Uniformly Continuous Research Articles

Modelling the effect of resource subsidy on a two-species predator-prey system under the influence of environmental noises

Our ecosystem is full of various types of food chain systems. Most of the species have more than one food source with various kinds of possible interactions. In this work, we have studied a predator-prey-subsidy model in a rapidly fluctuating environment. As per our consideration predator exploits both a prey population and an allochthonous resource which is provided as a subsidy to the system exogenously, with Holling Type II response functions. Furthermore, we have introduced Gaussian white noise to the main parameters subject to coupling of a prey-predator pair with its environment. We have examined the existence of unique global positive solutions for both the deterministic and stochastic systems. Boundedness, uniform continuity, global attractiveness in mean of solutions of the stochastic system are proved. Conditions for extinction of prey and predator population are derived and the importance of environmental noise as well as subsidy for extinction is discussed through mathematical results and numerical findings. We have derived conditions for persistence of the proposed stochastic system. The results of persistence and extinction have been justified by numerical simulations. Numerically we have found that presence of subsidy can be the cause of survival of predator even there is no prey available for consumption which fact is different from the studies of those systems where additional food is provided for predator. We have also observed that very high input of subsidy can be a cause for extinction of prey population.

Simulation Relations for Abstraction-based Robust Control of Hybrid Dynamical Systems

The paper addresses the problem of robust control of hybrid dynamical systems with respect to linear-time properties. First, three notions of robust controllers are formulated, that capture imperfect measurements, actuation errors and delays, and uncertain dynamics, respectively. Under mild assumptions of uniform continuity on the transition relation, robust controller synthesis problem with respect to uncertain dynamics is shown to be more general than those with respect to measurement errors, actuation errors and delays. Hence, the paper focuses on the latter notion of robust controllers. Next, foundations for abstraction-based robust controller synthesis are explored, and uniformly continuous alternating simulation relations are shown to preserve the existence of robust controllers with respect to uncertain dynamics.

A four-channel air pressure cooperative control system for 3-D bio-printing device

In recent years, the research and development of 3D bioprinting device for artificial blood vessels has attracted great attention of researchers. In this paper, the research object is the control system of 3D bioprinting device for artificial blood vessel and the four-channel air pressure cooperative control technology for 3D bioprinting device double nozzles printing is mainly studied. Through the scheme design, working principle innovation, hardware selection and air pressure output calibration experiment of four-way air pressure cooperative control system, the hardware platform of 3D bioprinting device air pressure control system is built. Then, by using the embedded TwinCAT NC PTP software platform based on EtherCAT Ethernet bus protocol, the four-channel air pressure cooperative control is programmed and field experiments are carried out, which realizes the printing process of uniform liquid output and good continuity, and meets the requirements of 3D bioprinting for printing continuity and forming quality. The study of four-channel air pressure cooperative control system in this course is a worthy of application and promotion control method.

Finite Dimensional Approximations of Hamilton--Jacobi--Bellman Equations in Spaces of Probability Measures

We prove that viscosity solutions of Hamilton--Jacobi--Bellman (HJB) equations, corresponding either to deterministic optimal control problems for systems of $n$ particles or to stochastic optimal control problems for systems of $n$ particles with a common noise, converge locally uniformly to the viscosity solution of a limiting HJB equation in the space of probability measures. We prove uniform continuity estimates for viscosity solutions of the approximating problems which may be of independent interest. We pay special attention to the case when the Hamiltonian is convex in the gradient variable and equations are of first order and provide a representation formula for the solution of the limiting first order HJB equation. We also propose an intrinsic definition of viscosity solution on the Wasserstein space.

Incentive compatible self-fulfilling mechanisms and rational expectations

This paper extends the exact equivalence result between the allocations realized by self-fulfilling mechanisms and rational expectations equilibrium allocations in Forges and Minelli (1997) to a large finite-agent replica economy where different replicas of the same agent are allowed to receive different private information. The first result states that the allocation realized by any incentive compatible self-fulfilling mechanism is an approximate rational expectations equilibrium allocation. Conversely, the second result states that, given any rational expectations equilibrium satisfying a “uniform continuity” condition on the equilibrium price, one can construct an approximate incentive compatible self-fulfilling mechanism whose equilibrium allocation coincides with the rational expectations equilibrium allocation for all non-monetary commodities.

Uniform Continuity of Fractal Interpolation Function

In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper. The result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.

Modelling the fear effect in a two-species predator–prey system under the influence of toxic substances

Effect of anti-predator defense due to fear of predator felt by prey and effect of toxic substance released by external sources on prey–predator system is a serious mater of concern in mathematical biology. In the proposed model we have discussed a prey–predator system in which both the species are infected by environmental toxicant. In our consideration prey species is directly infected by environmental toxicant and predator gets infected by consuming infected prey. Prey’s growth rate is assumed to be affected by fear of predator. In this work the proposed predator–prey model is analyzed in presence of environmental fluctuation, i.e., stochastic analysis of this model is discussed. Using Ito formula: positivity, boundedness, uniform continuity criterion and global attractivity of solutions of this system have been established. Conditions for which the prey as well as the predator goes extinct have been derived. Conditions for persistence of the system have also been discussed. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different levels of toxicant and different levels of fear have been demonstrated by depicting figures in numerical simulation using MATLAB.

Characterization of Absolute and Uniform Continuity

In the present paper, by considering nonlinear integral operators and using their approximations via regular summability methods, we obtain characterizations for some function spaces including the space of absolutely continuous functions, the space of uniformly continuous functions, and their other variants. We observe that Bell-type summability methods are quite effective to generalize and improve some related results in the literature. At the end of the paper, we discuss some special cases and applications.

Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels

In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between and the trace distance between quantum states. Namely, the pre-order attains a minimum over $\epsilon$-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance $\epsilon$, we obtain the path of majorization flow. This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which extends to the other functions, including the $\alpha$-Renyi entropy, for which we obtain novel bounds in the case $\alpha > 1$. We apply this technique to other Schur concave functions, such as the number of connected components of a certain random graph model, and the number of distinct realizations of a random variable. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes one at a time. We characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference after finitely many steps. Additionally, when the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction.

Non-regular g-measures and variable length memory chains

It is well-known that there always exists at least one stationary measure compatible with a continuous g-function g. Here we prove that if the set of discontinuities of a g-function g has null measure under a candidate measure obtained by some asymptotic procedure, then this candidate measure is compatible with g. We explore several implications of this result, and discuss comparisons with the literature concerning assumptions and examples. Important part of the paper is dedicated to the case of variable length memory chains, for which we obtain existence, uniqueness and weak-Bernoullicity (or β-mixing) under new assumptions. These results are specially designed for variable length memory models, and do not require uniform continuity. We also provide a further discussion on some related notions, such as random context processes, non-essential discontinuities and everywhere discontinuous stationary measures.

Some generalized definitions of uniform continuity for real valued functions

The purpose of this study is to give some uniform continuity definitions for real valued functions.

Analysis of a mutualism model with time-related coefficients in a stochastic environment

In this paper, a mutualism model with stochastic perturbations is considered and some of its coefficients are related to time. Under some assumptions, we make efforts to prove the existence and uniqueness of a positive solution, and the asymptotic behavior to the problem is discussed. Furthermore, we also prove the properties of stochastic boundedness, uniform continuity and stochastic permanence of this system. At last, some numerical simulations are introduced to illustrate our main results.

Analysis of Stochastic Predator-Prey Model with Disease in the Prey and Holling Type II Functional Response

A stochastic predator-prey model with disease in the prey and Holling type II functional response is proposed and its dynamics is analyzed. We discuss the boundedness of the dynamical system and find all feasible equilibrium solutions. For the stochastic systems, we obtain the conditions for the existence of the global unique solution, boundedness, and uniform continuity. We derive the conditions for extinction and permanence of species. Moreover, we construct appropriate Lyapunov functions and discuss the asymptotic stability of equilibria. To illustrate our theoretical findings, we have performed numerical simulations and presented the results.

Strong ISS implies strong iISS for time-varying impulsive systems

For time-invariant (nonimpulsive) systems, it is already well-known that the input-to-state stability (ISS) property is strictly stronger than integral input-to-state stability (iISS). Very recently, we have shown that under suitable uniform boundedness and continuity assumptions on the function defining system dynamics, ISS implies iISS also for time-varying systems. In this paper, we show that this implication remains true for impulsive systems, provided that asymptotic stability is understood in a sense stronger than usual for impulsive systems.

Strong convergence of quantum channels: Continuity of the Stinespring dilation and discontinuity of the unitary dilation

It is shown that a sequence {Φn} of quantum channels strongly converges to a quantum channel Φ0 if and only if there exist a common environment for all the channels and a corresponding sequence {Vn} of Stinespring isometries strongly converging to a Stinespring isometry V0 of the channel Φ0. A quantitative description of the above characterization of the strong convergence in terms of appropriate metrics on the sets of quantum channels and Stinespring isometries is given. As a result, the uniform selective continuity of the complementary operation with respect to the strong convergence is established. The discontinuity of the unitary dilation is shown by constructing a strongly converging sequence of quantum channels that cannot be represented as a reduction of a strongly converging sequence of unitary channels. The Stinespring representation of strongly converging sequences of quantum channels allows us to prove the lower semicontinuity of the entropic disturbance as a function of a pair (channel, input ensemble). Some corollaries of this property are considered.

Information-theoretic aspects of the generalized amplitude-damping channel

The generalized amplitude damping channel (GADC) is one of the sources of noise in superconducting-circuit-based quantum computing. It can be viewed as the qubit analogue of the bosonic thermal channel, and it thus can be used to model lossy processes in the presence of background noise for low-temperature systems. In this work, we provide an information-theoretic study of the GADC. We first determine the parameter range for which the GADC is entanglement breaking and the range for which it is anti-degradable. We then establish several upper bounds on its classical, quantum, and private capacities. These bounds are based on data-processing inequalities and the uniform continuity of information-theoretic quantities, as well as other techniques. Our upper bounds on the quantum capacity of the GADC are tighter than the known upper bound reported recently in [Rosati et al., Nat. Commun. 9, 4339 (2018)] for the entire parameter range of the GADC, thus reducing the gap between the lower and upper bounds. We also establish upper bounds on the two-way assisted quantum and private capacities of the GADC. These bounds are based on the squashed entanglement, and they are established by constructing particular squashing channels. We compare these bounds with the max-Rains information bound, the mutual information bound, and another bound based on approximate covariance. For all capacities considered, we find that a large variety of techniques are useful in establishing bounds.

Structure and Functions of Topological Metagroups

In this article, the structure of topological metagroups was investigated. Relations between topological and algebraic properties of metagroups were scrutinized. A uniform continuity of functions on them was studied. Smashed products of topological metagroups were investigated.

Bounds on continuous Scott rank

An analog of Nadel's effective bound for the continuous Scott rank of metric structures, developed by Ben Yaacov, Doucha, Nies, and Tsankov, will be established: Let $\mathscr{L}$ be a language of continuous logic with code $\hat{\mathscr{L}}$. Let $\Omega$ be a weak modulus of uniform continuity with code $\hat{\Omega}$. Let $\mathcal{D}$ be a countable $\mathscr{L}$-pre-structure. Let $\bar{\mathcal{D}}$ denote the completion structure of $\mathcal{D}$. Then $\mathrm{SR}_\Omega(\bar{D}) \leq \omega_1^{\hat{\mathscr{L}}\oplus\hat{\Omega}\oplus\mathcal{D}}$, the Church-Kleene ordinal relative to $\hat{\mathscr{L}}\oplus\hat{\Omega}\oplus\mathcal{D}$.

Compactness in quasi-Banach function spaces with applications to $$L^1$$ of the semivariation of a vector measure

We characterize the relatively compact subsets of the order continuous part $$E_a$$ of a quasi-Banach function space E showing that the strong connection between compactness, uniform absolute continuity, uniform integrability, almost order boundedness and L-weak compactness that appears in the classical setting of Lebesgue spaces remains almost invariant in this new context under mild assumptions. We also present a de la Vallee–Poussin type theorem in this context that allows us to locate each compact subset of $$E_a$$ as a compact subset of a smaller quasi-Banach Orlicz space $$E^\varPhi .$$ Our results generalize the previous known results for the Banach function spaces $$L^1(m)$$ and $$L^1_w(m)$$ associated to a vector measure m and moreover they can also be applied to the quasi-Banach function space $$L^1\left( \Vert m \Vert \right) $$ associated to the semivariation of m.

Relatively uniformly continuous semigroups on vector lattices

In this paper we introduce the notions of relatively uniform continuity and strong continuity with respect to the relatively uniform topology for semigroups on general vector lattices. These notions allow us to study semigroups on non-locally convex spaces such as Lp(R) for 0<p<1 and non-complete spaces such as Lip(R), UC(R), and Cc(R). We provide examples of relatively uniform continuous semigroups such as the (left) translation semigroup and the Ornstein-Uhlenbeck semigroup.