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 EDTReal sets
https://projecteuclid.org/euclid.tbilisi/1524232073
<strong>George Janelidze</strong>, <strong>Ross Street</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 23--49.</p><p><strong>Abstract:</strong><br/>
After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions: what is a set with half an element? what is a set with $\pi$ elements? The category of these extended positive real sets is equipped with a countable tensor product. We develop somewhat the theory of categories with countable tensors; we call the commutative such categories series monoidal and conclude by only briefly mentioning the non-commutative possibility called $\omega$- monoidal . We include some remarks on sets having cardinalities in $[−\infty, \infty]$.
</p>projecteuclid.org/euclid.tbilisi/1524232073_20180420094755Fri, 20 Apr 2018 09:47 EDTTopological $\ast$-autonomous categories, revisited
https://projecteuclid.org/euclid.tbilisi/1524232074
<strong>Michael Barr</strong>. <p><strong>Source: </strong>Tbilisi Mathematical Journal, Volume 10, Number 3, 51--64.</p><p><strong>Abstract:</strong><br/>
Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing object.
</p>projecteuclid.org/euclid.tbilisi/1524232074_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 EDT