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Abstract: In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum forces solutions to have higher integrability as well. In the present paper, we consider -obstacle problems of the nonhomogeneous anisotropic elliptic equations under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe. [Copyright &y& Elsevier]

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph with R red, G green, B blue edges, the number of rainbow triangles is at most 2 R G B , which is sharp. Second, we give a generalization of the Kruskal–Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality. [ABSTRACT FROM AUTHOR]

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset S of Z. Bhargava's factorials k ! S are invariants, constructed using the notion of p -orderings of S where p is a prime. This paper defines b -orderings of any nonempty subset S of Z for all integers b ≥ 2 , as well as "extreme" cases b = 1 and b = 0. It defines generalized factorials k ! S , T and generalized binomial coefficients ( k + ℓ k ) S , T as nonnegative integers, for all nonempty S and allowing only b in T ⊆ N. It computes b -ordering invariants when S is Z and when S is the set of all primes. [ABSTRACT FROM AUTHOR]

We provide generalizations of a class of stochastic Gronwall inequalities that has been studied by von Renesse and Scheutzow (2010), Scheutzow (2013), Xie and Zhang (2020) and Mehri and Scheutzow (2021). This class of stochastic Gronwall inequalities is a useful tool for SDEs. Our focus are convex generalizations of the Bihari-LaSalle type. The constants we obtain are sharp. In particular, we provide new sharp constants for the stochastic Gronwall inequalities. The proofs are connected to a domination inequality by Lenglart (1977), an inequality by Pratelli (1976) and a characterization of Lenglart's concept of domination via the Snell envelope. The inequalities we study appear for example in connection with exponential moments of solutions to path-dependent SDEs: For non-path-dependent SDEs, criteria for the finiteness of exponential moments are known. To be able to extend these proofs to the path-dependent case, a convex generalization of a stochastic Gronwall inequality seems necessary. Using the results of this paper, we obtain a criterion for the finiteness of exponential moments which is similar to that known for non-path-dependent SDEs. Stochastic Gronwall inequalities can also be applied to study other types of SDEs than path-dependent SDEs: An estimate of this paper is applied by Agresti and Veraar (2023) to prove global well-posedness for reaction-diffusion systems with transport noise. [ABSTRACT FROM AUTHOR]

Multiparametric quantum semigroups M q ˆ , p ˆ (n) are generalization of the one-parameter general linear semigroups M q (n) , where q ˆ = (q i j) and p ˆ = (p i j) are 2 n 2 parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain several classical identities such as Muir's identities, Newton's identities, Capelli-type identities, Cauchy-Binet's identity both for determinant and permanent as well as a rigorous proof of the MacMahon master equation for the quantum algebra of multiparametric Manin matrices. Some of the generalized identities are also lifted to multiparameter q -Yangians. [ABSTRACT FROM AUTHOR]

The study of graph discrepancy problems, initiated by Erdős in the 1960s, has received renewed attention in recent years. In general, given a 2-edge-coloured graph G , one is interested in embedding a copy of a graph H in G with large discrepancy (i.e. the copy of H contains significantly more than half of its edges in one colour). Motivated by this line of research, Gishboliner, Krivelevich and Michaeli considered an oriented version of graph discrepancy for Hamilton cycles. In particular, they conjectured the following generalisation of Dirac's theorem: if G is an oriented graph on n ≥ 3 vertices with δ (G) ≥ n / 2 , then G contains a Hamilton cycle with at least δ (G) edges pointing forwards. In this paper, we present a full resolution to this conjecture. [ABSTRACT FROM AUTHOR]

We present a formula for the class number of a multinorm one torus T L / k associated to any étale algebra L over a global field k. This is deduced from a formula for analogues of invariants introduced by T. Ono, which are interpreted as a generalization of Gauss genus theory. This paper includes the variants of Ono's invariant for arbitrary S -ideal class numbers and the narrow version, generalizing results of Katayama, Morishita, Ono and Sasaki. [ABSTRACT FROM AUTHOR]

We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be transferred between different graphs. It is hence important to study the GCNN transferability : the capacity of the network to have approximately the same repercussion on different graphs that represent the same phenomenon. Transferability ensures that GCNNs trained on certain graphs generalize if the graphs in the test set represent the same phenomena as the graphs in the training set. In this paper, we consider a model of transferability based on graphon analysis. Graphons are limit objects of graphs, and, in the graph paradigm, two graphs represent the same phenomenon if both approximate the same graphon. Our main contributions can be summarized as follows: 1) we prove that any fixed GCNN with continuous filters is transferable under graphs that approximate the same graphon, 2) we prove transferability for graphs that approximate unbounded graphon shift operators, which are defined in this paper, and 3) we obtain non-asymptotic approximation results, proving linear stability of GCNNs. This extends current state-of-the-art results which show asymptotic transferability for polynomial filters under graphs that approximate bounded graphons. • We study the generalization error of graph convolutional neural networks. • Graphs representing the same phenomenon are modeled via graphon analysis. • We show that networks can be transferred between graphs sampling the same phenomenon. • Our analysis allows working with generic continuous filters. • By introducing unbounded graphons, Euclidean CNNs are a special case of our analysis. [ABSTRACT FROM AUTHOR]

Hopf monads generalise Hopf algebras. They clarify several aspects of the theory of Hopf algebras and capture several related structures such as weak Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf algebras. In this paper we introduce a generalisation of Hopf monads, that we call slack Hopf monads. This generalisation retains a clean theory and is flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf monad is a colax magma monad T on a magma category C such that the forgetful functor U T : C T → C 'slackly' preserves internal Homs. We give a number of different descriptions of slack Hopf monads, and study special cases such as slack Hopf monads on cartesian categories and k -linear exact slack Hopf monads on Vect k , that is comagma algebras such that a modified fusion operator is invertible. In particular, we characterise quasi-Hopf algebras in terms of slackness. [ABSTRACT FROM AUTHOR]

Dlab–Ringel's standardization method gives a realization of a standardly stratified algebra. In this paper, we construct mixed stratified algebras, which are a generalization of standardly stratified algebras, following Dlab–Ringel's standardization method. Moreover, we study a Ringel duality of mixed stratified algebras from the viewpoint of stratifying systems. [ABSTRACT FROM AUTHOR]

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree δ ≥ ln 25 ⁡ Δ , the following holds: for every assignment L of lists of colours to the edges of G , such that | L (e) | ≥ (1 + o (1)) ⋅ max ⁡ { deg ⁡ (u) , deg ⁡ (v) } for each edge e = u v , there is an L -edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k -uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results. [ABSTRACT FROM AUTHOR]

The Higher-Order Ψ-calculus framework (HOΨ) by Parrow et al. is a generalisation of many first- and higher-order extensions of the π -calculus. In this paper we present a generic type system for HOΨ-calculi. It satisfies a subject reduction property and can be instantiated to yield both existing and new type systems for calculi, that can be expressed as HOΨ-calculi. In this paper, we consider the type system for termination in HO π by Demangeon et al. Moreover, we derive a new type system for the ρ -calculus of Meredith and Radestock and present a type system for non-interference for mobile code. [ABSTRACT FROM AUTHOR]

In this paper, we extend constructions and results for the Taylor complex to the generalized Taylor complex constructed by Herzog. We construct an explicit DG-algebra structure on the generalized Taylor complex and extend a result of Katthän on quotients of the Taylor complex by DG-ideals. We introduce a generalization of the Scarf complex for families of monomial ideals, and show that this complex is always a subcomplex of the minimal free resolution of the sum of these ideals. We also give an example of an ideal where the generalized Scarf complex strictly contains the standard Scarf complex. Moreover, we introduce the notion of quasitransverse monomial ideals, and prove a list of results relating to Golodness, Koszul homology, and other homological properties for such ideals. [ABSTRACT FROM AUTHOR]

• The "bilateral reduction" effect of Rényi entropy is theoretically studied and proved. • Relative entropy is extended by adding order α to generalized relative entropy. • Generalized relative entropy satisfies six sparse criteria. • Conversion between complexity measures and sparsity measures is realized. • New sparse measure cluster is proposed to enrich machine condition monitoring. Complexity measures and sparsity measures are capable of detecting and characterizing underlying dynamic changes in machine condition monitoring. Previous studies have developed Shannon entropy and Kullback-Leibler (KL) divergence and their generalization, namely Rényi entropy and Rényi divergence, but their theoretical properties as a function of sparsity and their transitions from complexity measures to sparsity measures are not fully explored and discussed. This paper continues conducting theoretical and experimental investigations on Rényi entropy and Rényi divergence and exploring their theoretical properties and transitions to enrich the domain of complexity and sparsity measures and their applications to machine condition monitoring. Specifically, this paper theoretically proves and verifies that Rényi entropy has a similar "bilateral reduction" effect exhibited by other complexity measures including correlation dimension, approximate entropy, fuzzy entropy and sample entropy. Inspired by relative entropy, which is a special case of KL divergence and a sparsity measure, and the relationship between KL divergence and Rényi divergence, in this paper, we attempt to generalize relative entropy by adding an order of α , to convert a series of complexity measures from Rényi entropy to a series of sparsity measures, which answers how to realize transitions from complexity measures to sparsity measures. More importantly, this paper is helpful to provide a guideline for correct uses of Rényi entropy, Rényi divergence, and generalized relative entropy for the quantification of impulsive transients caused by machine faults for machine condition monitoring. [ABSTRACT FROM AUTHOR]

Super-resolution image reconstruction techniques have advanced quickly, leading to the generation of a sizable number of super-resolution images using different super-resolution techniques. Nevertheless, accurately assessing the quality of super-resolution images remains a formidable challenge. This paper introduces a novel Multi-Frequency Cascade Transformers (MFCT) for evaluating super-resolution image quality (SR-IQA). In the first step, we develop a unique Frequency-Divided Module (FDM) to transform the super-resolution images into three different frequency bands. Subsequently, the Cascade Transformer Blocks (CAF) incorporating hierarchical self-attention mechanisms are employed to capture cross-window features for quality perception. Ultimately, the image quality scores from different frequency bands are fused to derive the overall image quality score. The experimental results show that, on the chosen SR-IQA databases, the proposed MFCT-based SR-IQA method can consistently outperforms all the compared Image Quality Assessment (IQA) models. Furthermore, a collection of thorough ablation studies demonstrates that, when compared to other earlier rivals, the newly proposed approach exhibits impressive generalization ability. The code will be available at https://github.com/kbzhang0505/MFCT. [Display omitted] This paper introduces a novel approach for evaluating super-resolution image quality. In the first step, we develop a unique Frequency-divided module (FDM) to transform the super-resolution images into three different frequency bands. Subsequently, the cascade transformer blocks (CAF) incorporating hierarchical self-attention mechanisms are employed to capture cross-window features for quality perception. Ultimately, the image quality scores obtained from different frequency bands are fused to derive the overall image quality score. • Frequency-divided module extracts perceptual features to quantify the quality of SR images. • Cascade transformer blocks captures cross-window features for quality perception. • Different frequency bands jointly estimate the overall quality score of an SR image. [ABSTRACT FROM AUTHOR]

Despite the considerable advances in deep learning for object recognition, there are still several factors that hinder the performance of deep learning models. One of these factors is domain shift, which occurs due to variations in the distribution of the testing and training data. This paper addresses the issue of compact feature clustering in domain generalization, with the aim of optimizing the embedding space from multi-domain data. Specifically, we propose a domain-aware triplet loss for domain generalization, which not only facilitates clustering of similar semantic features but also disperses features that arise from the domain. Unlike previous methods that focus on aligning distributions, our algorithm disperses domain information in the embedding space. Our approach is based on the assumption that embedding features can be clustered based on domain information, which is supported mathematically and empirically in this paper. Furthermore, in our investigation of feature clustering in domain generalization, we observe that the factors that influence the convergence of metric learning loss in domain generalization are more significant than the pre-defined domains. To address this issue, we utilize two methods to normalize the embedding space and reduce the internal covariate shift of the embedding features. Our ablation study illustrates the effectiveness of our algorithm. Additionally, our experiments on benchmark datasets, including PACS, VLCS, and Office-Home, demonstrate that our method outperforms related approaches that focus on domain discrepancy. Notably, our results on RegnetY-16GF are substantially better than state-of-the-art methods on the benchmark datasets. Our code is available at https://github.com/workerbcd/DCT. • We find that the feature embeddings may be clustered due to the domain gaps. • We visualize this domain clustering phenomenon and support it theoretically. • We propose a domain-aware triplet (D-Triplet) loss to generalize the model • D-Triplet loss can efficiently mitigate the domain-cluster issue on feature level. • We utilize various normalization techniques to prompt the convergence of D-Triplet. • The experimental results validate the effectiveness of our method. [Display omitted] [ABSTRACT FROM AUTHOR]

The Landau–Lifshitz–Baryakhtar (LLBar) equation is a generalisation of the Landau–Lifshitz–Gilbert and the Landau–Lifshitz–Bloch equations which takes into account contributions from nonlocal damping and is valid at moderate temperature below the Curie temperature. Therefore, it is used to explain some discrepancies between the experimental observations and the known theories in various problems on magnonics and magnetic domain-wall dynamics. In this paper, the existence and uniqueness of global weak, strong, and regular solutions to LLBar equation are proven. Hölder continuity of the solution is also discussed. [ABSTRACT FROM AUTHOR]

This paper provides algebraic and analytic, as well as numerical arguments why and how double preconditioning of the Gabor frame operator yields an efficient method to compute approximate dual (respectively tight) Gabor atoms for a given time-frequency lattice. We extend the definition of the approach to the continuous setting, making use of the so-called Banach Gelfand Triple, based on the Segal algebra (S 0 (R d) , ‖ ⋅ ‖ S 0 ) and show the continuous dependency of the double preconditioning operators on their parameters. The generalization allows to investigate the influence of the order of the two main single preconditioners (diagonal and convolutional). In the applied section we demonstrate the quality of double preconditioning over all possible lattices and adapt the method to approximate the canonical tight Gabor window, which yields a significant generalization of the FAB-method used in OFDM-applications. Finally, we demonstrate that our approach provides a way to efficiently compute approximate dual families for Gabor families which arise from a slowly varying pattern instead of a regular lattice. [ABSTRACT FROM AUTHOR]

In this paper, we give a Moser-type theorem for C ℓ -smooth hyperbolic-type degenerate Hamiltonian system with the following Hamiltonian H = 〈 ω , y 〉 + 1 2 v 2 − u 2 d + P (x , y , u , v) , (x , y , u , v) ∈ T n × R n × R 2 , which is associated with the standard symplectic structure, with d ≥ 1. Due to the difficulty coming from the degeneracy, our result is quite different from L. Chierchia and D. Qian's work [8] (non-degenerate case). An interesting phenomenon shown in degenerate case is the ℓ -regularity of above Hamiltonian system not only relies on the tori's dimension n but also strongly relies on the degenerate index d. Under arbitrary small perturbation P , we prove that if ℓ ≥ (5 d + 2) (8 τ + 3) , where τ > n − 1 , the above hyperbolic-type degenerate Hamiltonian system admits lower dimensional Diophantine tori which are proved to be of class C β for any β ≤ 8 τ + 2. Our result can be seen a generalization of paper [42] from analytic case to C ℓ -smooth case and can also be seen a generalization of paper [8] from non-degenerate case to degenerate case. [ABSTRACT FROM AUTHOR]

A generalization of the well–known Fibonacci sequence is the ℓ -Fibonacci sequence (F m (ℓ)) m whose first ℓ terms are 0 , ... , 0 , 1 and each term afterwards is the sum of the preceding ℓ terms. The k -Pell sequence (P n (k)) n , which is a generalization of the classical Pell sequence, can be defined similarly. In this paper, we find all coincidences between these two families of sequences. That is, we find all the solutions of the Diophantine equation P n (k) = F m (ℓ) in positive integers n , k , m , ℓ with k , ℓ ≥ 2. This paper continues and extends prior results which dealt with the above problem for some particular cases of k and ℓ. In particular, it extends the previous work [2] that found all Fibonacci numbers in the Pell sequence. [ABSTRACT FROM AUTHOR]

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of fractal codimension of a nilpotent contact point p , for λ = λ 0 , in smooth planar slow–fast systems X ϵ , λ when the contact order n λ 0 (p) of p is even, the singularity order s λ 0 (p) of p is odd and p has finite slow divergence, i.e., s λ 0 (p) ≤ 2 (n λ 0 (p) − 1). The fractal codimension of p is a generalization of the "traditional" codimension of a slow-fast Hopf point of Liénard type, introduced in (Dumortier and Roussarie (2009) [7]), and it is intrinsically defined, i.e., it can be directly computed without the need to first bring the system into its normal form. The intrinsic nature of the notion of fractal codimension stems from the Minkowski dimension of fractal sequences of points, defined near p using the so-called entry-exit relation, and slow divergence integral. We apply our method to a slow-fast Hopf point and read its degeneracy (i.e., the first nonzero Lyapunov quantity) as well as the number of limit cycles near such a Hopf point directly from its fractal codimension. We demonstrate our results numerically on some interesting examples by using a simple formula for computation of the fractal codimension. [ABSTRACT FROM AUTHOR]

We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper [6] , the first author proved that every tree has this property; and in another earlier paper with Maria Chudnovsky [2] , we proved that every cycle has this property. Here we give a common generalization. Say a "banana" is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentioned. [ABSTRACT FROM AUTHOR]

We study the nonlinear generalization of the classical multiview variety, which is a fundamental concept in computer vision. In this paper, we take the first comprehensive step to develop the nonlinear analogue of multiview varieties. To this end, we introduce a multigraded version of the saturated special fiber ring. By applying this tool, we are able to compute the multidegrees of several families of nonlinear multiview varieties. [ABSTRACT FROM AUTHOR]

The theory of morphisms being determined by objects was originally investigated by Auslander, and can be seen as the culmination part of Auslander-Reiten theory. This theory provides a more general frame for working with the Auslander-Reiten theory. In this paper, we will study the behavior of morphisms determined by objects under G -coverings in the sense of Asashiba, which is a generalization of Gabriel's Galois coverings. As an application, we reformulate Bautista and Liu's framework that normal G -coverings preserve sink maps and source maps. We also show that there is a G -covering between two relative stable categories, which unifies Asashiba, Hafezi, Vahed and Mahdavi's work on the stable categories. This is applied to the discussion on the existence of Serre functors by G -coverings of triangulated categories. [ABSTRACT FROM AUTHOR]

Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and gives a representation theoretic interpretation for Kohnert polynomials. [ABSTRACT FROM AUTHOR]

Let G be a K p -free graph. We say e is a K p -saturating edge of G if e ∉ E (G) and G + e contains a copy of K p. Denote by f p (n , m) the minimum number of K p -saturating edges that an n -vertex K p -free graph with m edges can have. Erdős and Tuza conjectured that f 4 (n , ⌊ n 2 / 4 ⌋ + 1) = (1 + o (1)) n 2 16. Balogh and Liu disproved this by showing f 4 (n , ⌊ n 2 / 4 ⌋ + 1) = (1 + o (1)) 2 n 2 33. They believed that a natural generalization of their construction for K p -free graph should also be optimal and made a conjecture that f p + 1 (n , ex (n , K p) + 1) = (2 (p − 2) 2 p (4 p 2 − 11 p + 8) + o (1)) n 2 for all integers p ≥ 3. The main result of this paper is to confirm the above conjecture of Balogh and Liu. [ABSTRACT FROM AUTHOR]

Lovász and Cherkassky discovered in the 1970s independently that if G is a finite graph with a given set T of terminal vertices such that G is inner Eulerian with respect to T , then the maximal number of edge-disjoint paths connecting distinct vertices in T is ∑ t ∈ T λ (t , T − t) where λ is the local edge-connectivity function. The optimality of a system of edge-disjoint T -paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Conjecture) together with the characterization of infinite Eulerian graphs due to Nash-Williams makes it possible to generalise the theorem for infinite graphs in a structural way. The aim of this paper is to formulate this generalisation and prove it for countable graphs. [ABSTRACT FROM AUTHOR]

There are a lot of effective, ineffective and explicit results concerning power values and polynomial values of binomial coefficients. Also, many papers deal with generalizations of these problems, involving polygonal numbers and pyramidal numbers. In this paper we prove effective and ineffective theorems concerning polynomial values of figurate numbers. Our results yield common extensions and generalizations of several previous theorems from the literature. [ABSTRACT FROM AUTHOR]

A well known generalisation of Dirac's theorem states that if a graph G on n ≥ 4 k vertices has minimum degree at least n / 2 then G contains a 2-factor consisting of exactly k cycles. This is easily seen to be tight in terms of the bound on the minimum degree. However, if one assumes in addition that G is Hamiltonian it has been conjectured that the bound on the minimum degree may be relaxed. This was indeed shown to be true by Sárközy. In subsequent papers, the minimum degree bound has been improved, most recently to (2 / 5 + ε) n by DeBiasio, Ferrara, and Morris. On the other hand no lower bounds close to this are known, and all papers on this topic ask whether the minimum degree needs to be linear. We answer this question, by showing that the required minimum degree for large Hamiltonian graphs to have a 2-factor consisting of a fixed number of cycles is sublinear in n. [ABSTRACT FROM AUTHOR]

We generalize the definition of overconvergent p -adic multiple polylogarithms and of p -adic cyclotomic multiple zeta values and we prove a bound on their norm. A byproduct of the proof is a characterization of these objects in terms of certain regularized p -adic iterated integrals. The generalization of the definition consists in replacing the underlying Frobenius structure by its iterations. The bound on the norms of overconvergent p -adic multiple polylogarithms that we obtain is a prerequisite for our subsequent papers on p -adic cyclotomic multiple zeta values. For a video summary of this paper, please visit https://youtu.be/Uxk_qbJxQo. [ABSTRACT FROM AUTHOR]

The h -restricted edge connectivity λ h is an important parameter to measure the reliability of massive parallel computing system. The λ h of G is the minimum cardinality of a set of edges in G , if any, whose deletion disconnects G , and every remaining component has minimum degree at least h. The arrangement graph is a generalization of the star graph, but its order is more flexible than that of the star graph. This paper investigates the h -restricted edge connectivity of the (n , k) -arrangement graph A n , k , and determines that λ 1 (A n , k) = 2 k (n − k) − 2 , λ 2 (A n , k) = 3 k (n − k) − 6 with k ≤ n − 2. • We study a generalized edge reliability of arrangement graph. • We investigate the h -restricted edge connectivity of the (n , k) -arrangement graph A n , k. • We determine that λ 1 (A n , k) = 2 k (n − k) − 2 , and λ 2 (A n , k) = 3 k (n − k) − 6 with k ≤ 2. [ABSTRACT FROM AUTHOR]

The aim of the paper is to develop the most general theory of one-sided fractions. The concepts of localizable set, localization of a ring and a module at a localizable set are introduced and studied. Localizable sets are generalization of Ore sets and denominator sets, and the localization of a ring/module at a localizable set is a generalization of localization of a ring/module at a denominator set. For a semiprime left Goldie ring, it is proven that the set of maximal left localizable sets that contain all regular elements is equal to the set of maximal left denominator sets (and they are explicitly described). For a semiprime Goldie ring, it is proven that the following five sets coincide: the maximal Ore sets, the maximal denominator sets, the maximal left or right or two-sided localizable sets that contain all regular elements (and they are explicitly described). [ABSTRACT FROM AUTHOR]

In this paper we study probabilistic Owen-Shapley spatial power indices, which are generalizations of the Owen-Shapley spatial power index (1977). We provide an explicit formula for calculating these spatial indices for unanimity games and give an axiomatic characterization of the family of probabilistic Owen-Shapley spatial power indices. We employ an equal power change property, a spatial dummy property, anonymity, a positional invariance property, and a positional continuity property. Some examples are also given. [ABSTRACT FROM AUTHOR]

We study the reachability problems of a quantum finite automaton. More precisely, we introduce quantum temporal logic (QTL) that specifies the time-dependent behavior of quantum finite automaton by presenting the time dependence of events temporal operators ◊ (eventually) and □ (always) and employing the projections on subspaces as atomic propositions. The satisfiability of QTL formulae corresponds to various reachability problems of matrix semigroups. We prove that the satisfiability problems for □ ∨ i m p i , ◊ □ ∨ i m p i and □ ◊ ∨ i m p i with atomic propositions p i are decidable. This result solves the open problem of Li and Ying 2014. Notably, the decidability of □ ◊ p can be interpreted as a generalization of Skolem-Mahler-Lech's celebrated theorem based on additive number theory. This paper's last part shows how the quantum finite automaton can model the general concurrent quantum programs, which may involve an arbitrary classical control flow. [ABSTRACT FROM AUTHOR]

We study the constitutive set K arising from a 2 × 2 system of conservation laws in one space dimension, endowed with one entropy and entropy-flux pair. The convexity properties of the set K relate to the well-posedness of the underlying system and the ability to construct solutions via convex integration. Relating to the convexity of K , in the particular case of the p -system, Lorent and Peng (2020) [21] show that K does not contain T 4 configurations. Recently, Johansson and Tione (2024) [14] showed that K does not contain T 5 configurations. In this paper, we provide a substantial generalization of Lorent-Peng, based on a careful analysis of the shock curves for a large class of 2 × 2 systems. We provide several sets of hypotheses on general systems which can be used to rule out the existence of T 4 configurations in the constitutive set K. In particular, our results show the nonexistence of T 4 configurations for every well-known 2 × 2 hyperbolic system of conservation laws for which both families of shocks verify the Liu entropy condition. [ABSTRACT FROM AUTHOR]

In this paper, we tackle the problem of detailed hand-object 3D reconstruction from monocular video with unknown objects , for applications where the required accuracy and level of detail is important, e.g. object hand-over in human–robot collaboration, or manipulation and contact point analysis. While the recent literature on this topic is promising, the accuracy and generalization abilities of existing methods are still lacking. This is due to several limitations, such as the assumption of known object class or model for a small number of instances, or over-reliance on off-the-shelf keypoint and structure-from-motion methods for object-relative viewpoint estimation, prone to complete failure with previously unobserved, poorly textured objects or hand-object occlusions. To address previous method shortcomings, we present a 2-stage pipeline superseding state-of-the-art (SotA) performance on several metrics. First, we robustly retrieve viewpoints relying on a learned pairwise camera pose estimator trainable with a low data regime, followed by a globalized Shonan pose averaging. Second, we simultaneously estimate detailed 3D hand-object shapes and refine camera poses using a differential renderer-based optimizer. To better assess the out-of-distribution abilities of existing methods, and to showcase our methodological contributions, we introduce the new SHOWMe benchmark dataset with 96 sequences annotated with poses, millimetric textured 3D shape scans, and parametric hand models, introducing new object and hand diversity. Remarkably, we show that our method is able to reconstruct 100% of these sequences as opposed to SotA Structure-from-Motion (SfM) or hand-keypoint-based pipelines, and obtains reconstructions of equivalent or better precision when existing methods do succeed in providing a result. We hope these contributions lead to further research under harder input assumptions. The dataset can be downloaded at https://download.europe.naverlabs.com/showme. • Object-agnostic hand-object 3D reconstruction from monocular hand-object motion video • Robust rigid-transformation estimation network that leverages large pre-trained model • Two-stage pipeline for 3D hand-object reconsruction • New hand-object dataset to benchmark hand-object 3D reconstruction • Rigid-transformation estimation performance analysis v/s object sizes and textures [ABSTRACT FROM AUTHOR]

In this paper, we introduce Kasparov's bivariant K-theory that is equivariant under symmetries of a C*-tensor category. It is motivated by some dualities in quantum group equivariant KK-theory, and the classification theory of inclusions of C*-algebras. The fundamental properties of the KK-theory, i.e., the existence of the Kasparov product, Cuntz's picture, universality, and triangulated category structure, hold in this generalization as well. Moreover, we further prove a new property specific to this theory; the invariance of KK-theory under weak Morita equivalence of the tensor categories. As an example, we study the Baum–Connes type property for 3-cocycle twists of discrete groups. [ABSTRACT FROM AUTHOR]

In this paper we generalize the theory of multiplicative G -Higgs bundles over a curve to pairs (G , θ) , where G is a reductive algebraic group and θ is an involution of G. This generalization involves the notion of a multiplicative Higgs bundle taking values in a symmetric variety associated to θ , or in an equivariant embedding of it. We also study how these objects appear as fixed points of involutions of the moduli space of multiplicative G -Higgs bundles, induced by the involution θ. [ABSTRACT FROM AUTHOR]

Arrowed Gelfand–Tsetlin patterns have recently been introduced to study alternating sign matrices. In this paper, we show that a (− 1) -enumeration of arrowed Gelfand–Tsetlin patterns can be expressed by a simple product formula. The numbers are up to 2 n a one-parameter generalization of the numbers 2 n (n − 1) / 2 ∏ j = 0 n − 1 (4 j + 2) ! (n + 2 j + 1) ! that appear in recent work of Di Francesco. A second result concerns the (− 1) -enumeration of arrowed Gelfand–Tsetlin patterns when excluding double-arrows as decoration in which case we also obtain a simple product formula. We are also able to provide signless interpretations of our results. The proofs of the enumeration formulas are based on a recent Littlewood-type identity, which allows us to reduce the problem to the evaluations of two determinants. The evaluations are accomplished by means of the LU-decompositions of the underlying matrices, and an extension of Sister Celine's algorithm as well as creative telescoping to evaluate certain triple sums. In particular, we use implementations of such algorithms by Koutschan, and by Wegschaider and Riese. [ABSTRACT FROM AUTHOR]

A normal element in a finite field extension F q n / F q is characterized by having linearly independent conjugates over F q. We consider the generalization of normal elements known as k -normal elements, where a subset of the conjugates are required to be linearly independent. In this paper, we provide an explicit combinatorial formula for counting the number of k -normal elements in a finite field extension motivated by an open problem proposed by Huczynska, Mullen, Panario, and Thomson in 2013. Furthermore, we use these results to establish new insights about F q -practical numbers. [ABSTRACT FROM AUTHOR]

In this paper we study commutative algebraic monoid structures on affine spaces over an arbitrary field of characteristic zero. We obtain full classification of such structures on A K 2 and A K 3 and describe some generalizations on A K n for any dimension n. [ABSTRACT FROM AUTHOR]

In this paper, we study fusion categories which contain a proper fusion subcategory with maximal rank. They are generalizations of near-group fusion categories. We first prove that they admit spherical structure. We then classify those which are non-degenerate or symmetric. Finally, we classify such fusion categories of rank 4. [ABSTRACT FROM AUTHOR]

Abstract In a recent paper Baker and Bowler introduced matroids over hyperfields, offering a common generalization of matroids, oriented matroids, and linear subspaces of based vector spaces. This paper introduces the notion of a topological hyperfield and explores the generalization of Grassmannians and realization spaces to this context, particularly in relating the (hyper)fields R and C to hyperfields arising in matroid theory and in tropical geometry. [ABSTRACT FROM AUTHOR]

Abstract The Riemann problem, and the associated generalized Riemann problem, are increasingly seen as the important building blocks for modern higher order Godunov-type schemes. In the past, building a generalized Riemann problem solver was seen as an intricately mathematical task because the associated Riemann problem is different for each hyperbolic system of interest. This paper changes that situation. The HLLI Riemann solver is a recently-proposed Riemann solver that is universal in that it is applicable to any hyperbolic system, whether in conservation form or with non-conservative products. The HLLI Riemann solver is also complete in the sense that if it is given a complete set of eigenvectors, it represents all waves with minimal dissipation. It is, therefore, very attractive to build a generalized Riemann problem solver version of the HLLI Riemann solver. This is the task that is accomplished in the present paper. We show that at second order, the generalized Riemann problem version of the HLLI Riemann solver is easy to design. Our GRP solver is also complete and universal because it inherits those good properties from original HLLI Riemann solver. We also show how our GRP solver can be adapted to the solution of hyperbolic systems with stiff source terms. Our generalized HLLI Riemann solver is easy to implement and performs robustly and well over a range of test problems. All implementation-related details are presented. Results from several stringent test problems are shown. These test problems are drawn from many different hyperbolic systems, and include hyperbolic systems in conservation form; with non-conservative products; and with stiff source terms. The present generalized Riemann problem solver performs well on all of them. Highlights • A universal HLLI Riemann solver for very general hyperbolic systems. • A spatial-temporally coupled GRP solver containing sufficient physics. • A useful building block to design higher order numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global regularity for a n-dimensional generalized liquid crystal system with fractional diffusion term ζ 2 u for the velocity field, where ζ u ˆ ( ξ ) = | ξ | α g ( | ξ | ) u ˆ ( ξ ) and g 2 ( | ξ | ) ≤ C 0 log ⁡ ( e + | ξ | ) for α ≥ 1 + n 2 . This paper provides a global regularity result for the generalized liquid crystal equations, which extends the result given by Jin, Zhu and Jin to the case β = 0 for n = 2 (Jin et al. (2015) [5] ). [ABSTRACT FROM AUTHOR]

Let f be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity S f (x) : = ∑ n ⩽ x f ([ x n ]) for x → ∞ , where [ t ] is the integral part of the real number t. This generalises some recent results of Bordellès, Dai, Heyman, Pan & Shparlinski and of Zhai (f = φ = the Euler function), and of Zhao & Wu (f = σ = the sum-of-divisors function). [ABSTRACT FROM AUTHOR]

In this paper we investigate the Kummer pairing associated to an arbitrary (one-dimensional) formal group. In particular, we obtain formulae describing the values of the pairing in terms of multidimensional p -adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. The results are a generalization to higher-dimensional local fields of Kolyvagin's explicit reciprocity laws. In particular, they constitute a generalization of the reciprocity laws of Artin-Hasse, Iwasawa and Wiles. [ABSTRACT FROM AUTHOR]

The value semigroup of a k -semiroot C k of a plane branch C allows us to recover part of the value semigroup Γ = 〈 v 0 , ... , v g 〉 of C , that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λ k of C k , that is an analytical invariant, and we show how it determine part of the set of values of differentials Λ of C. As a consequence, in a fixed topological class, we relate the Tjurina number τ of C with the Tjurina number of C k. In particular, we show that τ ≤ μ − 3 n g − 2 4 μ g − 1 where n g = g c d (v 0 , ... , v g − 1) , μ and μ g − 1 denote the Milnor number of C and C g − 1 respectively. If n g = 2 , we have that τ = μ − μ g − 1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister. [ABSTRACT FROM AUTHOR]

In an actual industrial environment, the complex working environment of mechanical equipment may lead to new faults in the target domain, called the open-set domain adaptation problem. Recently, open-set adaptive fault diagnosis has been extensively employed. However, most studies not only require pre-set fixed thresholds to identify unknown class features but also ignore the learning of discriminable features under specific tasks, which affects the diagnostic performance. Hence, this paper proposes a complementary weighted dual adversarial network combined with supervised contrastive learning (CWDAN-SCL) to address the open-set cross-different working fault diagnosis of bearings. Specifically, a novel complementary weighted adversarial learning strategy is designed using supervised classification and uncertainty measurement to effectively control the participation of target domain features in the domain adaptation process and achieve the alignment of shared class fault features between the source and target domains. Moreover, an adaptive unknown fault separation module is designed using an adversarial learning method to construct a hyperplane between shared and unknown class fault features in the target domain to identify unknown class faults accurately. Additionally, a supervised contrastive loss term is designed based on contrastive learning and label knowledge to improve the aggregation of fault features of the same class and enhance the model's generalization ability in target domain diagnosis tasks. Subsequently, the efficacy and advancement of the proposed method are substantiated through experimentation on two datasets. The experimental results illustrate that the average diagnostic performance of the proposed method is 91.73 %. This study contributes a dependable diagnostic approach for ascertaining the health status of rotating machinery equipment. [ABSTRACT FROM AUTHOR]