Equivalence Classes Of Functions Research Articles

Bounds on the Length of Terminal Stuck-Fault Test

A terminal stuck fault in a logic network is represented by one or more stuck-at-1 or stuck-at-0 faults on the n input lines or single output. It is shown that for n ? 5, a least upper bound on the test length is n + 1, and for n ≫ 5, an upper bound is 2n - 4. A greatest lower bound is 3, for all n ≫ 1. The upper bounds are based on a maximum size alternating 1-tree in the n-cube representation of the function. Of the more than 600 000 equivalence classes of functions of n variables, n ? 5, only one does not have an n-edge alternating 1-tree. An algorithm is proposed for constructing tests based on alternating 1-trees.

Well-Posed Problems for a Partial Differential Equation of Order $2m + 1$

containing the elliptic differential operator M of order 2m and the differential operator L of order <__ 2m. Hilbert space methods are used to formulate and solve an abstract form of the problem and to discuss existence, uniqueness, asymptotic behavior and boundary conditions of a solution. The formulation of a generalized problem is the objective of 1, and we shall have reason to consider two types of solutions, called weak and strong. Sufficient conditions on the operatorM are given for the existence and uniqueness ofa weak solution to the generalized problem. These conditions constitute elliptic hypotheses onM and are discussed briefly in 3. Similar assumptions on L lead to results on the asymptotic behavior of a weak solution. The case in whichM and L are equal and self-adjoint is discussed in 2, and it is here that the role of the coefficient 7 of the equation appears first. Special as it is, this is a situation that often arises in applications, and there has been considerable interest in this coefficient 7 [4], [25]. The weak and strong solutions are distinguished not only by regularity conditions but also by their associated boundary conditions. It first appears in 5 that it is possible to prescribe too many (independent) boundary conditions on a strong solution, but in the applications it is seen that the interdependence of these conditions is built into the assumptions on the domains ofthe operators.Two examples of applications appear in 6 with a discussion of the types of boundary conditions that are appropriate. 1. The generalized problem. LetG be anonempty open set in the n-dimensional real Euclidean space, R, whose boundaryG is an (n 1)-dimensional manifold with G lying on one side of it. C(G) is the space ofinfinitely differentiable functions on G, and C(G) is the linear subspace of C(G) consisting of functions with compact support in G. The Sobolev space Hm(G)= H is the Hilbert space of (equivalence classes of) functions in L2(G), all of whose distributional derivatives through order m belong to L2(G). The inner product and norm are given,

Michael A. Harrison. The number of transitivity sets of Boolean functions. Journal of the Society for Industrial and Applied Mathematics, t. 11 (1963), p. 806–828. - Michael A. Harrison. The number of equivalence classes of Boolean functions under groups containing negation. IEEE transactions on electronic computers, t. EC-12 (1963), p. 559–561. - Michael A. Harrison. On the number of classes

Michael A. Harrison. The number of transitivity sets of Boolean functions. Journal of the Society for Industrial and Applied Mathematics, t. 11 (1963), p. 806–828. - Michael A. Harrison. The number of equivalence classes of Boolean functions under groups containing negation. IEEE transactions on electronic computers, t. EC-12 (1963), p. 559–561. - Michael A. Harrison. On the number of classes

The Radon-Nikodym theorem and the mean convergence of Banach space valued martingales

In 1943, Phillips [3] extended the classical Radon-Nikodym Theorem to the setting of Banach space valued set functions. Later, in the early sixties, Chatterji [1 ] and Scalora [4] proved a theorem dealing with the norm convergence of martingales in the Lp (1 = p (x*)), l^p , is the Banach space of all measurable x*-valued (equivalence classes of) functions/ defined on s which satisfy ||/||P = (/!i||/||i'aM)1'!'< °°. By a partition it = {En}Z„ of s, we shall mean a finite set of disjoint sets in 2 such that U*_i P„ = fl. Definition. An additive set function F defined on 2 and having its values in i is said to be of /»-bounded variation (1 ?p < oo ) if

A Correspondence Between Equivalence Classes of Switching Functions and Group Codes

The correspondence defined below was used to convert Slepian's tabulation of the number of equivalence classes of (m, r) group codes <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</sup> into hitherto unpublished data of relevance to switching theory. Specifically, Table II lists the number of equivalence classes of switching functions of weight m≪ 20 or m≫ 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -20 in nine or fewer arguments under the group of linear transformations on its argument variables. The correspondence is established by means of an m×n binary matrix, all of whose rows are distinct. The rows of this matrix define m points at which a switching function of n arguments takes on unit value. If the rank of this matrix (over the 2-element field) is r, then its columns generate an r-dimensional subspace of binary m-tuples which, by definition, is the message set of an (m, r) group code.

The number of classes of linearly equivalent functions

Let V be a finite dimensional vector space over a finite field F. We call two functions f and g from V into some set linearly equivalent if there is a non-singular linear transformation L of V such that f=g○L. We determine the number of equivalence classes of functions from V into a set of given cardinality. Along the way, we obtain the number of orbits of L considered as a permutation on the points of V.

The power group enumeration theorem

The number of orbits of a permutation group was determined in a fundamental result of Burnside. This was extended in a classical paper by Pólya to a solution of the problem of enumerating the equivalence classes of functions with a given weight from a set X into a set Y, subject to the action of the permutation group A acting on X. A generalization by de Bruijn solved the counting problem when two permutation groups are involved, A acting on X and B on Y. Thus the Pólya formula is the special case of the de Bruijn result in which B is the identity group. The Power Group Enumeration Theorem achieves the same result using only one permutation group: the power group, B A , acting on the set Y X of functions. de Bruijn's method was used to count self-complementary graphs by R. C. Read and finite automata by M. Harrison. These results as well as the number of self-converse directed graphs and others are easily obtained by the proper use of the Power Group Enumeration Theorem.

Nonexistence of quasi-invariant measures on infinite-dimensional linear spaces

It was shown by V. N. Sudakov [7] that a locally convex topological linear space with a nontrivial quasi-invariant a-finite on its weakly measurable sets must be finite-dimensional. Sudakov's ingenious proof uses a theorem of Ulam [6, Footnote 3]. In some unpublished notes [8], Y. Umemara attempted to prove the same theorem, by utilizing Weil's converse to the existence of Haar measure [9, Appendix 1]. Umemara's proof was incorrect. However, the approach seemed a natural one. In the present article Weil's theorem will be utilized to prove a more general nonexistence theorem, from which Sudakov's theorem may immediately be deduced, but which also covers the case of a Borel on any metrizable linear space; for example, the equivalence classes of measurable functions on a space. Thanks are due to R. Dudley and J. Mermin for patient and critical listening, and to C. C. Moore for contributing an essential step to an earlier version of the proof. R will always denote the real line, and T the unit circle in the complex plane.

Topologies for Laplace transform spaces

In an earlier paper [2]; the author used equivalence classes of analytic functions to construct an imbedding space for Schwartz Distributions. The mechanism for constructing the mapping was the bilateral Laplace Transform, in this way the traditional approach to operational calculus was preserved. In that paper a topology was imposed on the imbedding space from the space of analytic functions. We now obtain some additional results about the possible topologies defined on the space of analytic functions.

Invertible Boolean Functions

A Boolean function has an inverse when every output is the result of one and only one input. There are 2n! Boolean functions of n variables which have an inverse. Equivalence classes of these functions are sets of equivalent functions in the sense that they are identical under a group operation on the input and output variables. This paper counts through five variables the number of equivalence classes of invertible Boolean functions under the group operation of complementation, permutation, and complementation and permutation, linear transformations and affine transformations. Lower bounds are given which experimentally give an asymptotic approximation. A representative function is given of each of the 52 classes of invertible Boolean functions of three variables under complementation and permutation. These are divided into three types of classes, 21 self-inverting functions, three functions have an inverse in the same class and 14 pairs of functions, each function of the pair in a different class. The four representative functions under the affine transformation are self-invertible.

Equivalence classes of functions over a finite field

Equivalence classes of functions over a finite field

On an extension of the concept conditional expectation

The purpose of the present note is in part to extend results of Sidak [3] to cases where the measure is not finite, perhaps not even r-finite. In particular, the conditional expectation of a random variable in L1 is obtained without the use of the Radon-Nikodym Theorem, which, indeed, does not apply with the required generality. It was found possible to extend at the same time results of [1 ] so as to omit the requirement that the measure be totally finite. For this reason the or-fields discussed in [3] are here replaced by c-lattices. In this connection it may be observed that E(XI ?) is the solution of the regression problem: given XCL2, choose Y in the class C of random variables in L2 measurable with respect to a c-field 4 so as to minimize E(XY)2; E(XI 4) is the projection in L2 of X on C. Certain problems of maximum likelihood estimation of ordered parameters have solutions which also solve the above regression problem, in which 4 is not a c-field but is a c-lattice, closed under countable union and countable intersection, but not necessarily under complementation (see references in [1 ]). Let (Q, 8, ,) be a measure space: S is a c-field of subsets of Q, and ,u is a measure, c-additive and complete, but not necessarily cr-finite. The symbol S will also be used to denote the class of all events: an event is an equivalence class of sets in 8, two sets being equivalent if their symmetric difference has measure 0. The symbol Q will denote also the equivalence class to which Q belongs. (Events will often be denoted by descriptions enclosed in brackets.) A random variable is an equivalence class of real-valued S-measurable functions on Q, two such functions representing the same random variable if they differ on a set of measure 0. If A is an event, Ac will denote the complement of A, and x(A) or XA will denote the indicator random variable of A. Let L1 denote the class of integrable random variables, L2 the class of square-integrable random variables. If 4 is a sub-cr-lattice (closed under countable union, countable intersection) of S containing 0 and Q, let a random variable X be termed ?-measurable if [X > a] E 4 for every real a. A family C of random variables will be called a convex cone if k_0, XCC, YCEC=kXC-C, X+YCC; a

Algebras of Riemann integrable functions

In the present paper we shall consider from two different standpoints the real Riemann integrable functions on a bounded closed interval I contained in En. Our results will be extendable to the complex case by standard arguments. (1) In ?1 we designate the real valued Riemann integrable functions by R(I). If R(I) is normed by Rflfl =sup.e lf(x) then it is known that R(I) is a Banach algebra. We show that R(I) can be characterized as a B(S, 1) space [1] which consists of all uniform limits of real finite linear combinations of characteristic functions of sets in I having Jordan content. This characterization immediately yields a representation of the dual space of R(I). (2) We consider in ?2 a collection of equivalence classes of functions in R(I): we define Roo(I) to be the collection of classes such that fi and f2 are in the same class if and only if fi -f2 vanishes everywhere except on a set of Lebesgue measure zero. If [f] is an element of Ro(I) we definle the norm II [fIII as follows. Let N designate the subsets of I of Lebesgue measure zero. Then if f i [f] let 11[fill = infsCN supx$I ff(x) I = fIf. It is obvious that this norm is induced by the norm of Loo(I) It is also clear that Roo(I) is isometrically isomorphic to a normed subalgebra of L,(I). Designating the Boolean ring of null Jordan sets by qj n N, we show that Ro,(I) =R(I)1/B (I, ,rJN) after first establishing that B(J, jgnN) (interpreted in the B(S, 1) sense) is a closed ideal in R(I). It is a pleasure to acknowledge the advice of Professor Fred B. Wright and Professor F. Quigley.

The Fourier Transform and Mean Convergence

At the undergraduate level the Fourier transform is customarily studied by pointwise convergence, and only absolutely integrable functions are considered. In view of the importance of convergence in the mean and of the invariance of the integral of the absolute square under the Fourier transform, it would seem desirable to have a treatment of the Fourier transform in terms of convergence in the mean, still using only Riemann integrals, since ordinarily no other integral is known to undergraduates. The present paper presents such a treatment of two fundamental theorems, the invariance of the integral of the absolute square and the inversion theorem. Let us say that a (complex-valued) function f on the real axis is squareintegrable if f is (Riemann) integrable over every interval and IfI 2 is integrable over the whole real axis. The Fourier theory makes no distinction between functions fi and f2 which are equal at so many places that the integral of fi -f2 2 iS 0 (or, what amounts to the same thing, the integral of If1-f2| is 0). Hence such functions will be called equivalent. Properly speaking, the range and the domain of the transform are not functions, but equivalence classes of functions. But we follow custom in saying that the transformation maps functions on functions, and we identify any two equivalent functions. If f is square-integrable we define its norm to be