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      Volume 129, Issue 4

      September 2019

    • On a conjecture of Fox–Kleitman and additive combinatorics

      S D ADHIKARI R BALASUBRAMANIAN S ELIAHOU D J GRYNKIEWICZ

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      Let $D_{k}$ denote the maximum degree of regularity of the equation $x_{1}+\cdots+x_{k}-y_{1}-\cdots-y_{k}=b_{k}$ as $b_{k}$ runs over the positive integers. The Fox and Kleitman conjecture, stating that $D_{k}$ should equal $2k-1$, has been confirmed by Schoen and Taczala (Moscow J. Combin. Number Theory 7 (2017) 79-93). Their proof is achieved by generalizing a theorem of Eberhard et al. (Ann. Math. 180 (2014) 621-652) on sets with doubling constant less than 4. Using much simpler methods and a result of Lev in additive combinatorics, our main result here is that the degree of regularity of the same equation for the specific value $b_{k} = c_{k-1} = \rm{lcm} \{i \colon i = 1, \dots, k-1\}$ is at least $k-1$. This shows in a simple and explicit way that $D_{k}$ behaves linearly in $k$.

    • On transformation of certain bilateral basic hypergeometric series and their applications

      D D SOMASHEKARA K N VIDYA

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      In this paper, we obtain new transformation and summation formulae forbilateral basic hypergeometric series and give a simple proof of Jacobi’s triple product identity. Further, we present applications of the main results.

    • Mean ergodicity of composition operators on Hardy space

      SHI-AN HAN ZE-HUA ZHOU

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      In this article, we investigate the mean ergodicity of composition operators acting on the Hardy space $H^{p}(\mathbb{D})$, $1\leq p \lt \infty$. Specifically, a composition operator $C_{\varphi}$ acting on $H^{p}(\mathbb{D})$ is mean ergodic if and only if $\varphi$ has an interior fixed point, in which case $C_{\varphi}$ is uniformly mean ergodic if and only if $\varphi$ is an elliptic automorphism of finite order or a non-automorphism that is not inner.

    • Determinant formulas of some Toeplitz–Hessenberg matrices with Catalan entries

      TARAS GOY MARK SHATTUCK

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      In this paper, we consider determinants of some families of Toeplitz–Hessenberg matrices having various translates of the Catalan numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Catalan numbers and multinomial coefficients. Combinatorial proofs may be given for several of the identities that are obtained.

    • A new existence result for the boundary value problem of $p$-Laplacian equations with sign-changing weights

      XIANGHUI XU

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      In this paper, a new existence result for the homogeneous Dirichlet boundary value problem of $p$-Laplacian equations with sign-changing weights which may not be in $L^{1}$ is presented. Our approach is based on the nonlinear alternative of Leray–Schauder type theorem.

    • Functional determinant of Laplacian on Cayley projective plane $\rm{P^{2}(Cay)}$

      RICHARD OLU AWONUSIKA

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      This paper presents explicit formulae for the determinants of the Laplacians on the Cayley projective plane $\rm{P^{2}(Cay)}$. The explicit calculation of this spectral invariant associated with $\rm{P^{2}(Cay)}$ is appearing for the first time in the literature.Other spectral invariants on $\rm{P^{2}(Cay)}$ – the Minakshisundaram–Pleijel coefficients and Minakshisundaram–Pleijel zeta functions are also explicitly discussed.

    • A priori $L^{\infty}(L^{2})$ error estimates for finite element approximations to parabolic integro-differential equations with discontinuous coefficients

      BHUPEN DEKA RAM CHARAN DEKA

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      Finite element treatment for parabolic integro-differential equations with discontinuous coefficients are presented in this work. Due to low global regularity of the solutions, the error analysis technique of the standard finite element method is difficult to adopt for interface problems. In this paper, convergence of continuous time Galerkin method for the spatially discrete scheme and backward difference scheme in time direction are discussed in $L^{\infty}(L^{2})$ norm for fitted finite element method with straight interface triangles. More precisely, optimal error estimates are derived in $L^{\infty}(L^{2})$ norm when initial data $u_{0} \in H^{3}\cap H_{0}^{1}(\Omega)$. The interface is assumed to be smooth for our purpose.

    • On $c$-capability and $n$-isoclinic families of a specific class of groups

      MARZIEH CHAKANEH AZAM KAHENI SAEED KAYVANFAR

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      Let $\chi$ denote the class of all groups $G$ such that $\phi(G)\cap Z(G)=1$. In this paper, it is shown that the converse of Baer's theorem holds for the groups in $\chi$. Then we prove that the existence of the isomorphism between the center factors of the groups in $\chi$ suffices for those groups to be isoclinic. We also prove that the isoclinism coincides with the $n$-isoclinism in $\chi$. Finally, we obtain a criterion for $c$-capability of finite groups in $\chi$.

    • Spectrum of some weighted composition operators

      MALIHE FARZI HAROMI MAHMOOD HAJI SHAABANI

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      In this paper, we determine the spectrum of power compact weighted composition operators $C_{\psi,\varphi}$, on the weighted Hardy spaces. Moreover, we find the spectrum and essential spectrum of $C_{\psi,\varphi}$ on $A^{2}_{\alpha}$ when $\varphi$ is a non-automorphic linear-fractional self-map of $\mathbb{D}$.

    • Finite groups with specific number of cyclic subgroups

      HEMANT KALRA

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      We classify all finite groups having exactly $6$, $7$ or $8$ cyclic subgroups. This gives a partial answer to the open problem posed by $\rm{T\breve{a}rn\breve{a}uceanu}$ (Amer. Math. Mon. 122 (2015) 275-276). As a consequence of our results, we also obtain an important result concerning the converse of Lagrange's theorem.

    • Means of some arithmetical functions on shifted smooth numbers in arithmetic progression

      KHADIJA MBARKI WALID WANNES

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      We provide asymptotic estimates for sums of some arithmetical functions taken over shifted smooth numbers in arithmetic progression related to the sum of digits function. Our results are based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to G Tenenbaum, Stephanie S Loiperdinger and Igor E Shparlinski.

    • On absolutely norm attaining operators

      D VENKU NAIDU G RAMESH

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      We give the necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss the structure of such operators in the case of self-adjoint and normal operators separately. Finally, we discuss several properties of absolutely norm attaining operators.

    • Primary decomposition and normality of certain determinantal ideals

      JOYDIP SAHA INDRANATH SENGUPTA GAURAB TRIPATHI

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      In this paper, we study primality and primary decomposition of certainideals which are generated by homogeneous degree 2 polynomials and occur naturally from determinantal conditions. Normality is derived from these results.

    • New weighted norm inequalities for Calderón–Zygmund operators with kernels of Dini’s type and their commutators

      XI HU JIANG ZHOU

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      In this paper, we introduce certain classes of Calderón–Zygmund operators with kernels of Dini’s type including pseudodifferential operators with smooth symbols. Applying a class of new weight functions, we establish some weighted norm inequalities for certain classes of Calderón–Zygmund operators with kernels of Dini’s type. In addition, new BMO spaces with respect to the class of new weight functions are introduced. Naturally, the pointwise, weighted strong type and endpoint $L$ log $L$ type estimates for the commutators with the new BMO functions are also obtained.

    • On a Ramanujan’s Eisenstein series identity of level fifteen

      E N BHUVAN K R VASUKI

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      On pages 255-256 of his second notebook, Ramanujan recorded an Eisenstein series identity of level 15 without offering a proof. Previously, Berndt (Ramanujan's Notebooks: Part III (1991) (New York: Springer)) proved this identity using the theory of modular forms. In this paper, we give an elementary proof of this identity. In the process, we also give an elementary proof of three Ramanujan's $P - Q$ identities of level 15. Further, using the $P - Q$ identities we prove four Ramanujantype Eisenstein series of level 15 due to Cooper and Ye (Trans. Am. Math. Soc. 368 (2016) 7883-7910), where they have proved using the theory of modular forms.

    • Heptavalent symmetric graphs of order $24 p$

      SONG-TAO GUO YUSEN WU

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      A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. In this paper, we classify connected heptavalent symmetric graphs of order $24p$ for each prime $p$. As a result, there are twelve sporadic such graphs: one for $p = 2$, four for $p = 3$, one for $p = 5$ and six for $p = 13$.

    • Nambu structures on Lie algebroids and their modular classes

      APURBA DAS SHILPA GONDHALI GOUTAM MUKHERJEE

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      We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $\Lambda$ as the modular class of the tangent Lie algebroid $TM$ with Nambu structure $\Lambda$. We show that many known properties of the modular class of a Nambu-Poisson manifold and that of a Lie algebroid extend to the setting of a Lie algebroid with Nambu structure. Finally, we prove that for a large class of Nambu-Poisson manifolds considered as tangent Lie algebroids with Nambu structures, the associated modular classes are closely related to Evens-Lu-Weinstein modular classes of Lie algebroids.

    • Results on the Hilbert coefficients and reduction numbers

      AMIR MAFI DLER NADERI

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      Let $(R,\frak{m})$ be a $d$-dimensional Cohen--Macaulay local ring, $I$ an $\frak{m}$-primary ideal and $J$ a minimal reduction of $I$. In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert coefficients. In addition, we also give some examples.

    • Abelian quotients of extriangulated categories

      JING HE PANYUE ZHOU

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      We prove that certain subquotient categories of extriangulated categories are abelian. As a particular case, if an extriangulated category $\mathcal{C}$ has a cluster-tilting subcategory $\mathcal{X}$, then $\mathcal{C}/\mathcal{X}$ is abelian. This unifies a result by Koenig and Zhu (Math. Z. 258 (2008) 143-160) for triangulated categories and a result by Demonet and Liu (J. Pure Appl. Algebra 217(12) (2013) 2282-2297) for exact categories.

    • Symplectic reduction of Sasakian manifolds

      INDRANIL BISWAS GEORG SCHUMACHER

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      When a complex semisimple group $G$ acts holomorphically on a K\"ahler manifold $(X,\, \omega)$ such that a maximal compact subgroup $K\, \subset\, G$ preserves the symplectic form $\omega$, a basic result of symplectic geometry says that the corresponding categorical quotient $X/G$ can be identified with the quotient of the zero-set of the moment map by the action of $K$. We extend this to the context of a semisimple group acting on a Sasakian manifold.

    • The convolution equation $\sigma*\mu=\mu$ on non-compact non-abelian semigroups

      ARUNAVA MUKHERJEA

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      In probability theory, often in connection with problems on weak convergence, and also in other contexts, convolution equations of the form $\sigma*\mu=\mu$ come up. Many years ago, Choqet and Deny (C. R. Acad. Sci. Paris 250 (1960) 799-801) studied these equations in locally compact abelian groups. Later, Szekely and Zeng (J. Theoret. Probab. 3(2) (1990) 361-365) studied these equations in abelian semigroups. Like in [2], the results in [7] are also complete. Thus, these equations are studied here for the first time on non-compact non-abelian semigroups. Our main results are Theorems 3.1 and 3.3 in section 3. They are new results as far as we know, and also the best possible under a minor condition. All semigroups in this paper are, unless otherwise mentioned, locally compact Hausdorff second countable topological semigroups. Theorems 3.1 and 3.3 hold for these semigroups.

    • On generalized parabolic Hitchin pairs

      SOURAV DAS

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      We define the Gieseker Hitchin pair data, construct the moduli space of such objects in rank and degree co-prime case and show that this moduli space is the normalization of the moduli of the Gieseker Hitchin pairs. We study the relation between the moduli of the Gieseker Hitchin pair data and the moduli of the generalized parabolic Hitchin pairs. We also prove that the Hitchin maps for both Gieseker Hitchin pair data and generalized parabolic Hitchin pairs are proper.

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