This is a new proof that $(\mathbf{M},Cr^\gamma,\delta)$-minimizing sets $S$ are pieces of $\mathcal{C}^{1,\gamma/2}$ curves, $0<\gamma\leqslant1$. To obtain this result, the almost monotonicity property is established for balls centered on $S$ or not. Furthermore it is proved that almost minimizing sets fulfill the epiperimetric inequality.

In this work, we define the quasi-Poisson Lie quasigroups, dual objects to the quasi-Poisson Lie groups and we establish the one-to-one correspondance up to isomorphism between the local quasi-Poisson Lie quasigroups and quasi-Lie bialgebras.

Dans ce travail, nous définissons les quasi-groupes de Lie quasi-Poisson, objets duaux des groupes de Lie quasi-Poisson et nous établissons une correspondance biunivoque à isomorphisme près entre les quasi-groupes de Lie quasi-Poisson locaux et les bigèbres quasi-Lie.

In this article, a single parametric class of modifications for Kovarik's method is proposed. It is proved that all methods in this class are quadratically convergent. Numerical comparison among methods of Kovarik, Petcu-Popa [5], and a special method in this class, chosen based on a specific value for the parameter, shows that Kovarik and Petcu-Popa's methods give almost similar convergence results. However, the special method converges faster and its iteration number is considerably lower than that of others. For Numerical experiments, there are used ten $n\times n$ test matrices with $n=5,10,20,50$, whose condition numbers vary in the interval [$2\,,\,8.1e146$].

Motivated by a recent paper by S. Ohno we calculate Hilbert-Schmidt norms of products of composition and differentiation operators on the Bergman space $A^2_\alpha,$ $\alpha>-1$ and the Hardy space $H^2$ on the unit disk. When the convergence of sequences $(\varphi_n)$ of symbols to a given symbol $\varphi$ implies the convergence of product operators $C_{\varphi_n}D^k$ is also studied. Finally, the boundedness and compactness of the operator $C_{\varphi}D^k: A^2_\alpha\to A^2_\alpha$ are characterized in terms of the generalized Nevanlinna counting function.

Let $A$ be a Banach algebra. We say that a pair $(\mathcal{G},\mathcal{U})$ is a (topologically Gelfand theory) Gelfand theory for $A$ if the following hold: (G1) $\mathcal{U}$ is a C*-algebra and $\mathcal{G}:A\to \mathcal{U}$ is a homomorphism which induces the (homeomorphism) bijection $\pi\mapsto \pi\circ \mathcal{G}$ from $\widehat{\mathcal{U}}$ onto $\widehat{\mathcal{A}}$; (G2) for every maximal modular left ideal $L$, $\mathcal{G}(A)\not\subseteq L$. We show that this definition is equivalent to the usual definition of gelfand theory in the commutative case. We prove that many properties of Gelfand theory of commutative Banach algebras remain true for Gelfand theories of arbitrary Banach algebras. We show that unital homogeneous Banach algebras and postliminal C*-algebras have unique Gelfand theories (up to an appropriate notion of uniqueness ).

The subject of this paper is the index theorem on odd-dimensional manifolds with boundary. Such a theorem has been formulated and proved by D. Freed and his proof is based on analysis by Calderon and Seeley. In this paper we prove this theorem using the heat kernel methods for boundary conditions of Dirichlet and Neumann type. Moreover, we also consider the Atiyah-Patodi-Singer spectral boundary condition which is not studied in Freed's paper. As a direct consequence of the method, we obtain some information about isospectral invariants of the boundary conditions. This proof does not use the cobordism invariance of the index and is generalized easily to the family case.

We establish the uniqueness and the blow-up rate of the large positive solution of the singular boundary problem $-\Delta_{p} u=\lambda u^{p-1}-b(x) u^q$ in $B_R(x_0)$, $u|_{\partial B_R(x_0)}=+\infty$, where $B_R(x_0)$ is a ball domain of radius $R$ centered at $x_0\in\mathbb{R}^N$, $N\geq3$, $\lambda>0$ and the potential function $b$ is a positive radially symmetric function. Our result extends the previous work by Ouyang and Xie from the case $p=2$ to the case $p>2$ and we prove that any large solution $u$ must satisfy $$\lim_{d(x)\rightarrow 0}\frac{u(x)}{K(b^{*}(\|x-x_{0}\|))^{-\beta}}=1,$$ where $d(x)= {\rm dist}(x, \partial B_{R}(x_{0}))$, $K$ is a constant defined by $$K:=\left[(p-1)[(\beta +1)C_{0}-1]\beta^{p-1}(C_{0}b_{0})^{(p-2)/2}\right]^{\frac{1}{q-p+1}},$$ with $$\beta:=\frac{p}{2(q-p+1)},\;q>p-1>1,\; b_{0}:=b(R)>0,\; C_{0}:=\lim_{r\rightarrow R}\frac{(B(r))^{2}}{b^{*}(r)b(r)}\geq 1$$ and $$B(r):=\int_{r}^{R} b(s)ds,\; b^{*}(r)=\int_{r}^{R}\int_{s}^{R} b(t)dt ds.$$

Two-dimensional informetrics is defined in the general context of sources that produce items and examples are given. These systems are called ``Information Production Processes'' (IPPs). They can be described by a size-frequency function $f$ or, equivalently, by a rank-frequency function $g$. If $f$ is a decreasing power law then we say that this function is the law of Lotka and it is equivalent with the power law $g$ which is called the law of Zipf. Examples in WWW are given. Next we discuss the scale-free property of $f$ also allowing for the interpretation of a Lotkaian IPP (i.e. for which $f$ is the law of Lotka) as a self-similar fractal. Then we discuss dynamical aspects of (Lotkaian) IPPs by introducing an item-transformation $\varphi$ and a source-transformation $\psi$. If these transformations are power functions we prove that the transformed IPP is Lotkaian and we present a formula for the exponent of the Lotka law. Applications are given on the evolution of WWW and on IPPs without low productive sources (e.g. sizes of countries, municipalities or databases). Lotka's law is then used to model the cumulative first citation distribution and examples of good fit are given. Finally, Lotka's law is applied to the study of performance indices such as the $h$-index (Hirsch) or the $g$-index (Egghe). Formulas are given for the $h$- and $g$-index in Lotkaian IPPs and applications are given.

In this paper, we extend the notion of similarity matrix, which has been used to define similarity between nodes of two graphs, to the case of colored graphs, where the coloring is either on the nodes or on the edges of both graphs. The proposed method tries to find the optimal matching between the nodes or edges of both graphs but only performs the comparison when their colors are the same. The proposed cost function nevertheless uses the connectivity between all nodes and edges of both graphs. We then also show how to extend this to the notion of low rank similarity matrix, by defining it as a constrained optimization problem.

After a quick review of the most used numerical indicators for evaluating research, we present an integrated model for ranking scientific publications together with authors and journals. Our model relies on certain adjacentcy matrices obtained from the relationship between papers, authors, and journals. These matrices are first normalized to obtain stochastic matrices and then are combined together using appropriate weights to form a suitable irreducible stochastic matrix whose dominant eigenvector provides the desired ranking. Our main contribution is a in-depth analysis of various strategies for choosing the weights, showing their probabilistic interpretation and showing how they affect the outcome of the ranking process. We also prove that, by solving an inverse eigenvector problem, we can determine a weighting strategy in which the relative importance of papers, authors, and journals is chosen by the final user of the ranking algorithm. The impact of the different weighting strategies is analyzed also by means of extensive experiments on large synthetic datasets.

Ranking has recently attracted much attention, conceptually as well as algorithmically and in its uses in major Internet search engines. A core goal of ranking and related techniques on the Web is to help people find what they are looking for -- and (depending on the application) to suggest to them things that they didn't know they were looking for, but might still find interesting. Any evaluation of these techniques should therefore consider such deployment scenarios. The paper starts from the observation that search engines and recommender systems generally provide users with some ranking on resources, and that standard evaluations rest on a comparison of this system output with an assumed mental representation of the user's ``true ranking''. This is followed by an overview of i) where and how ranking is used by the operators of a Web site or similar service, ii) how ranking is used by the end users of that site or service, and how such usage is measured, and iii) how and according to which criteria this usage and the success as well as the quality of ranking are measured. The paper demonstrates how an interdisciplinary approach can sharpen the view of challenges and promises of this user-oriented analysis of intelligent information access.