Jordan Type Research Articles

A note on Radon-Nikodým derivatives and similarity for completely bounded maps

We point out a relation between the Arveson's Radon-Nikodým derivative and known similarity results for completely bounded maps. We also consider Jordan type decompositions coming out from Wittstock's Decomposition Theorem and illustrate, by an example, the nonuniqueness of these decompositions.

Quasi-Jordan Algebras

In this article we introduce a new algebraic structure of Jordan type and we show several examples. This new structure, called “quasi-Jordan algebras,” appears in the study of the product where x, y are elements in a dialgebra (D, ⊣, ⊢). The quasi-Jordan algebras are a generalization of Jordan algebras where the commutative law is changed by a quasi-commutative identity and a special form of the Jordan identity is retained. We show a few results about the relationship between Jordan algebras and quasi-Jordan algebras. Also, we compare quasi-Jordan algebras with some structures. In particular, we find a special relation with Leibniz algebras. We attach a quasi-Jordan algebra to any ad-nilpotent element of index of nilpotence at most 3 in a Leibniz algebra.

Weakly tight functions, their Jordan type decomposition and total variation in effect algebras

In the present paper, we have studied envelopes of a function m defined on a subfamily E (containing 0 and 1) of an effect algebra L. The notion of a weakly tight function is introduced and its relation to tight functions is investigated; examples and counterexamples are constructed for illustration. A Jordan type decomposition theorem for a locally bounded real-valued weakly tight function m defined on E is established. The notions of total variation | m | on the subfamily E and m-atoms on a sub-effect algebra E (along with a few examples of m-atoms for null-additive as well as non null-additive functions) are introduced and studied. Finally, it is proved for a real-valued additive function m on a sub-effect algebra E that, m is non-atomic if and only if its total variation | m | is non-atomic.

Pseudo-atoms, atoms and a Jordan type decomposition in effect algebras

The aim of this paper is to introduce and investigate the concept of pseudo-atoms of a real-valued function m defined on an effect algebra L; a few examples of pseudo-atoms and atoms are given in the context of null-additive, null–null-additive and pseudo-null-additive functions and also, some fundamental results for pseudo-atoms under the assumption of null–null-additivity are established. The notions of total variation | m | , positive variation m + and negative variation m − of a real-valued function m on L are studied elaborately and it is proved for a modular measure m (which is of bounded total variation) defined on a D-lattice L that, m is pseudo-atomic (or atomic) if and only if its total variation | m | is pseudo-atomic (or atomic). Finally, a Jordan type decomposition theorem for an extended real-valued function m of bounded total variation defined on an effect algebra L is proved and some properties on decomposed parts of m such as continuity from below, pseudo-atomicity (or atomicity) and being measure, are discussed. A characterization for the function m to be of bounded total variation is established here and used in proving above-mentioned Jordan type decomposition theorem.

Some refined families of Jordan-type inequalities and their applications

In this paper, the classical Jordan's inequality is studied once again, a new sharpened and generalized version of Jordan's inequality is given by means of polynomial representation, and the result thus derived is then used to obtain several substantially more refined inequalities of Jordan type. Finally, an application of the results presented in this paper toward the improvement of the Yang Le inequality is considered.

Modules of constant Jordan type

We introduce the class of modules of constant Jordan type for a finite group scheme G over a field k of characteristic p > 0. This class is closed under taking direct sums, tensor products, duals, Heller shifts and direct summands, and includes endotrivial modules. It contains all modules in an Auslander-Reiten component which has at least one module in the class. Highly non-trivial examples are constructed using cohomological techniques. We offer conjectures suggesting that there are strong conditions on a partition to be the Jordan type associated to a module of constant Jordan type.

Jordan-type inequalities for differentiable functions and their applications

In this work, we extend Jordan’s inequality to obtain a new type of inequality involving functions and their higher-order derivatives. The result is then used to obtain some higher accurate inequalities of Jordan type.

A further refinement of a Jordan type inequality and its application

By introducing Taylor polynomials, a new sharpened and generalized version of Jordan’s inequality is established. The result is then used to obtain a substantially more refined inequality of Jordan type. Moreover, an application of the results presented here toward the improvement of the Yang Le inequality is also considered in this paper.

Generic and maximal Jordan types

For a finite group scheme G over a field k of characteristic p>0, we associate new invariants to a finite dimensional kG-module M. Namely, for each generic point of the projectivized cohomological variety $\text{Proj}\mathop{H}^{\bullet}(G,k)$ we exhibit a “generic Jordan type” of M. In the very special case in which G=E is an elementary abelian p-group, our construction specializes to the non-trivial observation that the Jordan type obtained by restricting M via a generic cyclic shifted subgroup does not depend upon a choice of generators for E. Furthermore, we construct the non-maximal support variety Γ(G) M , a closed subset of $\text{Proj}\mathop{H}^{\bullet}(G,k)$ which is proper even when the dimension of M is not divisible by p.

The convergence of partial sums of interpolating polynomials

For functions of ΛBV, we study the convergence of the partial sums of interpolating polynomials. An estimate is found for the Fourier–Lagrange coefficients of these functions. For functions in BV, convergence is shown at points of discontinuity if the order of the polynomial increases sufficiently rapidly compared to the order of the partial sum. A Dirichlet–Jordan type theorem is shown for functions of harmonic bounded variation, and this result is shown to be best possible.

A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality

A new generalized and sharp version of Jordan’s inequality is proved and it is applied in the improvement of the Yang Le inequality. Moreover, a mistake in the proof of sharpening Jordan’s inequality due to Zhu [S.H. Wu, On generalizations and refinements of Jordan type inequality, Octogon Math. Mag. 12(1) (2004) 267–272] is corrected.

Comparisons between the various types of neural networks with the data of wide range operational conditions of the Borssele NPP

This paper addresses a trend monitoring in operating nuclear power plant by use of two types of Recurrent Neural Networks (RNN). The interesting feature of the RNN is intrinsic dynamic memory that reflects the current output as well as the previous inputs and outputs are gradually quenched. The first one Elman type of RNN which has a feed-back from hidden layer to the input layer neurons while in the Jordan type, from the outputs of the neural net to the inputs of the neural net. In this paper the theoretical assessment of the both RNNs is given. Both topological structures including Back Propagation (BP) neural network were implemented to the Borssele NPP. Learning achieved from 30% to 100% nominal power at the starting period of the new core 30 September 2001. After learning period the reactor operation is followed by the neural network. Paper will present the reactor system, the real time data collection and the merits of the three types of the neural network applied while in the learning and continuous processing of the changing of the operational conditions.

Functions with bounded E-d-variation on undirected tree networks

In this paper we define and study functions with bounded \(E-d\)-variation on undirected tree networks. For these functions with bounded \(E-d\)-variation we establish a Jordan type theorem. We adopt the definition of network as metric space introduced by P. M. Dearing and R. L. Francis (1974).

Bases for Coordinate Rings of Conjugacy Classes of Nilpotent Matrices

A general conjecture is given for an explicit basis of the coordinate ring of the closure of the conjugacy class of a nilpotent matrix. This conjecture is proven when the partition given by the transpose Jordan type of the nilpotent matrix is a hook or has two parts.

Jordan block structure of two-dimensional quantum neural cellular network

We present a Jordan type spectral decomposition for the Hamiltonian operator of a two-dimensional (2D) Ising model driven by a local external field. The results reveal that the external field induces a dynamic contribution to the Jordan block structure, while the interaction between lattice spins contributes to the energy spectrum, tending to attract the system to a minimum energy state. Both interactions contributing to the principle vectors of the Hamiltonian operator reflect their correlation. Since we adopt a first-principle approach the results are useful to general 2D Ising spin systems subject to time-independent external fields irrespective of whether the system is in equilibrium.

The diffuse component of the global solar radiation at Qena / Egypt

Diffuse solar radiation on horizontal surfaces is estimated at Qena / Egypt. The basic procedure is to develop relationships of the widespread use Liu & Jordan types between the daily global horizontal radiation (G) and its diffuse component (D) using measured values of these two quantities. An error analysis has been done for the results of diffuse radiation calculated using the regression models obtained in this paper and those estimated from other known ones of the Liu & Jordan type, According to statistical evaluation of the various relationships, it is seen that our models provide the best estimation of the diffuse radiation, Effect of climatic conditions was considered in the discussion.

Atmospheric turbidity and the diffuse irradiance in Lagos, Nigeria

Values of the Angstrom turbidity coefficient, β, at 0.50μm wavelength have been determined in Lagos, a tropical coastal city in Nigeria, for 17months between 1990 and 1991 using a Volz sun photometer. The results obtained indicate high variability of aerosol loading, though the city, on average, experiences high turbidity for most parts of the year. An annual average of 0.299 with a standard deviation of 0.187 was found. On average, the month of January experienced the highest turbidity, with a mean value of 0.497, while November experienced the lowest aerosol loading on average with a value of 0.225. The aerosols influencing the city are both of maritime and Saharan origin, coupled with locally produced particulates from industries and automobile combustion. Possible effects of the Mount Pinatubo haze may be the cause of the increase of turbidity values in the second half of 1991 when compared with values for the same period in 1990. Addition of β in the Liu and Jordan type diffuse fraction correlation indicates little improvement in standard error.

Relaxation phenomena in dynamic magnetization processes of electrical iron sheets

A relaxation phenomenon consisting of at least three different stages has been found to exist in the dynamic magnetization processes of electrical iron sheets. Each stage is considered to originate from different kinds of relaxation processes: the eddy-current decay, the Richter type aftereffect and the Jordan type aftereffect.

Eigensets and power products of a bimatroid

We consider a bimatroid (linking system) which has a natural one-to-one corre-spondence between the row set and the column set. Such bimatroid can be regarded as a combinatorial counterpart of a matrix which is subject to similarity transformations. For such a bimatroid we can consider the power product, introduce the notion of an “eigenset,” which is a combinatorial analogue of an eigenvector, and define the “Jordan type;” in particular, the maximum size of an eigenset is characterized by the rank of the power products.

A Quantitative Dirichlet–Jordan Type Theorem for Orthogonal Polynomial Expansions

In 1985, Bojanic estimated the rate at which the partial sums of the orthogonal expansion of a function of bounded variation converge to the function. We generalize and sharpen his estimates when the function being expanded is an iterated integral of a function of bounded variation. The class of orthogonal polynomials considered include the Jacobi polynomials orthogonal on $[ - 1,1]$ with respect to $(1 - x)^\alpha (1 + x)^\beta $, $\alpha ,\beta \geqq - \frac{1}{2}$. For the Jacobi series, our method gives an asymptotic expression for the difference between the function and its partial sums at points of discontinuity of a high order derivative of the function. In particular, discontinuities in an even order derivative are shown not to affect the rate of convergence.