Two-valued Function Research Articles

Neural network based iterative learning control for magnetic shape memory alloy actuator with iteration-dependent uncertainties

The magnetic shape memory alloy based actuator (MSMA-BA) is an indispensable component mechanism for high-precision positioning systems as it possesses the advantages of high precision, low energy consumption, and large stroke. However, hysteresis is an intrinsic property of MSMA material, which seriously affects the positioning accuracy of MSMA-BA. In this study, we propose a multi meta-model approach incorporating the nonlinear auto-regressive moving average with exogenous inputs (NARMAX) and Bouc–Wen (BW) models to describe the complex dynamic hysteresis of MSMA-BA. In particular, the BW model is introduced into the NARMAX model as an exogenous variable function, and a wavelet neural network (WNN) is adopted to construct the nonlinear function of the multi meta-model. In addition, iterative learning control is combined with a WNN to improve its convergence speed. A two-valued function is employed in the controller design process, so as to make use of history iteration information in updating control input. The main contribution of this study is the convergence analysis of the proposed iteration learning controller with iteration-dependent uncertainties (non-strict repetition of the initial state and varying iteration length). The experiments conducted on the MSMA-BA illustrate the validity of the proposed control scheme.

Some results on majority bad number

A two-valued function [Formula: see text] defined on the vertices of a graph [Formula: see text] is a majority bad dominating function (MBDF), if the sum of its function values over at least half the closed neighborhoods is at most one. That is, for at least half the vertices [Formula: see text], [Formula: see text], where [Formula: see text] consists of v and every vertex adjacent to v. The majority bad domination number of a graph G, is denoted by [Formula: see text], and is defined as [Formula: see text]. In this paper, we determine some results on majority bad number.

Flexible Transmitter Network

Current neural networks are mostly built on the MP model, which usually formulates the neuron as executing an activation function on the real-valued weighted aggregation of signals received from other neurons. This letter proposes the flexible transmitter (FT) model, a novel bio-plausible neuron model with flexible synaptic plasticity. The FT model employs a pair of parameters to model the neurotransmitters between neurons and puts up a neuron-exclusive variable to record the regulated neurotrophin density. Thus, the FT model can be formulated as a two-variable, two-valued function, taking the commonly used MP neuron model as its particular case. This modeling manner makes the FT model biologically more realistic and capable of handling complicated data, even spatiotemporal data. To exhibit its power and potential, we present the flexible transmitter network (FTNet), which is built on the most common fully connected feedforward architecture taking the FT model as the basic building block. FTNet allows gradient calculation and can be implemented by an improved backpropagation algorithm in the complex-valued domain. Experiments on a broad range of tasks show that FTNet has power and potential in processing spatiotemporal data. This study provides an alternative basic building block in neural networks and exhibits the feasibility of developing artificial neural networks with neuronal plasticity.

Conventional and Improved Inclusion-Exclusion Derivations of Symbolic Expressions for the Reliability of a Multi-State Network

This paper deals with an emergent variant of the classical problem of computing the probability of the union of n events, or equivalently the expectation of the disjunction (ORing) of n indicator variables for these events, i.e., the probability of this disjunction being equal to one. The variant considered herein deals with multi-valued variables, in which the required probability stands for the reliability of a multi-state delivery network (MSDN), whose binary system success is a two-valued function expressed in terms of multi-valued component successes. The paper discusses a simple method for handling the afore-mentioned problem in terms of a standard example MSDN, whose success is known in minimal form as the disjunction of prime implicants or minimal paths of the pertinent network. This method utilizes the multi-state inclusion-exclusion (MS-IE) principle associated with a multi-state generalization of the idempotency property of the ANDing operation. The method discussed is illustrated with a detailed symbolic example of a real-case study, and it produces a more precise version of the same numerical value that was obtained earlier. The example demonstrates the notorious shortcomings and the extreme inefficiency that the MS-IE method suffers, but, on the positive side, it reveals the way to alternative methods, in which such a shortcoming is (partially) mitigated. A prominent and well known example of these methods is the construction of a multi-state probability-ready expression (MS-PRE). Another candidate method would be to apply the MS-IE principle to the union of fewer (factored or composite) paths that is converted (at minimal cost) to PRE form. A third candidate method, employed herein, is a novel method for combining the MS-PRE and MS-IE concepts together. It confines the use of MS-PRE to ‘shellable’ disjointing of ORed terms, and then applies MS-IE to the resulting partially orthogonalized disjunctive form. This new method makes the most of both MS-PRE and MS-IE, and bypasses the troubles caused by either of them. The method is illustrated successfully in terms of the same real-case problem used with the conventional MS-IE.

Self-organized wavy infection curve of COVID-19

Exploiting the SIQR model for COVID-19, I show that the wavy infection curve in Japan is the result of fluctuation of policy on isolation measure imposed by the government and obeyed by citizens. Assuming the infection coefficient be a two-valued function of the number of daily confirmed new cases, I show that when the removal rate of infected individuals is between these two values, the wavy infection curve is self-organized. On the basis of the infection curve, I classify the outbreak of COVID-19 into five types and show that these differences can be related to the relative magnitude of the transmission coefficient and the quarantine rate of infected individuals.

Efficient and robust bitstream processing in binarised neural networks

In the neural network context, used in a variety of applications, binarised networks, which describe both weights and activations as single-bit binary values, provide computationally attractive solutions. A lightweight binarised neural network system can be constructed using only logic gates and counters together with a two-valued activation function unit. However, binarised neural networks represent the weights and the neuron outputs with only one bit, making them sensitive to bit-flipping errors. Binarised weights and neurons are manipulated by the utilisation of bitstream processing with regard to stochastic computing to cope with this error sensitivity. Stochastic computing is shown to provide robustness for bit errors on data while being built on a hardware structure, whose implementation is simplified by a novel subtraction-free implementation of the neuron activation.

Majority bad number

A two-valued function [Formula: see text] defined on the vertices of a graph [Formula: see text] is a majority bad dominating function, briefly (MBDF) if the sum of its function values over at least half the closed neighborhoods is at most one. That is, for at least half the vertices [Formula: see text], [Formula: see text], where [Formula: see text] consists of [Formula: see text] and every vertex adjacent to [Formula: see text]. The majority bad domination number of a graph [Formula: see text] is denoted by [Formula: see text] and is defined as [Formula: see text]. In this paper, we initiate the study of majority bad number in graphs.

Signed Domination Number of the Directed Cylinder

In a digraph D = ( V ( D ) , A ( D ) ) , a two-valued function f : V ( D ) → { - 1 , 1 } defined on the vertices of D is called a signed dominating function if f ( N - [ v ] ) ≥ 1 for every v in D. The weight of a signed dominating function is f ( V ( D ) ) = ∑ v ∈ V ( D ) f ( v ) . The signed domination number γ s ( D ) is the minimum weight among all signed dominating functions of D. Let P m × C n be the Cartesian product of directed path P m and directed cycle C n . In this paper, the exact value of γ s ( P m × C n ) is determined for any positive integers m and n.

New Examples of Locally Algebraically Integrable Bodies

Any compact body with regular boundary in ℝN defines a two-valued function on the space of affine hyperplanes: the volumes of the two parts into which these hyperplanes cut the body. This function is never algebraic if N is even and is very rarely algebraic if N is odd: all known bodies defining algebraic volume functions are ellipsoids (and have been essentially found by Archimedes for N = 3). We demonstrate a new series of locally algebraically integrable bodies with algebraic boundaries in spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume functions coincide with algebraic ones in some open domains of the space of hyperplanes intersecting the body.

Differential uniformity and the associated codes of cryptographic functions

The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper, we consider more generally the codes associated with functions that have differential uniformity at least \begin{document}$ 4 $\end{document} . We emphasize, for such a function \begin{document}$ F $\end{document} , the role of codewords of weight \begin{document}$ 3 $\end{document} and \begin{document}$ 4 $\end{document} and of some cosets of its associated code \begin{document}$ C_F $\end{document} . We give some properties on codes associated with differential uniformity exactly \begin{document}$ 4 $\end{document} . We obtain lower bounds and upper bounds for the numbers of codewords of weight less than \begin{document}$ 5 $\end{document} of the codes \begin{document}$ C_F $\end{document} . We show that the nonlinearity of \begin{document}$ F $\end{document} decreases when these numbers increase. We obtain a precise expression to compute these numbers, when \begin{document}$ F $\end{document} is a plateaued or a differentially two-valued function. As an application, we propose a method to construct differentially \begin{document}$ 4 $\end{document} -uniform functions, with a large number of \begin{document}$ 2 $\end{document} -to- \begin{document}$ 1 $\end{document} derivatives, from APN functions.

Approximation of a two-valued function by an algebraic polynomial

We consider the minimax model of a nonlinear structure for approximating a two-valued function by an algebraic polynomial. We establish optimality conditions as a strong generalization of P. L. Chebyshev alternance optimality conditions in approximation of a function by a polynomial.

Majority reinforcement number

A two-valued function [Formula: see text] defined on the vertices of a graph [Formula: see text], is a majority dominating function if the sum of its function values over at least half the closed neighborhoods is at least one. That is, for at least half the vertices [Formula: see text], [Formula: see text], where [Formula: see text] consists of [Formula: see text] and every vertex adjacent to [Formula: see text]. The majority domination number of a graph [Formula: see text], denoted [Formula: see text], is the minimum value of [Formula: see text] over all majority dominating functions [Formula: see text] of [Formula: see text]. The majority reinforcement number of [Formula: see text], denoted by [Formula: see text], is defined to be the minimum cardinality [Formula: see text] of a set [Formula: see text] of edges such that [Formula: see text]. In this paper, we initiate the study of majority reinforcement number and determine the exact values of [Formula: see text] for paths and cycles.

Upper Bounds on the Signed $$(k,k)$$ ( k , k ) -Domatic Number of Digraphs

Let \(D\) be a simple digraph with vertex set \(V(D)\), and let \(f:V(D)\rightarrow \{-1,1\}\) be a two-valued function. If \(k\ge 1\) is an integer and \(\sum _{x\in N^-[v]}f(x)\ge k\) for each \(v\in V(D)\), where \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\), then \(f\) is a signed \(k\)-dominating function on \(D\). A set \(\{f_1,f_2,\ldots ,f_d\}\) of distinct signed \(k\)-dominating functions on \(D\) with the property that \(\sum _{i=1}^df_i(x)\le k\) for each \(x\in V(D)\), is called a signed \((k,k)\)-dominating family (of functions) on \(D\). The maximum number of functions in a signed \((k,k)\)-dominating family on \(D\) is the signed \((k,k)\)-domatic number of \(D\). In this article, we mainly present upper bounds on the signed \((k,k)\)-domatic number, in particular for regular digraphs.

Finding number of clusters in single‐step with similarity‐based information‐theoretic algorithm

A single-step algorithm is presented to find the number of clusters in a dataset. An almost two-valued function called cluster-boundary indicator is introduced with the use of similarity-based information-theoretic sample entropy and probability descriptions. This function finds inter-cluster boundary samples for cluster availability in a single iteration. Experiments with synthetic and anonymous real datasets show that the new algorithm outperforms its major counterparts statistically in terms of time complexity and the number of clusters found successfully.

Signed 2-independence of Cartesian product of directed paths

A two-valued function f: V(D)→{−1, 1} defined on the vertices of a digraph D=(V(D), A(D)) is called a signed 2-independence function (S2IF) if f(N−[v])≤1 for every v in D. The weight of a S2IF is f(V(D))=∑v∈V(D)f(v). The maximum weight of a S2IF of D is the signed 2-independence number (or the lower against number) of D. Let Pm×Pn be the Cartesian product of directed paths Pm and Pn. In this paper, we determine the exact values of for 1≤m≤5 and n≥1.

On Sampling Design Issues when Dealing with Zeros

We study the problem of identifying payment errors in a population of claims containing “zeros,” that is, claims that are error-free, and the advantage of using proportional stratified sampling versus simple random sampling. The classical model assumes that the payment, Y, is multiplied by a random variable, X, with probability p(Y) of being in error. Traditionally, p(Y) is assumed to be constant. In this article we focus on the model where p(Y) is a two-valued step function, but we also consider a model where p(Y) is a continuous exponential function. We assume that Y has a gamma distribution, a good approximation in many real-world populations of claim payments. We consider the use of proportional stratification with two strata and introduce a criterion of goodness, G, as the gain in reducing the coefficient of variation by use of proportional stratification relative to ignoring the strata and applying simple random sampling.

Upper bounds on the signed (k, k)-domatic number

Let G be a graph with vertex set V(G), and let f : V(G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and \({\sum_{x\in N[v]} f(x) \ge k}\) for each \({v \in V(G)}\) , where N[v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f1,f2, . . . ,fd} of distinct signed k-dominating functions on G with the property that \({\sum_{i=1}^d f_i(x) \le k}\) for each \({x \in V(G)}\) , is called a signed (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed (k, k)-dominating family on G is the signed (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed (k, k)-domatic number, in particular for regular graphs.

Bounds on the signed 2-independence number in graphs

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1,1} be a two-valued function. If P x2N[v] f(x) ≤ 1 for each v ∈ V (G), where N[v] is the closed neighborhood of v, then f is a signed 2independence function on G. The weight of a signed 2-independence function f is w(f) = P v2V (G) f(v). The maximum of weights w(f), taken over all signed 2-independence functions f on G, is the signed 2-independence number 2(G) of G. In this work, we mainly present upper bounds on 2(G), as for example 2(G) ≤ n −2⌈( G)/2⌉, and we prove the Nordhaus-Gaddum type inequality 2(G) + 2(G) ≤ n+ 1, where n is the order and ( G) is the maximum degree of the graph G. Some of our theorems improve well-known results on the signed 2-independence number.

NOTE ON UPPER BOUND SIGNED 2-INDEPENDENCE IN DIGRAPHS

Let D be a finite digraph with the vertex set V(D) and arc set A(D). A two-valued function defined on the vertices of a digraph D is called a signed 2-independence function if for every in D. The weight of a signed 2-independence function is $f(V(D))

Stabilization of control loops consisting of FOPDT process and parameter-dependent PID controller

In this paper, problem of stability analysis of the control loops consisting of first-order plus dead time (FOPDT) processes and proportional-integrative-derivative (PID) controllers is studied, where the controller coefficients are functions of one or more independent parameters. An effective procedure is presented to determine a stability region in the independent parameters space. This method does not require complex numerical calculations such as solving nonlinear equations. It is based on usage of a two-valued indicator function and by using that, a stability region is easily determined. In order to clarify that, why the stability region needs to be specified in the “independent parameters space” an optimal method is given to design the PID controller for the FOPDT processes, as an instance. In this optimal method the controller coefficients are obtained as the functions of a free parameter, where this parameter needs to be chosen by the designer such that it should be near to the maximum operating frequency of the system, besides on the other hand the closed-loop system to be stable. In the end, two illustrative examples are given in order to show the usefulness and effectiveness of the proposed method, and to compare the obtained stability regions with the whole stability regions.