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31
On pseudorandom generators in NC 0
 In Proceedings of 26th Mathematical Foundations of Computer Science
, 2001
"... Abstract. In this paper we consider the question of whether NC 0 circuits can generate pseudorandom distributions. While we leave the general question unanswered, we show • Generators computed by NC 0 circuits where each output bit depends on at most 3 input bits (i.e, NC 0 3 circuits) and with stre ..."
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Cited by 18 (0 self)
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Abstract. In this paper we consider the question of whether NC 0 circuits can generate pseudorandom distributions. While we leave the general question unanswered, we show • Generators computed by NC 0 circuits where each output bit depends on at most 3 input bits (i.e, NC 0 3 circuits) and with stretch factor greater than 4 are not pseudorandom. • A large class of “nonproblematic ” NC 0 generators with superlinear stretch (including all NC 0 3 generators with superlinear stretch) are broken by a statistical test based on a linear dependency test combined with a pairwise independence test. • There is an NC 0 4 generator with a superlinear stretch that passes the linear dependency test as well as kwise independence tests, for any constant k. 1
ZHANG’S CONJECTURE AND THE EFFECTIVE BOGOMOLOV CONJECTURE OVER FUNCTION FIELDS
, 2009
"... We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus ..."
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Cited by 16 (4 self)
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We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang’s Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.
An extremal problem on potentially Km − C4graphic sequences
 Journal of Combinatorial Mathematics and Combinatorial Computing
"... A sequence S is potentially Km −C4graphical if it has a realization containing a Km − C4 as a subgraph. Let σ(Km − C4, n) denote the smallest degree sum such that every nterm graphical sequence S with σ(S) ≥ σ(Km − C4, n) is potentially Km − C4graphical. In this paper, we prove that σ(Km − C4, n ..."
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Cited by 14 (11 self)
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A sequence S is potentially Km −C4graphical if it has a realization containing a Km − C4 as a subgraph. Let σ(Km − C4, n) denote the smallest degree sum such that every nterm graphical sequence S with σ(S) ≥ σ(Km − C4, n) is potentially Km − C4graphical. In this paper, we prove that σ(Km − C4, n) ≥ (2m − 6)n − (m − 3)(m − 2) + 2, for n ≥ m ≥ 4. We conjecture that equality holds for n ≥ m ≥ 4. We prove that this conjecture is true for m = 5. Key words: graph; degree sequence; potentially Km − C4graphic sequence AMS Subject Classifications: 05C07, 05C35 1
Proofs Without Syntax
 Annals of Mathematics
"... [M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatori ..."
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Cited by 11 (0 self)
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[M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graphtheoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h: C → G(φ), where G(φ) is a graph associated with φ and C is a coloured graph. The main theorem is soundness and completeness: φ is true if and only if there exists a combinatorial proof h: C → G(φ). 1.
An extremal problem on potentially Km − Pkgraphic sequences ∗
, 2004
"... A sequence S is potentially Km −Pk graphical if it has a realization containing a Km −Pk as a subgraph. Let σ(Km −Pk, n) denote the smallest degree sum such that every nterm graphical sequence S with σ(S) ≥ σ(Km−Pk, n) is potentially Km−Pk graphical. In this paper, we prove that σ(Km−Pk, n) ≥ (2m ..."
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Cited by 10 (10 self)
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A sequence S is potentially Km −Pk graphical if it has a realization containing a Km −Pk as a subgraph. Let σ(Km −Pk, n) denote the smallest degree sum such that every nterm graphical sequence S with σ(S) ≥ σ(Km−Pk, n) is potentially Km−Pk graphical. In this paper, we prove that σ(Km−Pk, n) ≥ (2m−6)n−(m−3)(m−2)+2, for n ≥ m ≥ k + 1 ≥ 4. We conjectured that equality holds for n ≥ m ≥ k + 1 ≥ 4. We proved that this conjecture is true for m = k + 1 = 5 and m = k + 2 = 5. Key words: graph; degree sequence; potentially Km − Pkgraphic sequence
THE TAU CONSTANT AND THE EDGE CONNECTIVITY OF A Metrized Graph
, 2009
"... The tau constant is an important invariant of a metrized graph, and it has applications in arithmetic properties of curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We discover identities which we call “contraction”, “deletion”, and ..."
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Cited by 9 (7 self)
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The tau constant is an important invariant of a metrized graph, and it has applications in arithmetic properties of curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We discover identities which we call “contraction”, “deletion”, and “contractiondeletion” identities on a metrized graph. By establishing a lower bound for the tau constant in terms of the edge connectivity, we prove that Baker and Rumely’s lower bound conjecture on the tau constant holds for metrized graphs with edge connectivity 5 or more. We show that proving this conjecture for 3regular graphs is enough to prove it for all graphs.
Subdivision surface watermarking
, 2006
"... This paper presents a robust nonblind watermarking scheme for subdivision surfaces. The algorithm works in the frequency domain, by modulating spectral coefficients of the subdivision control mesh. The compactness of the watermarking support (a coarse control mesh) has led us to optimize the trade ..."
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Cited by 8 (3 self)
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This paper presents a robust nonblind watermarking scheme for subdivision surfaces. The algorithm works in the frequency domain, by modulating spectral coefficients of the subdivision control mesh. The compactness of the watermarking support (a coarse control mesh) has led us to optimize the tradeoff between watermarking redundancy (which insures robustness) and imperceptibility by introducing two contributions: (1) Spectral coefficients are perturbed according to a new modulation scheme analysing the spectrum shape and (2) the redundancy is optimized by using error correcting codes coming from telecommunication theory. Since the watermarked surface can be attacked in a subdivided version, we have introduced an algorithm to retrieve the control polyhedron, starting from a subdivided, attacked version. Experiments have shown the high robustness of our scheme against geometry attacks such as noise addition, quantization or nonuniform scaling and also connectivity alterations such as remeshing or simplification.
THE SMALLEST DEGREE SUM THAT YIELDS POTENTIALLY CkGRAPHICAL SEQUENCE
, 2002
"... In this paper we consider a variation of the classical extremal problems. Let S be a nterm graphical sequence, and σ(S) be the sum of the terms in S. Let G be a graph. The problem is to determine the smallest even m such that any nterm graphical sequence S having σ(S) ≥ m has a realization contai ..."
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Cited by 8 (4 self)
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In this paper we consider a variation of the classical extremal problems. Let S be a nterm graphical sequence, and σ(S) be the sum of the terms in S. Let G be a graph. The problem is to determine the smallest even m such that any nterm graphical sequence S having σ(S) ≥ m has a realization containing G. Denote this value by σ(G, n). We show σ(C2m+1, n) = m(2n − m − 1) + 2, for m ≥ 3, n ≥ 3m; σ(C2m+2, n) = m(2n − m − 1) + 4, for m ≥ 3, n ≥ 5m − 2.
An extremal problem on potentially Kr+1 − Hgraphic sequences, accepted by Ars Combinatoria
 12 Chunhui Lai ⋃ and Yuzhen Sun, An extremal problem on potentially Kr+1 − (kP2 tK2)graphic sequences, submitted
"... Let Kk, Ck, Tk, and Pk denote a complete graph on k vertices, a cycle on k vertices, a tree on k + 1 vertices, and a path on k + 1 vertices, respectively. Let Km −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of Km). A sequence S is potentially Km − H ..."
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Cited by 7 (7 self)
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Let Kk, Ck, Tk, and Pk denote a complete graph on k vertices, a cycle on k vertices, a tree on k + 1 vertices, and a path on k + 1 vertices, respectively. Let Km −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of Km). A sequence S is potentially Km − Hgraphical if it has a realization containing a Km − H as a subgraph. Let σ(Km − H, n) denote the smallest degree sum such that every nterm graphical sequence S with σ(S) ≥ σ(Km − H, n) is potentially Km − Hgraphical. In this paper, we determine the values of σ(Kr+1−H, n) for n ≥ 4r+10, r ≥ 3, r+1 ≥ k ≥ 4 where H is a graph on k vertices which contains a tree on 4 vertices but not contains a cycle on 3 vertices. We also determine the values of σ(Kr+1 − P2, n) for n ≥ 4r + 8, r ≥ 3. Key words: graph; degree sequence; potentially Kr+1 − Hgraphic sequence AMS Subject Classifications: 05C07, 05C35 1
1 Optimal Scaling of Multicommodity Flows in Wireless Ad Hoc Networks: Beyond The GuptaKumar Barrier
"... Abstract—We establish a tight maxflow mincut theorem for multicommodity routing in random geometric graphs. We show that, as the number of nodes in the network n tends to infinity, the maximum concurrent flow (MCF) and the minimum cutcapacity scale as Θ(n 2 r 3 (n)/k) for a random choice of k≥Θ( ..."
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Cited by 3 (3 self)
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Abstract—We establish a tight maxflow mincut theorem for multicommodity routing in random geometric graphs. We show that, as the number of nodes in the network n tends to infinity, the maximum concurrent flow (MCF) and the minimum cutcapacity scale as Θ(n 2 r 3 (n)/k) for a random choice of k≥Θ(n) sourcedestination pairs, where r(n) is the communication range in the network. We exploit the fact that the MCF in a random geometric graph equals the capacity of an adhoc network under the protocol model and interferencefree communication to derive scaling laws for interferenceconstrained network capacity. We generalize all existing results reported to date by showing that the percommodity capacity of the network scales as Θ(1/r(n)k) for the singlepacket reception model suggested by Gupta and Kumar, and as Θ(nr(n)/k) for the multiplepacket reception model suggested by others. More importantly, we show that, if the nodes in the network are capable of multiplepacket transmission and reception, then it is feasible to achieve the optimal scaling of Θ ` n 2 r 3 (n)/k ´ , despite the presence of interference. This result provides an improvement of Θ ` nr 2 (n) ´ over the highest achieved capacity reported to date. In stark contrast to the conventional wisdom that has evolved from the GuptaKumar results, our results show that the capacity of adhoc networks can actually increase with n while the communication range tends to zero! I.