Equivalence Classes Of Functions Research Articles

Nikishin’s theorem and factorization through Marcinkiewicz spaces

Consider L 0 L^0 , the F F -space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin’s classical result on the factorization of sublinear continuous operators from a Banach space to L 0 L^0 , we prove a theorem that characterizes those maps from any quasi-metric space into L 0 L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces.

Functional equivalence and a cover relation for reaction systems

Reaction systems are a computational model originally introduced to formalize the interactions between biochemical reactions that are the basis of the functioning of the living cell. Subsets of reactions of a reaction system define result functions which leads to a concept of functional equivalence. This equivalence in turn induces a functional cover relation on the reactions of a reaction system which captures redundancies in the system. In this paper, the functional cover relation is transferred to functional equivalence classes of sets of reactions. We introduce so-called atoms as building blocks of reactions. Atoms provide a characterization of functional equivalence classes of sets of reactions and are used to prove that the functional cover relation on functional equivalence classes of sets of reactions is a lattice.

Paley Wiener Theorem on a Reductive Lie Group

Let G be a locally compact group, K a maximal compact subgroup of G and δ on arbitrary class of irreducible unitary representations of K. The spherical Grassmannian Gp,δ is an equivalence class of spherical functions of type δ−positive of height p. In this work, we give an extension of orbital integral with respect to δ, when G is reductive Lie group. Moreover, if the discret serie is not empty, we give an extension of Paley-Wiener theorem using a compact Cartan subgroup of G.

Gain total graphs and their spectra via G-phases and group representations

We introduce a definition for the total graph of a gain graph (Γ,ψ) on a group G by using G-phases, which are a generalization of the notion of orientation to gain graphs. Our construction is well-defined in the sense that gain graphs that are switching isomorphic have switching isomorphic total graphs. In particular, the switching equivalence class of the total graph does not depend on the particular choice of the G-phase associated with ψ. More precisely, we consider the left-right multiplication action of G on the space of all G-phases, proving that its orbits consist of sets of G-phases inducing the same switching equivalence class of gain functions on Γ or, equivalently, on its total graphs. Moreover, we prove that the stabilizers of this action are isomorphic to the centralizers of the sets of gains of closed walks in the associated gain graphs. Our construction is consistent with the existing notions of total graph for signed graphs and, in analogy with the signed case, we are able to explicitly compute the spectrum of the total graph of a regular gain graph over an arbitrary group G, where the spectrum is defined by means of a unitary representation of the gain group G.

Existence and uniqueness of solutions of nonlinear Cauchy‐type problems for first‐order fractional differential equations

New properties of the first‐order Riemann–Liouville fractional integrals are proved and first‐order fractional derivatives for the equivalence class of functions in are analyzed. Solutions, normal solutions, and generalized normal solutions of the initial value problems (IVPs) for nonlinear first‐order fractional differential equations (FDEs) with fraction are introduced. The equivalences in between the first‐order FDEs and the corresponding (Volterra) integral equations are established based on normal solutions and generalized normal solutions. New results on the existence and uniqueness of generalized normal solutions or nonnegative generalized normal solutions in of the IVPs are obtained for . These results are applied to study the existence and uniqueness of generalized normal nonnegative solutions in of the IVP for the nonlinear first‐order FDE with nonlinearities related to the population models with growth rates of Ricker type.

Functional equivalence classes in rats: Repeated olfactory discrimination reversals and delayed stimulus probes.

The simultaneous matching-to-sample procedures that are widely used to study stimulus equivalence in human participants have generally been unsuccessful in animals. However, functional equivalence classes have been demonstrated in pigeons and sea lions using a concurrent repeated reversal discrimination procedure. In this procedure, responding to one set of stimuli is reinforced but responding to a different set is not and the set associated with reinforcement is changed with multiple reversals during the experiment. The experiments reported here were designed to assess whether functional equivalence classes could be demonstrated in rats using similar techniques. Rats were initially trained with two sets of olfactory stimuli (six odors/set). Following many reversals, probe reversal sessions were conducted in which rats were exposed to a subset of the members of each set and, later in the session, the withheld stimuli were introduced. Responding to these delayed probe trials in accord with the reversed contingencies constituted transfer of function. There was some evidence of transfer in Experiment 1, but the effects were relatively weak and variable. Experiment 2 introduced procedural changes and found strong evidence of transfer of function consistent with the formation of functional equivalence classes. These procedures offer a promising strategy to study symbolic behavior in rodents.

On merge trees and discrete Morse functions on paths and trees

In this work we answer an open question asked by Johnson–Scoville. We show that each merge tree is represented by a discrete Morse function on a path. Furthermore, we present explicit constructions for two different but related kinds of discrete Morse functions on paths that induce any given merge tree. A refinement of the used methods allows us to define notions of equivalence of discrete Morse functions on trees which give rise to a bijection between equivalence classes of discrete Morse functions and isomorphism classes of certain labeled merge trees. We also compare our results to similar ones from the literature, in particular to work by Curry.

Defining the extent of gene function using ROC curvature.

Interactions between proteins help us understand how genes are functionally related and how they contribute to phenotypes. Experiments provide imperfect 'ground truth' information about a small subset of potential interactions in a specific biological context, which can then be extended to the whole genome across different contexts, such as conditions, tissues or species, through machine learning methods. However, evaluating the performance of these methods remains a critical challenge. Here, we propose to evaluate the generalizability of gene characterizations through the shape of performance curves. We identify Functional Equivalence Classes (FECs), subsets of annotated and unannotated genes that jointly drive performance, by assessing the presence of straight lines in ROC curves built from gene-centric prediction tasks, such as function or interaction predictions. FECs are widespread across data types and methods, they can be used to evaluate the extent and context-specificity of functional annotations in a data-driven manner. For example, FECs suggest that B cell markers can be decomposed into shared primary markers (10-50 genes), and tissue-specific secondary markers (100-500 genes). In addition, FECs suggest the existence of functional modules that span a wide range of the genome, with marker sets spanning at most 5% of the genome and data-driven extensions of Gene Ontology sets spanning up to 40% of the genome. Simple to assess visually and statistically, the identification of FECs in performance curves paves the way for novel functional characterization and increased robustness in the definition of functional gene sets. Code for analyses and figures is available at https://github.com/yexilein/pyroc. Supplementary data are available at Bioinformatics online.

A Characterization of the Vector Lattice of Measurable Functions

Given a probability measure space (X,Sigma ,mu ), it is well known that the Riesz space L^0(mu ) of equivalence classes of measurable functions f: X rightarrow mathbf {R} is universally complete and the constant function varvec{1} is a weak order unit. Moreover, the linear functional L^infty (mu )rightarrow mathbf {R} defined by f mapsto int f,mathrm {d}mu is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space E with a weak order unit e>0 which admits a strictly positive order continuous linear functional on the principal ideal generated by e is lattice isomorphic onto L^0(mu ), for some probability measure space (X,Sigma ,mu ).

Equivalence classes of Niho bent functions

Equivalence classes of Niho bent functions are described for all known types of hyperovals.

Complete solutions of inverse quantum orthogonal equivalence classes

In the year 1939, the mathematician G.H. Hardy proved that the only functions f which satisfy the classical orthogonality relation ∫01f(λmt)f(λnt)dt=0,m≠n,are the Bessel functions Jν(t) under certain constraints, where ν>−1 is the order of the Bessel function, and λm, λn are the nontrivial zeros of the Bessel function. By example, herein we present a complete equivalence class of inverse q-periodic functions, satisfying the inverse q-orthogonality relation ∫01f(λmt)f(λnt)d1qt=0,m≠n.The inverse q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points q−ℓ, with the step size at the point q−ℓ being q, ∀ℓ∈N0, where N0≔N∪{0}, and |q|≠1. An equivalence class of entire functions f∈Lq−12(0,1) that are inverse q-orthogonal with respect to their own nontrivial zeros is proven by invoking the Fundamental Theorem of Algebra, as the inverse q-periodic functions constitute an equivalence class of nonzero inverse q-periodic constants on the complex plane.

Characterisation of homogeneous fractional Sobolev spaces

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces mathcal {D}^{s,p} (mathbb {R}^n) and their embeddings, for s in (0,1] and pge 1. They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For s,p < n or s = p = n = 1 we show that mathcal {D}^{s,p}(mathbb {R}^n) is isomorphic to a suitable function space, whereas for s,p ge n it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey–Campanato inequality where the Gagliardo–Slobodeckiĭ seminorm controls from above a suitable Campanato seminorm.

Effects of preliminary class membership on subsequent stimulus equivalence class formation.

The present study examined the effects of including stimuli previously trained as members of functional classes or equivalence classes on subsequent equivalence class formation, and isolated the effects of preliminary training from those of the acquired function stimuli. Fifty-six adults were assigned to 1 of 5 conditions. The control group (CONT) received no preliminary training prior to the terminal phase. Participants in the other 4 groups learned two 3-member functional classes and two 3-member equivalence classes during the preliminary phase. The terminal equivalence phase trained two 5-member classes (A → B → C → D → E) comprising abstract forms; the C stimuli in the terminal phase were (a) from the preliminary functional classes for 1 group (ACQ-F), (b) from the preliminary equivalence classes for the second experimental group (ACQ-E), (c) pictures of everyday objects for the picture control group (PIC), and (d) novel, unfamiliar stimuli for the preliminary training control group (PRE-CONT). Class formation yields were 100% in the PIC condition and 11% in the CONT condition; however, low yields in the PRE-CONT, ACQ-F, and ACQ-E conditions were unexpected, suggesting that procedural variables in preliminary training account for more of the subsequent effects on class formation than the stimulus control properties of the acquired function stimuli.

On the Number of Equivalence Classes of Boolean and Invertible Boolean Functions

The number U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> of equivalence classes of Boolean functions of n variables and the number V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> of equivalence classes of vectorial Boolean functions of n variables under the action of four groups of transformations are considered. The four groups are the group S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ' of permutations of variables, the group G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> of permutations and complementations, the linear group GL(n, 2) and the affine group AGL(n, 2). Harrison obtained cycle indexes for these groups and the expressions for U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> and V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> in terms of the corresponding cycle index. He also tabulated the numbers U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> and the cycle indexes for n≤6 for S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ' and Gn, and for n≤5 for GL(n, 2) and AGL(n, 2). This bound was only recently slightly exceeded. Fripertinger implemented computation of cycle indexes for GL(n, q) and AGL(n, q); if q = 2 this implementation works for about n 5≤21. By introducing appropriate precomputed tables, we reduced the cycle index computation to evaluation of a sum over partitions of n for all the four groups. Using this more efficient procedure, we obtained values of U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> and the explicit cycle index expressions for larger values of n.

Computing the Number of the Equivalence Classes for Reversible Logic Functions

Reversible logic function classification plays an important role in reversible logic synthesis. This paper studies the calculation of the number of the equivalence classes for reversible logic functions. In order to do this work, an n-qubit reversible function is expressed as a permutation in the symmetric group $S_{2^{n}}$ , so that a universal formula for calculating the number of equivalence classes of reversible logic functions is derived based on group theory. Based on the calculation method of the number of conjugacy classes of permutation groups, an improved method for calculating the number of equivalence classes of reversible logic functions is obtained. In the experiments with GAP software on a laptop, we can compute the NPNP-equivalence classes for up to 13 qubits, LL-equivalence classes for up to 10 qubits and AA-equivalence classes for up to 10 qubits. Experiment results indicate that our scheme for calculating these equivalence classes of more than 6 qubits over previous published methods is a significant advancement.

Convergence rates in almost-periodic homogenization of higher-order elliptic systems

This paper concentrates on the quantitative homogenization of higher-order elliptic systems with almost-periodic coefficients in bounded Lipschitz domains. For almost-periodic coefficients in the sense of H. Weyl, we establish uniform local L 2 estimates for the approximate correctors. Under an additional assumption (1.8) on the frequencies of the coefficients, we derive the existence of true correctors as well as the O ( ε ) convergence rate in H m − 1 . As a byproduct, the large-scale Hölder estimate and a Liouville theorem are obtained for higher-order elliptic systems with almost-periodic coefficients in the sense of Besicovitch. Since (1.8) is not well-defined for equivalence classes of almost-periodic functions in the sense of H. Weyl or Besicovitch, we provide another condition yielding the O ( ε ) convergence rate under perturbations of the coefficients.

SAT-Based Generation of Optimum Circuits with Polymorphic Behavior Support

This paper presents a method for generating optimum multi-level implementations of Boolean functions based on Satisfiability (SAT) and Pseudo-Boolean Optimization (PBO) problems solving. The method is able to generate one or enumerate all optimum implementations, while the allowed target gate types and gates costs can be arbitrarily specified. Polymorphic circuits represent a newly emerging computation paradigm, where one hardware structure is capable of performing two or more different intended functions, depending on instantaneous conditions in the target operating environment. In this paper we propose the first method ever, generating provably size-optimal polymorphic circuits. Scalability and feasibility of the method are documented by providing experimental results for all NPN-equivalence classes of four-input functions implemented in AND–Inverter and AND–XOR–Inverter logics without polymorphic behavior support being used and for all pairs of NPN–equivalence classes of three-input functions for polymorphic circuits. Finally, several smaller benchmark circuits were synthesized optimally, both in standard and polymorphic logics.

On Computing Structural and Behavioral Complexities of Threshold Boolean Networks : Application to Biological Networks.

Various threshold Boolean networks (TBNs), a formalism used to model different types of biological networks (genes notably), can produce similar dynamics, i.e. share same behaviors. Among them, some are complex (according to Kolmogorov complexity), others not. By computing both structural and behavioral complexities, we show that most TBNs are structurally complex, even those having simple behaviors. For this purpose, we developed a new method to compute the structural complexity of a TBN based on estimates of the sizes of equivalence classes of the threshold Boolean functions composing the TBN.

Instantons on hyperk\xe4hler manifolds

An instanton (E, D) on a (pseudo-)hyperkähler manifold M is a vector bundle E associated with a principal G-bundle with a connection D whose curvature is pointwise invariant under the quaternionic structures of T_x M,~xin M, and thus satisfies the Yang–Mills equations. Revisiting a construction of solutions, we prove a local bijection between gauge equivalence classes of instantons on M and equivalence classes of certain holomorphic functions taking values in the Lie algebra of G^{mathbb {C}} defined on an appropriate mathrm {SL}_2({mathbb {C}})-bundle over M. Our reformulation affords a streamlined proof of Uhlenbeck’s compactness theorem for instantons on (pseudo-)hyperkähler manifolds.

String Amplitudes from Field-Theory Amplitudes and Vice Versa.

We present an integration-by-parts reduction of any massless tree-level string correlator to an equivalence class of logarithmic functions, which can be used to define a field-theory amplitude via a Cachazo-He-Yuan (CHY) formula. The string amplitude is then shown to be the double copy of the field-theory one and a special disk or sphere integral. The construction is generic as it applies to any correlator that is a rational function of correct SL(2) weight. By applying the reduction to open bosonic or heterotic strings, we get a closed-form CHY integrand for the (DF)^{2}+Yang-Mills+ϕ^{3} theory.