Randall Dougherty

Randall Dougherty
Institute for Defense Analyses | IDA · Center for Communications Research, La Jolla

About

55
Publications
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2,337
Citations
Citations since 2017
0 Research Items
573 Citations
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2017201820192020202120222023020406080100120140
2017201820192020202120222023020406080100120140
2017201820192020202120222023020406080100120140
Introduction

Publications

Publications (55)
Article
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Networks with a high degree of symmetry are useful models for parallel processor networks. In earlier papers, we defined several global communication tasks (universal exchange, universal broadcast, universal summation) that can be critical tasks when complex algorithms are mapped to parallel machines. We showed that utilizing the symmetry can make...
Conference Paper
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It is known that information inequalities on four random variables cannot be generated from a finite list. For the analogous case of linear rank variables, it is known that they can be generated from a finite list for up to five variables, but this is not known for six or more variables. Here we present partial results of computations on six-variab...
Article
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Two characteristic-dependent linear rank inequalities are given for eight variables. Specifically, the first inequality holds for all finite fields whose characteristic is not three and does not in general hold over characteristic three. The second inequality holds for all finite fields whose characteristic is three and does not in general hold ove...
Article
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Determining the achievable rate region for networks using routing, linear coding, or non-linear coding is thought to be a difficult task in general, and few are known. We describe the achievable rate regions for four interesting networks (completely for three and partially for the fourth). In addition to the known matrix-computation method for prov...
Article
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We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in...
Article
Casinos operate by generating sequences of outcomes which appear unpredictable, or random, to effective gamblers. We investigate relative notions of randomness for gamblers whose wagers are restricted to a finite set. Some sequences which appear unpredictable to gamblers using wager amounts in one set permit unbounded profits for gamblers using dif...
Article
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This paper explores the connection between network coding and matroid theory, a branch of mathematics that generalizes linear algebra and graph theory. ABSTRACT | Networks derived from matroids have played a fundamental role in proving theoretical results about the limits of network coding. In this tutorial paper, we review many con-nections betwee...
Article
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Any unconstrained information inequality in three or fewer random variables can be written as a linear combination of instances of Shannon's inequality I(A;B|C) >= 0 . Such inequalities are sometimes referred to as "Shannon" inequalities. In 1998, Zhang and Yeung gave the first example of a "non-Shannon" information inequality in four variables. Th...
Article
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Ranks of subspaces of vector spaces satisfy all linear inequalities satisfied by entropies (including the standard Shannon inequalities) and an additional inequality due to Ingleton. It is known that the Shannon and Ingleton inequalities generate all such linear rank inequalities on up to four variables, but it has been an open question whether add...
Article
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If beta and gamma are nonnegative integers and F is a field, then a polynomial collection {p<sub>1</sub>,<sub>hellip</sub> ,P<sub>beta</sub>} sube Z[alpha<sub>1,hellip</sub>, alpha<sub>gamma</sub>] is said to be solvable over F if there exist omega<sub>1hellip</sub>, omega<sub>gamma</sub> isin F such that for all i = 1,<sub>hellip</sub>, beta we ha...
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For measures on a Cantor space, the demand that the measure be \good" is a useful homogeneity condition. We examine the question of when a Bernoulli measure on the sequence space for an alphabet of size n is good. Complete answers are given for the n = 2 cases and the rational cases. Partial results are obtained for the general cases. 1. Introducti...
Article
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Let μ(r) be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights r and 1-r. It is a long-standing open problem to characterize-those r and s such that μ(r) and μ(s) are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending μ(r) to μ(s)). The (possibly)...
Article
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We define a class of networks, called matroidal networks, which includes as special cases all scalar-linearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that play an important role in proving results in the lit...
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We prove that for any finite directed acyclic network, there exists a corresponding multiple-unicast network, such that for every alphabet, each network is solvable if and only if the other is solvable, and, for every finite-field alphabet, each network is linearly solvable if and only if the other is linearly solvable. The proof is constructive an...
Conference Paper
Full-text available
All unconstrained information inequalities in three or fewer random variables are known to be "Shannon-type", in that they are nonnegative linear combinations of instances of the inequality I(A;B|C) ges 0. In 1998, Zhang and Yeung gave the first example of an information inequality in four variables that is not "Shannon-type". Here we give six new...
Article
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The coding capacity of a network is the supremum of ratios k/n, for which there exists a fractional (k,n) coding solution, where k is the source message dimension and n is the maximum edge dimension. The coding capacity is referred to as routing capacity in the case when only routing is allowed. A network is said to achieve its capacity if there is...
Conference Paper
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The well-studied Vámos matroid has provided a wealth of interesting theoretical results in matroid theory. We use the Vámos matroid to construct a new network, which we call the Vámos network. We then exploit the Vámos network to answer in the negative the open question as to whether Shannon-type information inequalities are in general sufficient f...
Article
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We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and rational, we present an algorithm for its computation, and we prove that every rational number in (0, 1] is the routing capacity of some solvable n...
Article
This paper consists of three parts supplementing the papers of K. Hauser 2002 and D. Mumford 2000: (1) There exist regular open sets of points in with paradoxical properties, which are constructed without using the axiom of choice or the continuum hypothesis. (2) There exist canonical universes of sets in which one can define essentially all object...
Conference Paper
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It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has...
Article
Full-text available
Let mu(r) be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights r and 1-r. It is a long-standing open problem to characterize those r and s such that mu(r) and mu(s) are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending mu(r) to mu(s)). The (possi...
Article
Full-text available
It is known that for every solvable multicast network, there exists a large enough finite-field alphabet such that a scalar linear solution exists. We prove: i) every binary solvable multicast network with at most two messages has a binary scalar linear solution; ii) for more than two messages, not every binary solvable multicast network has a bina...
Article
We present a 16-vertex tetrahedralization of S 3 (the 3-sphere) for which no topological bistellar flip other than a 1-to-4 flip (i.e., a vertex insertion) is possible. This answers a question of Altshuler et al. which asked if any two n-vertex tetrahedralizations of S 3 are connected by a sequence of 2-to-3 and 3-to-2 flips. The corresponding geom...
Article
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We show that, in any topological space, boolean combinations of open sets have a canonical representation as a nite union of locally closed sets. As an application, if M is a rst-order topological structure, then sets denable in M that are boolean combinations of open sets are boolean combinations of open denable sets. Let X be a topological space....
Article
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A Borel equivalence relation on a Polish space is countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy o...
Article
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Hausdorff's paradoxical decomposition of a sphere with countably many points removed (the main precursor of the Banach–Tarski paradox) actually produced a partition of this set into three pieces A,B,C such that A is congruent to B (i.e. there is an isometry of the set which sends A to B), B is congruent to C, and A is congruent to B ∪ C. While refi...
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A famous result of Hausdorff states that a sphere with countably many points removed can be partitioned into three pieces A, B, C such that A is congruent to B (i.e. there is an isometry of the sphere which sends A to B), B is congruent to C, and A is congruent to B ∪ C; this result was the precursor of the Banach–Tarski paradox. Later, R. Robinson...
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We show that, for 1≤p<q<∞, the relation of ℓ p -equivalence between infinite sequences of real numbers is Borel reducible to the relation of ℓ q -equivalence (i.e., the Borel cardinality of the quotient ℝ ℕ /ℓ p is no larger than that of ℝ ℕ /ℓ q ), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake...
Article
Results of Sierpiński and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is ‘narrow’ in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set ω × ω is partitioned into two pieces along the diagonal, then one pie...
Article
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We consider algebras with one binary operation · and one generator, satisfying the left distributive law a·(b·c) =(a·b)·(a·c); such algebras have been shown to have surprising connections with set-theoretic large cardinals and with braid groups. One can construct a sequence of finite left-distributive algebras An, and then take a limit to get an in...
Article
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Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set (omega \times omega) is partitioned into two pieces along the diagona...
Article
Larman showed that any closed subset of the plane with uncountable vertical cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that Larman's result is best possible: there exist closed sets with uncountable cross-sections which do not have more than aleph_1 disjoint Borel uniformizations, even if the continuum is much larger...
Article
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In this paper, we continue the study of a left-distributive algebra of elementary embeddings from the collection of sets of rank less than lambda to itself, as well as related finite left-distributive algebras (which can be defined without reference to large cardinals). In particular, we look at the critical points (least ordinals moved) of the ele...
Article
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We address the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs), both directed and undirected. The problem turns out to be closely related to the problem of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. In particular, we...
Article
We present a new algorithm for generating a logically rectangular curvilinear mesh for a region in the plane bounded by four smooth curves. The curvilinear grid is generated using a divide-and-conquer algorithm where at each step a single grid curve or line is generated using position and derivative information derived from previously generated cur...
Article
We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this an...
Article
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In 1924 Banach and Tarski, using ideas of Hausdorff, proved that there is a partition of the unit sphere S2 into sets A1,...,Ak,B1,..., Bl and a collection of isometries [sigma1,..., sigmak, rho1,..., rhol] so that [sigma1A1,..., sigmakAk] and [rho1B1,..., rholBl] both are partitions of S2. The sets in these partitions are constructed by using the...
Article
We consider algebras with one binary operation $\cdot$ and one generator ({\it monogenic}) and satisfying the left distributive law $a\cdot (b\cdot c)=(a\cdot b)\cdot (a\cdot c)$. One can define a sequence of finite left-distributive algebras $A_n$, and then take a limit to get an infinite monogenic left-distributive algebra~$A_\infty$. Results of...
Article
Dougherty, R., Critical points in an algebra of elementary embeddings, Annals of Pure and Applied Logic 65 (1993) 211-241.Given two elementary embeddings from the collection of sets of rank less than λ to itself, one can combine them to obtain another such embedding in two ways: by composition, and by applying one to (initial segments of) the other...
Article
Given a covariant or contravariant functor from the category of finite sets to itself, one can define a function from natural numbers to natural numbers by seeing how the functor maps cardinalities. In this paper we answer the question: what numerical functions arise in this way from functors? The sufficiency of the conditions we give is shown by s...
Article
A [55,16,19] binary Goppa code is used to construct [57,17,17], [58,17,18], [59,17,19], and [60,17,20] codes. The first two codes have smaller redundancy than previously known codes (linear or nonlinear) of the same length and minimum distance. The last two codes have parameters previously attained only by nonlinear codes
Article
We compute the covering radius of each binary cyclic code of length ≤ 64 (for both even and odd lengths) and redundancy ≤ 28 . We also compute the covering radii of their punctured codes and shortened codes. Thus we give exact covering radii of over six thousand codes. For each of these codes (except for certain composite codes), we also determine...
Article
We consider real-valued functions defined on the interval [0, 1]. We denote by Δ the set of derivatives; i.e., ƒ Є Δ iff there is a differentiable function F such that F' = ƒ. Any such F is a primitive of ƒ and is uniquely determined up to a constant. To normalize, we denote by F(x)=ƒ^x_0ƒ the primitive determined by F(0)=0. This is the original Ne...
Article
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The Hermite polynomials are simple, effective interpolants of discrete data. These interpolants can preserve local positivity, monotonicity, and convexity of the data if we restrict their derivatives to satisfy constraints at the data points. This paper describes the conditions that must be satisfied for cubic and quintic Hermite interpolants to pr...
Article
We characterize the closed sets E in the unit circle T which have the property that, for some nondecreasing h: (0, ∞) →(0, ∞) with h(0+) = 0, all the Hausdorff h-measure 0 closed sets F ⊆ E are sets of uniqueness (for trigonometric series). In conjunction with Körner's result on the existence of Helson sets of multiplicity, this implies the existen...
Article
Define the partial ordering ≤ on the Cantor space ω 2 by x≤y iff ∀nx(n)≤y(n) (this corresponds to the subset relation on the power set of ω). A set A⊆ ω 2 is monotone reducible to a set B⊆ ω 2 iff there is a monotone (i.e., x≤y⇒ f(x)≤f(y)) continuous function f: ω 2→ ω 2 such that x∈A iff f(x)∈B. In this paper, we study the relation of monotone red...
Article
A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for backgro...
Article
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We present an argument (due originally to R. C. Lyndon) which completes the proof of the following theorem: Every free group word which is not a proper power can represent any permutation of an innite set.
Article
Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expres...
Article
This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). We investigated the theory of wavelet transforms and their relation to Laboratory applications. The investigators have had considerable success in the past applying wavelet techniques to the numerical s...
Article
Thesis (Ph. D. in Mathematics)--University of California, Berkeley, May 1985. Includes bibliographical references (leaf 65).

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