BENFORD'S LAW FOR THE 3x + 1 FUNCTION

Benford's law (to base B) for an infinite sequence {x_k : k 1} of positive quantities x_k is the assertion that {log_B x_k: k 1} is uniformly distributed (mod 1). The 3x + 1 functionT(n) is given by T(n) = (3n + 1)/2 if n is odd, and T(n) = n/2if n is even. This paper studies the initial iterates x_k = T^(k)(x₀)for 1 k N of the 3x + 1 function, where N is fixed. It showsthat for most initial values x₀, such sequences approximatelysatisfy Benford's law, in the sense that the discrepancy ofthe finite sequence {log_B x_k : 1 k N} is small.     

Regularity of Rees Algebras

Let B = k[x₁, ..., x_n] be a polynomial ring over a field k,and let A be a quotient ring of B by a homogeneous ideal J.Let m denote the maximal graded ideal of A. Then the Rees algebraR = A[m t] also has a presentation as a quotient ring of thepolynomial ring k[x₁, ..., x_n, y₁, ..., y_n] by a homogeneousideal J^*. For instance, if A = k[x₁, ..., x_n], then Rk[x₁,...,x_n,y₁,...,y_n]/(x_iy_j–x_jy_i|i, j=1,...,n). In this paper we want to compare the homological propertiesof the homogeneous ideals J and J^*.     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.     

Closed-form Representations for Errors of Cubic Spline Approximations on Equally Spaced Knots

This paper considers estimating errors in the approximationof an arbitrary linear functional on C⁴[0, L] by use of cubicsplines on equally spaced knots. Using explicit formulas derivedfor the cubic spline approximations of f_k(x) = cos (kx/L), formulasare found for the cosine expansion of the Peano Kernel for theremainder functional.     

Morphisms Between Koszul Complexes II

Let K. denote the graded Koszul complex associated to the regularsequence (x₀, ..., x_n) in the graded polynomial ring A = k[x₀,..., x_n], |x_i| = 1 for all i, over an arbitrary field k. Let denote the Koszul complex associated to another regular sequence of homogeneous elements(p₀, ..., p_n) in A. In [5] we have studied ranks of graded chaincomplex morphisms with the property f₀ = id. Let ^k (respectively, '^k) denote the kernelof the Koszul differential d: K_k K_k–1 (respectively,), and let denote the restriction of f_k. The main result wasthat Rank . 1991 MathematicsSubject Classification 13D25.     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

The Range of Possible Values of f(x)

The purpose of this paper is to explain how to compute the rangeof possible values of a function of one variable, f(x), givenvalues of the function at n distinct points x₁ < x₂ <... < x_M–1 < x_M, and given a finite bound on thekth derivative of f: ||f^(k)|| L, 1 k n.     

Stability in the Sense of Bounded Average Power

An R^m-valued sequence (x_k): = (x_k : k = 1, 2, ...), e.g. generatedrecursively by x_k = f_k (x_k–_k, U_k), is called ‘averagepth power bounded’ if (1/K) is bounded uniformly in K= 1, 2,.... (The case p = 2 may correspond to ‘power’in the physical sense.) This is a notion of stability. Givenestimates of the form: f_k (x, u) < a x + ¶ _k conditionsare obtained on the coefficient sequence (a_k) and the inputestimates e_k:=¶_k (u_k) which ensure this form of stabilityfor the output (x_k). In particular, a condition (utilized inan application to adaptive control) is obtained which imposes(i) a bound b on (a_k) and a ‘sparsity measure’ m(K) on #{kK: a_k>} as K ( >1) (ii) average pth power boundednesson (e_k), and (iii) a growth condition on (e_k) related to b andm (•). This condition is sharp.     

Irregularities of Point Distribution Relative to Convex Polygons III

Suppose that P is a distribution of N points in the unit squareU=[0, 1]². For every x=(x₁, x₂)U, let B(x)=[0, x₁]x[0, x₂] denotethe aligned rectangle containing all points y=(y₁, y₂)U satisfying0y₁x₁ and 0y₂x₂. Denote by Z[P; B(x)] the number of points ofP that lie in B(x), and consider the discrepancy function D[P; B(x)]=Z[P; B(x)]–Nµ(B(x)), where µ denotes the usual area measure.     

The Analogue of Buchi's Problem for Rational Functions

Büchi's problem asked whether there exists an integer Msuch that the surface defined by a system of equations of theform has no integer pointsother than those that satisfy ±x_n = ± x₀ + n (the± signs are independent). If answered positively, itwould imply that there is no algorithm which decides, givenan arbitrary system Q = (q1,...,q_r) of integral quadratic formsand an arbitrary r-tuple B = (b₁,...,b_r) of integers, whetherQ represents B (see T. Pheidas and X. Vidaux, Fund. Math. 185(2005) 171–194). Thus it would imply the following strengtheningof the negative answer to Hilbert's tenth problem: the positive-existentialtheory of the rational integers in the language of additionand a predicate for the property ‘x is a square’would be undecidable. Despite some progress, including a conditionalpositive answer (depending on conjectures of Lang), Büchi'sproblem remains open. In this paper we prove the following: (A) an analogue of Büchi's problem in rings of polynomialsof characteristic either 0 or p 17 and for fields of rationalfunctions of characteristic 0; and (B) an analogue of Büchi's problem in fields of rationalfunctions of characteristic p 19, but only for sequences thatsatisfy a certain additional hypothesis. As a consequence we prove the following result in logic. Let F be a field of characteristic either 0 or at least 17 andlet t be a variable. Let L_t be the first order language whichcontains symbols for 0 and 1, a symbol for addition, a symbolfor the property ‘x is a square’ and symbols formultiplication by each element of the image of [t] in F[t].Let R be a subring of F(t), containing the natural image of[t] in F(t). Assume that one of the following is true: (i) R F[t]; (ii) the characteristic of F is either 0 or p 19. Then multiplication is positive-existentially definable overthe ring R, in the language L_t. Hence the positive-existentialtheory of R in L_t is decidable if and only if the positive-existentialring-theory of R in the language of rings, augmented by a constant-symbolfor t, is decidable.     

The tower of K-theory of truncated polynomial algebras

Let A be a regular noetherian F_p-algebra. The relative K-groupsK_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluatedby the author and Ib Madsen in terms of the big de Rham–Wittgroups W_rA^q of the ring A. In this paper, we evaluate the mapsof relative K-groups and Nil-groups induced by the canonicalprojection f: A[x]/(x^m) A[x]/(xⁿ). The result depends stronglyon the prime p. It generalizes earlier work by Stienstra onthe groups in degrees 2 and 3. Received February 28, 2007.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

The Hilbert Scheme Parameterizing Finite Length Subschemes of the Line with Support at the Origin

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A _k k[x]_(x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x]_(x).     

A Geometric Invariant for Metabelian Pro-p Groups

In [2] Bieri and Strebel introduced a geometric invariant forfinitely generated abstract metabelian groups that determineswhich groups are finitely presented. For a valuable survey oftheir results, see [6]; we recall the definition briefly inSection 4. We shall introduce a similar invariant for pro-pgroups. Let F be the algebraic closure of F_p and U be the formal powerseries algebra F[T], with group of units U^x. Let Q be a finitelygenerated abelian pro-p group. We write Z_p[Q] for the completedgroup algebra of Q over Z_p. Let T(Q) be the abelian group Hom(Q,U^x) of continuous homomorphisms from Q to U^x. We write 1 forthe trivial homomorphism. Each vT(Q) extends to a unique continuousalgebra homomorphism from Z_p[Q]to U.     

Extended conjugate points in the calculus of variations

This paper concerns a characterization of second-order optimalityconditions for the fixed-endpoint problem in the calculus ofvariations. The key new concept is a set S(x) with the propertythat S(x)=if and only if the second variation with respect tox, independently of non-singularity assumptions, is non-negativealong admissible variations. We show that, for this set of points,it may be much easier (and never more difficult) to prove itsnon-emptiness than directly finding variations that make thesecond variation negative. Earlier Loewen and Zheng, and Zeidan,introduced related sets C₁(x) and C₂(x), applicable to certainoptimal control problems, whose non-emptiness has been establishedmerely as a sufficient condition for the existence of negativesecond variations. These sets, when reduced to the problem weare considering, are related according to C₁(x) C₂(x) S(x).Contrary to the behaviour of S(x), verifying membership of C₁(x)or C₂(x) may be more difficult than verifying directly if thesecond-order condition holds. We provide several examples forwhich it is straightforward to prove that S(x) , but determiningthe sets C₁(x) or C₂(x) may be a very difficult or perhaps evena hopeless task.     

Integrals and Areas Expressed as Bilinear Forms of Discrete Defining Data

An ordered set of discrete data-pairs (x_k, y_k) is supposed givenfor k = 1(1)n, derived perhaps as a consequence of experiment,and indicating some form of relationship between x and y. Itis required that these data be interpolated in some systematicway so as to establish this relationship in the form of a rectifiablecurve in the (x, y) plane, and that y is then to be integratedwith respect to x (or vice versa), thereby defining an areain this plane. We shall here consider interpolation schemeswhere by this integral can be expressed as a bilinear form _jKC_jkY_jx_k,the "weighting" coefficients C_jk being numbers independent ofthe data values (though not necessarily of their number, n). The expressions obtained sepcialize to Gregory formulae, andti Simpson's Rule and other forms of Cotes formulae in suitablecontexts, but the information is primarily of use in dealingwith unequally-spaced data values, or in estumating the areawithin a closed curve. The simplicity of the numerical algorithmis clearly unaffected by the geometric complexity of the interpolatedcurve. Values for the weighting coefficients are proposed, and theresults of a numerical experiment are described which teststhe applicability of the formulae described.     

On Waring's Problem in Number Fields

Let K be an algebraic number field of degree n over the rationals,and denote by J_k the subring of K generated by the kth powersof the integers of K. Then G_K(k) is defined to be the smallests1 such that, for all totally positive integers vJ_k of sufficientlylarge norm, the Diophantine equation (1.1) is soluble in totally non-negative integers _i of K satisfying N(_i)<<N(v)^1/k (1is). (1.2) In (1.2) and throughout this paper, all implicit constants areassumed to depend only on K, k, and s. The notation G_K(k) generalizesthe familiar symbol G(k) used in Waring's problem, since wehave G_Q(k) = G(k). By extending the Hardy–Littlewood circle method to numberfields, Siegel [8, 9] initiated a line of research (see [1–4,11]) which generalized existing methods for treating G(k). Thistypically led to upper bounds for G_K(k) of approximate strengthnB(k), where B(k) was the best contemporary upper bound forG(k). For example, Eda [2] gave an extension of Vinogradov'sproof (see [13] or [15]) that G(k)(2+o(1))k log k. The presentpaper will eliminate the need for lengthy generalizations assuch, by introducing a new and considerably shorter approachto the problem. Our main result is the following theorem.     

A new transform method for evolution partial differential equations

We introduce a new transform method for solving initial-boundary-valueproblems for linear evolution partial differential equationswith spatial derivatives of arbitrary order. This method isillustrated by solving several such problems on the half-line{t > 0, 0 < x < }, and on the quarter-plane {t >0, 0 < x_j < , j = 1, 2}. For equations in one space dimensionthis method constructs q(x, t) as an integral in the complexk-plane involving an x-transform of the initial condition anda t-transform of the boundary conditions. For equations in twospace dimensions it constructs q(x₁, x₂, t) as an integral inthe complex (k₁, k₂)-planes involving an (x₁, x₂)-transformof the initial condition, an (x₂, t)-transform of the boundaryconditions at x₁ = 0, and an (x₁, t)-transform of the boundaryconditions at x₂ = 0. This method is simple to implement andyet it yields integral representations which are particularlyconvenient for computing the long time asymptotics of the solution.