scholarly journals A finiteness theorem on symplectic singularities

2016 ◽  
Vol 152 (6) ◽  
pp. 1225-1236 ◽  
Author(s):  
Yoshinori Namikawa
Keyword(s):  
Finite Number ◽  
Contact Structure ◽  
Finiteness Theorem ◽  
Maximal Weight ◽  
Fixed Dimension ◽  
Positive Integers ◽  
Boundedness Result ◽  
Symplectic Varieties

An affine symplectic singularity$X$with a good$\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers$N$and$d$, there are only a finite number of conical symplectic varieties of dimension$2d$with maximal weights$N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.

On Nilpotent Products of Cyclic Groups. II

1961 ◽  
Vol 13 ◽  
pp. 557-568 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Normal Subgroup ◽  
Finite Number ◽  
Free Product ◽  
Multiplication Table ◽  
Unique Representation ◽  
Cyclic Groups ◽  
Positive Integers

In a previous paper (18), G = F/Fn was studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. In that paper the following cases were completely treated:(a) F a free product of cyclic groups of order pαi, p a prime, αi positive integers, and n = 4, 5, … , p + 1.(b) F a free product of cyclic groups of order 2αi, and n = 4.In this paper, the following case is completely treated:(c) F a free product of cyclic groups of order pαi p a prime, αi positive integers, and n = p + 2.(Note that n = 2 is well known, and n — 3 was studied by Golovin (2).) By ‘'completely treated” is meant: a unique representation of elements of the group is given, and the order of the group is indicated. In the case of n = 4, a multiplication table was given.

Effective procedures in field theory

1956 ◽  
Vol 248 (950) ◽  
pp. 407-432 ◽  
Keyword(s):  
Finite Number ◽  
General Procedure ◽  
Finite Extension ◽  
Splitting Algorithm ◽  
Fundamental Field ◽  
Finite Procedure ◽  
Definition Of ◽  
Extension Ring ◽  
Positive Integers

Van der Waerden (1930 a , pp. 128- 131) has discussed the problem of carrying out certain field theoretical procedures effectively, i.e. in a finite number of steps. He defined an ‘explicitly given’ field as one whose elements are uniquely represented by distinguishable symbols with which one can perform the operations of addition, multiplication, subtraction and division in a finite number of steps. He pointed out that if a field K is explicitly given then any finite extension K' of K can be explicitly given, and that if there is a splitting algorithm for K , i.e. an effective procedure for splitting polynomials with coefficients in K into their irreducible factors in K [x], then(1) there is a splitting algorithm for K' . He observed in (1930 b ), however, that there was no general splitting algorithm applicable to all explicitly given fields K , or at least that such an algorithm would lead to a general procedure for deciding problems of the type ‘Does there exist an n such that E(n) ?’, where E is an arbitrarily given property of positive integers such that there is an algorithm for deciding for any n whether E(n) holds. In this paper we review these results in the light of the precise definition of algorithm (finite procedure) given by Church (1936), Kleene (1936) and Turing (1937) and discuss the existence of a number of field theoretical algorithms in explicit fields, and the effective construction of field extensions. We sharpen van der Waerden’s result on the non-existence of a general splitting algorithm by constructing (§7) a particular explicitly given field which has no splitting algorithm. We show (§7) that the result on the existence of a splitting algorithm for a finite extension field does not hold for inseparable extensions, i.e. we construct a particular explicitly given field K and an explicitly given inseparable algebraic extension K ( x ) such that K has a splitting algorithm but K ( x ) has not. (2) We note also (in §6) that there exist two isomorphic explicitly given fields, one of which possesses a splitting algorithm but the other of which does not. Thus the sort of properties of fields we are interested in depend not only on the abstract field but also on the particular representation chosen. It is necessary therefore to state rather carefully our definitions of explicit ring, extension ring, splitting algorithm, etc., and to introduce the concept of explicit isomorphism (3) and homomorphism. This occupies §§ 1,2 and 3. On the basis of these definitions we then discuss the existence of some fundamental field theoretical algorithms in explicit fields and their extension fields. This leads also to a classification of the types of extension fields which can be effectively constructed.

scholarly journals On the Closure of the Image of the Generalized Divisor Function

2017 ◽  
Vol 12 (2) ◽  
pp. 77-90 ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Finite Number ◽  
Real Number ◽  
Closed Form ◽  
Pairwise Disjoint ◽  
Arithmetic Function ◽  
Divisor Function ◽  
Closed Intervals ◽  
Topological Closure ◽  
Positive Integers ◽  
Sufficient Precision

Abstract For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , Iℓ. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , Iℓ and d1, . . . , dℓ, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2.

On the Lower Range of Perron's Modular Function

1969 ◽  
Vol 21 ◽  
pp. 808-816 ◽  
Author(s):  
J. R. Kinney ◽  
T. S. Pitcher
Keyword(s):  
Finite Number ◽  
Continued Fraction ◽  
Modular Function ◽  
Finite Collection ◽  
Continued Fraction Expansion ◽  
Lower Range ◽  
Image Position ◽  
Positive Integers

The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequalityis satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will writefor the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will writeIt is known (see 6) thatWhere

scholarly journals The speed of a random walk excited by its recent history

2016 ◽  
Vol 48 (1) ◽  
pp. 215-234 ◽  
Author(s):  
Ross G. Pinsky
Keyword(s):  
Random Walk ◽  
Finite Number ◽  
Threshold Level ◽  
Recent History ◽  
Threshold Levels ◽  
One Step ◽  
The Right ◽  
Positive Integers

Abstract Let N and M be positive integers satisfying 1≤ M≤ N, and let 0< p0 < p1 < 1. Define a process {Xn}n=0∞ on ℤ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first N steps, the process behaves like a random walk that jumps to the right with probability p0 and to the left with probability 1-p0. At subsequent steps the jump mechanism is defined as follows: if at least M out of the N most recent jumps were to the right, then the probability of jumping to the right is p1; however, if fewer than M out of the N most recent jumps were to the right then the probability of jumping to the right is p0. We calculate the speed of the process. Then we let N→ ∞ and M/N→ r∈[0,1], and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number l of threshold levels, (Mi,pi) i=1l, above the pre-threshold level p0, as well as for one model with l=N such thresholds.

scholarly journals A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

2018 ◽  
Vol 8 (1) ◽  
pp. 109-114
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Rational Solution ◽  
Computable Function ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions ◽  
Positive Integers

Abstract We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, . . . , xi, then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

Fractional parts of powers of rationals

1975 ◽  
Vol 77 (2) ◽  
pp. 269-279 ◽  
Author(s):  
A. Baker ◽  
J. Coates
Keyword(s):  
Finite Number ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Image Position ◽  
Upper Estimate ◽  
Positive Integers

Mahler (5) proved in 1957 that for any rational a/q, where a, q are relatively prime integers with a > q ≥ 2, and any ε > 0, there exist only finitely many positive integers n such that ∥(a/q)n∥ < e−εn; here ∥x∥ denotes the distance of x from the nearest integer taken positively. In particular there exist only finitely many n such thatand, as Mahler observed, this implies that the number g(k) occurring in Waring's problem is given byfor all but a finite number of values of k. It would plainly be of interest to establish a bound for the exceptional k and this would follow from an upper estimate for the integers n for which (1) holds. But Mahler's work was based on Ridout's generalization of Roth's theorem and, as is well known, the latter result is ineffective.

scholarly journals A FINITENESS THEOREM FOR DUAL GRAPHS OF SURFACE SINGULARITIES

2009 ◽  
Vol 20 (08) ◽  
pp. 1057-1068 ◽  
Author(s):  
PATRICK POPESCU-PAMPU ◽  
JOSÉ SEADE
Keyword(s):  
Finite Number ◽  
Dual Graph ◽  
Finite Graph ◽  
The Self ◽  
Surface Singularity ◽  
Finiteness Theorem ◽  
Minimal Resolution ◽  
Dual Graphs ◽  
Irreducible Components ◽  
Surface Singularities

Consider a fixed connected, finite graph Γ and equip its vertices with weights pi which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically Gorenstein surface singularity having Γ as the dual graph of the minimal resolution, the weights pi of the vertices being the arithmetic genera of the corresponding irreducible components. As a consequence we get that if Γ is not a cycle, then there is a finite number of possibilities of self-intersection numbers which one can attach to the vertices which are of valency ≥ 2, such that one gets the dual graph of the minimal resolution of a numerically Gorenstein surface singularity. Moreover, we describe precisely the situations when there exists an infinite number of possibilities for the self-intersections of the component corresponding to some fixed vertex of Γ.

Sums of Functions of Digits

1960 ◽  
Vol 12 ◽  
pp. 374-389 ◽  
Author(s):  
B. M. Stewart
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
The Other ◽  
Number Of Classes ◽  
Positive Integers

We generalize in several directions a paper by Porges (2) who considered the integer F(A) obtained from the positive integer .1 by taking the sum of the squares of the digits of A. Porges showed that if A > 99, then F(A) < A, so that under iteration of F(A) all the positive integers are divided into a finite number of classes, called orbits in the terminology of Isaacs (1), each containing a finite cycle. For his F(A) Porges showed there are only two orbits: one with the 1-cycle: 1 → 1 ; and the other with the interesting 8-cycle: 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.Consider the set Z of non-negative integers and choose as a base of enumeration any desired integer B ≧ 2 (not necessarily B = 10). Then only the “digits” 0, 1, 2, … , B — 1 are needed, in suitable multiplicity, to represent any A of Z.

Sums of Fractions with Bounded Numerators

1966 ◽  
Vol 18 ◽  
pp. 999-1003 ◽  
Author(s):  
B. M. Stewart ◽  
W. A. Webb
Keyword(s):  
Finite Number ◽  
General Problem ◽  
Unit Fractions ◽  
Finite Set ◽  
Positive Integers

The general problem considered in this paper is that of sums of a finite number of reduced fractions whose numerators are elements of a finite set S of integers, and whose denominators are distinct positive integers. Egyptian, or unit, fractions are merely the case S = ﹛1﹜. Problems concerning these fractions have been treated extensively. Another specific case S = ﹛1, — 1﹜ has been treated by Sierpinski (2).

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