Unique Limit Research Articles

Інтерполяційна задача у вагових просторах Пелі-Вінера

Yu. Lyubarskii and K. Seip K. (Revista Matematica Iberoamericana, 1997 (13), № 2) found the criterion for the existence of a uniqe solution of the interpolation problem f(λk) = bk in the Paley-Wiener spaces in terms of Makenhoupt’s (Ap) condition of entire functions of the exponential type at most π, where p is a real number more than 1, whose restriction to the real line coincides with the class of functions which module degree number p is integrable on R whith p-norm. These results make it possible to obtain the criterion of the unconditional basisness of the system of exponentials in the space of functions which module degree number p is an integrable on (–π; π). At the same time the sequence (λk) with a unique limit point at the infinity, for which mentioned interpolation problem has a uniqe solution is called a complete interpolating sequence in the Paley-Wiener spaces. For p = 2 those descriptions coincides with those given by Pavlov (1979), Nikolsky (1980), and Minkin (1982). We generalize these results to the weighted spaces of entire functions of the exponential type at most σ with the p-norm (there is a power function with exponent ω as a weight), where σ is a nonnegative real number, ω – real number more then –1. That is, we find conditions for the completeness of the interpolating sequence (λk) in the Paley-Wiener weighted space. Different forms of these conditions are considered. Among them there are Mackenhoupt’s conditions (continuous and discrete (Ap) conditions). There is proved that if (λk) is a complete interpolation sequence in the Paley-Wiener weighted space, then it is a relatively dense set in the space C. Also there constructed an example of a complete interpolating sequence for σ = π

Existence and Uniqueness of Limits at Infinity for Bounded Variation Functions

Existence and Uniqueness of Limits at Infinity for Bounded Variation Functions

Hopf bifurcation and limit cycle of the two‐variable Oregonator model for Belousov–Zhabotinsky reaction

This paper is concerned with a two‐variable Oregonator model for Belousov–Zhabotinsky Reaction. Llibre and Oliveira (2022) proved that Oregonator model with an unstable node or focus has at least one limit cycle in the positive quadrant, and the system has a unique stable limit cycle for some values of the parameters. It is shown in the present paper that the positive equilibrium is not a center, and that if the system has a positive equilibrium which is a weak focus, then its order is at most 2. There exist some parameter values such that system has small limit cycle around the positive equilibrium.

Anti $ T_{2} $-generalized topological spaces

In this paper, we investigated non strong hyperconnected generalized topological spaces. Ekici \cite{Eki11} and Devi \cite{Ren12} have provided the results of hyperconnectedness for strong generalized topological spaces. We generalized these results for arbitrary generalized topological spaces. Through the notion of hyperconnectedness of arbitrary generalized topological spaces, we constructed an example which fails Hausdorff characterization of topological spaces \lq\lq A first countable spaces is Hausdorff if and only if every convergent sequence has unique limit\rq\rq. This example also serves the purpose of constructing Anti Hausdorff Fr$\acute{e}$chet space in which every convergent sequence has unique limit required by Novak in \cite{Nov39}.

Group-wise monitoring of multivariate data with missing values

In this article, we aim to detect the changes in the process that generates multivariate data by monitoring their mean shift. To do this, we utilize a graphical tool known as a multivariate control chart. However, monitoring the mean of multivariate data poses two challenges: the grouped structure of individual observations and the presence of missing values. In this research, we introduce a novel method called HTC for monitoring group-wise multivariate data that includes missing values. HTC offers several advantages over existing methods. First, it is applicable to various types of dependence among individual observations within a group. Second, it provides a unique upper control limit (UCL) regardless of the missing data pattern. Lastly, HTC is computationally more efficient compared to resampling-based techniques. We conduct comprehensive numerical studies to evaluate the performance of the HTC method and compare it with the existing group-wise monitoring method, referred to as HTM. Compared to HTM, HTC achieves a higher true positive rate (TPR) while effectively controlling the in-control false alarm rate ( FAR 0 ) at a pre-determined level across various settings considered in our study. To illustrate its effectiveness, we applied HTC to monitoring multivariate environmental data collected from the manufacturing process of a semiconductor company in Korea.

A simulation study of the ability to detect power distribution perturbations in the texas A&M TRIGA reactor with self-powered neutron detectors

Given the variety of ways that nuclear reactor core power may be perturbed, reactor operators and developers are keen on understanding the accuracy and convergence time during which perturbations in reactor power distribution may be synthesized (i.e., inferred) from an array of in-core radiation detectors. A simulation study was conducted as described herein using a highly detailed model of the Texas A&M Training, Research, Isotopes, General Atomics Reactor, in which an array of self-powered neutron detectors (SPNDs) was considered for input to the power synthesis methodology. The core power synthesis is conducted using a point-based iterative method with an iterative loop built in to ensure working equation consistency. The forward problem of SPND response to simulated perturbations in reactor power was solved for Gaussian peak-type perturbations in the reactor power distribution. These perturbations varied in variance, amplitude, and core location to assess their impact on synthesis error and to determine the number of iterations required for convergence. A relation between the unique resolvability limit and perturbation width was identified such that the maximum synthesis error increased rapidly when the peak width went beneath this limit (a width approximating half the reactor’s fuel pin-to-pin pitch); this resolvability limit is specific to the SPND configuration and fuel segmentation considered herein. The synthesis error increased linearly with perturbation peak amplitude, whereas the convergence time increased nonlinearly. Perturbations located closer to the center of the core were synthesized more accurately, albeit with a higher number of required iterations. These findings provide a qualitative and quantitative understanding of the accuracy and speed at which different types of spatial power perturbations can be resolved in light-water reactors.

Limit cycles in a rotated family of generalized Liénard systems allowing for finitely many switching lines

Analytic rotated vector fields have four significant properties: as the rotated parameter $\alpha$ changes, the amplitude of each stable (or unstable) limit cycle varies monotonically, each semi-stable limit cycle bifurcates at most two limit cycles, the isolated homoclinic loop (if exists) disappears while a unique limit cycle with the same stability arises or no closed orbits arise oppositely, and a unique limit cycle arises near the weak focus (if exists). In this paper, we prove that the four properties remain true for a rotated family of generalized Liénard systems having finitely many switching lines. Furthermore, we discuss variational exponent and use it to formulate multiplicity of limit cycles. Then we apply our results to give exact number of limit cycles to a continuous piecewise linear system with three zones and answer to a question on the maximum number of limit cycles in an SD oscillator.

Continuous dependence of stationary distributions on parameters for stochastic predator–prey models

AbstractThis research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0.

Bifurcations of traveling wave solutions of a generalized Burgers–Fisher equation

We deal with the traveling wave solutions (including periodic wave solutions, solitary wave solutions and kink or antikink solutions) of a generalized Burgers–Fisher equation via studying the dynamics of a related Liénard system. We obtain necessary and sufficient conditions for the Liénard system to have a unique limit cycle, a homoclinic loop and a heteroclinic loop, respectively.

About sequentially open and closed subsets in product spaces

We remind two facts for topological spaces. The one is that in a Hausdorff space $X$ each sequence has a unique limit. This allows us to have a function from the set of all sequences in $X$ to $X$. Another is that in the first countable spaces some topological objects such as open subsets, closed subsets, closures and interiors of the sets, continuous functions and many others can be defined in terms of convergent sequences. 
 
 In this paper we compare these notions with their sequential versions in topological spaces. We will take the product spaces into account and give some results.

Existence and uniqueness of limits at infinity for homogeneous Sobolev functions

We establish the existence and uniqueness of limits at infinity along infinite curves outside a zero modulus family for functions in a homogeneous Sobolev space under the assumption that the underlying space is equipped with a doubling measure which supports a Poincaré inequality. We also characterize the settings where this conclusion is nontrivial. Secondly, we introduce notions of weak polar coordinate systems and radial curves on metric measure spaces. Then sufficient and necessary conditions for existence of radial limits are given. As a consequence, we characterize the existence of radial limits in certain concrete settings.

Collective motion driven by nutrient consumption

A classical problem describing the collective motion of cells, is the movement driven by consumption/depletion of a nutrient. Here we analyze one of the simplest such model written as a coupled Partial Differential Equation/Ordinary Differential Equation system which we scale so as to get a limit describing the usually observed pattern. In this limit the cell density is concentrated as a moving Dirac mass and the nutrient undergoes a discontinuity. We first carry out the analysis without diffusion, getting a complete description of the unique limit. When diffusion is included, we prove several specific a priori estimates and interpret the system as a heterogeneous monostable equation. This allow us to obtain a limiting problem which shows the concentration effect of the limiting dynamics.

Canard phenomena for a slow-fast predator-prey system with group defense of the prey

In this study, we consider that the predator reproduces much slower than the prey and propose a slow-fast predator-prey system with the group defense of the prey, which is described by the Ivlev-type functional response. We analyze the canard phenomena (relaxation oscillation and canard cycles) of our proposed model by applying the singular perturbation theory. Firstly, we apply the blow-up techniques and visualize the behavior near the special fold point, which allows us to prove the existence of the unique attractive limit cycle and imply the appearance of the canard phenomenon when crossing the fold point, and the numerical simulations verify the theoretical results. Then we further use the singular perturbation theory to investigate the existence and cyclicity of the canard cycle without head and the existence of the canard cycle with head. Our main results of the complex canard phenomenon reveal the fact that some certain species in the ecosystem suddenly burst back many years after being about extinct.

On limit measures and their supports for stochastic ordinary differential equations

This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin–Wentzell or Dembo–Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse–Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincaré–Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. Four nontrivial examples admitting a unique limit measure are provided, two of which possess infinite equivalent classes and their supports are Liapunov stable periodic orbits, one is supported on saddle-nodes.

On the cubic Kukles systems with an algebraic limit cycle of degree two

This paper is concerned with cubic Kukles system of the form x˙=−y, y˙=x+λy+a1x2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7y3, where λ,ai∈R, i=1,2,…,7, under the assumption that the above system has an algebraic periodic orbit of degree 2. It is shown that such an algebraic periodic orbit surrounds a center if and only if λ=0. If λ≠0, then it is the unique limit cycle. Moreover, we provide all the possibilities for the global phase portrait in the Poincaré disk of cubic Kukles system with an algebraic periodic orbit of degree 2.

The critical 2d Stochastic Heat Flow

We consider directed polymers in random environment in the critical dimension d = 2, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on {mathbb {R}}^2 with logarithmic correlations, which we call the Critical 2d Stochastic Heat Flow. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.

Hybrid Control of Self-Oscillating Resonant Converters

We describe parallel and series resonant converters (PRC and SRC) via a unified set of input-dependent coordinates whose dynamics are intrinsically hybrid. We then propose hybrid feedback showing a self-oscillating behavior whose amplitude and frequency can be adjusted by a reference input ranging from zero to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\pi $ </tex-math></inline-formula> . For any reference value in that range, we give a Lyapunov function certifying the existence of a unique nontrivial hybrid limit cycle whose basin of attraction is global except for the origin. Our results are confirmed by experimental results on an SRC prototype.

Dynamic analysis of a predator–prey model of Gause type with Allee effect and non-Lipschitzian hyperbolic-type functional response

In this work, we study a predator–prey model of Gause type, in which the prey growth rate is subject to an Allee effect and the action of the predator over the prey is determined by a generalized hyperbolic-type functional response, which is neither differentiable nor locally Lipschitz at the predator axis. This kind of functional response is an extension of the so-called square root functional response, used to model systems in which the prey have a strong herd structure. We study the behavior of the solutions in the first quadrant and the existence of limit cycles. We prove that, for a wide choice of parameters, the solutions arrive at the predator axis in finite time. We also characterize the existence of an equilibrium point and, when it exists, we provide necessary and sufficient conditions for it to be a center-type equilibrium. In fact, we show that the set of parameters that yield a center-type equilibrium, is the graph of a function with an open domain. We also prove that any center-type equilibrium is stable and it always possesses a supercritical Hopf bifurcation. In particular, we guarantee the existence of a unique limit cycle, for small perturbations of the system.

DETERMINATION OF THE ATTERBERG LIMITS USING A FALL CONE DEVICE ON LOW PLASTICITY SILTY SANDS

The Fall cone liquid limit testing procedure for low plasticity soil mixtures with sand, including the sample preparation procedure and the implementation of Fall cone plastic limit determination suggestions are covered within this research. A Fall cone apparatus was used in order to determine the liquid and plastic limits of soil types, for which the Casagrande cup and thread rolling methods proved inapplicable. Several issues are addressed concerning standardized sample mixture preparation and cup filling procedures for liquid limit testing, as well as the applicability of single measurements per moisture content and the effect of curing time on data gain quality. Both liquid and plastic limit testing results show a solid and expected linear trend of high precision. Liquid limit testing results correlate well with the existing data which suggests the Fall cone method as a unique liquid limit testing method for mixtures of low plasticity clays with sand. Plastic limit determination methods results show a deviation from values obtained with the classical Casagrande’s thread rolling method which could be caused by the bias in the tested soil type or apparatus. Test results are presented numerically and graphically and discussed with a focus on the given method applicability for determining Atterberg limits of low plasticity soil mixtures with sand.

Stability and Hopf bifurcation of an epidemiological model with effect of delay the awareness programs and vaccination: analysis and simulation

The present research is concerned with studying the media coverage delay impact and vaccine impact to control outbreaks the infectious disease in the population, we suggested that our mathematical modeling divide the population into three classes: susceptible persons S(t), vaccinated person V(t) and infected persons I(t). The susceptible individuals are also divided due to media programs into awareness and unawareness of disease danger and its mode of transmission. We studied the influence of time delay in response to the media program on awareness of the epidemic. We first discussed the existence of the equilibrium points of the system and obtain sufficient conditions for the local asymptotic stability of all equilibrium points. Further, the existence of Hopf bifurcation is shown near endemic equilibrium. It is interesting to note that there exists at least one limit cycle around the unstable endemic equilibrium. In particular, sufficient conditions for a unique stable limit cycle have been presented. Finally, we verified the feasibility of the results by numerical simulation and give a brief conclusion to generalize that the delay effect of media and vaccination can cause unexpected dynamic predictions for interacting populations.