682,453 research outputs found
On Foundation of the Generalized Nambu Mechanics
We outline the basic principles of canonical formalism for the Nambu
mechanics---a generalization of Hamiltonian mechanics proposed by Yoichiro
Nambu in 1973. It is based on the notion of Nambu bracket which generalizes the
Poisson bracket to the multiple operation of higher order on
classical observables and is described by Hambu-Hamilton equations of motion
given by Hamiltonians. We introduce the fundamental identity for the
Nambu bracket which replaces Jacobi identity as a consistency condition for the
dynamics. We show that Nambu structure of given order defines a family of
subordinated structures of lower order, including the Poisson structure,
satisfying certain matching conditions. We introduce analogs of action from and
principle of the least action for the Nambu mechanics and show how dynamics of
loops (-dimensional objects) naturally appears in this formalism. We
discuss several approaches to the quantization problem and present explicit
representation of Nambu-Heisenberg commutation relation for case. We
emphasize the role higher order algebraic operations and mathematical
structures related with them play in passing from Hamilton's to Nambu's
dynamical picture.Comment: 27 page
The Univalence Principle
The Univalence Principle is the statement that equivalent mathematical
structures are indistinguishable. We prove a general version of this principle
that applies to all set-based, categorical, and higher-categorical structures
defined in a non-algebraic and space-based style, as well as models of
higher-order theories such as topological spaces. In particular, we formulate a
general definition of indiscernibility for objects of any such structure, and a
corresponding univalence condition that generalizes Rezk's completeness
condition for Segal spaces and ensures that all equivalences of structures are
levelwise equivalences.
Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is
expressed in Voevodsky's Univalent Foundations (UF), extending previous work on
the Structure Identity Principle and univalent categories in UF. This enables
indistinguishability to be expressed simply as identification, and yields a
formal theory that is interpretable in classical homotopy theory, but also in
other higher topos models. It follows that Univalent Foundations is a fully
equivalence-invariant foundation for higher-categorical mathematics, as
intended by Voevodsky.Comment: A short version of this book is available as arXiv:2004.06572. v2:
added references and some details on morphisms of premonoidal categorie
Transformational Chair Design Analogous To Malaysia’s Rafflesia Plant Cellular Structure
The purpose of this paper is to provide a review of literature on Rafflesia cellular structure endemic to Malaysia Identity that is suitable to be applied in chair transformation design process which can help individual developing their understanding in plant analogy design. It is a critical and comprehensive review of a range of recently published literature sources (until May 2014) addressing various issues related to the transformation process design theory, Rafflesia Azlanii, and chair design. The sources are sorted into sections: morphological identification of Rafflesia, design for transformation and the analogy between R.A and chair. The paper provides insights about the characteristic and growth principle of R.A structure and form endemic to Malaysia resources; transformation process design theory that may influence the analogy between R.A cellular structure and chair design as a Malaysia identity product; and its benefits and guidelines to students/educators/designers of using R.A cellular structure as an analogy in developing chair design. It is hoped that the analysis, as captured in this paper, will highlight the different transformation process design theory in chair development. The paper will be of interest to researchers in the areas of tropical plant analogy design, enable learning environments, in general. Further, this paper demonstrates how the analysis of academic literature sources has been combined with commentaries and opinions on the journals and articles to develop this literature review. The finding is a very useful source of information and impartial advice which may be commercialized and influence learning and teaching strategies in higher and further education – specifically institutions that are considering the use of R.A cellular structure in chair design. This paper fulfils an identified information/resources of transformation design theory, design analogy of R.A cellular structure in chair design needs and offers practical help to students/educators/designers starting out a plant structure analogy design direction
The Broadest Necessity
In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given
The Logic of Opacity
We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models to show that some of these logics are non-degenerate
Gravity: A New Holographic Perspective
A general paradigm for describing classical (and semiclassical) gravity is
presented. This approach brings to the centre-stage a holographic relationship
between the bulk and surface terms in a general class of action functionals and
provides a deeper insight into several aspects of classical gravity which have
no explanation in the conventional approach. After highlighting a series of
unresolved issues in the conventional approach to gravity, I show that (i)
principle of equivalence, (ii) general covariance and (iii)a reasonable
condition on the variation of the action functional, suggest a generic
Lagrangian for semiclassical gravity of the form with
. The expansion of in terms of the
derivatives of the metric tensor determines the structure of the theory
uniquely. The zeroth order term gives the Einstein-Hilbert action and the first
order correction is given by the Gauss-Bonnet action. Any such Lagrangian can
be decomposed into a surface and bulk terms which are related holographically.
The equations of motion can be obtained purely from a surface term in the
gravity sector. Hence the field equations are invariant under the
transformation and gravity does not
respond to the changes in the bulk vacuum energy density. The cosmological
constant arises as an integration constant in this approach. The implications
are discussed.Comment: Plenary talk at the International Conference on Einstein's Legacy in
the New Millennium, December 15 - 22, 2005, Puri, India; to appear in the
Proceedings to be published in IJMPD; 16 pages; no figure
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