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The paper investigates the multiple rogue wave solutions associated with the generalized Hirota-Satsuma-Ito (HSI) equation and the newly proposed extended (3+1)-dimensional Jimbo-Miwa (JM) equation with the help of a symbolic computation technique. By incorporating a direct variable transformation and utilizing Hirota’s bilinear form, multiple rogue wave structures of different orders are obtained for both generalized HSI and JM equation. The obtained bilinear forms of the proposed equations successfully investigate the 1 s t , 2 n d and 3 r d -order rogue waves. The constructed solutions are verified by inserting them into original equations. The computations are assisted with 3D graphs to analyze the propagation dynamics of these rogue waves. Physical properties of these waves are governed by different parameters that are discussed in details.

This paper applies a powerful scheme, namely Bernoulli sub-equation function method, to some partial differential equations with high non-linearity. Many new travelling wave solutions, such as mixed dark-bright soliton, exponential and complex domain, are reported. Under a suitable choice of the values of parameters, wave behaviours of the results obtained in the paper – in terms of 2D, 3D and contour surfaces – are observed.

We prove that pseudorandom sets in the Grassmann graph have near-perfect expansion. This completes the last missing piece of the proof of the 2-to-2-Games Conjecture (albeit with imperfect completeness). The Grassmann graph has induced subgraphs that are themselves isomorphic to Grassmann graphs of lower orders. A set of vertices is called pseudorandom if its density within all such subgraphs (of constant order) is at most slightly higher than its density in the entire graph. We prove that pseudorandom sets have almost no edges within them. Namely, their edge-expansion is very close to 1.

The problem of determining whether a term is an instance of another in the simply typed lambda -calculus, i.e. of solving the equation a=b where a and b are simply typed lambda -terms and b is ground, is addressed. An algorithm that decides whether a matching problem in which all the variables are at most third order has a solution is given. The main idea is that if the problem a=b has a solution, then it also has a solution whose depth is bounded by some integer s depending only on the problem a=b, so a simple enumeration of the substitutions whose depth is bounded by s gives a decision algorithm. This result can also be used to bound the depth of the search tree in Huet's semi-decision algorithm and thus to turn it into an always-terminating algorithm. The problems that occur in trying to generalize the proof given to higher-order matching are discussed. >

A graph G is said to be fractional ( a , b , k ) -critical covered if for any W ⊆ V ( G ) with | W | = k , G − W is fractional [ a , b ] -covered. In this article, we demonstrate that a graph G of order n is fractional ( a , b , k ) -critical covered if G satisfies δ ( G ) ≥ ( t − 1 ) ( a + 1 ) + k , n ≥ ( a + b ) ( t ( a + b ) − 2 ) + b k + 2 b , and | N G ( x 1 ) ∪ N G ( x 2 ) ∪ ⋯ ∪ N G ( x t ) | ≥ a n + b k + 2 a + b for any independent subset { x 1 , x 2 , … , x t } of G , which is an improvement and extension of Yuan and Hao’s previous result (Yuan and Hao, 2020).

Let G be a connected graph of order n , A ( G ) the adjacency matrix of G and D ( G ) the diagonal matrix of the row-sums of A ( G ) . For every real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . In this paper, we obtain a factorization of the A α -characteristic polynomial of the some families of graphs and determine in these families the conditions about α so that A α ( G ) is positive semidefinite.

As a generation of the real-valued neural network (RVNN), complex-valued neural network (CVNN) is based on the complex-valued (CV) parameters and variables. The fractional-order (FO) CVNN with linear impulses and fixed time delays is discussed. By using the sign function, the Banach fixed point theorem, and two classes of activation functions, some criteria of uniform stability for the solution and existence and uniqueness for equilibrium solution are derived. Finally, three experimental simulations are presented to illustrate the correctness and effectiveness of the obtained results.

In this note, we show that linear dynamic controllers with integer coefficients are usually unstable. In fact, asymptotic stability can only be achieved if all controller eigenvalues are equal to zero. Moreover, for a fixed controller order, there exist only finitely many characteristic polynomials with integer coefficients that lead to marginally stable eigenvalues on the unit circle and we characterize these setups. The obtained results are, in particular, relevant to encrypted control, where (non-trivial) stable controllers with integer coefficients were on the ‘`wish list’'.

Investigated in the present paper is a fifth-order nonlinear evolution (FONLE) equation, known as a nonlinear water wave (NLWW) equation, with applications in the applied sciences. More precisely, a traveling wave hypothesis is firstly applied that reduces the FONLE equation to a 1D domain. The Kudryashov methods (KMs) are then adopted as leading techniques to construct specific wave structures of the governing model which are classified as W -shaped and other solitons. In the end, the effect of changing the coefficients of nonlinear terms on the dynamical features of W -shaped and other solitons is investigated in detail for diverse groups of the involved parameters.

Fix a positive integer $d$ and let $\Gamma_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $\Gamma_d$ were studied by Babai, Cameron and P\'{a}lfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we prove that there is a linear function $f$ such that the base size of any finite primitive group with point stabilizers in $\Gamma_d$ is at most $f(d)$. This generalizes a recent result of the first author on primitive groups with solvable point stabilizers. For non-affine primitive groups we obtain stronger results, assuming only that stabilizers of $c$ points lie in $\Gamma_d$. We also show that if $G$ is any permutation group of degree $n$ whose $c$-point stabilizers lie in $\Gamma_d$, then $|G| \leqslant ((1+o_c(1))d/e)^{n-1}$. This asymptotically extends and improves a $d^{n-1}$ upper bound on $|G|$ obtained by Babai, Cameron and P\'{a}lfy assuming $G \in \Gamma_d$., Comment: 19 pages; to appear in Journal of Algebra (special issue in memory of Jan Saxl)

In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed graph admits a homomorphism, of graphs with underlying graph of the form P n □ P m , of Cartesian products of signed paths, of Cartesian products of signed complete graphs and of Cartesian products of signed cycles.

Let $G$ be a finite group of order divisible by a prime $p$. The number of $p$-regular and $p'$-regular conjugacy classes of $G$ is at least $2\sqrt{p-1}$. Also, the number of $p$-rational and $p'$-rational irreducible characters of $G$ is at least $2\sqrt{p-1}$. Along the way we prove a uniform lower bound for the number of $p$-regular classes in a finite simple group of Lie type in terms of its rank and size of the underlying field., 46 pages

In this paper we study some geometrical questions about the polytope of bi-capacities. For this, we introduce the concept of pointed order polytope, a natural generalization of order polytopes. Basically, a pointed order polytope is a polytope that takes advantage of the order relation of a partially ordered set and such that there is a relevant element in the structure. We study which are the set of vertices of pointed order polytopes and sort out a simple way to determine whether two vertices are adjacent. We also study the general form of its faces. Next, we show that the set of bi-capacities is a special case of pointed order polytope. Then, we apply the results obtained for general pointed order polytopes for bi-capacities, allowing to characterize vertices and adjacency, and obtaining a bound for the diameter of this important polytope arising in Multicriteria Decision Making.

It is well known that the conventional fractional-order neural networks (FONNs) cannot generate nonconstant periodic oscillation. For this point, this article discusses a class of impulsive FONNs with piecewise Caputo derivatives (IPFONNs). By using the differential inclusion theory, the existence of the Filippov solutions for a discontinuous IPFONNs is investigated. Furthermore, some decision theorems are established for the existence and uniqueness of the (periodic) solution, global exponential stability, and impulsive control global stabilization to IPFONNs. This article achieves four key issues that were not solved in the previously existing literature: 1) the existence of at least one Filippov solution in a discontinuous IPFONN; 2) the existence and uniqueness of periodic oscillation in a nonautonomous IPFONN; 3) global exponential stability of IPFONNs; and 4) impulsive control global Mittag-Leffler stabilization for FONNs.

We deal with the monadic (second-order) theory of order. We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them. We prove (CH) that the monadic theory of the real order is undecidable. Our methods are model-theoretic, and we do not use automaton theory. This is a slightly corrected version of a very old work.

This work examines the impact of multiple vaccination strategies on the dynamics of COVID-19 in a population using the Atangana-Baleanu derivative. The existence and uniqueness of solution of the model is proven using Banach’s fixed point theorem. Local and global asymptotic stability of the equilibria of the model is also proven (under some conditions). Conditions for the existence of a unique or multiple equilibria are also derived and the model is shown to undergo backward bifurcation under certain scenarios. Using available data for the Pfizer, Moderna and Janssen vaccination programme for the city of Texas, United States of America from March 13, 2021 to June 29, 2021, the model is fitted using the three data sets. The three vaccination rates ν 1 , ν 2 and ν 3 corresponding to each vaccine as well as the effective contact rate for COVID-19 transmission, β , are estimated. Simulations of the model under different vaccination strategies are carried out. The results show that the three vaccination strategies not only cause significant reduction in the new asymptomatic and vaccinated symptomatic cases but also cause great decrease in the total number of vaccinated symptomatic individuals with severe COVID-19 illness.

‘‘The time-fractional generalized Burger–Fisher equation (TF-GBFE)” is used in various applied sciences and physical applications, including simulation of gas dynamics, financial mathematics, fluid mechanics, and ocean engineering. This equation represents a concept for the coordination of reaction systems, as well as advection, and conveys the understanding of dissipation. The Fractional Reduced Differential Transform Method (FRDTM) is used to evaluate “the time-fractional generalized Burger–Fisher equation (TF-GBFE).” To determine the method’s validity, when the solutions are obtained, they are correlated to exact solutions of α = 1 order, and even for various values of α . Three-dimensional graphs are used to depict the solutions. Additionally, the analysis of exact and FRDTM solutions indicates that the proposed approach is very accurate.

On being told that a piece of work he thought was his discovery had duplicated an earlier mathematician’s work, Larry Shepp once replied “Yes, but when I discovered it, it stayed discovered.” In this spirit we give discussion and probabilistic proofs of two related known results (Moon 1963, Joe 1988) on random tournaments which seem surprisingly unknown to modern probabilists. In particular, our proof of Moon’s theorem on mean score sequences seems more constructive than previous proofs. This provides a comparatively concrete introduction to a longstanding mystery, the lack of a canonical construction for a joint distribution in the representation theorem for convex order.

Value iteration is a fixed point iteration technique utilized to obtain the optimal value function and policy in a discounted reward Markov Decision Process (MDP). Here, a contraction operator is constructed and applied repeatedly to arrive at the optimal solution. Value iteration is a first order method and therefore it may take a large number of iterations to converge to the optimal solution. Successive relaxation is a popular technique that can be applied to solve a fixed point equation. It has been shown in the literature that, under a special structure of the MDP, successive over-relaxation technique computes the optimal value function faster than standard value iteration. In this work, we propose a second order value iteration procedure that is obtained by applying the Newton-Raphson method to the successive relaxation value iteration scheme. We prove the global convergence of our algorithm to the optimal solution asymptotically and show the second order convergence. Through experiments, we demonstrate the effectiveness of our proposed approach., Comment: Accepted for publication at IEEE Transactions on Automatic Control

This paper investigates the stability analysis and stabilization problems for fractional order T-S fuzzy systems via fuzzy Lyapunov functions method. A membership-function-dependent fuzzy Lyapunov function instead of the general quadratic Lyapunov function is employed to obtain the stability and stabilization criteria. Different from the general quadratic Lyapunov function, the fuzzy Lyapunov functions contain the product of three term functions. Since the general Leibniz formula cannot be satisfied for fractional derivative, the current results on the fractional derivative for the quadratic Lyapunov functions can not be extended to the fuzzy Lyapunov functions. Therefore, to estimate the fractional derivative of fuzzy Lyapunov functions, the fractional derivative rule for the product of three term functions is proposed. Based on the proposed fractional derivative rule, the corresponding stability and stabilization criteria are established, which extend the existing results. Finally, two simulation examples are presented to illustrate the effectiveness of the proposed theoretical analysis.

The current paper deals with the transmission of MERS-CoV model between the humans populace and the camels, which are suspected to be the primary source for the infection. The effect of time MERS-CoV disease transmission is explored using a non-linear fractional order in the sense of Caputo operator in this paper. The considered model is analyzed for the qualitative theory, uniqueness of the solution are discussed by using the Banach contraction principle. Stability analysis is investigated by the aid of Ulam-Hyres (UH) and its generalized version. Finally, we show the numerical results with the help of generalized Adams-Bashforth-Moulton Method (GABMM) are used for the proposed model, for supporting our analytical work.

Keating and Rudnick studied the variance of the polynomial von Mangoldt function Λ : F q [ t ] → C in arithmetic progressions and short intervals using two equidistribution results by Katz. Hall, Keating and Roditty-Gershon then generalised the result for arithmetic progressions for a von Mangoldt function Λ ρ attached to a Galois representation ρ : Gal ( F q ( t ) ‾ / F q ( t ) ) → GL m ( Q ‾ l ) . We employ a recent equidistribution result by Sawin in order to generalise the corresponding result for short intervals for Λ ρ .

In this study, we investigate the conditions for the existence of at least one and three positive solutions to nonlinear higher order multi-point fractional boundary value problems using Krasnosel'skii fixed point theorem and the five functionals fixed point theorem, respectively. © 2021 NSP Natural Sciences Publishing Cor. All Rights Reserved.

In this paper we give a new method of constructing partial order on the set of formal balls of a KM-fuzzy metric space, which overcomes a key defect that the partial order on the set of formal balls depends on the minimum t-norm ∧ in the case of [19] , [21] . Further, based on a continuous t-norm, we will establish a connection between KM-fuzzy metric space and domain theory, which is that a KM-fuzzy metric space is T-complete if and only if its poset of formal balls is a domain.

This paper investigates the finite-time delay-free stability criteria for fractional-order gene regulatory networks (FOGRNs) with feedback regulation time delays. The existence of unique equilibrium points of considered systems is established based on the Banach fixed point theorem and Cauchy–Schwarz inequality. Further, when the order γ satisfies 0 γ 1 and 1 γ 2 , some novel delay-free sufficient conditions are derived to ensure the finite-time stability for addressing fractional order systems by using the ideas of generalized Gronwall–Bellman inequality, equivalent norm techniques, and Laplace transform. In the end, the article comes up with two numerical cases to manifest the applicability and superiority of the obtained theoretical results.

In this article, developed the compact implicit difference method based Grunwald Letnikov formula (GLF) to compute the solution of the time-fractional diffusion-wave equation (TFDWE) describing wave propagation phenomenan having order α ( 1 α 2 ) . The fractional derivative is in Caputo sense. The theoretical analysis such as stability, consistency, convergence and solvability of the said scheme are discussed and proved that the scheme is conditionally stable and convergent with the order ( τ 2 + ( Δ x ) 4 ) . The numerical results compared with the recent existed method. The results of the numerical examples show that the GLF and the proposed method is very accurate and efficient for time fractional diffusion-wave equation.

By using the Banach fixed point theorem and Schauder fixed-point theorem for semilinear spaces, we study the existence of solutions to some class of boundary value problems for interval-valued differential equations on unbounded domains. Some sufficient conditions are provided in order to deduce the existence of solutions without switching points, and also for mixed solutions with a unique switching point. The influences of the range of the parameter in the boundary value condition has on the existence of solutions is also discussed. Finally, two examples are given to demonstrate the feasibility of the theorems.

Let $f$ be a positive multiplicative function and let $k\geq 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$, there exists a real number $c>0$ such that the $k$-tuples $(f(p+1),\ldots,f(p+k))$ are dense in the hypercube $[0,c]^k$ or in $[c,+\infty)^k$. In particular, the values $f(p+1),\ldots,f(p+k)$ can be put in any increasing order infinitely often. Our work generalises previous results of De Koninck and Luca., 11 pages; Corrected a few typos of the previous version

After giving a calculation formula for the g-derivative for a fuzzy-valued function with unilateral derivatives of the endpoint functions of level sets and characterizing the partial order relations between two fuzzy intervals by their generalized difference, we present the optimality conditions of weak LU-minimum solutions for fuzzy optimization problems by using the partial generalized differentiability and directional generalized differentiability. The optimization conditions obtained in this article extend the results in the literature. Examples are also given to illustrate the main theorems.

Let A,B be subsets of a finite abelian group G. Suppose that A+B does not contain a unique sum, i.e., there is no g∈G with a unique representation g=a+b, a∈A, b∈B. From such sets A,B, sparse linear systems over the rational numbers arise. We obtain a new determinant bound on invertible submatrices of the coefficient matrices of these linear systems. Under the condition that |A|+|B| is small compared to the order of G, these bounds provide essential information on the Smith Normal Form of these coefficient matrices. We use this information to prove that A and B admit coset partitions whose parts have properties resembling those of A and B. As a consequence, we improve previously known sufficient conditions for the existence of unique sums in A+B and show how our structural results can be used to classify sets A and B for which A+B does not contain a unique sum when |A|+|B| is relatively small. Our method also can be applied to subsets of abelian groups which have no unique differences. Ministry of Education (MOE) This research is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (RG27/18).

Let q be a power of the prime number p, let K = F q ( t ) , and let r ⩾ 2 be an integer. For points a , b ∈ K which are F q -linearly independent, we show that there exist positive constants N 0 and c 0 such that for each integer l ⩾ N 0 and for each generator τ of F q l / F q , we have that for all except N 0 values λ ∈ F q ‾ , the corresponding specializations a ( τ ) and b ( τ ) cannot have orders of degrees less than c 0 log ⁡ log ⁡ l as torsion points for the Drinfeld module Φ ( τ , λ ) : F q [ T ] ⟶ End F q ‾ ( G a ) (where G a is the additive group scheme), given by Φ T ( τ , λ ) ( x ) = τ x + λ x q + x q r .

A mathematical model describing the HIV/AIDS transmission dynamics in the existence of an aware community using fractional differential operator having Mittag–Leffler kernel is presented and investigated in this paper. By using the fixed point theorem, the existence and uniqueness conditions of the model are obtained. We have used a novel technique known as the iterative Laplace transform approach to obtain the approximate solution of the mathematical model of HIV/AIDS based on the Atagana-Baleanu operator.We investigate the necessary conditions for the disease control in order to determine the role of unaware infective in the spread of HIV/AIDS. The numerical simulations and plots are demonstrated for different values of fractional order. Moreover, we have compared the obtained numerical results based on the Atagana-Baleanu operator with the values obtained using the Caputo operator for the suggested model. We believe that Atangana-Baleanue fractional derivative and the suggested algorithm are expected to be used in future to formulate and analyse many generalised fractional models.

Objective: The objective of the study was to investigate the consequences of using CArm Cone-Beam computed Tomography (CBCT) on super-selective catheterization of Hepatic Artery (HA) branches during chemoembolization of hepatocellular carcinoma. Methods: Two groups of patients were created according to the dates of their treatment sessions. Group A and Group B included patients who had their treatment sessions in 2004 - 2005 and 2008 - 2010, respectively. The sessions performed in 2006 and 2007 were excluded to allow for the adoption and incorporation of CBCT imaging into clinical practice. All chemoembolized HA branches were categorized according to selection order (0-1, 2, or ≥3). Other procedure variables were documented. Results: A total of 58 and 183 sessions were included in Groups A and B, respectively, for 144 patients. C-arm CBCT was used in 2 (3%) sessions and 142 (78%) sessions in groups A and B, respectively. The average number of vessels treated was significantly higher in group B (1.8) compared to group A (1.3) (P < .0001). A shift to an increased selection order in group B (0-1, 44 [24%]; 2, 85 [46%]; ≥3, 54 [30%]) was more significant (P = .0004) than that in group A (0-1, 32 [55%]; 2, 18 [31%]; ≥3, 8 [14%]). The average duration of the procedure was significantly longer in group B (P = .0002). Conclusions: Using C-arm CBCT during chemoembolization has a positive impact on increasing the number and order of HA selected and chemoembolized. This comes at the expense of an increase in the duration of the procedure.

Two string links are equivalent up to 2n-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo n. Moreover, the set of the equivalence classes forms a finite group generated by elements of order n. The classification induces that if two string links are equivalent up to 2n-moves for every n > 0, then they are link-homotopic., Deux enlacements d’intervalles sont équivalents à 2n-mouvements et homotopie près si et seulement si leurs invariants d’homotopie de Milnor sont congrus modulo n. De plus, l’ensemble des classes d’équivalence forme un groupe fini engendré par des éléments d’ordre n. Cette classification implique que si deux enlacements d’intervalles sont équivalents à 2n-mouvements près pour tout n > 0, alors ils sont homotopes.

Testing the dependence between the response and the functional predictor in a functional linear model is of fundamental importance. In this paper, based on a U-statistic of order two, we develop a computationally more efficient test for lacking of dependence in functional linear regression model. By the martingale central limit theorem, we prove that the asymptotic normality of the proposed test statistic under some mild regularity conditions. Simulation results show that our proposed test can be tens or hundreds time faster than the FLUTE test by Hu et al. (2020) which uses a U-statistic of order four. We further demonstrate the superiority of our test by two real data applications.

Let A = F q [ T ] be the polynomial ring over F q , and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r ≥ 2 over F. For all but finitely many primes p ◁ A , one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ ⊗ F p of rank r over F p = A / p . The endomorphism ring E p = End F p ( ϕ ⊗ F p ) is an order in an imaginary field extension K of F of degree r. Let O p be the integral closure of A in K, and let π p ∈ E p be the Frobenius endomorphism of ϕ ⊗ F p . Then we have the inclusion of orders A [ π p ] ⊂ E p ⊂ O p in K. We prove that if End F alg ( ϕ ) = A , then for arbitrary non-zero ideals n , m of A there are infinitely many p such that n divides the index χ ( E p / A [ π p ] ) and m divides the index χ ( O p / E p ) . We show that the index χ ( E p / A [ π p ] ) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r = 2 case we describe an algorithm for computing the orders A [ π p ] ⊂ E p ⊂ O p , and give some computational data.

The cumulative incidence function (CIF) plays an important role in the comparison of competing risks in a competing risks model. Its value at time t is the probability of failure by time t from a particular type of risk in the presence of other risks. In this paper we consider the estimation of two CIFs, F 1 and F 2 , corresponding to two competing risks when the ratio R ( t ) ≡ F 1 ( t ) / F 2 ( t ) is nondecreasing in t > 0 . First, we derive their nonparametric maximum likelihood estimators (NPMLE) of these CIFs in the continuous case under this order constraint and show that they are inconsistent. We then develop projection-type estimators that are uniformly strongly consistent and study the weak convergence of the resulting processes. Through simulations, we compare the finite sample performance of the NPMLEs and our estimators and show that our estimators outperform them in general in terms of mean square error at all the scenarios that we consider. We also develop a test for the presence of this order constraint and extend all these results to the censored case. To illustrate the applicability of the theory we develop, we provide a real life example.

In this paper, we present a novel scheme for solving a time-fractional initial–boundary value problem, where the equation contains a sum of Caputo derivatives with orders between 0 and 1. In order to overcome the difficulty of initial layer, we introduce a change of variable in the temporal direction and investigate the regularity of the solutions of the resulting system. A modified L 1 approximation is used to approximate the Caputo derivatives and a standard Galerkin-Spectral method is applied to approximate the spatial derivatives. Unconditional stability and convergence of the fully-discrete scheme are proved by applying a novel discrete fractional Gronwall inequality. Finally, numerical examples are given to confirm our theoretical results.

Periodic Nonuniform Samplings of order N (PNSN) are interleavings of periodic samplings. For a base period T, simple algorithms can be used to reconstruct functions of spectrum included in an union of N intervals δk of length 1/T. In this paper we study the behavior of these algorithms when applied to any function. We prove that they result in N (or less) foldings on , each of δk holding at most one folding.

A novel nonlinear fractional order damped Pantographs equation involving input delay is formulated. We explored the controllability of the underlying nonlinear damped dynamical Pantograph equation of fractional order with input delay. The solution of the problem under consideration has initially expressed in the form of the Mittag–Leffler function in two different domains. The controllability Grammians matrices for both cases have been defined. Controllability in the linear case has been shown and applied Schaefer’s fixed point theorem as well as the Arzela-Ascoli theorem to derive necessary conditions for the controllability of the nonlinear case.

In 1973 Schwenk [7] proved that almost every tree has a cospectral mate. Inspired by Schwenk's result, in this paper we study the spectrum of two families of trees. The p-sun of order 2 p + 1 is a star K 1 , p with an edge attached to each pendant vertex, which we show to be determined by its spectrum among connected graphs. The ( p , q ) -double sun of order 2 ( p + q + 1 ) is the union of a p-sun and a q-sun by adding an edge between their central vertices. We determine when the ( p , q ) -double sun has a cospectral mate and when it is determined by its spectrum among connected graphs. Our method is based on the fact that these trees have few distinct eigenvalues and we are able to take advantage of their nullity to shorten the list of candidates.

Among the great bunch of additional properties which fuzzy implications may satisfy, the so-called invariance with respect to T-powers highlights due to its applications in approximate reasoning. This property ensures the invariance of the truth value of the fuzzy implication when both the antecedent and the consequent are modified using the same quantifier modeled by using powers of t-norms. In order to find the fuzzy implications that satisfy this property, a new class of fuzzy implications named T-power based implications has been proposed. In this paper we study when this kind of fuzzy implication satisfies the Modus Ponens property with respect to a continuous t-norm T ♯ . Then a characterization theorem on this topic in terms of residual implication generated from a summand of T ♯ is presented.

In this paper, we present ergodicity criteria for 1-Lipschitz functions on Z p , in terms of the van der Put coefficients as well as the inherent data associated with the function. These criteria are applied to provide sufficient conditions for ergodicity of the 1-Lipschitz p-adic functions with special features, such as everywhere/uniform differentiability with respect to the Mahler expansion. In particular, the ergodicity criteria are obtained for certain 1-Lipschitz functions on Z 2 and Z 3 , which are known as B -functions, in terms of the Mahler and van der Put expansions. These functions are locally analytic functions of order 1 (and therefore contain polynomials). For arbitrary primes p ≥ 5 , an ergodicity criterion of B -functions on Z p is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on Z p that realize a given unicyclic permutation modulo p. Thus, a complete description of ergodic polynomials modulo p μ , which are reduced from all ergodic B -functions on Z p , is provided where μ = μ ( p ) = 3 for p ∈ { 2 , 3 } and μ = 2 for p ≥ 5 .

We consider the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. In this situation it may happen that there is no existence and uniqueness theorem for balanced solutions of a given initial value problem. We describe the set of solutions the equation does have and establish that the adjoint of the minimal operator is still the maximal operator, even though unique continuation of balanced solutions fails.

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p × p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v , satisfy u p = v p = 1 p (the p × p identity matrix) and the commutation relation u v = ζ v u , where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d × d unitary matrices such that u p = v p = 1 d and u v = ζ v u , then there exists a unital completely positive linear map ϕ : M p ( C ) → M d ( C ) such that ϕ ( u ) = u and ϕ ( v ) = v . We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails. There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63] . This standard construction is generalised herein to the case p ≥ 3 , producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.