ВЫПУКЛЫЕ РОМБОДОДЕКАЭДРЫ И ПАРАМЕТРИЧЕСКИЕ BR-МНОЖЕСТВА

Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary

Bounded remainder sets

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

In this paper we give a simple, explicit construction of polytopal bounded remainder sets of all possible volumes, for any irrational rotation on the $d$ dimensional adelic torus $\mathbb{A}^d/\mathbb{Q}^d$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument that reduces the existence argument to the case of an irrational rotation on the torus $\mathbb{R}^d/\mathbb{Q}^d$.

This paper solves the problem of computing the best grasp in a discrete point set based on a popular grasp quality measure, namely the largest origin-centered ball contained in the grasp wrench set. So far, the solution to this problem is very limited. Noticing that the quality measure for a grasp is equal to the minimum value of the support function of its grasp wrench set over all directions and its computation together with force closure test can be fulfilled by evaluating the support function in a sequence of directions, we can quickly determine that a new grasp is worse whenever its support function in the sequence of directions or any other specific direction is less than the quality value of the current best grasp and avoid further computation. Furthermore, we enumerate candidate grasps in the point set in an adaptive way such that grasps that are more likely to outperform the current best grasp will be checked first, which helps find the best grasp earlier and significantly reduce the number of candidate grasps to be fully examined. With the aid of the adaptive enumeration and the quick comparison of grasps, the proposed algorithm takes tens of seconds to several hours on a normal PC to compute the best grasp in tens to hundreds of points on 3-D objects and it is two to several orders of magnitude faster than the brute-force search.

In this paper we give a broad unified framework via group actions for constructing complexity functions of infinite words x = x 0 x 1 x 2 ⋯ ∈ A N x=x_0x_1x_2\cdots \in \mathbb {A}^{\mathbb {N}} with values in a finite set A . \mathbb {A}. Factor complexity, Abelian complexity and cyclic complexity are all particular cases of this general construction. We consider infinite sequences of permutation groups ω = ( G n ) n ≥ 1 \omega = (G_n)_{n\geq 1} with each G n ⊆ S n . G_n \subseteq S_n. Associated with every such sequence is a complexity function p ω , x : N → N p_{\omega ,x}:\mathbb {N} \rightarrow \mathbb {N} which counts, for each length n , n, the number of equivalence classes of factors of x x of length n n under the action of G n G_n on A n \mathbb {A}^n given by g ∗ ( u 1 u 2 ⋯ u n ) = u g − 1 ( 1 ) u g − 1 ( 2 ) ⋯ u g − 1 ( n ) . g* (u_1u_2\cdots u_n)=u_{g^{-1}(1)}u_{g^{-1}(2)}\cdots u_{g^{-1}(n)}. Each choice of ω = ( G n ) n ≥ 1 \omega =(G_n)_{n\geq 1} defines a unique complexity function which reflects a different combinatorial property of a given infinite word. For instance, an infinite word x x has bounded Abelian complexity if and only if x x is k k -balanced for some positive integer k , k, while bounded cyclic complexity is equivalent to x x being ultimately periodic. A celebrated result of G.A. Hedlund and M. Morse states that every aperiodic infinite word x ∈ A N x\in \mathbb {A}^{\mathbb {N}} contains at least n + 1 n+1 distinct factors of each length n . n. Moreover x ∈ A N x\in \mathbb {A}^{\mathbb {N}} has exactly n + 1 n+1 distinct factors of each length n n if and only if x x is a Sturmian word, i.e., binary, aperiodic and balanced. We prove that this characterisation of aperiodicity and Sturmian words extends to this general framework.

Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The Abelian subshift \(\mathcal A_{\varvec{x}}\) of an infinite word \(\varvec{x}\) is the set of infinite words \(\varvec{y}\) such that, for each factor u of \(\varvec{y}\), there exists a factor v of \(\varvec{x}\) which is Abelian equivalent to u. The notion of Abelian subshift gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which \(\mathcal A_{\varvec{x}}\) equals the shift orbit closure \(\varOmega _{\varvec{x}}\). On the other hand, the Abelian subshift of the Thue-Morse word contains uncountably many minimal subshifts. In this paper we undertake a general study of Abelian subshifts. In particular, we characterize the Abelian subshifts of recurrent aperiodic balanced words and the Abelian subshifts of ternary words having factor complexity \(n+2\) for all \(n\ge 1\).

This paper contains a summary table of the author’s computation of the Gaussian primes of the form a + i a + i . For the values x = 1000 , 10000 , 100000 , 180000 x = 1000, 10000, 100000, 180000 , and 500000 (500000) 14000000, the following values are tabulated: G ( x ) G(x) , the numbers of Gaussisn primes a + i a + i with a ≦ 14000000 ; π 1 ( x ) a \leqq 14000000;{\pi _1}(x) , the number of primes ≦ x \leqq x congruent to 1 m o d 4 1 mod 4 ; π 3 ( x ) {\pi _3}(x) , the number of primes ≦ x \leqq x congruent to 3 m o d 4 3 mod 4 ; and G ( x ) / π 3 ( x ) G(x)/{\pi _3}(x) .

A 39,350 bp cosmid containing DNA of Saccharomyces cerevisiae chromosome XII was sequenced by making use of ordered sub-clones of 1 kb insert-length selected from a physical clone map. In a first analysis, 96 clones were sequenced from both ends (10 gels) with two standard sequencing primers covering 91% of the total sequence (49% double-stranded). After selection of another eight clones six gaps of a total of 1.8 kb and several single-stranded stretches remained. These gaps were closed by 86 primer walks leading to an overall redundancy of 4.4 per base and a total of 292 sequencing reactions. The number of walking primers can be reduced significantly by more uniform clone lengths and longer sequencing reads, thus, the total amount of sequencing reactions can approach the minimum value achieved with primer walking strategies, with only very few walking primers needed for gap closure.

Familial isolated hyperparathyroidism: clinical and genetic characteristics of 36 kindreds.

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

Nanog, Oct4, and Sox2 form the core of a transcription factor network that maintains embryonic stem cells in the pluripotent state in both humans and mice. These critical factors have been implicated as both positive and negative regulators of transcription, varying by promoter and differentiation state of the cell. The Mediator complex, a ubiquitous conserved complex of approximately 30 subunits, facilitates transcription by coordinating RNA polymerase II binding to target promoters via gene-specific activators and can be divided into several functional subcomplexes. Med12 is part of a subcomplex of four proteins associated with the core Mediator complex and has been found to function both in repressing and activating transcription when recruited to target promoters. We identified an interaction between Med12 and Nanog and present evidence of involvement of Med12 in regulation of Nanog function. Gene expression analysis of embryonic stem cells knocked down for Med12 showed a similarity to Nanog knockdown, with increased expression of Nanog-repressed targets and decreased expression of Nanog-activated targets. Using chromatin immunoprecipitation, we found that Med12 and Nanog co-occupied Nanog target promoters in embryonic stem cells and that Med12 dissociated from target promoters upon differentiation with kinetics similar to Nanog. Our results indicate that Nanog and Med12 function in concert to regulate Nanog target genes and identify a novel role for Med12 in embryonic stem cell regulation.

In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true.Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]).The results illustrate an intimate connection between problems in computational theory, such as “P=NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

With the availability of microcomputers, mathematics classes have the opportunity to explore topics in number theory that have previously been ignored or treated superficially. The exploration of problems in number theory permits a further understanding of the BASIC language by introducing new computer functions. One topic that couples number theory to programming is palindromes.