Euclidean combinatorial optimization problems are considered as discrete optimization problems on a set of combinatorial configurations mapped into an arithmetic Euclidean space. Modern methods of ...Euclidean combinatorial optimization are reviewed. The properties of the corresponding images of combinatorial sets are described. A theory of continuous functional representations and convex extensions is proposed for solving this class of problems. Areas of practical application and promising research areas are indicated.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
A class of problems of vector Euclidean combinatorial optimization is considered as problems of discrete optimization on the set of combinatorial configurations mapped into the Euclidean space. The ...properties of the graphs of combinatorial configurations used to describe the new method are given. A two-stage method for solving problems of vector Euclidean combinatorial optimization on combinatorial configurations of permutations is proposed. The results of the numerical experiment and their analysis are presented.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We discuss algebraic similarity of the Wilson’s renormalization groups in the Euclidean and
p
-adic spaces. Automodel Hamiltonians have identical form in both cases in the framework of perturbation ...theory. Fermionic
p
-adic model has exact renormalization group solution which generates a list of non-trivial conjectures for the Euclidean case.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
This series of papers is devoted to the investigation ot the extent to which the accuracy oi operation of multidimensional scaling can be put onto a quantitative footing. In thls third and final ...paper, four probabilistic models for the generation of euclidean-distance-like dissimilarity function are proposed; these models reflect some of the ways in which dissimilarities actually arise. and allow such effects as dependence between dissimilarities to be studied. Using these models. simulation experiments are carried out to assess the response of both classical and ordinal (non-metric) scaling to errors, with procrustes statistics being used to measure accuracy of recovery. A further scaling method. leait bquarer scaling. is discussed briefly and shown to display empirically a useful combination of properties. as is a technique used to preprocess the dissimilarity matrix.
•This paper introduces the conformal model of the 3D space based solely on elementary linear algebra.•No knowledge of Clifford algebra is required.•Matrix representations of isometries of the 3D ...space are presented
Motivated by questions on orthogonality of isometries, we present a new construction of the conformal model of the 3D space using just elementary linear algebra. In addition to pictures that can help the readers to understand the conformal model, our approach allows to obtain matrix representation of isometries that can be useful, for example, in applications of computational geometry, including computer graphics, robotics, and molecular geometry.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
•The INDSCAL and MFA gave very similar results for two components.•The MFA was slightly better than INDSCAL for estimating the consensus configuration for more than 2 dimensions.•Using singular ...vectors can give a more relevant measure of similarity of configurations.•It is important to consider more than two consensus components.
In this paper a general framework is proposed for understanding and analysing more than two consensus components in projective mapping (also known as Napping®) studies. Focus is on how two models, multiple factor analysis (MFA) and individual differences scaling (INDSCAL) based on the weighted Euclidean model (WEM), relate to each other and to the general framework. The stability of the consensus configurations of both methods are compared. The relations between the results of the two methods are investigated using the RV coefficient and an alternative index called SMI which gives equal weight to the axes regardless of the relative size of the singular values. The methods are tested and compared using three datasets and simulations.
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•Violation of the Saint–Venant compatibility condition is considered.•Equation for the Airy stress function in the non-Euclidean model is obtained.•The Airy stress function is calculated through the ...incompatibility function.•Residual stress are determined in the framework of the non-Euclidean model.
We introduced the Airy stress function in the non-Euclidean model of a continuous medium, for which the Saint-Venant compatibility condition for deformations is not satisfied. For this function, an inhomogeneous biharmonic equation is obtained whose right side is determined by the incompatibility of deformations. The internal stresses was shown to be composed of elastic stresses and stresses determined by the incompatibility of deformations. The theoretical results obtained are used to analyse experimental data on the study of residual stresses.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The paper proposes a non-Euclidean approach based on physical mesomechanics and a scale classification for modeling the hierarchical block structure of the Erath’s subsurface as a defects-containing ...continuum whose main element is an opening mode crack initiated by shear under multiaxial compression. It is shown that such shear-induced opening mode fracture in the subsurface block structure results in incompatibility between its elements, which favors the non-Euclidean description of the block structure on different scales. The efficiency of the approach is demonstrated on the example of mesostructures of the first two scales classified.
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•Strain energy density method can be used to investigate the zonal disintegration.•The critical condition of zonal disintegration in deep rock mass is defined.•Stress-fields have wave feature when ...external load is more than critical value.•Rock mass displays elastic behavior when external load is less than critical value.
The deformation and failure modes of the deep rock mass are different from those of the shallow rock mass. Zonal disintegration phenomenon occurs in the deep rock mass, while loosened zone, plastic zone and elastic zone appear successively in the shallow rock mass. It is known that it is necessary to adopt the different support system for the different deformation and failure modes of rock mass. Therefore, it is very significant to define the critical condition of zonal disintegration. In this paper, a non-Euclidean model is proposed to investigate the critical condition of zonal disintegration in deep rock mass. A bulk free energy function of deep rock mass is introduced to describe the effects of microcracks. The standard formalism of non-equilibrium thermodynamics is used to obtain an equation for the non-Euclidean parameter. The relationship between non-Euclidean parameter and stress is established. The critical value of stress for zonal disintegration is derived using strain energy density approach. When the external load is less than the critical value of stress, the mechanical behaviors of rock materials can be simulated using the classical elastic model. In other words, when the external load is less than the critical value, zonal disintegration phenomenon does not occur. When the external load exceeds the critical value of stress, the mechanical behaviors of rock mass can be simulated using the non-Euclidean model, and zonal disintegration phenomenon occurs.
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Rock mass has a complex internal structure. Under the action of external loads, incompatible deformation develops in rock mass that makes the concept of elementary volume and the Saint Venant's ...condition of compatible deformation problematic. The incompatibility of the deformation indicates the non-coincidence of the internal and the external metrics and the breaking of the Euclidean structure of the space. Therefore, the use of differential geometry to describe the incompatible deformation is natural. At present, the expression of non-Euclidean models is not concise and compact; the constitutive relation is not complete and the evolution of incompatibility parameters is absent. In this study, with the help of the orthogonal frame method, we used the generalized distortion tensor, torsion tensor, and Riemann tensor to describe the incompatible deformations. The used geometric parameters have clear physical meaning and can be used as thermodynamic variables. By constructing Helmholtz free energy and using irreversible thermodynamics, we obtained the constitutive equations. For completing the constitutive equations, the evolution equations of the used geometric parameters are derived. In this manner, the non-Euclidean description of the incompatible deformation extends the classical models of deformable bodies.
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