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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Kaplan classes of a certain family of functions

100%
Ignaciuk S., Parol M.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2020
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tom 74
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nr 2
EN
We give the complete characterization of members of Kaplan classes of products of power functions with all zeros symmetrically distributed in \(\mathbb{T} := \{z \in\mathbb{C} : |z| = 1\}\) and weakly monotonic sequence of powers. In this way we extend Sheil-Small’s theorem. We apply the obtained result to study univalence of antiderivative of these products of power functions.
2
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Modelling tumour-immunity interactions with different stimulation functions

100%
Zhivkov P., Waniewski J.
International Journal of Applied Mathematics and Computer Science
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2003
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tom 13
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nr 3
307-315
EN
Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.
3
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Equivariant maps of joins of finite G-sets and an application to critical point theory

100%
Rozpłoch-Nowakowska D.
Annales Polonici Mathematici
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1991-1992
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tom 56
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nr 2
195-211
EN
A lower estimate is proved for the number of critical orbits and critical values of a G-invariant C¹ function $f:S^n → ℝ$, where G is a finite nontrivial group acting freely and orthogonally on $ℝ^{n+1} \ {0}$. Neither Morse theory nor the minimax method is applied. The proofs are based on a general version of Borsuk's Antipodal Theorem for equivariant maps of joins of G-sets.
4
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Critical points of asymptotically quadratic functions

100%
Fečkan M.
Annales Polonici Mathematici
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1995
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tom 61
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nr 1
63-76
EN
Existence results for critical points of asymptotically quadratic functions defined on Hilbert spaces are studied by using Morse-Conley index and pseudomonotone mappings. Applications to differential equations are given.
5
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Smooth approximations without critical points

88%
Hájek P., Johanis M.
Open Mathematics
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2003
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tom 1
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nr 3
284-291
EN
In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.
6
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Generalized Newton and NCP-methods: convergence, regularity, actions

75%
Kummer B.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2000
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tom 20
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nr 2
209-244
EN
Solutions of several problems can be modelled as solutions of nonsmooth equations. Then, Newton-type methods for solving such equations induce particular iteration steps (actions) and regularity requirements in the original problems. We study these actions and requirements for nonlinear complementarity problems (NCP's) and Karush-Kuhn-Tucker systems (KKT) of optimization models. We demonstrate their dependence on the applied Newton techniques and the corresponding reformulations. In this way, connections to SQP-methods, to penalty-barrier methods and to general properties of so-called NCP-functions are shown. Moreover, direct comparisons of the hypotheses and actions in terms of the original problems become possible. Besides, we point out the possibilities and bounds of such methods in dependence of smoothness.
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