scholarly journals Some results on porous set relating to ratio sets

2017 ◽  
pp. 171-177
Author(s):  
D. K. Ganguly ◽  
Dhananjoy Halder
Keyword(s):  
Porous Set

scholarly journals A generalizedσ-porous set with a small complement

2005 ◽  
Vol 2005 (5) ◽  
pp. 535-541
Author(s):  
Jaroslav Tišer
Keyword(s):  
Banach Space ◽  
Measure Zero ◽  
Porous Set

We show that in every Banach space, there is ag-porous set, the complement of which is ofℋ1-measure zero on everyC1curve.

The structure of l∞ in some generalized Orlicz sequence spaces

2004 ◽  
Vol 41 (4) ◽  
pp. 457-465
Author(s):  
J. Ewert ◽  
Z. Lewandowska
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Porous Set ◽  
Orlicz Sequence Spaces

In this paper we consider the structure of l∞ in the modular sequence space T(A, {fn}) defined in [2]. We obtain the conditions when l∞ = T(A, {fn}). We prove that if l∞ ≠ T(A, {fn}), then the space l∞ is an Fσ, σ-strorigly porous set in T(A, {fn}).

scholarly journals A porosity result in convex minimization

2005 ◽  
Vol 2005 (3) ◽  
pp. 319-326
Author(s):  
P. G. Howlett ◽  
A. J. Zaslavski
Keyword(s):  
Banach Space ◽  
Minimization Problem ◽  
Metric Space ◽  
Convex Functions ◽  
Convex Sets ◽  
Reflexive Banach Space ◽  
Complete Metric Space ◽  
Baire Category ◽  
Countable Intersection ◽  
Porous Set

We study the minimization problemf(x)→min,x∈C, wherefbelongs to a complete metric spaceℳof convex functions and the setCis a countable intersection of a decreasing sequence of closed convex setsCiin a reflexive Banach space. Letℱbe the set of allf∈ℳfor which the solutions of the minimization problem over the setCiconverge strongly asi→∞to the solution over the setC. In our recent work we show that the setℱcontains an everywhere denseGδsubset ofℳ. In this paper, we show that the complementℳ\ℱis not only of the first Baire category but also aσ-porous set.

scholarly journals Trajectory of the turning point is dense for a co-σ-porous set of tent maps

2000 ◽  
Vol 165 (2) ◽  
pp. 95-123
Author(s):  
Karen Brucks ◽  
Zoltán Buczolich
Keyword(s):  
Turning Point ◽  
Porous Set ◽  
Tent Maps

Fr échet Differentiability of Real-Valued Functions

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Variational Principles ◽  
Mean Value ◽  
Lipschitz Functions ◽  
Monotone Functions ◽  
Porous Set ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
The Mean ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter shows that cone-monotone functions on Asplund spaces have points of Fréchet differentiability and that the appropriate version of the mean value estimates holds. It also proves that the corresponding point of Fréchet differentiability may be found outside any given σ‎-porous set. This new result considerably strengthens known Fréchet differentiability results for real-valued Lipschitz functions on such spaces. The avoidance of σ‎-porous sets is new even in the Lipschitz case. The chapter first discusses the use of variational principles to prove Fréchet differentiability before analyzing a one-dimensional mean value problem in relation to Lipschitz functions. It shows that results on existence of points of Fréchet differentiability may be generalized to derivatives other than the Fréchet derivative.

Porosity and ε‎-Fr échet differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Porous Set ◽  
Delta Set ◽  
Lipschitz Maps ◽  
Finite Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Null Set

This chapter demonstrates that the results about smallness of porous sets, and so also of sets of irregularity points of a given Lipschitz function, can be used to show existence of points of (at least) ε‎-Fréchet differentiability of vector-valued functions. The approach involves combining this new idea with the basic notion that points of ε‎-Fréchet differentiability should appear in small slices of the set of Gâteaux derivatives. The chapter obtains very precise results on existence of points of ε‎-Fréchet differentiability for Lipschitz maps with finite dimensional range. The main result applies when every porous set is contained in the unions of a σ‎-directionally porous (and hence Haar null) set and a Γ‎ₙ-null Gsubscript Small Delta set.

Convergence field of Abel’s summation method

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Peter Letavaj
Keyword(s):  
Metric Space ◽  
Linear Subspace ◽  
Summation Method ◽  
Matrix Transformation ◽  
Matrix Transformations ◽  
Regular Matrix ◽  
Porous Set ◽  
Linear Metric Space ◽  
Bounded Sequences

AbstractLet F(A) denote the set of all bounded sequences summable by Abel’s method. It is known, that F(A) is a linear subspace of the linear metric space (S, ρ) of all bounded sequences endowed with the sup metric. It is shown in [KOSTYRKO, P.: Convergence fields of regular matrix transformations 2, Tatra Mt. Math. Publ. 40 (2008), 143–147] that the convergence field of a regular matrix transformation is a σ-porous set. We show that F(A) is very porous in S.

scholarly journals Existence of solutions of minimization problems with an increasing cost function and porosity

2003 ◽  
Vol 2003 (11) ◽  
pp. 651-670 ◽  
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Banach Space ◽  
Cost Function ◽  
Minimization Problem ◽  
Closed Subset ◽  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Ordered Banach Space ◽  
Porous Set ◽  
Minimization Problems ◽  
Semicontinuous Functions

We consider the minimization problemf(x)→min,x∈K, whereKis a closed subset of an ordered Banach spaceXandfbelongs to a space of increasing lower semicontinuous functions onK. In our previous work, we showed that the complement of the set of all functionsf, for which the corresponding minimization problem has a solution, is of the first category. In the present paper we show that this complement is also aσ-porous set.

Some properties of porouscontinuous functions

2014 ◽  
Vol 64 (3) ◽  
Author(s):  
Ján Borsík ◽  
Juraj Holos
Keyword(s):  
Porous Set ◽  
Quasicontinuous Functions

AbstractThe notion of porouscontinuous function will be introduced on the base of porous set and relations between porouscontinuous, continuous and quasicontinuous functions will be investigated.

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