Displaying similar documents to “The equation - Δ 𝑢 - λ 𝑢 | 𝑥 | 2 = | 𝑢 | 𝑝 + 𝑐 𝑓 ( 𝑥 ) : The optimal power”

The Rothberger property on C p ( Ψ ( 𝒜 ) , 2 )

Daniel Bernal-Santos (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 𝒰 n : n ω of open covers of X , there exists U n 𝒰 n for each n ω such that X = n ω U n . For any n ω , necessary and sufficient conditions are obtained for C p ( Ψ ( 𝒜 ) , 2 ) n to have the Rothberger property when 𝒜 is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family 𝒜 for which the space C p ( Ψ ( 𝒜 ) , 2 ) n is Rothberger for all n ω .

Coincidence for substitutions of Pisot type

Marcy Barge, Beverly Diamond (2002)

Bulletin de la Société Mathématique de France

Similarity:

Let ϕ be a substitution of Pisot type on the alphabet 𝒜 = { 1 , 2 , ... , d } ; ϕ satisfies theif for every i , j 𝒜 , there are integers k , n such that ϕ n ( i ) and ϕ n ( j ) have the same k -th letter, and the prefixes of length k - 1 of ϕ n ( i ) and ϕ n ( j ) have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if d = 2 and provide a partial result for d 2 .

Involutivity degree of a distribution at superdensity points of its tangencies

Silvano Delladio (2021)

Archivum Mathematicum

Similarity:

Let Φ 1 , ... , Φ k + 1 (with k 1 ) be vector fields of class C k in an open set U N + m , let 𝕄 be a N -dimensional C k submanifold of U and define 𝕋 : = { z 𝕄 : Φ 1 ( z ) , ... , Φ k + 1 ( z ) T z 𝕄 } where T z 𝕄 is the tangent space to 𝕄 at z . Then we expect the following property, which is obvious in the special case when z 0 is an interior point (relative to 𝕄 ) of 𝕋 : If z 0 𝕄 is a ( N + k ) -density point (relative to 𝕄 ) of 𝕋 then all the iterated Lie brackets of order less or equal to k Φ i 1 ( z 0 ) , [ Φ i 1 , Φ i 2 ] ( z 0 ) , [ [ Φ i 1 , Φ i 2 ] , Φ i 3 ] ( z 0 ) , ... ( h , i h k + 1 ) belong to T z 0 𝕄 . Such a property has been proved in [9] for k = 1 and its proof in the...

Further generalized versions of Ilmanen’s lemma on insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2021)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

The author proved in 2018 that if G is an open subset of a Hilbert space, f 1 , f 2 : G continuous functions and ω a nontrivial modulus such that f 1 f 2 , f 1 is locally semiconvex with modulus ω and f 2 is locally semiconcave with modulus ω , then there exists f C loc 1 , ω ( G ) such that f 1 f f 2 . This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of n ). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to L p spaces, p [ 2 , ) . We...

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

Adel Mahmoud Gomaa (2017)

Czechoslovak Mathematical Journal

Similarity:

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation x ˙ ( t ) = ( t ) x ( t ) + f ( t , x ( t ) ) , t ( P ) where { ( t ) : t } is a family of linear operators from a Banach space E into itself and f : × E E . By L ( E ) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0 , we let C ( [ - d , 0 ] , E ) be the Banach space of continuous functions from [ - d , 0 ] into E and f d : [ a , b ] × C ( [ - d , 0 ] , E ) E . Let ^ : [ a , b ] L ( E ) be a strongly measurable and Bochner integrable operator on [ a , b ] and for t [ a , b ] define τ t x ( s ) = x ( t + s ) for each s [ - d , 0 ] . We prove that, under certain...

Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster (2019)

Czechoslovak Mathematical Journal

Similarity:

We consider the equation - y ' ( x ) + q ( x ) y ( x - ϕ ( x ) ) = f ( x ) , x , where ϕ and q ( q 1 ) are positive continuous functions for all x and f C ( ) . By a solution of the equation we mean any function y , continuously differentiable everywhere in , which satisfies the equation for all x . We show that under certain additional conditions on the functions ϕ and q , the above equation has a unique solution y , satisfying the inequality y ' C ( ) + q y C ( ) c f C ( ) , where the constant c ( 0 , ) does not depend on the choice of f .

On the n -fold symmetric product of a space with a σ - ( P ) -property c n -network ( c k -network)

Luong Q. Tuyen, Ong V. Tuyen (2020)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We study the relation between a space X satisfying certain generalized metric properties and its n -fold symmetric product n ( X ) satisfying the same properties. We prove that X has a σ - ( P ) -property c n -network if and only if so does n ( X ) . Moreover, if X is regular then X has a σ - ( P ) -property c k -network if and only if so does n ( X ) . By these results, we obtain that X is strict σ -space (strict -space) if and only if so is n ( X ) .

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

Similarity:

Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...