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ON CERTAIN PROPERTY OF THE NORMS BY MODULARS BY
"... Let $R $ be a universally continuous semiordered linear space. A functional $m(a)(a\in R) $ is said to be a modularl) on $R $ if it satisfies the following modular conditions: (1) $ 0\leqq m(a)\leqq\infty $ for all $a\in R $; (2) if $m(\xi a)=0 $ for all $\xi>0 $ , then $a=0 $; (3) for any $a\i ..."
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Cited by 2 (0 self)
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Let $R $ be a universally continuous semiordered linear space. A functional $m(a)(a\in R) $ is said to be a modularl) on $R $ if it satisfies the following modular conditions: (1) $ 0\leqq m(a)\leqq\infty $ for all $a\in R $; (2) if $m(\xi a)=0 $ for all $\xi>0 $ , then $a=0 $; (3) for any $a\in R $ there exists $a>0 $ such that $ m(aa)<\infty $; (4) for every $a\in R, $ $m(\xi a) $ is a convex function of $\xi $; (5) $a\leqqb $ implies $m(a)\leqq m(b) $; (6) $a\wedge b=\backslash 0 $ implies $m(a+b)=m(a)+m(b) $; (7) $0\leqq a_{l}\uparrow aRC.4 $ implies $m(a)=\sup_{R\in\Lambda}m(a_{\lambda}) $. In $R $ , we define functionals $a, $ $\Verta\Vert(a\in R) $ as follows $a=\inf_{\xi>0}\frac{1+m(\tilde{\sigma}a)}{\xi} $. $\Verta_{1}^{1}=\inf_{m(\text{\’{e}} a)\leq 1}\frac{1}{\xi} $. Then it is easily seen that both $a $ and $\Verta\Vert $ are norms on $R $ and $_{1}a\Vert\leqqa\leqq 2\Verta\Vert $ for all $a\in R $. $a $ is said to be the first norm by $m$ and $\Verta\Vert $ is said to be the second norm by $m $. Let $\overline{R}^{m} $ be the modular
ON CERTAIN PROPERTIES OF MODULAR CONVERGENCE By
"... Let $R $ be a universally continuous semiordered linear space. A functional $m(a)(a\in R) $ is said to be a modular on $R $ if it satisfies the following modular conditions: (1) $ 0\leqq m(a)\leqq+\infty $ for all $a\in R $; (2) if $m(\hat{\sigma}a)=0 $ for all $\xi\geqq 0 $ , then $a=0 $; (3) for ..."
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Let $R $ be a universally continuous semiordered linear space. A functional $m(a)(a\in R) $ is said to be a modular on $R $ if it satisfies the following modular conditions: (1) $ 0\leqq m(a)\leqq+\infty $ for all $a\in R $; (2) if $m(\hat{\sigma}a)=0 $ for all $\xi\geqq 0 $ , then $a=0 $; (3) for any $a\in R $ there exists $a>0 $ such that $ m(aa)<+\infty $; (4) for every $a\in R, $ $m(\xi a) $ is a convex function of $\xi $; (5) $a\leqqb $ implies $m(a)\leqq m(b) $; (6) $a\leftrightarrow b=0 $ implies $m(a+b)=m(a)+m(b) $; (7) $0\leqq a_{l}\uparrow_{l\in\Lambda}a_{\backslash} $ implies $m(a)=\sup_{\lambda\in\Lambda}m(a_{\lambda}) $. Throughout the paper we use the notations and terminologies used in [2]. Here $w\lim_{\nu\rightarrow\infty}a_{\nu}=a $ or $w\lim_{\nu\rightarrow\infty}a_{\nu}=a $ for $a, $ $a_{\nu}\in R(\nu=1,2,3, \cdots)$ means $\lim_{\nu\rightarrow\infty}\overline{\sigma}(a_{\nu}a)=0 $ or $\lim_{\nu\rightarrow\infty} $ a $(a_{\nu}a)=0 $ respectively for any $\overline{a}\in\overline{R}^{m1)} $. If $\Phi(u) $ is a real convex function, defined for $u\geqq 0 $ , such that $\Phi(0)=0$ and $\Phi(u)\geqq 0 $ for $u>0 $ , but $\Phi(u) $ not identically zero or infinity for $u>0 $,
On certain properties of cosmological models
 Zh. Eksper. Teoret. Fiz
, 1972
"... Abstract. It is shown that in homogeneous cosmological models the Einstein equations can be reduced, on the basis of scale invariance, to systems with friction. The formalism involving friction permits one to investigate the problem of isotropization of the solutions in the Bianchi model IX at late ..."
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Cited by 1 (1 self)
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at late development stages. The possibility of a statistical description of the properties of the model is discussed. A number of investigations of the last decade have been devoted to the question (first explicitly formulated by Landau) of the singularities of the solutions of Einstein’s equations
On Certain Properties of Intersection Grammars
"... Dedicated to the Professor Noam Chomsky who combated for persecuted czech mathematics against bolshevik injury in ČSSR for a long time on radio stations Free Europe, BBC London and Voice of America. Abstract: This paper links up to the papers [1], [2] and [3]. The author deals with the intersections ..."
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} m s=0 be a finite sequence of strings from the set V ∗ such that s = s0, si−1 ⇒ si (R) for every i with the property 1 ≤ i ≤ m and sm = t. Then we say that the sequence {si} m s=0 is the s − derivative of the string t in the set R of the length m. 1.5 Remark For m = 0 the sequence {si} m s=0 from 1
Functions and their basic properties
 JOURNAL OF FORMALIZED MATHEMATICS
, 2003
"... The definitions of the mode Function and the graph of a function are introduced. The graph of a function is defined to be identical with the function. The following concepts are also defined: the domain of a function, the range of a function, the identity function, the composition of functions, the ..."
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Cited by 1317 (31 self)
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, the 11 function, the inverse function, the restriction of a function, the image and the inverse image. Certain basic facts about functions and the notions defined in the article are proved.
Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes
 J. COMP. PHYS
, 1981
"... Several numerical schemes for the solution of hyperbolic conservation laws are based on exploiting the information obtained by considering a sequence of Riemann problems. It is argued that in existing schemes much of this information is degraded, and that only certain features of the exact solution ..."
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Cited by 950 (2 self)
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are worth striving for. It is shown that these features can be obtained by constructing a matrix with a certain “Property U.” Matrices having this property are exhibited for the equations of steady and unsteady gasdynamics. In order to construct them, it is found helpful to introduce “parameter vectors
The adaptive LASSO and its oracle properties
 Journal of the American Statistical Association
"... The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain sc ..."
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Cited by 649 (10 self)
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The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain
Certain Properties of Generalized Fibonacci Sequence
, 2014
"... In this study, we present certain properties of Generalized Fibonacci sequence. Generalized Fibonacci sequence is defined by recurrence relation 1 2, 2k k kF pF qF k − − = + ≥ with 0 1,F a F b = =. This was introduced by Gupta, Panwar and Sikhwal. We shall use the Induction method and Binet’s for ..."
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In this study, we present certain properties of Generalized Fibonacci sequence. Generalized Fibonacci sequence is defined by recurrence relation 1 2, 2k k kF pF qF k − − = + ≥ with 0 1,F a F b = =. This was introduced by Gupta, Panwar and Sikhwal. We shall use the Induction method and Binet’s
A Simple Proof of the Restricted Isometry Property for Random Matrices
 CONSTR APPROX
, 2008
"... We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmical ..."
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Cited by 620 (63 self)
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We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided
Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties
, 2001
"... Variable selection is fundamental to highdimensional statistical modeling, including nonparametric regression. Many approaches in use are stepwise selection procedures, which can be computationally expensive and ignore stochastic errors in the variable selection process. In this article, penalized ..."
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Cited by 902 (60 self)
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functions are symmetric, nonconcave on (0, ∞), and have singularities at the origin to produce sparse solutions. Furthermore, the penalty functions should be bounded by a constant to reduce bias and satisfy certain conditions to yield continuous solutions. A new algorithm is proposed for optimizing
Results 1  10
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1,819,792