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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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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 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]

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ -summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module. [ABSTRACT FROM AUTHOR]

Generalizing to out-of-distribution (OOD) data is a challenging task for existing deep learning approaches. This problem largely comes from the common but often incorrect assumption of statistical learning algorithms that the source and target data come from the same i.i.d. distribution. To tackle the limited variability of domains available during training, as well as domain shifts at test time, numerous approaches for domain generalization have focused on generating samples from new domains. Recent studies on this topic suggest that feature statistics from instances of different domains can be mixed to simulate synthesized images from a novel domain. While this simple idea achieves state-of-art results on various domain generalization benchmarks, it ignores structural information which is key to transferring knowledge across different domains. In this paper, we leverage the ability of humans to recognize objects using solely their structural information (prominent region contours) to design a Structural-Aware Feature Stylization method for domain generalization. Our method improves feature stylization based on mixing instance statistics by enforcing structural consistency across the different style-augmented samples. This is achieved via a multi-task learning model which classifies original and augmented images while also reconstructing their edges in a secondary task. The edge reconstruction task helps the network preserve image structure during feature stylization, while also acting as a regularizer for the classification task. Through quantitative comparisons, we verify the effectiveness of our method upon existing state-of-the-art methods on PACS, VLCS, OfficeHome, DomainNet and Digits-DG. The implementation is available at this repository. • Feature stylization alone led to losing structural domain-invariant information. • Image contours contain important domain-agnostic structural information. • Enforcing structural consistency while stylization boost model's generalization [ABSTRACT FROM AUTHOR]

In this paper we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of Grothendieck polynomials including canonical stable Grothendieck polynomials due to Yeliussizov, refined Grothendieck polynomials due to Chan and Pflueger, and refined dual Grothendieck polynomials due to Galashin, Liu, and Grinberg. We give Jacobi–Trudi-like formulas, combinatorial models, Schur expansions, Schur positivity, and dualities of these polynomials. [ABSTRACT FROM AUTHOR]

• A consistency association is established among different conditions signals. • Tool structure and process parameters are introduced to the model structure design. • The model's generalization boundary in tool wear monitoring is studied. • The proposed method is generalized and interpretable on extensive experiments. Cross-conditions tool wear monitoring has a wide application prospect in manufacturing. However, the data distribution discrepancies caused by the inconsistency of process elements restrict the generalization of models under cross-conditions or even similar conditions. The existing methods based on diversity enhancement make it difficult to effectively establish the correlation between the source domain and target domains, which limits the improvement of model generalization. Therefore, this paper details the cause of data distribution discrepancies and proposes a discontinuous physical property-constrained single-source domain generalization network for milling tool wear monitoring. Firstly, a spatial attention mechanism is introduced to weight key signal segments adaptively. Secondly, the generation module is constrained by a standard sample and is used to generate diverse samples with physical properties. Thirdly, extensive recognition experiments on an open dataset and machining experiments with three distribution discrepancy levels were conducted to verify the effectiveness of the proposed method. Finally, feature visualizations provide consistency and interpretability. [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]

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]

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 C be a set of m by n matrices over F q such that the rank of A − B is at least d for all distinct A , B ∈ C . Suppose that m ⩽ n . If # C = q n ( m − d + 1 ) , then C is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary 1 < d < m − 1 . One was found by Delsarte (1978) [8] and Gabidulin (1985) [10] independently, and it was later generalized by Kshevetskiy and Gabidulin (2005) [16] . We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016) [22] , and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family. [ABSTRACT FROM AUTHOR]

The punctured binary Reed–Muller code is cyclic and was generalised into the punctured generalised Reed–Muller code over GF ( q ) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed–Muller code and the binary Reed–Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] for all m ≥ 2 . Their extended codes have parameters [ 3 m , 3 m − 1 − 2 m , 5 ] for all m ≥ 2 , and are also optimal. [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 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 give a generalization of the problem that was posed by I. Raşa in [10] and was proved based on a probabilistic approach by Mrowiec, Rajba and Wąsowicz in [8] . We end this paper by proposing two open problems related to Raşa's proposed inequality. [ABSTRACT FROM AUTHOR]

Hexagons with four-lobed regions called shamrocks removed from their center were introduced in their 2013 paper by Ciucu and Krattenthaler, where product formulas for the number of their lozenge tilings were provided. In analogy with the plane partitions which they generalize, we consider the problem of enumerating the lozenge tilings which are invariant under some symmetries of the underlying region. This leads to six symmetry classes besides the base case of requiring no symmetry. In this paper we provide product formulas for two of these symmetry classes (namely, the ones generalizing cyclically symmetric, and cyclically symmetric and transpose complementary plane partitions). [ABSTRACT FROM AUTHOR]

In the present paper we introduce a notion of ( ω , t ) -convexity as a natural generalization of the notion of usual t -convexity, t -strongly convexity, approximate t -convexity, delta t -convexity and many other. The main result of this paper establishes the necessary and sufficient conditions under which an ( ω , t ) -convex map can be supported at a given point by an ( ω , t ) -affine support function. Several applications of this support theorem are presented. For instance, new characterizations of inner product spaces among normed spaces involving the notion of ( ω , t ) -convexity are given. [ABSTRACT FROM AUTHOR]