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In a recent paper [JUKIĆ, D.: A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set, J. Comput. Appl. Math. 375 (2020)], a new existence level was introduced and then was used to obtain a necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set. In this paper, we determined that existence level for the residual sum of squares of the power-law regression with an unknown location parameter, and so we obtained a necessary and sufficient condition which guarantee the existence of the least squares estimate.

In this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence.

This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator ℓ p → X {\ell_{p}\to X} has the BAP when X has it and 0 < p ≤ 1 {0 , which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

The paper deals with the nonlinear tensor diffusion which yields a coherence improvement. It is very appropriate for images with flow-like structures. Two convolutions are used in the construction of diffusion tensor for such a model, see [Drblíková, O.—Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. Anal. 46 (2007), 37–60], [Weickert, J.: Coherence-enhancing diffusion filtering, Int. J. Comput. Vis. 31 (1999), 111–127]. In this paper we introduce the third supplemental convolution in order to enhance the diffusion strategy. First, we briefly present the classical coherence enhancing model and explain our modification. Then the discrete scheme is provided. The core of the paper consists in numerical experiments. Benefits of the additional convolution are discussed and illustrated in the figures.

As a continuation of a previous paper, we prove Perelman’s assertion, that is, for ancient solutions to the Ricci flow with bounded nonnegative curvature operator, uniformly bounded entropy is equivalent to κ-noncollapsing on all scales. We also establish an equality between the asymptotic entropy and the asymptotic reduced volume, which is a result similar to a paper by Xu (2017), where he assumes the Type I curvature bound.

In this paper, a new relation for the set of interval-valued fuzzy relations is introduced. This relation is an interval order for the family of intervals and for the family of interval-valued fuzzy relations in a given set, it has the reflexivity property. Consequences of considering such a relation are studied in the context of operations on interval-valued fuzzy relations. A new transitivity property, namely possible T-transitivity is studied (pos-T-transitivity for short). This transitivity property is connected with the new relation proposed in this paper. Preservation of this type of transitivity by some operations is also discussed.

The aim of this paper is to introduce some new classes of functions termed as somewhat (μ, λ)-continuous functions, somewhat (μ, λ)-open functions and hardly (μ, λ)-open functions. Some properties of these newly defined functions are discussed in this paper. Relationships among these functions are also being studied.

The purpose of the paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial. The results of the paper improve and generalize the recent results due to X. B. Zhang and J. F. Xu [19]. We also solve an open problem as posed in the last section of [19].

In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class 𝓢 of univalent functions in the unit disc.

In this paper, we introduce the notion of fuzzy costs in a more general context, in universal algebra by the use of coset terms. We study the structure of fuzzy cosets by applying the general theory of algebraic fuzzy systems. Fuzzy cosets generated by a fuzzy set are characterized in different ways. It is also proved that the class of fuzzy cosets determined by an element forms an algebraic closure fuzzy set system. Finally, we give a set of necessary and sufficient conditions for a given variety of algebras to be congruence permutable by applying the theory of fuzzy cosets.

In this paper, we use some methods of combinatorial set theory, in particular, the ones related to the construction of independent families of sets and also some modified version of the notion of small sets originally introduced by Riečan and Neubrunn, to give an abstract and generalized formulation of a remarkable theorem of Kakutani and Oxtoby related to a nonseparable invariant extension of the Lebesgue measure in spaces with transformation groups.

Having only two parameters, the Gamma-Lindley distribution does not provide enough flexibility for analyzing different types of lifetime data. From this perspective, in order to further enhance its flexibility, we set forward in this paper a new class of distributions named Generalized Gamma-Lindley distribution with four parameters. Its construction is based on certain mixtures of Gamma and Lindley distributions. The truncated moment, as a characterization method, has drawn a little attention in the statistical literature over the great popularity of the classical methods. We attempt to prove that the Generalized Gamma-Lindley distribution is characterized by its truncated moment of the first order statistics. This method rests upon finding a survival function of a distribution, that is a solution of a first order differential equation. This characterization includes as special cases: Gamma, Lindley, Exponential, Gamma-Lindley and Weighted Lindley distributions. Finally, a simulation study is performed to help the reader check whether the available data follow the underlying distribution.

The Tribonacci sequence is a generalization of the Fibonacci sequence which starts with 0,0,1 and each term afterwards is the sum of the three preceding terms. Here, we show that the only Tribonacci numbers that are concatenations of two repdigits are 13,24,44,81. This paper continues a previous work that searched for Fibonacci numbers which are concatenations of two repdigits.

The present paper introduces and studies the concepts of K-outer approximation and K-inner approximation for a monotone function μ defined on a D-poset P, by a subfamily K of P. Some desirable properties of K-approximable functions are established and it is shown that the family of all elements of P that possess K-approximation, forms a lattice and is closed under orthosupplementation. We have proved that a submodular measure on a suitable subfamily of P having K-outer approximation can be extended to a function that has K-outer approximation, and a tight function that has K-inner approximation can be extended to a function having K-inner approximation.

The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

In this paper, we introduce a shrinking projection method of an inertial type with self-adaptive step size for finding a common element of the set of solutions of a split generalized equilibrium problem and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step size incorporated helps to overcome the difficulty of having to compute the operator norm, while the inertial term accelerates the rate of convergence of the proposed algorithm. Under standard and mild conditions, we prove a strong convergence theorem for the problems under consideration and obtain some consequent results. Finally, we apply our result to solve split mixed variational inequality and split minimization problems, and we present numerical examples to illustrate the efficiency of our algorithm in comparison with other existing algorithms. Our results complement and generalize several other results in this direction in the current literature.

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in {\mathbb{R}^{n}} , whose level set {\{u=0\}} is contained in a half-space. If {n\leq 3} , we prove that the solution must be one-dimensional. In dimension {n\geq 4} , we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

In this paper, the notion of an operator \gamma on a supra topological space (X,\mu ) is studied and then utilized to analyze supra \gamma -open sets. The notions of {\mu }_{\gamma }-g.closed sets on the subspace are introduced and investigated. Furthermore, some new {\mu }_{\gamma }-separation axioms are formulated and the relationships between them are shown. Moreover, some characterizations of the new functions via operator \gamma on \mu are presented and investigated. Finally, we give some properties of {S}_{(\gamma ,\beta )}-closed graph and strongly {S}_{(\gamma ,\beta )}-closed graph.

In this paper, we show that the Diophantine equation a x +(a +2) y = z 2, where a ≡ 5 (mod 42) and a ∈ ℕ has no solution in non-negative integers.

In this paper we propose an efficient multivariate encryption scheme based on permutation polynomials over finite fields. We single out a commutative group ℒ(q, m) of permutation polynomials over the finite field F q m. We construct a trapdoor function for the cryptosystem using polynomials in ℒ(2, m), where m =2 k for some k ≥ 0. The complexity of encryption in our public key cryptosystem is O(m 3) multiplications which is equivalent to other multivariate public key cryptosystems. For decryption only left cyclic shifts, permutation of bits and xor operations are used. It uses at most 5m 2+3m – 4 left cyclic shifts, 5m 2 +3m + 4 xor operations and 7 permutations on bits for decryption.

We outline the transition from classical probability space (Ω, A, p) to its "divisible" extension, where (as proposed by L. A. Zadeh) the σ-field A of Boolean random events is extended to the class 𝓜(A) of all measurable functions into [0,1] and the σ-additive probability measure p on A is extended to the probability integral ∫(·) dp on 𝓜(A). The resulting extension of (Ω, A,p) can be described as an epireflection reflecting A to 𝓜(A) and p to ∫(·) dp. The transition from A to 𝓜(A), resembling the transition from whole numbers to real numbers, is characterized by the extension of two-valued Boolean logic on A to multivalued Łukasiewicz logic on 𝓜(A) and the divisibility of random events: for each random event u ∈ 𝓜(A) and each positive natural number n we have u/n ∈ 𝓜(A) and ∫(u/n) dp = (1/n) ∫u dp. From the viewpoint of category theory, objects are of the form 𝓜(A), morphisms are observables from one object into another one and serve as channels through which stochastic information is conveyed. We study joint random experiments and asymmetrical stochastic dependence/independence of one constituent experiment on the other one. We present a canonical construction of conditional probability so that observables can be viewed as conditional probabilities. In the present paper we utilize various published results related to "quantum and fuzzy" generalizations of the classical theory, but our ultimate goal is to stress mathematical (categorical) aspects of the transition from classical to what we call divisible probability.

We introduce and investigate central lifting property (CLP) for orthomodular lattices as a property whereby all central elements can be lifted modulo every p-ideal. It is shown that prime ideals, maximal ideals and finite p-ideals have CLP. Also Boolean algebras, simple chain finite orthomodular lattices, subalgebras of an orthomodular lattices generated by two elements and finite orthomodular lattices have CLP. The main results of the present paper include the investigation of CLP for principal p-ideals and finite direct products of orthomodular lattices.

In this paper we investigate Fibonacci type sequences defined by kth order linear recurrence. Based on their companion matrix and its graph interpretation we determine multinomial and binomial formulas for these sequences. Moreover we present a graphical rule for calculating the words of these sequences from the Pascal’s triangle.

In this paper we introduce a generalization of Jacobsthal hybrid numbers – J(r, n)-Jacobsthal hybrid numbers. We give some of their properties: character, Binet’s formula, a summation formula and a generating function.

We study points of joint continuity of multi-variable separately quasi-continuous functions which are continuous with respect to one variable. We will show that the assumption of complete regularity in the paper of V. Maslyuchenko et al in ”Tatra Mt. Math. Pub. 68, (2017), 47–58” can be removed in some situations. Moreover, we use a topological game argument to prove that the set of points of continuity of functions with closed graph into ωΔ-spaces is a Gδ subset of its domain provided that its domain is a W-space.

In the paper some properties of compositions of ϱ-lower continuous functions are presented. We will show conditions for a function and for a homeomorphism f : I → I under which g ◦ f is ϱ-lower continuous for each ϱ-lower continuous function g : I → 𝕉. Some relevant properties will be discussed.

In this paper, we study the evolution of the Weyl curvature tensor W of hypersurfaces in 𝕉 n +1 under the mean curvature flow. We find a bound for the Weyl curvature tensor of hypersurfaces during the evolution in terms of time. As a consequence, we suppose that the initial hypersurface is conformally flat, i.e., W =0 at t = 0 and then we find an upper estimate for W during the evolution in terms of time.

In this paper we investigate some fundamental properties of unital topology on a lattice ordered group with order unit. We show that some essential properties of order unit norm on a vector lattice with order unit, are valid for unital l-groups. For instance we show that for an Archimedean Riesz space G with order unit u, the unital topology and the strong link topology are the same.

This paper is devoted to the study of the class of decomposable double MS-algebras. Necessary and sufficient conditions for a decomposable MS-algebra to be a decomposable double MS-algebra are deduced. We construct decomposable double MS-algebras by means of decomposable MS-quadruples and we prove that there exists a one-to-one correspondence between decomposable double MS-algebras and decomposable MS-quadruples. Moreover, a construction of decomposable K 2-algebras (Stone algebras) by means of K 2-quadruples (Stone quadruples) is given. We conclude by introducing and characterizing isomorphisms of decomposable double MS-algebras in terms of decomposable MS-quadruples.

In this paper we define the monadic pseudo BE-algebras and investigate their properties. We prove that the existential and universal quantifiers of a monadic pseudo BE-algebra form a residuated pair. Special properties are studied for the particular case of monadic bounded commutative pseudo BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is proved that the quantifiers on bounded commutative pseudo BE-algebras are also quantifiers on the corresponding pseudo MV-algebras. The monadic deductive systems and monadic congruences of monadic pseudo BE-algebras are defined and their properties are studied. It is proved that, in the case of a monadic distributive commutative pseudo BE-algebra there is a one-to-one correspondence between monadic congruences and monadic deductive systems, and the monadic quotient pseudo BE-algebra algebra is also defined.

The purpose of this paper is to introduce a new type of φ -implicit relation in S - metric spaces and to prove a general fixed point for a pair of weakly compatible mappings, which generalize Theorems 1, 2, 4 [23], Theorems 1-7 [13], Corollary 2.19 [13], Theorems 2.2, 2.4 [19], Theorems 3.2, 3.3, 3.4 [20] and other known results.

In the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.

In this paper a new class of sets termed as ω ∗ μ-open sets has been introduced and studied. Using these concept, a unified theory for decomposition of (μ, λ)-continuity has been given.

In this paper, we find the condition on a function ω and a weight v which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ℳ p , ω ⁢ ( ℝ n , v ) {{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} and generalized weighted central Morrey spaces ℳ ˙ p , ω ⁢ ( ℝ n , v ) {\dot{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} , when v belongs to the Muckenhoupt A ∞ {A_{\infty}} -class.

In this paper, we give some necessary and sufficient conditions under which the topology generated by a partial metric is equivalent to the topology generated by a suitably defined metric. Next, we study some new extensions of the Generalized Banach Contraction Principle to partial metric spaces. Moreover, we draw a particular attention to the space of all sequences showing, in particular, that some well-known fixed point theorems for ultrametric spaces, can be used for operators acting in that space. We illustrate our considerations by suitable examples and counterexamples.

In this paper, a kind of new analogue of Phillips operators based on (p,q)-integers is introduced. The moments of the operators are established. Then some local approximation for the above operators is discussed. Also, the rate of convergence and weighted approximation by these operators by means of modulus of continuity are studied. Furthermore, the Voronovskaja type asymptotic formula is investigated.

The purpose of the paper is to show that noncanonical operator $$\begin{array}{} \displaystyle \mathcal {L}\,y=\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)' \end{array}$$ can be easily written in essentially unique canonical form $$\begin{array}{} \displaystyle \mathcal {L}\,y = q_3(t)\left(q_2(t)\left(q_1(t)\left(q_0(t)y(t)\right)'\right)'\right)' \end{array}$$ such that $$\begin{array}{} \displaystyle \int\limits^\infty \frac{1}{q_i(s)}\,\text{d}{s}=\infty, \quad i=1,2. \end{array}$$ The canonical representation is applied for examination of the third order noncanonical equations $$\begin{array}{} \displaystyle \left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'+p(t)y(\tau(t))=0. \end{array}$$

The purpose of this paper is to find common fixed point results for two families of multivalued mappings fulfilling generalized rational type A–dominated contractive conditions on a closed ball in complete dislocated b-metric spaces. Some new fixed point results with graphic contractions on a closed ball for two families of multi-graph dominated mappings on dislocated b-metric space have been established. An application to the unique common solution of two families of nonlinear integral equations is presented to show the novelty of our results.

In this paper, we break the index set N into disjoint subsets S and its complement, and propose two S-type exclusion sets that all the eigenvalues do not belong to them. Furthermore, we establish new S-type eigenvalue inclusion sets, which can reduce computations and obtain more accurate numerical results. At the same time, we give two criteria for identifying nonsingular tensors. Finally, new S-type eigenvalue inclusion sets are shown to be sharper than existing results via two examples.

The paper deals with two ideals of subsets of ℕ {\mathbb{N}} defined with the help of density points and its category analogue. We present basic properties of these ideals and show that they are incomparable under inclusion.

In this paper we introduce a new cryptographic system which is based on the idea of encryption due to [McEliece, R. J. A public-key cryptosystem based on algebraic coding theory, DSN Progress Report. 44, 1978, 114–116]. We use the McEliece encryption system with a new linear error-correcting code, which was constructed in [Hannusch, C.—Lakatos, P.: Construction of self-dual binary 22 k, 22 k−1, 2 k-codes, Algebra and Discrete Math. 21 (2016), no. 1, 59–68]. We show how encryption and decryption work within this cryptosystem and we give the parameters for key generation. Further, we explain why this cryptosystem is a promising post-quantum candidate.

In the present paper we shall introduce some generalized difference Cesàro sequence spaces of fuzzy real numbers defined by Musielak-Orlicz function and -convergence. We make an e ort to study some topological and algebraic properties of these sequence spaces. Furthermore, some inclusion relations between these sequence spaces are establish.

Fully homomorphic encryption (FHE) has been among the most popular research topics of the last decade. While the bootstrapping-based, public key cryptosystems that follow Gentry’s original design are getting more and more efficient, their performance is still far from being practical. This leads to several attempts to construct symmetric FHE schemes that would not be as inefficient as their public key counterparts. Unfortunately, most such schemes were also based on (randomized) linear transformations, and shown to be completely insecure. One such broken scheme was the Matrix Operation for Randomization and Encryption (MORE). In a recent paper, Hariss, Noura and Samhat propose Enhanced MORE, which is supposed to improve over MORE’s weaknesses. We analyze Enhanced MORE, discuss why it does not improve over MORE, and show that it is even less secure by presenting a highly efficient ciphertext-only decryption attack. We implement the attack and confirm its correctness.

The most commonly used methods for solving classical (historical) ciphers are based on global optimization (meta-heuristic methods). Despite the fact that global optimization is a well-studied problem, in the case of classical ciphers, there are still many open questions such as the construction of fitness functions or efficient transformation of the cryptanalysis (breaking attempt) to an optimization problem. Therefore the transformation of a cryptanalytical task to an optimization problem and the choice of a suitable fitness function form an important part of the topic. In this paper, we focus on the simple columnar transposition in depth. Our main contribution is a detailed analysis and comparison of different fitness functions, fitness landscape analysis and solving experiments.

In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

After the establishment of the finite embeddability property for integral and commutative residuated ordered monoids by Blok and Van Alten in 2002, the finite embeddability property for some other types of residuated ordered algebraic structures have been extensively studied. The main purpose of this paper is to construct a finite quantale X̂F from a finite partial subalgebra F of an increasing quantum B-algebra X so that F can be embedded into X̂F , that is, the class of increasing quantum B-algebras has the finite embeddability property.

The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI -simple and DLO-semiring and prove that every EMV-semiring is a GI -simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

In this paper, we study the Hardy type space H ℒ p ⁢ ( ℝ n ) {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} by means of local maximal functions associated with the heat semigroup e - t ⁢ ℒ {e^{-t\mathcal{L}}} generated by - ℒ {-\mathcal{L}} , where ℒ = - Δ + μ {\mathcal{L}=-\Delta+\mu} is the generalized Schrödinger operator in ℝ n {{\mathbb{R}^{n}}} ( n ≥ 3 {n\geq 3} ) and μ ≢ 0 {\mu\not\equiv 0} is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Via the equivalence of the norms between various local maximal functions, we show that the norms ∥ f ∥ H ℒ p ⁢ ( ℝ n ) p {\lVert f\rVert_{H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})}^{p}} and ∥ f ∥ H m p , q ⁢ ( ℝ n ) p {\lVert f\rVert_{H_{m}^{p,q}({\mathbb{R}^{n}})}^{p}} are equivalent for n n + δ ′ < p ≤ 1 ≤ q ≤ ∞ {\frac{n}{n+\delta^{\prime}} ( p ≠ q {p\neq q} ) with some δ ′ > 0 {\delta^{\prime}>0} . As applications, we prove that Calderón–Zygmund operators related to the auxiliary function m ⁢ ( x , μ ) {m(x,\mu)} are bounded from H ℒ p ⁢ ( ℝ n ) {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} for n n + γ 1 < p ≤ 1 {\frac{n}{n+\gamma_{1}} with γ 1 > 0 {\gamma_{1}>0} . In particular, we show that the Riesz transform ∇ ( - Δ + μ ) - 1 2 {\nabla(-\Delta+\mu)^{-\frac{1}{2}}} , which is a special example of the above Calderón–Zygmund operators, is bounded from H ℒ p ⁢ ( ℝ n ) {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into H p ⁢ ( ℝ n ) {{H}^{p}({\mathbb{R}^{n}})} for n n + γ 1 < p ≤ 1 {\frac{n}{n+\gamma_{1}} with 0 < γ 1 < 1 {0 .

In this paper, we investigate the relationships between fixed points of meromorphic functions, and their higher order differences and shifts, and generalize the case of fixed points into the more general case for first order difference and shift. Concretely, some estimates on the order and the exponents of convergence of special points of meromorphic functions and their differences and shifts are obtained.