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),

     

On number system constructions

We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M ⁻¹) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M ⁻¹. We shall prove further that if the polynomial f(x) = c ₀ + c ₁ x + ··· + c _k x ^k ∈ Z[x], c _k = 1 satisfies the condition |c ₀| > 2 Σ _i=1 ^k |c _i | then there is a suitable digit set D for which (Z ^k , M, D) is a number system, where M is the companion matrix of f(x). The research was supported by OTKA-T043657 and Bolyai Fellowship Committee.     

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 real hypersurfaces with η-parallel curvature tensor in complex space forms

The purpose of this paper is to give a complete classification of real hypersurfces M in complex space forms M _n(c), c≠0 in terms of an η-parallel curvature tensor and a certain commutative condition defined on the distribution T ₀={X∈T _x M| X⊥ξ} of M in M _n(c).     

Non-spectrality of planar self-affine measures with three-elements digit set

The self-affine measure μM,D associated with an affine iterated function system {?d(x)=M−1(x+d)}_d∈D is uniquely determined. The problems of determining the spectrality or non-spectrality of a measure μM,D have been 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 for an expanding integer matrix M∈M₂(Z) and the three-elements digit set D given by

     

Global hypoellipticity for first-order operators on closed smooth manifolds

The main goal of this paper is to address global hypoellipticity issues for the class of first-order pseudo-differential operators L = D_t + C(t, x,D_x), where (t, x) ∈ T × M, T is the one-dimensional torus, M is a closed manifold, and C(t, x,D_x) is a first-order pseudo-differential operator on M, smoothly depending on the periodic variable t. In the case of separation of variables, when C(t, x,D_x) = a(t)p(x,D_x) + ib(t)q(x,D_x), we give necessary and sufficient conditions for the global hypoellipticity of L. In particular, we show that the famous (P) condition of Nirenberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of L.     

Some asymptotic formulas on generalized divisor functions,II

Let A be an infinite sequence of positive integers a₁ < a₂ <… and put $\text{f}{}_{\mathrm{A}}\text{(x) =}\text{Σ}{}_{\mathrm{a\in A,\; a\le x}}\text{(}\text{1}\text{a}\text{)}$, $\text{D}{}_{\mathrm{A}}\text{(x) =}\text{max}{}_{\mathrm{1\le n\le x}}\text{Σ}{}_{\mathrm{<mtext>a\in A,</mtext><mtext>a</mtext><mtext>n</mtext>}}\text{1}$. In Part I, it was proved that $\text{lim}{}_{\mathrm{x\to +\infty }}\text{sup}\text{D}{}_{\mathrm{A}}\text{(x)}\text{f}{}_{\mathrm{A}}\text{(x)}\text{= +∞}$. In this paper, this theorem is sharpened by estimating D_A(x) in terms of f_A(x). It is shown that lim_x→+∞sup D_A(x) exp(?c₁(logf_A(x))²) = +∞ and that this assertion is not true if c₁ is replaced by a large constant c₂.     

On the homeomorphism groups of manifolds and their universal coverings

Let H _c (M) stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold M. It is shown that H _c (M) is perfect and simple under mild assumptions on M. Next, conjugation-invariant norms on H _c (M) are considered and the boundedness of H _c (M) and its subgroups is investigated. Finally, the structure of the universal covering group of H _c (M) is studied.     

On the parity of the partition function

Let N be the set of all positive integers and D a subset of N. Let p(D,n) be the number of partitions of n with parts in D and let |D(x)| denote the number of elements of D not exceeding x. It is proved that if D is an infinite subset of N such that p(D,n) is even for all n?n₀, then |D(x)|?logx/log2−logn₀/log2. Moreover, if D is an infinite subset of N such that p(D,n) is odd for all n?n₀ and , then |D(x)|?logx/log2−logn₀/log2. These lower bounds are essentially the best possible.     

On the dimension of the space of ℝ-places of certain rational function fields

We prove that for every n ∈ ? the space M(K(x ₁, …, x _n ) of ?-places of the field K(x ₁, …, x _n ) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x ₁, …, x _n )) ≤ n. For n = 2 the space M(K(x ₁, x ₂)) has covering and integral dimensions dimM(K(x ₁, x ₂)) = dim_? M(K(x ₁, x ₂)) = 2 and the cohomological dimension dim _G M(K(x ₁, x ₂)) = 1 for any Abelian 2-divisible coefficient group G.     

Analytic extension of non quasi-analytic Whitney jets of Roumieu type

Let (M_r)_{r∈? 0} be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi-analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function ? on ?ⁿ belonging to the non quasi-analytic (M_r)-class of Roumieu type, there is an element g of the same class which is analytic on ?ⁿ F and such that D^α ?(x) = D^αg(x) for every σ ∈ ?₀ ⁿ SBAP and x ∈ F.     

How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?

 Let D be a semicomplete multipartite digraph, with partite sets V ₁, V ₂,…, V _c, such that |V ₁|≤|V ₂|≤…≤|V _c|. Define f(D)=|V(D)|−3|V _c|+1 and . We define the irregularity i(D) of D to be max|d ⁺(x)−d ⁻(y)| over all vertices x and y of D (possibly x=y). We define the local irregularity i _l(D) of D to be max|d ⁺(x)−d ⁻(x)| over all vertices x of D and we define the global irregularity of D to be i _g(D)=max{d ⁺(x),d ⁻(x) : x∈V(D)}−min{d ⁺(y),d ⁻(y) : y∈V(D)}. In this paper we show that if i _g(D)≤g(D) or if i _l(D)≤min{f(D), g(D)} then D is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from [6]. Our result also gives support to the conjecture from [6] that all diregular c-partite tournaments (c≥4) are pancyclic, and it is used in [9], which proves this conjecture for all c≥5. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, D, with i _g(D)=i(D)=i _l(D)=g(D)+?≤f(D)+1. Revised: September 17, 1998     

Homomorphisms of matrix semigroups over division rings from dimension two to four

Let D be an arbitrary division ring and M_n(D) the multiplicative semigroup of all n×n matrices over D. We study non-degenerate, injective homomorphisms from M₂(D) to M₄(D). In particular, we present a structural result for the case when D is the ring of quaternions.     

Local behaviour of first passage probabilities

Suppose that S is an asymptotically stable random walk with norming sequence c _n and that T _x is the time that S first enters (x, ∞), where x?≥ 0. The asymptotic behaviour of P(T ₀?=?n) has been described in a recent paper of Vatutin and Wachtel (Probab. Theory Relat. Fields 143:177–217, 2009), and here we build on that result to give three estimates for P(T _x ?=?n), which hold uniformly as n → ∞ in the regions x?=?o(c _n ), x?=?O(c _n ), and x/c _n → ∞, respectively.     

Comments of the paper, “stochastic integral equations applied to telecommunications traffic without delay”

Let F be a univariate distribution with negative expectation, and let M denote the distribution of the positive maxima of a random walk generated by a sequence of independent observations from F. We consider the Laplace transforms of 1?F(x) and 1?M(x). A relation between the transforms yields some known results on the moments and the regularly varying properties of the two distributions.     

A nonlinear parabolic-Sobolev equation

Let M and L be (nonlinear) operators in a reflexive Banach space B for which Rg(M + L) = B and $\text{¦(Mx ? My) + α(Lx ? Ly)¦ ? | mx ? My |}$ for all α > 0 and pairs x, y in D(M) ∩ D(L). Then there is a unique solution of the Cauchy problem (Mu(t))′ + Lu(t) = 0, Mu(0) = v₀. When M and L are realizations of elliptic partial differential operators in space variables, this gives existence and uniqueness of generalized solutions of boundary value problems for nonlinear partial differential equations of mixed parabolic-Sobolev type.     

Coloring some classes of mixed graphs

We consider the coloring problem for mixed graphs, that is, for graphs containing edges and arcs. A mixed coloring c is a coloring such that for every edge [x_i,x_j], c(x_i)≠c(x_j) and for every arc (x_p,x_q), c(x_p)<c(x_q). We will analyse the complexity status of this problem for some special classes of graphs.     

Iterative methods for linear complementarity problems with upperbounds on primary variables

This paper is concerned with iterative methods for the linear complementarity problem (LCP) of findingx and y in Rⁿ such thatc+Dx+y≥0, b-x≥0, andx ^T (c+Dx+y)=y ^T (b-x)=0 whenb>0. This is the LCP (M, q) withM=( _{l 0} ^{D t} ), which is in turn equivalent to a linear variational inequality over a rectangle. This type of problem arises, for example, from quadratic programming (QP) problems with upper and lower bounds ifD is symmetric, and from multicommodity market equilibrium problems with institutional price controls imposed ifD is not necessarily symmetric. Iterative methods for the LCP (M, q) with symmetricM are well developed and studied using QP applications. For nonsymmetric cases withM being either an H-matrix with positive diagonals, or a Z-matrix, there exists a robust iterative method with guaranteed convergence. This paper extends this algorithm so that the LCP (M, q) withM=( _{l 0} ^{D t} ) which is neither symmetric, nor an H-matrix with positive diagonals, nor a Z-matrix, can be processed when onlyD notM satisfies such properties. The case whereD is nonsymmetric is explicitly discussed.     

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom _R (−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x ₁, …, x _n is a regular sequence on M contained in α, then H _{(x1, …,xnR} ⁿ D(H _a ⁿ (M))) is a homomorphic image of D(M), where H _b ⁱ (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H _{(x1, …,xn)R} ⁿ (D(H _a ⁿ (M)))) ⋟ D(D(M)).