The principles of integrative nursing and caring science align with the unitary paradigm in a way that can inform and shape nursing knowledge, patient care delivery across populations and settings, and new healthcare policy. The proposed policies may transform the healthcare system in a way that supports nursing praxis and honors the discipline’s unitary paradigm. This call to action provides a distinct and hopeful vision of a healthcare system that is accessible, equitable, safe, patient-centered, and affordable. In these challenging times, it is the unitary paradigm and nursing wisdom that offer a clear path forward.

Unitary transformations in an angular momentum Hilbert space H(2j + 1), are considered. They are expressed as a finite sum of the displacement operators (which play the role of SU(2j + 1) generators) with the Weyl function as coefficients. The Chinese remainder theorem is used to factorize large qudits in the Hilbert space H(2j + 1) in terms of smaller qudits in Hilbert spaces H(2ji + 1). All unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits.

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Nous étudions les représentations complexes, induites par l'induction parabolique, du groupe U(5), défini sur un corps local non-archimedean de caractéristique 0. C'est Qp ou une extension finie de Qp .On parle des 'corps p-adiques'. Soit F un corps p-adique. Soit E : F une extension de corps de degré 2. Soit Gal(E : F ) = {id, σ}le groupe de Galois. On écrit σ(x) = overline{x} forall x ∈ E. Soit | |p la norme p-adique de E. Soient E* = E {0} et E 1 = {x ∈ E | xoverline{x}= 1} .U (5) a trois sous-groupes paraboliques propres. Soit P0 le sous-groupe parabolique minimal et soientP1 et P2 les deux sous-groupes paraboliques maximaux. Soient M0 , M1 et M2 les sous-groupes de Levi standards et soient N0 , N1 et N2 des sous-groupes unipotents de U (5). On a la décomposition de Levi Pi = Mi Ni , i ∈{0, 1, 2} .M0 = E* × E* × E 1 est le sous-groupe de Levi minimal, M1 = GL(2, E) × E 1 et M2 = E* × U(3) sont les sous-groupes de Levi maximaux.On considère les représentations des sous-groupes de Levi, et on les étend trivialement au sous-groupes unipotents pour obtenir des représentations des sous-groupes paraboliques. On exécute une procédure appelée 'l'induction parabolique' pour obtenir les représentations de U (5). Nous considérons les représentations de M0 , puis les représentations non-cuspidales, induites à partir de M1 et M2 . Cela veut dire que la représentation du facteur GL(2, E) de M1 est un sous-quotient propre d'une représentation induite de E* × E* à GL(2, E). La représentation du facteur U (3) de M2 est un sous-quotient propre d'une représentation induite de E* × E 1 à U(3). Un exemple pour M1 est | det |α χ(det) StGL2 * λ' , où α ∈ R, χ est un caractère unitaire de E* , StGL2 est la représentation Steinberg de GL(2, E) et λ' est un caractère de E 1 . Un exemple pour M2 est| |α χ λ (det) StU (3) , où α ∈ R, χ est un caractère unitaire de E* , λ' est un caractère unitaire de E 1et StU (3) est la représentation Steinberg de U(3). On remarque que λ' est unitaire.Ensuite on considère les représentations cuspidales de M1 .On détermine les droites et les points de réductibilité des représentations de U(5) et on détermine les sous-quotients irréductibles. Ensuite, sauf quelque cas particuliers, on détermine le dual unitaire de U(5)par rapport au quotients de Langlands. Les représentations complexes, paraboliquement induites, de U(3) sur un corps p-adique sont classifiées par Charles David Keys dans [Key84], les représentations complexes, paraboliquement induites, de U(4)sur un corps p-adique sont classifiées par Kazuko Konno dans [Kon01]. / We study the parabolically induced complex representations of the unitary group in 5 variables - U(5)- defined over a non-archimedean local field of characteristic 0. This is Qp or a finite extension of Qp ,where p is a prime number. We speak of a 'p-adic field'.Let F be a p-adic field. Let E : F be a field extension of degree two. Let Gal(E : F ) = {id, σ}. We write σ(x) = overline{x} forall x ∈ E. Let | |p denote the p-adic norm on E. Let E* := E {0} and let E 1 := {x ∈ E | x overline{x} = 1} .U(5) has three proper parabolic subgroups. Let P0 denote the minimal parabolic subgroup and P1 andP2 the two maximal parabolic subgroups. Let M0 , M1 and M2 denote the standard Levi subgroups and let N0 , N1and N2 denote unipotent subgroups of U(5). One has the Levi decomposition Pi = Mi Ni , i ∈ {0, 1, 2} .M0 = E* × E* × E 1 is the minimal Levi subgroup, M1 = GL(2, E) × E 1 and M2 = E* × U (3) are the two maximal parabolic subgroups.We consider representations of the Levi subgroups and extend them trivially to the unipotent subgroups toobtain representations of the parabolic groups. One now performs a procedure called 'parabolic induction'to obtain representations of U (5).We consider representations of M0 , further we consider non-cuspidal, not fully-induced representationsof M1 and M2 . For M1 this means that the representation of the GL(2, E)− part is a proper subquotientof a representation induced from E* × E* to GL(2, E). For M2 this means that the representation of theU (3)− part of M2 is a proper subquotient of a representation induced from E* × E 1 to U (3).As an example for M1 , take | det |α χ(det) StGL2 * λ' , where α ∈ R, χ is a unitary character of E* , StGL2 is the Steinberg representation of GL(2, E) and λ' is a character of E 1 . As an example forM2 , take | |α χ λ' (det) StU (3) , where α ∈ R, χ is a unitary character of E* , λ' is a character of E 1 andStU (3) is the Steinberg representation of U (3). Note that λ' is unitary.Further we consider the cuspidal representations of M1 .We determine the points and lines of reducibility of the representations of U(5), and we determinethe irreducible subquotients. Further, except several particular cases, we determine the unitary dual ofU(5) in terms of Langlands-quotients.The parabolically induced complex representations of U(3) over a p-adic field have been classied byCharles David Keys in [Key84], the parabolically induced complex representations of U(4) over a p-adicfield have been classied by Kazuko Konno in [Kon01].An aim of further study is the classication of the induced complex representations of unitary groupsof higher rank, like U (6) or U (7). The structure of the Levi subgroups of U (6) resembles the structureof the Levi subgroups of U (4), the structure of the Levi groups of U (7) resembles those of U (3) and ofU (5).Another aim is the classication of the parabolically induced complex representatioins of U (n) over ap-adic field for arbitrary n. Especially one would like to determine the irreducible unitary representations.

Discusses certain one-parameter subgroups of the low-rank unitary groups called root subgroups. Unitary groups also have representations of Lie type which means they consist of transformations that act as automorphisms of an underlying Lie algebra, in this case the special linear algebra. Exploring this definition of the unitary groups, we find a correlation, via exponentiation, to the basis elements of Lie algebra.

The analysis of similarity of matrices over fields, as well as integral domains which are not fields, is a classical problem in Linear Algebra and has received considerable attention. A related problem is that of simultaneous similarity of matrices. Many interesting algebraic questions that arise in such problems are discussed by Shmuel Friedland[1]. A special case of this problem is that of Simultaneous Unitary Similarity of hermitian matrices, which we describe as follows: Given a collection of m ordered pairs of similar n×n hermitian matrices denoted by {(Hl,Dl)}ml=1, 1. determine if there exists a unitary matrix U such that UHl U∗ = Dl for all l, 2. and in the case where a U exists, find such a U, (where U∗is the transpose conjugate of U ).The problem is easy for m =1. The problem is challenging for m > 1.The problem stated above is the algorithmic version of the problem of classifying hermitian matrices upto unitary similarity. Any problem involving classification of matrices up to similarity is considered to be “wild”[2]. The difficulty in solving the problem of classifying matrices up to unitary similarity is a indicator of, the toughness of problems involving matrices in unitary spaces [3](pg, 44-46 ).Suppose in the statement of the problem we replace the collection {(Hl,Dl)}ml=1, by a collection of m ordered pairs of complex square matrices denoted by {(Al,Bl) ml=1, then we get the Simultaneous Unitary Similarity problem for square matrices. Suppose we consider k ordered pairs of complex rectangular m ×n matrices denoted by {(Yl,Zl)}kl=1, then the Simultaneous Unitary Equivalence problem for rectangular matrices is the problem of finding whether there exists a m×m unitary matrix U and a n×n unitary matrix V such that UYlV ∗= Zl for all l and in the case they exist find them. In this thesis we describe algorithms to solve these problems. The Simultaneous Unitary Similarity problem for square matrices is challenging for even a single pair (m = 1) if the matrices involved i,e A1,B1 are not normal. In an expository article, Shapiro[4]describes the methods available to solve this problem by arriving at a canonical form. That is A1 or B1 is used to arrive at a canonical form and the matrices are unitarily similar if and only if the other matrix also leads to the same canonical form. In this thesis, in the second chapter we propose an iterative algorithm to solve the Simultaneous Unitary Similarity problem for hermitian matrices. In each iteration we either get a step closer to “the simple case” or end up solving the problem. The simple case which we describe in detail in the first chapter corresponds to finding whether there exists a diagonal unitary matrix U such that UHlU∗= Dl for all l. Solving this case involves defining “paths” made up of non-zero entries of Hl (or Dl). We use these paths to define an equivalence relation that partitions L = {1,…n}. Using these paths we associate scalars with each Hl(i,j) and Dl(i,j)denoted by pr(Hl(i,j)) and pr(Dl(i,j)) (pr is used to indicate that these scalars are obtained by considering products of non-zero elements along the paths from i,j to their class representative). Suppose i (I Є L)belongs to the class[d(i)](d(i) Є L) we denote by uisol a modulus one scalar expressed in terms of ud(i) using the path from i to d( i). The free variable ud(i) can be chosen to be any modulus one scalar. Let U sol be a diagonal unitary matrix given by U sol = diag(u1 sol , u2 sol , unsol ). We show that a diagonal U such that U HlU∗ = Dl exists if and only if pr(Hl(i, j)) = pr(Dl(i, j))for all l, i, j and UsolHlUsol∗= Dl. Solving the simple case sets the trend for solving the general case. In the general case in an iteration we are looking for a unitary U such that U = blk −diag(U1,…, Ur) where each Ui is a pi ×p (i, j Є L = {1,… , r}) unitary matrix such that U HlU ∗= Dl. Our aim in each iteration is to get at least a step closer to the simple case. Based on pi we partition the rows and columns of Hl and Dl to obtain pi ×pj sub-matrices denoted by Flij in Hl and Glij in D1. The aim is to diagonalize either Flij∗Flij Flij∗ and a get a step closer to the simple case. If square sub-matrices are multiples of unitary and rectangular sub-matrices are zeros we say that the collection is in Non-reductive-form and in this case we cannot get a step closer to the simple case. In Non- reductive-form just as in the simple case we define a relation on L using paths made up of these non-zero (multiples of unitary) sub-matrices. We have a partition of L. Using these paths we associate with Flij and (G1ij ) matrices denoted by pr(F1ij) and pr(G1ij) respectively where pr(F1ij) and pr(G1ij) are multiples of unitary. If there exist pr(Flij) which are not multiples of identity then we diagonalize these matrices and move a step closer to the simple case and the given collection is said to be in Reduction-form. If not, the collection is in Solution-form. In Solution-form we identify a unitary matrix U sol = blk −diag(U1sol , U2 sol , …, Ur sol )where U isol is a pi ×pi unitary matrix that is expressed in terms of Ud(i) by using the path from i to[d(i)]( i Є [d(i)], d(i) Є L, Ud(i) is free). We show that there exists U such that U HlU∗ = Dl if and only if pr((Flij) = pr(G1ij) and U solHlU sol∗ = Dl. Thus in a maximum of n steps the algorithm solves the Simultaneous Unitary Similarity problem for hermitian matrices. In the second chapter we also relate the Simultaneous Unitary Similarity problem for hermitian matrices to the simultaneous closed system evolution problem for quantum states. In the third chapter we describe algorithms to solve the Unitary Similarity problem for square matrices (single ordered pair) and the Simultaneous Unitary Equivalence problem for rectangular matrices. These problems are related to the Simultaneous Unitary Similarity problem for hermitian matrices. The algorithms described in this chapter are similar in flow to the algorithm described in the second chapter. This shows that it is the fact that we are looking for unitary similarity that makes these forms possible. The hermitian (or normal)nature of the matrices is of secondary importance. Non-reductive-form is the same as in the hermitian case. The definition of the paths changes a little. But once the paths are defined and the set L is partitioned the definitions of Reduction-form and Solution-form are similar to their counterparts in the hermitian case. In the fourth chapter we analyze the worst case complexity of the proposed algorithms. The main computation in all these algorithms is that of diagonalizing normal matrices, partitioning L and calculating the products pr((Flij) = pr(G1ij). Finding the partition of L is like partitioning an undirected graph in the square case and partitioning a bi-graph in the rectangular case. Also, in this chapter we demonstrate the working of the proposed algorithms by running through the steps of the algorithms for three examples. In the fifth and the final chapter we show that finding if a given collection of ordered pairs of normal matrices is Simultaneously Similar is same as finding if the collection is Simultaneously Unitarily Similar. We also discuss why an algorithm to solve the Simultaneous Similarity problem, along the lines of the algorithms we have discussed in this thesis, may not exist. (For equations pl refer the pdf file)

ITC/USA 2005 Conference Proceedings / The Forty-First Annual International Telemetering Conference and Technical Exhibition / October 24-27, 2005 / Riviera Hotel & Convention Center, Las Vegas, Nevada / Multiple-input multiple output (MIMO) communication systems can have dramatically higher throughput than single-input, single-output systems. Unfortunately, it can be difficult to find the space-time codes these systems need to achieve their potential. Previously published results located good codes by minimizing the maximum correlation between transmitted signals. This paper shows how this min-max method may produce sub-optimal codes. A new method which sorts codes based on the union bound of pairwise error probabilities is presented. This new technique can identify superior MIMO codes, providing higher system throughput without increasing the transmitted power or bandwidth requirements.

Thesis (MSc (Physics))--University of Stellenbosch, 2005. / The goal of this thesis is the development and implementation of a non-perturbative solution method for Wegner’s flow equations. We show that a parameterization of the flowing Hamiltonian in terms of a scalar function allows the flow equation to be rewritten as a nonlinear partial differential equation. The implementation is non-perturbative in that the derivation of the PDE is based on an expansion controlled by the size of the system rather than the coupling constant. We apply this method to the Lipkin model and obtain very accurate results for the spectrum, expectation values and eigenstates for all values of the coupling and in the thermodynamic limit. New aspects of the phase structure, made apparent by this non-perturbative treatment, are also investigated. The Dicke model is treated using a two-step diagonalization procedure which illustrates how an effective Hamiltonian may be constructed and subsequently solved within this framework.

Thesis (MSc)--Stellenbosch University, 2003. / ENGLISH ABSTRACT: This thesis presents an overview of the flow equations recently introduced by Wegner. The little known mathematical framework is established in the initial chapter and used as a background for the entire presentation. The application of flow equations to the Foldy-Wouthuysen transformation and to the elimination of the electron-phonon coupling in a solid is reviewed. Recent flow equations approaches to the Lipkin model are examined thoroughly, paying special attention to their utility near the phase change boundary. We present more robust schemes by requiring that expectation values be flow dependent; either through a variational or self-consistent calculation. The similarity renormalization group equations recently developed by Glazek and Wilson are also reviewed. Their relationship to Wegner's flow equations is investigated through the aid of an instructive model. / AFRIKAANSE OPSOMMING: Hierdie tesis bied 'n oorsig van die vloeivergelykings soos dit onlangs deur Wegner voorgestel is. Die betreklik onbekende wiskundige raamwerk word in die eerste hoofstuk geskets en deurgans as agtergrond gebruik. 'n Oorsig word gegee van die aanwending van die vloeivergelyking vir die Foldy-Wouthuysen transformasie en die eliminering van die elektron-fonon wisselwerking in 'n vastestof. Onlangse benaderings tot die Lipkin model, deur middel van vloeivergelykings, word ook deeglik ondersoek. Besondere aandag word gegee aan hul aanwending naby fasegrense. 'n Meer stewige skema word voorgestel deur te vereis dat verwagtingswaardes vloei-afhanklik is; óf deur gevarieerde óf self-konsistente berekenings. 'n Inleiding tot die gelyksoortigheids renormerings groep vergelykings, soos onlangs ontwikkel deur Glazek en Wilson, word ook aangebied. Hulle verwantskap met die Wegner vloeivergelykings word bespreek aan die hand van 'n instruktiewe voorbeeld.

In this thesis, we prove that local genericity implies globally genericity for the quasi-split unitary group U3 for a quadratic extension of number fields E/F. We follow [Fli1992] and [GJR2001] closely, using the relative trace formula approach. Our main result is the existence of smooth transfer for the relative trace formulae in [GJR2001], which is circumvented there. The basic idea is to compute the Mellin transform of Shalika germ functions and show that they are equal in the unitary case and the general linear case.