Mild Extra Conditions Research Articles

Non-Projective Two-Weight Codes.

It has been known since the 1970's that the difference of the non-zero weights of a projective Fq-linear two-weight code has to be a power of the characteristic of the underlying field. Here, we study non-projective two-weight codes and, e.g., show the same result under mild extra conditions. For small dimensions we give exhaustive enumerations of the feasible parameters in the binary case.

Threshold dynamics of a switching diffusion SIR model with logistic growth and healthcare resources.

In this article, we have constructed a stochastic SIR model with healthcare resources and logistic growth, aiming to explore the effect of random environment and healthcare resources on disease transmission dynamics. We have showed that under mild extra conditions, there exists a critical parameter, i.e., the basic reproduction number $ R_0^s $, which completely determines the dynamics of disease: when $ R_0^s < 1 $, the disease is eradicated; while when $ R_0^s > 1 $, the disease is persistent. To validate our theoretical findings, we conducted some numerical simulations using actual parameter values of COVID-19. Both our theoretical and simulation results indicated that (1) the white noise can significantly affect the dynamics of a disease, and importantly, it can shift the stability of the disease-free equilibrium; (2) infectious disease resurgence may be caused by random switching of the environment; and (3) it is vital to maintain adequate healthcare resources to control the spread of disease.

Threshold dynamics of stochastic H7N9 model with Markov switching and hybrid strategy

This article aims to study the impact of different environmental states on the transmission dynamics of avian influenza A (H7N9) for poultry by using an epidemic model with general incidence rate and Markov switching. The model couples the essential roles of quarantine and hybrid strategy of vaccination and elimination. By establishing the basic reproduction number R0 as a sharp threshold for the spread of H7N9. When R0<1, the disease almost surely dies out exponentially. We use the compactness of space to prove that when R0>1, the disease almost surely persistent, and this method can be applied to other infectious disease models considering vaccination. In addition, under a mild extra condition, the ergodicity of the Markov process is proved. Our research indicates that: (i) theoretical results and sensitivity analysis show that implementing hybrid strategy is an effective method to control H7N9 propagation; (ii) under the influence of environmental switching, numerical simulations find that the transmission state of H7N9 will neutralize multiple regimes, leading to the possibility of multi wave transmission. This can be used to explain the multiple outbreaks of H7N9 in China.

Polynomial ergodicity of an SIRS epidemic model with density‐dependent demographics

AbstractIn this paper, a stochastic susceptible‐infective‐recovered‐susceptible (SIRS) model with density‐dependent demographics is proposed to study the dynamics of transmission of infectious diseases under stochastic environmental fluctuations. We demonstrate that the position of the basic reproduction number with respect to 1 is the threshold between extinction and persistence of the disease under mild extra conditions. That is, under mild extra conditions, when , the disease is eradicated with probability 1; when , the disease is persistent almost surely and the Markov process has a unique stationary distribution and is polynomial ergodic. As an application, we use the 2017 influenza A data from Western Asia to estimate the parameter values of the model and based on that investigate the effect of random noises on the dynamics of the model. Our study reveals that the basic reproduction number is negatively correlated with the noise intensity for the infected but positively correlated with that for the susceptible population, which are different from the findings obtained in the existing literature.

Epidemic threshold of a COVID-19 model with gaussian white noise and semi-Markov switching

In this study, we investigate the COVID-19 propagation dynamics using a stochastic SIQR model with Gaussian white noise and semi-Markovian switching, focusing on the impacts of Gaussian white noise and semi-Markovian switching on the propagation dynamics of COVID-19. It is suggested that the fate of COVID-19 is entirely determined by the basic reproduction number R0, under mild extra conditions. By making sensitivity analysis on R0, we found that the effect of quarantine rate on R0 was more significant compared to transmission rate. Our results demonstrate that: (i) The presence of Gaussian white noise, while reducing the basic reproduction number R0 of COVID-19, also poses more challenges for the prediction and control of COVID-19 propagation. (ii) The conditional holding time distribution has a significant effect on the kinetics of COVID-19. (iii) The semi-Markov switching and Gaussian white noise can support irregular recurrence of COVID-19 outbreaks.

Instanton sheaves on projective schemes

A h–instanton sheaf on a closed subscheme X of some projective space endowed with an ample and globally generated line bundle OX(h) is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with h–instanton sheaves relating them to Ulrich sheaves. Moreover, we study h–instanton sheaves on smooth curves and surfaces, cyclic n–folds, Fano 3–folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to h–instanton bundles on varieties satisfying some mild extra technical conditions.

Constructible graphs and pursuit

A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite graphs the situation is not well understood.One of our aims in this paper is to give a graph that is cop-win but not constructible. This is the first known such example. We also show that every countable ordinal arises as the rank of some constructible graph, answering a question of Evron, Solomon and Stahl. In addition, we give a finite constructible graph for which there is no construction order whose associated domination map is a homomorphism, answering a question of Chastand, Laviolette and Polat.Lehner showed that every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set). Our other main aim is to investigate how this notion relates to the notion of ‘locally constructible’ (every finite graph is contained in a finite constructible subgraph). We show that, under mild extra conditions, every locally constructible graph is a weak cop win. But we also give an example to show that, in general, a locally constructible graph need not be a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. We also give some open problems.

Constructible Graphs And Pursuit

A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite graphs the situation is not well understood. One of our aims in this paper is to give a graph that is cop-win but not constructible. This is the first known such example. We also show that every countable ordinal arises as the rank of some constructible graph, answering a question of Evron, Solomon and Stahl. In addition, we give a finite constructible graph for which there is no construction order whose associated domination map is a homomorphism, answering a question of Chastand, Laviolette and Polat. Lehner showed that every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set). Our other main aim is to investigate how this notion relates to the notion of `locally constructible' (every finite graph is contained in a finite constructible subgraph). We show that, under mild extra conditions, every locally constructible graph is a weak cop win. But we also give an example to show that, in general, a locally constructible graph need not be a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. We also give some open problems.

Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?

Let M and N be two unital J B ∗ -algebras and let U ( M ) and U ( N ) denote the sets of all unitaries in M and N, respectively. We prove that the following statements are equivalent: M and N are isometrically isomorphic as (complex) Banach spaces; M and N are isometrically isomorphic as real Banach spaces; there exists a surjective isometry Δ : U ( M ) → U ( N ) . We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry Δ : U ( M ) → U ( N ) , we can find a surjective real linear isometry Ψ : M → N which coincides with Δ on the subset e i M s a . If we assume that M and N are J B W ∗ -algebras, then every surjective isometry Δ : U ( M ) → U ( N ) admits a (unique) extension to a surjective real linear isometry from M onto N. This is an extension of the Hatori–Molnár theorem to the setting of J B ∗ -algebras.

The impact of hospital resources and environmental perturbations to the dynamics of SIRS model

Incorporating the environmental perturbations and available resources of the public health system, we construct both deterministic and stochastic models of SIRS type. The deterministic model exhibits very rich dynamics, such as Hopf bifurcation and backward bifurcation which leads to the co-existence of the stable disease-free state and a stable endemic equilibrium. For the stochastic model, we show that under mild extra conditions, if the basic reproduction number is less than one, then the disease will be eradicated almost surely, and if the basic reproduction number is greater than one, the stochastic model will admit a unique ergodic stationary distribution, which implies that the disease persists almost surely. Part of our numerical simulations indicate that: (i) The introduction of environmental perturbations may drift the endemic equilibrium to the disease-free equilibrium, or vice versa; (ii) Increasing available resources is necessary in order to mitigate the infections.

Rings arising from finite tight hypergroups

The notion of a hypergroup (in the sense of [8]) provides a far reaching and meaningful generalization of the concept of a group. Specific classes of hypergroups have given rise to challenging questions and interesting connections to geometric and group theoretic topics; cf. [4], [9], [10], and [14]. The present article is a continuation of a study of residually thin hypergroups which was initiated in [4]. One of the main results of [4] says that finite tight hypergroups, that is hypergroups all elements h of which satisfy hh⁎h={h}, are residually thin; cf. [4, Theorem 7.7]. (Tight hypergroups occur in the study of metathin association schemes; cf. [3,5–7,12,13].) In the present article, we take advantage of this result in order to construct, for each finite tight hypergroup satisfying a mild extra condition, an associative ring. The rings which we find generalize the construction of scheme rings for a certain class of association schemes, including those which correspond to finite groups (group rings).

A stochastic SIRS epidemic model with non-monotone incidence rate under regime-switching

In this paper, we propose and discuss a stochastic SIRS epidemic model with non-monotone incidence rate under regime-switching. First of all, we show that there is a unique positive solution, which is a prerequisite for analyzing the long-term behavior of the stochastic model. Then, a threshold dynamic determined by the basic reproduction number R0s is established: the disease can be eradicated almost surely if R0s<1 and under mild extra conditions, whereas if R0s>1, the densities of the distributions of the solution can converge in L1 to an invariant density by using the Markov semigroups theory. Finally, based on realistic parameters obtained from previous literatures, numerical simulations have been performed to verify our analytical results.

A stochastic SIS epidemic model with saturating contact rate

In this paper, we formulate and investigate the dynamics of a SIS epidemic model with saturating contact rate and its corresponding stochastic differential equation version. For the deterministic epidemic model, we show that the disease-free equilibrium is global asymptotically stable if the basic reproduction number is less than unity; and if the basic reproduction number is greater than unity, model (1.2) admits a unique endemic equilibrium which is locally asymptotically stable by analyzing the corresponding characteristic equations. For the stochastic epidemic model, the existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. The basic reproduction number R0s is proved to be a sharp threshold which determines whether there is an endemic outbreak or not. If R0s<1 and under mild extra conditions, the disease can be eradicated almost surely whereas if R0s>1, it has a stationary distribution which leads to the stochastic persistence of the disease. Finally, numerical simulations are presented to illustrate our theoretical results.

Threshold dynamics of a stochastic SIS epidemic model with nonlinear incidence rate

In this paper, we study a stochastic SIS epidemic model with nonlinear incidence rate. By employing the Markov semigroups theory, we verify that the reproduction number R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α) can be used to govern the threshold dynamics of the studied system. If R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α)>1, we show that there is a unique stable stationary distribution and the densities of the distributions of the solutions can converge in L1 to an invariant density. If R0−σ2Λ2f2(Λμ,0)2μ2(μ+γ+α)<1, under mild extra conditions, we establish sufficient conditions for extinction of the epidemic. Our results show that larger white noise can lead to the extinction of the epidemic while smaller white noise can ensure the existence of a stable stationary distribution which leads to the stochastic persistence of the epidemic.

A model of HBV infection with intervention strategies: dynamics analysis and numerical simulations.

In this paper, we analyze the effect of environment noise on the transmission dynamics of a stochastic hepatitis B virus (HBV) infection model with intervention strategies. By using the Markov semigroups theory, we define the stochastic basic reproduction number and find it can be used to govern disease extinction or persistence. When it is less than one, under a mild extra condition, the stochastic system has a disease-free equilibrium and the disease is predicted to die out with probability one. When it is greater than one, under mild extra conditions, the model admits a stationary distribution which means the persistence of the disease. Thus, we observe that larger intensity of noise (resulting in a smaller stochastic basic reproduction number) can suppress the emergence of hepatitis B outbreak. Numerical simulations are also carried out to investigate the influence of information intervention strategies that may change individual behavior and protect the susceptible from infection. Our analysis shows that the environmental noise can greatly a ect the long-term behavior of the system, highlighting the importance of the role of intervention strategies in the control of hepatitis B.

A stochastic differential equation SIS epidemic model incorporating Ornstein–Uhlenbeck process

In this paper, based on the results of Gray et al. (2011), we propose a new SDE SIS model incorporating mean-reverting Ornstein–Uhlenbeck process, and prove that the stochastic basic reproduction number R0s can be used to identify the stochastic extinction and persistence for the SDE mode: if R0s<1 under mild extra conditions, the disease will be extinct a.s., while if R0s>1, the disease will persist a.s. Epidemiologically, we find that smaller speed of reversion or bigger intensity of volatility can suppress the disease outbreak. Thus, in order to control the spread of the disease, we must decrease the speed of reversion or increase the intensity of volatility.

7. On Nonarchimedean Banach Fields

We study the problem of whether a commutative nonarchimedean Banach ring which is algebraically a field can be topologized by a multiplicative norm. This can fail in general, but it holds for uniform Banach rings under some mild extra conditions. Notably, any perfectoid ring whose underlying ring is a field is a perfectoid field.

Strongly self-absorbing C*-dynamical systems. II

This is a continuation of the study of strongly self-absorbing actions of locally compact groups on C-algebras. Given a strongly self-absorbing action \gamma: G \curvearrowright \mathcal{D} , we establish permanence properties for the class of separable C*-dynamical systems absorbing \gamma tensorially up to cocycle conjugacy. Generalizing results of both Toms–Winter and Dadarlat–Winter, it is proved that the desirable equivariant analogues of the classical permanence properties hold in this context. For the permanence with regard to equivariant extensions, we need to require a mild extra condition on \gamma , which replaces K_1 -injectivity assumptions in the classical theory. This condition turns out to be automatic for equivariantly Jiang–Su absorbing C*-dynamical systems, yielding a large class of examples. It is left open whether this condition is redundant for all strongly self-absorbing actions, and we do consider examples that satisfy this condition but are not equivariantly Jiang–Su absorbing.

On Commutative Nonarchimedean Banach Fields

We study the problem of whether a commutative nonarchimedean Banach ring which is algebraically a field can be topologized by a multiplicative norm. This can fail in general, but it holds for uniform Banach rings under some mild extra conditions. Notably, any perfectoid ring whose underlying ring is a field is a perfectoid field.

From the divergence between two measures to the shortest path between two observables

We consider two independent and stationary measures over $\unicode[STIX]{x1D712}^{\mathbb{N}}$, where $\unicode[STIX]{x1D712}$ is a finite or countable alphabet. For each pair of $n$-strings in the product space we define $T_{n}^{(2)}$ as the length of the shortest path connecting one of them to the other. Here the paths are generated by the underlying dynamic of the measures. If they are ergodic and have positive entropy we prove that, for almost every pair of realizations $(\mathbf{x},\mathbf{y})$, $T_{n}^{(2)}/n$ is concentrated in one, as $n$ diverges. Under mild extra conditions we prove a large-deviation principle. We also show that the fluctuations of $T_{n}^{(2)}$ converge (only) in distribution to a non-degenerate distribution. These results are all linked to a quantity that computes the similarity between those two measures. This is the so-called divergence between two measures, which is also introduced. Several examples are provided.