• Volume 119, Issue 3

June 2009,   pages  267-410

• A Finer Classification of the Unit Sum Number of the Ring of Integers of Quadratic Fields and Complex Cubic Fields

The unit sum number, $u(R)$, of a ring 𝑅 is the least 𝑘 such that every element is the sum of 𝑘 units; if there is no such 𝑘 then $u(R)$ is 𝜔 or $\infty$ depending on whether the units generate 𝑅 additively or not. Here we introduce a finer classification for the unit sum number of a ring and in this new classification we completely determine the unit sum number of the ring of integers of a quadratic field. Further we obtain some results on cubic complex fields which one can decide whether the unit sum number is 𝜔 or $\infty$. Then we present some examples showing that all possibilities can occur.

• Remarks on some Zero-Sum Theorems

In the present paper, we give a new proof of a weighted generalization of a result of Gao in a particular case. We also give new methods for determining the weighted Davenport constant and another similar constant for some particular weights.

• Orthogonality and Hecke Operators

In this article we analyze orthogonality relations between old forms and the connection to the theory of Hecke operators.

• Last Multipliers on Lie Algebroids

In this paper we extend the theory of last multipliers as solutions of the Liouville’s transport equation to Lie algebroids with their top exterior power as trivial line bundle (previously developed for vector fields and multivectors). We define the notion of exact section and the Liouville equation on Lie algebroids. The aim of the present work is to develop the theory of this extension from the tangent bundle algebroid to a general Lie algebroid (e.g. the set of sections with a prescribed last multiplier is still a Gerstenhaber subalgebra). We present some characterizations of this extension in terms of Witten and Marsden differentials.

• Submanifolds Weakly Associated with Graphs

We establish an interesting link between differential geometry and graph theory by defining submanifolds weakly associated with graphs. We prove that, in a local sense, every submanifold satisfies such an association, and other general results. Finally, we study submanifolds associated with graphs either in low dimensions or belonging to some special families.

• Meet and Join Matrices in the Poset of Exponential Divisors

It is well-known that $(\mathbb{Z}_+,|)=(\mathbb{Z}_+,GCD,LCM)$ is a lattice, where $|$ is the usual divisibility relation and $GCD$ and $LCM$ stand for the greatest common divisor and the least common multiple of positive integers.

The number $d=\prod^r_{k=1}p^{d^{(k)}}_k$ is said to be an exponential divisor or an 𝑒-divisor of $n=\prod^r_{k=1}p^{n^{(k)}}_k(n &gt;1)$, written as $d|_e n$, if $d^{(k)}|n^{(k)}$ for all prime divisors $p_k$ of 𝑛. It is easy to see that $(\mathbb{Z}_+\backslash\{1\},|_e)$ is a poset under the exponential divisibility relation but not a lattice, since the greatest common exponential divisor $(GCED)$ and the least common exponential multiple $(LCEM)$ do not always exist.

In this paper we embed this poset in a lattice. As an application we study the $GCED$ and $LCEM$ matrices, analogues of $GCD$ and $LCM$ matrices, which are both special cases of meet and join matrices on lattices.

• Smooth Maps of a Foliated Manifold in a Symplectic Manifold

Let 𝑀 be a smooth manifold with a regular foliation $\mathcal{F}$ and a 2-form 𝜔 which induces closed forms on the leaves of $\mathcal{F}$ in the leaf topology. A smooth map $f:(M,\mathcal{F})\longrightarrow(N, \sigma)$ in a symplectic manifold $(N, \sigma)$ is called a foliated symplectic immersion if 𝑓 restricts to an immersion on each leaf of the foliation and further, the restriction of $f^∗\sigma$ is the same as the restriction of 𝜔 on each leaf of the foliation.

If 𝑓 is a foliated symplectic immersion then the derivative map $Df$ gives rise to a bundle morphism $F:TM\longrightarrow TN$ which restricts to a monomorphism on $T\mathcal{F}\subseteq TM$ and satisfies the condition $F^∗\sigma=\omega$ on $T\mathcal{F}$. A natural question is whether the existence of such a bundle map 𝐹 ensures the existence of a foliated symplectic immersion 𝑓. As we shall see in this paper, the obstruction to the existence of such an 𝑓 is only topological in nature. The result is proved using the ℎ-principle theory of Gromov.

• On Infinitesimal Conformal Transformations of the Tangent Bundles with the Synectic Lift of a Riemannian Metric

The purpose of the present article is to investigate some relations between the Lie algebra of the infinitesimal fibre-preserving conformal transformations of the tangent bundle of a Riemannian manifold with respect to the synectic lift of the metric tensor and the Lie algebra of infinitesimal projective transformations of the Riemannian manifold itself.

• Divisors, Measures and Critical Functions

In [4] we have introduced a new distance between Galois orbits over $\mathbb{Q}$. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].

• New Characterizations of Fusion Frames (Frames of Subspaces)

In this article, we give new characterizations of fusion frames, on the properties of their synthesis operators, on the behavior of fusion frames under bounded operators with closed range, and on erasures of subspaces of fusion frames. Furthermore we show that every fusion frame is the image of an orthonormal fusion basis under a bounded surjective operator.

• Riesz Isomorphisms of Tensor Products of Order Unit Banach Spaces

In this paper we formulate and prove an order unit Banach space version of a Banach–Stone theorem type theorem for Riesz isomorphisms of the space of vector-valued continuous functions. Similar results were obtained recently for the case of lattice-valued continuous functions in [5] and [6].

• Moment Convergence Rates in the Law of the Logarithm for Dependent Sequences

Let $\{X_n;n\geq 1\}$ be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $S_n=\sum^n_{k=1}X_k,M_n=\max_{k\leq n}|S_k|,n\geq 1$. Suppose $\sigma^2=EX^2_1+2\sum^\infty_{k=2}EX_1X_k(0 &lt; \sigma &lt; \infty)$. In this paper, the exact convergence rates of a kind of weighted infinite series of $E\{M_n-\sigma\varepsilon\sqrt{n\log n}\}_+$ and $E\{|S_n|-\sigma\varepsilon\sqrt{n\log n}\}_+$ as $\varepsilon\searrow 0$ and $E\{\sigma\varepsilon\sqrt{\frac{\pi^2 n}{8\log n}}-M_n\}_+$ as $\varepsilon\nearrow\infty$ are obtained.

• Strong Ideal Convergence in Probabilistic Metric Spaces

In the present paper we introduce the concepts of strongly ideal convergent sequence and strong ideal Cauchy sequence in a probabilistic metric (PM) space endowed with the strong topology, and establish some basic facts. Next, we define the strong ideal limit points and the strong ideal cluster points of a sequence in this space and investigate some properties of these concepts.

• # Proceedings – Mathematical Sciences

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