Volume 67, Issue 3

In this paper, we study local Hilbert algebras and give some theorems that characterize local Hilbert algebras. Also, we prove that, (i) H is a local Hilbert algebra if and only if the set of all dense elements of H is H ∖ {0}. (ii) H is a local Hilbert algebra if and only if the set of all regular elements of H is a local Boolean algebra. In a local Hilbert algebra H with supremum, we prove that: (i) If F is a Boolean filter of H , then F is a prime filter. (ii) F is a Boolean filter iff F is a maximal filter iff F = H ∖ {0}.

Following [BOTUR, M.—CHAJDA, I.—HALAŠ, R.: Are basic algebras residuated structures? , Soft Comput. 14 (2010), 251—255] we discuss the connections between left-residuated partially ordered groupoids and the so-called basic algebras, which are a non-commutative and non-associative generalization of MV-algebras and orthomodular lattices.

We state and prove the first law of cubology of the Rubik’s Revenge and provide necessary and sufficient conditions for a randomly assembled Rubik’s Revenge to be solvable.

For orthomodular lattices the word problem is still not decided, an important obstacle is the lack of distributivity. But also the absence of the associativity law for operations other than the lattice operations meet and join causes difficulties. In this paper we treat some aspects of associativity in orthomodular lattices. There are six out of 96 orthomodular operations which are associative. We search for orthomodular lattice operations which fulfil the associative identity under some conditions. Specifically we assume that some arguments commute and identify, for each operation, sufficient conditions under which a triple of elements fulfils the respective associativity identity.

Given k ≥ 2, let a n be the sequence defined by the recurrence a n = α 1 a n –1 + … + α k a n – k for n ≥ k , with initial values a 0 = a 1 = … = a k –2 = 0 and a k –1 = 1. We show under a couple of assumptions concerning the constants α i that the ratio annan− 1n− 1 $\frac{\sqrt[n]{a_n}}{\sqrt[n-1]{a_{n-1}}}$ is strictly decreasing for all n ≥ N , for some N depending on the sequence, and has limit 1. In particular, this holds in the cases when all of the α i are unity or when all of the α i are zero except for the first and last, which are unity. Furthermore, when k = 3 or k = 4, it is shown that one may take N to be an integer less than 12 in each of these cases.

The concept of conditionally independent sets is introduced in this article. Its link with the concept of the conditional probability and with the concept of a system of independent sets is described here. Furthermore, particular example of a system of conditionally independent sets is given and different ways to create the system of conditionally independent sets are described.

In this paper, we propose a novel encryption scheme based on the concepts of the commutative law of the k -th order linear recurrences over the finite field 𝔽 q for k > 2. The proposed encryption scheme is an ephemeral-static, which is useful in situations like email where the recipient may not be online. The security of the proposed encryption scheme depends on the difficulty of solving some Linear Feedback Shift Register (LFSR) problems. It has also the property of semantic security. For k = 2, we propose another encryption scheme by using the commutative law of Lucas and Fibonacci sequences having also the same security as proposed for k > 2.

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational sides. We discuss the problem of finding such quadrilaterals where the ratio of the radii of the circumcircle and incircle is rational. We show that this problem can be formulated in terms of a family of elliptic curves given by E a : y 2 = x 3 + ( a 4 − 4 a 3 − 2 a 2 − 4 a + 1) x 2 + 16 a 4 x which have, in general, (ℤ/8ℤ), and in rare cases (ℤ/2ℤ × ℤ/8ℤ) as torsion subgroups. We show the existence of infinitely many elliptic curves E a of rank at least two with torsion subgroup ℤ/8ℤ, parametrised by the points of an elliptic curve of rank at least one, and give five particular examples of rank 5. We, also, show the existence of a subfamily of E a whose torsion subgroup is ℤ/2ℤ × ℤ/8ℤ.

We show that for Dirichlet characters χ , χ 1 , …, χ s mod p m the sum ∑ x1=1pm… ∑ xs=1pmχ 1(x1)… χ s(xs)χ (A1x1+⋯ +Asxs+Bx1w1⋯ xsws) $$\begin{array}{} \displaystyle\sum\limits_{x_1=1}^{p^m} \dots \sum\limits_{x_s=1}^{p^m}\chi_1(x_1)\dots \chi_s(x_s) \chi(A_1x_1 + \dots + A_sx_s + Bx_1^{w_1}\cdots x_s^{w_s}) \end{array} $$ has a simple evaluation when m is sufficently large, for m ≥ 2 if p∤ 2A1⋯ AsB(1− w1− ⋯ − ws). $$p\nmid 2A_1\cdots A_sB(1-w_1-\cdots -w_s). $$

In this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X.—BUGAJEWSKI, D.: On formal Laurent series , Bull. Braz. Math. Soc. 42 (2011), 415–437], is not continuous. We also show some applications of fixed point theorems to some nonlinear mappings defined on the space of formal power series or on the space of formal Laurent series. Finally, in the second part of the paper, we propose another approach to study of dot product and multiplication of formal Laurent series, in particular establishing integral representation of dot product and convolution representation of multiplication.

We study b1′ $b_{1}'$ ( M ), the co-rank of the fundamental group of a smooth closed connected manifold M . We calculate this value for the direct product of manifolds. We characterize the set of all possible combinations of b1′ $b_{1}'$ ( M ) and the first Betti number b 1 ( M ) by explicitly constructing manifolds with any possible combination of b1′ $b_{1}'$ ( M ) and b 1 ( M ) in any given dimension. Finally, we apply our results to the topology of Morse form foliations. In particular, we construct a manifold M and a Morse form ω on it for any possible combination of b1′ $b_{1}'$ ( M ), b 1 ( M ), m ( ω ), and c ( ω ), where m ( ω ) is the number of minimal components and c ( ω ) is the maximum number of homologically independent compact leaves of ω .

In this paper, we introduce the notion of generalized hyperaction of hypermonoid (GHS-act) on a set, establish its basic properties and prove some fundamental results. We also define the notion of prime, semiprime, irreducible and maximal GHS-subacts and characterize hypermonoid in terms of these GHS-subacts.

In the paper we present some properties of functions which are ϱ -upper continuous in O’Malley’s sense. Connections with the standard ϱ -upper continuous functions are studied and the maximal additive class and the maximal multiplicative class for this family of functions are described.

The proofs of uniqueness theorems, presented here, allow to extend the earlier results. For example, the following hold: let μ and ν be two finitely additive probabilities on a structure L̃ (for example, L̃ is a pseudo-effect algebra), and let μ be convex-ranged; if there exists an element a ∈ L̃ with 0 < μ ( a ) < 1 and such that μ ( a ) = μ ( b ) ⇒ ν ( a ) = ν ( b ) whenever b ∈ L̃ , then ν = μ .

In this paper, we will consider value sharing problems on a transcendental entire function f with its differential-difference polynomials. We establish some results that can be viewed as the differential-difference analogues of Brück conjecture.

In this paper we derive oscillation criteria for the second order half-linear neutral differential equation [r(t)Φ (z′(t))]′+c(t)Φ (x(σ (t)))=0,z(t)=x(t)+b(t)x(τ (t)), $$ \displaystyle \Bigl[r(t)\Phi(z'(t))\Bigr]'+c(t)\Phi(x(\sigma(t)))=0, \quad z(t)=x(t)+b(t)x(\tau(t)), $$ where Φ( t ) = | t | p −2 t , p ≥ 2, is a power type nonlinearity. We improve recent results published in the literature by obtaining better oscillation constants and removing the usual condition σ ( τ ( t )) = τ ( σ ( t )). Two methods (comparison method and Riccati equation method) are used.

In the present paper we prove a separation theorem for t -Wright convex and t -Wright concave functionals. We also consider the problem of supporting the t -Wright concave functions by t -Wright affine ones. As a consequence of these results we prove some decomposition theorems for t -Wright convex functions.

The following notion of bounded index for complex entire functions was presented by Lepson. function f ( z ) is of bounded index if there exists an integer N independent of z , such that max{l:0≤ l≤ N}|f(l)(z)|l!≥ |f(n)(z)|n!for alln. $$ \max\limits_{\{l: 0\leq l\leq N\}} \left \{ \frac{|{f^{(l)}(z)}|}{l!}\right \} \geq \frac{|{f^{(n)}(z)}|}{n!}\quad\text{for all}\,\, n. $$ The main goal of this paper is extend this notion to holomorphic bivariate function. To that end, we obtain the following definition. A holomorphic bivariate function is of bounded index, if there exist two integers M and N such that M and N are the least integers such that max{(k,l):0,0≤ k,l≤ M,N}|f(k,l)(z,w)|k!l!≥ |f(m,n)(z,w)|m!n!for allmandn. $$ \max\limits_{\{(k,l): 0,0\leq k, l\leq M, N\}} \left \{ \frac{|{f^{(k,l)}(z,w)}|}{k!\,l!}\right \} \geq \frac{|{f^{(m,n)}(z,w)}|}{m!\,n!}\quad\text{for all}\,\, m \, \text{and}\,\, n. $$ Using this notion we present necessary and sufficient conditions that ensure that a holomorphic bivariate function is of bounded index.

In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set γ of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve γ . Complete asymptotic expansions will be constructed if that order is constant along γ , and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested.

Let 𝓜(Ω, μ ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ ). Let B ⊂ 𝓜(Ω, μ ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p -negative type when viewed as a metric subspace of L p (Ω, μ ) if and only if B is an affinely independent subset of 𝓜(Ω, μ ) (when 𝓜(Ω, μ ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ ) has strict p -negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p -negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ ) that have p -negative type but not q -negative type for any q > p . Such sets preclude the existence of certain types of isometry into L p -spaces.

The present article is devoted to the study of conditions on a hemi-slant submanifold of a nearly Kaehler manifold under which the submanifold is a warped product submanifold.

In multivariate models the test usually used for a linear hypothesis is the Wilks lambda test. There does not seem to be a need for any alternative. However, if the covariance matrix is parameterized by a few parameters only, and the parameters can be estimated, the problem arises whether a “plug-in” test, which approximates the test in a situation where the covariance matrix is totally known, is more suitable. A partial answer to this question is given in the paper. The sensitivity approach is used in order to check how this principle works in a situation where a plug-in procedure can be compared with another, already established, procedure.

The health care costs have been rapidly rising in recent years and the public health insurance companies are highly interested in prescription policies of general practitioners. In our contribution, a complex model for prescription behaviour of Belgian general practitioners is built using the structural equation modelling (SEM) framework. The model involves a large number of prescribed medicament groups as well as doctors’ and patients’ characteristics. As one of the results, a relatively small number of medicament groups, which effectively describe the prescription behaviour of a given doctor, is obtained. These indicators are consequently used in a generalized linear model for predicting the drug expenses per patient. Such a model can be used as a useful guideline for the expenses’ assessment of a particular practitioner.

A characterization of the congruence lattice of a principal p -algebra satisfying a certain condition in terms of congruence pairs is given.