Band 15, Heft 1

The method of brackets is a collection of heuristic rules, some of which have being made rigorous, that provide a flexible, direct method for the evaluation of definite integrals. The present work uses this method to establish classical formulas due to Frullani which provide values of a specific family of integrals. Some generalizations are established.

In this paper, we study the dynamics of a reaction-diffusion Cellular Nonlinear Network (RD-CNN) nodel in which the reaction term is represented by Brusselator cell. We investigate the RD-CNN dynamics by means of describing function method. Comparison with classical results for Brusselator equation is provided. Then we introduce a new RD-CNN model with memristor coupling, for which the edge of chaos regime in the parameter space is determined. Numerical simulations are presented for obtaining dynamic patterns in the RD-CNN model with memristor coupling.

Teaching and learning calculus are notoriously difficult and the didactic solutions may involve resorting to intuitive but vague definitions or informal gestures offered as proofs. The teaching literature is rife with examples of metaphors, adverb manipulations and descriptions of what happens “just before” the limit. It is then difficult to leave the domain of the mental image, thus losing the training in rigour. The author (with Karel Hrbacek and Olivier Lessmann) has endeavoured a radically different approach with the objective of training students to prove theorems while preserving both intuition and mathematical rigour. Hence we change the mathematical setting rather than the didactic setting. The result (which is a by-product of nonstandard analysis) has been used in several high schools in Geneva – Switzerland – for over ten years.

In this paper, we define and study the hyper S -posets over an ordered semihypergroup in detail. We introduce the hyper version of a pseudoorder in a hyper S -poset, and give some related properties. In particular, we characterize the structure of factor hyper S -posets by pseudoorders. Furthermore, we introduce the concepts of order-congruences and strong order-congruences on a hyper S -poset A , and obtain the relationship between strong order-congruences and pseudoorders on A . We also characterize the (strong) order-congruences by the ρ -chains, where ρ is a (strong) congruence on A . Moreover, we give a method of constructing order-congruences, and prove that every hyper S -subposet B of a hyper S -poset A is a congruence class of one order-congruence on A if and only if B is convex. In the sequel, we give some homomorphism theorems of hyper S -posets, which are generalizations of similar results in S -posets and ordered semigroups.

In this paper we obtain the description of the Leibniz algebras whose subalgebras are ideals.

The aim of this work is to introduce the notion of weak altering distance functions and prove new fixed point theorems in metric spaces endowed with a transitive binary relation by using weak altering distance functions. We give some examples which support our main results where previous results in literature are not applicable. Then the main results of the paper are applied to the multidimensional fixed point results. As an application, we apply our main results to study a nonlinear matrix equation. Finally, as numerical experiments, we approximate the definite solution of a nonlinear matrix equation using MATLAB.

Matrix mathematics provides a powerful tool set for addressing statistical problems, in particular, the theory of matrix ranks and inertias has been developed as effective methodology of simplifying various complicated matrix expressions, and establishing equalities and inequalities occurred in statistical analysis. This paper describes how to establish exact formulas for calculating ranks and inertias of covariances of predictors and estimators of parameter spaces in general linear models (GLMs), and how to use the formulas in statistical analysis of GLMs. We first derive analytical expressions of best linear unbiased predictors/best linear unbiased estimators (BLUPs/BLUEs) of all unknown parameters in the model by solving a constrained quadratic matrix-valued function optimization problem, and present some well-known results on ordinary least-squares predictors/ordinary least-squares estimators (OLSPs/OLSEs). We then establish some fundamental rank and inertia formulas for covariance matrices related to BLUPs/BLUEs and OLSPs/OLSEs, and use the formulas to characterize a variety of equalities and inequalities for covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. As applications, we use these equalities and inequalities in the comparison of the covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. The work on the formulations of BLUPs/BLUEs and OLSPs/OLSEs, and their covariance matrices under GLMs provides direct access, as a standard example, to a very simple algebraic treatment of predictors and estimators in linear regression analysis, which leads a deep insight into the linear nature of GLMs and gives an efficient way of summarizing the results.

The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind fourth hybrid power mean of the quartic Gauss sums and Kloosterman sums, and give an exact computational formula for it.

In this paper, we prove that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically ( C , 1, 1) summable to the same number. We construct an example that the converse of this statement is not true in general. We obtain that the statistically ( C , 1, 1) summable double sequence of fuzzy numbers is convergent and statistically convergent to the same number under the slowly oscillating and statistically slowly oscillating conditions in certain senses, respectively.

Let G be a group with identity e . Let R be a G -graded commutative ring and M a graded R -module. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring.

In this paper, we discuss the approximation operators apr¯NS ${\overline {apr} _{NS}}$ and apr¯S ${\overline {apr} _S}$ which are based on NS ( U ) and S . We not only obtain some properties of NS ( U ) and S , but also give examples to show some special properties. We also study sufficient and necessary conditions when they become closure operators. In addition, we give general and topological characterizations of the covering for two types of covering-based upper approximation operators being closure operators.

In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO (4), we construct bi-integrable and tri-integrable couplings associated with SO (4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.

This paper is concerned with a competition and cooperation model of two enterprises with multiple delays and feedback controls. With the aid of the difference inequality theory, we have obtained some sufficient conditions which guarantee the permanence of the model. Under a suitable condition, we prove that the system has global stable periodic solution. The paper ends with brief conclusions.

In the context of the atomic poset, we propose several new methods of constructing the complete lattice and the algebraic lattice, and the mutual decision of relationship between atomic posets and complete lattices (algebraic lattices) is studied.

Restricted ( s , t )-Wythoff’s game, introduced by Liu et al. in 2014, is an impartial combinatorial game. We define and solve a class of games obtained from Restricted ( s , t )-Wythoff’s game by adjoining to it some subsets of its P -positions as additional moves. The results show that under certain conditions they are equivalent to one case in which only one P -position is adjoined as an additional move. Furthermore, two winning strategies of exponential and polynomial are provided for the games.

A new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensor are obtained. It is proved that the new lower and upper bounds improve the corresponding bounds provided by He and Huang (J. Inequal. Appl., 2014, 2014, 114) and Zhao and Sang (J. Inequal. Appl., 2016, 2016, 268). Finally, two numerical examples are given to verify the theoretical results.

For an odd and squarefree level N , Kohnen proved that there is a canonically defined subspace S κ + 1 2 n e w ( N ) ⊂ S κ + 1 2 ( N ) , and S κ + 1 2 n e w ( N ) and S 2 k n e w ( N ) $S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\subset S_{\kappa+\frac{1}{2}}(N),\,\,{\text{and}}\,\,S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\,\,{\text{and}}\,\,S_{2k}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$ are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product a g ( m ) a g ( n ) ¯ $a_{g}(m)\overline{a_{g}(n)}$ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral weight and of level 4 N in terms of certain cycle integrals of the corresponding form f of integral weight. To this end he first constructed Shimura and Shintani lifts, and then combining these lifts with the multiplicity one theorem he deduced the formula in [2, Theorem 3]. In this paper we will prove that there is a Hecke equivariant isomorphism between the spaces S 2 k + ( p ) and S k + 1 2 ( p ) . $S_{2k}^{+}(p)\,\,{\text{and}}\,\,\mathbb{S}_{k+\frac{1}{2}}(p).$ We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].

This paper studies quantile linear regression models with response data missing at random. A quantile empirical-likelihood-based method is proposed firstly to study a quantile linear regression model with response data missing at random. It follows that a class of quantile empirical log-likelihood ratios including quantile empirical likelihood ratio with complete-case data, weighted quantile empirical likelihood ratio and imputed quantile empirical likelihood ratio are defined for the regression parameters. Then, a bias-corrected quantile empirical log-likelihood ratio is constructed for the mean of the response variable for a given quantile level. It is proved that these quantile empirical log-likelihood ratios are asymptotically χ 2 distribution. Furthermore, a class of estimators for the regression parameters and the mean of the response variable are constructed, and the asymptotic normality of the proposed estimators is established. Our results can be used directly to construct the confidence intervals (regions) of the regression parameters and the mean of the response variable. Finally, simulation studies are conducted to assess the finite sample performance and a real-world data set is analyzed to illustrate the applications of the proposed method.

Radii of convexity, starlikeness, lemniscate starlikeness and close-to-convexity are determined for the convex combination of the identity map and a normalized convex function F given by f ( z ) = α z +(1− α ) F ( z ).

Let A = PQ T , where P and Q are two n × 2 complex matrices of full column rank such that Q T P is singular. We solve the quadratic matrix equation AXA = XAX . Together with a previous paper devoted to the case that Q T P is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.

In this paper, we use filters of an EQ -algebra E to induce a uniform structure ( E , 𝓚), and then the part 𝓚 induce a uniform topology 𝒯 in E . We prove that the pair ( E , 𝒯) is a topological EQ -algebra, and some properties of ( E , 𝒯) are investigated. In particular, we show that ( E , 𝒯) is a first-countable, zero-dimensional, disconnected and completely regular space. Finally, by using convergence of nets, the convergence of topological EQ -algebras is obtained.

In calculus, an indefinite integral of a function f is a differentiable function F whose derivative is equal to f . The main goal of the paper is to generalize this notion of the indefinite integral from the ring of real functions to any ring. We also investigate basic properties of such generalized integrals and compare them to the well-known properties of indefinite integrals of real functions.

In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.

While defining the anti-Ramsey number Erdős, Simonovits and Sós mentioned that the extremal colorings may not be unique. In the paper we discuss the uniqueness of the colorings, generalize the idea of their construction and show how to use it to construct the colorings of the edges of complete split graphs avoiding rainbow 2-connected subgraphs. These colorings give the lower bounds for adequate anti-Ramsey numbers.

The group of fractions of a semigroup S , if exists, can be written as G = SS −1 . If S is abelian, then G must be abelian. We say that a semigroup identity is transferable if being satisfied in S it must be satisfied in G = SS −1 . One of problems posed by G.Bergman in 1981 asks whether the group G must satisfy every semigroup identity which is satisfied in S , that is whether every semigroup identity is transferable. The first non-transferable identities were constructed in 2005 by S.V.Ivanov and A.M. Storozhev. A group G is called Hopfian if each epimorphizm G → G is the automorphism. The residually finite groups are Hopfian, however there are many problems concerning the Hopfian property e.g. of infinite Burnside groups, of finitely generated relatively free groups [11, Problem 15]. We prove here that if G = SS −1 is an n -generator group of fractions of a relatively free semigroup S , satisfying m -variable ( m < n ) non-transferable identity, then G is the non-Hopfian group.

We are interested in the evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.

An element in a ring R with identity is said to be strongly nil clean if it is the sum of an idempotent and a nilpotent that commute, R is said to be strongly nil clean if every element of R is strongly nil clean. Let C ( R ) be the center of a ring R and g ( x ) be a fixed polynomial in C ( R )[ x ]. Then R is said to be strongly g ( x )-nil clean if every element in R is a sum of a nilpotent and a root of g ( x ) that commute. In this paper, we give some relations between strongly nil clean rings and strongly g ( x )-nil clean rings. Various basic properties of strongly g ( x ) -nil cleans are proved and many examples are given.

In this paper we prove stability-type theorems for functional equations related to spherical functions. Our proofs are based on superstability-type methods and on the method of invariant means.

The convolution sum, ∑ (l,m)∈ N02α l+β m=nσ (l)σ (m), $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n . Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where ( a , b ) = (1, 11), (1, 13).

Translation surfaces in the Galilean 3-space G 3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G 3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.

In the present paper, we have established the complete convergence for weighted sums of pairwise independent random variables, from which the rate of convergence of moving average processes is deduced.

The aim of the paper is to present the binomial transformation formulae of Fibonacci numbers scaled by complex multipliers. Many of these new and nontrivial relations follow from the fundamental properties of the so-called delta-Fibonacci numbers defined by Wituła and Słota. The paper contains some original relations connecting the values of delta-Fibonacci numbers with the respective values of Chebyshev polynomials of the first and second kind.

We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.

This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.

In this paper, we compute explicitly the oscillation constant for certain half-linear second-order differential equations having different periodic coefficients. Our result covers known result concerning half-linear Euler type differential equations with α —periodic positive coefficients. Additionally, our result is new and original in case that the least common multiple of these periods is not defined. We give an example and corollaries which illustrate cases that are solved with our result.

In this paper, we give the necessary and sufficient conditions for non-null curves with non-null normals in 4-dimensional Semi-Euclidian space with indeks 2 to be osculating curves. Also we give some examples of non-null osculating curves in E24 $\mathbb{E}_{2}^{4}$ .

The aim of this paper is to present some new and essential facts about group 𝒢 generated by the family of convergent permutations, i.e. the permutations on ℕ preserving the convergence of series of real terms. We prove that there exist permutations preserving the sum of series which do not belong to 𝒢. Additionally, we show that there exists a family G (possessing the cardinality equal to continuum) of groups of permutations on ℕ such that each one of these groups is different than 𝒢 and is composed only from the permutations preserving the sum of series. This result substantially strengthens some old Pleasants’ result.

Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.

In this paper, we define the supersolvable order of hyperplanes in a supersolvable arrangement, and obtain a class of inductively free arrangements according to this order. Our main results improve the conclusion that every supersolvable arrangement is inductively free. In addition, we assert that the inductively free arrangement with the required induction table is supersolvable.

This paper is concerned with the existence and uniqueness of asymptotically almost automorphic solutions to differential equations with piecewise constant argument. To study that, we first introduce several notions about asymptotically almost automorphic type functions and obtain some properties of such functions. Then, on the basis of a systematic study on the associated difference system, the existence and uniqueness theorem is established. Compared with some earlier results, we do not assume directly that the Green’s function is a Bi -almost automorphic type function.

In this paper, we give a complete classification of the finite groups G whose second maximal subgroups are cyclic

Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = ( N 1 , N 2 ), provided that L = ( L 1 , L 2 ) satisfies dim Ker L 1 = ∞ and dim Ker L 2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed.

In this paper, we investigate the existence of positive solutions for Hadamard type fractional differential system with coupled nonlocal fractional integral boundary conditions on an infinite domain. Our analysis relies on Guo-Krasnoselskii’s and Leggett-Williams fixed point theorems. The obtained results are well illustrated with the aid of examples.

We calibrate Heston stochastic volatility model to real market data using several optimization techniques. We compare both global and local optimizers for different weights showing remarkable differences even for data (DAX options) from two consecutive days. We provide a novel calibration procedure that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods. We test and compare several simulation schemes using the parameters obtained by calibration to real market data. Next to the known schemes (log-Euler, Milstein, QE, Exact scheme, IJK) we introduce also a new method combining the Exact approach and Milstein (E+M) scheme. Test is carried out by pricing European call options by Monte Carlo method. Presented comparisons give an empirical evidence and recommendations what methods should and should not be used and why. We further improve the QE scheme by adapting the antithetic variates technique for variance reduction.

In this paper, we use the estimate for trigonometric sums and the properties of the congruence equations to study the computational problem of one kind sixth power mean of the three-term exponential sums. As a conclusion, we give an exact computational formula for it.

In this article, we survey the existence, uniqueness and convergence of a common best proximity point for a cyclic pair of mappings, which is equivalent to study of a solution for a nonlinear programming problem in the setting of uniformly convex Banach spaces. Finally, we provide an extension of Edelstein’s fixed point theorem in strictly convex Banach spaces. Examples are given to illustrate our main conclusions.

The main purpose of this paper is to investigate the uniqueness of meromorphic functions that share two finite sets in the k -punctured complex plane. It is proved that there exist two sets S 1 , S 2 with ♯ S 1 = 2 and ♯ S 2 = 5, such that any two admissible meromorphic functions f and g in Ω must be identical if E Ω ( S j , f ) = E Ω ( S j , g )( j = 1,2).

For a square-free positive integer N , we study the normalizer of Γ Δ ( N ) in PSL 2 (ℝ) and investigate the group structure of its quotient by Γ Δ ( N ) under certain conditions.

If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics [ x 1 x 2 ], [ x 2 x 3 ] and [ x 3 x 1 ] in X . The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ -neighborhood of the union of the two other sides, for every geodesic triangle T in X . Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

We firstly generalize the fuzzy way-below relation on an L -poset, and consider its continuity by means of this relation. After that, we introduce a kind of stratified L -generalized convergence structure on an L -poset. In terms of that, L -fuzzy Scott topology and fuzzy Scott topology are considered, and the properties of fuzzy Scott topology are discussed in detail. At last, we investigate the Scott convergence of stratified L -filters on an L -poset, and show that an L -poset is continuous if and only if the Scott convergence on it coincides with the convergence with respect to the corresponding topological space.

This paper establishes conditions that secure the almost sure upper and lower bounds for a particular normalized weighted sum of independent nonnegative random variables. These random variables do not possess a finite first moment so these results are not typical. These mild conditions allow us to show that the almost sure upper limit is infinity while the almost sure lower bound is one.

In this paper, completely regular endomorphisms of the join of split graphs are investigated. We give conditions under which all completely regular endomorphisms of the join of two split graphs form a monoid.

This study presents an efficient branch and bound algorithm for globally solving the minimax fractional programming problem (MFP). By introducing an auxiliary variable, an equivalent problem is firstly constructed and the convex relaxation programming problem is then established by utilizing convexity and concavity of functions in the problem. Other than usual branch and bound algorithm, an adapted partition skill and a practical reduction technique performed only in an unidimensional interval are incorporated into the algorithm scheme to significantly improve the computational performance. The global convergence is proved. Finally, some comparative experiments and a randomized numerical test are carried out to demonstrate the efficiency and robustness of the proposed algorithm.

By using the properties of Kloosterman sum and Dirichlet character, an optimal upper bound estimate of incomplete Cochrane sum is given.

In this paper we study the iterated order and oscillations of the solutions to some complex linear differential equations in angular domains. Our theorems improve some recent results.

The nonlinear diffusion equation of the ideal barotropic gas through a porous medium is considered. If the diffusion coefficient is degenerate on the boundary, then the solutions may be controlled by the initial value completely, the well-posedness of the solutions may be obtained without any boundary condition.

In this study, a term operation Sheffer stroke is presented in a given basic algebra 𝒜 and the properties of the Sheffer stroke reduct of 𝒜 are examined. In addition, we qualify such Sheffer stroke basic algebras. Finally, we construct a bridge between Sheffer stroke basic algebras and Boolean algebras.

This paper gives extensions and improvements of Sherman’s inequality for n -convex functions obtained by using new identities which involve Green’s functions and Fink’s identity. Moreover, extensions and improvements of Majorization inequality as well as Jensen’s inequality are obtained as direct consequences. New inequalities between geometric, logarithmic and arithmetic means are also established.

We characterize complete atomistic lattices whose classification lattices are geometric. This implies an proper solution to a problem raised by S. Radeleczki in 2002.

Let x , y be two distinct real numbers. An { x , y }-matrix is a matrix whose entries are either x or y . We determine the possible numbers of x ’s in an { x , y }-matrix with a given rank. Our proof is constructive.

Some new error bounds for the linear complementarity problems are obtained when the involved matrices are weakly chained diagonally dominant B -matrices. Numerical examples are given to show the effectiveness of the proposed bounds.

We study the vector-valued B -singular integral operators associated with the Laplace-Bessel differential operator △ B=∑ k=1n− 1∂ 2∂ x k 2+(∂ 2∂ x n 2+2vxn∂ ∂ x n),v>0. $$\triangle_{B}=\sum\limits_{k=1}^{n-1}\frac{\partial^{2}}{\partial x_{k}^{2}}+(\frac{\partial^{2}}{\partial x_{n}^{2}}+\frac{2v}{x_{n}}\frac{\partial}{\partial x_{n}}) , v>0.$$ We prove the boundedness of vector-valued B -singular integral operators A from Lp,v(R+n,H1)toLp,v(R+n,H2), $L_{p,v}(\mathbb{R}_{+}^{n}, H_{1}) \,{\rm to}\, L_{p,v}(\mathbb{R}_{+}^{n}, H_{2}),$ 1 < p < ∞, where H 1 and H 2 are separable Hilbert spaces.

Let p be an odd prime. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one ā with 1 ≤ ā ≤ p − 1 such that a · ā ≡ 1 mod p . Let N ( p ) denote the set of all primitive roots a mod p with 1 ≤ a ≤ p − 1 in which a and ā are of opposite parity. The main purpose of this paper is using the analytic method and the estimate for the hybrid exponential sums to study the solvability of the congruence a + b ≡ 1 mod p with a , b ∈ N ( p ), and give a sharper asymptotic formula for the number of the solutions of the congruence equation.

Our goal is to state and prove the almost sure central limit theorem for maxima ( M n ) of X 1 , X 2 , ..., X n , n ∈ ℕ, where ( X i ) forms a stochastic process of identically distributed r.v.’s of the continuous type, such that, for any fixed n , the family of r.v.’s ( X 1 , ..., X n ) has the Archimedean copula C Ψ .

In this paper, for multiple different chaotic systems with unknown bounded disturbances and fully unknown parameters, a more general synchronization method called modified function projective multi-lag combined synchronization is proposed. This new method covers almost all of the synchronization methods available. As an advantage of the new method, the drive system is a linear combination of multiple chaotic systems, which makes the signal hidden channels more abundant and the signal hidden methods more flexible. Based on the finite-time stability theory and the sliding mode variable structure control technique, a dual-stage adaptive variable structure control scheme is established to realize the finite-time synchronization and to tackle the parameters well. The detailed theoretical derivation and representative numerical simulation is put forward to demonstrate the correctness and effectiveness of the advanced scheme.

The aim of this paper is to prove some fixed point results for generalized φ -weak contraction mapping and study a new concept of stability which is called comparably almost T -stable by using iterative schemes in CAT (0) spaces.

Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p ( k )-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.

Let A n be an alternating group of degree n . Some authors have proved that A 10 , A 147 and A 189 cannot be OD-characterizable. On the other hand, others have shown that A 16 , A 23+4 , and A 23+5 are OD -characterizable. We will prove that the alternating groups A p + d except A 10 , are OD -characterizable, where p is a prime and d is a prime or equals to 4. This result generalizes other results.

In this paper, the notions of Jordan homomorphism and Jordan derivation of inverse semirings are introduced. A few results of Herstein and Bre š ar on Jordan homomorphisms and Jordan derivations of rings are generalized in the setting of inverse semirings.

As a generalization of the class of inverse semigroups, the class of Ehresmann semigroups is introduced by Lawson and investigated by many authors extensively in the literature. In particular, Gomes and Gould construct a fundamental Ehresmann semigroup C E from a semilattice E which plays for Ehresmann semigroups the role that T E plays for inverse semigroups, where T E is the Munn semigroup of a semilattice E . From a varietal perspective, Ehresmann semigroups are derived from reduction of inverse semigroups. In this paper, from varietal perspective Ehresmann semigroups are extended to generalized Ehresmann semigroups derived instead from normal orthodox semigroups (i.e. regular semigroups whose idempotents form normal bands) with an inverse transversal. We present here a semigroup C ( I ,Λ, E ∘ ) from an admissible triple ( I , Λ, E ∘ ) that plays for generalized Ehresmann semigroups the role that C E from a semilattice E plays for Ehresmann semigroups. More precisely, we show that a semigroup is a fundamental generalized Ehresmann semigroup whose admissible triple is isomorphic to ( I , Λ, E ∘ ) if and only if it is (2,1,1,1)-isomorphic to a quasi - full (2,1,1,1)-subalgebra of C ( I ,Λ, E ∘ ) . Our results generalize and enrich some results of Fountain, Gomes and Gould on weakly E-hedges semigroups and Ehresmann semigroups.

Let λ denote any one of the spaces ℓ ∞ and ℓ p and λ ( Ť ) be the domain of the band matrix Ť . We study ℓ p ( Ť ) for 1 ≤ p ≤ ∞ and give some inclusions and its topological properties. Also, we define the alpha −, beta − and gamma − duals of the space ℓ p ( Ť ). Finally, we give some matrix mappings.

A recursion formula for derivatives of Chebyshev polynomials is replaced by an explicit formula. Similar formulae are derived for scaled Fibonacci numbers.

In this paper, we study chain conditions on composite Hurwitz series rings and composite Hurwitz polynomial rings. More precisely, we characterize when composite Hurwitz series rings and composite Hurwitz polynomial rings are Noetherian, S -Noetherian or satisfy the ascending chain condition on principal ideals.

We consider vertex colorings where the number of colors given to specified subgraphs is restricted. In particular, given some fixed graph F and some fixed set A of positive integers, we consider (not necessarily proper) colorings of the vertices of a graph G such that, for every copy of F in G , the number of colors it receives is in A . This generalizes proper colorings, defective coloring, and no-rainbow coloring, inter alia. In this paper we focus on the case that A is a singleton set. In particular, we investigate the colorings where the graph F is a star or is 1-regular.

The method of brackets is an efficient method for the evaluation of alarge class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients a n have meromorphic representations for n ∈ ℂ, but might vanish or blow up when n ∈ ℕ. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

This paper presents a branch-delete-bound algorithm for effectively solving the global minimum of quadratically constrained quadratic programs problem, which may be nonconvex. By utilizing the characteristics of quadratic function, we construct a new linearizing method, so that the quadratically constrained quadratic programs problem can be converted into a linear relaxed programs problem. Moreover, the established linear relaxed programs problem is embedded within a branch-and-bound framework without introducing any new variables and constrained functions, which can be easily solved by any effective linear programs algorithms. By subsequently solving a series of linear relaxed programs problems, the proposed algorithm can converge the global minimum of the initial quadratically constrained quadratic programs problem. Compared with the known methods, numerical results demonstrate that the proposed method has higher computational efficiency.

Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [Math. Comput. Modelling, 1993, 17, 89-95]. Here we introduce a variation of the geodetic problem and call it strong edge geodetic problem. We illustrate how this problem is evolved from social transport networks. It is shown that the strong edge geodetic problem is NP-complete. We derive lower and upper bounds for the strong edge geodetic number and demonstrate that these bounds are sharp. We produce exact solutions for trees, block graphs, silicate networks and glued binary trees without randomization.

Let ( M 3 , g ) be an almost Kenmotsu 3-manifold such that the Reeb vector field is an eigenvector field of the Ricci operator. In this paper, we prove that if g represents a Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M 3 is locally isometric to either the hyperbolic space ℍ 3 (−1) or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. In particular, when g represents a gradient Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M 3 is locally isometric to either ℍ 3 (−1) or ℍ 2 (−4) × ℝ.

We prove uniqueness theorems of meromorphic functions, which show how two meromorphic functions are uniquely determined by their two finite shared sets. This answers a question posed by Gross. Moreover, some examples are provided to demonstrate that all the conditions are necessary.

Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ be any fourth-order character mod p . In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G ( k , p ) = τ k ( ψ )+ τ k ( ψ ), and give an interesting fourth-order linear recurrence formula for it, where τ ( ψ ) denotes the classical Gauss sums.

This paper deals with a stochastic two-species competitive model. Some very verifiable criteria on the global stability of the positive equilibrium of the deterministic system are established. An example with its computer simulations is given to illustrate our main theoretical findings.

A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al . (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S -type E -eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al . (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S -type upper bound for the Z -spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

In this paper we will show some relations between generalized versions of strong sequences introduced by Efimov in 1965 and independent families. We also show some inequalities between cardinal invariants associated with these both notions.

The main purpose of this paper is to prove that the boundedness of the commutator Mκ ,b∗ $\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator Mκ ∗ $\mathcal{M}_{\kappa}^{*} $ and RBMO ( μ ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of Mκ ∗ $\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that Mκ ,b∗ $\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces L p ( μ ) for 1 < p < ∞, bounded from the space L log L ( μ ) to the weak Lebesgue space L 1,∞ ( μ ), and is bounded from the atomic Hardy spaces H 1 ( μ ) to the weak Lebesgue spaces L 1,∞ ( μ ).

A general linear model can be given in certain multiple partitioned forms, and there exist submodels associated with the given full model. In this situation, we can make statistical inferences from the full model and submodels, respectively. It has been realized that there do exist links between inference results obtained from the full model and its submodels, and thus it would be of interest to establish certain links among estimators of parameter spaces under these models. In this approach the methodology of additive matrix decompositions plays an important role to obtain satisfactory conclusions. In this paper, we consider the problem of establishing additive decompositions of estimators in the context of a general linear model with partial parameter restrictions. We will demonstrate how to decompose best linear unbiased estimators (BLUEs) under the constrained general linear model (CGLM) as the sums of estimators under submodels with parameter restrictions by using a variety of effective tools in matrix analysis. The derivation of our main results is based on heavy algebraic operations of the given matrices and their generalized inverses in the CGLM, while the whole contributions illustrate various skillful uses of state-of-the-art matrix analysis techniques in the statistical inference of linear regression models.

We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and 3 × 3 matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations.

According to the classification by Kac, there are eight Cartan series and five exceptional Lie superalgebras in infinite-dimensional simple linearly compact Lie superalgebras of vector fields. In this paper, the Hom-Lie superalgebra structures on the five exceptional Lie superalgebras of vector fields are studied. By making use of the ℤ-grading structures and the transitivity, we prove that there is only the trivial Hom-Lie superalgebra structures on exceptional simple Lie superalgebras. This is achieved by studying the Hom-Lie superalgebra structures only on their 0-th and (−1)-th ℤ-components.

We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.

All finite-dimensional solvable Leibniz algebras L , having N = NF n ⊕ Fm1 $\begin{array}{} F_{m}^{1} \end{array} $ as the nilradical and the dimension of L equal to n + m +3 (the maximal dimension) are described. NF n and Fm1 $\begin{array}{} F_{m}^{1} \end{array} $ are the null-filiform and naturally graded filiform Leibniz algebras of dimensions n and m , respectively. Moreover, we show that these algebras are rigid.

The generating functions of divisor functions are quasimodular forms of weight 2 and their products belong to a space of quasimodular forms of higher weight. In this article, we evaluate the convolution sums ∑ al+bm=nlσ (l)σ (m) $$\begin{array}{} \displaystyle\sum\limits_{al+bm=n}\,l\sigma(l)\sigma(m) \end{array} $$ for all positive integers a , b and n with ab ≤ 9 and gcd( a , b ) = 1.

In this paper, we study a kind of soft rough semigroups according to Shabir’s idea. We define the upper and lower approximations of a subset of a semigroup. According to Zhan’s idea over hemirings, we also define a kind of new C -soft sets and CC -soft sets over semigroups. In view of this theory, we investigate the soft rough ideals (prime ideals, bi-ideals, interior ideals, quasi-ideals, regular semigroups). Finally, we give two decision making methods: one is for looking a best a parameter which is to the nearest semigroup, the other is to choose a parameter which keeps the maximum regularity of regular semigroups.

In this paper, we present several new and generalized Hermite-Hadamard type inequalities for s-convex as well as s-concave functions via classical and Riemann-Liouville fractional integrals. As applications, we provide new error estimations for the trapezoidal formula.

An edge-magic total labeling of an ( n , m )-graph G = ( V , E ) is a one to one map λ from V ( G ) ∪ E ( G ) onto the integers {1,2,…, n + m } with the property that there exists an integer constant c such that λ ( x ) + λ ( y ) + λ ( xy ) = c for any xy ∈ E ( G ). It is called super edge-magic total labeling if λ ( V ( G )) = {1,2,…, n }. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edge-magic total labeling, called super edge-magic deficiency of a graph G , is denoted by μ s ( G ) [4]. If such vertices do not exist, then deficiency of G will be + ∞. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].

The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.

This article presents a new method of solving partial differential equations. The method is an improvement of the previously reported compact finite difference quasilinearization method (CFDQLM) which is a combination of compact finite difference schemes and quasilinearization techniques. Previous applications of compact finite difference (FD) schemes when solving parabolic partial differential equations has been solely on discretizing the spatial variables and another numerical technique used to discretize temporal variables. In this work we attempt, for the first time, to use the compact FD schemes in both space and time. This ensures that the rich benefits of the compact FD schemes are carried over to the time variable as well, which improves the overall accuracy of the method. The proposed method is tested on four nonlinear evolution equations. The method produced highly accurate results which are portrayed in tables and graphs.

The past decades have witnessed several well-known beautiful conclusions on four con-cyclic points. With highly promising research value, we profoundly studied circumscribed ellipses of convex quadrilaterals in this paper. Using tools of parallel projective transformation and analytic geometry, we derived several theorems including the proof of the existence of circumscribed ellipses of convex quadrilaterals, the properties of its minimal coverage area, and locus center, respectively. This simple approach lays a solid foundation for its application to three-dimensional situations, which is namely the circumscribed quadric surface of a solid figure and its wide-range utility in construction engineering. Meanwhile, we have a new insight into innate connection of conic sections as well as a taste of beauty and harmony of geometry.

For a given endomorphism φ on a connected Lie group G this paper studies several subgroups of G that are intrinsically connected with the dynamic behavior of φ .

In this paper, we obtained that when k = 455, every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of unlike powers of primes and k powers of 2.

A significant connection between certain second-order differential subordination and subordination of f ′( z ) is obtained. This fundamental result is next applied to investigate the convexity of analytic functions defined in the open unit disk. As a consequence, criteria for convexity of functions defined by integral operators are determined. Connections are also made to earlier known results.

In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., N=p13+…+pj3 $\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with |pi− (N/j)1/3|≤ N1/3− δ +ε (1≤ i≤ j), $\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some 0< δ ≤ 190. $\begin{array}{} 0 \lt \delta\leq\frac{1}{90}. \end{array} $ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

The θ -commutators of elements of a group with respect to an automorphism are introduced and their properties are investigated. Also, corresponding to θ -commutators, we define the θ -non-commuting graphs of groups and study their correlations with other notions. Furthermore, we study independent sets in θ -non-commuting graphs, which enable us to evaluate the chromatic number of such graphs.

This paper studies the quasi-maximum likelihood estimator (QMLE) for the generalized autoregressive conditional heteroscedastic (GARCH) model based on the Laplace (1,1) residuals. The QMLE is proposed to the parameter vector of the GARCH model with the Laplace (1,1) firstly. Under some certain conditions, the strong consistency and asymptotic normality of QMLE are then established. In what follows, a real example with Laplace and normal distribution is analyzed to evaluate the performance of the QMLE and some comparison results on the performance are given. In the end the proofs of some theorem are presented.

In this paper, we study second-order nonlinear discrete Robin boundary value problem with parameter dependence. Applying invariant sets of descending flow and variational methods, we establish some new sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions of the system when the parameter belongs to appropriate intervals. In addition, an example is given to illustrate our results.

The γ * -relation defined on a general hyperring R is the smallest strongly regular relation such that the quotient R / γ * is a ring. In this note we consider a particular class of hyperrings, where we define a new equivalence, called ε m∗ $\varepsilon^{*}_{m} $, smaller than γ * and we prove it is the smallest strongly regular relation on such hyperrings such that the quotient R / ε m∗ $\varepsilon^{*}_{m} $ is a ring. Moreover, we introduce the concept of m -idempotent hyperrings, show that they are a characterization for Krasner hyperfields, and that ε m∗ $\varepsilon^{*}_{m} $ is a new exhibition for γ * on the above mentioned subclass of m -idempotent hyperrings.

Due to discontinuity on the boundary, traditional Fourier approximation does not work efficiently for d −variate functions on [0, 1] d . In this paper, we will give a recursive method to reconstruct/approximate functions on [0, 1] d well. The main process is as follows: We reconstruct a d −variate function by using all of its ( d −1)–variate boundary functions and few d –variate Fourier coefficients. We reconstruct each ( d −1)–variate boundary function given in the preceding reconstruction by using all of its ( d −2)–variate boundary functions and few ( d −1)–variate Fourier coefficients. Continuing this procedure, we finally reconstruct each univariate boundary function in the preceding reconstruction by using values of the function at two ends and few univariate Fourier coefficients. Our recursive method can reconstruct multivariate functions on the unit cube with much smaller error than traditional Fourier methods.

In this paper, by using the theory of calculus on time scales and some mathematical methods, several nabla dynamic inequalities on time scales are established. As an application, we apply the obtained results to a logistic integrodifferential equation on time scales and sufficient conditions for the permanence of the equation are derived. Finally, numerical examples together with their simulations are presented to illustrate the feasibility and effectiveness of the results.

In this paper, we calculate the number of spanning trees in the sequence of Dürer graphs with a special feature that it has two alternate states. Using the electrically equivalent transformations, we obtain the weights of corresponding equivalent graphs and further derive relationships for spanning trees between the Dürer graphs and transformed graphs. By algebraic calculations, we obtain a closed-form formula for the number of spanning trees with regard to iteration step. Finally we compare the entropy of our graph with other studied graphs and see that its value of entropy lies in the interval of those of graphs with average degree being 3 and 4.

The ordering of normal linear experiments with respect to quadratic estimation, introduced by Stępniak in [Ann. Inst. Statist. Math. A 49 (1997), 569-584], is extended here to the experiments involving the nuisance parameters. Typical experiments of this kind are induced by allocations of treatments in the blocks. Our main tool, called quotient of information matrices, may be interesting itself. It is known that any orthogonal allocation of treatments in blocks is optimal with respect to linear estimation of all treatment contrasts. We show that such allocation is, however, not optimal for quadratic estimation.

In 1859, Cayley introduced the ordered Bell numbers which have been used in many problems in number theory and enumerative combinatorics. The ordered Bell polynomials were defined as a natural companion to the ordered Bell numbers (also known as the preferred arrangement numbers). In this paper, we study Fourier series of functions related to higher-order ordered Bell polynomials and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions.

In this paper, we discuss the properties of simple modules over Auslander regular rings with global dimension at most 3. Using grade theory, we show the right projective dimension of ExtΛ 1(S,Λ ) $\begin{array}{} \text{Ext}_{{\it\Lambda}}^{1}(S,\ {\it\Lambda}) \end{array} $ is equal to 1 for any simple Λ -module S with gr S = 1. As a result, we give some equivalent characterization of diagonal Auslander regular rings.

In this paper, the weighted multilinear p -adic Hardy operators are introduced, and their sharp bounds are obtained on the product of p -adic Lebesgue spaces, and the product of p -adic central Morrey spaces, the product of p -adic Morrey spaces, respectively. Moreover, we establish the boundedness of commutators of the weighted multilinear p -adic Hardy operators on the product of p -adic central Morrey spaces. However, it’s worth mentioning that these results are different from that on Euclidean spaces due to the special structure of the p -adic fields.

This paper considers the guaranteed cost finite-time boundedness of positive switched nonlinear systems with D -perturbation and time-varying delay. Firstly, the definition of guaranteed cost finite-time boundedness is introduced. By using the Lyapunov-Krasovskii functional and average dwell time (ADT) approach, an output feedback controller is designed and sufficient conditions are obtained to ensure the corresponding closed-loop systems to be guaranteed cost finite-time boundedness (GCFTB). Such conditions can be solved by linear programming. Finally, two examples are provided to show the effectiveness of the proposed method.

In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a modified quasi-boundary value method to construct approximate stable solutions for the original ill-posed boundary value problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. Moreover, numerical tests are presented to illustrate the accuracy and efficiency of this method.

The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H -function are also presented.

In this paper, a criterion for the existence of large linear algebras consisting, except for zero, of one-to-one operators on an infinite dimensional Banach space is provided. As a consequence, it is shown that every separable infinite dimensional Banach space supports a commutative infinitely generated free linear algebra of operators all of whose nonzero members are one-to-one. In certain cases, the assertion holds for nonseparable Banach spaces.

There are just a few general criteria on spaceability. This survey paper is the history of one of the first ones. Let I 1 and I 2 be arbitrary operator ideals and E and F be Banach spaces. The spaceability of the set of operators I 1 ( E , F )\ I 2 ( E , F ) is studied. Before stating the criterium, the paper summarizes the main results about lineability and spaceability of differences between particular operator ideals obtained in recent years. They are the seed of the ideas contained in the general criterium.

Here we proved the existence of a closed vector space of sequences - any nonzero element of which does not comply with Schur’s property, that is, it is weakly convergent but not norm convergent. This allows us to find similar algebraic structures in some subsets of functions.

This expository paper focuses on the study of extreme surjective functions in ℝ ℝ . We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, κ -everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The algebraic structure of the sets of surjective functions we show here is studied using the concept of lineability. In the final sections of this work we also reveal unexpected connections between the different degrees of extreme surjectivity given above and other interesting sets of functions such as the space of additive mappings, the class of mappings with a dense graph, the class of Darboux functions and the class of Sierpiński-Zygmund functions in ℝ ℝ .

We introduce several notions of linear dynamics for multivalued linear operators (MLO’s) between separable Fréchet spaces, such as hypercyclicity, topological transitivity, topologically mixing property, and Devaney chaos. We also consider the case of disjointness, in which any of these properties are simultaneously satisfied by several operators. We revisit some sufficient well-known computable criteria for determining those properties. The analysis of the dynamics of extensions of linear operators to MLO’s is also considered.

During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C 0 -semigroups of linear and continuous operators. We will review some of these notions and we will discuss basic properties of the dynamics of C 0 -semigroups. We will also study in detail the dynamics of the translation C 0 -semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C 0 -semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models.

In Hörmander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the operators corresponding to these problems are isomorphisms between appropriate Hörmander spaces. The regularity of the functions which form these spaces is characterized by a pair of number parameters and a function parameter varying regularly at infinity in the sense of Karamata. Owing to this function parameter, the Hörmander spaces describe the regularity of functions more finely than the anisotropic Sobolev spaces.

This paper investigates the inverse problem of finding the time-dependent diffusion coefficient in a quasilinear parabolic equation with the nonlocal boundary and integral overdetermination conditions. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown. Finally, some numerical experiments are presented.

We prove that homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable, has only trivial solution in the case when the characteristic determinant of the problem is nonzero. In another, opposite case, we prove the existence of nontrivial solutions of the problem, and we propose a differential-symbol method of constructing them.

In this paper, an optimal vibration control problem for a nonlinear plate is considered. In order to obtain the optimal control function, wellposedness and controllability of the nonlinear system is investigated. The performance index functional of the system, to be minimized by minimum level of control, is chosen as the sum of the quadratic 10 functional of the displacement. The velocity of the plate and quadratic functional of the control function is added to the performance index functional as a penalty term. By using a maximum principle, the nonlinear control problem is transformed to solving a system of partial differential equations including state and adjoint variables linked by initial-boundary-terminal conditions. Hence, it is shown that optimal control of the nonlinear systems can be obtained without linearization of the nonlinear term and optimal control function can be obtained analytically for nonlinear systems without linearization.

We consider a commutative algebra 𝔹 over the field of complex numbers with a basis { e 1 , e 2 } satisfying the conditions (e12+e22)2=0,e12+e22≠ 0. $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ Let D be a bounded simply-connected domain in ℝ 2 . We consider (1-4)-problem for monogenic 𝔹-valued functions Φ( xe 1 + ye 2 ) = U 1 ( x , y ) e 1 + U 2 ( x , y ) i e 1 + U 3 ( x , y ) e 2 + U 4 ( x , y ) i e 2 having the classic derivative in the domain D ζ = { xe 1 + ye 2 : ( x , y ) ∈ D }: to find a monogenic in D ζ function Φ, which is continuously extended to the boundary ∂ D ζ , when values of two component-functions U 1 , U 4 are given on the boundary ∂ D . Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions.

We consider the inverse problem of determining how the physiological structure of a harvested population evolves in time, and of finding the time-dependent effort to be expended in harvesting, so that the weighted integral of the density, which may be, for example, the total number of individuals or the total biomass, has prescribed dynamics. We give conditions for the existence of a unique, global, weak solution to the problem. Our investigation is carried out using the method of characteristics and a generalization of the Banach fixed-point theorem.

In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

We have investigated the behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities in bounded and unbounded domains. We found exponents of the solution’s decreasing rate near the boundary singularities.

The purpose of this paper is to present some existence results for coupled fixed point of a ( φ , ψ ) —contractive condition for mixed monotone operators in metric spaces endowed with a directed graph. Our results generalize the results obtained by Jain et al. in (International Journal of Analysis, Volume 2014, Article ID 586096, 9 pages). Moreover, we have an application to some integral system to support the results.

In the present article we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with L 1 -right-hand sides in a bounded domain of ℝ n ( n ⩾ 2) . This class is described by the presence of a set of exponents q 1 ,…, q n and a set of weighted functions ν 1 ,…, ν n in growth and coercitivity conditions on coefficients of the equations. The exponents q i characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions ν i characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to investigate the existence of entropy solutions of the problem under consideration.

Some nonlinear integro-differential equations of fourth order with variable exponents of the nonlinearity are considered. The initial-boundary value problem for these equations is investigated and the existence theorem for the problem is proved.

We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.

In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations (− Δ )su+V(x)u=λ f(x,u)inRN, $$ {( - \Delta )^s}u + V(x)u = \lambda f(x,u)\,\,{\rm in}\,\,{\mathbb{R}^N}, $$ where (− Δ )su(x)=2limε → 0∫ RN∖ Bε (X)u(x)− u(y)|x− y|N+2sdy,x∈ RN $ {( - \Delta )^s}u(x) = 2\lim\limits_{\varepsilon \to 0} \int_ {{\mathbb{R}^N}\backslash {B_\varepsilon }(X)} {{u(x) - u(y)} \over {|x - y{|^{N + 2s}}}}dy,\,\,x \in {\mathbb{R}^N} $ is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

In the paper the conditions of the existence and uniqueness of the solution for the inverse problem for higher order ultraparabolic equation are obtained. The equation contains two unknown functions of spatial and time variables in its right-hand side. The overdetermination conditions of the integral type are used.

A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3 D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞( ε ). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε → 0. Namely, we derive the limit problem ( ε = 0) in the corresponding graph, define other terms of the asymptotic approximation and prove energetic and uniform pointwise estimates. These estimates allow us to observe the impact of the aneurysm on some properties of the solution.

This paper reviews the existing literature on the combination of metaheuristics with machine learning methods and then introduces the concept of learnheuristics, a novel type of hybrid algorithms. Learnheuristics can be used to solve combinatorial optimization problems with dynamic inputs (COPDIs). In these COPDIs, the problem inputs (elements either located in the objective function or in the constraints set) are not fixed in advance as usual. On the contrary, they might vary in a predictable (non-random) way as the solution is partially built according to some heuristic-based iterative process. For instance, a consumer’s willingness to spend on a specific product might change as the availability of this product decreases and its price rises. Thus, these inputs might take different values depending on the current solution configuration. These variations in the inputs might require from a coordination between the learning mechanism and the metaheuristic algorithm: at each iteration, the learning method updates the inputs model used by the metaheuristic.

Nowadays, nature–inspired metaheuristic algorithms are most powerful optimizing algorithms for solving the NP–complete problems. This paper proposes three approaches to find near–optimal Golomb ruler sequences based on nature–inspired algorithms in a reasonable time. The optimal Golomb ruler (OGR) sequences found their application in channel–allocation method that allows suppression of the crosstalk due to four–wave mixing in optical wavelength division multiplexing systems. The simulation results conclude that the proposed nature–inspired metaheuristic optimization algorithms are superior to the existing conventional and nature–inspired algorithms to find near–OGRs in terms of ruler length, total optical channel bandwidth, computation time, and computational complexity. Based on the simulation results, the performance of proposed different nature–inspired metaheuristic algorithms are being compared by using statistical tests. The statistical test results conclude the superiority of the proposed nature–inspired optimization algorithms.

It is well known that cyclic codes are very useful because of their applications, since they are not computationally expensive and encoding can be easily implemented. The relationship between cyclic codes and invariant subspaces is also well known. In this paper a generalization of this relationship is presented between monomial codes over a finite field 𝔽 and hyperinvariant subspaces of 𝔽 n under an appropriate linear transformation. Using techniques of Linear Algebra it is possible to deduce certain properties for this particular type of codes, generalizing known results on cyclic codes.

Critical Infrastructures in public administration would be compromised by Advanced Persistent Threats (APT) which today constitute one of the most sophisticated ways of stealing information. This paper presents an effective, learning based tool that uses inductive techniques to analyze the information provided by firewall log files in an IT infrastructure, and detect suspicious activity in order to mark it as a potential APT. The experiments have been accomplished mixing real and synthetic data traffic to represent different proportions of normal and anomalous activity.

The approach to convolutional codes from the linear systems point of view provides us with effective tools in order to construct convolutional codes with adequate properties that let us use them in many applications. In this work, we have generalized feedback equivalence between families of convolutional codes and linear systems over certain rings, and we show that every locally Brunovsky linear system may be considered as a representation of a code under feedback convolutional equivalence.