Full-factor matrix model of accuracy of dimensions performed on multi-purpose CNC machines

Introduction. One of the main reasons that modern multi-purpose CNC machines do not use the capabilities of multi-tool processing is the lack of recommendations for design in this direction and, accordingly, for adjustment schemes. The study of the possibilities of multi-tool processing on multi-purpose machines is the subject of the work. The purpose of research: The problem of developing full-factor matrix models of dimensional accuracy and its sensitivity to the machining process is considered to increase the machining efficiency while ensuring machining accuracy using the technological capabilities of multi-tool machining on modern multi-purpose CNC machines. For this purpose, full-factor matrix models of the size scattering fields performed on multi-tool double-carriage adjustments have been developed, taking into account the cases of processing parts with dimensions that differ sharply in different directions, which are often encountered in practice, and in this case, the significant influence of the turns of the workpiece on the processing error, especially in directions with sharply different overall dimensions. Results of research: The developed accuracy models make it possible to calculate not only plane-parallel displacements of the technological system for double-carriage adjustments, but also angular displacements around base points, take into account the combined effect of many factors – a complex characteristic of the subsystems of the technological system (plane-parallel matrix of compliance and angular matrix of compliance), the geometry of the cutting tool , the amount of bluntness of the tool, cutting conditions, etc. As a result, based on the developed accuracy models, it is possible to obtain several ways to control multi-tool machining, including improving the structure of multi-tool adjustments, calculating the limiting values of cutting conditions. Based on the developed full-factor matrix models, it became possible to develop recommendations for the design of adjustments and the creation of an automated design system for multi-tool machining for a group of modern multi-purpose CNC lathes. Scope of the results: The results obtained can be used to create mathematical support for the design of operations in CAD-systems provided for multi-tool multi-carriage machining performed on multi-purpose machines. Conclusions: The developed models and methodology for simulating the machining accuracy make it possible to increase the accuracy and efficiency of simultaneous machining, to predict the machining accuracy within the specified conditions.

Emergent fractal phase in energy stratified random models

We study the effects of partial correlations in kinetic hopping terms of long-range disordered random matrix models on their localization properties. We consider a set of models interpolating between fully-localized Richardson’s model and the celebrated Rosenzweig-Porter model (with implemented translation-invariant symmetry). In order to do this, we propose the energy-stratified spectral structure of the hopping term allowing one to decrease the range of correlations gradually. We show both analytically and numerically that any deviation from the completely correlated case leads to the emergent non-ergodic delocalization in the system unlike the predictions of localization of cooperative shielding. In order to describe the models with correlated kinetic terms, we develop the generalization of the Dyson Brownian motion and cavity approaches basing on stochastic matrix process with independent rank-one matrix increments and examine its applicability to the above set of models.

Matrix model partition function by a single constraint

In the recent study of Virasoro action on characters, we discovered that it gets especially simple for peculiar linear combinations of the Virasoro operators: particular harmonics of $${\hat{w}}$$ w ^ -operators. In this letter, we demonstrate that even more is true: a singlew-constraint is sufficient to uniquely specify the partition functions provided one assumes that it is a power series in time-variables. This substitutes the previous specifications in terms of two requirements: either a string equation imposed on the KP/Toda $$\tau $$ τ -function or a pair of Virasoro generators. This mysterious single-entry definition holds for a variety of theories, including Hermitian and complex matrix models, and also matrix models with external matrix: the unitary and cubic Kontsevich models. In these cases, it is equivalent to W-representation and is closely related to super integrability. However, a similar single equation that completely determines the partition function exists also in the case of the generalized Kontsevich model (GKM) with potential of higher degree, when the constraint algebra is a larger W-algebra, and neither W-representation, nor superintegrability are understood well enough.

Matrix Description of Non-Linear Properties of Materials or Structural Components—Idea and Application Examples

Numerical methods are widely used in structural analysis problems. In the cases of the most complex and practical problems, they are often the only way to obtain solutions, as analytical methods prove ineffective. The motivation for this paper was the desire to extend the scope of numerical methods to cover the problems of creating constitutive models of structural materials. The aim of this research was to develop a matrix or numerical discrete constitutive model of materials. It presents the general assumptions of the developed method for modeling the physical properties of materials. The matrix model is only useful with an appropriate numerical algorithm. Such an algorithm was created and described in this paper. Based on its findings, computer software was developed to perform numerical simulations. Presented calculation examples confirmed the effectiveness of the developed method to create constitutive matrix models of various typical materials, such as steel, but also, e.g., hyper-elastic materials. It also presents the usefulness of constitutive matrix models for simulations of simple stress states and analyses of structural elements such as reinforced concrete. All presented examples involved the physical nonlinearity of the materials. It is proved that the developed matrix constitutive model of materials is efficient and quite versatile. In complex analyses of structures made of nonlinear materials, it can be used as an effective alternative to classical constitutive or analytical models based on elementary mathematical functions.

Target space entanglement in Matrix Models

We study target space entanglement in gauged multi-matrix models as models of entanglement between groups of D-branes separated by a planar entangling surface, paying close attention to the implementation of gauge invariance. We open with a review of target space entanglement between identical particles, which shares some important features (specifically a gauged permutation symmetry) with our main problem. For our matrix models, we implement a gauge fixing well-adapted to the entangling surface. In this gauge, we map the matrix model problem to that of entanglement of a U(1) gauge theory on a complete or all-to-all lattice. Matrix elements corresponding to open strings stretching across the entangling surface in the target space lead to interesting contributions to the entanglement entropy.

Applying graph-matrix models in the strategic analysis of production capacity of iron and steel enterprises

Subject. The article addresses the development of a methodology for seeking reserves to increase production capacity through the enhanced balance of production system and bringing the latter to practical use at iron and steel enterprises. Objectives. The purpose is to design a methodology to search for reserves to enhance production capacity of industrial enterprises at the strategic level, using the graph-matrix models of integrated steel works. Methods. The study draws on methods of analysis and synthesis, principles of consistency and complexity, the graph theory, and matrix calculus. Results. The paper offers a method for assessing and analyzing reserves to increase production capacity through enhancing the balance of the production system. Conclusions. If used, the presented approach will help improve the quality of strategic management of production facilities in the iron and steel industry, and, as a result, the profitability and competitiveness of ironworks.