Partitions and functional Santaló inequalities

We give a direct proof of a functional Santaló inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santaló inequality. The argument combines a logarithmic form of the Prékopa-Leindler inequality and a partition theorem of Yao and Yao. Received: 21 August 2008     

On a conditional Gołab-Schinzel equation

Let We show that for every function satisfying the conditional equation

Homogeneous symmetric means of two variables

Summary.  Let be given continuous functions on the interval I such that g ≠ 0, and is strictly monotonic (thus invertible) on I. Taking an increasing nonconstant function μ on [0, 1]

Deriving Behavior of Boolean Bioregulatory Networks from Subnetwork Dynamics

In the well-known discrete modeling framework developed by R. Thomas, the structure of a biological regulatory network is captured in an interaction graph, which, together with a set of Boolean parameters, gives rise to a state transition graph describing all possible dynamical behaviors. For complex networks the analysis of the dynamics becomes more and more difficult, and efficient methods to carry out the analysis are needed. In this paper, we focus on identifying subnetworks of the system that govern the behavior of the system as a whole. We present methods to derive trajectories and attractors of the network from the dynamics suitable subnetworks display in isolation. In addition, we use these ideas to link the existence of certain structural motifs, namely circuits, in the interaction graph to the character and number of attractors in the state transition graph, generalizing and refining results presented in [10]. Lastly, we show for a specific class of networks that all possible asymptotic behaviors of networks in that class can be derived from the dynamics of easily identifiable subnetworks.      

On the Yao-Yao partition theorem

The Yao-Yao partition theorem states that for any probability measure μ on having a density which is continuous and bounded away from 0, it is possible to partition into 2ⁿ regions of equal measure for μ in such a way that every affine hyperplane of avoids at least one of the regions. We give a constructive proof of this result and extend it to slightly more general measures. Received: 21 August 2008     

A Math Query Language with an Expanded Set of Wildcards

Math search is a new area of research with many enabling technologies but also many challenges. Some of the enabling technologies include XML, XPath, XQuery, and MathML. Some of the challenges involve enabling search systems to recognize mathematical symbols and structures. Several math search projects have made considerable progress in meeting those challenges. One of the remaining challenges is the creation and implementation of a math query language that enables the general users to express their information needs intuitively yet precisely. This paper will present such a language and detail its features. The new math query language offers an alternative way to describe mathematical expressions that is more consistent and less ambiguous than conventional mathematical notation. In addition, the language goes beyond the Boolean and proximity query syntax found in standard text search systems. It defines a powerful set of wildcards that are deemed important for math search. These wildcards provide for more precise structural search and multi-levels of abstractions. Three new sets of wildcards and their implementation details will also be discussed.      

Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra

We identify two noncommutative structures naturally associated with countable directed graphs. They are formulated in the language of operators on Hilbert spaces. If G is a countable directed graphs with its vertex set V(G) and its edge set E(G), then we associate partial isometries to the edges in E(G) and projections to the vertices in V(G). We construct a corresponding von Neumann algebra as a groupoid crossed product algebra of an arbitrary fixed von Neumann algebra M and the graph groupoid induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph groupoid of G has its binary operation, called admissibility. This has concrete local parts , for all e ∈ E(G). We characterize of , induced by the local parts of , for all e ∈ E(G). We then characterize all amalgamated free blocks of . They are chracterized by well-known von Neumann algebras: the classical group crossed product algebras , and certain subalgebras (M) of operator-valued matricial algebra . This shows that graph von Neumann algebras identify the key properties of graph groupoids. Received: December 20, 2006. Revised: March 07, 2007. Accepted: March 13, 2007.     

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Derivation of the material momentum equation from the energy balance

Based on a recent paper of Yavari et al, we show how the equation of material momentum can be obtained from invariance of the energy balance under translations in the material space. This approach further highlights the duality between the physical and material space.      

Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006     

Knots and Unknots in Superfluid Turbulence

The study of vortex lines in a inviscid, incompressible Euler fluid dates back to the times of Lord Kelvin. Vortex lines are highly idealised mathematical objects which at first sight do not seem relevant to vortices and turbulence in real fluids. In this article I show that superfluid vortices are good physical realizations of these classical mathematical objects, and provide us with a convenient context to investigate topological aspects of turbulence. Received: March 2007     

A new approach to L p estimates for Calderón-Zygmund type singular integrals

In this paper we give a new proof for L^p estimates of the Calderón-Zygmund type singular integrals. Our approach is completely free from harmonic analysis. This work was partially supported by the innovation foundation of Shanghai university under Grant A10-0101-08-905.