Some results on starlike trees and sunlike graphs

A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike treesG andH are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, letG be a simple graph of ordern with vertex setV(G)={1,2, …,n} and letH={H ₁,H ₂, ...H _n } be a family of rooted graphs. According to [2], the rooted productG(H) is the graph obtained by identifying the root ofH _i with thei-th vertex ofG. In particular, ifH is the family of the paths $P_{k_1 } , P_{k_2 } , ..., P_{k_n } $ with the rooted vertices of degree one, in this paper the corresponding graphG(H) is called the sunlike graph and is denoted byG(k ₁,k ₂, …,k _n ). For any (x ₁,x ₂, …,x _n ) ∈I _* ⁿ , whereI _*={0,1}, letG(x ₁,x ₂, …,x _n ) be the subgraph ofG which is obtained by deleting the verticesi ₁, i₂, …,i _j ∈ V(G) (0≤j≤n), provided that $x_{i_1 } = x_{i_2 } = ... = x_{i_j } = 0$ . LetG(x ₁,x ₂,…, x _n] be the characteristic polynomial ofG(x ₁,x ₂,…, x _n ), understanding thatG[0, 0, …, 0] ≡ 1. We prove that $$G[k_1 , k_2 ,..., k_n ] = \Sigma _{x \in ^{I_ * ^n } } \left[ {\Pi _{i = 1}^n P_{k_i + x_i - 2} (\lambda )} \right]( - 1)^{n - (\mathop \Sigma \limits_{i = 1}^n x_i )} G[x_1 , x_2 , ..., x_n ]$$ where x=(x ₁,x ₂,…,x _n );G[k ₁,k ₂,…,k _n ] andP _n (γ) denote the characteristic polynomial ofG(k ₁,k ₂,…,k _n ) andP _n , respectively. Besides, ifG is a graph with λ₁(G)≥1 we show that λ₁(G)≤λ₁(G(k ₁,k ₂, ...,k _n )) < for all positive integersk ₁,k ₂,…,k _n , where λ₁ denotes the largest eigenvalue.     

Two Hadamard matrices of order 956 of Goethals-Seidel type

No Hadamard matrices of order 956 appear in the literature. In this note we construct two such matrices. They are both of Goethals-Seidel type. As a consequence of this and of a result of Seberry and Yamada, we conclude that there exist 4-Williamson type matrices of order 1913.This work was supported by NSERC of Canada Grants A-5285 and EQP0107539.     

Weakly compactly generated Banach spaces and the strong law of large numbers

IfB is a weakly compactly generated Banach space andf: (S,S, ) satisfies the strong law of large numbers, thenf=f ₁+f ₂, wheref ₁ is Bochner -integrable andf ₂ is Pettis -integrable with Pettis norm 0. The decomposition is unique.     

On general relations in relativistic kinematics and some of their applications

Two vector fields are considered, a timelike one,u , and an arbitrary one, . The relative expansion and rotation are defined with respect to these fields, their mutual relations are studied, and some general formulas obtained. Applications are made, first to vector fields which are mutually nonrotating in a plane-symmetric metric, then to the electromagnetic field of an arbitrary magnetohydrodynamic fluid.     

The elliptical gaussian wave transformation due to diffraction by an elliptical hologram

Realized as an interferogram of a spherical and a cylindrical wave, the elliptical hologram is treated as a plane diffracting grating which produces Fresnel diffraction of a simple astigmatic gaussian incident wave. It is shown that if the principal axes of the incident beam coincide with the principal axes of the hologram, the diffracted wave field is composed of three different astigmatic gaussian waves, with their waists situated in parallel but distinct planes. The diffraction pattern, observed on a transverse screen, is the result of the interference of the three diffracted wave components. It consists of three systems of overlapped second-order curves, whose shape depends on the distance of the observation screen from the hologram, as well as on the parameters of the incident wave beam and the hologram. The results are specialized for gratings in the form of circular and linear holograms and for the case of a stigmatic gaussian incident wave, as well as for the normal plane-wave incidence on the three mentioned types of hologram.     

Density of states for intermediate valence and Kondo systems

We calculate the density of states for the nondegenerate Anderson model for various values ofu=U/ andn _f using the perturbation theory withu as the expansion parameter. Summing all the -independent self-energy diagrams, we use the Friedel sum rule and Ward identities to express the physical quantities in terms of the remaining -dependent part of the self-energy, which we evaluate to the 2nd order. The results for the spin and charge susceptibilities obtained in such a way compare rather well with the Bethe-ansatz results. The density of states exhibits different features in different parts of the parameter space. In Kondo region (u>1,n _f 1, i.e., – _f ~U/2), we obtain a many-body resonance (half-width T _K ) around the Fermi level and two broad peaks () at about _f +n _f U and _f +U. In the VF region (u>1, and | _f |) we obtain only two peaks (), one at about _f and one between _f +n _f U and _f +U. The consequences regarding the shape of the photoemission and inverse photoemission spectra of Ce intermetallics are discussed.     

On a problem connected with zeros of ζ(s) on the critical line

The existence of zeros ofZ ^(k)(t) in short intervals of the type [T, T+H] is established, whereHT ^a(k)logT, . Hitherto the sharpest bounds for the constanta(k) are obtained by employing a certain exponential averaging technique and the estimation of the relevant exponential sums. Bounds for are also derived, under the assumption that orZ(t) does not vanish in certain short intervals.     

Restoration of universality for the rod-to-coil transition scaling in the infinite-dimensionality limit: Exact results for directed walks

We derive scaling forms for the thermodynamic and correlation quantities for the turn-weighted fully and partially directed self-avoiding walks on the hypercubic lattices ind2. In the grand canonical (fixed fugacity per step) ensemble, the conformational rod-to-coil transition sets up in the regimew¯N=O(1), wherew is the weight of each 90° turn and¯N is the (fugacity-dependent) average number of steps. Contrary to the conventional critical phenomena wisdom, the scaling functions for the two different walk models, directed and partially directed, become universal only in the limitd.     

Difference equations in statistical mechanics. I. Cluster statistics models

A review and some new results are presented for several cluster statistics models, solutions of which can be reduced to difference equations. Mathematical techniques suitable for solving these equations are surveyed.