Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S⊂ℝ ^k+m be a compact semi-algebraic set defined by P ₁≥0,…,P _ℓ ≥0, where P _i ∈ℝ[X ₁,…,X _k ,Y ₁,…,Y _m ], and deg (P _i )≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ ^k+m onto ℝ ^m . We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m) ^{O(q ℓ)}. We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is . For fixed q and ℓ, both the bounds are polynomial in k+m. The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship.     

The <Emphasis Type="Italic">f</Emphasis>-Vector of the Descent Polytope

For a positive integer n and a subset S⊆[n−1], the descent polytope DP  _S is the set of points (x ₁,…,x _n ) in the n-dimensional unit cube [0,1] ⁿ such that x _i ≥x _i+1 if i∈S and x _i ≤x _i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F _S (t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.     

Sausages are good packings

LetB ^d be thed-dimensional unit ball and, for an integern, letC _n ={x ¹,...,x ⁿ } be a packing set forB ^d , i.e.,|x ⁱ −x ^j |≥2, 1≤i<j≤n. We show that for every a dimensiond(ρ) exists such that, ford≥d(ρ),V(conv(C _n )+ρB ^d )≥V(conv(S _n )+ρB ^d ), whereS _n is a “sausage” arrangement ofn balls, holds. This gives considerable improvement to Fejes Tóth's “sausage” conjecture in high dimensions. Further, we prove that, for every convex bodyK and ρ<1/32d ⁻²,V(conv(C _n )+ρK)≥V(conv(S _n )+ρK), whereC _n is a packing set with respect toK andS _n is a minimal “sausage” arrangement ofK, holds.     

Equational Noetherianness of rigid soluble groups

A group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > … > G _m  > G _m + 1 = 1, whose quotients G _i /G _i + 1 are Abelian and are torsion free as right Z[G/G _i ]-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any n, every system of equations in x ₁, …, x _n over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being Noetherian on G ⁿ , which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed. It is proved that every rigid group is equationally Noetherian. Supported by RFBR (project No. 09-01-00099) and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project No. 2.1.1.419). Translated from Algebra i Logika, Vol. 48, No. 2, pp. 258–279, March–April, 2009.     

Decomposition of ranges of vector measures

The following conditions on a zonoidZ, i.e., a range of a non-atomic vector measure, are equivalent: (i) the extreme set containing 0 in its relative interior is a parallelepiped; (ii) the zonoidZ determines them-range of any non-atomic vector measure with rangeZ, where them-range of a vector measure μ is the set ofm-tuples (μ(S ₁), …, μ(S _m), whereS ₁, …S _m are disjoint measurable sets and (iii) there is avector measure space (X, Σ, μ) such that any finite factorization ofZ, Z =ΣZ _i , in the class of zonoids could be achieved by decomposing (X, Σ). In the case of ranges of non-atomic probability measures (i) is automatically satisfied, so (ii) and (iii) hold. Partially supported by NSF grant MCS-79-06634     

On the Stanley depth of squarefree Veronese ideals

Let K be a field and S=K[x ₁,…,x _n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth (M), and conjectured that depth (M)≤sdepth (M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M=I/J with J⊂I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in S. In particular, if I _n,d is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤d≤n<5d+4, then sdepth (I _n,d )=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I _n,d )≤⌊(n−d)/(d+1)⌋+d.     

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

Transversals of additive Latin squares

LetA={a ₁, …,a _k} andB={b ₁, …,b _k} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈S _ksuch that the sums α _i +b _i , 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon’s result to the groups (ℤ _p ) ^a and in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order. Supported by Hungarian research grants OTKA F030822 and T029759. Supported by the Catalan Research Council under grant 1998SGR00119. Partially supported by the Hungarian Research Foundation (OTKA), grant no. T029132.     

Finitary reconstruction of a measure preserving transformation

This paper considers thefinitary reconstruction of an ergodic measure preserving transformationT of a complete separable metric spaceX from a single trajectoryx, Tx, …, or more generally, from a suitable reconstruction sequence x=x ₁,x ₂, … withx _i∈X. Ann-sample reconstruction is a functionT _n: X ⁿ⁺¹ →X; the map (·;x ₁, …,x _n)is treated as an estimate ofT(·) based on then initial elements of x. Given a reference probability measureμ ₀ and constantM>1, functionsT ₁,T ₂, … are defined, and it is shown that for everyμ with 1/M≤dμ/dμ ₀≤M, everyμ-preserving transformationT, and every reconstruction sequence x forT, the estimates (·;x ₁, …,x _nconverge toT in the weak topology. For the family of interval exchange transformations of [0, 1] a simple family of estimates is described and shown to be consistent both pointwise and in the strong topology. However, it is also shown that no finitary estimation scheme is consistent in the strong topology for the family of all ergodic Lebesgue measure preserving transformations of the unit interval, even if x is assumed to be a generic trajectory ofT. Supported in part by NSF Grant DMS-9501926.     

Positive values of non-homogeneous indefinite quadratic forms of type (1, 4)

Let Г_r,n—r denote the infimum of all number Г > 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c₁; c₂,…, c_n, there exist integersx ₁,x₂,…,x_n satisfying 0 < Q(x₁+c₁,x₂ + c₂,…,x_n + c_n) ≤(Г|Z > |)^1/n. All the values of Г_r,n—r are known except for г_1,4. Earlier it was shown that 8 ≤Г_1,4 ≤16. Here we improve the upper bound to get Г_1,4 < 12.     

Pseudo-differential operators on Sobolev and Lipschitz spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.     

Additive Latin transversals and group rings

LetA={a ₁, …,a _k} and {b ₁, …,b _k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata _i+b _i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ _p ^r andZ _p ^rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|. This work has been supported partly by NSFC grant number 19971058 and 10271080.     

The nearness problems for symmetric centrosymmetric with a special submatrix constraint

We say that X=[x_ij]_i,j=1ⁿX=[x_{ij}]_{i,j=1}^n is symmetric centrosymmetric if x _ij  = x _ji and x _{n − j + 1,n − i + 1}, 1 ≤ i,j ≤ n. In this paper we present an efficient algorithm for minimizing ||AXA ^T  − B|| where ||·|| is the Frobenius norm, A ∈ ℝ ^m×n , B ∈ ℝ ^m×m and X ∈ ℝ ^n×n is symmetric centrosymmetric with a specified central submatrix [x _ij ] _{p ≤ i,j ≤ n − p} . Our algorithm produces a suitable X such that AXA ^T  = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

Polytopes with prescribed contents of (n−1)-facets

If ann-dimensional polytope has facets of areaA ₁,A ₂, …,A _m, then 2A _i <A ₁+…+A _m fori=1,…,m. We show here that conversely these inequalities also ensure the existence of a polytope having these areas.     

On exactlym times covers

In this paper it is shown that if every integer is covered bya ₁+n ₁ℤ,…,a _k +n _k ℤ exactlym times then for eachn=1,…,m there exist at least ( _n ^m ) subsetsI of {1,…k} such that ∑ _{i ∈ I} 1/n _i equalsn. The bound ( _n ^m ) is best possible. Research supported by the National Nature Science Foundation of P.R. of China.     

On samples of independent random vectors in spaces of infinitely increasing dimension

We study the asymptotic behavior of a set of random vectors ξ_i, i = 1,..., m, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate the asymptotics of the distribution of the random vectors, the consistency of the sets M _m⁽ⁿ⁾ = ξ₁,..., ξ_m and X _n^λ = x ∈ X _n: ρ(x) ≤ λn, and the mutual location of pairs of vectors. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1706–1711, December, 1998.     

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     

相关序的进一步研究

Let (X1,X2,…,Xn) and (Y1,Y2,…Yn) be real random vectors with the same marginal distributions,if (X1,X2,…,Xn)≤c(Y1,Y2,…Yn), it is showed in this paper that ∑i=1^n Xi≤cx∑i=1^n Yi and max1≤k≤n∑i=1^k Xi≤icx max1≤k≤n∑i=1^k Yi hold. Based on this fact,a more general comparison theorem is obtained.