This work provides new adequate conditions for difference equations with forcing, positive and negative terms to have non-oscillatory solutions. A few mathematical inequalities the properties of discrete fractional calculus serve as fundamental foundation our approach. To help establish main results, an analogous representation equation, called a Volterra-type summation is constructed. Two nume...

The theory of the calculus of variations was recently extended to the more general time scales setting, both for delta and nabla integrals. The primary purpose of this paper is to further extend the theory on time scales, by establishing some basic diamond-alpha dynamic integral inequalities. We prove generalized versions of Hölder, Cauchy-Schwarz, Minkowski, and Jensen’s inequalities. For the ...

In this study, we are concerned with introducing Weyl-Titchmarsh theory for a class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian time scales. After developing fundamental properties of solutions and regular spectral problems, we introduce the corresponding maximal and minimal operators for the system. Matrix disks are constructed and proved to be nested and converge ...

A series of silyl, germyl and alkyl substituted trifluoroacetylfurans has been synthesized under Friedel-Crafts electrophilic acylation conditions. Biological investigations have demonstrated that germyl derivatives of trifluoroacetylfuran are more toxic than the silicon analogues. 5-Triethylgermyl-2- trifluoroacetylfuran was the most toxic compound (CD(nabla), 11.2 mg kg(-1), i.p. for white mi...

We study the algebra of semigroups Laplacians on Weyl algebra. consider first-order partial differential operators $\nabla^\pm_i$ forming Lie $[\nabla^\pm_j,\nabla^\pm_k]= i\mathcal{R}^\pm_{jk}$ and $[\nabla^+_j,\nabla^-_k] =i\frac{1}{2}(\mathcal{R}^+_{jk}+\mathcal{R}^-_{jk})$ with some anti-symmetric matrices $\mathcal{R}^\pm_{ij}$ define corresponding $\Delta_\pm=g_\pm^{ij}\nabla^\pm_i\nabla^...

We present the concept of a disjunctive basis as a generic framework for normal forms in modal logic based on coalgebra. Disjunctive bases were defined in previous work on completeness for modal fixpoint logics, where they played a central role in the proof of a generic completeness theorem for coalgebraic mu-calculi. Believing the concept has a much wider significance, here we investigate it m...

This paper deals with classical solution of system $ \begin{array}{*{20}{l}}{\begin{eqnarray} { } \left\{ \begin{array}{ll} u_t = \epsilon_1\Delta u-\chi_{1}\nabla\cdot(u\nabla v)-\chi_{2}\nabla\cdot(u\nabla z)+\mu_1 u(1-u-a_1w), &\\ 0 \Delta v-v+w^{\sigma_1}, w_t \epsilon_2\Delta w-\chi_3\nabla\cdot(w\nabla z)-\chi_4\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), z-z+u^{\sigma_2} & \end{arra...