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Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted $\mathscr{N}$). When K is an imaginary quadratic extension, $\mathscr{N}$ is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in $\mathscr{N}$ can be written as $\alpha/\overline{\alpha}$ for some $\alpha \in \mathscr{O}_K$, which yields another ordering of $\mathscr{N}$ given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that $\mathscr{N}$ is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that $\mathscr{N}$ is equidistributed with respect to norm, under the map β ↦ log|β|(mod log|ϵ2|) where ϵ is a fundamental unit of $\mathscr{O}_K$. [ABSTRACT FROM AUTHOR]

Let C be a graph and ƒ ∈ C0(G). It is proved that P(ƒ)¯ = G if R(ƒ)¯ = G and P(ƒ) ≠ ∅. This result generalizes several corresponding results given in [3] and [10]. [ABSTRACT FROM AUTHOR]

The article presents several mathematical problems and solutions. These include works on positive integers as proposed by Erwin Just from Bronx Community College of the City University of New York, the usual gamma function as proposed by M. L. Glasser from Clarkson University, and partitioning values of a polynomial into sets of equal sum as proposed by Marian Tevita from the National College in Romania. Also explored are oddly triangulated polygons and the characterization of permutation matrices

A numerical algorithm for calculating the generalized Mittag-Leffler Eα,≈(z) function for arbitrary complex argument z and real parameters α > 0 and β ϵ ℝ is presented. The algorithm uses the Taylor series, the exponentially improved asymptotic series, and integral representations to obtain optimal stability and accuracy of the algorithm. Special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm. [ABSTRACT FROM AUTHOR]

In a paper of 2003, Külshammer, Olsson and Robinson defined ℓ-blocks for the symmetric groups, where ℓ>1 is an arbitrary integer. In this paper, we give a definition for the defect group of the principal ℓ-block. We then check that, in the Abelian case, we have an analogue of one of Broué's conjectures. [ABSTRACT FROM PUBLISHER]

We investigate the class of ±1 polynomials evaluated at q defined as: A(q) = {ϵ0 + ϵ1q + … + ϵmqm : ϵi Є {-1, 1}} and usually called spectrum, and show that, if q is the root of the polynomial xn - xn-1 - … - xk+1 + xk + xk-1 + … + x + 1 between 1 and 2, and n > 2k + 3, then A(q) is discrete, which means that it does not have any accumulation points. [ABSTRACT FROM AUTHOR]

Abstract: In 1988 Manickam and Singhi conjectured that for every positive integer and every , every set of real numbers whose sum is non-negative contains at least subsets of size whose sums are non-negative. In this paper we make use of Hall’s matching theorem in order to study some numbers related to this conjecture. [Copyright &y& Elsevier]

Let T 2 k+1 be the set of trees on 2 k+1 vertices with nearly perfect matchings and α( T) be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T 2 k+1. Specifically, 10 trees T 2, T 3,⋯, T 11 and two classes of trees T(1) and T(12) in T 2 k+1 are introduced. It is shown in this paper that for each tree T , T ∈ T(1) and T , T ∈ T(12) and each i, j with 2 ≤ i < j ≤ 11, α( T ) = α( T ) > α( T i ) > α( T j ) > α( T ) = α( T ). It is also shown that for each tree T with T ∈ T 2 k+1 ( T(1) ∪ { T 2,T 3,⋯, T 11} ∪ T(12)), α( T ) > α( T). [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop a method for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth” sums , , . The proof of this result is completely combinatorial and elementary. [Copyright &y& Elsevier]

Abstract: In this paper we analyze the status of some ‘unbelievable results’ presented in the paper ‘On Some Contradictory Computations in Multi-Dimensional Mathematics’ [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] published in Nonlinear Analysis, a journal indexed in the Science Citation Index. Among some of the unbelievable results ‘proved’ in the paper we can find statements like: (i) a rotation , , is inconsistent with arithmetic, (ii) complex number theory is inconsistent. Besides these ‘results’ of mathematical nature, [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] offers also a ‘proof’ that Special Relativity is inconsistent. Now, we are left with only two options (a) the results of [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] are correct and in this case we need a revolution in Mathematics (and also in Physics) or (b) the paper is a potpourri of nonsense. We show that option (b) is the correct one. All ‘proofs’ appearing in [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] are trivially wrong, being based on a poor knowledge of advanced calculus notions. There are many examples (some of them discussed in [A.L.T. Carvalho, W.A. Rodrigues Jr., The non sequitur mathematics and physics of the ‘new electrodynamics’ proposed by the AIAS group, Random Oper. Stoch. Equs. 9 (2001) 161–206. http://arxiv.org/abs/physics/0302016; E. Capelas de Oliveira, W.A. Rodrigues Jr., Dotted and undotted algebraic spinor fields in general relativity, Int. J. Mod. Phys. D 13 (2004) 1637–1659. http://arxiv.org/abs/math-ph/0407024; W.A. Rodrigues Jr., E. Capelas de Oliveira, Clifford valued differential forms and some issues on gravitation, electromagnetism and “unified” theories, Int. J. Mod. Phys. D 13 (2004) 1879–1915. http://arxiv.org/abs/math-ph/0407025; W.A. Rodrigues Jr., Q.A.G. Souza, An ambiguous statement called ‘tetrad postulate’ and the correct field equations satisfied by the tetrad fields, Int. J. Mod. Phys. D 14 (2005) 2095–2150. http://arxiv.org/abs/math-ph/0411085; W.A. Rodrigues Jr., A comment on Emergent Gravity, http://arxiv.org/abs/gr-qc/0602111]) of completely wrong papers using non sequitur Mathematics in the Physics literature. Taking into account also that a paper like [L.A.V. Carvalho, On some contradictory computations in multi-dimensional mathematics, Nonlinear Anal. 63 (2005) 725–734] appeared in a Mathematics journal we think that it is time for editors and referees of scientific journals to become more careful in order to avoid the dissemination of nonsense. [Copyright &y& Elsevier]

Abstract: Let and be sets of nonnegative integers with . Let be a family of subsets of with for each and for any . We prove that when we have the conditions that and ’s are consecutive. We also prove the same bound under the condition instead of the above conditions. Finally, an observation gives us a bound of on when . [Copyright &y& Elsevier]

The article presents several proposed mathematical problems and solutions. Raimond Struble of Santa Monica, California proposed a sequence of complex numbers while Manuel Kauers of the Research Institute for Symbolic Computation offers a problem on Fibonnaci number. Nobuhisa Abe of NBU Attached Senior High School, Saiki, Japan proposes a problem related to integers. Solutions to other mathematical problems are also discussed including secondary inradii, polynomials with three distinct zeros summing to zero, a cousin of Gronwall's inequality and partial fractions.

Let σ( k, n) be the smallest even integer such that each n–term positive graphic sequence with term sum at least σ( k, n) can be realized by a graph containing a clique of k + 1 vertices. Erdős et al. (Graph Theory, 1991, 439–449) conjectured that σ( k, n) = ( k −1)(2 n− k)+2. Li et al. (Science in China, 1998, 510–520) proved that the conjecture is true for k ≥ 5 and $$ n \geqslant {\left( {\begin{array}{*{20}c} {k} \\ {2} \\ \end{array} } \right)} + 3 $$ , and raised the problem of determining the smallest integer N( k) such that the conjecture holds for n ≥ N( k). They also determined the values of N( k) for 2 ≤ k ≤ 7, and proved that $$ {\left\lceil {\frac{{5k - 1}} {2}} \right\rceil } \leqslant N{\left( k \right)} \leqslant {\left( {\begin{array}{*{20}c} {k} \\ {2} \\ \end{array} } \right)} + 3 $$ for k ≥ 8. In this paper, we determine the exact values of σ( k, n) for n ≥ 2 k+3 and k ≥ 6. Therefore, the problem of determining σ( k, n) is completely solved. In addition, we prove as a corollary that $$ N{\left( k \right)} = {\left\lceil {\frac{{5k - 1}} {2}} \right\rceil } $$ for k ≥ 6. [ABSTRACT FROM AUTHOR]

Assume that m ⩾ 2, p is a prime number, (m,p(p - 1)) = 1, -1 ∉ (p) ⊂ (∤/m∤)* and [(∤/m∤)* : (p)] = 4. In this paper, we calculate the value of Gauss sum G(X) = Σx∈F*qX(x)ζT(x)P over ...q, where q = pf, f = φ(m)/4, X is a multiplicative character of ...q and T is the trace map from ...q to ...p. Under our assumptions, G(X) belongs to the decomposition field K of p in ℚ(ζm) and K is an imaginary quartic abelian number field. When the Galois group Gal(K/ℚ) is cyclic, we have studied this cyclic case in another paper: "Gauss sums of index four: (1) cyclic case" (accepted by Acta Mathematica Sinica, 2003). In this paper we deal with the non-cyclic case. [ABSTRACT FROM AUTHOR]

Baranov and Zhilinskii (1999) have shown a classification theorem for diagonal direct limits of simple Lie algebras. In this work, we will transfer their results to diagonal direct limits of certain matrix groups, and show that homotopy groups are significant invariants for specific classes of direct limit groups. Communicated by B. Allison [ABSTRACT FROM AUTHOR]

We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primal-dual schema and the local ratio technique. Recently, primal-dual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primal-dual algorithm. This was done in the case of the $2$-approximation algorithms for the feedback vertex set problem and in the case of the first primal-dual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447--478]. In this paper we answer this question by showing that the two paradigms are equivalent. [ABSTRACT FROM AUTHOR]

We call a matrix $A$ nearly totally unimodular if it can be obtained from a totally unimodular matrix $\tilde{A}$ by adding to each row of $\tilde{A}$ an integer multiple of some fixed row $a^{\transp}$ of $\tilde{A}$. For an integer vector $b$ and a nearly totally unimodular matrix $A$, we denote by $P_{A,b}$ the integer hull of the set $\{x\in\mathbb{R}^n\mid Ax\leq b\}$. We show that $P_{A,b}$ has the integer decomposition property and that we can find a decomposition of a given integer vector $x\in kP_{A,b}$ in polynomial time. An interesting special case that plays a role in many cyclic scheduling problems is when $A$ is a circular-ones matrix. In this case, we show that given a nonnegative integer $k$ and an integer vector $x$, testing if $x\in kP_{A,b}$ and finding a decomposition of $x$ into $k$ integer vectors in $P_{A,b}$ can be done in time $O(n(n+m)+\text{size}(x))$, where $A$ is an $m\times n$ matrix. We show that the method unifies some known results on coloring circular arc graphs and edge coloring nearly bipartite graphs. It also gives an efficient algorithm for a packet scheduling problem for smart antennas posed by Amaldi, Capone, and Malucelli in [Fourth ALIO/EURO Workshop on Applied Combinatorial Optimization, Pucón, Chile, 2002]; [Proceedings of the Second Cologne-Twente Workshop on Graphs and Combinatorial Optimization, Vol. 1, 2003, pp. 1--4]. [ABSTRACT FROM AUTHOR]

Presents the mathematical theory in the lesser-known delta-function in number theory. Number of contexts required in analytic number theory; Versions of the classical sieve of Eratorsthenes that produces primes; Presentation of a number of results concerning the delta function; Expressions of the two values of delta function.

Presents a calendar of mathematical problems for the month of January 2004. Ways to express 100 as the sum of two or more consecutive positive integers; Probability of rolling a fair die 5 times; Sum of three integers whose base is ten.

Earlier work of the author on prime values of [10nα] for almost all α > 0 is complemented by the construction of ‘prime‐like’ sets A for which [10nα] ∈ A has only finitely many solutions for almost all α. [ABSTRACT FROM PUBLISHER]

The article presents a curious proof of Fermat's little theorem. Any formal power series can be represented in a unique way as a formal product where the coefficients being used are integers. A binomial theorem is not used in the proof of Fermat's little theorem. The author proves that any infinite sequence of integers has an inverse sequence of integers wherein all such sequences split into mutually inverse pairs. An inverse of a certain sequence for which the infinite product has a rich mathematical content is also expressed.

The article traces the contributions of Pierre Hérigone in making mathematics algebraic rather than geometric. It discusses the algebraic concepts and notation of Hérigone which can be found in his volume on algebra. It introduces his universal symbolic language applicable to all branches of mathematics. It presents his algebraic procedures and subsections on identities, equations, geometric constructions and ambiguous equations. His influence on later mathematicians including Isaac Barrow and Gottfried Wilhelm von Leibniz is also discussed.

The article explores the proof of the infinitude of primes from mathematician Hillel Furstenberg. It avoids topological language and exhibits the real reason that Furstenberg's approach works as viewed by the author. It denotes that if m and r are integers, a set of integers congruent to r mod m exists which refers as arithmetic progression. The author proves that a finite intersection of AP is either empty or infinite and if S is any collection of sets, then a finite intersection of finite unions of sets in S is also a finite union of finite intersections of sets in S.