The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Displaying similar documents to “Estimating the critical determinants of a class of three-dimensional star bodies”

Rigidity of critical circle mappings I

Edson de Faria, Welington de Melo (1999)

Journal of the European Mathematical Society

Similarity:

We prove that two C 3 critical circle maps with the same rotation number in a special set 𝔸 are C 1 + α conjugate for some α > 0 provided their successive renormalizations converge together at an exponential rate in the C 0 sense. The set 𝔸 has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C critical circle maps with the same rotation number that are not C 1 + β conjugate for any β > 0 . The class of rotation numbers for which such examples exist...

On the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2

Ruizhou Tong (2021)

Czechoslovak Mathematical Journal

Similarity:

Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 x , p ± 3 ( mod 8 ) , the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2 has no positive integer solution except when p = 3 or p is of the form p = 2 a 0 2 + 1 , where a 0 > 1 is an odd positive integer. (2) if 2 x , 2 y , y 2 , 4 , then the Diophantine equation ( 2 x - 1 ) ( p y - 1 ) = 2 z 2 has no positive integer solution.

On some Diophantine equations involving balancing numbers

Euloge Tchammou, Alain Togbé (2021)

Archivum Mathematicum

Similarity:

In this paper, we find all the solutions of the Diophantine equation B 1 p + 2 B 2 p + + k B k p = B n q in positive integer variables ( k , n ) , where B i is the i t h balancing number if the exponents p , q are included in the set { 1 , 2 } .

Diophantine equations involving factorials

Horst Alzer, Florian Luca (2017)

Mathematica Bohemica

Similarity:

We study the Diophantine equations ( k ! ) n - k n = ( n ! ) k - n k and ( k ! ) n + k n = ( n ! ) k + n k , where k and n are positive integers. We show that the first one holds if and only if k = n or ( k , n ) = ( 1 , 2 ) , ( 2 , 1 ) and that the second one holds if and only if k = n .

A note on the article by F. Luca “On the system of Diophantine equations a ² + b ² = ( m ² + 1 ) r and a x + b y = ( m ² + 1 ) z ” (Acta Arith. 153 (2012), 373-392)

Takafumi Miyazaki (2014)

Acta Arithmetica

Similarity:

Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying A + B ( - 1 ) = ( m + ( - 1 ) ) r . We prove that the Diophantine equation | A | x + | B | y = ( m ² + 1 ) z has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever r > 10 74 or m > 10 34 . Our result is an explicit refinement of a theorem due to F. Luca.

Padovan and Perrin numbers as products of two generalized Lucas numbers

Kouèssi Norbert Adédji, Japhet Odjoumani, Alain Togbé (2023)

Archivum Mathematicum

Similarity:

Let P m and E m be the m -th Padovan and Perrin numbers respectively. Let r , s be non-zero integers with r 1 and s { - 1 , 1 } , let { U n } n 0 be the generalized Lucas sequence given by U n + 2 = r U n + 1 + s U n , with U 0 = 0 and U 1 = 1 . In this paper, we give effective bounds for the solutions of the following Diophantine equations P m = U n U k and E m = U n U k , where m , n and k are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.

A remark on a Diophantine equation of S. S. Pillai

Azizul Hoque (2024)

Czechoslovak Mathematical Journal

Similarity:

S. S. Pillai proved that for a fixed positive integer a , the exponential Diophantine equation x y - y x = a , min ( x , y ) > 1 , has only finitely many solutions in integers x and y . We prove that when a is of the form 2 z 2 , the above equation has no solution in integers x and y with gcd ( x , y ) = 1 .

On the Diophantine equation j = 1 k j F j p = F n q

Gökhan Soydan, László Németh, László Szalay (2018)

Archivum Mathematicum

Similarity:

Let F n denote the n t h term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation F 1 p + 2 F 2 p + + k F k p = F n q in the positive integers k and n , where p and q are given positive integers. A complete solution is given if the exponents are included in the set { 1 , 2 } . Based on the specific cases we could solve, and a computer search with p , q , k 100 we conjecture that beside the trivial solutions only F 8 = F 1 + 2 F 2 + 3 F 3 + 4 F 4 , F 4 2 = F 1 + 2 F 2 + 3 F 3 , and F 4 3 = F 1 3 + 2 F 2 3 + 3 F 3 3 satisfy the title equation.

On k -Pell numbers which are sum of two Narayana’s cows numbers

Kouèssi Norbert Adédji, Mohamadou Bachabi, Alain Togbé (2025)

Mathematica Bohemica

Similarity:

For any positive integer k 2 , let ( P n ( k ) ) n 2 - k be the k -generalized Pell sequence which starts with 0 , , 0 , 1 ( k terms) with the linear recurrence P n ( k ) = 2 P n - 1 ( k ) + P n - 2 ( k ) + + P n - k ( k ) for n 2 . Let ( N n ) n 0 be Narayana’s sequence given by N 0 = N 1 = N 2 = 1 and N n + 3 = N n + 2 + N n . The purpose of this paper is to determine all k -Pell numbers which are sums of two Narayana’s numbers. More precisely, we study the Diophantine equation P p ( k ) = N n + N m in nonnegative integers k , p , n and m .

Critical points of the Moser-Trudinger functional on a disk

Andrea Malchiodi, Luca Martinazzi (2014)

Journal of the European Mathematical Society

Similarity:

On the unit disk B 1 2 we study the Moser-Trudinger functional E ( u ) = B 1 e u 2 - 1 d x , u H 0 1 ( B 1 ) and its restrictions E | M Λ , where M Λ : = { u H 0 1 ( B 1 ) : u H 0 1 2 = Λ } for Λ > 0 . We prove that if a sequence u k of positive critical points of E | M Λ k (for some Λ k > 0 ) blows up as k , then Λ k 4 π , and u k 0 weakly in H 0 1 ( B 1 ) and strongly in C loc 1 ( B ¯ 1 { 0 } ) . Using this fact we also prove that when Λ is large enough, then E | M Λ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

A note on the weighted Khintchine-Groshev Theorem

Mumtaz Hussain, Tatiana Yusupova (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let W ( m , n ; ψ ̲ ) denote the set of ψ 1 , ... , ψ n –approximable points in m n . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions ψ ̲ . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of m and n . It can not be removed for m = n = 1 as Duffin–Schaeffer provided the counter example. We deal with the only remaining case m = 2 and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem. ...

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

Similarity:

We study the leading order behaviour of positive solutions of the equation - Δ u + ϵ u - | u | p - 2 u + | u | q - 2 u = 0 , x N , where N 3 , q > p > 2 and when ϵ > 0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p , q and N . The behavior of solutions depends sensitively on whether p is less, equal or bigger than the critical Sobolev exponent 2 * = 2 N N - 2 . For p < 2 * the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p > 2 * the solution asymptotically...

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

Similarity:

The Littlewood conjecture in Diophantine approximation claims that inf q 1 q · q α · q β = 0 holds for all real numbers α and β , where · denotes the distance to the nearest integer. Its p -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that inf q 1 q · q α · | q | p = 0 holds for every real number α and every prime number p , where | · | p denotes the p -adic absolute value normalized by | p | p = p - 1 . We survey the known results on these conjectures and highlight recent developments. ...

Mersenne numbers as a difference of two Lucas numbers

Murat Alan (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let ( L n ) n 0 be the Lucas sequence. We show that the Diophantine equation L n - L m = M k has only the nonnegative integer solutions ( n , m , k ) = ( 2 , 0 , 1 ) , ( 3 , 1 , 2 ) , ( 3 , 2 , 1 ) , ( 4 , 3 , 2 ) , ( 5 , 3 , 3 ) , ( 6 , 2 , 4 ) , ( 6 , 5 , 3 ) where M k = 2 k - 1 is the k th Mersenne number and n > m .

Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let 𝔻 denote the unit disk { z : | z | < 1 } in the complex plane . In this paper, we study a family of polynomials P with only one zero lying outside 𝔻 ¯ .  We establish  criteria for P to satisfy implying that each of P and P '   has exactly one critical point outside 𝔻 ¯ .

Complete solution of the Diophantine equation x y + y x = z z

Mihai Cipu (2019)

Czechoslovak Mathematical Journal

Similarity:

The triples ( x , y , z ) = ( 1 , z z - 1 , z ) , ( x , y , z ) = ( z z - 1 , 1 , z ) , where z , satisfy the equation x y + y x = z z . In this paper it is shown that the same equation has no integer solution with min { x , y , z } > 1 , thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.