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Cazanave has identified the algebraic homotopy class of a rational function of 1 variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question posed by Eisenbud for polynomial functions in 1 variable. Finally, we announce results answering this question for functions in an arbitrary number of variables. [ABSTRACT FROM AUTHOR]

Suppose { X k } k ∈ Z is a sequence of bounded independent random matrices with common dimension d × d and common expectation E [ X k ] = X. Under these general assumptions, the normalized random matrix product Z n = (I d + 1 n X n) (I d + 1 n X n − 1) ⋯ (I d + 1 n X 1) converges to e X as n → ∞. Normalized random matrix products of this form arise naturally in stochastic iterative algorithms, such as Oja's algorithm for streaming Principal Component Analysis. Here, we derive nonasymptotic concentration inequalities for such random matrix products. In particular, we show that the spectral norm error satisfies ‖ Z n − e X ‖ = O ((log ⁡ (n)) 2 log ⁡ (d / δ) / n) with probability exceeding 1 − δ. This rate is sharp in n , d , and δ , up to logarithmic factors. The proof relies on two key points of theory: the Matrix Bernstein inequality concerning the concentration of sums of random matrices, and Baranyai's theorem from combinatorial mathematics. Concentration bounds for general classes of random matrix products are hard to come by in the literature, and we hope that our result will inspire further work in this direction. [ABSTRACT FROM AUTHOR]

We determine the region of singular values of the commutator X Y − Y X when X and Y vary over the set of norm one rank one matrices. This answers in affirmative a conjecture raised in Wenzel (2015) [7]. [ABSTRACT FROM AUTHOR]

Let G be a weakly connected directed graph with asymmetric graph Laplacian L. Consensus and diffusion are dual dynamical processes defined on G by x ˙ = − L x for consensus and p ˙ = − p L for diffusion. In this survey, we consider both these processes as well their discrete time analogues. We define a basis of row vectors { γ ¯ i } i = 1 k of the left null-space of L and a basis of column vectors { γ i } i = 1 k of the right null-space of L in terms of the partition of G into strongly connected components. This allows for complete characterization of the asymptotic behavior of both diffusion and consensus — discrete and continuous — in terms of these eigenvectors. As an application of these ideas, we present a treatment of the pagerank algorithm that is dual to the usual one. We further show that the teleporting (see below) feature usually included in the algorithm is not strictly necessary. Together with [1] , this is a complete and self-contained overview of the asymptotics of consensus and diffusion on digraphs. Many of the ideas presented here can be found scattered in the literature, though mostly outside mainstream mathematics and not always with complete proofs. This paper seeks to remedy this by providing a compact and accessible survey. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we extend the concept of absolutely Cesàro boundedness to the fractional case. We construct a weighted shift operator belonging to this class of operators, and we prove that if T is an absolutely Cesàro bounded operator of order α with 0 < α ≤ 1 , then ‖ T n ‖ = o (n α) , generalizing the result obtained for α = 1. Moreover, if α > 1 , then ‖ T n ‖ = O (n). We apply such results to get stability properties for the Cesàro means of bounded operators. [ABSTRACT FROM AUTHOR]

Abstract We consider the problem of updating the SVD when augmenting a "tall thin" matrix, i.e., a rectangular matrix A ∈ R m × n with m ≫ n. Supposing that an SVD of A is already known, and given a matrix B ∈ R m × n ′ , we derive an efficient method to compute and efficiently store the SVD of the augmented matrix [ A B ] ∈ R m × (n + n ′). This is an important tool for two types of applications: in the context of principal component analysis, the dominant left singular vectors provided by this decomposition form an orthonormal basis for the best linear subspace of a given dimension, while from the right singular vectors one can extract an orthonormal basis of the kernel of the matrix. We also describe two concrete applications of these concepts which motivated the development of our method and to which it is very well adapted. [ABSTRACT FROM AUTHOR]

Abstract For each non-negative integer n let A n be an n + 1 by n + 1 Toeplitz matrix over a finite field, F , and suppose for each n that A n is embedded in the upper left corner of A n + 1. We study the structure of the sequence ν = { ν n : n ∈ Z + } , where ν n = null (A n) is the nullity of A n. For each n ∈ Z + and each nullity pattern ν 0 , ν 1 , ... , ν n , we count the number of strings of Toeplitz matrices A 0 , A 1 , ... , A n with this pattern. As an application we present an elementary proof of a result of D. E. Daykin on the number of n × n Toeplitz matrices over G F (2) of any specified rank. [ABSTRACT FROM AUTHOR]

A n -set of equi-isoclinic planes in R r is a set of n planes spanning R r each pair of which has the same non-zero angle arccos ⁡ λ . We prove that for any odd integer k ≥ 3 such that 2 k = p α + 1 , p an odd prime, α a positive integer the maximum number of equi-isoclinic planes with angle arccos ⁡ 1 2 k − 2 in R 2 k − 1 is equal to 2 k − 1 . It is shown that the solution of this geometric problem is obtained by the construction of complex symmetric conference matrices of order 2 k − 1 , and that all these constructions are performed by use of the Legendre symbol of the Galois field G F ( p α ) . [ABSTRACT FROM AUTHOR]

Let A and B be nonnegative square matrices of the same order. Denote by ‖ ⋅ ‖ and ρ ( ⋅ ) the spectral norm and the spectral radius respectively. We prove the following inequalities: ‖ A ∘ B ‖ ≤ ‖ A ∘ A ‖ 1 2 ‖ B ∘ B ‖ 1 2 ; ‖ A ∘ B ‖ ≤ ρ 1 2 ( A T B ∘ B T A ) ≤ ρ 1 2 ( A T B ∘ A T B ) ≤ ρ ( A T B ) , where ∘ denotes the Hadamard product. This interpolates the inequality ‖ A ∘ B ‖ ≤ ρ ( A T B ) due to Huang. Some spectral norm inequalities for an arbitrarily finite number of nonnegative square matrices are also obtained, which refine some other results of Huang. [ABSTRACT FROM AUTHOR]

In [1] , the set of refined inertias H n for sign patterns is expanded to H n ⁎ for zero-nonzero patterns. Also, it is proved that there are no zero-nonzero patterns of even order 2 m that require H 2 m ⁎ . In this paper, it is shown that there are no zero-nonzero patterns of odd order n ≥ 5 that require H n ⁎ , which together with the previous result prove that there are no zero-nonzero patterns of order n ≥ 4 that require H n ⁎ . [ABSTRACT FROM AUTHOR]

Let M and N be arbitrary von Neumann algebras. For any a in M or in N , let Δ λ ( a ) denote the λ -Aluthge transform of a . Suppose that M has no abelian direct summand. We prove that every bijective map Φ : M → N satisfying Φ ( Δ λ ( a ∘ b ⁎ ) ) = Δ λ ( Φ ( a ) ∘ Φ ( b ) ⁎ ) , for all a , b ∈ M , (for a fixed λ ∈ [ 0 , 1 ] ), maps the hermitian part of M onto the hermitian part of N (i.e. Φ ( M s a ) = N s a ) and its restriction Φ | M s a : M s a → N s a is a Jordan isomorphism. If we also assume that Φ ( x + i y ) = Φ ( x ) + Φ ( i y ) for all x , y ∈ M s a , then there exists a central projection p c in M such that Φ | p c M is a complex linear Jordan ⁎ -isomorphism and Φ | ( 1 − p c ) M is a conjugate linear Jordan ⁎ -isomorphism. Given two complex Hilbert spaces H and K with dim ( H ) ≥ 2 , we also show that every bijection Φ : B ( H ) → B ( K ) satisfying Φ ( Δ λ ( a b ⁎ ) ) = Δ λ ( Φ ( a ) Φ ( b ) ⁎ ) , for all a , b ∈ B ( H ) , must be a complex linear or a conjugate linear ⁎ -isomorphism. [ABSTRACT FROM AUTHOR]

For each non-negative integer n let A n be an n + 1 by n + 1 Toeplitz matrix over a finite field, F , and suppose for each n that A n is embedded in the upper left corner of A n + 1. We study the structure of the sequence ν = { ν n : n ∈ Z + } , where ν n = null (A n) is the nullity of A n. For each n ∈ Z + and each nullity pattern ν 0 , ν 1 , ... , ν n , we count the number of strings of Toeplitz matrices A 0 , A 1 , ... , A n with this pattern. As an application we present an elementary proof of a result of D. E. Daykin on the number of n × n Toeplitz matrices over G F (2) of any specified rank. [ABSTRACT FROM AUTHOR]

It is proved that regularity of an interval matrix implies singularity of four related interval matrices. The result is used to prove that for each nonsingular point matrix A , either A or A − 1 can be brought to a singular matrix by perturbing only the diagonal entries by an amount of at most 1 each. As a consequence, the notion of a diagonally singularizable matrix is introduced. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The distance Laplacian matrix of G is defined as D L ( G ) = T r ( G ) − D ( G ) , where D ( G ) is the distance matrix and T r ( G ) = diag ( t r v 1 , t r v 2 , … , t r v n ) is the diagonal matrix of vertex transmissions of G . The largest eigenvalue of D L ( G ) is called the distance Laplacian spectral radius of G . In this paper, we obtain a graft transformation of a connected graph, which increases its distance Laplacian spectral radius. Using this transformation, we prove a conjecture involving the distance Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

We prove stability results for a class of Gabor frames in L 2 ( R ) . We consider window functions in the Sobolev spaces H 0 1 ( R ) and B-splines of order p ≥ 1 . Our results can be used to describe the effect of the timing jitters in the p -order hold models of signal reconstruction. [ABSTRACT FROM AUTHOR]

We prove a result concerning the behavior of the spectral radius of a hypergraph under relocations of edges. We determine the unique hypergraphs with maximum spectral radius among connected k -uniform hypergraphs with fixed number of pendant edges, the unique k -uniform hypertrees with respectively maximum, second maximum and third maximum spectral radius, the unique k -uniform unicyclic hypergraphs ( k -uniform linear unicyclic hypergraphs, respectively) with respectively maximum and second maximum spectral radius. We also determine the unique hypergraphs with maximum spectral radius among k -uniform unicyclic hypergraphs with given girth. [ABSTRACT FROM AUTHOR]

Gaussian elimination with no pivoting and block Gaussian elimination are attractive alternatives to the customary but communication intensive Gaussian elimination with partial pivoting 1 provided that the computations proceed safely and numerically safely , that is, run into neither division by 0 nor numerical problems. Empirically, safety and numerical safety of GENP have been consistently observed in a number of papers where an input matrix was pre-processed with various structured multipliers chosen ad hoc. Our present paper provides missing formal support for this empirical observation and explains why it was elusive so far. Namely we prove that GENP is numerically unsafe for a specific class of input matrices in spite of its pre-processing with some well-known and well-tested structured multipliers, but we also prove that GENP and BGE are safe and numerically safe for the average input matrix pre-processed with any nonsingular and well-conditioned multiplier. This should embolden search for sparse and structured multipliers, and we list and test some new classes of them. We also seek randomized pre-processing that universally (that is, for all input matrices) supports (i) safe GENP and BGE with probability 1 and/or (ii) numerically safe GENP and BGE with a probability close to 1. We achieve goal (i) with a Gaussian structured multiplier and goal (ii) with a Gaussian unstructured multiplier and alternatively with Gaussian structured augmentation. We consistently confirm all these formal results with our tests of GENP for benchmark inputs. We have extended our approach to other fundamental matrix computations and keep working on further extensions. 1 Hereafter we use the acronyms GENP , BGE , and GEPP . [ABSTRACT FROM AUTHOR]

W.E. Roth (1952) proved that the matrix equation A X − X B = C has a solution if and only if the matrices [ A C 0 B ] and [ A 0 0 B ] are similar. A. Dmytryshyn and B. Kågström (2015) extended Roth's criterion to systems of matrix equations A i X i ′ M i − N i X i ″ σ i B i = C i ( i = 1 , … , s ) with unknown matrices X 1 , … , X t , in which every X σ is X , X ⊤ , or X ⁎ . We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations. [ABSTRACT FROM AUTHOR]

The columns of an m × n ACI-matrix over a field F are independent affine subspaces of F m . An ACI-matrix has constant rank ρ if all its completions have rank ρ . Huang and Zhan (2011) [4] characterized the m × n ACI-matrices of constant rank when | F | ≥ min ⁡ { m , n + 1 } . We complete their result characterizing the m × n ACI-matrices of constant rank over arbitrary fields. Quinlan and McTigue (2014) [8] proved that every partial matrix of constant rank ρ has a ρ × ρ submatrix of constant rank ρ if and only | F | ≥ ρ . We obtain an analogous result for ACI-matrices over arbitrary fields by introducing the concept of complete irreducibility. [ABSTRACT FROM AUTHOR]

We prove that for each positive integer k , all irreducible k -potent ray pattern classes allow k -potence. We construct all irreducible real (complex) k -potent matrices in an irreducible sign (ray) k -potent sign (ray) pattern class. For a given reducible sign (ray) pattern that is sign (ray) k -potent for some positive integer k , we determine necessary conditions for that pattern to allow k -potence. This generalizes recent work by Lee and Park (2015) [2] who determined necessary conditions for a sign idempotent sign pattern to allow idempotence, and who showed that every irreducible, sign k -potent sign pattern must allow k -potence. [ABSTRACT FROM AUTHOR]

An ( n , m ) -graph is referred to be a graph with n vertices and m edges. Let Δ ( G ) and δ ( G ) be the maximum and minimum degree of a graph G and let μ ( G ) and q ( G ) be the Laplacian and signless Laplacian spectral radius of G , respectively. In this paper, we prove that for two connected nonregular ( n , m ) -graphs G and G ′ , if Δ ( G ) ≥ 2 m − ( n − 1 ) δ ( G ′ ) δ ( G ′ ) + 1 + δ ( G ′ ) and Δ ( G ) > Δ ( G ′ ) + δ ( G ′ ) − 1 , then μ ( G ) > μ ( G ′ ) and q ( G ) > q ( G ′ ) . Also, we obtain that for two connected nonregular ( n , m ) -graphs G and G ′ , if Δ ( G ) ≥ 2 m − ( n − 1 ) δ ( G ′ ) δ ( G ′ ) + 1 + 1 and Δ ( G ) > Δ ( G ′ ) , then μ ( G ) > μ ( G ′ ) − δ ( G ′ ) + 1 and q ( G ) > q ( G ′ ) − δ ( G ′ ) + 1 . [ABSTRACT FROM AUTHOR]

In this paper, we propose new lower and upper bounds on the linear extension complexity of regular n -gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size r of a polytope P , and (ii) a rank- r nonnegative factorization of a slack matrix of the polytope P . The lower bound is based on an improved bound for the rectangle covering number (also known as the boolean rank) of the slack matrix of the n -gons. The upper bound is a slight improvement of the result of Fiorini, Rothvoss and Tiwary (2012) [9] . The difference with their result is twofold: (i) our proof uses a purely algebraic argument while Fiorini et al. used a geometric argument, and (ii) we improve the base case allowing us to reduce their upper bound 2 ⌈ log 2 ⁡ ( n ) ⌉ by one when 2 k − 1 < n ≤ 2 k − 1 + 2 k − 2 for some integer k . We conjecture that this new upper bound is tight, which is suggested by numerical experiments for small n . Moreover, this improved upper bound allows us to close the gap with the best known lower bound for certain regular n -gons (namely, 9 ≤ n ≤ 13 and 21 ≤ n ≤ 24 ) hence allowing for the first time to determine their extension complexity. [ABSTRACT FROM AUTHOR]

Let C n denote the cycle with n vertices, and let τ ( G ) be the number of triangles of G . If G is a graph with n vertices, then μ 1 ( G ) ≥ μ 2 ( G ) ≥ ⋯ ≥ μ n ( G ) denote the signless Laplacian spectrum of G . Two graphs are said to be Q -cospectral if they have the same signless Laplacian spectrum. A graph G is said to be Q − D S if there is no other non-isomorphic graph H such that G and H are Q -cospectral. In this paper, we prove that “Let G be a graph with n ≥ 12 vertices and 1 ≤ μ n ( G ) ≤ μ 2 ( G ) ≤ 5 < n < μ 1 ( G ) or let G be a graph with n ≥ 10 vertices and 0 < μ n ( G ) ≤ μ 2 ( G ) ≤ 4 < n < μ 1 ( G ) . If G and H are Q -cospectral, then G and H share the same degree sequences and τ ( G ) = τ ( H ) ”. As a consequence of our results, we show that all multi-wheel graphs K 1 ∨ ( C q 1 ∪ C q 2 ∪ ⋯ ∪ C q t ) are Q − D S . [ABSTRACT FROM AUTHOR]

For any two complex numbers a and b , V i r ( a , b ) is a central extension of W ( a , b ) which is universal in the case ( a , b ) ≠ ( 0 , 1 ) , where W ( a , b ) is the Lie algebra with basis { L n , W n | n ∈ Z } and relations [ L m , L n ] = ( n − m ) L m + n , [ L m , W n ] = ( a + n + b m ) W m + n , [ W m , W n ] = 0 . In this paper, we construct and classify a class of non-weight modules over the algebra V i r ( a , b ) which are free U ( C L 0 ⊕ C W 0 ) -modules of rank 1. It is proved that such modules can only exist for a ∈ Z . [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with maximum degree Δ and minimum degree δ . Let ( d ) = ( d 1 , d 2 , … , d n ) and ( d ⁎ ) = ( d 1 ⁎ , d 2 ⁎ , … , d n ⁎ ) be the sequences of degrees and conjugate degrees of G . We define π = ∑ i = 1 n d i and π ⁎ = ∑ i = 1 n d i ⁎ , and prove that π ⁎ ≤ L E L ≤ I E ≤ π where LEL and IE are, respectively, the Laplacian-energy-like invariant and the incidence energy of G . Moreover, we prove that π − π ⁎ > ( δ / 2 ) ( n − Δ ) for a certain class of graphs. Finally, we compare the energy of G and π , and present an upper bound for the Laplacian energy in terms of degree sequence. [ABSTRACT FROM AUTHOR]

In 1962, Erdős proved a theorem on the existence of Hamilton cycles in graphs with given minimum degree and number of edges. Significantly strengthening in case of balanced bipartite graphs, Moon and Moser proved a corresponding theorem in 1963. In this paper we establish several spectral analogues of Moon and Moser's theorem on Hamilton paths in balanced bipartite graphs and nearly balanced bipartite graphs. One main ingredient of our proofs is a structural result of its own interest, involving Hamilton paths in balanced bipartite graphs with given minimum degree and number of edges. [ABSTRACT FROM AUTHOR]

It seems to be of recurring interest in the literature to give alternative proofs for the fact that the determinant of a symplectic matrix is one. We state four short and elementary proofs for symplectic matrices over general fields. Two of them seem to be new. [ABSTRACT FROM AUTHOR]

We define and study three different types of upper (and lower) triangular blocked tensors, which are all generalizations of the triangular blocked matrices, and are also generalizations of reducible or weakly reducible tensors. We study some common properties, as well as different properties of these three types of triangular blocked tensors. We obtain the formulas for the determinants, characteristic polynomials and spectra of the first and second type triangular blocked tensors, and give an example to show that these formulas no longer hold for the third type triangular blocked tensors. We prove that the product of any two ( n 1 , ⋯ , n r ) -upper (or lower) triangular blocked tensors of the first or second or third type is still an ( n 1 , ⋯ , n r ) -upper (or lower) triangular blocked tensor of the same type. We also prove that, if an ( n 1 , ⋯ , n r ) -upper triangular blocked tensor of the first or second or third type has a left k -inverse, then its unique left k -inverse is still an ( n 1 , ⋯ , n r ) -upper triangular blocked tensor of all the three types. Also if it has a right k -inverse, then all of its right k -inverses are still ( n 1 , ⋯ , n r ) -upper triangular blocked tensors of all the three types. By showing that the left k -inverse (if any) of a weakly irreducible nonsingular M -tensor is a positive tensor, we show that the left k -inverse (if any) of a first or second or third type normal ( n 1 , ⋯ , n r ) -upper triangular blocked nonsingular M -tensor is an ( n 1 , ⋯ , n r ) -upper triangular blocked tensor of all the three types all of whose diagonal blocks are positive tensors. We also show that every order m dimension n tensor is permutation similar to some third type normal upper triangular blocked tensor (all of whose diagonal blocks are irreducible). We give an example to show that this is not true for the first type normal upper triangular blocked tensor. [ABSTRACT FROM AUTHOR]

A correlation matrix is a positive semi-definite Hermitian matrix with all diagonals equal to 1. The minimum of the permanents on singular correlation matrices is conjectured to be given by the matrix Y n , all of whose non-diagonal entries are − 1 / ( n − 1 ) . Also, Frenzen–Fischer proved that per Y n approaches to e / 2 as n → ∞ . In this paper, we analyze some immanants of Y n , which are the generalizations of the determinant and the permanent, and we generalize these results to some other immanants and conjecture most of those converge to 1. [ABSTRACT FROM AUTHOR]

The matrix equation A X − X B = C has a solution if and only if the matrices [ A C 0 B ] and [ A 0 0 B ] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X − A X B = C over a field has a solution if and only if the matrices [ A C 0 I ] and [ I 0 0 B ] are simultaneously equivalent to [ A 0 0 I ] and [ I 0 0 B ] . We extend these criteria to the matrix equations A X − X ˆ B = C and X − A X ˆ B = C over the skew field of quaternions with a fixed involutive automorphism q ↦ q ˆ . [ABSTRACT FROM AUTHOR]

For a matrix A whose off-diagonal entries are nonpositive, its nonnegative invertibility (namely, that A is an invertible M -matrix) is equivalent to A being a P -matrix, which is necessary and sufficient for the unique solvability of the linear complementarity problem defined by A . This, in turn, is equivalent to the statement that A is strictly semimonotone. In this paper, an analogue of this result is proved for singular symmetric Z -matrices. This is achieved by replacing the inverse of A by the group generalized inverse and by introducing the matrix classes of strictly range semimonotonicity and range column sufficiency. A recently proposed idea of P # -matrices plays a pivotal role. Some interconnections between these matrix classes are also obtained. [ABSTRACT FROM AUTHOR]

An n × n complex matrix A is called coninvolutory if A ¯ A = I n and skew-coninvolutory if A ¯ A = − I n (which implies that n is even). We prove that each matrix of size n × n with n > 1 is a sum of 5 coninvolutory matrices and each matrix of size 2 m × 2 m is a sum of 5 skew-coninvolutory matrices. We also prove that each square complex matrix is a sum of a coninvolutory matrix and a condiagonalizable matrix. A matrix M is called condiagonalizable if M = S ¯ − 1 D S in which S is nonsingular and D is diagonal. [ABSTRACT FROM AUTHOR]

Yuan and Leng (2007) gave a generalization of the matrix form of the Brunn–Minkowski inequality. In this note, we first give a simple proof of this inequality, and then show a generalization of this to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. [ABSTRACT FROM AUTHOR]

We show that the d -cube is determined by the spectrum of its distance matrix. [ABSTRACT FROM AUTHOR]

For any algebraically closed field F and any two square matrices A , B over F , Sylvester (1884) [8] and Cecioni (1910) [1] showed that A X = X B implies X = 0 if and only if A and B have no common eigenvalue. It is proved that a third equivalent statement is that, for any given polynomials f , g in F [ t ] , there exists h in F [ t ] such that f ( A ) = h ( A ) and g ( B ) = h ( B ) . Corresponding results hold also for any finite set of square matrices over F , and these lead to a new property of all associative rings and algebras (even over arbitrary fields) with 1. [ABSTRACT FROM AUTHOR]

We obtain some conditions about the weighted shift and the composition operators on ℓ p to be ( m , q ) -isometries. In particular, we prove that if T is a strict ( 2 , q ) isometry, then p = q , whether T is a weighted shift or a composition operator on ℓ p . Moreover, we characterize the admissible weights for the weighted shift to be a strict ( 3 , p ) -isometry on ℓ p . The case of strict ( 4 , p ) -isometry is also studied. [ABSTRACT FROM AUTHOR]

We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite graphs. Furthermore, we analyze sequences of polynomials that are determinants of polynomially sized matrices consisting only of zeros, ones, and variables. We show that these are exactly the sequences in the complexity class of constant free polynomially sized (weakly) skew circuits. [ABSTRACT FROM AUTHOR]

We prove that for an algebraic curvature tensor on a pseudo-Euclidean space, the Jordan–Osserman condition implies the Rakić duality principle, and that the Osserman condition and the duality principle are equivalent in the semisimple case. [ABSTRACT FROM AUTHOR]

Inspired by the theoretical results on optimal preconditioning stated by Ng, Chan, and Tang in the framework of Reflective boundary conditions (BCs), in this paper we present analogous results for Anti-Reflective BCs. Here a key technical difficulty is represented by the non-orthogonal character of the Anti-Reflective transform and indeed the proof proposed by Ng, Chan, and Tang does not work. Nevertheless, in both cases, the optimal preconditioner is the blurring matrix associated to the symmetrized Point Spread Function (PSF). The geometrical idea on which our proof is based is very simple and general, so it may be useful in the future to prove theoretical results for new proposed BCs. Numerical tests show that the optimal preconditioning strategy is effective when using both preconditioned conjugate gradient methods and recently introduced nonstationary preconditioned iterations. [ABSTRACT FROM AUTHOR]

We prove two results with regard to reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”). First we show that phase retrievable nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Specifically, if nonlinear analysis maps α , β : H ˆ → R m are injective, with α ( x ) = ( | 〈 x , f k 〉 | ) k = 1 m and β ( x ) = ( | 〈 x , f k 〉 | 2 ) k = 1 m , where { f 1 , … , f m } is a frame for a Hilbert space H and H ˆ = H / T 1 , then α is bi-Lipschitz with respect to the class of “natural metrics” D p ( x , y ) = min φ ⁡ ‖ x − e i φ y ‖ p , whereas β is bi-Lipschitz with respect to the class of matrix-norm induced metrics d p ( x , y ) = ‖ x x ⁎ − y y ⁎ ‖ p . Second we prove that reconstruction can be performed using Lipschitz continuous maps. That is, there exist left inverse maps (synthesis maps) ω , ψ : R m → H ˆ of α and β respectively, that are Lipschitz continuous with respect to appropriate metrics. Additionally, we obtain the Lipschitz constants of ω and ψ in terms of the lower Lipschitz constants of α and β , respectively. Surprisingly, the increase in both Lipschitz constants is a relatively small factor, independent of the space dimension or the frame redundancy. [ABSTRACT FROM AUTHOR]

A method for generating all rational generalized matrices on indefinite real inner product spaces isomorphic to R p , q is presented. The proposed method is based on the proof of a weak version of the Cartan–Dieudonné theorem, handled using Clifford algebras. It is shown that all rational B -orthogonal matrices in an indefinite inner product space ( X , B ) are products of simple matrices with rational entries. [ABSTRACT FROM AUTHOR]

Let R be a unital associative algebra over a field K of characteristic zero, and let f be a multilinear polynomial of degree m over K . If m ≤ 3 , we prove that all traceless matrices can be written as the sum of two values of f evaluated over M n ( R ) with n ≥ 2 . If m = 4 , we prove the same result for n ≥ 3 . [ABSTRACT FROM AUTHOR]

A matrix A ∈ M n ( R ) with coefficients in any ring R is a quasi-permutation matrix if each row and each column has at most one nonzero element. It is shown that a singular quasi-permutation matrix with coefficients in a domain is a product of idempotent matrices. As an application, we prove that a nonnegative singular matrix having nonnegative von Neumann inverse (also known as generalized inverse) is a product of nonnegative idempotent matrices. [ABSTRACT FROM AUTHOR]

Unextendible sets of Mutually Unbiased Bases (MUBs) are examined from the point of view of complementary subalgebras. We show, that the linear span of less than d + 1 factors of M d ⊗ M d does not contain pure states, and therefore some complementary decompositions give rise to unextendible sets of MUBs. We provide some new complementary decompositions, and thus prove strong unextendibility of some set of MUBs. [ABSTRACT FROM AUTHOR]

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type Tr f ( Φ ( A p ) 1 / 2 Ψ ( B q ) Φ ( A p ) 1 / 2 ) and symmetric (anti-) norm functions of the form ‖ f ( Φ ( A p ) σ Ψ ( B q ) ) ‖ , where Φ and Ψ are positive linear maps, σ is an operator mean, and f ( x γ ) with a certain power γ is an operator monotone function on ( 0 , ∞ ) . Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on ( 0 , ∞ ) so that we prove joint concavity/convexity of more trace functions of Lieb type. [ABSTRACT FROM AUTHOR]

Let A and B be unital primitive Banach algebras with minimal idempotents. We prove that every surjective spectral isometry from A onto B is of the form λ Ψ where λ ∈ C with | λ | = 1 and Ψ is either an isomorphism or an anti-isomorphism from A onto B . As an application we show that, for all Banach spaces X and Y , the spectral nearisometries and the approximate spectrum-preserving maps from L ( X ) onto L ( Y ) are perturbations of actual spectral isometries and spectrum-preserving maps, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we explore various matrix-theoretic aspects of the hidden Z class and demonstrate how the concept of principal pivot transform can be effectively used to extend many existing results. In fact, we revisit various results obtained for hidden Z class by Mangasarian, Cottle and Pang in context of solving linear complementarity problems as linear programs. We identify hidden Z -matrices of special category and discuss the number of solutions of the associated linear complementarity problems. We also present game theoretic interpretation of various results related to hidden Z class and obtain proofs following the game theoretic approach of Raghavan for a subclass of Z -matrices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the free sets and the free subsemimodules in a semimodule over a commutative semiring S . First, we discuss some properties of the free sets and give a sufficient condition for a nonempty finite set to be free in a finitely generated free S -semimodule and obtain a relation between free set and linear independent set in an S -semimodule. Then we consider the free subsemimodules and prove that the rank of any free subsemimodule of a finitely generated S -semimodule M does not exceed that of M . Also, we give some equivalent descriptions for a commutative semiring S to have the property that all nonzero subsemimodules of any finitely generated free S -semimodule are free. Partial results obtained in the paper develop and generalize the corresponding results for modules over rings and linear spaces over fields. [ABSTRACT FROM AUTHOR]

In this paper we shall study non-commutative central moment inequalities with a focus on whether the commutative bounds are tight in the non-commutative case as well. We prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case. As an application, we shall present some lower estimates of the spread of Hermitian and normal matrices as well. [ABSTRACT FROM AUTHOR]

We prove that the Bernoulli numbers satisfy some special lower triangular Toeplitz systems of linear equations. One of these systems has a strong link with eleven Ramanujan's linear equations satisfied by the first eleven Bernoulli numbers. [ABSTRACT FROM AUTHOR]