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We deduce new properties of the orbicyclic function $E$ of several variables investigated in a recent paper by V. A. Liskovets. We point out that the function $E$ and its connection to the number of solutions of certain linear congruences occur in the literature in a slightly different form. We investigate another similar function considered by Deitmar, Koyama and Kurokawa by studying analytic properties of some zeta functions of Igusa type. Simple number theoretic proofs for some known properties are also given., Comment: 10 pages

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.

This article first introduces neural networks and their characteristics. Based on a comparison of the structure and function of biological neurons and artificial neurons, it focuses on the structure, classification, activation rules, and learning rules of neural network models. Based on the existing literature, this article adds a distributed time lag term of the neural network system. In the actual problem, history has a very important influence on the current change situation, and it is not only at a specific time in the past. It has an impact on the current state change rate. Therefore, based on the existing literature that only has discrete time lags, this paper adds distributed time lags. Such neural network systems can better reflect real-world problems. In this paper, we use three different inequality scaling methods to study the existence, uniqueness, and global asymptotic stability of a class of neural network systems with mixed delays and uncertain parameters. First, using the principle of homeomorphism, a new upper-norm norm is introduced for the correlation matrix of the neural network, and enough conditions for the existence of unique equilibrium points in several neural network systems are given. Under these conditions, the appropriate Lyapunov is used. Krasovskii functional, we prove that the equilibrium point of the neural network system is globally robust and stable. Numerical experiments show that the stability conditions of the neural network system we obtained are feasible, and the conservativeness of the stability conditions of the neural network system is reduced. Finally, some applications and problems of neural network models in psychology are briefly discussed.

The digital and physical methods of establishing minimal curved surfaces are the basis for realizing the design of the minimal curved surface modeling structure. Based on this research background, the paper showed an artistic surface modeling method based on nonlinear over-limit difference equations. The article combines parameter optimization and 3D modeling methods to model the constructed surface modeling. The research found that the nonlinear out-of-limit difference equation proposed in the paper is more accurate than the standard fractional differential equation algorithm. For this reason, the method can be extended and applied to the design of artistic surface modeling.

Several recent papers have examined a rational polyhedron Pm whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing m. A combinatorial description of the faces of Pm was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of Pm containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of Pm . In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of Pm .

The global arrangement of the degrees of freedom in a standard Argyris finite element method (FEM) enforces C 2 {C^{2}} at interior vertices, while solely global C 1 {C^{1}} continuity is required for the conformity in H 2 {H^{2}} . Since the Argyris finite element functions are not C 2 {C^{2}} at the midpoints of edges in general, the bisection of an edge for mesh-refinement leads to non-nestedness: the standard Argyris finite element space A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} associated to a triangulation 𝒯 {\mathcal{T}} with a refinement 𝒯 ^ {\widehat{\mathcal{T}}} is not a subspace of the standard Argyris finite element space A ′ ⁢ ( 𝒯 ^ ) {A^{\prime}(\widehat{\mathcal{T}})} associated to the refined triangulation 𝒯 ^ {\widehat{\mathcal{T}}} . This paper suggests an extension A ⁢ ( 𝒯 ) {A(\mathcal{T})} of A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} that allows for nestedness A ⁢ ( 𝒯 ) ⊂ A ⁢ ( 𝒯 ^ ) {A(\mathcal{T})\subset A(\widehat{\mathcal{T}})} under mesh-refinement. The extended Argyris finite element space A ⁢ ( 𝒯 ) {A(\mathcal{T})} is called hierarchical, but is still based on the concept of the Argyris finite element as a triple ( T , P 5 ⁢ ( T ) , ( Λ 1 , … , Λ 21 ) ) {(T,P_{5}(T),(\Lambda_{1},\dots,\Lambda_{21}))} in the sense of Ciarlet. The other main results of this paper are the optimal convergence rates of an adaptive mesh-refinement algorithm via the abstract framework of the axioms of adaptivity and uniform convergence of a local multigrid V-cycle algorithm for the effective solution of the discrete system.

The paper is devoted to numerical research of the medical ultrasound tomography problem. This problem consists in finding small inclusions in the breast tissue by boundary measurements of the acoustic waves generated by sources located on the boundary. For medical diagnostic, it is important to recover the image of the acoustical medium and to determine the values of velocity, attenuation and density. In the paper, we describe a numerical experiment of visualization of several inclusions using the energy version of the reverse time migration (RTM). Certainly, the RTM image does not separate velocity and attenuation inclusions. However, kinematic and amplitude analysis gives a possibility to estimate values of the velocity and attenuation. As a result, we split the RTM image into two ones. Note that in this paper we consider the scalar value of sound velocity.

The independent super high-rise building weakens and reduces the cooperation of functional subsystems. Implanting aerial courtyards could improve the aerial environment of that corresponding floor and enable the plane courtyards to be stereoscopic. Constructing multiple aerial courtyard-centred functional clusters of higher floors can optimise the overall space structural relationship. The paper uses the space syntax theory to discuss how to construct holistically linked aerial courtyards in a super high-rise building and applies the Grasshopper-based space syntax arithmetic logical unit to analyse and calculate the aerial courtyard of super high-rise building in different space structure modes, and also proposes design optimisation strategies based on these analyses. The paper points out that by setting aerial courtyard between vertical transportation space and functional space, establishing a direct connection between adjacent aerial courtyards, arranging vertical transportation space dispersedly and so on, it is possible to promote vertical cooperative effect, enhance integration and intelligibility of entire space structure, and provide guidelines and foundation for the construction of vertical cities.

In this paper, we prove some common fixed point theorems satisfying contractive type mapping in the setting of b-metric spaces. The presented theorem is an extension the results of M. Sarwar and M. U. Rahman [23] as well as a generalization of many well-known results in the literature through the context of b-metric spaces. Also, we present a few examples to illustrate the validity of the results obtained in the paper. Finally, results are applied to find the solution for an integral equation.

An approximate method for analysis and synthesis of moving rigid bodies (prisms) in the airflow without using numerical methods of space-time programming techniques is described by applying a fluid (air)–rigid solid body interaction concept for engineering applications through a straightforward mathematical model. The interaction of rigid body (prism) and air is encountered in different cases: moving body (prism) in the air; stationary bodies (prism) in the airflow; moving body (prism) in the airflow. The complicated task of rigid body (prism) and air interaction is simplified by using superposition principles, i.e., by taking into account the upstream and downstream rigid body (prism) and air interaction phenomenon, which has been found to be different under varying speeds. Numerical results obtained for various forms of prisms are shown for constant air–speed, where the steady state Reynolds-averaged Navier–Stokes (RANS) equation is solved by using k-ε realizable turbulence model. A detailed explanation to support the proposed approximate method is given by using numerical results obtained in ANSYS computations. All equations are formed based on laws of classical mechanics; the interaction of viscous forces is neglected in forming the mathematical model. Numerical results for different model prisms are compared and the theoretical results discussed in detail. The mathematical model in the present paper is applicable only to bodies that undergo a rectilinear translation motion. In the final part of the present paper, the proposed method is used in the synthesis and optimization task of energy extraction by considering the motion of a variable parameter prism in the airflow.

In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.

In this paper, at first we study strong Sheffer stroke NMV-algebra. For getting more results and some classification, the notions of filters and subalgebras are introduced and studied. Finally, by a congruence relation, we construct a quotient strong Sheffer stroke NMV-algebra and isomorphism theorems are proved., Ege University Scientific Research Projects Directorate [20772], We wish to thank the reviewers for excellent suggestions that have been incorporated into the paper. This study is supported by Ege University Scientific Research Projects Directorate with the project number 20772.

With ever increasing complexity and intelligence of the modern cities, protecting key public facilities and important targets from any damage is a major challenge for the security sector. In all types of anti-terrorism prediction models, the prediction of attack behaviour is indispensable. Therefore, the attack behaviour model plays an important role in the anti-terrorism security system. This paper builds the attacker’s behaviour model, and carries out the prediction about the possible attack behaviour by the attacker model based on random strategy. According to the attack strategies, analysis and construction of the attack tree and attack graph are carried out based on the state-based stochastic model. The paper describes the security system in detail taking use of the state-based stochastic model method, so as to clarify the state distribution and the transfer relationship between the states of various security resources after threatened by attacks. At the same time, this paper applies the state-based stochastic model to establish the attacker model through the impact of attack on the security system.

In this paper the goal is to explain predictions from complex machine learning models. One method that has become very popular during the last few years is Shapley values. The original development of Shapley values for prediction explanation relied on the assumption that the features being described were independent. If the features in reality are dependent this may lead to incorrect explanations. Hence, there have recently been attempts of appropriately modelling/estimating the dependence between the features. Although the previously proposed methods clearly outperform the traditional approach assuming independence, they have their weaknesses. In this paper we propose two new approaches for modelling the dependence between the features. Both approaches are based on vine copulas, which are flexible tools for modelling multivariate non-Gaussian distributions able to characterise a wide range of complex dependencies. The performance of the proposed methods is evaluated on simulated data sets and a real data set. The experiments demonstrate that the vine copula approaches give more accurate approximations to the true Shapley values than their competitors.

The aim of this paper is to characterize the solutions Φ : G → M 2(ℂ) of the following matrix functional equations Φ ( x y ) + Φ ( σ ( y ) x ) 2 = Φ ( x ) Φ ( y ) , x , y , ∈ G , {{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, and Φ ( x y ) − Φ ( σ ( y ) x ) 2 = Φ ( x ) Φ ( y ) , x , y , ∈ G , {{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, where G is a group that need not be abelian, and σ : G → G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.

In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2 -discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2 -discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2 -discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

Summary This paper provides estimation and hypothesis testing procedures for experiments in split-plot designs. These experiments have been shown to have a convenient orthogonal block structure when properly randomized. Due to this property, the analysis of experimental data can be carried out in a relatively simple manner. Relevant simplification procedures are indicated. According to the adopted approach, the analysis of variance and hypothesis testing procedures can be performed directly, rather than by combining the results of analyses based on some stratum submodels. The practical application of the presented theory is illustrated by examples of real experiments in appropriate split-plot designs. The present paper is the fourth in the planned series of publications on the analysis of experiments with orthogonal block structure.

The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in a delicate way. The properties of the transmission eigenvalues have been extensively and intensively studied over the years, whereas the intrinsic properties of the transmission eigenfunctions are much less studied. Recently, in a series of papers, several intriguing local and global geometric structures of the transmission eigenfunctions are discovered. Moreover, those longly unveiled geometric properties produce some interesting applications of both theoretical and practical importance to direct and inverse scattering problems. This paper reviews those developments in the literature by summarizing the results obtained so far and discussing the rationales behind them. There are some side results of this paper including the general formulations of several types of transmission eigenvalue problems, some interesting observations on the connection between the transmission eigenvalue problems and several challenging inverse scattering problems, and several conjectures on the spectral properties of transmission eigenvalues and eigenfunctions, with most of them are new to the literature.

Aspace X is said to have the absolutely strongly star -𝒤-Hurewicz (ASS𝒤H) property if for each sequence (𝒰 n : n ∈ 𝕅)of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ 𝕅) of finite subsets of Y such that for each x ∈ X, {n ∈ 𝕅 : x ∉ St(Fn , 𝒰 n )}∈ 𝒤, where 𝒤 is the proper admissible ideal of 𝕅. In this paper, we investigate the relationship between the ASS𝒤H property and other related properties and study the topological properties of the ASS𝒤H property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASS𝒤H properties.

In this paper, we consider the following nonlinear Schrödinger equation: i u t + Δ g u + i a ( x ) u − | u | p − 1 u = 0 ( x , t ) ∈ M × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) x ∈ M , $$\begin{array}{} \displaystyle \begin{cases}iu_t+{\it\Delta}_g u+ia(x)u-|u|^{p-1}u=0\qquad (x,t)\in \mathcal{M} \times (0,+\infty), \cr u(x,0)=u_0(x)\qquad x\in \mathcal{M},\end{cases} \end{array}$$ (0.1) where (𝓜, g) is a smooth complete compact Riemannian manifold of dimension n(n = 2, 3) without boundary. For the damping terms −a(x)(1 − Δ)−1a(x)ut and i a ( x ) ( − Δ ) 1 2 a ( x ) u , $\begin{array}{} \displaystyle ia(x)(-{\it\Delta})^{\frac12}a(x)u, \end{array}$ the exponential stability results of system (0.1) have been proved by Dehman et al. (Math Z 254(4): 729-749, 2006), Laurent. (SIAM J. Math. Anal. 42(2): 785-832, 2010) and Cavalcanti et al. (Math Phys 69(4): 100, 2018). However, from the physical point of view, it would be more important to consider the stability of system (0.1) with the damping term ia(x)u, which is still an open problem. In this paper, we obtain the exponential stability of system (0.1) by Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments.

In this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.

We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of the four basic logical notions. This is what we are doing here, as well as introducing a challenging fifth logical notion. We first recall the context and origin of what are here called Tarski-Lindenbaum logical notions. In the second part, we present these notions in the simple case of a binary relation. In the third part, we examine in which sense these are considered as logical notions contrasting them with an example of a nonlogical relation. In the fourth part, we discuss the formulations of the four logical notions in natural language and in first-order logic without equality, emphasizing the fact that two of the four logical notions cannot be expressed in this formal language. In the fifth part, we discuss the relations between these notions using the theory of the square of opposition. In the sixth part, we introduce the notion of variety corresponding to all non-logical notions and we argue that it can be considered as a logical notion because it is invariant, always referring to the same class of structures. In the seventh part, we present an enigma: is variety formalizable in first-order logic without equality? There follow recollections concerning Jan Woleński. This paper is dedicated to his 80th birthday. We end with the bibliography, giving some precise references for those wanting to know more about the topic.

This paper gives an outline of truthmaker semantics for natural language against the background of standard possible-worlds semantics. It develops a truthmaker semantics for attitude reports and deontic modals based on an ontology of attitudinal and modal objects and on a semantic function of clauses as predicates of such objects. The semantics is applied to factive verbs and response-stance verbs as well as to cases of modal concord. The paper also presents new motivations for ‘object-based truthmaker semantics’ from intensional transitive verbs such as need, look for, own, and buy and gives an outline of their semantics based on a further development of truthmaker semantics.

Linear regression with measurement error in the covariates is a heavily studied topic, however, the statistics/econometrics literature is almost silent to estimating a multi-equation model with measurement error. This paper considers a seemingly unrelated regression model with measurement error in the covariates and introduces two novel estimation methods: a pure Bayesian algorithm (based on Markov chain Monte Carlo techniques) and its mean field variational Bayes (MFVB) approximation. The MFVB method has the added advantage of being computationally fast and can handle big data. An issue pertinent to measurement error models is parameter identification, and this is resolved by employing a prior distribution on the measurement error variance. The methods are shown to perform well in multiple simulation studies, where we analyze the impact on posterior estimates arising due to different values of reliability ratio or variance of the true unobserved quantity used in the data generating process. The paper further implements the proposed algorithms in an application drawn from the health literature and shows that modeling measurement error in the data can improve model fitting., Comment: 29 Pages, 8 Tables, 2 Figures

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method.

Let X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point x ∈ X has the next properties: (1) ω ( x , f ) ⊆ Per ( f ) ¯ \omega (x,f) \subseteq \overline {Per(f)} , where Per( f ) is the set of periodic points of f ; (2) ω(x, f ) is either a periodic orbit or a minimal Cantor set. In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3) Ω ( f ) = Per ( f ) ¯ \Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f. The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.

This paper described the volatility characteristic of the rate of return of financial asset by using QR-GARCH model, through introducing EVT model and constructing the extreme risk measure model based on QR-GARCH-EVT. In this paper, HS300 index data test was applied to show that under 5% significance level, and QR-GARCH-EVT model can effectively measure the risk value of the sample, but under 1% significance level. QR-GARCH-EVT model will underestimate the risk value of the sample to a certain degree, but generally speaking, compared with other models, the risk value measured by QR-GARCH-EVT model has a higher accuracy to enhance effectiveness.

In this paper we establish a new critical point theorem for a class of perturbed differentiable functionals without satisfying the Palais–Smale condition. We prove the existence of at least one critical point to such functionals, provided that the perturbation is sufficiently small. The main abstract result of this paper is applied both to perturbed nonhomogeneous equations in Orlicz–Sobolev spaces and to nonlocal problems in fractional Sobolev spaces.

Research background: A lot of research has been directed at describing empirical distributions by using a theoretical model. In the literature there are proposals for various types of mathematical functions. In 2010 Zenga proposed a new three-parameter model for economic size distribution which possesses interesting statistical properties which can be used to model income, wealth and financial variables. Purpose: The aim of this paper is to apply the Zenga model to income distributions in Poland by voivodeship. Research methodology: The basis for the calculations presented in the paper has been based on the individual data coming from a random sample obtained within a Household Budget Survey conducted by the central Statistical Office in the year 2014. The parameters estimates of the Zenga distribution were obtained by means of the D’Addario’s invariants methods, mainly with the Pietra index. Results: The results of the conducted approximations, presented in the paper confirmed the good consistency of the Zenga distribution with the empirical income distribution in Poland, both in total and for households. Novelty: The study contributed to the application of a new three-parameter income distribution model to describe income distributions in Poland.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere ℂ ⁢ ℙ 1 ∖ { 0 , 1 , ∞ } {\mathbb{CP}^{1}\setminus\{0,1,\infty\}} in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of ℂ ⁢ ℙ 1 ∖ { a 1 , … , a n } {\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}} , n ≥ 3 {n\geq 3} , as well, and the metric on ℂ ⁢ ℙ 1 ∖ { 0 , 1 3 , - 1 6 ± 3 6 ⁢ i } {\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}} will be given as a concrete example of our results.

In this paper, we look closely at the issue of contaminated data sets, where apart from legitimate (proper) patterns we encounter erroneous patterns. In a typical scenario, the classification of a contaminated data set is always negatively influenced by garbage patterns (referred to as foreign patterns). Ideally, we would like to remove them from the data set entirely. The paper is devoted to comparison and analysis of three different models capable to perform classification of proper patterns with rejection of foreign patterns. It should be stressed that the studied models are constructed using proper patterns only, and no knowledge about the characteristics of foreign patterns is needed. The methods are illustrated with a case study of handwritten digits recognition, but the proposed approach itself is formulated in a general manner. Therefore, it can be applied to different problems. We have distinguished three structures: global, local, and embedded, all capable to eliminate foreign patterns while performing classification of proper patterns at the same time. A comparison of the proposed models shows that the embedded structure provides the best results but at the cost of a relatively high model complexity. The local architecture provides satisfying results and at the same time is relatively simple.

The paper presents several approaches to gauge repeatability and reproducibility (GRR) analysis regarding non-replicable measurements. Measurement systems have to deal with processes in which, by the nature of the measured object or by the type of measurement itself, measurements are not repeatable. In these cases, each sample unit can be measured only once. Such situations are referred as nonreplicable measurement systems. The aim of the paper is to map out the current approaches being used in GRR analysis in various cases of non-replicable tests and compare each other in order to find out the suitable use of analysis application. Approaches used are subject to critical analysis so that its review can serve a useful base for analysis of different non-replicable tests. At present, it is desirable to bring the improving actions in order to obtain the results of high quality from such kind of measurements. Since different non-replicable tests can measure a different quality characteristic, it is valuable to bring the appropriate designs for various tests. Subsequently, this review will serve an outline how to proceed in analyzing the results obtained by non-replicable tests. Specifically, GRR analysis works with two known designs named as “Crossed” and “Nested” design, which statistical software normally use. Doubtfully, crossed design is suggested to use at certain cases and nested at other specific cases. This is assessed and improving actions designed.

In this paper, we develop a Lyapunov stability framework for fractionally damped mechanical systems. In particular, we study the asymptotic stability of a linear single degree-of-freedom oscillator with viscous and fractional damping. We prove that the total mechanical energy, including the stored energy in the fractional element, is a Lyapunov functional with which one can prove stability of the equilibrium. Furthermore, we develop a strict Lyapunov functional for asymptotic stability, thereby opening the way to a nonlinear stability analysis beyond an eigenvalue analysis. A key result of the paper is a Lyapunov stability condition for systems having negative viscous damping but a sufficient amount of positive fractional damping. This result forms the stepping stone to the study of Hopf bifurcations in fractionally damped mechanical systems. The theory is demonstrated on a stick-slip oscillator with Stribeck friction law leading to an effective negative viscous damping.

In this paper, we develop the elementary theory of inverse semigroups to the cases of type B semigroups. The main aim of this paper is to study proper type B semigroups. We introduce first the concept of a left admissible triple. After obtaining some basic properties of left admissible triple, we give the definition of a Q-semigroup and get a structure theorem of Q-semigroup. In particular, we introduce the notion of an admissible triple and give some characterization of proper type B semigroups. It is proved that an arbitrary Q-semigroup with an admissible triple is an E-unitary type B semigroup.

The paper presents an first type boundary value problem for a Schrödinger equation. The aim of paper is to give the existence and uniqueness theorems of the boundary value problem using Galerkin’s method. Also, a priori estimate for its solution is given.

A general computational model covering many types of frictional contact interfaces between visco-elastic bodies is considered for some cases physically relevant in numerical analysis of contact in civil engineering structures. The relations between mechanical quantities and internal parameters of the model are illustrated in a couple of simplified examples including cohesive contact combined with Coulomb friction and/or interface plasticity. The computations are implemented a semi-implicit time discretisation, quadratic programming algorithms, and the boundary-element method.

The object of the present paper is to study generalized weakly symmetric and generalized weakly Ricci symmetric Kenmotsu manifolds whose metric tensor is η-Ricci soliton. The paper also aims to bring out curvature conditions for which η-Ricci solitons in Kenmotsu manifolds are sometimes shrinking or expanding and some other time remain steady. The existence of each generalized weakly symmetric and generalized weakly Ricci symmetric Kenmotsu manifold is ensured by an example.

The main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

The research background of the paper covers the development of a country, that can be measured in various ways. Simple indicators, like GDP and also complex indicators such as HDI (human development index), can be used to measure country development. However, usually countries are divided into groups via setting some arbitrary levels of final measure. What is more, the composite (complex) indices have some problems and errors. The main purpose of the paper is the assessment of the development of the selected European OECD countries with the application of the linear ordering and ensemble clustering of symbolic data as well as comparison of the ensemble clustering with a single model. Research methodology covers linear ordering with the application of multidimensional scaling for a visualisation of results and ensemble clustering for symbolic data. The results are compared according to adjusted Rand and silhouette indices. The obtained results show that ensemble clustering for symbolic data can be a useful tool in country development analysis and allows reaching better results than a single model. The novelty of the proposed approach is to use a cluster analysis to obtain the clusters of countries with similar variables’ values (indicators of development) and the application of multidimensional scaling for symbolic data in order to visualise linear ordering results.

The main theme of the paper is the Concept of Military Leadership in the Slovenian Armed Forces, which currently represents the highest substantive and guiding normative act in the field of military leadership in the Slovenian Armed Forces (hereinafter referred to as the SAF). Due to the enormous importance of this field in working with people and the aim of influencing the change of the concept, the paper analysed and compared the concepts of military leadership in other selected armed forces and looked for similarities and divergences, especially in two segments - substantive and normative. On the one hand, we have shown the substantive obsolescence and inadequate normative rank of the Concept of Military Leadership in the Slovenian Armed Forces and, on the other hand, the necessity of substantive updating with concrete proposals and arguments for the development of the Doctrine of Military Leadership in the SAF. Key words Leadership, armed forces, Concept of Military Leadership in the SAF

The presented paper is strictly connected, among others, with the paper On the sum of some alternating series, Comp. Math. Appl. (2011), written by Wituła and Słota. A problem concerning the form of determinants formulated in the cited paper is solved here. Next, the obtained result is adapted to solve some system of linear equations and the description of the sum of alternating series.

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one. The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h(d) formula in real quadratic fields claims that we have h ( d ) . log ε d = Δ . ℒ ( 1 , χ d ) h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d > 0 and the fundamental unit ɛ d of ℚ ( d ) {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ-function and fundamental unit ɛ d are significant and necessary tools for determining the structure of real quadratic fields. The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 (mod4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element w d , fundamental unit ɛ d , and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

This paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix $M$ of size $n$ over a field $F$ with the property that there exists $\alpha\in F$ such that $M$ is similar to $diag(\alpha \cdot Id_k, M_1)$, where $M_1$ is cyclic and $0\leq k\leq n$). While a previous paper dealt with the Weil representations of finite classical groups, which play a key role in the general picture, the present paper provides a conclusive answer for all cross-characteristic projective irreducible representations of the finite quasi-simple groups of Lie type and their automorphism groups., Comment: To appear on Journal of Group Theory

This paper basically completes a project to enumerate permutations avoiding a triple T of 4-letter patterns, in the sense of classical pattern avoidance, for every T. There are 317 symmetry classes of such triples T and previous papers have enumerated avoiders for all but 14 of them. One of these 14 is conjectured not to have an algebraic generating function. Here, we find the generating function for each of the remaining 13, and it is algebraic in each case.

AG-groupoids (non-associative structure) are basic structures in Flocks theory. This theory mainly focuses on distance optimization, motion replication, and leadership maintenance with a wide range of applications in physics and biology. In this paper, we define some new types of fuzzy ideals of AG-groupoids called (α, β)-fuzzy bi-ideals, (α, β)-fuzzy interior ideals, (β̄, ᾱ)-fuzzy bi-ideals, and (β̄, ᾱ)-fuzzy interior ideals, where α, β∈{∈γ, qδ, ∈γ∨qδ, ∈γ∧qδ} and ᾱ, β̄∈{⋶γ, q̄δ, ⋶γ∨q̄δ, ⋶γ∧q̄δ}, with α≠∈γ∧qδ and β̄≠⋶γ∧q̄δ. An important milestone achieved by this paper is providing the connection between classical algebraic structures (ordinary bi-ideals, interior ideals) and new types of fuzzy algebraic structures [(∈γ, ∈γ∨qδ)-fuzzy bi-ideals, (∈γ, ∈γ∨qδ)-fuzzy interior ideals]. Special attention is given to (∈γ, ∈γ∨qδ)-fuzzy bi-ideals and (⋶γ, ⋶γ∨q̄δ) -fuzzy bi-ideals.

This paper concerns the existence and multiplicity of solutions for the Schrődinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent. As a particular case, we study the following degenerate Kirchhoff-type nonlocal problem: ‖ u ‖ λ ( θ − 1 ) p [ λ ( − Δ ) p s u + V ( x ) | u | p − 2 u ] = | u | p s ⋆ − 2 u + f ( x , u ) i n ℝ N , ‖ u ‖ λ = ( λ ∫ ℝ ∫ 2 N | u ( x ) − u ( y ) | p | x − y | N + p s d x d y + ∫ ℝ N V ( x ) | u | p d x ) 1 / p $$\begin{align}& \left\| u \right\|_{\lambda }^{\left( \theta -1 \right)p}\left[ \lambda \left( -\Delta \right)_{p}^{s}u+V\left( x \right){{\left| u \right|}^{p-2}}u \right]={{\left| u \right|}^{p_{s}^{\star }-2}}u+f\left( x,u \right)\,in\,{{\mathbb{R}}^{N}}, \\ & {{\left\| u \right\|}_{\lambda }}={{\left( \lambda \int\limits_{\mathbb{R}}{\int\limits_{2N}{\frac{{{\left| u\left( x \right)-u\left( y \right) \right|}^{p}}}{{{\left| x-y \right|}^{N+ps}}}}dxdy+\int\limits_{{{\mathbb{R}}^{N}}}{V\left( x \right){{\left| u \right|}^{p}}dx}} \right)}^{{1}/{p}\;}} \\ \end{align}$$ where ( − Δ ) p s $\left( -\Delta \right)_{p}^{s}$ is the fractional p–Laplacian with 0 < s < 1 < p < N/s, p s ⋆ = N p / ( N − p s ) $p_{s}^{\star }={Np}/{\left( N-ps \right)}\;$ is the critical fractional Sobolev exponent, λ > 0 is a real parameter, 1 < θ ≤ p s ⋆ / p , $1 and f : ℝ N × ℝ → ℝ is a Carathéodory function satisfying superlinear growth conditions. For θ ∈ ( 1 , p s ⋆ / p ) , $\theta \in \left( 1,{p_{s}^{\star }}/{p}\; \right),$ by using the concentration compactness principle in fractional Sobolev spaces, we show that if f(x, t) is odd with respect to t, for any m ∈ ℕ+ there exists a Λm > 0 such that the above problem has m pairs of solutions for all λ ∈ (0, Λm ]. For θ = p s ⋆ / p , $\theta ={p_{s}^{\star }}/{p}\;,$ by using Krasnoselskii’s genus theory, we get the existence of infinitely many solutions for the above problem for λ large enough. The main features, as well as the main difficulties, of this paper are the facts that the Kirchhoff function is zero at zero and the potential function satisfies the critical frequency inf x ∈ℝ V(x) = 0. In particular, we also consider that the Kirchhoff term satisfies the critical assumption and the nonlinear term satisfies critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p–Laplacian case.

In the present paper, we introduce a certain subclass 𝒦q(λ, γ, h) of analytic functions by means of a quasi-subordination. Sharp bounds of the Fekete-Szegő functional for functions belonging to the class 𝒦q(λ, γ, h) are obtained. The results presented in the paper give improved versions for the certain subclasses involving the quasi-subordination and majorization.

This paper deals with the methodology for practical application of nonparametric control charts. This topic is very important for two reasons: firstly nonparametric control charts are very effective instruments for the realization of the statistical process monitoring phase I due to their robustness against various deviations from the data assumptions that must be met when applying model-based control charts. Secondly nonparametric control charts have very weak SW support and also they are not taught in the frame of training courses not even of the university study programmes. For that reason the practitioners do not know them and do not use them. The paper offers the proposal how to practically apply these control charts which is based on the complex simulation study of various nonparametric control charts performance when various data assumptions have not been met. The study has covered these nonparametric control charts: Shewhart sign control chart, nonparametric EWMA and nonparametric CUSUM control charts, nonparametric progressive mean control chart, control chart based on Mood statistics and robust median absolute deviation control chart. All charts have been studied in condition of not normally distributed data, autocorrelated data and data with nonconstant distribution parameters. The simulations were realized for statistically stable (IC – in control) and also statistically unstable (OC – out of control) processes. For the evaluation of the control charts performance median run length, 0.05-quantile, and 0.95-quantile were used.