Projection Methods of Solving the Wiener-Hopf Equation and Its Discrete Analogue

Face recognition is the most efficient image analysis application, and the reduction of dimensionality is an essential requirement. The curse of dimensionality occurs with the increase in dimensionality, the sample density decreases exponentially. Dimensionality Reduction is the process of taking into account the dimensionality of the feature space by obtaining a set of principal features. The purpose of this manuscript is to demonstrate a comparative study of Principal Component Analysis and Linear Discriminant Analysis methods which are two of the highly popular appearance-based face recognition projection methods. PCA creates a flat dimensional data representation that describes as much data variance as possible, while LDA finds the vectors that best discriminate between classes in the underlying space. The main idea of PCA is to transform high dimensional input space into the function space that displays the maximum variance. Traditional LDA feature selection is obtained by maximizing class differences and minimizing class distance.

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Mutual influence of the faces contact area and the pre-fracture zone near the tip of the interfacial crack

Strength Fracture and Complexity ◽

10.3233/sfc-204003 ◽

2020 ◽

Vol 13 (3) ◽

pp. 143-161

Author(s):

M.V. Dudyk

Keyword(s):

Contact Area ◽

Fracture Zone ◽

Mutual Influence ◽

Interfacial Crack ◽

Normal Displacement ◽

Crack Model ◽

Hopf Equation ◽

Resistant Material ◽

Order Of Magnitude ◽

Crack Faces

BACKGROUND: Under plane strain conditions, a crack model was developed on a plane interface between two different materials, which assumes the existence near its tip of the faces contact area and a narrow lateral pre-fracture zone in a less crack-resistant material of the composite compound. The pre-fracture zone is modeled by the line of normal displacement rupture, on which the normal stress is equal to the tensile strength of the material. Assuming that the dimensions of the pre-fracture zone and the contact zone have the same order of magnitude and are significantly smaller than the crack length, the problem is reduced to the vector Wiener–Hopf equation. METHODS: An approximate method for solving the vector Wiener–Hopf equation was developed, which was used to obtain the equations for determining the sizes of the pre-fracture zone and the contact faces area. The pre-fracture zone orientation was determined from the condition of the potential energy maximum accumulated in the zone. Numerical calculations of the indicated parameters and analysis of their dependences on the configuration and module of external load are executed. RESULTS: A significant mutual influence of the pre-fracture zone and crack faces contact on their sizes and orientation of the zone was revealed.

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Bounds on the Lattice Point Enumerator via Slices and Projections

Discrete & Computational Geometry ◽

10.1007/s00454-021-00310-7 ◽

2021 ◽

Author(s):

Ansgar Freyer ◽

Martin Henk

Keyword(s):

Convex Body ◽

Lattice Point ◽

Geometric Mean ◽

Discrete Analogue ◽

Lattice Points ◽

General Context ◽

General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

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