Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2014
Volume
Vol. 34
Number
No. 3
Description
Journal Volume
Opuscula Mathematica
Vol. 34 (2014)
Projects
Pages
Articles
A note on on-line Ramsey numbers for quadrilaterals
(2014) Cyman, Joanna; Dzido, Tomasz
We consider on-line Ramsey numbers defined by a game played between two players, Builder and Painter. In each round Builder draws an the edge and Painter colors it either red or blue, as it appears. Builder’s goal is to force Painter to create a monochromatic copy of a fixed graph $H$ in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number $\widetilde{r}(H)$ of the graph $H$. An asymmetric version of the on-line Ramsey numbers $\widetilde{r}(G,H)$ is defined accordingly. In 2005, Kurek and Ruciński computed $\widetilde{r}(C_3)$. In this paper, we compute $\widetilde{r}(C_4,C_k)$ for $3 \le k \le 7$. Most of the results are based on computer algorithms but we obtain the exact value $\widetilde{r}(C_4)$ and do so without the help of computer algorithms.
On the existence of positive periodic solutions for totally nonlinear neutral differential equations of the second-order with functional delay
(2014) Essel, Emmanuel K.; Yankson, Ernest
We prove that the totally nonlinear second-order neutral differential equation $\frac{d^2}{dt^2}x(t)+p(t)\frac{d}{dt}x(t)+q(t)h(x(t))$ $=\frac{d}{dt}c(t,x(t-\tau(t)))+f(t,\rho(x(t)),g(x(t-\tau(t))))$ has positive periodic solutions by employing the Krasnoselskii-Burton hybrid fixed point theorem.
On reflectionless equi-transmitting matrices
(2014) Kurasov, Pavel Borisovič; Ogik, Rao; Rauf, Amar
Reflectionless equi-transmitting unitary matrices are studied in connection to matching conditions in quantum graphs. All possible such matrices of size 6 are described explicitly. It is shown that such matrices form 30 six-parameter families intersected along 12 five-parameter families closely connected to conference matrices.
Existence and controllability results for damped second order impulsive functional differential systems with state-dependent delay
(2014) Arjunan, M. Mallika; Nadaf, N. Y.
In this paper, we investigate the existence and controllability of mild solutions for a damped second order impulsive functional differential equation with state-dependent delay in Banach spaces. The results are obtained by using Sadovskii’s fixed point theorem combined with the theories of a strongly continuous cosine family of bounded linear operators. Finally, an example is provided to illustrate the main results.
Conjugate functions, Lp-norm like functionals, the generalized Hölder inequality, Minkowski inequality and subhomogeneity
(2014) Matkowski, Janusz
For $h:(0,\infty )\rightarrow \mathbb{R}$, the function $h^{\ast }\left( t\right) :=th(\frac{1}{t})$ is called $(∗)$-conjugate to $h$. This conjugacy is related to the Hölder and Minkowski inequalities. Several properties of $(∗)$-conjugacy are proved. If $\varphi$ and $\varphi ^{\ast }$ are bijections of $\left(0,\infty \right)$ then $(\varphi ^{-1}) ^{\ast }=\left( \left[ \left( \varphi ^{\ast }\right) ^{-1}\right] ^{\ast }\right) ^{-1}$. Under some natural rate of growth conditions at $0$ and $infty$, if $\varphi$ is increasing, convex, geometrically convex, then $\left[ \left( \varphi^{-1}\right) ^{\ast }\right] ^{-1}$ has the same properties. We show that the Young conjugate functions do not have this property. For a measure space $(\Omega ,\Sigma ,\mu )$ denote by $S=S(\Omega ,\Sigma ,\mu )$ the space of all $\mu$-integrable simple functions $x:\Omega \rightarrow \mathbb{R}$. Given a bijection $\varphi :(0,\infty )\rightarrow (0,\infty )$, define $\mathbf{P}_{\varphi }:S\rightarrow \lbrack 0,\infty )$ by $\mathbf{P}_{\varphi }(x):=\varphi ^{-1}\bigg( \int\limits_{\Omega (x)}\varphi \circ \left\vert x\right\vert d\mu \bigg),$ where $\Omega(x)$ is the support of $x$. Applying some properties of the $(∗)$ operation, we prove that if $\int\limits_{\Omega }xy\leq \mathbf{P}_{\varphi }(x)\mathbf{P}_{\psi }(y)$ where $\varphi ^{-1}$ and $\psi ^{-1}$ are conjugate, then $\varphi$ and $\psi$ are conjugate power functions. The existence of nonpower bijections $\varphi$ and $\psi$ with conjugate inverse functions $\psi =\left[ ( \varphi ^{-1}) ^{\ast}\right] ^{-1}$ such that $\mathbf{P}_{\varphi }$ and $\mathbf{P}_{\psi }$ are subadditive and subhomogeneous is considered.