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**A**: The author would like to thank David Gabai and Sergio Fenley for several helpful discussions about this work**B**: A special thanks is owed to the referee whose exceptionally detailed and patient reading resulted in many corrections and improvements to the article. The author is indebted to Peter Ozsváth for his... | CAB | ACB | BAC | BAC | Selection 2 |
**A**: I am very grateful to him for many helpful comments and for suggesting the definition of the size of a loose chart in Remark 1.13 (personal communication, December 3, 2020)**B**: Acknowledgments: The idea for this paper originated in discussions with Georgios Dimitroglou Rizell**C**: I am also grateful to the re... | BAC | ABC | BCA | ACB | Selection 1 |
**A**: In Definition 2.1.10 we recalled the definition of the homotopy colimit as a derived functor**B**: After introducing some definitions we briefly explain how this method provides a good theory of homotopy colimits, see also [Rie14, Chapters 4,5] and [Shu06, Section 7].
**C**: Here, we will present an alternative ... | CBA | CBA | BCA | ACB | Selection 4 |
**A**: See Appendix A for examples of expanding Thurston maps.
**B**: On the other hand, an expanding Thurston map may be obstructed, i.e., not topologically conjugate to a rational map; in particular, it can have periodic critical points**C**: Expanding Thurston maps give natural topological models for investigating t... | CBA | BAC | ABC | ABC | Selection 1 |
**A**: In the subsequent hierarchy, the search is refined to a narrower region, yielding a new estimate**B**: This iterative process ultimately produces an accurate estimate of the dominant frequency**C**: While ML-QCELS has proven effective for single-phase estimation, it is not suitable for multiple-phase estimation.... | CAB | CAB | ABC | ACB | Selection 3 |
**A**: This leaves open the question of what happens in the intermediate volume regimes**B**: On one side, as noted above, one should in general not expect existence of optimal sets. But on the other side, at least for the cases when the optimal measure is uniformly distributed over a sphere, our results provide existe... | ABC | BCA | BAC | BAC | Selection 2 |
**A**: The key idea behind mixed-subtree quasi-isometries is that they are quasi-isometries which are defined iteratively for vertices further and further away from the root. Moreover, at each step, the allowed choices are in some sense independent from the choices for earlier vertices.**B**:
In this section, we descr... | ABC | ABC | ACB | CAB | Selection 4 |
**A**: Our approach is partly inspired by Shen and Wang [MR4308060], though there are several key differences. Shen and Wang use conformal transformations and compare the boundary integral (1.4) of the underlying domain with that of the punctured disk**B**: Subsequently, techniques involving the Schwarzian derivative a... | CAB | CBA | BCA | ABC | Selection 4 |
**A**: It was noticed in [WG15] that the presence of grazing boundaries may cause a breakdown in the diffusive asymptotics within the L∞superscript𝐿L^{\infty}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT framework**B**: We refer to [Wu21, Ouy24] for recent development of the diffusion limit of the neutron trans... | BCA | CBA | CBA | CAB | Selection 1 |
**A**: Theorem 3.4 can also be proved by either appropriately modifying the proof of Proposition 2.6.12 in [2] or by the methods described in Section 6 of [19]**B**: We are grateful to both David Blecher and Hannes Thiel for pointing these references to us**C**: An alternative way is to use
Corollary 4.1.7 in [7]. This... | ACB | ABC | CAB | ACB | Selection 2 |
**A**: We first prove Theorem 1.2 when π𝜋\piitalic_π is supercuspidal, by combining a local-global argument and an involution method similar to that of [Xue21, Section 4]**B**: If π𝜋\piitalic_π is a discrete series representation, then π𝜋\piitalic_π is a segment of the form
**C**: Theorem 1.2 is proved in Section 5,... | ACB | ABC | ABC | BCA | Selection 4 |
**A**: There is no active deformation along each link, neither in length nor in curvature, and elongation is produced by new material points appearing at the tips of the filament.
Notice that this does not imply that a new part of the body is generated: for instance, one can consider, as in vine robots, a thin tubular ... | BCA | ACB | ABC | CAB | Selection 1 |
**A**: Regarding random regular graphs, the results displayed in Figs. 22b, 22c, 22e, 22f, 22g, 22h, 22i and 22j indicate that using a partition size of 6 results in the best combination of efficiency and accuracy.
To summarize, employing a larger partition leads to a larger environment size, generally reducing the con... | BAC | CBA | ABC | BCA | Selection 3 |
**A**: P. Neff acknowledges support in the framework of the DFG-Priority Programme 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials”, Neff 902/10-1, Project-No**B**: 440935806.**C**:
Patrizio Neff is grateful for the helpful discussions with Michael A. Slawinski (Memor... | CAB | BCA | ABC | BAC | Selection 2 |
**A**: The action can be extracted from [37]*5.3**B**: The following irreducible Yetter-Drinfeld modules over W𝑊Witalic_W have a finite-dimensional associated Nichols algebra, [2]*Theorem 3.1**C**: The grading of all of them is supported in C𝐶Citalic_C, thus to understand the braiding, it is enough to consider the C�... | ABC | BAC | BCA | ABC | Selection 2 |
**A**: In the final section, we discuss the global strong solutions to the system (1.1). Due to technical difficulties described above, we are able to prove the regularity of the weak solutions in the class (1.3) in periodic domains only (Theorem 4.2). Moreover, using the abstract theory developed in [3, 4], we prove t... | CAB | BAC | ABC | BAC | Selection 1 |
**A**: The DMD is able to extract dynamically relevant flow features from time-resolved experimental or numerical data by generalizing the global stability modes and approximating the eigenvalues and eigenfunctions of the Koopman operator [14]**B**: Both methods use the snapshots of the fully discretized PDEs in time. ... | ACB | BCA | BAC | BAC | Selection 2 |
**A**: It can be easily verified that these two groups of equations are independent and do not imply (or deny) one another**B**: there must exist solutions to this system of equations, if fact there will be a unique solution for every x∈L𝑥𝐿x\in Litalic_x ∈ italic_L.
**C**: Therefore, by dimensionality arguments in th... | CBA | ABC | BCA | ACB | Selection 4 |
**A**:
This paper describes a general algorithm that solves the above problem when there exists a representation of τ𝜏\tauitalic_τ as a series with rational terms. The algorithm is described in §2, and its basic properties are addressed**B**: The complexity of the algorithm is analysed in §3. Application to specific ... | ABC | ACB | CBA | CBA | Selection 1 |
**A**: Finally I would also like to thank the anonymous referee for many helpful comments and suggestions.**B**: Acknowledgments: Most of the results of this paper are part of my PhD thesis**C**: I am grateful to Alessandra Sarti for her constant support, for the supervision of this work and for many useful discussions... | BCA | BCA | CBA | CAB | Selection 4 |
**A**: What we need is a set of starting points that becomes dense in the support [0,1]dsuperscript01𝑑[0,1]^{d}[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, such as a fine grid, where, for the theory, the grid size would tend to zero**B**: While the algorithms above are defined by using the data as sta... | BAC | CBA | ACB | ACB | Selection 1 |
**A**: Thus in some sense at the moment the only motivation for the equivalences (1.2.1) and (1.2.2) comes from mathematical physics.
**B**: Let us note that the q=1𝑞1q=1italic_q = 1 case discussed above is a special case of a very general set of conjectures due to D. Ben-Zvi, Y. Sakellaridis and A. Venkatesh; those c... | BCA | CAB | BCA | BAC | Selection 2 |
**A**: In the proof of Theorem 2.18, it is shown that the definition of a combinatorial HHS implies W𝑊Witalic_W is connected; see Sections 1 and 5.2 of [BHMS20]**B**: We emphasis that there is no a priori requirement that either X𝑋Xitalic_X or W𝑊Witalic_W is connected**C**: We will take advantage of this in Section ... | CBA | BAC | BCA | BCA | Selection 2 |
**A**: Figure 6 illustrates 8 of them**B**: For n=3𝑛3n=3italic_n = 3 there are 16 rooted labeled forests**C**: Figure 7 illustrates how the rooted labeled forest can be visualized when the exceptional sequence is embedded in the Auslander-Reiten quiver of the path algebra.
| BAC | ABC | CBA | ABC | Selection 1 |
**A**:
In its minimal genus thickened surface, the knot must have all regions simply connected, or otherwise we could find a reducing annulus**B**: In particular, as in Figure 22, each region has either a clockwise rotation or a counterclockwise rotation, and two regions that share an edge must have opposite rotations... | ABC | CBA | ACB | CBA | Selection 3 |
**A**: If the two sizes are close to each other and not too large, then agreement on both opinions is possible. Otherwise, either no agreement is possible, or the process converges to an agreement towards a single opinion, which is that of the largest stubborn community. The case in which the two stubborn communities h... | ACB | BAC | BAC | CAB | Selection 4 |
**A**: The results we present allow both to simplify the computations appearing in [Sau20] about volumes of strata of flat surfaces with rational singularities and to give an explicit formula for the orbifold Euler characteristic of minimal strata of abelian differentials based on results from [CMZ20a]**B**: This last ... | CAB | ACB | BCA | CBA | Selection 3 |
**A**: The empirical eigenvalues are simple to compute, so it is simple to validate this assumption with a diagnostic plot. Figure 2 verifies polynomial decay of the empirical eigenvalues in the real world application of Section 6; the Project STAR data have a low effective dimension as required by Assumptions 5.2 and ... | CAB | BCA | CBA | CAB | Selection 3 |
**A**: Mainly cosmetic changes to arguments in loc. cit. are necessary for us, but for completeness, we produce the necessary arguments in this section when slight changes are needed.
**B**: In proving the weak Frobenius structure theorem, we note that many arguments appearing in [GS21] Section 6666 describing moduli s... | CAB | CBA | ACB | ABC | Selection 1 |
**A**:
Coxeter groups generated by reflections on codimension 1 faces of a right-angled polyhedron P𝑃Pitalic_P, as defined in [DJ02, p**B**: the tiling of the hyperbolic plane by compact regular pentagons). The nerve is simplicially isomorphic to the dual of the boundary of the polyhedron P𝑃Pitalic_P, so it is again... | CAB | CAB | ACB | BAC | Selection 3 |
**A**: 1111 but, since M𝑀Mitalic_M is graded, the latter case cannot occur**B**: Then the**C**: Therefore,
(S+,Nφ)subscript𝑆subscript𝑁𝜑\left(S_{+},N_{\varphi}\right)( italic_S start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_N start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT ) is of right-class 3333 | CBA | CAB | ACB | BCA | Selection 3 |
**A**: Hanneke (2021); Blanchard (2022) adopt a universal learning approach and analyze algorithms which are able to achieve vanishing long-term average loss for general data-generation processes**B**: Connection to online and batch learning in non-stationary environments.
Our work also relate broadly to papers studyin... | BCA | BAC | ABC | ABC | Selection 2 |
**A**: The TOP shares constraints similar to the BPP, with additional constraints (for details refer to Subsection 3.2) to ensure that the number of trolleys needed to build each PCB does not exceed the capacity of the assembly line otherwise the problem will be either infeasible or will need to change trolleys while b... | ABC | BCA | BAC | ACB | Selection 2 |
**A**: In Examples 1 and 1 the step-size was reduced at most once for the direction computed to be accepted by the descent criteria (13)-(18).**B**: This is expected and consistent with Proposition 4.3 for the strictly convex case**C**:
We highlight that Algorithm 2 never reduced the step-size in order to achieve cond... | BAC | CBA | CAB | ACB | Selection 2 |
**A**:
Since this work, there have been numerous developments in the field of log Gromov-Witten theory**B**: On another front, the more flexible construction of punctured log curves has been developed, which allows for negative contact orders of marked points with boundary divisors. This is useful for cutting and glui... | CBA | CBA | ABC | ACB | Selection 4 |
**A**: In Theorem 1.1, we have given a doubly exponential upper bound on the d𝑑ditalic_d-dimensional Ramsey numbers**B**: It would be very interesting to close the gap. In particular, it would be good to know whether there is a simple exponential upper bound, or whether the numbers grow more quickly.
**C**: From below... | BCA | BAC | CAB | ACB | Selection 4 |
**A**: We provided a convergence analysis and showed that in each step the resourcefulness of the iterates increase**B**: We illustrated our approach for the special case of the geometric measure, allowing us to identify interesting quantum states, discover novel AME states, and characterize highly entangled subspaces ... | CBA | BCA | ABC | CAB | Selection 2 |
**A**: In addition to the properties of these functions established by Körner, we will need certain uniform bounds on the functions**B**: Actually the functions we will use are not quite the same ones used by Körner. Here is our modification of [20, Lemma 1.2].**C**:
As mentioned in the introduction, we will use the f... | BCA | ABC | ABC | BAC | Selection 1 |
**A**: We suppose that this fact might be caused by the intrinsic errors. More numerical experiments should be performed to have a clearer picture of the method’s positive and negative features.
**B**: In conclusion, RCDT is usually better at capturing the geometric features and reducing the number of modes required to... | BCA | CBA | BCA | BCA | Selection 2 |
**A**: Motivated by the works [2, 3], we extend the tensor CUR (TCUR) approximation [18] to a pair and a triplet of tensors based on the t-product and we refer to as the generalized TCUR (GTCUR). As real-world datasets often have multidimensional structure, it is more favorable to generalize the ideas proposed in [2, 1... | CAB | CAB | ABC | ACB | Selection 4 |
**A**: In-splits for topological graphs correspond to factorisations of the range map**B**: On the other hand, out-splits for topological graphs correspond to a factorisation of the source map, which defines the right-module structure of the graph correspondence. This makes the noncommutative analogy for out-splits mor... | ACB | CAB | CAB | CAB | Selection 1 |
**A**: In this section, we make two observations about this problem**B**: As far as these authors can tell, this problem is currently open**C**: The first observation is that a positive answer to the previous problem implies a positive answer to the first-order free group factor alternative:
| ACB | BAC | CAB | ACB | Selection 2 |
**A**: Trigonometric functions are prevalent in stochastic dynamical systems of motion**B**: The exponential function is directly related to trigonometric functions as well as to characteristic functions of distributions**C**: We combine the methodology proposed in (Jasour et al., 2021) with Prob-Solvable loops to obt... | CBA | ACB | ABC | ACB | Selection 3 |
**A**: Conventions**B**: In this paper the coefficient ring ΛΛ\Lambdaroman_Λ will be assumed to be of finite characteristic not equal to 2222 and all the schemes considered will have the property that their residue characteristics are prime to the characteristic of ΛΛ\Lambdaroman_Λ**C**: Whenever we consider complex al... | ACB | BAC | CAB | ABC | Selection 4 |
**A**: In this article, we introduce a function that assigns a non-negative integer to a pair of Legendrian representatives of a given knot type. Roughly speaking, this function measures the obstruction to converting a topological isotopy between two Legendrian representatives into a Legendrian isotopy**B**: We show th... | CAB | BCA | ACB | ABC | Selection 3 |
**A**: On the other hand the uniqueness of radially symmetric solutions of (1.4) which satisfies (1.7) is proved in [HuK3] using a careful analysis of the asymptotic expansion of the solution
f𝑓fitalic_f near the origin**B**: This method also may not work in other problems**C**: Hence our approach in this paper | ABC | CAB | ACB | BCA | Selection 1 |
**A**: close to the configuration of minimal energy, then the phenomenology is different and KAM tori can persist in the presence of small perturbations, see [19, 26].**B**: Indeed, if we consider a system close to the ground state, i.e**C**:
Before proceeding, we note that the above restriction to positive temperatur... | ABC | BCA | CBA | BCA | Selection 3 |
**A**: This relies on Lemma 3.2 from [6], restated as Lemma 2.1 below, to control collisions**B**: Then, we control differences of solutions (contraction estimates) as in Proposition 2.3. With these alone, we can construct solutions to the Vlasov-Poisson-Landau system (1) as the limit of smooth approximate solutions gi... | ACB | ACB | BCA | CBA | Selection 3 |
**A**: The new genus bound proves that both these knots, that have the same LG𝐿𝐺LGitalic_L italic_G, are of genus greater or equal to 2**B**: This is significant since both the Alexander invariant and the Levine-Tristram signature fail to detect genus for the pair.**C**:
One example where the bound given by Theorem... | CAB | CBA | BCA | ACB | Selection 3 |
**A**: The criterion will be used in the proofs of both Theorem 3 and Theorem 1**B**: Because the latter proof requires truncating at a random time, we give the criterion in terms of a geometric random variable τ𝜏\tauitalic_τ rather than a fixed T𝑇Titalic_T.**C**:
In this section, we give a criterion for directional... | ACB | ACB | BCA | CAB | Selection 3 |
**A**: However, it is important that we consider Kummer surfaces and not just abelian surfaces, for the following reason**B**: In contrast, for K3 surfaces, it can happen that the algebraic part of the Brauer group consists only of constant elements (and so does not contribute to any Brauer–Manin obstruction), while th... | BCA | ACB | CAB | ABC | Selection 2 |
**A**:
After that, we organized the construction of any finite rank generalized Fuchsian Schottky group in the upper-half plane with orientation preserving and orientation reversing isometries as side-pairing transformations**B**: The Fenchel-Nielsen coordinates in Tiemüller space for any arbitrary rank Fuchsian Schot... | CBA | BAC | CAB | ACB | Selection 4 |
**A**: Fortunately, various kinds of metric viscosity solutions have been provided in [2, 14, 15, 29, 30, 33] on metric spaces, and these metric viscosity solutions only depend on the metric structure**B**: However, the classical theory of viscosity solution relies heavily on the differential structure, hindering its e... | CBA | ACB | CBA | BAC | Selection 4 |
**A**: 1695/22 by B. A**B**:
Natural Sciences and Engineering Research Council of Canada (NSERC) via the Canada Research Chair (CRC) of D.V.S.; the Fonds de Recherche du Québec Nature et Technologies (FRQNT) via Institut Transdisciplinaire d’ Information Quantique Transdisciplinary Institute for Quantum Information (I... | CBA | BCA | CAB | BAC | Selection 4 |
**A**: Stability near fold points in the bifurcation diagrams of Proposition 7 assuming stability does not change away from the neighbourhood of the fold where it was computed, indicated as ⋆⋆\star⋆**B**: Conventions: solid lines are attractors, dashed lines are saddles, dot-dashed lines are totally repelling. On dotte... | ACB | BCA | CAB | ABC | Selection 2 |
**A**:
Part of the work has been done when CD was a visiting reseacher at the Institut des Hautes Études and she would like to thank Laure Saint-Raymond for her support and hospitality**B**: Frank for helpful suggestions, leading to our consideration of Lorentz norms in Theorem 2. We also thank Hajer Bahouri, Morris B... | ABC | ABC | ACB | BCA | Selection 3 |
**A**: Finally, in § 3.4 and § 3.5 we study the symmetry properties of the extensions and the relation with the self-adjoint realization of the Dirac operator with an AB flux, respectively.**B**: This section is devoted to the proofs of the results previously stated.
The first result we prove is the classification of s... | CAB | CBA | BAC | BCA | Selection 1 |
**A**: I would like to thank Oded Carmon for many discussions about Speh representations and matrix Kloosterman sums**B**: I would also like to thank Ofir Gorodetsky for encouraging me to write this paper**C**: Finally, I would like to thank the anonymous referee for their careful reading of this manuscript and their c... | CBA | ABC | CAB | BAC | Selection 2 |
**A**: Moreover, again if q=0𝑞0q=0italic_q = 0 and m=0𝑚0m=0italic_m = 0,
the rate function IMD(x;0)subscript𝐼MD𝑥0I_{\mathrm{MD}}(x;0)italic_I start_POSTSUBSCRIPT roman_MD end_POSTSUBSCRIPT ( italic_x ; 0 ) in Proposition 3.3 is not well-defined (because there is a denominator**B**: the costant random variable equa... | ABC | CBA | CAB | ACB | Selection 2 |
**A**:
The inequalities in Theorem 1.1 are optimal**B**: Moreover, Theorem 1.1 is global in the sense that the completeness assumption on Mnsuperscript𝑀𝑛M^{n}italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT cannot be dropped**C**: See §3§3\S{3}§ 3 for details. | BAC | ACB | ABC | ACB | Selection 3 |
**A**: She also thanks Dr. Anand Sawant for several insightful comments, which led to Theorem 3.16. Additionally, the author is grateful to the anonymous referee for their careful reading and various suggestions that improved the exposition of the paper.
**B**: Chetan Balwe, for suggesting this problem, his constant gu... | ACB | ABC | ACB | CBA | Selection 4 |
**A**:
It is always possible to obtain many P.O**B**: actions from ordered global actions using restrictions**C**: The question we want to answer is that when we can, starting from a P.O. action α𝛼\alphaitalic_α, give rise to a global ordered action β𝛽\betaitalic_β such that a restriction of β𝛽\betaitalic_β is equi... | CAB | BCA | BCA | ABC | Selection 4 |
**A**: Remarkably, it encapsulates the trajectories of the concentrations of each species, in each grid cell, in function of the other two species through time, serving as a central hub for their dynamic behavior. This emphasizes the pivotal role played by the attractor in dictating the enduring patterns and long-term ... | BCA | CAB | BCA | CBA | Selection 4 |
**A**: The theorems obtained arise from the study of equivalents polynomial systems that we established from the results of the previous section**B**: We give the complete proofs of theorems 4 and 7. By applying the same technique on the other algebras with the equivalent polynomial systems we obtain the proofs of theo... | BCA | CAB | CBA | CAB | Selection 1 |
**A**:
Let (X,d)𝑋𝑑\left(X,d\right)( italic_X , italic_d ) be a complete, locally compact, separable metric space which is a length space**B**: In particular, by the Hopf-Rinow theorem for length spaces (see e.g**C**: [37, p. 9]), each pair of points can be joined by a minimizing geodesic, i. e. a rectifiable curve w... | ABC | ACB | ACB | BAC | Selection 1 |
**A**: energy consumption**B**: The 𝐐𝐐{\bf Q}bold_Q and 𝐑𝐑{\bf R}bold_R matrices we used in LQR design are exactly
the same as the ones used in [15]**C**: The target of robust pole assignment is to have a similar settling time to the LQR design but with fewer oscillations—this is of particular importance to space t... | ACB | ACB | BAC | ABC | Selection 4 |
**A**: They proved that the length of the longest run of heads in n𝑛nitalic_n tosses of a fair coin will approximately be equal to logn𝑛\log nroman_log italic_n. The question of fluctuation for such stochastic processes was first studied by Chung and Hunt [Chung-Hunt], and later by Chung and Erdős [Erdos1], and Csák... | BCA | CBA | ABC | ACB | Selection 1 |
**A**: We note that Lemma 3.2 does not require anything from the host graph G𝐺Gitalic_G, and Lemma 3.7 only utilises some of the isoperimetric profile of G𝐺Gitalic_G, yet does not utilise the product structure of G𝐺Gitalic_G**B**: Still, in this proof, the bound on the number of sets with ‘bad expansion’ from Lemma ... | ACB | CAB | CAB | BCA | Selection 1 |
**A**: Identical feature variance is assumed across all agents, the relaxation of which requires feature statistics-aware power control**B**: Symbol-level synchronization and sufficient channel coherence time across all agents are assumed to facilitate AirComp design. However, under high mobility, such conditions may n... | CBA | CAB | CAB | BCA | Selection 4 |
**A**: It enhances the so-called CH-partitioning from [3, 7], which divides the original problem into many smaller subproblems, solves them with the QPU, and finally takes the largest stable set across the subproblems as the final solution**B**: In order to improve solutions returned by the QPU, we combine the simple C... | CBA | ABC | BCA | ABC | Selection 3 |
**A**: This approach leads to the popular DDPM sampler (Ho et al.,, 2020; Nichol and Dhariwal,, 2021)**B**: The above result motivates the following paradigm: we can construct the forward process (1.1) by time-discretizing the diffusion process (1.3), and construct the reverse process (1.2) by discretizing the reverse-... | ACB | BAC | CAB | ABC | Selection 2 |
**A**:
As is evident from Lemma 3.1, we must establish an adequate bound over the minor arcs**B**: We will instead make use of the assumption that C𝐶Citalic_C dominates the function**C**: For this we make use of the arguments of [3] without assuming our matrix is highly nonsingular | CAB | CAB | ACB | CAB | Selection 3 |
**A**: Similarly, Ebrahimi et al**B**: [16] and Motalebi et al**C**: [31] explored zero-inflated and hurdle models to better capture the inherent sparsity in social and biological networks.
Furthermore, Dong et al. [15] and Motalebi et al. [32] specifically focused on adapting stochastic block models to account for exc... | BAC | ACB | ABC | CAB | Selection 3 |
**A**: So, by Lemma 2.12 we conclude z⋅T(W)=g(W)T(W)⋅𝑧𝑇𝑊𝑔𝑊𝑇𝑊z\cdot T(W)=g(W)T(W)italic_z ⋅ italic_T ( italic_W ) = italic_g ( italic_W ) italic_T ( italic_W ) for some rational function g𝑔gitalic_g. To compute explicitly g(W)𝑔𝑊g(W)italic_g ( italic_W ) we first consider W𝑊Witalic_W such that T(W)∈Cst�... | BCA | CBA | CAB | ACB | Selection 1 |
**A**: I would like to thank my MSc advisor (currently my Phd advisor), Eitan Sayag, for teaching me a lot of mathematics and also for many helpful discussions about my research and his many helpful comments during the writing of this text**B**:
Most of the results presented in this work were obtained as part of my MS... | BCA | BCA | BCA | BAC | Selection 4 |
**A**: The large deviation principle for the number of edges in the graph in Theorem 2.3 suggests that also a central limit theorem should also hold**B**: This is further exemplified by the nice limiting generating function for the number of edges in Lemma 2.10. Unfortunately, while being suggestive, it seems hard to t... | CBA | CAB | BCA | ABC | Selection 3 |
**A**: A common guideline is that its diameter should be at least as large as the signal wavelength; otherwise, the pressure waves does not effectively interact with the device.**B**: The size of the lens has a significant impact on its performance for two main reasons.
First, a larger device interacts with a broader a... | BCA | BAC | CAB | ABC | Selection 3 |
**A**: Recall (1.2).
**B**: To support that idea, we consider a random traps environment (see e.g. [4, 12]), which is simpler to study than the random conductances environment, while preserving the “trapping effect” it has on the random walk**C**: We provide an analogous result to Theorem 1.5 under the following assump... | CBA | CAB | ACB | BAC | Selection 2 |
**A**: We have addressed several key challenges of modeling such extreme conditions through extensive numerical simulations. First, we quantified the impact of mesh resolution on energy stability and mass conservation, and our results show that higher interfacial mesh resolution is crucial for maintaining energy stabil... | ACB | ACB | BCA | ABC | Selection 4 |
**A**: Using a density we assume u,v𝑢𝑣\displaystyle u,vitalic_u , italic_v smooth**B**:
Let 1/2≤s<112𝑠1\displaystyle 1/2\leq s<11 / 2 ≤ italic_s < 1, u∈W2,p(Ω)𝑢superscript𝑊2𝑝Ω\displaystyle u\in W^{2,p}(\Omega)italic_u ∈ italic_W start_POSTSUPERSCRIPT 2 , italic_p end_POSTSUPERSCRIPT ( roman_Ω ) with ‖u‖W2,p≤1su... | ACB | BCA | BAC | CAB | Selection 3 |
**A**: Indeed, a coarse space X𝑋Xitalic_X that has a μ𝜇\muitalic_μ-PS is coarsely equivalent to itself with the zero measure 00, but it does not have a 00-PS**B**: To avoid such situation, we impose an additional condition to coarsely equivalent maps. In this section, we prove the following theorem:**C**:
However, f... | ACB | BCA | CBA | CAB | Selection 2 |
**A**: Then the Poisson structure π𝜋\piitalic_π on the homogeneous space is obtained by canonical projection from the corresponding ΠΠ\Piroman_Π on SO(2,1) given by Eq. 19**B**: Moreover, when the stronger condition
**C**: In that case, we have a PH lightcone, in which the additional Poisson bracket π𝜋\piitalic_π is ... | ACB | BCA | CBA | CBA | Selection 1 |
**A**: Such formulation is not stable for large degrees rather being very slow as shown by [30]. Thus, the need for customizable cap-like domains with efficient harmonic bases persists.
**B**: From the perspective of surface harmonics, the choice of the parameterization domain is constrained by the availability of a se... | BAC | CAB | CBA | BAC | Selection 2 |
**A**: An example of the nontrivial thermal physics of causal diamonds is provided by the recent analysis of entanglement of a bipartite system [28]**B**: Specifically, it was found that if the system starts in a maximally entangled state prepared from an inertial perspective, entanglement is degraded due to the presen... | BCA | BAC | BAC | ACB | Selection 1 |
**A**: One can see that, although this is not defined for k=4𝑘4k=4italic_k = 4, the integrand in that case is in fact the pointwise conformal invariant guaranteed by**B**: The second term is a pointwise invariant, and of course the third gives
a multiple of the Euler characteristic**C**: The first term gives a multipl... | CAB | CBA | BCA | ABC | Selection 2 |
**A**: An inverse semigroup with zero S𝑆Sitalic_S is said primitive if ⪯precedes-or-equals\preceq⪯ is equality in S∗superscript𝑆S^{*}italic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT**B**: A congruence is said 0-restricted if the class of 0 only contains 0**C**: A congruence is said primitive if the quotient by th... | ACB | BAC | ABC | CAB | Selection 2 |
**A**:
One could then proceed by defining the Liouvillian as the infinitesimal generator of the unitary propagator in Eq. (43), and equivalently formulate all results in the next sections within this representation**B**: We shall briefly comment on this point later on in the paper**C**: In particular, see Remark 3.10. | ABC | CBA | CAB | CBA | Selection 1 |
**A**: It follows that the curve at infinity is irreducible and that the mapping (3.27) is surjective. In particular, (3.27) defines a rational parametrisation of the curve at infinity of the Segre surface**B**: The image of the mapping (3.27) is an irreducible subset of the curve at infinity. If the curve at infinity ... | CAB | ACB | BCA | BAC | Selection 4 |
**A**: If the arrangement has a real realization, then let’s send the four degree-three points to the vertices of the unit square, (0,0,1),(0,1,1),(1,0,1),(1,1,1)001011101111(0,0,1),(0,1,1),(1,0,1),(1,1,1)( 0 , 0 , 1 ) , ( 0 , 1 , 1 ) , ( 1 , 0 , 1 ) , ( 1 , 1 , 1 )**B**: By Assumption 2, every collinear triple has two... | ABC | CAB | BAC | ACB | Selection 4 |
**A**: This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no**B**: The work is supported by the state assignment of MIPT (project FSMG-2023-0013). The author is a Young Russian Mathematics award winn... | CBA | ACB | BAC | BAC | Selection 2 |
**A**: Section 3 provides explicit coordinate equations for conformally Hamiltonian vector fields on cotangent bundles, introduces the scaled cotangent lift, and showes that cotangent lifted diffeomorphisms are conformally symplectic**B**: These elements are essential for understanding the geometric framework of our st... | CBA | CAB | ACB | BCA | Selection 4 |
**A**:
To prove that (ii) and (iii) of Sec. 2.1.1, we need to introduce a potential function, defined everywhere locally on the cones**B**: There exists an explicit construction of a potential function ΦΦ\Phiroman_Φ defined everywhere locally on the cones**C**: This is obtained as follows. Consider the Koszul–Vinberg ... | ABC | BAC | CAB | BCA | Selection 1 |
**A**: For the most complicated inequality we have explained in detail how to obtain the uniform continuity constants in Item 2 (see page 5.4).**B**: The python code can be found in the GitHub repository [Rod24b]; both files that check the inequality and files that output a 3D plot of the function f(x,y)𝑓𝑥𝑦f(x,y)it... | BAC | BAC | CBA | CAB | Selection 3 |
**A**: This appears in stark contrast with the more renowned arithmetical and projective hierarchies, where separation happens “everywhere”. However, by no means is this a reason to dismiss (our definition of) the hierarchy, or discourage the study thereof. One need not look far to find a well-studied hierarchy of the ... | CBA | CAB | ACB | BCA | Selection 2 |
**A**:
We remark that there have been a number of generalizations of Vinogradov’s three primes theorem to various subsets of primes**B**: We also mention recent work of Green [5] establishing a variant of Waring’s problem using only k𝑘kitalic_k-th powers of integers with restricted digits.**C**: To highlight just a s... | ACB | CBA | BAC | BCA | Selection 1 |
**A**: And the results in Section 4.4 and Section 4.5 apply**B**:
By Corollary 5.10, the assumption of Proposition 4.5 is satisfied**C**: For simplicity, we will use Corollary 5.10 tacitly in the remainder of this subsection to invoke the results in Section 4.4 and Section 4.5. | ACB | BAC | ACB | ABC | Selection 2 |
**A**:
An alternative but equivalent path to study these properties is given by Conformal Renormalization**B**: We shortly review this connection below.**C**: This scheme is based on the idea that CG — a four-derivative gravity theory which is invariant under Weyl rescalings of the metric — is free of IR divergences f... | CBA | ACB | BAC | BCA | Selection 2 |
**A**: However, this is not true for infinite state spaces. Hence we need some conditions that guarantee geometric ergodicity for which we refer the reader to [13, Theorem 9].
Under geometric ergodicity, the mixing time is upper-bounded by**B**: The key difference is that in geometric ergodicity, the constant N𝑁Nitali... | CBA | ACB | CAB | ABC | Selection 3 |
**A**: To counter these characteristics, our methods take this notion of geometric latent spaces further: we enforce our own manifold geometries in the latent space with properties we choose. In particular, we develop steady-state flows of manifolds that are nondegenerate, of sufficient measure, and canonical. Our meth... | ABC | CAB | ACB | BCA | Selection 2 |
**A**: Much of existing multi-agent BO literature studies batch BO, in which a central coordinator has access to each agent’s acquired information [21], [22]**B**: It then computes the sampling decisions for all agents, and communicates these decisions to each agent. These decisions are disseminated in batches, allowin... | BCA | CAB | ABC | CBA | Selection 1 |
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