A Case of Monotone Ratio Growth for Quadratic-Like Mappings

Waldemar Pałuba

Displaying similar documents to “A Case of Monotone Ratio Growth for Quadratic-Like Mappings”

High-order phase transitions in the quadratic family

Daniel Coronel, Juan Rivera-Letelier (2015)

Journal of the European Mathematical Society

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We give the first example of a transitive quadratic map whose real and complex geometric pressure functions have a high-order phase transition. In fact, we show that this phase transition resembles a Kosterlitz-Thouless singularity: Near the critical parameter the geometric pressure function behaves as $x \mapsto \exp (- x^{- 2})$ near $x = 0$ , before becoming linear. This quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.

Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $- [a + b (\int_{Ω} {| \nabla u | ² d x)}^{m} {] Δ u = f (x, u) + | u |}^{2 * - 2} u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

On cusps and flat tops

Neil Dobbs (2014)

Annales de l’institut Fourier

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Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to $C^{1 + ϵ}$ . The critical points are not required to verify a non-flatness condition, so the results are applicable to $C^{1 + ϵ}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

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We describe all sets $A \subseteq_{p}$ which represent the quadratic residues $R \subseteq_{p}$ in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.

Quadratic differentials $(A (z - a) (z - b) / {(z - c)}^{2}) d z^{2}$ and algebraic Cauchy transform

Mohamed Jalel Atia, Faouzi Thabet (2016)

Czechoslovak Mathematical Journal

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We discuss the representability almost everywhere (a.e.) in $ℂ$ of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials $A (z - a) (z - b) {(z - c)}^{- 2} d z^{2}$ . More precisely, we give a necessary and sufficient condition on the complex numbers $a$ and $b$ for these quadratic differentials to have finite critical...

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

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The relative class number $H_{d} (f)$ of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of $_{f}$ and $_{K}$ , where $_{K}$ denotes the ring of integers of K and $_{f}$ is the order of conductor f given by $ℤ + f_{K}$ . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...

Monotonicity of the period function for some planar differential systems. Part I: Conservative and quadratic systems

A. Raouf Chouikha (2005)

Applicationes Mathematicae

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We first examine conditions implying monotonicity of the period function for potential systems with a center at 0 (in the whole period annulus). We also present a short comparative survey of the different criteria. We apply these results to quadratic Loud systems $(L_{D, F})$ for various values of the parameters D and F. In the case of noncritical periods we propose an algorithm to test the monotonicity of the period function for $(L_{D, F})$ . Our results may be viewed as a contribution to proving (or disproving)...

The invertibility of the isoparametric mappings for triangular quadratic Lagrange finite elements

Josef Dalík (2012)

Applications of Mathematics

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A reference triangular quadratic Lagrange finite element consists of a right triangle $\hat{K}$ with unit legs $S_{1}$ , $S_{2}$ , a local space $\hat{ℒ}$ of quadratic polynomials on $\hat{K}$ and of parameters relating the values in the vertices and midpoints of sides of $\hat{K}$ to every function from $\hat{ℒ}$ . Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping $ℱ_{h} = (F_{1}, F_{2}) \in \hat{ℒ} \times \hat{ℒ}$ . We explicitly describe such invertible isoparametric mappings $ℱ_{h}$ for which the images $ℱ_{h} (S_{1})$ , $ℱ_{h} (S_{2})$ of the segments...

On asymptotic critical values and the Rabier Theorem

Zbigniew Jelonek (2004)

Banach Center Publications

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Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f ₁, . . ., f_{m}) : X \to k^{m}$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with...

Positivity of quadratic base change $L$ -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

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We show that certain quadratic base change $L$ -functions for $Gl (2)$ are non-negative at their center of symmetry.

Mapping hierarchy for dendrites

J. J. Charatonik, W. J. Charatonik, J. R. Prajs

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AbstractLet a family S of spaces and a class F of mappings between members of S be given. For two spaces X and Y in S we define $Y \leq_{F} X$ if there exists a surjection f ∈ F of X onto Y. We investigate the quasi-order $\leq_{F}$ in the family of dendrites, where F is one of the following classes of mappings: retractions, monotone, open, confluent or weakly confluent mappings. In particular, we investigate minimal and maximal elements, chains and antichains in the quasi-order $\leq_{F}$ , and characterize spaces...

Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

Y. Akdim, J. Bennouna, M. Mekkour, H. Redwane (2012)

Applicationes Mathematicae

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We study the problem ∂b(x,u)/∂t - div(a(x,t,u,Du)) + H(x,t,u,Du) = μ in Q = Ω×(0,T), ${b (x, u) |}_{t = 0} = b (x, u ₀)$ in Ω, u = 0 in ∂Ω × (0,T). The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on H(x,t,u,Du). The critical growth condition on H is only with respect to Du and not with respect to u. The datum μ is assumed to be in $L ¹ (Q) + L^{p^{'}} (0, T; W^{- 1, p^{'}} (Ω))$ and b(x,u₀) ∈ L¹(Ω).

Distortion bounds for $C^{2 + η}$ unimodal maps

Mike Todd (2007)

Fundamenta Mathematicae

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We obtain estimates for derivative and cross-ratio distortion for $C^{2 + η}$ (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that $g_{T} : T \to ℝ$ is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove...

Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Jia-Feng Liao, Jiu Liu, Peng Zhang, Chun-Lei Tang (2016)

Annales Polonici Mathematici

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We study the following singular elliptic equation with critical exponent ⎧ $- Δ u = Q (x) u^{2 * - 1} + λ u^{- γ}$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

Bulk superconductivity in Type II superconductors near the second critical ﬁeld

Soren Fournais, Bernard Helffer (2010)

Journal of the European Mathematical Society

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We consider superconductors of Type II near the transition from the ‘bulk superconducting’ to the ‘surface superconducting’ state. We prove a new $L^{\infty}$ estimate on the order parameter in the bulk, i.e. away from the boundary. This solves an open problem posed by Aftalion and Serfaty [AS].

Real $C^{k}$ Koebe principle

Weixiao Shen, Michael Todd (2005)

Fundamenta Mathematicae

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We prove a $C^{k}$ version of the real Koebe principle for interval (or circle) maps with non-flat critical points.

Tameness criterion for posets with zero-relations and three-partite subamalgams of tiled orders

Stanisław Kasjan (2002)

Colloquium Mathematicae

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A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if $Λ^{•}$ is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form $q_{Λ^{•}}$ associated with $Λ^{•}$ in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28]...

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator $\Delta^{2}$ in a bounded domain $\Omega\subset\mathbb{R}^{n}$ . We state some existence results when $n\geq 8$ . Moreover, we consider $5\leq n\leq 7$ , expecially when $\Omega$ is a ball in $\mathbb{R}^{n}$ .

Capturing forms in dense subsets of finite fields

Brandon Hanson (2013)

Acta Arithmetica

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An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field $_{q}$ . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size...

The sum of divisors of a quadratic form

Lilu Zhao (2014)

Acta Arithmetica

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We study the sum τ of divisors of the quadratic form m₁² + m₂² + m₃². Let $S ₃ (X) = \sum_{1 \leq m ₁, m ₂, m ₃ \leq X} τ (m ₁ ² + m ₂ ² + m ₃ ²)$ . We obtain the asymptotic formula S₃(X) = C₁X³logX + C₂X³ + O(X²log⁷X), where C₁,C₂ are two constants. This improves upon the error term $O_{ε} (X^{8 / 3 + ε})$ obtained by Guo and Zhai (2012).

On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu Lei, Chang-Mu Chu, Hong-Min Suo, Chun-Lei Tang (2015)

Annales Polonici Mathematici

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In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧ $- (a + b \int_{Ω} | \nabla u | ² d x) Δ u = u ⁵ + λ u^{q - 1} / {| x |}^{β}$ in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

Grothendieck and Witt groups in the reduced theory of quadratic forms

Andrzej Sładek (1980)

Annales Polonici Mathematici

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Abstract. Let F be a formally real field. Denote by G(F) and $G_{t} (F)$ the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group $G (F) / G_{t} (F)$ . This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for $G (F) / G_{t} (F)$ , which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the...

Critical points via $Γ$ -convergence: general theory and applications

Jerrard, Robert L., Peter Sternberg (2009)

Journal of the European Mathematical Society

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