We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy tubular neighbourhood, and spinning can be encoded on a computer used to calculate ambient isotopy invariants continuous embeddings N ↪ M one topological manifold into another. More specifically, we describe an algorithm for computing homology H n ( X ,...

The chromatic sum Σ(G) of a graph G is the smallest sum of colors among of proper coloring with the natural number. In this paper, we introduce a necessary condition for the existence of graph homomorphisms. Also, we present Σ(G) < χf (G)|G| for every graph G.

We consider the complexity of oriented homomorphism and two of its variants, namely strong oriented homomorphism and pushable homomorphism, for planar graphs with large girth. In each case, we consider the smallest target graph such that the corresponding homomorphism is NP-complete. These target graphs T4, T5, and T6 have 4, 5, and 6 vertices, respectively. For i ∈ {4, 5, 6} and for every g, w...

In the early 1980's, Johnson defined a homomorphism $\mathcal{I}_{g}^1\to\bigwedge^3 H_1(S_{g},\mathbb{Z})$, where $\mathcal{I}_{g}^1$ is Torelli group of closed, connected and oriented surface genus $g$ with boundary component $S_g$ corresponding without component. This known as homomorphism. We study map induced by on rational homology groups apply it to abelian cycles determined disjoint bou...

A graphs of rank n (homotopy equivalent to a wedge of n circles) without “separating edges” has a canonical n-dimensional compact C manifold thickening. This implies that the canonical homomorphism φ : Out(Fn) → GL(n,Z) is trivial in rational cohomology in the stable range answering a question raised in [6]. Another consequence of the construction is the existence of higher Reidemeister torsion...

We propose cryptanalysis of the First Domingo-Ferrer’s algebraic privacy homomorphism E :Zn → (Zp ×Zq)d where n= pq. We show that the scheme can be broken by (d+ 1) known plaintexts in O(d3 log2 n) time. Even when the modulus n is kept secret, it can be broken by 2(d + 1) known plaintexts in O(d4 logdn+ d3 log2 n+ ε(m)) time with overwhelming probability.  2005 Elsevier B.V. All rights reserved.

In this paper we consider a iterative system of two-point boundary value problems with integral conditions having n singularities and involve an increasing homeomorphism, positive homomorphism operator. By applying Hölder’s inequality Krasnoselskii’s cone fixed point theorem in Banach space, derive sufficient for the existence denumerably many solutions. Finally provide example to check validity o...

This is a survey of the author's recent results on the Kadison and Halmos similarity problems and the closely connected notion of " length " of an operator algebra. We start by recalling a well known conjecture formulated by Kadison [Ka] in 1955. Kadison's similarity problem. Let A be a unital C *-algebra and let u: A → B(H) (H Hilbert) be a unital homomorphism (i.e. we have u(1) = 1 and u(ab) ...

Consider a group G such that there is no homomorphism f : G → {±1}. In that case, how close can we come to such a homomorphism? We show that if f has zero expectation, then the probability that f(xy) = f(x)f(y), where x, y are chosen uniformly and independently from G, is at most 1/2(1 + 1/ √ d), where d is the dimension of G’s smallest nontrivial irreducible representation. For the alternating...

Given a right LCM semigroup S $S$ and homomorphism N : → [ 1 , + ∞ ) $N\colon S\rightarrow [1,+\infty )$ we use the groupoid approach to study KMS β $_\beta$ -states on C ∗ ( $C^*(S)$ with respect dynamics induced by $N$ . We establish necessary sufficient conditions for existence uniqueness of -states. As an application, show that condition obtained so-called generalized scales is as well. Our...