Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

Lower bounds on the number of maximal subgroups in a finite group

For a finite group G, let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. When G is a cyclic group, an elementary calculation proves that |m(G)| = |π(G)|. In this paper, we prove lower bounds on |m(G)| when G is not cyclic. In general, ${|m(G)| \geq |\pi(G)|+p}$ | m ( G ) | ≥ | π ( G ) | + p , where ${p \in \pi(G)}$ p ∈ π ( G ) is the smallest prime that divides |G|. If G has a noncyclic Sylow subgroup and ${q \in \pi(G)}$ q ∈ π ( G ) is the smallest prime such that ${Q \in {\rm syl}_q(G)}$ Q ∈ syl q ( G ) is noncyclic, then ${|m(G)| \geq |\pi(G)|+q}$ | m ( G ) | ≥ | π ( G ) | + q . Both lower bounds are best possible.     

Degree Sum Conditions for Cyclability in Bipartite Graphs

We denote by G[X, Y] a bipartite graph G with partite sets X and Y. Let d _G (v) be the degree of a vertex v in a graph G. For G[X, Y] and ${S \subseteq V(G),}$ we define ${\sigma_{1,1}(S):=\min\{d_G(x)+d_G(y) : (x,y) \in (X \cap S,Y) \cup (X, Y \cap S), xy \not\in E(G)\}}$ . Amar et al. (Opusc. Math. 29:345–364, 2009) obtained σ _1,1(S) condition for cyclability of balanced bipartite graphs. In this paper, we generalize the result as it includes the case of unbalanced bipartite graphs: if G[X, Y] is a 2-connected bipartite graph with |X| ≥ |Y| and ${S \subseteq V(G)}$ such that σ _1,1(S) ≥ |X| + 1, then either there exists a cycle containing S or ${|S \cap X| > |Y|}$ and there exists a cycle containing Y. This degree sum condition is sharp.     

A Spanning Tree with High Degree Vertices

Let G be a connected graph, let ${X \subset V(G)}$ and let f be a mapping from X to {2, 3, . . .}. Kaneko and Yoshimoto (Inf Process Lett 73:163–165, 2000) conjectured that if |N _G (S) ? X| ≥ f (S) ? 2|S| + ω _G (S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ . In this paper, we show a result with a stronger assumption than this conjecture; if |N _G (S) ? X| ≥ f (S) ? 2|S| + α(S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ .     

On sum sets of sidon sets,II

LetG be a finite abelian group,G?{Z _n, Z₂?Z_2n}. Then every sequenceA={g ₁,...,g_t} of $t = \frac{{4\left| G \right|}}{3} + 1$ elements fromG contains a subsequenceB?A, |G|=|G| such that $\sum\nolimits_{g_i \in B^{g_i } } { = 0 (in G)} $ . This bound, which is best possible, extends recent results of [1] and [22] concerning the celebrated theorem of Erdös-Ginzburg-Ziv [21].     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

A sharp weightedL 2-estimate for the solution to the time-dependent Schrödinger equation

For Ξ∈R ⁿ ,t∈R andf∈S(R ⁿ ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys ₀ such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R ⁿ ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|²) is replaced by exp(it|ξ|^a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL ²(R ⁿ ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

Convergence rates in the law of large numbers for random variables on partially ordered sets

Let (A, ≤) be a partially ordered set, {X _α} a collection of i. i. d. random variables, indexed byA. Let \(S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta \) , |α|=card {β∈A, β∈α}. We study the convergence rates ofS _α/|α|. We derive for a large class of partially ordered sets theorems, like the following one: For suitabler, t with 1/2< <r/t≤1:E|X| ^t M (|X| ^t/r )<∞ andEX=μ if and only if $$S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta $$ for all ε>0, where \(M(x) = \sum _{j< x} d(j)\) withd(j)=card {α∈A, |α|=j}.     

Singular integrals along Lipschitz curves with holomorphic kernels

If γ(x)=x+iA(x),tan ^?1‖A′‖_∞<ω<π/2,S _ω ⁰ ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S _ω ⁰ and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C ₀(γ), ?z∈suppf, where C_c(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:

1. T can be extended to be a bounded operator on L²(γ);
2. there exists a function ?₁ ∈H ^∞(S _ω ⁰ ) such that ?′₁(z)=?(z)+?(-z), ?z∈S _ω ⁰ ?z∈S _w ⁰ .

     

An estimate for the sum of Legendre symbols

For the sum S of the Legendre symbols of a polynomial of odd degree n ≥ 3 modulo primes p ≥ 3, Weil’s estimate |S| ≤ (n ? 1) $ \sqrt p $ and Korobov’s estimate $$ \left| S \right| \leqslant (n - 1)\sqrt {p - \frac{{(n - 3)(n - 4)}} {4}} forp \geqslant \frac{{n^2 + 9}} {2} $$ are well known. In this paper, we prove a stronger estimate, namely, $$ \left| S \right| < (n - 1)\sqrt {p - \frac{{(n - 3)(n + 1)}} {4}} $$ .     

On reconstructing separable reducedp-groups with a given socle

Let \(\bar B^* \) be a separable reduced (abelian)p-group which is torsion complete. We ask whether for \(G \subseteq \bar B^* \) there is \(H \subseteq _{pr} \bar B^* ,H[p] = G[p]\) ,H[p]=G[p],H not isomorphic toG. IfG is the sum of cyclic groups or is torsion complete, the answer is easily no. For otherG, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character ofG is equal to Min _n<ω|p ⁿG|, i.e., $$\mathop {Min}\limits_{n< \omega } |p^n G| = \mathop {Min}\limits_m \mathop \Sigma \limits_{n > m} |(p^n G)[p]/(p^{n + 1} G)[p]|$$ Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.     

The sharp threshold for percolation on expander graphs

We consider a random subgraph G _n (p) of a finite graph family G _n = (V _n , E _n ) formed by retaining each edge of G _n independently with probability p. We show that if G _n is an expander graph with vertices of bounded degree, then for any c _n ∈ (0, 1) satisfying $c_n \gg {1 \mathord{\left/ {\vphantom {1 {\sqrt {\ln n} }}} \right. \kern-0em} {\sqrt {\ln n} }}$ and $\mathop {\lim \sup }\limits_{n \to \infty } c_n < 1$ , the property that the random subgraph contains a giant component of order c _n |V _n | has a sharp threshold.     

Minimum degree of bipartite graphs and the existence ofk-factors

LetG be a bipartite graph with bipartition (X, Y) andk a positive integer. If (i) $$\left| X \right| = \left| Y \right|,$$ (ii) $$\delta (G) \geqslant \left\lceil {\frac{{\left| X \right|}}{2}} \right\rceil \geqslant k,$$ \(\left| X \right| \geqslant 4k - 4\sqrt k + 1\) when |X| is odd and |X| ≥ 4k ? 2 when |X| is even, thenG has ak-factor.     

Generalization of some polynomial inequalities not vanishing in a disk

If $P(z) = \sum\limits_{\nu = 0}^n {c_\nu z^\nu } $ is a polynomial of degree n, then for |β| ≤ 1, it was proved in [4] that $\left| {zP'(z) + n\frac{\beta } {2}P(z)} \right| \leqslant n\left| {1 + \frac{\beta } {2}} \right|\mathop {\max }\limits_{|z| = 1} |P(z)|,|z| = 1 $ In this paper, first we generalize the above result for the s ^th derivative of polynomials and next we improve the above inequality for polynomials with restricted zeros.     

Radial functions and regularity of solutions to the Schrödinger equation

Letf be a radial function and setT ^* f(x)=sup_0<t<1 |T _t f(x)|, x ∈ ?ⁿ, n≥2, where(T_t f)^ (ξ)=e ^it|ξ|a \(\hat f\) (ξ),a > 1. We show that, ifB is the ball centered at the origin, of radius 100, then \(\int\limits_B {|T^ * f(x)|} dx \leqslant c(\int {|\hat f(\xi )|^2 (l + |\xi |^s )ds} )^{1/2} \) if and only ifs≥1/4.     

A priori estimate of gradient of a solution of a certain differential inequality and quasiconformal mappings

We prove a global estimate for the gradient of the solution of the Poisson differential inequality |Δu(x)| ≤ a|Du(x)|² + b, x ∈ B ⁿ , where a, b < ∞ and $u|_{S^{n - 1} } \in C^{1,\alpha } (S^{n - 1} ,\mathbb{R}^m )$ . If m = 1 and $a \le (n + 1)/({\left| u \right|_\infty }4n\sqrt n )$ , then |Du| is a priori bounded. This generalizes some similar results due to S. Bernstein [4] and E. Heinz [10] for the plane. An application of these results yields the main result, namely that a quasiconformal mapping of the unit ball onto a domain with C ² smooth boundary satisfying the Poisson differential inequality is Lipschitz continuous. This extends some results of the author, Mateljevi?, and Pavlovi? from the complex plane to ? ⁿ .     

Schwache Lösungen des Frankl-Morawetz-Problems im ℝ3

In a bounded simple connected region G ? ?³ we consider the equation $$L\left[ u \right]: = k\left( z \right)\left( {u_{xx} + u_{yy} } \right) + u_{zz} + d\left( {x,y,z} \right)u = f\left( {x,y,z} \right)$$ where k(z)? 0 whenever z ? 0.G is surrounded forz≥0 by a smooth surface Γ₀ with S:=Γ₀ ? {(x,y,z)|=0} and forz<0 by the characteristic \(\Gamma _2 :---(x^2 + y^2 )^{{\textstyle{1 \over 2}}} + \int\limits_z^0 {(---k(t))^{{\textstyle{1 \over 2}}} dt = 0} \) and a smooth surface Γ₁ which intersect the planez=0 inS and where the outer normal n=(n_x, n_y, n_z) fulfills \(k(z)(n_x^2 + n_y^2 ) + n_z^2 |_{\Gamma _1 } > 0\) . Under conditions on Γ₁ and the coefficientsk(z), d(x,y,z) we prove the existence of weak solutions for the boundary value problemL[u]=f inG with \(u|_{\Gamma _0 \cup \Gamma _1 } = 0\) . The uniqueness of the classical solution for this problem was proved in [1].