linear functions
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2022 ◽  
Vol 0 (0) ◽  
pp. 0
Suman Dutta ◽  
Subhamoy Maitra ◽  
Chandra Sekhar Mukherjee

<p style='text-indent:20px;'>Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation spectrum, and the autocorrelation spectrum. We introduce the existing 2-fold Forrelation formulation with bent duality-based promise problems as desirable instantiations. Next, we concentrate on the 3-fold version through two approaches. First, we judiciously set up some of the functions in 3-fold Forrelation so that given oracle access, one can sample from the Walsh Spectrum of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Using this, we obtain improved results than what one can achieve by exploiting the Deutsch-Jozsa algorithm. In turn, it has implications in resiliency checking. Furthermore, we use a similar idea to obtain a technique in estimating the cross-correlation (and thus autocorrelation) value at any point, improving upon the existing algorithms. Finally, we tweak the quantum algorithm with the superposition of linear functions to obtain a cross-correlation sampling technique. This is the first cross-correlation sampling algorithm with constant query complexity to the best of our knowledge. This also provides a strategy to check if two functions are uncorrelated of degree <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula>. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>.</p>

Ivan Wolansky ◽  

Deep learning is a type of machine learning (ML) that is growing in importance in the medical field. It can often perform better than traditional ML models on different metrics, and it can handle non-linear problems due to activation functions. Activation functions are different non-linear functions that are used to restrict the values propagated to an interval. In deep learning, information propagates forward, passing through different layers of weights and activation functions, before reaching the final layer. Then a cost function is evaluated and propagated back through the network to adjust weights. A convolutional neural network (CNN) is a form of deep learning that is used primarily in imaging. CNNs perform significantly well with grid-like inputs because they learn shapes well. CNNs compute dot products between layers and kernels in a convolutional layer, prior to pooling, which outputs summary statistics. CNNs are better than trivial neural networks for imaging due to a number of reasons, like sparse interaction and equivariance of translation

2021 ◽  
pp. 123-138
Татьяна Михайловна Леденева

Данная статья представляет результаты исследования аддитивных генераторов в форме дробно-линейных функций. Определены ограничения на коэффициенты возрастающих и убывающих генераторов. Показано, что обратные функции при выполнении определенных условий также являются аддитивными генераторами. Для каждого случая найдены соответствующие треугольные нормы или конормы. Установлено, что треугольные нормы и конормы, полученные на основе дробно-линейных функций, а также двойственные им имеют одинаковую структуру. Определены значения параметров, при которых получаются известные семейства. По сути, предложено новое семейства двойственных треугольных норм и конорм, которое обобщает известные семейства. This article presents the results of the study of additive generators in the form of fractional linear functions. Restrictions on the coefficients of increasing and decreasing generators are determined. It is shown that inverse functions under certain conditions are also additive generators. For each case, the corresponding triangular norms or conorms are found. It is established that triangular norms and conorms obtained on the basis of fractional linear functions, as well as dual ones, have the same structure. The values of the parameters at which the known families are obtained are determined. In fact, a new family of dual triangular norms and conorms is proposed, which generalizes the known families.

2021 ◽  
Borzou Rostami ◽  
Masoud Chitsaz ◽  
Okan Arslan ◽  
Gilbert Laporte ◽  
Andrea Lodi

The economies of scale in hub location is usually modeled by a constant parameter, which captures the benefits companies obtain through consolidation. In their article “Single allocation hub location with heterogeneous economies of scale,” Rostami et al. relax this assumption and consider hub-hub connection costs as piecewise linear functions of the flow amounts. This spoils the triangular inequality property of the distance matrix, making the classical flow-based model invalid and further complicates the problem. The authors tackle the challenge by building a mixed-integer quadratically constrained program and by developing a methodology based on constructing Lagrangian function, linear dual functions, and specialized polynomial-time algorithms to generate enhanced cuts. The developed method offers a new strategy in Benders-type decomposition through relaxing a set of complicating constraints in subproblems when such relaxation is tight. The results confirm the efficacy of the solution methods in solving large-scale problem instances.

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2432
Nabil Abdoun ◽  
Safwan El Assad ◽  
Thang Manh Hoang ◽  
Olivier Deforges ◽  
Rima Assaf ◽  

In this paper, we propose, implement and analyze an Authenticated Encryption with Associated Data Scheme (AEADS) based on the Modified Duplex Construction (MDC) that contains a chaotic compression function (CCF) based on our chaotic neural network revised (CNNR). Unlike the standard duplex construction (SDC), in the MDC there are two phases: the initialization phase and the duplexing phase, each contain a CNNR formed by a neural network with single layer, and followed by a set of non-linear functions. The MDC is implemented with two variants of width, i.e., 512 and 1024 bits. We tested our proposed scheme against the different cryptanalytic attacks. In fact, we evaluated the key and the message sensitivity, the collision resistance analysis and the diffusion effect. Additionally, we tested our proposed AEADS using the different statistical tests such as NIST, Histogram, chi-square, entropy, and correlation analysis. The experimental results obtained on the security performance of the proposed AEADS system are notable and the proposed system can then be used to protect data and authenticate their sources.

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3205
Robin Dee ◽  
Armin Fügenschuh ◽  
George Kaimakamis

We describe the problem of re-balancing a number of units distributed over a geographic area. Each unit consists of a number of components. A value between 0 and 1 describes the current rating of each component. By a piecewise linear function, this value is converted into a nominal status assessment. The lowest of the statuses determines the efficiency of a unit, and the highest status its cost. An unbalanced unit has a gap between these two. To re-balance the units, components can be transferred. The goal is to maximize the efficiency of all units. On a secondary level, the cost for the re-balancing should be minimal. We present a mixed-integer nonlinear programming formulation for this problem, which describes the potential movement of components as a multi-commodity flow. The piecewise linear functions needed to obtain the status values are reformulated using inequalities and binary variables. This results in a mixed-integer linear program, and numerical standard solvers are able to compute proven optimal solutions for instances with up to 100 units. We present numerical solutions for a set of open test instances and a bi-criteria objective function, and discuss the trade-off between cost and efficiency.

2021 ◽  
Vol 4 ◽  
pp. 1-8
Maja Kalinic ◽  
Jukka M. Krisp

Abstract. Traffic congestion is a dynamic spatial and temporal process and as such might not be possible to model with linear functions of various dependent variables. That leaves a lot of space for non-linear approximates, such as neutral networks and fuzzy logic. In this paper, the focus is on the fuzzy logic as a possible approach for dealing with the problems of measuring traffic congestion. We investigate the application of this framework on a selected case study, and use floating car data (FCD) collected in Augsburg, Germany. A fuzzy inference system is built to detect degrees of congestion on a federal highway B17. With FCD, it is possible to obtain local speed information on almost all parts of the network. This information, together with collected vehicle location, time and heading, can be further processed and transformed into valuable information in the form of trip routes, travel times, etc. Initial results are compared with traditional method of expressing levels of congestion on a road network e.g. Level of Service – LOS. The fuzzy model, with segmented mean speed and travel time parameters, performed well and showed to be promising approach to detect traffic congestions. This approach can be further improved by involving more input parameters, such as density or vehicle flow, which might reflect traffic congestion event even more realistically.

Xiaole Huang ◽  
Wennian Xu ◽  
Yu Ding ◽  
Dong Xia ◽  
Shiyuan Xiong ◽  

Vegetation-growing Concrete (VC), as a new type of cemented soil, is usually used for plants growing on the surface of high and steep rocky slopes. With the widespread application of VC substrate, a pressing problem arises to ensure its durability under wetting and drying conditions. To explore the greatest possible impact on the mechanical properties and microstructure features of VC substrate, an experimental program including triaxial test, SEM analysis, and ultrasonic testing was implemented. The results showed that wetting and drying cycles can significantly decrease more than 40-percent of peak strength, 60-percent of residual strength, and 50-percent of cohesion for VC substrate under ultimate conditions. The fundamental cause of reduction in mechanical performance was found to be the weakening of the bond between soil particles. And it was discovered that structural damage increased as the number of wetting and drying cycles increased but at a slower rate. Based on the tested results, linear functions between the loss extent parameters of mechanical performance and the structural damage variable were established for the VC substrate. Finally, the action mechanisms of wetting and drying cycles for VC substrate were discussed, and the main influential factors were proposed.

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