A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbounded fanin circuits which are denoted Finite Field ZP (n) Circuits, where each node computes either multiple sums or products of integers modulo a prime P (n). In particular, it is proved that all functions computed by Threshold Circuits of size S(n) n and depth D(n) can also be computed by ZP (n) Circuits of size O(S(n) log S(n)+nP (n) log P (n)) and depth O(D(n)). Furthermore, it is shown that all functions computed by ZP (n) Circuits of size S(n) and depth D(n) can be computed by Threshold Circuits of size O ( 1 2 (S(n) log P (n)) 1+) and depth O ( 1 5 D(n)). These are the main results of this paper. There are many useful and quite surprising consequences of this result. For example, integer reciprocal can be computed in size n O(1) and depth O(1). More generally, anyanalytic function with a convergent rational polynomial power series (such as sine, cosine, exponentiation, square root, and logarithm) can be computed within accuracy 2,nc, for any constant c, by Threshold Circuits of

It has been conjectured that every configuration C of convex objects in 3-space with disjoint interiors can be taken apart by translation with two hands: that is, some proper subset of C can be translated to infinity without disturbing its complement. We show that the conjecture holds for five or fewer objects and give a counterexample with six objects. We extend the counterexample to a configuration that cannot be taken apart with two hands using arbitrary isometries (rigid motions).

Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations on Cartesian grids and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on the distance to the singularity with a regularization parameter proportional to the mesh size. The first algorithm is based on a tensor product of regularized one-dimensional delta functions.

We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite. 1

This dissertation describes cost-effective design and implementation of scalable web based high performance multimedia-on-demand (MOD) servers and services. An important aspect of this dissertation has been prototyping, deploying MOD applications, services and servers and learning from this experience. The three main components of this dissertation are (1) Web based interactive MOD services, (2) innovative enhancements to a server node operating system (OS) to support such MOD services, and (3) design and prototyping of a scalable server architecture and associated data layout and scheduling schemes to support a large number of independent, concurrent clients. We first describe design and prototyping of two example multimedia-on-demand services, namely interactive recording service for content crea...

We examine a primality testing algorithm presented in [Man99] and the related conjecture in [Bha01]. We show that this test is stronger than most of the popular tests today: the Fermat test, the Solovay Strassen test and a strong form of the Fibonacci test. From this, we show the correctness of the algorithm based on a widely believed conjecture, the Extended Riemann Hypothesis. We also show that any n which is accepted by the algorithm must be an odd square-free number. Thus, it is arguably the simplest and yet the strongest test for primality. Based on our computations and results proved in this paper we feel that unlike other tests, this test is very promising as the related conjecture seems provable.

We introduce FMG (Fraenkel-Mostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax — de Bruijn indices, FM sets, and name-carrying syntax — have a relation generalising to all sets and not only sets of syntax trees. We also give syntaxfree accounts of Barendregt representatives, scope extrusion, and other phenomena associated to α-equivalence. Our presentation uses a novel presentation based not on a theory but on a concrete model U.

Consider the following two questions from introductory group theory 1: and What is the converse of Lagrange's theorem? Is Z3 a subgroup of Z6?

Derived schema components are an important aspect of traditional semantic data modeling. In this paper, we address the issue of defining such schema constructs in the context of object-oriented database (OODB) part hierarchies. In particular, we present the concept of derived attribute defined with respect to value propagation across a part relationship between two object classes. Three different types of value propagation, namely, invariant, transformational, and cumulative, allow for a high degree of expressiveness in the definition of such derived attributes. We also present the notion of a generalized derived attribute, which may be defined in terms of simultaneous value propagations across many part relationships. The ambiguity problem of multiple value propagation in part hierarchies is solved by this latter construct, an analogue of which is not applicable in ordinary OODB IS-A hierarchies. It allows for the representation of such common expressions as the weight of the whole is...

Data James Wen Department of Computer Science Brown University Providence, Rhode Island 02912 CS-95-20 June 1995 Exploiting Orthogonality in Three Dimensional Graphics for Visualizing Abstract Data James Wen Department of Computer Science Brown University Providence, RI 02912 ABSTRACT Using three dimensional graphics to visualize abstract data presents an interesting challenge because, by definition, there is no physically intuitive counterpart to abstract data. Relatively little work has been done in exploring how to best utilize the third dimension to increase visualization capabilities beyond simply increasing the volume of data presented. This paper presents a design principle that enriches the amount of semantic information that could be visualized by treating the third dimension not so much as one of three equivalent dimensions in three-space but as an independent channel of information. KEYWORDS: information visualization, abstract data, scientific visualization, ...