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Homogeneous multivariate polynomials with the halfplane property
 Adv. in Appl. Math
"... A polynomial P in n complex variables is said to have the “halfplane property” (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right halfplane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A ..."
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Cited by 38 (4 self)
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A polynomial P in n complex variables is said to have the “halfplane property” (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right halfplane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an runiform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the halfplane property is necessarily the set of bases of a matroid. Conversely, we ask: For which matroids M does the basis generating polynomial P B(M) have the halfplane property? Not all matroids have the halfplane property, but we find large classes that do: all sixthrootofunity matroids, and a subclass of transversal (or cotransversal) matroids that we call “nice”. Furthermore, the class of matroids with the halfplane property is closed under minors, duality, direct sums, 2sums, series and parallel connection, fullrank matroid union, and some special cases of principal truncation, principal extension, principal cotruncation and principal coextension. Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the halfplane property: a determinant construction (exploiting “energy” arguments), and a permanent construction (exploiting the Heilmann–Lieb theorem on matching polynomials). We conclude with a list of open questions. KEY WORDS: Graph, matroid, jump system, abstract simplicial complex, spanning tree, basis, generating polynomial, reliability polynomial, Brown–Colbourn conjecture,
On the Chromatic Roots of Generalized Theta Graphs
 J. COMBINATORIAL THEORY, SERIES B
, 2000
"... The generalized theta graph \Theta s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : : ; s k 1. We prove that the roots of the chromatic polynomial (\Theta s 1 ;:::;s k ; z) of a kary generalized theta graph all lie in the disc jz \Gamma 1 ..."
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Cited by 15 (4 self)
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The generalized theta graph \Theta s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : : ; s k 1. We prove that the roots of the chromatic polynomial (\Theta s 1 ;:::;s k ; z) of a kary generalized theta graph all lie in the disc jz \Gamma 1j [1 + o(1)] k= log k, uniformly in the path lengths s i . Moreover, we prove that \Theta 2;:::;2 ' K 2;k indeed has a chromatic root of modulus [1 + o(1)] k= log k. Finally, for k 8 we prove that the generalized theta graph with a chromatic root that maximizes jz \Gamma 1j is the one with all path lengths equal to 2; we conjecture that this holds for all k.
Asymptotic Properties Of HeineStieltjes And Van Vleck Polynomials
 J. APPROX. THEORY
"... We study the the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lamé differential equations, when their coeffients satisfy ertain asymptotic conditions. The limit distribution is described by an equilibrium measure in presence of an external field, generated by ch ..."
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Cited by 12 (4 self)
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We study the the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lamé differential equations, when their coeffients satisfy ertain asymptotic conditions. The limit distribution is described by an equilibrium measure in presence of an external field, generated by charges at the singular points of the equation. Moreover, a case of nonpositive charges is onsidered, which leads to an equilibrium with a nonconvex external field.
The Cost Distribution of QueueMergesort, Optimal Mergesorts, and PowerofTwo Rules
"... Queuemergesort is recently introduced by Golin and Sedgewick as an optimal variant of mergesorts in the worst case. In this paper, we present a complete analysis of the cost distribution of queuemergesort, including the best, average and variance cases. The asymptotic normality of its cost is also ..."
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Cited by 9 (5 self)
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Queuemergesort is recently introduced by Golin and Sedgewick as an optimal variant of mergesorts in the worst case. In this paper, we present a complete analysis of the cost distribution of queuemergesort, including the best, average and variance cases. The asymptotic normality of its cost is also established under the uniform permutation model. We address the corresponding optimality problems and show that if we fix the merging scheme then the optimal mergesort as far as the average number of comparisons is concerned is to divide as evenly as possible at each recursive stage (topdown mergesort). On the other hand, the variance of queuemergesort reaches asymptotically the minimum value. We also characterize a class of mergesorts with the latter property. A comparative discussion is given on the probabilistic behaviors of topdown mergesort, bottomup mergesort and queuemergesort. We derive an "invariance principle" for asymptotic linearity of divideandconquer recurrences based on general "poweroftwo" rules of which the underlying dividing rule of queuemergesort is a special case. These analyses reveal an interesting algorithmic feature for general "poweroftwo" rules.
UNIMODULARITY OF ZEROS OF SELFINVERSIVE POLYNOMIALS
, 2011
"... Abstract. We generalise a necessary and sufficient condition given by Cohn for all the zeros of a selfinversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closel ..."
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Cited by 3 (1 self)
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Abstract. We generalise a necessary and sufficient condition given by Cohn for all the zeros of a selfinversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang as well as by Lalín and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lalín and Rogers on computational evidence. 1.
Bifurcation of the ACT map
, 709
"... In this paper, we study the ArneodoCoulletTresser map F(x,y,z) = (ax − b(y − z),bx + a(y − z),cx − dx k + ez) where a,b,c,d,e are real with bd ̸ = 0 and k> 1 is an integer. We obtain stability regions for fixed points of F and symmetric period2 points while c and e vary as parameters. Varying ..."
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In this paper, we study the ArneodoCoulletTresser map F(x,y,z) = (ax − b(y − z),bx + a(y − z),cx − dx k + ez) where a,b,c,d,e are real with bd ̸ = 0 and k> 1 is an integer. We obtain stability regions for fixed points of F and symmetric period2 points while c and e vary as parameters. Varying a and e as parameters, we show that there is a hyperbolic invariant set on which F is conjugate to the full shift on two or three symbols. We also show that chaotic behaviors of F while c and d vary as parameters and F is near an antiintegrable limit. Some numerical results indicates F has Hopf bifurcation, strange attractors, and nested structure of invariant tori. 1
Abstract Efficient Algorithms for Computing the Nearest Polynomial with Constrained Roots ∗
"... Continuous changes of the coefficients of a polynomial move the roots continuously. We consider the problem finding the minimal perturbations to the coefficients to move a root to a given locus, such as a single point, the real or imaginary axis, the unit circle, or the right half plane. We measure ..."
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Continuous changes of the coefficients of a polynomial move the roots continuously. We consider the problem finding the minimal perturbations to the coefficients to move a root to a given locus, such as a single point, the real or imaginary axis, the unit circle, or the right half plane. We measure minimality in both the Euclidean distance to the coefficient vector and maximal coefficientwise change in absolute value (infinity norm), either with entirely real or with complex coefficients. If the locus is a piecewise parametric curve, we can give efficient, i.e., polynomial time algorithms for the Euclidean norm; for the infinity norm we present an efficient algorithm when a root of the minimally perturbed polynomial is constrained to a single point. In terms of robust control, we are able to compute the radius of stability in the Euclidean norm for a wide range of convex open domains of the complex plane. 1