A criterion for generating full matrix algebras

Let M_n(F) be the algebra of n×n matrices over a field F, and let A∈M_n(F) have characteristic polynomial c(x)=p₁(x)p₂(x)?p_r(x) where p₁(x),…,p_r(x) are distinct and irreducible in F[x]. Let X be a subalgebra of M_n(F) containing A. Under a mild hypothesis on the p_i(x), we find a necessary and sufficient condition for X to be M_n(F).     

On the enumeration of certain weighted graphs

We enumerate weighted simple graphs with a natural upper bound condition on the sum of the weight of adjacent vertices. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that the generating function for connected bipartite simple graphs is of the form p₁(x)/(1-x)^m+1. For nonbipartite simple graphs, we get a generating function of the form p₂(x)/(1-x)^m+1(1+x)^l. Here m is the number of vertices of the graph, p₁(x) is a symmetric polynomial of degree at most m, p₂(x) is a polynomial of degree at most m+l, and l is a nonnegative integer. In addition, we give computational results for various graphs.     

A polynomial dual of partitions

Let P(x) = Σ_i=0ⁿa_ixⁱ be a nonnegative integral polynomial. The polynomial P(x) is m-graphical, and a multi-graph G a realization of P(x), provided there exists a multi-graph G containing exactly P(1) points where a_i of these points have degree i for 0≤i≤n. For multigraphs G, H having polynomials P(x), Q(x) and number-theoretic partitions (degree sequences) π, ?, the usual product P(x)Q(x) is shown to be the polynomial of the Cartesian product G × H, thus inducing a natural product π? which extends that of juxtaposing integral multiple copies of ?. Skeletal results are given on synthesizing a multi-graph G via a natural Cartesian product G₁ × … × G_k having the same polynomial (partition) as G. Other results include an elementary sufficient condition for arbitrary nonnegative integral polynomials to be graphical.     

Polynomials arising in factoring generalized Vandermonde determinants II: A condition for monicity

In our previous paper [1], we observed that generalized Vandermonde determinants of the form V_n;ν(x₁,…,x_s) = |x_i^μk|, 1 ≤ i, k ≤ n, where the x_i are distinct points belonging to an interval [a, b] of the real line, the index n stands for the order, the sequence μ consists of ordered integers 0 ≤ μ₁ < μ₂ < … < μ_n, can be factored as a product of the classical Vandermonde determinant and a Schur function. On the other hand, we showed that when x = x_n, the resulting polynomial in x is a Schur function which can be factored as a two-factors polynomial: the first is the constant ∏_i=1ⁿ⁻¹ x_i^μ₁ times the monic polynomial ∏_i=1ⁿ⁻¹ (x − x_i, while the second is a polynomial P_M(x) of degree M = m_n−1 − n + 1. In this paper, we first present a typical application in which these factorizations arise and then we discuss a condition under which the polynomial P_M (x) is monic.     

On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

On R-robustly entropy-expansive diffeomorphisms

For a homoclinic class H(p _f ) of f ?? Diff¹(M), f?OH(p _f ) is called R-robustly entropy-expansive if for g in a locally residual subset around f, the set ?? _? (x) = {y ?? M: dist(g ⁿ (x), g ⁿ (y)) ?? g3 (?n ?? ?)} has zero topological entropy for each x ?? H(p _g ). We prove that there exists an open and dense set around f such that for every g in it, H(p _g ) admits a dominated splitting of the form E ?? F ₁ ?? ... ?? F _k ?? G where all of F _i are one-dimensional and non-hyperbolic, which extends a result of Pacifico and Vieitez for robustly entropy-expansive diffeomorphisms. Some relevant consequences are also shown.     

On the quantum filtration of the Khovanov-Rozansky cohomology

We prove the quantum filtration on the Khovanov-Rozansky link cohomology Hp with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to Hp that is invariant under Reidemeister moves, whose E₁ term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology Hn. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the self-linking number of transversal links in standard contact S³.     

Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR(pe,d) is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x-1)m+pg(x). Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR(pe,d) is equal to pe-1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.     

Generalized varieties of sums of powers

Let X ? P^N be an irreducible, non-degenerate variety. The generalized variety of sums of powers V S P_H^X(h) of X is the closure in the Hilbert scheme Hilb_h (X) of the locus parametrizing collections of points {x₁,..., x_h} such that the (h -1)-plane >x₁,..., x_h> passes through a fixed general point p ∈ P^N. When X = V_dⁿ is a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S P_H^X(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S P_H^X(h) are inherited from the birational geometry of variety X itself.     

Cyclotomy and addition sets

In this paper, we develop methods to solve the polynomial congruence θ(x)θ(x^g) ≡ d + λ(1 + x +… + x^p?1) (mod x^p ? 1), where p is an odd prime and θ(x) is a polynomial with nonnegative integral coefficients. Using these methods, we construct some new addition sets that are the unions of index classes for some primes p. We also establish the nonexistence of both the (95, 10, 1, 18)-addition set and the (95, 10, 1, 56)-addition set.     

Eigenvalue estimate for the weighted p-Laplacian

Let M be an n-dimensional closed manifold with metric g, dμ = e ^h(x) dV(x) be the weighted measure and ? _{μ, p} be the weighted p-Laplacian. In this article, we get the lower bound estimate of the first nonzero eigenvalue for the weighted p-Laplacian when the m-dimensional Bakry-émery curvature has a positive lower bound.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.     

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal solution R(x)∈Uq(g)^⊗2 of the Quantum Dynamical Yang-Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang-Baxter Equation. It can be built from the standard R-matrix and from the solution F(x)∈Uq(g)^⊗2 of the Quantum Dynamical coCycle Equation as . F(x) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation.Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g=sl(n+1)(n?1) only, there could exist an element M(x)∈Uq(sl(n+1)) such that the dynamical gauge transform RJ of R(x) by M(x),

RJ=M₁⁻¹(x)M₂(xqh₁⁾⁻¹R(x)M₁(xqh₂)M₂(x),     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

Detecting rigid convexity of bivariate polynomials

Given a polynomial x∈Rⁿ?p(x) in n=2 variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set P={x:p(x)?0} containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial p(x) is hyperbolic with respect to the origin. The problem boils down to checking whether a univariate polynomial matrix is positive semidefinite, an optimization problem that can be solved with eigenvalue decomposition. When the variety C={x:p(x)=0} is an algebraic curve of genus zero, a second algorithm based on Bézoutians is proposed to detect whether P has an LMI representation and to build such a representation from a rational parametrization of C. Finally, some extensions to positive genus curves and to the case n>2 are mentioned.     

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L_p(R^d) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ R^d > 1) on the variable Lebesgue spaces L_p(·)(R^d), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator M_s^γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator M_s^γδ is bounded on the space L_p(·)(R^d) (or the maximal operator is of weak-type (p(·), p(·))).     

Stably average shadowable homoclinic classes

Let p be a hyperbolic periodic saddle of a diffeomorphism of f on a closed smooth manifold M, and let H_f(p) be the homoclinic class of f containing p. In this paper, we show that if H_f(p) is locally maximal and every hyperbolic periodic point in H_f(p) is uniformly far away from being nonhyperbolic, and H_f(p) has the average shadowing property, then H_f(p) is hyperbolic.     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.