On the Semi-Local Convergence of a Traub-Type Method for Solving Equations

The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results.

On involutive division on monoids

We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.

Anthropocentrism in the Structure and Description of Language

Although the connection between language, meaning and culture has been dealt with on several occasions in Serbianlinguistics, we thought it appropriate to point out several examples of the lexical-grammatical structure of the Serbian language that illustrate the anthropocentric view of relations in the world immediately surrounding man. In order to realise our intention, we opted for verb constructions in which we can see the semantic transition from the original to the secondary domains, where the original domain primarily indicates the state of affairs typical of man or living beings in general.

Application of Floquet Theory to Human Gait Kinematics and Dynamics

In this work, the lower extremity physiological parameters are recorded during normal walking gait, and the dynamical systems theory is applied towards its stability analysis. The human walking gait pattern of kinematic and dynamical data is approximated to periodic behavior. The embedding dimension analysis of the kinematic variable's time trace and use of Taken's theorem allows us to compute a reduced-order time series that retains the essential dynamics. In conjunction with Floquet Theory, this approach can help study the system's stability characteristics. The Lyapunov-Floquet (L-F) Transformation application results in constructing an invariant manifold resembling the form of a simple oscillator system. It is also demonstrated that the simple oscillator system, when re-mapped back to the original domain, reproduces the original system's time evolution (hip angle or knee angle, for example). A re-initialization procedure is suggested that improves the accuracy between the processed data and actual data. The theoretical framework proposed in this work is validated with the experiments using a motion capture system.

Method and Algorithm for Implementation of Decoding Operation in the Threshold Cryptomodule of Secret Separation Using a Minimally Redundant Modular Number System

The article presents a new development of method and algorithm for performing secret separation in a threshold cryptomodule with masking transformation of the decoding operation. To solve this problem a recursive binary exponent division scheme and computational technology on the ranges of large numbers of the table-adder type, based on minimally redundant modular arithmetic (MRMA) are applied. A distinctive feature of the developed approach is usage the secret-original domain of finite residue rings for modules that have the form of powers of the number 2. This significantly reduces the complexity of the resulting decoding MRMA-procedure. Decomposition of scalable residues into large modules allows you to efficiently map the computational process being implemented to sets of easily implemented data extraction operations from table memory and their summation, providing a high level of performance, uniformity, and unification of basic structures. In terms of speed, the created MIMA decoding algorithm surpasses non-redundant analogues by at least l(19l-3)/(22l-6) times (l is the number of subscribers restoring the secret original). When l = 7...40 a (6.15...34.65)-fold increase in productivity is achieved.

POLYSEMY OF RUSSIAN ADJECTIVES IN THE COGNITIVE ASPECT

Russian polysemous adjectives is one of the important issues in the study of the lexical semantics of the Russian language. Its essence lies in the methods of understanding the semantic content of adjectives and establishing internal connections between their meanings. Cognitive metaphor as a key working mechanism can clearly explain the phenomenon of Russian adjectives polysemy. This study uses the theory of cognitive metaphor that contains many working mechanisms affecting the change in the lexical semantics of adjectives. Among them, semantic transfer between cognitive domains is one of the most typical and active mechanisms. The present study is based on two original cognitive domains: the domain of space and the domain of physical perception. Under the cognitive effect of transferring from the original domain to other domains, adjectives acquire different meanings. This research focuses upon Russian polysemous spatial adjectives and adjectives of physical perception...

On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains

The $p$-Laplacian has attracted more and more attention in data analysis disciplines in the past decade. However, there is still a knowledge gap about its behavior, which limits its practical application. In this paper, we are interested in its iterative behavior in domains contained in two-dimensional Euclidean space. Given a connected set $\varOmega _0 \subset \mathbb{R}^2$, define a sequence of sets $(\varOmega _n)_{n=0}^{\infty }$ where $\varOmega _{n+1}$ is the subset of $\varOmega _n$ where the first eigenfunction of the (properly normalized) Neumann $p$-Laplacian $ -\varDelta ^{(p)} \phi = \lambda _1 |\phi |^{p-2} \phi $ is positive (or negative). For $p=1$, this is also referred to as the ratio cut of the domain. We conjecture that these sets converge to the set of rectangles with eccentricity bounded by 2 in the Gromov–Hausdorff distance as long as they have a certain distance to the boundary $\partial \varOmega _0$. We establish some aspects of this conjecture for $p=1$ where we prove that (1) the 1-Laplacian spectral cut of domains sufficiently close to rectangles is a circular arc that is closer to flat than the original domain (leading eventually to quadrilaterals) and (2) quadrilaterals close to a rectangle of aspect ratio $2$ stay close to quadrilaterals and move closer to rectangles in a suitable metric. We also discuss some numerical aspects and pose many open questions.

Representation of Kinetics Models in Batch Flotation as Distributed First-Order Reactions

Four kinetic models are studied as first-order reactions with flotation rate distribution f(k): (i) deterministic nth-order reaction, (ii) second-order with Rectangular f(k), (iii) Rosin–Rammler, and (iv) Fractional kinetics. These models are studied because they are considered as alternatives to the first-order reactions. The first-order representation leads to the same recovery R(t) as in the original domain. The first-order R∞-f(k) are obtained by inspection of the R(t) formulae or by inverse Laplace Transforms. The reaction orders of model (i) are related to the shape parameters of first-order Gamma f(k)s. Higher reaction orders imply rate concentrations at k ≈ 0 in the first-order domain. Model (ii) shows reverse J-shaped first-order f(k)s. Model (iii) under stretched exponentials presents mounded first-order f(k)s, whereas model (iv) with derivative orders lower than 1 shows from reverse J-shaped to mounded first-order f(k)s. Kinetic descriptions that lead to the same R(t) cannot be differentiated between each other. However, the first-order f(k)s can be studied in a comparable domain.

AN OVERLAPPING SCHWARZ METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER CONVECTION-DIFFUSION TYPE

In this paper, we have constructed an iterative numerical method based on an overlapping Schwarz procedure with uniform mesh for singularly perturbed fourth-order of convection-diffusion type. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the method produces numerical approximations which converge in the maximum norm to the exact solution. We prove that, when appropriate subdomains are used the method produces convergence of almost second-order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results.

On an improved convergence analysis of a two-step Gauss-Newton type method under generalized Lipschitz conditions

We present a local convergence analysis of a two-step Gauss-Newton method under the generalized and classical Lipschitz conditions for the first- and second-order derivatives. In contrast to earlier works, we use our new idea using a center average Lipschitz conditions through which, we define a subset of the original domain that also contains the iterates. Then, the remaining average Lipschitz conditions are at least as tight as the corresponding ones in earlier works. This way, we obtain: weaker sufficient convergence criteria, larger radius of convergence, tighter error estimates and more precise information on the location of the solution. These advantages are obtained under the same computational effort, since the new Lipschitz functions are special cases of the ones in earlier works. Finally, we give a numerical example that confirms the theoretical results, and compares favorably to the results from previous works.