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The authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by Cabrer et al., give access to a viable method for studying admissible rules within relevance logic, specifically for extensions of the deductive system R -mingle. This paper builds on the work already done on the theory of natural dualities for Sugihara algebras. Its purpose is to provide an integrated suite of multisorted duality theorems of a uniform type, encompassing finitely generated quasivarieties and varieties of both Sugihara algebras and Sugihara monoids, and embracing both the odd and the even cases. The overarching theoretical framework of multisorted duality theory developed here leads on to amenable representations of free algebras. More widely, it provides a springboard to further applications.

A partition π of a set S is a collection B 1 , B 2 , …, B k of non-empty disjoint subsets, alled blocks, of S such that ⋃i=1kBi=S.$\begin{array}{}\displaystyle\bigcup _{i=1}^kB_i=S.\end{array}$ We assume that B 1 , B 2 , …, B k are listed in canonical order; that is in increasing order of their minimal elements; so min B 1 < min B 2 < ⋯ < min B k . A partition into k blocks can be represented by a word π = π 1 π 2 ⋯ π n , where for 1 ≤ j ≤ n , π j ∈ [ k ] and ⋃i=1n{πi}=[k],$\begin{array}{}\displaystyle\bigcup _{i=1}^n \{\pi_i\}=[k],\end{array}$ and π j indicates that j ∈ B π j . The canonical representations of all set partitions of [ n ] are precisely the words π = π 1 π 2 ⋯ π n such that π 1 = 1, and if i < j then the first occurrence of the letter i precedes the first occurrence of j . Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [ n ]. These squares can overlap and their bases are not necessarily on the x -axis. We determine the generating function P ( x , y , q ) for the number of set partitions of [ n ] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [ n ].

Let S be a set of states of a physical system and let p ( s ) be the probability of an occurrence of an event when the system is in the state s ∈ S . The function p from S to [0, 1] is called a numerical event, multidimensional probability or, alternatively, S -probability. Given a set of numerical events which has been obtained by measurements and not supposing any knowledge of the logical structure of the events that appear in the physical system, the question arises which kind of logic is inherent to the system under consideration. In particular, does one deal with a classical situation or a quantum one? In this survey several answers are presented. Starting by associating sets of numerical events to quantum logics we study structures that arise when S -probabilities are partially ordered by the order of functions and characterize those structures which indicate that one deals with a classical system. In particular, sequences of numerical events are considered that give rise to Bell-like inequalities. At the center of all studies there are so called algebras of S -probabilities, subsets of these and their generalizations. A crucial feature of these structures is that order theoretic properties can be expressed by the addition and subtraction of real functions entailing simplified algorithmic procedures. The study of numerical events and algebras of S -probabilities goes back to a cooperation of E. G. Beltrametti and M. J. Mączyński in 1991 and has since then resulted in a series of subsequent papers of physical interest the main results of which will be commented on and put in an appropriate context.

Building on similar notions for MV-algebras, polyhedral DMV-algebras are defined and investigated. For such algebras dualities with suitable categories of polyhedra are established, and the relation with finitely presented Riesz MV-algebras is investigated. Via hull-functors, finite products are interpreted in terms of hom-functors, and categories of polyhedral MV-algebras, polyhedral DMV-algebras and finitely presented Riesz MV-algebras are linked together. Moreover, the amalgamation property is proved for finitely presented DMV-algebras and Riesz MV-algebras, and for polyhedral DMV-algebras.

We introduce new classes of functions related to metric-preserving functions, b-metrics, and ultrametrics. We investigate their properties and compare them to those of metric-preserving functions.

In this paper, we give several characterizations for boundedness, essential norm and compactness of generalized weighted composition operators from Bloch type spaces to n th weighted type spaces.

Levels of fast escaping sets were discussed by Rippon and Stallard. Here we have defined a set B R ( f ) analogous to 0 th level of fast escaping set by using maximum term and formation of spider’s web structure has been discussed for this set.

Utilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane f n ( z ) + P ( z , f , f ′ , … , f ( t ) ) = P 1 e α 1 z + P 2 e α 2 z + P 3 e α 3 z , $$\begin{array}{}\displaystylef^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z},\end{array}$$ where P j and α i are nonzero constants for j = 1, 2, 3, such that | α 1 | > | α 2 | > | α 3 | and P ( z , f , f ′, …, f ( t ) is an algebraic differential polynomial in f ( z ) of degree no greater than n – 1.

A class of third order differential equations with a sublinear neutral term of the type a ( t ) b ( t ) x ( t ) + p ( t ) x α ( τ ( t ) ) ′ ′ ′ + q ( t ) x γ ( σ ( t ) ) = 0 $$\begin{array}{}\displaystyle\left( a(t)\left( b(t)\left[ x(t)+p(t)x^{\alpha }(\tau (t))\right] ^{\prime}\right) ^{\prime }\right) ^{\prime }+q(t)x^{\gamma }(\sigma (t))=0\end{array}$$ is considered. Some oscillation criteria are presented to improve and complement those in the literature. Several examples are established to illustrate the main results.

This paper is concerned with the existence and nonexistence of positive solutions for the eigenvalue problems of nonlinear Hadamard fractional differential equations with p -Laplacian operator. By applying the properties of the Green function and Guo-Krasnosel’skii fixed point theorem on cones, some existence and nonexistence results of positive solutions are obtained based on different eigenvalue intervals. Finally, some examples are presented to demonstrate the feasibility of our main results.

Let G be an IN-group and 0 < p , q < ∞. In this paper, we determine necessary and sufficient conditions for the existence of the convolution of functions, in the amalgam space L(p,q)π$\begin{array}{}L^{\pi}_{(p,q)}\end{array}$( G ).

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 note we will prove that the trace of a commutator of the discrete Cesáro operator and a compact operator is zero when the commutator belongs to the trace-class.

In this paper, we study Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold. First, we prove that if a Kenmotsu metric is a Yamabe soliton, then it has constant scalar curvature. Examples has been provided on a larger class of almost Kenmotsu manifolds, known as β -Kenmotsu manifold. Next, we study quasi Yamabe soliton on a complete Kenmotsu manifold M and proved that it has warped product structure with constant scalar curvature in a region Σ where ∣ Df ∣ ≠ 0.

In this paper, we prove that every non-Riemannian 4-th root metric of isotropic scalar curvature has vanishing scalar curvature. Then, we show that every 4-th root metric of weakly isotropic flag curvature has vanishing scalar curvature. Finally, we find the necessary and sufficient condition under which the conformal change of a 4-th root metric is of isotropic scalar curvature.

Characterizations of pseudoultrametric-preserving functions and semimetric-preserving functions are found. The structural properties of pseudoultrametrics which can be represented as a composition of an ultrametric and ultrametric-pseudoultrametric-preserving function are obtained. A dual form of Pongsriiam-Termwuttipong characterization of the ultrametric-preserving functions is described. We also introduce a concept of k -separating family of functions and use it to characterize the ultrametric spaces.

Considering the natural topology or 𝓢-density topology on the domain and on the range we obtain different families of continuous functions f : ℝ → ℝ. In this paper we compare these families in porosity terms. In particular, we obtain strengthening of some recent results by J. Hejduk, A. Loranty, R. Wiertelak.

In this paper, we propose a new three-parameter modified Burr XII distribution based on the standard Burr XII distribution and the composition technique developed by [14]. Among others, we show that this technique has the ability to significantly increase the flexibility of the former Burr XII distribution, with respect to the density and hazard rate shapes. Also, complementary theoretical aspects are studied as shapes, asymptotes, quantiles, useful expansion, moments, skewness, kurtosis, incomplete moments, moments generating function, stochastic ordering, reliability parameter and order statistics. Then, a Monte Carlo simulation study is carried out to assess the performance of the maximum likelihood estimates of the modified Burr XII model parameters. Finally, three applications to real-life data sets are presented, with models comparisons. The results are favorable for the new modified Burr XII model.

The purpose of this paper is to study a chi-square-type distribution who degrees of freedom are geometric random variables in connection with weak limiting distributions of geometric random sums of squares of independent, standard normal distributed random variables. Some characteristics of chi-square-type random variables with geometrically distributed degrees of freedom including probability density function, probability distribution function, mean and variance are calculated. Some asymptotic behaviors of chi-square-type random variables with geometrically distributed degrees of freedom are also established via weak limit theorems for normalized geometric random sums of squares of independent, standard normal distributed random variables. The rates of convergence in desired weak limit theorems also estimated through Trotter’s distance. The received results are extensions and generalizations of several known results.

We give an alternative lower bound for the numerical radii of Hilbert space operators. As a by-product, we find conditions such that ω 0 R S 0 = ∥ R ∥ + ∥ S ∥ 2 $$\begin{array}{}\displaystyle\omega\left(\left[\begin{array}{cc}0 & R \\S & 0\end{array}\right]\right)=\frac{\Vert R \Vert +\Vert S\Vert }{2}\end{array}$$ where R , S ∈ 𝔹(𝓗).