Rigid realizations of graphs on small grids

A framework (G,p) is a straight line realization of a graph G=(V,E) in , given by a map . We prove that if (G,p) is an infinitesimally rigid framework then there is an infinitesimally rigid framework (G,q) for which the points q(v), vV(G), are distinct points of the k×k grid, where . We also show that such a framework on G can be constructed in O(|V|³) time.     

A note on the duality mapping of a locally uniformly convex Banach space

Let X be a real locally uniformly convex Banach space with normalized duality mapping J:X→2^X*. The purpose of this note is to show that for every R>0 and every x₀X there exists a function , which is nondecreasing and such that (r)>0 for r>0,(0)=0 and

for all . Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.     

disjoint cycles containing specified independent vertices

Let k1 be an integer and G be a graph of order n3k satisfying the condition that σ₂(G)n+k-1. Let v₁,…,v_k be k independent vertices of G, and suppose that G has k vertex-disjoint triangles C₁,…,C_k with v_iV(C_i) for all 1ik.Then G has k vertex-disjoint cycles such that

(i) for all 1ik.

(ii) , and

(iii) At least k-1 of the k cycles are triangles.

The condition of degree sum σ₂(G)n+k-1 is sharp.

Sharp bounds for the first non-zero Stekloff eigenvalues

Let (M,,) be an n(2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problems

where Δ is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first non-zero eigenvalues of the above problems will be denoted by p₁ and q₁, respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c, then with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius , here λ₁ denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q₁nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius . Finally, we show that q₁A/V and that if the equality holds and if there is a point x₀∂M such that the mean curvature of ∂M at x₀ is no less than A/{nV}, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively.     

Every 4-connected line graph of a quasi claw-free graph is hamiltonian connected

Let G be a graph. For u,vV(G) with dist_G(u,v)=2, denote J_G(u,v)={wN_G(u)∩N_G(v)|N_G(w)N_G(u)N_G(v){u,v}}. A graph G is called quasi claw-free if J_G(u,v)≠ for any u,vV(G) with dist_G(u,v)=2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected.     

A new kind of graph coloring

A proper k-coloring C¹,C²,…,C^k of a graph G is called strong if, for every vertex uV(G), there exists an index i{1,2,…,k} such that u is adjacent to every vertex of Cⁱ. We consider classes of strongly k-colorable graphs and show that the recognition problem of is NP-complete for every k4, but it is polynomial-time solvable for k=3. We give a characterization of in terms of forbidden induced subgraphs. Finally, we solve the problem of uniqueness of a strong 3-coloring.     

Theorems about the attractor for incompressible non-Newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces g(x,t)=g(x,t,t/) possessing the average g⁰(x,t) as →0⁺, where 0<₀<1. Firstly, with assumptions (A₁)–(A₅) on the functions g(x,t,ξ) and g⁰(x,t), we prove that the Hausdorff distance between the uniform attractors and in space H, corresponding to the oscillating equations and the averaged equation, respectively, is less than O() as →0⁺. Then we establish that the Hausdorff distance between the uniform attractors and in space V is also less than O() as →0⁺. Finally, we show for each [0,₀].     

Complete solutions to the Oberwolfach problem for an infinite set of orders

Let n3 and let F be a 2-regular graph of order n. The Oberwolfach problem OP(F) asks for a 2-factorisation of K_n if n is odd, or of K_n−I if n is even, in which each 2-factor is isomorphic to F. We show that there is an infinite set of primes congruent to such that OP(F) has a solution for any 2-regular graph F of order . We also show that for each of the infinitely many with prime, OP(F) has a solution for any 2-regular graph F of order n.     

Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Let I be a finite interval, , and 1p∞. Given a set M, of functions defined on I, denote by the subset of all functions yM such that the s-difference is nonnegative on I, τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm x^(r)_Lp1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in L_q for s>r+1 and (r,p,q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈n^-2.     

Engel graph associated with a group

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph as follows: Take G L(G) as vertices of and join two distinct vertices x and y whenever [x,_ky]≠1 and [y,_kx]≠1 for all positive integers k. We call , the Engel graph of G. In this paper we study the graph theoretical properties of .     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.     

On groups with conjugacy classes of distinct sizes

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following:

1. , for 1a5, a≠2.

2. A₈.

3. PSL(3,4)^e, for 1e10.

4. A₅×PSL(3,4)^e, for 1e10.

Based on this result, we virtually show that if G is an ah-group with π(G) 2,3,5,7 , then F(G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S₃, then the non-abelian socle of G is either trivial or isomorphic to one of the following:

1. , for 3a5.

2. PSL(3,4)^e, for 1e10.

Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z₂S_n. These results are of independent interest and are located in appendices for greater autonomy.     

Orthonormal polynomials with exponential-type weights

Let and let w_ρ(x)|x|^ρexp(-Q(x)), where and is an even function. In this paper we consider the properties of the orthonormal polynomials with respect to the weight , obtaining bounds on the orthonormal polynomials and spacing on their zeros. Moreover, we estimate A_n(x) and B_n(x) defined in Section 4, which are used in representing the derivative of the orthonormal polynomials with respect to the weight .     

Periodic Schur functions and slit discs

A simply connected domain is called a slit disc if minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2π. The conformal mapping of onto , (0)=0, ^′(0)>0, extends to a continuous function on mapping it onto . A finite union E of closed non-intersecting arcs e_k on is called rational if for every k, ν_E(e_k) being the harmonic measures of e_k at ∞ for the domain . A compact E is rational if and only if there is a rational slit disc such that . A compact E essentially supports a measure with periodic Verblunsky parameters if and only if for a rationally placed . For any tuple (α₁,…,α_g+1) of positive numbers with ∑_kα_k=1 there is a finite family of closed non-intersecting arcs e_k on such that ν_E(e_k)=α_k. For any set and any >0 there is a rationally placed compact such that the Lebesgue measure |EE^*| of the symmetric difference EE^* is smaller than .     

Region of variability for close-to-convex functions-II

For a complex number α with let be the class of analytic functions f in the unit disk with f(0)=0 satisfying in , for some convex univalent function in . For any fixed , and we shall determine the region of variability V(z₀,α,λ) for f(z₀) when f ranges over the class

In the final section we graphically illustrate the region of variability for several sets of parameters z₀ and α.     

Logarithmic Sobolev Inequalities for Pinned Loop Groups

LetGbe a connected compact type Lie group equipped with anAd_G-invariant inner product on the Lie algebra ofG. Given this data there is a well known left invariant “H¹-Riemannian structure” on =(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”ν_T(g₀, ·) associated to the Laplace–Beltrami operator on (G). HereT>0,g₀∈(G), andν_T(g₀,·) is a certain probability measure on (G). For fixedg₀∈(G) andT>0, we use the measureν_T(g₀,·) and the Riemannian structure on (G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.     

Weight distribution of some reducible cyclic codes

Let q=p^m where p is an odd prime, m3, k1 and gcd(k,m)=1. Let Tr be the trace mapping from to and . In this paper we determine the value distribution of following two kinds of exponential sums

and

where is the canonical additive character of . As an application, we determine the weight distribution of the cyclic codes and over with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where h₁(x), h₂(x) and h₃(x) are the minimal polynomials of π⁻¹, π⁻² and π^−(pk+1) over , respectively, for a primitive element π of .     

A proof of a conjecture on the Randić index of graphs with given girth

The Randić index R(G) of a graph G is defined by , where is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche, Hansen and Zheng proposed the following conjecture: For any connected graph on n≥3 vertices with Randić index R and girth g,

with equalities if and only if . This paper is devoted to giving a confirmative proof to this conjecture.     

Sharp tridiagonal pairs

Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of -linear transformations A:V→V and A^*:V→V that satisfies the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^*V_iV_i-1+V_i+V_i+1 for 0id, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^* such that for 0iδ, where and ; (iv) there is no subspace W of V such that AWW, A^*WW, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0id the dimensions of coincide. We say the pair A,A^* is sharp whenever dimV₀=1. A conjecture of Tatsuro Ito and the second author states that if is algebraically closed then A,A^* is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A,A^* is sharp and using the data we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by or or . We prove that if the isomorphism class of Φ is determined by the parameter array then there exists a nondegenerate symmetric bilinear form , on V such that Au,v=u,Av and A^*u,v=u,A^*v for all u,vV.     

Additive maps preserving Jordan zero-products on nest algebras

Let and be nest algebras associated with the nests and on Banach Spaces. Assume that and are complemented whenever N_-=N and M_-=M. Let be a unital additive surjection. It is shown that Φ preserves Jordan zero-products in both directions, that is Φ(A)Φ(B)+Φ(B)Φ(A)=0AB+BA=0, if and only if Φ is either a ring isomorphism or a ring anti-isomorphism. Particularly, all unital additive surjective maps between Hilbert space nest algebras which preserves Jordan zero-products are characterized completely.