Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

On the equation of state and atomic mean square displacement of crystals

We have presented a Green's function method for the calculation of the atomic mean square displacement (MSD) for an anharmonic Hamil toni an . This method effectively sums a whole class of anharmonic contributions to MSD in the perturbation expansion in the high temperature limit. Using this formalism we have calculated the MSD for a nearest neighbour fcc Lennard Jones solid. The results show an improvement over the lowest order perturbation theory results, the difference with Monte Carlo calculations at temperatures close to melting is reduced from 11% to 3%. We also calculated the MSD for the Alkali metals Nat K/ Cs where a sixth neighbour interaction potential derived from the pseudopotential theory was employed in the calculations. The MSD by this method increases by 2.5% to 3.5% over the respective perturbation theory results. The MSD was calculated for Aluminum where different pseudopotential functions and a phenomenological Morse potential were used. The results show that the pseudopotentials provide better agreement with experimental data than the Morse potential. An excellent agreement with experiment over the whole temperature range is achieved with the Harrison modified point-ion pseudopotential with Hubbard-Sham screening function. We have calculated the thermodynamic properties of solid Kr by minimizing the total energy consisting of static and vibrational components, employing different schemes: The quasiharmonic theory (QH), ).2 and).4 perturbation theory, all terms up to 0 ().4) of the improved self consistent phonon theory (ISC), the ring diagrams up to o ().4) (RING), the iteration scheme (ITER) derived from the Greens's function method and a scheme consisting of ITER plus the remaining contributions of 0 ().4) which are not included in ITER which we call E(FULL). We have calculated the lattice constant, the volume expansion, the isothermal and adiabatic bulk modulus, the specific heat at constant volume and at constant pressure, and the Gruneisen parameter from two different potential functions: Lennard-Jones and Aziz. The Aziz potential gives generally a better agreement with experimental data than the LJ potential for the QH, ).2, ).4 and E(FULL) schemes. When only a partial sum of the).4 diagrams is used in the calculations (e.g. RING and ISC) the LJ results are in better agreement with experiment. The iteration scheme brings a definitive improvement over the).2 PT for both potentials.

Phonon spectra and temperature variation of thermodynamic properties of fcc metals via Finnis-Sinclair type many body potentials: Sutton-Chen and improved Sutton-Chen models

Volume(density)-independent pair-potentials cannot describe metallic cohesion adequately as the presence of the free electron gas renders the total energy strongly dependent on the electron density. The embedded atom method (EAM) addresses this issue by replacing part of the total energy with an explicitly density-dependent term called the embedding function. Finnis and Sinclair proposed a model where the embedding function is taken to be proportional to the square root of the electron density. Models of this type are known as Finnis-Sinclair many body potentials. In this work we study a particular parametrization of the Finnis-Sinclair type potential, called the "Sutton-Chen" model, and a later version, called the "Quantum Sutton-Chen" model, to study the phonon spectra and the temperature variation thermodynamic properties of fcc metals. Both models give poor results for thermal expansion, which can be traced to rapid softening of transverse phonon frequencies with increasing lattice parameter. We identify the power law decay of the electron density with distance assumed by the model as the main cause of this behaviour and show that an exponentially decaying form of charge density improves the results significantly. Results for Sutton-Chen and our improved version of Sutton-Chen models are compared for four fcc metals: Cu, Ag, Au and Pt. The calculated properties are the phonon spectra, thermal expansion coefficient, isobaric heat capacity, adiabatic and isothermal bulk moduli, atomic root-mean-square displacement and Gr\"{u}neisen parameter. For the sake of comparison we have also considered two other models where the distance-dependence of the charge density is an exponential multiplied by polynomials. None of these models exhibits the instability against thermal expansion (premature melting) as shown by the Sutton-Chen model. We also present results obtained via pure pair potential models, in order to identify advantages and disadvantages of methods used to obtain the parameters of these potentials.

Marginal/minority popular music : the concept of the "third space" and the case for "hybridities" of cultures/identities /

The topic of this thesis is marginaVminority popular music and the question of identity; the term "marginaVminority" specifically refers to members of racial and cultural minorities who are socially and politically marginalized. The thesis argument is that popular music produced by members of cultural and racial minorities establishes cultural identity and resists racist discourse. Three marginaVminority popular music artists and their songs have been chosen for analysis in support of the argument: Gil Scott-Heron's "Gun," Tracy Chapman's "Fast Car" and Robbie Robertson's "Sacrifice." The thesis will draw from two fields of study; popular music and postcolonialism. Within the area of popular music, Theodor Adorno's "Standardization" theory is the focus. Within the area of postcolonialism, this thesis concentrates on two specific topics; 1) Stuart Hall's and Homi Bhabha's overlapping perspectives that identity is a process of cultural signification, and 2) Homi Bhabha's concept of the "Third Space." For Bhabha (1995a), the Third Space defines cultures in the moment of their use, at the moment of their exchange. The idea of identities arising out of cultural struggle suggests that identity is a process as opposed to a fixed center, an enclosed totality. Cultures arise from historical memory and memory has no center. Historical memory is de-centered and thus cultures are also de-centered, they are not enclosed totalities. This is what Bhabha means by "hybridity" of culture - that cultures are not unitary totalities, they are ways of knowing and speaking about a reality that is in constant flux. In this regard, the language of "Otherness" depends on suppressing or marginalizing the productive capacity of culture in the act of enunciation. The Third Space represents a strategy of enunciation that disrupts, interrupts and dislocates the dominant discursive construction of US and THEM, (a construction explained by Hall's concept of binary oppositions, detailed in Chapter 2). Bhabha uses the term "enunciation" as a linguistic metaphor for how cultural differences are articulated through discourse and thus how differences are discursively produced. Like Hall, Bhabha views culture as a process of understanding and of signification because Bhabha sees traditional cultures' struggle against colonizing cultures as transforming them. Adorno's theory of Standardization will be understood as a theoretical position of Western authority. The thesis will argue that Adorno's theory rests on the assumption that there is an "essence" to music, an essence that Adorno rationalizes as structure/form. The thesis will demonstrate that constructing music as possessing an essence is connected to ideology and power and in this regard, Adorno's Standardization theory is a discourse of White Western power. It will be argued that "essentialism" is at the root of Western "rationalization" of music, and that the definition of what constitutes music is an extension of Western racist "discourses" of the Other. The methodological framework of the thesis entails a) applying semiotics to each of the three songs examined and b) also applying Bhabha's model of the Third Space to each of the songs. In this thesis, semiotics specifically refers to Stuart Hall's retheorized semiotics, which recognizes the dual function of semiotics in the analysis of marginal racial/cultural identities, i.e., simultaneously represent embedded racial/cultural stereotypes, and the marginal raciaVcultural first person voice that disavows and thus reinscribes stereotyped identities. (Here, and throughout this thesis, "first person voice" is used not to denote the voice of the songwriter, but rather the collective voice of a marginal racial/cultural group). This dual function fits with Hall's and Bhabha's idea that cultural identity emerges out of cultural antagonism, cultural struggle. Bhabha's Third Space is also applied to each of the songs to show that cultural "struggle" between colonizers and colonized produces cultural hybridities, musically expressed as fusions of styles/sounds. The purpose of combining semiotics and postcolonialism in the three songs to be analyzed is to show that marginal popular music, produced by members of cultural and racial minorities, establishes cultural identity and resists racist discourse by overwriting identities of racial/cultural stereotypes with identities shaped by the first person voice enunciated in the Third Space, to produce identities of cultural hybridities. Semiotic codes of embedded "Black" and "Indian" stereotypes in each song's musical and lyrical text will be read and shown to be overwritten by the semiotic codes of the first person voice, which are decoded with the aid of postcolonial concepts such as "ambivalence," "hybridity" and "enunciation."

Kierkegaard : the temporality of becoming a self

Introduction In Difference and Repetition, Deleuze compares and contrasts Kierkegaard's and Nietzsche's ideas of repetition. He argues that neither of them really give a representation of repetition. Repetition for them is a sort of selective task: the way in which they determine what is ethical and eternal. With Nietzsche, it is a theater of un belie f. ..... Nietzsche's leading idea is to found the repetition in the etemal return at once on the death of God and the dissolution of the self But it is a quite different alliance in the theater of faith: Kierkegaard dreams of alliance between a God and a self rediscovered. I Repetition plays a theatrical role in their thinking. It allows them to dramatically stage the interplay of various personnae. Deleuze does give a positive account ofKierkegaard's "repetition"; however, he does not think that Kierkegaard works out a philosophical model, or a representation of what repetition is. It is true that in the book Repetition, Constantin Constantius does not clearly and fully work out the concept of repetition, but in Sickness Unto Death, Kierkegaard gives a full explanation of the self and its temporality which can be connected with repetition. When Sickness Unto Death is interpreted according to key passages from Repetition and The Concept of Anxiety, a clear philosophical concept of repetition can be established. In my opinion, Kierkegaard's philosophy is about the task of becoming a self, and I will be attempting to show that he does have a model of the temporality of self-becoming. In Sickness Unto Death, Kierkegaard explains his notions of despair with reference to sin, self, self-becoming, faith, and repetition. Despair is a sickness of the spirit, of the self, and accordingly can take three forms: in despair not to be conscious of having a self (not despair in the strict sense); in despair not to will to be oneself; in despair to will to be oneself2 In relation to this definition, he defines a self as "a relation that relates itself to itself and in relating itself to itself relates to another.''3 Thus, a person is a threefold relationship, and any break in that relationship is despair. Despair takes three forms corresponding to the three aspects of a self s relation to itself Kierkegaard says that a selfis like a house with a basement, a first floor, and a second floor.4 This model of the house, and the concept of the stages on life's way that it illustrates, is central to Kierkegaard's philosophy. This thesis will show how he unpacks this model in many of his writings with different concepts being developed in different texts. His method is to work with the same model in different ways throughout his authorship. He assigns many of the texts to different pseudonyms, but in this thesis we will treat the model and the related concepts as being Kierkegaard's and not only the pseudonyms. This is justified as our thesis will show this modelremains the same throughout Kierkegaard's work, though it is treated in different ways by different pseudonyms. According to Kierkegaard, many people live in only the basement for their entire lives, that is, as aesthetes ("in despair not to be conscious of having a self'). They live in despair of not being conscious of having a self They live in a merely horizontal relation. They want to get what they desire. When they go to the first floor, so to speak, they reflect on themselves and only then do they begin to get a self In this stage, one acquires an ideology of the required and overcomes the strict commands of the desired. The ethical is primarily an obedience to the required whereas the aesthetic is an obedience to desire. In his work Fear and Trembling (Copenhagen: 1843), Johannes de Silentio makes several observations concerning this point. In this book, the author several times allows the desired ideality of esthetics to be shipwrecked on the required ideality of ethics, in order through these collisions to bring to light the religious ideality as the ideality that precisely is the ideality of actuality, and therefore just as desirable as that of esthetics and not as impossible as the ideality of ethics. This is accomplished in such a way that the religious ideality breaks forth in the dialectical leap and in the positive mood - "Behold all things have become new" as well as in the negative mood that is the passion of the absurd to which the concept "repetition" corresponds.s Here one begins to become responsible because one seeks the required ideality; however, the required ideality and the desired ideality become inadequate to the ethical individual. Neither of them satisfy him ("in despair not to will to be oneself'). Then he moves up to the second floor: that is, the mystical region, or the sphere of religiousness (A) ("despair to will to be oneself). Kiericegaard's model of a house, which is connected with the above definition ofdespair, shows us how the self arises through these various stages, and shows the stages of despair as well. On the second floor, we become mystics, or Knights of Infinite Resignation. We are still in despair because we despair ofthe basement and the first floor, however, we can be fiill, free persons only ifwe live on all the floors at the same time. This is a sort of paradoxical fourth stage consisting of all three floors; this is the sphere of true religiousness (religiousness (B)). It is distinguished from religiousness (A) because we can go back and live on all the floors. It is not that there are four floors, but in the fourth stage, we live paradoxically on three at once. Kierkegaard uses this house analogy in order to explain how we become a self through these stages, and to show the various stages of despair. Consequently, I will be explaining self-becoming in relation to despair. It will also be necessary to explain it in relation to faith, for faith is precisely the overcoming of despair. After explaining the becoming of the self in relation to despair and faith, I will then explain its temporality and thereby its repetition. What Kierkegaard calls a formula, Deleuze calls a representation. Unfortunately, Deleuze does not acknowledge Kierkegaard's formula for repetition. As we shall see, Kierkegaard clearly gives a formula for despair, faith, and selfbecoming. When viewed properly, these formulae yield a formula for repetition because when one hasfaith, the basement, firstfloor, and secondfloor become new as one becomes oneself The self is not bound in the eternity ofthe first floor (ethical) or the temporality of the basement (aesthete). I shall now examine the two forms of conscious despair in such a way as to point out also a rise in the consciousness of the nature of despair and in the consciousness that one's state is despair, or, what amounts to the same thing and is the salient point, a rise in the consciousness of the self The opposite to being in despair is to have faith. Therefore, the formula set forth above, which describes a state in which there is not despair at all, is entirely correct, and this formula is also the formula for faMi in ^elating itself to itself and in willing to be itself, the self rests transparently in the power that established it.

Learning styles and achievement in postsecondary classrooms

The purpose of this study was to investigate the relationship between learning styles and academic achievement in postsecondary education. It was the intent of the study to establish if there was a relationship between student learning style, teacher style, learner/teacher matching and/or mismatching, student gender and age, to the academic grades of students. This study was basically a replication of a study completed by Mary J. Thompson and Terrance P. O'Brien in 1991 on two campuses of a southeast community college in the United States. In the present study, 243 students and 18 teachers from two different campuses of a community college in the Province of Ontario participated in the research. All participants were administered the Gregorc Style Delineator and students identified by program, age and gender. Data were tested by two analysis of variance (ANOVA) models. In the first ANOVA model considered in this study, significant main effects were manifested in regard to the teaching style, age group and gender. With the exception of gender, these findings were very similiar to those of the original study. Duncan's multiple range test revealed that Concrete Sequential (CS) teachers assigned significantly lower grades than did teachers dominant in any of the other three learning styles. Post hoc testing revealed that students 25 years of age and older received significantly higher grades than did younger students. Female students also received significantly higher grades than did male students. In the second ANOVA model student/teacher learning style match/mismatch did emerge as a significant main effect. However, Duncan's multiple range test and Chi square analysis did not substantiate the relationship. Forty-eight references are cited.

Certificate of Acceptance and Initiation to Canadian Order Chosen Friends - Thomas Cowan

A certificate of initiation and acceptance to the Canadian Order Chosen Friends, Thomas Cowan. The certificate reads "This certifies that evidence has been received that Thomas Cowan has been accepted and initiated by the Council name below, and has thus become a member of the Canadian Order of Chosen Friends, and entitled to all the rights and privileges of membership and a benefit of not exceeding one thousand dollars from the relief fund of said order, which shall in case of death be paid to Annie Cowan his wife in the manner and subject to the conditions set forth in the laws governing said relief fund and in the application for membership. This certificate to be in force and binding when accepted in writing by the said member, with the acceptance attested by the Councilor and Recorder and the seal of the Subordinate Council affixed, so long as said member shall comply with the requirements of the Constitution, Laws and Regulations now in force or hereafter adopted for the government of the Order: otherwise, and also in the case of granting of a new certificate, to be null and void. In witness whereof, we have hereunto attached our signatures, and affixed the seal of the Grand Council of the Canadian Order of Chosen Friends. Dated the Twenty Seventh day of July, A.D. 1891." The front and back of the certificate are available for viewing.

The Implementations of Professional Learning Communities (PLCs) in Beijing and Ontario Schools

Since the knowledge-based economy has become a fashion over the last few decades, the concept of the professional learning community (PLC) has started being accepted by educational institutions and governments as an effective framework to improve teachers’ collective work and collaboration. The purpose of this research was to compare and contrast the implementations of PLCs between Beijing schools and Ontario schools from principals’ personal narratives. In order to discover the lessons and widen the scope to understand the PLC, this research applied qualitative design to collect the data from two principal participants in each location by semistructured interviews. Four themes emerged: (a) structure and technology, (b) identity and climate, (c) task and support, and (d) change and challenge. This research found that the root of the characteristics of the PLCs in Beijing and Ontario was the different existing teaching and learning systems as well as the test systems. Teaching Research Groups (TRGs) is one of the systems that help Chinese to organize routine time and input resources to improve teachers’ professional development. However, Canadian schools lack a similar system that guarantees the time and resources. Moreover, standardized test plays different roles in China and Canada. In China, standardized tests, such as the college entrance examination, are regarded as the important purpose of education, whereas Ontario principals saw the Education Quality and Accountability Office (EQAO) as a tool rather than a primary purpose. These two main differences influenced principals’ beliefs, attitudes, strategies, and practices. The implications based on this discovery provide new perspectives for principals, teachers, policy makers, and scholars to widen and deepen the research and practice of the PLC.

Niagara Region Municipality Records 1962-1963, 1969-1976, 1980

Niagara Region Municipality Records 1962-1963, 1969-1976, 1980