liangzid/robench2024b_all_setphysicsSCP-s · Datasets at Hugging Face

shuffled_text stringlengths 268 3.92k	A stringclasses 6 values	B stringclasses 6 values	C stringclasses 6 values	D stringclasses 6 values	label stringclasses 4 values
A: Based on EqsB: (58), (61), (62), and the time translational symmetry, the line element on the thin shell ΣΣ\Sigmaroman_Σ is a constant over time, i.e.,C: Since the system has a time translational symmetry, we can extend these results to all other times	CAB	ABC	BCA	BAC	Selection 3
A: Table 5: Mass differences in the charmonium spectrum in MeV compared to experimental values (calculated from 0954-3899-37-7A-075021 ; the value for the 1P hyperfine splitting is from Olsen:2012xn )B: For the results of this paper, the first error denotes the statistical uncertainty and the second error denotes the uncertainty from setting the lattice scale666For spin-dependent quantities the indirect contribution of the scale setting uncertainty to the kappa tuning uncertainty is sizable. Our scale setting error only accounts for the direct uncertainty associated with the setting of the lattice scale..C: Bars denote spin-averaged values	BCA	ACB	BCA	CAB	Selection 2
A: Then, in the mean-field limit, a corresponding Gross-Pitaevskii equation is introduced and numerically solved for different initial conditions Raghavan ; Ostrovskaya ; Ananikin . 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 B: 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. in situations when the local interaction energy is much larger than the single-particle tunneling energy. For example, it was shown that for initially imbalanced occupations the dynamics is heavily affected by strong interactions DuttaS . 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.C: 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	CAB	ABC	ACB	BCA	Selection 4
A: SUSY has found its way into many applications, including differential geometry [33], the study of coherent states [7, 12] and BCS theory [34]B: In SUSY, one assumes that the ground state energy of the quantum mechanical system is zero since ultimately one only measures energy changes, not absolute energy values, and seeks to write a Hamiltonian asC: The theory of supersymmetric quantum mechanics (SUSY) generated insights from the factorization method for more general quantum mechanical systems [11]	CAB	CBA	CBA	BCA	Selection 4
A: Then this is an equilibrium of having maximal entropy hence is unstable in light ofB: precisely Rinitial=−G⁢m2/2⁢Esubscript𝑅initial𝐺superscript𝑚22𝐸R_{\rm initial}=-Gm^{2}/2Eitalic_R start_POSTSUBSCRIPT roman_initial end_POSTSUBSCRIPT = - italic_G italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_EC: volume is fine-tuned i.e	ABC	BAC	CBA	BAC	Selection 3
A: The uncontracted indices are particularly important in the description of gauge fields inside the bulk and hence will become important for the extension towards double field theory. In a holographic code there are two types of uncontracted indices, namely the bulk indices and the boundary indicesB: All other indices are contracted between tensors arising on different layers of tilings. The bulk and the boundary indices are however not separated. Because the code is essentially an isometric embedding of the bulk Hilbert space into the boundary Hilbert space, the two indices are related. Each polygon provides an isometry from incoming and bulk indices to outgoing indices. C: These expansions form isometric encoding maps which will encode a certain number of logical qubits into the emerging physical qubits. Holographic quantum error correction codes are then implemented by contracting perfect tensors taking into account the geometry of the bulk space (in this case hyperbolic) and its tiling by corresponding polygons	BAC	ACB	ACB	BCA	Selection 4
A: both cases can occurB: This is obvious for flat manifolds admitting aC: τ⁢(Rg)≠0𝜏subscript𝑅𝑔0\tau(R_{g})\not=0italic_τ ( italic_R start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ) ≠ 0 or τ⁢(Rg)=0𝜏subscript𝑅𝑔0\tau(R_{g})=0italic_τ ( italic_R start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ) = 0 i.e	BCA	CBA	CBA	ACB	Selection 1
A: As discussed in the introduction of Section IV, here we do not have analogues of ‘in and an out’ states. The main variable for the transition are the angles at the sphere ΔΔ\Deltaroman_Δ which belong neither to the ‘past’ nor to the ‘future’ of the transitionB: Also, it is not clear what would serve here as the ‘measurement’ or ‘collapse’ and indeed we would not here have an analogue of a ‘time’ operator. Therefore, the analogy of the black to white hole transition with a quantum mechanical tunneling or the flight time problem is far from perfect. C: We also note that the problem we study here seems reminiscent of the arrival time problem Aharonov:1997md ; Delgado:1997tj ; Grot:1996xu .101010We thank an anonymous referee for pointing us to relevant literature. It is not clear to us whether a good analogy to this well studied problem and the problem we study here can be arrived at	CAB	ABC	CAB	BCA	Selection 4
A: If the exponent in (61)B: Unfortunately, that is not the case hereC: Kd/4⁢(12⁢\|Mi\|⁢Ωi⁢B2)subscript𝐾d412subscript𝑀isubscriptΩisuperscript𝐵2K_{\rm d/4}(\tfrac{1}{2}\|M_{\rm i}\|\Omega_{\rm i}B^{2})italic_K start_POSTSUBSCRIPT roman_d / 4 end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG \| italic_M start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT \| roman_Ω start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )	BCA	ACB	BAC	CBA	Selection 4
A: in the proof of Proposition 2.3 of [Be], can not easily be generalized to the twisted caseB: To prove an analogous result, we need to assume that ΓΓ\Gammaroman_Γ stabilizes a Borel subalgebra of 𝔤𝔤\mathfrak{g}fraktur_g, and use Lemma 2.5 cruciallyC: It will be interesting to see if this assumption can be removed.	BCA	CAB	ABC	BCA	Selection 3
A: By dimensional analysis, this is always possible, as there are at most two propagators expanded beyond leading order that contribute to the m∞4superscriptsubscript𝑚4m_{\infty}^{4}italic_m start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT-termB: This choice of momenta simplifies future integrations, as with it one only needs to consider the two regions P≫Qmuch-greater-than𝑃𝑄P\gg Qitalic_P ≫ italic_Q and Q≫Pmuch-greater-than𝑄𝑃Q\gg Pitalic_Q ≫ italic_P to obtain the double-logarithm coefficient. The full expression corresponding to eq. (10) in the main text, that is, the two-loop HTL diagrams re-expanded to two self-energy insertions to obtain possible double-logarithm corrections at N3LO, is C: When performing the expansions in m∞2superscriptsubscript𝑚2m_{\infty}^{2}italic_m start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, we label the momenta using linear changes of variables P↦−P−Qmaps-to𝑃𝑃𝑄P\mapsto-P-Qitalic_P ↦ - italic_P - italic_Q and Q↦−P−Qmaps-to𝑄𝑃𝑄Q\mapsto-P-Qitalic_Q ↦ - italic_P - italic_Q when necessary, so that no factors of ΠT/L⁢(P+Q)subscriptΠ𝑇𝐿𝑃𝑄\Pi_{T/L}(P+Q)roman_Π start_POSTSUBSCRIPT italic_T / italic_L end_POSTSUBSCRIPT ( italic_P + italic_Q ) appear	ACB	BCA	ACB	ACB	Selection 2
A: In fact, in Ref.Mitra (1993), the author constructed a fundamental field theory of semions only, as certain generalized Grassman number valued field theoryB: If anyons may be the excitation modes of a fundamental field, the field cannot be real, complex, or even Grassmann number valuedC: Unfortunately, more general constructions, in particular of non-Abelian anyons, is yet available and is our undergoing work.	BAC	ABC	BCA	CAB	Selection 1
A: But by Alaoglu’s theorem (e.gB: [24, p. 484]) the unitC: \mu_{0}}\frac{V_{0}}{V_{1}}\Big{)}\frac{\mu_{1}}{V_{1}}divide start_ARG italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ↦ - ∫ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT roman_log ( divide start_ARG roman_d italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG roman_d italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG divide start_ARG italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) divide start_ARG italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG is continuous	BCA	ACB	ABC	ACB	Selection 1
A: the BV operator is compatible with a product and a bracket, both given by counting pairs of pants but with asymptotic markers treated differently.B: [1]), i.eC: In characteristic zero, symplectic cohomology SHSH{\operatorname{SH}}roman_SH is known to have the structure of a BV-algebra (see e.g	ABC	CBA	CAB	CAB	Selection 2
A: However, this method can not be applied in arbitrary high-dimensional mixed statesB: As we know, a star on a Bloch sphere can represent a pure state, and a set of stars on a Bloch sphere can represent a high-dimensional pure statesC: Our result is not easy to be generalized to arbitrary high-dimensional mixed-spin systems, such as a mixed-spin (s,1)𝑠1(s,1)( italic_s , 1 ) system. Because we can not directly use an effective pseudo spin 1111 to describe it.	BCA	CBA	BAC	CAB	Selection 3
A: , ψ\|Γ=Cevaluated-at𝜓Γ𝐶\psi\|_{\Gamma}=Citalic_ψ \| start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT = italic_C (constant)), corresponding to havingB: italic_e C: the boundary condition (∇∥ψ)\|Γ=0evaluated-atsubscript∇parallel-to𝜓Γ0\left(\nabla_{\parallel}\psi\right)\|_{\Gamma}=0( ∇ start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT italic_ψ ) \| start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT = 0 (i.e.,formulae-sequence𝑖𝑒i.e.,italic_i	CBA	BAC	CAB	BCA	Selection 1
A: In [39] the authors applied a version of para-controlled calculus to the stochastic wave equation setting, and obtained almost sure local well-posedness for a quadratic wave equation with additive white noise on 𝕋3superscript𝕋3\mathbb{T}^{3}blackboard_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPTB: Finally, we would like to mention two recent results of Gubinelli-Koch-Oh [39] and Bringmann [15]C: This relied on several new ingredients, including the analysis of a random operator (which is different from and unrelated to the random averaging operator introduced in the current paper).	BAC	CBA	CBA	ACB	Selection 1
A: The choices of branch and labeling will depend on the value of m∈ℂ∗𝑚superscriptℂm\in\mathbb{C}^{}italic_m ∈ blackboard_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT in the following way:B: For the definition of Stokes data, we need to fix a branch of the Log and a labeling of the sectors determined by anti-Stokes raysC*: The reason for the choices made below will become apparent in subsequent sections, where we start comparing and matching the twistor coordinates for harmonic bundles with the twistor coordinates of the Ooguri-Vafa space	BCA	BCA	CAB	BAC	Selection 3
A: The authors thank the Parker Solar Probe team, especially the FIELDS and SWEAP teams for their supportB: The FIELDS experiment on the Parker Solar Probe spacecraft was designed and developed under NASA contract NNN06AA01CC: The authors wish to acknowledge helpful conversations with Dr. Ivan Vasko.	ACB	CAB	ABC	BAC	Selection 3
A: square rootsB: As a result the expression under squareC: Indeed, we have bS=−5subscript𝑏𝑆5b_{S}=-5italic_b start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = - 5, bI=4subscript𝑏𝐼4b_{I}=4italic_b start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT = 4 and D=bS2−bI2=(3)2𝐷superscriptsubscript𝑏𝑆2superscriptsubscript𝑏𝐼2superscript32D=b_{S}^{2}-b_{I}^{2}=(3)^{2}italic_D = italic_b start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_b start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( 3 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT	CBA	CBA	CAB	ACB	Selection 4
A: The experiments used six gases, ammonia, acetaldehyde, acetone, ethylene, ethanol, and toluene, presented in arbitrary order and at variable concentrations. Chemical interferents were also presented to the sensors between batches, and the time between presentations varied, both of which contributed to further sensor variability. The dataset thus exemplifies sensor variance due to contamination and variable odor concentration in a controlled setting. B: Every batch contains between 161161161161 to 3,60036003{,}6003 , 600 samples, and each sample is represented by a 128-dimensional feature vector; 8 features each from 16 metal oxide-based gas sensors. These features summarizing the time series sensor responses are the raw and normalized steady-state features and the exponential moving average of the increasing and decaying transients taken at three different alpha valuesC: Experiments in this paper used the gas sensor drift array dataset [7]. The data consists of 10 sequential collection periods, called batches	CAB	BAC	ABC	CBA	Selection 4
A: In Figs. (3) and (4) we have demonstrated a possible way to minimize error and achieve high fidelity quantum gate operations using several bounded control pulses (e.g., π𝜋\piitalic_π, CORPSE, SCORPE, symmetric and asymmetric pulses) acting on a one qubit system in presence of random telegraph noises. Here we have compared the fidelities obtained from the pulses discussed aboveB: More precisely, symmetric pulse (see Fig. 3(a)) provide large fidelity for the small energy amplitudes of noise strength which may be useful for the experiments that perform at low temperatures. On the other hand, CORPE pulse (see Fig. 3(b,c)) provide large fidelity for the large energy amplitudes of noise strength which may be useful for the experiments that perform at high temperatures. Finally in Fig. 7, I have shown that when π𝜋\piitalic_π pulse acts in x direction, CORPSE pulse acts in y direction and SCORPSE pulse acts in z-direction in presence of arbitrary low and high temperature measurements noise conditions, large fidelity recovery can be achieved and may consider for implementing in future for electronic circuits design to minimize error.C: We conclude that in the limit of vanishing noise correlation time, π𝜋\piitalic_π-pulse can be used for the measurement of achieving high fidelity due to its small gate operation time. In the limit of large correlation time, two pulses namely symmetric pulse and CORPSE pulse were identified to achieve recovery of high fidelity	CAB	CBA	BAC	ACB	Selection 4
A: We have obtained and studied the thermodynamics of black brane solutions in asymptotically Lifshitz spacetimes in κ𝜅\kappaitalic_κ-deformed Horndeski gravity in four dimensionsB: The main property of these solutions is that they can assume arbitrary values of the critical exponents z𝑧zitalic_z associated with Lorentz symmetry violationC: Usually, when asymptotically Lifshitz spacetimes are considered in Horndeski gravity, the critical exponents typically acquire some specific fixed values for which some solution can be found. What we have shown here is that the inclusion of the κ𝜅\kappaitalic_κ-deformation in Horndesky theory releases these constraints on the critical exponents allowing them to assume arbitrary values and then finding solutions in this general case. This is a consequence of the auxiliary equation (17) that comes from the κ𝜅\kappaitalic_κ-algebra leading to the solution of the equations of motion for any value of the critical exponent.	ACB	BAC	ABC	BCA	Selection 3
A: 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. (1992, 1994); Cronin et alB: (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.C: (2009); Bollinger et al. (1996); Döring et al	ABC	CAB	ABC	ACB	Selection 4
A: These were introduced in joint work with Sven Meinhardt in [DM20], as part of a project to realise the cohomological Hall algebras defined by Kontsevich and Soibelman [KS11] as positive halves of generalised YangiansB: The construction of the BPS Lie algebra for arbitrary symmetric quivers with potential is recalled in §2.4C: Note that the BPS Lie algebra is defined by a quite different perverse filtration, on vanishing cycle cohomology of a different Calabi–Yau category.	ACB	CBA	BAC	BCA	Selection 3
A: (2023) and CMS Hayrapetyan et al. (2024) experiments.B: Amongst its many applications, we note three in particular. First, it facilitates studies of the J/ψ⁢J/ψ𝐽𝜓𝐽𝜓J/\psi J/\psiitalic_J / italic_ψ italic_J / italic_ψ system, where resonance-like structures were reported by the LHCb experiment Aaij et alC: (2020c) and confirmed by both the ATLAS Aad et al	CAB	CBA	ACB	ABC	Selection 1
A: 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)B: The chain with respect to which one is considering representation stability there is of course still the chain of symmetric groups. Thus, representation stability with respect to a chain of diagrammatically defined algebras is not considered in [1] (Barter, Entova-Aizenbud, Heidersdorf).C: 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. 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	ABC	ACB	BCA	BCA	Selection 2
A: (2012); Wu et al. (2018); Chen et al. (2017); Karliner et al. (2017); Bai et al. (2019); Wang (2017); Richard et alB: Even before the discovery, fully-heavy multiquark states have already been studied Iwasaki (1975); Chao (1981); Ader et al. (1982); Zouzou et al. (1986); Heller and Tjon (1985); Lloyd and Vary (2004); Barnea et al. (2006); Vijande et al. (2009); Berezhnoy et alC: (2017); Anwar et al. (2018); Debastiani and Navarra (2019); Richard et al. (2018); Esposito and Polosa (2018); Wang and Di (2019); Liu et al. (2019b); Wang et al. (2019); Chen (2020); Lundhammar and Ohlsson (2020).	CBA	ACB	BAC	ABC	Selection 3
A: The resultsB: For the model presented in Ref. Farzan:2015doa we have defined By≡B1−y⁢B2−(3−y)⁢B3subscript𝐵𝑦subscript𝐵1𝑦subscript𝐵23𝑦subscript𝐵3B_{y}\equiv B_{1}-yB_{2}-(3-y)B_{3}italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ≡ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_y italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - ( 3 - italic_y ) italic_B start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT where y𝑦yitalic_y is an arbitrary constantC: studied in the literature	CAB	CBA	CAB	BAC	Selection 2
A: A guiding principle, on which we will elaborate, asserts that, when many valleys are present, chaos reins: the system is fragile, with small perturbations causing profound changes in the form of the ground state.B: In such systems, the ground state, namely the value x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X that minimizes H𝐻Hitalic_H, may be joined in the landscape by a host of near ground states, resting at the bases of various local valleys, at more or less removed locationsC: The presence of multiple competing near-minimizers has significance in problems concerning the response of the system to perturbations of its law suffered by changes in parameters such as temperature or in the realization of the disorder	CAB	ACB	ACB	BAC	Selection 1
A: 1, ae=3ae[l⁢o⁢c]a_{e}=3\,a{}_{e}^{[loc]}italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = 3 italic_a start_FLOATSUBSCRIPT italic_e end_FLOATSUBSCRIPT start_POSTSUPERSCRIPT [ italic_l italic_o italic_c ] end_POSTSUPERSCRIPT leading to ce=13⁢ce[l⁢o⁢c]subscript𝑐𝑒13superscriptsubscript𝑐𝑒delimited-[]𝑙𝑜𝑐c_{e}=\frac{1}{\sqrt{3}}\,c_{e}^{[loc]}italic_c start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 3 end_ARG end_ARG italic_c start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_l italic_o italic_c ] end_POSTSUPERSCRIPT per c∝a−1/2proportional-to𝑐superscript𝑎12c\propto a^{-1/2}italic_c ∝ italic_a start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT,B: In FigC: to ae>ae[l⁢o⁢c]subscript𝑎𝑒superscriptsubscript𝑎𝑒delimited-[]𝑙𝑜𝑐a_{e}>a_{e}^{[loc]}italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT > italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_l italic_o italic_c ] end_POSTSUPERSCRIPT	CBA	BAC	BAC	BAC	Selection 1
A: 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 alB: (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. (2016), and adiabatically scanning through a quantum phase transition Lee (2006); Huang et al. (2018).C: (1997); Moore et al. (1999), quantum nondemolition measurement Appel et al. (2009); Kuzmich et al. (1998); Louchet-Chauvet et al. (2010); Hammerer et al. (2010), one-axis twisting Schleier-Smith et al	BAC	BAC	ACB	BCA	Selection 3
A: The analytical and the numerical results for the Einstein-de Sitter universe are identical, thus confirming our numerical implementation. Despite the difference in the cosmological background evolution between the left and central panels compared to the right panel, the results are quite similar. B: It is opposed there to the numerical result for an Einstein-de Sitter universe in the centre panel for comparison, and to the functional derivative of D+subscript𝐷D_{+}italic_D start_POSTSUBSCRIPT + end_POSTSUBSCRIPT obtained by evaluating (28) numerically as described in the following subsection, shown in the bottom panelC: This analytic result is shown in the top panel of Fig. 4	BAC	ACB	BAC	CBA	Selection 4
A: 1b). The following impossibility of certain clean algorithms is a corollary of Theorem 3.B: In other words, the ancilla output is independent of the unknown oracleC: Clean algorithms enable ancillae recycling and preserve coherences if used inside an interferometer (Fig	CBA	CAB	ACB	BCA	Selection 2
A: Yanbing Zhu: Review and editing (equal); Conceptualization (supporting). Evan J. Read: Conceptualization (lead); Review and editing (equal); Supervision (equal)B: Peter Schindler: Writing – original draft (lead); Review and editing (equal); Software (lead); Methodology (lead); Visualization (lead); Investigation (lead); Data curation (lead); Conceptualization (supporting); Supervision (equal); Resources (lead). Evan RC: Antoniuk: Review and editing (equal); Methodology (supporting); Conceptualization (supporting). Gowoon Cheon: Review and editing (equal); Conceptualization (supporting)	CBA	ABC	BAC	CAB	Selection 4
A: blowup profile, such that the Wick-ordered L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff does not exclude this blowup profile for any cutoff size K>0𝐾0K>0italic_K > 0B: See, in particular, Lemma 3.4 and the proof of (3.42)C: We also mention related works [33, 13, 53, 12, 46, 54] on the non-normalizability (and other issues)	BAC	CAB	ABC	ACB	Selection 3
A: We performed hard X-ray diffraction experiments at the BW5 beamline at HASYLAB, Deutsches Elektronen-Synchrotron, Hamburg, GermanyB: The sample was cut into a thin slice (d=1.4𝑑1.4d=1.4italic_d = 1.4 mm) in order to reduce the absorption, meaning that 10 % of the incoming beam was transmitted through the sample. C: Here we looked for a signature of charge-density waves	ACB	CAB	CAB	BCA	Selection 1
A: We strongly emphasize that there is a great deal of room for improvement on the techniques described hereB: We have not, for example, carefully assessed the most effective method for laying out the grid of data on which we store information about adiabatic backreaction and waveform amplitudes, nor have we thoroughly investigated efficient methods for interpolating these data to off-grid points (e.g., [16]). These important points will be studied in future work, as we begin assessing how to take this framework and use it to develop EMRI waveforms in support of LISA data analysis and science studies.C: In Sec. VI, we present various important technical details describing how we implement this framework for the results we present in this paper	ACB	CBA	ABC	BCA	Selection 4
A: The reason is that although an increase of beam current will enhance the radiation power, the coexisting increase in modulation depth and the fixed dispersion strength result in the over-compressed and a smaller bunching factor, which is not conducive to the radiation power growth. This feedback mechanism makes radiation of DEHG not naturally sensitive to the beam current. Fig. 3(c, d) display the radiation power versus seed laser power and electron beam current in nominal HGHG and DEHG, respectively. One can see that in nominal HGHG, the radiated power increases as the electron beam current risesB: Fig. 3(a, b) present the modulation depth as functions of the seed laser power and electron beam current. For nominal HGHG, the modulation depth is related to the seed laser intensity, but not to the electron beam current. For DEHG, however, the modulation depth is correlated with both the seed laser power and the electron beam current. This phenomenon makes the radiation of DEHG more tolerant to the variation of currentC: In DEHG, however, the radiation power does not increase with current in a certain area due to the feedback mechanism. Power stable area can be regarded as a region in which the radiation power decreases by no more than 5% due to jitters in the electron beam current and seed laser power. In this case, the power stable area of DEHG is 28% larger than that of nominal HGHG. Further simulations reveal that this quantity becomes larger when the frequency up-conversion amplitude is lower. When the radiation frequency is at 9th of the seed laser, the power stable area of the DEHG is twice as large as that of the nominal HGHG. These results indicate the great stability of DEHG.	ACB	CBA	CBA	BAC	Selection 4
A: This paper is also based in part on work supported by NSF under grant DMS-1440140 while CK was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Fall 2019. FR was supported by NSERC and a Canada research chair. B: The authors are grateful to Rafe Mazzeo, Richard Melrose and Michael Singer for helpful discussions, as well as to an anonymous referee for useful suggestionsC: CK was supported by NSF Grant number DMS-1811995	ABC	BAC	CAB	CBA	Selection 3
A: So, besides the increase of the derivative of the inversion temperature, the increase of the electric charge Q𝑄Qitalic_Q also implies a small displacement of these curves in the direction of increasing pressure, in accordance with EqB: (32). In this panel, one also sees that for small values of the temperature and pressure the curves of constant charge intersect each other. C: In the right panel of Fig. 1, we also see that these non-zero values for the pressure increase with increasing electric charge Q𝑄Qitalic_Q	ACB	BAC	ABC	BCA	Selection 4
A: The above process, as shown in Fig. 1, involves multiple runs, completing the Maxwell Demon-assisted EPR steering. B: After the operation of the qubit, the demon informs Alice of its specific operation when a classical information channel is established between Alice and Bob. At last, Alice declares her result to Bob, based on the demon’s informationC: Once the demon knows Bob’s choice, it will immediately perform a corresponding operation on the qubit before it is measured by Bob. The qubits sent by Alice are all prepared in the same pure state and will be transformed into eigenstates of Bob’s chosen measurement setting by the demon	CAB	CBA	CAB	ACB	Selection 2
A: We shall provide a detailed answer to this question in this paperB: Statement 1) below corresponds to Theorems 1.3, 1.4, and Statement 2) corresponds to Theorem 1.5.C: Let us already paraphrase two special cases of the main statements (which are summarised in section 1.2)	ACB	CAB	BCA	CAB	Selection 1
A: Any given two n-qubit states are then SLOCC-equivalent if a complete set of SLIP measures has the same values for both of them Viehmann et al. (2011)B: SLIP measures provide not only a convenient method for entanglement classification but also its practical detection. Indeed, it was shown that almost all SLOCC equivalence classes can be distinguished by ratios of such measures Gour and Wallach (2013)C: For more than four qubits, however, the size of such a set grows exponentially, making it intractable to use this approach to discriminate SLOCC-equivalent states with more than four qubits Love et al. (2007).	CBA	BCA	ABC	BAC	Selection 4
A: Figure 1B: The two types of universality: The first histogram shows the normalised gaps of the two middle eigenvalues in the spectrum of 5000500050005000 complex Wigner matrices of size 100×100100100100\times 100100 × 100C: The second histogram shows the empirical normalised bulk eigenvalue gaps of a single complex Wigner matrix of size 5000×5000500050005000\times 50005000 × 5000. Both distributions asymptotically approach the Gaudin-Mehta distribution p2subscript𝑝2p_{2}italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT drawn as solid lines, see Section 2.3.	CBA	ACB	ABC	CAB	Selection 3
A: Our main result is that the infected and uninfected bacteria form self-similar travelling waves, which retreat before the expanding phage front and which grow exponentially in timeB: The phage also form a self-similar front, which does not grow exponentially, but this is only in the case where superinfection (where a single bacterium can be simultaneously infected by multiple phage) is permitted; without superinfection the phage wave also grows and changes shape as it developsC: The speeds of these various waves depend on the species tracked (bacteria or phage) and on whether the front or peak of the wave is tracked: the viral wave is retarded, while the wavefront of infected bacteria is advanced, compared to the case without bacterial growth. The advanced speed of the infected bacterial wave does not stem from the initial conditions, as is usual in FKPP theory, but is instead controlled dynamically by the shape of the phage wavefront in a novel selection mechanism. Interestingly, the varying wave speed also causes a non-monotonic variation in the width of the infectious wave, which is narrowest at intermediate growth rates.	BAC	CBA	CBA	ABC	Selection 4
A: An exemplary Gaussian noise model can be thought to be analogous to probabilistically mixing white noise to a state in the finite-dimensional formalism (egB: Note 1C: the Werner state). The coefficients are chosen to be Gaussian, since several noise models can be approximated by the same and it is easier to handle both theoretically and experimentally, thereby being applicable in quantum information processing tasks like continuous-variable quantum key distribution,[108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118]). We also find that qualitatively similar results can be obtained, if, instead of the coefficients coming from a Gaussian or more specifically half-Gaussian distribution, the local states that get mixed with the initial states are thermal, i.e.,	BCA	CAB	BAC	CAB	Selection 3
A: The corresponding symmetry algebra has a basis given by (5.5)B: and ξμsuperscript𝜉𝜇\xi^{\mu}italic_ξ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT are given by the Killing solution (4.2)C: This algebra coincides with that of the linear wave equation when the infinite-dimensional subalgebra is factored out.	BCA	ABC	BAC	CAB	Selection 3
A: Section 3 is devoted to a study of PPT square conjecture. In particular, we will present the general approach to the problemB: In Section 2 we give definitions of the concepts just introduced. Furthermore, we will define and discuss the associated classes of positive mapsC: Keeping in mind that entanglement breaking maps are so important in Quantum Information, in Section 4 we will give a detailed analysis of the relevant tensor cones and associated classes of maps. This will be done for finite dimensional cases.	CAB	CAB	BAC	ACB	Selection 3
A: Therefore, Gould’s original formalism requires substantial modification to properly account for the physics of this extreme environment. Such corrections have gradually been introducedB: Much of this attention is generated by the possibility that DM will thermalise [41, 57, 58] and annihilate within these objects, leading to a potentially observable level of heating [45, 46, 47, 48, 59, 38]. For NSs, additional consequences include the possibility of induced collapse to black holes [60, 43, 61, 62, 63, 44, 64, 65, 66] in the case of asymmetric or non-annihilating DM, or modifications to the rate and gravitational wave signatures of mergers [67, 65, 66, 68]. It is important to note that the DM capture process in NSs differs significantly from that in the Sun, because of the extreme conditions presentC: Ref. [53] presented a new formalism that consistently incorporated many of the relevant physical effects. 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,	BCA	CAB	BAC	CAB	Selection 3
A: 2191 (2022) 1, 012015, contribution to: DERELİ-FS-2021, may have some questinable parts, like the reflection coefficient being a ratio of two infinite numbers • Since it was published in a separate form, we could not publish an Addendum or Correction to this paper. B: The method we used to calculate these coefficients in the first version of this paper, published within the conference proceedings by IOP, in J.Phys.Conf.SerC: Our aim to revise our work is to show that, as also stated in the first form of this paper, to choose another method in the calculation of the reflection coefficients	ABC	CBA	CAB	CAB	Selection 2
A: We find that neural étendue expansion also enables higher fidelity étendue expanded 3D color hologramsB: 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. C: Finally, we also investigate 3D étendue expanded holograms	CBA	BCA	CAB	CBA	Selection 2
A: The numerical procedure we had originally developed in Zeh et alB: the U𝑈Uitalic_U band, which is very vagueC: (2004) assumes that for ν>ν⁢(U⁢band)𝜈𝜈Uband\nu>\nu(\rm U\leavevmode\nobreak\ band)italic_ν > italic_ν ( roman_U roman_band ) the SN flux scales ∝ν−3proportional-toabsentsuperscript𝜈3\propto\nu^{-3}∝ italic_ν start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT,	BAC	ACB	BCA	CAB	Selection 1
A: Many metrics are possible. We can for instance define a measure of energy use (energy efficiency), the degree of removal of specific compounds (selectivity), or a measure of how long membranes remain functional. B: After the setpoints of a process have been defined, metrics (or, equivalently, ‘performance indicators’) come into playC: They describe optimized operation	BCA	CAB	BCA	BCA	Selection 2
A: 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. B: 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 3C: We visualize the two features obtained from experiments on Belem in a 2-D plane as in Figure 8 right. The blue dash line is the classification boundary	ACB	ACB	CAB	ABC	Selection 3
A: The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, IncB: This research has made use of the CIRADA cutout service at URL cutouts.cirada.ca, operated by the Canadian Initiative for Radio Astronomy Data Analysis (CIRADA)C: CIRADA is funded by a grant from the Canada Foundation for Innovation 2017 Innovation Fund (Project 35999), as well as by the Provinces of Ontario, British Columbia, Alberta, Manitoba and Quebec, in collaboration with the National Research Council of Canada, the US National Radio Astronomy Observatory and Australia’s Commonwealth Scientific and Industrial Research Organisation.	CBA	ABC	BAC	CBA	Selection 2
A: The thermalization deteriorates the Fock states according to the map (S7)B: The vertical axis quantifies the maximal mean number of thermal phonons that preserves the presented quantum aspects. Note, the quantum non-Gaussianity and genuine one-phonon quantum non-Gaussianity are identical properties, and therefore their depth is the same for \|1⟩ket1\|1\rangle\| 1 ⟩.C: Figure S4: The thermal depth of the genuine n𝑛nitalic_n-phonon quantum non-Gaussianity (green points), the quantum non-Gaussianity (blue points) and negativity in the Wigner function (red points) that are exhibited by the ideal Fock states	BAC	BCA	CAB	BAC	Selection 2
A: Our random restriction lemma shows that if one randomly fixes most of the inputs to a quantum query algorithm, then the algorithm’s behavior on the unrestricted inputs can be approximated by a “simple” function (say, a small decision tree or small DNF formula)B: Notably, our proof of Theorem 7 does not appeal to Raz-Tal at all, but instead relies on a new random restriction lemma for the acceptance probabilities of quantum query algorithmsC: We then use this random restriction lemma to generalize the usual random restriction proof that, for example, Parity∉𝖠𝖢𝟢Paritysuperscript𝖠𝖢0\textsc{Parity}\not\in\mathsf{AC^{0}}Parity ∉ sansserif_AC start_POSTSUPERSCRIPT sansserif_0 end_POSTSUPERSCRIPT [Hås87].	CAB	CAB	ACB	BAC	Selection 4
A: In CoauthorshipsNet, node means scientist and weights mean coauthorship, where weights are assigned by the original papers. For this network, there is no ground truth about nodes labels, and the numbers of communities are unknown. The CoauthorshipsNet has 1589 nodes, however its adjacency matrix is disconnectedB: Among the 1589 nodes, there are totally 396 disconnected components, and only 379 nodes fall in the largest connected component. For convenience, we use CoauthorshipsNet1589 to denote the original network, and CoauthorshipsNet379 to denote the giant component. To find the number of communities for CoauthorshipsNet, we plot the leading 40 eigenvalues of their adjacency matricesC: Results shown in Figure 8 suggest that the number of communities is 2, where [27] also applies the idea of eigengap to estimate the number of communities for real-world networks. Note that though CoauthorshipsNet1589 is disconnected, we can still apply nDFA and DFA on it since there is no requirement on network connectivity when applying DFA and nDFA. Note that since the overall embeddedness is defined for adjacency matrix that is connected, it is not applicable for CoauthorshipsNet1589.	ABC	BAC	CBA	BAC	Selection 1
A: Another important ingredient of any resource theory is the set of free operations. Free operations satisfy a necessary condition that they do not create resource states from free states, i.e., if ΛΛ\Lambdaroman_Λ is a free operation, thenB: In entanglement theory the free states are separable states [49, 5]. Every state which is not free is called resource stateC: An important ingredient of every quantum resource theory is the set of free states ℱℱ\mathcal{F}caligraphic_F. Typically, this set corresponds to all states which can be easily prepared	BCA	ABC	ABC	CBA	Selection 4
A: This hole effect is corrected employing first-order time-independent perturbation theory Mougeot (2018), as proposed by Vatai. B: Vatai (1970), the vacancy created in a subshell by the capture process has a significant influence on all the orbitalsC: As underlined by Vatai in Ref	BAC	CAB	CBA	BCA	Selection 3
A: Input Ω,KΩ𝐾\Omega,Kroman_Ω , italic_KB: The above analysis gives rise to the following algorithm called Ideal DFSP (short for Distribution-Free SP algorithm) in the oracle case with known population adjacency matrix ΩΩ\Omegaroman_ΩC: Output: ΠΠ\Piroman_Π.	ACB	BAC	ACB	BCA	Selection 2
A: (2021b) and disagrees with results from zoom-ins from Jiang et al. (2019), although both studies focused on more massive galaxiesB: The null correlation with halo spin agrees with Jiang et al. (2019). For the largest, most massive haloes in the ‘FB-S’ galaxies, there may be a slight dependence on the halo spin (bottom right panel). This could be due to the transition from dispersion to rotationally supported galaxies, where the formation of discs becomes increasingly significant (Fall &C: However, there is no such distinction in the GHSR relations when separated into quartiles by either concentration or spin. The null result with concentration agrees with the semi-empirical model from Zanisi et al	BAC	ACB	CAB	BCA	Selection 4
A: In this paper we are interested in the subset of the pionless EFT Lagrangian relevant for beta transitions, that is in the quartic interactions between a proton, a neutron, an electron, and a neutrino.555Interactions with more than two nucleon fields also contribute to nuclear beta transitions (A¿1), in particular the ones with four nucleons and two leptons are referred to as two-body currents in the literatureB: We organize these interactions asC: We comment on this issue in Appendix D	CBA	CBA	ACB	BCA	Selection 3
A: In both cases, the differences between the predictions and the calculated energies are less than 1 mHaB: It is reassuring to note that the short chain length predictions are most accurate throughout the prediction interval, which is a consequence of training the Gaussian processes on short chains. Prediction errors grow with the lengths of the chains because the generalization error increases with system size. This is reflected in the larger confidence intervals that accompany the larger chain length predictions.C: Figure 3 depicts the differences between the energies computed with the UCCSD(T) (left) and AFQMC (right) methods, and their respective GPR predictions	ACB	BAC	BCA	CAB	Selection 3
A: The proof builds, in particular, on the χχ\upchiroman_χ-independence result of [MS20], the dimensional reduction isomorphism of [TK22], and the interpretation of deformed Calabi–Yau completions of total spaces of line bundles of [KM21]B: It is a cohomological version of the χχ\upchiroman_χ-independence conjecture for Gopakumar–Vafa invariants of Calabi–Yau threefolds (see [MT18, Tod17] for details), and is indeed obtained as a special case of the χχ\upchiroman_χ-independence result for general local curves in Calabi–Yau threefolds, also proved in [KK21].C: The following result was very recently proved by Naoki Koseki and Tasuki Kinjo	BAC	ABC	CAB	BCA	Selection 4
A: Community structure can greatly influence disease dynamics on networks [26, 30]B: Salathé and Jones [38] illustrated that changes in community structure are correlated with changes in disease quantities for susceptible–infected–recovered (SIR) dynamics on a networkC: One of their findings is that outbreak duration can achieve a maximum at intermediate modularity values. Inspired by Salathé and Jones [38], we use our adaptation of InfoMap to study an example contagion process that illustrates how absorption in disease dynamics affects community structure, which in turn affects disease spread. We investigate the association between changes in the effective community structure that is induced by the node-absorption rates with disease quantities such as outbreak size and duration.	CBA	BCA	ABC	ACB	Selection 3
A: Here, we develop an efficient heuristicB: The general QNR problem can be formulated in terms of hypergraph flows and solved using LP (see Appendix A)C: Although polynomial-time and provably optimal, the LP-based approach has a very high time-complexity for it to be practically useful	ABC	CBA	CBA	CAB	Selection 4
A: From a mathematical point of view, the Legendre transform we have introduced in Eq. (23) is at the level of probability distributions, and it does not rely on asymptoticsB: Such a microscopic Legendre transform has previously appeared in other fields such as information geometry [47, 48] and the foundations of statistical mechanics [58], and we have formulated the Legendre transform in Eq. (23) by borrowing ideas from information geometry and replacing information theoretic quantities with thermodynamic ones, such as the Shannon entropy with the thermodynamic EPC: This formulation extends the existing connections between information geometry and thermodynamics [59, 60, 61, 62]. Detailed explorations of this geometric picture of stochastic thermodynamics is left for future work.	ABC	BAC	CBA	BAC	Selection 1
A: This stochastic reproductive number plays a pivotal role in delineating the dynamics of extinction and persistence of the coronavirus (disease). Specifically, our attention is directed towards analyzing the infected class (4), i.e. B: The existence of the virus-free equilibrium and endemic equilibrium hinges on computing the basic reproductive numberC: In this section, our focus shifts to establishing the basic reproductive number for the stochastic counterpart	ABC	BAC	CAB	CBA	Selection 3
A: We would like to thank Frédéric Faure, Benjamin Delarue, Stéphane Nonnenmacher, Tobias Weich and Luchezar Stoyanov for very interesting discussionsB: Finally, we would like to thank the anonymous referee for his comments and suggestions that helped to improve the manuscript. The first author is supported from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme with agreement No. 725967.C: Thanks are also due to Colin Guillarmou for pointing out to us that we could use Fried’s result for the proof of Theorem 3	BAC	BCA	BCA	ACB	Selection 4
A: 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]B: 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]C: This NFL phase was later revisited and further reformulated using dimensional regularization and renormalization group (RG) techniques by Moon et al. [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. From recent analytical works, we do have some other examples of nodal NFLs [50, 51] as well.	BCA	CBA	BAC	ABC	Selection 4
A: (2022). This scheme, compatible with a two-dimensional architecture, offers a substantial reduction in the physical resource cost of fault-tolerant quantum computing against the surface code—the archetype for fault-tolerant quantum computing Fowler et al. (2012); Litinski (2019a, b); Fowler and Gidney (2019); Gidney and Fowler (2019). Compared with the most resource efficient mode of surface-code quantum computation Litinski (2019a), our scheme reduces the space-time overhead as a function of the code distance by approximately a factor of 3333. Even accounting for the color code’s lower error threshold using existing decoders, the space-time overhead is reduced by 10% at a physical error rate of 10−3superscript10310^{-3}10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT and 50% at 10−4superscript10410^{-4}10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT. These results elevate consideration of the color-code architecture as a promising pathway for large-scale fault-tolerant quantum computers and motivate further work towards better color-code decoders Kubica and Delfosse (2023); Sahay and Brown (2022); Gidney and Jones (2023); Zhang et alB: In this manuscript we propose a scheme for fault-tolerant quantum computing based on the color code Bombin and Martin-Delgado (2006) where we perform logical operations using an ancillary lattice of qubits as a resource to make Pauli measurements between logical data qubits and sources of magic states Bravyi and Kitaev (2005); Litinski (2019a); Chamberland et al. (2022); Chamberland and Campbell (2022); Cohen et al. (2022) using lattice surgery Horsman et al. (2012); Landahl and Ryan-Anderson (2014); Brown et al. (2017); Litinski and Oppen (2018); Cohen et alC: (2023) and syndrome extraction circuits Fowler (2011); Landahl et al. (2011); Stephens (2014a); Chamberland et al. (2020); Beverland et al. (2021); Bombin et al. (2023); McEwen et al. (2023).	ACB	BAC	CBA	CAB	Selection 2
A: An alternative notion that incorporates the 3⁢D−limit-from3𝐷3D-3 italic_D -compatibility property of a map, is the so-called Yang-Baxter propertyB: The maps that satisfy the Yang-Baxter property will be called Yang-Baxter mapsC: Note that if a 3⁢D−limit-from3𝐷3D-3 italic_D -compatible map is quadrirational, its companion map is a Yang-Baxter map.	BAC	CBA	ABC	CBA	Selection 3
A: For example, when using the information in the linear point [45, 46, 47, 48], no reconstruction is requiredB: Also, the estimated α𝛼\alphaitalic_α from the traditional BAO methods and from the linear point approach may conceptually differ, and a comparison is beyond the scope of this work. C: Also note that the BAO reconstruction procedure is not always required for extracting geometric information in the galaxy clustering	ACB	CAB	BCA	CBA	Selection 3
A: Our results are shown in Table 1. Most of the ordered alloys we studied order in the bcc-derived structures B2 and D03.[43, 44, 45] The B2 structure is a bcc-derived structure with two atoms in the primitive unit cell. This structure is therefore present in alloys with the ratio of 1–1 of two elementsB: On the other hand, D03 structure contains four atoms in the primitive unit cell, so it is present in ordered binary alloys with the 1–3 ratio of constituent elements, or the 1–1–2 ratio in the case of ternary alloys. Among the ordered alloys we studied, the only ones that are not in the bcc-derived structure are Al, FeAl3, CoAl3, Co, andC: Now we present our results for ordered Fe-Co-Al alloys. We start by discussing the computed lattice constants and crystal structure of these alloys	BAC	CAB	BCA	CAB	Selection 3
A: We provide evidence for power law decaying autocorrelations, and for a pattern of hierarchical relaxation when quenched from extremal initial states into the different equilibrium phases. In Sec. V we study the large deviations statistics of dynamical observables by means of numerical MPS. As in other constrained models, the phase transitions at the LD level underpin the slow dynamics and fluctuations seen in typical relaxation trajectories.B: In Sec. IV we study the relaxation dynamicsC: As in the case of the quantum model [8], the stochastic Fredkin spin chain exhibits slow dynamics	CBA	ACB	BCA	CAB	Selection 4
A: EqB: (41) can be written eν=(1+A⁢r2)nsuperscript𝑒𝜈superscript1𝐴superscript𝑟2𝑛e^{\nu}=(1+Ar^{2})^{n}italic_e start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT = ( 1 + italic_A italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT.C: field equations	CAB	BCA	ACB	CAB	Selection 2
A: Why this happens is examined in the Exploration subsectionB: But it shows that BO can be much faster, even if care must be taken when choosing the models and priors.C: By inspection of Fig. 2 one can see that the selection examples have many local optima, and so would suffer from this. On the London data one of the GP models does much better than either baseline, but the other two perform worse	BCA	CAB	CBA	ABC	Selection 1
A: The in-situ proof-of-principle in nature is however still to be shown. We have already demonstrated that cosmic ray air showers create particle cascades in ice with properties very similar to neutrino-induced cascades. Therefore, the radar detection of these in-ice cosmic-ray-induced particle cascades would show the proof-of-principle of the method in nature. B: This indicates that this method could be used to detect neutrino-induced particle cascades in iceC: It was recently shown at the Stanford Linear Accelerator Center (SLAC) that high-energy particle cascades in dense media can be probed using the radar echo technique [7]	BCA	BAC	CBA	ACB	Selection 3
A: Accordingly, the back-propagation of the ghost propagator’s non-analyticities into the gluon DSE at least requires a perturbative three-loop computation. While certainly being challenging, this may be within the technical range of perturbative computations in the CF model, and is very desirableB: In a massive extension of Yang-Mills theory, complex conjugate poles may occur in the gluon propagator at one-loop. This implies that their impact on the ghost propagator may be visible at two-loop in the ghost gap equationC: The back-propagation of the additional cuts poses a major obstruction that only can be circumvented by intricate relations between the complex structures of propagators and that of the vertices, in particular the ghost-gluon vertex. Signs for the latter gathered in perturbation theory at least require a three-loop analysis of the ghost DSE as argued in Section IV.2. Such an analysis, while highly desirable, has not been undertaken yet in the literature.	ABC	BAC	BCA	BCA	Selection 2
A: Reconstructing an approximation of the data from a few set of kPCA components is non trivial problem and its presentation is here omitted (see Mendez, 2023b )B: Interested readers are referred to Bakir et al., (1999)C: We close this section with an application, illustrated in Figure 4.	ACB	BCA	ACB	ABC	Selection 4
A: In Sec. VI, we show that the anomaly equation at the edge can be derived variationally by adding a boundary chiral boson action, which is non-linearly coupled to the edge density, to the topological fluid action. In Section VII, we offer a heuristic boundary layer interpretation of the edge anomaly equation. The paper concludes with a discussion and outlook in Section VIII.B: In Section IV, we demonstrate the duality between the topological fluid action and the composite boson (CSGL) action. Section V discusses the incompatibility of the no-slip condition with the FQH edge dynamicsC: The paper is organized as follows: we start with a brief review of the superfluid dynamics of the composite boson model in Section II. In Section III, we introduce a variational formulation for these fluid dynamic equations, by including additional topological terms to the hydrodynamic action	BCA	CAB	BAC	CBA	Selection 4
A: A dispersive dipole magnet allows the reconstruction of the energy spectrum. The X-rays produced in the beam-target interaction are measured with profile monitors and spectrometers (not shown).B: The beam after the interaction passes through an optional profile monitor which allows to measure the transverse momentum spread and deflection. It is thereafter imaged with a quadrupole triplet onto a scintillating screenC: Figure 5: Sketch of the E-336 experimental setup at FACET-II. The electron bunch is focused on a target with a hole array	ABC	ABC	CBA	BAC	Selection 3
A: Our dataset includes 31 points from the Hubble dataset, 1048 points from the Pantheon SN samples, and 6 points from the BAO datasetB: To investigate the observational characteristics of our cosmological model, we leverage the latest cosmic Hubble, SN observations, and BAOC: Employing Bayesian analysis, we utilize the likelihood function and the Markov Chain Monte Carlo (MCMC) method implemented in the emcee Python library [65].	CBA	BAC	BCA	BCA	Selection 2
A: 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). 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)B: 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. Applying this new framework, we quantify the roles of classical and non-classical crystallization mechanisms at different precipitation rates and supersaturation levels. This model framework can also be applied to other crystal systems and tested with other paired element and isotope measurements.C: 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	ACB	CBA	BCA	CAB	Selection 1
A: In Section 2, we review the Lie point symmetry analysis of (1.1)B: Solutions of the invariance condition as a Riccati equation in the independent variable I⁢(x)𝐼𝑥I(x)italic_I ( italic_x ) has been discussed. Section 3 is devoted to five examples of PDEs in connection with the study of Brownian motion and mathematical finance. The results of Section 2 are applied to them to construct solutions of physical interest. C: The structure of this paper is as follows	CAB	BCA	ABC	CAB	Selection 2
A: The local effects typically amount to a relatively small contribution compared to the terms in (12). Moreover, (12) is sufficient to obtain a true understanding of the nature of RSDs in the angular power spectrum. However, these effects need to be taken into account in the light of upcoming precision cosmological era. B: In reality, RSDs do exist in the large-scale structure among several effects other than the matter density; these include the Doppler and the potential effects (neglecting integral effects, which are negligible at z≲ 1less-than-or-similar-to𝑧1z\,{\lesssim}\,1italic_z ≲ 1, being the z𝑧zitalic_z of interest in this work)C: These effects together (apart from density and RSDs) are, henceforth, termed “local effects.” They surface in the overdensity in redshift space through the redshift and the volume perturbations, accordingly	BAC	CAB	BCA	BCA	Selection 2
A: The proofs of these results are given in the subsequent sections.B: The next section briefly recalls only the essential part of [8] used in the sequelC: The main results obtained in the present paper are Theorems 3.1 and 4.1 from sections 3 and 4 respectively	ABC	CAB	BCA	ABC	Selection 2
A: The VTX is described in detail in Refs. [22, 18]B: It is composed of two arms, each with \|η\|<1𝜂1\|\eta\|<1\| italic_η \| < 1 and Δ⁢ϕ≈ 0.8⁢πΔitalic-ϕ0.8𝜋\Delta\phi\,{\approx}\,0.8\piroman_Δ italic_ϕ ≈ 0.8 italic_π coverageC: Each arm has four layers around the	BCA	ACB	ABC	BCA	Selection 3
A: AGN can also exhibit short-term variability for periods of hours or days, but we caution that the detection of a high-energy neutrino alert is a process that requires a substantial fluence at the IceCube detector, even after accounting for the significant Eddington bias associated with cosmic neutrino detection (Strotjohann et al., 2019)B: Such highly luminous rapid neutrino flares are not well motivated theoretically, it is therefore unlikely that short AGN flares are indicators of neutrino production. C: The corresponding neutrino flux that is required is inversely proportional to the duration of neutrino emission, and therefore associating a neutrino detection with a temporary electromagnetic signature lasting hours or days would imply an extremely high average neutrino flux for the duration of that signature	BCA	ACB	BAC	BCA	Selection 2
A: Fig. 4 for the pTsubscript𝑝𝑇p_{T}italic_p start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT range 1.0<pT<1.21.0subscript𝑝𝑇1.21.0<\mbox{$p_{T}$}<1.21.0 < italic_p start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT < 1.2 GeV/c𝑐citalic_cB: The four panels correspond to four different centrality selectionsC: Each panel	ABC	BCA	CAB	CAB	Selection 1
A: In this section we discuss examples with concrete codesB: They serve several purposesC: Firstly, they show that invariants we proposed are nontrivial, calculable and yield what is expected on physical grounds in models which are already well understood. Secondly, they support our physical interpretation of mathematical objects and the conjecture that braiding is non-degenerate. Finally, the last example illustrates certain technical complication that does not arise for codes with	BAC	ABC	BAC	ACB	Selection 2
A: After preparing the initial ground state, the system will evolve according to the quench profile (2)B: To this end, we render the neural network parameters to be functions of time. Then, the parameters will be computed at every time step with the time-dependent VMC method, by minimizing the distances δ𝛿\deltaitalic_δ between the exact time evolution and the approximate variational evolution C: Therefore, the wave function should also depend on time	BAC	CAB	ACB	BAC	Selection 3
A: In order to show the effect of different space intervals and time intervals, two groups of comparison tests have been doneB: Table 2 shows the result of different hℎhitalic_h. C: First, we choose different space interval hℎhitalic_h to observe the performance of our model	CAB	ACB	BCA	BAC	Selection 2
A: The microcentrifuge tube was then agitated for 1 week on an orbital shaker at 100 rpm (PSU-10i Grant Bio, UK). B: In brief, 5 mM oleic acid vesicles were prepared in a buffer that contained 100 mM Na-bicine (pH 8.3) and up to 500 mM sucrose by adding the appropriate amount of oleate micellesC: Vesicles were prepared by the self-assembly method [1]	CBA	ABC	BCA	ABC	Selection 1
A: To be sure, our prescription has only a limited range of validity; nevertheless, it serves as a useful illustration of several related techniques for computing and understanding holographic entanglement.B: If we treat the limits correctly, we find a simple picture of holographic entanglement entropy: in certain regimes, it is just the logarithm of a scalar propagatorC: The order-of-limits issue discussed above is a classic one Fischetti:2014zja ; Headrick:abc	ACB	ABC	CAB	CBA	Selection 4
A: At low noise, to achieve a quantum semantic fidelity of 0.70.70.70.7, QSC requires around 50%percent5050\%50 % quantum communication resources compared to semantic-agnostic QCNs using pruning data compression without any semantic concept extractionB: This demonstrate the advantages of QSC accurately sending and reconstructing semantic information.C: In Figure 3, we show the quantum semantic fidelity achieved against the amount of quantum communication resources used for \|𝒳\|=500𝒳500\lvert\mathcal{X}\rvert=500\| caligraphic_X \| = 500	CBA	CBA	BCA	ABC	Selection 3
A: solutions (2019) The non ideal behavior illustrated here occurs at frequencies smaller than this by at least two and at most eight orders magnitude. B: However, the remaining deviation from ideal behavior is caused by the equivalent series inductance that typically produces self-resonance frequencies above 100100100100 MHzC: These effects are mitigated when using a vacuum capacitor	ABC	CBA	CAB	ACB	Selection 2
A: This approach to screw theory via special vector fields has the advantage of being independent of any reduction pointB: We stress that in this work the screws will always be elements of a 6-dimensional real vector space, hence our notion of screw is equivalent to that of torseur, and conforms with the terminology used by Dimentberg [9]. This terminology differs from the original terminology used by Sir R. Ball.C: It goes back, at least, to Lovell III’s PhD thesis [14] and has long been used in French engineering universities where screws are called torseurs [24, 4, 25, 19, 6]	ACB	ABC	CBA	ABC	Selection 1

**A**: Based on Eqs**B**: (58), (61), (62), and the time translational symmetry, the line element on the thin shell ΣΣ\Sigmaroman_Σ is a constant over time, i.e.,**C**: Since the system has a time translational symmetry, we can extend these results to all other times

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**A**: Table 5: Mass differences in the charmonium spectrum in MeV compared to experimental values (calculated from 0954-3899-37-7A-075021 ; the value for the 1P hyperfine splitting is from Olsen:2012xn )**B**: For the results of this paper, the first error denotes the statistical uncertainty and the second error denotes the uncertainty from setting the lattice scale666For spin-dependent quantities the indirect contribution of the scale setting uncertainty to the kappa tuning uncertainty is sizable. Our scale setting error only accounts for the direct uncertainty associated with the setting of the lattice scale..**C**: Bars denote spin-averaged values

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**A**: Then, in the mean-field limit, a corresponding Gross-Pitaevskii equation is introduced and numerically solved for different initial conditions Raghavan ; Ostrovskaya ; Ananikin . 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 **B**: 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. in situations when the local interaction energy is much larger than the single-particle tunneling energy. For example, it was shown that for initially imbalanced occupations the dynamics is heavily affected by strong interactions DuttaS . 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.**C**: 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

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**A**: SUSY has found its way into many applications, including differential geometry [33], the study of coherent states [7, 12] and BCS theory [34]**B**: In SUSY, one assumes that the ground state energy of the quantum mechanical system is zero since ultimately one only measures energy changes, not absolute energy values, and seeks to write a Hamiltonian as**C**: The theory of supersymmetric quantum mechanics (SUSY) generated insights from the factorization method for more general quantum mechanical systems [11]

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**A**: Then this is an equilibrium of having maximal entropy hence is unstable in light of**B**: precisely Rinitial=−G⁢m2/2⁢Esubscript𝑅initial𝐺superscript𝑚22𝐸R_{\rm initial}=-Gm^{2}/2Eitalic_R start_POSTSUBSCRIPT roman_initial end_POSTSUBSCRIPT = - italic_G italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_E**C**: volume is fine-tuned i.e

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**A**: The uncontracted indices are particularly important in the description of gauge fields inside the bulk and hence will become important for the extension towards double field theory. In a holographic code there are two types of uncontracted indices, namely the bulk indices and the boundary indices**B**: All other indices are contracted between tensors arising on different layers of tilings. The bulk and the boundary indices are however not separated. Because the code is essentially an isometric embedding of the bulk Hilbert space into the boundary Hilbert space, the two indices are related. Each polygon provides an isometry from incoming and bulk indices to outgoing indices. **C**: These expansions form isometric encoding maps which will encode a certain number of logical qubits into the emerging physical qubits. Holographic quantum error correction codes are then implemented by contracting perfect tensors taking into account the geometry of the bulk space (in this case hyperbolic) and its tiling by corresponding polygons

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**A**: both cases can occur**B**: This is obvious for flat manifolds admitting a**C**: τ⁢(Rg)≠0𝜏subscript𝑅𝑔0\tau(R_{g})\not=0italic_τ ( italic_R start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ) ≠ 0 or τ⁢(Rg)=0𝜏subscript𝑅𝑔0\tau(R_{g})=0italic_τ ( italic_R start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ) = 0 i.e

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**A**: As discussed in the introduction of Section IV, here we do not have analogues of ‘in and an out’ states. The main variable for the transition are the angles at the sphere ΔΔ\Deltaroman_Δ which belong neither to the ‘past’ nor to the ‘future’ of the transition**B**: Also, it is not clear what would serve here as the ‘measurement’ or ‘collapse’ and indeed we would not here have an analogue of a ‘time’ operator. Therefore, the analogy of the black to white hole transition with a quantum mechanical tunneling or the flight time problem is far from perfect. **C**: We also note that the problem we study here seems reminiscent of the arrival time problem Aharonov:1997md ; Delgado:1997tj ; Grot:1996xu .101010We thank an anonymous referee for pointing us to relevant literature. It is not clear to us whether a good analogy to this well studied problem and the problem we study here can be arrived at

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**A**: If the exponent in (61)**B**: Unfortunately, that is not the case here**C**: Kd/4⁢(12⁢|Mi|⁢Ωi⁢B2)subscript𝐾d412subscript𝑀isubscriptΩisuperscript𝐵2K_{\rm d/4}(\tfrac{1}{2}|M_{\rm i}|\Omega_{\rm i}B^{2})italic_K start_POSTSUBSCRIPT roman_d / 4 end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG | italic_M start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT | roman_Ω start_POSTSUBSCRIPT roman_i end_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )

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**A**: in the proof of Proposition 2.3 of [Be], can not easily be generalized to the twisted case**B**: To prove an analogous result, we need to assume that ΓΓ\Gammaroman_Γ stabilizes a Borel subalgebra of 𝔤𝔤\mathfrak{g}fraktur_g, and use Lemma 2.5 crucially**C**: It will be interesting to see if this assumption can be removed.

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**A**: By dimensional analysis, this is always possible, as there are at most two propagators expanded beyond leading order that contribute to the m∞4superscriptsubscript𝑚4m_{\infty}^{4}italic_m start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT-term**B**: This choice of momenta simplifies future integrations, as with it one only needs to consider the two regions P≫Qmuch-greater-than𝑃𝑄P\gg Qitalic_P ≫ italic_Q and Q≫Pmuch-greater-than𝑄𝑃Q\gg Pitalic_Q ≫ italic_P to obtain the double-logarithm coefficient. The full expression corresponding to eq. (10) in the main text, that is, the two-loop HTL diagrams re-expanded to two self-energy insertions to obtain possible double-logarithm corrections at N3LO, is **C**: When performing the expansions in m∞2superscriptsubscript𝑚2m_{\infty}^{2}italic_m start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, we label the momenta using linear changes of variables P↦−P−Qmaps-to𝑃𝑃𝑄P\mapsto-P-Qitalic_P ↦ - italic_P - italic_Q and Q↦−P−Qmaps-to𝑄𝑃𝑄Q\mapsto-P-Qitalic_Q ↦ - italic_P - italic_Q when necessary, so that no factors of ΠT/L⁢(P+Q)subscriptΠ𝑇𝐿𝑃𝑄\Pi_{T/L}(P+Q)roman_Π start_POSTSUBSCRIPT italic_T / italic_L end_POSTSUBSCRIPT ( italic_P + italic_Q ) appear

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**A**: In fact, in Ref.Mitra (1993), the author constructed a fundamental field theory of semions only, as certain generalized Grassman number valued field theory**B**: If anyons may be the excitation modes of a fundamental field, the field cannot be real, complex, or even Grassmann number valued**C**: Unfortunately, more general constructions, in particular of non-Abelian anyons, is yet available and is our undergoing work.

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**A**: But by Alaoglu’s theorem (e.g**B**: [24, p. 484]) the unit**C**: \mu_{0}}\frac{V_{0}}{V_{1}}\Big{)}\frac{\mu_{1}}{V_{1}}divide start_ARG italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ↦ - ∫ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT roman_log ( divide start_ARG roman_d italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG roman_d italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG divide start_ARG italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) divide start_ARG italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG is continuous

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**A**: the BV operator is compatible with a product and a bracket, both given by counting pairs of pants but with asymptotic markers treated differently.**B**: [1]), i.e**C**: In characteristic zero, symplectic cohomology SHSH{\operatorname{SH}}roman_SH is known to have the structure of a BV-algebra (see e.g

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**A**: However, this method can not be applied in arbitrary high-dimensional mixed states**B**: As we know, a star on a Bloch sphere can represent a pure state, and a set of stars on a Bloch sphere can represent a high-dimensional pure states**C**: Our result is not easy to be generalized to arbitrary high-dimensional mixed-spin systems, such as a mixed-spin (s,1)𝑠1(s,1)( italic_s , 1 ) system. Because we can not directly use an effective pseudo spin 1111 to describe it.

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**A**: , ψ|Γ=Cevaluated-at𝜓Γ𝐶\psi|_{\Gamma}=Citalic_ψ | start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT = italic_C (constant)), corresponding to having**B**: italic_e **C**: the boundary condition (∇∥ψ)|Γ=0evaluated-atsubscript∇parallel-to𝜓Γ0\left(\nabla_{\parallel}\psi\right)|_{\Gamma}=0( ∇ start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT italic_ψ ) | start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT = 0 (i.e.,formulae-sequence𝑖𝑒i.e.,italic_i

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**A**: In [39] the authors applied a version of para-controlled calculus to the stochastic wave equation setting, and obtained almost sure local well-posedness for a quadratic wave equation with additive white noise on 𝕋3superscript𝕋3\mathbb{T}^{3}blackboard_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT**B**: Finally, we would like to mention two recent results of Gubinelli-Koch-Oh [39] and Bringmann [15]**C**: This relied on several new ingredients, including the analysis of a random operator (which is different from and unrelated to the random averaging operator introduced in the current paper).

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**A**: The choices of branch and labeling will depend on the value of m∈ℂ∗𝑚superscriptℂm\in\mathbb{C}^{*}italic_m ∈ blackboard_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT in the following way:**B**: For the definition of Stokes data, we need to fix a branch of the Log and a labeling of the sectors determined by anti-Stokes rays**C**: The reason for the choices made below will become apparent in subsequent sections, where we start comparing and matching the twistor coordinates for harmonic bundles with the twistor coordinates of the Ooguri-Vafa space

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**A**: The authors thank the Parker Solar Probe team, especially the FIELDS and SWEAP teams for their support**B**: The FIELDS experiment on the Parker Solar Probe spacecraft was designed and developed under NASA contract NNN06AA01C**C**: The authors wish to acknowledge helpful conversations with Dr. Ivan Vasko.

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**A**: square roots**B**: As a result the expression under square**C**: Indeed, we have bS=−5subscript𝑏𝑆5b_{S}=-5italic_b start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = - 5, bI=4subscript𝑏𝐼4b_{I}=4italic_b start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT = 4 and D=bS2−bI2=(3)2𝐷superscriptsubscript𝑏𝑆2superscriptsubscript𝑏𝐼2superscript32D=b_{S}^{2}-b_{I}^{2}=(3)^{2}italic_D = italic_b start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_b start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( 3 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

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**A**: The experiments used six gases, ammonia, acetaldehyde, acetone, ethylene, ethanol, and toluene, presented in arbitrary order and at variable concentrations. Chemical interferents were also presented to the sensors between batches, and the time between presentations varied, both of which contributed to further sensor variability. The dataset thus exemplifies sensor variance due to contamination and variable odor concentration in a controlled setting. **B**: Every batch contains between 161161161161 to 3,60036003{,}6003 , 600 samples, and each sample is represented by a 128-dimensional feature vector; 8 features each from 16 metal oxide-based gas sensors. These features summarizing the time series sensor responses are the raw and normalized steady-state features and the exponential moving average of the increasing and decaying transients taken at three different alpha values**C**: Experiments in this paper used the gas sensor drift array dataset [7]. The data consists of 10 sequential collection periods, called batches

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**A**: In Figs. (3) and (4) we have demonstrated a possible way to minimize error and achieve high fidelity quantum gate operations using several bounded control pulses (e.g., π𝜋\piitalic_π, CORPSE, SCORPE, symmetric and asymmetric pulses) acting on a one qubit system in presence of random telegraph noises. Here we have compared the fidelities obtained from the pulses discussed above**B**: More precisely, symmetric pulse (see Fig. 3(a)) provide large fidelity for the small energy amplitudes of noise strength which may be useful for the experiments that perform at low temperatures. On the other hand, CORPE pulse (see Fig. 3(b,c)) provide large fidelity for the large energy amplitudes of noise strength which may be useful for the experiments that perform at high temperatures. Finally in Fig. 7, I have shown that when π𝜋\piitalic_π pulse acts in x direction, CORPSE pulse acts in y direction and SCORPSE pulse acts in z-direction in presence of arbitrary low and high temperature measurements noise conditions, large fidelity recovery can be achieved and may consider for implementing in future for electronic circuits design to minimize error.**C**: We conclude that in the limit of vanishing noise correlation time, π𝜋\piitalic_π-pulse can be used for the measurement of achieving high fidelity due to its small gate operation time. In the limit of large correlation time, two pulses namely symmetric pulse and CORPSE pulse were identified to achieve recovery of high fidelity

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**A**: We have obtained and studied the thermodynamics of black brane solutions in asymptotically Lifshitz spacetimes in κ𝜅\kappaitalic_κ-deformed Horndeski gravity in four dimensions**B**: The main property of these solutions is that they can assume arbitrary values of the critical exponents z𝑧zitalic_z associated with Lorentz symmetry violation**C**: Usually, when asymptotically Lifshitz spacetimes are considered in Horndeski gravity, the critical exponents typically acquire some specific fixed values for which some solution can be found. What we have shown here is that the inclusion of the κ𝜅\kappaitalic_κ-deformation in Horndesky theory releases these constraints on the critical exponents allowing them to assume arbitrary values and then finding solutions in this general case. This is a consequence of the auxiliary equation (17) that comes from the κ𝜅\kappaitalic_κ-algebra leading to the solution of the equations of motion for any value of the critical exponent.

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**A**: 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. (1992, 1994); Cronin et al**B**: (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.**C**: (2009); Bollinger et al. (1996); Döring et al

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**A**: These were introduced in joint work with Sven Meinhardt in [DM20], as part of a project to realise the cohomological Hall algebras defined by Kontsevich and Soibelman [KS11] as positive halves of generalised Yangians**B**: The construction of the BPS Lie algebra for arbitrary symmetric quivers with potential is recalled in §2.4**C**: Note that the BPS Lie algebra is defined by a quite different perverse filtration, on vanishing cycle cohomology of a different Calabi–Yau category.

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**A**: (2023) and CMS Hayrapetyan et al. (2024) experiments.**B**: Amongst its many applications, we note three in particular. First, it facilitates studies of the J/ψ⁢J/ψ𝐽𝜓𝐽𝜓J/\psi J/\psiitalic_J / italic_ψ italic_J / italic_ψ system, where resonance-like structures were reported by the LHCb experiment Aaij et al**C**: (2020c) and confirmed by both the ATLAS Aad et al

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**A**: 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)**B**: The chain with respect to which one is considering representation stability there is of course still the chain of symmetric groups. Thus, representation stability with respect to a chain of diagrammatically defined algebras is not considered in [1] (Barter, Entova-Aizenbud, Heidersdorf).**C**: 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. 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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**A**: (2012); Wu et al. (2018); Chen et al. (2017); Karliner et al. (2017); Bai et al. (2019); Wang (2017); Richard et al**B**: Even before the discovery, fully-heavy multiquark states have already been studied Iwasaki (1975); Chao (1981); Ader et al. (1982); Zouzou et al. (1986); Heller and Tjon (1985); Lloyd and Vary (2004); Barnea et al. (2006); Vijande et al. (2009); Berezhnoy et al**C**: (2017); Anwar et al. (2018); Debastiani and Navarra (2019); Richard et al. (2018); Esposito and Polosa (2018); Wang and Di (2019); Liu et al. (2019b); Wang et al. (2019); Chen (2020); Lundhammar and Ohlsson (2020).

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**A**: The results**B**: For the model presented in Ref. Farzan:2015doa we have defined By≡B1−y⁢B2−(3−y)⁢B3subscript𝐵𝑦subscript𝐵1𝑦subscript𝐵23𝑦subscript𝐵3B_{y}\equiv B_{1}-yB_{2}-(3-y)B_{3}italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ≡ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_y italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - ( 3 - italic_y ) italic_B start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT where y𝑦yitalic_y is an arbitrary constant**C**: studied in the literature

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Selection 2

**A**: A guiding principle, on which we will elaborate, asserts that, when many valleys are present, chaos reins: the system is fragile, with small perturbations causing profound changes in the form of the ground state.**B**: In such systems, the ground state, namely the value x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X that minimizes H𝐻Hitalic_H, may be joined in the landscape by a host of near ground states, resting at the bases of various local valleys, at more or less removed locations**C**: The presence of multiple competing near-minimizers has significance in problems concerning the response of the system to perturbations of its law suffered by changes in parameters such as temperature or in the realization of the disorder

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Selection 1

**A**: 1, ae=3ae[l⁢o⁢c]a_{e}=3\,a{}_{e}^{[loc]}italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = 3 italic_a start_FLOATSUBSCRIPT italic_e end_FLOATSUBSCRIPT start_POSTSUPERSCRIPT [ italic_l italic_o italic_c ] end_POSTSUPERSCRIPT leading to ce=13⁢ce[l⁢o⁢c]subscript𝑐𝑒13superscriptsubscript𝑐𝑒delimited-[]𝑙𝑜𝑐c_{e}=\frac{1}{\sqrt{3}}\,c_{e}^{[loc]}italic_c start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 3 end_ARG end_ARG italic_c start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_l italic_o italic_c ] end_POSTSUPERSCRIPT per c∝a−1/2proportional-to𝑐superscript𝑎12c\propto a^{-1/2}italic_c ∝ italic_a start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT,**B**: In Fig**C**: to ae>ae[l⁢o⁢c]subscript𝑎𝑒superscriptsubscript𝑎𝑒delimited-[]𝑙𝑜𝑐a_{e}>a_{e}^{[loc]}italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT > italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT [ italic_l italic_o italic_c ] end_POSTSUPERSCRIPT

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**A**: 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**B**: (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. (2016), and adiabatically scanning through a quantum phase transition Lee (2006); Huang et al. (2018).**C**: (1997); Moore et al. (1999), quantum nondemolition measurement Appel et al. (2009); Kuzmich et al. (1998); Louchet-Chauvet et al. (2010); Hammerer et al. (2010), one-axis twisting Schleier-Smith et al

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Selection 3

**A**: The analytical and the numerical results for the Einstein-de Sitter universe are identical, thus confirming our numerical implementation. Despite the difference in the cosmological background evolution between the left and central panels compared to the right panel, the results are quite similar. **B**: It is opposed there to the numerical result for an Einstein-de Sitter universe in the centre panel for comparison, and to the functional derivative of D+subscript𝐷D_{+}italic_D start_POSTSUBSCRIPT + end_POSTSUBSCRIPT obtained by evaluating (28) numerically as described in the following subsection, shown in the bottom panel**C**: This analytic result is shown in the top panel of Fig. 4

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Selection 4

**A**: 1b). The following impossibility of certain clean algorithms is a corollary of Theorem 3.**B**: In other words, the ancilla output is independent of the unknown oracle**C**: Clean algorithms enable ancillae recycling and preserve coherences if used inside an interferometer (Fig

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Selection 2

**A**: Yanbing Zhu: Review and editing (equal); Conceptualization (supporting). Evan J. Read: Conceptualization (lead); Review and editing (equal); Supervision (equal)**B**: Peter Schindler: Writing – original draft (lead); Review and editing (equal); Software (lead); Methodology (lead); Visualization (lead); Investigation (lead); Data curation (lead); Conceptualization (supporting); Supervision (equal); Resources (lead). Evan R**C**: Antoniuk: Review and editing (equal); Methodology (supporting); Conceptualization (supporting). Gowoon Cheon: Review and editing (equal); Conceptualization (supporting)

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Selection 4

**A**: blowup profile, such that the Wick-ordered L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff does not exclude this blowup profile for any cutoff size K>0𝐾0K>0italic_K > 0**B**: See, in particular, Lemma 3.4 and the proof of (3.42)**C**: We also mention related works [33, 13, 53, 12, 46, 54] on the non-normalizability (and other issues)

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Selection 3

**A**: We performed hard X-ray diffraction experiments at the BW5 beamline at HASYLAB, Deutsches Elektronen-Synchrotron, Hamburg, Germany**B**: The sample was cut into a thin slice (d=1.4𝑑1.4d=1.4italic_d = 1.4 mm) in order to reduce the absorption, meaning that 10 % of the incoming beam was transmitted through the sample. **C**: Here we looked for a signature of charge-density waves

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Selection 1

**A**: We strongly emphasize that there is a great deal of room for improvement on the techniques described here**B**: We have not, for example, carefully assessed the most effective method for laying out the grid of data on which we store information about adiabatic backreaction and waveform amplitudes, nor have we thoroughly investigated efficient methods for interpolating these data to off-grid points (e.g., [16]). These important points will be studied in future work, as we begin assessing how to take this framework and use it to develop EMRI waveforms in support of LISA data analysis and science studies.**C**: In Sec. VI, we present various important technical details describing how we implement this framework for the results we present in this paper

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Selection 4

**A**: The reason is that although an increase of beam current will enhance the radiation power, the coexisting increase in modulation depth and the fixed dispersion strength result in the over-compressed and a smaller bunching factor, which is not conducive to the radiation power growth. This feedback mechanism makes radiation of DEHG not naturally sensitive to the beam current. Fig. 3(c, d) display the radiation power versus seed laser power and electron beam current in nominal HGHG and DEHG, respectively. One can see that in nominal HGHG, the radiated power increases as the electron beam current rises**B**: Fig. 3(a, b) present the modulation depth as functions of the seed laser power and electron beam current. For nominal HGHG, the modulation depth is related to the seed laser intensity, but not to the electron beam current. For DEHG, however, the modulation depth is correlated with both the seed laser power and the electron beam current. This phenomenon makes the radiation of DEHG more tolerant to the variation of current**C**: In DEHG, however, the radiation power does not increase with current in a certain area due to the feedback mechanism. Power stable area can be regarded as a region in which the radiation power decreases by no more than 5% due to jitters in the electron beam current and seed laser power. In this case, the power stable area of DEHG is 28% larger than that of nominal HGHG. Further simulations reveal that this quantity becomes larger when the frequency up-conversion amplitude is lower. When the radiation frequency is at 9th of the seed laser, the power stable area of the DEHG is twice as large as that of the nominal HGHG. These results indicate the great stability of DEHG.

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Selection 4

**A**: This paper is also based in part on work supported by NSF under grant DMS-1440140 while CK was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Fall 2019. FR was supported by NSERC and a Canada research chair. **B**: The authors are grateful to Rafe Mazzeo, Richard Melrose and Michael Singer for helpful discussions, as well as to an anonymous referee for useful suggestions**C**: CK was supported by NSF Grant number DMS-1811995

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Selection 3

**A**: So, besides the increase of the derivative of the inversion temperature, the increase of the electric charge Q𝑄Qitalic_Q also implies a small displacement of these curves in the direction of increasing pressure, in accordance with Eq**B**: (32). In this panel, one also sees that for small values of the temperature and pressure the curves of constant charge intersect each other. **C**: In the right panel of Fig. 1, we also see that these non-zero values for the pressure increase with increasing electric charge Q𝑄Qitalic_Q

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Selection 4

**A**: The above process, as shown in Fig. 1, involves multiple runs, completing the Maxwell Demon-assisted EPR steering. **B**: After the operation of the qubit, the demon informs Alice of its specific operation when a classical information channel is established between Alice and Bob. At last, Alice declares her result to Bob, based on the demon’s information**C**: Once the demon knows Bob’s choice, it will immediately perform a corresponding operation on the qubit before it is measured by Bob. The qubits sent by Alice are all prepared in the same pure state and will be transformed into eigenstates of Bob’s chosen measurement setting by the demon

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Selection 2

**A**: We shall provide a detailed answer to this question in this paper**B**: Statement 1) below corresponds to Theorems 1.3, 1.4, and Statement 2) corresponds to Theorem 1.5.**C**: Let us already paraphrase two special cases of the main statements (which are summarised in section 1.2)

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Selection 1

**A**: Any given two n-qubit states are then SLOCC-equivalent if a complete set of SLIP measures has the same values for both of them Viehmann et al. (2011)**B**: SLIP measures provide not only a convenient method for entanglement classification but also its practical detection. Indeed, it was shown that almost all SLOCC equivalence classes can be distinguished by ratios of such measures Gour and Wallach (2013)**C**: For more than four qubits, however, the size of such a set grows exponentially, making it intractable to use this approach to discriminate SLOCC-equivalent states with more than four qubits Love et al. (2007).

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Selection 4

**A**: Figure 1**B**: The two types of universality: The first histogram shows the normalised gaps of the two middle eigenvalues in the spectrum of 5000500050005000 complex Wigner matrices of size 100×100100100100\times 100100 × 100**C**: The second histogram shows the empirical normalised bulk eigenvalue gaps of a single complex Wigner matrix of size 5000×5000500050005000\times 50005000 × 5000. Both distributions asymptotically approach the Gaudin-Mehta distribution p2subscript𝑝2p_{2}italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT drawn as solid lines, see Section 2.3.

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Selection 3

**A**: Our main result is that the infected and uninfected bacteria form self-similar travelling waves, which retreat before the expanding phage front and which grow exponentially in time**B**: The phage also form a self-similar front, which does not grow exponentially, but this is only in the case where superinfection (where a single bacterium can be simultaneously infected by multiple phage) is permitted; without superinfection the phage wave also grows and changes shape as it develops**C**: The speeds of these various waves depend on the species tracked (bacteria or phage) and on whether the front or peak of the wave is tracked: the viral wave is retarded, while the wavefront of infected bacteria is advanced, compared to the case without bacterial growth. The advanced speed of the infected bacterial wave does not stem from the initial conditions, as is usual in FKPP theory, but is instead controlled dynamically by the shape of the phage wavefront in a novel selection mechanism. Interestingly, the varying wave speed also causes a non-monotonic variation in the width of the infectious wave, which is narrowest at intermediate growth rates.

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Selection 4

**A**: An exemplary Gaussian noise model can be thought to be analogous to probabilistically mixing white noise to a state in the finite-dimensional formalism (eg**B**: Note 1**C**: the Werner state). The coefficients are chosen to be Gaussian, since several noise models can be approximated by the same and it is easier to handle both theoretically and experimentally, thereby being applicable in quantum information processing tasks like continuous-variable quantum key distribution,[108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118]). We also find that qualitatively similar results can be obtained, if, instead of the coefficients coming from a Gaussian or more specifically half-Gaussian distribution, the local states that get mixed with the initial states are thermal, i.e.,

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Selection 3

**A**: The corresponding symmetry algebra has a basis given by (5.5)**B**: and ξμsuperscript𝜉𝜇\xi^{\mu}italic_ξ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT are given by the Killing solution (4.2)**C**: This algebra coincides with that of the linear wave equation when the infinite-dimensional subalgebra is factored out.

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Selection 3

**A**: Section 3 is devoted to a study of PPT square conjecture. In particular, we will present the general approach to the problem**B**: In Section 2 we give definitions of the concepts just introduced. Furthermore, we will define and discuss the associated classes of positive maps**C**: Keeping in mind that entanglement breaking maps are so important in Quantum Information, in Section 4 we will give a detailed analysis of the relevant tensor cones and associated classes of maps. This will be done for finite dimensional cases.

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Selection 3

**A**: Therefore, Gould’s original formalism requires substantial modification to properly account for the physics of this extreme environment. Such corrections have gradually been introduced**B**: Much of this attention is generated by the possibility that DM will thermalise [41, 57, 58] and annihilate within these objects, leading to a potentially observable level of heating [45, 46, 47, 48, 59, 38]. For NSs, additional consequences include the possibility of induced collapse to black holes [60, 43, 61, 62, 63, 44, 64, 65, 66] in the case of asymmetric or non-annihilating DM, or modifications to the rate and gravitational wave signatures of mergers [67, 65, 66, 68]. 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**C**: Ref. [53] presented a new formalism that consistently incorporated many of the relevant physical effects. 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,

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Selection 3

**A**: 2191 (2022) 1, 012015, contribution to: DERELİ-FS-2021, may have some questinable parts, like the reflection coefficient being a ratio of two infinite numbers • Since it was published in a separate form, we could not publish an Addendum or Correction to this paper. **B**: The method we used to calculate these coefficients in the first version of this paper, published within the conference proceedings by IOP, in J.Phys.Conf.Ser**C**: Our aim to revise our work is to show that, as also stated in the first form of this paper, to choose another method in the calculation of the reflection coefficients

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Selection 2

**A**: We find that neural étendue expansion also enables higher fidelity étendue expanded 3D color holograms**B**: 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. **C**: Finally, we also investigate 3D étendue expanded holograms

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Selection 2

**A**: The numerical procedure we had originally developed in Zeh et al**B**: the U𝑈Uitalic_U band, which is very vague**C**: (2004) assumes that for ν>ν⁢(U⁢band)𝜈𝜈Uband\nu>\nu(\rm U\leavevmode\nobreak\ band)italic_ν > italic_ν ( roman_U roman_band ) the SN flux scales ∝ν−3proportional-toabsentsuperscript𝜈3\propto\nu^{-3}∝ italic_ν start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT,

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Selection 1

**A**: Many metrics are possible. We can for instance define a measure of energy use (energy efficiency), the degree of removal of specific compounds (selectivity), or a measure of how long membranes remain functional. **B**: After the setpoints of a process have been defined, metrics (or, equivalently, ‘performance indicators’) come into play**C**: They describe optimized operation

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Selection 2

**A**: 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. **B**: 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**C**: We visualize the two features obtained from experiments on Belem in a 2-D plane as in Figure 8 right. The blue dash line is the classification boundary

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Selection 3

**A**: The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc**B**: This research has made use of the CIRADA cutout service at URL cutouts.cirada.ca, operated by the Canadian Initiative for Radio Astronomy Data Analysis (CIRADA)**C**: CIRADA is funded by a grant from the Canada Foundation for Innovation 2017 Innovation Fund (Project 35999), as well as by the Provinces of Ontario, British Columbia, Alberta, Manitoba and Quebec, in collaboration with the National Research Council of Canada, the US National Radio Astronomy Observatory and Australia’s Commonwealth Scientific and Industrial Research Organisation.

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Selection 2

**A**: The thermalization deteriorates the Fock states according to the map (S7)**B**: The vertical axis quantifies the maximal mean number of thermal phonons that preserves the presented quantum aspects. Note, the quantum non-Gaussianity and genuine one-phonon quantum non-Gaussianity are identical properties, and therefore their depth is the same for |1⟩ket1|1\rangle| 1 ⟩.**C**: Figure S4: The thermal depth of the genuine n𝑛nitalic_n-phonon quantum non-Gaussianity (green points), the quantum non-Gaussianity (blue points) and negativity in the Wigner function (red points) that are exhibited by the ideal Fock states

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Selection 2

**A**: Our random restriction lemma shows that if one randomly fixes most of the inputs to a quantum query algorithm, then the algorithm’s behavior on the unrestricted inputs can be approximated by a “simple” function (say, a small decision tree or small DNF formula)**B**: Notably, our proof of Theorem 7 does not appeal to Raz-Tal at all, but instead relies on a new random restriction lemma for the acceptance probabilities of quantum query algorithms**C**: We then use this random restriction lemma to generalize the usual random restriction proof that, for example, Parity∉𝖠𝖢𝟢Paritysuperscript𝖠𝖢0\textsc{Parity}\not\in\mathsf{AC^{0}}Parity ∉ sansserif_AC start_POSTSUPERSCRIPT sansserif_0 end_POSTSUPERSCRIPT [Hås87].

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Selection 4

**A**: In CoauthorshipsNet, node means scientist and weights mean coauthorship, where weights are assigned by the original papers. For this network, there is no ground truth about nodes labels, and the numbers of communities are unknown. The CoauthorshipsNet has 1589 nodes, however its adjacency matrix is disconnected**B**: Among the 1589 nodes, there are totally 396 disconnected components, and only 379 nodes fall in the largest connected component. For convenience, we use CoauthorshipsNet1589 to denote the original network, and CoauthorshipsNet379 to denote the giant component. To find the number of communities for CoauthorshipsNet, we plot the leading 40 eigenvalues of their adjacency matrices**C**: Results shown in Figure 8 suggest that the number of communities is 2, where [27] also applies the idea of eigengap to estimate the number of communities for real-world networks. Note that though CoauthorshipsNet1589 is disconnected, we can still apply nDFA and DFA on it since there is no requirement on network connectivity when applying DFA and nDFA. Note that since the overall embeddedness is defined for adjacency matrix that is connected, it is not applicable for CoauthorshipsNet1589.

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Selection 1

**A**: Another important ingredient of any resource theory is the set of free operations. Free operations satisfy a necessary condition that they do not create resource states from free states, i.e., if ΛΛ\Lambdaroman_Λ is a free operation, then**B**: In entanglement theory the free states are separable states [49, 5]. Every state which is not free is called resource state**C**: An important ingredient of every quantum resource theory is the set of free states ℱℱ\mathcal{F}caligraphic_F. Typically, this set corresponds to all states which can be easily prepared

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Selection 4

**A**: This hole effect is corrected employing first-order time-independent perturbation theory Mougeot (2018), as proposed by Vatai. **B**: Vatai (1970), the vacancy created in a subshell by the capture process has a significant influence on all the orbitals**C**: As underlined by Vatai in Ref

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Selection 3

**A**: Input Ω,KΩ𝐾\Omega,Kroman_Ω , italic_K**B**: The above analysis gives rise to the following algorithm called Ideal DFSP (short for Distribution-Free SP algorithm) in the oracle case with known population adjacency matrix ΩΩ\Omegaroman_Ω**C**: Output: ΠΠ\Piroman_Π.

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Selection 2

**A**: (2021b) and disagrees with results from zoom-ins from Jiang et al. (2019), although both studies focused on more massive galaxies**B**: The null correlation with halo spin agrees with Jiang et al. (2019). For the largest, most massive haloes in the ‘FB-S’ galaxies, there may be a slight dependence on the halo spin (bottom right panel). This could be due to the transition from dispersion to rotationally supported galaxies, where the formation of discs becomes increasingly significant (Fall &**C**: However, there is no such distinction in the GHSR relations when separated into quartiles by either concentration or spin. The null result with concentration agrees with the semi-empirical model from Zanisi et al

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Selection 4

**A**: In this paper we are interested in the subset of the pionless EFT Lagrangian relevant for beta transitions, that is in the quartic interactions between a proton, a neutron, an electron, and a neutrino.555Interactions with more than two nucleon fields also contribute to nuclear beta transitions (A¿1), in particular the ones with four nucleons and two leptons are referred to as two-body currents in the literature**B**: We organize these interactions as**C**: We comment on this issue in Appendix D

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Selection 3

**A**: In both cases, the differences between the predictions and the calculated energies are less than 1 mHa**B**: It is reassuring to note that the short chain length predictions are most accurate throughout the prediction interval, which is a consequence of training the Gaussian processes on short chains. Prediction errors grow with the lengths of the chains because the generalization error increases with system size. This is reflected in the larger confidence intervals that accompany the larger chain length predictions.**C**: Figure 3 depicts the differences between the energies computed with the UCCSD(T) (left) and AFQMC (right) methods, and their respective GPR predictions

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Selection 3

**A**: The proof builds, in particular, on the χχ\upchiroman_χ-independence result of [MS20], the dimensional reduction isomorphism of [TK22], and the interpretation of deformed Calabi–Yau completions of total spaces of line bundles of [KM21]**B**: It is a cohomological version of the χχ\upchiroman_χ-independence conjecture for Gopakumar–Vafa invariants of Calabi–Yau threefolds (see [MT18, Tod17] for details), and is indeed obtained as a special case of the χχ\upchiroman_χ-independence result for general local curves in Calabi–Yau threefolds, also proved in [KK21].**C**: The following result was very recently proved by Naoki Koseki and Tasuki Kinjo

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Selection 4

**A**: Community structure can greatly influence disease dynamics on networks [26, 30]**B**: Salathé and Jones [38] illustrated that changes in community structure are correlated with changes in disease quantities for susceptible–infected–recovered (SIR) dynamics on a network**C**: One of their findings is that outbreak duration can achieve a maximum at intermediate modularity values. Inspired by Salathé and Jones [38], we use our adaptation of InfoMap to study an example contagion process that illustrates how absorption in disease dynamics affects community structure, which in turn affects disease spread. We investigate the association between changes in the effective community structure that is induced by the node-absorption rates with disease quantities such as outbreak size and duration.

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Selection 3

**A**: Here, we develop an efficient heuristic**B**: The general QNR problem can be formulated in terms of hypergraph flows and solved using LP (see Appendix A)**C**: Although polynomial-time and provably optimal, the LP-based approach has a very high time-complexity for it to be practically useful

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Selection 4

**A**: From a mathematical point of view, the Legendre transform we have introduced in Eq. (23) is at the level of probability distributions, and it does not rely on asymptotics**B**: Such a microscopic Legendre transform has previously appeared in other fields such as information geometry [47, 48] and the foundations of statistical mechanics [58], and we have formulated the Legendre transform in Eq. (23) by borrowing ideas from information geometry and replacing information theoretic quantities with thermodynamic ones, such as the Shannon entropy with the thermodynamic EP**C**: This formulation extends the existing connections between information geometry and thermodynamics [59, 60, 61, 62]. Detailed explorations of this geometric picture of stochastic thermodynamics is left for future work.

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Selection 1

**A**: This stochastic reproductive number plays a pivotal role in delineating the dynamics of extinction and persistence of the coronavirus (disease). Specifically, our attention is directed towards analyzing the infected class (4), i.e. **B**: The existence of the virus-free equilibrium and endemic equilibrium hinges on computing the basic reproductive number**C**: In this section, our focus shifts to establishing the basic reproductive number for the stochastic counterpart

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Selection 3

**A**: We would like to thank Frédéric Faure, Benjamin Delarue, Stéphane Nonnenmacher, Tobias Weich and Luchezar Stoyanov for very interesting discussions**B**: Finally, we would like to thank the anonymous referee for his comments and suggestions that helped to improve the manuscript. The first author is supported from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme with agreement No. 725967.**C**: Thanks are also due to Colin Guillarmou for pointing out to us that we could use Fried’s result for the proof of Theorem 3

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Selection 4

**A**: 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]**B**: 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]**C**: This NFL phase was later revisited and further reformulated using dimensional regularization and renormalization group (RG) techniques by Moon et al. [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. From recent analytical works, we do have some other examples of nodal NFLs [50, 51] as well.

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Selection 4

**A**: (2022). This scheme, compatible with a two-dimensional architecture, offers a substantial reduction in the physical resource cost of fault-tolerant quantum computing against the surface code—the archetype for fault-tolerant quantum computing Fowler et al. (2012); Litinski (2019a, b); Fowler and Gidney (2019); Gidney and Fowler (2019). Compared with the most resource efficient mode of surface-code quantum computation Litinski (2019a), our scheme reduces the space-time overhead as a function of the code distance by approximately a factor of 3333. Even accounting for the color code’s lower error threshold using existing decoders, the space-time overhead is reduced by 10% at a physical error rate of 10−3superscript10310^{-3}10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT and 50% at 10−4superscript10410^{-4}10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT. These results elevate consideration of the color-code architecture as a promising pathway for large-scale fault-tolerant quantum computers and motivate further work towards better color-code decoders Kubica and Delfosse (2023); Sahay and Brown (2022); Gidney and Jones (2023); Zhang et al**B**: In this manuscript we propose a scheme for fault-tolerant quantum computing based on the color code Bombin and Martin-Delgado (2006) where we perform logical operations using an ancillary lattice of qubits as a resource to make Pauli measurements between logical data qubits and sources of magic states Bravyi and Kitaev (2005); Litinski (2019a); Chamberland et al. (2022); Chamberland and Campbell (2022); Cohen et al. (2022) using lattice surgery Horsman et al. (2012); Landahl and Ryan-Anderson (2014); Brown et al. (2017); Litinski and Oppen (2018); Cohen et al**C**: (2023) and syndrome extraction circuits Fowler (2011); Landahl et al. (2011); Stephens (2014a); Chamberland et al. (2020); Beverland et al. (2021); Bombin et al. (2023); McEwen et al. (2023).

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**A**: An alternative notion that incorporates the 3⁢D−limit-from3𝐷3D-3 italic_D -compatibility property of a map, is the so-called Yang-Baxter property**B**: The maps that satisfy the Yang-Baxter property will be called Yang-Baxter maps**C**: Note that if a 3⁢D−limit-from3𝐷3D-3 italic_D -compatible map is quadrirational, its companion map is a Yang-Baxter map.

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**A**: For example, when using the information in the linear point [45, 46, 47, 48], no reconstruction is required**B**: Also, the estimated α𝛼\alphaitalic_α from the traditional BAO methods and from the linear point approach may conceptually differ, and a comparison is beyond the scope of this work. **C**: Also note that the BAO reconstruction procedure is not always required for extracting geometric information in the galaxy clustering

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**A**: Our results are shown in Table 1. Most of the ordered alloys we studied order in the bcc-derived structures B2 and D03.[43, 44, 45] The B2 structure is a bcc-derived structure with two atoms in the primitive unit cell. This structure is therefore present in alloys with the ratio of 1–1 of two elements**B**: On the other hand, D03 structure contains four atoms in the primitive unit cell, so it is present in ordered binary alloys with the 1–3 ratio of constituent elements, or the 1–1–2 ratio in the case of ternary alloys. Among the ordered alloys we studied, the only ones that are not in the bcc-derived structure are Al, FeAl3, CoAl3, Co, and**C**: Now we present our results for ordered Fe-Co-Al alloys. We start by discussing the computed lattice constants and crystal structure of these alloys

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**A**: We provide evidence for power law decaying autocorrelations, and for a pattern of hierarchical relaxation when quenched from extremal initial states into the different equilibrium phases. In Sec. V we study the large deviations statistics of dynamical observables by means of numerical MPS. As in other constrained models, the phase transitions at the LD level underpin the slow dynamics and fluctuations seen in typical relaxation trajectories.**B**: In Sec. IV we study the relaxation dynamics**C**: As in the case of the quantum model [8], the stochastic Fredkin spin chain exhibits slow dynamics

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**A**: Eq**B**: (41) can be written eν=(1+A⁢r2)nsuperscript𝑒𝜈superscript1𝐴superscript𝑟2𝑛e^{\nu}=(1+Ar^{2})^{n}italic_e start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT = ( 1 + italic_A italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT.**C**: field equations

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**A**: Why this happens is examined in the Exploration subsection**B**: But it shows that BO can be much faster, even if care must be taken when choosing the models and priors.**C**: By inspection of Fig. 2 one can see that the selection examples have many local optima, and so would suffer from this. On the London data one of the GP models does much better than either baseline, but the other two perform worse

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**A**: The in-situ proof-of-principle in nature is however still to be shown. We have already demonstrated that cosmic ray air showers create particle cascades in ice with properties very similar to neutrino-induced cascades. Therefore, the radar detection of these in-ice cosmic-ray-induced particle cascades would show the proof-of-principle of the method in nature. **B**: This indicates that this method could be used to detect neutrino-induced particle cascades in ice**C**: It was recently shown at the Stanford Linear Accelerator Center (SLAC) that high-energy particle cascades in dense media can be probed using the radar echo technique [7]

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**A**: Accordingly, the back-propagation of the ghost propagator’s non-analyticities into the gluon DSE at least requires a perturbative three-loop computation. While certainly being challenging, this may be within the technical range of perturbative computations in the CF model, and is very desirable**B**: In a massive extension of Yang-Mills theory, complex conjugate poles may occur in the gluon propagator at one-loop. This implies that their impact on the ghost propagator may be visible at two-loop in the ghost gap equation**C**: The back-propagation of the additional cuts poses a major obstruction that only can be circumvented by intricate relations between the complex structures of propagators and that of the vertices, in particular the ghost-gluon vertex. Signs for the latter gathered in perturbation theory at least require a three-loop analysis of the ghost DSE as argued in Section IV.2. Such an analysis, while highly desirable, has not been undertaken yet in the literature.

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**A**: Reconstructing an approximation of the data from a few set of kPCA components is non trivial problem and its presentation is here omitted (see Mendez, 2023b )**B**: Interested readers are referred to Bakir et al., (1999)**C**: We close this section with an application, illustrated in Figure 4.

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Selection 4

**A**: In Sec. VI, we show that the anomaly equation at the edge can be derived variationally by adding a boundary chiral boson action, which is non-linearly coupled to the edge density, to the topological fluid action. In Section VII, we offer a heuristic boundary layer interpretation of the edge anomaly equation. The paper concludes with a discussion and outlook in Section VIII.**B**: In Section IV, we demonstrate the duality between the topological fluid action and the composite boson (CSGL) action. Section V discusses the incompatibility of the no-slip condition with the FQH edge dynamics**C**: The paper is organized as follows: we start with a brief review of the superfluid dynamics of the composite boson model in Section II. In Section III, we introduce a variational formulation for these fluid dynamic equations, by including additional topological terms to the hydrodynamic action

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**A**: A dispersive dipole magnet allows the reconstruction of the energy spectrum. The X-rays produced in the beam-target interaction are measured with profile monitors and spectrometers (not shown).**B**: The beam after the interaction passes through an optional profile monitor which allows to measure the transverse momentum spread and deflection. It is thereafter imaged with a quadrupole triplet onto a scintillating screen**C**: Figure 5: Sketch of the E-336 experimental setup at FACET-II. The electron bunch is focused on a target with a hole array

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**A**: Our dataset includes 31 points from the Hubble dataset, 1048 points from the Pantheon SN samples, and 6 points from the BAO dataset**B**: To investigate the observational characteristics of our cosmological model, we leverage the latest cosmic Hubble, SN observations, and BAO**C**: Employing Bayesian analysis, we utilize the likelihood function and the Markov Chain Monte Carlo (MCMC) method implemented in the emcee Python library [65].

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Selection 2

**A**: 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). 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)**B**: 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. Applying this new framework, we quantify the roles of classical and non-classical crystallization mechanisms at different precipitation rates and supersaturation levels. This model framework can also be applied to other crystal systems and tested with other paired element and isotope measurements.**C**: 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

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**A**: In Section 2, we review the Lie point symmetry analysis of (1.1)**B**: Solutions of the invariance condition as a Riccati equation in the independent variable I⁢(x)𝐼𝑥I(x)italic_I ( italic_x ) has been discussed. Section 3 is devoted to five examples of PDEs in connection with the study of Brownian motion and mathematical finance. The results of Section 2 are applied to them to construct solutions of physical interest. **C**: The structure of this paper is as follows

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Selection 2

**A**: The local effects typically amount to a relatively small contribution compared to the terms in (12). Moreover, (12) is sufficient to obtain a true understanding of the nature of RSDs in the angular power spectrum. However, these effects need to be taken into account in the light of upcoming precision cosmological era. **B**: In reality, RSDs do exist in the large-scale structure among several effects other than the matter density; these include the Doppler and the potential effects (neglecting integral effects, which are negligible at z≲ 1less-than-or-similar-to𝑧1z\,{\lesssim}\,1italic_z ≲ 1, being the z𝑧zitalic_z of interest in this work)**C**: These effects together (apart from density and RSDs) are, henceforth, termed “local effects.” They surface in the overdensity in redshift space through the redshift and the volume perturbations, accordingly

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**A**: The proofs of these results are given in the subsequent sections.**B**: The next section briefly recalls only the essential part of [8] used in the sequel**C**: The main results obtained in the present paper are Theorems 3.1 and 4.1 from sections 3 and 4 respectively

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**A**: The VTX is described in detail in Refs. [22, 18]**B**: It is composed of two arms, each with |η|<1𝜂1|\eta|<1| italic_η | < 1 and Δ⁢ϕ≈ 0.8⁢πΔitalic-ϕ0.8𝜋\Delta\phi\,{\approx}\,0.8\piroman_Δ italic_ϕ ≈ 0.8 italic_π coverage**C**: Each arm has four layers around the

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Selection 3

**A**: AGN can also exhibit short-term variability for periods of hours or days, but we caution that the detection of a high-energy neutrino alert is a process that requires a substantial fluence at the IceCube detector, even after accounting for the significant Eddington bias associated with cosmic neutrino detection (Strotjohann et al., 2019)**B**: Such highly luminous rapid neutrino flares are not well motivated theoretically, it is therefore unlikely that short AGN flares are indicators of neutrino production. **C**: The corresponding neutrino flux that is required is inversely proportional to the duration of neutrino emission, and therefore associating a neutrino detection with a temporary electromagnetic signature lasting hours or days would imply an extremely high average neutrino flux for the duration of that signature

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**A**: Fig. 4 for the pTsubscript𝑝𝑇p_{T}italic_p start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT range 1.0<pT<1.21.0subscript𝑝𝑇1.21.0<\mbox{$p_{T}$}<1.21.0 < italic_p start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT < 1.2 GeV/c𝑐citalic_c**B**: The four panels correspond to four different centrality selections**C**: Each panel

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**A**: In this section we discuss examples with concrete codes**B**: They serve several purposes**C**: Firstly, they show that invariants we proposed are nontrivial, calculable and yield what is expected on physical grounds in models which are already well understood. Secondly, they support our physical interpretation of mathematical objects and the conjecture that braiding is non-degenerate. Finally, the last example illustrates certain technical complication that does not arise for codes with

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Selection 2

**A**: After preparing the initial ground state, the system will evolve according to the quench profile (2)**B**: To this end, we render the neural network parameters to be functions of time. Then, the parameters will be computed at every time step with the time-dependent VMC method, by minimizing the distances δ𝛿\deltaitalic_δ between the exact time evolution and the approximate variational evolution **C**: Therefore, the wave function should also depend on time

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**A**: In order to show the effect of different space intervals and time intervals, two groups of comparison tests have been done**B**: Table 2 shows the result of different hℎhitalic_h. **C**: First, we choose different space interval hℎhitalic_h to observe the performance of our model

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**A**: The microcentrifuge tube was then agitated for 1 week on an orbital shaker at 100 rpm (PSU-10i Grant Bio, UK). **B**: In brief, 5 mM oleic acid vesicles were prepared in a buffer that contained 100 mM Na-bicine (pH 8.3) and up to 500 mM sucrose by adding the appropriate amount of oleate micelles**C**: Vesicles were prepared by the self-assembly method [1]

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Selection 1

**A**: To be sure, our prescription has only a limited range of validity; nevertheless, it serves as a useful illustration of several related techniques for computing and understanding holographic entanglement.**B**: If we treat the limits correctly, we find a simple picture of holographic entanglement entropy: in certain regimes, it is just the logarithm of a scalar propagator**C**: The order-of-limits issue discussed above is a classic one Fischetti:2014zja ; Headrick:abc

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Selection 4

**A**: At low noise, to achieve a quantum semantic fidelity of 0.70.70.70.7, QSC requires around 50%percent5050\%50 % quantum communication resources compared to semantic-agnostic QCNs using pruning data compression without any semantic concept extraction**B**: This demonstrate the advantages of QSC accurately sending and reconstructing semantic information.**C**: In Figure 3, we show the quantum semantic fidelity achieved against the amount of quantum communication resources used for |𝒳|=500𝒳500\lvert\mathcal{X}\rvert=500| caligraphic_X | = 500

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**A**: solutions (2019) The non ideal behavior illustrated here occurs at frequencies smaller than this by at least two and at most eight orders magnitude. **B**: However, the remaining deviation from ideal behavior is caused by the equivalent series inductance that typically produces self-resonance frequencies above 100100100100 MHz**C**: These effects are mitigated when using a vacuum capacitor

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Selection 2

**A**: This approach to screw theory via special vector fields has the advantage of being independent of any reduction point**B**: We stress that in this work the screws will always be elements of a 6-dimensional real vector space, hence our notion of screw is equivalent to that of torseur, and conforms with the terminology used by Dimentberg [9]. This terminology differs from the original terminology used by Sir R. Ball.**C**: It goes back, at least, to Lovell III’s PhD thesis [14] and has long been used in French engineering universities where screws are called torseurs [24, 4, 25, 19, 6]

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Datasets:

liangzid
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robench2024b_all_setphysicsSCP-s

Collection including liangzid/robench2024b_all_setphysicsSCP-s

robench-2024b