Results on the existence of solutions to a new class of impulsive singular fractional differential systems with multiple base points are established. The assumptions imposed on the nonlinearities, see ((C) and (D) in Theorem 3.1), are weaker than known ones, (i.e., (A) in Introduction section). The analysis relies on the well known fixed point theorems. An example is given to illustrate the efficiency of the main theorems. The investigation shows that these results and methods are helpful for study in the nonlinear area and the numerical simulation, especially for study in the the numerical solution of a fractional differential equation with multiple base points with or without impulse effects. A section “Conclusions” is given with future work research directions.

In this study a relation between the Laplace transform and the generalized Hankel-Clifford transform is established. The relation between distributional generalized Hankel-Clifford transform and distributional one sided Laplace transform is developed. The results are verified by giving illustrations. The relation between fractional Laplace and fractional generalized Hankel-Clifford transformation is also established. Further inversion theorem considering fractional Laplace and fractional generalized Hankel-Clifford transformation is proved in Zemanian space.

Let $L$ be a finitely generated free Lie $p$-algebra and $\langle a\rangle$ an ideal generated by $a\in L$. It is proved that $L/\langle a \rangle$ is free if and only if $\langle a\rangle$ is primitive (i.e. $a$ belongs to some set of free generators of $L$). Earlier analogues theorems were proved for some objects, for example, for groups, Lie algebras, free algebras and so on.

Let $G$ be a $\left(p,q\right)$ graph. Let $f:V\left(G\right)\to \left\{1,2,\dots ,p\right\}$ be a function. For each edge $uv$, assign the label $\left|f(u)-f(v)\right|$. $f$ is called a difference cordial labeling if $f$ is an injective map and $\left|e_{f} \left(0\right)-e_{f} \left(1\right)\right|\leq 1$ where $e_{f} \left(1\right)$ and $e_{f} \left(0\right)$ denote the number of edges labeled with $1$ and not labeled with $1$ respectively. A graph which admits a difference cordial labeling is called a difference cordial graph. In this paper, we investigate the difference cordiality of torus grids $C_{m}\times C_{n}$, $K_{m}\times P_{2}$, prism, book, mobius ladder, Mongolian tent and $n$-cube.

Let $L/K$ be any separable extension of complete discrete valued fields of degree $p$. This work, is a study of some “standard over-extensions” of $L/K$, with the description of their Galois groups. The second target, which is the aim of this work, concerns the Galois closure of $L/K$. The study of the normal case has been done in some former work.

K.K.Dewan and A.Ahuja, Growth of polynomials with prescribed zeros, II, Tbilisi Math. J. 8 (2) (2015), 69-74, De Gruyter Open,

DOI 10 1515/tmj - 2015 - 0011.

The authors have requested that their paper be withdrawn because main results of the text were published in Mathematick Vesnik,

S.M. Pukhta, Growth of polynomials with prescribed zeros, Journal of Mathematick Vesnik 65 (1) (2013), 137-142.

In the article, the authors introduce the new concepts “co-ordinated $(\alpha, \mbox{QC})$-, $(\alpha,\mbox{JQC})$-, $(\alpha, \mbox{CJ})$- and $(\alpha, \mbox{J})$-convex functions”, establish some Hermite-Hadamard's type integral inequalities for the co-ordinated $(\alpha, \mbox{QC})$-, $(\alpha,\mbox{JQC})$-, $(\alpha, \mbox{CJ})$- and $(\alpha, \mbox{J})$-convex functions.

In this paper, the Authors establish a new identity for twice differentiable functions. Afterwards some new inequalities are presented related to perturbed trapezoid inequality for the classes of functions whose second derivatives of absolute values are convex, $s$-convex and $tgs$-convex. Last of all, applications to special means have also been presented.

In this paper, we derive a new lemma including third-order derivative of a function via fractional integrals. Using this lemma, we establish some new fractional estimates for Hermite-Hadamard type inequalities for convex functions. Several special cases are also discussed. Some applications to special means of real numbers are also discussed. The ideas and techniques used in this paper may stimulate future investigations regarding Hermite-Hadamard type of inequalities and its application in different areas.

In this paper we study characterizations of odd and even dimensional mixed super quasi-Einstein manifold and we give three and four dimensional examples (both Riemannian and Lorentzian) of mixed super quasi-Einstein manifold to show the existence of such manifold. Also in the last section we give the examples of warped product on mixed super quasi-Einstein manifold.

Let $H^{m}$ be a Heisenberg group provided with a norm $\rho$. A mapping $f:H^{m}\rightarrow H^{m}$ is called preserving the distance $n$ if for all $x,y$ of $H^{m}$ with $\rho(x^{-1}y)=n$ then $\rho(f(x)^{-1}f(y))=n$. We obtain some results for the Aleksandrov problem in the Heisenberg group.

The aim of this paper is to determine Hyers-Ulam-Rassias Stability results concerning the quadratic functional equation, $f(2x \,+\,y)\,+ \, f(2x\,-\,y)\, =\,2f(x\,+\,y)\,+\,2f(x\,-\,y)\,+\,4f(x)\,-\,2f(y)$ in intuitionistic fuzzy Banach spaces.

A semiring $S$ is said to be a $t$-$k$-simple semiring if it has no non-trivial proper left $k$-ideal and no non-trivial proper right $k$-ideal. We introduce the notion of $t$-$k$-simple semirings and characterize the semirings in $\mathbb{SL^{+}}$, the variety of all semirings with a semilattice additive reduct, which are distributive lattices of $t$-$k$-simple subsemirings. A semiring $S$ is a distributive lattice of $t$-$k$-simple subsemirings if and only if every $k$-bi-ideal in $S$ is completely semiprime $k$-ideal. Also the semirings for which every $k$-bi-ideal is completely prime has been characterized.

This communication deals with first introducing the exponential type hybrid $B-(b,\rho,\theta,\tilde{p},\tilde{r})$- invexities and then establishing a class of the $\epsilon-$ efficiency conditions applying to multiobjective fractional programming problems. The exponential type hybrid $B-(b,\rho,\theta,\tilde{p},\tilde{r})$-invexities encompass most of the existing generalized higher order invexities as well as the exponential type generalized invexities, including the Antczak type first order $B-(\tilde{p},\tilde{r})-$invexities. To the best of our knowledge, the obtained results seem to be most advanced on generalized invexities available in the literature, while offer more suitable applications to other fields and beyond.

In this paper, we have considered the fractional diffusion and fractional diffusion-wave equations in which the time derivative is a fractional derivative in the Caputo form and have obtained their numerical solutions by collocation method using cubic B-spline base functions. In the solution process, for the fractional diffusion equation $L1$ discretizaton formula of the fractional derivative is applied, and for the fractional diffusion-wave equation $L2$ discretizaton formula of the fractional derivative is applied. Accuracy of the proposed method is discussed by computing the error norms $L_{2}$ and $L_{\infty}$ . A stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

We develop new closed form representations of sums of alternating harmonic numbers and reciprocal binomial coefficients.

A double sequence $\{x_{k,l}\}$ is quasi-Cauchy if given an $\epsilon \gt 0$ there exists an $N \in {\bf N}$ such that $$\max_{r,s= 1\mbox{ and/or } 0} \left \{|x_{k,l} - x_{k+r,l+s}|\right \} \lt \epsilon .$$ We study continuity type properties of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$.

In the present manuscript, a cubic B-spline finite element collocation method has been used to obtain numerical solutions of the nonlinear time fractional gas dynamics equation. While the Caputo form is used for the time fractional derivative appearing in the equation, the $L1$ discretization formula is applied to the equation in terms of time. It has been seen that the results of the present study are in agreement with the those of exact solution. Therefore, the present method can be used as an alternative and efficient one to find out the numerical solutions of both linear and nonlinear fractional differential equations available in the literature. In order to control the accuracy and efficiency of the present method, the error norms $L_{2}$ and $L_{\infty}$ have been calculated. It is evident that the newly obtained numerical solutions by the present method can be computed easily with the implementation and effectiveness of the approach used in the article.

The objective of the present paper is to study of indefinite trans-Sasakian manifold with a semi-symmetric metric connection. We have found the relations between curvature tensors, Ricci curvature tensors and scalar curvature of indefinite trans-Sasakian manifolds with semi-symmetric metric connection and with metric connection. Also, we have proved some results on quasi-projectively flat and $\varphi -$projectively flat manifolds with respect to semi-symmetric metric connection. It is shown that the manifold satisfying $\overset{\_}{R.}\overset{\_}{S}$ $=0$ is an $\eta -$Einstein manifold if $\alpha =0$ and $\beta =constant.$ It is also proved that the manifold satisfying $\overset{\_}{P}.\overset{\_}{S}=0$ is an $\eta -$ Einstein manifold if $\alpha =0$ and $\beta =constant.$ Finally, we have obtained the conditions for the manifold with semi-symmetric metric connection to be conformally flat and $\xi -$conformally flat.

This paper is concerned with four-point boundary value problems of the one-dimensional generalized Lane-Emden systems on whole lines. The Green's functions $G(t,s)$ for the problem $-(\rho(t)x'(t))'=0$ with boundary conditions $\lim\limits_{t\to-\infty}x(t)-kx(\xi)= \lim\limits_{t\to +\infty}x(t)-l x(\eta)=0$ and $\lim\limits_{t\to-\infty}x(t)-kx(\xi)= \lim\limits_{t\to +\infty}\rho(t)x'(t)-l \rho(\eta)x'(\eta)=0$ are obtained respectively. We proved that $G(t,s)\ge 0$ under some assumptions which actually generalize a corresponding result in [J. Math. Anal. Appl. 305 (2005) 253-276]. Sufficient conditions to guarantee the existence of positive solutions of this kind of models are established. Examples are given at the end of the paper.

For every positive integer $k$ and every graph $G=(V,E)$ with minimum degree at least $k$, a vertex set $S$ is a $k$-tuple total dominating set (resp. $k$-tuple dominating set) of $G$, if for every vertex $v\in V$, $|N_{G}(v)\cap S|\geq k$, (resp. $|N_{G}[v]\cap S|\geq k$). The $k$-tuple total domination number $\gamma _{\times k,t}(G)$ (resp. $k$-tuple domination number $\gamma _{\times k}(G)$) is the minimum cardinality of a $k$-tuple total dominating set (resp. $k$-tuple dominating set) of $G$.

In this paper, we first prove that if $m$ is a positive integer, then for which graphs $G$, $\gamma _{\times k,t}(G)=m$ or $\gamma _{\times k}(G)=m$, and give a necessary and sufficient condition for $\gamma _{\times k,t}(G)=\gamma _{\times (k+1)}(G)$. Then we show that if $G$ is a graph of order $n$ with $\delta (G)\geq k+1\geq 2$, then $\gamma _{\times k,t}(G)$ has the lower bound$\ 2\gamma _{\times (k+1)}(G)-n$ and characterize graphs that equality holds for them. Finally we present two upper bounds for the $k$-tuple total domination number of a graph in terms of its order, minimum degree and $k$.

In this paper, fuzzy fractional diffusion equations (FFDEs) and fuzzy fractional wave equations (FFWEs), subjected to initial and boundary conditions are considered. As these equations have significant applications in physics and engineering, a methodical spectral-tau scheme is utilized to obtain efficient solutions of FFDE and FFWE. For this purpose, shifted Chebyshev polynomials (SCPs) together with its operational matrix of integration in Riemann-Liouville sense and operation matrix of derivative in Caputo sense are employed to approximate the fuzzy-valued functions, their integral and differential terms, respectively. The proposed method is applied to some illustrative examples considered under generalized Hukuhara partial differentiability ($gH_{P}$-differentiability). Graphical results are included with error bar plots of each example that show the efficiency and convergence of the method towards the exact solution.

In this paper, we characterize metrically compact weighted composition operators from Nevanlinna type spaces to weighted Bloch type spaces.

In this paper we establish new inequalities of weighted version of Hermite-Hadamard type inequality for functions whose derivatives absolute values are $m-$ convex. Also we obtain some Fejér type inequalities for quasi-convex functions.