eng
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
Khayyam Journal of Mathematics
2423-4788
2423-4788
2015-08-01
1
2
125
150
10.22034/kjm.2015.13161
13161
Minimal Usco and Minimal Cusco Maps
Lubica Hola
1
Dusan Holy
2
Academy of Sciences, Institute of Mathematics Stefˇ anikova 49, 81473 Bratislava,´ Slovakia
Department of Mathematics and Computer Science, Faculty of Education, Trnava University, Priemyselna 4, 918 43 Trnava, Slovakia´
The main aim of this paper is to present a survey of known results concerning minimal usco and minimal cusco maps. We give characterizations of minimal usco and minimal cusco maps in the class of all set-valued maps using quasicontinuous selections. If X is a topological space and Y is a Banach space, there is a bijection between the space of minimal usco maps from X to Y and the space of minimal cusco maps from X to Y. We study this bijection with respect to various topologies on underlying spaces. Some new results are also given.
http://www.kjm-math.org/article_13161_9345c4bd651f19c82ac0fcbc44d079f6.pdf
Quasicontinuous function
minimal usco map
minimal cusco map
subcontinuous function
Selection
eng
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
Khayyam Journal of Mathematics
2423-4788
2423-4788
2015-08-01
1
2
151
163
10.22034/kjm.2015.13162
13162
Cayley Graphs under Graph Operations II
Nasrin Malekmohammadi
1
Ali Reza Ashrafi
2
Department of Pure Mathematics, University of Kashan, Kashan, P.O. Box 87317-51167, Iran
Department of Pure Mathematics, University of Kashan, Kashan, P.O. Box 87317-51167, Iran
The aim of this paper is to investigate the behavior of Cayley graphs under some graph operations. It is proved that the NEPS, corona, hierarchical, strong, skew and converse skew products of Cayley graphs are again Cayley graphs under some conditions.
http://www.kjm-math.org/article_13162_30ac021e4108183dc343b8ef0daeb6bb.pdf
Cayley graph
corona
Hierarchical product
skew product
converse skew product
NEPS
strong product
eng
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
Khayyam Journal of Mathematics
2423-4788
2423-4788
2015-08-01
1
2
164
173
10.22034/kjm.2015.13163
13163
Statistical Ergodic Theorems for Markov Semigroups in Spaces with Mixed Norm
Inomjon Ganiev
1
Sanobar Sadaddinova
2
Umarjon Ganiev
3
Department of Science in Engineering, Faculty of Engineering, International Islamic University Malaysia, P.O. Box 10, 50728 Kuala-Lumpur, Malaysia
Department of Mathematics, Tashkent University of Information Technologies , Tashkent, Uzbekistan
Department of Physics Fergana Medical College, Fergana, Uzbekistan
This paper describes the semigroups generated by the Markov processes in spaces with mixed norm and proves analogues of statistical ergodic theorems for such semigroups.
http://www.kjm-math.org/article_13163_f79e1ed5cb57130e6a0ef97d8461f321.pdf
Statistical ergodic theorem
Markov semigroup
mixed norm
eng
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
Khayyam Journal of Mathematics
2423-4788
2423-4788
2015-08-01
1
2
174
184
10.22034/kjm.2015.13164
13164
Exponential Stability and Instability in Multiple Delays Difference Equations
S. Almutairy
1
M. Alshammari
2
Y. Raffoul
3
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316 USA;
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316 USA;
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316 USA;
We use Lyapunov functionals and obtain sufficient conditions that guarantee exponential stability of the zero solution of the difference equation with multiple delays begin{equation*} x(t+1) = a(t)x(t)+sum^{k}_{j=1}b_j(t)x(t-h_j). end{equation*} The novelty of our work is the relaxation of the condition $|a(t)| <1$, in spite of the presence of multiple delays. Using a slightly modified Lyapunov functional, we obtain necessary conditions for the unboundedness of all solutions and for the instability of the zero solution. We provide an example as an application to our obtained results.
http://www.kjm-math.org/article_13164_48e6c2523d83b480c06970173928b701.pdf
exponential stability
Instability
Lyapunov functional
eng
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
Khayyam Journal of Mathematics
2423-4788
2423-4788
2015-08-01
1
2
185
218
10.22034/kjm.2015.13165
13165
Lipschitz Tensor Product
M.G. Cabrera-Padilia
1
J.A. Chavez-Dominguez
2
A. Jimenez-Vargas
3
M. Viliegas-Vallecillos
4
Departamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.
Department of Mathematics, University of Oklahoma, Norman, Oklahoma, 7 3019, United States.
Departamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.
Departamento de Matematicas´ , Universidad de Cadiz´ , 11510 Puerto Real, Spain.
Inspired by ideas of R. Schatten in his celebrated monograph [23] on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product $Xboxtimes E$ of a pointed metric space $X$ and a Banach space $E$ as a certain linear subspace of the algebraic dual of $text{Lip}0(X,E^*)$. We prove that $leftlangle text{Lip}0(X,E^*),Xboxtimes Erightrangle$ forms a dual pair.We prove that $Xboxtimes E$ is linearly isomorphic to the linear space of all finite-rank continuous linear operators from $(X^#,tau_p)$ into $E$, where $X^#$ denotes the space $text{Lip}0(X,mathbb{K})$ and $tau_p$ is the topology of pointwise convergence of $X^#$. The concept of Lipschitz tensor product of elements of $X^#$ and $E^*$ yields the space $X^#⧆ E^*$ as a certain linear subspace of the algebraic dual of $Xboxtimes E$. To ensure the good behavior of a norm on $Xboxtimes E$ with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz cross-norm on $Xboxtimes E$ is defined. We show that the Lipschitz injective norm $varepsilon$, the Lipschitz projective norm $pi$ and the Lipschitz $p$-nuclear norm $d_p$ $(1leq pleqinfty)$ are uniform dualizable Lipschitz cross-norms on $Xboxtimes E$. In fact, $varepsilon$ is the least dualizable Lipschitz cross-norm and $pi$ is the greatest Lipschitz cross-norm on $Xboxtimes E$. Moreover, dualizable Lipschitz cross-norms $alpha$ on $Xboxtimes E$ are characterized by satisfying the relation $varepsilonleqalphaleqpi$.<br />In addition, the Lipschitz injective (projective) norm on $Xboxtimes E$ can be identified with the injective (respectively, projective) tensor norm on the Banach-space tensor product of the Lipschitz-free space over $X$ and $E$, but this identification does not hold for the Lipschitz $2$-nuclear norm and the corresponding Banach-space tensor norm. In terms of the space $X^#⧆ E^*$, we describe the spaces of Lipschitz compact (finite-rank, approximable) operators from $X$ to $E^*$.
http://www.kjm-math.org/article_13165_7d5b50b35f614ef5dfef49db33a649f2.pdf
Lipschitz map
tensor product
$p$-summing operator
duality
Lipschitz compact operator
eng
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
Khayyam Journal of Mathematics
2423-4788
2423-4788
2015-08-01
1
2
219
229
10.22034/kjm.2015.13166
13166
$n$-Dual Spaces Associated to a Normed Space
Yosafat E.P. Pangalela
1
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand
For a real normed space $X$, we study the $n$-dual space of $left( X,leftVert cdot rightVert right) $ and show that the space is a Banach space. Meanwhile, for a real normed space $X$ of dimension $dgeq n$ which satisfies property ($G$), we discuss the $n$-dual space of $left( X,leftVert cdot,ldots ,cdot rightVert _{G}right) $, where $% leftVert cdot ,ldots ,cdot rightVert _{G}$ is the Gähler $n$-norm. We then investigate the relationship between the $n$-dual space of $% left( X,leftVert cdot rightVert right) $ and the $n$-dual space of $% left( X,leftVert cdot,ldots ,cdot rightVert _{G}right) $. We use this relationship to determine the $n$-dual space of $left( X,leftVert cdot ,ldots ,cdot rightVert _{G}right) ~$and show that the space is also a Banach space.
http://www.kjm-math.org/article_13166_e95026b5f6197b67bbfb945b4b49545b.pdf
$n$-dual spaces
$n$-normed spaces
bounded linear functionals
eng
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
Khayyam Journal of Mathematics
2423-4788
2423-4788
2015-08-01
1
2
230
242
10.22034/kjm.2015.13167
13167
Toeplitz and Hankel Operators on a Vector-valued Bergman Space
Namita Das
1
Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar, 751004,, Odisha, India
In this paper, we derive certain algebraic properties of Toeplitz and Hankel operators defined on the vector-valued Bergman spaces $L_a^{2, mathbb{C}^n}(mathbb{D})$, where $mathbb{D}$ is the open unit disk in $mathbb{C}$ and $ngeq 1.$ We show that the set of all Toeplitz operators $T_{Phi}, Phiin L_{M_n}^{infty}(mathbb{D})$ is strongly dense in the set of all bounded linear operators ${mathcal L}(L_a^{2, mathbb{C}^n}(mathbb{D}))$ and characterize all finite rank little Hankel operators.
http://www.kjm-math.org/article_13167_cd3e86baa83715a9bf967ed60c149d34.pdf
Bergman space
Toeplitz operators
little Hankel operators
strong-operator topology
finite rank operators
eng
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
Khayyam Journal of Mathematics
2423-4788
2423-4788
2015-08-01
1
2
243
252
10.22034/kjm.2015.13168
13168
On the Degree of Approximation of Functions Belonging to the Lipschitz Class by (E, q)(C, α, β) Means
Xhevat Z. Krasniqi
1
University of Prishtina “Hasan Prishtina”, Faculty of Education, Department of Mathematics and Informatics, Avenue “Mother Theresa” 5, Prishtine, Kosovo.
In this paper two generalized theorems on the degree of approximation of conjugate functions belonging to the Lipschitz classes of the type $text{Lip}alpha $, $0<alpha leq 1$, and $W(L_{p},xi (t))$ are proved. The first one gives the degree of approximation with respect to the $L_{infty}$-norm, and the second one with respect to $L_{p}$-norm, $pgeq 1$. In addition, a correct condition in proving of the second mentioned theorem is employed.
http://www.kjm-math.org/article_13168_e62c9334109527d4f323a4bd46193542.pdf
Lipschitz classes
Fourier series
summability
degree of approximation