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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]

Batched linear solvers, which solve many small related but independent problems, are increasingly important for highly parallel processors such as graphics processing units (GPUs). GPUs need a substantial amount of work to keep them operating efficiently and it is not an option to solve smaller problems one-by-one. Because of the small size of each problem, the task of implementing a parallel partitioning scheme and mapping the problem to hardware is not trivial. In recent history, significant attention has been given to batched dense linear algebra. However, there is also an interest in utilizing sparse iterative solvers in a batched form. An example use case is found in a gyrokinetic Particle-In-Cell (PIC) code used for modeling magnetically confined fusion plasma devices. The collision operator has been identified as a bottleneck, and a proxy app has been created for facilitating optimizations and porting to GPUs. The current collision kernel linear solver does not run on the GPU—a major bottleneck. As these matrices are sparse and well-conditioned, batched iterative sparse solvers are an attractive option. A batched sparse iterative solver capability has recently been developed in the Ginkgo library. In this paper, we describe how Ginkgo 's batched solver technology can integrate into the XGC collision kernel and accelerate the simulation process. Comparisons for the solve times on NVIDIA V100 and A100 GPUs and AMD MI100 GPUs with one dual-socket Intel Xeon Skylake CPU node with 40 cores are presented for matrices from the collision kernel of XGC. Further, the speedups observed for the overall collision kernel are presented in comparison to different modern CPUs on multiple supercomputer systems. The results suggest that Ginkgo 's batched sparse iterative solvers are well suited for efficient utilization of the GPU for this problem, and the performance portability of Ginkgo in conjunction with Kokkos (used within XGC as the heterogeneous programming model) allows seamless execution on exascale-oriented heterogeneous architectures. • Fast batched sparse iterative linear solvers for modern graphics processing units. • Implementation of different batched sparse matrix formats. • Automatic tuning of shared memory utilization on the GPU. • Strategy for integration into the plasma simulation code XGC via Kokkos. • Performance results on various CPUs and V100, A100 and MI100 GPUs. [ABSTRACT FROM AUTHOR]

Machine learning and artificial intelligence (AI) applications often rely on performing many small matrix operations—in particular general matrix–matrix multiplication (GEMM). These operations are usually performed in a reduced precision, such as the 16-bit floating-point format (i.e., half precision or FP16). The GEMM operation is also very important for dense linear algebra algorithms, and half-precision GEMM operations can be used in mixed-precision linear solvers. Therefore, high-performance batched GEMM operations in reduced precision are significantly important, not only for deep learning frameworks, but also for scientific applications that rely on batched linear algebra, such as tensor contractions and sparse direct solvers. This paper presents optimized batched GEMM kernels for graphics processing units (GPUs) in FP16 arithmetic. The paper addresses both real and complex half-precision computations on the GPU. The proposed design takes advantage of the Tensor Core technology that was recently introduced in CUDA-enabled GPUs. With eight tuning parameters introduced in the design, the developed kernels have a high degree of flexibility that overcomes the limitations imposed by the hardware and software (in the form of discrete configurations for the Tensor Core APIs). For real FP16 arithmetic, performance speedups are observed against cuBLAS for sizes up to 128, and range between 1. 5 × and 2. 5 ×. For the complex FP16 GEMM kernel, the speedups are between 1. 7 × and 7 × thanks to a design that uses the standard interleaved matrix layout, in contrast with the planar layout required by the vendor's solution. The paper also discusses special optimizations for extremely small matrices, where even higher performance gains are achievable. • The use of FP16 arithmetic is proven to be useful for numerical linear algebra. • The use of Tensor Cores is restricted by some programming model limitations. • The proposed kernel builds a flexible abstraction layer over the tensor cores. • Comprehensive auto-tuning is important for small sizes. • Interleaved matrix layouts outperform planar layouts for half-complex matrices. [ABSTRACT FROM AUTHOR]

In this paper we investigate the image of the center Z of the distribution algebra D i s t (G L (m | n)) of the general linear supergroup over a ground field of positive characteristic under the Harish-Chandra morphism h : Z → D i s t (T) obtained by the restriction of the natural map D i s t (G L (m | n)) → D i s t (T). We define supersymmetric elements in D i s t (T) and show that each image h (c) for c ∈ Z is supersymmetric. The central part of the paper is devoted to a description of a minimal set of generators of the algebra of supersymmetric elements over Frobenius kernels T r. [ABSTRACT FROM AUTHOR]

This paper studies separating invariants of finite groups acting on affine varieties through automorphisms. Several results, proved by Serre, Dufresne, Kac–Watanabe and Gordeev, and Jeffries and Dufresne exist that relate properties of the invariant ring or a separating subalgebra to properties of the group action. All these results are limited to the case of linear actions on vector spaces. The goal of this paper is to lift this restriction by extending these results to the case of (possibly) non-linear actions on affine varieties. Under mild assumptions on the variety and the group action, we prove that polynomial separating algebras can exist only for reflection groups. The benefit of this gain in generality is demonstrated by an application to the semigroup problem in multiplicative invariant theory. Then we show that separating algebras which are complete intersections in a certain codimension can exist only for 2-reflection groups. Finally we prove that a separating set of size n + k − 1 (where n is the dimension of X ) can exist only for k -reflection groups. Several examples show that most of the assumptions on the group action and the variety that we make cannot be dropped. [ABSTRACT FROM AUTHOR]

In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra gl m | n ( C ) . The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac–Moody Lie superalgebra gl m | n ˆ ( C ) on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter μ , and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter μ is nonzero. [ABSTRACT FROM AUTHOR]

In this paper we investigate the Sturmian theory for general (possibly uncontrollable) linear Hamiltonian systems by means of the Lidskii angles, which are associated with a symplectic fundamental matrix of the system. In particular, under the Legendre condition we derive formulas for the multiplicities of the left and right proper focal points of a conjoined basis of the system, as well as the Sturmian separation theorems for two conjoined bases of the system, in terms of the Lidskii angles. The results are new even in the completely controllable case. As the main tool we use the limit theorem for monotone matrix-valued functions by Kratz (1993). The methods allow to present a new proof of the known monotonicity property of the Lidskii angles. The results and methods can also be potentially applied in the singular Sturmian theory on unbounded intervals, in the oscillation theory of linear Hamiltonian systems without the Legendre condition, in the comparative index theory, or in linear algebra in the theory of matrices. [ABSTRACT FROM AUTHOR]

The resultant theory plays a crucial role in computational algebra and algebraic geometry. The theory has two aspects: algebraic and geometric. In this paper, we focus on the algebraic aspect. One of the most important and well known algebraic properties of the resultant is that it is equal to the determinant of the Sylvester matrix. In 2008, Odagiri proved that a similar property holds over the tropical semiring if one replaces subtraction with addition. The tropical semiring belongs to a large family of algebraic structures called commutative idempotent semiring. In this paper, we prove that the same property (with subtraction replaced with addition) holds over an arbitrary commutative idempotent semiring. [ABSTRACT FROM AUTHOR]

Linear algebra operations have been widely used in big data analytics and scientific computations. Many works have been done on optimizing linear algebra operations on GPUs with regular-shaped input. However, few works focus on fully utilizing GPU resources when the input is not regular-shaped. Current optimizations do not consider fully utilizing the memory bandwidth and computing power; therefore, they can only achieve sub-optimal performance. In this paper, we propose two efficient algorithms – TSM2R and TSM2L – for two classes of tall-and-skinny matrix–matrix multiplications on GPUs. Both of them focus on optimizing linear algebra operation with at least one of the input matrices tall-and-skinny. Specifically, TSM2R is designed for a large regular-shaped matrix multiplying a tall-and-skinny matrix, while TSM2L is designed for a tall-and-skinny matrix multiplying a small regular-shaped matrix. We implement our proposed algorithms and test on several modern NVIDIA GPU micro-architectures. Experiments show that, compared to the current state-of-the-art works, (1) TSM2R speeds up the computation by 1.6x on average and improves the memory bandwidth utilization and computing power utilization by 18.1% and 20.5% on average, respectively, when the regular-shaped matrix size is relatively large or medium; and (2) TSM2L speeds up the computation by 1.9x on average and improves the memory bandwidth utilization by up to 9.3% on average when the regular-shaped matrix size is relatively small. • Few works focus on optimizing GEMM on GPUs for the irregular-shaped input. • Current optimizations do not fully utilize the memory bandwidth and computing power. • We propose two efficient algorithms for two classes of tall-and-skinny GEMM on GPUs. • Our optimizations speedup GEMM by 1.1x ∼ 3.5x for various tall-and-skinny inputs. [ABSTRACT FROM AUTHOR]

Let E be the extraspecial p -group of order p 3 and exponent p where p is an odd prime. We determine the mod p cohomology H ⁎ ( X , F p ) of a summand X in the stable splitting of p -completed classifying space BE . In the previous paper (Hida and Yagita (2014) [7] ), we determined this cohomology modulo nilpotence. In this paper, we consider the whole of the cohomology. Moreover, we consider the stable splittings of BG for some finite groups with Sylow p -subgroup E related to the three dimensional linear group L 3 ( p ) . [ABSTRACT FROM AUTHOR]

This paper introduces hybrid LU–QR algorithms for solving dense linear systems of the form A x = b . Throughout a matrix factorization, these algorithms dynamically alternate LU with local pivoting and QR elimination steps based upon some robustness criterion. LU elimination steps can be very efficiently parallelized, and are twice as cheap in terms of floating-point operations, as QR steps. However, LU steps are not necessarily stable, while QR steps are always stable. The hybrid algorithms execute a QR step when a robustness criterion detects some risk for instability, and they execute an LU step otherwise. The choice between LU and QR steps must have a small computational overhead and must provide a satisfactory level of stability with as few QR steps as possible. In this paper, we introduce several robustness criteria and we establish upper bounds on the growth factor of the norm of the updated matrix incurred by each of these criteria. In addition, we describe the implementation of the hybrid algorithms through an extension of the PaRSEC software to allow for dynamic choices during execution. Finally, we analyze both stability and performance results compared to state-of-the-art linear solvers on parallel distributed multicore platforms. A comprehensive set of experiments shows that hybrid LU–QR algorithms provide a continuous range of trade-offs between stability and performances. [ABSTRACT FROM AUTHOR]

The (q , Q) -current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic q -Schur algebras. In this paper, we study the (q , Q) -current algebra U q (sl n 〈 Q 〉 [ x ]) associated with the special linear Lie algebra sl n. In particular, we classify finite dimensional simple U q (sl n 〈 Q 〉 [ x ]) -modules. [ABSTRACT FROM AUTHOR]

Finding connected components is one of the most widely used operations on a graph. Optimal serial algorithms for the problem have been known for half a century, and many competing parallel algorithms have been proposed over the last several decades under various different models of parallel computation. This paper presents a class of parallel connected-component algorithms designed using linear-algebraic primitives. These algorithms are based on a PRAM algorithm by Shiloach and Vishkin and can be designed using standard GraphBLAS operations. We demonstrate two algorithms of this class, one named LACC for Linear Algebraic Connected Components, and the other named FastSV which can be regarded as LACC's simplification. With the support of the highly-scalable Combinatorial BLAS library, LACC and FastSV outperform the previous state-of-the-art algorithm by a factor of up to 12x for small to medium scale graphs. For large graphs with more than 50B edges, LACC and FastSV scale to 4K nodes (262K cores) of a Cray XC40 supercomputer and outperform previous algorithms by a significant margin. This remarkable performance is accomplished by (1) exploiting sparsity that was not present in the original PRAM algorithm formulation, (2) using high-performance primitives of Combinatorial BLAS, and (3) identifying hot spots and optimizing them away by exploiting algorithmic insights. [ABSTRACT FROM AUTHOR]

This paper examines linear independence of shearlet systems. This property has already been studied for wavelets and other systems such as, for instance, for Gabor systems. In fact, for Gabor systems this problem is commonly known as the HRT conjecture. In this paper we present a proof of linear independence of compactly supported separable shearlet systems. For this, we employ a sampling strategy to utilize the structure of an implicitly given underlying oversampled wavelet system as well as the shape of the supports of the shearlet elements. [ABSTRACT FROM AUTHOR]

Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular generator polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of generator polynomial matrices of GQC codes, including a relation formula between generator polynomial matrices and parity-check polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Gröbner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all generator polynomial matrices with given cyclic intervals. [ABSTRACT FROM AUTHOR]

Basic Linear Algebra Subprograms (BLAS) are the building blocks for various numerical algorithms and are widely used in scientific computations. However, some linear algebra applications need more precision than the standard double precision available in most existing BLAS libraries. In this paper, we implement and evaluate multiple-precision scalar and vector BLAS functions on graphics processing units (GPUs). We use the residue number system (RNS) to represent arbitrary length floating-point numbers. The non-positional nature of RNS enables parallelism in multiple-precision arithmetic and makes RNS a good tool for high-performance computing applications. We first present new data-parallel algorithms for multiplying and adding RNS-based floating-point representations. Next, we suggest algorithms for multiple-precision vectors specially designed for parallel computations on GPUs. Using these algorithms, we develop and evaluate four GPU-accelerated multiple-precision BLAS functions, ASUM, DOT, SCAL, and AXPY. It is shown through experiments that in many cases, the implemented functions achieve significantly better performance compared to existing multiple-precision software for CPU and GPU. • Residue number system enables parallelism in arithmetic with multiple precision. • New algorithms for arbitrary length floating-point numbers are presented. • Multiple-precision ASUM, DOT, SCAL, and AXPY are implemented using CUDA and evaluated. • We show that multiple-precision BLAS can be accelerated using GPU. [ABSTRACT FROM AUTHOR]

In this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable somebody to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a negligible overall overhead. We first provide quadratic time and space non-interactive certificates saving the logarithmic factors of previously known ones. Then we propose interactive certificates for the same problems whose Monte Carlo verification complexity requires a small constant number of matrix-vector multiplications, a linear space, and a linear number of extra field operations, with a linear number of interactions. As an application we also give an interactive protocol, certifying the determinant or the signature of dense matrices, faster for the Prover than the best previously known one. Finally we give linear space and constant round certificates for the row or column rank profiles. [ABSTRACT FROM AUTHOR]

Abstract: In the literature there are two ways of endowing an affine ind-variety with a topology. One possibility is due to Shafarevich and the other to Kambayashi. In this paper we specify a large class of affine ind-varieties where these two topologies differ. We give an example of an affine ind-variety that is reducible with respect to Shafarevichʼs topology, but irreducible with respect to Kambayashiʼs topology. Moreover, we give a counter-example of a supposed irreducibility criterion given in Shafarevich (1981) [Sha81] which is different from the counter-example given by Homma in Kambayashi (1996) [Kam96]. We finish the paper with an irreducibility criterion similar to the one given by Shafarevich. [Copyright &y& Elsevier]

Abstract: A module M is called an automorphism-invariant module if every isomorphism between two essential submodules of M extends to an automorphism of M. This paper introduces the notion of dual of such modules. We call a module M to be a dual automorphism-invariant module if whenever and are small submodules of M, then any epimorphism with small kernel lifts to an endomorphism φ of M. In this paper we give various examples of dual automorphism-invariant module and study its properties. In particular, we study abelian groups and prove that dual automorphism-invariant abelian groups must be reduced. It is shown that over a right perfect ring R, a lifting right R-module M is dual automorphism-invariant if and only if M is quasi-projective. [Copyright &y& Elsevier]

Abstract: Let q be a power of 2, n be a positive integer, and let be the finite field with elements. In this paper, we consider the existence of some specific elements in . The main results obtained in this paper are listed as follows: [(1)] There is an element ξ in such that both ξ and are primitive elements of if , and n is an odd number no less than 13 and . [(2)] For , and any odd n, there is an element ξ in such that ξ is a primitive normal element and is a primitive element of if either , and , or , and , . [Copyright &y& Elsevier]

Abstract: Two vectors in are called twins if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers M with the property that each integral vector with length has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible. [Copyright &y& Elsevier]

Abstract This paper has two main parts. First, we give a classification of the ℓ -blocks of finite special linear and unitary groups SL n (ϵ q) in the non-defining characteristic ℓ ≥ 3. Second, we describe how the ℓ -weights of SL n (ϵ q) can be obtained from the ℓ -weights of GL n (ϵ q) when ℓ ∤ gcd (n , q − ϵ) , and verify the Alperin weight conjecture for SL n (ϵ q) under the condition ℓ ∤ gcd (n , q − ϵ). As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent ℓ -block of SL n (ϵ q) if ℓ ∤ gcd (n , q − ϵ). [ABSTRACT FROM AUTHOR]

Abstract As abstractions of Terwilliger algebras, generalized Terwilliger algebras and varietal Terwilliger algebras have been studied in [5,6,17]. In general, it is difficult to determine the structure of a varietal Terwilliger algebra. As proposed in the concluding remarks in [4] , in this paper we study the structures of those varietal Terwilliger algebras arising from the wreath products of rank two association schemes. In particular, we give the explicit formulas for their central primitive idempotents. Applications to Terwilliger algebras of association schemes are also discussed, and an affirmative answer to the Terwilliger conjecture mention in [4] is given. [ABSTRACT FROM AUTHOR]

Abstract Starting from our previous papers Angelini et al. (2018c) and Angelini et al. (2018a) , we prove the existence of a non-empty Euclidean open subset whose elements are polynomial vectors with 4 components, in 3 variables, degrees, respectively, 2 , 3 , 3 , 3 and rank 6, which are not identifiable over C but are identifiable over R. This result has been obtained via computer-aided procedures suitably adapted to investigate the number of Waring decompositions for general polynomial vectors over the fields of complex and real numbers. Furthermore, by means of the Hessian criterion (Chiantini et al., 2014), we prove identifiability over C for polynomial vectors in many cases of sub-generic rank. [ABSTRACT FROM AUTHOR]

Abstract Many codes have been developed to study highly relativistic, magnetized flows around and inside compact objects. Depending on the adopted formalism, some of these codes evolve the vector potential A , and others evolve the magnetic field B = ∇ × A directly. Given that these codes possess unique strengths, sometimes it is desirable to start a simulation using a code that evolves B and complete it using a code that evolves A. Thus transferring the data from one code to another would require an inverse curl algorithm. This paper describes two new inverse curl techniques in the context of Cartesian numerical grids: a cell-by-cell method, which scales approximately linearly with the numerical grid, and a global linear algebra approach, which has worse scaling properties but is generally more robust (e.g., in the context of a magnetic field possessing some nonzero divergence). We demonstrate these algorithms successfully generate smooth vector potential configurations in challenging special and general relativistic contexts. Highlights • Our solution consists of a numerical implementation of the "inverse curl" operator. • We use two different methods that have different strengths and weaknesses. • The cell-by-cell technique is very fast and scales with the number of gridpoints. • The global linear algebra method is slower but has better symmetry properties. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we find the number of classes of conjugate π-Hall subgroups in all finite almost simple groups. We also complete the classification of π-Hall subgroups in finite simple groups and correct some mistakes from our previous paper. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra sitting inside an associative algebra A and any associative algebra we introduce and study the algebra , which is the Lie subalgebra of generated by . In many examples A is the universal enveloping algebra of . Our description of the algebra has a striking resemblance to the commutator expansions of used by M. Kapranov in his approach to noncommutative geometry. To each algebra we associate a “noncommutative algebraic” group which naturally acts on by conjugations and conclude the paper with some examples of such groups. [Copyright &y& Elsevier]

Abstract: Let π denote a partition into parts . In a 2006 paper we defined BG-rank as This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let denote the number of partitions of n with . Here, we provide a combinatorial proof that by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by into five equal classes. This proof uses the orbit construction from our previous paper and a new identity for the BG-rank. Let denote the number of t-cores of n with . We find eta-quotient representations for when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients , . [Copyright &y& Elsevier]

Abstract: Let be a class of unital C*-algebras. The class of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in is studied (Lin considered the case that consists of finite-dimensional C*-algebras or the tensor products of such with ). In particular, the question is considered whether, for any simple separable , there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in , such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.) [Copyright &y& Elsevier]

Abstract: This paper gives a direct proof for the coincidence of the following two central elements in the universal enveloping algebra of the orthogonal Lie algebra: an element recently given by A. Wachi in terms of the column-determinant in a way similar to the Capelli determinant, and an element given by T. Umeda and the author in terms of the symmetrized determinant. The fact that these two elements actually coincide was shown by A. Wachi, but his observation was based on the following two non-trivial results: (i) the centrality of the first element, and (ii) the calculation of the eigenvalue of the second element. The purpose of this paper is to prove this coincidence of two central elements directly without using these (i) and (ii). Conversely this approach provides us new proofs of (i) and (ii). A similar discussion can be applied to the symplectic Lie algebras. [Copyright &y& Elsevier]

Abstract: This paper deals with the sub-supersolution method for the -Laplacian equations. A sub-supersolution principle for the Dirichlet problems involving the -Laplacian is established. It is proved that the local minimizers in the topology are also local minimizers in the topology for given energy functionals. A strong comparison theorem for the -Laplacian equations is presented. Some applications of the abstract theorems obtained in this paper to the eigenvalue problems for the -Laplacian equations are given. [Copyright &y& Elsevier]

Abstract: In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A (where the combinatorics is based on Young tableaux, for instance) to arbitrary type; our approach is type-independent. The main results of this paper are: (1) a combinatorial description of the crystal graphs corresponding to the irreducible representations (this result includes a transparent proof, based on the Yang–Baxter equation, of the fact that the mentioned description does not depend on the choice involved in our model); (2) a combinatorial realization (which is the first direct generalization of Schützenberger''s involution on tableaux) of Lusztig''s involution on the canonical basis exhibiting the crystals as self-dual posets; (3) an analog for arbitrary root systems, based on the Yang–Baxter equation, of Schützenberger''s sliding algorithm, which is also known as jeu de taquin (this algorithm has many applications to the representation theory of the Lie algebra of type A). [Copyright &y& Elsevier]

Abstract: Many papers study polynomial tractability for multivariate problems. Let be the minimal number of information evaluations needed to reduce the initial error by a factor of for a multivariate problem defined on a space of -variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that is bounded by a polynomial in and and this holds for all . In this paper we study generalized tractability by verifying when can be bounded by a power of for all , where can be a proper subset of . Here is a tractability function, which is non-decreasing in both variables and grows slower than exponentially to infinity. In this article we consider the set for some and . We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on such that generalized tractability holds for linear tensor product problems. We show a number of examples for which polynomial tractability does not hold but generalized tractability does. [Copyright &y& Elsevier]

Abstract: Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras under injections which preserve the root spaces. It is known that a root-reductive Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of finite-dimensional simple Lie algebras as well as copies of the three simple infinite-dimensional root-reductive Lie algebras , , and . As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra were introduced and studied in [K.-H. Neeb, I. Penkov, Cartan subalgebras of , Canad. Math. Bull. 46 (2003) 597–616]. In the present paper we refine and extend the results of [K.-H. Neeb, I. Penkov, Cartan subalgebras of , Canad. Math. Bull. 46 (2003) 597–616] to the case of a general root-reductive Lie algebra . We prove that the Cartan subalgebras of are the centralizers of maximal toral subalgebras and that they are nilpotent and self-normalizing. We also give an explicit description of all Cartan subalgebras of the simple Lie algebras , , and . We conclude the paper with a characterization of the set of conjugacy classes of Cartan subalgebras of the Lie algebras , , , and with respect to the group of automorphisms of the natural representation which preserve the Lie algebra. [Copyright &y& Elsevier]

Abstract: Let k be a field and let H be a rigid braided Hopf k-algebra. In this paper we continue the study of the theory of braided Hopf crossed products began in [J.A. Guccione, J.J. Guccione, Theory of braided Hopf crossed products, J. Algebra 261 (2003) 54–101]. First we show that to have an H-braided comodule algebra is the same that to have an -braided module algebra, where is a variant of , and then we study the maps and , that appear in the Morita context introduced in the above cited paper. [Copyright &y& Elsevier]

Abstract: In this paper we complete a distributional Fourier analysis developed by Howell in a serie of papers. We investigate convolution operators in the corresponding distribution spaces. [Copyright &y& Elsevier]

Abstract: This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53–64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119–171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173–177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91–117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have -forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17–22, 2001, TIFR, Mumbai, 2001, pp. 469–490] of D. Morris (Witte) on a completely different topic. We will answer that question too. [Copyright &y& Elsevier]

Abstract: In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor Nr from the category of special -rings into the category of special -rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant''s paper [Free Lie algebras and formal power series, J. Algebra 253(1) (2002) 167–188] to the case of graded Lie (super)algebras with a group action by applying the Euler–Poincaré principle. [Copyright &y& Elsevier]

Abstract: In this paper we present new randomized and deterministic algorithms for the classical problem of broadcasting in radio networks with unknown topology. We consider directed n-node radio networks with specified eccentricity D (maximum distance from the source node to any other node). Bar-Yehuda et al. presented an algorithm that for any n-node radio network with eccentricity D completes the broadcasting in time, with high probability. This result is almost optimal, since as it has been shown by Kushilevitz and Mansour and Alon et al., every randomized algorithm requires expected time to complete broadcasting. Our first main result closes the gap between the lower and upper bound: we describe an optimal randomized broadcasting algorithm whose running time complexity is , with high probability. In particular, we obtain a randomized algorithm that completes broadcasting in any n-node radio network in time , with high probability. The main source of our improvement is a better “selecting sequence” used by the algorithm that brings some stronger property and improves the broadcasting time. Two types of “selecting sequences” are considered: randomized and deterministic ones. The algorithm with a randomized sequence is easier (more intuitive) to analyze but both randomized and deterministic sequences give algorithms of the same asymptotic complexity. Next, we demonstrate how to apply our approach to deterministic broadcasting, and describe a deterministic oblivious algorithm that completes broadcasting in time , which improves upon best known algorithms in this case. The fastest previously known algorithm had the broadcasting time of , it was non-oblivious and significantly more complicated; our algorithm can be seen as a natural extension of our randomized algorithm. In this part of the paper we assume that each node knows the eccentricity D. Finally, we show how our randomized broadcasting algorithm can be used to improve the randomized complexity of the gossiping problem. [Copyright &y& Elsevier]

Abstract: In this paper the authors investigate the representation type of the blocks of the relative (parabolic) category for complex semisimple Lie algebras. A complete classification of the blocks corresponding to regular weights is given. The main results of the paper provide a classification of the blocks in the “mixed” case when the simple roots corresponding to the singular set and S do not meet. [Copyright &y& Elsevier]

Abstract: This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of -categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on simplicial presheaves over a Grothendieck site of Joyal and Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspondence between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition of étale K-theory of ring spectra, extending the étale K-theory of commutative rings. [Copyright &y& Elsevier]

Abstract: In Part I of this paper, we introduced a method of making two isomorphic intervals of a bounded lattice congruence equivalent. In this paper, we make one interval dominate another one. Let L be a bounded lattice, let and be intervals of L, and let φ be a homomorphism of onto . We construct a bounded (convex) extension K of L such that a congruence Θ of L has an extension to K iff implies that , for , in which case, Θ has a unique extension to K. This result presents a lattice K whose congruence lattice is derived from the congruence lattice of L in a new way, different from the one presented in Part I. The main technical innovation is the -Boolean triple construction, which owes its origin to the Boolean triple construction of G. Grätzer and F. Wehrung. [Copyright &y& Elsevier]

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated single-, double-, and triple-integral representations. Some known or new consequences of the general results presented here, involving such classical orthogonal polynomials as the Jacobi, Laguerre, Hermite, and Bessel polynomials, and various other relatively less familiar hypergeometric polynomials, are also considered. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. [Copyright &y& Elsevier]

In an earlier paper the authors classified the nonsolvable primitive linear groups of prime degree over C. The present paper deals with the classification of the nonsolvable imprimitive linear groups of prime degree (equivalently, the irreducible monomial groups of prime degree). If G is a monomial group of prime degree r, then there is a projection π of G onto a transitive group H of permutation matrices with a kernel A consisting of diagonal matrices. The transitive permutation groups of prime degree are known, so the classification reduces to (i) determining the possible diagonal groups A for a given group H of permutation matrices; (ii) describing the possible extensions which might occur for given A and H; and (iii) determining when two of these extensions are conjugate in the general linear group. We prove that for given nonsolvable H there is a finite set Φ(r,H) of diagonal groups such that all monomial groups G with π(G)=H can be determined in a simple way from the monomial groups which are extensions of A∈Φ(r,H) by H, and calculate Φ(r,H) in many cases. We also show how the problem of determining conjugacy in the general case is reduced to solving this problem when A∈Φ(r,H). In general, the results hold over any algebraically closed field with modifications required in the case of a few small characteristics. [Copyright &y& Elsevier]

This paper deals with a stochastic stability concept for discrete-time Markovian jump linear systems. The random jump parameter is associated to changes between the system operation modes due to failures or repairs, which can be well described by an underlying finite-state Markov chain. In the model studied, a fixed number of failures or repairs is allowed, after which, the system is brought to a halt for maintenance or for replacement. The usual concepts of stochastic stability are related to pure infinite horizon problems, and are not appropriate in this scenario. A new stability concept is introduced, named stochastic τ-stability that is tailored to the present setting. Necessary and sufficient conditions to ensure the stochastic τ-stability are provided, and the almost sure stability concept associated with this class of processes is also addressed. The paper also develops equivalences among second order concepts that parallels the results for infinite horizon problems. [Copyright &y& Elsevier]

In the past decades, the construction of de Bruijn sequences has been extensively studied. Let ϕ be the one-to-one map from F 2 [ x ] to the set of all linear Boolean functions and the symbol “⁎” denote the product of two Boolean functions f ( x 0 , x 1 , . . . , x n ) and g ( x 0 , x 1 , . . . , x m ) given by f ( g ( x 0 , x 1 , . . . , x m ) , g ( x 1 , x 2 , . . . , x m + 1 ) , . . . , g ( x n , x n + 1 , . . . , x n + m ) ) . Then in this paper, a feedback shift register (FSR) with the characteristic function ( l + 1 ) ⁎ ϕ ( p ( x ) ) is considered to construct de Bruijn sequences for the first time, where ( l + 1 ) is an affine Boolean function and p ( x ) is a primitive polynomial over F 2 of degree n > 2 with ϕ − 1 ( l ) not divisible by p ( x ) . In specific, we determine the cycle structure and the adjacency graphs of this type of FSRs. As an example, we present the adjacency graph of FSRs with characteristic functions of the form ( ( x 0 + x 1 + x 2 + x 3 + 1 ) ⁎ ϕ ( p ( x ) ) ) and calculate the total number of de Bruijn sequences constructed from these FSRs. Note that it is the first time that the FSRs with affine characteristic functions are considered to construct de Bruijn sequences, and thus a new class of de Bruijn sequences are derived. [ABSTRACT FROM AUTHOR]

Integrative analysis of disparate data blocks measured on a common set of experimental subjects is a major challenge in modern data analysis. This data structure naturally motivates the simultaneous exploration of the joint and individual variation within each data block resulting in new insights. For instance, there is a strong desire to integrate the multiple genomic data sets in The Cancer Genome Atlas to characterize the common and also the unique aspects of cancer genetics and cell biology for each source. In this paper we introduce Angle-Based Joint and Individual Variation Explained capturing both joint and individual variation within each data block. This is a major improvement over earlier approaches to this challenge in terms of a new conceptual understanding, much better adaption to data heterogeneity and a fast linear algebra computation. Important mathematical contributions are the use of score subspaces as the principal descriptors of variation structure and the use of perturbation theory as the guide for variation segmentation. This leads to an exploratory data analysis method which is insensitive to the heterogeneity among data blocks and does not require separate normalization. An application to cancer data reveals different behaviors of each type of signal in characterizing tumor subtypes. An application to a mortality data set reveals interesting historical lessons. Software and data are available at GitHub https://github.com/MeileiJiang/AJIVE_Project . [ABSTRACT FROM AUTHOR]

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. As their natural analogues, there are constructions of nil Lie p -algebras over a field of characteristic 2 [40] and arbitrary positive characteristic [52] . In characteristic zero, similar examples of Lie algebras do not exist by a result of Martinez and Zelmanov [32] . The second author constructed analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of arbitrary characteristic, the virtue of that construction is that the Lie superalgebras have clear monomial bases [41] . That Lie superalgebras have slow polynomial growth and are graded by multidegree in the generators. In particular, a self-similar Lie superalgebra Q is Z 3 -graded by multidegree in 3 generators, its Z 3 -components lie inside an elliptic paraboloid in space, the components are at most one-dimensional, thus, the Z 3 -grading of Q is fine. An analogue of the periodicity is that homogeneous elements of the grading Q = Q 0 ¯ ⊕ Q 1 ¯ are ad-nilpotent. In particular, Q is a nil finely graded Lie superalgebra, which shows that an extension of the mentioned result of Martinez and Zelmanov [32] to the Lie superalgebras of characteristic zero is not valid. But computations with Q are rather technical. In this paper, we construct a similar but simpler and “smaller” example. Namely, we construct a 2-generated fractal Lie superalgebra R over arbitrary field. We find a clear monomial basis of R and, unlike many examples studied before, we find also a clear monomial basis of its associative hull A , the latter has a quadratic growth. The algebras R and A are Z 2 -graded by multidegree in the generators, positions of their Z 2 -components are bounded by pairs of logarithmic curves on plane. The Z 2 -components of R are at most one-dimensional, thus, the Z 2 -grading of R is fine. As an analogue of periodicity, we establish that homogeneous elements of the grading R = R 0 ¯ ⊕ R 1 ¯ are ad-nilpotent. In case of N -graded algebras, a close analogue to being simple is being just infinite. Unlike previous examples of Lie superalgebras, we are able to prove that R is just infinite, but not hereditary just infinite. Our example is close to the smallest possible example, because R has a linear growth with a growth function γ R ( m ) ≈ 3 m , as m → ∞ . Moreover, R is of finite width 4 ( char K ≠ 2 ). In case char K = 2 , we obtain a Lie algebra of width 2 that is not thin. Thus, we have got a more handy analogue of the Grigorchuk and Gupta-Sidki groups. The constructed Lie superalgebra R is of linear growth, of finite width 4, and just infinite. It also shows that an extension of the result of Martinez and Zelmanov [32] to the Lie superalgebras of characteristic zero is not valid. [ABSTRACT FROM AUTHOR]

In this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold. Assuming some extra nondegenericity and using the result of Hong, complete normal form descriptions can be given, and in low dimensions, we obtain a complete classification without any extra assumptions. [ABSTRACT FROM AUTHOR]

For 0 < p ≤ ∞ , let F φ p be the Fock space induced by a weight function φ satisfying d d c φ ≃ ω 0 . In this paper, given p ∈ ( 0 , 1 ] we introduce the concept of weakly localized operators on F φ p , we characterize the compact operators in the algebra generated by weakly localized operators. As an application, for 0 < p < ∞ we prove that an operator T in the algebra generated by bounded Toeplitz operators with BMO symbols is compact on F φ p if and only if its Berezin transform satisfies certain vanishing property at ∞. In the classical Fock space, we extend the Axler–Zheng condition on linear operators T , which ensures T is compact on F α p for all possible 0 < p < ∞ . [ABSTRACT FROM AUTHOR]