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54
ON THRESHOLD CIRCUITS AND POLYNOMIAL COMPUTATION
"... 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 unbound ..."
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Cited by 51 (1 self)
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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, any analytic 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
PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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Cited by 26 (2 self)
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
Discretization of Dirac Delta Functions in Level Set Methods
 J. Comput. Phys
, 2004
"... 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 ..."
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Cited by 26 (2 self)
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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 onedimensional delta functions.
Objects That Cannot Be Taken Apart With Two Hands
 Proc. of the 9th ACM Symp. on Computational Geometry
, 1993
"... It has been conjectured that every configuration C of convex objects in 3space 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 fe ..."
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Cited by 23 (2 self)
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It has been conjectured that every configuration C of convex objects in 3space 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).
Project Mars: Scalable, High Performance, Web Based MultimediaOnDemand (MOD) Services And Servers
, 1998
"... This dissertation describes costeffective design and implementation of scalable web based high performance multimediaondemand (MOD) servers and services. An important aspect of this dissertation has been prototyping, deploying MOD applications, services and servers and learning from this experie ..."
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Cited by 12 (0 self)
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This dissertation describes costeffective design and implementation of scalable web based high performance multimediaondemand (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 multimediaondemand services, namely interactive recording service for content crea...
A General Mathematics of Names
 Information and Computation
, 2007
"... We introduce FMG (FraenkelMostowski 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 namecarrying sy ..."
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Cited by 7 (4 self)
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We introduce FMG (FraenkelMostowski 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 namecarrying 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 αequivalence. Our presentation uses a novel presentation based not on a theory but on a concrete model U.
Towards a deterministic polynomialtime Primality Test
, 2002
"... 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 ..."
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Cited by 6 (1 self)
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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 squarefree 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.
Students' use and misuse of mathematical theorems: the case
 of Lagrange's Theorem . For the Learning of Mathematics
, 1996
"... 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? ..."
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Cited by 6 (2 self)
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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?
The most influential paper Gerard Salton never wrote
 Library Trends
"... Gerard Salton is often credited with developing the vector space model (VSM) for information retrieval (IR). Citations to Salton give the impression that the VSM must have been articulated as an IR model sometime between 1970 and 1975. However, the VSM as it is understood today evolved over a longer ..."
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Cited by 5 (0 self)
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Gerard Salton is often credited with developing the vector space model (VSM) for information retrieval (IR). Citations to Salton give the impression that the VSM must have been articulated as an IR model sometime between 1970 and 1975. However, the VSM as it is understood today evolved over a longer time period than is usually acknowledged, and an articulation of the model and its assumptions did not appear in print until several years after those assumptions had been criticized and alternative models proposed. An often cited overview paper titled “A Vector Space Model for Information Retrieval ” (alleged to have been published in 1975) does not exist, and citations to it represent a confusion of two 1975 articles, neither of which were overviews of the VSM as a model of information retrieval. Until the late 1970s, Salton did not present vector spaces as models of IR generally but rather as models of specific computations. Citations to the phantom paper reflect an apparently widely held misconception that the operational features and explanatory devices now associated with the VSM must have been introduced at the same time it was first proposed as an IR model.