Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.

Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.

Quantum systems with variables in the ring Z(d) are considered, and the concepts of weak mutually unbiased bases and mutually unbiased projectors are discussed. The lines through the origin in the Z(d) x Z(d) phase space, are classified into maximal lines (sets of d points), and sublines (sets of d(i) points where d(i)vertical bar d). The sublines are intersections of maximal lines. It is shown that there exists a duality between the properties of lines (resp., sublines), and the properties of weak mutually unbiased bases (resp., mutually unbiased projectors).

The negative effects of the Russian invasion of Ukraine in 2022 has been felt on a global scale.This makes it more relevant than it has been in a long time to understand how to achievesuccessful mediation and negotiation between Russia and post-soviet states. By examining theRusso-Georgian War of 2008 and the Russo-Ukrainian War that began in 2022, and thencomparing the two cases. This paper aims to investigate what factors contribute to successfulmediation in these two cases, and to hypothesise what obstacles exist for achieving successfulmediation in the war between Russia and Ukraine. The method used for this purpose isprocess-tracing, and the theoretical framework that has been used is Zartmans ripeness theoryand its formula for the concepts of mutually enticing opportunity, mutually hurting stalemate,ripe moment, and way out. The results of this study shows that a ripe moment existed in theconflict between Russia and Georgia despite the weak grounds for the existence of a mutuallyhurting stalemate. The study also concluded that the ceasefire agreement between the twocountries included a strong presence of one of the subcategories of the mutually enticingopportunity concept. Furthermore, the study also showed that a mutually hurting stalemate doesnot exist in the war between Russia and Ukraine, but simultaneously indicated strong grounds fora mutually hurting stalemate developing in the near future. Finally, the study identified fourdifferent obstacles for successful mediation between Russia and Ukraine. These related to theexistence of a mutually hurting stalemate, the motivations of the two conflict parties, and thechallenge of developing a mutually enticing opportunity.

Thesis advisor: Susan Michalczyk / Turbulent times always elicit change. Outside forces are constantly shifting, only to affect the lives of the characters that are at their mercy. Only by adapting to these changes can one truly survive in this world. However, unlike these forces of nature, God, or chance, people have to think about changing before they can carry through with it. Questions plague their minds and options torment their souls. What course of action is best for them, and which allows them to maintain some sense of self? These collected stories focus on people in turmoil. People in pain. People who must change. What happens to a man who will truly die if he loses the love of his life? Should you take the advice of a psychic, abandoning your own free will in the process? When the larger forces in the world are trying to destroy you, do you let them, or fight back against literally impossible odds? In the end, everyone must come to a decision. Stagnation and inaction themselves can decide your fate. As Neil Gaiman once said: Everything changes. But nothing is truly lost. / Thesis (BS) — Boston College, 2006. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: College Honors Program.

In a time when interdependence in business becomes more prevalent and necessary to maintain and sustain competitive advantage, understanding the mechanisms by which businesses relate and collaboratively adapt become central to collaborative growth and mutual success. Learning becomes central to the adaptive process. Interorganizational learning is an often challenging result of collaborative efforts. The more different the organizations are from one another, the more challenging the adaptive process and interorganizational learning. This writing addresses some of the complexities involved in collaborative learning of organizations with divergent goals through the lens of exploration exploitation phenomena. It further addresses how interorganizational learning happens between organizations that are private and public in nature. This writing is a case study that answers the question of how organizations working in collaboratives learn from each other to attain mutually beneficial results by examining two such entities in a government and private partnership. This study extends concepts of interorganizational learning as well as provides guidelines for business entities seeking to attain or sustain learning organizations. It also provides a framework from which government entities may work synergistically with private enterprise to provide competitive service to their respective demographic.

There has being an enormous work on finite quantum systems with variables in Zd, especially on mutually unbiased bases. The reason for this is due to its wide areas of application. We focus on partial ordering of weak mutually un-biased bases. In it, we studied a partial ordered relation which exists between a subsystem ^(q) and a larger system ^(d) and also, between a subgeometry Gq and larger geometry Gd. Furthermore, we show an isomorphism between: (i) the set {Gd} of subgeometries of a finite geometry Gd and subsets of the set {D(d)} of divisors of d. (ii) the set {hd} of subspaces of a finite Hilbert space Hd and subsets of the set {D(d)} of divisors of d. (iii) the set {Y(d)} of subsystems of a finite quantum system ^(d) and subsets of the set {D(d)} of divisors of d. We conclude this work by showing a duality between lines in finite geometry Gd and weak mutually unbiased bases in finite dimensional Hilbert space Hd.

There has being an enormous work on finite quantum systems with variables in Zd, especially on mutually unbiased bases. The reason for this is due to its wide areas of application. We focus on partial ordering of weak mutually un-biased bases. In it, we studied a partial ordered relation which exists between a subsystem ^(q) and a larger system ^(d) and also, between a subgeometry Gq and larger geometry Gd. Furthermore, we show an isomorphism between: (i) the set {Gd} of subgeometries of a finite geometry Gd and subsets of the set {D(d)} of divisors of d. (ii) the set {hd} of subspaces of a finite Hilbert space Hd and subsets of the set {D(d)} of divisors of d. (iii) the set {Y(d)} of subsystems of a finite quantum system ^(d) and subsets of the set {D(d)} of divisors of d. We conclude this work by showing a duality between lines in finite geometry Gd and weak mutually unbiased bases in finite dimensional Hilbert space Hd.

yes / Quantum systems with variables in Z(d) are considered, and three different structures are studied. The first is weak mutually unbiased bases, ... The second is maximal lines through the origin in the Z(d)×Z(d) phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. For simplicity, the case where d=p1×p2, where p1,p2 are odd prime numbers different from each other, is considered. / The full text will be available 12 months after publication

Yes / A quantum system (n) with variables in Z(n), where n = Qpi (with pi prime numbers), is considered. The non-near-linear geometry G(n) of the phase space Z(n) × Z(n), is studied. The lines through the origin are factorized in terms of ‘prime factor lines’ in Z(pi)×Z(pi). Weak mutually unbiased bases (WMUB) which are products of the mutually unbiased bases in the ‘prime factor Hilbert spaces’ H(pi), are also considered. The factorization of both lines and WMUB is analogous to the factorization of integers in terms of prime numbers. The duality between lines and WMUB is discussed. It is shown that there is a partial order in the set of subgeometries of G(n), isomorphic to the partial order in the set of subsystems of (n).