Equivalence Classes Of Functions Research Articles

Small systems convergence and metrizability

The abstract notion of convergence of functions with respect to a small system has its roots in the concept of convergence of functions in measure. The second author has shown that convergence of functions with respect to a small system generates a Fréchet topology. In this paper we show that convergence with respect to a small system is equivalent to convergence with respect to a certain complete pseudometric (or metric if we consider equivalence classes of functions with respect to the small system).

The Connectedness of the Group of Automorphisms of L 1 (0, 1)

For each of the radical Banach algebras L1(0, 1) and L1 (w) an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of L1(0, 1) and L1(w) endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if 111 11 IP denotes the norm of B(LP(O, 1), L1 (O, 1)), 1 d)-, where cv E R, ) 0, and D is a derivation, and where (eAdeDe Ad) denotes the extension by continuity of eAdeDe Ad from a dense subalgebra of 11 (w) to 11 (w). 0. Introduction. Suppose in the Banach space L1 (0, 1) we define the product * by rx (A*g)(x)=J f(x-y)g(y)dy (f,gEL1(0,1), a.e. xE(0,1)). o With this product L1(0, 1) becomes a radical Banach algebra [12]. In [12] Kamowitz and Scheinberg studied the derivations and automorphisms of L1(0, 1), and asked whether the group of automorphisms is connected. We prove that this group is connected in the bounded strong operator topology (BSO) and for topologies induced by the norm of B(LP(O, 1), L1 (0, 1)), where 1 < p < oo. The class of weighted convolution algebras has been studied by several authors from different viewpoints [1, 3, 4 and 5]. Suppose w is a continuous and positive function on R+ with w(0) = 1 and w(s + t) < w(s)w(t), and let Ll(w) be the Banach space of all equivalence classes of Lebesgue measurable functions f with r°° llf 11 = J If (x)l w(x) dx < oo, o with convolution product * defined by rx (f * g)(x) = j f(x-y)g(y) dy; o L1(w) is a Banach algeora. We prove that the group of automorphisms of Ll(w) endowed with the topology (BSO) is connected. Received by the editors March 3, 1986 and, in revised form, June 20, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46H99; Secondary 43A20, 43A22.

Equivalence Classes of Length Functions on Groups

Equivalence Classes of Length Functions on Groups

Operator and algebraic methods for NMR spectroscopy. II. NMR projection operators and spin functions

We outline double coset and restricted character cycle index methods which generate the equivalence classes of NMR spin functions and the irreducible representations in the representation spanned by spin functions in any equivalence class. Elegant operator methods are developed by which the projection operators of NMR groups can be obtained in terms of the projection operators of composing groups. Thus, the projection operators of NMR groups of many molecules can be obtained without the knowledge of their character tables. Projection operators thus obtained are applied on the set of spin functions in an equivalence class of NMR spin functions (rather than the whole set of spin functions) which are generated by double coset methods to obtain symmetry-adapted NMR spin functions in the composite particle representation. Methods are illustrated with several examples of molecules containing as many as 230 NMR spin functions. The methods developed here are quite general so that they can be applied to NMR of nuclei with multispin states and they do not require the character tables of NMR groups. The techniques outlined here are illustrated with 2, 2, 3, 3-tetramethyl butane for which the symmetry-adapted NMR spin couplings are obtained in the composite particle representation. The NMR Hamiltonian matrix of this molecule which is of order 218×218 is factored into 26×26 matrix by the composite particle treatment. This is further factored into 8 1×1 matrices and 6 4×4, 4 2×2, and 3 8×8 matrices in the symmetry-adapted basis.

Equivalence classes of functions on finite sets

By using Pólya′s theorem of enumeration and de Bruijn′s generalization of Pólya′s theorem, we obtain the numbers of various weak equivalence classes of functions in RD relative to permutation groups G and H where RD is the set of all functions from a finite set D to a finite set R, G acts on D and H acts on R. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn′s theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non‐isomorphic fm‐graphs relative to a given group.

On the number of equivalence classes of Boolean functions under a transformation group (Corresp.)

An asymptotic estimate is established for the number of equivalence classes of Boolean functions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> variables under transformations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x) \rightarrow f(Ax+b)+ L(x)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x=(x_{l}, \cdots ,x_{n}), A</tex> is a nonsingular <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> matrix, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> is a vector, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L(x)</tex> is a linear function.

Some considerations to the fundamental theory of infinite delay equations

In recent years we see an increasing interest in infinite delay equations. The main reason is that equations of this type become more and more important for different applications. Regardless of the specific problem one has to deal with it is in most cases necessary to establish some fundamental theory as, for instance, existence, uniqueness of solutions and continuous dependence on initial data. Also in the last few years we find an increasing number of papers where the general theory of linear and nonlinear semigroups or evolution operators is applied to functional differential equations. Of fundamental importance for all approaches is the right choice of the state space which in most cases is a Banach space of functions or of equivalence classes of functions. For equations with bounded delays this in general is not a difficult problem. But for infinite delay equations the choice of an appropriate state space is no more trivial. It was natural to investigate which properties of the state space are sufficient in order to establish the fundamental theory for infinite delay equations. Up to now the most thorough discussion of this problem is contained in [12]. The set of axioms given there seems to be in more or less final form. Other systems of axioms are just slight modifications of those given in [12] (cf., for instance, [18, 211) or consider more special cases (as in [6, 17). Section 1 of this paper can be considered as a discussion of the axioms given in [12] which results in a somewhat streamlined version of these axioms. It should be mentioned here that state spaces for equations with bounded retardation are included as special case, of course. In Section 2 we prove existence, uniqueness and continuous dependence of solutions concentrating on the nonstandard situation where the right-hand side of the equation is not defined for elements in the state space but only for representatives of elements in the state space. Such situations occur for instance if difference-differential equations are considered in a space like UP

On the Number of Fanout-Free Functions and Unate Cascade Functions

In this correspondence, the number of functions and the number of equivalence classes of functions realized by fanout-free networks and cascades of AND'S, OR'S, and inverters are presented. For fanout-free functions, recursive formulas for T ND (n) and φ ND (n), the number of n-p-n-equivalence classes of n-variable functions and p-equivalence classes of n-variable functions, respectively, are derived. For unate cascade functions, a recursive formula for ψ ND (n), the number of n-variable functions, and formulas for U ND (n) and ψ ND (n), the number of n-p -equivalence classes and p-equivalence classes, respectively, are derived. Some asymptotic properties of ψ ND (n), U ND (n), and ψ ND (n) are also examined and it is shown that ψ ND )/ψ(n)→ 1/√2, U ND (n)/U(n)→ 1/2, and ψ ND (n)/ψ(n)→ 1/√2 as n →∞, where ψ(n) is the number of distinct unate cascade functions of up to n variables, and U(n) and ψ(n) are the number of distinct n-p-n-and p-equivalence classes of unate cascade functions of up to n variables, respectively.

An efficient method of determining functional equivalence classes of a network with reconvergent fan-out

This paper is a study of the effects of the faults on tho functional operation of a combinational logic circuit. The conditions whereby two different faults can produce tho sancio functional output arc investigated. In this approach two fault graphs of the circuits arc drawn. By manipulating these fault graphs the faults which are functionally equivalent can be obtained. An algorithm for determining the functionally equivalent classes of faults in a combinational circuit is presented. The unique feature of the algorithm is that it produces tho true functional equivalence (not structural equivalence) even for the circuit with reconvergent fan-out with unequal parity.

Equivalence, linearization, and decomposition of holomorphic operator functions

This paper discusses the problem of classifying holomorphic operator functions up to equivalence. A survey is given in §1 of the main results about equivalence classes of holomorphic matrix functions and holomorphic Fredholm-operator functions. In §2, it is shown that given a holomorphic function A on a bounded domain Ω into a space of bounded linear operators between two Banach spaces, it is possible to extend the operators A( λ) (for each λ ϵ Ω) by an identity operator I Z in such a way that the extended operator function A(·) ⊕ I Z is equivalent on Ω to a linear function of λ, T − λI. Other versions of this “linearization by extension” are described, including the cases of entire functions and polynomials (where Ω = C ). As an application of these results, we consider the operator function equation A 2(λ) Z 2(λ) + Z 1(λ) A 1(λ) = C(λ), λ ϵ Ω , (∗) and explicitly construct the solutions Z 1 and Z 2. The formulas for Z 1 and Z 2 seem to be new, even when A 1, A 2 and C are matrix polynomials. The existence of solutions of ( ∗) makes it possible to analyze an operator function A whose spectrum decomposes into pairwise disjoint compact subsets σ 1, …, σ n of Ω. In this case, a suitable extension of A is equivalent on Ω to a direct sum of operator functions, A 1, …, A n , such that the spectrum of A i is σ i ( i = 1, …, n). In the final section of the paper, we discuss the relation between local and global equivalence on Ω, and show that there exist operator functions A and B which are locally equivalent on Ω, but admit no extensions (of the sort considered in this paper) which are globally equivalent on Ω.

Dilations of Positive Contractions on Lp Spaces*

Throughout this article p denotes a fixed number such that 1 ≤ p &lt; ∞. The definition of a real Lp space associated with a measure space is well known. These spaces are Banach Spaces and, with the usual partial ordering of (equivalence classes of) functions, also Banach Lattices.

Equivalence classes of functions over a finite field

Equivalence classes of functions over a finite field

Lp(G)-isotone measures

Let G be a locally compact topological group with left Haar measure, m; let M(G) denote the bounded regular Borel measures on G and let Lp(G) denote the equivalence classes of pth power integrable functions on G with respect to the left Haar measure.

On the Modeling of Systems for Identification. Part II: Time-Varying Systems

Certain Banach spaces, denoted $\mathcal{L}_T^p $, of equivalence classes of functions of a real variable are introduced and investigated. These are to be used as system input and output spaces for systems operating for all time. Some basic properties of causal systems with finite memory are established. The concept of forming time-interval truncations of a time-varying system is formalized and investigated. Trajectories of such truncations are studied. It is proved that under certain reasonable conditions, the trajectories of a class of systems are generated by a strongly continuous semigroup of linear operators. It is also shown that $\varepsilon $-representations of the truncations can be generated by an induced semigroup. Thus, the evolution in time of classes of, in general, nonlinear, time-varying systems, is described in the framework of a linear dynamical theory.

The Probability of a Correct Output from a Combinational Circuit

The paper discusses two methods to evaluate the signal reliability of the output of logical circuits. It is known that faults present in a circuit will not always cause the output of the circuit to be incorrect. Given the probability of faults occurring in the circuit and the probabilities of the input combinations, it is possible to determine the likelihood of the output being correct. The signal reliability of the output is thus defined as the probability that the circuit output is correct. The first method evaluates the contribution of each fault to the reliability of the circuit and requires the enumeration of the behavior of each fault in the entire fault set. The use of McCluskey and Clegg's characterization of faulty networks by evaluating the functional equivalence classes of the network is a way to reduce the amount of computation involved. Lower bounds can be obtained by considering a restricted fault set, for example, the single fault set. The second method uses a probabilistic model of logical circuits and consists of straightforward operations which can easily be automated. The method also yields the signal reliability and has the capability of very easily specifying the individual fault probabilities of all the circuit lines independently.

Conjugacy correspondences: a unified view

As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel’s correspondence among convex functions and Rockafellar’s extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new correspondences are brought to light as further special cases. Various questions are answered concerning the lower and upper closures and effective domain of the resulting equivalence class, as well as the effect the correspondence has on the related purely convex function and the subdifferential mapping.

Equivalence classes of functions of finite Markov chains

A matrical representation of a Markov chain consists of the initial vector and transition matrix of the chain, along with matrices that specify which observable response occurs for each state. The likelihood function based on a Markov model can be stated in a general way using the components of the model's matrical representation. It follows directly from that statement that two models are equivalent in likelihood if they are related through matrix operations that constitute a change of basis of the matrical representation. Two necessary properties of a change matrix associating two Markov models that are members of the same equivalence class with respect to likelihood are derived. Examples are provided, involving use of the results in analyzing identifiability of Markov models, including a useful application of diagonalization that provides a connection between the problem of identifiability and the eigenvalue problem.

Positive Contractions on L 1 -Spaces

Let T be a positive linear operator on L1 of a probability space such that II T1j i: 1. In this note we consider the following question: Under what condition is T multiplicative on Let (X, F, m) be a probability space and L1=L1(X, F, m) the Banach space of equivalence classes of integrable complex-valued functions on X. Let T be a positive linear operator on L1 such that II TIIl< 1, and define G(T)={f E L1; If I = I Tf I = 1). The main purpose of this note is to prove the following: THEOREM. If G(T) is total in L1, i.e., the linear manifold generated by G(T) is dense in L1 in the norm topology, then for any bounded functions f and g we have T(fg)= (Tf)(Tg). As a corollary of the Theorem we have the following result; essentially the same idea has been used by Halmos [2, p. 45] to find a necessary and sufficient condition for a unitary operator on L2=L2(X, Y, m) to be induced by an automorphism of the measure algebra (F(m), m) associated with (X, F, m). COROLLARY. A necessary and sufficient condition that T be induced by a homomorphism of the measure algebra (f(m), m) into itself is that G(T) be total in L1. For the proof of the theorem we need some lemmas. LEMMA 1. Let K be a compact Hausdorff space and C(K) the Banach algebra of all complex-valued continuous functions f on K wvith norm lIf IIU=sup{If(s)I; s e K}. Let U be a positive linear operator on C(K) such that U1=1. Let f,geC(K), IfI=lgi=1, and IUfI=IUgi=1. Then U(fg)=(Uf)(Ug)Received by the editors February 28, 1973 and, in revised form, May 10, 1973. AMS (MOS) subject classifications (1970). Primary 47B55, 47A35; Secondary 28A65, 46J10.

Function spaces with a projective limit structure

Vector spaces of functions and equivalence classes of functions for which a natural projective limit structure exists are studied in a systematic manner. The theory is illustrated by a series of examples arising from specific applications to the stability of feedback systems and the theory of hereditary differential systems.

Generation of Representative Functions of the NPN Equivalence Classes of Unate Boolean Functions

An algorithm is described which generates the set of representative functions of the negation and/or permutation of variables and negation of the function (NPN) equivalence classes of unate Boolean functions. The algorithm is based upon integer programming techniques. The set of representative functions for the NPN equivalence classes of unate functions of six or fewer variables was obtained using this algorithm. The set of pseudothreshold functions of six or fewer variables was also tabulated by using this algorithm as a basis.