Independence in hypergraphs

Suppose an integral function (|A|)q₁ defined on the subsets of edges of a hypergraph (X,u,) satisfies the following two conditions: 1) any set W u such that |A|(|A|) for any AW is matroidally independent; 2) if W is an independent set, then there exists a unique partitionW=T₁+ T₂+...+T_v such that |T ⁱ |=(|T ⁱ |),i1:v, and for any AW, |A|(|A|) there exists a T_i such that AT_i. The form of such a function is found, in terms of parameters of generalized connected components, hypercycles, and hypertrees.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 196–204, 1982.     

A generalized sandwich theorem

It is proved for an arbitrary commutative ring A with identity and any integer n3 that if H is a subgroup of GL_n(A) normalized by E _n(A,q), then there is an ideal of A such that E _n(A,) H GL _n (A, (:q⁴⁰).Furthermore, is uniquely determined up to a certain equivalence relation on the set of ideals of A. The result extends a theorem of Bak, by removing a stability condition he uses on A.     

The monadic theory of (ω 2, <) may be complicated

Summary Assume ZFC is consistent then for everyB there is a generic extension of the ground world whereB is recursive in the monadic theory of ₂.The second authro would like to thank the United States — Israel Binational Science Foundation for partially supporting this research. Publ. 411     

Determinants in net subgroups

Suppose R is a commutative ring with 1, =( _ij ) is a fixedD-net of ideals of R of ordern, and G is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for a homomorphismdet of the subgroup G() into a certain Abelian group (). Let I be the index set {1...,n}. For each subset I let ()= _{ij ji} , wherei, ranges over all indices in and j independently over the indices in the complement I ((I) is the zero ideal). Letdet (a) denote the principal minor of order ||n of the matrixa G () corresponding to the indices in , and let' () be the Cartesian product of the multiplicative groups of the quotient rings R/() over all subsets I. The homomorphismdet is defined as follows: It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det coincides with the subgroup E() generated by all transvections in G(). For these R is also definedTm det .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982.     

Exceptional Family of Elements and Feasibility for Nonlinear Complementarity Problems

In this paper, we introduce the new concept of -exceptional families of elements and (,)-exceptional families of elements for continuous functions, and utilize these notions for the study of the feasibility of nonlinear complementarity problems in R ⁿ and an infinite-dimensional Hilbert space H without the assumption K ^* K.     

Patching ideal families on℘ kλ

Ideal families defined on a cardinalk often exhibit reflection properties. IfC k is a club, for example, thenC is a club-in- club-in-k often. In this paper we generalize this notion to ideal families defined on _k and exhibit some examples.     

Separately harmonic and subharmonic functions

Let u(x, y) be defined in B ₁×B ₂ where B ₁ ^m and B ₂ ⁿ , and assume that u(x, ·) harmonic for every fixed x and u(·, y) is subharmonic for every fixed y. We show that if u(·, y) is, in addition, C ² for each y then u is subharmonic in B ₁×B ₂ in both variables jointly.     

On the Scrimger varieties of lattice-ordered groups

The containmentS _nL _n is proper. (Here n2 is a positive integer,S _n is the small variety of lattice-ordered groups of Scrimger andL _n is the variety of lattice-ordered groups defined by the law aⁿbⁿ=bⁿaⁿ.) Jo E. Smith proved this result for n a composite integer. In this note we proveS _nL _n when n is prime.Presented by L. Fuchs.     

A generalization of the Heine limit for functions which converge on a base

. , , . . . .     

Orthogonal systems invariant under an automorphism

, , - ( ) . , - , ( ) .     

L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

On the immersion of metrics close to immersible metrics

For a C^r,-immersion z:X E, r 2, 0 < < 1, of an n-dimensional (n 1) simply-connected C^r+2,-manifold X into Euclidean space E, the metric I(z) induced by z has a neighborhood in C^r,-topology in which every metric from a given subbundle of metrics is C^r,-immersible into E. In particular, it is proved that metric ds ₀ ² of the Riemannian product of p spheres of dimensions ₁, , _p 2 has a neighborhood in C^2,-topology from which any conformally equivalent metric to ds ₀ ² , is immersible into E with dimE = ₁ + + _p + p. The proofs are based on the investigation of a varied system of Gauss—Codazzi—Ricci equations for an infinitely small deformation of surface z(X) in E with a prescribed variation of the metric.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 49–67, 1992.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Extremal Eigenvalues of the Laplacian in a Conformal Class of Metrics: The `Conformal Spectrum'

Let M be a compact connected manifold of dimension n endowed witha conformal class C ofRiemannian metrics of volume one. For any integer k 0, we consider the conformal invariant _k ^c (C) defined as the supremum of the k-th eigenvalue _k (g) of the Laplace–Beltrami operator _g , where g runs over C.First, we give a sharp universal lower bound for _k ^c (C) extending to all k a result obtained by Friedlander andNadirashvili for k = 1. Then, we show that the sequence \{ _k ^c (C)\}, that we call `conformal spectrum',is strictly increasing and satisfies, k 0, _k+1 ^c (C) ^n/2– _k ^c (C) ^n/2 n ^n/2 _n , where _n is the volume of the n-dimensionalstandard sphere.When M is an orientable surface of genus , we also considerthe supremum _k ^top()of _k (g) over theset of all the area one Riemannian metrics on M, and study thebehavior of _k ^top() in terms of .     

Ergodic properties of certain surjective cellular automata

We consider one-dimensional cellular automata, i.e. the mapsT:P P (P is a finite set with more than one element) which are given by (Tx) _i =F(x _i+l , ...,x _i+r ),x=(x _i )P for some integerslr and a mappingFP ^r–l+1P. We prove that ifF is right- (left-) permutative (in Hedlund's terminology) and 0l<r (resp.l<r0), then the natural extension of the dynamical system (P , , ,T) is a Bernoulli automorphism ( stands for the (1/p, ..., 1/p)-Bernoulli measure on the full shiftP ). Ifr<0 orl>0 andT is surjective, then the natural extension of the system (P , , ,T) is aK-automorphism. We also prove that the shift ²-action on a two-dimensional subshift of finite type canonically associated with the cellular automatonT is mixing, ifF is both right and left permutative. These results answer some questions raised in [SR].     

On properties of the probabilistic constrained linear programming problem and its dual

In this paper, the two problems inf{inf{cx:x R ⁿ,A ₁ xy,A ₂ xb}:y suppF R ^m,F(y)p} and sup{inf{uy:y suppF R ^m,F(y)p}+vb:uA ₁+vA ₂=c, (u,v0} are investigated, whereA ₁,A ₂,b,c are given matrices and vectors of finite dimension,F is the joint probability distribution of the random variables ₁,..., _m, and 0<p<1. The first problem was introduced as the deterministic equivalent and the second problem was introduced as the dual of the probabilistic constrained linear programming problem inf{cx:P(A ₁ x)p,A ₂ xb}.b}. Properties of the sets and the functions involved in the two problems and regularity conditions of optimality are discussed.     

On Domination Number of 4-Regular Graphs

Let G be a simple graph. A subset S V is a dominating set of G, if for any vertex v V – S there exists a vertex u S such that uv E(G). The domination number, denoted by (G), is the minimum cardinality of a dominating set. In this paper we prove that if G is a 4-regular graph with order n, then (G) ⁴/₁₁ n     

On the newly generalized absolute Riesz summability of orthogonal series

[3] , [1], ¦C,,(t)¦ _k - . , , ¦R, , (t)¦ _k - ¦R, , (t)¦ _k - . (t)=1,k=1 [7], [9].

---

---

This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant#T 016 393.