Waclaw Marzantowicz - Academia.edu (original) (raw)

Papers by Waclaw Marzantowicz

Topology and its Applications, 2021

Abstract We estimate the number of simplices required for triangulations of compact Lie groups. A... more Abstract We estimate the number of simplices required for triangulations of compact Lie groups. As in the previous work [11] , our approach combines the estimation of the number of vertices by means of the covering type via a cohomological argument from [10] , and application of the recent versions of the Lower Bound Theorem of combinatorial topology. For the exceptional Lie groups, we present a complete calculation using the description of their cohomology rings given by the first and third authors. For the infinite series of classical Lie groups of growing dimension d, we estimate the growth rate of number of simplices of the highest dimension, which extends to the case of simplices of (fixed) codimension d − i .

Journal of Fixed Point Theory and Applications, 2020

The partition invariant π(K) of a simplicial complex K ⊆ 2 [m] is the minimum integer ν, such tha... more The partition invariant π(K) of a simplicial complex K ⊆ 2 [m] is the minimum integer ν, such that for each partition A1 • • • Aν = [m] of [m], at least one of the sets Ai is in K. A complex K is r-unavoidable if π(K) ≤ r. We say that a complex K is almost r-non-embeddable in R d if, for each continuous map f : |K| → R d , there exist r vertex disjoint faces σ1, • • • , σr of |K|, such that f (σ1) ∩ • • • ∩ f (σr) = ∅. One of our central observations (Theorem 2.1), summarizing and extending results of Schild et al. is that interesting examples of (almost) r-nonembeddable complexes can be found among the joins K = K1 * • • • * Ks of r-unavoidable complexes.

Wiadomości Matematyczne, 2015

Mathematica

In this paper, we prove the existence of at least three solutions for a differential inclusion pr... more In this paper, we prove the existence of at least three solutions for a differential inclusion problem involving the p-Laplacian with nonhomogeneous and nonsmooth Neumann boundary conditions. We use a three critical points theorem for locally Lipschitz functions given by A. Kristály et al. [J. Glob. Optim. 46, No. 1, 49–62 (2010; Zbl 1188.90252)].

Wydawnictwo Naukowe PWN eBooks, Mar 1, 2022

Rendiconti del Seminario Matematico della Università di Padova, 1987

L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova... more L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Topology and its Applications, Apr 1, 2008

A natural number m is called a homotopy minimal period of a map f : X → X if every map g homotopi... more A natural number m is called a homotopy minimal period of a map f : X → X if every map g homotopic to f has periodic points of minimal period m. In this paper we give a description for the sets of homotopy minimal periods of maps of all compact solvmanifolds of dimension three. Techniques based on the notion of a model solvmanifold are different than those previously used to study tori, compact nilmanifolds, and special NR-solvmanifolds.

Topology and its Applications, Nov 1, 2018

We describe a unified approach to estimating the dimension of f −1 (A) for any G-equivariant map ... more We describe a unified approach to estimating the dimension of f −1 (A) for any G-equivariant map f : X → Y and any closed G-invariant subset A ⊆ Y in terms of connectivity of X and dimension of Y , where G is either a cyclic group of order p k , a p-torus (p a prime), or a torus.

Simon Stevin, Dec 1, 2017

Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that... more Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that V G = W G = {0}, then a G-equivariant map S(V) → S(W) exists provided that dim V H ≤ dim W H for any closed subgroup H ⊆ G. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when G is a torus.

Mathematica Scandinavica, Dec 1, 2010

Topological Methods in Nonlinear Analysis, Jun 1, 2009

Journal of the London Mathematical Society, Jan 28, 2004

Nonlinear Analysis-theory Methods & Applications, Nov 1, 2005

Let G be a compact Lie group. In this paper, combining a short exact sequence obtained by Balanov... more Let G be a compact Lie group. In this paper, combining a short exact sequence obtained by Balanov and Krawcewicz with some additional topological techniques, we complete the computation of the first equivariant stem G st 1. Using the exact sequence and a property of nonabelian connected compact Lie groups, whose proof was suggested to us by R. Oliver, we show that this group is finite if and only if G is finite.

Topological Methods in Nonlinear Analysis, Mar 1, 2009

Suppose f : M → M on a compact manifold. Let m be a natural number. One of the most important que... more Suppose f : M → M on a compact manifold. Let m be a natural number. One of the most important questions in the topological theory of periodic points is whether the Nielsen-Jiang periodic number N Fm(f) is a sharp lower bound on #Fix(g m) over all g ∼ f. This question has a positive answer if dim M ≥ 3 but in general a negative answer for self maps of compact surfaces. However, we show the answer to be positive when M = K is the Klein bottle. As a consequence, we reconfirm a result of Llibre and compute the set HPer(f) of homotopy minimal periods on the Klein bottle.

arXiv (Cornell University), Apr 5, 2017

Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that... more Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that V G = W G = {0}, then a G-equivariant map S(V) → S(W) exists provided that dim V H ≤ dim W H for any closed subgroup H ⊆ G. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when G is a torus.

arXiv (Cornell University), Oct 12, 2012

We give a complete description of the behaviour of the sequence of displacements η n (z) = Φ n (x... more We give a complete description of the behaviour of the sequence of displacements η n (z) = Φ n (x)−Φ n−1 (x) mod 1, z = exp(2πix), along a trajectory {ϕ n (z)}, where ϕ is an orientation preserving circle homeomorphism and Φ : R → R its lift. If the rotation number ̺(ϕ) = p q is rational then η n (z) is asymptotically periodic with semi-period q. This convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. If ̺(ϕ) / ∈ Q then the values of η n (z) are dense in a set which depends on the map γ (semi-)conjugating ϕ with the rotation by ̺(ϕ) and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if ϕ is C 1-diffeomorphism and show approximation of the displacement distribution by sample displacements measured along a trajectory of any other circle homeomorphism which is sufficiently close to the initial homeomorphism ϕ. Finally, we prove that even for the irrational rotation number ̺ the displacement sequence exhibits some regularity properties.

arXiv (Cornell University), Apr 11, 2013

We analyze properties of the firing map, which iterations give information about consecutive spik... more We analyze properties of the firing map, which iterations give information about consecutive spikes, for periodically driven linear integrate-and-fire models. By considering locally integrable (thus in general not continuous) input functions, we generalize some results of other authors. In particular we prove theorems concerning continuous dependence of the firing map on the input in suitable function spaces. Using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism, we provide also a complete description of the regularity properties of the sequence of interspike-intervals and behaviour of the interspike-interval distribution. Our results allow to explain some facts concerning this distribution observed numerically by other authors. These theoretical findings are illustrated by carefully chosen computational examples.

arXiv (Cornell University), Jun 16, 2011

In mathematical biology and the theory of electric networks the firing map of an integrate-and-fi... more In mathematical biology and the theory of electric networks the firing map of an integrate-and-fire system is a notion of importance. In order to prove useful properties of this map authors of previous papers assumed that the stimulus function f of the systemẋ = f (t, x) is continuous and usually periodic in the time variable. In this work we show that the required properties of the firing map for the simplified modelẋ = f (t) still hold if f ∈ L 1 loc (R) and f is an almost periodic function. Moreover, in this way we prepare a formal framework for next study of a discrete dynamics of the firing map arising from almost periodic stimulus that gives information on consecutive resets (spikes).

Osaka Journal of Mathematics, Dec 1, 2002

Bulletin Of The Brazilian Mathematical Society, New Series, Jul 1, 2005

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere... more A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere S n , n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : S n → S n commutes with a free homeomorphism g : S n → S n of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points.

Topology and its Applications, 2021

Abstract We estimate the number of simplices required for triangulations of compact Lie groups. A... more Abstract We estimate the number of simplices required for triangulations of compact Lie groups. As in the previous work [11] , our approach combines the estimation of the number of vertices by means of the covering type via a cohomological argument from [10] , and application of the recent versions of the Lower Bound Theorem of combinatorial topology. For the exceptional Lie groups, we present a complete calculation using the description of their cohomology rings given by the first and third authors. For the infinite series of classical Lie groups of growing dimension d, we estimate the growth rate of number of simplices of the highest dimension, which extends to the case of simplices of (fixed) codimension d − i .

Journal of Fixed Point Theory and Applications, 2020

The partition invariant π(K) of a simplicial complex K ⊆ 2 [m] is the minimum integer ν, such tha... more The partition invariant π(K) of a simplicial complex K ⊆ 2 [m] is the minimum integer ν, such that for each partition A1 • • • Aν = [m] of [m], at least one of the sets Ai is in K. A complex K is r-unavoidable if π(K) ≤ r. We say that a complex K is almost r-non-embeddable in R d if, for each continuous map f : |K| → R d , there exist r vertex disjoint faces σ1, • • • , σr of |K|, such that f (σ1) ∩ • • • ∩ f (σr) = ∅. One of our central observations (Theorem 2.1), summarizing and extending results of Schild et al. is that interesting examples of (almost) r-nonembeddable complexes can be found among the joins K = K1 * • • • * Ks of r-unavoidable complexes.

Wiadomości Matematyczne, 2015

Mathematica

In this paper, we prove the existence of at least three solutions for a differential inclusion pr... more In this paper, we prove the existence of at least three solutions for a differential inclusion problem involving the p-Laplacian with nonhomogeneous and nonsmooth Neumann boundary conditions. We use a three critical points theorem for locally Lipschitz functions given by A. Kristály et al. [J. Glob. Optim. 46, No. 1, 49–62 (2010; Zbl 1188.90252)].

Wydawnictwo Naukowe PWN eBooks, Mar 1, 2022

Rendiconti del Seminario Matematico della Università di Padova, 1987

L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova... more L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Topology and its Applications, Apr 1, 2008

A natural number m is called a homotopy minimal period of a map f : X → X if every map g homotopi... more A natural number m is called a homotopy minimal period of a map f : X → X if every map g homotopic to f has periodic points of minimal period m. In this paper we give a description for the sets of homotopy minimal periods of maps of all compact solvmanifolds of dimension three. Techniques based on the notion of a model solvmanifold are different than those previously used to study tori, compact nilmanifolds, and special NR-solvmanifolds.

Topology and its Applications, Nov 1, 2018

We describe a unified approach to estimating the dimension of f −1 (A) for any G-equivariant map ... more We describe a unified approach to estimating the dimension of f −1 (A) for any G-equivariant map f : X → Y and any closed G-invariant subset A ⊆ Y in terms of connectivity of X and dimension of Y , where G is either a cyclic group of order p k , a p-torus (p a prime), or a torus.

Simon Stevin, Dec 1, 2017

Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that... more Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that V G = W G = {0}, then a G-equivariant map S(V) → S(W) exists provided that dim V H ≤ dim W H for any closed subgroup H ⊆ G. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when G is a torus.

Mathematica Scandinavica, Dec 1, 2010

Topological Methods in Nonlinear Analysis, Jun 1, 2009

Journal of the London Mathematical Society, Jan 28, 2004

Nonlinear Analysis-theory Methods & Applications, Nov 1, 2005

Let G be a compact Lie group. In this paper, combining a short exact sequence obtained by Balanov... more Let G be a compact Lie group. In this paper, combining a short exact sequence obtained by Balanov and Krawcewicz with some additional topological techniques, we complete the computation of the first equivariant stem G st 1. Using the exact sequence and a property of nonabelian connected compact Lie groups, whose proof was suggested to us by R. Oliver, we show that this group is finite if and only if G is finite.

Topological Methods in Nonlinear Analysis, Mar 1, 2009

Suppose f : M → M on a compact manifold. Let m be a natural number. One of the most important que... more Suppose f : M → M on a compact manifold. Let m be a natural number. One of the most important questions in the topological theory of periodic points is whether the Nielsen-Jiang periodic number N Fm(f) is a sharp lower bound on #Fix(g m) over all g ∼ f. This question has a positive answer if dim M ≥ 3 but in general a negative answer for self maps of compact surfaces. However, we show the answer to be positive when M = K is the Klein bottle. As a consequence, we reconfirm a result of Llibre and compute the set HPer(f) of homotopy minimal periods on the Klein bottle.

arXiv (Cornell University), Apr 5, 2017

Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that... more Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that V G = W G = {0}, then a G-equivariant map S(V) → S(W) exists provided that dim V H ≤ dim W H for any closed subgroup H ⊆ G. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when G is a torus.

arXiv (Cornell University), Oct 12, 2012

We give a complete description of the behaviour of the sequence of displacements η n (z) = Φ n (x... more We give a complete description of the behaviour of the sequence of displacements η n (z) = Φ n (x)−Φ n−1 (x) mod 1, z = exp(2πix), along a trajectory {ϕ n (z)}, where ϕ is an orientation preserving circle homeomorphism and Φ : R → R its lift. If the rotation number ̺(ϕ) = p q is rational then η n (z) is asymptotically periodic with semi-period q. This convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. If ̺(ϕ) / ∈ Q then the values of η n (z) are dense in a set which depends on the map γ (semi-)conjugating ϕ with the rotation by ̺(ϕ) and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if ϕ is C 1-diffeomorphism and show approximation of the displacement distribution by sample displacements measured along a trajectory of any other circle homeomorphism which is sufficiently close to the initial homeomorphism ϕ. Finally, we prove that even for the irrational rotation number ̺ the displacement sequence exhibits some regularity properties.

arXiv (Cornell University), Apr 11, 2013

We analyze properties of the firing map, which iterations give information about consecutive spik... more We analyze properties of the firing map, which iterations give information about consecutive spikes, for periodically driven linear integrate-and-fire models. By considering locally integrable (thus in general not continuous) input functions, we generalize some results of other authors. In particular we prove theorems concerning continuous dependence of the firing map on the input in suitable function spaces. Using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism, we provide also a complete description of the regularity properties of the sequence of interspike-intervals and behaviour of the interspike-interval distribution. Our results allow to explain some facts concerning this distribution observed numerically by other authors. These theoretical findings are illustrated by carefully chosen computational examples.

arXiv (Cornell University), Jun 16, 2011

In mathematical biology and the theory of electric networks the firing map of an integrate-and-fi... more In mathematical biology and the theory of electric networks the firing map of an integrate-and-fire system is a notion of importance. In order to prove useful properties of this map authors of previous papers assumed that the stimulus function f of the systemẋ = f (t, x) is continuous and usually periodic in the time variable. In this work we show that the required properties of the firing map for the simplified modelẋ = f (t) still hold if f ∈ L 1 loc (R) and f is an almost periodic function. Moreover, in this way we prepare a formal framework for next study of a discrete dynamics of the firing map arising from almost periodic stimulus that gives information on consecutive resets (spikes).

Osaka Journal of Mathematics, Dec 1, 2002

Bulletin Of The Brazilian Mathematical Society, New Series, Jul 1, 2005

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere... more A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere S n , n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : S n → S n commutes with a free homeomorphism g : S n → S n of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points.