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A class of topologically free locally convex spaces and related group Hopf algebras
No full textA locally convex space E will be called p-reflexive provided the evaluation map from E to (E')' is a topological linear isomorphism, where the dual space in each case is given the topology of precompact convergence. Let [special characters omitted] denote the category of p-reflexive spaces and continuous linear maps. [special characters omitted] is shown to be complete, cocomplete, and self-dual. Let Sets denote the category of sets and functions and ν:[special characters omitted] → Sets, the forgetful functor. Theorem. There exist functors Hom: [special characters omitted] × [special characters omitted] → [special characters omitted] and ⊗: [special characters omitted] × [special characters omitted] → [special characters omitted] such that (1) ν∘ Hom = hom ; and (2) for all A, B, C ∈ ob [special characters omitted], Hom(A ⊗ B, C) is naturally isomorphic to Hom(A, Hom(B, C)). Thus [special characters omitted] can be extended to be a closed, symmetric, monoidal category in the sense of Eilenberg and Kelly. Let [special characters omitted] denote the category of k-spaces and continuous maps, and let c: [special characters omitted] × [special characters omitted] → Sets denote the functor such that if X ∈ ob[special characters omitted] and E ∈ ob[special characters omitted], c(X,E) equals the set of continuous functions from X to E. Theorem. There exist functors C: [special characters omitted] × [special characters omitted] → [special characters omitted] and M: [special characters omitted] → [special characters omitted] such that (1) ν ∘ C = c , and (2) for all X ∈ ob[special characters omitted] and E ∈ ob[special characters omitted], C(X, E) is naturally isomorphic to Hom(M(X), E). It is known that [special characters omitted] can be extended to be a closed, symmetric, monoidal category. Let ☆: [special characters omitted] × [special characters omitted] → [special characters omitted] denote the product functor. Theorem. If X, Y ∈ ob[special characters omitted], then M(X☆Y) is naturally isomorphic to M(X) ⊗ M(Y). M can be extended to be a "strong," symmetric, monoidal functor. M will transform any algebraic structure that occurs at the level of [special characters omitted] to an analogous algebraic structure at the level of [special characters omitted]. Theorem. Let [special characters omitted] and [special characters omitted] be categories such that either (a) [special characters omitted] = k-spaces and [special characters omitted] = [special characters omitted]-coalgebras; (b) [special characters omitted] = k-monoids and [special characters omitted] = [special characters omitted]-bialgebras; or (c) [special characters omitted] = k-groups and [special characters omitted] = [special characters omitted]-Hopf algebras. Then M can be regarded as a functor from [special characters omitted] to [special characters omitted]. Furthermore, there exists a functor Ω: [special characters omitted] → [special characters omitted] such that if G ∈ ob[special characters omitted] and H ∈ ob[special characters omitted], then [special characters omitted](G, Ω(H)) is naturally isomorphic to [special characters omitted] (M(G),H). M is an adjoint of Ω. Loosely speaking, a [special characters omitted]-algebra is a p-reflexive space with an associative multiplication and a unit; a [special characters omitted]-coalgebra is a p-reflexive space with a coassociative comultiplication and a counit; a [special characters omitted]-bialgebra is a p-reflexive space which simultaneously has the structure of a [special characters omitted]-algebra and a [special characters omitted]-coalgebra with the property that the comultiplication and counit maps are [special characters omitted]-algebra morphisms; and a [special characters omitted]-Hopf algebra is a [special characters omitted]-bialgebra which admits an antipode. k-monoids and k-groups are k-spaces which satisfy the axioms of topological monoids and topological groups respectively, except that the k-space product is used in place of the usual product
Buddhists in Vietnam's Contemporary Politics
No full textThe Role of the Vietnamese Buddhists in Contemporary Vietnam Politics is presented factually and analytically here in the light of political science methodology, with the aim toward generalized prediction on the future relationship between religion and politics. The dissertation consists of three parts: Part 1 examines the assets constituting the political power of the Buddhists. such as the social legitimacy and the internal structural strength of South Vietnam's Unified Buddhist Church. Part 2 discusses the political participation and activities of the Buddhists. The findings obtained from the factual analyses in those two parts are: a) there is a direct casual relationship between the South Vietnam Buddhists' relative political strength and their political engagement. b) The Buddhist leaders aspirations and proposed solutions for them were more or less political in nature. c) The strategy of their movement inside South Vietnam put them on the world political map and among the world Buddhist organizations. d) The Buddhists in Vietnam, Ceylon and Burma had similarities. e) Nationalism was used by the Buddhists and the political leaders as political weapons in Asia. f) Since the Buddhist events in 1963, the foreign actors have continued their intervention. g) Therefore, some Buddhist priests with intense political sense have emphasized that the Buddhists can only have important role in a non-aligned country which is completely independent from interference by big powers. Part 3 on prescriptive proposals for the Vietnamese Buddhists consists of delineation of the clear role of the Buddhists in the nation and in relation with other religions, and the role of the state in religious affairs — which should be based on modern progressive concepts, and adaptable to the realities of the situation in Vietnam, and consequently should be easily accepted by all political actors in the national life. In essence, that is the solution of separation of church and state, each with its separate sphere of influence and balance of power between the religious leaders and secular leaders, a solution that is originating from a humane social movement in the nation and satisfying the deeply held aspiration of the Buddhists on freedom of religion and at the same time consolidating political stability
The impact of modern physics on the humanities
No full textOne of the central ideas of modern physics is the probabilistic interpretation of nature in which causuality is not present. Quantum Mechanics is the present day outgrowth of this new interpretation of nature. Set forms based upon the deterministic success of Newtonian mechanics were present in Classical music, representational art and in literature. Indeterminate forms in art, literature and music emerged from such physics ideas in the 20 th century as the Uncertainty Principle and Quantum Mechanics in which causuality is not present. Aleatoric music, in which there is no predetermined performance is one example of this. Another art form affected by indeterminacy is painting in which is non-representational such as that of Franz Klein. Drama in which there is some non-predetermined acting and dialogue is also an example of this effect modern physics has on the arts
Extensions of Morley's theorem
No full textA review of the literature since 1899, when Morley offered his original hypothesis, is presented. Various authors have given proofs of the theorem, shown the existence of 27 such triangles, and established relationships to other mathematical concepts: isogonal conjugates, Lemoine point, Hessian points and axis, circles of Apollonius, Brianchon hexagons. The author explores the various intersections of the n-sectors of the angles of a general triangle, showing, incidentally, the uniqueness of the Morley equilateral triangle. Turning to the quadrilateral, the intersections of the tri-sectors of: the angles of a parallelogram, rectangle, trapezoid, and kite-shaped quadrilateral are examined, as well as those of the n-sectors. An important by-product of the author's investigation of the p-sided polygon confirms the uniqueness of the equilateral triangle of Morley's theorem. The author concludes with a discussion of the associated isogonal attributes and thoughts for further study
The Influence of the Sympathetic System on Overdrive Suppression of the Ventricular Pacemaker
No full textBackground Subsidiary mammalian cardiac pacemakers are normally kept under inhibition because they are driven by the sinus node at a rate that is faster than their own intrinsic rate. Failure of this inhibition can result in cardiac dysrhythmias and death. The mechanism by which this inhibition is brought about is still under study. The degree to which the sympathetic system is able to influence both the suppression and the initiation of ventricular automaticity following over-drive remains unclear. The purpose of this study is to examine the role of neural and hormonal sympathetic influences on ventricular overdrive suppression and the possible role of potassium as an associated mediator. Methods Mongrel open-chest dogs with surgically induced atrioventricular block (in vivo) were used to study ventricular drive-suppression relationships and associated changes in coronary sinus blood effluent potassium concentrations under conditions of stellate ganglion isolation and stimulation, chemically induced hypotension with amyl nitrite, intravenous infusion of noradrenaline, beta receptor blockade with propranolol, and chronic catacholamine depletion by pretreatment with reserpine. Small canine hearts were studied using a modified Langendorf isolated perfused heart preparation (in vitro) in a manner analogous to those studies performed in the intact animals. Results Factors that reduce sympathetic influences such as bilateral stellate ganglionectomy, beta blockade, catecholamine depletion, and perfusing an isolated heart with blood free crystalloid were observed to reduce ventricular automaticity while at the same time increasing the effect of over-driving the ventricular pacemaker. Enhancing sympathetic factors by stellate ganglion stimulation, induced hypotension, and noradrenaline infusion had the opposite effect. Increasing the rate of overdrive enhanced the inibitory effects and delayed recovery, whereas reducing the rate of overdrive had the opposite effect. Regardless of the methods employed to alter the overdrive suppression relationships, in both in vivo and in vitro hearts, the coronary sinus effluent potassium concentration rose during overdrive and slightly after it's cessation and then fell below baseline during the recovery period. Conclusions Sympathetic neural and humoral factors significantly affect the overdrive suppression of ventricular pacemakers over a physiologic range, as demonstrated by their shifting of overdrive-suppression relationships used in the present models. The role of extracellular potassium concentration appears to be related, but the exact mechanism remains unclear
