On impulsive semidynamical systems: Relation between topological entropy and various types of expansivity

In this paper, we generalize the notion of expansivity, C-W expansivity and N-expansivity for impulsive dynamical systems and we prove C-W expansivity implies positive [Formula: see text]-topological entropy. Also, we define new version of topological entropy for impulsive semiflows and we can bound this topological entropy of expansive impulsive semiflows from lower by the exponential growth rate of periodic orbits.

A 3D extension of pantographic geometries to obtain metamaterial with semi-auxetic properties

In this work we present a three-dimensional extension of pantographic structures and describe its properties after homogenization of the unit cell. Here we rely on a description involving only the first gradient of displacement, as the semi-auxetic property is effectively described by first-order stiffness terms. For a homogenization technique, discrete asymptotic expansion is used. The material shows two positive ([Formula: see text]) and one negative Poisson’s ratios ([Formula: see text]). If, on the other hand, we assume inextensible Bernoulli beams and perfect pivots, we find a vanishing stiffness matrix, suggesting a purely higher gradient material.

Payoff-dependence learning ability resolves social dilemmas

Understanding the appearance and maintenance of cooperation behavior is one of the most interesting challenges in natural and social sciences. Evolutionary game is a useful tool to study this issue. Here, we consider a basic strategy updating rule: the probability of a player updating its strategy is affected by the learning ability, which is determined by payoffs and an aspiration parameter [Formula: see text]. For positive [Formula: see text], learning ability is directly proportional to player’s own payoff. When [Formula: see text] equals 0, it returns to traditional situation. It is found that increasing the value of [Formula: see text] can promote the cooperation. With the increase of [Formula: see text], the player’s learning ability is continuously enhanced, and the probability of changing strategies is also increased. This paper verifies the influence of the introduced selection parameter [Formula: see text] on the cooperation rate from different aspects. We tested this hypothesis through the Monte Carlo simulation, and demonstrated that introducing [Formula: see text] changed the network of interaction effectively, therefore changing the effect of the adoption of the strategy on the uncertainty of cooperation evolution. This paper analyzed the results of the payoff-dependence learning ability of different players when they imitate the strategies of their opponents, which can effectively promote the evolution of cooperation.

Norm approximation by Taylor polynomials in Hardy and Bergman spaces

For positive [Formula: see text] and real [Formula: see text] let [Formula: see text] denote the weighted Bergman spaces of the unit ball [Formula: see text] introduced in [R. Zhao and K. Zhu, Theory of Bergman Spaces on the Unit Ball in [Formula: see text], Mémoires de la Société Mathématique de France, Vol. 115 (2008)]. It is well known that, at least in the case [Formula: see text], all functions in [Formula: see text] can be approximated in norm by their Taylor polynomials if and only if [Formula: see text]. In this paper we show that, for [Formula: see text] with [Formula: see text], we always have [Formula: see text] as [Formula: see text], where [Formula: see text] and [Formula: see text] is the [Formula: see text]th Taylor polynomial of [Formula: see text]. We also show that for every [Formula: see text] in the Hardy space [Formula: see text], [Formula: see text], we always have [Formula: see text] as [Formula: see text], where [Formula: see text]. This generalizes and improves a result in [J. McNeal and J. Xiong, Norm convergence of partial sums of [Formula: see text] functions, Internat. J. Math. 29 (2018) 1850065, 10 pp.].

General Schwarz Lemmata and their applications

We prove estimates interpolating the Schwarz Lemmata of Royden–Yau and the ones recently established by the author. These more flexible estimates provide additional information on (algebraic) geometric aspects of compact Kähler manifolds with nonnegative holomorphic sectional curvature, nonnegative [Formula: see text] or positive [Formula: see text].

Black hole production in the presence of a maximal momentum in horizon wave function formalism

We study the Horizon Wave Function (HWF) description of a generalized uncertainty principle (GUP) black hole in the presence of two natural cutoffs as a minimal length and a maximal momentum. This is motivated by a metric which allows the existence of sub-Planckian black holes, where the black hole mass [Formula: see text] is replaced by [Formula: see text]. Considering a wave-packet with a Gaussian profile, we evaluate the HWF and the probability that the source might be a (quantum) black hole. By decreasing the free parameter, the general form of probability distribution, [Formula: see text], is preserved, but this resulted in reducing the probability for the particle to be a black hole accordingly. The probability for the particle to be a black hole grows when the mass is increasing slowly for larger positive [Formula: see text], and for a minimum mass value it reaches to [Formula: see text]. In effect, for larger [Formula: see text] the magnitude of [Formula: see text] and [Formula: see text] increases, matching with our intuition that either the particle ought to be more localized or more massive to be a black hole. The scenario undergoes a change for some values of [Formula: see text] significantly, where there is a minimum in [Formula: see text], so this expresses that every particle can have some probability of decaying to a black hole. In addition, for sufficiently large [Formula: see text], we find that every particle could be fundamentally a quantum black hole.

Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation

It is shown that the uniform radius of spatial analyticity [Formula: see text] of solutions at time [Formula: see text] to the KdV equation cannot decay faster than [Formula: see text] as [Formula: see text] given initial data that is analytic with fixed radius [Formula: see text]. This improves a recent result of Selberg and da Silva, where they proved a decay rate of [Formula: see text] for arbitrarily small positive [Formula: see text]. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear [Formula: see text] estimates associated with the KdV equation.

The equation (w + x + y + z)(1/w + 1/x + 1/y + 1/z) = n

Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Compos. 61(203) (1993) 117–130] investigated the Diophantine problem of representing integers [Formula: see text] in the form [Formula: see text] for rationals [Formula: see text]. For fixed [Formula: see text], the equation represents an elliptic curve, and the existence of solutions depends upon the rank of the curve being positive. They observed that the corresponding equation in four variables, the title equation here (representing a surface), has infinitely many solutions for each [Formula: see text], and remarked that it seemed plausible that there were always solutions with positive [Formula: see text] when [Formula: see text]. This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when [Formula: see text] is of the form [Formula: see text], [Formula: see text], where [Formula: see text]. Computations within our range seem to indicate that solutions exist for all other values of [Formula: see text].

DOMINIONS AND PRIMITIVE POSITIVE FUNCTIONS

LetA≤Bbe structures, and${\cal K}$a class of structures. An elementb∈BisdominatedbyArelative to${\cal K}$if for all${\bf{C}} \in {\cal K}$and all homomorphismsg,g':B → Csuch thatgandg'agree onA, we havegb=g'b. Our main theorem states that if${\cal K}$is closed under ultraproducts, thenAdominatesbrelative to${\cal K}$if and only if there is a partial functionFdefinable by a primitive positive formula in${\cal K}$such thatFB(a1,…,an) =bfor somea1,…,an∈A. Applying this result we show that a quasivariety of algebras${\cal Q}$with ann-ary near-unanimity term has surjective epimorphisms if and only if$\mathbb{S}\mathbb{P}_n \mathbb{P}_u \left( {\mathcal{Q}_{{\text{RSI}}} } \right)$has surjective epimorphisms. It follows that if${\cal F}$is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by${\cal F}$has surjective epimorphisms.