Vali-e-Asr university of Rafsanjan
Wavelet and Linear Algebra
2383-1936
2
1
2015
09
01
On the characterization of subrepresentations of shearlet group
1
9
14265
EN
V.
Atayi
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
R. A.
Kamyabi-Gol
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Journal Article
2015
09
06
We regard the shearlet group as a semidirect product group and show that its standard representation is,typically, a quasiregu- lar representation. As a result we can characterize irreducible as well as square-integrable subrepresentations of the shearlet group.
https://wala.vru.ac.ir/article_14265_9bf627ecedb9c35cc07168835acddb42.pdf
Vali-e-Asr university of Rafsanjan
Wavelet and Linear Algebra
2383-1936
2
1
2015
09
01
Cyclic wavelet systems in prime dimensional linear vector spaces
11
24
14266
EN
A.
Ghaani Farashahi
Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics,
University of Vienna
Journal Article
2015
09
07
Finite affine groups are given by groups of translations and di- lations on finite cyclic groups. For cyclic groups of prime order we develop a time-scale (wavelet) analysis and show that for a large class of non-zero window signals/vectors, the generated full cyclic wavelet system constitutes a frame whose canonical dual is a cyclic wavelet frame.
https://wala.vru.ac.ir/article_14266_785b6fb99f7fe9032e4a04c9dc31a079.pdf
Vali-e-Asr university of Rafsanjan
Wavelet and Linear Algebra
2383-1936
2
1
2015
09
01
On the distance from a matrix polynomial to matrix polynomials with two prescribed eigenvalues
25
38
14267
EN
E.
Kokabifar
Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.
G.B.
Loghmani
Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.
A. M.
Nazari
Department of Mathematics, Faculty of Science, Arak University, Arak,
Islamic Republic of Iran.
S. M.
Karbassi
Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd,
Islamic Republic of Iran.
Journal Article
2015
09
07
<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">Consider an <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>n </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">× <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>n </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">matrix polynomial <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>P</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">λ<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">). A spectral norm distance from <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>P</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">λ<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">) to the set of <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>n </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">× <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>n </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">matrix polynomials that have <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">a given scalar <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">µ <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∈ <span style="font-family: MSBM10; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">C <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">as a multiple eigenvalue was introduced <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and obtained by Papathanasiou and Psarrakos. They computed <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">lower and upper bounds for this distance, constructing an associated perturbation of <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>P</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">λ<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">). In this paper, we extend this result <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">to the case of two given distinct complex numbers <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">µ<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">µ<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">2<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">. <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">First, we compute a lower bound for the spectral norm distance <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">from <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>P</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">λ<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">) to the set of matrix polynomials that have <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">µ<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, µ<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">2 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">as <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">two eigenvalues. Then we construct an associated perturbation <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">of <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>P</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">λ<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">) such that the perturbed matrix polynomial has two given <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">scalars <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">µ<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">µ<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">2 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">in its spectrum. Finally, we derive an upper <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">bound for the distance by the constructed perturbation of <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>P</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">λ<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">). <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">Numerical examples are provided to illustrate the validity of the <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">method.</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>
https://wala.vru.ac.ir/article_14267_031c86cfbf3947aad1230b028a5506b5.pdf
Vali-e-Asr university of Rafsanjan
Wavelet and Linear Algebra
2383-1936
2
1
2015
09
01
G-dual function-valued frames in L2(0,∞)
39
47
14268
EN
M. A.
Hasankhanifard
Vali-e-Asr university of Rafsanjan
M. A.
Dehghan
Vali-e-Asr university of Rafsanjan
Journal Article
2015
09
07
In this paper, g-dual function-valued frames in L2(0;1) are in- troduced. We can achieve more reconstruction formulas to ob- tain signals in L2(0;1) by applying g-dual function-valued frames in L2(0;1).
https://wala.vru.ac.ir/article_14268_9b2b81f3b7b0c0ef1d67b31101cdd174.pdf
Vali-e-Asr university of Rafsanjan
Wavelet and Linear Algebra
2383-1936
2
1
2015
09
01
Schur multiplier norm of product of matrices
49
54
14269
EN
M.
Khosravi
Shahid Bahonar university of Kerman
A.
Sheikhhosseini
Shahid Bahonar university of Kerman
Journal Article
2015
09
07
<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">For <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>A </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∈ <span style="font-family: MSBM10; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">M <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>n</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, the Schur multiplier of <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>A </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">is defined as <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>A</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">) <span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">= <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>A </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">◦ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">for all <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∈ <span style="font-family: MSBM10; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">M <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>n </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and the spectral norm of <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>A </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">can be state <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">as <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>A</em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∥ <span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">= <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">sup<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X</em><span style="font-family: txsyc; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">,<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0 <span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>A </em><span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">◦<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∥ ∥<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">. The other norm on <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>A </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">can be defined <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">as <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>A</em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: rtxmi; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">ω <span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">= <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">sup<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X</em><span style="font-family: txsyc; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">,<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0 <span style="font-family: rtxmi; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">ω<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: rtxmi; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">ω <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>S</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">( <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 5pt; color: #000000; font-style: normal; font-variant: normal;"><em>A</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">) )) <span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">= <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">sup<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X</em><span style="font-family: txsyc; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">,<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0 <span style="font-family: rtxmi; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">ωω <span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>A </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">◦<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">) )<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, where <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ω<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>A</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">) stands <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">for the numerical radius of <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>A</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">. In this paper, we focus on the <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">relation between the norm of Schur multiplier of product of matrices and the product of norm of those matrices. This relation is <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">proved for Schur product and geometric product and some applications are given. Also we show that there is no such relation <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">for operator product of matrices. Furthermore, for positive definite matrices <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>A </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>B </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">with <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>A</em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: rtxmi; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">ω <span style="font-family: MSAM10; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">⩽ <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">1 and <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>B</em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∥<span style="font-family: rtxmi; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">ω <span style="font-family: MSAM10; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">⩽ <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">1, we show <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">that <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>A</em><span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">♯<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>B </em><span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">= <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>n</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>I </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">− <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>Z</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">)<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1<span style="font-family: rtxmi; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">/<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">2<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>C</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>I </em><span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">+ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>Z</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">)<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1<span style="font-family: rtxmi; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">/<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">2<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">for some contraction <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>C </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">Hermitian contraction <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>Z</em><span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">.</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>
https://wala.vru.ac.ir/article_14269_e9a1dc6d7c67b98d454cd7225318629e.pdf
Vali-e-Asr university of Rafsanjan
Wavelet and Linear Algebra
2383-1936
2
1
2015
09
01
Ultra Bessel sequences in direct sums of Hilbert spaces
55
64
14270
EN
M. R.
Abdollahpour
University
of Mohaghegh Ardabili
A.
Rahimi
University of Maragheh
Journal Article
2015
09
07
<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">In this paper, we establish some new results in ultra Bessel sequences and ultra Bessel sequences of subspaces. Also, we investigate ultra Bessel sequences in direct sums of Hilbert spaces. <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">Specially, we show that <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">{<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">( <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>g</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">)<span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">}<span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∞ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: rtxr; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">=<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">is a an ultra Bessel sequence <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">for Hilbert space <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">H ⊕ K <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">if and only if <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">{ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">}<span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∞ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: rtxr; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">=<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">{<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>g</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">}<span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∞ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: rtxr; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">=<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">are ultra <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">Bessel sequences for Hilbert spaces <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">H <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">K<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">respectively.</span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>
https://wala.vru.ac.ir/article_14270_489abf58eb663e969ea22c4d90360acb.pdf
Vali-e-Asr university of Rafsanjan
Wavelet and Linear Algebra
2383-1936
2
1
2015
09
01
Some relations between ε-directional derivative and ε-generalized weak subdifferential
65
80
14591
EN
A.
Mohebi
Shahid Bahonar university of Kerman
H.
Mohebi
Shahid Bahonar university of Kerman
Journal Article
2015
10
11
<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">In this paper, we study <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ε<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">-generalized weak subdi<span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">erential for <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">vector valued functions defined on a real ordered topological <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">vector space <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X</em><span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">. <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">We give various characterizations of <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ε<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">-generalized <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">weak subdi<span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">erential for this class of functions. It is well known <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">that if the function <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">: <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">→ <span style="font-family: MSBM10; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">R <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">is subdi<span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">erentiable at <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0 <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∈ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X</em><span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">then <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">has a global minimizer at <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">if and only if 0 <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∈ <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∂ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">)<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">. <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">We show that a similar result can be obtained for <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ε<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">-generalized <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">weak subdi<span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">erential. Finally, we investigate some relations between <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ε<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">-directional derivative and <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ε<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">-generalized weak subdifferential. In fact, in the classical subdi<span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">erential theory, it is well <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">known that if the function <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">: <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">→ <span style="font-family: MSBM10; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">R <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">is subdi<span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">erentiable at <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0 <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∈ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X </em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and it has directional derivative at <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">in the direction <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>u </em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∈ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>X</em><span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">then the relation <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f </em><span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">′<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>u</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">) <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">≥ ⟨<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>u</em><span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∗<span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">⟩<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∀ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∗ <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∈ <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">∂ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">(<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>x</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">0<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">) is <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">satisfied. We prove that a similar result can be obtained for <span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ε<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">- <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">generalized weak subdi<span style="font-family: rtxr; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">ff<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">erential.</span></span></span></span></span></span></span></span></span></span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>
https://wala.vru.ac.ir/article_14591_7255b9cf0db6154ec39af397e9141d48.pdf