Realizability of combinatorial types of convex polyhedra over fields

It is shown that the minimal subfield of the field of real numbers over which all real combinatorial types of convex polyhedra can be realized is the field of all real algebraic numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 123, pp. 203–207, 1983.     

Littlewood Pisot numbers

A Pisot number is a real algebraic integer, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is a Littlewood polynomial, one with {+1,-1}-coefficients, and shows that they form an increasing sequence with limit 2. It is known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Littlewood polynomials. Finally, we prove that every reciprocal Littlewood polynomial of odd degree n?3 has at least three unimodular roots.     

Face numbers of generalized balanced Cohen-Macaulay complexes

A common generalization of two theorems on the face numbers of Cohen-Macaulay (CM, for short) simplicial complexes is established: the first is the theorem of Stanley (necessity) and Bj?rner-Frankl-Stanley (sufficiency) that characterizes all possible face numbers of a-balanced CM complexes, while the second is the theorem of Novik (necessity) and Browder (sufficiency) that characterizes the face numbers of CM subcomplexes of the join of the boundaries of simplices.     

Length of the Sum and Product of Algebraic Numbers

In the present paper, we consider products of lengths of algebraic numbers whose sum or product is a chosen algebraic number. These products are used to construct a new height function for algebraic numbers. With the help of this function, a metric on the set of all algebraic numbers, which induces the discrete topology, is introduced.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 854–860.Original Russian Text Copyright ©2005 by A. Dubickas, C. J. Smyth.     

Nonnegatively realizable spectra with two positive eigenvalues

Boyle and Handelman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316.] characterized all lists of n complex numbers that can be the nonzero spectrum of a nonnegative matrix. This article presents a constructive proof of this result in the special case when the lists are real and contain two positive numbers and n ? 2 negative numbers. A bound for the number of zeros that needs to be added to the list to achieve a nonnegative realization is presented in this case.     

On three problems for sequences

If ξ∈ (0,1) and A=a_n, n?? is a sequence of real numbers define S_n(ξ,A)∶=Σ{a_k∶:k=[nξ]+1 to n}, n??, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence S_n (ξ,A), n??, for all ξ in an appropriate set K of real numbers, that the sequence a_n, n??, converges to zero. It was shown that such a conclusion is possible if K={ξ,1?ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper:

1. does the convergence of S_n (ξ,A), n??, for a single irrational number ξ imply a_n→0.
2. does the convergence of S_n(ξ,A), n??, for finitely many rational numbers ξ∈ (0, 1) imply a_n→0.
3. does the convergence of S_n (ξ,A), n??, for all rational numbers ξ∈ (0,1) imply a_n→0?

     

A new statistic on Dyck paths for counting 3-dimensional Catalan words

A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path D, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly D. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.     

Fast compact prime number sieves (among others)

A parameterized family of algorithms is presented for the problem of finding all prime numbers up to a limit N. Many previous algorithms for this problem are shown to correspond to members of this family. A particular choice of parameters leads to an algorithm that requires only Θ(N) additions and that runs in bits. The small storage requirement is made possible by a provably compact method of storing all the primes up to some limit.     

Multiperfect numbers on lines of the Pascal triangle

A number n is said to be multiperfect (or multiply perfect) if n divides its sum of divisors σ(n). In this paper, we study the multiperfect numbers on straight lines through the Pascal triangle. Except for the lines parallel to the edges, we show that all other lines through the Pascal triangle contain at most finitely many multiperfect numbers. We also study the distribution of the numbers σ(n)/n whenever the positive integer n ranges through the binomial coefficients on a fixed line through the Pascal triangle.     

Enumeration of difference graphs

A difference graph is a bipartite graph G = (X, Y; E) such that all the neighborhoods of the vertices of X are comparable by inclusion. We enumerate labeled and unlabeled difference graphs with or without a bipartition of the vertices into two stable sets. The labeled enumerations are expressed in terms of combinatorial numbers related to the Stirling numbers of the second kind.     

Comparison of fuzzy numbers using a fuzzy distance measure

A new approach for ranking fuzzy numbers based on a distance measure is introduced. A new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first, and then it is used to formulate the distance measure for fuzzy numbers. The approach is illustrated by numerical examples, showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches.     

Counting finite posets and topologies

A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with n points and k antichains for n11 and all relevant integers k. Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with n points for n14. Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the numbers of finite topological spaces with n points and k open sets for n11 and all k, and then the numbers of all topologies on n14 points satisfying various degrees of separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds 10²³.     

Raney and Catalan

Raney’s lemma is often used in a counting argument to prove the formula for (generalized) Catalan numbers. It ensures the existence of “good” cyclic shifts of certain sequences, i.e. cyclic shifts for which all partial sums are positive.We introduce a simple algorithm that finds these cyclic shifts and also those with a slightly weaker property. Moreover it provides simple proofs of lemma’s of Raney type.A similar clustering procedure is also used in a simple proof of a theorem on probabilities of which many well-known results (e.g. on lattice paths and on generalized Catalan numbers) can be derived as corollaries. The theorem generalizes generalized Catalan numbers. In the end it turns out to be equivalent to a formula of Raney.     

From Bruhat intervals to intersection lattices and a conjecture of Postnikov

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w∈Sn is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups.A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.     

Blaschke-Süss type theorems for closed plane curves

A C¹-class of plane closed curves is considered, which contains all ovals and rosettes. This class is divided into some subclasses for which greatest lower bounds of numbers of antipodal and arc-antipodal sets are determined.     

Automorphisms of the lattice of all subalgebras of the semiring of polynomials in one variable

In this paper, we describe automorphisms of the lattice $ {\mathbb A} $ of all subalgebras of the semiring ?⁺[x] of polynomials in one variable over the semifield ?⁺ of nonnegative real numbers. It is proved that any automorphism of the lattice A is generated by an automorphism of the semiring ?⁺[x] that is induced by a substitution x ? px for some positive real number p. It follows that the automorphism group of the lattice $ {\mathbb A} $ is isomorphic to the group of all positive real numbers with multiplication. A technique of unigenerated subalgebras is applied.     

Gruppentopologien auf nichtabelschen abzählbaren Gruppen

Group topologies of countable groups are constructed. They form a dense subset of the lattice of all group topologies. Every such group topology induces an injective mapping from the set of all filters of the natural numbers, which are finer than the filter of those sets of natural numbers having finite complements, into the lattice of group topologies. A sufficient condition is given, which groups can be topologized. It is shown, that every group can be topologized by a non-linear group topology, if it can be topologized by a group topology.     

Lorentz变换的四元数表示——纪念李国平院士吴新谋教授诞辰100周年

为一般Lorentz变换给出了一种新的形式简单的四元数表示. 其特点是所用四元数的分量要么是实数, 要么是纯虚数. 与以往的向量-张量表示和八元数表示(双四元数)相比, 有其明显的优点.     

Division in a binary representation for complex numbers

Computer operations involving complex numbers, essential in such applications as Fourier transforms or image processing, are normally performed in a ‘divide-and-conquer’ approach dealing separately with real and imaginary parts. A number of proposals have treated complex numbers as a single unit but all have foundered on the problem of the division process without which it is impossible to carry out all but the most basic arithmetic. This paper resurrects an early proposal to express complex numbers in a single ‘binary’ representation, reviews basic complex arithmetic and is able to provide a fail-safe procedure for obtaining the quotient of two complex numbers expressed in the representation. Thus, while an outstanding problem is solved, recourse is made only to readily accessible methods. A variety of extensions to the work requiring similar basic techniques are also identified. An interesting side-line is the occurrence of fractal structures, and the power of the ‘binary’ representation in analysing the structure is briefly discussed.     

Primes are nonnegative values of a polynomial in 10 variables

A polynomial with integer coefficients of 10 variables is constructed, whose set of all nonnegative values (for positive integer values of the variables) is precisely the set of all prime numbers.Translated by Louise Guy and James P. Jones, The University of Calgary, Calgary, CanadaTranslated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 68, pp. 62–82, 1977.