Solution: Let G(V,E) denote the graph in question; we construct a new graph H(W,F ) in which an ordinary single particle random walk corresponds to the two particle random walk on G. Take W = V × V = {(v1, v2)|v1, v2 ∈ V } and F = {((v1, v2), (w1, w2))|(v1, w1), (v2, w2) ∈ E}. The idea here is that a uniform random walk on H encodes the state of a two particle random walk on G (in the same way ...

and Applied Analysis 3 Step 1. Let F {a} be singleton. Then, there are u ∈ U and v ∈ V such that ‖uv‖ < M, and ‖ u, v, a − a‖ < M 1 . 2.2 Letting w uv ◦ uv, then ‖ uv ◦ uv, a − a‖ ‖ u, v, u, v, a − a − u, v, a − a ‖ < . 2.3 Step 2. Let F {a1, a2}. There is a u1, v1 ∈ U × V such that ‖ u1, v1, a1 − a1‖ < / 1 M , and for u1, v1, a2 − a2 ∈ A there is a u2, v2 ∈ U × V such that ‖ u2, v2, u1, v1, a2...

The purpose of this work is to give a continuous convex function, for which we can characterize the subdifferential, in order to reformulate a variational inequality problem: find u = (u1,u2) ∈ K such that for all v = (v1,v2) ∈ K, ∫ Ω∇u1∇(v1 −u1)+ ∫ Ω∇u2∇(v2−u2)+(f ,v−u)≥ 0 as a system of independent equations, where f belongs to L2(Ω)×L2(Ω) and K = {v ∈H1 0(Ω)×H 0(Ω) : v1 ≥ v2 a.e. in Ω}. 2000...

Let V = V1 ⊕ V2 be a finite-dimensional vector space over an infinite locally-finite field F . Then V admits the torus action of G = F • by defining (v1 ⊕ v2) = v1g−1 ⊕ v2g. If K is a field of characteristic different from that of F , then G acts on the group algebra K[V ] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we show that, for almost ...

Let S* lim represent a subclass of analytic functions f defined in the unit disk such that zf?(z)/f(z) lies interior region bounded by limacon which is given equation [(u ? 1)2 + v2 1/4]2 2[(u 1 1/2)2 v2] = 0. For this class, we obtain structural formula, inclusion results and some radii problems for subclasses starlike functions. Furthermore, sufficient conditions coefficient bounds class

and is also a sufficient condition for avoiding interpenetration of matter. The constitutive properties of hyperelastic materials are completely determined by the stored energy function W (F ) :M + → [0,∞), which — due to frame indifference — has to be invariant under rotations. For isotropic elastic materials W (F )=Φ(v1,v2,v2), where Φ is a symmetric function of the principal stretches v1,v2,...

We consider uniqueness theorems in classical analysis having the form (+) ∀u ∈ U, v1, v2 ∈ Vu ( G(u, v1) = 0 = G(u, v2) → v1 = v2 ) , where U, V are complete separable metric spaces, Vu is compact in V and G : U × V → IR is a constructive function. If (+) is proved by arithmetical means from analytical assumptions (++) ∀x ∈ X∃y ∈ Yx∀z ∈ Z ( F (x, y, z) = 0 ) only (where X, Y, Z are complete sep...

Model modulation index Figure 1 (a) An input image is convolved with a model V1 front-end with units tuned to different orientations and spatial frequencies. The first V2 stage computes derivatives over the activation map of V1 across space, as well as orientation and spatial frequency. We modelled “V2 simple cells” as the rectified output of these filters. The final V2 stage pools and rectifie...

For a class V of algebras, denote by Conc V the class of all (∨, 0)semilattices isomorphic to the semilattice Conc A of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1;V2) the smallest cardinality of a (∨, 0)-semilattice in Conc V1 which is not in Conc V2 if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that i...

Let D1 and D2 be two irreducible bounded symmetric domains in the complex spaces V1 and V2 respectively. Let E be the Euclidean metric on V2 and h the Bergman metric on V1. The Bloch constant b(D1,D2) is defined to be the supremum of E(f ′(z)x, f ′(z)x) 1 2 /hz(x, x)1/2, taken over all the holomorphic functions f : D1 → D2 and z ∈ D1, and nonzero vectors x ∈ V1. We find the constants for all th...