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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

Droplet impact and penetration into the paper-like medium are essential physical phenomena in variety of natural and industrial processes. The lattice Boltzmann model coupled with random-walk-based stochastic scheme is presented to calculate the interactions between inertial dominated droplet and fibrous medium. Results show that the droplet spreading regime is independent of surface wettability, volume fraction, and Ohnesorge number at very early impact stage. For fibrous medium with volume fraction ~ 50% and wettability 85°, three stages of penetration process are observed. Whereas for low-volume-fraction fibrous medium, the phases of droplet spreading and penetration are coupled which results in a small spreading width for hydrophilic wettability. It is found that the spaces of spreading width and penetration rate are divided by the curve of $$ Oh = 0.038 $$ and $$ Re = 144 $$ . The effects of Ohnesorge and Reynolds numbers largely influence the behaviors of droplet spreading and penetration process. The in-depth insight of inkjet droplet impact onto the paper-like fibrous medium is beneficial for improving printed products in Paper Electronics.

In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

In this paper, we consider precoder designs for multiuser multiple-input-single-output (MISO) broadcasting channels. Instead of using a traditional linear zero-forcing (ZF) precoder, we propose a generalized ZF (GZF) precoder in conjunction with successive dirty-paper coding (DPC) for data-transmissions, namely, the GZF-DP precoder, where the suffix \lq{}DP\rq{} stands for \lq{}dirty-paper\rq{}. The GZF-DP precoder is designed to generate a band-shaped and lower-triangular effective channel $\vec{F}$ such that only the entries along the main diagonal and the $\nu$ first lower-diagonals can take non-zero values. Utilizing the successive DPC, the known non-causal inter-user interferences from the other (up to) $\nu$ users are canceled through successive encoding. We analyze optimal GZF-DP precoder designs both for sum-rate and minimum user-rate maximizations. Utilizing Lagrange multipliers, the optimal precoders for both cases are solved in closed-forms in relation to optimal power allocations. For the sum-rate maximization, the optimal power allocation can be found through water-filling, but with modified water-levels depending on the parameter $\nu$. While for the minimum user-rate maximization that measures the quality of the service (QoS), the optimal power allocation is directly solved in closed-form which also depends on $\nu$. Moreover, we propose two low-complexity user-ordering algorithms for the GZF-DP precoder designs for both maximizations, respectively. We show through numerical results that, the proposed GZF-DP precoder with a small $\nu$ ($\leq\!3$) renders significant rate increments compared to the previous precoder designs such as the linear ZF and user-grouping based DPC (UG-DP) precoders., Comment: 31 pages, 13 figures, submitted to IEEE Transactions on Wireless Communications in Aug. 2016

Based on symbolic and numeric manipulations, a model simplification technique is proposed in this paper for the linear fractional representation (LFR) and for the differential algebraic representation introduced by Trofino and Dezuo (2013). This representation is needed for computational Lyapunov stability analysis of uncertain rational nonlinear systems. The structure of the parameterized rational Lyapunov function is generated from the linear fractional representation (LFR) of the system model. The developed method is briefly compared to the n-D order reduction technique known from the literature. The proposed model transformations does not affect the structure of Lyapunov function candidate, preserves the well-posedness of the LFR and guarantees that the resulting uncertainty block is at most the same dimensional as the initial one. The applicability of the proposed method is illustrated on two examples.

A theory to analyze complex scenarios facing threats with competing factors and limited resources has been introduced. The scenarios are modeled as closed systems. Hamilton's principle of stationary action is used to conceive a theory in which competing factors dispute available resources to minimize undesirable outcomes. The result indicates that the minimum response is obtained by a combination of the competing factors weighted by their corresponding criticalities. The theory has been applied to the COVID-19 pandemic with two competing factors: Health and Economy. As main result, to minimize the total number of deaths, the recommendation is to balance the emphasis on both factors. This implies to give more emphasis to the economic factor, by avoiding restrict interventions like lockdowns and business closures. The model may evolve from a qualitative to a quantitative status, allowing for computational simulations aimed at validations and forecasting. As such, this approach may become a useful tool for strategic decision-making regarding resources allocations to reduce guessing in scenarios full of uncertainties.

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models., Comment: 36 pages, 8 figures

We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{\zeta(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{\eta(s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form., Comment: 17 pages, no figures

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker–Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker–Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker–Planck equation from a Chapman–Kolmogorov equation, but no proof was offered that a Chapman–Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker–Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker–Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker–Planck pdes and Chapman–Kolmogorov equations for stochastic processes with finite memory.

In this paper we analyze the population dynamics of bacteria competing by anti-bacterial toxins (bacteriocins). Three types of bacteria involved in these dynamics can be distinguished: toxin producers, resistant bacteria and sensitive bacteria. Their interplay can be regarded as a R ock- S cissors- P aper - game (RSP). Here, this is modeled by a reasonable three-dimensional Lotka- Volterra ($L$V) type differential equation system. In contrast to earlier approaches to modeling the RSP game such as replicator equations, all interaction terms have negative signs because the interaction between the three different types of bacteria is purely competitive, either by toxification or by competition for nutrients. The model allows one to choose asymmetric parameter values. Depending on parameter values, our model gives rise to a stable steady state, a stable limit cycle or a heteroclinic orbit with three fixed points, each fixed point corresponding to the existence of only one bacteria type. An alternative model, the May - Leonard model, leads to coexistence only under very restricted conditions. We carry out a comprehensive analysis of the generic stability conditions of our model, using, among other tools, the Volterra-Lyapunov method. We also give biological interpretations of our theoretical results, in particular, of the predicted dynamics and of the ranges for parameter values where different dynamic behavior occurs. For example, one result is that the intrinsic growth rate of the producer is lower than that of the resistant while its growth yield is higher. This is in agreement with experimental results for the bacterium Listeria monocytogenes.

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral ∫ab ∫-11 f(x, y)/x-y dxdy, where (a,b) ≡ (-1,1) or (a,b) ≡ (a,-1) or (a,b) ≡ (1, b) and f(x,y) is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

The class of generalized reflectionless Schrodinger potentials was introduced by Marchenko-Lundina and was analyzed by Kotani. We state and prove various results concerning those stationary ergodic processes of Schrodinger potentials which are contained in this class.

*Correspondence: wadade@se.kanazawa-u.ac.jp 3Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan Full list of author information is available at the end of the article The purpose of this note is to clarify the novelty of the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’ which was published in the J. Inequal. Appl. :,  []. After this paper was published, the authors were informed of the references [–], and [], the results of which partly overlap with those of []. In the paper [], the authors established the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces H n p p,q(R); see Section  in [] for the precise definition of H n p p,q(R). The main theorem in [] is stated as follows. Theorem A [, Theorem .] Let n ∈ N,  < p

V. G. Maz’ya and J. Nagel found for certain classes of weighted Sobolev norms (defined using the Fourier transform) equivalent Slobodeckij-type difference representations. We extend these considerations to a wider class of anisotropic norms which arise in the theory of Markov processes. In particular we show that these Sobolev norms are equivalent to Dirichlet norms.

The authors would like to correct the proof of Theorem 3(a) in the original paper “On the asymptotic behaviour of the 2D Navier-Stokes equations with Navier friction conditions towards Euler equations.” [ZAMM 89(10), 810-822, 2009].

We point out errors and oversights in a paper by Giordano, Laforgia, and Pecaric [3] on inequalities involving Bessel functions.

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

We rebut Kowalenko's claims in 2010 that he proved the irrationality of Euler's constant, and that his rational series for it is new., Comment: 3 pages including a referee's report; added Goldbach's theorem and a reference for it

A hypergeometric transformation formula is developed that simultaneously simplifies and generalizes arguments and identities in a previous paper of Rao et al. [An entry of Ramanujan on hypergeometric series in his notebooks, J. Comput. Appl. Math. 173(2) (2005) 239–246].

The two papers in this issue deal with differential equations, one with the numerical solution of partial differential equations, and the other one with analytic solutions for ordinary differential equations. In his paper "From Functional Analysis to Iterative Methods", Robert Kirby is concerned with linear systems arising from discretizations of partial differential equations (PDEs). Specifically, the PDEs are elliptic and describe boundary value problems; the discretizations are done via finite elements, and at issue is the convergence rate of iterative methods for solving the linear systems. The author's approach is to go back to the underlying variational problem in a Hilbert space, and to make ample use of the Riesz representation theorem. This point of view results in short and elegant proofs, as well as the construction of efficient preconditioners. The general theory is illustrated with two concrete model problems of PDEs for convection diffusion and planar elasticity. This paper will appeal to anybody who has an interest in the numerical solution of PDEs. In 1963 the mathematician/meteorologist Edward Lorenz formulated a system of three coupled nonlinear ordinary differential equations, whose long-term behavior is described by an attractor with fractal structure. You can see a beautiful rendition of the thus named Lorenz attractor on the cover of this issue. Although it is "easy" to plot solutions of the Lorenz system, it is much harder to determine them mathematically. This is what motivated the paper "Complex Singularities and the Lorenz Attractor" by Divakar Viswanath and Sonmez Sahutoglu. Their idea is to allow the time variable to be complex, rather than real; to focus on singular solutions; and to express these singular solutions in terms of so-called psi series. After all is said and done, the authors end up with a two-parameter family of complex solutions to the Lorenz system. This a highly readable and very enjoyable paper, with concrete steps for future research, and connections to thunderstorms and analytic function theory.

In the referenced paper an analytical approach was introduced, which allows one to demonstrate the instability in linearly stable systems, specifically, in a classical three-body problem. These considerations are disproved here.

We disagree with Bucy’s comments (Bucy 1994) on our (Udwadia &; Kalaba 1992) paper. The reasons are as follows. 1. Just as the Gibbs-Appell equations (which use quasi-coordinates) are not the same as the Lagrange equations (which use Lagrange multipliers), the equations of motion obtained by us are not the same as the Gibbs-Appell equations. Some of the steps required to obtain the Gibbs-Appell equations are (Pars 1979; Whittaker 1937): (1) choice of quasi-coordinates; (2) elimination of certain quasicoordinates in terms of other preferred quasi-coordinates; (3) setting up of the Gibbs function; and (4) differentiation of the Gibbs function with respect to the preferred quasi-coordinates. None of these steps is used in obtaining our equations of motion. 2. Yet, the equations of motion obtained by us are equivalent to the Lagrange equations with multipliers (Kalaba et al. 1995) and to the Gibbs-Appell equations (Udwadia & Kalaba 1996). By equivalent we mean each set of equations implies the other. In fact all these different sets of equations are simply different, yet equivalent, ways of stating D’Alembert’s principle for bilinear constraints. Thus what applies to one set of equations applies to the others. Any deficiency in the equations derived in our paper (say regarding rank changes of the matrix A) is therefore present in the Gibbs-Appell equations and Lagrange’s equations as well, because of their equivalence.

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.

Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas’minskii theory and Lyapunov function approach, we construct a critical value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S$$\end{document}R0S corresponding to the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0$$\end{document}R0 of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S>1$$\end{document}R0S>1. Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker–Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S>1$$\end{document}R0S>1. It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted.

In view of the facts in the infection and propagation of COVID-19, a stochastic reaction–diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b 1, …, b n }.

If ƒ(z) is an entire function of order larger than one that satisfies certain additional conditions, characterizations of the growth of its Fourier transform are given, in terms of the growth of ƒ(z) . The results obtained are extensions of theorems of Paley and Wiener, and are a refinement of prior work of Gel'fand and Silov.

Summary Kennett has shown how the equations of elastodynamics may be derived from a variational principle (see also Lanczos, Germain) and attempted to extend this method to the ‘Minor Matrix’ system (Gilbert & Backus). Unfortunately such a direct extension is impossible as will be shown below. However in the case of a medium composed of a series of homogeneous layers, it is possible to construct a stationary expression by a simple transformation. Our discussion will be limited to P-SV or Rayleigh waves in such a medium. References to the equations of Kennett will be denoted, e. g. (K- 4. 1).

An iterative method is proposed for solving the electromagnetic deposition inside lossy inhomogeneous dielectric bodies. The technique uses the conventional method of moments to formulate the problem in matrix form. The resulting system of linear equations is solved iteratively by the method of conjugate gradients. The main advantage of the method is that the iterative procedure does not require the storage of any matrix, thus offering the possibility of solving larger problems compared to conventional inversion or Gaussian elimination schemes. Another important advantage is that monotonic convergence to a solution is ensured and accomplished within a fixed number of iterations, not exceeding the total number of basis functions, independently of the initial guess for the solution. Preliminary examples involving two-dimensional cylinders of fat and muscle are illustrated. The iterative method is expendable and applicable to the three-dimensional case.

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In this paper, we revisit the fundamental problem of computing MIMO capacity under per-antenna power constraint (PAPC). Unlike the sum power constraint counterpart which likely admits water-filling-like solutions, MIMO capacity with PAPC has been largely studied under the framework of generic convex optimization. The two main shortcomings of these approaches are (i) their complexity scales quickly with the problem size, which is not appealing for large-scale antenna systems, and/or (ii) their convergence properties are sensitive to the problem data. As a starting point, we first consider a single user MIMO scenario and propose two provably-convergent iterative algorithms to find its capacity, the first method based on fixed-point iteration and the other based on alternating optimization and minimax duality. In particular, the two proposed methods can leverage the water-filling algorithm in each iteration and converge faster, compared to current methods. We then extend the proposed solutions to multi-user MIMO systems with dirty paper coding (DPC) based transmission strategies. In this regard, capacity regions of Gaussian broadcast channels with PAPC are also computed using closed-form expressions. Numerical results are provided to demonstrate the outperformance of the proposed solutions over existing approaches. European Commission - European Regional Development Fund Science Foundation Ireland

In this paper we analyze two stochastic versions of one of the simplest classes of contagion models, namely so-called SIS models. Several formulations of such models, based on stochastic differential equations, have been recently discussed in literature, mainly with a focus on the existence and uniqueness of stationary distributions. With applicability in view, the present paper uses the Fokker-Planck equations related to SIS stochastic differential equations, not only in order to derive basic facts, but also to derive explicit expressions for stationary densities and further characteristics related to the asymptotic behaviour. Two types of models are analyzed here: The first one is a version of the SIS model with external parameter noise and saturated incidence. The second one is based on the Kramers-Moyal approximation of the simple SIS Markov chain model, which leads to a model with scaled additive noise. In both cases we analyze the asymptotic behaviour, which leads to limiting stationary distributions in the first case and limiting quasistationary distributions in the second case. Finally, we use the derived properties for analyzing the decision problem of choosing the cost-optimal level of treatment intensity.

Considering the influence of environmental capacity and forgetting on rumor spreading, we improve the traditional SIR (susceptible–infected–removed) rumor propagation model and give two dynamic models of rumor propagation in heterogeneous environment and homogeneous environment, respectively. The main purpose of this paper is to make a dynamic analysis of rumor propagation models. In the spatial heterogeneous environment, we have analyzed the uniform persistence of the rumor propagation model and the asymptotic behavior of the positive equilibrium point when the diffusion rate of rumor–susceptible tends to zero. In the spatial homogeneous environment, we discuss the stability of rumor propagation model. Further, optimal control and the necessary optimality conditions are obtained by using the maximum principle. Finally, we study the Hopf bifurcation phenomenon through inducing time delay in the reaction–diffusion model. In addition, the existence of Hopf bifurcation is verified and the influence of diffusion coefficients is studied by numerical simulations.

In this paper, we have considered a deterministic epidemic model with logistic growth rate of the susceptible population, non-monotone incidence rate, nonlinear treatment function with impact of limited hospital beds and performed control strategies. The existence and stability of equilibria as well as persistence and extinction of the infection have been studied here. We have investigated different types of bifurcations, namely Transcritical bifurcation, Backward bifurcation, Saddle-node bifurcation and Hopf bifurcation, at different equilibrium points under some parametric restrictions. Numerical simulation for each of the above-defined bifurcations shows the complex dynamical phenomenon of the infectious disease. Furthermore, optimal control strategies are performed using Pontryagin's maximum principle and strategies of controls are studied for two infectious diseases. Lastly using efficiency analysis we have found the effective control strategies for both cases.