Ideal einstein,conformally flat and semi-symmetric immersions

Recently, B. Y. Chen introduced a new intrinsic invariant of a manifold, and proved that everyn-dimensional submanifold of real space formsR ^m (ε) of constant sectional curvature ε satisfies a basic inequality δ(n ₁,…,n _k )≤c(n ₁,…,n _k )H ²+b(n ₁,…,n _k )ε, whereH is the mean curvature of the immersion, andc(n ₁,…,n _k ) andb(n ₁,…,n _k ) are constants depending only onn ₁,…,n _k ,n andk. The immersion is calledideal if it satisfies the equality case of the above inequality identically for somek-tuple (n ₁,…,n _k ). In this paper, we first prove that every ideal Einstein immersion satisfyingn≥n ₁+…+n _k +1 is totally geodesic, and that every ideal conformally flat immersion satisfyingn≥n ₁+…+n _k +2 andk≥2 is also totally geodesic. Secondly we completely classify all ideal semi-symmetric hypersurfaces in real space forms. The author was supported by the NSFC and RFDP.     

An optimal version of an inequality involving the third symmetric means

Let (GA) _n ^[k](a), A _n (a), G _n (a) be the third symmetric mean of k degree, the arithmetic and geometric means of a ₁, …, a _n (a _i > 0, i = 1, …, n), respectively. By means of descending dimension method, we prove that the maximum of p is k−1/n−1 and the minimum of q is n/n−1(k−1/k) ^k/n so that the inequalities {fx505-1} hold.     

Refined enumerations of alternating sign matrices: monotone (<Emphasis Type="Italic">d</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-trapezoids with prescribed top and bottom row

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to n×n alternating sign matrices when prescribing (1,2,…,n) as bottom row of the array. We define monotone (d,m)-trapezoids as monotone triangles with m rows where the d−1 top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row (k ₁,…,k _n ) is given by a polynomial α(n;k ₁,…,k _n ) in the k _i ’s. The main purpose of this paper is to show that the number of monotone (d,m)-trapezoids with prescribed top and bottom row appears as a coefficient in the expansion of a specialisation of α(n;k ₁,…,k _n ) with respect to a certain polynomial basis. This settles a generalisation of a recent conjecture of Romik et al. (Adv. Math. 222:2004–2035, 2009). Among other things, the result is used to express the number of monotone triangles with bottom row (1,2,…,i−1,i+1,…,j−1,j+1,…,n) (which is, by the standard bijection, also the number of n×n alternating sign matrices with given top two rows) in terms of the number of n×n alternating sign matrices with prescribed top and bottom row, and, by a formula of Stroganov for the latter numbers, to provide an explicit formula for the first numbers. (A formula of this type was first derived by Karklinsky and Romik using the relation of alternating sign matrices to the six-vertex model.)     

Deformation of certain quadratic algebras and the corresponding quantum semigroups

LetV be a finite-dimensional vector space. Given a decompositionV⊗V=⊕ _i=1,…n I _i , definen quadratic algebrasQ(V, J _(m)) whereJ _(m)=⊕ _i≠m I _i . There is also a quantum semigroupM(V; I ₁, …,I _n ) which acts on all these quadratic algebras. The decomposition determines as well a family of associative subalgebras of End (V ^⊗k ), which we denote byA _k =A _k (I ₁,…,I _n ),k≥2. In the classical case, whenV⊗V decomposes into the symmetric and skewsymmetric tensors,A _k coincides with the image of the representation of the group algebra of the symmetric groupS _k in End(V ^⊗k ). LetI _i,h be deformations of the subspacesI _i . In this paper we give a criteria for flatness of the corresponding deformations of the quadratic algebrasQ(V, J _(m),h ) and the quantum semigroupM(V;I _1,h ,…,I _n,h ). It says that the deformations will be flat if the algebrasA _k (I ₁, …,I _n ) are semisimple and under the deformation their dimension does not change. Usually, the decomposition intoI _i is defined by a given semisimple operatorS onV⊗V, for whichI _i are its eigensubspaces, and the deformationsI _i,h are defined by a deformationS _h ofS. We consider the cases whenS _h is a deformation of Hecke or Birman-Wenzl symmetry, and also the case whenS _h is the Yang-Baxter operator which appears by a representation of the Drinfeld-Jimbo quantum group. Applying the flatness criteria we prove that in all these cases we obtain flat deformations of the quadratic algebras and the corresponding quantum semigroups. Partially supported by a grant from the Israel Science Foundation administered by the Israel Academy of Sciences.     

Non-<Emphasis Type="Italic">φ</Emphasis>-admissible normal subgroups of free burnside groups

In the present paper for arbitrary automorphism φ of the free Bunside group B(m, n) and for any odd number n ≥ 1003 a sufficient condition for existence of non-φ-admissible normal subgroup of B(m, n) was found. In particular, if automorphism φ is normal, then for any basis {a ₁, a ₂, …, a _m } of the group B(m, n) there is an integer k such that for each i the elements a _i and φ(a i) ^k are conjugates.     

Completely monotone multisequences, symmetric probabilities and a normal limit theorem

Let G _n,k be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc _n(β₁,β₂,…, β _k ), β₁,β₂,…, β_k = 0,1,2,…, β₁+β₂ + … +β _k ≤n,c _n(0,0,…, 0) = 1 and whenever β₀ ≤n - (β₁ + β₂ + … + β _k ) where Δc _n(β₁,β₂,…, β _k ) =c _n(β₁ + 1, β₂,…, β _k )+c _n(β₁,β₂+1,…, β _k )+…+c _n (β₁,β₂,…, β _k +1) -c _n(β₁,β₂,…, β _k ). Further, let Π _n,k be the set of all symmetric probabilities on {0,1,2,…,k} ⁿ . We establish a one-to-one correspondence between the sets G _n,k and Π _n,k and use it to formulate and answer interesting questions about both. Assigning to G _n,k the uniform probability measure, we show that, asn→∞, any fixed section {it{c_n}(β₁,β₂,…, β _k ), 1 ≤ Σβ _i ≤m}, properly centered and normalized, is asymptotically multivariate normal. That is, converges weakly to MVN[0, Σ _m ]; the centering constantsc ₀(β₁, β₂,…, β _k ) and the asymptotic covariances depend on the moments of the Dirichlet (1, 1,…, 1; 1) distribution on the standard simplex inR ^k.     

Disjoint homotopic paths and trees in a planar graph

In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ², a subset {I ₁, …,I _p} of the faces ofG, and pathsC ₁, …,C _k inG, with endpoints on the boundary ofI ₁ ∪ … ∪I _p; find: pairwise disjoint simple pathsP ₁, …,P _k inG so that, for eachi=1, …,k, P _i is homotopic toC _i in the space ℝ²\(I ₁ ∪ … ∪I _p). Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ² and pairwise disjoint setsW ₁, …,W _k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT ₁, …,T _k ofG whereT _i (i=1, …, k).     

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.     

Realization of a sum of sequences by a sum graph

It is shown that the realizability of the sequences ϕ=(a ₁,…, _a ), ψ=(b ₁,…,b _n ) and ϕ+ψ is a sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor ifb _i ≦1 fori=1,…,n. The condition is not sufficient in general. A necessary and sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor is given for the case that ϕ is realizable by a star and isolated vertices.     

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.     

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.     

The bieri-strebel invariant and homological finiteness conditions for metabelian groups

A group Γ has type F P_n if a trivial ℤΓ-module ℤ has a projective resolution P:…P_n → … → P₁ → P₀ → ℤ in which ℤΓ-module P_n,…P₁, P₀ are finitely generated. Let the finitely generated group Γ be a split extension of the Abelian group M by an Abelian group Q, suppose M is torsion free, and assume Γ∈F P_m, m≥2. Then the invariant ∑ _c ^M is m-tame. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 194–218, March–April, 1997.     

Successions in integer partitions

A partition of an integer n is a representation n=a ₁+a ₂+⋅⋅⋅+a _k , with integer parts 1≤a ₁≤a ₂≤…≤a _k . For any fixed positive integer p, a p-succession in a partition is defined to be a pair of adjacent parts such that a _i+1−a _i =p. We find generating functions for the number of partitions of n with no p-successions, as well as for the total number of such successions taken over all partitions of n. In the process, various interesting partition identities are derived. In addition, the Hardy-Ramanujan asymptotic formula for the number of partitions is used to obtain an asymptotic estimate for the average number of p-successions in the partitions of n. This material is based upon work supported by the National Research Foundation under grant number 2053740.     

A bounded N-tuplewise independent and identically distributed counterexample to the CLT

Summary. A sequence of random variables X ₁,X ₂,X ₃,… is said to be N-tuplewise independent if X i ₁,X i ₂,…,X i _N are independent whenever (i ₁,i ₂,…,i _N ) is an N-tuple of distinct positive integers. For any fixed N∈ℤ⁺, we construct a sequence of bounded identically distributed N-tuplewise independent random variables which fail to satisfy the central limit theorem. Received: 17 May 1996 / In revised form: 28 January 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.     

The increase of sums and products dependent on (y 1, …,y n ) by rearrangement of this set

LetF(u, v) be a symmetric real function defined forα<u, v<β and assume thatG(u, v, w)=F(u, v)+F(u, w)−F(v, w) is decreasing inv andw foru≦min (u, v). For any set (y)=(y ₁, …,y _n ),α<y _i <β, given except in arrangement Σ _i ^=1/n F(y _i ,y _i+1) wherey _n+1=y ₁) is maximal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. Examples are given and an additional result is proved on the productΠ _i ^=1/n [(y2i−1y2i) ^m +α ₁(y _2i−1 y _2i ) ^m−1+ … +a _m ] wherea _k ≧0 and where the set (y)=(y ₁, ..,y _n ),y _i ≧0 is given except in arrangement. The problems considered here arose in connection with a theorem by A. Lehman [1] and a lemma of Duffin and Schaeffer [2]. This paper is part of the author’s Master of Science dissertation at the Technion-Israel Institute of Technology. The author wishes to thank Professor B. Schwarz and Professor E. Jabotinsky for their help in the preparation of this paper.     

A proof of snevily’s conjecture

We prove Snevily’s conjecture, which states that for any positive integer k and any two k-element subsets {a ₁, …, a _k } and {b ₁, …, b _k } of a finite abelian group of odd order there exists a permutation π ∈ S _k such that all sums a _i + b _π(i) (i ∈ [1, k]) are pairwise distinct.     

Correlated equilibria of games with many players

Let G _m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G _m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2⁻ⁿ; and (2) for each 1<c<m, the solution set contains c ⁿ correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c ₁, …, c _l be independent and symmetric random vectors in R ^k, l≥k. Then the probability that the convex hull of c ₁, …, c _l intersects R ^k ₊ is greater than or equal to . Received: December 1998/Final version: March 2000     

A superadditivity and submultiplicativity property for cardinalities of sumsets

For finite sets of integers A ₁,…,A _n we study the cardinality of the n-fold sumset A ₁+…+ A _n compared to those of (n−1)-fold sumsets A ₁+…+A _i−1+A _i+1+…+A _n . We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets.     

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.