Displaying similar documents to “Characterization of automorphisms of Radford's biproduct of Hopf group-coalgebra”

Automorphism group of green algebra of weak Hopf algebra corresponding to Sweedler Hopf algebra

Liufeng Cao, Dong Su, Hua Yao (2023)

Czechoslovak Mathematical Journal

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Let r ( 𝔴 2 0 ) be the Green ring of the weak Hopf algebra 𝔴 2 0 corresponding to Sweedler’s 4-dimensional Hopf algebra H 2 , and let Aut ( R ( 𝔴 2 0 ) ) be the automorphism group of the Green algebra R ( 𝔴 2 0 ) = r ( 𝔴 2 0 ) . We show that the quotient group Aut ( R ( 𝔴 2 0 ) ) / C 2 S 3 , where C 2 contains the identity map and is isomorphic to the infinite group ( * , × ) and S 3 is the symmetric group of order 6.

Higgs bundles and representation spaces associated to morphisms

Indranil Biswas, Carlos Florentino (2015)

Archivum Mathematicum

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Let G be a connected reductive affine algebraic group defined over the complex numbers, and K G be a maximal compact subgroup. Let X , Y be irreducible smooth complex projective varieties and f : X Y an algebraic morphism, such that π 1 ( Y ) is virtually nilpotent and the homomorphism f * : π 1 ( X ) π 1 ( Y ) is surjective. Define f ( π 1 ( X ) , G ) = { ρ Hom ( π 1 ( X ) , G ) A ρ factors through f * } , f ( π 1 ( X ) , K ) = { ρ Hom ( π 1 ( X ) , K ) A ρ factors through f * } , where A : G GL ( Lie ( G ) ) is the adjoint action. We prove that the geometric invariant theoretic quotient f ( π 1 ( X , x 0 ) , G ) / / G admits a deformation retraction to f ( π 1 ( X , x 0 ) , K ) / K . We also show that the space of conjugacy classes of n almost commuting...

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

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Let p be an odd prime, and let a be an integer not divisible by p . When m is a positive integer with p 1 ( mod 2 m ) and 2 is an m th power residue modulo p , we determine the value of the product k R m ( p ) ( 1 + tan ( π a k / p ) ) , where R m ( p ) = { 0 < k < p : k is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y , then k R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

Automorphism group of representation ring of the weak Hopf algebra H 8 ˜

Dong Su, Shilin Yang (2018)

Czechoslovak Mathematical Journal

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Let H 8 be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra H 8 ˜ based on H 8 , then we investigate the structure of the representation ring of H 8 ˜ . Finally, we prove that the automorphism group of r ( H 8 ˜ ) is just isomorphic to D 6 , where D 6 is the dihedral group with order 12.

On a sequence formed by iterating a divisor operator

Bellaouar Djamel, Boudaoud Abdelmadjid, Özen Özer (2019)

Czechoslovak Mathematical Journal

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Let be the set of positive integers and let s . We denote by d s the arithmetic function given by d s ( n ) = ( d ( n ) ) s , where d ( n ) is the number of positive divisors of n . Moreover, for every , m we denote by δ s , , m ( n ) the sequence d s ( d s ( ... d s ( d s ( n ) + ) + ... ) + ) m -times = d s ( n ) for m = 1 , d s ( d s ( n ) + ) for m = 2 , d s ( d s ( d s ( n ) + ) + ) for m = 3 , We present classical and nonclassical notes on the sequence ( δ s , , m ( n ) ) m 1 , where , n , s are understood as parameters.

Positive solutions for concave-convex elliptic problems involving p ( x ) -Laplacian

Makkia Dammak, Abir Amor Ben Ali, Said Taarabti (2022)

Mathematica Bohemica

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We study the existence and nonexistence of positive solutions of the nonlinear equation - Δ p ( x ) u = λ k ( x ) u q ± h ( x ) u r in Ω , u = 0 on Ω where Ω N , N 2 , is a regular bounded open domain in N and the p ( x ) -Laplacian Δ p ( x ) u : = div ( | u | p ( x ) - 2 u ) is introduced for a continuous function p ( x ) > 1 defined on Ω . The positive parameter λ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive...

On Kneser solutions of the n -th order nonlinear differential inclusions

Martina Pavlačková (2019)

Czechoslovak Mathematical Journal

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The paper deals with the existence of a Kneser solution of the n -th order nonlinear differential inclusion x ( n ) ( t ) - A 1 ( t , x ( t ) , ... , x ( n - 1 ) ( t ) ) x ( n - 1 ) ( t ) - ... - A n ( t , x ( t ) , ... , x ( n - 1 ) ( t ) ) x ( t ) for a.a. t [ a , ) , where a ( 0 , ) , and A i : [ a , ) × n , i = 1 , ... , n , are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

Traceability in { K 1 , 4 , K 1 , 4 + e } -free graphs

Wei Zheng, Ligong Wang (2019)

Czechoslovak Mathematical Journal

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A graph G is called { H 1 , H 2 , , H k } -free if G contains no induced subgraph isomorphic to any graph H i , 1 i k . We define σ k = min i = 1 k d ( v i ) : { v 1 , , v k } is an independent set of vertices in G . In this paper, we prove that (1) if G is a connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and σ 3 ( G ) n - 1 , then G is traceable, (2) if G is a 2-connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and | N ( x 1 ) N ( x 2 ) | + | N ( y 1 ) N ( y 2 ) | n - 1 for any two distinct pairs of non-adjacent vertices { x 1 , x 2 } , { y 1 , y 2 } of G , then G is traceable, i.e., G has a Hamilton path, where K 1 , 4 + e is a graph obtained by joining a pair of non-adjacent vertices in a K 1 , 4 .

The bicrossed products of H 4 and H 8

Daowei Lu, Yan Ning, Dingguo Wang (2020)

Czechoslovak Mathematical Journal

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Let H 4 and H 8 be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through H 8 and H 4 (equivalently, any bicrossed product between the Hopf algebras H 8 and H 4 ) must be isomorphic to one of the following four Hopf algebras: H 8 H 4 , H 32 , 1 , H 32 , 2 , H 32 , 3 . The set of all matched pairs ( H 8 , H 4 , , ) is explicitly described, and then the associated bicrossed product is given by generators and relations.

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | Ω = 0 , u &gt; 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 &lt; p &lt; , p - 1 &lt; q p * - 1 , λ &gt; 0 , and 0 &lt; δ &lt; 1 . As usual, p * = N p N - p if 1 &lt; p &lt; N , p * ( p , ) is arbitrarily large if p = N , and p * = if p &gt; N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

The structures of Hopf * -algebra on Radford algebras

Hassan Suleman Esmael Mohammed, Hui-Xiang Chen (2019)

Czechoslovak Mathematical Journal

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We investigate the structures of Hopf * -algebra on the Radford algebras over . All the * -structures on H are explicitly given. Moreover, these Hopf * -algebra structures are classified up to equivalence.

Persistence of Coron’s solution in nearly critical problems

Monica Musso, Angela Pistoia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the problem - Δ u = u N + 2 N - 2 + λ in Ω ε ω , u &gt; 0 in Ω ε ω , u = 0 on Ω ε ω , where Ω and ω are smooth bounded domains in N , N 3 , ε &gt; 0 and λ . We prove that if the size of the hole ε goes to zero and if, simultaneously, the parameter λ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

Copies of l p n ’s uniformly in the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X )

Dumitru Popa (2017)

Czechoslovak Mathematical Journal

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We study the presence of copies of l p n ’s uniformly in the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X ) . By using Dvoretzky’s theorem we deduce that if X is an infinite-dimensional Banach space, then Π 2 ( C [ 0 , 1 ] , X ) contains λ 2 -uniformly copies of l n ’s and Π 1 ( C [ 0 , 1 ] , X ) contains λ -uniformly copies of l 2 n ’s for all λ > 1 . As an application, we show that if X is an infinite-dimensional Banach space then the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X ) are distinct, extending the well-known result that the spaces Π 2 ( C [ 0 , 1 ] , X ) and 𝒩 ( C [ 0 , 1 ] , X ) are distinct.

Subclasses of typically real functions determined by some modular inequalities

Leopold Koczan, Katarzyna Trąbka-Więcław (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ : = { z : | z | < 1 } , normalized by f ( 0 ) = f ' ( 0 ) - 1 = 0 and such that Im z Im f ( z ) 0 for z Δ . Moreover, let us denote: T ( 2 ) : = { f T : f ( z ) = - f ( - z ) for z Δ } and T M , g : = { f T : f M g in Δ } , where M > 1 , g T S and S consists of all analytic functions, normalized and univalent in Δ .We investigate  classes in which the subordination is replaced with the majorization and the function g is typically real but does not necessarily univalent, i.e. classes { f T : f M g in Δ } , where M > 1 , g T , which we denote...

H 2 convergence of solutions of a biharmonic problem on a truncated convex sector near the angle π

Abdelkader Tami, Mounir Tlemcani (2021)

Applications of Mathematics

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We consider a biharmonic problem Δ 2 u ω = f ω with Navier type boundary conditions u ω = Δ u ω = 0 , on a family of truncated sectors Ω ω in 2 of radius r , 0 < r < 1 and opening angle ω , ω ( 2 π / 3 , π ] when ω is close to π . The family of right-hand sides ( f ω ) ω ( 2 π / 3 , π ] is assumed to depend smoothly on ω in L 2 ( Ω ω ) . The main result is that u ω converges to u π when ω π with respect to the H 2 -norm. We can also show that the H 2 -topology is optimal for such a convergence result.

On behavior of solutions to a chemotaxis system with a nonlinear sensitivity function

Senba, Takasi, Fujie, Kentarou

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In this paper, we consider solutions to the following chemotaxis system with general sensitivity τ u t = Δ u - · ( u χ ( v ) ) in Ω × ( 0 , ) , η v t = Δ v - v + u in Ω × ( 0 , ) , u ν = u ν = 0 on Ω × ( 0 , ) . Here, τ and η are positive constants, χ is a smooth function on ( 0 , ) satisfying χ ' ( · ) > 0 and Ω is a bounded domain of 𝐑 n ( n 2 ). It is well known that the chemotaxis system with direct sensitivity ( χ ( v ) = χ 0 v , χ 0 > 0 ) has blowup solutions in the case where n 2 . On the other hand, in the case where χ ( v ) = χ 0 log v with 0 < χ 0 1 , any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness...