Riesz transformations of distributions and a generalized Hardy space

LeiE(ℝⁿ) be the space of all functions on ℝⁿ which can continue to the entire holomorphic functions on ℂⁿ. We define Riesz transformation R_j of distributions as a linear transformation of the quotient spaceD′(ℝⁿ)/E(ℝⁿ) to itself, j=1,2,..., n. These generalized Riesz transformations share the same properties with the classical ones, such as . As applications we generalize further a theorem of F. & M. Riesz generalized by Stein and Weiss, and then define a generalized Hardy space, of which some properties are studied.     

Lower Bounds For Multidimensional Zero Sums

Let f(n, d) denote the least integer such that any choice of f(n, d) elements in contains a subset of size n whose sum is zero. Harborth proved that (n-1)2 ^d +1 f(n,d) (n-1)n ^d +1. The upper bound was improved by Alon and Dubiner to c _d n. It is known that f(n-1) = 2n-1 and Reiher proved that f(n-2) = 4n-3. Only for n = 3 it was known that f(n,d) > (n-1)2 ^d +1, so that it seemed possible that for a fixed dimension, but a sufficiently large prime p, the lower bound might determine the true value of f(p,d). In this note we show that this is not the case. In fact, for all odd n 3 and d 3 we show that .     

Elementary characterizations of generalized weighted Morrey-Campanato spaces

Let α∈ (0,∞), p, q ∈ [1,∞), s be a nonnegative integer, and ω∈ A1(Rn) (the class of Muckenhoupt's weights). In this paper, we introduce the generalized weighted Morrey-Campanato space L(α, p, q, s, ω; Rn) and obtain its equivalence on different p ∈ [1,β) and integers s ≥ nα (the integer part of nα), where β = (1q - α)-1 when α 1q or β = ∞ when α≥ 1q. We then introduce the generalized weighted Lipschitz space ∧(α, q, ω; Rn) and prove that L(α, p, q, s, ω; Rn)  ∧(α, q, ω; Rn) when α∈ (0,∞), s ≥ nα , and p ∈ [1,β).     

Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}      5. On sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers on sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers      Pavel Pražák  Pavel Trojovský 《Mathematica Slovaca》2010,60(6):751-770 New results about some sums s n (k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Y, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s n (k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.      6. Greedy Algorithm Computing Minkowski Reduced Lattice Bases with Quadratic Bit Complexity of Input Vectors      Hao CHEN  Liqing XU 《数学年刊B辑(英文版)》2011,32(6):857-862 The authors present an algorithm which is a modification of the Nguyen-Stehle greedy reduction algorithm due to Nguyen and Stehle in 2009. This algorithm can be used to compute the Minkowski reduced lattice bases for arbitrary rank lattices with quadratic bit complexity on the size of the input vectors. The total bit complexity of the algorithm is $O(n^2 \cdot (4n!)^n \cdot (\tfrac{{n!}} {{2^n }})^{\tfrac{n} {2}} \cdot (\tfrac{4} {3})^{\tfrac{{n(n - 1)}} {4}} \cdot (\tfrac{3} {2})^{\tfrac{{n^2 (n - 1)}} {2}} \cdot \log ^2 A) $O(n^2 \cdot (4n!)^n \cdot (\tfrac{{n!}} {{2^n }})^{\tfrac{n} {2}} \cdot (\tfrac{4} {3})^{\tfrac{{n(n - 1)}} {4}} \cdot (\tfrac{3} {2})^{\tfrac{{n^2 (n - 1)}} {2}} \cdot \log ^2 A) , where n is the rank of the lattice and A is maximal norm of the input base vectors. This is an O(log2 A) algorithm which can be used to compute Minkowski reduced bases for the fixed rank lattices. A time complexity n! · 3 n (log A) O(1) algorithm which can be used to compute the successive minima with the help of the dual Hermite-Korkin-Zolotarev base was given by Blomer in 2000 and improved to the time complexity n! · (log A) O(1) by Micciancio in 2008. The algorithm in this paper is more suitable for computing the Minkowski reduced bases of low rank lattices with very large base vector sizes.      7. Designs of ϕ-optimal control for second-order processes      Der-Shin Chang  Yuang-Chin Chiang 《Annals of the Institute of Statistical Mathematics》1980,32(1):275-281 Consider a realization of the process on the intervalT=[0,1] for functionsf 1(t),f 2(t),...,f n (t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f k (t)} k=1 n to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD s-optimal design problems are discussed.      8. On rigidity of Clifford torus in a unit sphere      Yi-wen Xu  Zhi-yuan Xu 《高校应用数学学报(英文版)》2011,26(1):121-126 We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S n+1 satisfying Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n + δ(n), then S ≡ n, i.e., M is one of the Clifford torus S~k ((k/n)~1/2 ) ×S~...      9. Limit theorems for the critical age-dependent branching process with infinite variance      Martin I. Goldstein  Fred M. Hoppe 《Stochastic Processes and their Applications》1977,5(3):297-305 Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0+. Then we find, as t → ∞, (P{Z(t)> 0})αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + uα)?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.      10. The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor      Lutz?VolkmannEmail author 《Combinatorica》2004,24(3):531-540 For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).      11. An extremal set theoretical characterization of some steiner systems      Peter Frankl 《Combinatorica》1983,3(2):193-199 Letn, k, t be integers,n>k>t≧0, and letm(n, k, t) denote the maximum number of sets, in a family ofk-subsets of ann-set, no two of which intersect in exactlyt elements. The problem of determiningm(n, k, t) was raised by Erdős in 1975. In the present paper we prove that ifk≦2t+1 andk−t is a prime, thenm(n, k, t)≦( t n )( k 2k-t-1 )/( t 2k-t-1 ). Moreover, equality holds if and only if an (n, 2k−t−1,t)-Steiner system exists. The proof uses a linear algebraic approach.      12. Roman and Total Domination      《Quaestiones Mathematicae》2013,36(6):749-757 Abstract A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑u ∈ V (G) f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.      13. The upper traceable number of a graph      Futaba Okamoto  Ping Zhang  Varaporn Saenpholphat 《Czechoslovak Mathematical Journal》2008,58(1):271-287 For a nontrivial connected graph G of order n and a linear ordering s: v 1, v 2, …, v n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t +(G) of G is t +(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t +(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.      14. On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series      Matthias Beck  Alan Stapledon 《Mathematische Zeitschrift》2010,264(1):195-207 For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)d + 1=?m 3 0 g(m)  tm{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracUnh(t)(1- t)d+1 = ?m 3 0g(nm)  tm{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n d , U n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.      15. New degree bounds for polynomial threshold functions      Ryan O’Donnell  Rocco A. Servedio 《Combinatorica》2010,30(3):327-358 A real multivariate polynomial p(x 1, …, x n ) is said to sign-represent a Boolean function f: {0,1} n →{−1,1} if the sign of p(x) equals f(x) for all inputs x∈{0,1} n . We give new upper and lower bounds on the degree of polynomials which sign-represent Boolean functions. Our upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of superconstant depth. Our lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968, improving results of Minsky and Papert. The lower bounds are proved constructively; we give explicit dual solutions to the necessary linear programs.      16. AA-Rings      《代数通讯》2013,41(10):3853-3860 Abstract Let R be a ring with identity such that R +, the additive group of R, is torsion-free of finite rank (tffr). The ring R is called an E-ring if End(R +) = {x ? ax : a ∈ R} and is called an A-ring if Aut(R +) = {x ? ux : u ∈ U(R)}, where U(R) is the group of units of R. While E-rings have been studied for decades, the notion of A-rings was introduced only recently. We now introduce a weaker notion. The ring R, 1 ∈ R, is called an AA-ring if for each α ∈ Aut(R +) there is some natural number n such that α n  ∈ {x ? ux : u ∈ U(R)}. We will find all tffr AA-rings with nilradical N(R) ≠ {0} and show that all tffr AA-rings with N(R) = {0} are actually E-rings. As a consequence of our results on AA-rings, we are able to prove that all tffr A-rings are indeed E-rings.      17. Arithmetic of semigroups of series in multiplicative systems      I. P. Il’inskaya 《Ukrainian Mathematical Journal》2009,61(7):1113-1122 We study the arithmetic of a semigroup MP\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?n = 0￥ ancn(x)    ( an 3 0,?n = 0￥ an = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { cn }n = 0￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup MP\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R n are true. We describe the class I0(MP)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in MP\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.      18. SOME ENDPOINT ESTIMATES FOR COMMUTATORS OF θ-TYPE CALDERON-ZYGMUND OPERATORS      Xiaoli Chen  Jie Sun  Dongxiang Chen 《分析论及其应用》2009,25(2):175-181 In this note, the authors prove that the commutator generated by θ-type Calderón-Zygmund operator T and a Lipschitz function is bounded from L p (R n ) into (R n ) and also maps from (R n ) into BMO (R n ). Supported by NSFC(10571014), NSFC(10571156), the Doctor Foundation of Jxnu (2443), the Natural Science Foundation of Jiangxi province(2008GZS0051).      19. Graceful labellings of paths      Rohan Cattell 《Discrete Mathematics》2007,307(24):3161-3176 It is easily shown that every path has a graceful labelling, however, in this paper we show that given almost any path P with n vertices then for every vertex v∈V(P) and for every integer i∈{0,…,n-1} there is a graceful labelling of P such that v has label i. We show precisely when these labellings can also be α-labellings. We then extend this result to strong edge-magic labellings. In obtaining these results we make heavy use of π-representations of α-labellings and review some relevant results of Kotzig and Rosa.      20. Blow up of subcritical quantities at the first singular time of the mean curvature flow      Nam Q. Le 《Geometriae Dedicata》2011,151(1):361-371 Consider a family of smooth immersions F(·,t) : Mn? \mathbbRn+1{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò0tòMs\frac|A|n + 2 log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.

On sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers on sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers

New results about some sums s _n (k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Y, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s _n (k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.     

Greedy Algorithm Computing Minkowski Reduced Lattice Bases with Quadratic Bit Complexity of Input Vectors

The authors present an algorithm which is a modification of the Nguyen-Stehle greedy reduction algorithm due to Nguyen and Stehle in 2009. This algorithm can be used to compute the Minkowski reduced lattice bases for arbitrary rank lattices with quadratic bit complexity on the size of the input vectors. The total bit complexity of the algorithm is $O(n^2 \cdot (4n!)^n \cdot (\tfrac{{n!}} {{2^n }})^{\tfrac{n} {2}} \cdot (\tfrac{4} {3})^{\tfrac{{n(n - 1)}} {4}} \cdot (\tfrac{3} {2})^{\tfrac{{n^2 (n - 1)}} {2}} \cdot \log ^2 A) $O(n^2 \cdot (4n!)^n \cdot (\tfrac{{n!}} {{2^n }})^{\tfrac{n} {2}} \cdot (\tfrac{4} {3})^{\tfrac{{n(n - 1)}} {4}} \cdot (\tfrac{3} {2})^{\tfrac{{n^2 (n - 1)}} {2}} \cdot \log ^2 A) , where n is the rank of the lattice and A is maximal norm of the input base vectors. This is an O(log² A) algorithm which can be used to compute Minkowski reduced bases for the fixed rank lattices. A time complexity n! · 3 ⁿ (log A) ^O(1) algorithm which can be used to compute the successive minima with the help of the dual Hermite-Korkin-Zolotarev base was given by Blomer in 2000 and improved to the time complexity n! · (log A) ^O(1) by Micciancio in 2008. The algorithm in this paper is more suitable for computing the Minkowski reduced bases of low rank lattices with very large base vector sizes.     

Designs of ϕ-optimal control for second-order processes

Consider a realization of the process on the intervalT=[0,1] for functionsf ₁(t),f ₂(t),...,f _n (t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f _k (t)} _k=1 ⁿ to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD _s-optimal design problems are discussed.     

On rigidity of Clifford torus in a unit sphere

We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S n+1 satisfying Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n + δ(n), then S ≡ n, i.e., M is one of the Clifford torus S~k ((k/n)~1/2 ) ×S~...     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

An extremal set theoretical characterization of some steiner systems

Letn, k, t be integers,n>k>t≧0, and letm(n, k, t) denote the maximum number of sets, in a family ofk-subsets of ann-set, no two of which intersect in exactlyt elements. The problem of determiningm(n, k, t) was raised by Erdős in 1975. In the present paper we prove that ifk≦2t+1 andk−t is a prime, thenm(n, k, t)≦( _t ⁿ )( _k ^2k-t-1 )/( _t ^2k-t-1 ). Moreover, equality holds if and only if an (n, 2k−t−1,t)-Steiner system exists. The proof uses a linear algebraic approach.     

Roman and Total Domination

Abstract

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number ^γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑_{u ∈ V (G)} f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if ^γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.     

The upper traceable number of a graph

For a nontrivial connected graph G of order n and a linear ordering s: v ₁, v ₂, …, v _n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t ⁺(G) of G is t ⁺(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t ⁺(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.     

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)^{d + 1}=?_{m 3 0} g(m)  t^m{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracU_nh(t)(1- t)^d+1 = ?_{m 3 0}g(nm)  t^m{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n _d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n _d , U _n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.     

New degree bounds for polynomial threshold functions

A real multivariate polynomial p(x ₁, …, x _n ) is said to sign-represent a Boolean function f: {0,1} ⁿ →{−1,1} if the sign of p(x) equals f(x) for all inputs x∈{0,1} ⁿ . We give new upper and lower bounds on the degree of polynomials which sign-represent Boolean functions. Our upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of superconstant depth. Our lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968, improving results of Minsky and Papert. The lower bounds are proved constructively; we give explicit dual solutions to the necessary linear programs.     

AA-Rings

Abstract

Let R be a ring with identity such that R ⁺, the additive group of R, is torsion-free of finite rank (tffr). The ring R is called an E-ring if End(R ⁺) = {x ? ax : a ∈ R} and is called an A-ring if Aut(R ⁺) = {x ? ux : u ∈ U(R)}, where U(R) is the group of units of R. While E-rings have been studied for decades, the notion of A-rings was introduced only recently. We now introduce a weaker notion. The ring R, 1 ∈ R, is called an AA-ring if for each α ∈ Aut(R ⁺) there is some natural number n such that α ⁿ  ∈ {x ? ux : u ∈ U(R)}. We will find all tffr AA-rings with nilradical N(R) ≠ {0} and show that all tffr AA-rings with N(R) = {0} are actually E-rings. As a consequence of our results on AA-rings, we are able to prove that all tffr A-rings are indeed E-rings.     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.     

SOME ENDPOINT ESTIMATES FOR COMMUTATORS OF θ-TYPE CALDERON-ZYGMUND OPERATORS

In this note, the authors prove that the commutator generated by θ-type Calderón-Zygmund operator T and a Lipschitz function is bounded from L ^p (R ⁿ ) into (R ⁿ ) and also maps from (R ⁿ ) into BMO (R ⁿ ). Supported by NSFC(10571014), NSFC(10571156), the Doctor Foundation of Jxnu (2443), the Natural Science Foundation of Jiangxi province(2008GZS0051).     

Graceful labellings of paths

It is easily shown that every path has a graceful labelling, however, in this paper we show that given almost any path P with n vertices then for every vertex v∈V(P) and for every integer i∈{0,…,n-1} there is a graceful labelling of P such that v has label i. We show precisely when these labellings can also be α-labellings. We then extend this result to strong edge-magic labellings. In obtaining these results we make heavy use of π-representations of α-labellings and review some relevant results of Kotzig and Rosa.     

Blow up of subcritical quantities at the first singular time of the mean curvature flow

Consider a family of smooth immersions F(·,t) : Mⁿ? \mathbbRⁿ⁺¹{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò₀^tò_Ms\frac|A|^{n + 2} log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.