Integer Components Research Articles

A symmetry relationship for a class of partitions

Let S( n, k, v) denote the number of vectors ( a 0,…, a n−1 ) with nonnegative integer components that satisfy a 0 + … + a n − 1 = k and Σ i=0 n−1 ia i ≡ v (mod n). Two proofs are given for the relation S( n, k, v) = S( k, n, v). The first proof is by algebraic enumeration while the second is by combinatorial construction.

Accelerating Greenberg's metho̧d for the computation of knapsack functions

Greenberg [4] described an interesting procedure based on dynamic programming for generating the knapsack function Φ( s) = max{ cx: ax ⩽ s, x ⩾ 0, integer} for all arguments s ⩽ b, where c, a, and x are vectors with nonnegative integer components and b is a positive integer. The principle search effort can be associated with finding the minimum of a string D of k elements. It is shown here that a rank ordering of the components of a yields sequences D such that a search for the minimum in D over the first m elements is sufficient. In general, m is smaller than k and the calculation of Φ is accelerated. Extensions of the basic procedure to the case where the components of x have upper-bounds is also described.

Generation of all integer points for given sets of linear inequalities

We propose to give a computationally feasible procedure for the generation of all the integer points satisfying a given set of inequalities. Five different systems of inequalities will be considered. In order to generate all of these integer points, one requires a particular set of integer points, called fundamental points, and a set of linearly independent vectors with integer components. The number of these fundamental points is given by a simple formula. We show how to generate the fundamental points and the required vectors. We give an application concerning the localization of the integer optimum of a linear objective function subject to constraints which geometrically define a cone or a parallelotope.

A naturally ordered enumeration of compositions

A composition of the positive integer M, with N-parts, is a vector of N non-negative integer components the sum of which is M. The paper presents a transformation that “enumerates” (assigns serial numbers) the set of all possible compositions most densely so that their “natural” ordering is preserved. The transformation is useful for the efficient retrieval and compact storage of probabilities, or functions on probability spaces, in finite memories.

On B 2-sequences of vectors

We shall prove a generalization of a classic inequality of Erdös and Turán on B 2 -sequences of integers to B 2 -sequences of vectors with integer components. We improve a recent result by the author when the dimension tends to infinity.

More on the generalized Macaulay theorem

Let k 1 ⩽ k 2 ⩽ … ⩽ k n be given positive integers and let F denote the set of vectors ( l 1, …, l n ) with integer components satisfying 0 ⩽ l i ⩽ k i , i = 1, 2, …, n. If H is a subset of F, let ( l) H denote the subset of H consisting of those vectors with component sum l, and let C(( l) H) denote the smallest [( l) H] elements of ( l) F. The generalized Macaulay theorem due to the author and B. Lindström [3] shows that | Gamma;(( C)( l)( H)|, ⩾ | Γ( C(( l) H))|, where Γ(( l) H) is the setof vectors in F obtainable by subtracting l from a single component of a vector in ( l) H. A method is given for computing [ Γ( C(( l) H)] in this paper. It is analogous to the method for computing | Γ( C( l) H))| in the k 1 = … = k n = 1 case which has been given independently by Katona [4] and Kruskal [5].

A note on the Ehrenfest multiurn model

Denote by SN,r the set of r-tuples n having non-negative integer components summing to N, and by ei the r-tuple having a one in the ith position and zeros elsewhere. For m and n belonging to SN r, we let P(n, t|m, s) represent the transition probability of moving from state m at time s to state n at time t. Bartlett's conservative process [1] assumes that for n–ei + ej and n ∈ SN,r and Δt > 0 where the λij(t) are non-negative continuous functions of t.

A note on the Ehrenfest multiurn model

Denote by SN, r the set of r-tuples n having non-negative integer components summing to N, and by e i the r-tuple having a one in the ith position and zeros elsewhere. For m and n belonging to SN r, we let P( n , t| m , s) represent the transition probability of moving from state m at time s to state n at time t. Bartlett's conservative process [1] assumes that for n – ei + e j and n ∈ SN, r and Δt > 0 where the λ ij (t) are non-negative continuous functions of t.

A generalization of a combinatorial theorem of macaulay

Let E denote the set of all vectors of dimension n ( n≥2) with non-negative integral components. E is ordered in the lexicographic order. Let E v denote the subset of all vectors in E with component sum v. If H v denotes any subset of E v let LH v denote the set of the → H v → last elements in E v , where → H v → is the number of elements of H v . Let PH v denote the set of all vectors of E v+1 , which are obtained by the addition of 1 to a component of a vector in H v . In [3] Macaulay proved the inclusion P( LH v ) ⊂ L( PH v ). Sperner gave a shorter proof in [4]. Let k 1 ≤ k 2 ≤ … ≤ k n be given positive integers and let F denote the set of all vectors ( a 1,…, a n ) with integer components and 0 ≤ a i ≤ k i i = 1,…, n. We shall prove Macaulay's inclusion for subsets H v of F v even if the operators P and L are restricted to operate in F. This will follow from our theorem. As another application we prove a generalization of the main result in [2]. By a different method Katona proved the theorem when k 1 = k 2 = … = k n = 1 (see [1, Theorem 1]).

A note on a theorem of J. N. Franklin

where A is a d X d matrix with real components and b is a fixed d-dimensional real column vector. In a recent paper [1] J. N. Franklin proved the following THEOREM. If all the components of the matrix A are rational integers, then the sequence of d-dimensional vectors x n (n = 0, 1, 2, *.) defined by (1) is equidistributed modulo one for almost all initial vectors x if A has no eigenvalue equal to zero or a root of unity; when b = 0, the sequence is equidistributed modulo one for almost all x if and only if A has no eigenvalue equal to zero or a root of unity. Franklin's proof of this result makes use of the criterion of H. Weyl [3] for the equidistribution modulo one of sequences of vectors and the individual ergodic theorem due to F. Riesz as well. The purpose of this note is to give a simple proof of the theorem of Franklin on the basis of the criterion of Weyl only. By the way, we note that the above theorem has some applications in the theory of normal numbers. As a sample of this we may mention the following. J. E. Maxfield [2] introduced the notion of normal d-tuples (or, d-dimensional vectors) with real components to scale r, where r ? 2 is a fixed integer, and showed that almost all d-tuples are normal to scale r (and hence normal to scale r for every r > 2, i.e. absolutely normal). Indeed, a d-tuple x is normal to scale r if and only if the sequence rnx (n = 0, 1, 2, * * * ) is equidistributed modulo one (cf. [2, Theorem 5]): thus, the conclusion that almost all d-tuples are normal to scale r is an immediate consequence of the theorem of Franklin with an appropriate diagonal matrix A and b = 0. Now, let x n (n = 0, 1, 2, ***) be a sequence of real d-dimensional vectors detined' 6y (IC)', wfiere 6 is a fixed realf d-diimensiknai vectorand' we suppose Gao 6Y components of the d X d matrix A are integers. Let h be a d-dimensional vector with integer components and put

The organization of a cyclical computing process using a parametric representation of special form

The mathematical description of the solution of a problem usually contains variables denoted by different letters.** Each variable may be a certain set of numbers. The elements of this set are called the components of the variable. A one-to-one correspondence is established between all the components of a given variable and a certain set of vectors with integer components. In this correspondence all the vectors corresponding to the components of a given variable have the same dimension.

Integral Representation of Semi-Groups of Unbounded Self-Adjoint Operators

1. In this paper we obtain the spectral integral representation for semi-groups {Tx} of, in general, unbounded self-adjoint operators in Hilbert space. The index x ranges over a locally compact semi-group e which can be imbedded in a locally compact group ( which satisfies some other additional conditions which will be listed in Section 2. Our result generalizes theorems by E. Hille [6], B. v. Sz. Nagy [9], A. Devinatz [2], A. E. Nussbaum [11] and a recent theorem by A. Devinatz and the author [3]. In all these previous results it was assumed either that x ranges over a semi-group which is a Cartesian product I.+ x E+ where I+ is the set of vectors in rn-dimensional euclidean space with non-negative integer components and E + is the set of vectors in n-dimensional euclidean space with non-negative components (cf. A. Devinatz and A. E. Nussbaum loc. cit.) or that the operators Tx are uniformly bounded (cf. A. E. Nussbaum [11]). The method of proof consists, in essence, in imbedding the semi-group of operators in an algebra of unbounded operators (cf. J. M. G. Fell and J. L. Kelley [5]) and an adaptation to our situation of a method used by R. S. Phillips [12] in his demonstration of M. H. Stone's theorem on the integral representation of groups of unitary operators (cf. [11]). It must be remarked, however, that the imbedding of the semi-group of self-adjoint operators in an algebra of unbounded operators is only possible by virtue of a theorem concerning the permutability of self-adjoint operators proved by the present author in collaboration with A. Devinatz and J. von Neumann [4]. 2. We start with the definition of a locally compact full semi-group. (cf. [11]. Note that we do not assume here that X is abelian). DEFINITION 1. A semi-group (5 is called a locally compact full semigroup if (a) e can be imbedded in a locally compact group ( (b) e is locally compact in the relative topology of (, and (c) every non-empty bounded open set in (5 has non-zero measure with respect to the left (or right) Haar measure of (. Throughout this paper we shall always assume that e is a locally compact full semi-group.

Extensions of the lemma of Haar in the calculus of variations

where DXj denotes partial differentiation with respect to Xj. The sum in (1) is taken over all vectors i with O^iy^my, where m i s a fixed vector with positive integer components. For the domain of the functional L it is convenient to take the class of all functions v of class C and having support compact on G (i.e., compact and contained in G). Then L(v) is well defined when the coefficients Ai axe all locally integrable in G. Also the following notations are meaningful (with exceptional sets of measure zero) for a locally integrable function ƒ :

On the Permutability of Normal Operators

1. In this paper we obtain a theorem about the permutability of (in general unbounded) normal operators. This theorem is analogous to a theorem proved by the present authors in collaboration with J. von Neumann [3] concerning the permutability of self-adjoint operators. As we shall point out later it is not possible to generalize precisely the conditions given in [3] to the case of normal operators. Nevertheless, the proof of the theorem given here, with essential modifications, follows the proof given in [3]. We give our theorem and proof in Section 2. In Section 3 we apply this theorem on the permutability of normal operators to obtain the integral representation for a semi-group {N,} of normal operators (in general unbounded), where the index x ranges over a semi-group which is a cartesian product I+ X E where I+ is the set of vectors in m-dimensional euclidean space with non-negative integer components and Et is the set of vectors in n-dimensional euclidean space with non-negative components. This theorem contains as special cases theorems by E. Hille [6], B. v.Sz. Nagy [9, pp. 74-76], A. Devinatz [2] and a theorem recently published by R. Getoor [5]. It should be pointed out that our theorem represents an essential generalization of these previous results. For the case where { Nx ; x > 0 } is a one parameter semi-group of unbounded normal operators, the method of Nagy [9, pp. 74-75] may be pushed through to obtain the integral representation, and only in this case. However, with the use of our permutability theorem we are able to materially simplify the proof for the one parameter case. Moreover, in the unbounded case, even with the use of our permutability theorem, there remain some difficulties for multi-parameter semi-groups which do not allow us to use immediately the theorem for the one-parameter case. If the index x ranges over a more general Abelian topological semi-group than we consider, the representation problem is still open, although for semi-groups of normal operators which are uniformly bounded a method introduced by R. Phillips [8] may be used (cf. also A. E. Nussbaum [7]). 2. We start with the definition of permutability of two (in general unbounded) normal operators defined in a Hilbert space & (cf. Nagy [9] p. 50). DEFINITION 1. Let N1 and N2 be normal operators and {K1(z)} and {K2(Z)I their corresponding canonical resolutions of the identity. We say that N1 and N2 permute if Kl(zi)K2(z2) = K2(z2)Kl(zi) for each complex z1 and z2 It should be pointed out here that this definition is, by a theorem of B. Fuglede [4], equivalent to the usual definition of permutability in the case where N1 and N2 are bounded (N1N2 = N2N1) or in the case where N1 is bounded and N2 is unbounded (N1N2 _ N2N1). DEFINITION 2. We say that the operators N1, N2 and N have the property P if they are normal operators and if N = N1N2 = N2N1 .