AppliedMath

35 pages, 2277 KiB

Open AccessEditor’s ChoiceArticle

Measuring the Risk of Vulnerabilities Exploitation

by Maria de Fátima Brilhante, Dinis Pestana, Pedro Pestana and Maria Luísa Rocha

AppliedMath 2024, 4(1), 20-54; https://doi.org/10.3390/appliedmath4010002 - 24 Dec 2023

Cited by 2 | Viewed by 2053

Modeling the vulnerabilities lifecycle and exploitation frequency are at the core of security of networks evaluation. Pareto, Weibull, and log-normal models have been widely used to model the exploit and patch availability dates, the time to compromise a system, the time between compromises, [...] Read more.

Modeling the vulnerabilities lifecycle and exploitation frequency are at the core of security of networks evaluation. Pareto, Weibull, and log-normal models have been widely used to model the exploit and patch availability dates, the time to compromise a system, the time between compromises, and the exploitation volumes. Random samples (systematic and simple random sampling) of the time from publication to update of cybervulnerabilities disclosed in 2021 and in 2022 are analyzed to evaluate the goodness-of-fit of the traditional Pareto and log-normal laws. As censoring and thinning almost surely occur, other heavy-tailed distributions in the domain of attraction of extreme value or geo-extreme value laws are investigated as suitable alternatives. Goodness-of-fit tests, the Akaike information criterion (AIC), and the Vuong test, support the statistical choice of log-logistic, a geo-max stable law in the domain of attraction of the Fréchet model of maxima, with hyperexponential and general extreme value fittings as runners-up. Evidence that the data come from a mixture of differently stretched populations affects vulnerabilities scoring systems, specifically the common vulnerabilities scoring system (CVSS). Full article

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19 pages, 9516 KiB

Open AccessEditor’s ChoiceArticle

Modeling and Visualizing the Dynamic Spread of Epidemic Diseases—The COVID-19 Case

by Loukas Zachilas and Christos Benos

AppliedMath 2024, 4(1), 1-19; https://doi.org/10.3390/appliedmath4010001 - 20 Dec 2023

Viewed by 1373

Abstract

Our aim is to provide an insight into the procedures and the dynamics that lead the spread of contagious diseases through populations. Our simulation tool can increase our understanding of the spatial parameters that affect the diffusion of a virus. SIR models are [...] Read more.

Our aim is to provide an insight into the procedures and the dynamics that lead the spread of contagious diseases through populations. Our simulation tool can increase our understanding of the spatial parameters that affect the diffusion of a virus. SIR models are based on the hypothesis that populations are “well mixed”. Our model constitutes an attempt to focus on the effects of the specific distribution of the initially infected individuals through the population and provide insights, considering the stochasticity of the transmission process. For this purpose, we represent the population using a square lattice of nodes. Each node represents an individual that may or may not carry the virus. Nodes that carry the virus can only transfer it to susceptible neighboring nodes. This important revision of the common SIR model provides a very realistic property: the same number of initially infected individuals can lead to multiple paths, depending on their initial distribution in the lattice. This property creates better predictions and probable scenarios to construct a probability function and appropriate confidence intervals. Finally, this structure permits realistic visualizations of the results to understand the procedure of contagion and spread of a disease and the effects of any measures applied, especially mobility restrictions, among countries and regions. Full article

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30 pages, 1226 KiB

Open AccessEditor’s ChoiceArticle

Max-C and Min-D Projection Auto-Associative Fuzzy Morphological Memories: Theory and an Application for Face Recognition

by Alex Santana dos Santos and Marcos Eduardo Valle

AppliedMath 2023, 3(4), 989-1018; https://doi.org/10.3390/appliedmath3040050 - 8 Dec 2023

Viewed by 1199

Abstract

Max-C and min-D projection auto-associative fuzzy morphological memories (max-C and min-D PAFMMs) are two-layer feedforward fuzzy morphological neural networks designed to store and retrieve finite fuzzy sets. This paper addresses the main features of these auto-associative memories: unlimited absolute [...] Read more.

Max-C and min-D projection auto-associative fuzzy morphological memories (max-C and min-D PAFMMs) are two-layer feedforward fuzzy morphological neural networks designed to store and retrieve finite fuzzy sets. This paper addresses the main features of these auto-associative memories: unlimited absolute storage capacity, fast retrieval of stored items, few spurious memories, and excellent tolerance to either dilative or erosive noise. Particular attention is given to the so-called Zadeh’ PAFMM, which exhibits the most significant noise tolerance among the max-C and min-D PAFMMs besides performing no floating-point arithmetic operations. Computational experiments reveal that Zadeh’s max-C PFAMM, combined with a noise masking strategy, yields a fast and robust classifier with a strong potential for face recognition tasks. Full article

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48 pages, 663 KiB

Open AccessEditor’s ChoiceReview

Interval Quadratic Equations: A Review

by Isaac Elishakoff and Nicolas Yvain

AppliedMath 2023, 3(4), 909-956; https://doi.org/10.3390/appliedmath3040048 - 1 Dec 2023

Viewed by 1459

Abstract

In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an [...] Read more.

In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an interval variable. The four methods reviewed here in order to solve this problem are: (i) the method of classic interval analysis used by Elishakoff and Daphnis, (ii) the direct method based on minimizations and maximizations also used by the same authors, (iii) the method of quantifier elimination used by Ioakimidis, and (iv) the interval parametrization method suggested by Elishakoff and Miglis and again based on minimizations and maximizations. We will also compare the results yielded by all these methods by using the computer algebra system Mathematica for computer evaluations (including quantifier eliminations) in order to conclude which method would be the most efficient way to solve problems relevant to interval quadratic equations. Full article

28 pages, 8158 KiB

Open AccessEditor’s ChoiceArticle

Some Comments about Zero and Non-Zero Eigenvalues from Connected Undirected Planar Graph Adjacency Matrices

by Daniel A. Griffith

AppliedMath 2023, 3(4), 771-798; https://doi.org/10.3390/appliedmath3040042 - 1 Nov 2023

Viewed by 2287

Abstract

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies [...] Read more.

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies approximating the eigenvalues of massively large versions of these two aforementioned matrices. The proposed solutions outlined in this paper essentially are a reformulated multiple linear regression analysis for the first problem and a matrix inertia refinement adapted to existing work for the second problem. Full article

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12 pages, 315 KiB

Open AccessEditor’s ChoiceArticle

Terracini Loci for Maps

by Edoardo Ballico

AppliedMath 2023, 3(3), 690-701; https://doi.org/10.3390/appliedmath3030036 - 17 Sep 2023

Viewed by 1399

Abstract

Let X be a smooth projective variety and

f : X \to P^{r}

a morphism birational onto its image. We define the Terracini loci of the map f. Most results are only for the case

dim X = 1

. With [...] Read more.

Let X be a smooth projective variety and

f : X \to P^{r}

a morphism birational onto its image. We define the Terracini loci of the map f. Most results are only for the case

dim X = 1

. With this new and more flexible definition, it is possible to prove strong nonemptiness results with the full classification of all exceptional cases. We also consider Terracini loci with restricted support (solutions not intersecting a closed set

B ⊊ X

or solutions containing a prescribed

p \in X

). Our definitions work both for the Zariski and the euclidean topology and we suggest extensions to the case of real varieties. We also define Terracini loci for joins of two or more subvarieties of the same projective space. The proofs use algebro-geometric tools. Full article

15 pages, 306 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Optimal Statistical Analyses of Bell Experiments

by Richard D. Gill

AppliedMath 2023, 3(2), 446-460; https://doi.org/10.3390/appliedmath3020023 - 16 May 2023

Viewed by 2351

Abstract

We show how both smaller and more reliable p-values can be computed in Bell-type experiments by using statistical deviations from no-signalling equalities to reduce statistical noise in the estimation of Bell’s S or Eberhard’s J. Further improvement was obtained by using [...] Read more.

We show how both smaller and more reliable p-values can be computed in Bell-type experiments by using statistical deviations from no-signalling equalities to reduce statistical noise in the estimation of Bell’s S or Eberhard’s J. Further improvement was obtained by using the Wilks likelihood ratio test based on the four tetranomially distributed vectors of counts of the four different outcome combinations, one 4-vector for each of the four setting combinations. The methodology was illustrated by application to the loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich, at NIST, and in Vienna, respectively, and also to the earlier (1998) Innsbruck experiment of Weihs et al. and the recent (2022) Munich experiment of Zhang et al., which investigates the use of a loophole-free Bell experiment as part of a protocol for device-independent quantum key distribution (DIQKD). Full article

11 pages, 423 KiB

Open AccessEditor’s ChoiceArticle

Analytical Approximation of the Jackknife Linking Error in Item Response Models Utilizing a Taylor Expansion of the Log-Likelihood Function

by Alexander Robitzsch

AppliedMath 2023, 3(1), 49-59; https://doi.org/10.3390/appliedmath3010004 - 5 Jan 2023

Cited by 6 | Viewed by 2227

Abstract

Linking errors in item response models quantify the dependence on the chosen items in means, standard deviations, or other distribution parameters. The jackknife approach is frequently employed in the computation of the linking error. However, this jackknife linking error could be computationally tedious [...] Read more.

Linking errors in item response models quantify the dependence on the chosen items in means, standard deviations, or other distribution parameters. The jackknife approach is frequently employed in the computation of the linking error. However, this jackknife linking error could be computationally tedious if many items were involved. In this article, we provide an analytical approximation of the jackknife linking error. The newly proposed approach turns out to be computationally much less demanding. Moreover, the new linking error approach performed satisfactorily for datasets with at least 20 items. Full article

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