Displaying similar documents to “Relative Borsuk-Ulam Theorems for Spaces with a Free ℤ₂-action”

A note on the theorems of Lusternik-Schnirelmann and Borsuk-Ulam

T. E. Barros, C. Biasi (2008)

Colloquium Mathematicae

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Let p be a prime number and X a simply connected Hausdorff space equipped with a free p -action generated by f p : X X . Let α : S 2 n - 1 S 2 n - 1 be a homeomorphism generating a free p -action on the (2n-1)-sphere, whose orbit space is some lens space. We prove that, under some homotopy conditions on X, there exists an equivariant map F : ( S 2 n - 1 , α ) ( X , f p ) . As applications, we derive new versions of generalized Lusternik-Schnirelmann and Borsuk-Ulam theorems.

Addition theorems and related geometric problems of group representation theory

Ekaterina Shulman (2013)

Banach Center Publications

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The Levi-Civita functional equation f ( g h ) = k = 1 n u k ( g ) v k ( h ) (g,h ∈ G), for scalar functions on a topological semigroup G, has as the solutions the functions which have finite-dimensional orbits in the right regular representation of G, that is the matrix elements of G. In considerations of some extensions of the L-C equation one encounters with other geometric problems, for example: 1) which vectors x of the space X of a representation g T g have orbits O(x) that are “close” to a fixed finite-dimensional subspace?...

Maps into the torus and minimal coincidence sets for homotopies

D. L. Goncalves, M. R. Kelly (2002)

Fundamenta Mathematicae

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Let X,Y be manifolds of the same dimension. Given continuous mappings f i , g i : X Y , i = 0,1, we consider the 1-parameter coincidence problem of finding homotopies f t , g t , 0 ≤ t ≤ 1, such that the number of coincidence points for the pair f t , g t is independent of t. When Y is the torus and f₀,g₀ are coincidence free we produce coincidence free pairs f₁,g₁ such that no homotopy joining them is coincidence free at each level. When X is also the torus we characterize the solution of the problem in terms of the...

Pairs of square-free values of the type n 2 + 1 , n 2 + 2

Stoyan Dimitrov (2021)

Czechoslovak Mathematical Journal

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We show that there exist infinitely many consecutive square-free numbers of the form n 2 + 1 , n 2 + 2 . We also establish an asymptotic formula for the number of such square-free pairs when n does not exceed given sufficiently large positive number.

Free locally convex spaces and L -retracts

Rodrigo Hidalgo Linares, Oleg Okunev (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study the relation of L -equivalence defined between Tychonoff spaces, that is, we study the topological isomorphisms of their respective free locally convex spaces. We introduce the concept of an L -retract in a Tychonoff space in terms of the existence of a special kind of simultaneous extensions of continuous functions, explore the relation of this concept with the Dugundji extension theorem, and find some conditions that allow us to identify L -retracts in various classes of topological...

Formally self-referential propositions for cut free classical analysis and related systems

G. Kreisel, G. Takeuti

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CONTENTSIntroduction............................................................................................................................................................................................................ 5  I. Results on self-referential propositions............................................................................................................................. 11    1. Definitions of some principal metamathematical notions......................................................................

Some remarks providing discontinuous maps on some C p ( X ) spaces

S. Moll (2008)

Banach Center Publications

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Let X be a completely regular Hausdorff topological space and C p ( X ) the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space C p ( X ) , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of C p ( X ) .

Consecutive square-free values of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1

Ya-Fang Feng (2023)

Czechoslovak Mathematical Journal

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We show that for any given integer k there exist infinitely many consecutive square-free numbers of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1 . We also establish an asymptotic formula for 1 x , y , z H such that x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1 are square-free. The method we used in this paper is due to Tolev.

The range of non-linear natural polynomials cannot be context-free

Dömötör Pálvölgyi (2020)

Kybernetika

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Suppose that some polynomial f with rational coefficients takes only natural values at natural numbers, i. e., L = { f ( n ) n } . We show that the base- q representation of L is a context-free language if and only if f is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma.

Equalizers and coactions of groups

Martin Arkowitz, Mauricio Gutierrez (2002)

Fundamenta Mathematicae

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If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group f G * H be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism p f = p | f : f G . A right inverse (section) G f of p f is called a coaction on G. In this paper we study f and the sections of p f . We consider the following topics: the structure of f as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and...

Computation of some examples of Brown's spectral measure in free probability

Philippe Biane, Franz Lehner (2001)

Colloquium Mathematicae

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We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include u + u where uₙ and u are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form S α + i S β where S α and S β are free semicircular elements of variance α and β.

A note on certain partial sum operators

Marek Bożejko, Gero Fendler (2006)

Banach Center Publications

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We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of L p functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative L p spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.

Large free subgroups of automorphism groups of ultrahomogeneous spaces

Szymon Głąb, Filip Strobin (2015)

Colloquium Mathematicae

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We consider the following notion of largeness for subgroups of S . A group G is large if it contains a free subgroup on generators. We give a necessary condition for a countable structure A to have a large group Aut(A) of automorphisms. It turns out that any countable free subgroup of S can be extended to a large free subgroup of S , and, under Martin’s Axiom, any free subgroup of S of cardinality less than can also be extended to a large free subgroup of S . Finally, if Gₙ are countable...

Prescribing endomorphism algebras of n -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

Non-orbit equivalent actions of 𝔽 n

Adrian Ioana (2009)

Annales scientifiques de l'École Normale Supérieure

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For any 2 n , we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group 𝔽 n . These actions arise as diagonal products between a generalized Bernoulli action and the action 𝔽 n ( 𝕋 2 , λ 2 ) , where 𝔽 n is seen as a subgroup of SL 2 ( ) .

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G , then for every subset B of G with | B | > | G | / k 1 / 3 we have B 3 = G . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if k 2 , then G has a...

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

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A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, U α : α , such that U α U β whenever β ≤ α for all α , β . The class of all metrizable topological groups is a proper subclass of the class T G of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group G T G is metrizable, and hence G is strictly angelic. We deduce from...