Characterization of measures by potentials

Let μ be a measure in a Banach spaceE, f be an even function onR. We consider the potentialg(a)=f _E f(‖x?a‖)dμ(x). The question is as follows: For whichf does the potentialg determine μ uniquely? In this article we give answers in the cases whereE=l _∞ ⁿ and wheref(t)=|t| ^p andE is a finite dimensional Banach space with symmetric analytic norm. Calculating the Fourier transform of the functionf(‖x‖ _∞) we give a new proof of the J. Misiewicz's result that the functionf(‖x‖ _∞) is positive definite only iff is a constant function.     

Bipartite anti‐Ramsey numbers of cycles

We determine the maximum number of colors in a coloring of the edges of K_m,n such that every cycle of length 2k contains at least two edges of the same color. One of our main tools is a result on generalized path covers in balanced bipartite graphs. For positive integers q≤ a, let g(a,q) be the maximum number of edges in a spanning subgraph G of K_a,a such that the minimum number of vertex‐disjoint even paths and pairs of vertices from distinct partite sets needed to cover V(G) is q. We prove that g(a,q) = a² ? aq + max {a, 2q ? 2}. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 9–28, 2004     

Some results on λ‐designs with two block sizes

A λ‐design is a family ?? = {B₁, B₂, …, B_v} of subsets of X = {1, 2, …, v} such that |B_i∩B_j| = λ for all i≠jand not all B_i are of the same size. The only known example of λ‐designs (called type‐1 designs) are those obtained from symmetric designs by a certain complementation procedure. Ryser [J Algebra 10 (1968), 246–261] and Woodall [Proc London Math Soc 20 (1970), 669–687] independently conjectured that all λ‐designs are type‐1. Let g = gcd(r ? 1, r^* ? 1), where rand r^* are the two replication numbers. Ionin and Shrikhande [J Combin Comput 22 (1996), 135–142; J Combin Theory Ser A 74 (1996), 100–114] showed that λ‐designs with g = 1, 2, 3, 4 are type‐1 and that the Ryser–Woodall conjecture is true for λ‐designs on p + 1, 2p + 1, 3p + 1, 4p + 1 points, where pis a prime. Hein and Ionin [Codes and Designs—Proceedings of Conference honoring Prof. D. K. Ray‐Chaudhuri on the occasion of his 65th birthday, Ohio State University Mathematical Research Institute Publications, 10, Walter de Gruyter, Berlin, 2002, pp. 145–156] proved corresponding results for g = 5 and Fiala [Codes and Designs—Proceedings of Conference honoring Prof. D. K. Ray‐Chaudhuri on the occasion of his 65th birthday, Ohio State University Mathematical Research Institute Publications, 10, Walter de Gruyter, Berlin, 2002, pp. 109–124; Ars Combin 68 (2003), 17–32; Ars Combin, to appear] for g = 6, 7, and 8. In this article, we consider λ designs with exactly two block sizes. We show that in this case, the conjecture is true for g = 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, and for g = 10, 14, 18, 22 with v≠4λ ? 1. We also give two results on such λ‐designs on v = 9p + 1 and 12p + 1 points, where pis a prime. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:95‐110, 2011     

Gitterpunkte unter algebraischen Kurven

Lattice points below Algebraic Curves. A generalization of the classical circle problem is treated. An asymptotic formula for the number of lattice points in a region whose boundary is an algebraic curve is obtained. This gives a mean value formula for the number of representations of the positive integers in the formn=g(x,y), whereg is a polynomial with coefficients 0 and leading terma _d0x^d*a_0dy^d. The caseg(x,y)=p₁(x)+p₂(y) was considered inKuba andNowak [4], andKuba [5]. The discrete Hardy-Littlewood method is used along with Rouché's theorem.

     

List‐coloring the square of a subcubic graph

The square G² of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G²) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list‐chromatic number of G² equals the chromatic number of G², that is, χ_l(G²) = χ(G²) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χ_l(G²) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χ_l(G²) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χ_l(G²) ≤ 7. Dvo?ák, ?krekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χ_l(G²) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χ_l(G²) ≤ 6. All of our proofs can be easily translated into linear‐time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008     

p-Helson sets, 1<p<2

Ap-Helson set is defined to be a closed subsetE of the circle groupT with the property that every continuous function onE can be extended to the full circle in such a way that this extension has its sequence of Fourier coefficients inl ^p. For 1<p<2, the union of two such sets is again ap-Helson set. It is shown that thep-Helson sets (p>1) differ from the Helson sets and also that the notion really depends on the indexp. An analogue of H. Helson’s result is given: ap-Helson set supports no nonzero measure with Fourier-Stieltjes transform inl ^q, 1/p+1/q=1.     

Second‐order properties of the point process of nodes in a stationary Voronoi tessellation

In this paper we derive representation formulae for the second factorial moment measure of the point process of nodes and the second moment of the number of vertices of the typical cell associated with a stationary normal Voronoi tessellation in ?^d . In case the Voronoi tessellation is generated by a stationary Poisson process with intensity λ > 0 the corresponding pair correlation function g_V,λ (r) can be expressed by a weighted sum of d +2 (numerically tractable) multiple parameter integrals. The asymptotic variance of the number of nodes in an increasing cubic domain as well as the second moment of the number of vertices of the typical Poisson Voronoi cell are calculated exactly by means of these parameter integrals. The existence of a (d ? 1)st‐order pole of g_V,λ (r) at r = 0 is proved and the exact value of lim_{r →0} r^{d –1} g_V,λ (r) is determined. In the particular cases d = 2 and d = 3 the graph of g_V,1(r) including its local extreme points, the points of level 1 of gV, 1(r) and other characteristics are computed by numerical integration. Furthermore, an asymptotically exact confidence interval for the intensity of nodes is obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extreme Points and Minimal Outer Area Problem for Meromorphic Univalent Functions

For 0 < p < 1, letS_pdenote the class of functionsf(z) meromorphic univalent in the unit disk with the normalizationf(0) = 0,f′(0) = 1, andf(p) = ∞. LetS_p(a) be the subclass ofS_pwith the fixed residuea. In this note we determine the extreme points of the classS_p(a). As an application, we solve the problem of minimizing the outer area overS_p(a), which was posed by S. Zemyan (J. Analyse Math.39, 1981, 11–23).     

Partitions of large multipartites with congruence conditions. II

For 1 ≦ l ≦ j, let $\text{a}$_l = ?_h=1^q(l){a_lh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the a_lh are positive integers such that j > 1, a_l1 < … < a_lq(2) ≦ M, and let $\text{a}$_l^′ = $\text{a}$_l ∪ {0}. Let p(n : $\text{B}$) be the number of partitions of n = (n₁,…,n_j) where, for 1 ≦ l ≦ j, the lth component of each part belongs to $\text{B}$_l and let p¹(n : $\text{B}$) be the number of partitions of n into different parts where again the lth component of each part belongs to $\text{B}$_l. Asymptotic formulas are obtained for p(n : $\text{a}$), p¹(n : $\text{a}$) where all but one n_l tend to infinity much more rapidly than that n_l, and asymptotic formulas are also obtained for p(n : $\text{a}$′), p¹(n ; $\text{a}$′), where one n_l tends to infinity much more rapidly than every other n_l. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the n_l tend to infinity at approximately the same rate.     

Application of the τ‐function theory of Painlevé equations to random matrices: PV,PIII, the LUE,JUE, and CUE

With 〈·〉 denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is for I = (0, s) and I = (s, ∞), where χ = 1 for λ_l ∈ I and χ = 0 otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that ?_N(I; a, μ) is a τ‐function associated with the Hamiltonian therein, and so can be characterized as the solution of a certain second‐order second‐degree differential equation, or in terms of the solution of certain difference equations. The cases μ = 0 and μ = 2 are of particular interest, because they correspond to the cumulative distribution and density function, respectively, for the smallest and largest eigenvalue. In the case I = (s, ∞), ?_N(I; a, μ) is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group, and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard‐edge and soft‐edge scaled limits of ?_N(I; a, μ). In particular, in the hard‐edge scaled limit it is shown that the limiting quantity ?^hard((O, s); a, μ) can be evaluated as a τ‐function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation. © 2002 Wiley Periodicals, Inc.     

Minimum permanents of doubly stochastic matrices with One fixed entry

For n  3 and sufficiently small a  0, the minimum value of the permanent function restricted on n × n doubly stochastic matrices with at least one entry equal to a is obtained. For n = 3, the explicit form of the function p is derived where p(a) = min{per(C):C = (c_ij )?Ω₃ c ₁₁ = a}a? [0, 1].     

An extension of the hardy-littlewood-pólya inequality

The Hardy-Littlewood-Pólya （HLP） inequality [1] states that if a∈l^p,b∈l^q and p>1,q>1,1/p + 1/q>1, λ=2-（1/p+1/q）,then Σ[（a_rb_s）/︱r-s︱^λ]（r≠s）≤C‖a‖_P‖b‖_q.In this article, we prove the HLP inequality in the case where λ= 1, p = q = 2 with a logarithm correction, as conjectured by Ding [2]:Σ[（a_rb_s）/︱r-s︱^λ]（r≠s,1≤r,s≤N）≤（2㏑N+1）‖a‖₂‖b‖₂.In addition, we derive an accurate estimate for the best constant for this inequality.     

Strong Haagerup inequalities with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on , which improves and generalizes previous results by Kemp–Speicher (in the scalar case) and Buchholz and Parcet–Pisier (in the non-holomorphic setting). Namely the norm of an element of the form ∑_{i=(i1,…,id)}a_iλ(g_i1g_id) is less than , where M₀,…,M_d are d+1 different block-matrices naturally constructed from the family (a_i)_iI^d for each decomposition of I^dI^l×I^d−l with l=0,…,d. It is also proved that the same inequality holds for the norms in the associated non-commutative L_p spaces when p is an even integer, pd and when the generators of the free group are more generally replaced by *-free -diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.     

AN EIGENSPACE CHARACTERIZATION OF LORENTZ-IMPROVING MEASURES ON COMPACT GROUPS

Abstract

A measure μ on a compact group is called Lorentz-improving if for some 1 > p > ∞ and 1 → q ₁ > q ₂ ∞ μ *L (p, q ₂) ? L(p, q ₁). Let T _μ denote the operator on L ² defined by T _μ(f) = μ * f. Lorentz-improving measures are characterized in terms of the eigenspaces of T _μ, if T _μ is a normal operator, and in terms of the eigenspaces of |T _μ| otherwise. This result generalizes our recent characterization of Lorentz-improving measures on compact abelian groups and is modelled after Hare's characterization of L ^p -improving measures on compact groups.     

A New Class of Perfect Fréchet Spaces

This paper deals with a new class of perfect Frèchet spaces which can be obtained by interpolation of echelon spaces: l_p,q[a_m,n]. We determine the reflexive, Montel, Schwartz, totally reflexive, totally Montel and nuclear spaces l_p,q[a_m,n]. We also derive results on closed subspaces on the spaces (l_p,q)(N).     

Precise Asymptotic Formulas for Semilinear Eigenvalue Problems

. We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | ^p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L²-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = l_n (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L² norm of the eigenfunction associated with l \lambda .     

Lieb–Thirring inequalities for higher order differential operators

We derive Lieb–Thirring inequalities for the Riesz means of eigenvalues of order γ ≥ 3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators, in dimensions greater than one. For the critical case γ = 1 – 1/(2l) in dimension d = 1 with l ≥ 2 we prove the inequality L⁰_l,γ,d < L_l,γ,d , which holds in contrast to current conjectures. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Mean Value of the Index of Composition of an Integer

For each integer n l(n)=[(log n)/(log g(n))]\lambda(n)={{\rm log}\, n\over{\rm log}\, \gamma(n)} be the index of composition of n, where g(n)=?_p|np\gamma(n)=\prod_{p\vert n}p . For convenience, we write ?_{x ￡ n ￡ x+?x}l(n)\sum_{x\le n\le x+\sqrt{x}}\lambda(n) and ?_{n ￡ x}l(n)\sum_{n\le x}\lambda(n) , as well as for ?_{x ￡ n ￡ x+?x}1/l(n)\sum_{x\le n\le x+\sqrt{x}}1/\lambda(n) and ?_{n ￡ x}1/l(n)\sum_{n\le x}1/\lambda(n) . Finally we study the sum of running over shifted primes.     

Number of points on certain hyperelliptic curves defined over finite fields

For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums _l(v), ψ_l(v), and ψ_2l(v), and the Jacobsthal–Whiteman sums and , over finite fields F_q such that . These results are obtained only in terms of q and l. We apply these results pertaining to the Jacobsthal sums, to determine, for each integer n1, the exact number of F_qⁿ-rational points on the projective hyperelliptic curves aY²Z^e−2=bX^e+cZ^e (abc≠0) (for e=l,2l), and aY²Z^l−1=X(bX^l+cZ^l) (abc≠0), defined over such finite fields F_q. As a consequence, we obtain the exact form of the ζ-functions for these three classes of curves defined over F_q, as rational functions in the variable t, for all distinct cases that arise for the coefficients a,b,c. Further, we determine the exact cases for the coefficients a,b,c, for each class of curves, for which the corresponding non-singular models are maximal (or minimal) over F_q.