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Theory of Computing Systems © 1998 *Springer-Verlag* *New* *York* *Inc*.

"... where K denotes simple Kolmogorov entropy (i.e., the very first version of Kolmogorov complexity having been introduced by Kolmogorov himself) and KP denotes prefix entropy (self-delimiting complexity by the terminology of Li and Vitanyi [1]). It turns out that from (1) the following well-known geom ..."

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where K denotes simple Kolmogorov entropy (i.e., the very first version of Kolmogorov complexity having been introduced by Kolmogorov himself) and KP denotes prefix entropy (self-delimiting complexity by the terminology of Li and Vitanyi [1]). It turns out that from (1) the following well-known geometric fact can be inferred: |V | 2 ≤|Sxy|·|Syz|·|Sxz|, where V is a set in three-dimensional space, Sxy,Syz,Sxz are its three two-dimensional projections, and |W | is the volume (or the area) of W. Inequality (2), in its turn, is a corollary of the well-known Cauchy–Schwarz inequality. So the connection between geometry and Kolmogorov complexity works in both directions.

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© *Springer-Verlag* *New* *York* *Inc*. 1996 Standardized Nomenclature for Alu Repeats

, 1995

"... Short interspersed elements (SINEs) may be found in the genomes of a wide variety of mammals (Deininger and Batzer 1993). The Alu family of SINEs is one of the most successful mobile genetic elements, having arisen to a copy number in excess of 500,000 within the human genome in approximately 65 mil ..."

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Short interspersed elements (SINEs) may be found in the genomes of a wide variety of mammals (Deininger and Batzer 1993). The Alu family of SINEs is one of the most successful mobile genetic elements, having arisen to a copy number in excess of 500,000 within the human genome in approximately 65 million years of primate evolution. (For reviews see Deininger 1989; Okada 1991; Schmid and Maraia 1992; Deininger and Batzer 1993.) Alu sequences are thought to be ancestrally derived

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9 1987 *Springer-Verlag* *New* *York* *Inc*. The Set LCS Problem

"... Abstract. An efficient algorithm is presented that solves a generalization of the Longest Common Subsequence problem, in which one of the two input strings contains sets of symbols which may be permuted. This problem arises from a music application. Key Words. Subsequence, Common subsequence, Dynami ..."

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Abstract. An efficient algorithm is presented that solves a generalization of the Longest Common Subsequence problem, in which one of the two input strings contains sets of symbols which may be permuted. This problem arises from a music application. Key Words. Subsequence, Common subsequence, Dynamic programming.

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© 1988 *Springer-Verlag* *New* *York* *inc*.ry Simplified Voronoi Diagrams*

"... Abstract. We are interested in Voronoi diagrams as a tool in robot path planning, where the search for a path in an r-dimensional space may be simplified to a search on an (r- l)-dimensional Voronoi diagram. We define a Voronoi diagram V based on a measure of distance which is not a true metric. Thi ..."

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Abstract. We are interested in Voronoi diagrams as a tool in robot path planning, where the search for a path in an r-dimensional space may be simplified to a search on an (r- l)-dimensional Voronoi diagram. We define a Voronoi diagram V based on a measure of distance which is not a true metric. This formulation has lower algebraic complexity than the usual definition, which is a considerable advantage in motion-planning problems with many degrees of freedom. In its simplest form, the measure of distance between a point and a polytopc is the maximum of the distances of the point from the half-spaces which pass through faces of the polytope. More generally, the measure is defined in configuration spaces which represent rotation. The Voronoi diagram defined using this distance measure is no longer a strong deformation retract of free space, but it has the following useful property: any path through free space which starts and ends on the diagram can be continuously deformed so that it lies entirely on the diagram. Thus it is still complete for motion planning, but it has lower algebraic complexity than a diagram based on the Euclidean metric. 1.

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APPROXIMATION © 2002 *Springer-Verlag* *New* *York* *Inc*. On a Problem of Daubechies

"... Abstract. We solve a problem posed by Daubechies [12] by showing the nonexistence of orthonormal wavelet bases with good time-frequency localization associated with irrational dilation factors. 1. ..."

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Abstract. We solve a problem posed by Daubechies [12] by showing the nonexistence of orthonormal wavelet bases with good time-frequency localization associated with irrational dilation factors. 1.

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© 2001 *Springer-Verlag* *New* *York* *Inc*. The Honeycomb Conjecture

"... Abstract. This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. 1. ..."

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Abstract. This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. 1.

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© *Springer-Verlag* *New* *York* *Inc*. 1998 Relics from the RNA World

, 1997

"... Abstract. An RNA world is widely accepted as a probable stage in the early evolution of life. Two implications are that proteins have gradually replaced RNA as the main biological catalysts and that RNA has not taken on any major de novo catalytic function after the evolution of protein synthesis, t ..."

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Abstract. An RNA world is widely accepted as a probable stage in the early evolution of life. Two implications are that proteins have gradually replaced RNA as the main biological catalysts and that RNA has not taken on any major de novo catalytic function after the evolution of protein synthesis, that is, there is an essentially irreversible series of steps RNA → RNP → protein. This transition, as expected from a consideration of catalytic perfection, is essentially complete for reactions when the substrates are small molecules. Based on these principles we derive criteria for identifying RNAs in modern organisms that are relics from the RNA world and then examine the function and phylogenetic distribution of RNA for such remnants of the RNA world. This allows an estimate of the minimum complexity of the last riboorganism—the stage just preceding the advent of genetically encoded protein synthesis. Despite the constraints placed on its size by a low fidelity of replication (the Eigen limit), we conclude that the genome of this organism reached a considerable level of complexity that included several RNA-processing steps. It would include a large protoribosome with many smaller RNAs involved in its assembly, pre-tRNAs and tRNA processing, an ability for recombination of RNA, some RNA editing, an ability to copy to the end of each RNA strand, and some transport functions. It is harder to recognize specific metabolic reactions that must have existed but synthetic and bio-energetic functions would be necessary. Overall, this requires that such an organism maintained a multiple

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© 2003 *Springer-Verlag* *New* *York* *Inc*. Multicolorings of Series-Parallel Graphs

"... Abstract. Let G be a graph, and let each vertex v of G have a positive integer weight ω(v). A multicoloring of G is to assign each vertex v a set of ω(v) colors so that any pair of adjacent vertices receive disjoint sets of colors. This paper presents an algorithm to find a multicoloring of a given ..."

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Abstract. Let G be a graph, and let each vertex v of G have a positive integer weight ω(v). A multicoloring of G is to assign each vertex v a set of ω(v) colors so that any pair of adjacent vertices receive disjoint sets of colors. This paper presents an algorithm to find a multicoloring of a given series-parallel graph G with the minimum number of colors in time O(nW), where n is the number of vertices and W is the maximum weight of vertices in G.

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© 2000 *Springer-Verlag* *New* *York* *Inc*. The Geometry of the Phase Diffusion Equation

, 1998

"... Summary. The Cross-Newell phase diffusion equation, ¿.jEkj/2T D ¡r ¢.B.jEkj / ¢Ek/; Ek D r2; and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equatio ..."

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Summary. The Cross-Newell phase diffusion equation, ¿.jEkj/2T D ¡r ¢.B.jEkj / ¢Ek/; Ek D r2; and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order “self-dual ” equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. 1.

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© 1999 *Springer-Verlag* *New* *York* *Inc*. Relative Equilibria of Molecules

, 1997

"... Summary. We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0. Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. W ..."

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Summary. We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0. Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectively to the critical points of a function h on the two-sphere which is invariant under an action of the symmetry group of the equilibrium point. From this it follows that for each rotation axis of the equilibrium configuration there is a bifurcating family of relative equilibria for which the molecule rotates about that axis. In addition, for each reflection plane there is a family of relative equilibria for which the molecule rotates about an axis perpendicular to the plane. We also show that if the equilibrium is nondegenerate and stable, then the minima, maxima, and saddle points of h correspond respectively to relative equilibria which are (orbitally) Liapounov stable, linearly stable, and linearly unstable. The stabilities of the bifurcating branches of relative equilibria are computed explicitly for XY2, X3, and XY4 molecules. These existence and stability results are corollaries of more general theorems on relative equilibria of G-invariant Hamiltonian systems that bifurcate from equilibria with finite isotropy subgroups as the momentum is varied. In the general case, the function h is defined on the Lie algebra dual g ∗ and the bifurcating relative equilibria correspond to critical points of the restrictions of h to the coadjoint orbits in g ∗.