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Wyświetlanie 1-10 z 60
Tytuł:
Twice Differentiable Characterizations of Convexity Notions for Functions on Full Dimensional Convex Sets
Autorzy:
Stein, Oliver
Powiązania:
https://bibliotekanauki.pl/articles/1373546.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Jagielloński. Wydawnictwo Uniwersytetu Jagiellońskiego
Tematy:
convexity
strict convexity
strong convexity
true convexity
differentiable characterization
full dimensional convex set
Opis:
We derive $C^2$−characterizations for convex, strictly convex, as well as strongly convex functions on full dimensional convex sets. In the cases of convex and strongly convex functions this weakens the well-known openness assumption on the convex sets. We also show that, in a certain sense, the full dimensionality assumption cannot be weakened further. In the case of strictly convex functions we weaken the well-known sufficient $C^2$−condition for strict convexity to a characterization. Several examples illustrate the results.
Źródło:
Schedae Informaticae; 2012, 21; 55-63
0860-0295
2083-8476
Pojawia się w:
Schedae Informaticae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On approximately Breckner s-convex functions
Autorzy:
Burai, P.
Hazy, A.
Juhasz, T.
Powiązania:
https://bibliotekanauki.pl/articles/205651.pdf
Data publikacji:
2011
Wydawca:
Polska Akademia Nauk. Instytut Badań Systemowych PAN
Tematy:
convexity
approximate convexity
s-convexity
regularity properties of generalized convex functions
Opis:
The main goal of this paper is to consider the regularity and convexity properties of a given type of approximately generalized convex functions, namely approximately Breckner s-convex functions (see the origin of the definition in Breckner, 1978). Our main result is a Bernstein-Doetsch type one. It is proved that the local boundedness of such a type of function from above at a point of its domain implies approximate convexity and stronger regularity properties of the function in question on the whole domain.
Źródło:
Control and Cybernetics; 2011, 40, 1; 91-99
0324-8569
Pojawia się w:
Control and Cybernetics
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Decomposability of Abstract and Path-Induced Convexities in Hypergraphs
Autorzy:
Malvestuto, Francesco Mario
Moscarini, Marina
Powiązania:
https://bibliotekanauki.pl/articles/31339339.pdf
Data publikacji:
2015-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
convex hull
hypergraph convexity
path-induced convexity
con- vex geometry
Opis:
An abstract convexity space on a connected hypergraph H with vertex set V (H) is a family C of subsets of V (H) (to be called the convex sets of H) such that: (i) C contains the empty set and V (H), (ii) C is closed under intersection, and (iii) every set in C is connected in H. A convex set X of H is a minimal vertex convex separator of H if there exist two vertices of H that are separated by X and are not separated by any convex set that is a proper subset of X. A nonempty subset X of V (H) is a cluster of H if in H every two vertices in X are not separated by any convex set. The cluster hypergraph of H is the hypergraph with vertex set V (H) whose edges are the maximal clusters of H. A convexity space on H is called decomposable if it satisfies the following three properties: (C1) the cluster hypergraph of H is acyclic, (C2) every edge of the cluster hypergraph of H is convex, (C3) for every nonempty proper subset X of V (H), a vertex v does not belong to the convex hull of X if and only if v is separated from X in H by a convex cluster. It is known that the monophonic convexity (i.e., the convexity induced by the set of chordless paths) on a connected hypergraph is decomposable. In this paper we first provide two characterizations of decomposable convexities and then, after introducing the notion of a hereditary path family in a connected hypergraph H, we show that the convexity space on H induced by any hereditary path family containing all chordless paths (such as the families of simple paths and of all paths) is decomposable.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 3; 493-515
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Convexity ranks in higher dimensions
Autorzy:
Kojman, Menachem
Powiązania:
https://bibliotekanauki.pl/articles/1205064.pdf
Data publikacji:
2000
Wydawca:
Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:
convexity
convexity number
Polish vector space
continuum hypothesis
Cantor-Bendixson degree
Opis:
A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists $α < ω_1$ so that ϱ(x) < α for all x ∈ S. Classification of countably convex closed subsets of Polish linear spaces follows then easily. A similar classification (by a different rank function) was previously known for closed subset of $ℝ^2$ [3]. As an application of ϱ to Banach space geometry, it is proved that for every $α < ω_1$, the unit sphere of C(ωα) with the sup-norm has rank α. Furthermore, a countable compact metric space K is determined by the rank of the unit sphere of C(K) with the natural sup-norm: Theorem. If $K_1,K_1$ are countable compact metric spaces and $S_i$ is the unit sphere in $C(K_i)$ with the sup-norm, i = 1,2, then $ϱ(S_1) = ϱ(S_2)$ if and only if $K_1$ and $K_2$ are homeomorphic. Uncountably convex closed sets are also studied in dimension n > 2 and are seen to be drastically more complicated than uncountably convex closed subsets of $ℝ^2$
Źródło:
Fundamenta Mathematicae; 2000, 164, 2; 143-163
0016-2736
Pojawia się w:
Fundamenta Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The generalized Day norm. Part I. Properties
Autorzy:
Budzyńska, Monika
Grzesik, Aleksandra
Kot, Mariola
Powiązania:
https://bibliotekanauki.pl/articles/747286.pdf
Data publikacji:
2017
Wydawca:
Uniwersytet Marii Curie-Skłodowskiej. Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej
Tematy:
Asymptotic normal structure
Day norm
local uniform convexity
normal structure
Opial property
strict convexity
uniform convexity in every direction
Opis:
In this paper we introduce a modification of the Day norm in \(c_0(\Gamma)\) and investigate properties  of this norm.
Źródło:
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica; 2017, 71, 2
0365-1029
2083-7402
Pojawia się w:
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets
Autorzy:
Pallaschke, Diethard
Urbański, Ryszard
Powiązania:
https://bibliotekanauki.pl/articles/1373556.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Jagielloński. Wydawnictwo Uniwersytetu Jagiellońskiego
Tematy:
nonsmooth optimization
generalized convexity
Opis:
The quasidifferential calculus developed by V.F. Demyanov and A.M. Rubinov provides a complete analogon to the classical calculus of differentiation for a wide class of nonsmooth functions. Although this looks at the first glance as a generalized subgradient calculus for pairs of subdifferentials it turns out that, after a more detailed analysis, the quasidifferential calculus is a kind of Fréchet-differentiation whose gradients are elements of a suitable Minkowski–Rådström–Hörmander space. One aim of the paper is to point out this fact. The main results in this direction are Theorem 1 and Theorem 5. Since the elements of the Minkowski–Rådström–Hörmander space are not uniquely determined, we focus our attention in the second part of the paper to smallest possible representations of quasidifferentials, i.e. to minimal representations. Here the main results are two necessary minimality criteria, which are stated in Theorem 9 and Theorem 11.
Źródło:
Schedae Informaticae; 2012, 21; 107-125
0860-0295
2083-8476
Pojawia się w:
Schedae Informaticae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Generalized Levinsons inequality and exponential convexity
Autorzy:
Pecaric, J.
Praljak, M.
Witkowski, A.
Powiązania:
https://bibliotekanauki.pl/articles/255037.pdf
Data publikacji:
2015
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
Levinson's inequality
exponential convexity
Opis:
We give a probabilistic version of Levinson's inequality under Mercer's assumption of equal variances for the family of 3-convex functions at a point. We also show that this is the largest family of continuous functions for which the inequality holds. New families of exponentially convex functions and related results are derived from the obtained inequality.
Źródło:
Opuscula Mathematica; 2015, 35, 3; 397-410
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Minimax theorems for ϕ−convex functions with applications
Autorzy:
Bednarczuk, E. M.
Syga, M.
Powiązania:
https://bibliotekanauki.pl/articles/206451.pdf
Data publikacji:
2014
Wydawca:
Polska Akademia Nauk. Instytut Badań Systemowych PAN
Tematy:
abstract convexity
ϕ−convexity
fi-conjugation
convexlikeness
minimax theorems
joint ϕ-convexlikeness
Opis:
We investigateminimax theorems for ϕ−convex functions. As an application we provide a formula for the ϕ- conjugation of the pointwise maximum of ϕ- convex functions.
Źródło:
Control and Cybernetics; 2014, 43, 3; 421-437
0324-8569
Pojawia się w:
Control and Cybernetics
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals
Autorzy:
Czerwińska, Malgorzata M.
Kaminska, Anna H.
Powiązania:
https://bibliotekanauki.pl/articles/746224.pdf
Data publikacji:
2017
Wydawca:
Polskie Towarzystwo Matematyczne
Tematy:
Symmetric spaces of measurable operators
unitary matrix spaces
rearrangement invariant spaces
k-extreme points
k-convexity
complex extreme points
complex convexity
monotonicity
(local) uniform (complex and real) convexity
p-convexity
Opis:
This is a review article of geometric properties of noncommutative symmetric spaces of measurable operators \(E(\mathcal{M},\tau)\), where \(\mathcal{M}\) is a semifinite von Neumann algebra with a faithful, normal, semifinite trace \(\tau\), and \(E\) is a symmetric function space. If \(E\subset c_0\) is a symmetric sequence space then the analogous properties in the unitary matrix ideals \(C_E\) are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Pólya, Köthe duality, the spaces \(L_p\left(\mathcal{M},\tau\right)\), \(1\leq p < \infty\), the identification of \(C_E\) and \(G(B(H), \operatorname{tr})\) for some symmetric function space \(G\), the commutative case when \(E\) is identified with \(E(\mathcal{N}, \tau)\) for \(\mathcal{N}\) isometric to \(L_\infty\) with the standard integral trace, trace preserving \(*\)-isomorphisms between \(E\) and a \(*\)-subalgebra of \(E\left(\mathcal{M},\tau\right)\), and a general method for removing the assumption of non-atomicity of \(\mathcal{M}\). The main results on geometric properties are given in separate sections. We present the results on (complex) extreme points, (complex) strict convexity, strong extreme points and midpoint local uniform convexity, \(k\)-extreme points and \(k\)-convexity, (complex or local) uniform convexity, smoothness and strong smoothness, (strongly) exposed points, (uniform) Kadec−Klee properties, Banach−Saks properties, Radon−Nikodym property and stability in the sense of Krivine−Maurey. We also state some open problems.
Źródło:
Commentationes Mathematicae; 2017, 57, 1
0373-8299
Pojawia się w:
Commentationes Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
New estimate for the curvature of an order-convex set and related questions
Autorzy:
Ramazanov, Ali B.
Powiązania:
https://bibliotekanauki.pl/articles/2050043.pdf
Data publikacji:
2020
Wydawca:
Polska Akademia Nauk. Instytut Badań Systemowych PAN
Tematy:
gradient
estimates
curvature
convexity
algorithm
Opis:
It is well known that in discrete optimization problems, gradient (local) algorithms do not always guarantee an optimal solution. Therefore, the problem arises of finding the accuracy of the gradient algorithm. This is a fairly well-known problem and numerous publications have been devoted to it. In establishing accuracy, various approaches are used. One of these approaches is to obtain guaranteed estimates of the accuracy of the gradient algorithm in terms of the curvature of the admissible domain. With this approach, it is required to find the curvatures of the admissible region. Since finding the exact value of curvature is a difficult problem to solve, curvature estimates in terms of more or less simply calculated parameters of the problem are relevant. A new improved bound for the curvature of an order-convex set is found and is presented in this paper in terms of the steepness and parameters of strict convexity of the function.
Źródło:
Control and Cybernetics; 2020, 49, 2; 233-240
0324-8569
Pojawia się w:
Control and Cybernetics
Dostawca treści:
Biblioteka Nauki
Artykuł
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