Unbounded Component Research Articles

Bifurcation analysis of fractional Kirchhoff problems with singular exponential nonlinearity

This paper deals with the bifurcation results of (weak) solutions for the Kirchhoff fractional p-Laplacian equation { M ( x , [ u ] s , p p ) ( − Δ ) p s u = f ( λ , x , u ) in Ω , u = 0 in R N ∖ Ω , where ( − Δ ) p s denotes the fractional operator, with sp = N, and the nonlinearity f exhibits the singular exponential growth at infinity. Moreover, the existence of unbounded components of (weak) solutions emanating from the trivial solution, is treated via the fix point result and the global bifurcation theorem due to Rabinowitz. Finally, the main feature of this paper consists of the existence of positive solutions to the above equation for λ small enough.

Bifurcation and regularity analysis of the Schrödinger-Poisson equation

Abstract The aim of this paper is to present bifurcation results for (weak) solutions of the Schrödinger-Poisson system in R 3 , involving subcritical and critical nonlinearities and using the global bifurcation theorem. Furthermore, we establish the existence of unbounded components of (weak) solutions, which bifurcate from trivial solutions and from infinity, respectively. The novelties of the paper lie in the appearance of the indefinite nonlinearity and the critical nonlinear term. Finally, we also analyze the regularity result that guarantees the L ∞ -bound of the (weak) solutions of the nonlinear Schrödinger-Poisson system under consideration.

Fallen Sanctuary: A Higher-Order and Leakage-Resilient Rekeying Scheme

This paper presents a provably secure, higher-order, and leakage-resilient (LR) rekeying scheme named LR Rekeying with Random oracle Repetition (LR4), along with a quantitative security evaluation methodology. Many existing LR primitives are based on a concept of leveled implementation, which still essentially require a leak-free sanctuary (i.e., differential power analysis (DPA)-resistant component(s)) for some parts. In addition, although several LR pseudorandom functions (PRFs) based on only bounded DPA-resistant components have been developed, their validity and effectiveness for rekeying usage still need to be determined. In contrast, LR4 is formally proven under a leakage model that captures the practical goal of side-channel attack (SCA) protection (e.g., masking with a practical order) and assumes no unbounded DPA-resistant sanctuary. This proof suggests that LR4 resists exponential invocations (up to the birthday bound of key size) without using any unbounded leak-free component, which is the first of its kind. Moreover, we present a quantitative SCA success rate evaluation methodology for LR4 that combines the bounded leakage models for LR cryptography and a state-of-the-art information-theoretical SCA evaluation method. We validate its soundness and effectiveness as a DPA countermeasure through a numerical evaluation; that is, the number of secure calls of a symmetric primitive increases exponentially by increasing a security parameter under practical conditions.

Uniqueness of Unbounded Component for Level Sets of Smooth Gaussian Fields

Abstract For a large family of stationary continuous Gaussian fields $f$ on ${\mathbb {R}}^{d}$, including the Bargmann–Fock and Cauchy fields, we prove that there exists at most one unbounded connected component in the level set $\{f=\ell \}$ (as well as in the excursion set $\{f\geq \ell \}$) almost surely for every level $\ell \in {\mathbb {R}}$, thus proving a conjecture proposed by Duminil-Copin, Rivera, Rodriguez, and Vanneuville. As the fields considered are typically very rigid (e.g., analytic almost surely), there is no sort of finite energy property available and the classical approaches to prove uniqueness become difficult to implement. We bypass this difficulty using a soft shift argument based on the Cameron–Martin theorem.

Single and double toral band Fatou components in meromorphic dynamics

We analyze the existence and types of unbounded Fatou components for elliptic functions and other meromorphic functions with doubly periodic Julia sets. We show that apart from Herman rings and Siegel disks, all types of dynamics can occur in these domains, which are called toral bands. We show that toral bands are not necessarily periodic, and we give results about the number of distinct residue classes of critical points in each toral band.

Global bifurcation of positive solutions for a class of superlinear first-order differential systems

We are concerned with the first-order differential system of the form $$\left\{ \begin{array}{ll} u'(t)+a(t)u(t)=\lambda b(t) f(v(t-\tau(t))), &t\in\mathbb{R},\\ v'(t)+a(t)v(t)=\lambda b(t)g(u(t-\tau(t))), &t\in\mathbb{R},\\ \end{array} \right. $$ where $\lambda\in\mathbb{R}$~is a parameter. $a,b\in C(\mathbb{R},[0,+\infty))$ are two $\omega$-periodic functions such that $\int_0^\omega a(t)\text{d}t>0$,~$\int_0^\omega b(t)\text{d}t>0$,~$\tau\in C(\mathbb{R},\mathbb{R})$ is an $\omega$-periodic function. The nonlinearities~$f,g\in C(\mathbb{R},(0,+\infty))$~are two nondecreasing continuous functions and satisfy superlinear conditions at infinity.~By using the bifurcation theory,~we will show the existence of an unbounded component of positive solutions, which is bounded in positive $\lambda$-direction.

The Accuracy of the Gaussian Tail and Dual Dirac Model in Jitter Histogram and Probability Density Functions

Jitter can be generally modeled as a superposition of an unbounded random component that follows a Gaussian distribution and a bounded deterministic component. It is usually assumed that the probability distribution of total jitter for very large values of jitter tends to a purely Gaussian distribution. The standard deviation of this Gaussian distribution is identical to the standard deviation of the random component and the mean value of the distribution is related to the deterministic component. A mathematical justification for this assumption is, however, lacking in the literature. In this work, a general asymptotic expression is derived for the tail of the total jitter distribution. It is shown that, to first order, the tail of the distribution can be expressed as a Gaussian function divided by a power function of jitter. Asymptotic expressions for the cumulative distribution function and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> -scale are also derived. The implications of these results for the accuracy of broadly accepted tail fitting routines are discussed.

A Polyhedral Homotopy Algorithm for Real Zeros

We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients satisfying certain concavity conditions. It operates entirely over the real numbers and tracks the optimal number of solution paths. In more technical terms; we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba.

Boundedness of the nodal domains of additive Gaussian fields

We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets { f ≤ ℓ } \{f \le \ell \} of additive planar Gaussian fields are bounded almost surely at the critical level ℓ c = 0 \ell _c = 0 . Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension d ≥ 3 d \ge 3 the excursion sets have unbounded components at all levels.

Functions with no unbounded Fatou components

For a transcendental entire function with sufficiently small growth, Baker raised the question whether it has no unbounded Fatou components. We have shown that if the function is of order strictly less than half, minimal type, then it has no unbounded Fatou components. This, in particular gives a partial answer to Baker's question. In addition, we have addressed Wang's question on Fejér gaps. Certain results about functions with Fabry gaps and of infinite order have also been generalized.

Non-triviality in a totally asymmetric one-dimensional Boolean percolation model on a half-line

It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space is covered a.s., or the covered volume fraction is strictly less than one. The aim of this work is to demonstrate that there is a third possibility in a Boolean model with totally asymmetric grains on a half-line: a.s. there is no unbounded component, but the covered volume fraction is one. An explicit condition is given characterizing the existence of an unbounded occupied component.

The annealed spectral sample of Voronoi percolation

In this paper, we introduce and study the annealed spectral sample of Voronoi percolation, which is a continuous and finite point process in R2 whose definition is mostly inspired by the spectral sample of Bernoulli percolation introduced in (Acta Math. 205 (2010) 19–104) by Garban, Pete and Schramm. We show a clustering effect as well as estimates on the full lower tail of this spectral object. Our main motivation is the study of two models of dynamical critical Voronoi percolation in the plane. In the first model, the Voronoi tiling does not evolve in time while the colors of the cells are resampled at rate 1. In the second model, the centers of the cells move according to (independent) long range stable Lévy processes but the colors do not evolve in time. We prove that for these two dynamical processes there exist almost surely exceptional times with an unbounded monochromatic component.

Relative equilibria, stability and bifurcations in Hamiltonian galactic tidal models

ABSTRACT We study special solutions of Galactic Tidal models and the existence, stability and bifurcations of relative equilibria. Precisely, we found four equilibria and rectilinear, as well as circular solutions, previous to any manipulation of the original system. Moreover, by averaging of the Keplerian energy and making use of Reeb's theorem, we find four periodic orbits. The stability of these relative equilibria is analysed and determined in several cases that involve a relation between the Keplerian energy and the gravitational parameter. Further averaging on the angle giving the orbital plane node provides the Matese–Whitman model. This Hamiltonian is expressed as a function of the invariants of the twice-reduced space, which is given by three connected components. For the three connected components become attached to each other by two singular points. Although only the bounded component is endowed with physical meaning, we provide a full analysis of the stability and bifurcations for all the equilibria. By doing this, we get an insight into the bifurcation process. Precisely, we observe that for the singular leaf of the reduced space, some of the equilibria in the bounded and unbounded components switch their locations.

Inversion of operator pencils on Banach space using Jordan chains when the generalized resolvent has an isolated essential singularity

We assume that the generalized resolvent for a bounded linear operator pencil mapping one Banach space onto another has an isolated essential singularity at the origin and is analytic on some annular region of the complex plane centred at the origin. In such cases the resolvent operator can be represented on the annulus by a convergent Laurent series and the spectral set has two components—a bounded component inside the inner boundary of the annulus and an unbounded component outside the outer boundary. In this paper we prove that the complementary spectral separation projections on the domain space are uniquely determined by the respective generating subspaces for the associated infinite-length generalized Jordan chains of vectors and that the domain space is the direct sum of these two subspaces. We show that the images of the generating subspaces under the mapping defined by the pencil provide a corresponding direct sum decomposition for the range space and that this is simply the decomposition defined by the complementary spectral separation projections on the range space. If the domain space has a Schauder basis we show that the separated systems of fundamental equations are reduced to two semi-infinite systems of matrix equations which can be solved recursively to obtain a basic solution and thereby determine the Laurent series coefficients for the resolvent operator on the given annular region.

The critical threshold for Bargmann–Fock percolation

In this article, we study the excursions sets $\mathcal{D}_p=f^{-1}([-p,+\infty[)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann-Fock field. More precisely, $f$ is the centered Gaussian field on $\mathbb{R}^2$ with covariance $(x,y) \mapsto \exp(-\frac{1}{2}|x-y|^2)$. In~\cite{bg_16}, Beffara and Gayet prove that, if $p \leq 0$, then a.s. $\mathcal{D}_p$ has no unbounded component. We show that conversely, if $p>0$, then a.s. $\mathcal{D}_p$ has a unique unbounded component. As a result, the critical level of this percolation model is $0$. We also prove exponential decay of crossing probabilities under the critical level. To show these results, we develop several tools including a KKL-type result for biased Gaussian vectors (based on the analogous result for product Gaussian vectors by Keller, Mossel and Sen in~\cite{keller2012geometric}) and a sprinkling inspired discretization procedure. These intermediate results hold for more general Gaussian fields, for which we prove a discrete version of our main result.

Existence of Positive Solutions to Singular Boundary Value Problems Involving φ-Laplacian

This paper is concerned with the existence of positive solutions to singular Dirichlet boundary value problems involving φ -Laplacian. For non-negative nonlinearity f = f ( t , s ) satisfying f ( t , 0 ) ≢ 0 , the existence of an unbounded solution component is shown. By investigating the shape of the component depending on the behavior of f at ∞ , the existence, nonexistence and multiplicity of positive solutions are studied.

Typical long-time behavior of ground state-transformed jump processes

We consider a class of Lévy-type processes derived via a Doob transform from Lévy processes conditioned by a control function called potential. These ground state transformed processes (also called [Formula: see text]-processes) have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise upper envelopes for the almost sure long-time behavior of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.

The size of the boundary in first-passage percolation

First-passage percolation is a random growth model defined using i.i.d. edge-weights $(t_{e})$ on the nearest-neighbor edges of $\mathbb{Z}^{d}$. An initial infection occupies the origin and spreads along the edges, taking time $t_{e}$ to cross the edge $e$. In this paper, we study the size of the boundary of the infected (“wet”) region at time $t$, $B(t)$. It is known that $B(t)$ grows linearly, so its boundary $\partial B(t)$ has size between $ct^{d-1}$ and $Ct^{d}$. Under a weak moment condition on the weights, we show that for most times, $\partial B(t)$ has size of order $t^{d-1}$ (smooth). On the other hand, for heavy-tailed distributions, $B(t)$ contains many small holes, and consequently we show that $\partial B(t)$ has size of order $t^{d-1+\alpha }$ for some $\alpha >0$ depending on the distribution. In all cases, we show that the exterior boundary of $B(t)$ [edges touching the unbounded component of the complement of $B(t)$] is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $B(t)$, we show the inequality $\#\partial B(t)\leq (\log t)^{C}t^{d-1}$ for all large $t$.

Addendum to ‘Statistical thermodynamics and magnetic moments of Landau quantized group VI dichalcogenides’

This addendum addresses the issue of convergence/existence of the basic statistical thermodynamic functions of Dirac-like materials, which involve an apparently unbounded negative energy spectrum component.

Baker's conjecture for functions with real zeros

Baker's conjecture states that a transcendental entire function of order less than $1/2$ has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1. Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1.