Positive Rational Numbers Research Articles

On the equivalence of types

Types over a discrete valued field (K,v) are computational objects that pa- rameterize certain families of monic irreducible polynomials in Kv(x), where Kv is the completion of K at v. Two types are considered to be equivalent if they encode the same family of prime polynomials. In this paper, we characterize the equivalence of types in terms of certain data supported by them. based on the choice of certain key polynomials φi ∈ K(x) and positive rational numbers νi. Then, given an irreducible polynomial f ∈ K(x), he characterized all extensions of v to the field L := K(x)/(f) as limits of sequences of inductive valuations on K(x) whose value at f grows to infinity. In the case K = Q, Ore's p-regularity condition is satisfied when all valuations on L extending the p-adic valuation are sufficiently close to valuations onK(x) that may be obtained from µ0 by a single augmentation step. In 1999, J. Montes carried out Ore's program in its original formulation (3, 10). He introduced types as computational objects which are able to construct MacLane's valuations and the higher residual polynomial operators foreseen by Ore. These ideas made the whole theory constructive and well-suited to computational applications, and led to the design of several fast algorithms to perform arithmetic tasks in global fields (2, 4, 5, 7, 11). In 2007, M. Vaquie reviewed and generalized MacLane's work to non-discrete valuations. The introduction of the graded algebra Gr(µ) of a valuation µ led him to a more elegant presentation of the theory. In the papers (1) and (6), which deal only with discrete valuations, the ideas of Montes were used to develop a constructive treatment of Vaquie's approach, which included the computation of generators of the graded algebras and a thorough revision and simplification of the algorithmic applications.

Scattering by infinitely rising one-dimensional potentials

Infinitely rising one-dimensional potentials constitute impenetrable barriers which reflect totally any incident wave. However, the scattering by such kind of potentials is not structureless: resonances may occur for certain values of the energy. Here we consider the problem of scattering by the members of a family of potentials Va(x)=−sgn(x)|x|a, where sgn represents the sign function and a is a positive rational number. The scattering function and the phase shifts are obtained from global solutions of the Schrödinger equation. For the determination of the Gamow states, associated to resonances, we exploit their close relation with the eigenvalues of the PT-symmetric Hamiltonians with potentials VaPT(x)=−isgn(x)|x|a. Calculation of the time delay in the scattering at real energies is used to characterize the resonances. As an additional result, the breakdown of the PT-symmetry of the family of potentials VaPT for a<3 may be conjectured.

Free monoids and forests of rational numbers

The Calkin–Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each such number occurs in the tree exactly once and in the form a/b, where a and b are relatively prime positive integers. This tree is associated with the matrices L1=(1011) and R1=(1101), which freely generate the monoid SL2(N0) of 2×2 matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices Lu and Rv that freely generate submonoids of GL2(N0), there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.

Points at rational distances from the vertices of certain geometric objects

We consider various problems related to finding points in Q2 and in Q3 which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in Q2, and a cube or tetrahedron in Q3. In particular, as one of several results, we prove that the set of positive rational numbers a such that there exist infinitely many rational points in the plane which lie at rational distance from the four vertices of the rectangle with vertices (0,0), (0,1), (a,0), and (a,1), is dense in R+.

Bounds and constructions for optimal [formula omitted]-OOCs

Let W={w1,w2,…,wr} be an ordering of a set of r integers greater than 1, Λa=(λa(1),λa(2),…,λa(r)) be an r-tuple of positive integers, λc be a positive integer, and Q=(q1,q2,…,qr) be an r-tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code ((n,W,Λa,λc,Q)-OOC) for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Some work had been done on the constructions of optimal (n,{3,4},Λa,1,Q)-OOCs with unequal auto- and cross-correlation constraints. In this paper, we focus our main attention on (n,{3,5},Λa,1,Q)-OOCs, where Λa∈{(1,2),(2,1),(2,2)}. Tight upper bounds on the maximum code size of an (n,{3,5},Λa,1,Q)-OOC are obtained, and infinite classes of optimal balanced (n,{3,5},Λa,1)-OOCs are constructed.

Analytic continuation and asymptotics of Dirichlet series with partitions

The analytic continuation of a family of Dirichlet series whose coefficients are partition functions having parts in a finite set is established. The singularities arising from this continuation are shown to be finitely many simple poles, where the residues are computable positive rational numbers. Then we give several applications. First, special values of the continuation are studied in terms of algebraic combinations of the Hurwitz zeta function. Second, a transformation formula is obtained for exponentially decaying series involving these partition functions using special values of the continuation at negative integers. Finally, asymptotic behavior and orthogonality of certain convolutions with the normalization of these partition functions are investigated in the mean value sense with precise error terms.

The cone spanned by maximal Cohen-Macaulay modules and an application

The aim of this paper is to define the notion of the Cohen- Macaulay cone of a Noetherian local domain R R and to present its applications to the theory of Hilbert-Kunz functions. It has been shown by the second author that with a mild condition on R R , the Grothendieck group G 0 ( R ) ¯ \overline {G_0(R)} of finitely generated R R -modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R R is the cone in G 0 ( R ) ¯ R \overline {G_0(R)}_{\mathbb R} spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the Cohen-Macaulay cone in this paper. As an application, various examples of Hilbert-Kunz functions in the polynomial type will be produced. Precisely, for any given integers ϵ i = 0 , ± 1 \epsilon _i = 0, \pm 1 ( d / 2 &gt; i &gt; d d/2 &gt; i &gt; d ), we shall construct a d d -dimensional Cohen-Macaulay local ring R R (of characteristic p p ) and a maximal primary ideal I I of R R such that the function ℓ R ( R / I [ p n ] ) \ell _R(R/I^{[p^n]}) is a polynomial in p n p^n of degree d d whose coefficient of ( p n ) i (p^n)^i is the product of ϵ i \epsilon _i and a positive rational number for d / 2 &gt; i &gt; d d/2 &gt; i &gt; d . The existence of such ring is proved by using Segre products to construct a Cohen-Macaulay ring such that the Chow group of the ring is of certain simplicity and that test modules exist for it.

Motion of a motor-structure non-ideal system

Abstract In this paper the generalization to dynamics of a nonlinear oscillator excited with a non-ideal source is studied. The model of the structure-motor system is generalized by assuming that the driving torque is a nonlinear function of the angular velocity and the oscillator is with strong nonlinearity. The oscillator-motor system is assumed as a non-ideal one where not only the motor affects the motion of the oscillator but also vibrations of the oscillator have an influence on the motor motion. The model of the motor-structure system is described with two coupled strong nonlinear differential equations. An improved asymptotic analytic method based on the averaging procedure is developed for solving such a system of strong nonlinear differential equations. The steady state motion and its stability is studied. Results available the discussion of the Sommerfeld effect. A new procedure for determination of parameters of the non-ideal system for which the Sommerfeld effect does not exist is developed. For these critical values of the parameter the Sommerfeld effect is suppressed. As a special case the truly nonlinear oscillator where the order of the nonlinearity is a positive rational number is investigated. In the paper the influence of the order of nonlinearity on the dynamic properties of the system are analyzed. For the motor, with the torque which is the cubic function of the angular velocity, and the truly nonlinear oscillator, with certain order of nonlinearity, the numerical calculation is done. Analytically obtained results are compared with numerical ones. Results show good agreement.

A forest of linear fractional transformations

The Calkin–Wilf tree is a rooted infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form a/b, where a and b are relatively prime positive integers. For every 2 × 2 matrix with nonnegative integral coordinates and nonzero determinant, it is possible to construct an analogous tree with this root. If the root is the identity matrix, then the tree consists of all matrices with determinant 1, and this tree possesses the basic properties of the Calkin–Wilf tree of positive rational numbers. The set of all matrices with nonnegative integral coordinates and nonzero determinant decomposes into a forest of rooted infinite binary trees.

Global finite-time stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers

In this paper, we will present new results on global finite-time stabilization for a class of switched strict-feedback nonlinear systems, whose subsystems have chained integrators with the powers of positive odd rational numbers (i.e., numerators and denominators of the powers are all positive odd integers but not necessarily relatively prime). All the powers in each equation of subsystems of the switched systems can be different. Based on the technique of adding a power integrator, the global finite-time stabilizers of individual subsystems are first systematically constructed to guarantee global finite-time stability of the closed-loop smooth switched system under arbitrary switchings, and then a co-design of stabilizers and a state-dependent switching law is proposed to achieve global finite-time stabilization of the closed-loop non-smooth switched systems. In the controller design, a common coordinate transformation of all subsystems is exploited to avoid using individual coordinate transformations for individual subsystems. We also give some sufficient conditions that enable our design by characterizing the powers of the chained integrators of the considered switched systems. Numerical examples are provided to demonstrate the effectiveness of the proposed results.

Sums of digits in q-ary expansions

Let sq(n) denote the sum of the digits of a number n expressed in base q. We study here the ratio [Formula: see text] for various values of q and α. In 1978, Kenneth B. Stolarsky showed that [Formula: see text] and that [Formula: see text] using an explicit construction. We show that for α = 2 and q ≥ 2, the above ratio can in fact be any positive rational number. We also study what happens when α is a rational number that is not an integer, terminating the resulting expression by using the floor function.

Groupoid Cardinality and Egyptian Fractions

SummaryTwo very new questions about the cardinality of groupoids reduce to very old questions concerning the ancient Egyptians’ method for writing fractions. First, the question of whether any positive real number is the groupoid cardinality of some groupoid reduces to the question of whether any positive rational number has an Egyptian fraction decomposition. Second, the question of how many nonequivalent groupoids have a given cardinality can be answered via the number of distinct Egyptian fraction decompositions.

The development of rational number knowledge: Old topic, new insights

The development of rational number knowledge has been studied extensively by mathematics education researchers, cognitive-developmental psychologists and, more recently, by neuroscientists as well. Building on a rich body of prior research, and some exciting new ideas, the target articles re-visit several topics, with a view to refine and deepen our understanding of how rational number understanding develops. The effect of prior natural number knowledge-either positive or adverse-on rational number learning is highlighted by all contributors. I draw on the articles to discuss five different aspects of the whole, or natural, number bias.

On Salem numbers, expansive polynomials and Stieltjes continued fractions

A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This association makes use of Bertin-Boyd's Theorem A (1995) of interlacing of conjugates on the unit circle; in this context, a Salem number $\beta$ is produced and coded by an m-tuple of positive rational numbers characterizing the (SITZ) Stieltjes continued fraction of the corresponding Hurwitz quotient (alternant). The subset of Stieltjes continued fractions over a Salem polynomial having simple roots, not cancelling at $\pm 1$, coming from monic expansive polynomials of constant term equal to their Mahler measure, has a semigroup structure. The sets of corresponding generalized Garsia numbers inherit this semi-group structure.

On the multiplicative group generated by $${\{ \frac{[\sqrt{2}n]}{n} | n \in \mathbb{N} \}}$$ { [ 2 n ] n | n ∈ N }

We prove that the multiplicative group generated by $${\{ \frac{[\sqrt{2}n]}{n} | n \in \mathbb{N} \}}$$ is the group of positive rational numbers. It is proved that if a completely additive function f satisfying f $${([\sqrt{2}n]) - f(n) \rightarrow C (n \rightarrow \infty)}$$ for some real number C, then $${f(n) = A{\text{log}} n}$$ , where $${A = \frac{2C}{log 2}}$$ .

Wall-crossing in genus zero quasimap theory and mirror maps

For each positive rational number , the theory of -stable quasimaps to certain GIT quotients W==G developed in [CKM14] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to ! 0. For > 1 one obtains the usual Gromov{Witten theory of W==G, while the other theories are new. However, they are all expected to contain the same information and, in particular, the numerical invariants should be related by wall-crossing formulas. In this paper we analyze the genus zero picture and nd that the wall-crossing in this case signicantly generalizes toric mirror symmetry (the toric cases correspond to abelian groups G). In particular, we give a geometric interpretation of the mirror map as a generating series of quasimap invariants. We prove our wall-crossing formulas for all targets W==G which admit a torus action with isolated xed points, as well as for zero loci of sections of homogeneous vector bundles on such W==G.

Further results on balanced [formula omitted]-OOCs

Let W={w1,…,wr} be an ordering of a set of r integers greater than 1, Λa=(λa(1),…,λa(r)) be an r-tuple of positive integers, λc be a positive integer, and Q=(q1,…,qr) be an r-tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code ((n,W,Λa,λc,Q)-OOC) for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Most existing works on variable-weight optical orthogonal codes assume that λa(1),…,λa(r)=λc=1. In this paper, new balanced (n,{3,4},Λa,1)-OOCs are constructed, where Λa∈{(1,2),(2,1),(2,2)}.

Asymptotics of Landau Constants with Optimal Error Bounds

We study the asymptotic expansion for the Landau constants $G_n$ $$\pi G_n\sim \ln N + \gamma+4\ln 2 + \sum_{s=1}^\infty \frac {\beta_{2s}}{N^{2s}},~~n\rightarrow \infty, $$ where $N=n+3/4$, $\gamma=0.5772\cdots$ is Euler's constant, and $(-1)^{s+1}\beta_{2s}$ are positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative $n$. Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form $$ \ln N+\gamma+4\ln 2+\sum_{s=1}^{2m}\frac{\beta_{2s}}{N^{2s}}< \pi G_n < \ln N+\gamma+4\ln 2+\sum_{s=1}^{2k-1}\frac{\beta_{2s}}{N^{2s}}$$ for all $n=0,1,2,\cdots$, $m=1,2,\cdots$, and $k=1,2,\cdots$. The results are proved by approximating the coefficients $\beta_{2s}$ with the Gauss hypergeometric functions involved, and by using the second order difference equation satisfied by $G_n$, as well as an integral representation of the constants $\rho_k=(-1)^{k+1}\beta_{2k}/(2k-1)!$.

Convexity with respect to families of means

In this paper we investigate continuity properties of functions $f:\mathbb{R}_+\to\mathbb{R}_+$ that satisfy the $(p,q)$-Jensen convexity inequality $$ f\big(H_p(x,y)\big)\leq H_q(f(x),f(y)) \qquad(x,y>0), $$ where $H_p$ stands for the $p$th power (or H\"older) mean. One of the main results shows that there exist discontinuous multiplicative functions that are $(p,p)$-Jensen convex for all positive rational number $p$. A counterpart of this result states that if $f$ is $(p,p)$-Jensen convex for all $p\in P\subseteq\mathbb{R}_+$, where $P$ is a set of positive Lebesgue measure, then $f$ must be continuous.

Layered tropical mathematics

Generalizing supertropical algebras, we present a “layered” structure, “sorted” by a semiring which permits varying ghost layers, and indicate how it is more amenable than the “standard” supertropical construction in factorizations of polynomials, description of varieties, and for mathematical analysis and calculus, in particular with respect to multiple roots of polynomials. This gives rise to a significantly better understanding of the tropical resultant and discriminant. Explicit examples and comparisons are given for various sorting semirings such as the natural numbers and the positive rational numbers, and we see how this theory relates to some recent developments in the tropical literature.