Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates

In this paper, two delayed SEIR epidemic models with continuous and impulsive vaccination and saturating incidence are investigated. The dynamical behaviors of the disease are analyzed. For continuous vaccination, we obtain a basic reproductive number _R1$R_1$ and prove that if _R1≤1$R_1\le 1$ then the disease-free equilibrium is globally attractive and if _R1>1$R_1>1$ then the disease is permanent by using the Lyapunov functional method. For impulsive vaccination, we obtain two thresholds R^∗$R^{\ast }$ and _R∗$R_{\ast }$ and prove that if R^∗<1$R^{\ast }<1$ then the disease-free periodic solution is globally attractive and if _R∗>1$R_{\ast }>1$ then the disease is permanent by using the comparison theorem of impulsive differential equation and the Lyapunov functional method. Lastly, we compared the effects of two vaccination strategies.     

Weighted power means of q-rung orthopair fuzzy information and their applications in multiattribute decision making

Weighted power means with weights and exponents serving as their parameters are generalizations of arithmetic means. Taking into account decision makers' flexibility in decision making, each attribute value is usually expressed by a $q$-rung orthopair fuzzy value ($q$-ROFV, $q\ge 1$), where the former indicates the support for membership, the latter support against membership, and the sum of their $q$th powers is bounded by one. In this paper, we propose the weighted power means of $q$-rung orthopair fuzzy values to enrich and flourish aggregations on $q$-ROFVs. First, the $q$-rung orthopair fuzzy weighted power mean operator is presented, and its boundedness is precisely characterized in terms of the power exponent. Then, the $q$-rung orthopair fuzzy ordered weighted power mean operator is introduced, and some of its fundamental properties are investigated in detail. Finally, a novel multiattribute decision making method is explored based on developed operators under the $q$-rung orthopair fuzzy environment. A numerical example is given to illustrate the feasibility and validity of the proposed approach, and it is shown that the power exponent is an index suggesting the degree of the optimism of decision makers.     

Rational Solutions of Riccati-like Partial Differential Equations

When factoring linear partial differential systems with a finite-dimensional solution space or analysing symmetries of nonlinear ODEs, we need to look for rational solutions of certain nonlinear PDEs. The nonlinear PDEs are called Riccati-like because they arise in a similar way as Riccati ODEs. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie’s system { ∂_xu + u² + a₁u + a₂v + a₃, ∂_yu + uv + b₁u + b₂v + b₃, ∂_xv + uv + c₁u + c₂v + c₃, ∂_yv + v² + d₁u + d₂v + d₃},where a₁, . . . , d₃are rational functions of x and y.     

Ein Verfahren zur Identifizierung von Zeichen,deren Wiedergabe stationären statischen Störungen unterworfen ist

Zusammenfassung SeiE={e ₁, ...,e _m } eine Menge zufälliger Ereignissee _i , die mit den WahrscheinlichkeitenP _i auftreten. Auf ein Ereignise _i folgt ein Ereignis aus einer weiteren Ereignismenge mit einer vone _i bedingten Wahrscheinlichkeit. wird ein Zufallsvektor x zugeordnet. Es wird angenommen, daß fürP _i , den Mittelwert und die zentralen Momente zweiter Ordnung bezüglich der bedingten Wahrscheinlichkeit Schätzwerte erhalten werden können. Ferner wird die Hypothese gemacht, daß die bedingte Wahrscheinlichkeit durch eine Dichtefunktionf _i (d _i ) beschrieben werden kann, welche nur von einer positiv definiten quadratischen Formd _i ² (x) abhängt. Dann wird gezeigt, daß für beliebigef _i die quadratische Form durch den Mittelwert und die zentralen Momente zweiter Ordnung bestimmt ist. Weiterhin wird angenommen, daß eine Schätzfunktion für die eindimensionale Dichtefunktionf _i (d) erhalten werden kann. Wird irgendein x vorgelegt, so kann dann nach Wahrscheinlichkeit entschieden werden, welchese _i vor x erfolgt ist.

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Summary LetE={e ₁, ...,e _m } be a set of random eventse _i occuring with probabilitiesP _i . An evente _i is followed by an event of another set of events with a probability conditional one _i . To is assigned a random vector x. It is assumed that estimates are obtainable forP _i , the mean, and the central second order moments with respect to the conditional probability. Further, the hypothesis is made that the conditional probability can be described by a density functionf _i (d _i ) which depends only on a positive definite quadratic formd _i ² (x). Then it is shown that for arbitraryf _i the quadratic form is determined by the mean and the central second order moments. Let an estimate for the one-dimensional density functionf _i (d) be known. Then, if any x is presented it can be decided in probability whiche _i preceded x.

     

Digital Steiner sets and Matheron semi-groups

There are various ways to define digital convexity in Zⁿ$Z^n$. The proposed approach focuses on structuring elements (and not the sets under study), whose digital versions should allow to construct hierarchies of operators satisfying Matheron semi-groups law _γλγμ=_γmax(λ,_μ)${\gamma }_{\lambda }{\gamma }_{\mu }={\gamma }_{\max (\lambda \text{,}\mu )}$, where λ$\lambda$ is a size factor. In Rⁿ$R^n$ the convenient class is the Steiner one. Its elements are Minkowski sums of segments. We prove that it admits a digital equivalent when the segments of Zⁿ$Z^n$ are Bezout. The conditions under which the Steiner sets are convex in Zⁿ$Z^n$, and are connected, are established. The approach is then extended to structuring elements that vary according to the law of perspective, and also to anamorphoses, so that the digital Steiner class and its properties can extend to digital spaces as a sphere or a torus.     

Cyclic codes over R = Fp + uFp +?+ uFp with length pn

In this paper, all cyclic codes with length p^sn, (n prime to p) over the ring R = F_p + uF_p +?+ u^k−1F_p are classified. It is first proved that Tor_j(C) is an ideal of , so that the structure of ideals over extension ring S_u^k(m,ω)=GR(u^k,m)[ω]/〈ω^ps-1〉 is determined. Then, an isomorphism between R[X]/〈X^N − 1〉 and a direct sum ⊕_h∈IS_u^k(m_h,ω) can be obtained using discrete Fourier transform. The generator polynomial representation of the corresponding ideals over F_p + uF_p +?+ u^k−1F_p is calculated via the inverse isomorphism. Moreover, torsion codes, MS polynomial and inversion formula are described.     

Edge-bipancyclicity of the k-ary n-cubes with faulty nodes and edges

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. In this paper, we investigate the edge-bipancyclicity of k-ary n-cubes with faulty nodes and edges. It is proved that every healthy edge of the faulty k-ary n-cube with f_v faulty nodes and f_e faulty edges lies in a fault-free cycle of every even length from 4 to kⁿ − 2f_v (resp. kⁿ − f_v) if k ? 4 is even (resp. k ? 3 is odd) and f_v + f_e ? 2n − 3. The results are optimal with respect to the number of node and edge faults tolerated.