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Relativistic electromagnetic plasma waves are described by a dynamical equation that can be solved not only in terms of plane waves, but for several different accelerating wavepacket solutions. Depending on the spatial and temporal dependence of the plasma frequency, different kinds of accelerating solution can be obtained, for example, in terms of Airy or Weber functions. Also, we show that an arbitrary accelerated wavepacket solution is possible, for example, for a system with a luminal plasma slab.

This paper is devoted to establishing several new formulas relating Bernoulli and Stirling numbers of both kinds.

This paper describes a path integral formulation of the free energy principle. The ensuing account expresses the paths or trajectories that a particle takes as it evolves over time. The main results are a method or principle of least action that can be used to emulate the behaviour of particles in open exchange with their external milieu. Particles are defined by a particular partition, in which internal states are individuated from external states by active and sensory blanket states. The variational principle at hand allows one to interpret internal dynamics - of certain kinds of particles - as inferring external states that are hidden behind blanket states. We consider different kinds of particles, and to what extent they can be imbued with an elementary form of inference or sentience. Specifically, we consider the distinction between dissipative and conservative particles, inert and active particles and, finally, ordinary and strange particles. Strange particles can be described as inferring their own actions, endowing them with apparent autonomy or agency. In short - of the kinds of particles afforded by a particular partition - strange kinds may be apt for describing sentient

It is an established assumption that pattern-based models are good at precision, while learning based models are better at recall. But is that really the case? I argue that there are two kinds of recall: d-recall, reflecting diversity, and e-recall, reflecting exhaustiveness. I demonstrate through experiments that while neural methods are indeed significantly better at d-recall, it is sometimes the case that pattern-based methods are still substantially better at e-recall. Ideal methods should aim for both kinds, and this ideal should in turn be reflected in our evaluations.

Erné weakened the concept of sobriety in order to extend the theory of sober spaces and locally hypercompact spaces to situations where directed joins were missing, and introduced and discussed three kinds of non-sober spaces: cut spaces, weakly sober spaces, and quasisober spaces. Three other kinds of non-sober spaces, namely $\mathsf{DC}$ space, $\mathsf{RD}$ space and $\mathsf{WD}$ space, were introduced and investigated by Xu, Shen, Xi and Zhao. All these six kinds of spaces are strictly weaker than sober spaces. In this paper, it is shown that none of the category of all $\mathsf{DC}$ spaces, that of all $\mathsf{RD}$ spaces, that of all $\mathsf{WD}$ spaces, that of all quasisober spaces, that of all weakly spaces and that of all cut spaces is reflective in the category of all $T_0$ spaces with continuous mappings.

We present a general method to obtain asymptotic power series for three kinds of sequences. And we give recurrence relations for determining the coefficients of asymptotic power series for these sequences. As applications, we show how these theoretical results can be used to deduce approximation formulas for some well-known sequences and some integrals with a parameter.

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\ a_i ot=a_{i+1} \ \text{for}\ 1\le i<n\}$$ and $$C(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_i\in A_i\ (1\le i\le n),\ \text{and}\ a_i ot=a_{i+1} \ \text{for}\ 1\le i<n,\ \text{and}\ a_n ot=a_1\}$$ recently introduced by the second author, when $G$ is the additive group of a field we obtain lower bounds for $|L(A_1,\ldots,A_n)|$ and $|C(A_1,\ldots,A_n)|$ via the polynomial method. Moreover, when $G$ is torsion-free and $A_1=\cdots=A_n$, we determine completely when $|L(A_1,\ldots,A_n)|$ or $|C(A_1,\ldots,A_n)|$ attains its lower bound.

We demonstrate the possibility of classifying causal systems into kinds that share a common structure without first constructing an explicit dynamical model or using prior knowledge of the system dynamics. The algorithmic ability to determine whether arbitrary systems are governed by causal relations of the same form offers significant practical applications in the development and validation of dynamical models. It is also of theoretical interest as an essential stage in the scientific inference of laws from empirical data. The algorithm presented is based on the dynamical symmetry approach to dynamical kinds. A dynamical symmetry with respect to time is an intervention on one or more variables of a system that commutes with the time evolution of the system. A dynamical kind is a class of systems sharing a set of dynamical symmetries. The algorithm presented classifies deterministic, time-dependent causal systems by directly comparing their exhibited symmetries. Using simulated, noisy data from a variety of nonlinear systems, we show that this algorithm correctly sorts systems into dynamical kinds. It is robust under significant sampling error, is immune to violations of normality

Based on the MFT arguments, a general description for discontinuous phase transitions in the presence temporal disorder is considered. Our analysis extends the recent findings [Phys. Rev. E {\bf 98}, 032129 (2018)] by considering other kinds of phase transitions beyond the absorbing ones. The theory is exemplified in one of the simplest (nonequilibrium) order disorder (discontinuous) phase transition with "up-down" $Z_2$ symmetry: the inertial majority vote (IMV) model for two kinds of temporal disorder. As for the APT case, the temporal disorder does not suppress the occurrence of discontinuous phase transitions, but remarkable differences emerge when compared with the pure case. A comparison between the distinct kinds of temporal disorder is also performed beyond the MFT for random-regular (RR) complex topologies.

In this paper we follow the general approach, proposed earlier by the first author, which is derived from the invariant theory field and provides a way of obtaining of the polynomial identities for any arbitrary polynomial family. We introduce the notion of Chebyshev derivations of the first and second kinds, which is based on the polynomial algebra, and corresponding specific differential operators. We derive the elements of their kernels and prove that any element of the kernel of the derivations defines a polynomial identity satisfied by the Chebyshev polynomials of the first and second kinds. Combining elementary methods and combinatorial techniques, we obtain several new polynomial identities involving the Chebyshev polynomials of the both kinds and a special case of the Jacobi polynomials. Using the properties of the generalised hypergeometric function, we specify the Chebyshev polynomials of the first and second kinds via the generalised hypergeometric function and, as a consequence, derive the corresponding identities involving the generalised hypergeometric function and the Chebyshev polynomials of the first and second kinds.

The purpose of this paper is to make a comprehensive connection between the basic results and properties derived from the two kinds of topologies (namely the $(ε,λ)-$topology introduced by the author and the stronger locally $L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et. al) for a random locally convex module. First, we give an extremely simple proof of the known Hahn-Banach extension theorem of $L^{0}-$linear functions as well as its continuous variants. Then we give the essential relations between the hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J. Funct. Anal.256(2009)3996--4029] and a basic strict separation theorem in [Guo et. al, Nonlinear Anal. 71(2009)3794--3804]: in the process obtain a useful and surprising fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies! Based on the relation between the two kinds of completeness, we go on to present the central part of this paper: we prove that most of the previously established deep results of random conjugate spaces of random normed modules under the $(ε,λ)-$topology are still valid under the locally $L

In this paper, we focus on the construction of high order volume preserving in- tegrators for divergence-free vector fields: the monomial basis, the exponential basis and tensor product of the monomial and the exponential basis. We first prove that the commutators of elementary divergence-free vector fields (EDFVF) of those three kinds are still divergence-free vector fields of the same kind. Assuming then there is only diagonal part of divergence-free vector field of the monomial basis, for those three kinds of divergence-free vector fields, we construct high order volume-preserving inte- grators using the multi-commutators for EDFVFs. Moreover, we consider the ordering of the EDFVFs and their commutators to reduce the error of the schemes, showing by numerical tests that the strategy in [9] works very well.

This paper studies the weighted Hardy inequalities on the discrete intervals with four different kinds of boundary conditions. The main result is the uniform expression of the basic estimate of the optimal constant with the corresponding boundary condition. Firstly, one-side boundary condition is considered, which means that the sequences vanish at the right endpoint (ND-case). Based on the dual method, it can be translated into the case vanishing at left endpoint (DN-case). Secondly, the condition is the case that the sequences vanish at two endpoints (DD-case). The third type of condition is the generality of the mean zero condition (NN-case), which is motivated from probability theory. To deal with the second and the third kinds of inequalities, the splitting technique is presented. Finally, as typical applications, some examples are included.

In this paper, we numerically study quantum walks on two kinds of two-dimensional graphs: cylindrical strip and Mobius strip. The two kinds of graphs are typical two-dimensional topological graph. We study the crossing property of quantum walks on these two models. Also, we study its dependence on the initial state, size of the model. At the same time, we compare the quantum walk and classical walk on these two models to discuss the difference of quantum walk and classical walk.

In this article, we study on the separable N=2 solutions of Sine Gordon equation. From the original symmetry,we get two kinds of N=2 separable solutions. we find these two kinds are related to Landen transformation

This paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled Schrödinger equations. The first one concerns with the equivalence of the minimal norm and the minimal time control problems. (The minimal time control problems are also called the first type of optimal time control problems.) The targets of the aforementioned two kinds of problems are the origin of the state space. The second one deals with the equivalence of three optimal control problems which are optimal target control problems, optimal norm control problems and the second type of optimal time control problems. These two theorems were estabilished for heat equations in [18] and [17] respectively.

Efficient operation sequences to couple and interchange quantum information between quantum dot spin qubits of different kinds are derived using exchange interactions. In the qubit encoding of a single-spin qubit, a singlet-triplet qubit, and an exchange-only (triple-dot) qubit, some of the single-qubit interactions remain on during the entangling operation; this greatly simplifies the operation sequences that construct entangling operations. In the ideal setup, the gate operations use the intra-qubit exchange interactions only once. The limitations of the entangling sequences are discussed, and it is shown how quantum information can be converted between different kinds of quantum dot spin qubits.

According to different topological configurations, we suggest that there are two kinds of extreme black holes in the nature. We find that the Euler characteristic plays an essential role to classify these two kinds of extreme black holes. For the first kind of extreme black holes, Euler characteristic is zero, and for the second kind, Euler characteristic is two or one provided they are four dimensional holes or two dimensional holes respectively.