Given graphs G and H, a mapping f : V (G) → V (H) is a homomorphism if (f(u), f(v)) is an edge of H for every edge (u, v) of G. In this paper, we initiate the study of computational complexity of locally injective homomorphisms called partial covers of graphs. We motivate the study of partial covers by showing a correspondence to generalized (2,1)-colorings of graphs, the notion stemming from a...

With the help of a simple variational procedure it is possible to convert the partial sums of order N of many divergent series expansions f(g) = ∑∞ n=0 ang n into partial sums ∑N n=0 bng −ωn, where 0 < ω < 1 is a parameter that parametrizes the approach to the large-g limit. The latter are partial sums of a strong-coupling expansion of f(g) which converge against f(g) for g outside a certain di...

In this article, we study the following problem $$\Delta(w(x)\Delta u) = \ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n}=0 on } \quad\partial B,$$ where $B$ is unit ball of $\mathbb{R}^{4}$ and $ w(x)$ a singular weight logarithm type. The reaction source $f(x,u)$ radial function with respect to $x$ it critical view exponential inequality Adams' existence result proved b...

We consider the boundary value problem $$\displaylines{ - \Delta u + c(x) = \alpha m(x) u^+ \beta u^- +f(x,u), \quad x \in \Omega, \cr \frac{\partial u}{\partial \eta} \sigma (x) =\alpha \rho u^+- +g(x,u), \partial }$$ where \((\alpha, \beta) \mathbb{R}^2\), \(c, m L^\infty (\Omega)\), \(\sigma, (\partial\Omega)\), and nonlinearities f g are bounded continuous functions. study asymmetric (Fucik...

In many-valued logic finite basic sets are considered. We only have to consider the set Ek := {0, 1, . . . , k−1} with k ≥ 3 being fixed in the rest of this paper. The set Pk := {f (n) | f (n) : E k → Ek, n ≥ 1} is the set of all total functions on Ek. Let D ⊆ E k , n ≥ 1 and f (n) : D → Ek. Then f is called an n-ary partial function on Ek with domain D. We also write dom(f) = D. Let P̃ (n) k be...

"Using the theory of Young measures, we prove existence solutions to a strongly quasilinear parabolic system \[\frac{\partial u}{\partial t}+A(u)=f,\] where $A(u)=-\text{div}\,\sigma(x,t,u,Du)+\sigma_0(x,t,u,Du)$, $\sigma(x,t,u,Du)$ and $\sigma_0(x,t,u,Du)$ are satisfy some conditions $f\in L^{p'}(0,T;W^{-1,p'}(\Omega;\R^m))$."

The main goal of the present paper is to study the local solvability of semilnear partial differential operators of the form F(u) = P(D)u + f (x, Q1(D)u, ......, QM (D)u), where P(D), Q1(D), ..., QM (D) are linear partial differntial operators of constant coefficients and f (x, v) is a C∞ function with respect to x and an entire function with respect to v. Under suitable assumptions on the nonl...

⋆ When there are several independent variables it is easy to make mistakes in taking partial derivatives. The fundamental rule is: always know which set of independent variables is in use, so that you are sure which are being held fixed during the process of taking a partial derivative. ⋆ A simple example will illustrate the problems that await the careless. Start with f(x, y) = 2x + y; ∂f/∂x =...

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from...

Quaternionic analysis is a branch of classical referring to different generalizations the Cauchy-Riemann equations quaternion skew field \(\mathbb H\) context. In this work we deals with H-\)valued \((\theta , u)-\)hyperholomorphic functions related elements kernel Helmholtz operator parameter \(u \in \mathbb H\), just in same way as usual quaternionic set harmonic functions. Given domain \(\Om...