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We use Regev's double centralizer theorem from [7] to study invariants of matrices with group actions. This theory then has applications to trace identities and Poincaré series. [ABSTRACT FROM AUTHOR]

Abstract: This appendix collects some material on the cohomology of discrete subgroups of Lie groups and the theory of their representation varieties to provide background for the results of Sin and Thompson (2010, 2013) in [22,23]. [Copyright &y& Elsevier]

In 1935, the introduction of the norm of a group by Reinhold Baer is a turning point in group theory. In fact, Baer proved that there is a very strong relationship between the structure of the norm and that of the whole group (see [1] , [2] , [3] , [4] , [5]). Since then, the norm has been playing a very significant roles in many aspects of group theory and its applications: it has been used in [43] to describe the connection between Hopf–Galois structures and skew braces; it has been used in [23] to describe some special types of profinite groups; and it has been fundamental in the theory of subgroup lattices of groups (see [40]). In this paper, we weaken the original definition of norm by taking into account only those subgroups that are "large" in some sense. Depending on the chosen concept of largeness, the resulting norm can have an impact on the structure of the whole group that is even greater than that of Baer's norm. This is exactly what happens with the non-polycyclic norm , and in fact, Theorem 4.17 gives a precise description of generalized soluble groups in which the non-polycyclic norm is non-Dedekind (and can be considered as the main result of the paper). Other times, the resulting norms have their own peculiar behaviour; this is the case if "large" means "infinite", "having infinite rank", "being non-Černikov", or "having cardinality m " for some given uncountable cardinal number m. [ABSTRACT FROM AUTHOR]

This paper completes the investigation of finite ▪-groups of characteristic p in terms of their Baumann components we began in [5] and [6]. In this paper we define for each finite p -group B a non-trivial characteristic subgroup ▪ and for each finite ▪-group G of characteristic p with ▪, subnormal subgroups of G called Baumann blocks of G. We prove that ▪, where ▪ is the normal subgroup generated by the Baumann blocks of G. Moreover, we give the exact structure of the Baumann blocks of G and show that any two distinct Baumann blocks centralize each other. [ABSTRACT FROM AUTHOR]

A group G is called residually finite if for every non-trivial element g ∈ G , there exists a finite quotient Q of G such that the element g is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient Q depending on the complexity of the element g , for example by using the word norm ‖ g ‖ G if the group G is assumed to be finitely generated. The residual finiteness growth RF G : N → N is then defined as the smallest function such that if ‖ g ‖ G ≤ r , there exists a morphism φ : G → Q to a finite group Q with | Q | ≤ RF G (r) and φ (g) ≠ e Q. Although upper bounds have been established for several classes of groups, exact asymptotics for the function RF G are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals log k for some k ∈ N , where the value k is given by an explicit expression. As an application, we show that for every m ≥ 1 and every 1 ≤ k ≤ m , there exists a group G containing a normal abelian subgroup of rank m and with RF G ≈ log k. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let p be a prime. In this paper, we study the structure of finite groups with a large number of p -regular conjugacy classes or, equivalently, a large number of irreducible p -modular representations. We prove sharp lower bounds for this number in terms of p and the p ′ -part of the order of G which ensure that G is p -solvable. A bound for the p -length is obtained which is sharp for odd primes p. We also prove a new best possible criterion for the existence of a normal Sylow p -subgroup in terms of these quantities. [ABSTRACT FROM AUTHOR]

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements invariably generate S n is bounded away from zero by an absolute constant for all n. Subsequently, Eberhard, Ford and Green have shown that the probability that three randomly selected elements invariably generate S n tends to zero as n → ∞. In this paper, we prove an analogous result for the finite classical groups. More precisely, let G r (q) be a finite classical group of rank r over F q. We show that for q large enough, the probability that four randomly selected elements invariably generate G r (q) is bounded away from zero by an absolute constant for all r , and for three elements the probability tends to zero as q → ∞ and r → ∞. We use the fact that most elements in G r (q) are separable and the well-known correspondence between classes of maximal tori containing separable elements in classical groups and conjugacy classes in their Weyl groups. [ABSTRACT FROM AUTHOR]

The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H. The main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic edge groups. [ABSTRACT FROM AUTHOR]

The main goal of the paper is to present a general model theoretic framework to understand a result of Shalev on probabilistically finite nilpotent groups. We prove that a suitable group where the equation [ x 1 , ... , x k ] = 1 holds on a wide set, in a model theoretic sense, is an extension of a nilpotent group of class less than k by a uniformly locally finite group. In particular, this result applies to amenable groups, as well as to suitable model-theoretic families of definable groups such as groups in simple theories and groups with finitely satisfiable generics. [ABSTRACT FROM AUTHOR]

This is the author's second paper treating the double coset problem for classical groups. Let G be an algebraic group over an algebraically closed field K. The double coset problem consists of classifying the pairs H , J of closed connected subgroups of G with finitely many (H , J) -double cosets in G. The critical setup occurs when H is reductive and J is a parabolic subgroup. Assume that G is a classical group, H is simple and J is a maximal parabolic P k , the stabilizer of a totally singular k -space. We show that if there are finitely many (H , P k) -double cosets in G , then the triple (G , H , k) belongs to a finite list of candidates. Most of these candidates have k = 1 or k = 2. The case k = 1 was solved in [23] and here we deal with k = 2. We solve this case by determining all faithful irreducible self-dual H -modules V , such that H has finitely many orbits on totally singular 2-spaces of V. [ABSTRACT FROM AUTHOR]

The normaliser problem takes as input subgroups G and H of the symmetric group S n , and asks one to compute N G (H). The fastest known algorithm for this problem is simply exponential, whilst more efficient algorithms are known for restricted classes of groups. In this paper, we will focus on groups with many orbits. We give a new algorithm for the normaliser problem for these groups that performs many orders of magnitude faster than previous implementations in GAP. We also prove that the normaliser problem for the special case G = S n is at least as hard as computing the group of monomial automorphisms of a linear code over any field of fixed prime order. [ABSTRACT FROM AUTHOR]

Highlights • Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. • M. Herzog et al. in [J. Algebra, 2018] give two new criteria for solvability of finite groups. • They proved that if G is a group of order n and ψ (G) ≥ ψ (C n) / 6.68 , then G is solvable. • They conjectured that: Conjecture. If G is a group of order n and ψ (G) > 211 1617 ψ (C n) , then G is solvable. • As the main result of this paper we prove the validity of this conjecture. Abstract Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. In [M. Herzog, et al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following conjecture: Conjecture. If G is a group of order n and ψ (G) > 211 ψ (C n) / 1617 , where C n is the cyclic group of order n, then G is solvable. In this paper we prove the validity of this conjecture. [ABSTRACT FROM AUTHOR]

• New operators for splitting degenerate subspaces of symmetry group C 6v are proposed. • The operators are applied to the eigenvalue vibration analysis of two structural systems. • The computation of repeating frequencies of C 6v -symmetric systems is greatly simplified. • Obtained results for natural frequencies agree exactly with existing results in the literature. • New insights on the symmetries of vibration modes in degenerate subspaces are revealed. In considerations of the linear vibration of symmetric systems, group theory allows the space of the eigenvalue problem to be decomposed into independent subspaces that are spanned by symmetry-adapted freedoms. These problems usually feature one or more degenerate subspaces (i.e. subspaces that contain repeating solutions). For such subspaces, the associated idempotents, as calculated from the character table of the symmetry group, are not capable of full decomposition of the subspace. In this paper, and based on group theory, simple algebraic operators that fully decompose the two degenerate subspaces of structural problems belonging to the C 6v symmetry group are proposed. The operators are applied to the vibration of a spring-mass system, for which the results for natural frequencies are found to agree exactly with results from the literature. Their application to the vibration of a hexagonal plane grid reveals new insights on the character of the modes of degenerate subspaces. The overall conclusion is that, for problems belonging to the C 6v symmetry group, the proposed operators allow the mixed modes of degenerate subspaces to be separated into two distinct symmetry categories, and are very effective in simplifying the actual computation of the repeating eigenvalues of these subspaces. [ABSTRACT FROM AUTHOR]

The N th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large N th linear complexity. However, in this paper we show that for q -automatic sequences over F q the converse is not true. We prove that any (not ultimately periodic) q -automatic sequence over F q has N th linear complexity of order of magnitude N . For some famous sequences including the Thue–Morse and Rudin–Shapiro sequence we determine the exact values of their N th linear complexities. These are non-trivial examples of predictable sequences with N th linear complexity of largest possible order of magnitude. [ABSTRACT FROM AUTHOR]

Recent works on the vulnerability of automatic speaker verification (ASV) systems confirm that malicious spoofing attacks using synthetic speech can provoke significant increase in false acceptance rate. A reliable detection of synthetic speech is key to develop countermeasure for synthetic speech based spoofing attacks. In this paper, we targeted that by focusing on three major types of artifacts related to magnitude, phase and pitch variation, which are introduced during the generation of synthetic speech. We proposed a new approach to detect synthetic speech using score-level fusion of front-end features namely, constant Q cepstral coefficients (CQCCs), all-pole group delay function (APGDF) and fundamental frequency variation (FFV). CQCC and APGDF were individually used earlier for spoofing detection task and yielded the best performance among magnitude and phase spectrum related features, respectively. The novel FFV feature introduced in this paper to extract pitch variation at frame-level, provides complementary information to CQCC and APGDF. Experimental results show that the proposed approach produces the best stand-alone spoofing detection performance using Gaussian mixture model (GMM) based classifier on ASVspoof 2015 evaluation dataset. An overall equal error rate of 0.05% with a relative performance improvement of 76.19% over the next best-reported results is obtained using the proposed method. In addition to outperforming all existing baseline features for both known and unknown attacks, the proposed feature combination yields superior performance for ASV system (GMM with universal background model/i-vector) integrated with countermeasure framework. Further, the proposed method is found to have relatively better generalization ability when either one or both of copy-synthesized data and limited spoofing data are available a priori in the training pool. [ABSTRACT FROM AUTHOR]

In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket { − , − } and Turaev cobracket δ on the K -span of homotopy classes of free loops on Σ. Applying an expansion θ : K π → K 〈 x 1 , … , x n 〉 yields an algebraic description of the operations { − , − } and δ in terms of non-commutative variables x 1 , … , x n . If Σ is a surface of genus g = 0 the lowest degree parts { − , − } − 1 and δ − 1 are canonically defined (and independent of θ ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31] . It was conjectured by the second and the third authors that one can define an expansion θ such that { − , − } = { − , − } − 1 and δ = δ − 1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24] , Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2] ). [ABSTRACT FROM AUTHOR]

Suppose G is a compact Lie group, H is a closed subgroup of G , and the homogeneous space G / H is connected. The paper investigates the Ricci flow on a manifold M diffeomorphic to [ 0 , 1 ] × G / H . First, we prove a short-time existence and uniqueness theorem for a G -invariant solution g ( t ) satisfying the boundary condition II ( g ( t ) ) = F ( t , g ∂ M ( t ) ) and the initial condition g ( 0 ) = g ˆ . Here, II ( g ( t ) ) is the second fundamental form of ∂ M , g ∂ M is the metric induced on ∂ M by g ( t ) , F is a smooth map and g ˆ is a metric on M . Second, we study Perelman's F -functional on M . Our results show, roughly speaking, that F is non-decreasing on a G -invariant solution to the modified Ricci flow, provided that this solution satisfies boundary conditions inspired by a paper of Gianniotis. [ABSTRACT FROM AUTHOR]

The thin residue is an important concept in the theory of association schemes. Association schemes whose thin residues are thin have been studied in several papers. In this paper we construct a new class of association schemes whose thin residues are thin, and give a characterization of their automorphism groups. In particular, we give a formula for the order of the automorphism group, and a necessary and sufficient condition under which the association scheme is Schurian. Schur rings are used as a tool in our approach. [ABSTRACT FROM AUTHOR]

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6] ). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Z p 5 for all odd primes p and Z p 4 for all odd primes p such that p ≡ 3 mod 4 . Although counterexamples in low dimensional groups over cyclic rings Z n were previously known they were usually for non-prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis. [ABSTRACT FROM AUTHOR]

Let G be a finite solvable group, given through a refined consistent polycyclic presentation, and α an automorphism of G , given through its images of the generators of G. In this paper, we discuss algorithms for computing the order of α as well as the cycle length of a given element of G under α. We give correctness proofs and discuss the theoretical complexity of these algorithms. Along the way, we carry out detailed complexity analyses of several classical algorithms on finite polycyclic groups. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSpace -complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSpace -complete. Additionally, we provide simpler constructions for the uniform word problems of these classes. For the uniform word problem for automaton groups (without rational constraints), we show NL -hardness. Finally, we investigate a question asked by Cain about a better upper bound for the length of a word on which two distinct elements of an automaton semigroup must act differently. A detailed listing of the contributions of this paper can be found at the end of this paper. [ABSTRACT FROM AUTHOR]

Ulf Rehmann and Jun Morita, in their 1989 paper A Matsumoto Type Theorem for Kac–Moody Groups , gave a presentation of K 2 ( A , F ) for any generalised Cartan matrix A and field F . The purpose of this paper is to use this presentation to compute K 2 ( A , F ) more explicitly in the case when A is hyperbolic. In particular, we shall show that these K 2 ( A , F ) can always be expressed as a product of quotients of K 2 ( F ) and K 2 ( 2 , F ) . Along the way, we shall also prove a similar result in the case when A has an odd entry in each column. [ABSTRACT FROM AUTHOR]

This paper mainly studies totally Abelian operators in the context of analytic Toeplitz operators on both the Hardy and Bergman space. When the symbol is a meromorphic function on C , we establish the connection between the totally Abelian property of these operators and geometric properties of their symbol curves. It is found that winding numbers and multiplicities of self-intersection of symbol curves play an important role in this topic. Techniques of group theory, complex analysis, geometry and operator theory are intrinsic in this paper. As a byproduct, under a mild condition we provide an affirmative answer to a question raised in [2] , and also construct some examples to show that the answer is negative if the associated conditions are weakened. [ABSTRACT FROM AUTHOR]

The classification of Curtis–Tits amalgams with connected, triangle free, simply-laced diagram over a field of size at least 4 was completed in [3] . Orientable amalgams are those arising from applying the Curtis–Tits theorem to groups of Kac–Moody type, and indeed, their universal completions are central extensions of those groups of Kac–Moody type. The paper [2] exhibits concrete (matrix) groups as completions for all Curtis–Tits amalgams with diagram A ˜ n − 1 . For non-orientable amalgams these groups are symmetry groups of certain unitary forms over a ring of skew Laurent polynomials. In the present paper we generalize this to all amalgams arising from the classification above and, under some additional conditions, exhibit their universal completions as central extensions of twisted groups of Kac–Moody type. [ABSTRACT FROM AUTHOR]

If one restricts an irreducible representation V λ of GL 2 n to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of λ are even (resp. the conjugate partition λ ′ is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered q , t -generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the q = 0 limit (Hall–Littlewood), and provided direct combinatorial arguments for these identities; this approach led to various generalizations and a finite-dimensional analog of a recent summation identity of Warnaar. In this paper, we reformulate some of these results using p -adic representation theory; this parallels the representation-theoretic interpretation in the Schur case. The nonzero values of the identities are interpreted as certain p -adic measure counts. This approach provides a p -adic interpretation of these identities (and a new identity), as well as independent proofs. As an application, we obtain a new Littlewood summation identity that generalizes a classical result due to Littlewood and Macdonald. Finally, our p -adic method also leads to a generalized integral identity in terms of Littlewood–Richardson coefficients and Hall polynomials. [ABSTRACT FROM AUTHOR]

In general, extended Runge–Kutta–Nyström (ERKN) methods are more effective than traditional Runge–Kutta–Nyström (RKN) methods in dealing with oscillatory Hamiltonian systems. However, the theoretical analysis for ERKN methods, such as the order conditions, the symplectic conditions and the symmetric conditions, becomes much more complicated than that for RKN methods. Therefore, it is a bottleneck to construct high-order ERKN methods efficiently. In this paper, we first establish the ERKN group Ω for ERKN methods and the RKN group G for RKN methods, respectively. We then rigorously show that ERKN methods are a natural extension of RKN methods, that is, there exists an epimorphism η of the ERKN group Ω onto the RKN group G . This epimorphism gives a global insight into the structure of the ERKN group by the analysis of its kernel and the corresponding RKN group G . Meanwhile, we establish a particular mapping φ of G into Ω so that each image element is an ideal representative element of the congruence class in Ω. Furthermore, an elementary theoretical analysis shows that this map φ can preserve many structure-preserving properties, such as the order, the symmetry and the symplecticity. From the epimorphism η together with its section φ , we may gain knowledge about the structure of the ERKN group Ω via the RKN group G . In light of the theoretical analysis of this paper, we obtain high-order structure-preserving ERKN methods in an effective way for solving oscillatory Hamiltonian systems. Numerical experiments are carried out and the results are very promising, which strongly support our theoretical analysis presented in this paper. [ABSTRACT FROM AUTHOR]

In a prequel to this paper [1] a notion of curvature on the integers was introduced, based on a formal patching technique. In this paper, which is essentially independent of its prequel, we introduce another notion of curvature on the integers, based on “algebraization of Frobenius lifts by correspondences.” Our main results are vanishing/non-vanishing theorems for this new type of curvature in the case of “Chern connections” attached to classical groups. [ABSTRACT FROM AUTHOR]

The spread of a group G , written s (G) , is the largest k such that for any nontrivial elements x 1 , ... , x k ∈ G there exists y ∈ G such that G = 〈 x i , y 〉 for all i. Burness, Guralnick and Harper recently classified the finite groups G such that s (G) > 0 , which involved a reduction to almost simple groups. In this paper, we prove an asymptotic result that determines exactly when s (G n) → ∞ for a sequence of almost simple groups (G n). We apply probabilistic and geometric ideas, but the key tool is Shintani descent, a technique from the theory of algebraic groups that provides a bijection, the Shintani map, between conjugacy classes of almost simple groups. We provide a self-contained presentation of a general version of Shintani descent, and we prove that the Shintani map preserves information about maximal overgroups. This is suited to further applications. Indeed, we also use it to study μ (G) , the minimal number of maximal overgroups of an element of G. We show that if G is almost simple, then μ (G) ⩽ 3 when G has an alternating or sporadic socle, but in general, unlike when G is simple, μ (G) can be arbitrarily large. [ABSTRACT FROM AUTHOR]

ı quantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an ı quantum group, and study their properties along the lines of the representation theory of weight modules over a quantum group. In several cases, we classify the finite-dimensional irreducible classical weight modules by a highest weight theory. [ABSTRACT FROM AUTHOR]

Axial algebras are non-associative algebras generated by semisimple idempotents, known as axes, whose definition is motivated by examples of algebras related to group theory and theoretical physics. In this paper, we construct an axial algebra of Monster type over the ring R [ t ]. By specialising the parameter t to a value in R , we construct an infinite family of pairwise non-isomorphic axial algebras of Monster type. This family is the first of its kind and has a number of important and unexpected properties. Its existence has important implications for the general theory of axial algebras. In particular, this result shows that the algebraic variety of 3-generated axial algebras of Monster type is of positive dimension. [ABSTRACT FROM AUTHOR]

Let Γ be a discrete group. A C*-algebra A is an exotic C*-algebra (associated to Γ) if there exist proper surjective C*-quotients C ⁎ ( Γ ) → A → C r ⁎ ( Γ ) which compose to the canonical quotient C ⁎ ( Γ ) → C r ⁎ ( Γ ) . In this paper, we show that a large class of exotic C*-algebras has poor local properties. More precisely, we demonstrate the failure of local reflexivity, exactness, and local lifting property. Additionally, A does not admit an amenable trace and, hence, is not quasidiagonal and does not have the WEP when A is from the class of exotic C*-algebras defined by Brown and Guentner (see [8] ). In order to achieve the main results of this paper, we prove a result which implies the factorization property for the class of discrete groups which are algebraic subgroups of locally compact amenable groups. [ABSTRACT FROM AUTHOR]

Let F be a non-archimedean local field of characteristic zero with residual characteristic p . In this paper, we first present a simple proof and construction of the local Langlands correspondence for simple supercuspidal representations of GL n ( F ) , when p ∤ n . Our proof relies on the existence of the local Langlands correspondence for GL n ( F ) , due to Harris/Taylor and Henniart. We then prove Jacquet's conjecture on the local converse problem for GL n ( F ) in the case of simple supercuspidal representations, following the strategy of Jiang, Nien and Stevens. Recently, Reeder and Yu constructed a family of epipelagic supercuspidal representations for semi-simple p -adic groups. In the last part of our paper, we show that the analogous construction for GL n ( F ) produces only simple supercuspidals. In the process, we conclude that the constructed epipelagic supercuspidals of Reeder and Yu may not necessarily exhaust all epipelagic supercuspidals of a p -adic group. [ABSTRACT FROM AUTHOR]

Let A → B be a homomorphism of commutative rings. The squaring operation is a functor Sq B / A from the derived category D ( B ) of complexes of B -modules into itself. This operation is needed for the definition of rigid complexes (in the sense of Van den Bergh), that in turn leads to a new approach to Grothendieck duality for rings, schemes and even DM stacks. In our paper with J.J. Zhang from 2008 we introduced the squaring operation, and explored some of its properties. Unfortunately some of the proofs in that paper had severe gaps in them. In the present paper we reproduce the construction of the squaring operation. This is done in a more general context than in the first paper: here we consider a homomorphism A → B of commutative DG rings . Our first main result is that the square Sq B / A ( M ) of a DG B-module M is independent of the resolutions used to present it . Our second main result is on the trace functoriality of the squaring operation . We give precise statements and complete correct proofs. In a subsequent paper we will reproduce the remaining parts of the 2008 paper that require fixing. This will allow us to proceed with the other papers, mentioned in the bibliography, on the rigid approach to Grothendieck duality. The proofs of the main results require a substantial amount of foundational work on commutative and noncommutative DG rings, including a study of semi-free DG rings , their lifting properties, and their homotopies. This part of the paper could be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov–Kuznetsov (GZK) equations of form u t + ( F ( u ) ) x x x + ( G ( u ) ) x y y + ( H ( u ) ) x = 0 . As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov–Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we consider coupled wave–wave, Petrovsky–Petrovsky and wave–Petrovsky systems in N -dimensional open bounded domain with complementary frictional damping and infinite memory acting on the first equation. We prove that these systems are well-posed in the sense of semigroups theory and provide a weak stability estimate of solutions, where the decay rate is given in terms of the general growth of the convolution kernel at infinity and the arbitrary regularity of the initial data. We finish our paper by considering the uncoupled wave and Petrovsky equations with complementary frictional damping and infinite memory, and showing a strong stability estimate depending only on the general growth of the convolution kernel at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we pursue the investigations started in [18] where the authors provide a construction of the Dunkl intertwining operator for a large subset of the set of regular multiplicity values. More precisely, we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function. In particular, we obtain a formula for the corresponding Dunkl kernel and another representation of the generalized Bessel function already derived in [7] . When the multiplicity function is everywhere constant, our computations give a solution to the problem of counting the number of factorizations of an element from a dihedral group into a fixed number of (non-necessarily simple) reflections. In the remainder of the paper, we supply another method to derive the Dunkl kernel associated with dihedral systems from the corresponding generalized Bessel function. This time, we use the shift principle together with multiple combinations of Dunkl operators in the directions of the vectors of the canonical basis of R 2 . When the dihedral system is of order six and only in this case, a single combination suffices to get the Dunkl kernel and agrees up to an isomorphism with the formula recently obtained by Amri [2, Lemma 1] in the case of a root system of type A 2 . We finally derive an integral representation for the Dunkl kernel associated with the dihedral system of order eight. [ABSTRACT FROM AUTHOR]

Hooks are prominent in representation theory (of symmetric groups) and they play a role in number theory (via cranks associated to Ramanujan's congruences). A partition of a positive integer n has a Young diagram representation. To each cell in the diagram there is an associated statistic called hook length, and if a number t is absent from the diagram then the partition is called a t -core. A partition is an ( s , t ) -core if it is both an s - and a t -core. Since the work of Anderson on ( s , t ) -cores, the topic has received growing attention. This paper expands the discussion to multiple-cores. More precisely, we explore ( s , s + 1 , … , s + k ) -core partitions much in the spirit of a recent paper by Stanley and Zanello. In fact, our results exploit connections between three combinatorial objects: multi-cores, posets and lattice paths (with a novel generalization of Dyck paths). Additional results and conjectures are scattered throughout the paper. For example, one of these statements implies a curious symmetry for twin-coprime ( s , s + 2 ) -core partitions. [ABSTRACT FROM AUTHOR]

Text Let Z n be the ring of residue classes modulo n and Z n ⁎ be the group of its units. In 1926, Brauer obtained an explicit formula for the number of solutions of the linear congruence x 1 + ⋯ + x k ≡ c ( mod n ) with x 1 , x 2 , … , x k ∈ Z n ⁎ . In 2009, for any c ∈ Z n , Sander gave a formula for the number of representations of c as the sum of two units, the sum of two nonunits, and the sum of two mixed pairs, respectively. In this paper, we extend Sander's results to the quadratic case. We also pose some problems for further research. Video For a video summary of this paper, please visit http://youtu.be/PjC2lhc6Cs0 . [ABSTRACT FROM AUTHOR]

In earlier papers it was shown that the generic tropical variety of an ideal can contain information on algebraic invariants as for example the depth in a direct way. The existence of generic tropical varieties has so far been proved in the constant coefficient case for the usual notion of genericity. In this paper we generalize this existence result to include the case of non-constant coefficients in certain settings. Moreover, we extend the notion of genericity to arbitrary closed subvarieties of the general linear group. In addition to including the concept of genericity on algebraic groups this yields structural results on the tropical variety of an ideal under an arbitrary linear coordinate change. [ABSTRACT FROM AUTHOR]

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order 2 s for every s ⩾ 3 has been wide open. This question is answered in the affirmative in this paper via the construction of a connected tetravalent half-arc-transitive graph with vertex stabilizer D 8 2 × C 2 m for each integer m ⩾ 1 , where D 8 2 is the direct product of two copies of the dihedral group of order 8 and C 2 m is the direct product of m copies of the cyclic group of order 2. The graphs constructed have surprisingly many significant properties in various contexts. [ABSTRACT FROM AUTHOR]

Frobenius groups are an object of fundamental importance in finite group theory. As such, several generalizations of these groups have been considered. Some examples include: A Frobenius–Wielandt group is a triple (G , H , L) where H / L is almost a Frobenius complement for G ; A Camina pair is a pair (G , N) where N is almost a Frobenius kernel for G ; A Camina triple is a triple (G , N , M) where (G , N) and (G , M) are almost Camina pairs. In this paper we study triples (G , N , M) where (G , N) and (G , M) are almost Frobenius groups. [ABSTRACT FROM AUTHOR]

Leibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. In this paper we define subinvariant subalgebras of Leibniz algebras and study their properties. It is shown that the signature results on subinvariance in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

We continue the project started in [8] to describe the structure of the finite groups G of characteristic p in terms of their Baumann components and the conjugacy class Bau p (G). The reduction theorem proved in [8] allows to assume that G has a unique Baumann component. In this paper we use this property to determine the isomorphism type of G / O p (G) and the action of G on Ω 1 (Z (O p (G))). In addition, we prove reduction theorems which allow to focus on groups G which satisfy G / O p (G) ≅ SL n (q) , Sp 2 n (q) or G 2 (q) and O p (G) ⩽ B for B ∈ Bau p (G). [ABSTRACT FROM AUTHOR]

In this paper, an explicit construction of a countable parafree Lie algebra with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over Z is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial H 2. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup, central series, and ultimately nilpotence to a supercharacter theoretic setting, and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories. We then use these concepts to show that the upper and lower annihilator series of J can be described in terms of certain central series for the algebra group G = 1 + J defined by S , when S is the algebra group supercharacter theory defined by Diaconis–Isaacs. [ABSTRACT FROM AUTHOR]

In a previous paper, we reported the pure rotational spectrum of the CH 3 + –He ion observed in a 22-pole ion trap by IR-MW double resonance spectroscopy [Töpfer et al. (2018)]. In that work the K = 1 rotational levels in the ground state of the ion were observed to be split, i.e. the K -type doubling was observed. This finding would be consistent with a slightly asymmetric rotor of C s symmetry, although ab-initio calculations predict that the equilibrium structure of the ion is of C 3 v symmetry. In the present study we inspect the energy-level correlation-diagram between the semi-rigid symmetric-top (C 3 v), the free internal-rotor (C 3 v (M) ), and the semi-rigid asymmetric-top models (C s), by using group theory, especially the correlation and reverse correlation in the symmetry representations. We find that the lowest K = 1 rotational levels of the CH 3 + –He ion can be of A 1 ⊕ A 2 symmetry rather than of E symmetry, and thus they can be split depending on the nature of the C–He bending vibration. [Display omitted] • A splitting has been reported for the K = 1 rotational state of CH 3 + –He in a previous paper. • This splitting is caused by the large amplitude motion of He in this symmetric top. • Group theory and correlation diagrams are used to explain this phenomenon. • We find that the lowest K = 1 level may split into levels of symmetry A 1 and A 2. [ABSTRACT FROM AUTHOR]

The aim of this paper is the geometric study of the symplectic operators which are a perturbation of the identity by a Hilbert–Schmidt operator. This subgroup of the symplectic group was introduced in Pierre de la Harpe's classical book of Banach–Lie groups. Throughout this paper we will endow the tangent spaces with different Riemannian metrics. We will use the minimal curves of the unitary group and the positive invertible operators to compare the length of the geodesic curves in each case. Moreover we will study the completeness of the symplectic group with the geodesic distance. [ABSTRACT FROM AUTHOR]

We associate with every group G a set τ ( G ) of subgroups of G with 1 ∈ τ ( G ) . If H ∈ τ ( G ) , then we say that H is a τ-subgroup of G . If θ ( τ ( G ) ) ⊆ τ ( θ ( G ) ) for each epimorphism θ : G → G ⁎ , then we say that τ is a subgroup functor . We say also that a subgroup functor τ is: hereditary provided H ∈ τ ( E ) whenever H ≤ E ≤ G and H ∈ τ ( G ) ; regular provided for any group G , whenever H ∈ τ ( G ) is a p -group and N is a minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p ; Φ -regular (respectively Φ -quasiregular ) provided for any primitive group G , whenever H ∈ τ ( G ) is a p -group and N is a (respectively abelian) minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p . Let K ≤ H be subgroups of G and τ a subgroup functor. Then we say that: the pair ( K , H ) satisfies the F -supplement condition in G if G has a subgroup T such that H T = G and H ∩ T ⊆ K Z F ( T ) ; H is F τ -supplemented in G if for some τ -subgroup S ¯ of G ¯ contained in H ¯ the pair ( S ¯ , H ¯ ) satisfies the F -supplement condition in G ¯ , where G ¯ = G / H G and H ¯ = H / H G . In this paper we study the structure of a group G under the condition that some primary subgroups of G are F τ -supplemented in G . In particular, we prove the following result. Theorem A. Let F be a saturated formation containing the class U of all supersoluble groups, E a normal subgroup of G with G / E ∈ F , X = E or X = F ⁎ ( E ) , and τ a regular or hereditary Φ -regular subgroup functor. Suppose that every τ-subgroup of G contained in X is subnormally embedded in G. If every maximal subgroup of every non-cyclic Sylow subgroup of X is U τ -supplemented in G, then G ∈ F . Moreover, in the case when τ is regular, then every chief factor of G below E is cyclic. The results in this paper not only cover and unify a long list of some known results but also cause a wide series of new results. [ABSTRACT FROM AUTHOR]

In this paper, we shall introduce h -expansiveness and asymptotical h -expansiveness for actions of sofic groups. By definition, each h -expansive action of a sofic group is asymptotically h -expansive. We show that each expansive action of a sofic group is h -expansive, and, for any given asymptotically h -expansive action of a sofic group, the entropy function (with respect to measures) is upper semi-continuous and hence the system admits a measure with maximal entropy. Observe that asymptotically h -expansive property was first introduced and studied by Misiurewicz for Z -actions using the language of tail entropy. And thus in the remaining part of the paper, we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of Misiurewicz's ideas when the group is amenable. It turns out that these two definitions are equivalent in this setting. [ABSTRACT FROM AUTHOR]