Linear Transformation

We prove some results about nilpotent linear transformations. As an application we solve some cases of Albert's problem on the solvability of nilalgebras. More precisely, we prove the following results: commutative power-associative nilalgebras of dimension n and nilindex n − 1 or n − 2 are solvable; commutative power-associative nilalgebras of dimension 7 are solvable.

With the rapid development of network and multimedia technologies, the digital information has been applied to many areas in real-world
applications. Communication has become a very important aspect in today's life. So, security plays an important role in transferring the data.

One such way to secure information is cryptography. In cryptography we hide the information from unauthorized users by employing various techniques, encryption is one such technique where we transform the data
into a form understandable only by the authorized users. We need to hide the data for privacy purpose and for ensuring data received at the
authenticated user end is not modied .We have several encryption and decryption algorithms for encrypting the data at sender end and decrypting
the same at receiver side ensuring secure data transfer .

This paper presents synthetically the most recent models for description of the anisotropic plastic behavior. The first section gives an overview of the classical models. Further, the discussion is focused on the anisotropic formulations developed on the basis of the theories of linear transformations and tensor representations, respectively. Those models are applied to different types of materials: body centered, faced centered and hexagonalclose packed metals. A brief review of the experimental methods used for characterizing and modeling the anisotropic plastic behavior of metallic sheets and tubes under biaxial loading is presented together with the models and methods developed for predicting and establishing the limit strains. The capabilities of some commercial programs specially designed for the computation of forming limit curves (FLC) are also analyzed.

A new dynamic harmonic estimator is presented as an extension of the fast Fourier transform (FFT), which assumes a fluctuating complex envelope at each harmonic. This estimator is able to estimate harmonics that are time varying inside the observation window. The extension receives the name "Taylor-Fourier transform (TFT)" since it is based on the McLaurin series expansion of each complex envelope. Better estimates of the dynamic harmonics are obtained due to the fact that the Fourier subspace is contained in the subspace generated by the Taylor-Fourier basis. The coefficients of the TFT have a physical meaning: they represent instantaneous samples of the first derivatives of the complex envelope, with all of them calculated at once through a linear transform. The Taylor-Fourier estimator can be seen as a bank of maximally flat finite-impulse-response filters, with the frequency response of ideal differentiators about each harmonic frequency. In addition to cleaner harmonic phasor estimates under dynamic conditions, among the new estimates are the instantaneous frequency and first derivatives of each harmonic. Two examples are presented to evaluate the performance of the proposed estimator.

The paper analyses the problem of ranking accuracy in multiple criteria decision-making (MCDM) methods. The methodology for measuring the accuracy of determining the relative significance of alternatives as a function of the criteria values is developed. An algorithm of the Technique for the Order Preference by Similarity to Ideal Solution (TOPSIS) that applies criteria values' transformation through a normalization of vectors and the linear transformation is considered. A computational experiment is presented, to compare the results of a multiple criteria analysis and the ranking accuracy in a particular situation.

In this paper, we investigate the impact of several portfolio selection models based on different tracking error measures, performance measures, and risk measures. In particular, mimicking the theory of ideal probability metrics, we examine ideal financial risk measures in order to solve portfolio choice problems.

A ring R is said to be clean if every element of R is a sum of an idempotent and a unit. The class of clean rings is quite large and includes, for instance, semiperfect rings (and thus finite rings), and rings of linear transformations of vector spaces. We prove that the endomorphism ring of every continuous (or discrete) module is clean. Lemma 2.1. Let S = End(M R ) and f, e ∈ S, where e is an idempotent with A = Ker(e) and B = Im(e). Then f −e ∈ U(S) if and only if there exists an R-module decomposition M = C ⊕D such that f (A) ⊆ C, (1 − f )(B) ⊆ D, and both f : A → C and 1 − f : B → D are isomorphisms. Proof. First suppose that f = e + u for some unit u ∈ S. Put C = uA and D = uB. As f (1 − e) = (e + u)(1 − e) = u(1 − e) and (1 − f )e = (e − f )e = −ue, f : A → C and 1 − f : B → D are isomorphisms.

Novel bond-level molecular descriptors are proposed, based on linear maps similar to the ones defined in algebra theory. The kth edge-adjacency matrix (E k ) denotes the matrix of bond linear indices (non-stochastic) with regard to canonical basis set. The kth stochastic edge-adjacency matrix, ES k , is here proposed as a new molecular representation easily calculated from E k . Then, the kth stochastic bond linear indices are calculated using ES k as operators of linear transformations. In both cases, the bond-type formalism is developed. The kth non-stochastic and stochastic total linear indices are calculated by adding the kth non-stochastic and stochastic bond linear indices, respectively, of all bonds in molecule. First, the new bond-based molecular descriptors (MDs) are tested for suitability, for the QSPRs, by analyzing regressions of novel indices for selected physicochemical properties of octane isomers (first round). General performance of the new descriptors in this QSPR studies is evaluated with regard to the well-known sets of 2D/3D MDs. From the analysis, we can conclude that the non-stochastic and stochastic bond-based linear indices have an overall good modeling capability proving their usefulness in QSPR studies. Later, the novel bond-level MDs are also used for the description and prediction of the boiling point of 28 alkyl-alcohols (second round), and to the modeling of the specific rate constant (log k), partition coefficient (log P), as well as the antibacterial activity of 34 derivatives of 2-furylethylenes (third round). The comparison with other approaches (edge- and vertices-based connectivity indices, total and local spectral moments, and quantum chemical descriptors as well as E-state/biomolecular encounter parameters) exposes a good behavior of our method in this QSPR studies. Finally, the approach described in this study appears to be a very promising structural invariant, useful not only for QSPR studies but also for similarity/diversity analysis and drug discovery protocols.

A coupled theory of continuum damage mechanics and finite strain plasticity (with small elastic strains) is formulated in the Eulerian reference system. The yield function used is of the von Mises type and incorporates both isotropic and kinematic hardening. An explicit matrix representation is derived for the damage effect tensor for a general state of deformation and damage. Although the theory is applicable to anisotropic d.-unage, the matrix representation is restricted to isotropy.