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The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce…
A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…
This paper investigates what can be inferred about an arbitrary continuous probability distribution from a finite sample of $N$ observations drawn from it. The central finding is that the $N$ sorted sample points partition the real line…
We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure…
An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give…
A stochastic algorithm is proposed, finding some elements from the set of intrinsic $p$-mean(s) associated to a probability measure $\nu$ on a compact Riemannian manifold and to $p\in[1,\infty)$. It is fed sequentially with independent…
We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully…
This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition…
Categories of partitions are combinatorial structures arising from the representation theory of certain compact quantum groups and are linked to classical diagram algebras such as the Temperley-Lieb algebra. In this paper, we present…
Geometric embeddings have recently received attention for their natural ability to represent transitive asymmetric relations via containment. Box embeddings, where objects are represented by n-dimensional hyperrectangles, are a particularly…
A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some…
Raimi's classical theorem establishes a partition of the natural numbers with a remarkable unavoidability property: for every finite coloring of $\mathbb{N}$, there is a color class whose translate meets both parts of the partition in…
We compute the integer cohomology rings of the ``polygon spaces'' introduced in [Hausmann,Klyachko,Kapovich-Millson]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we…
We propose a novel method for building fuzzy clusters of large data sets, using a smoothing numerical approach. The usual sum-of-squares criterion is relaxed so the search for good fuzzy partitions is made on a continuous space, rather than…
We introduce the notion of combinatorial encoding of continuous dynamical systems and suggest the first examples, which are the most interesting and important, namely, the combinatorial encoding of a Bernoulli process with continuous state…
This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts. (i) The first part considers the foundation of the probe and enclosure methods for an…
This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn's problem, the…
By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…
Model-based clustering is a powerful tool that is often used to discover hidden structure in data by grouping observational units that exhibit similar response values. Recently, clustering methods have been developed that permit…
We prove some general recursions for the numbers of representations of positive integers as a sum x+y, x in X, y in Y, where X,Y are increasing sequences. In particular, we obtain recursions for the number of the Goldbach, Lemoine-Levy,…