More results on Ramsey—Turán type problems

The paper deals with common generalizations of classical results of Ramsey and Turán. The following is one of the main results. Assumek≧2, ε>0,G _n is a sequence of graphs ofn-vertices and at least 1/2((3k?5) / (3k?2)+ε)n ² edges, and the size of the largest independent set inG _n iso(n). LetH be any graph of arboricity at mostk. Then there exists ann ₀ such that allG _n withn>n ₀ contain a copy ofH. This result is best possible in caseH=K _2k .     

Fractal relaxed Dirichlet problems

Given aself similar fractal K ? ? ⁿ of Hausdorff dimension α>n?2, andc ₁>0, we give an easy and explicit construction, using the self similarity properties ofK, of a sequence of closed sets? _h such that for every bounded open setΩ?? ⁿ and for everyf ∈ L²(Ω) the solutions to $$\left\{ \begin{gathered} - \Delta u_h = f in \Omega \backslash \varepsilon _h \hfill \\ u_h = 0 on \partial (\Omega \backslash \varepsilon _h ) \hfill \\ \end{gathered} \right.$$ converge to the solution of the relaxed Dirichlet boundary value problem $$\left\{ \begin{gathered} - \Delta u + uc_1 \mathcal{H}_{\left| K \right.}^\alpha = f in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ (H _∣ ^α denotes the restriction of the α-dimensional Hausdorff measure toK). The condition α>n?2 is strict.     

Asymptotic estimates for the principal eigenvalue of the laplacian in a geodesic ball

LetM be a compact Riemannian manifold and letB _ε be a geodesic ball of radiusε with center0 ∈ M. We investigate the asymptotic behavior ofλ _ε , the principal eigenvalue of the Laplace-Beltrami operator on \(M/\bar B_\varepsilon\) with homogeneous Dirichlet boundary conditions. We prove thatλ _ε ~Cφ _n (ε) wheren = dimM, φ ₂ (ε)=(logε ^?1)^?1 andφ _n (ε) = (n-2)ε ^n-2 (n>2). In the case whereM is a model the constantC is explicitly evaluated.     

On subharmonic functions dominated by certain functions

Given two kinds of functionsf(X) andh(y) defined on them-dimensional Euclidean spaceR ^m (m≧1) and the set of positive real numbers respectively, we give an estimation of growth of subharmonic functionsu(P) defined onR ^m+n (n≧1) such that $$u(P) \leqq f\left( X \right)h\left( {\left\| Y \right\|} \right)$$ for anyP=(X, Y),X ∈R ^m, Y ∈R ⁿ, where ‖Y ‖ denotes the usual norm ofY. Using an obtained result, we give a sharpened form of an ordinary Phragmén-Lindelöf theorem with respect to the generalized cylinderD ×R ⁿ, with a bounded domainD inR ^m.     

Многомерные вариаци и множеств и их контин генции

LetE be a compact set inR ⁿ (n≧2), and denote byV ₀(E) the number of the components ofE. Letp=1,2, ...,n?1;k=0,1, ...,p, and $$V_k (E;n,p) = \int\limits_{\Omega _k^n } {V_0 (E \cap \tau )^{{{(n - p)} \mathord{\left/ {\vphantom {{(n - p)} {(n - k)}}} \right. \kern-\nulldelimiterspace} {(n - k)}}} d\mu _\tau ,}$$ whereΩ _k ⁿ is the set of all (n-k)-dimensional hyperplanesτ?R ⁿ anddμ _τ is the Haar measure on the spaceΩ _k ⁿ ; furthermore, let $$V_n (E;n,n - 1) = mes_n E.$$ . Theorem. Let E?Rⁿ, p=1, 2 ..., n?1, V_p+1(E;n,p)=0, and V_k(E; n, p)<∞ for k= =0,1, ..., p. Then the contingency of the set E at a point x∈E coincides with a certain p-dimensional hyperplane for almost all points x∈E in the sense of Hausdorff p-measure.     

On the Brunn-Minkowski theorem

Let K ₁ and K ₂ be n-dimensional convex bodies. If V denotes volume the Brunn-Minkowski theorem in its simplest form states that \(V(K_1 + K_2 )^{1/n} \geqslant V(K_1 )^{1/n} + V(K_2 )^{1/n} \) , and that equality holds if and only if K ₁ and K ₂ are homothetic. We consider the following associated stability problem: If \(V(K_1 + K_2 )^{1/n} \) differs not more than ε from \(V(K_1 )^{1/n} + V(K_2 )^{1/n} \) how close is K ₁ (in terms of the Hausdorff metric) to a nearest homothetic copy of K ₂? Several theorems that answer questions of this kind are proved. These results can also be expressed as inequalities that are stronger than the original Brunn-Minkowski inequality. Furthermore, some consequences concerning stability statements for other geometric inequalities are discussed.     

Spin invariants of multivectors

LetCl(p, q) be a real universal Clifford algebra which is isomorphic to a full matrix algebra ?(2m). In this paper we show that on the linear subspaceCl ^k(p, q) ofk-vectors the determinant can be written as a product of two polynomialsd _i of degreem and that on the subset ofdecomposable k-vectors we have det=±Q ^m for some quadratic formQ. The polynomialsd _i andQ are examples of a spin invariant, the latter being defined as a functionJ:Cl ^k (p,q) → ? for whichJ(sus^?1)=J(u) for allu ∈Cl ^k(p, q) ands ∈Spin(p, q). In the last section we identify the ‘fundamental’ spin invariants on the bivector spacesCl ²(p, p) forp=2 andp=3.     

On the eigenvalues of the fredholm operator

We prove that if ω(t, x, K ₂ ^(m) )?c(x)ω(t) for allxε[a, b] andx ε [0,b-a] wherec ∈L ¹(a, b) and ω is a modulus of continuity, then λ _n =O(n ^?m-1/2ω(1/n)) and this estimate is unimprovable.     

Analytic factorizations and completely bounded maps

We prove an analytic factorization theorem in the setting of the recently developed theory of operator spaces. We especially obtain the following result: LetA be aC ^*-algebra andH be a Hilbert space. Let π be an element ofH ^∞ (CB(A, B(H))), i.e. a bounded analytic function valued in the space of completely bounded maps fromA intoB(H). Then there exist a Hilbert spaceK, a representation π:A→B(K), ?₁ ∈₁ H ^∞ (B(H,K)) and ∈₂ H ^∞ (B(K,H)) such that ‖ε₁‖∞‖∈₂‖∞ ≤ ‖∈‖∞ and: $\forall z \in D, \forall a \in A, \varphi (z)(a) = \varphi _2 (z)\pi (a)\varphi _1 (z).$ We also prove an analogous result for completely bounded multilinear maps. The last part of the paper is devoted to a new proof of Pisier's theorem about gamma-norms.     

Chaotic behaviour of the map x ↦ ω(x, f)

Let K(2^?) be the class of compact subsets of the Cantor space 2^?, furnished with the Hausdorff metric. Let f ∈ C(2^?). We study the map ω _f : 2 ^? → K(2^?) defined as ω _f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω _f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω _f is everywhere discontinuous on a subsystem is dense in C(2^?). The relationships between the continuity of ω _f and some forms of chaos are investigated.     

A lower bound on the Kobayashi metric near a point of finite type in ℂ n

Let Ω be a bounded domain in ? ⁿ andbΩ smooth pseudoconvex near z₀ ∈bΩ of finite type. Then there are constantsc>0 and ε′>0 such that the Kobayashi metric,K _Ω(z; X), satisfiesK _Ω(z; X)≥c|X|δ(z)^?t′ for allX ∈T _z ^1,0 ? ⁿ in a neighborhood ofz ₀. Here δ(z) denotes the distance fromz tobΩ. As an application, we prove the Hölder continuity of proper holomorphic maps onto pseudoconvex domains.     

On restricted colourings ofK n

Given a sample graphH and two integers,n andr, we colourK _n byr colours and are interested in the following problem. Which colourings of the subgraphs isomorphic to H in K _n must always occur (and which types of colourings can occur whenK _n is coloured in an appropriate way)? These types of problems include theRamsey theory, where we ask: for whichn andr must a monochromaticH occur. They also include theanti-Ramsey type problems, where we are trying to ensure a totally multicoloured copy ofH, that is, anH each edge of which has different colour.     

Singular sets for curvature equation of order k

We consider the removability of singular sets for the curvature equations of the form Hk[u]=ψ, which is determined by the kth elementary symmetric function, in an n-dimensional domain Ω. We prove that, for 1?k?n−1 and a compact set K whose (n−k)-dimensional Hausdorff measure is zero, any generalized solution to the curvature equation on Ω?K is always extendable to a generalized solution on the whole domain Ω.     

On finite groups with a sylowp-subgroup of type (m,n)

A finitep-groupP is of type (m, n) ifP has nilpotency classm-1,P/P'≌Z _p ⁿ ×Z _p ⁿ and all the lower central factorsK _i (P)/K _i+1 (P) are cyclic of orderp ⁿ . Our main result on finite groups with a Sylowp-subgroup of type (m, n) is (Theorem 4.1): Let G be a finite group with a Sylow p-subgroup P of type (m, n), n≧2 p≧3, m≧(n+5)(p?1)+1. For H≦G denote \(\bar H = HO_{p'} (G)/O_{p'} (G)\) . If O_p (G) is not cyclic and P'₁ ≠ 1, then \(\bar P \Delta \bar G\) and \(\bar G = \bar P \cdot \bar T\) is a semidirect product of \(\bar P\) and \(\bar T\) , where \(\bar T\) is cyclic of order t, t|p-1. Here P₁ is the subgroup defined in section 0. This theorem easily yields that under its assumptionsN _G (P)/O ^P (N _G (P))≌G/O ^P (G), it gives information on the conjugacy pattern ofp-elements ofG and gives information on the structure ofp-local subgroups ofG (Theorems 4.2, 4.3 and 4.4).     

Idempotence preserving maps on spaces of triangular matrices

SupposeF is an arbitrary field. Let |F| be the number of the elements ofF. LetT _n (F) be the space of allnxn upper-triangular matrices overF. A map Ψ: T _N (F) → T _N (F) is said to preserve idempotence ifA - λ B is idempotent if and only if Ψ(A) - λΨ(B) is idempotent for anyA, B ∈ T _n (F) and λ ∈ F. It is shown that: when the characteristic ofF is not 2, |F|>3 and n ≥ 3, Ψ:T _n (F) → T _n (F) is a map preserving idempotence if and only if there exists an invertible matrixP τ T _n (F) such that either ?(A) = PAP ^?1 for everyA ∈ T _n (F) or Ψ(A) = PJA ^t JP ^?1 for everyA ∈ T _n (F), whereJ = ∑ _n=1 ⁿ E _i,n+1?i and E_ij is the matrix with 1 in the (i,j)th entry and 0 elsewhere.     

A compact updating formula for quasi-Newton minimization algorithms

Quasi-Newton algorithms minimize a functionF(x),x ∈R ⁿ, searching at any iterationk along the directions _k=?H _kg_k, whereg _k=?F(x _k) andH _k approximates in some sense the inverse Hessian ofF(x) atx _k. When the matrixH is updated according to the formulas in Broyden's family and when an exact line search is performed at any iteration, a compact algorithm (free from the Broyden's family parameter) can be conceived in terms of the followingn ×n matrix: $$H{_R} = H - Hgg{^T} H/g{^T} Hg,$$ which can be viewed as an approximating reduced inverse Hessian. In this paper, a new algorithm is proposed which uses at any iteration an (n?1)×(n?1) matrixK related toH _R by $$H_R = Q\left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & K \\ \end{array} } \right]Q$$ whereQ is a suitable orthogonaln×n matrix. The updating formula in terms of the matrixK incorporated in this algorithm is only moderately more complicated than the standard updating formulas for variable-metric methods, but, at the same time, it updates at any iteration a positive definite matrixK, instead of a singular matrixH _R. Other than the compactness with respect to the algorithms with updating formulas in Broyden's class, a further noticeable feature of the reduced Hessian algorithm is that the downhill condition can be stated in a simple way, and thus efficient line searches may be implemented.     

The size of chordal,interval and threshold subgraphs

Given a graphG withn vertices andm edges, how many edges must be in the largest chordal subgraph ofG? Form=n ²/4+1, the answer is 3n/2?1. Form=n ²/3, it is 2n?3. Form=n ²/3+1, it is at least 7n/3?6 and at most 8n/3?4. Similar questions are studied, with less complete results, for threshold graphs, interval graphs, and the stars on edges, triangles, andK ₄'s.     

Existence, uniqueness and stability ofC m solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f ⁿ (x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ?,G∈C ^m (J ⁿ⁺¹, ?) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.     

Decrease rate of the probabilities ofε-deviations for the means of stationary processes

The asymptotic behavior asn → ∞ of the normed sumsσn =n ^?1 Σ _k ^=0n?1 Xk for a stationary processX = (X _n ,n ∈ ?) is studied. For a fixedε > 0, upper estimates for P(sup _k≥n ¦σ _k ¦ ≥ε) asn → ∞ are obtained.     

Certain finite dimensional maps and their application to hyperspaces

In [8] Y. Sternfeld and this author gave a positive answer to the following longstanding open problem: Is the hyperspace (=the space of all subcontinua endowed with the Hausdorff metric) of a 2-dimensional continuum infinite dimensional? This result was improved in [9] where it was shown that for every positive integer numbern a 2-dimensional continuum contains a 1-dimensional subcontiuum with hyperspace of dimension ≥n. And it was asked there: Does a 2-dimensional continuum contain a 1-dimensional subcontinuum with infinite dimensional hyperspace? In this note we answer this question in the positive. Our proof applies maps with the following properties. A real valued mapf on a compactumX is called a Bing map if every continuum that is contained in a fiber off is hereditarily indecomposable.f is called ann-dimensional Lelek map if the union of all non-trivial continua which are contained in the fibers off isn-dimensional. It is shown that for dimX=n+1 the maps which are both Bing andn-dimensional Lelek maps form a denseG _σ-subset of the function spaceC(X, I)