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1
Content available remote New estimate for the curvature of an order-convex set and related questions
EN
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.
EN
Some developments of the second-order characterizations of convex functions are investigated by using the coderivative of the subdifferential mapping. Furthermore, some applications of the second-order subdifferentials in optimization problems are studied.
3
Content available remote Approximations by multivariate sublinear and Max-product operators under convexity
EN
Here we search quantitatively under convexity the approximation of multivariate function by general multivariate positive sublinear operators with applications to multivariate Max-product operators. These are of Bernstein type, of Favard-Szász-Mirakjan type, of Baskakov type, of sampling type, of Lagrange interpolation type and of Hermite-Fejér interpolation type. Our results are both: under the presence of smoothness and without any smoothness assumption on the function to be approximated which fulfills a convexity assumption.
4
Content available remote Multivariate and abstract approximation theory for Banach space valued functions
EN
Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.
5
Content available remote On coefficient problems of an operator with respect to symmetric point
EN
In this research work, we study the properties of a certain differential subordination involving an operator with respect to a symmetric point. We establish coefficient estimates as our main results.
6
Content available remote L*-operators of convexity
EN
A notion of L^*-spaces is investigated as a generalization of convex subspaces. This gives some topological extensions for celebrated theorems due to Maynard Smith, Brouwer and Nash.
PL
W pracy jest badane pojęcie uogólnionej wypukłości, które umożliwia otrzymanie bardzo prostych dowodów twierdzenia Maynarda Smitha o istnieniu strategii ewolucyjnych w modelach biologicznych oraz twierdzenia o sygnaturach dających znaczne rozszerzenia twierdzenia Nasha o równowadze.
EN
In the paper there are determined, for some classes defined by coefficient or analytic conditions, the sets of complex parameter γ, for which all the functions of the appropriate family have some geometrical properties. There are also provided the examples of the mappings showing that some inclusions between classes are impossible or confirming that sets of the parameter γ cannot be extended in some cases without loss of these geometric properties.
8
Content available Immunization and convex interest rate shifts
EN
An important issue in immunization theory is the form of the interest rate process under which immunization is feasible. This paper generalizes Fisher and Weil immunization result to convex interest rate shifts, and examines the practical significance of this generalization. We examine the features of a linear factor model that are consistent with a convex shift. In particular, we show that a specific two factor linear model is sufficient and necessary for a convex shift. This two factor specification allows parallel and damped yield curve shifts, which in combination can twist the yield curie.
EN
We derive C2-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 C2-condition for strict convexity to a characterization. Several examples illustrate the results.
10
Content available A characterization of convex φ-functions
EN
The properties of four elements (LPFE) and (UPFE), introduced by Isac and Persson, have been recently examined in Hilbert spaces, Lp-spaces and modular spaces. In this paper we prove a new theorem showing that a modular of form ρφ(∫)= ∫ Ω φ (t,/∫(t)/)dμ(t) satisfies both (LPFE) and (UPFE) if and only if φ is convex with respect to its second variable. A connection of this result with the study of projections and antiprojections onto latticially closed subsets of the modular space Lφ is also discussed.
11
Content available Probabilistic characterization of strong convexity
EN
Strong convexity is considered for real functions defined on a real interval. Probabilistic characterization is given and its geometrical sense is explained. Using this characterization some inequalities of Jensen-type are obtained.
12
Content available remote Modes, modals, and barycentric algebras: a brief survey and an additivity theorem
EN
Modes are idempotent and entropic algebras.Modals are both join semi lattices and modes,where the mode structure distributes over the join.Barycentric algebras are equipped with binary operations from the open unit interval,satisfying idempo tence,skew commutativity,and skew associativity.The article aims to give a brief survey of these structures and some of their applications.Special attention is devoted to hierar chical statistical mechanics and the modeling of complex systems.An additivity theorem for the entropy of independent combinations of systems is proved.
13
Content available On approximately Breckner s-convex functions
EN
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.
14
Content available remote Generalized approximate midconvexity
EN
Let X be a normed space and V ⊂ X a convex set with nonempty interior. Let α : [0, ∞) → [0, ∞) be a given nondecreasing function. A function ƒ : V → R is α(⋅)-midconvex if ƒ [wzór]. In this paper we study α(⋅)-midconvex functions. Using a version of Bernstein-Doetsch theorem we prove that if ƒ is α(⋅)-midconvex and locally bounded from above at every point then ƒ(rx + (1 - r)y) ≤ rƒ(x) + (1 - r)ƒ(y) + Pα(r, ¦¦x - y¦¦) for x,y ∈ V and r ∈ [0,1], where Pα : [0,1] x [0,∞) → [0,∞) is a specific function dependent on α. We obtain three different estimations of Pα. This enables us to generalize some results concerning paraconvex and semiconcave functions.
PL
W pracy przedstawiono efekty ekonomiczne uzyskane w wyniku opracowania i wdrożenia komputerowego systemu kształtowania szczeliny walcowniczej z wykorzystaniem sztucznej sieci neuronowej oraz asymetrycznego walcowania w dwóch ostatnich klatkach układu ciągłego walcowni gorącej blach. Zastosowanie i wdrożenie nowej technologii wpłynęło na zmniejszenie falistości produkcji towarowej w wybraku użytecznym, niedowalcowań, wypukłości pasma, wskaźnika zużycia walców oraz pozwoliło rozszerzyć asortyment walcowanych blach cienkich.
EN
The paper presents economic effects achieved as a result of development and implementation of a computer system for roll gap control using an artificial neural network and asymmetrical rolling in the last two stands of the continuous hotplate rolling mill system. The application and implementation of the new technology has contributed to reduction ofwaviness of commodity production in the usable reject, misrolling, strip convexity and roll wear rate and has enabled to expand the rolled sheet range.
16
Content available remote Valuation and metric on a lattice effect algebra
EN
In the present paper we have derived some properties of the pseudo-metric introduced by Riećanova on a lattice effect algebra E corresponding to a valuation u on E, which turns out to be a metric if u happens to be faithful. Using these properties we have been able to prove that this metric is complete. Also it is observed that the resulting metric space is convex if and only if w is non-atomic if and only if E is atomless.
17
Content available remote On multivalently analytic and multivalently meromorphic functions
EN
In the present investigation, some theorems involving certain inequalities between multivalently analytic functions in the unit disk U and multivalently meromorphic functions in the punctured unit disk D are stated. AIso certain interesting consequences of main results are given.
EN
We generalize the embedding order to shuffle on trajectories and examine its properties. We give general characterizations of reflexivity, transitivity, anti-symmetry and other properties of binary relations, and investigate the associated decision problems. We also examine the property of convexity of a language with respect to these generalized embedding relations.
19
Content available remote On some generalization of coefficient conditions for complex harmonic mappings
EN
Let h = u + iv, where u, v are real harmonic functions in the unit disc A. Such functions are called complex mappings harmonic in A. The function h may be written in the form h = f + g, where f,g are functions holomorphic in the unit disc, of course. Studies of complex harmonic functions were initiated in 1984 by J. Clunie and T. Sheil-Small ([2]) and were continued by many others mathematicians. We can find some papers on functions harmonic in A, satisfying certain coefficient conditions, e.g. [1], [4], [6], [7], [8]. We investigate some more general problem, i. e. a coefficient inequality with any fixed sequence of real positive numbers.
20
Content available remote Rotundity, smoothness and duality
EN
The duality between smoothness and rotundity of functions is studied in a nonlinear abstract framework. Here smoothness is enlarged to subdifferentiability properties and rotundity is formulated by means of approximation properties.
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