskedSetence|> We show that the preparation of the quantum gas is efficient and robust to small misalignment.
The paper is organized as follows. <|MaskedSetence|> We end with a final discussion in Sec. IV..
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**A**: After describing in some detail the production of the BEC in a hybrid trap in Sec. II, we will describe the loading process to the shell trap in Sec. III.3 and give its performances.
**B**:
In this paper, we propose and demonstrate a new approach to prepare a quantum gas in a shell trap.
**C**: This strategy minimizes the displacement of the cloud during the transfer from one trap to the other, hence its subsequent excitation.
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As an application for these general results we apply our findings on two holographic models of interest. <|MaskedSetence|> This background was studied as a holographic dual for a series of condensed matter systems and it was used in the holographic study of Fermi surfaces [12]. <|MaskedSetence|> [13], as another application of our general results. <|MaskedSetence|> In this model, the entropy is a linear function of the temperature, for low T𝑇Titalic_T. So, our general results allow us to obtain the diffusion coefficient and mean square displacement of fermionic systems which are relevant to the study of Fermi surfaces and Fermi liquid in the holographic set up.
This work is organized as follows. In Section II, we start setting the class of metrics explored here and calculating the Hawking temperature. From the Nambu-Goto action we find the general equation of motion in the linear regime. Then we solve it using a patching method. In Section III, we find a general expression for the linear response function and for the diffusion coefficient in terms of the general metric elements..
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**A**: This is a top-down model with finite chemical potential which describes dual Fermi liquids with massless charged fermionic modes.
**B**: First, we consider the hyperscaling-Lifshitz model at finite temperature and zero chemical potential for bosons and fermions.
**C**: Second, we discuss a charged dilatonic AdS black hole presented in Ref.
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