A Markov-Nikolskii type inequality for absolutely monotone polynomials of order <Emphasis Type="Italic">K</Emphasis>

A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q′(x) ≥ 0, …, Q(k)(x) ≥ 0, for all x ε I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in L _p [−1, 1], p > 0, is established. One may guess that the right Markov factor is cn ²/k, and this indeed turns out to be the case. Similarly sharp results hold in the case of higher derivatives and Markov-Nikolskii type inequalities. There is also a remarkable connection between the right Markov inequality for absolutely monotone polynomials of order k in the supremum norm and essentially sharp bounds for the largest and smallest zeros of Jacobi polynomials. This is discussed in the last section of the paper.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

On distance sets of large sets of integer points

Distance sets of large sets of integer points are studied in dimensions at least 5. To any ε > 0 a positive integer Q _ɛ is constructed with the following property; If A is any set of integer points of upper density at least ε, then all large multiples of Q _ɛ ² occur as squares of distances between the points of the set A. Research supported in part by NSF Grant DMS-0456490.     

Combinatorial aspects of free associative algebras and cohomologies of Lie <Emphasis Type="Italic">p</Emphasis>-algebras with one defining relation

Let R and Q be elements of a free, associative, finitely generated algebra A‹X› over a field k. Assume that a leading homogeneous part Q _v does not have two-sided divisors and RQ = QR and R _v = Q ^t _v . In this paper, the solutions of the equation Σ _i x _i Ry _i = z in A‹X› are found and with their help, the identity theorem and Freiheitssatz for a finitely generated associative algebra k‹X; R = 0›with one defining relation are proved. As a consequence, similar theorems for Lie p-algebras with one defining relation are proved; these results are applied to the proof of the periodicity of cohomologies of Lie p-algebras with one defining relation. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

Liftings for Haar measure on {0,1} k

We callE ⊆ {0,1} ^k projective if for some countableA ⊆κ there is anE _A ⊆ {0, 1} ^A such thatE=E _A ×{0,1} ^k\A andE _A is a projective subset of the Cantor set {0, 1} ^A . We construct a model where Haar measure on {0,1} ^k has no projective lifting (and in particular no Baire lifting) for anyκ≥ω. Research partially supported by NATO Science Fellowship. The first author would like to thank the Mathematics Department at the University of Essex for its hospitality during the academic year 1988/89 while part of this research was being carried out. This research was initiated while the second author was a postdoctoral fellow at the University of Toronto. Its completion was supported by NSF grant DMS-8505550.     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

On the Stabilizer of the Automorphism Group of a 4—valent Vertex—transitive Graph with Odd—prime—power Order

Let X be a 4-valent connected vertex-transitive graph with odd-prime-power order p^κ（κ≥1） and let A be the full automorphism group of X.In this paper,we prove that the stabilizer Av of a vertex v in A is a 2-group if p≠5,or a {2,3}-group if p=5.Furthermore,if p=5|Av| is not divisible by 3^2.As a result ,we show that any 4-valent connected vertex-transitive graph with odd-prime-power order p^κ(κ≥1） is at most 1-arc-transitive for p≠5 and 2-arc-transitive for p=5.     

Successive projection method for solving the unbalanced Procrustes problem

We present a successive projection method for solving the unbalanced Procrustes problem: given matrix A∈Rn×n and B∈Rn×k, n>k, minimize the residual‖AQ-B‖F with the orthonormal constraint QTQ = Ik on the variant Q∈Rn×k. The presented algorithm consists of solving k least squares problems with quadratic constraints and an expanded balance problem at each sweep. We give a detailed convergence analysis. Numerical experiments reported in this paper show that our new algorithm is superior to other existing methods.     

Generalized Smash Products

In this paper．we study the ring #(D．B)and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all(B．D)-Hopf modules BM^D．Cai and Chen have proved this result in the case B=D=A．Secondly they have proved that if A has a nonzero left integral then A#A^*rat is a dense subring of Endk(A)．We prove that #(A,A) is a dense subring of Endk(Q),where Q is a certain subspace of #(A．A)under the condition that the antipode is bijective(see Theorem18).This condition is weaker than the condition that A has a nonzero integral.It is well known the antipode is bijective in case A has a nonzero integral．Furthermore if A has nonzero left integral,Q can be chosen to be A(see Corollary 19)and #(A,A)is both left and right primitive.Thus A#A^*rat #(A,A)-Endk(A)．Moreover we prove that the left singular ideal of the ring #(A,A)is zero．A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional,namely the ring #(A,A)has a finite uniform dimension．     

Almost completely decomposable groups with primary regulator quotients and their endomorphism rings

Let X be a block-rigid almost completely decomposable group of ring type with regulator A and p-primary regulator quotient X/A such that p ^l = exp X/A with natural l > 1. From the well-known fact p ^l End A ⊂ End X ⊂ End A it follows that End X = End X ∪ End A and p ^l End A = End X ∪ p ^l End A. Generalizing these, we determine the chain End X = ɛ _A ^(l) ⊂ ɛ _A ^(l−1) ⊂ ɛ _A ^(l−2) ⊂ ⋯ ⊂ ɛ _A ⁽¹⁾ ⊂ ɛ _A ⁽⁰⁾ = End A, satisfying p ^l−k ɛ _A ^(k) = End X ∪p ^l−k End A, and construct groups X _k ^′ and such that ɛ _A ^(k) = Hom , where k = 1, 2,..., l − 1. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 17–38, 2006.     

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

Streaming Algorithms for Line Simplification

We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p ₀,p ₁,p ₂,… in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k ²) additional storage.     

On the Schur multiplier of augmented algebras

Letk be a field, andA a finitely generatedk-algebra, with augmentation. Suppose there is a presentation ofA 0→I→R→A→0 whereR is a finitely generated freek-algebra andI is non-zero. IfA is infinite dimensional overk, Lewin proved thatR/I ² is not finitely presented. A stronger statement would be that the ‘Schur multiplier’ ofR/I ² is not finite dimensional. In the case thatA is an augmented domain, we prove this stronger statement, and some related statements.     

Dual space,carleson measure and sequence interpolation onA p (ϕ)(O<p<1)

LetD={z∈Σ:|z|<1} and ϕ be a normal function on [0,1). Forp∈(0,1) such a function ϕ is used to define a Bergman spaceA ^p (ϕ) onD with weight ϕ ^p (|·|)/(1-|·|²). In this paper, the dual space ofA ^p (ϕ) is given, four characteristics of Carleson measure onA ^p (ϕ) are obtained. Moreover, as an application, three sequence interpolation theorems inA ^p (ϕ) are derived. Supported by the Doctoral Program Foundation of Institute of Higher Education, P.R. China.     

Schur multiplier characterization of a class of infinite matrices

Let B _w (ℓ ^p ) denote the space of infinite matrices A for which A(x) ∈ ℓ ^p for all x = {x _k } _k=1^∞ ∈ ℓ ^p with |x _k | ↘ 0. We characterize the upper triangular positive matrices from B _w (ℓ ^p ), 1 < p < ∞, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.     

Equivalence of quotient Hilbert modules

LetM be a Hilbert module of holomorphic functions over a natural function algebraA(Ω), where Ω ⊆ ℂ ^m is a bounded domain. LetM ₀ ⊆M be the submodule of functions vanishing to orderk on a hypersurfaceZ ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modulesQ =M ⊖M ₀ The invariants are given explicitly in the particular case ofk = 2.     

On the number of elements of given order in a finitep-group

A. Kulakoff [9] proved that forp>2 the numberN _k =N _k (G) of solutions of the equationx ^{p k} =e in a non-cyclicp-groupG is divisible byp ^k+1. This result is a generalization of the well-known theorem of G. A. Miller asserting that the numberC _k =C _k (G) of cyclic subgroups of orderp ^k >p>2 is divisible byp. In this note we show that, as a rule: (1) ifk>1, thenN _k ≡0(modp ^k+p ); (2) ifk>2, thenC _k ≡0(modp ^p ). These facts are generalizations of many results from [1–5,8,9].     

Whitney theorem of interpolatory type for k-monotone functions

One of the main results of this paper is the following Whitney theorem of interpolatory type for k-monotone functions (i.e., functions f such that divided differences f[x ₀,…, x _k ] are nonnegative for all choices of (k + 1) distinct points x ₀,…, x _k .