Unlimited Order Research Articles

Nonlinear thermodynamic phase field theory with application to fracture and dynamic inelastic phenomena in ceramic polycrystals

A geometrically nonlinear phase field theory accounting for dissipation, rate effects, nonlinear thermoelasticity, fracture, and other structural changes is constructed in the context of continuum thermodynamics. First, a general framework accommodating arbitrary strain energy potentials, inelastic deformation kinematics, and unlimited order parameters is formulated. Next, the framework is specialized to account for deformation physics pertinent to crystalline ceramics and minerals deformed at high rates and high pressures. Notably, a logarithmic elastic strain tensor referred to an intermediate material configuration enters the nonlinear thermoelastic potential. Order parameters represent fractures, solid–solid phase transformations, deformation twinning, or slip of partial dislocations. Thermodynamically consistent kinetics manifest in equations reminiscent of Ginzburg–Landau dynamics, wherein viscosity coefficients are most generally state- and rate-dependent. Pressure-dependent strength commensurate with frictional resistance is enabled in alternative kinetic equations for dynamic fracture with irreversibility constraints. Linearization of the model suitable for moderate volume changes but small deviatoric elastic strain and rotation is undertaken. The theory is applied to study deformation and failure of polycrystalline forms of boron carbide (B4C), titanium diboride (TiB2), and a B4C-TiB2 ceramic composite. Solutions are derived and evaluated numerically for uniaxial stress tension and compression, uniaxial strain compression, and planar shock compression. The latter analysis yields relationships among viscosity coefficients, gradient regularization lengths, and characteristics of steady plastic waveforms. Results give new insight into high-rate deformation mechanisms previously speculated in these materials.

The development of the probability theory of autogenous grinding ore

The aim is to develop a general theory of autogenous grinding, which takes into account the impact of large pieces as the bodies destroying themselves, as well as grinding and grinded bodies. The most difficult process of mechanical destruction of substance – the process of autogenous grinding ore materials – is considered in the framework of probability theory, developed by authors. This theory includes probabilistic exposure of concentration, steric, activation and frequency factors and the likeness of the kinetics of consecutive irreversible reactions of unlimited order exact solution of the system of differential equations for the rate of destruction and accumulation of each fraction. Created a general mathematical model, which takes into account the impact of large pieces as autodisable, grinded and grinding bodies. Calculations based on the developed mathematical model theoretically substantiate, known from practice, formation of “pieces of critical size” which is not able to serve either as grinding or as grinded bodies and to be withdrawn from the process. Thanks to the “relay” transfer of functions of the grinding bodies from the larger fractions to the forming as a result of their self-destruction and the destruction is formed a more uniform distribution of fractions, characteristic for these mills. Therefore, probabilistic model of autogenous grinding can use to analyze, forecast and manage of the process.

HP a-priori error estimates for a non-dissipative spectral discontinuous Galerkin method to solve the Maxwell equations in the time domain

. In this paper, we present the hp-convergence analysis of a non-dissipative high-order discontinuous Galerkin method on unstructured hexahedral meshes using a mass-lumping technique to solve the time-dependent Maxwell equations. In particular, we underline the spectral convergence of the method (in the sense that when the solutions and the data are very smooth, the discretization is of unlimited order). Moreover, we see that the choice of a non-standard approximate space (for a discontinuous formulation) with the absence of dissipation can imply a loss of spatial convergence. Finally we present a numerical result which seems to confirm this property.

Commerce on the Stock and Commodity Exchanges [Die Börsenverkehr

A stock or commodity exchange is a market in which the purchase and sale of large quantities of goods and money, of stocks and commercial bills of exchange, take place between professional businessmen. Those who wish to buy or sell the stocks or types of commodities available on the exchange for example, farmers who wish to sell grain, or large millers who wish to purchase it; capitalists who wish to purchase stocks as a financial investment; and others who wish to sell securities because they need cash are most readily assured, at any given time, of finding sellers or buyers on this tremendous market, and at a price most favorable to themselves. In order to do so, they give their order to an agent on commission [Kommissiondr] who is represented on the exchange. The person placing the order thereby either designates the price at the upper limit at which they are prepared to buy or sell (they "limit" it for example, to buy 10,000 rubles at a price not over, but naturally if possible under, 210 marks for each 100 rubles); or, if they really wish to buy or to sell in any case, they place an "unlimited order" (for example, to buy 10,000 rubles at the "best price," that is at the cheapest-possible price which the agent can attain on the market). How many commodities or securities of a specific kind are offered to, or sought by, the bearer of such an order at any given time, and what sort of price is offered or demanded for them, naturally depend upon a diverse range of unanticipatible causes that influence the motivations of the participants to buy or to sell. It is something that is impossible to outline here in detail and that we have therefore only attempted to sketch quickly.

COSY INFINITY version 8

The latest version of the particle optics code COSY INFINITY is presented. Using Differential Algebraic (DA) methods, the code allows the computation of aberrations of arbitrary field arrangements to in principle unlimited order. Besides providing a general overview of the code, several recent techniques developed for specific applications are highlighted. These include new features for the direct utilization of detailed measured fields as well as rigorous treatment of remainder bounds.

Structural adaptation of parsimonious higher-order neural classifiers

We exploit the potential of parsimonious higher-order neural classifiers for reduction of hardware expenses, speedup in learning, and robust generalization. Specifically, our neuron model allows for computation of input products of potentially unlimited order. Structural adaptation of the topology is achieved by two alternative algorithms, that ultimately allocate resources for the relevant nonlinear interactions only. At the same time, the problem of combinatorial explosion of higher-order terms is kept in check. The first algorithm, being a deterministic pruning variant, starts with the ultimate higher-order neuron, and performs an iterated process of weight elimination. The second algorithm, implementing a stochastic search, explores the space of sparse topologies. It starts with a randomly allocated set of higher-order terms, and modifies resource allocation, while keeping the size of the architecture fixed. Two challenging classification benchmarks were chosen to demonstrate the excellent performance of the presented approach: first, the two-spirals separation problem, and second the left-/right-shift classification problem for binary strings. Our simulation results show that the proposed model may be a powerful tool for a variety of hard classification problems.

Macromolecular configurations simulated by random walks with limited orders of non-self-intersections

Computer computations have been carried out for various orders of self-avoiding random walks used to simulate flexible macromolecules. (The order of a walk is defined as the number of steps in the longest loop excluded by reason of the self-avoiding feature.) Calculations have been carried out for walks of orders up through 10 on a simple square lattice in two dimensions and for walks of orders up through 12 on a tetrahedral lattice in three dimensions. The calculations established precisely the parameters associated with the asymptotic behavior of long chains subject to the model assumed. The results are in agreement with earlier theoretical predictions as to form, and they serve to identify the factors contributing to increasing chain dimensions with length of walk. Asymptotically, the mean square end-to-end separation is linear in the number of steps for any self-avoiding walk of finite order. The behavior of completely self-avoiding walks (i.e., those of unlimited order) is represented very closely by the envelope of asymptotes corresponding to walks of different orders. In addition, the largest symmetric and antisymmetric eigenvalues associated with the transition matrix governing unit increments in contour chain length have been evaluated. Those eigenvalues determine an upper limit to the asymptotic behavior of long model chains.

New Scheme for the Construction of Phase Shifts with Application to Nuclear Magnetic Resonance

A semiempirical scheme is developed for the construction of phase shifts of unlimited order for the investigation of properties depending on the electron redistribution in alloys. The method utilizes a generalized free-electron potential around the solute atom, residual-resistivity data, and the Friedel sum rule. Application to the Knight shifts in a series of dilute alloys with liquid copper and solid silver as hosts leads to much better agreement with experiment than obtained previously.

On Subvariants, i.e. Semi-Invariants to Binary Quantics of an Unlimited Order

On Subvariants, i.e. Semi-Invariants to Binary Quantics of an Unlimited Order