Tbilisi Mathematical Journal Articles (Project Euclid)
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The latest articles from Tbilisi Mathematical Journal on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2018 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Fri, 20 Apr 2018 09:47 EDTFri, 20 Apr 2018 09:47 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Wreaths, mixed wreaths and twisted coactions
https://projecteuclid.org/euclid.tbilisi/1524232072
<strong>Ross Street</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 1--22.</p><p><strong>Abstract:</strong><br/>
Distributive laws between two monads in a 2-category $\mathscr K$, as defined by Jon Beck in the case $\mathscr{K} = \mathrm{Cat}$, were pointed out by the author to be monads in a 2-category $\mathrm{Mnd}\mathscr{K}$ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category $\mathrm{EM}\mathscr{K}$ of monads with different 2-cells from $\mathrm{Mnd}\mathscr{K}$.
Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others; they are comonads in $\mathrm{Mnd}\mathscr{K}$. Actually, as pointed out by John Power and Hiroshi Watanabe, there are a number of dual possibilities for mixed distributive laws.
It is natural then to consider mixed wreaths as we do in this article; they are comonads in $\mathrm{EM}\mathscr{K}$. There are also mixed opwreaths: comonads in the Kleisli construction completion $\mathrm{Kl}\mathscr{K}$ of $\mathscr{K}$. The main example studied here arises from a twisted coaction of a bimonoid on a monoid. A wreath determines a monad structure on the composite of the two endomorphisms involved; this monad is called the wreath product. For mixed wreaths, corresponding to this wreath product, is a convolution operation analogous to the convolution monoid structure on the set of morphisms from a comonoid to a monoid. In fact, wreath convolution is composition in a Kleisli-like construction. Walter Moreira’s Heisenberg product of linear endomorphisms on a Hopf algebra, is an example of such convolution, actually involving merely a mixed distributive law. Monoidality of the Kleisli-like construction is also discussed.
</p>projecteuclid.org/euclid.tbilisi/1524232072_20180420094755Fri, 20 Apr 2018 09:47 EDTBundle functors and fibrations
https://projecteuclid.org/euclid.tbilisi/1524232075
<strong>Anders Kock</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 65--82.</p><p><strong>Abstract:</strong><br/>
We give an account of bundle-functors and star-bundle-functors (known from differential geometry) in terms of fibered categories.
</p>projecteuclid.org/euclid.tbilisi/1524232075_20180420094755Fri, 20 Apr 2018 09:47 EDTHilsum–Skandalis maps as Frobenius adjunctions with application to geometric morphisms
https://projecteuclid.org/euclid.tbilisi/1524232076
<strong>Christopher Townsend</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 83--120.</p><p><strong>Abstract:</strong><br/>
Hilsum-Skandalis maps, from differential geometry, are studied in the context of a cartesian category. It is shown that Hilsum-Skandalis maps can be represented as stably Frobenius adjunctions. This leads to a new and more general proof that Hilsum-Skandalis maps represent a universal way of inverting essential equivalences between internal groupoids.
To prove the representation theorem, a new characterisation of the connected components adjunction of any internal groupoid is given. The characterisation is that the adjunction is covered by a stable Frobenius adjunction that is a slice and whose right adjoint is monadic. Geometric morphisms can be represented as stably Frobenius adjunctions. As applications of the study we show how it is easy to recover properties of geometric morphisms, seeing them as aspects of properties of stably Frobenius adjunctions.
</p>projecteuclid.org/euclid.tbilisi/1524232076_20180420094755Fri, 20 Apr 2018 09:47 EDTEquilogical spaces and algebras for a double-power monad
https://projecteuclid.org/euclid.tbilisi/1524232077
<strong>Giulia Frosoni</strong>, <strong>Giuseppe Rosolini</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 121--139.</p><p><strong>Abstract:</strong><br/>
We investigate the algebras for the double-power monad on the Sierpisnki space in the category $\mathcal{Equ}$ of equilogical spaces, a cartesian closed extension of $\mathcal{Top}_0$ introduced by Scott, and the relationship of such algebras with frames. In particular, we focus our attention on interesting subcategories of $\mathcal{Equ}$. We prove uniqueness of the algebraic structure for a large class of equilogical spaces, and we characterize the algebras for the double-power monad in the category of algebraic lattices and in the category of continuous lattices, seen as full subcategories of $\mathcal{Equ}$.
We also analyse the case of algebras in the category $\mathcal{Top}_0$ of $\mathrm{T}_0$-spaces, again seen as a full subcategoy of $\mathcal{Equ}$, proving that each algebra for the double-power monad in $\mathcal{Top}_0$ has an underlying sober, compact, connected space.
</p>projecteuclid.org/euclid.tbilisi/1524232077_20180420094755Fri, 20 Apr 2018 09:47 EDTTriposes, exact completions, and Hilbert's ε-operator
https://projecteuclid.org/euclid.tbilisi/1524232078
<strong>Maria Emilia Maietti</strong>, <strong>Fabio Pasquali</strong>, <strong>Giuseppe Rosolini</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 141--166.</p><p><strong>Abstract:</strong><br/>
Triposes were introduced as presentations of toposes by J.M.E. Hyland, P.T. Johnstone and A.M. Pitts. They introduced a construction that, from a tripos $P:\mathcal{C}^\mathrm{op} \rightarrow \mathbf{Pos}$, produces an elementary topos $\mathcal{T}_P$ in such a way that the fibration of the subobjects of the topos $\mathcal{T}_P$ is freely obtained from $P$. One can also construct the “smallest” elementary doctrine made of subobjects which fully extends $P$, more precisely the free full comprehensive doctrine with comprehensive diagonals $P_\mathrm{cx}:\mathcal{Prd}_P\,^\mathrm{op}\rightarrow \mathbf{Pos}$ on $P$. The base category has finite limits and embeds into the topos $\mathcal{T}_P$ via a functor $K:\mathcal{Prd}_P \rightarrow \mathcal{T}_P$ determined by the universal property of $P_\mathrm{cx}$ and which preserves finite limits. Hence it extends to an exact functor $K^\mathrm{ex}:(\mathcal{Prd}_{P})_\mathrm{ex/lex} \rightarrow \mathcal{T}_P$ from the exact completion of $\mathcal{Prd}_P$.
We characterize the triposes $P$ for which the functor $K^\mathrm{ex}$ is an equivalence as those $P$ equipped with a so-called $\varepsilon$-operator. We also show that the tripos-to-topos construction need not preserve $\varepsilon$-operators by producing counterexamples from localic triposes constructed from well-ordered sets.
A characterization of the tripos-to-topos construction as a completion to an exact category is instrumental for the results in the paper and we derived it as a consequence of a more general characterization of an exact completion related to Lawvere's hyperdoctrines.
</p>projecteuclid.org/euclid.tbilisi/1524232078_20180420094755Fri, 20 Apr 2018 09:47 EDTOn fixed-point theorems in synthetic computability
https://projecteuclid.org/euclid.tbilisi/1524232079
<strong>Andrej Bauer</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 167--181.</p><p><strong>Abstract:</strong><br/>
Lawvere's fixed point theorem captures the essence of diagonalization arguments. Cantor's theorem, Gödel's incompleteness theorem, and Tarski's undefinability of truth are all instances of the contrapositive form of the theorem. It is harder to apply the theorem directly because non-trivial examples are not easily found, in fact, none exist if excluded middle holds.
We study Lawvere's fixed-point theorem in synthetic computability, which is higher-order intuitionistic logic augmented with the Axiom of Countable Choice, Markov's principle, and the Enumeration axiom, which states that there are countably many countable subsets of $\mathbb{N}$. These extra-logical principles are valid in the effective topos, as well as in any realizability topos built over Turing machines with an oracle, and suffice for an abstract axiomatic development of a computability theory.
We show that every countably generated $\omega$-chain complete pointed partial order ($\omega$cppo) is countable, and that countably generated $\omega$cppos are closed under countable products. Therefore, Lawvere's fixed-point theorem applies and we obtain fixed points of all endomaps on countably generated $\omega$cppos. Similarly, the Knaster-Tarski theorem guarantees existence of least fixed points of continuous endomaps. To get the best of both theorems, namely that all endomaps on domains ($\omega$cppos generated by a countable set of compact elements) have least fixed points, we prove a synthetic version of the Myhill-Shepherdson theorem: every map from an $\omega$cpo to a domain is continuous. The proof relies on a new fixed-point theorem, the synthetic Recursion Theorem, so called because it subsumes the classic Kleene-Rogers Recursion Theorem. The Recursion Theorem takes the form of Lawvere's fixed point theorem for multi-valued endomaps.
</p>projecteuclid.org/euclid.tbilisi/1524232079_20180420094755Fri, 20 Apr 2018 09:47 EDTThe construction of $\pi_0$ in Axiomatic Cohesion
https://projecteuclid.org/euclid.tbilisi/1524232080
<strong>Matías Menni</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 183--207.</p><p><strong>Abstract:</strong><br/>
We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of $\pi_0$ as the image of a natural map to a product of discrete spaces. A particular case of this construction produces, out of a local and hyperconnected geometric morphism $p : \mathcal{E} \rightarrow \mathcal{S}$, an idempotent monad $\pi_0 : \mathcal{E} \rightarrow \mathcal{E}$ such that, for every $X$ in $\mathcal{E}$, $\pi_{0}X = 1$ if and only if $(p^*\Omega)^! : (p^*\Omega)^1 \rightarrow (p^*\Omega)^X$ is an isomorphism. For instance, if $\mathcal{E}$ is the topological topos (over $\mathcal{S} = Set$), the $\pi_0$-algebras coincide with the totally separated (sequential) spaces. To illustrate the connection with classical topology we show that the $\pi_0$-algebras in the category of compactly generated Hausdorff spaces are exactly the totally separated ones. Also, in order to relate the construction above with the axioms for Cohesion we prove that, for a local and hyperconnected $p : \mathcal{E} \rightarrow \mathcal{S}$, $p$ is pre-cohesive if and only if $p^* : \mathcal{S} \rightarrow \mathcal{E}$ is cartesian closed. In this case, $p_! = p_* \pi_0 : \mathcal{E} \rightarrow \mathcal{S}$ and the category of $\pi_0$-algebras coincides with the subcategory $p^* : \mathcal{S} \rightarrow \mathcal{E}$.
</p>projecteuclid.org/euclid.tbilisi/1524232080_20180420094755Fri, 20 Apr 2018 09:47 EDTFunctoriality of modified realizability
https://projecteuclid.org/euclid.tbilisi/1524232081
<strong>Peter Johnstone</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 209--222.</p><p><strong>Abstract:</strong><br/>
We study the notion of modified realizability topos over an arbitrary Schönfinkel algebra. In particular we show that such toposes are induced by subsets of the algebra which we call right pseudo-ideals, and which generalize the right ideals (or right absorbing sets) previously considered. We also investigate the notion of compatibility with right pseudo-ideals which ensures that quasi-surjective (applicative) morphisms of Schönfinkel algebras yield geometric morphisms between these toposes.
</p>projecteuclid.org/euclid.tbilisi/1524232081_20180420094755Fri, 20 Apr 2018 09:47 EDTQuantalic topological theories
https://projecteuclid.org/euclid.tbilisi/1524232082
<strong>Walter Tholen</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 223--237.</p><p><strong>Abstract:</strong><br/>
The paper proposes the notions of topological platform and quantalic topological theory for the presentation and investigation of categories of interest beyond the realm of algebra. These notions are nevertheless grounded in algebra, through the notions of monad and distributive law. The paper shows how they entail previously proposed concepts with similar goals.
</p>projecteuclid.org/euclid.tbilisi/1524232082_20180420094755Fri, 20 Apr 2018 09:47 EDTEnriched and internal categories: an extensive relationship
https://projecteuclid.org/euclid.tbilisi/1524232083
<strong>Thomas Cottrell</strong>, <strong>Soichiro Fujii</strong>, <strong>John Power</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 239--254.</p><p><strong>Abstract:</strong><br/>
We consider an extant infinitary variant of Lawvere's finitary definition of extensivity of a category $\mathcal V$. In the presence of cartesian closedness and finite limits in $\mathcal V$, we give two characterisations of the condition in terms of a biequivalence between the bicategory of matrices over $\mathcal V$ and the bicategory of spans over discrete objects in $\mathcal V$. Using the condition, we prove that $\mathcal{V}﹣\mathrm{Cat}$ and the category $\mathrm{Cat}_\mathrm{d}(\mathcal{V})$ of internal categories in $\mathcal V$ with a discrete object of objects are equivalent. Our leading example has $\mathcal{V} = \mathrm{Cat}$, making $\mathcal{V}﹣\mathrm{Cat}$ the category of all small 2-categories and $\mathrm{Cat}_\mathrm{d}(\mathcal{V})$ the category of small double categories with discrete category of objects. We further show that if $\mathcal V$ is extensive, then so are $\mathcal{V}﹣\mathrm{Cat}$ and $\mathrm{Cat}(\mathcal{V})$, allowing the process to iterate.
</p>projecteuclid.org/euclid.tbilisi/1524232083_20180420094755Fri, 20 Apr 2018 09:47 EDTComprehensive factorisation systems
https://projecteuclid.org/euclid.tbilisi/1524232084
<strong>Clemens Berger</strong>, <strong>Ralph M. Kaufmann</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 255--277.</p><p><strong>Abstract:</strong><br/>
We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems exist for the categories of topological spaces, simplicial sets, small multicategories and Feynman categories. In each case comprehensive factorisation induces a natural notion of universal covering, leading to a Galois-type definition of fundamental group for based objects of the category.
</p>projecteuclid.org/euclid.tbilisi/1524232084_20180420094755Fri, 20 Apr 2018 09:47 EDTAlgebra and local presentability: how algebraic are they? (A survey)
https://projecteuclid.org/euclid.tbilisi/1524232085
<strong>Jiří Adámek</strong>, <strong>Jiří Rosický</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 279--295.</p><p><strong>Abstract:</strong><br/>
This is a survey of results concerning the algebraic hulls of two 2-categories: $\mathbf{VAR}$, the 2- category of finitary varieties, and $\mathbf{LFP}$, the 2-category of locally finitely presentable categories.
</p>projecteuclid.org/euclid.tbilisi/1524232085_20180420094755Fri, 20 Apr 2018 09:47 EDTOn the concrete representation of discrete enriched abstract clones
https://projecteuclid.org/euclid.tbilisi/1524232086
<strong>Marcelo Fiore</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 297--328.</p><p><strong>Abstract:</strong><br/>
We consider discrete enriched abstract clones and provide two constructions investigating their representation as discrete enriched clones of operations on objects in concrete enriched categories over the enriching category. Our first construction embeds a discrete enriched abstract clone into the concrete discrete enriched clone of operations on an object in the enriching category. Our second construction refines the given embedding by introducing a monoid action and restricting attention to the concrete discrete enriched clone of its equivariant operations. As in the classical theory of abstract clones, our main focus is on discrete enriched abstract clones with finite arities. However, we also consider discrete enriched abstract clones with countable arities to show that the representation theory of the former is conceptually explained by that of the latter.
</p>projecteuclid.org/euclid.tbilisi/1524232086_20180420094755Fri, 20 Apr 2018 09:47 EDTGeneralization of value distribution and uniqueness of certain types of difference polynomialshttps://projecteuclid.org/euclid.tbilisi/1524276027<strong>Harina P. Waghamore</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 1--13.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the distribution of zeros as well as the uniqueness problems of certain type of differential polynomials sharing a small function with finite weight. The result obtained improves and generalizes the recent results.
</p>projecteuclid.org/euclid.tbilisi/1524276027_20180420220037Fri, 20 Apr 2018 22:00 EDTStability of the general form of quadratic-quartic functional equations in non-Archimedean $\mathcal{L}$-fuzzy normed spaceshttps://projecteuclid.org/euclid.tbilisi/1524276028<strong>Abasalt Bodaghi</strong>, <strong>Pasupathi Narasimman</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 15--29.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce and obtain the general solution of a new generalized mixed quadratic and quartic functional equation and investigate its stability in non-Archimedean $\mathcal{L}$-fuzzy normed spaces.
</p>projecteuclid.org/euclid.tbilisi/1524276028_20180420220037Fri, 20 Apr 2018 22:00 EDTSome sequence spaces of Invariant means and lacunary defined by a Musielak-Orlicz function over $n$-normed spaceshttps://projecteuclid.org/euclid.tbilisi/1524276029<strong>Sunil K. Sharma</strong>, <strong>Kuldip Raj</strong>, <strong>Ajay K. Sharma</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 31--47.</p><p><strong>Abstract:</strong><br/>
In the present paper we introduce some sequence spaces combining lacunary sequence, invariant means over $n$-normed spaces defined by Musielak-Orlicz function $\mathcal{M} = (M_{k})$. We study some topological properties and also prove some inclusion results between these spaces.
</p>projecteuclid.org/euclid.tbilisi/1524276029_20180420220037Fri, 20 Apr 2018 22:00 EDTOn $\lambda$-pseudo bi-starlike and $\lambda$-pseudo bi-convex functions with respect to symmetrical pointshttps://projecteuclid.org/euclid.tbilisi/1524276030<strong>S. Sümer Eker</strong>, <strong>Bilal Şeker</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 49--57.</p><p><strong>Abstract:</strong><br/>
In this paper, defining new interesting classes, $\lambda$-pseudo bi-starlike functions with respect to symmetrical points and $\lambda$-pseudo bi-convex functions with respect to symmetrical points in the open unit disk $\mathbb U$, we obtain upper bounds for the initial coefficients of functions belonging to these new classes.
</p>projecteuclid.org/euclid.tbilisi/1524276030_20180420220037Fri, 20 Apr 2018 22:00 EDTA new extended generalized Burr-III family of distributionshttps://projecteuclid.org/euclid.tbilisi/1524276031<strong>Farrukh Jamal</strong>, <strong>Mohammad A. Aljarrah</strong>, <strong>M. H. Tahir</strong>, <strong>M. Arslan Nasir</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 59--78.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a new extended generalized Burr III family of distributions in the so-called T-BurrIII{Y} family by using the quantile functions of a few popular distributions. We derive the general mathematical properties of this extended family including explicit expressions for the quantile function, Shannon entropy, moments and mean deviations. Three new Burr III sub-families are then investigated, and four new extended Burr III models are discussed. The density of Burr III extended distributions can be symmetric, left-skewed, right-skewed or reversed-J shaped, and the hazard rate shapes can be increasing, decreasing, bathtub and upside-down bathtub. The potentiality of the newly generated distributions is demonstrated through applications to censored and complete data sets.
</p>projecteuclid.org/euclid.tbilisi/1524276031_20180420220037Fri, 20 Apr 2018 22:00 EDTSome different type integral inequalities concerning twice differentiable generalized relative semi-$(r; m, h)$-preinvex mappingshttps://projecteuclid.org/euclid.tbilisi/1524276032<strong>Artion Kashuri</strong>, <strong>Rozana Liko</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 79--97.</p><p><strong>Abstract:</strong><br/>
In this article, we first present some integral inequalities for Gauss-Jacobi type quadrature formula involving generalized relative semi-$(r; m, h)$-preinvex mappings. And then, a new identity concerning twice differentiable mappings defined on $m$-invex set is derived. By using the notion of generalized relative semi-$(r; m, h)$-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Hermite-Hadamard, Ostrowski and Simpson type inequalities via fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article.
</p>projecteuclid.org/euclid.tbilisi/1524276032_20180420220037Fri, 20 Apr 2018 22:00 EDTSome approximation properties of generalized integral type operatorshttps://projecteuclid.org/euclid.tbilisi/1524276033<strong>Alok Kumar</strong>, <strong> Vandana</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 99--116.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce and study the Stancu type generalization of the integral type operators defined in (1.1). First, we obtain the moments of the operators and then prove the Voronovskaja type asymptotic theorem and basic convergence theorem. Next, the rate of convergence and weighted approximation for the above operators are discussed. Then, weighted $L_p$-approximation and pointwise estimates are studied. Further, we study the $A$-statistical convergence of these operators. Lastly, we give better estimations of the above operators using King type approach.
</p>projecteuclid.org/euclid.tbilisi/1524276033_20180420220037Fri, 20 Apr 2018 22:00 EDTExistence of solution for a coupled system of Urysohn-Stieltjes functional integral equationshttps://projecteuclid.org/euclid.tbilisi/1524276034<strong>A. M. A. El-Sayed</strong>, <strong>M. M. A. Al-Fadel</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 117--125.</p><p><strong>Abstract:</strong><br/>
We present an existence theorem for at least one continuous solution for a coupled system of nonlinear functional (delay) integral equations of Urysohn-Stieltjes type.
</p>projecteuclid.org/euclid.tbilisi/1524276034_20180420220037Fri, 20 Apr 2018 22:00 EDTDifferential and integral equations associated with some hybrid families of Legendre polynomialshttps://projecteuclid.org/euclid.tbilisi/1524276035<strong>M. Riyasat</strong>, <strong>S. A. Wani</strong>, <strong>S. Khan</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 127--139.</p><p><strong>Abstract:</strong><br/>
The article aims to explore some new classes of differential and integral equations for some hybrid families of Legendre polynomials. Beginning with the recurrence relations and shift operators, the authors derived the differential, integro-differential and partial differential equations for the hybrid Legendre-Appell polynomials. Certain examples are framed for the hybrid Legendre-Bernoulli, Legendre-Euler and Legendre-Genocchi polynomials to show the applications of main results. Further, the homogeneous Volterra integral equations for the hybrid Legendre-Appell and other hybrid families of special polynomials are derived. The inclusion of integral equations is a bonus to this article.
</p>projecteuclid.org/euclid.tbilisi/1524276035_20180420220037Fri, 20 Apr 2018 22:00 EDTFekete-Szegö problem and Second Hankel Determinant for a class of bi-univalent functionshttps://projecteuclid.org/euclid.tbilisi/1524276036<strong>N. Magesh</strong>, <strong>J. Yamini</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 141--157.</p><p><strong>Abstract:</strong><br/>
In this paper we define a subclass of bi-univalent functions. Further, we find the estimates on the bounds $|a_{2}|$ and $|a_{3}|$, the Fekete-Szegö inequalities and the second Hankel determinant inequality for defined class of bi-univalent functions.
</p>projecteuclid.org/euclid.tbilisi/1524276036_20180420220037Fri, 20 Apr 2018 22:00 EDTSome results on $p$-calculushttps://projecteuclid.org/euclid.tbilisi/1524276037<strong>A. Neamaty</strong>, <strong>M. Tourani</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 159--168.</p><p><strong>Abstract:</strong><br/>
Our aim is to present some new properties of functions in $p$-calculus. The effects of a convex or monotone function on the $p$-derivative and vice versa and also the behavior of $p$-derivative in a neighborhood of a local extreme point are expressed. Moreover, mean value theorems for $p$-derivatives and $p$-integrals are proved.
</p>projecteuclid.org/euclid.tbilisi/1524276037_20180420220037Fri, 20 Apr 2018 22:00 EDTOn Soft Supra Compactness in Supra Soft Topological Spaceshttps://projecteuclid.org/euclid.tbilisi/1524276038<strong>A. M. Abd El-latif</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 1, 169--178.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce anew form of soft supra compact spaces namely, soft supra compact spaces, soft supra closed spaces, soft supra lindelof spaces and soft supra generalized compactness. Furthermore, we study its several properties and characterizations in detail. Also, the invariance of these kinds of soft supra compact spaces under some types of soft mapping and their hereditary properties are also investigated.
</p>projecteuclid.org/euclid.tbilisi/1524276038_20180420220037Fri, 20 Apr 2018 22:00 EDTOn algebraic $K$-functors of crossed group rings and its applicationshttps://projecteuclid.org/euclid.tbilisi/1529460017<strong>Giorgi Rakviashvili</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 1--15.</p><p><strong>Abstract:</strong><br/>
Let $R[\pi, \sigma, \rho]$ be a crossed group ring. An induction theorem is proved for the functor $G_{0}^{R}(R[\pi, \sigma, \rho])$ and the Swan-Gersten higher algebraic $K$-functors $K_{i}(R[\pi,\sigma,\rho])$. Using this result, a theorem on reduction is proved for the discrete normalization ring $R$ with the field of quotients $K$: I f $P$ and $Q$ are finitely generated $R[\pi, \sigma, \rho]$-projective modules and $K\bigotimes_{R}P\simeq K\bigotimes_{R}Q$ as $K[\pi, \sigma, \rho]$-modules, then $P\simeq Q.$ Under some restrictions on $n=(\pi:1)$ it is shown that finitely generated $R[\pi,\sigma,\rho]$-projective modules are decomposed into the direct sum of left ideals of the ring $R[\pi,\sigma,\rho]$. More stronger results are proved when $\sigma=id$.
</p>projecteuclid.org/euclid.tbilisi/1529460017_20180619220033Tue, 19 Jun 2018 22:00 EDTHarmonic numbers operational matrix for solving fifth-order two point boundary value problemshttps://projecteuclid.org/euclid.tbilisi/1529460019<strong>Y. H. Youssri</strong>, <strong>W. M. Abd-Elhameed</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 17--33.</p><p><strong>Abstract:</strong><br/>
The principal purpose of this paper is to present and implement two numerical algorithms for solving linear and nonlinear fifth-order two point boundary value problems. These algorithms are developed via establishing a new Galerkin operational matrix of derivatives. The nonzero elements of the derived operational matrix are expressed explicitly in terms of the well-known harmonic numbers. The key idea for the two proposed numerical algorithms is based on converting the linear or nonlinear fifth-order two BVPs into systems of linear or nonlinear algebraic equations by employing Petrov-Galerkin or collocation spectral methods. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the two proposed algorithms.
</p>projecteuclid.org/euclid.tbilisi/1529460019_20180619220033Tue, 19 Jun 2018 22:00 EDTOn skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$https://projecteuclid.org/euclid.tbilisi/1529460020<strong>Mohammad Ashraf</strong>, <strong>Ghulam Mohammad</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 35--45.</p><p><strong>Abstract:</strong><br/>
In the present paper, we study skew cyclic codes over the ring $F_{q}+vF_{q}+v^2F_{q}$, where $v^3=v,~q=p^m$ and $p$ is an odd prime. The structural properties of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ have been studied by using decomposition method. By defining a Gray map from $F_{q}+vF_{q}+v^2F_{q}$ to $F_{q}^3$, it has been proved that the Gray image of a skew cyclic code of length $n$ over $F_{q}+vF_{q}+v^2F_{q}$ is a skew $3$-quasi cyclic code of length $3n$ over $F_{q}$. Further, it is shown that the skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ are principally generated. Finally, the idempotent generators of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ have also been studied.
</p>projecteuclid.org/euclid.tbilisi/1529460020_20180619220033Tue, 19 Jun 2018 22:00 EDTCrossed semimodules of categories and Schreier 2-categorieshttps://projecteuclid.org/euclid.tbilisi/1529460021<strong>Sedat Temel</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 47--57.</p><p><strong>Abstract:</strong><br/>
The main idea of this paper is to introduce the notion of a Schreier 2-category and of a crossed semimodule over categories and to prove the categorical equivalence between their categories.
</p>projecteuclid.org/euclid.tbilisi/1529460021_20180619220033Tue, 19 Jun 2018 22:00 EDTA matrix application on absolute weighted arithmetic mean summability factors of infinite serieshttps://projecteuclid.org/euclid.tbilisi/1529460022<strong>Şebnem Yildiz</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 59--65.</p><p><strong>Abstract:</strong><br/>
In this present paper, we have generalized a main theorem dealing with $|\bar{N}, p_{n}|_{k}$ summability of non-decreasing sequences to $|A, p_{n}|_{k}$ summability method by using almost increasing sequences and taking normal matrices in place of weighted mean matrices.
</p>projecteuclid.org/euclid.tbilisi/1529460022_20180619220033Tue, 19 Jun 2018 22:00 EDTUniqueness for the difference monomials of $\rm{P}$-adic entire functionshttps://projecteuclid.org/euclid.tbilisi/1529460023<strong>Chao Meng</strong>, <strong>Gang Liu</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 67--76.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to discuss the uniqueness of p-adic difference monomials $f^{n}f(z+c)$. The results obtained in this paper are the p-adic analogues and supplements of the theorems given by Qi, Yang and Liu [Uniqueness and periodicity of meromorphic functions concerning the difference operator, Comput. Math. Appl. 60(2010), 1739-1746], Wang, Han and Wen [Uniqueness theorems on difference monomials of entire functions, Abstract Appl. Anal. 2012(2012), Article ID 407351], Yang and Hua [Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22(1997), 395-406].
</p>projecteuclid.org/euclid.tbilisi/1529460023_20180619220033Tue, 19 Jun 2018 22:00 EDTApproximate $n$-dimensional additive functional equation in various Banach spaceshttps://projecteuclid.org/euclid.tbilisi/1529460024<strong>Abasalt Bodaghi</strong>, <strong>Mohan Arunkumar</strong>, <strong>Elumalai Sathya</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 77--96.</p><p><strong>Abstract:</strong><br/>
In this paper, we modify the Cauchy additive functional equation and find all solutions of this new functional equation. Then, we study generalized Ulam-Hyers stability of such functional equation in various Banach spaces via Hyers' method.
</p>projecteuclid.org/euclid.tbilisi/1529460024_20180619220033Tue, 19 Jun 2018 22:00 EDTGeneral solution and stability of quattuorvigintic functional equation in matrix paranormed spaceshttps://projecteuclid.org/euclid.tbilisi/1529460025<strong>J. M. Rassias</strong>, <strong>R. Murali</strong>, <strong>M. J. Rassias</strong>, <strong>V. Vithya</strong>, <strong>A. A. Raj</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 97--109.</p><p><strong>Abstract:</strong><br/>
In this current work, we acquire the general solution and Hyers-Ulam stability for a new form of Quattuorvigintic functional equation in Matrix Paranormed Space by using the direct and fixed point methods.
</p>projecteuclid.org/euclid.tbilisi/1529460025_20180619220033Tue, 19 Jun 2018 22:00 EDTLaguerre-based Hermite-Bernoulli polynomials associated with bilateral serieshttps://projecteuclid.org/euclid.tbilisi/1529460026<strong>Waseem Ahmad Khan</strong>, <strong>Serkan Araci</strong>, <strong>Mehmet Acikgoz</strong>, <strong>Ayhan Esi</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 111--121.</p><p><strong>Abstract:</strong><br/>
In the paper, we define Laguerre-based Hermite-Bernoulli polynomial with its generating function, and investigate certain properties. From this generating function, we derive summation formulas and related bilateral series associated with the newly introduced generating function. Some of whose special cases are also presented. Relevant connections of some results presented here with those involving simpler known partly unilateral and partly bilateral representations are also obtained.
</p>projecteuclid.org/euclid.tbilisi/1529460026_20180619220033Tue, 19 Jun 2018 22:00 EDTA sinc-Gauss-Jacobi collocation method for solving Volterra's population growth model with fractional orderhttps://projecteuclid.org/euclid.tbilisi/1529460027<strong>Abbas Saadatmandi</strong>, <strong>Ali Khani</strong>, <strong>Mohammad-Reza Azizi</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 123--137.</p><p><strong>Abstract:</strong><br/>
A new sinc-Gauss-Jacobi collocation method for solving the fractional Volterra's population growth model in a closed system is proposed. This model is a nonlinear fractional Volterra integro-differential equation where the integral term represents the effects of toxin. The fractional derivative is considered in the Liouville-Caputo sense. In the proposed method, we first convert fractional Volterra's population model to an equivalent nonlinear fractional differential equation, and then the resulting problem is solved using collocation method. The proposed collocation technique is based on sinc functions and Gauss-Jacobi quadrature rule. In this approach, the problem is reduced to a set of algebraic equations. The obtained numerical results of the present method are compared with some well-known results in the literature to show the applicability and efficiency of the proposed method.
</p>projecteuclid.org/euclid.tbilisi/1529460027_20180619220033Tue, 19 Jun 2018 22:00 EDTAn application of Perov type results in gauge spaceshttps://projecteuclid.org/euclid.tbilisi/1530842676<strong>Lakshmi Narayan Mishra</strong>, <strong>Animesh Gupta</strong>, <strong> Vandana</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 139--151.</p><p><strong>Abstract:</strong><br/>
In this paper we present Perov type fixed point theorems for contractive mappings in Gheorghiu’s sense on spaces endowed with a family of vector valued pseudo-metrics. Applications to systems of integral equations are given to illustrate the theory. The examples also prove the advantage of using vector valued pseudo-metrics and matrices that are convergent to zero, for the study of systems of equations.
</p>projecteuclid.org/euclid.tbilisi/1530842676_20180705220447Thu, 05 Jul 2018 22:04 EDTCertain subclasses of bi-univalent functions associated with the Chebyshev polynomials based on Hohlov operatorhttps://projecteuclid.org/euclid.tbilisi/1530842677<strong>G. Murugusundaramoorthy</strong>, <strong>K. Vijaya</strong>, <strong>H. Ö. Güney</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 2, 153--166.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce and investigate two new subclasses of the function class $\Sigma$ biunivalent functions in the open unit disk, which are associated with the Hohlov operator, and satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-MacLaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses by using Chebyshev polynomials. Several new consequences of these results are also pointed out.
</p>projecteuclid.org/euclid.tbilisi/1530842677_20180705220447Thu, 05 Jul 2018 22:04 EDTMajorizatiuon and Zipf-Mandelbrot lawhttps://projecteuclid.org/euclid.tbilisi/1538532023<strong>Naveed Latif</strong>, <strong>Đilda Pečarić</strong>, <strong>Josip Pečarić</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 1--27.</p><p><strong>Abstract:</strong><br/>
In this paper we show how the Zipf-Mandelbrot law is connected to the theory of majorization. Firstly we consider the Csiszár $f$-divergence for the Zipf-Mandelbrot law and then develop important majorization inequalities for these divergences. We also discuss some special cases for our generalized results by using the Zipf-Mandelbrot law. As applications, we present the majorization inequalities for various distances obtaining by some special convex functions in the Csiszár $f$-divergence for Z-M law like the Rényi $\alpha$-order entropy for Z-M law, variational distance for Z-M law, the Hellinger distance for Z-M law, $\chi^{2}$-distance for Z-M law and triangular discrimination for Z-M law. At the end, we give important applications of the Zipf's law in linguistics and obtain the bounds for the Kullback-Leibler divergence of the distributions associated to the English and the Russian languages.
</p>projecteuclid.org/euclid.tbilisi/1538532023_20181002220035Tue, 02 Oct 2018 22:00 EDTExistence of a pair of new recurrence relations for the Meixner-Pollaczek polynomialshttps://projecteuclid.org/euclid.tbilisi/1538532024<strong>E. I. Jafarov</strong>, <strong>A. M. Jafarova</strong>, <strong>S. M. Nagiyev</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 29--39.</p><p><strong>Abstract:</strong><br/>
We report on existence of pair of new recurrence relations (or difference equations) for the Meixner-Pollaczek polynomials. Proof of the correctness of these difference equations is also presented. Next, we found that subtraction of the forward shift operator for the Meixner-Pollaczek polynomials from one of these recurrence relations leads to the difference equation for the Meixner-Pollaczek polynomials generated via $\cosh$ difference differentiation operator. Then, we show that, under the limit $\varphi \to 0$, new recurrence relations for the Meixner-Pollaczek polynomials recover pair of the known recurrence relations for the generalized Laguerre polynomials. At the end, we introduced differentiation formula, which expresses Meixner-Pollaczek polynomials with parameters $\lambda>0$ and $0 \lt \varphi \lt \pi$ via generalized Laguerre polynomials.
</p>projecteuclid.org/euclid.tbilisi/1538532024_20181002220035Tue, 02 Oct 2018 22:00 EDTA note to establish the Hyers-Ulam stability for a nonlinear integral equation with Lipschitzian kernelhttps://projecteuclid.org/euclid.tbilisi/1538532025<strong>Mohammad Saeed Khan</strong>, <strong>Dinu Teodorescu</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 41--45.</p><p><strong>Abstract:</strong><br/>
In the stability theory, the nonlinear equations were not so much investigated. In this note we consider the stability of a nonlinear integral equation with Lipschitzian kernel. The approach is based on monotonicity properties of a nonlinear operator.
</p>projecteuclid.org/euclid.tbilisi/1538532025_20181002220035Tue, 02 Oct 2018 22:00 EDTThe transmuted Gompertz-G family of distributions: properties and applicationshttps://projecteuclid.org/euclid.tbilisi/1538532026<strong>Hesham Reyad</strong>, <strong>Farrukh Jamal</strong>, <strong>Soha Othman</strong>, <strong>G. G. Hamedani</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 47--67.</p><p><strong>Abstract:</strong><br/>
We introduce and study a new class of continuous distributions called the transmuted Gompertz-G family which extends the Gompertz class proposed by Alizadeh et al. (2016a). Explicit expressions for the ordinary and incomplete moments, generating function, probability weighted moment, Lorenz and Bonferroni curves, order statistics, Rényi and Shanon entropies, stress strength model moment of residual and reversed residual life and characterizations for the new family are investigated. We discuss the maximum likelihood estimates for the model parameters. The performance of the new family is assesed by means of two applications.
</p>projecteuclid.org/euclid.tbilisi/1538532026_20181002220035Tue, 02 Oct 2018 22:00 EDTA nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectrahttps://projecteuclid.org/euclid.tbilisi/1538532027<strong>Michael Ching</strong>, <strong>John E. Harper</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 69--79.</p><p><strong>Abstract:</strong><br/>
The aim of this short paper is to prove a $\mathsf{TQ}$-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can be thought of as a $\mathsf{TQ}$-homology analog for structured ring spectra of Dror's generalized Whitehead theorem for topological spaces; here $\mathsf{TQ}$-homology is short for topological Quillen homology. We also prove retract theorems for the $\mathsf{TQ}$-completion and homotopy completion of nilpotent structured ring spectra.
</p>projecteuclid.org/euclid.tbilisi/1538532027_20181002220035Tue, 02 Oct 2018 22:00 EDTConnection problems and matrix representations for certain hybrid polynomialshttps://projecteuclid.org/euclid.tbilisi/1538532028<strong>Subuhi Khan</strong>, <strong>Tabinda Nahid</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 81--93.</p><p><strong>Abstract:</strong><br/>
In this paper, we deal with the connection and duplication problems associated with the hybrid Sheffer family. The hybrid Sheffer polynomials are also studied via matrix approach. The properties of these polynomials are established using simple matrix operations. Examples providing the corresponding results for certain members of the hybrid Sheffer family are considered. This article is first attempt in the direction of obtaining connection and duplication coefficients and matrix representations for the hybrid polynomials.
</p>projecteuclid.org/euclid.tbilisi/1538532028_20181002220035Tue, 02 Oct 2018 22:00 EDTLogarithmic-Sheffer polynomials of the second kindhttps://projecteuclid.org/euclid.tbilisi/1538532029<strong>Paolo Emilio Ricci</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 95--106.</p><p><strong>Abstract:</strong><br/>
In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them have been studied. In this article new sets of logarithmic-Sheffer polynomials are introduced. Connection with Bell numbers are shown.
</p>projecteuclid.org/euclid.tbilisi/1538532029_20181002220035Tue, 02 Oct 2018 22:00 EDTCommon fixed points of set mappings endowed with directed graphshttps://projecteuclid.org/euclid.tbilisi/1538532030<strong>Jamshaid Ahmad</strong>, <strong>Ahmed Al-Rawashdeh</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 107--123.</p><p><strong>Abstract:</strong><br/>
The aim of this article is to obtain common fixed points of set mappings defined on a family of sets endowed with a graph satisfying $\Theta-$contraction. Our results generalize and extend various results in the existing literature. We also provide some non trivial examples to support the validity of our main results. Some applications to the construction of common fixed points of set mappings in $\varepsilon$-chainable metric space are also discussed.
</p>projecteuclid.org/euclid.tbilisi/1538532030_20181002220035Tue, 02 Oct 2018 22:00 EDTCertain results on para-Kenmotsu manifolds equipped with $M$-projective curvature tensorhttps://projecteuclid.org/euclid.tbilisi/1538532031<strong>Abhishek Singh</strong>, <strong>Shyam Kishor</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 125--132.</p><p><strong>Abstract:</strong><br/>
The purpose of the article is to study the certain results on para-Kenmotsu manifolds equipped with $M$-projective curvature tensor. Here we investigate para-Kenmotsu manifolds satisfying some curvature conditions $\widetilde{M}\cdot R=0,$ $\widetilde{M}\cdot Q=0$ and $Q\cdot \widetilde{M}=0,$ where $R$, $Q$ and $\widetilde{M}$ respectively denote the Riemannian curvature tensor, Ricci operator and $M$-projective curvature tensor.
</p>projecteuclid.org/euclid.tbilisi/1538532031_20181002220035Tue, 02 Oct 2018 22:00 EDTMarcinkiewicz integrals with rough kernel associated with Schrödinger operators and commutators on generalized vanishing local Morrey spaceshttps://projecteuclid.org/euclid.tbilisi/1538532032<strong>Ferit Gürbüz</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 133--156.</p><p><strong>Abstract:</strong><br/>
Let $L=-\Delta+V\left( x\right) $ be a Schrödinger operator, where $\Delta$ is the Laplacian on ${\mathbb{R}^{n}}$, while nonnegative potential $V\left(x\right)$ belonging to the reverse Hölder class. In this paper, using the some conditions on $\varphi\left(x.r\right) $, we dwell on the boundedness of Marcinkiewicz integrals with rough kernel associated with schrödinger operators and commutators generated by these operators and local Campanato functions both on generalized local Morrey spaces and on generalized vanishing local Morrey spaces, respectively. As an application of the above results, the boundedness of parametric Marcinkiewicz integral and its commutator both on generalized local Morrey spaces and on generalized vanishing local Morrey spaces is also obtained.
</p>projecteuclid.org/euclid.tbilisi/1538532032_20181002220035Tue, 02 Oct 2018 22:00 EDTMultiple solutions of critical singular degenerate elliptic system with concave-convex nonlinearitieshttps://projecteuclid.org/euclid.tbilisi/1538532033<strong>Chang-Mu Chu</strong>, <strong>Lin Li</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 157--174.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to study a class of critical singular degenerate elliptic system with concave-convex nonlinearities and sign-changing weight functions. The existence and multiplicity of nontrivial nonnegative solutions are obtained by the variational.
</p>projecteuclid.org/euclid.tbilisi/1538532033_20181002220035Tue, 02 Oct 2018 22:00 EDTSome estimations of summation-integral-type operatorshttps://projecteuclid.org/euclid.tbilisi/1538532034<strong>Vishnu Narayan Mishra</strong>, <strong>Rishikesh Yadav</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 3, 175--191.</p><p><strong>Abstract:</strong><br/>
In this paper, the Szász-Mirakjan-Kantorovich type operators are studied with the help of direct result and weighted approximation properties. Some basic lemmas, theorems are given and proved, at last, we discuss the rate of convergence and a comparison takes place with the Szász-Mirakjan-Kantorovich operators by graphical representations.
</p>projecteuclid.org/euclid.tbilisi/1538532034_20181002220035Tue, 02 Oct 2018 22:00 EDTDouble absolute indexed matrix summability with its applicationshttps://projecteuclid.org/euclid.tbilisi/1546570881<strong>B. B. Jena</strong>, <strong>S. K. Paikray</strong>, <strong>U. K. Misra</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 1--18.</p><p><strong>Abstract:</strong><br/>
The well established theory of summability of simple series has been brought to a high degree of development; however the extension of this theory to multiple series is still in its infancy. As regards to the double series, in the proposed paper a result on absolute indexed matrix summability with an additional parameter of doubly infinite lower triangular matrix has been established that generalizes a theorem of E. Savaş and B. E. Rhoades [10] (see E. Savaş and B. E. Rhoades, Double absolute summability factor theorems and applications, Nonlinear Anal. 69 (2008), 189-200). Furthermore, some concluding remarks and applications are presented in support of our result.
</p>projecteuclid.org/euclid.tbilisi/1546570881_20190103220131Thu, 03 Jan 2019 22:01 ESTSemi-slant Riemannian maps from almost contact metric manifolds into Riemannian manifoldshttps://projecteuclid.org/euclid.tbilisi/1546570882<strong>Rajendra Prasad</strong>, <strong>Sushil Kumar</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 19--34.</p><p><strong>Abstract:</strong><br/>
Firstly, a generalization of Riemannian submersions, slant submersions and semi-slant submersions, we introduce semi-slant Riemannian maps from almost contact metric manifolds onto Riemannian manifolds. In this paper, we obtain some results on such maps by taking the vertical structure vector field. Among them, we study integrability of distributions and the geometry of foliations. Further, we find the necessary and sufficient conditions for semi-slant Riemannian maps to be harmonic and totally geodesic. We, also investigate some decomposition theorems and provide some examples to show the existence of the maps.
</p>projecteuclid.org/euclid.tbilisi/1546570882_20190103220131Thu, 03 Jan 2019 22:01 ESTNon-global solutions for a class of fourth-order wave equationshttps://projecteuclid.org/euclid.tbilisi/1546570883<strong>Tarek Saanouni</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 35--42.</p><p><strong>Abstract:</strong><br/>
Adopting the so-called concavity method, we establish a finite time blow-up result for a class of fourth-order non-linear wave equations with positive energy.
</p>projecteuclid.org/euclid.tbilisi/1546570883_20190103220131Thu, 03 Jan 2019 22:01 ESTA note on the new set operator $\psi_{r}$https://projecteuclid.org/euclid.tbilisi/1546570884<strong>Arife Atay</strong>, <strong>Hasan İlhan Tutalar</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 43--52.</p><p><strong>Abstract:</strong><br/>
Recently many published works made on local function used in ideal topological spaces can be found in related literature. "Semi Local Functions in Ideal Topological Spaces", "Closure Local Functions", and "$()^p$ and $\psi_p$-Operator" can be mentioned among such works those aim to define such functions. In general, the researchers prefer using the generalized open sets instead of topology in ideal topological spaces. Obtaining a Kuratowski closure operator with the help of local functions is an important detail in ideal topological space. However, it is not possible to obtain a Kuratowski closure operator from many of these local functions proposed by the above mentioned works. In order to address the lack of such an operator, the goal of this paper is to introduce another local function to give possibility of obtaining a Kuratowski closure operator. On the other hand, regular local functions defined for ideal topological spaces have not been found in the current literature. Regular local functions for the ideal topological spaces has been described within this work. Moreover, with the help of regular local functions Kuratowski closure operators $cl_I^{*r}$ and $\tau^{*r}$ topology are obtained. Many theorems in the literature have been revised according to the definition of regular local functions.
</p>projecteuclid.org/euclid.tbilisi/1546570884_20190103220131Thu, 03 Jan 2019 22:01 ESTModified Apostol-Euler numbers and polynomials of higher orderhttps://projecteuclid.org/euclid.tbilisi/1546570885<strong>Maged G. Bin-Saad</strong>, <strong>Ali Z. Bin-Alhag</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 53--66.</p><p><strong>Abstract:</strong><br/>
The purpose of this work is to give modified definitions of Apostol-Euler polynomials and numbers of higher order. We establish their elementary properties, sums, explicit relations, integrals and differential relations. A number of new results which introduced are generalization of known results and their special cases lead to the corresponding formulas of the classical Euler numbers and polynomials.
</p>projecteuclid.org/euclid.tbilisi/1546570885_20190103220131Thu, 03 Jan 2019 22:01 ESTUniqueness results related to L-functions and certain differential polynomialshttps://projecteuclid.org/euclid.tbilisi/1546570886<strong>Pulak Sahoo</strong>, <strong>Samar Halder</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 67--78.</p><p><strong>Abstract:</strong><br/>
In this paper, using the idea of weighted sharing we investigate the uniqueness problem of a meromorphic function and an $L$-function when certain differential polynomials generated by them share a nonzero finite value or have the same fixed points. Our results improve the recent results due to Liu-Li-Yi [Proc. Japan Acad. Ser. A, 93 (2017), 41-46].
</p>projecteuclid.org/euclid.tbilisi/1546570886_20190103220131Thu, 03 Jan 2019 22:01 ESTBlending type approximation by Stancu-Kantorovich operators associated with the inverse Pólya-Eggenberger distributionhttps://projecteuclid.org/euclid.tbilisi/1546570887<strong>M. Mursaleen</strong>, <strong>A. A. H. Al-Abied</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 79--91.</p><p><strong>Abstract:</strong><br/>
In this paper, we give some approximation properties by Stancu-Kantorovich operators based on inverse Pólya-Eggenberger distribution in the polynomial weighted space introduced in the literature and obtain convergence properties of these operators by using Korovkin's theorem. We discuss the direct result and Voronovskaja type asymptotic formula.
</p>projecteuclid.org/euclid.tbilisi/1546570887_20190103220131Thu, 03 Jan 2019 22:01 EST$I$-Convergence of triple difference sequence spaces over $n$-normed spacehttps://projecteuclid.org/euclid.tbilisi/1546570888<strong>Tanweer Jalal</strong>, <strong>Ishfaq Ahmad Malik</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 93--102.</p><p><strong>Abstract:</strong><br/>
The main objective of this paper is to study triple difference sequence spaces over $n$-normed space via the sequence of modulus functions. Some algebraic and topological properties of the newly constructed spaces are also established.
</p>projecteuclid.org/euclid.tbilisi/1546570888_20190103220131Thu, 03 Jan 2019 22:01 ESTOperator approach for orthogonality in linear spaceshttps://projecteuclid.org/euclid.tbilisi/1546570889<strong>Mahdi Iranmanesh</strong>, <strong>Maryam Saeedi Khojasteh</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 103--112.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce the operator approach for orthogonality in linear spaces. In particular, we represent the concept of orthogonal vectors using an operator associated with them, in normed linear spaces.
</p>projecteuclid.org/euclid.tbilisi/1546570889_20190103220131Thu, 03 Jan 2019 22:01 ESTPerturbed fourth-order Kirchho-type problemshttps://projecteuclid.org/euclid.tbilisi/1546570890<strong>Shapour Heidarkhani</strong>, <strong>Shahin Moradi</strong>, <strong>Giuseppe Caristi</strong>, <strong>Bin Ge</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 113--143.</p><p><strong>Abstract:</strong><br/>
We establish the existence of at least three distinct weak solutions for a perturbed nonlocal fourth-order Kirchhoff-type problem with Navier boundary conditions under appropriate hypotheses on nonlinear terms. Our main tools are based on variational methods and some critical points theorems. We give some examples to illustrate the obtained results.
</p>projecteuclid.org/euclid.tbilisi/1546570890_20190103220131Thu, 03 Jan 2019 22:01 ESTStructural properties for $(m,n)$-quasi-hyperideals in ordered semihypergroupshttps://projecteuclid.org/euclid.tbilisi/1546570891<strong>Ahsan Mahboob</strong>, <strong>Noor Mohammad Khan</strong>, <strong>Bijan Davvaz</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 145--163.</p><p><strong>Abstract:</strong><br/>
In this paper, we first introduce the notion of an $(m,n)$-quasi-hyperideal in an ordered semihypergroup and, then, study some properties of $(m,n)$-quasi-hyperideals for any positive integers $m$ and $n$. Thereafter, we characterize the minimality of an $(m,n)$-quasi-hyperideal in terms of $(m,0)$-hyperideals and $(0,n)$-hyperideals respectively. The relation $\mathcal{Q}_m^n$ on an ordered semihypergroup is, then, introduced for any positive integers $m$ and $n$ and proved that the relation $\mathcal{Q}_m^n$ is contained in the relation $\mathcal{Q}=\mathcal{Q}_1^1$. We also show that, in an $(m,n)$-regular ordered semihypergroup, the relation $\mathcal{Q}_m^n$ coincides with the relation $\mathcal{Q}$. Finally, the notion of an $(m,n)$-quasi-hypersimple ordered semihypergroup is introduced and some properties of $(m,n)$-quasi-hypersimple ordered semihypergroups are studied. We further show that, on any $(m,n)$-quasi-hypersimple ordered semihypergroup, the relations $\mathcal{Q}_m^n$ and $\mathcal{Q}$ are equal and are universal relations.
</p>projecteuclid.org/euclid.tbilisi/1546570891_20190103220131Thu, 03 Jan 2019 22:01 ESTOn the second radical elements of lattice moduleshttps://projecteuclid.org/euclid.tbilisi/1546570892<strong>Narayan Phadatare</strong>, <strong>Vilas Kharat</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 11, Number 4, 165--173.</p><p><strong>Abstract:</strong><br/>
Let $L$ be a $C$-lattice and $M$ be a lattice module over $L$. For a non-zero element $N\in M$, join of all second elements $X$ of $M$ with $X\leq N$ is called the second radical of $N$, and it is denoted by $\sqrt[s]{N}$. In this paper, we study some properties of second radical of elements of $M$ and obtain some related results.
</p>projecteuclid.org/euclid.tbilisi/1546570892_20190103220131Thu, 03 Jan 2019 22:01 EST