Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.     

Uniform separation of points and measures and representation by sums of algebras

LetX andY _i, 1 ≦i ≦k, be compact metric spaces, and letρ _i:X →Y _i be continuous functions. The familyF={ρ _i} _i ^1/k is said to be ameasure separating family if there exists someλ > 0 such that for every measureμ inC(X)*, ‖μ ^o ρ _i ^?1 ‖ ≧λ ‖μ ‖ holds for some 1 ≦i ≦k.F is auniformly (point) separating family if the above holds for the purely atomic measures inC(X)*. It is known that fork ≦ 2 the two concepts are equivalent. In this note we present examples which show that fork ≧ 3 measure separation is a stronger property than uniform separation of points, and characterize those uniformly separating families which separate measures. These properties and problems are closely related to the following ones: letA ₁,A ₂, ...,A _k be closed subalgebras ofC(X); when isA ₁ +A ₂ + ... +A _k equal to or dense inC(X)?     

An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U _f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U _f -algebra. For any U-algebraAsetU _A =U _i εI({i}×(A ^K _i—domf _i ^A )), where (K _i) _i εI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U _f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ?N such that there exists a bijective mapping σ:U _{F(N, U)} →N ?M satisfying σ((i, α)) ? α(K _i ) for all (i, α) ∈U _{F (N, U)}, and, for the structureg=(g _i)_iεI defined by ,g _i : =f _i ^{F(N, U)} ∪ {(α, σ((i, α))) | (i, α ∈U _{F(N, U)}} id _M induces an isomorphism betweenF(M, U _f ), and (F(N, U)g).     

Primitive words,free factors and measure preservation

Let F _k be the free group on k generators. A word w ∈ F _k is called primitive if it belongs to some basis of F _k . We investigate two criteria for primitivity, and consider more generally subgroups of F _k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H ≤ J ≤ F _k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F _k is primitive. Again let w ∈ F _k and consider the word map w: G × … × G → G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G × … × G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation, and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and, in particular, prove the conjecture for k = 2. It was asked whether the primitive elements of F _k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F ₂.     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

Homotopy type of smooth weighted complete intersections

If the total degree d has no prime divisors less than(n+3)/2,then we prove that the homotopy type of complex odd dimensional smooth weighted complete intersection Xn(d;w) is determined by the dimension n,the total degree d,the Euler characteristic and the Kervaire invariant,provided that the weights w =(ω0,...,ωn+r) is pairwise relatively prime.     

Doeblin's and Harris' theory of Markov processes

Our notation and definitions are taken from (Chung, K. L.: The general theory of Markov processes according to Doeblin. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 2, 230–254 (1964)). A closed set H is called recurrent in the sense of Harris if there exists a σ-finite measure ? such that for E=H, ?(E) >0 implies Q(x, E)=1 for all tx?H. Theorem 1. Let X be absolutely essential and indecomposable. Then there exists a closed set B?X. such that B contains no acountable disjoint collection of perpetuable sets if and only if X=H+1 where H is recurrent in the sense of Harris and I is either inessential or improperly essential. Theorem 2. If there exists no uncountable disjoint collection of closed sets, then there exists a countable disjoint collection {D_n} _n=1 ^∞ of absolutely essential and indecomposable closed sets such that \(I = X - \sum\nolimits_{n = 1}^\infty {D_n } \) . Under the additional assumption that Suslin's Conjecture holds, Theorem 2 was proved by Jamison (Jamison, B.: A Result in Doeblin's Theory of Markov Chains implied by Suslin's Conjecture. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 287–293 (1972)).     

Finite union theorem with restrictions

The aim of this paper is to prove the following extension of the Folkman-Rado-Sanders Finite Union Theorem: For every positive integersr andk there exists a familyL of sets having the following properties:

1. ifS ₁,S ₂, ...,S _{k + 1} are distinct pariwise disjoint elements ofL then there exists nonemptyI ? {1, 2, ...,k + 1} with ∪ _i∈I S _i ?L
2. ifL =L ₁ ?...?L _r is an arbitrary partition then there existsj ≤ r and pairwise disjoint setsS ₁,S ₂, ...,S _k ∈L _j , such thatL _i∈I S _i ∈L _j for every nonemptyI ? {1, 2, ...,k}.

     

Exact solution of the hypergraph Turán problem for k-uniform linear paths

A k-uniform linear path of length ?, denoted by ? _? ^(k) , is a family of k-sets {F ₁,...,F _? such that |F _i ∩ F _i+1|=1 for each i and F _i ∩ F _bj = \(\not 0\) whenever |i?j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex _k (n, H), is the maximum number of edges in a k-uniform hypergraph \(\mathcal{F}\) on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex _k (n, P _? ^(k) exactly for all fixed ? ≥1, k≥4, and sufficiently large n. We show that $ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$ . The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that $ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$ , and describe the unique extremal family. Stability results on these bounds and some related results are also established.     

Simultaneous Domination in Graphs

Let \({F_1, F_2, \ldots, F_k}\) be graphs with the same vertex set V. A subset \({S \subseteq V}\) is a simultaneous dominating set if for every i, 1 ≤ i ≤ k, every vertex of F _i not in S is adjacent to a vertex in S in F _i ; that is, the set S is simultaneously a dominating set in each graph F _i . The cardinality of a smallest such set is the simultaneous domination number. We present general upper bounds on the simultaneous domination number. We investigate bounds in special cases, including the cases when the factors, F _i , are r-regular or the disjoint union of copies of K _r . Further we study the case when each factor is a cycle.     

Spaces of Riemannian metrics on open manifolds

We define for the set M of metrics on an open manifold M ⁿ suitable uniform structures, obtain completed spaces ^b,m M or M ^r (I, B _k ), respectively and calculate for each component of M ^r (I, B _k ) the infinitedimensional geometry. In particular, we show that the sectional curvature is non positive.     

Set-coloring of Edges and Multigraph Ramsey Numbers

Edge-colorings of multigraphs are studied where a generalization of Ramsey numbers is given. Let ${M_n^{(r)}}$ be the multigraph of order n, in which there are r edges between any two different vertices. Suppose q ₁, q ₂, . . . , q _k and r are positive integers, and q _i ≥ 2(1 ≤ i ≤ k), k > r. Let the multigraph Ramsey number ${f^{(r)} (q_1 ,q_2 , \ldots ,q_k )}$ be the minimum positive integer n such that in any k-edge coloring of ${M_n^{(r)}}$ (every edge is colored with one among k given colors, and edges between the same pair of vertices are colored with different colors), there must be ${i \in \{1,2,\ldots,k\}}$ such that ${M_n^{(r)}}$ has such a complete subgraph of order q _i , of which all the edges are in color i. By Ramsey’s theorem it is easy to show ${f^{(r)} (q_1 ,q_2 , \ldots ,q_k )}$ exists for given q ₁ ,q ₂, . . . , q _k and r. Lower and upper bounds for some multigraph Ramsey numbers are given.     

Large Deviations for Proportions of Observations Which Fall in Random Sets Determined by Order Statistics

Let {X _n :n?≥?1} be independent random variables with common distribution function F and consider $K_{h:n}(D)=\sum_{j=1}^n1_{\{X_j-X_{h:n}\in D\}}$ , where h?∈?{1,...,n}, X _1:k ?≤???≤?X _k:k are the order statistics of the sample X ₁,...,X _k and D is some suitable Borel set of the real line. In this paper we prove that, if F is continuous and strictly increasing in the essential support of the distribution and if $\lim_{n\to\infty}\frac{h_n}{n}=\lambda$ for some λ?∈?[0,1], then $\{K_{h_n:n}(D)/n:n\geq 1\}$ satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics $\{X_{h_n:n}:n\geq 1\}$ . We also present results for the special case of Bernoulli distributed random variables with mean p?∈?(0,1), and we see that the large deviation principle holds only for p?≥?1/2. We discuss further almost sure convergence of $\{K_{h_n:n}(D)/n:n\geq 1\}$ and some related quantities.     

A test for identities satisfied in lattices of submodules

SupposeR is ring with 1, andH?(R) denotes the variety of modular lattices generated by the class of lattices of submodules of allR-modules. An algorithm using Mal'cev conditions is given for constructing integersm≧0 andn≧1 from any given lattice polynomial inclusion formulad∈e. The main result is thatd∈e is satisfied in every lattice inH?(R) if and only if there existsx inR such that (m·1)x=n·1 inR, where 0·1=0 andk·1=1+1...+1 (k times) fork≧1. For example, this “divisibility” condition holds form=2 andn=1 if and only if 1+1 is an invertible element ofR, and it holds form=0 andn=12 if and only if the characteristic ofR divides 12. This result leads to a complete classification of the lattice varietiesH?(R),R a ring with 1. A set of representative rings is constructed, such that for each ringR there is a unique representative ringS satisfyingH?(R)=H?(R). There is exactly one representative ring with characteristick for eachk≧1, and there are continuously many representative rings with characteristic zero. IfR has nonzero characteristic, then all free lattices inH?(R) have recursively solvable word problems. A necessary and sufficient condition onR is given for all free lattices inH?(R) to have recursively solvable word problems, ifR is a ring with characteristic zero. All lattice varieties of the formH?(R) are self-dual. A varietyH?(R) is a congruence variety, that is, it is generated by the class of congruence lattices of all members of some variety of algebras. A family of continuously many congruence varieties related to the varietiesH?(R) is constructed.     

On lattices of congruences of relational systems and universal algebras

Let \(\mathfrak{X}\) =〈X;R〉 be a relational system.X is a non-empty set andR is a collection of subsets ofX ^α, α an ordinal. The system of equivalence relations onX having the substitution property with respect to members ofR form a complete latticeC( \(\mathfrak{X}\) ) containing the identity but not necessarilyX×X. It is shown that for any relational system (X;R) there is a groupoid definable onX whose congruence lattice isC( \(\mathfrak{X}\) )U{X×X} . Theorem 2 and Corollary 2 contain some interesting combinatorial pecularities associated with oriented complete graphs and simple groupoids.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Torsion and Abelianization in Equivariant Cohomology

Let X be a topological space upon which a compact connected Lie group G acts. It is well known that the equivariant cohomology H ^* _G (X; Q) is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology H ^* _T (X; Q), where T is a maximal torus of G. This relationship breaks down for coefficient rings k other than Q. Instead, we prove that under a mild condition on k the algebra H ^* _G (X; k) is isomorphic to the subalgebra of H ^* _T (X; k) annihilated by the divided difference operators.     

Embeddings into power series rings

Let T be an ordered ring without divisors of zero, and letA be the set of archimedean subgroups of T generated by a Banaschewski functionτ. Let_XΠ_Δ R be the power series ring of the real numbers ? over the totally ordered semigroup Δ of archimedean classes of T, and letχ be the usual Banaschewski function on_XΠ_Δ R. The following are equivalent:

1. τ satisfies the additional condition; for convex subgroups P,Q of T, where
2. There exists a one-to-one homomorphism Γ:T→_XΠ_Δ R of ordered rings such that for every convex subgroup Q of_XΠ_Δ R, there exists a convex subgroup P of T such that \(\Gamma (P) \subseteq Q\) and \(\Gamma (\tau (P)) \subseteq \chi (Q)\) .