Continuation Theorem Research Articles

Dynamical Relation of Poisson Spike Trains In Hodkin-Huxley Neural Ion Current Model and Formation of Non-Canonical Bases, Islands, and G-quadruplexes in DNA, mRNA, and RNA at or near the Transcription.

Ground breaking application of mathematics and biochemistry to explain formation of non-canonical bases, islands, G-quadruplex structures, and analog bases in DNA and mRNA at or near the transcription with connection to neural networks is implemented using statistical and stochastic methods apparatus with the addition of quantum principles. As a result the usual transience of Poisson spike trains (PST) becomes very instrumental tool for finding periodical type of solutions to Fokker-Plank (FP) stochastic differential equation (SDE). The present study develops new multidimensional methods of finding solutions to SDE. This is based on more rigorous approach to mathematical apparatus through Kolmogorov-Chentsov continuity theorem (KCCT) that allows the stochastic processes with jumps under certain conditions to have -Holder continuous modification, which is used as basis for finding analogous parallels in neural networks and dynamics of formation of CpG and non-CpG islands (CpGI or non CpGI), non-canonical bases, and structures involving G-quadruplexes during DNA transcription.

On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response

UDC 517.9 In this study, the two-dimensional predator-prey system with Beddington–DeAngelis type functional response with impulses is considered in a periodic environment. For this special case, necessary and sufficient conditions are found for the considered system when it has at least one -periodic solution. This result is mainly based on the continuation theorem in the coincidence degree theory and to get the globally attractive -periodic solution of the given system, an inequality is given as the necessary and sufficient condition by using the analytic structure of the system.

Periodic Solution of Impulsive Predator-Prey Model with Stage Structure for the Prey undercrowding effect

The theoretical aim of this paper is to study the existence of periodic solution in a predator-prey model with stage structure for the prey under the corresponding impulsive perturbations and undercrowding effect. For this purpose, Mawhin’s continuation theorem has been applied. As the result, the researcher of the paper found a sufficient condition of the existence of periodic solution. Finally, under impulsive control, numerical simulations are given to substantiate this theoretical result. Meanwhile, the phase portrait of this numerical simulations can illustrate various dynamical behaviors of the system.

The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications

We root this tribute to Nicholas Yannelis in Chapter II of his 1983 Rochester Ph.D. dissertation, and in his 1983 paper with Prabhakar: this work strengthens the lower semicontinuity assumption of Michael’s continuous selection theorem to open lower sections, and leads to correspondences defined on a paracompact space with values on a Hausdorff linear topological space. We move beyond the literature to provide a necessary and sufficient condition for upper semi-continuous local and global selections of correspondences, and apply our result to four domains of Yannelis’ contributions: Berge’s maximum theorem, the Gale–Nikaido–Debreu lemma, the Sonnenschein–Shafer non-transitive setting, and the Anderson–Khan–Rashid approximate existence theorem. The last also resonates with Chapter VI of Yannelis’ dissertation, and allows a more general framing of the pioneering application of the paracompactness condition to his current and ongoing work in mathematical economics.

Stability analysis for quaternion-valued inertial memristor-based neural networks with time delays

Quaternion-valued inertial memristor-based neural networks with time-varying delays are found to be effective in demonstrating complex dynamic behaviors. Considering the stability of the periodic solution for a novel neural network, we construct a class quaternion-valued inertial memristor-based neural network model by combining with the cellular neural networks and the inertial item with time-varying delay. Based on the continuation theorem of Mawhin’s coincidence degree theory, differential inclusion theory, and inequality techniques, some sufficient criteria for the existence of periodic solutions are obtained. Meanwhile, through constructing a novel Lyapunov functional method, several sufficient conditions have been derived to ensure the global exponential stability of periodic solutions. At last, an example is provided to illustrate the obtained theoretical results of the quaternion-valued inertial memristor-based neural networks model.

Erratum to “The Hölder continuous subsolution theorem for complex Hessian equations”

Erratum to “The Hölder continuous subsolution theorem for complex Hessian equations”

Anti-periodic behavior for quaternion-valued delayed cellular neural networks

In this manuscript, quaternion-valued delayed cellular neural networks are studied. Applying the continuation theorem of coincidence degree theory, inequality techniques and a Lyapunov function approach, a new sufficient condition that guarantees the existence and exponential stability of anti-periodic solutions for quaternion-valued delayed cellular neural networks is presented. The obtained results supplement some earlier publications that deal with the anti-periodic solutions of quaternion-valued neural networks with distributed delay or impulse or state-dependent delay or inertial term. Computer simulations are displayed to check the derived analytical results.

Multiple results to $$\phi $$-Laplacian singular Liénard equation and applications

In this paper, we investigate the existence of a positive periodic solution for the following $$\phi $$ -Laplacian generalized Lienard equation with a singularity: $$\begin{aligned} (\phi (u'))'+f(t,u)u'+ \frac{b(t)}{u^\rho }=h(t)u^m, \end{aligned}$$ where $$\rho $$ is a positive constant and m is a constant. Our proof is based on the Manasevich–Mawhin continuation theorem, and the results are applicable to weak and strong singularities of attractive type (or repulsive type, attractive-repulsive type). Besides, these results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. As applications, we study the existence of positive periodic solutions for a Rayleigh–Plesset equation. Finally, three examples are given to show applications of these theorems.

Families of periodic orbits: Closed 1-forms and global continuability

We investigate global continuation of periodic orbits of a differential equation depending on a parameter, assuming that a closed 1-form satisfying certain properties exists. We begin by extending the global continuation theory of Alexander, Alligood, Mallet-Paret, Yorke, and others to this situation, formulating a new notion of global continuability and a new global continuation theorem tailored for this situation. In particular, we show that the existence of such a 1-form ensures that local continuability of periodic orbits implies global continuability. Using our general theory, we then develop continuation-based techniques for proving the existence of periodic orbits. In contrast to previous work, a key feature of our results is that existence of periodic orbits can be proven (i) without finding trapping regions for the dynamics and (ii) without establishing a priori upper bounds on the periods of orbits. We illustrate the theory in examples inspired by the synthetic biology literature.

Stress intensity factors for cusp-type crack problem under mechanical and thermal loading

Abstract The general solutions of the stress intensity factors (SIFs) for a cusp-type crack problem under remote uniform mechanical and thermal loads are presented in this work. According to the complex variable theory and the method of conformal mapping, a symmetric airfoil crack is mapped onto a unit circle, and both the temperature and stress potentials are used to solve the relevant boundary-value problems. By introducing the auxiliary function and applying the analytical continuation theorem, the SIFs at the cusp-type crack tip can be analytically determined. The obtained SIF results are dependent on the geometric configurations of the cusp-type crack components and the magnitudes of the mechanical and thermal loads. For some combinations of combined loads, the SIF is maximized, and the system has a high risk of damage.

General Mapping Between Complex Spatial and Temporal Frequencies by Analytical Continuation

This article introduces a general technique for intermapping the complex spatial frequency (or propagation constant) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma =\alpha +j\beta $ </tex-math></inline-formula> and the temporal frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega = \omega _{\text {r}}+j\omega _{\text {i}}$ </tex-math></inline-formula> of arbitrary electromagnetic media and structures. This technique is based on the analytic property of complex functions that describe physical phenomena; it invokes the analytic continuity theorem to assert the unicity of the mapping function and builds this function from known data within a restricted domain of its analyticity by curve fitting to a generic polynomial expansion. It is not only applicable to canonical problems admitting an analytical solution but also to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">general</i> medium or structure problems, from eigen- or driven-mode full-wave simulation results. The proposed technique is demonstrated for several systems, namely, an unbounded lossy medium, a dielectric-filled rectangular waveguide, a periodically loaded transmission line, a 1-D photonic crystal, and a series-fed patch (SFP) leaky-wave antenna (LWA), and it is validated either by analytical results or by full-wave simulated results.

Uniqueness and increasing stability in electromagnetic inverse source problems

In this paper we study the uniqueness and the increasing stability in the inverse source problem for electromagnetic waves in homogeneous and inhomogeneous media from boundary data at multiple wave numbers. For the unique determination of sources, we consider inhomogeneous media and use tangential components of the electric field and magnetic field at the boundary of the reference domain. The proof relies on the Fourier transform with respect to the wave numbers and the unique continuation theorems. To study the increasing stability in the source identification, we consider homogeneous media and measure the absorbing data or the tangential component of the electric field at the boundary of the reference domain as additional data. By using the Fourier transform with respect to the wave numbers, explicit bounds for analytic continuation, Huygens' principle and bounds for initial boundary value problems, increasing (with larger wave numbers intervals) stability estimate is obtained.

Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for -quasiconformal harmonic mappings of unit disk

UDC 517.51 We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem, and coefficient estimate for the classes of -quasiconformal harmonic mappings from the unit disk onto itself.

Effective estimation of some oscillatory integrals related to infinitely divisible distributions

We present a practical framework to prove, in a simple way, two-term asymptotic expansions for Fourier integrals $$\begin{aligned} {{\mathcal {I}}}(t) = \int _{{\mathbb {R}}}(\mathrm{e}^{it\phi (x)}-1) \mathop {}\!\mathrm {d}\mu (x), \end{aligned}$$where \(\mu \) is a probability measure on \({{\mathbb {R}}}\) and \(\phi \) is measurable. This applies to many basic cases, in link with Levy’s continuity theorem. We present applications to limit laws related to rational continued fraction coefficients.

Decidable fan theorem and uniform continuity theorem with continuous moduli

AbstractThe uniform continuity theorem states that every pointwise continuous real‐valued function on the unit interval is uniformly continuous. In constructive mathematics, is strictly stronger than the decidable fan theorem , but Loeb [17] has shown that the two principles become equivalent by encoding continuous real‐valued functions as type‐one functions. However, the precise relation between such type‐one functions and continuous real‐valued functions (usually described as type‐two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus of a continuous real‐valued function on [0, 1], and show that real‐valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that is equivalent to the statement that every pointwise continuous real‐valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that is equivalent to a similar principle for real‐valued functions on the Cantor space . These results extend Berger's [2] characterisation of for integer‐valued functions on and unify some characterisations of in terms of functions having continuous moduli.

APPLICATION OF MULTIDIMENSIONAL TIME MODEL TO DYNAMICAL RELATION OF POISSON SPIKE TRAINS IN NEURAL ION CURRENT MODELS AND FORMATION OF NON-CANONICAL BASES, ISLANDS, AND G-QUADRUPLEXES IN DNA, MRNA, AND RNA AT OR NEAR THE TRANSCRIPTION

Ground breaking application of mathematics and biochemistry to explain formation of non-canonical bases, islands, G-quadruplex structures, and analog bases in DNA and mRNA at or near the transcription with connection to neural networks is implemented using statistical and stochastic methods apparatus with the addition of quantum principles. As a result the usual transience of Poisson spike trains (PST) becomes very instrumental tool for finding periodical type of solutions to Fokker-Plank (FP) stochastic differential equation (SDE). The present study develops new multidimensional methods of finding solutions to SDE. This is based on more rigorous approach to mathematical apparatus through Kolmogorov-Chentsov continuity theorem (KCCT) that allows the stochastic processes with jumps under certain conditions to have γ-Holder continuous modification, which is used as basis for finding analogous parallels in dynamics of formation of CpG and non-CpG islands (CpGI or non CpGI), repeats of G-quadruplexes, and non canonical bases during DNA (de)- methylation and neural networks.

Equidistribution of random walks on compact groups II. The Wasserstein metric

We consider a random walk Sk with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum ∑ k=1Nf(Sk) with Hölder continuous test functions f, including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided that the distribution of Sk converges to the Haar measure in the p-Wasserstein metric fast enough. As an example, we construct discrete random walks on an irrational lattice on the torus Rd/Zd, and find their precise rate of convergence to uniformity in the p-Wasserstein metric. The proof uses a new Berry–Esseen type inequality for the p-Wasserstein metric on the torus, and the simultaneous Diophantine approximation properties of the lattice. These results complement the first part of this paper on random walks with an absolutely continuous component and quantitative ergodic theorems for Borel measurable test functions.

A Kolmogorov-type theorem for stochastic fields

We generalize the Kolmogorov continuity theorem and prove the continuity of a class of stochastic fields with the parameter. As an application, we derive the continuity of solutions for nonlocal stochastic parabolic equations driven by non-Gaussian Lévy noises.

Controllability for semilinear heat equation with globally Lipschitz nonlinearities and memory term

In this paper, we investigate the controllability for a semilinear heat equation with memory and internal control in a bounded domain of Rn. The semilinearity has gradient terms and is globally Lipschitzian. Since memory term does not allow applying Fabre’s unique continuation theorem directly, we made certain adjustments to the terms of the equation. Additionally, the proof of the controllability uses a variant of a classic fixed point method and a technique due to Zuazua and Fabre-Puel-Zuazua.

Degree, quaternions and periodic solutions.

The paper computes the Brouwer degree of some classes of homogeneous polynomials defined on quaternions and applies the results, together with a continuation theorem of coincidence degree theory, to the existence and multiplicity of periodic solutions of a class of systems of quaternionic valued ordinary differential equations. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.