• Volume 120, Issue 1

February 2010,   pages  1-130

• On the General Dedekind Sums and its Reciprocity Formula

In this paper, we prove an interesting reciprocity formula for a certain case of a general Dedekind sums using analytic methods and the Fourier expansion of the Bernoulli polynomials.

• Dirichlet Expression for $L(1, \chi)$ with General Dirichlet Character

In the famous work of Dirichlet on class number formula, $L(s, \chi)$ at $s=1$ has been expressed as a finite sum, where $L(s, \chi)$ is the Dirichlet 𝐿-series of a real Dirichlet character. We show that this expression with obvious modification is valid for the general primitive Dirichlet character 𝜒.

• On the Complexity of Labeled Oriented Trees

We define a notion of complexity for labeled oriented trees (LOTs) related to the bridge number in knot theory and prove that LOTs of complexity 2 are aspherical. We also present a class of LOTs of higher complexity which is aspherical, give an upper bound for the complexity of labeled oriented intervals and study the complexity of torus knots.

• Semisymmetric Cubic Graphs of Order $16p^2$

An undirected graph without isolated vertices is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper, we inquire the existence of connected semisymmetric cubic graphs of order $16p^2$. It is shown that for every odd prime 𝑝, there exists a semisymmetric cubic graph of order $16p^2$ and its structure is explicitly specified by giving the corresponding voltage rules generating the covering projections.

• 𝑛-Colour even Self-Inverse Compositions

An 𝑛-colour even self-inverse composition is defined as an 𝑛-colour self-inverse composition with even parts. In this paper, we get generating functions, explicit formulas and recurrence formulas for 𝑛-colour even self-inverse compositions. One new binomial identity is also obtained.

• On the Matlis Duals of Local Cohomology Modules and Modules of Generalized Fractions

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring with non-zero identity, $\mathfrak{a}$ a proper ideal of 𝑅 and 𝑀 a finitely generated 𝑅-module with $\mathfrak{a}M\neq M$. Let $D(-):=\mathrm{Hom}_R(-,E)$ be the Matlis dual functor, where $E:=E(R/\mathfrak{m})$ is the injective hull of the residue field $R/\mathfrak{m}$. In this paper, by using a complex which involves modules of generalized fractions, we show that, if $x_1,\ldots,x_n$ is a regular sequence on 𝑀 contained in $\mathfrak{a}$, then $H^n_{(x_1,\ldots,x_n)R}(D(H^n_{\mathfrak{a}}(M)))$ is a homomorphic image of $D(M)$, where $H^i_{\mathfrak{b}}(-)$ is the 𝑖-th local cohomology functor with respect to an ideal $\mathfrak{b}$ of 𝑅. By applying this result, we study some conditions on a certain module of generalized fractions under which $D(H^n_{(x_1,\ldots,x_n)R}(D(H^n_{\mathfrak{a}}(M))))\cong D(D(M))$.

• On the Iwasawa Algebra Associated to a Normal Element of $\mathbb{C}_p$

Given a prime number 𝑝 and the Galois orbit $O(x)$ of a normal element 𝑥 of $\mathbb{C}_p$, the topological completion of the algebraic closure of the field of 𝑝-adic numbers, we study the Iwasawa algebra of $O(x)$ with scalars drawn from $\mathbb{Q}_p$ and relate it with $\mathbb{Q}_p$-distributions and functionals.

• Relatively Hyperbolic Extensions of Groups and Cannon-Thurston Maps

Let $1\to(K, K_1)\to(G, N_G(K_1))\to(\mathcal{Q}, \mathcal{Q}_1)\to 1$ be a short exact sequence of pairs of finitely generated groups with $K_1$ a proper non-trivial subgroup of 𝐾 and 𝐾 strongly hyperbolic relative to $K_1$. Assuming that, for all $g\in G$, there exists $k_g\in K$ such that $gK_1g^{-1}=k_gK_1k^{-1}_g$, we will prove that there exists a quasi-isometric section $s:\mathcal{Q}\to G$. Further, we will prove that if 𝐺 is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon–Thurston map for the inclusion $i:\Gamma_K\to\Gamma_G$.

• A Note on the Tangent Bundle of $G/P$

Let 𝑃 be a parabolic subgroup of a complex simple linear algebraic group 𝐺. We prove that the tangent bundle $T(G/P)$ is stable.

• A Heat Kernel Version of Cowling-Price Theorem for the Laguerre Hypergroup

In this paper, we prove a heat kernel version of Cowling–Price theorem for the Laguerre hypergroup.

• John Disks, the Apollonian Metric, and Min-Max Properties

The main results of this paper are characterizations of John disks–the simply connected proper subdomains of the complex plane that satisfy a twisted double cone connectivity property. One of the characterizations of John disks is an analog of a result due to Gehring and Hag who found such a characterization for quasidisks. In both situations the geometric condition is an estimate for the domain’s hyperbolic metric in terms of its Apollonian metric. The other characterization is in terms of an arc min-max property.

• Homomorphisms between $C^\ast$-Algebra Extensions

In this paper we consider the question when a homomorphism between two extension algebras preserves the essential ideal in the corresponding extension. Some conditions of two essential extensions being isomorphic are given. We also describe the relationship between the induced extensions and the Kasparov products and give the completely positive liftings of the induced extensions.

• Nuclearity for Dual Operator Spaces

In this short paper, we study the nuclearity for the dual operator space $V^∗$ of an operator space 𝑉. We show that $V^∗$ is nuclear if and only if $V^{∗∗∗}$ is injective, where $V^{∗∗∗}$ is the third dual of 𝑉. This is in striking contrast to the situation for general operator spaces. This result is used to prove that $V^{∗∗}$ is nuclear if and only if 𝑉 is nuclear and $V^{∗∗}$ is exact.

• Symmetries, Integrals and Solutions of Ordinary Differential Equations of Maximal Symmetry

Second-and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, respectively contact, symmetry are examined for the mappings they produce in solutions and fundamental first integrals. The properties of the `exceptional symmetries’, i.e. those not considered to be generic to scalar equations of maximal symmetry, can be recast into a form which is applicable to all such equations of maximal symmetry. Some properties of these symmetries are demonstrated.

• # Proceedings – Mathematical Sciences

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