Displaying similar documents to “QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations”

On the uniqueness of periodic decomposition

Viktor Harangi (2011)

Fundamenta Mathematicae

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Let a , . . . , a k be arbitrary nonzero real numbers. An ( a , . . . , a k ) -decomposition of a function f:ℝ → ℝ is a sum f + + f k = f where f i : is an a i -periodic function. Such a decomposition is not unique because there are several solutions of the equation h + + h k = 0 with h i : a i -periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the ( a , . . . , a k ) -decomposition is essentially unique. We characterize those periods for which essential...

Probabilistic properties of a Markov-switching periodic G A R C H process

Billel Aliat, Fayçal Hamdi (2019)

Kybernetika

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In this paper, we propose an extension of a periodic G A R C H ( P G A R C H ) model to a Markov-switching periodic G A R C H ( M S - P G A R C H ), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically and of weakly periodically stationary solutions. We establish necessary and sufficient conditions ensuring the existence of higher order moments. We further provide closed-form expressions for calculating the even-order moments as well...

Existence and global attractivity of periodic solutions in a higher order difference equation

Chuanxi Qian, Justin Smith (2018)

Archivum Mathematicum

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Consider the following higher order difference equation x ( n + 1 ) = f ( n , x ( n ) ) + g ( n , x ( n - k ) ) , n = 0 , 1 , where f ( n , x ) and g ( n , x ) : { 0 , 1 , } × [ 0 , ) [ 0 , ) are continuous functions in x and periodic functions in n with period p , and k is a nonnegative integer. We show the existence of a periodic solution { x ˜ ( n ) } under certain conditions, and then establish a sufficient condition for { x ˜ ( n ) } to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also...

On the topology of polynomials with bounded integer coefficients

De-Jun Feng (2016)

Journal of the European Mathematical Society

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For a real number q > 1 and a positive integer m , let Y m ( q ) : = i = 0 n ϵ i q i : ϵ i 0 , ± 1 , ... , ± m , n = 0 , 1 , ... . In this paper, we show that Y m ( q ) is dense in if and only if q < m + 1 and q is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Anne-Laure Dalibard (2011)

Journal of the European Mathematical Society

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This article investigates the long-time behaviour of parabolic scalar conservation laws of the type t u + div y A ( y , u ) - Δ y u = 0 , where y N and the flux A is periodic in y . More specifically, we consider the case when the initial data is an L 1 disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in L 1 norm like a self-similar profile for large times. The proof uses a time and space change of variables...

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.

Existence and multiplicity of solutions for a fractional p -Laplacian problem of Kirchhoff type via Krasnoselskii’s genus

Ghania Benhamida, Toufik Moussaoui (2018)

Mathematica Bohemica

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We use the genus theory to prove the existence and multiplicity of solutions for the fractional p -Kirchhoff problem - M Q | u ( x ) - u ( y ) | p | x - y | N + p s d x d y p - 1 ( - Δ ) p s u = λ h ( x , u ) in Ω , u = 0 on N Ω , where Ω is an open bounded smooth domain of N , p > 1 , N > p s with s ( 0 , 1 ) fixed, Q = 2 N ( C Ω × C Ω ) , λ > 0 is a numerical parameter, M and h are continuous functions.

L p inequalities for the growth of polynomials with restricted zeros

Nisar A. Rather, Suhail Gulzar, Aijaz A. Bhat (2022)

Archivum Mathematicum

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Let P ( z ) = ν = 0 n a ν z ν be a polynomial of degree at most n which does not vanish in the disk | z | < 1 , then for 1 p < and R > 1 , Boas and Rahman proved P ( R z ) p ( R n + z p / 1 + z p ) P p . In this paper, we improve the above inequality for 0 p < by involving some of the coefficients of the polynomial P ( z ) . Analogous result for the class of polynomials P ( z ) having no zero in | z | > 1 is also given.

Cobham's theorem for substitutions

Fabien Durand (2011)

Journal of the European Mathematical Society

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The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then a sequence x A , where A is a finite alphabet, is both α -substitutive and β -substitutive if and only if x is ultimately...

Weakly nonlinear stochastic CGL equations

Sergei B. Kuksin (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: d d t u + i - Δ + V ( x ) u = ν Δ u - γ R | u | 2 p u - i γ I | u | 2 q u + ν η ( t , x ) . ( * ) The force η is white in time and smooth in x ; the potential V ( x ) is typical. We are concerned with the limiting, as ν 0 , behaviour of solutions on long time-intervals 0 t ν - 1 T , and with behaviour of these solutions under the double limit t and ν 0 . We show that these two limiting behaviours may be described in terms of solutions for the( * ) which is a well...

One-dimensional symmetry of periodic minimizers for a mean field equation

Chang-Shou Lin, Marcello Lucia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider on a two-dimensional flat torus T defined by a rectangular periodic cell the following equation Δ u + ρ e u T e u - 1 | T | = 0 , T u = 0 . It is well-known that the associated energy functional admits a minimizer for each ρ 8 π . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting λ 1 ( T ) to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever ρ min { 8 π , λ 1 ( T ) | T | } . Our results hold more generally for solutions that...

Periodic solutions for a class of non-autonomous Hamiltonian systems with p ( t ) -Laplacian

Zhiyong Wang, Zhengya Qian (2024)

Mathematica Bohemica

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We investigate the existence of infinitely many periodic solutions for the p ( t ) -Laplacian Hamiltonian systems. By virtue of several auxiliary functions, we obtain a series of new super- p + growth and asymptotic- p + growth conditions. Using the minimax methods in critical point theory, some multiplicity theorems are established, which unify and generalize some known results in the literature. Meanwhile, we also present an example to illustrate our main results are new even in the case p ( t ) p = 2 . ...

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...

Geometric rigidity of × m invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

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Let μ be a probability measure on [ 0 , 1 ] which is invariant and ergodic for T a ( x ) = a x 𝚖𝚘𝚍 1 , and 0 < 𝚍𝚒𝚖 μ < 1 . Let f be a local diffeomorphism on some open set. We show that if E and ( f μ ) E μ E , then f ' ( x ) ± a r : r at μ -a.e. point x f - 1 E . In particular, if g is a piecewise-analytic map preserving μ then there is an open g -invariant set U containing supp μ such that g U is piecewise-linear with slopes which are rational powers of a . In a similar vein, for μ as above, if b is another integer and a , b are not powers of a common integer, and if ν is...

The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner (2011)

Studia Mathematica

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We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots t j = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let V n , ν be the space of piecewise linear continuous functions on the torus with knots t j : 0 j N - 1 . Finally, let P n , ν be the orthogonal projection operator from L²([0,1)) onto V n , ν . The main result is l i m n , ν = 1 | | P n , ν : L L | | = s u p n , 0 ν n | | P n , ν : L L | | = 2 + ( 33 - 18 3 ) / 13 . This shows in particular that the Lebesgue constant of the classical...

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.

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...

Solutions of an advance-delay differential equation and their asymptotic behaviour

Gabriela Vážanová (2023)

Archivum Mathematicum

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The paper considers a scalar differential equation of an advance-delay type y ˙ ( t ) = - a 0 + a 1 t y ( t - τ ) + b 0 + b 1 t y ( t + σ ) , where constants a 0 , b 0 , τ and σ are positive, and a 1 and b 1 are arbitrary. The behavior of its solutions for t is analyzed provided that the transcendental equation λ = - a 0 e - λ τ + b 0 e λ σ has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.

Complete monotonicity of the remainder in an asymptotic series related to the psi function

Zhen-Hang Yang, Jing-Feng Tian (2024)

Czechoslovak Mathematical Journal

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Let p , q with p - q 0 , σ = 1 2 ( p + q - 1 ) and s = 1 2 ( 1 - p + q ) , and let 𝒟 m ( x ; p , q ) = 𝒟 0 ( x ; p , q ) + k = 1 m B 2 k ( s ) 2 k ( x + σ ) 2 k , where 𝒟 0 ( x ; p , q ) = ψ ( x + p ) + ψ ( x + q ) 2 - ln ( x + σ ) . We establish the asymptotic expansion 𝒟 0 ( x ; p , q ) - n = 1 B 2 n ( s ) 2 n ( x + σ ) 2 n as x , where B 2 n ( s ) stands for the Bernoulli polynomials. Further, we prove that the functions ( - 1 ) m 𝒟 m ( x ; p , q ) and ( - 1 ) m + 1 𝒟 m ( x ; p , q ) are completely monotonic in x on ( - σ , ) for every m 0 if and only if p - q [ 0 , 1 2 ] and p - q = 1 , respectively. This not only unifies the two known results but also yields some new results.

Maximum number of limit cycles for generalized Liénard polynomial differential systems

Aziza Berbache, Ahmed Bendjeddou, Sabah Benadouane (2021)

Mathematica Bohemica

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We consider limit cycles of a class of polynomial differential systems of the form x ˙ = y , y ˙ = - x - ε ( g 21 ( x ) y 2 α + 1 + f 21 ( x ) y 2 β ) - ε 2 ( g 22 ( x ) y 2 α + 1 + f 22 ( x ) y 2 β ) , where β and α are positive integers, g 2 j and f 2 j have degree m and n , respectively, for each j = 1 , 2 , and ε is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center x ˙ = y , y ˙ = - x using the averaging theory of first and second order.

On square functions associated to sectorial operators

Christian Le Merdy (2004)

Bulletin de la Société Mathématique de France

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We give new results on square functions x F = 0 F ( t A ) x 2 d t t 1 / 2 p associated to a sectorial operator A on L p for 1 &lt; p &lt; . Under the assumption that A is actually R -sectorial, we prove equivalences of the form K - 1 x G x F K x G for suitable functions F , G . We also show that A has a bounded H functional calculus with respect to . F . Then we apply our results to the study of conditions under which we have an estimate ( 0 | C e - t A ( x ) | 2 d t ) 1 / 2 q M x p , when - A generates a bounded semigroup e - t A on L p and C : D ( A ) L q is a linear mapping.