Non-spectral problem for a class of planar self-affine measures

The self-affine measure μM,D corresponding to an expanding matrix M∈Mn(R) and a finite subset D⊂Rn is supported on the attractor (or invariant set) of the iterated function system {?d(x)=M⁻¹(x+d)}_d∈D. The spectral and non-spectral problems on μM,D, including the spectrum-tiling problem implied in them, have received much attention in recent years. One of the non-spectral problem on μM,D is to estimate the number of orthogonal exponentials in L²(μM,D) and to find them. In the present paper we show that if a,b,c∈Z, |a|>1, |c|>1 and ac∈Z?(3Z),

     

Polynomially bounded solutions to the Loewner differential equation in several complex variables

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z,t)=etAz+?, where A∈L(Cn,Cn) has the property m(A)>0. Here m(A)=min{R〈A(z),z〉:‖z‖=1}. We also give sufficient conditions for g(z,t)=L(f(z,t)) to be polynomially bounded, where f(z,t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.     

Linear-time certifying algorithms for near-graphical sequences

A sequence 〈d₁,d₂,…,d_n〉 of non-negative integers is graphical if it is the degree sequence of some graph, that is, there exists a graph G on n vertices whose ith vertex has degree d_i, for 1≤i≤n. The notion of a graphical sequence has a natural reformulation and generalization in terms of factors of complete graphs.If H=(V,E) is a graph and g and f are integer-valued functions on the vertex set V, then a (g,f)-factor of H is a subgraph G=(V,F) of H whose degree at each vertex v∈V lies in the interval [g(v),f(v)]. Thus, a (0,1)-factor is just a matching of H and a (1, 1)-factor is a perfect matching of H. If H is complete then a (g,f)-factor realizes a degree sequence that is consistent with the sequence of intervals 〈[g(v₁),f(v₁)],[g(v₂),f(v₂)],…,[g(v_n),f(v_n)]〉.Graphical sequences have been extensively studied and admit several elegant characterizations. We are interested in extending these characterizations to non-graphical sequences by introducing a natural measure of “near-graphical”. We do this in the context of minimally deficient (g,f)-factors of complete graphs. Our main result is a simple linear-time greedy algorithm for constructing minimally deficient (g,f)-factors in complete graphs that generalizes the method of Hakimi and Havel (for constructing (f,f)-factors in complete graphs, when possible). It has the added advantage of producing a certificate of minimum deficiency (through a generalization of the Erdös-Gallai characterization of (f,f)-factors in complete graphs) at no additional cost.     

On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

We consider the set Σ(R,C) of all m×n matrices having 0-1 entries and prescribed row sums R=(r₁,…,rm) and column sums C=(c₁,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.     

Normality criteria of meromorphic functions with multiple zeros

Take positive integers n,k?2. Let F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k. If, for each pair (f,g) in F, fn(f(k)) and gn(g(k)) share a non-zero complex number a ignoring multiplicity, then F is normal in D.     

Accretive operators and Cassels inequality

Let a,b>0 and let Z∈M_n(R) such that Z lies into the operator ball of diameter [aI,bI]. Then for all positive definite A∈M_n(R),

     

On the decomposition of the tensor algebra of the classical Lie algebras

Let $\text{g}$_n be the Lie algebra gl_n(C), let S($\text{g}$_n) be the symmetric algebra of $\text{g}$_n, and let T($\text{g}$_n) be the tensor algebra of $\text{g}$_n. In a recent paper, R. K. Gupta studied certain sequences of representations R_∞ = (R_n)^∞_{n = 1}, where R_n is a representation of $\text{g}$_n. These sequences have the property that every irreducible representation occurring in S($\text{g}$_n) is in exactly one of these sequences. Fixing f, she considers s(R_∞, f) which is the limit on n of the multiplicity of R_n in S^f($\text{g}$_n), the fth-graded piece of S($\text{g}$_n). She and R. P. Stanley independently showed that the limit s(R_∞, f) exists and is given by an amazingly elegant formula. They call s(R_∞, f) the stable multiplicity of R_n in S^f($\text{g}$_n). In this paper, an entirely different approach is used to extend the above result in several directions. Appropriately defined sequences R_∞ for all of the classical Lie algebras $\text{g}$_n are studied, and a simple formula for the stable multiplicity m(R)_∞, ψ, f, $\text{g}$_∞) of R_n in the ψ-isotypic component of T^f($\text{g}$_n), where ψ is any irreducible character of the symmetric group tS_f, is obtained. As in the work of Gupta and Stanley, the expressions for m(R)_∞, ψ, f, $\text{g}$_∞) are amazingly simple. Special cases include the stable decomposition of the tensor algebra, the symmetric algebra and the exterior algebra of $\text{g}$_n. As a byproduct of our proof, a “stable” decomposition of every isotypic component of T($\text{g}$_n) is obtained. This combinatorial decomposition is in some sense a generalization of Kostant's decomposition of S($\text{g}$_n) into direct sum of the harmonics and the ideal generated by the invariants of positive degree. To be precise, for f <n the combinatorial decomposition of T^f($\text{g}$_n) projects onto Kostant's decomposition of S^f($\text{g}$_n).     

A note on approximation properties of q-Durrmeyer operators

In this paper, the approximation properties of q-Durrmeyer operators D_n,q(f;x) for f∈C[0,1] are discussed. The exact class of continuous functions satisfying approximation process lim_n→∞D_n,q(f;x)=f(x) is determined. The results of the paper provide an elaboration of the previously-known ones on operators D_n,q.     

Homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems , where t∈R, u∈Rn and W₁,W₂∈C¹(R×Rn,R) and f∈C(R,Rn) are not necessary periodic in t. This result generalizes and improves some existing results in the literature.     

Minimal zero sum sequences of length four over finite cyclic groups

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Let G be a finite cyclic group. Every sequence S over G can be written in the form S=(n₁g)⋅…⋅(nlg) where g∈G and n₁,…,nl∈[1,ord(g)], and the index ind(S) of S is defined to be the minimum of (n₁+?+nl)/ord(g) over all possible g∈G such that 〈g〉=〈supp(S)〉. The problem regarding the index of sequences has been studied in a series of papers, and a main focus is to determine sequences of index 1. In the present paper, we show that if G is a cyclic of prime power order such that gcd(|G|,6)=1, then every minimal zero-sum sequence of length 4 has index 1.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=BC7josX_xVs.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production

This article provides sharp constructive upper and lower bound estimates for the Boltzmann collision operator with the full range of physical non-cut-off collision kernels (γ>−n and s∈(0,1)) in the trilinear L²(Rn) energy 〈Q(g,f),f〉. These new estimates prove that, for a very general class of g(v), the global diffusive behavior (on f) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works (Gressman and Strain, 2010 [15], 2011 [16]). We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non-cut-off Boltzmann collision operator in the energy space L²(Rn).     

Strong commutativity preserving maps on Lie ideals

Let A be a prime ring and let R be a noncentral Lie ideal of A. An additive map f:R→A is called strong commutativity preserving (SCP) on R if [f(x),f(y)]=[x,y] for all x,y∈R. In this paper we show that if f is SCP on R, then there exist λ∈C, λ²=1 and an additive map μ:R→Z(A) such that f(x)=λx+μ(x) for all x∈R where C is the extended centroid of A, unless charA=2 and A satisfies the standard identity of degree 4.     

On the integrality of nth roots of generating functions

Motivated by the discovery that the eighth root of the theta series of the E₈ lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f∈R (where R=1+xZ?x?) can be written as f=gn for g∈R, n?2. Let Pn:={gn|g∈R} and let . We show among other things that (i) for f∈R, f∈Pn⇔f (mod μn)∈Pn, and (ii) if f∈Pn, there is a unique g∈Pn with coefficients mod μn/n such that f≡gn (mod μn). In particular, if f≡1 (mod μn) then f∈Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E₈ in R⁸) is in Pn if n is of the form i²j³k⁵ (i?3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed-Muller code of length m² is in Pr² (and similarly that the theta series of the Barnes-Wall lattice BWm² is in Pm²). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f∈Pn (n?2) with coefficients restricted to the set {1,2,…,n}.     

Properties of convergence for the q-Meyer-König and Zeller operators

In this paper, we discuss properties of convergence for the q-Meyer-König and Zeller operators Mn,q. Based on an explicit expression for Mn,q(t²,x) in terms of q-hypergeometric series, we show that for qn∈(0,1], the sequence (Mn,qn(f))_n?1 converges to f uniformly on [0,1] for each f∈C[0,1] if and only if limn→∞qn=1. For fixed q∈(0,1), we prove that the sequence (Mn,q(f)) converges for each f∈C[0,1] and obtain the estimates for the rate of convergence of (Mn,q(f)) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-König and Zeller operators for fixed 0<q<1. If 0<q<1, f∈C¹[0,1], we show that the rate of convergence for the Meyer-König and Zeller operators is o(qn) if and only if

     

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),     

An algorithm for unimodular completion over Laurent polynomial rings

We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field K. More precisely, given n?3 and a unimodular vector V=^t(v₁,…,v_n)∈Rⁿ (that is, such that 〈v₁,…,v_n〉=R), the algorithm computes a matrix M in M_n(R) whose determinant is a monomial such that MV=^t(1,0,…,0), and thus M^-1 is a completion of V to an invertible matrix.     

Classification of pinched positive scalar curvature manifolds

Let (Mn,g), n?3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg?n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition

     

On the root closedness of continuous function algebras

For a compact Hausdorff space X, C(X) denotes the algebra of all complex-valued continuous functions on X. For a positive integer n, we say that C(X) is n-th root closed if, for each f∈C(X), there exists g∈C(X) such that f=gn. It is shown that, for each integer m?2, there exists a compact Hausdorff space Xm such that C(Xm) is m-th root closed, but not n-th root closed for each integer n relatively prime to m. This answers a question posed by Countryman Jr. [R.S. Countryman Jr., On the characterization of compact Hausdorff X for which C(X) is algebraically closed, Pacific J. Math. 20 (1967) 433-438] et al.     

An extension of Birkhoff’s theorem with an application to determinants

Let M_n(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈M_n(C), that is, J=J^*,J²=I, let us consider Cⁿ endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cⁿ. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j₁,?,j_n)=I_r⊕-I_n-r. For T=(|t_ik|²)∈M_n(R), the matrices of the form T=(|t_ik|²j_ij_k), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.