Journal of Mathematics Research
http://www.ccsenet.org/journal/index.php/jmr
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http://www.ccsenet.org/journal/index.php/jmr/article/view/41158
In this paper we show a regularity theorem for local minima of scalar integral functionals of the Calculus of Variations with nonstandard general growth conditions. Let us consider functionals in the following form<br />\begin{equation*}<br />\mathcal{F}\left[ u,\Omega \right] =\int\limits_{\Omega }f\left( x,u\left(x\right) ,\nabla u\left( x\right) \right),dx<br />\end{equation*}<br />where $f$: $\Omega \times\mathbb{R} \times\mathbb{R}^{N}\rightarrow\mathbb{R}$ is a Carath\'{e}odory function\ satisfying the inequalities<br />\begin{equation*}<br />\Phi \left( \left\vert z\right\vert \right) -c_{1}\leq f\left( x,s,z\right)\leq c_{2}\left[ 1+\left( \Phi ^{\ast }\left( \left\vert z\right\vert\right) \right) ^{\beta }+\left( \Phi ^{\ast }\left( \left\vert s\right\vert\right) \right) ^{\beta }\right]<br />\end{equation*}<br />for each $z\in\mathbb{R}^{N}$, $s\in\mathbb{R}$ and for $\mathcal{L}^{N}$-a. e. $x\in \Omega $, where $c_{1}$ and $c_{2}$ are two positive real constants, with $c_{1}<c_{2}$, $\Omega $ is an open subset of $\mathbb{R}^{N}$, $N\geq 2$, $\Phi \in \triangle _{2}^{m}\cap \nabla _{2}^{r}$ [Definition 6 and Definition 8], $1\leq r<m<N$ and the function $\Phi ^{\ast}$ is the Sobolev conjugate of $\Phi $ [Definition 12], $\beta $ is a positive real number that we will opportunely fix [Hypothesis $H_{1,f}$].Tiziano Granucci2014-10-092014-10-096Cordiality of a Star of the Complete Graph and a Cycle Graph $C(N\cdot K_{N})$
http://www.ccsenet.org/journal/index.php/jmr/article/view/39732
In this paper we prove that a star of $K_{n}$ and a cycle of $n$ copies of $K_{n}$ are cordial. We also get condition for maximum value of $e_{f}(1)-e_{f}(0)$ and highest negative value of $e_{f}(1)-e_{f}(0)$ in $K_{n}$, where $f$ is the binary vertex labeling function on the vertex set of $K_{n}$.<br /><br />V. J. KaneriaH. M. MakadiaMeera Meghpara2014-10-092014-10-096Fit States on Girard Algebras
http://www.ccsenet.org/journal/index.php/jmr/article/view/41159
Recently Weber proposed to define ``weakly additive" states on a Girard algebra by the additivity only on its sub-$MV$-algebras and characterized such states on the canonical Girard algebra extensions of any finite $MV$-chain. In the present paper, we take another viewpoint: the arguable sub-$MV$-algebras are replaced by suitable substructures coming from author, H\"{o}hle and Weber's own previous investigations. We propose a new notion of \emph{fit} states on a Girard algebra by the additivity on the mentioned substructures and consider such states on the ``non-effectible" Girard algebra ``$n$-extensions" (= canonical extensions when $n=1$) of $MV$-chains restricting ourselves to ones having less than six nontrivial elements. Our fit states appear as solutions of certain inconsistent systems of linear equations. They have extensive enough domains of the additivity-in any comparable case more extensive than Weber's states have.Remigijus Petras Gylys2014-10-092014-10-096Backward Ornstein-Uhlenbeck Transition Operators and Mild Solutions of Non-Autonomous Hamilton-Jacobi Equations in Banach Spaces
http://www.ccsenet.org/journal/index.php/jmr/article/view/41409
In this paper we revisit the mild-solution approach to second-order semi-linear PDEs of Hamilton-Jacobi type in infinite-dimensional spaces. We show that a well-known result on existence of mild solutions in Hilbert spaces can be easily extended to non-autonomous Hamilton-Jacobi equations in Banach spaces. The main tool is the regularizing property of Ornstein-Uhlenbeck transition evolution operators for stochastic Cauchy problems in Banach spaces with time-dependent coefficients.Rafael Serrano2014-10-202014-10-206One-Parameter Equations of Spherical Conics and Its Applications
http://www.ccsenet.org/journal/index.php/jmr/article/view/38845
If we transform definitions of the conics in Euclidean plane on sphere, we obtain spherical conics. To calculate the E. Study Map of the spherical conics, we have to find one parameter equations of them. We had done this before in (Altunkaya, Yayl{\i}, Hac{\i}saliho\u{g}lu, \& Arslan, 2011). In this paper, we not only developed the results that we have found before, but also calculated the E. Study Map of the spherical conics when they are great circles by using the theorems in (Hac{\i}saliho\u{g}lu, 1977).Bülent AltunkayaYusuf YayliH. Hilmi HacisalihogluFahrettin Arslan2014-10-202014-10-206