486 research outputs found
Computer Simplification of Engineering Systems Formulas
No full textCurrently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA (the 3 M’s). These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Gröbner Basis Algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations.
On the other hand, most of the computation involved in linear control theory is performed on matrices and these do not commute. A typical issue of IEEE TAC is full of A B C D type linear systems and computations with the A B C D’s or partitions of them into block matrices. The 3 M’s are weak in the area of non-commutative operations. They allow a user to declare an operation to be non-commutative, but provide very few commands for manipulating such operations and no powerful algorithmic tools.
It is the purpose of this article to report on applications of a powerful tool: a non-commutative version of the Gröbner Basis Algorithm. The commutative version of this algorithm is implemented on each of the three M’s. It has many applications ranging from solving systems of equations to computations involving polynomial ideals. The non-commutative version is relatively new [Mora]. Our application to the simplification of expressions which occur in systems theory is unique. We will describe the Gröbner Basis for several elementary situations which arise in systems theory. These give (in a sense to be made precise) a “complete” set of simplifying rules for formulas which arise in these situations. We have found that this process elucidates the nature of simplifying rules and provides a practical means of simplifying some types of complex expressions.
The research required the use of software suited for computing with non-commuting symbolic expressions. Most of the research was performed using a special purpose system developed for the project by J. Wavrik. This system uses a new approach to the development of mathematical software. It provides the flexibility needed for experimentation with algorithms, data representation, and data analysis.
In another direction, Helton, Miller and Stankus have written packages for Mathematica called NCAlgebra which extend many of Mathematica’s commands to symbolic expressions in non-commutative algebras. We have incorporated in these packages some of the results on simplification described in this paper
Assessment of Dynamic Spectrum Allocation in Realistic Mobile Networks
No full textSince the advent of radio communications, radio spectrum has been increasingly getting crowded with different kinds of applications. Different radio communications systems have been developed for various purposes and multiple actors became interested in using these systems at the same time and space. Such situation inevitably led to the point when practically entire usable radio spectrum became occupied by different actors. To alleviate this problem, Dynamic Spectrum (Re)Allocation (DSA) has been proposed, which is a branch of frequency spectrum management that aims to improve spectrum usage efficiency and end-user experience by introducing more flexibility to spectrum usage. This thesis aims to provide additional insight into DSA applicability and effectiveness in a typical realistic cellular network, in intra-operator scenario, taking into account 2G and 4G radio technologies, with the aim of improving 4G performance without adverse impact to 2G. We use realistic dynamic system level simulations to assess DSA performance in the selected cellular network areas that can be classified as urban, suburban and rural. Our simulation results show that DSA is capable of improving 4G throughput without adverse impact to 2G performance in all simulated areas. Among the simulated areas, urban area benefits from DSA most, as significant throughput gains for 4G are achieved without adverse impact to 2G performance, while simulations show that spectrum refarming is clearly not an option for this type of area. However, throughput gains for 4G in urban area are limited during the busy-hours. Suburban and rural areas indicate benefits from DSA too, however the difference between DSA and spectrum refarming in these areas is diminishing. Hence, with reasonable half-rate timeslot tolerance for 2G voice calls, spectrum refarming could be an option in the simulated suburban and rural areas.Electrical EngineeringNetwork Architectures and ServicesElectrical Engineering, Mathematics and Computer Scienc
Isosymmetric Linear Transformations on Complex Hilbert Space
No full textWe explore the elementary operator theory of the equation (0.1) ∑ cm,nT*nTm=0 for cm,n E C, cm,n nonzero for only finitely many m, n and Ta bounded linear transformation on a complex Hilbert space in Chapters 1 and 2. We explore the equation
(0.2) T*2T - T*T2 + T - T* = 0 in greater depth in Chapters 4 and 5.
Chapter 1 explores the algebraic and C*-algebraic aspects of the equation (0.1) and both the spectral picture of and growth conditions on the resolvent of the operator T satisfying (0.1).
Chapter 2 explores the implications of Rosenblum\u27s Theorem to the study of (0.1). These implications are sufficient in some cases to completely classify a solution to (0.1) given information about the spectrum of T. Chapter 2 also recalls a few definitions and results from the theory of von Neumann algebras which will be used in the rest of the paper.
Chapter 3 guarantees the existence of maximal invariant subspaces M for an operator T such that T restricted to M is a member of a fixed family of operators. This provides an approach to completely solving the equation (0.1) for T for certain choices of c.
In Chapters 4 and 5, we study operators T satisfying (0.2). These operators are termed isosymmetries. The results of Chapters 1, 2 and 3 do not solve equation (0.2).
Chapter 4 gives the elementary operator theory of isosymmetries.
Chapter 5 classifies several collections of isosymmetries. Indeed, if T is an isosymmetry and T is hyponormal, T is a contraction, Im(T) ≥ 0 or Im(T) ≤ 0, then T is subnormal and the minimal normal extension of T has the same properties. If T*T ≥ 1, then T is the restriction to an invariant subspace of a direct integral of rank one perturbations of the unilateral shift. If the spectrum of T equals its boundry, then T has the form of a direct integral of 1 x 1 and 2 x 2 matrices. These constraints arise naturally from the analysis of Chapter 4
Land forces soldier’s systems development (needs and opportunities).
No full textNowadays. the topic of the thesis is relevant, because it concerns the fast developing and one of the most important projects in Lithuanian armed forces called the „Modern Soldier“. The focus of the project is towards the individual equipment and weaponry. Also, in this thesis, the author will concentrate on the individual equipment of Lithuanian armed forces soldier. Lithuanian armed forces soldier equipment was not analysed by Lithuanian authors. Due to that, it is relevant to assess the situation and the perspectives of the new Lithuanian soldier’s equipment in the context of other foreign countries. Methodology – author has chosen the empiric qualitative research, because it helps to understand the problem of the thesis and it is the most comprehensive. In addition, in the theoretical part of the thesis, a comparison between the different countries and their soldiers’ equipment was made. The evaluation of the results showed that the worst logistic situation is in the Lithuanian Military Academy and Lithuanian Volunteer Forces. The suggestion would be that the elements of the equipment should not be bought separately, but rather together as a whole. Due to that, it would allow to avoid the problem of incompatibility. All systems and elements should be modern and updated. Hypothesis: Lithuanian armed forces soldier’s equipment fits the requirements of other foreign countries through complex elements and used technologies
Leasing Agreement Regulation in Commercial Law
No full textBachelor’s paper deals with regulation of leasing agreement in Latvia’s Commercial law. Objective of study is to consider questions concerning basic principles of leasing in general and to raise understanding of current leasing regulation in Latvia. Paper also gives review of Worlds developed countries experience in leasing regulation and their legislation features in these aspects. It is important to pay attention to new concepts of legal leasing regulation, to understand the further evolution of leasing and to propose useful suggestions to improve current situation.
Summing up the studies verities and coming to conclusions it could be recognized that Latvia’s strategy in building legal framework for leasing is cautious without drastic changes and novelty based on sound practice. This will help us step by step introducing needed changes to get for the best practice in legal leasing regulation.
Author gives some proposals to law makers concerning Commercial law, to finalize obtained cognitions. Given proposals basically supplement already proposed draft law with additional party’s rights and duties and suggests expanding conditions for secondary leases
Computer Assistance In Discovering Formulas And Theorems In System Engineering
No full textIf one reads a typical article on A,B,C,D systems in the control transactions, one finds that most of the algebra involved is non commutative rather than commutative. Thus, for symbolic computing to have much impact on linear systems research, one needs a program which will do non-commuting operations. Mathematica, Macsyma and Maple do not. We have a package, NCAlgebra, which runs under Mathematica which does the basic operations, block matrix manipulations and other things. The package might be seen as a competitor to a yellow pad. Like Mathematica the emphasis is on interaction with the program and flexibility.
The issue now is what types of “intelligence” to put in the package. [HSW] (CDC94) focused on procedures for simplifying complicated expressions automatically. In this talk we turn to a much more adventurous pursuit which is in a primitive stage. This is a highly computer assisted method for discovering certain types of theorems.
At the beginning of “discovering” a theorem, an engineering problem is often presented as a large system of matrix equations. The point is to isolate and to minimize what the user must do by running heavy algorithms. Often when viewing the output of the algorithm, one can see what additional hypothesis should be added to produce a useful theorem and what the relevant matrix quantities are.
Rather than use the word “algorithm”, we call our method a strategy since it allows for modest human intervention. We are under the impression that many theorems in engineering systems might be derivable in this way
Classification of Hereditary Matrices
No full textA classical approach used to obtain basic facts in the theory of square matrices involves an analysis of the relationship between polynomials p in one variable and square matrices T such that p(T) = 0. We consider matrices and operators which satisfy a different type of polynomial constraint. Let H be a complex Hilbert space, T be a bounded linear transformation of H, T* be the adjoint of T, and C[x, y] be the algebra of polynomials in x and y with complex coefficients. For a polynomial p E C[x, y] in two variables with complex coefficients, define p(T) = Σm, n ≥, 0 p ^(m, n)T* nTm, where p ^ (m, n) is the coefficient of ynxm in the expansion of p in a power series about the point (0, 0). T is called a root of p if and only if p(T) = 0. Note that if p E C[x, y] is a polynomial in the single variable x, then the definition of p(T) given here agrees with the classical definition. In this paper, we study the relationships which p(T) = 0 forces between p and T when T is an algebraic operator (i.e., there exists n ≥ 1 and complex numbers a0, …, an − 1 such that 0 = a0 + a1T + … + an − 1Tn − 1 + Tn). The classification starts with the following observation: Suppose p E C[x, y] and an algebraic operator T E L(H) satisfy p(T) = 0. Then certain subspaces of H which are invariant for T must be orthogonal or certain coefficients of p must vanish. This leads to the notions of a graph attached to each p E C[x, y] and a graph attached to each square matrix T. For diagonalizable T, a necessary and sufficient graph theoretic condition for solving p(T) = 0 is given. For nondiagonalizable T, this condition is necessary, but not sufficient. The use of these graphs does, however, reduce the problem to the problem of solving the equation p(T) = 0 for T with exactly one or two eigenvalues. For T with one eigenvalue, we give a necessary and sufficient condition for solving p(T) = 0. This leaves the case of solving p(T) = 0 when T has exactly two eigenvalues. This problem mixes algebra involving polynomials with matrix theory. We show that it is equivalent to the purely algebraic problem of determining if equations of the form Σ(i, j) E ECi, jXi + r, j + s = 0 have solutions of finite support with certain nonvanishing properties. We call these equations bi-Hankel equations subordinate to a given subset E of the lattice of integer pairs {(i, j) : 0 ≤ i ≤ m − 1, 0 ≤ j ≤ n − 1}. It turns out that there is an algorithm (which uses Gröbner bases) for determining if the type of solution we seek exists and for computing it
Computer Assistance In Discovering Formulas And Theorems In System Engineering II
No full text[HSWcdc94] focused on procedures for simplifying complicated expressions automatically. [HScdc95] turned to the adventurous pursuit of developing a highly computer assisted method for “discovering” certain types of formulas and theorems.
It is often the case that some variables in the formulation of a problem are not the natural “coordinates” for solution of the problem. Gröbner Basis Algorithms, which lie at the core of our method, are very good at eliminating unknowns, but have no way of finding good changes of variables. This paper gives a way of incorporating changes of variables into our method.
As an example, we “discover” the DGKF equations of H∞ control
Toral Algebraic Sets and Function Theory on Polydisks
No full textA toral algebraic set A is an algebraic set in Cn whose intersection with Tn is sufficiently large to determine the holomorphic functions on A. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set
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