Kinematic measure of segments lying in a domain

The paper proves a formula for calculation of the kinematic measure K(D, l) of set of segments with constant length l, entirely contained in a bounded convex domainDof the Euclidean space. The obtained formula permits to find an explicit form for the kinematic measure K(D, l) for the domains D with known chord length distribution. In particular, application of the obtained formula gives explicit expressions for K(D, l) in the disc, regular triangle, rectangle and regular pentagon.     

Rang maximal pourT_{P^n }

Letk be an algebraically closed field,P ⁿ the n-dimensional projective space overk andT _P ⁿ the tangent vector bundle ofP ⁿ . In this paper I prove the following result: for every integerl, for every non-negative integers, ifZ _s is the union ofs points in sufficiently general position inP ⁿ , then the restriction mapH ⁰(P ⁿ ,T _P ⁿ (l)) →H ⁰(Z _s,T _P ⁿ (l)_{|z s} ) has maximal rank. This result implies that the last non-trivial term of the minimal free resolution of the homogeneous ideal ofZ _s is the conjectured one by the Minimal Resolution Conjecture of Anna Lorenzini (cf. [Lo]).     

CM values and central L-values of elliptic modular forms

Let f be a holomorphic cusp form of weight l on SL₂(Z) and Ω an algebraic Hecke character of an imaginary quadratic field K with Ω((α)) = (α/|α|) ^l for ${\alpha\in K^{\times}}Let f be a holomorphic cusp form of weight l on SL₂(Z) and Ω an algebraic Hecke character of an imaginary quadratic field K with Ω((α)) = (α/|α|) ^l for a ? K^×{\alpha\in K^{\times}}. Let L(f, Ω; s) be the Rankin-Selberg L-function attached to (f, Ω) and P(f, Ω) an “Ω-averaged” sum of CM values of f. In this paper, we give a formula expressing the central L-values L(f, Ω; 1/2) in terms of the square of P(f, Ω).     

Shape sensitivity analysis of stress intensity factors by Generalized J-integrals

We consider an elastic plate with the non-deformed shape Ω_Σ := Ω \ Σ, where Ω is a domain bounded by a smooth closed curve Γ and Σ ⊂ Ω is a curve with the end points {γ₁, γ₂}. If the force g is given on the part Γ_N of Γ, the displacement u is fixed on Γ_D := Γ \ Γ_N and the body force f is given in Ω, then the displacement vector u(x) = (u₁(x), u₂(x)) has unbounded derivatives (stress singularities) near γ_k, k = 1, 2    u(x) = ∑²_{k, l=1} K_l(γ_k)r^1/2_kS^C_kl(θ_k) + u_R(x)     near γ_k. Here (r_k, θ_k) denote local curvilinear polar co-ordinates near γ_k, k = 1, 2, S^C_kl (θ_k) are smooth functions defined on [−π, π] and u_R(x) ∈ {H²(near γ_k)}². The constants K_l(γ_k),   l = 1, 2, which are called the stress intensity factors at γ_k (abbr. SIFs), are important parameters in fracture mechanics. We notice that the stress intensity factors K_l(γ_k) (l = 1, 2;  k = 1, 2) are functionals K_l(γ_k) = K_l(γ_k; ℒ︁, Ω, Σ) depending on the load ℒ︁, the shape of the plate Ω and the shape of the crack Σ. We say that the crack Σ is safe, if K_l(γ_k; Ω)² + K₂(γ_k; Ω)² < RẼ. By a small change of Ω the shape Σ can change to a dangerous one, i.e. we have K₁(γ_k; Ω)² + K₂(γ_k; Ω)² ⩾ RẼ. Therefore it is important to know how K_l(γ_k) depends on the shape of Ω. For this reason, we calculate the Gâteaux derivative of K_l(γ_k) under a class of domain perturbations which includes the approximation of domains by polygonal domains and the Hadamard's parametrization Γ(τ) := {x + τϕ(x)n(x);  x ∈ Γ}, where ϕ is a function on Γ and n is the outward unit normal on Γ. The calculations are quite delicate because of the occurrence of additional stress singularities at the collision points {γ₃, γ₄} = Γ _D ∩ Γ _N. The result is derived by the combination of the weight function method and the Generalized J-integral technique (abbr. GJ-integral technique). The GJ-integrals have been proposed by the first author in order to express the variation of energy (energy release rate) by extension of a crack in a 3D-elastic body. This paper begins with the weak solution of the crack problem, the weight function representation of SIF's, GJ-integral technique and finish with the shape sensitivity analysis of SIF's. GJ-integral J_ω(u; X) is the sum of the P-integral (line integral) P_ω(u, X) and the R-integral (area integral) R_ω(u, X). With the help of the GJ-integral technique we derive an R-integral expression for the shape derivative of the potential energy which is valid for all displacement fields u ∈ H¹. Using the property that the GJ-integral vanishes for all regular fields u ∈ H² we convert the R-integral expression for the shape derivative to a P-integral expression. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Light subgraphs in planar graphs of minimum degree 4 and edge‐degree 9

A graph H is light in a given class of graphs if there is a constant w such that every graph of the class which has a subgraph isomorphic to H also has a subgraph isomorphic to H whose sum of degrees in G is ≤ w. Let be the class of simple planar graphs of minimum degree ≥ 4 in which no two vertices of degree 4 are adjacent. We denote the minimum such w by w(H). It is proved that the cycle C_s is light if and only if 3 ≤ s ≤ 6, where w(C₃) = 21 and w(C₄) ≤ 35. The 4‐cycle with one diagonal is not light in , but it is light in the subclass consisting of all triangulations. The star K_1,s is light if and only if s ≤ 4. In particular, w(K_1,3) = 23. The paths P_s are light for 1 ≤ s ≤ 6, and heavy for s ≥ 8. Moreover, w(P₃) = 17 and w(P₄) = 23. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 261–295, 2003     

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

In this short paper we investigate the relation between higher Chern classes and reduced power operations in motivic cohomology. More precisely, we translate the well-known arguments [5] into the context of motivic cohomology and define higher Chern classes c_p,q : K _p(X) → H^2q-p (X,Z(q)) → H^2q-p(X, Z/l(q)), where X is a smooth scheme over the base field k, l is a prime number and char(k) ≠ l. The same approach produces the classes for K-theory with coefficients as well. Let further Pⁱ : H^m(X, Z/l(n)) → H^m+2i(l-1) (X, Z/l(n + i(l - 1))) denote the ith reduced power operation in motivic cohomology, constructed in [2]. The main result of the paper looks as follows.     

Sensitivity analysis of nondifferentiable sums of singular values of rectangular matrices

Let K(x) be an s-by-r rectangular matrix depending on a parameter x ε E and denote by g(x) the sum of its m largest singular values (1 ≤ m ≤ Min{s,r}). If K(x) depends affinely on x, then g is a nondifferentiable convex function. In this paper we consider first the affine case and give some formulas for the conjugate, subdifferential, and ε-subdifferential of g. These formulas are then used to obtain perturbation bounds for g(x). We study next the nonaffine case and discuss some questions related with the regularity, generalized subdifferentiability, and directional differentiability of g.     

Off‐diagonal and asymptotic results on the Ramsey number r(K2,m,K2,n)

An upper bound on the Ramsey number r(K_2,n‐s,K_2,n) where s ≥ 2 is presented. Considering certain r(K_2,n‐s,K_2,n)‐colorings obtained from strongly regular graphs, we additionally prove that this bound matches the exact value of r(K_2,n‐s,K_2,n) in infinitely many cases if holds. Moreover, the asymptotic behavior of r(K_2,m,K_2,n) is studied for n being sufficiently large depending on m. We conclude with a table of all known Ramsey numbers r(K_2,m,K_2,n) where m,n ≤ 10. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 252–268, 2003     

A Structure Preserving Generalization of Independence,Domination and Irredundance

Abstract

The now famous inequality chain ir≤γ≤i≤β ≤ Γ ≤ IR, where ir and IR denote the lower and upper irredundance numbers of a graph, γ and Γ the lower and upper domination numbers, i the independent domination number and β the independence number of a graph, may be seen as the culmination of a process by which we start with independence (a hereditary property of vertex sets); we characterize maximal independence by domination (a superhereditary property of vertex sets), and then characterize minimal domination by irredundance (again a hereditary property). In this paper we generalize independent, dominating and irredundant sets of a graph G to what we will call s-dominating, s-independent and s-irredundant functions (for s a positive integer), which are functions of the type f : V (G) → N, in such a way that the maximal 1-independent, the minimal 1- dominating and the maximal 1-irredundant functions are the characteristic functions of the maximal independent, the minimal dominating and the maximal irredundant sets of G respectively. In addition, we would want to preserve those properties of and relationships between independence, domination and irredundance needed to extend the inequality chain ir≤γ≤i≤β ≤ Γ ≤ IR to one for s-dominating, s-independent and s-irredundant functions by a process similar to that described above.     

Un Nouveau ℓ (K) Qui Possede la Propriete de Grothendieck

Using the continuum hypothesis, we construct a compact spaceK such that ℓ(K) possesses the Grothendieck property, but such that the unit ball of ℓ(K)′ does not containβ N, and hence, in particular, such thatl ^∞(N) is neither a subspace nor quotient of ℓ(K). In particular,K does not contain a convergent sequence but does not containβ N.      

Lower bounds for lower Ramsey numbers

For any graph G, let i(G) and μ;(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers m and n, the lower Ramsey number s(m, n) is the largest integer p so that every graph of order p has i(G) ≤ m or μ;(G) ≤ n. In this paper we give several new lower bounds for s (m, n) as well as determine precisely the values s(1, n).     

Contact Elements on Fibered Manifolds

For every product preserving bundle functor T ^μ on fibered manifolds, we describe the underlying functor of any order (r, s, q), s ≥ r ≤ q. We define the bundle K_k,l^r,s,q YK_{k,l}^{r,s,q} Y of (k, l)-dimensional contact elements of the order (r, s, q) on a fibered manifold Y and we characterize its elements geometrically. Then we study the bundle of general contact elements of type μ. We also determine all natural transformations of K_k,l^r,s,q YK_{k,l}^{r,s,q} Y into itself and of T( K_k,l^r,s,q Y )T\left( {K_{k,l}^{r,s,q} Y} \right) into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms from Y to K_k,l^r,s,q YK_{k,l}^{r,s,q} Y .     

On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001     

Proper S(t,K,v)'s for t ≥ 3,v ≤ 16, |K| > 1 and Their Extensions

We characterize the proper t-wise balanced designs t-(v,K,1) for t ≥ 3, λ = 1 and v ≤ 16 with at least two block sizes. While we do not examine extensions of S(3,4,16)'s, we do determine all other possible extensions of S(3,K,v)'s for v ≤ 16. One very interesting extension is an S(4, {5,6}, 17) design.©1995 John Wiley & Sons, Inc.     

Some Bipartite Ramsey Numbers

Consider a complete bipartite graph K(s, s) with p = 2s points. Let each line of the graph have either red or blue colour. The smallest number p of points such that K(s, s) always contains red K(m, n) or blue K(m, n) is called bipartite Ramsey number denoted by rb(K(m, n), K(m, n)). In this paper, we show that

ASYMPTOTIC BOUNDS FOR IRREDUNDANT RAMSEY NUMBERS

Abstract

The irredundant Ramsey number s(m,n) is the smallest N such that in every red-blue colouring of the edges of K_N , either the blue graph contains an m-element irredundant set or the red graph contains an n-element irredundant set. We prove an asymptotic lower bound for s(m, n).     

On the mean square of Lerch zeta-functions

Using the approximate functional equation for L(l,a, s) = ?_n=0^￥ [(e(ln))/((n+a)^s)] L(\lambda,\alpha, s) = \sum\limits_{n=0}^{\infty} {e(\lambda n)\over (n+\alpha)^s} , we prove for fixed parameters $ 0<\lambda,\alpha\leq 1 $ 0<\lambda,\alpha\leq 1 asymptotic formulas for the mean square of L(l,a,s) L(\lambda,\alpha,s) inside the critical strip. This improves earlier results of D. Klusch and of A. Laurin)ikas.     

Circular choosability of graphs

This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) χ_{c, l}(G) of a graph G and prove that they are equivalent. Then we prove that for any graph G, χ_{c, l}(G) ≥ χ_l(G) ? 1. Examples are given to show that this bound is sharp in the sense that for any ? 0, there is a graph G with χ_{c, l}(G) > χ_l(G) ? 1 + ?. It is also proved that k‐degenerate graphs G have χ_{c, l}(G) ≤ 2k. This bound is also sharp: for each ? < 0, there is a k‐degenerate graph G with χ_{c, l}(G) ≥ 2k ? ?. This shows that χ_{c, l}(G) could be arbitrarily larger than χ_l(G). Finally we prove that if G has maximum degree k, then χ_{c, l}(G) ≤ k + 1. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 210–218, 2005     

Improper choosability of graphs and maximum average degree

Improper choosability of planar graphs has been widely studied. In particular, ?krekovski investigated the smallest integer g_k such that every planar graph of girth at least g_k is k‐improper 2‐choosable. He proved [9] that 6 ≤ g₁ ≤ 9; 5 ≤ g₂ ≤ 7; 5 ≤ g₃ ≤ 6; and ? k ≥ 4, g_k = 5. In this article, we study the greatest real M(k, l) such that every graph of maximum average degree less than M(k, l) is k‐improper l‐choosable. We prove that if l ≥ 2 then . As a corollary, we deduce that g₁ ≤ 8 and g₂ ≤ 6, and we obtain new results for graphs of higher genus. We also provide an upper bound for M(k, l). This implies that for any fixed l, . © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 181–199, 2006