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<|MaskedSetence|> <|MaskedSetence|> It shows that the operator Op(a)Op𝑎\text{Op}(a)Op ( italic_a ) belongs to the ∗*∗-algebra C∞(AdU)superscript𝐶Ad𝑈C^{\infty}(\text{Ad}\,U)italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( Ad italic_U ) of smooth elements for the representation AdUAd𝑈\text{Ad}\,UAd italic_U, for every a∈ℬ𝒞(ℝ2n)𝑎superscriptℬ𝒞superscriptℝ2𝑛a\in\mathcal{B}^{\mathcal{C}}(\mathbb{R}^{2n})italic_a ∈ caligraphic_B start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT ). In particular, every element of ℬJ𝒞superscriptsubscriptℬ𝐽𝒞\mathcal{B}_{J}^{\mathcal{C}}caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT is contained in C∞(AdU)superscript𝐶Ad𝑈C^{\infty}(\text{Ad}\,U)italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( Ad italic_U ), so we may equip ℬJ𝒞superscriptsubscriptℬ𝐽𝒞\mathcal{B}_{J}^{\mathcal{C}}caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT with the subspace topology induced by the usual Fréchet topology of C∞(AdU)superscript𝐶Ad𝑈C^{\infty}(\text{Ad}\,U)italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( Ad italic_U ), which will be denoted by τℬJ𝒞,C∞subscript𝜏superscriptsubscriptℬ𝐽𝒞superscriptC\tau_{\mathcal{B}_{J}^{\mathcal{C}},\text{C}^{\infty}}italic_τ start_POSTSUBSCRIPT caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT , C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT. <|MaskedSetence|> Then (3.23) combined with (3.22) shows that
(3.24). | **A**:
h∈ℝℎℝh\in\mathbb{R}italic_h ∈ blackboard_R, 1⩽k⩽n1𝑘𝑛1\leqslant k\leqslant n1 ⩽ italic_k ⩽ italic_n, combined with the estimate (3.23), give (3.22) in the case |α|+|β|=1𝛼𝛽1|\alpha|+|\beta|=1| italic_α | + | italic_β | = 1.
**B**: The equality for general α,β∈ℕn𝛼𝛽superscriptℕ𝑛\alpha,\beta\in\mathbb{N}^{n}italic_α , italic_β ∈ blackboard_N start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT follows from an iteration of this procedure.
**C**: Also, injectivity of the map L:f⟶Lf:𝐿⟶𝑓subscript𝐿𝑓L\colon f\longrightarrow L_{f}italic_L : italic_f ⟶ italic_L start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT allows us to equip ℬJ𝒞superscriptsubscriptℬ𝐽𝒞\mathcal{B}_{J}^{\mathcal{C}}caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT with a Fréchet space topology τℬsubscript𝜏ℬ\tau_{\mathcal{B}}italic_τ start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT induced by the natural topology of the function algebra ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) defined by the family (3.4) of ∗*∗-norms.
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<|MaskedSetence|> Organization
This paper is organized as follows. In Section 2 we fix notations and present some results in commutative algebra that are needed in the paper. In Section 3 we give the nonsingularity conditions which guarantee that (1.3) holds for some d𝑑ditalic_d.
In Section 4 we establish the effective degree bound (1.7) under 1. In Section 5 we apply the main result to gradient type SOS. In Section 6 provide some examples satisfying 1 and make some discussions on this assumption. <|MaskedSetence|> <|MaskedSetence|> | **A**: In appendix we present some deferred proofs..
**B**: In Section 7 we conclude and provide some perspectives on future research.
**C**:
1.4.
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<|MaskedSetence|> Thus it is natural to define the relative class of a given tropical disc to be that of the holomorphic disc which intersects the toric divisors in accordance with unbounded edges of the tropical disc. Throughout, we will frequently use the terms such as relative classes, symplectic areas, Maslov indices, etc., for tropical discs in this spirit.
We will sometimes need to impose further constraints to tropical discs. <|MaskedSetence|> <|MaskedSetence|> For this reason, we define generalized Maslov index of a constrained tropical disc by
. | **A**: On the other hand, the intersection patterns of a holomorphic disc with toric divisors completely determine its topological type, and in particular its symplectic area and Maslov index.
**B**: Namely, we fix a set of generic points in ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, and consider the counting invariant concerning tropical discs that pass through these points.
**C**: Obviously, the constraints make discs more rigid, i.e., they behave like discs with lower Maslov indices.
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<|MaskedSetence|> Botnan et al. <|MaskedSetence|> <|MaskedSetence|> Asashiba et al. provided a criterion for determining whether or not a given multiparameter persistence module is interval decomposable without having to explicitly compute indecomposable decompositions
[1].
Dey and Xin proposed an efficient algorithm for decomposing. | **A**: introduced notions of signed barcode and rank decomposition for encoding the rank invariant of multiparameter persistence modules as a linear combination of rank invariants of indicator modules [10].
**B**: In their paper, Möbius inversion was utilized for computing the rank decomposition, characterizing the generalized persistence diagram in terms of rank decompositions.
**C**: Other related work.
McCleary and Patel utilized the Möbius inversion formula for establishing a functorial pipeline to summarize simplicial filtrations over finite lattices into persistence diagrams [46].
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Organization
The rest of the paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> In Section 4, we show how to choose this sum of squares so that (P1), (P3) are satisfied and finish the proof of Theorem 3 and Theorem 4. In Section 5, we extend our proof technique to the upper bounds to obtain Theorem 5 and Theorem 6. <|MaskedSetence|> | **A**: In Section 2, we introduce some notations and cover the necessary preliminaries on orthogonal (Gegenbauer) polynomials.
**B**: Finally, we give a proof of some technical statements in Appendix A..
**C**: In Section 3, we present closed form expressions of the Christoffel-Darboux kernel and use them to obtain kernels whose associated operators satisfy (P2) and whose eigenvalues are given by the coefficients of a univariate sum of squares in an appropriate basis of orthogonal polynomials.
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Remark \the\thmnumbering.
It was claimed in [GK, Example 8.15] that the canonical metric on any line bundle of an abelian variety is locally the tensor product of a smooth and a formal metric. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Therefore, it remains an open question whether canonical metrics are -metrics. As this paper clearly shows, it is better to consider a more general notion of -metrics replacing smoothness by weak smoothness.. | **A**: However, this was done under the incorrect assumption that canonical tropicalization maps of abelian varieties are smooth.
**B**: The latter means that locally there is a smooth tropicalization map such that the first Chern current is induced by a (1,1)11(1,1)( 1 , 1 )-form with piecewise smooth coefficients on the tropical chart (see [GK, 9.13]).
**C**: This was used in [GK, Example 9.17] to prove that such canonical metrics are -metrics.
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The above theorem leaves us with an intriguing question for harmonic maps with a higher degree. <|MaskedSetence|> There we have shown that similar quantitative stability for higher degree ones is only true in the local sense. <|MaskedSetence|> More precisely, given a harmonic map (or a compact set of harmonic maps), there is a local stability result near it. <|MaskedSetence|> | **A**: We have addressed a similar question for half-harmonic maps in [9].
**B**: In this paper, we shall prove that a similar phenomenon happens here.
**C**: The bound in general will depend on the given harmonic map (or the compact set).
To that end, let us introduce some notations..
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In particular, real nilpotent Lie algebras with an ad-invariant metric are classified up to dimension 10101010 in [17]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Lie algebras with a nice basis, called nice Lie algebras in this paper, possess a strong algebraic structure, which can be encoded in a special kind of directed graph. Nice nilpotent Lie algebras have been classified up to dimension 6666 in [13, 20] and up to dimension 9999 in [7] (see also [16] for the particular case where the Nikolayevsky derivation is simple and the root matrix is surjective); these classifications show that in small dimension most, but not all, nilpotent Lie algebras are nice.
A natural question arises: does every nilpotent Lie algebra carrying ad-invariant metrics fulfill the nice condition?. | **A**: Nice bases are mostly studied in the context of Einstein Riemannian metrics on solvmanifolds.
**B**: In [6], we showed that the irreducible Lie algebras appearing in this classification admit a unique ad-invariant metric (up to sign); as a step in the proof, we proved that all of them admit a nice basis.
**C**: They were first introduced in [19], and received their (now widely accepted) name in [23].
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> In contrast, the Brownian transport map allows us to prove estimates on the Lipschitz constant of the transport map in expectation, which is what is needed to make the connection with the Kannan–Lovász–Simonovits conjecture; cf. Theorem 1.4. (We remark however that the heat flow map has its own advantages, as explained in [54, p.3].)
. | **A**:
(b) The second way to apply the Hessian estimate is to use it within the context of the heat flow transport map of Kim and Milman [37].
**B**: This approach avoids the issues mentioned in part (a).
**C**: On the other hand, the usage of this transport map is only suitable if we want to prove pointwise estimates on the Lipschitz constant of the transport map.
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<|MaskedSetence|> <|MaskedSetence|> This result was extended to the subanalytic category by A. <|MaskedSetence|> The author gives in [31] a construction of locally bi-Lipschitz trivial stratifications for definable sets in polynomially bounded o-minimal structures (expanding ℝℝ\mathbb{R}blackboard_R) for which the local trivializations are in addition definable. The stratifications constructed in [22, 21, 31] can be required to be compatible with finitely many definable sets, and therefore, to refine a given stratification. Hence, the theorem below follows from [22, Theorem 1.4], [21, Theorem 2.62.62.62.6], or [31, Corollary 1.6.81.6.81.6.81.6.8].
. | **A**: Parusiński [22], to polynomially bounded o-minimal structures expanding ℝℝ\mathbb{R}blackboard_R in [21], and to polynomially bounded o-minimal structures expanding an arbitrary real closed field in [15].
**B**:
T.
**C**: Mostowski [20, Proposition 1.21.21.21.2] proved that every complex analytic set admits a locally bi-Lipschitz trivial stratification.
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<|MaskedSetence|> Previous results on proper permutations
Proper permutations were first introduced in [BHY20]. We highlight that our definition differs slightly from the original definition. The original definition of properness was motivated by the study of Levi-spherical Schubert varieties in GLn/B𝐺subscript𝐿𝑛𝐵GL_{n}/Bitalic_G italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT / italic_B. <|MaskedSetence|> This introduces a difference of 1111 on the right hand side of Definition 1.3 as compared to [BHY20, Definition 1]; this is due to the fact that the Levi subgroups in SLn𝑆subscript𝐿𝑛SL_{n}italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT have dimension one less than the corresponding Levi subgroups in GLn𝐺subscript𝐿𝑛GL_{n}italic_G italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. <|MaskedSetence|> | **A**:
1.3.
**B**: This updated definition is considerably more natural in the general type setting.
.
**C**: To study Levi-spherical Schubert varieties in G/B𝐺𝐵G/Bitalic_G / italic_B, for G𝐺Gitalic_G a simple Lie group, requires a definition of properness that corresponds to Levi-spherical Schubert varieties in SLn/B𝑆subscript𝐿𝑛𝐵SL_{n}/Bitalic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT / italic_B (SLn𝑆subscript𝐿𝑛SL_{n}italic_S italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT being a simple Lie group).
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<|MaskedSetence|> Ref. [16] showed that space-time block coding (STBC) with single polarization outperforms STBC with dual polarization in Rayleigh and Ricean fading channels. <|MaskedSetence|> It is noteworthy that the extent of benefit from dual-polarized antennas depends on the associated schemes to exploit the characteristics of polarized wireless channel [15, 16, 17, 1, 6]. <|MaskedSetence|> | **A**: A MIMO system with dual-polarized antenna elements can have lower spatial diversity but higher spatial multiplexing gain than a conventional MIMO system with single-polarized antennas, particularly, in Ricean fading channels with high K𝐾Kitalic_K-factor [17].
**B**: Various channel sounding campaigns and channel models provide insights into the characteristics of wireless channel polarization [26, 21, 22, 20, 27, 28, 23, 29, 30].
.
**C**:
Various other aspects of polarization in MIMO systems have been investigated as well.
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Several other results address the non-existence of a second bound state for a Schrödinger equation in one dimension, and they could potentially be applicable here. For instance, Calogero [14] established various results of the kind.
One well-known result by Calogero requires the monotonicity of the potential function, and it is difficult to incorporate such a result even when the front solution has a monotone profile because the location of the zero cannot be controlled. <|MaskedSetence|> See, for instance, Proposition 5.7 and Lemma 6.16 for details. There are other results that generalize the findings in [37]. <|MaskedSetence|> <|MaskedSetence|> | **A**: See, for instance, [46].
**B**: While any of these results could be useful here, the choice of [37] (or [5]) is made because it is analytically simpler and has an interesting physical interpretation as the distance to a piece-wise constant shock.
.
**C**: Nevertheless, such a Calogero bound will play a significant role later when complementing a computer-assisted proof.
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In particular, 𝒫0=:𝒫\mathcal{P}_{0}=:\mathcal{P}caligraphic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = : caligraphic_P, the well known Carathéodory class. <|MaskedSetence|> Further, let 𝒮𝒮\mathcal{S}caligraphic_S be a subclass of 𝒜1subscript𝒜1\mathcal{A}_{1}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT consisting of all univalent functions. <|MaskedSetence|> In particular, 𝒮∗(0)=:𝒮∗\mathcal{S}^{*}(0)=:\mathcal{S}^{*}caligraphic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 0 ) = : caligraphic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. <|MaskedSetence|> In [2], Bucka and Ciozda generalized the above classes and introduced the class
. | **A**: Suppose 𝒜psubscript𝒜𝑝\mathcal{A}_{p}caligraphic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT be the subclass of ℋ[0,p]ℋ0𝑝\mathcal{H}[0,p]caligraphic_H [ 0 , italic_p ] consisting of functions normalized by f(p)(0)=1.superscript𝑓𝑝01f^{(p)}(0)=1.italic_f start_POSTSUPERSCRIPT ( italic_p ) end_POSTSUPERSCRIPT ( 0 ) = 1 .
**B**: Further, Brannan
and Kirvan [1] and Stankiewicz [14] independently introduced the class 𝒮𝒮∗(β)𝒮superscript𝒮𝛽\mathcal{SS}^{*}(\beta)caligraphic_S caligraphic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_β ), consisting of functions f∈𝒮𝑓𝒮f\in\mathcal{S}italic_f ∈ caligraphic_S such that |arg(zf′(z))/f(z)|<βπ/2𝑧superscript𝑓′𝑧𝑓𝑧𝛽𝜋2|\arg(zf^{\prime}(z))/f(z)|<\beta\pi/2| roman_arg ( italic_z italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) ) / italic_f ( italic_z ) | < italic_β italic_π / 2, for some β∈(0,1]𝛽01\beta\in(0,1]italic_β ∈ ( 0 , 1 ] and these functions are said to be strongly starlike functions of order β𝛽\betaitalic_β.
**C**: A function f∈𝒮𝑓𝒮f\in\mathcal{S}italic_f ∈ caligraphic_S is said to be starlike of order α𝛼\alphaitalic_α if Re(zf′(z)/f(z))>αRe𝑧superscript𝑓′𝑧𝑓𝑧𝛼\operatorname{Re}(zf^{\prime}(z)/f(z))>\alpharoman_Re ( italic_z italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) / italic_f ( italic_z ) ) > italic_α, for some α∈[0,1)𝛼01\alpha\in[0,1)italic_α ∈ [ 0 , 1 ) and the class of all such functions is denoted by 𝒮∗(α)superscript𝒮𝛼\mathcal{S}^{*}(\alpha)caligraphic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_α ).
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<|MaskedSetence|> In Section 2 we start with the definition of a standard set, along with some basic definitions and properties on the set of monomials in a polynomial ring. <|MaskedSetence|> <|MaskedSetence|> In Section 4 we use strongly Lech-independence to analyze the colength of powers of ideals and derive inequalities on multiplicities.
2. standard sets in a polynomial ring. | **A**: In Section 3 we define strongly Lech-independence and expansion property and prove some equivalent conditions.
**B**: The paper is organized in the following way.
**C**: There are also some examples showing the relation between strongly Lech-independence and other notions.
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We can reverse the procedure to construct an arbitrary meander from prime ones.
Unfortunately, there are meanders that can be constructed in different ways. For instance, consider a direct snake of order (3,0)30(3,0)( 3 , 0 ). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> For this purpose, we need to introduce new class of meanders.. | **A**: That is, if we consider only the sequences of insertions in which no direct snake is inserted into another snake (neither direct nor inverse), each meander can be constructed in a unique way.
It is convenient to consider another way of constructing meanders.
**B**: In fact, the insertion of a direct snake into other snakes is the only obstacle to the uniqueness of the construction.
**C**: The insertion of another direct snake of the same order at any point results in a direct snake of order (5,0)50(5,0)( 5 , 0 ).
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In the present paper, we examine absorbing random walks on graphs in which different nodes can have different absorption rates, inducing an “effective” network structure that is reflected only partially by the edge weights of a network. Many notions of network community structure arise from the analysis of random walks [12, 25], and we expect different types of random walks to yield different community structures [17, 18]. A “community” in a network is a tightly knit set of nodes that is connected sparsely to other tightly knit sets of nodes [12, 33]. Communities are a common feature of many real-world networks, and community structure influences dynamical processes such as the spread of infectious diseases [42] and online content [15, 45]. For example, community structure can affect the size and duration of a disease outbreak [38]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> These absorption-scaled graphs are related to their associated absorbing random walks by a generalized inverse (the so-called “absorption inverse”) and a fundamental matrix [16]. We use absorption inverses and results from [16] to study the absorption-scaled graphs that are associated with our adaptations of InfoMap.. | **A**: In our adaptation, we apply InfoMap to absorption-scaled graphs, which account for absorption by scaling the edge weights of a network [16].
**B**: There is intense interest in understanding how community structure and node characteristics combine to influence contagions on networks [24, 34, 37].
We develop community-detection algorithms that account for node-absorption rates.
**C**: We adapt the widely-used community-detection algorithm InfoMap [35, 36, 41] to absorbing random walks and thereby account for heterogeneous node-absorption rates in the detected communities.
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<|MaskedSetence|> Organization of the paper
In Section 2, we include the definitions of adjacency matrices of hypergraphs. <|MaskedSetence|> <|MaskedSetence|> The proof for the correctness of our algorithms for Theorem 1.7 and Corollary 1.9 are given in Section 5. The proof of Theorem 1.6, as well as the proofs of many auxiliary lemmas and useful lemmas in the literature, are provided in the supplemental materials.. | **A**: The concentration results for the adjacency matrices are provided in Section 3.
**B**: The algorithms for partial recovery are presented in Section 4.
**C**:
1.4.
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A major difference between the models of adaptive walks/flights and adaptive dynamics is that the former assume a fitness landscape that is random but fixed in time, while in the latter case the fitness landscape is dynamic and depends on the current resident traits. As mentioned before, the notion of local fitness maxima can nevertheless be translated. <|MaskedSetence|> We study this special case in a number of examples. Overall, the results of this paper can be seen as a validation of certain types of adaptive walks or flights, deriving their macroscopic dynamics from a microscopic, individual-based model.
The remainder of this paper is structured as follows: In Chapter 2, we rigorously define the individual-based model of adaptive dynamics, for which we derive our limit theorems. We introduce key quantities, like the fitness of a trait, and recapitulate the most important results of [14] that lead to a metastable state on the lnK𝐾\ln Kroman_ln italic_K-time scale. Finally, we heuristically derive the limit behaviour on longer time scales and present the formal convergence results, starting with a single metastable transition in Section 2.3 and treating the full jump process in Section 2.4. <|MaskedSetence|> <|MaskedSetence|> A combinatorial result on excursions of subcritical birth death processes and the complete version of the results from [14] are stated in Appendix A, for the convenience of the reader.
. | **A**: Moreover, if equal competition between all traits is assumed in the adaptive dynamics model, the fitness landscape can again be regarded as fixed.
**B**: Chapter 3 is devoted to the discussion of a number of examples that highlight different aspects of the complicated limiting dynamics in an easy set up.
**C**: The proofs of the main results of this paper can be found in Chapter 4.
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<|MaskedSetence|> <|MaskedSetence|> ℒ≥n(X)subscriptℒabsent𝑛𝑋\mathcal{L}_{\geq n}(X)caligraphic_L start_POSTSUBSCRIPT ≥ italic_n end_POSTSUBSCRIPT ( italic_X )) denote the set of words of length n𝑛nitalic_n
(resp. <|MaskedSetence|> We also let ℒ(x)ℒ𝑥\mathcal{L}(x)caligraphic_L ( italic_x ) denote the set
of factors of a sequence x𝑥xitalic_x. Thus ℒ(X)=∪x∈Xℒ(x)ℒ𝑋subscript𝑥𝑋ℒ𝑥\mathcal{L}(X)=\cup_{x\in X}\mathcal{L}(x)caligraphic_L ( italic_X ) = ∪ start_POSTSUBSCRIPT italic_x ∈ italic_X end_POSTSUBSCRIPT caligraphic_L ( italic_x ).
. | **A**: ≥nabsent𝑛\geq n≥ italic_n) in ℒ(X)ℒ𝑋\mathcal{L}(X)caligraphic_L ( italic_X ).
**B**: set of all factors of the sequences x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X.
**C**: We let
ℒn(X)subscriptℒ𝑛𝑋\mathcal{L}_{n}(X)caligraphic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_X ) (resp.
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Environmental fluctuations have emerged as a significant factor in the study of diseases, particularly in the context of the coronavirus. Consequently, it becomes crucial to investigate the impact of random disturbances on epidemic models. Disease spread is inherently stochastic, and the introduction of stochastic noise can notably influence the likelihood of disease extinction during the early stages of an outbreak. While ordinary differential equation (ODE) models provide specific sample solutions, employing a stochastic differential equation (SDE) model allows for the exploration of the stochastic distribution of disease dynamics.
In the current landscape, various stochastic epidemic models have been extensively explored. Notably, articles such as those in the [9, 10, 11, 12, 13, 14, 15, 16, 17] series have delved into stochastic epidemic models influenced by Lévy noise. Motivated by this body of research, we introduce the assumption that the contact rate is perturbed by Lévy noise, emphasizing the need to employ Lévy processes for disease protection and control. The resulting model is detailed in (4), as elaborated in Section 4. <|MaskedSetence|> <|MaskedSetence|> If the reproduction number exceeds 1, it indicates that each existing infection leads to more than one new infection, resulting in exponential growth of the disease within the population. <|MaskedSetence|> This research contributes to the broader understanding of stochastic epidemic modeling and its implications for disease dynamics.. | **A**: Our objective is to derive the basic reproduction number, a critical determinant governing the extinction or persistence of the disease (infection).
**B**: The reproduction number represents the average number of secondary infections produced by a single infected individual in a susceptible population, making it an essential metric for understanding disease spread dynamics.
**C**: Conversely, if the reproduction number is less than 1, the disease is likely to diminish and eventually disappear, as each infected person infects, on average, fewer than one new individual.
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<|MaskedSetence|> <|MaskedSetence|> The first step computes the cluster mean. The second step minimizes the within-cluster distance. Just like k𝑘kitalic_k-means clustering, the two-step optimization is then iterated until convergence. <|MaskedSetence|> | **A**: If μ𝜇\muitalic_μ is given and fixed, the identification of clusters C𝐶Citalic_C can be done easily by assigning each network to the closest mean.
**B**: Thus the topological clustering algorithm can be written as the two-step optimization similar to the expectation maximization (EM) algorithm often used in variational inferences and likelihood methods (Bishop, 2006).
**C**: Such process converges locally.
Theorem 3.4
.
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Next, we present a test case to illustrate the performance of the proposed approach under disturbance. We consider a scenario where an adversary injects a cyberattack in the form of a disturbance to the battery module to induce overdischarge. The disturbance is injected at 700s700𝑠700s700 italic_s as current drain from the module which forces the State-of-Charge (SOC) of the battery to reach zero. <|MaskedSetence|> 3. <|MaskedSetence|> Furthermore, the overdischarge proceeds to induce a battery failure through increased heating of the cell [37]. The increased heat generation due to additional current drain and subsequent heating due to overdischarge is shown in Fig. <|MaskedSetence|> | **A**: 4.
Figure 3: State-of-Charge (SOC) of the battery module under nominal condition and under disturbance through battery overdischarge.
.
**B**: It can be seen that the disturbance was initiated around 700s700𝑠700s700 italic_s, and consequently, after 1098s1098𝑠1098s1098 italic_s the modules goes into the overdischarge mode by crossing zero SOC.
**C**: The SOC evolution under nominal scenario and under disturbance are shown in Fig.
| CAB | CBA | CBA | CBA | Selection 2 |
Problems involving the optimization of pathways are critical for a number of applications across different fields. The general problem stemming from optimizing collision-free paths for robots in the realm of numerical analysis is studied in [1]. In graph theory, Dijkstra’s algorithm is used to look at the shortest paths on polyhedral surfaces, noting applications in robotics, geographic information systems, and route finding in [5]. <|MaskedSetence|> An approach based in finding the shortest path on a polyhedral surface is used to model a network design problem with applications in telecommunications and transportation in [6]. In this paper, rather than find an algorithm to be applied, we provide our polyhedra with a coordinate system so that exact formulae can be derived. <|MaskedSetence|> Specifically, we will restrict our view to the paths lying along the surfaces of cubes and tetrahedra. <|MaskedSetence|> | **A**: With this goal in mind, we will define new net substructures, and using them identify where these shortest paths can lie as well as the lengths of these paths (for any two given points).
.
**B**: The same type of problem relating to robotics and motion planning is explored by using sequence trees in [3].
**C**: As a result, once these formulae are obtained, numerical calculations can be easily performed by anyone wishing to apply these findings.
In order to develop these formulae, we will utilize nets to calculate the shortest distance between points along the surface of convex regular polyhedra.
| BCA | BCA | BCA | BCA | Selection 3 |
<|MaskedSetence|> Influence of spatial correlations has been investigated in other contexts such as localization of directed polymers with space-time noise [31], Brownian motion in correlated Poisson potential [32, 33, 39] as well as Anderson localization, both by mathematicians [26, 27, 34] and physicists [2, 14, 46]. In our model however, the extreme value statistics of the gaps are more relevant than the correlation structure itself. <|MaskedSetence|> <|MaskedSetence|> Heavy tails also play a decisive role in trap models, see e.g. [4, 15].
. | **A**: Those are related to the heavy tails of the gap distribution.
**B**:
Since the obstacles are drawn according to a renewal process, our work provides an exemple of localization in a correlated disordered environment.
**C**: Influence of heavy tails in localization phenomena has been considered in the context of directed polymers [3, 5, 16, 25, 48] as well as in the parabolic Anderson model [6, 8, 12, 21, 35, 47].
| BCA | BAC | BAC | BAC | Selection 3 |
<|MaskedSetence|> See, e.g. <|MaskedSetence|> In particular, the Faber-Krahn inequality [12, 17, 18] states that the (Lebesgue) principal frequency of a domain K𝐾Kitalic_K of a fixed Lebesgue measure is minimized when K𝐾Kitalic_K is an Euclidean ball. <|MaskedSetence|> [22]) states that, conversely, the torsional rigidity of a domain K𝐾Kitalic_K of a fixed Lebesgue measure is maximized when K𝐾Kitalic_K is an Euclidean ball.
. | **A**: The result of Saint-Venant (see e.g.
**B**:
In the case when μ𝜇\muitalic_μ is the Lebesgue measure and L=Δ,𝐿ΔL=\Delta,italic_L = roman_Δ , these quantities have been studied extensively, and are intimately tied with the subject of isoperimetric inequalities.
**C**: Kawohl [13], Pólya and Szegö [22],
Burchard [5], Lieb and Loss [20], Kesavan [14], or Vazquez [24].
| BCA | BCA | CAB | BCA | Selection 1 |
<|MaskedSetence|> In the subsequent part, we present a synthesis scheme to allocate excitation and measurement signals for identifiability of an acyclic network model set. <|MaskedSetence|> In [6], all the vertices in a directed network are measured, and the allocation of excitations is based on decomposing the underlying graph of the network into edge-disjoint pseudotrees, whose roots are supposed to be excited for generic identifiability. <|MaskedSetence|> Because we are handling acyclic networks here, the disjoint pseudotrees can be relaxed to disjoint directed trees, as explained next.
. | **A**: In this work, we extend the work in [6] to the partial excitation and measurement case.
**B**: [16] for the definition.
**C**: This will lead to alternative and more compact identifiability conditions.
To this end, we resort to a graph covering approach as introduced in [6] for general (cyclic) dynamic networks.
| BCA | BCA | BCA | BAC | Selection 2 |
<|MaskedSetence|> Section 2 introduces the necessary background of differential privacy and presents the estimation of the noisy bi-degree sequence based on the moment equation, and obtains unified asymptotic properties for the differentially private estimation as the number of nodes goes to infinity. Section 3 applies our theoretical results in Section 2 to the Probit model. <|MaskedSetence|> <|MaskedSetence|> All proofs are deferred to the Appendix section.
. | **A**: Some further discussion is given in Section 5.
**B**:
The rest of this article is organized as follows.
**C**: Section 4 carries out the simulations under the Probit model and a real data analysis.
| BCA | BAC | BCA | BCA | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> This group is called the special Euclidean group and is denoted by SE(2)𝑆𝐸2SE(2)italic_S italic_E ( 2 ). In many applications, the congruence with respect to other groups is considered. <|MaskedSetence|> If a light source can be considered to be infinitely far away (like a sun), then the shadows are related by an affine transformation. See [13] for an excellent exposition of the roles played by projective, (special) affine, and (special) Euclidean transformations in computer vision. Starting in the 19th century, it was widely accepted that Euclidean geometry, although the most intuitive, is not the only possible consistent geometry, and that congruence can be defined relative to other transformation groups [14].
. | **A**: For example, two shadows cast by the same object onto two different planes by blocking the rays of light emitted from a lamp are related by a projective transformation.
**B**: However, since a reflection changes the orientation of an object, a group of orientation-preserving rigid motions, consisting of rotations and translations only, is often considered.
**C**:
To a human eye, two figures look the same if they are related by a rigid motion.
| CBA | BCA | CBA | CBA | Selection 1 |
A substantial review of variants of coordinate descent algorithms can be found in [4, Section 6.5.1]. The cyclic selection of coordinates is normally assumed to ensure convergence of the algorithm. <|MaskedSetence|> <|MaskedSetence|> Obviously, this is not guaranteed for each instance of the algorithm. <|MaskedSetence|> | **A**: Particularly, it is shown in [31] that randomization leads to faster convergence in terms of expectation.
**B**: On the other hand, the use of an irregular order is then considered by researchers to accelerate convergence.
**C**: The Gauss-Southwell method leads to faster convergence at the cost of extra computations and evaluations of gradients during the selection of coordinates which can be an issue in large-scale problems [25].
.
| BAC | CBA | BAC | BAC | Selection 1 |
<|MaskedSetence|> R. Graham and J. M. <|MaskedSetence|> Biquard [6].
Similar results, yet unpublished, has been obtained simultaneously by J. <|MaskedSetence|> | **A**: Lee [19, 23].
The general case of asymptotically symmetric metrics have been covered by O.
**B**: Roth in his Ph.D thesis for the complex hyperbolic case [25].
.
**C**: They all admit a sphere at infinity, which is endowed with a particular geometric structure closely related to the Riemannian metric of these spaces; they are called conformal infinities.
Asymptotically hyperbolic Einstein metrics with prescribed conformal infinity have been built by C.
| CAB | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> We start with a slight reorganization and minor refinement of ideas introduced by Lempert and Szőke in [LS-2014], and then we introduce definitions that are already suggested or implied in [LS-2014]. For example, in [LS-2014] the introduction of a ∂¯¯\bar{\partial}over¯ start_ARG ∂ end_ARG-operator to define the holomorphic structure of a smooth (quasi)-Hilbert field is not abstracted, but introduced in the ‘direct image’ constructions of [LS-2014, Part II] for the bundle of holomorphic sections. And the discussion of subfields is not as explicit, but many of the elements are hinted at in both [LS-2014] and [B-2009]. <|MaskedSetence|> | **A**: In fact, [B-2009] considers subfields in the case of trivial families of complex manifolds..
**B**:
2.
**C**: Abstract Theory of Berndtsson-Lempert-Szőke Fields
In this section we discuss the general geometric structure with which ℋℋ{\mathscr{H}}script_H and ℒℒ{\mathscr{L}}script_L are eventually endowed.
| BCA | BCA | CAB | BCA | Selection 1 |
<|MaskedSetence|> In the next Section we formulate main results of the paper. In Section 2 we set up the notation. Section 3 contains all complete proofs, for instance, in §3.1 we have considered an important special case — a level computation for exterior squares of elementary groups. <|MaskedSetence|> <|MaskedSetence|> We would like to express our sincere gratitude to our scientific adviser Nikolai Vavilov for formulating the problem and for a constant support, without which this paper would never have been written. The authors are grateful to Alexei Stepanov for carefully reading our original manuscript and for numerous remarks and corrections. Also, we would like to thank an anonymous referee for bringing our attention to the paper [32].
. | **A**: Finally, a level reduction for exterior powers is proved in §3.5.
Acknowledgment.
**B**: In §3.2–§3.4 we develop a technique for an arbitrary general exterior power.
**C**: The present paper is organized as follows.
| CBA | CAB | CBA | CBA | Selection 1 |
<|MaskedSetence|> Outline of the paper
In section 2 we recall some basic facts of T-varieties and the combinatorial framework to describe them. In particular, we recall the definitions of p-divisors and of divisorial fans. In Section 3 we restrict to the complexity one case and we describe the invariant subvarieties induced by the polyhedra in the slices of the divisorial fan. <|MaskedSetence|> For this we mainly follow [UN]. <|MaskedSetence|> | **A**: 1.1.
**B**: Then we recall the classification of T-invariant Cartier divisors in terms of Cartier divisorial support functions, following the work in [AIPSV] and [PS].
.
**C**: It turns out that these invariant subvarieties generate the pseudoeffective cone of the T-variety and we give a list of its generators (Theorem 3.3).
| ACB | ACB | ABC | ACB | Selection 4 |
Note that this discrepancy between Bayesian and frequentist measures differs considerably from the situation in standard statistical inference with non-adaptive samples. For a fixed-sample problem, the Bernstein–von Mises theorem describes the asymptotic equivalence of Bayesian and frequentist inference. However, in an adaptive sampling scheme, underestimation of an arm due to the randomness of the empirical mean results in a smaller number of samples of the arm in the future. <|MaskedSetence|> <|MaskedSetence|> This is notable because it enables solutions that extend beyond the conventional one- or two-step lookahead. <|MaskedSetence|> | **A**: We demonstrate several instances in which it is feasible to perform exact analyses of dynamic programming regardless of the need to project the evolution of the posterior over extended future periods.
**B**: Bayesian algorithms are robust up to a polynomially small underestimation, whereas frequentist algorithms are robust up to an exponentially small underestimation.
Our results offer analytical innovation by establishing foundational principles for a formal analysis that yields exact solutions in dynamic programming.
**C**: Bayesian and frequentist BAI algorithms are both robust against such randomness but with different confidence levels.
| BCA | CBA | CBA | CBA | Selection 4 |
As well as the results of Čulík [6], Guy [10] and Roman [19] already discussed, there has been some more recent work that determines exact values for Zarankiewicz numbers. Goddard, Henning and Oellermann [9] and Collins, Riasanovsky, Wallace and Radziszowski [4] determined exact values for various small parameters. Damásdi, Héger and Szőnyi [7] found a number of interesting exact values and bounds on Z2,2(m,n)subscript𝑍22𝑚𝑛Z_{2,2}(m,n)italic_Z start_POSTSUBSCRIPT 2 , 2 end_POSTSUBSCRIPT ( italic_m , italic_n ) relating to finite projective planes and other designs.
The rest of the paper is arranged as follows. In Section 2 we establish the new upper bounds and prove Theorem 1.1 and Corollary 1.2. <|MaskedSetence|> <|MaskedSetence|> Theorem 1.5 is proved in Section 6. <|MaskedSetence|> | **A**: Section 4 is devoted to the cases where ⌊Aλk(m,n)⌋subscriptsuperscript𝐴𝑘𝜆𝑚𝑛\lfloor A^{k}_{\lambda}(m,n)\rfloor⌊ italic_A start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_m , italic_n ) ⌋ is achieved, while Section 5 is devoted to the cases where ⌊Bλk+1(m,n)⌋subscriptsuperscript𝐵𝑘1𝜆𝑚𝑛\lfloor B^{k+1}_{\lambda}(m,n)\rfloor⌊ italic_B start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_m , italic_n ) ⌋ or ⌊Bλk(m,n)⌋subscriptsuperscript𝐵𝑘𝜆𝑚𝑛\lfloor B^{k}_{\lambda}(m,n)\rfloor⌊ italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_m , italic_n ) ⌋ is achieved and concludes by completing the proofs of Theorem 1.3 and Corollary 1.4.
**B**: Section 3 details some preliminary results we require for our proof of Theorem 1.3 which then takes place in Sections 4 and 5.
**C**: We conclude with some discussion of our work and possible future directions in Section 7..
| BAC | BAC | BAC | CAB | Selection 3 |
2}-1}^{\prime}lk ( italic_p , italic_M ) = italic_P ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_x , italic_q , italic_y , italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, …,p2′,p1′)\dots,p_{2}^{\prime},p_{1}^{\prime})… , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). Then, no edge of the form qz𝑞𝑧qzitalic_q italic_z can be identified with an edge in the path P(pn1,pn1−1,…,p1,p,p1′,p2′,…,pn2′)𝑃subscript𝑝subscript𝑛1subscript𝑝subscript𝑛11…subscript𝑝1𝑝superscriptsubscript𝑝1′superscriptsubscript𝑝2′…superscriptsubscript𝑝subscript𝑛2′P(p_{n_{1}},p_{n_{1}-1},\dots,p_{1},p,p_{1}^{\prime},p_{2}^{\prime},\dots,p_{n%
_{2}}^{\prime})italic_P ( italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Consequently, the union of st(p,M)st𝑝𝑀\mbox{st}\,(p,M)st ( italic_p , italic_M ) and st(q,M)st𝑞𝑀\mbox{st}\,(q,M)st ( italic_q , italic_M ) is a 2-ball (cf. Figure 2 (b)𝑏(b)( italic_b )), contradicting the fact that the Möbius strip M𝑀Mitalic_M is the same as the union of st(p,M)st𝑝𝑀\mbox{st}\,(p,M)st ( italic_p , italic_M ) and st(q,M)st𝑞𝑀\mbox{st}\,(q,M)st ( italic_q , italic_M ).
. | **A**: Here, x,y,pi,pj′,qk𝑥𝑦subscript𝑝𝑖superscriptsubscript𝑝𝑗′subscript𝑞𝑘x,y,p_{i},p_{j}^{\prime},q_{k}italic_x , italic_y , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, and ql′superscriptsubscript𝑞𝑙′q_{l}^{\prime}italic_q start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are non-singular vertices, 1≤i≤n11𝑖subscript𝑛11\leq i\leq n_{1}1 ≤ italic_i ≤ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, 1≤j≤n21𝑗subscript𝑛21\leq j\leq n_{2}1 ≤ italic_j ≤ italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, 1≤k≤m11𝑘subscript𝑚11\leq k\leq m_{1}1 ≤ italic_k ≤ italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, 1≤l≤m21𝑙subscript𝑚21\leq l\leq m_{2}1 ≤ italic_l ≤ italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.
**B**: If we identify any edge from the paths P(pn1,pn1−1,…,p1)𝑃subscript𝑝subscript𝑛1subscript𝑝subscript𝑛11…subscript𝑝1P(p_{n_{1}},p_{n_{1}-1},\dots,p_{1})italic_P ( italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) or P(p1′,p2′,…,pn2′)𝑃superscriptsubscript𝑝1′superscriptsubscript𝑝2′…superscriptsubscript𝑝subscript𝑛2′P(p_{1}^{\prime},p_{2}^{\prime},\dots,p_{n_{2}}^{\prime})italic_P ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) with an edge from the paths P(qm1,qm1−1P(q_{m_{1}},q_{m_{1}-1}italic_P ( italic_q start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT, …,q1)\dots,q_{1})… , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) or P(q1′,q2′,…,qm2′)𝑃superscriptsubscript𝑞1′superscriptsubscript𝑞2′…superscriptsubscript𝑞subscript𝑚2′P(q_{1}^{\prime},q_{2}^{\prime},\dots,q_{m_{2}}^{\prime})italic_P ( italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , … , italic_q start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), then the identified edge will be an interior edge of M𝑀Mitalic_M, leading to a contradiction.
**C**: Let lk(q,M)=P(q1,q2,…,qm1,x,p,y,qm2′,qm2−1′\mbox{lk}\,(q,M)=P(q_{1},q_{2},\dots,q_{m_{1}},x,p,y,q_{m_{2}}^{\prime},q_{m_{%
2}-1}^{\prime}lk ( italic_q , italic_M ) = italic_P ( italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_q start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_x , italic_p , italic_y , italic_q start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_q start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, …,q2′,q1′)\dots,q_{2}^{\prime},q_{1}^{\prime})… , italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ).
| CAB | ACB | CAB | CAB | Selection 1 |
In this section we equip an S𝑆Sitalic_S-adic shift with an invariant measure and study the existence of measurable eigenvalues. <|MaskedSetence|> In particular
we state results from [BKMS13], [Men91] and [CDHM03] which are couched in terms of Bratteli diagrams, or cutting-and-stacking transformations, but which we rephrase in our language. Our main result in this section is Theorem 5.7. <|MaskedSetence|> <|MaskedSetence|> | **A**: The combination of strong primitivity and finitary ensures that our systems are of exact finite rank,
a condition which has been extensively studied in the literature [Men91, Bos92], and one which is useful in the proof of Theorem 5.7.
.
**B**: For this, Theorem 2.7 is particularly useful, as, with the conditions that the directive sequence is recognizable and everywhere growing, it allows us to use and compare results concerning Bratteli-Vershik systems.
**C**: It requires a strengthening of the notion of straightness to that of strong straightness, in Definition 5.5.
| ACB | BCA | BCA | BCA | Selection 3 |
<|MaskedSetence|> Because with the Steiner system there is an association between pilot sequences and user IDs, observing a certain pilot sequence in a given slot automatically indicates which user is active and where to look for its remaining replicas. <|MaskedSetence|> One possibility is to look for the correlation between signals in different slots and combine those with the highest correlation score.
However, such a solution is not perfect as it might miss some of the replicas or introduce false positives. <|MaskedSetence|> Clearly, the downside of this solution is the introduction of overhead.
. | **A**: This is not the case when Random selection scheme is used so additional procedures might be required.
**B**:
Another caveat is that, clearly, the receiver must know where each replica of each user is located in order to perform combining through MRC.
**C**: Furthermore, it entails exhaustive search and, hence, high complexity.
Alternatively, a unique ID that can be decoded independently of the rest of the payload could be added to each packet or, each slot could be preceded by an activity-indication phase.
| BAC | BAC | BCA | BAC | Selection 4 |
<|MaskedSetence|> In the course of proving Theorem 6.8 we will prove a bundle construction-type theorem, Theorem 3.15, which gives an equivalence between a category of plotwise defined cocycles on X𝑋Xitalic_X and the category of diffeological principal G𝐺Gitalic_G-bundles over X𝑋Xitalic_X. This result may be of independent interest, and thus we prove it in Section 3, before we need to introduce any abstract machinery.
It is our view that by applying tools from higher topos theory can be beneficial to the still young subject of diffeological spaces. <|MaskedSetence|> <|MaskedSetence|> Pulling these definitions over to diffeological spaces and analyzing the results are the subject of future work.. | **A**: Furthermore, higher topos theory already has definitions for higher principal bundles (called bundle gerbes) and connections on such objects inherently built into it.
**B**:
Thus ∞\infty∞-stack cohomology of diffeological spaces also encompasses nonabelian cohomology.
**C**: In particular we believe that while the machinery of ∞\infty∞-stack cohomology may come from an abstract framework, it can ultimately output important and down-to-earth results.
| BCA | CBA | BCA | BCA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> Then it follows from Lemma 2.12 that there exists a holomorphic function f~jsubscript~𝑓𝑗\tilde{f}_{j}over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT on {Ψ1<0}∩DjsubscriptΨ10subscript𝐷𝑗\{\Psi_{1}<0\}\cap D_{j}{ roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < 0 } ∩ italic_D start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT such that Cj=∫{Ψ1<0}∩Dj|f~j|2subscript𝐶𝑗subscriptsubscriptΨ10subscript𝐷𝑗superscriptsubscript~𝑓𝑗2C_{j}=\int_{\{\Psi_{1}<0\}\cap D_{j}}|\tilde{f}_{j}|^{2}italic_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∫ start_POSTSUBSCRIPT { roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < 0 } ∩ italic_D start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT | over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and (f~j−f)o∈I(p0Ψ1)osubscriptsubscript~𝑓𝑗𝑓𝑜𝐼subscriptsubscript𝑝0subscriptΨ1𝑜(\tilde{f}_{j}-f)_{o}\in I(p_{0}\Psi_{1})_{o}( over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_f ) start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT ∈ italic_I ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT for any j∈ℤ>0𝑗subscriptℤabsent0j\in\mathbb{Z}_{>0}italic_j ∈ blackboard_Z start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT. <|MaskedSetence|> Then we can apply Lemma 2.9 and the diagonal method to extract a subsequence of {f~j}j∈ℤ>0subscriptsubscript~𝑓𝑗𝑗subscriptℤabsent0\{\tilde{f}_{j}\}_{j\in\mathbb{Z}_{>0}}{ over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_j ∈ blackboard_Z start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT (also denoted by {f~j}j∈ℤ>0subscriptsubscript~𝑓𝑗𝑗subscriptℤabsent0\{\tilde{f}_{j}\}_{j\in\mathbb{Z}_{>0}}{ over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_j ∈ blackboard_Z start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT) compactly convergent to a holomorphic function f~0subscript~𝑓0\tilde{f}_{0}over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT on {Ψ1<0}subscriptΨ10\{\Psi_{1}<0\}{ roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < 0 }, which satisfies (f~0−f)o∈I(p0Ψ)osubscriptsubscript~𝑓0𝑓𝑜𝐼subscriptsubscript𝑝0Ψ𝑜(\tilde{f}_{0}-f)_{o}\in I(p_{0}\Psi)_{o}( over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_f ) start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT ∈ italic_I ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_Ψ ) start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT. As Cjsubscript𝐶𝑗C_{j}italic_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is increasing with respect to j𝑗jitalic_j, using Fatou’s Lemma and the definition of G(0;Ψ1,I+(Ψ1)o,f)𝐺0subscriptΨ1subscript𝐼subscriptsubscriptΨ1𝑜𝑓G(0;\Psi_{1},I_{+}(\Psi_{1})_{o},f)italic_G ( 0 ; roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT , italic_f ), we obtain that. | **A**: holds for any j∈ℤ>0𝑗subscriptℤabsent0j\in\mathbb{Z}_{>0}italic_j ∈ blackboard_Z start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT and r1∈(0,1]subscript𝑟101r_{1}\in(0,1]italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ ( 0 , 1 ].
Without loss of generality, assume that there exists r0∈(0,1]subscript𝑟001r_{0}\in(0,1]italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ ( 0 , 1 ] such that 1r02∫{Ψ1≤2logr0}|f|2<+∞1superscriptsubscript𝑟02subscriptsubscriptΨ12subscript𝑟0superscript𝑓2\frac{1}{r_{0}^{2}}\int_{\{\Psi_{1}\leq 2\log r_{0}\}}|f|^{2}<+\inftydivide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT { roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ 2 roman_log italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } end_POSTSUBSCRIPT | italic_f | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < + ∞.
**B**: Note that supj∫{Ψ1<0}∩Dj|f~j|2<+∞subscriptsupremum𝑗subscriptsubscriptΨ10subscript𝐷𝑗superscriptsubscript~𝑓𝑗2\sup_{j}\int_{\{\Psi_{1}<0\}\cap D_{j}}|\tilde{f}_{j}|^{2}<+\inftyroman_sup start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT { roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < 0 } ∩ italic_D start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT | over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < + ∞.
**C**: Following from inequality (4.2), we get that supjCj<+∞subscriptsupremum𝑗subscript𝐶𝑗\sup_{j}C_{j}<+\inftyroman_sup start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT < + ∞.
Lemma 2.7 tells us that there exists p0∈(1,2)subscript𝑝012p_{0}\in(1,2)italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ ( 1 , 2 ) such that I+(Ψ1)o=I(p0Ψ1)osubscript𝐼subscriptsubscriptΨ1𝑜𝐼subscriptsubscript𝑝0subscriptΨ1𝑜I_{+}(\Psi_{1})_{o}=I(p_{0}\Psi_{1})_{o}italic_I start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT = italic_I ( italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_Ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT.
| ACB | ACB | ACB | ABC | Selection 2 |
<|MaskedSetence|> If p≡7(mod8)𝑝annotated7pmod8p\equiv 7\pmod{8}italic_p ≡ 7 start_MODIFIER ( roman_mod start_ARG 8 end_ARG ) end_MODIFIER, there is only one down 2222-isogeny from E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. If E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT corresponds to the form (q,r,r2+p4q)𝑞𝑟superscript𝑟2𝑝4𝑞(q,r,\frac{r^{2}+p}{4q})( italic_q , italic_r , divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_p end_ARG start_ARG 4 italic_q end_ARG ) with q𝑞qitalic_q satisfying (1)1(\ref{e1})( ), then E2subscript𝐸2E_{2}italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT corresponds to the form (q,4r,4r2+4pq)𝑞4𝑟4superscript𝑟24𝑝𝑞(q,4r,\frac{4r^{2}+4p}{q})( italic_q , 4 italic_r , divide start_ARG 4 italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 italic_p end_ARG start_ARG italic_q end_ARG ).
If p≡3(mod8)𝑝annotated3pmod8p\equiv 3\pmod{8}italic_p ≡ 3 start_MODIFIER ( roman_mod start_ARG 8 end_ARG ) end_MODIFIER, then there are three down 2222-isogenies from E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Denote them by E2,isubscript𝐸2𝑖E_{2,i}italic_E start_POSTSUBSCRIPT 2 , italic_i end_POSTSUBSCRIPT with i∈{1,2,3}𝑖123i\in\{1,2,3\}italic_i ∈ { 1 , 2 , 3 }. If E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT corresponds to the form (q,r,r2+p4q)𝑞𝑟superscript𝑟2𝑝4𝑞(q,r,\frac{r^{2}+p}{4q})( italic_q , italic_r , divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_p end_ARG start_ARG 4 italic_q end_ARG ), by Proposition 4.4, there are three forms with discriminant −16p16𝑝-16p- 16 italic_p in the non-principal genus that can be derived from (q,r,r2+p4q)𝑞𝑟superscript𝑟2𝑝4𝑞(q,r,\frac{r^{2}+p}{4q})( italic_q , italic_r , divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_p end_ARG start_ARG 4 italic_q end_ARG ). <|MaskedSetence|> On the contrary, suppose that the curve E2,isubscript𝐸2𝑖E_{2,i}italic_E start_POSTSUBSCRIPT 2 , italic_i end_POSTSUBSCRIPT corresponds to the form (qi,4ri,4ri2+4pqi)subscript𝑞𝑖4subscript𝑟𝑖4superscriptsubscript𝑟𝑖24𝑝subscript𝑞𝑖(q_{i},4r_{i},\frac{4r_{i}^{2}+4p}{q_{i}})( italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , 4 italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , divide start_ARG 4 italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 italic_p end_ARG start_ARG italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) with qisubscript𝑞𝑖q_{i}italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT satisfying (1)1(\ref{e1})( ). For every i𝑖iitalic_i, the form (qi,4ri,4ri2+4pqi)subscript𝑞𝑖4subscript𝑟𝑖4superscriptsubscript𝑟𝑖24𝑝subscript𝑞𝑖(q_{i},4r_{i},\frac{4r_{i}^{2}+4p}{q_{i}})( italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , 4 italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , divide start_ARG 4 italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 italic_p end_ARG start_ARG italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) can be derived from (qi,ri,ri2+p4qi)subscript𝑞𝑖subscript𝑟𝑖superscriptsubscript𝑟𝑖2𝑝4subscript𝑞𝑖(q_{i},r_{i},\frac{r_{i}^{2}+p}{4q_{i}})( italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , divide start_ARG italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_p end_ARG start_ARG 4 italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ). <|MaskedSetence|> | **A**: Moreover, each form corresponds to one of {E2,i}i∈{1,2,3}subscriptsubscript𝐸2𝑖𝑖123\{E_{2,i}\}_{i\in\{1,2,3\}}{ italic_E start_POSTSUBSCRIPT 2 , italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ { 1 , 2 , 3 } end_POSTSUBSCRIPT.
**B**:
In fact, the curve E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is on the surface and E2subscript𝐸2E_{2}italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is on the floor.
**C**: It follows that the three forms (qi,ri,ri2+p4qi),i∈{1,2,3}subscript𝑞𝑖subscript𝑟𝑖superscriptsubscript𝑟𝑖2𝑝4subscript𝑞𝑖𝑖123(q_{i},r_{i},\frac{r_{i}^{2}+p}{4q_{i}}),i\in\{1,2,3\}( italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , divide start_ARG italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_p end_ARG start_ARG 4 italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) , italic_i ∈ { 1 , 2 , 3 } are equivalent in C(−p)𝐶𝑝C(-p)italic_C ( - italic_p ) and the curve E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT corresponds to every one of these three forms..
| BAC | BAC | ABC | BAC | Selection 4 |
Biological sciences are the field where SSPs have been most investigated over the past three decades, raising several challenges in both methods and applications. <|MaskedSetence|> [2019], which shows how large scale genomic data provide a fertile ground for SSPs. Although sequencing technologies have advanced the understanding of genome biology, observed samples may not be perfectly representative of the molecular heterogeneity or species composition of the underlying DNA library, often providing a poor representation due to low-abundance molecules that are hard to sample. <|MaskedSetence|> <|MaskedSetence|> [2019] identified three major questions of interest:
Q1)
. | **A**: This is testified by the work of Deng et al.
**B**: Deng et al.
**C**: Due to the impossibility of sequencing DNA libraries up to complete saturation, it is common to make use of the observed samples, typically collected under suitable budget constraints, to infer the molecular heterogeneity of additional unobserved samples from the library, as well as of the library itself.
| ACB | CAB | ACB | ACB | Selection 3 |
<|MaskedSetence|> Its long intervals terminate at 8.58.58.58.5 and start from 10.510.510.510.5. <|MaskedSetence|> <|MaskedSetence|> It also contains a Red join gadget in [1.5,4.5]1.54.5[1.5,4.5][ 1.5 , 4.5 ]. Its long intervals terminate at 1.51.51.51.5 and start from 4.54.54.54.5.
. | **A**:
The fifth buffer contains the second part of the second switch gadget in [8.5,11.5]8.511.5[8.5,11.5][ 8.5 , 11.5 ].
**B**: Its long intervals terminate at {12,15.5}1215.5\{12,15.5\}{ 12 , 15.5 } and start from {17.5,21}17.521\{17.5,21\}{ 17.5 , 21 }.
**C**: It also contains the third switch gadget in [12,21]1221[12,21][ 12 , 21 ].
| ACB | ACB | BCA | ACB | Selection 1 |
<|MaskedSetence|> There are several topological complications to be addressed, but the outcome is relatively satisfactory: the invariants are well-defined and finite in many interesting cases and interrelate to shed a light on the structure theory of certain natural classes of topological groups.
In Section 2 we introduce box spaces and define asymptotic dimension of locally compact (second countable) groups and show that the invariant is finite for large classes of residually compact groups. In Section 3 we define the Hirsch length for topological groups and prove the Hirsch formula, which is critical for inductive arguments based on the Hirsch length. <|MaskedSetence|> Same is done in Section 4 for topologically elementary amenable groups. <|MaskedSetence|> | **A**: A small final epilogue in Section 5 is devoted to finiteness of the asymptotic dimension of totally disconnected, locally compact, second countable groups..
**B**:
Here instead of finitely generated discrete groups one has to work with compactly generated locally compact groups (usually second countable, to make sure a plig metric exists).
**C**: This section also deals with a natural extension of virtually nilpotent groups in the topological realm, which are shown to be residually compact and so of finite asymptotic dimension.
| BCA | BCA | BCA | BCA | Selection 4 |
<|MaskedSetence|> Our notion relies on [Ken92] notion of bifilteredness, together with a convenient notion of bicompact objects (Definition 3.1.1) enjoying the analogous property of compact objects against bifiltered bicolimits; we define finitely bi-accessible categories as those having bifiltered bicolimits and an essentially small subcategory of bicompact objects generating them under bifiltered bicolimits; finitely bipresentable 2-categories are as those that are moreover bicocomplete - but similarly to the one dimensional case, this amounts to having weighted bilimits. In particular, a handy criterion to detect finite bipresentability of a 2222-category is offered in Theorem 3.4.3, which is a 2222-dimensional analog of a classical characterization through strong generators. <|MaskedSetence|> <|MaskedSetence|> Though most pseudofunctors cannot be expressed as conical bicolimits of representables (requiring either σ𝜎\sigmaitalic_σ-bicolimits or non trivial weights to encode 2-dimensional data),. | **A**: We then prove that categories of flat pseudofunctors are bi-accessible (Corollary 4.2.6) and bipresentable (Theorem 4.3.6) if their domain admits finite weighted bilimits, the latter result being part of a categorification of the well known Gabriel-Ulmer duality established in Section 5.3.
**B**: To fix this, we introduce here relaxed notions of bi-accessible (Definition 3.2.1) and bipresentable (Definition 3.3.1) 2-categories and connect them to the recent advance of [DDS18] on the theory of flat pseudofunctors.
**C**:
A subtle part of this work is understanding the exact relation between different classes of 2-dimensional (co)-limits involved here, in particular with the formalism of σ𝜎\sigmaitalic_σ-colimits (a class intermediate between bi and lax) involved in [DDS18].
| BAC | BAC | BAC | BCA | Selection 2 |
Proof.
Let (V1,V2,V3,V4)subscript𝑉1subscript𝑉2subscript𝑉3subscript𝑉4(V_{1},V_{2},V_{3},V_{4})( italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) be an ℳ𝒢𝒫ℳ𝒢𝒫\mathcal{MGP}caligraphic_M caligraphic_G caligraphic_P-partition corresponding to a gp-set S𝑆Sitalic_S of ℳ(G)ℳ𝐺\mathcal{M}(G)caligraphic_M ( italic_G ) not containing the root u∗superscript𝑢u^{*}italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. It follows from Lemma 1 that |S|≥n+1𝑆𝑛1|S|\geq n+1| italic_S | ≥ italic_n + 1, so we can assume that n1>n4subscript𝑛1subscript𝑛4n_{1}>n_{4}italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_n start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. Let u𝑢uitalic_u be the universal vertex of G𝐺Gitalic_G. <|MaskedSetence|> <|MaskedSetence|> By Condition 3, there cannot be vertices of V2subscript𝑉2V_{2}italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in two different cliques of W𝑊Witalic_W, so V4subscript𝑉4V_{4}italic_V start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT is non-empty and we would have |S|≤n𝑆𝑛|S|\leq n| italic_S | ≤ italic_n, a contradiction. If u∈V2𝑢subscript𝑉2u\in V_{2}italic_u ∈ italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, then, assuming that W1subscript𝑊1W_{1}italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT contains a vertex v𝑣vitalic_v of V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, Definition 5 shows that G𝐺Gitalic_G contains no further vertices of V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, so V4=∅subscript𝑉4V_{4}=\emptysetitalic_V start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = ∅ and Lemma 12 shows that |W1|=1subscript𝑊11|W_{1}|=1| italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = 1 and |S|=n+1𝑆𝑛1|S|=n+1| italic_S | = italic_n + 1. <|MaskedSetence|> | **A**: If u∈V1𝑢subscript𝑉1u\in V_{1}italic_u ∈ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, then by Condition 1 of Definition 5 there are no further vertices in V1∪V3subscript𝑉1subscript𝑉3V_{1}\cup V_{3}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.
**B**: If u∈V3𝑢subscript𝑉3u\in V_{3}italic_u ∈ italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, then by Definition 5 V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is empty, contradicting |S|>n𝑆𝑛|S|>n| italic_S | > italic_n.
**C**: Hence we can assume that u∈V4𝑢subscript𝑉4u\in V_{4}italic_u ∈ italic_V start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT.
.
| BAC | ACB | BAC | BAC | Selection 4 |
Our approach to formulating risk-averse MDPs is grounded in the understanding that law-invariant convex risk measures can be interpreted as functionals defined on the space of probabilities over ℝℝ\mathbb{R}blackboard_R. This perspective has been effectively employed in various contexts to further the development of risk measure theory (cf. <|MaskedSetence|> Notably, [61] is a seminal contribution that systematically investigates static risk measures and DRMs from a distributional standpoint. In a similar vein, we explore DRMs at the level of distributions, conceptualizing them as nested compositions of state-dependent law-invariant convex risk measures. It is important to emphasize that previous studies on DRMs at the distributional level, such as in [61, 6], primarily focused on static one-step risk measures in characterization or construction. Our approach, which allows the risk measures to vary according to the state, introduces additional complexity. A significant advantage of this distributional-level construction is that it automatically ensures MDPs with identical distributions are treated as equivalent in terms of risk when assessed under the proposed DRMs. <|MaskedSetence|> Furthermore, our framework provides an appropriate foundation for balancing various assumptions, including a weakly continuous transition kernels, while still ensuring the attainment of the optimal outcome. <|MaskedSetence|> It’s important to note that, although the construction above may seem like a straightforward alteration of existing frameworks, it involves some unique technical aspects that have not been previously discussed. For simplicity, we consider bounded costs, which allows for conditional risk mappings that contain essential supremum as a major ingredient – a feature that is often omitted otherwise.
The main contributions of this paper can be summarized as follows:. | **A**: This, in turn, allows for greater flexibility in risk-averse modeling.
**B**: [61, 1, 26, 32]).
**C**: Moreover, it seamlessly integrates latent costs and random actions through the concept of regular conditional distributions.
| BCA | ACB | BCA | BCA | Selection 3 |
We note that the junta approximation results of [10] and [13], as well as the results for sets from [33] and [12] came together with stability results that characterized the approximate structure of families that are close to extremal. <|MaskedSetence|> <|MaskedSetence|> And, of course, this opens the way to exploring other Turán-type problems for subfamilies of ‘nice’ families, such as the family of all permutations.
Lastly, let us return to intersecting families of k𝑘kitalic_k-sets. On the one extreme, we have the Erdős–Ko–Rado theorem, which is tight for families of sets containing a fixed element. On the other extreme, we have intersecting families that are regular (all elements of the ground set are contained in the same number of sets) or even symmetric (its group of automorphisms is transitive). Note that if a family is symmetric, then it is regular, but not vice versa. <|MaskedSetence|> | **A**: Our general results in Section 3 and 4 allow for similar results in a sparse setting that have no analogues via the junta method.
**B**: We show the following result that answers a question of Narayanan.
.
**C**: Our approach also gives such structural results, moreover, it allows finding structure in families of much smaller size than that suitable for the use of the junta method.
| CAB | CAB | CAB | CAB | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Traveling along the curve beginning at the base point in the direction of the orientation. An ordered pair (a,b)𝑎𝑏(a,b)( italic_a , italic_b ) of two crossings a𝑎aitalic_a and b𝑏bitalic_b are called parallel if and only if smoothing two crossings along the orientation (Figure 1) produces a three-component curve Cab(1),Cab(2),Cab(3)subscriptsuperscript𝐶1𝑎𝑏subscriptsuperscript𝐶2𝑎𝑏subscriptsuperscript𝐶3𝑎𝑏C^{(1)}_{ab},C^{(2)}_{ab},C^{(3)}_{ab}italic_C start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT. For each parallel pair a,b𝑎𝑏a,bitalic_a , italic_b, we define the components’ order Cab(1),Cab(2),Cab(3)subscriptsuperscript𝐶1𝑎𝑏subscriptsuperscript𝐶2𝑎𝑏subscriptsuperscript𝐶3𝑎𝑏C^{(1)}_{ab},C^{(2)}_{ab},C^{(3)}_{ab}italic_C start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT and words w𝑤witalic_w (Definition 3); we define a map ΔΔ\Deltaroman_Δ by
. | **A**: Figure 1.
**B**: The label L𝐿Litalic_L (resp. R𝑅Ritalic_R) indicates “left” (resp. “right”).
Let C𝐶Citalic_C be a curve with a base point.
**C**: Smoothing.
| BCA | ACB | ACB | ACB | Selection 3 |
There are additional complications in real-world implementations of Elo. <|MaskedSetence|> <|MaskedSetence|> Sometimes rating updates are batched. That is, one accumulates their pot winnings and losses over several games, and updates their rating once at the end. <|MaskedSetence|> All these details and even more complications are outlined thoroughly by different organizations implementing Elo; one may read about them in, for example, the FIDE Handbook [4].
. | **A**: For legibility and practicality, fractional and negative rating points are avoided by scaling and shifting points up and rounding to the nearest integer, and by imposing an artificial floor on possible ratings (by gifting a player points if they would otherwise dip below the floor).
The total size of the pot K𝐾Kitalic_K may also vary depending on various factors, such as how many games each player has played before.
**B**: For example, K𝐾Kitalic_K may be large for a new player to facilitate faster convergence of their rating to their true skill level, and may decrease over time to reduce arbitrary fluctuations for experienced players.
**C**: Often times all the games played at a single tournament are batched together in that manner.
| ABC | ABC | CBA | ABC | Selection 4 |
This has been studied in [wu] under some restrictions. The results were then generalized using the same approach in [parsur]. <|MaskedSetence|> <|MaskedSetence|> In a [guhsur], the authors generalize the main Theorem 7.9 in some special cases.
For a detailed account on the notions of this section, we refer the reader to [involutions]. We will use the notation from Section 1.2. <|MaskedSetence|> | **A**: In both cases, the authors show a Hasse principle for homogeneous varieties under unitary groups.
**B**: Assume, moreover, that k𝑘kitalic_k is a local field and chark≠2.char𝑘2{\mathrm{char}\ k\neq 2}.roman_char italic_k ≠ 2 ..
**C**: We give here a brief summary of the interpretation of these results in our setting, without claim to originality.
| ACB | ACB | ACB | BCA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> However none of these works comes with an explicit convergence rate. The work [stoltz2018longtime] takes a
very nice approach, producing a perturbative expansion in ϵitalic-ϵ\epsilonitalic_ϵ of the invariant measure of the slow-fast system, and therefore proving convergence of the invariant measure of the slow-fast system to the invariant measure of the averaged equation.
The analysis in [stoltz2018longtime] is made possible by the fact that, due to the specific form of the Langevin dynamics, an explicit expression for the invariant measure of the slow-fast system is a priori known. In our approach, there is no need to have such knowledge and indeed, while our assumptions do imply that the slow-fast system has an invariant measure, this fact is never used in the analysis. <|MaskedSetence|> | **A**: The only other UiT results we are aware of are those in [ilyin1998global] (and references therein), for deterministic systems; for stochastic dynamics, aside from the work [barr2020fast], we are only aware of [cheng2023second], which deals with Stochastic Partial Differential equations, and of [stoltz2018longtime], which is inspired by problems in molecular dynamics and treats specifically the case of Langevin dynamics on compact state space, hence the drifts of the SDEs are in gradient form and the diffusion coefficients are constant.
**B**: So the class of SDEs we consider here is truly general.
.
**C**: To the best of our knowledge, there exist truly very few UiT averaging theorems in the literature, even when searching beyond the case of slow-fast systems of SDEs and looking in the multiscale literature for PDEs, SPDEs and ODEs.
| CAB | BCA | CAB | CAB | Selection 4 |
Recall that R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG is finitely generated as an R𝑅Ritalic_R-module, and also it is noetherian and local, because R𝑅Ritalic_R is complete (for example R¯=k[[x4,y4,x3y,xy3,x2y2]]¯𝑅𝑘delimited-[]superscript𝑥4superscript𝑦4superscript𝑥3𝑦𝑥superscript𝑦3superscript𝑥2superscript𝑦2\overline{R}=k[[x^{4},y^{4},x^{3}y,xy^{3},x^{2}y^{2}]]over¯ start_ARG italic_R end_ARG = italic_k [ [ italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_y , italic_x italic_y start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ]. Note that x2y2=(x3y)2x4superscript𝑥2superscript𝑦2superscriptsuperscript𝑥3𝑦2superscript𝑥4x^{2}y^{2}=\frac{(x^{3}y)^{2}}{x^{4}}italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG ( italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_y ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG belongs to the fraction field of R𝑅Ritalic_R, and it is the root of f(T)=T3−x4T∈R[T]𝑓𝑇superscript𝑇3superscript𝑥4𝑇𝑅delimited-[]𝑇f(T)=T^{3}-x^{4}T\in R[T]italic_f ( italic_T ) = italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_T ∈ italic_R [ italic_T ]. Since k[[x4,y4,x3y,xy3,x2y2]]𝑘delimited-[]superscript𝑥4superscript𝑦4superscript𝑥3𝑦𝑥superscript𝑦3superscript𝑥2superscript𝑦2k[[x^{4},y^{4},x^{3}y,xy^{3},x^{2}y^{2}]]italic_k [ [ italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_y , italic_x italic_y start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ] is the invariant ring, it is normal, and so becomes the integral closure of R𝑅Ritalic_R). Suppose on the way of contradiction that R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG is tor-rigid. This allows us to apply [4, 4.3] to deduce that each R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG-regular sequence is an R𝑅{R}italic_R-regular sequence. <|MaskedSetence|> <|MaskedSetence|> Thus, a,b𝑎𝑏a,bitalic_a , italic_b is an R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG-regular sequence.
By the mentioned result of Auslander, a,b𝑎𝑏a,bitalic_a , italic_b is an R𝑅{R}italic_R-regular sequence. <|MaskedSetence|> | **A**: By
Serre’s characterization of normality, see [24, Theorem 23.8], we know R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG satisfies Serre’s condition (S2)subscript𝑆2(S_{2})( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) (for the definition, see [24, page 183]).
**B**: Now, let a,b𝑎𝑏a,bitalic_a , italic_b
be a system of parameter for R𝑅Ritalic_R, and so a parameter sequence for R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG (for example, in the example set a:=x4,b:=y4formulae-sequenceassign𝑎superscript𝑥4assign𝑏superscript𝑦4a:=x^{4},b:=y^{4}italic_a := italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_b := italic_y start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT).
**C**: So,.
| BAC | BAC | BAC | CAB | Selection 2 |
2. Basic definitions and main theorem
In this section, we state our main theorem. <|MaskedSetence|> <|MaskedSetence|> To an n×n𝑛𝑛n\times nitalic_n × italic_n matrix A,𝐴A,italic_A , we can associate a weighted digraph D(A),𝐷𝐴D(A),italic_D ( italic_A ) , with vertex set [n]delimited-[]𝑛[n][ italic_n ] and for each ordered pair (i,j),𝑖𝑗(i,j),( italic_i , italic_j ) , there is an edge directed from i𝑖iitalic_i to j𝑗jitalic_j with weight aijsubscript𝑎𝑖𝑗a_{ij}italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT. A linear subdigraph γ𝛾\gammaitalic_γ of D(A)𝐷𝐴D(A)italic_D ( italic_A ) is a spanning collection of pairwise vertex-disjoint cycles. A loop around a single vertex is also considered to be a cycle of length 1.11.1 . The weight of a linear subdigraph γ,𝛾\gamma,italic_γ , written as w(γ)𝑤𝛾w(\gamma)italic_w ( italic_γ ) is the product of the weights of all its edges. The number of cycles contained in γ𝛾\gammaitalic_γ is denoted by c(γ).𝑐𝛾c(\gamma).italic_c ( italic_γ ) . <|MaskedSetence|> Now the cycle-decomposition of permutations yields the following description of det(A),det𝐴{\rm det}(A),roman_det ( italic_A ) , namely
. | **A**: See [4] for details.
**B**: The sign of a linear subdigraph γ𝛾\gammaitalic_γ is (−1)n+c(γ),superscript1𝑛𝑐𝛾(-1)^{n+c(\gamma)},( - 1 ) start_POSTSUPERSCRIPT italic_n + italic_c ( italic_γ ) end_POSTSUPERSCRIPT , where n𝑛nitalic_n is the order of the matrix A.𝐴A.italic_A .
**C**: Before that, let us briefly describe a well-known graphical interpretation of determinant.
| CAB | CAB | CAB | CAB | Selection 1 |
Now gXng−1=−Xn⇒gXn=−Xng⇒g2,1=0andg1,1=−g2,2𝑔subscript𝑋𝑛superscript𝑔1subscript𝑋𝑛⇒𝑔subscript𝑋𝑛subscript𝑋𝑛𝑔⇒subscript𝑔210andsubscript𝑔11subscript𝑔22gX_{n}g^{-1}=-X_{n}\Rightarrow gX_{n}=-X_{n}g\Rightarrow g_{2,1}=0\ \hbox{and}%
\ g_{1,1}=-g_{2,2}italic_g italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = - italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⇒ italic_g italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = - italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g ⇒ italic_g start_POSTSUBSCRIPT 2 , 1 end_POSTSUBSCRIPT = 0 and italic_g start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT = - italic_g start_POSTSUBSCRIPT 2 , 2 end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> This is a contradiction. Hence, X𝑋Xitalic_X is not strongly AdSL(2,ℍ)subscriptAdSL2ℍ{\rm Ad}_{{\rm SL}(2,\mathbb{H})}roman_Ad start_POSTSUBSCRIPT roman_SL ( 2 , blackboard_H ) end_POSTSUBSCRIPT-real. <|MaskedSetence|> | **A**: But gXsg−1=−Xs⇒g1,1𝐢=−𝐢g1,1𝑔subscript𝑋𝑠superscript𝑔1subscript𝑋𝑠⇒subscript𝑔11𝐢𝐢subscript𝑔11gX_{s}g^{-1}=-X_{s}\Rightarrow g_{1,1}\mathbf{i}=-\mathbf{i}g_{1,1}italic_g italic_X start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = - italic_X start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ⇒ italic_g start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT bold_i = - bold_i italic_g start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT.
**B**: ∎
.
**C**: Therefore, g2=I2superscript𝑔2subscriptI2g^{2}=\mathrm{I}_{2}italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = roman_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT implies g1,12=1superscriptsubscript𝑔1121g_{1,1}^{2}=1italic_g start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1, i.e., g1,1=±1subscript𝑔11plus-or-minus1g_{1,1}=\pm 1italic_g start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT = ± 1.
| CAB | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> Another possibility is stacking of infinitely many copies of a certain code to create a code with higher dimension. <|MaskedSetence|> In this section we discuss stacking and coarse-graining (in particular how they affect invariants of a code), but compactifications are postponed to future work. Moreover, we explain that the choice of n𝑛nitalic_n (which has to be a common multiple of qubit dimensions) does not matter and that the whole theory reduces to the case when n𝑛nitalic_n is a prime power.
. | **A**: For example, one may “compatify” some (even all) spatial directions, i.e. replace ΛΛ\Lambdaroman_Λ by a quotient group.
**B**: Finally, one has coarse-graining, which does not change the code, but forgets about some of its translation symmetry.
**C**: 4 Operations on Pauli stabilizer codes
One Pauli stabilizer code may give rise to various other codes.
| CAB | CAB | ABC | CAB | Selection 2 |
<|MaskedSetence|> Theorem 1. <|MaskedSetence|> Lemma 2.1). <|MaskedSetence|> For experts familiar with the ways ReLU neural network functions can degenerate, Lemma 3.12 should provide a clear explanation for why most ReLU neural network functions are not PL Morse.
. | **A**: Indeed, one should view the analogues of critical cells (both Morse and non-Morse) in the PL category as an appropriate subset of the flat cells.
**B**:
Unsurprisingly, the key to understanding how the topology of sublevel sets changes as one varies the threshold are the PL analogues of points where the gradient of the function vanishes, cf.
**C**: These are the so-called flat or constant cells (Definition 3.7), which map to nontransversal thresholds (cf.
| BCA | BCA | ABC | BCA | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> We are also grateful to D. Kubrak, S. Mondal, S. Petrov, and A. Prikhodko for many useful conversations. We thank A. Langer for several useful comments on an earlier draft. <|MaskedSetence|> Finally, we would like to thank S. Naprienko for creating the website Thuses.com, without which this collaboration would likely not have taken place. BZ acknowledges funding through the Max Planck Institute for
Mathematics in Bonn, Germany, during the preparation of this work.
. | **A**: In particular, we thank him for the suggestion to use the results from [BMS19] to attack Theorem 1.1.1 and for explaining the relevant background.
**B**: We thank the referee for many useful suggestions which greatly improved this paper.
**C**:
We are very grateful to B. Bhatt for many fruitful discussions.
| CAB | CAB | CAB | CBA | Selection 1 |
We note here that the model above does not include a latent period, however since the results in this paper relate to the final outcome of an SIR epidemic they are insensitive to quite general assumptions concerning a latent period (see e.g. <|MaskedSetence|> Moreover, the results of the present paper (suitably modified) carry over to a model in which very general two-type point processes, representing the times they make global and local contacts, are assigned to infectious individuals; so long as each of the contacts is made with an individual chosen uniformly from the population or community. <|MaskedSetence|> <|MaskedSetence|> . | **A**: In particular, all results in this paper apply without change to corresponding SEIR epidemics.
**B**: Adjusting our results to allow for
a fixed number of initial infectives spread in some specified way through the communities is also straightforward, but it is rather lengthy to describe.
**C**: Ball, 1986).
| CAB | CAB | CAB | BCA | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> Seq2seq strategy was proposed in [16], where the PINN learns to predict the solution at each time step, instead of all times. <|MaskedSetence|> We take the prediction at t=𝑡absentt=italic_t =d𝕋𝕋\mathbb{T}blackboard_T by using the model of the first sequence and use this as the initial condition to make a prediction in the next sequence, and so on. which can be shown in Figure 2.. | **A**: Note that the only data available of the first sequence is from the PDE itself, i.e., just the initial condition.
**B**: In complex cases, this can be more difficult to learn.
**C**:
(3.5)
The original PINN approach trains the NN model to predict the entire space-time at once.
| ABC | CBA | CBA | CBA | Selection 3 |
We experiment with Voronoi diagrams in which the cells in the center of the diagram tend to be smaller. A point is sampled by first choosing a random cell and then choosing a uniform point on its boundary. <|MaskedSetence|> <|MaskedSetence|> Further details of the data generation process may be found in Appendix B.
We compare the performances of the proposed filtration against that of the distance-to-measure filtration. <|MaskedSetence|> | **A**: We further inject additive noise.
**B**: The sample points are shown in Figure 9, the persistence diagrams are shown in Figure 11, and the significant loops found by oracle and subsample bootstrapping are shown in Figure 10.
.
**C**: This results in a higher sampling density on boundaries of smaller cells.
| CAB | CAB | CAB | CAB | Selection 3 |
Our work is related to a line of recent work on the sample efficiency of reinforcement learning for POMDPs. In detail, Azizzadenesheli et al. (2016); Guo et al. <|MaskedSetence|> (2021) establish sample complexity guarantees for searching the optimal policy in POMDPs whose models are identifiable and can be estimated by spectral methods. <|MaskedSetence|> (2016) and Guo et al. (2016) add extra assumptions such that efficient exploration of the POMDP can always be achieved by running arbitrary policies. In contrast, the upper bound confidence (UCB) method is used in Xiong et al. (2021) for adaptive exploration. However, they require strictly positive state transition and observation emission kernels to ensure fast convergence to the stationary distribution. The more related work is Jin et al. (2020a), which considers undercomplete POMDPs, in other words, the observations are more than the latent states. <|MaskedSetence|> For example, Kwon et al. (2021) considers latent POMDPs, where each process has only one latent state, and the proposed algorithm efficiently infers the latent state using a short trajectory. Kozuno et al. (2021) considers POMDPs having tree-structured states with their positions in certain partitions being the observations. Compared with general POMDPs, these specially structures reduce the complexity of finding the optimal actions, and the corresponding algorithms use techniques closer to those for MDPs. Also, the aforementioned literature only consider tabular POMDPs.. | **A**: Their proposed algorithm can attain the optimal policy without estimating the exact model, but an observable component (Jaeger, 2000; Hsu et al., 2012), which is the same for our algorithm design, while only applies to tabular POMDPs.
In a broader context of reinforcement learning with partial observability, our work is related to several recent works on POMDPs with special structures.
**B**: However, Azizzadenesheli et al.
**C**: (2016); Xiong et al.
| CBA | BCA | CBA | CBA | Selection 4 |
<|MaskedSetence|> Since 𝒟𝒟{\mathscr{D}}script_D is a convex neighbourhood of the origin, it follows that tH∈𝒟¯𝑡𝐻¯𝒟tH\in{\overline{\mathscr{D}}}italic_t italic_H ∈ over¯ start_ARG script_D end_ARG is minimal for 0≤t≤10𝑡10\leq t\leq 10 ≤ italic_t ≤ 1. For t=1𝑡1t=1italic_t = 1 we have H∈∂𝒟𝐻𝒟H\in\partial{\mathscr{D}}italic_H ∈ ∂ script_D so that there exists γ≠0𝛾0\gamma\neq 0italic_γ ≠ 0 in ΓΓ\Gammaroman_Γ such that |H+γ|=|H|𝐻𝛾𝐻|H+\gamma|=|H|| italic_H + italic_γ | = | italic_H |. Thus Y=H+γ𝑌𝐻𝛾Y=H+\gammaitalic_Y = italic_H + italic_γ is a nontrivial focal equivalent to H𝐻Hitalic_H and by Lemma 2.4 it follows that (1+ϵ)H1italic-ϵ𝐻(1+\epsilon)H( 1 + italic_ϵ ) italic_H has norm greater than Y+ϵH𝑌italic-ϵ𝐻Y+\epsilon Hitalic_Y + italic_ϵ italic_H, for ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, while exp((1+ϵ)H)=exp(Y+ϵH)1italic-ϵ𝐻𝑌italic-ϵ𝐻\exp((1+\epsilon)H)=\exp(Y+\epsilon H)roman_exp ( ( 1 + italic_ϵ ) italic_H ) = roman_exp ( italic_Y + italic_ϵ italic_H ). <|MaskedSetence|> <|MaskedSetence|> | **A**: By using that Ad(K)Ad𝐾\mathrm{Ad}(K)roman_Ad ( italic_K ) conjugates every X∈𝔰𝑋𝔰X\in\mathfrak{s}italic_X ∈ fraktur_s to some H∈𝔱𝐻𝔱H\in\mathfrak{t}italic_H ∈ fraktur_t, it follows from Proposition 6.2 that the cut locus of S𝑆Sitalic_S is contained in the adjoint orbit of the cut locus of Tp𝑇𝑝Tpitalic_T italic_p, which proves item (i)..
**B**: It follows that exp(tH)𝑡𝐻\exp(tH)roman_exp ( italic_t italic_H ) is not a minimizing geodesic for t>1𝑡1t>1italic_t > 1, which proves our claim.
For item (i), from Proposition 6.2 it follows that the cut locus of Tp𝑇𝑝Tpitalic_T italic_p is contained in the cut locus of S𝑆Sitalic_S, and so its adjoint orbit, since Ad(K)Ad𝐾\mathrm{Ad}(K)roman_Ad ( italic_K ) acts by isometries in 𝔰𝔰\mathfrak{s}fraktur_s.
**C**: Let H𝐻Hitalic_H be in the tangent cut locus of T𝑇Titalic_T, then tH∈𝒟¯𝑡𝐻¯𝒟tH\in{\overline{\mathscr{D}}}italic_t italic_H ∈ over¯ start_ARG script_D end_ARG for 0≤t≤10𝑡10\leq t\leq 10 ≤ italic_t ≤ 1, and tH∉𝒟¯𝑡𝐻¯𝒟tH\not\in{\overline{\mathscr{D}}}italic_t italic_H ∉ over¯ start_ARG script_D end_ARG for t>1𝑡1t>1italic_t > 1 so that H∈∂𝒟𝐻𝒟H\in\partial{\mathscr{D}}italic_H ∈ ∂ script_D.
Reciprocally, let H𝐻Hitalic_H be in the boundary of 𝒟𝒟{\mathscr{D}}script_D.
| CBA | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> It was defined by Björner, Korte and Lovász, and they give two equivalent definitions (greedoid, , Theorem 6.1). One of them is topological, and uses the (abstract) dual complex of the greedoid. <|MaskedSetence|> We will give two proofs for our main theorem, each based on these two definitions. <|MaskedSetence|> | **A**: Hence we repeat here both definitions.
.
**B**: Another definition uses activities.
**C**:
One important invariant of a greedoid is the greedoid polynomial.
| CBA | CBA | CBA | BCA | Selection 3 |
Among the available approaches, the concept of control invariant set is one of the most exploited historically, since it ensures the existence of some feedback law able to steer the closed-loop trajectories of the uncertain system within a prescribed state set 25, 6, 8, 37. This is traditionally achieved by associating a control Lyapunov function (CLF) with the invariant set design, which for polytopic systems has been proven to be universal, namely the stabilization of the linear uncertain system and the existence of a polyhedral CLF can be used interchangeably 7. With a specific focus on discrete-time polytopic systems, an admissible control policy that actually makes a polyhedral CLF a suitable Lyapunov candidate for the closed-loop system is typically synthesized in two ways: through a variable structure 25, 46, 47, or a (minimal) selection-based controller 3. We will also refer to these policies as traditional stabilizing controllers for linear uncertain systems.
Once fixed feasible control inputs at the vertices of the invariant set have been computed, a variable structure controller either takes a convex combination of those values by exploiting the vertex reconstruction of any state belonging to such a set, or coincides with a purely linear gain stemming from a triangulation, i.e., a simplicial partition 16, of the underlying set. These methods therefore require one to solve a linear program (LP) online or to generate a lookup table to identify the region in which the current state resides. If the simplicial partition-based implementation is considered, then one has also to account for the complexity of the resulting invariant set, which is typically high 6, 8, 49, 10, 2, 9. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> While solving a numerical optimization problem online provides flexibility and performance guarantees, the real-time computational efforts required complicate its application in polytopic linear systems characterized by high sampling rates.. | **A**: As a common drawback affecting both the implementations, however, fixing the input values at the vertices may result in poor control performance for the stabilization task.
A more sophisticated control method coincides with the selection-based policy.
**B**: These methods can therefore require significant memory to store the vectors and/or matrices describing every simplicial partition and associated linear control gain.
**C**: By requiring the online resolution of a nonlinear optimization problem, parametric in the current measured state, this method directly enforces a certain degree of contraction possessed by the CLF at every control step.
| BAC | BAC | CBA | BAC | Selection 1 |
The classical Keller-Segel model assumes that the density diffusion is not affected by the nonlocal behaviour of the organisms. <|MaskedSetence|> Then, the trajectories of the population of organisms are better described by the so called Lévy flights than Brownian motion (see [8, 9]). Lévy flights behaviour has been suggested
in numerous biological contexts, including immune cells, ecology, and human
populations (c.f. [10] and references for a deeper discussion). This consideration motivates the substitution of the classical diffusion in the Keller-Segel system (1.1) by a fractional diffusion. <|MaskedSetence|> <|MaskedSetence|> In addition, taking into account that the behavior of most biological systems has memory properties, which are neglected when an integer-order time derivative is assumed, we also assume a time variation in a fractional framework. This introduces a
nonlocal delay in time for the moving population [12].. | **A**: This last consideration has been point out relevant in the analysis of the propagation of chaos for some aggregation-diffusion models [11].
**B**: However, in many situations found in nature, organisms develop alternative search strategies, particularly when chemoattractants, food, or other targets are sparse or rare.
**C**: On the other hand, regarding to the flux by chemotaxis, it is also relevant to consider that the attraction force be replaced by a less singular interaction kernel.
| ACB | BCA | BCA | BCA | Selection 2 |
Section 3 consists of the technical results on (relative) cyclic (co)homology of Lie algebras that we shall need in the sequel. In Subsection 3.1 we outline stable and unimodular stable AYD modules over Lie algebras, followed by the cyclic homology of a Lie algebra with coefficients in a stable AYD module in Subsection 3.2. Finally, in Subsection 3.3, we recall the cyclic cohomology of a Lie algebra, with unimodular stable AYD module coefficients.
Section 4 is where we prove the two van Est type isomorphisms for the classical Hopf algebras. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: We recall in Subsection 4.1 the Hopf-cyclic homology of the coordinate algebra of functions on an algebraic group with coefficients in an SAYD module.
**B**: Then, in Subsection 4.2 we prove our first van Est isomorphisms between the Hopf-cyclic (co)homology of the coordinate algebra of functions on an algebraic group, and the relative Hopf-cyclic (co)homology of the universal enveloping algebra of its Lie algebra relative to the universal enveloping algebra of the Lie algebra of a maximal compact subgroup.
.
**C**: Here we also recalled the Hopf-cyclic cohomology of the coordinate Hopf algebra of an algebraic group, but this time with the coefficients in an SAYD contra-module.
| ACB | ABC | ACB | ACB | Selection 3 |
This is due to the fact that the argument of our paper [17] is based on the Dirichlet-to-Neumann (DN) machinery. There is a possibility to modify the approach by invoking Neumann-to-Dirichlet maps instead, which would have two advantages: one could consider all rates of vertex volume decay in (iii), and certain geometric smoothness requirements could be somewhat relaxed.
Nevertheless, in this paper we stick with the DN version of the approach, in order to align the exposition with that of [17]. <|MaskedSetence|> <|MaskedSetence|> (In non-dimensional terms this corresponds to the wavelength being of the order of some positive power of ε𝜀\varepsilonitalic_ε). <|MaskedSetence|> The complexity of the dimension reduction process for these models increases along the sequence. While an initial result regarding the high-frequency situation is presented below, we postpone the full analysis to a future publication.. | **A**: The alternative strategy will be followed up elsewhere, both in the present context and in the setting of [17].
The above results of course imply the Hausdorff spectral convergence, at the same time yielding a sharp estimate on its rate.
**B**: Moreover, in contrast to [56], our approach allows one to consider “high-frequency” regimes, i.e., setups where the spectral parameter (which in the wave propagation context may represent the square of the frequency) is no longer constrained to a compact set but is still constrained by some negative power of the small parameter ε𝜀\varepsilonitalic_ε.
**C**: This leads to a sequence of “effective”, dimensionally reduced, models of the thin structure, which are sequentially applicable for a set of (asymptotic) frequency intervals.
| ABC | BCA | ABC | ABC | Selection 1 |
We have benefited from discussions and correspondence with J. Kollár, M. Mustaţă, and M. Popa on higher du Bois singularities while preparing [FL22]. The second author had several related discussions with M. Kerr and M. <|MaskedSetence|> We also thank J. de Jong, M. Saito and C. <|MaskedSetence|> <|MaskedSetence|> Mustaţă, M. Popa informed us of some their recent work ([MP22b], [CDM22]) related to Theorem 1.6 and Conjecture 1.8. We are grateful to them for these communications. M. Saito kindly provided us with proofs of Proposition 1.9(ii) and Conjecture 1.8 in the hypersurface case, and agreed to include those as an appendix to our paper. Finally, we would like the referees for a very carefully reading of the first version of this paper and for many helpful comments.
. | **A**: Schnell for some further comments related to this paper.
**B**: After circulating an earlier version of this paper, M.
**C**: Saito while preparing previous joint work.
| ABC | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> First, we consider a protocol that requires significance in two independent trials—we call this the standard protocol. <|MaskedSetence|> Second, we consider the case where only one trial is required, but at the 0.010.010.010.01 level for a two-sided test. <|MaskedSetence|> Lastly, we consider a protocol where two trials are conducted and approval is granted if either trial yields a two-sided p𝑝pitalic_p-value below 0.050.050.050.05, so that the probability of approving a placebo is 0.04940.04940.04940.0494. We call this the high-discretion accelerated protocol. This protocol is looser than typical FDA behavior, but in extreme cases approvals with this low level of statistical evidence do occur. Notably, aducanumab (Aduhelm) was approved after two trials, one with a statistically significant result and one with a statistically insignificant result (FDA Center for Drug Evaluation and
Research, 2020).. | **A**: The probability that a placebo drug is approved with this protocol is 0.0252=0.000625superscript0.02520.0006250.025^{2}=0.0006250.025 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0.000625.
**B**:
To facilitate formal analysis, we analyze three simplified statistical protocols that track the FDA evaluatory process.
**C**: We call this the modernized protocol, in view of the FDA Modernization Act of 1997.
We assume the probability that a placebo drug is approved with the modernized protocol is 0.005.0.0050.005.0.005 .
| BAC | BCA | BAC | BAC | Selection 4 |
The structure of the paper is as follows. In Section 2 we recall some basic definitions about rings and modules, and prove that the notion of rank is well defined. <|MaskedSetence|> In Section 4 we obtain several results of screw theory by using the formalism of 𝔻𝔻\mathbb{D}blackboard_D-module geometry. In Section 5 we obtain results that, a posteriori, can be interpreted as a manifestation of the transference principle. <|MaskedSetence|> We end the paper with some conclusions.
As for notation and terminologies, throughout the work we use, unless otherwise stated, the Einstein summation convention. <|MaskedSetence|> | **A**: Again, they are obtained by using the formalism of 𝔻𝔻\mathbb{D}blackboard_D-module geometry.
**B**: Vectors.
**C**: In Section 3 we show how the Euclidean space emerges out of 𝔻𝔻\mathbb{D}blackboard_D-module geometry, in particular, Theorem 3 is proved.
| CAB | ACB | CAB | CAB | Selection 3 |
Theorem 1 predicts that in certain situations FNO has a systematic bias. Namely, if one trains FNO on an insufficiently fine grid, activation functions introduce distortions that FNO will learn to mitigate. When the grid is sufficiently refined, aliasing errors disappear, but since FNO was trained to mitigate them, it predicts output functions that differ from targets it was trained on. The precise discrepancy is hard to analyze but one can expect that for operators with smoother output aliasing will be milder. This hypothesis is confirmed by our experiments. <|MaskedSetence|> For both examples, FNO still produces good approximations, but one should be cautious using FNO for more complex problems, because in these applications FNO tends to be less accurate [Pat+22], [Gra+22].
Our solution to the problem of aliasing errors is to decouple interpolation from the mapping between functional spaces. It is described in Section 4. <|MaskedSetence|> One simply needs to train (or fine-tune) on a sufficiently fine grid. <|MaskedSetence|> | **A**: However, it is possible to decrease aliasing error even when FNO is used for interpolation.
**B**: Indeed, the right graph in Figure 2a shows that for the integration operator, which performs smoothing, aliasing leads to an approximately two-fold error increase, while for the differentiation operator the increase is five-fold.
**C**: The decrease of aliasing error in this scenario is illustrated in the left plot of Figure 2a..
| BAC | CAB | BAC | BAC | Selection 4 |
where K(f)(g)=∑n∈ℕ⟨f(n),g(n)⟩𝐾𝑓𝑔subscript𝑛ℕ𝑓𝑛𝑔𝑛K(f)(g)=\sum_{n\in\mathbb{N}}\left<f(n),g(n)\right>italic_K ( italic_f ) ( italic_g ) = ∑ start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT ⟨ italic_f ( italic_n ) , italic_g ( italic_n ) ⟩ for each f∈ℓ1(ℕ,X**)𝑓superscriptℓ1ℕsuperscript𝑋absentf\in\ell^{1}(\mathbb{N},X^{**})italic_f ∈ roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_N , italic_X start_POSTSUPERSCRIPT * * end_POSTSUPERSCRIPT ) and g∈ℓ∞(ℕ,X*)𝑔superscriptℓℕsuperscript𝑋g\in\ell^{\infty}(\mathbb{N},X^{*})italic_g ∈ roman_ℓ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_N , italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ). Clearly K𝐾Kitalic_K is an isometry(not necessarily surjective). <|MaskedSetence|> <|MaskedSetence|> Thus, we have the direct sum decomposition of Banach spaces ℓ1(ℕ,X)**=Range(P)⊕Range(I−P)superscriptℓ1superscriptℕ𝑋absentdirect-sumRange𝑃Range𝐼𝑃\ell^{1}(\mathbb{N},X)^{**}=\text{Range}(P)\oplus\text{Range}(I-P)roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_N , italic_X ) start_POSTSUPERSCRIPT * * end_POSTSUPERSCRIPT = Range ( italic_P ) ⊕ Range ( italic_I - italic_P ). <|MaskedSetence|> Hence, we have the following Banach space decomposition
(3.1). | **A**: Let P=K∘i*:ℓ1(ℕ,X)**→ℓ1(ℕ,X)**:𝑃𝐾superscript𝑖→superscriptℓ1superscriptℕ𝑋absentsuperscriptℓ1superscriptℕ𝑋absentP=K\circ i^{*}:\ell^{1}(\mathbb{N},X)^{**}\to\ell^{1}(\mathbb{N},X)^{**}italic_P = italic_K ∘ italic_i start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT : roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_N , italic_X ) start_POSTSUPERSCRIPT * * end_POSTSUPERSCRIPT → roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_N , italic_X ) start_POSTSUPERSCRIPT * * end_POSTSUPERSCRIPT.
**B**: One can easily verify that the map P𝑃Pitalic_P is a projection i.e P2=Psuperscript𝑃2𝑃P^{2}=Pitalic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_P and the range of P𝑃Pitalic_P is ℓ1(ℕ,X**)superscriptℓ1ℕsuperscript𝑋absent\ell^{1}(\mathbb{N},X^{**})roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_N , italic_X start_POSTSUPERSCRIPT * * end_POSTSUPERSCRIPT ).
**C**: But Range(I−P)=i(c0(ℕ,X*))⟂Range𝐼𝑃𝑖superscriptsubscript𝑐0ℕsuperscript𝑋perpendicular-to\text{Range}(I-P)=i\left(c_{0}(\mathbb{N},X^{*})\right)^{\perp}Range ( italic_I - italic_P ) = italic_i ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_N , italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) ) start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT.
| CBA | ABC | ABC | ABC | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> What makes the monopole-dimer model less physical is that configurations have a signed weight and they cannot be interpreted as energies anymore. On the other hand, the partition function here can be expressed as a determinant. Moreover, it is a perfect square for a 2m×2n2𝑚2𝑛2m\times 2n2 italic_m × 2 italic_n grid graph. A combinatorial interpretation of the square root is given in [Ayy20].
. | **A**:
In another direction, a signed version of the monomer-dimer model called the monopole-dimer model has been
introduced [Ayy15] for planar graphs.
**B**: Configurations of the monopole-dimer model can be thought of as superpositions of two monomer-dimer configurations having monomers (called monopoles there) at the same locations.
**C**: Thus, one ends up with even loops and isolated vertices.
| ABC | BAC | ABC | ABC | Selection 4 |
After completing the proofs of Theorems 1.3 and 1.4, we were informed that Fujino has also proved these results see [Fuj22b]. We note that Fujino’s approach is based on [BCHM10] whereas our approach is inspired by [CL10]. <|MaskedSetence|> In Subsection 2.4 we prove two important results, namely Theorem 2.29 and 2.36. These two results work as our main tools for testing whether a (1,1)11(1,1)( 1 , 1 ) class α𝛼\alphaitalic_α is nef or not, see Remark 2.2 for more details. <|MaskedSetence|> <|MaskedSetence|> In Part 3, we prove Theorem 1.1 (in Section 7) and Theorem 1.2 (in Section 8).. | **A**: Part 2 of the article is devoted to proving finite generation as in [CL10].
**B**: We prove Theorem 1.3 and 1.4 in Section 4 of this part.
**C**: Another possible approach can be found in [Pau12], which is particularly suited to the analytic context.
This article is organized in the following manner: In Part 1, we collect and prove various preliminary results.
| CAB | CAB | CAB | BCA | Selection 2 |
Here and throughout the paper we use L2(D)superscript𝐿2𝐷L^{2}(D)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) and H1(D)superscript𝐻1𝐷H^{1}(D)italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_D ) to denote the standard Lebesgue and Sobolev spaces of functions on D𝐷Ditalic_D. Moreover, H01(D)superscriptsubscript𝐻01𝐷H_{0}^{1}(D)italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_D ) is the subspace of functions in H1(D)superscript𝐻1𝐷H^{1}(D)italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_D ) with vanishing trace on ∂D𝐷\partial D∂ italic_D, its dual space is indicated by H−1(D)superscript𝐻1𝐷H^{-1}(D)italic_H start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_D ). The inner product in L2(D)superscript𝐿2𝐷L^{2}(D)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) is denoted by (.,.)0,D(.,.)_{0,D}( . <|MaskedSetence|> ) start_POSTSUBSCRIPT 0 , italic_D end_POSTSUBSCRIPT, the norms in L2(D)superscript𝐿2𝐷L^{2}(D)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ), H1(D)superscript𝐻1𝐷H^{1}(D)italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_D ), H−1(D)superscript𝐻1𝐷H^{-1}(D)italic_H start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_D ) are denoted by ∥.∥0,D\|.\|_{0,D}∥ . ∥ start_POSTSUBSCRIPT 0 , italic_D end_POSTSUBSCRIPT, ∥.∥1,D\|.\|_{1,D}∥ . <|MaskedSetence|> ∥ start_POSTSUBSCRIPT - 1 , italic_D end_POSTSUBSCRIPT, respectively. <|MaskedSetence|> For D=Ω𝐷ΩD=\Omegaitalic_D = roman_Ω we drop the subscript D𝐷Ditalic_D.
The Euclidean vector norm in ℝdsuperscriptℝ𝑑\mathbb{R}^{d}blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is indicated by ∥.∥\|.\|∥ . ∥ (without any subscript).
. | **A**: ∥ start_POSTSUBSCRIPT 1 , italic_D end_POSTSUBSCRIPT,∥.∥−1,D\|.\|_{-1,D}∥ .
**B**: , .
**C**: The same symbols are used for inner product and norms
in the vector-valued versions of the spaces.
| BAC | BAC | ABC | BAC | Selection 2 |
<|MaskedSetence|> This exploration is carried out on a different nonlinear model, highlighting our approach’s flexibility. <|MaskedSetence|> q=1𝑞1q=1italic_q = 1. <|MaskedSetence|> A data-set is simulated according to a logistic growth model such as:
. | **A**: The other parameters are assumed to be shared among individuals and they are estimated jointly.
**B**: For ease of presentation, we consider variable selection in a one-dimensional setting, i.e.
**C**:
5.2.1 Model
Now, the performance of SAEMVS is studied under a variety of scenarios.
| BCA | CBA | CBA | CBA | Selection 4 |
To avoid the difficulties in analysis in the metric level and due to the interests from mirror symmetry, there are replacements for the original proposal from algebraic geometry and symplectic geometry. On the algebraic side, Kontsevich–Soibelman [2006-Kontsevich-affine-structures-and-non-archimedean-analytic-spaces] and Gross–Siebert programs [2011-Gross-Siebert-from-real-affine-geometry-to-complex-geometry] introduced the notion of scattering diagrams for understanding the quantum correction.
On the symplectic side, Fukaya proposed the family Floer homology [F] and later studied by Tu [T2], Abouzaid [A1, A2, A3], and Yuan [Y2]. There is another approach using relative symplectic cohomology [GV].
Both approaches are designed to capture the quantum corrections in the mirror construction and both have big success in explaining mirror symmetry in an intrinsic way. <|MaskedSetence|> Until very recently, T. Collins, A. Jacob, and the third author constructed the first non-trivial example111In the sense that a priori without knowing the existence of elliptic fibration structure after hyperKähler rotation, which is a technical part in analysis. of genuine special Lagrangian fibrations [2021-Collins-Jacob-Lin-special-lagrangian-submaniflds-of-log-calabi-yau-manifolds]. <|MaskedSetence|> <|MaskedSetence|> | **A**: We refer the readers to [Gross]*Section 7 and [CL]*Section 1 and references therein.
.
**B**: However, it is still unclear how the two approaches are related to the original conjecture explicitly, again due to the lacking of the existence of genuine SYZ fibrations.
**C**: In this paper, we will study the complex affine structures on the base of the genuine special Lagrangian fibrations in these examples and explain that there can be subtle differences between the SYZ bases and the affine manifold with singularities used in the Gross–Siebert program (in this case is worked out by Carl–Pumperla–Siebert), which is a long-termed folklore conjecture.
| BCA | BCA | BCA | BCA | Selection 3 |
<|MaskedSetence|> It is not hard to see that the L𝐿Litalic_L-subdifferential is L𝐿Litalic_L-monotone [20, Proposition 1.9]. <|MaskedSetence|> Several authors have investigated conditions under which it is [10, 11, 18, 19, 17, among others]. Here we show that under Assumption 1 the result is also true. <|MaskedSetence|> | **A**: Our proof generalises the proof by [13].
.
**B**:
Our next result concerns the maximal monotonicity of the abstract subdifferential operator.
**C**: However, [4, example 3.1] showed that it is not generally maximal L𝐿Litalic_L-monotone.
| BCA | ABC | BCA | BCA | Selection 4 |
Related Work
Operations of AMoD systems are relevant to several mathematical problems, including dispatching, routing, scheduling, and rebalancing of service fleets. <|MaskedSetence|> Within the context of AMoD systems, fleet operations have been examined primarily using queuing-theoretic models [13] [14], simulation-based approaches [15], and dynamic fluid models [16] [17]. Particularly, Zhang et al. <|MaskedSetence|> Sayarshad and Chow [18] developed a queuing-based formulation for the relocation of MoD vehicles and solved the problem using a Lagrangian decomposition heuristic. Salazar et al. [17] proposed a congestion-aware routing scheme for AMoD fleets and formulated a convex quadratic program based on a piecewise linear approximation for the Bureau of Public Roads (BPR) model. <|MaskedSetence|> For instance, Chen et al. [19] proposed a decentralized cooperative cruising strategy for a fleet of autonomous taxis to maximize the total pickups during communication shutdowns. Duan et al. [20] combined centralized and decentralized dispatching strategies to serve both immediate and reservation requests, where the decentralized dispatcher enables each vehicle to reserve capacity for long-term requests.
. | **A**: More recently, AMoD fleet operations have been extended to a decentralized manner.
**B**: Examples include the VRP and its variants [4, 5, 6][8], where distance-minimized routes are to be planned to satisfy a set of transportation requests while considering different constraints.
**C**: [16] addressed the optimal coordination of AMoD systems by devising a model predictive control algorithm, wherein the vehicle scheduling and routing problem was solved as a mixed integer linear program.
| BCA | BCA | BCA | BAC | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> The focus of his undergraduate and master studies was statistics and discretization techniques and reduction order models for partial differential equations. He obtained his PhD in Applied Mathematics at Emory University in Atlanta (GA) in May 2018. The main topic of his doctorate work was the development of efficient and resilient linear solvers for upcoming computing architectures moving towards exascale. Upon graduation, Max joined the Oak Ridge National Laboratory (ORNL) as a Postdoctoral Researcher Associate in the Scientific Computing Group at the National Center for Computational Sciences (NCCS). Since 2020 Max has been a Data Scientist in the Scalable Algorithms and Coupled Physics Group in the Advanced Computing Methods for Engineered Systems Section of the Computational Sciences and Engineering Division at ORNL. Max’s research focuses on the development of surrogate models for material sciences, scalable hyper parameter optimization techniques for deep learning models, and acceleration of computational methods for physics applications. <|MaskedSetence|> | **A**: Massimiliano (Max) Lupo Pasini obtained his Bachelor of Science and Master of Science in Mathematical Engineering at the Politecnico di Milano in Milan, Italy.
**B**: He is currently the lead of the Artificial Intelligence for Scientific Discovery thrust of the ORNL Artificial Intelligence Initiative..
**C**:
Author Biography
{biography}
Massimiliano Lupo Pasini.
| CAB | CAB | BCA | CAB | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> This is formulated precisely in Proposition 3.8(3). Using this result we then proceed to proving an analogue of Lemma 2.9. <|MaskedSetence|> The strategy of proof is similar to that carried out in [11, §6.2], using results on 1-h-minimal fields, but working in mixed characteristic requires a little more care.
While the results of Section 3.2 may be of independent interest, the results of Section 3.1 are more technical. Readers unfamiliar with the analogous results from [11] may wish to first read Section 4.1 for a more detailed discussion of the underlying geometric motivation. . | **A**: Section 3.2 is dedicated to the second main property that we need: local linearity of definable functions in K/𝒪𝐾𝒪K/{\mathcal{O}}italic_K / caligraphic_O, culminating in Corollary 3.32.
**B**:
We prove two geometric properties that play a crucial role in the sequel.
**C**: The first, comprising the heart of Section 3.1, is an analogue of Fact 2.8, that can be interpreted as asserting the existence of generic neighbourhoods (of generic points) with respect to the above topology.
| BCA | BCA | BAC | BCA | Selection 4 |
<|MaskedSetence|> More specifically, in these settings we derive novel upper and lower bounds for the limiting acceptance probabilities that clarify in which contexts the abc posterior is still well–defined for n𝑛nitalic_n large enough. When this is the case, it is further possible to obtain informative supersets for the support of such a posterior. These show that, when relaxing standard convergence assumptions employed in state–of–the–art theoretical studies, the control established by a fixed abc threshold on the discrepancy among the empirical distributions does not necessarily translate, asymptotically, into the same control on the discrepancy among the corresponding truths, but rather yields an upper bound equal to the sum between the abc threshold and a multiple of the Rademacher complexity, namely a measure of richness of the class of functions that uniquely identify the chosen ips; see Section 3.1.
The above results clarify the fundamental relation among the limiting behavior of abc posteriors and the learning properties of the chosen discrepancy, when measured via Rademacher complexity. Moreover, the bounds derived clarify that a sufficient condition to recover a limiting pseudo–posterior with the same threshold–control on the discrepancy among the truths as the one enforced on the corresponding empirical distributions, is that the selected discrepancy has a Rademacher complexity vanishing to zero in the large–data limit. As proved in Section 3.2, this setting also allows constructive derivations of novel, informative and uniform concentration bounds for discrepancy–based abc posteriors in the challenging regime where the threshold shrinks towards zero as both m𝑚mitalic_m and n𝑛nitalic_n diverge. This is facilitated by the existence of meaningful upper bounds for the Rademacher complexity of popular abc discrepancies, along with the availability of constructive conditions for the derivation of these bounds (e.g., Sriperumbudur et al., 2012) which leverage fundamental connections among such a complexity measure and other key quantities in statistical learning theory, such as the Vapnik–Chervonenkis (vc) dimension and the notion of uniform Glivenko–Cantelli classes (see e.g., Wainwright, 2019, Chapter 4). <|MaskedSetence|> <|MaskedSetence|> settings, leveraging results in Mohri & Rostamizadeh (2008) on Rademacher complexity under β𝛽\betaitalic_β–mixing processes (e.g., Doukhan, 1994).. | **A**: The former advantage is illustrated within Section 4 through a specific focus on mmd with routinely–implemented bounded and unbounded kernels, whereas the latter is clarified in Section 6, where we extend the theory from Section 3 to non–i.i.d.
**B**: This yields an improved understanding of the factors that govern the concentration of discrepancy–based abc posteriors under a unified perspective that further allows to (i) quantify rates of concentration and (ii) directly translate any advancement on Rademacher complexity into novel abc theory.
**C**:
Crucially, the theoretical framework we introduce allows to prove informative theoretical results even in yet–unexplored settings that possibly lack those convergence guarantees assumed in available literature.
| CBA | CBA | ABC | CBA | Selection 4 |
The simulation results are summarized in Fig. 4. <|MaskedSetence|> <|MaskedSetence|> To capture the dynamics of the evolution of the thin diffuse interface, the mesh size should be much smaller than ϵitalic-ϵ\epsilonitalic_ϵ, the width of the diffuse interface. Traditional numerical methods often use an adaptive or moving mesh approach to overcome this difficulty. <|MaskedSetence|> The number of parameters of the neural network can be much smaller than the number of samples needed to resolve the diffuse interface. Consequently, the dimension of the optimization problem in the NN-based scheme is much smaller than in the FEM scheme without using adaptive or moving meshes.
. | **A**: In contrast, a neural network-based numerical scheme has a mesh-free feature.
**B**: Numerical simulation of phase-field type models is often challenging.
**C**: It clearly shows that our method can achieve comparable results with the FEM.
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<|MaskedSetence|> Outline
In Section 2, we provide basic preliminaries on Deligne’s Hodge theory and the perverse Leray filtration. Section 3 introduces the notion of a hybrid LG model, the primary focus of this paper, and examines its topological properties. Section 4 explores the Hodge theory of hybrid LG models, introducing two different LG Hodge numbers and extending the conjecture of Katzarkov-Kontsevich-Pantev[KKP17]. <|MaskedSetence|> We also explain how the mirror P=W conjecture can be deduced from this relative version. <|MaskedSetence|> Additionally, we provide a combinatorial description of the perverse Leray filtration, which holds independent interest.
. | **A**: In the Appendix, we review the theory of cohomological mixed Hodge complexes, which will be employed in Section 4.
**B**: Section 5 contains speculative discussions and proposes a relative version of homological mirror symmetry for the extended Fano/LG correspondence.
**C**: 1.5.
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<|MaskedSetence|> Hladký, Král, and Norine [13] confirmed a conjecture of Baker [2] relating the ranks of a divisor on a graph and on a tropical curve, and provided a purely combinatorial algorithm for computing the rank of a divisor on a tropical curve, which can be considered simply as a metric graph, see [18, 12]. <|MaskedSetence|> Furthermore, by a corollary of the Riemann-Roch theorem, the rank can be computed in polynomial time for divisors of degree greater than 2g−22𝑔22g-22 italic_g - 2, where g𝑔gitalic_g denotes the cyclotomic number of the graph (also called the genus). Baker and Shokrieh [4] provided an algorithm that can efficiently check whether the rank of a divisor on a graph is at least c𝑐citalic_c for any fixed constant c𝑐citalic_c. <|MaskedSetence|> | **A**:
Previous work
From the positive side, Luo [16] introduced the notion of rank-determining sets of metric graphs, and verified the existence of finite rank-determining sets constructively.
**B**: For multigraphs with a constant number of vertices, Manjunath [17] gave a polynomial time algorithm that computes the rank of a divisor.
**C**: Cori and le Borgne [10] provided a linear time algorithm for determining the rank of a divisor on a complete graph..
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<|MaskedSetence|> This problem has been studied by L. <|MaskedSetence|> <|MaskedSetence|> Kriventsov in [5], where the authors have proved the existence of a solution u𝑢uitalic_u for the problem and the regularity of its jump set. Another similar problem, in a non-linear context, has been deepened by D. Bucur and A. Giacomini in [4] with a boundedness constraint.
. | **A**:
with v∈SBV(ℝn)𝑣SBVsuperscriptℝ𝑛v\in\operatorname{SBV}(\mathbb{R}^{n})italic_v ∈ roman_SBV ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) and v=1𝑣1v=1italic_v = 1 in ΩΩ\Omegaroman_Ω.
**B**: A.
**C**: Caffarelli and D.
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<|MaskedSetence|> <|MaskedSetence|> However, in the case that K𝐾Kitalic_K is of three-genus one and is algebraically slice, the invariant is determined by the Levine-Tristram signature functions of certain knots formed as simple closed curves on a genus one Seifert surface. <|MaskedSetence|> The paper [MR3096507] presents a more recent exposition. We isolate the result we need. In this statement, σK(ω)subscript𝜎𝐾𝜔\sigma_{K}(\omega)italic_σ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ( italic_ω ) denotes the Tristram-Levine signature function defined on the unit circle in ℂ*superscriptℂ{\mathbb{C}}^{*}blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT.
. | **A**: Computing σ1τ(Bq,χ0)subscript𝜎1𝜏subscript𝐵𝑞superscript𝜒0\sigma_{1}\tau(B_{q},\chi^{0})italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_τ ( italic_B start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , italic_χ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ).
In general, there are few methods available for computing σ1τ(K,χ)subscript𝜎1𝜏𝐾𝜒\sigma_{1}\tau(K,\chi)italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_τ ( italic_K , italic_χ ).
**B**:
2.2.
**C**: This is a consequence of results related to companionship proved independently by Cooper [CooperThesis], Gilmer [MR711523], and Litherland [MR780587].
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Now we prove (a). Suppose that n0=dk0minursubscript𝑛0superscriptsubscript𝑑superscriptsubscript𝑘0urn_{0}=d_{k_{0}^{\min}}^{\mathrm{ur}}italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ur end_POSTSUPERSCRIPT is not a vertex by contradiction, i.e. <|MaskedSetence|> <|MaskedSetence|> By definition, we have dk′ur≤n1<dk0minursuperscriptsubscript𝑑superscript𝑘′ursubscript𝑛1superscriptsubscript𝑑superscriptsubscript𝑘0urd_{k^{\prime}}^{\mathrm{ur}}\leq n_{1}<d_{k_{0}^{\min}}^{\mathrm{ur}}italic_d start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ur end_POSTSUPERSCRIPT ≤ italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_d start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ur end_POSTSUPERSCRIPT, and hence k′<k0minsuperscript𝑘′superscriptsubscript𝑘0k^{\prime}<k_{0}^{\min}italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT. <|MaskedSetence|> | **A**: the index i=n0𝑖subscript𝑛0i=n_{0}italic_i = italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT would be part of a segment (n1,n2)subscript𝑛1subscript𝑛2(n_{1},n_{2})( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) with n1<n0<n2subscript𝑛1subscript𝑛0subscript𝑛2n_{1}<n_{0}<n_{2}italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT over which NP(G(wk~,−))NP𝐺subscript𝑤~𝑘\mathrm{NP}\left(G\left(w_{\tilde{k}},-\right)\right)roman_NP ( italic_G ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_k end_ARG end_POSTSUBSCRIPT , - ) ) is a straight line.
**B**: By Proposition 3.6(2), (n1,n2)subscript𝑛1subscript𝑛2(n_{1},n_{2})( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) is a maximal near-Steinberg range, say (n1,n2)=nSwk~,k′subscript𝑛1subscript𝑛2subscriptnSsubscript𝑤~𝑘superscript𝑘′(n_{1},n_{2})=\mathrm{nS}_{w_{\tilde{k}},k^{\prime}}( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = roman_nS start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT over~ start_ARG italic_k end_ARG end_POSTSUBSCRIPT , italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT for some k′∈𝒦superscript𝑘′𝒦k^{\prime}\in\mathcal{K}italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_K.
**C**: Combined with (3.12), this strict inequality implies that
vp(k′−k0min)<m≤vp(k~−k0min),subscript𝑣𝑝superscript𝑘′superscriptsubscript𝑘0min𝑚subscript𝑣𝑝~𝑘superscriptsubscript𝑘0minv_{p}(k^{\prime}-k_{0}^{\mathrm{min}})<m\leq v_{p}(\tilde{k}-k_{0}^{\mathrm{%.
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Our pseudo-stabilization technique discussed in Appendix A is essential to the scalability of the JFP method and thus also for its application to practical computational problems. We emphasize that high-precision computations are required only for the computation of fractional integration matrices and not for the solution of the resulting linear systems. <|MaskedSetence|> <|MaskedSetence|> The JFP method is used in Section 6 to solve a variety of FIEs444Julia code for the examples in Section 6 are available at https://github.com/putianyi889/JFP-demo, including an FDE and fractional PDE reformulated as FIEs, and comparisons are made with the sum space method. <|MaskedSetence|> Appendix A is devoted to the above-mentioned pseudo-stabilization of an algorithm for computing fractional integration matrices.
. | **A**: We conclude the paper with a summary and a discussion of topics for future work.
**B**: In addition, we do not require any of the inputs to the FIE or FDE (e.g., variable coefficients, the function on the right-hand side of the equation, boundary conditions, etc.) to be computable to high-precision accuracy.
The following is an outline of the paper: We introduce the basic constituents of the JFP method in Sections 2 and 3 (matrix representations of operators on quasimatrices, high-precision floating-point numbers, the JFP basis, etc.).
**C**: We then focus on the properties (Section 4) and computation (Section 5) of fractional integration operators and matrices acting on the JFP basis.
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On the other hand, processes with Moran type interactions are natural models for finite populations with either variety-increasing or variety-reducing effects such as genetic drift, genetic mutations and natural selection. First introduced by Moran [57], the Moran model describes the evolution of N𝑁Nitalic_N genes such that at an exponential rate, two particles are chosen uniformly at random from the population, one of which is killed and the other splits in two. <|MaskedSetence|> <|MaskedSetence|> More precisely, when the system is initiated from N𝑁Nitalic_N particles, each particle evolves according to an independent copy of a given Markov process, X𝑋Xitalic_X, until either a (binary) branching or killing event happens. <|MaskedSetence|> If such a branching event occurs, with a probability that may depend on the configuration of the whole system, another particle is removed from the system a selection mechanism. Similarly, if a killing event occurs, with a probability which may also depend on the configuration of the whole system, another particle is duplicated via a resampling mechanism. We refer to this model as the binary branching model with Moran interactions, or BBMMI for short.. | **A**: Thus, the independence between particles is lost.
We refer the reader to [33] and references therein for an overview of this model and of its extensions.
**B**: This type of resampling has since been employed in a range of particle system models to numerically solve Feynman-Kac models, see [15, 25, 26, 66] and references therein.
The model we consider in this paper provides a combination of these two types of dynamics: (natural) branching and killing, as well as Moran type interactions.
**C**: Here, binary refers to the fact that the particle is replaced by exactly two independent copies of itself.
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However, the study of linear convergence rates of DYS has been limited. Bredies, Chenchene, Lorenz, and Naldi studied the convergence of DYS under a wider range of stepsizes [4] and Aragón-Artacho and Torregrosa-Belén also proved the convergence of the DYS iteration over a wider range of stepsizes [1]. Dao and Phan analyzed a modified version of DYS iterations [16], and Pedregosa analyzed sublinear convergence rates of DYS iterations [33], but they did not analyze linear rates of convergence. Condat and Richtárik analyzed linear convergence rates of a randomized version of Primal-Dual Davis–Yin (PDDY) [14]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> SRG has also found applications in control theory: Chaffey, Forni, and Rodolphe utilized the SRG to analyze input-output properties of feedback systems [11, 10], and Chaffey and Sepulchre furthermore used it as an experimental tool to determine properties of a given model[13, 9, 12].. | **A**: Ryu, Taylor, Bergeling, and Giselsson [39] and Wang, Fazlyab, Chen, and Preciado [45] formulated the problem of computing tight contraction factors as SDPs using the performance estimation problem (PEP) and integral quadratic constraint (IQC) approaches, but these are numerical results, not analytic convergence proofs in the classical sense.
To the best of our knowledge, the only prior works providing analytic linear convergence rates of DYS iterations are the arXiv version of the original DYS paper [18] and [14].
**B**: Follow-up work has extended the theory and applied it to analyze nonlinear operators: Huang, Ryu, and Yin characterized the SRG of normal matrices [25], Pates further characterized the SRG of linear operators using the Toeplitz–Hausdorff theorem [31], and Huang, Ryu, and Yin [26] established the tightness of the averagedness coefficient of the composition of averaged operators [29] and the DYS operator [18].
**C**: The latter work [14] considers randomized updates, which makes the setup distinct from ours.
On the other hand, the theory of scaled relative graph (SRG) by Ryu, Hannah, and Yin [38] allows one to analyze the convergence of operator splitting methods by mapping the action of a multi-valued nonlinear operator to the extended complex plane, analogous to how the spectrum maps the action of a linear operator to the complex plane.
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We introduce basic notation and briefly recall required properties of Bregman divergences in Section 2. <|MaskedSetence|> <|MaskedSetence|> In particular, information geometry is employed in a way that indicates how other convex programs could be handled in the same way, in principle. While we indicate multilevel extensions in Sections 3.2 and 4.7, respectively, we mainly focus on the core problem in this paper, that is two-level geometric optimization. <|MaskedSetence|> We conclude in Section 6.
. | **A**: The core part of this paper, Section 4, generalizes this scheme to a Riemannian setting.
**B**: Numerical experiments are reported in Section 5 for a range of problem instances and compared to a recent state-of-the-art method [HRX21].
**C**: Section 3 summarizes the basic scheme of two-level optimization in Euclidean spaces.
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Given the above result, it may seem that, similarly to the case of monotone networks with ReLU activations, the class of monotone networks with threshold activations is too limited, in the sense that it cannot approximate any monotone function with a constant depth (allowing the depth to scale with the dimension was considered in [12], see below). <|MaskedSetence|> Any continuous function over a bounded domain can be approximated by a depth-2222 network [3, 11, 22] and this universality result holds for networks with threshold or ReLU as activation functions. <|MaskedSetence|> We establish a depth separation result for monotone threshold networks and show that monotone networks can interpolate arbitrary monotone data sets by slightly increasing the number of layers. <|MaskedSetence|> As noted, this is in sharp contrast to general neural networks, where adding extra layers can affect the efficiency of the representation [16], but does not change the expressive power.
. | **A**: Thereafter, a simple argument shows that monotone networks of bounded depth are universal approximators of monotone functions.
**B**: Our first main result supports the contrary to this belief.
**C**: One reason for such a belief is that, for non-monotone networks, depth 2222 suffices to ensure universality.
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The evolution of the estimation error {‖-Θtθoptt‖} is shown in Fig. 12 for each value of b listed in (102). Only ξct in (103) was used as the probing signal for these experiments. We see in Fig. 12 that tracking performance deteriorates as the rate of change of {θoptt} increases. When >t/T2, the signal {θoptt} is nearly static for each period and hence we expect the rate of change of {θoptt} to have little effect on the AAD.
This is confirmed by the results in shown in Fig. 12: the steady state estimation errors are roughly the same for all values of b.
Fig. 13 shows the evolution {Γ(Θt),Γ(Θ1Ft),Γ(Θ2Ft)} as functions of time for ∈η{5,15}. <|MaskedSetence|> <|MaskedSetence|> When >t/T2, the estimates {Θ1Ft} only yield lower objective values than {Θ2Ft} for very small portions of the run. <|MaskedSetence|> | **A**: That is, a second order filter is preferable when the rate of change of {θoptt} is small..
**B**: We see in Fig. 13 that the filtered estimates {Θ1Ft} yield lower objective values than {Θ2Ft} for ≤t/T2.
**C**: Again, the selected probing signal was ξct and the signal {θoptt} had period T0 with =b3 for both choices of η.
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<|MaskedSetence|> GKAD21220144, 2021GXNSFFA196004 and GKAD23026237, the NNSF of China Grant Nos. 12001478, 12101143 and 12371312, the China Postdoctoral Science Foundation funded project No. 2022M721560, the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2023ZK13, and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH. <|MaskedSetence|> <|MaskedSetence|> | **A**: It is also supported by the project cooperation between Guangxi Normal University and Yulin Normal University.
**B**: The second author is supported by the grant from the National Program for Research of the National Association of Technical Universities - GNAC ARUT 2023, ID: 220235047, ”Sisteme cuplate de inegalităţi hemivariaţionale şi aplicaţii”..
**C**:
Acknowledgment
This project has received funding from the Natural Science Foundation of Guangxi Grant Nos.
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In the classical case of matroids independent sets need three axioms. For q𝑞qitalic_q-matroids, the straightforward q𝑞qitalic_q-analogue of these three axioms are not strong enough to get a q𝑞qitalic_q-matroid with a semimodular rank function. Therefore, a fourth axiom was added (see [10]).
This unexpected fourth axiom raises some questions. Why would we need this extra axiom for independent spaces, bases, and spanning spaces, but not for things like dependent spaces, circuits, flats, and hyperplanes? Can’t we find a better way to describe the axioms for independent spaces, bases, and spanning spaces, using only three axioms?
In this paper we present a positive answer to this question. We propose two ways to define independent spaces, bases, and spanning spaces with only three axioms. <|MaskedSetence|> <|MaskedSetence|>
As an application of this restriction of the number of axioms, we prove two cryptomorphisms. <|MaskedSetence|> The second one is a cryptomorphism between independent spaces and bases. This was done in [10], but we believe there was a gap in the proof that we will fix here.. | **A**: The second one is an alternative for the third axiom that is still a q𝑞qitalic_q-analogue of the classical case, but that obliterates the need for the fourth axiom.
**B**: The first one is a direct cryptomorphism between independent sets and circuits that was not shown before.
**C**: The first one is to remove the third axiom, because it is implied by the fourth one (where we have to be a bit careful for independent spaces to pick the right variation of the fourth axiom).
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<|MaskedSetence|> It is structured as follows. <|MaskedSetence|> <|MaskedSetence|> DG in the QMC framework is presented in Section 5 and the corresponding parametric regularity analysis is considered in Section 6. Numerical results, which confirm our theoretical findings, are given in Section 7, while a short conclusion wraps up our exposition.
. | **A**: Section 3 describes randomly shifted lattice rules, and Section 4 gives a brief overview over the analysis of conforming FE methods.
**B**: Notations and preliminaries are introduced in Section 2.
**C**: However, since discontinuous Galerkin methods are non-conforming, we cannot directly apply the existing QMC theory.
This paper tries to bridge this theoretical gap.
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Structure of the paper. In Section 2, we first recall fractal structures and self-affine structures. Then we estimate the lower bound of the Hausdorff dimension of the associated fractal set. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: In Section 3, we generalize the lattice point counting results in [LSST20, Section 3] to higher dimensional cases.
**B**: In Section 4, we construct a fractal set contained in the set of weighted singular vectors and prove Theorem 1.1 by estimating the Hausdorff dimension of the fractal set.
**C**:
.
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