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J M Mccarthy - One of the best experts on this subject based on the ideXlab platform.

  • Kinematic Synthesis of Spatial Serial Chains Using Clifford Algebra Exponentials
    Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 2006
    Co-Authors: Alba Perez-gracia, J M Mccarthy
    Abstract:

    AbstractThis article presents a formulation of the design equations for a spatial serial chain that uses the Clifford Algebra exponential form of its kinematics equations. This is the even Clifford...

  • sizing a serial chain to fit a task trajectory using Clifford Algebra exponentials
    International Conference on Robotics and Automation, 2005
    Co-Authors: Alba Perez, J M Mccarthy
    Abstract:

    In this paper we formulate the “generalized inverse kinematics problem” for a spatial serial chain, where the goal is to determine values for structural parameters as well as for the joint parameters. The kinematics equations of the chain are formulated first using matrix exponentials and then cast into a form based on exponentials in a Clifford Algebra. These equations contain the coordinates of the joint axes explicitly and have a systematic structure that can be exploited in their solution. As an example we fit a seven degree-of-freedom CCS chain to a 12 position task trajectory. In this problem, we can also specify desired values for the first two joint angles, and compute the structural parameters and the remaining joint angles.

  • The quartic singularity surfaces of planar platforms in the Clifford Algebra of the projective plane
    Mechanism and Machine Theory, 1998
    Co-Authors: C.l. Collins, J M Mccarthy
    Abstract:

    Abstract In this paper, we study the workspace and singular configurations of a planar platform supported by three linearly actuated legs, the 3-RPR parallel manipulator. The constraint equations of the platform are formulated in the Clifford Algebra of the projective plane, C + (P 2 ) , which yields a manifold defining its set of reachable positions and orientations. We compute the Jacobian of these equations and derive the Algebraic equation of the surface of points in C + (P 2 ) for which this Jacobian is singular, called the singularity surface of the manipulator. For the general planar platform manipulator this surface is a quartic surface with a double line. For the special case of the “proportional” planar platform, the surface factors into two planes and a circular hyperboloid. For the special case of the “in-line” planar platform, this surface reduces to a quartic ruled surface. Further special cases of this surface are examined and found to consist of pairs of hyperbolic paraboloids.

D S Shirokov - One of the best experts on this subject based on the ideXlab platform.

Anthony Joseph - One of the best experts on this subject based on the ideXlab platform.

  • zhelobenko invariants bernstein gelfand gelfand operators and the analogue kostant Clifford Algebra conjecture
    Transformation Groups, 2012
    Co-Authors: Anthony Joseph
    Abstract:

    Let \( \mathfrak{g} \) be a complex simple Lie Algebra and \( \mathfrak{h} \) a Cartan subAlgebra. The Clifford Algebra C(\( \mathfrak{g} \)) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2 m + 1 is just the zero weight vector of the simple (2 m + 1)-dimensional module of the principal s-triple obtained from the Langlands dual \( {\mathfrak{g}^\vee } \). Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford Algebra conjecture is a question concerning the Harish-Chandra map for the enveloping Algebra U(\( \mathfrak{g} \)) composed with evaluation at the half sum ρ of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a “generalized Harish-Chandra” map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric Algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.

  • zhelobenko invariants bernstein gelfand gelfand operators and the analogue kostant Clifford Algebra conjecture
    arXiv: Representation Theory, 2011
    Co-Authors: Anthony Joseph
    Abstract:

    Let g be a complex simple Lie Algebra and h a Cartan subAlgebra. The Clifford Algebra C(g) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2m+1 is just the zero weight vector of the simple 2m+1-dimensional module of the principal s-triple obtained from the Langlands dual. Bazlov settled this conjecture positively in type A. The Kostant conjecture was reformulated (Alekseev-Bazlov-Rohr) in terms of the Harish-Chandra map for the enveloping Algebra U(g) composed with evaluation at the half sum of the positive roots. Here an analogue of the Kostant conjecture is settled by replacing the Harish-Chandra map by a "generalized Harish-Chandra" map which had been studied notably by Zhelobenko. The proof involves a symmetric Algebra version of the Kostant conjecture (settled in works of Alekseev-Bazlov-Rohr), the Zhelobenko invariants in the adjoint case and surprisingly the Bernstein-Gelfand-Gelfand operators introduced in their study of the cohomology of the flag variety.

Marco Budinich - One of the best experts on this subject based on the ideXlab platform.

  • The Extended Fock Basis of Clifford Algebra
    Advances in Applied Clifford Algebras, 2012
    Co-Authors: Marco Budinich
    Abstract:

    We investigate the properties of the Extended Fock Basis (EFB) of Clifford Algebras [1] with which one can replace the traditional multivector expansion of $${\mathcal{C} \ell(g)}$$ with an expansion in terms of simple (also: pure) spinors. We show that a Clifford Algebra with 2 m generators is the direct sum of 2^ m spinor subspaces S characterized as being left eigenvectors of Γ; furthermore we prove that the well known isomorphism between simple spinors and totally null planes holds only within one of these spinor subspaces. We also show a new symmetry between spinor and vector spaces: similarly to a vector space of dimension 2 m that contains totally null planes of maximal dimension m , also a spinor space of dimension 2^ m contains “totally simple planes”, subspaces made entirely of simple spinors, of maximal dimension m .

  • the extended fock basis of Clifford Algebra
    arXiv: Mathematical Physics, 2010
    Co-Authors: Marco Budinich
    Abstract:

    We investigate the properties of the Extended Fock Basis (EFB) of Clifford Algebras introduced in [1]. We show that a Clifford Algebra can be seen as a direct sum of multiple spinor subspaces that are characterized as being left eigenvectors of \Gamma. We also show that a simple spinor, expressed in Fock basis, can have a maximum number of non zero coordinates that equals the size of the maximal totally null plane (with the notable exception of vectorial spaces with 6 dimensions).

  • on computational complexity of Clifford Algebra
    Journal of Mathematical Physics, 2009
    Co-Authors: Marco Budinich
    Abstract:

    After a brief discussion of the computational complexity of Clifford Algebras, we present a new basis for even Clifford Algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford Algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich [“A spinorial formulation of the maximum clique problem of a graph,” J. Math. Phys. 47, 043502 (2006)].

  • on computational complexity of Clifford Algebra
    arXiv: Mathematical Physics, 2009
    Co-Authors: Marco Budinich
    Abstract:

    After a brief discussion of the computational complexity of Clifford Algebras, we present a new basis for even Clifford Algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford Algebra formulation of the NP-complete problem of the maximum clique of a graph introduced in a previous paper.

E Conte - One of the best experts on this subject based on the ideXlab platform.

  • quantum cognition a new demonstration of quantum collapse using mathematical formulation of quantum mechanics by using Clifford Algebra
    viXra, 2017
    Co-Authors: E Conte
    Abstract:

    Starting with 2010 we gave demonstration of Von Neumann postulates of measurements in quantum mechanics by using Clifford Algebra. In this paper we give proof by adding a further demonstration following our previous results on the logical origins of quantum mechanics and on the Algebraic nature of mental entities intended as abstract elements of Clifford Algebra

  • what is the reason to use Clifford Algebra in quantum cognition part ii information cognition and the principle of existence are intrinsically structured in the quantum model of reality
    Neuroquantology, 2013
    Co-Authors: E Conte
    Abstract:

    The thesis of this paper is that Information, Cognition and a Principle of Existence are intrinsically structured in the quantum model of reality. There is an intrinsic “factor of knowledge” that supports its structure. We reach such evidence by using the Clifford Algebra. In detail we analyze quantization in some traditional cases of interest in quantum mechanics and, in particular, in quantum harmonic oscillator, orbital angular momentum and hydrogen atom. We adopt the basic von Neumann results that projection operators represent logical statements and we demonstrated that are intrinsically structured in the cases of quantization that are taken in consideration. NeuroQuantology | March 2013 | Volume 11 | Issue 1| Page 34-46

  • an investigation on the basic conceptual foundations of quantum mechanics by using the Clifford Algebra
    arXiv: General Physics, 2011
    Co-Authors: E Conte
    Abstract:

    We review our approach to quantum mechanics adding also some new interesting results. We start by giving proof of two important theorems on the existence of the ) (Si A and 1 ,± i N Clifford Algebras. This last Algebra gives proof of the von Neumann basic postulates on the quantum measurement explaining thus in an Algebraic manner the wave function collapse postulated in standard quantum theory. In this manner we reach the objective to expose a self-consistent version of quantum mechanics. In detail we realize a bare bone skeleton of quantum mechanics recovering all the basic foundations of this theory on an Algebraic framework. We give proof of the quantum like Heisenberg uncertainty relations using only the basic support of the Clifford Algebra. In addition we demonstrate the well known phenomenon of quantum Mach Zender interference using the same Algebraic framework, as well as we give Algebraic proof of quantum collapse in some cases of physical interest by direct application of the theorem that we derive to elaborate the 1 ,± i N Algebra. We also discuss the problem of time evolution of quantum systems as well as the changes in space location, in momentum and the linked invariance principles. We are also able to re-derive the basic wave function of standard quantum mechanics by using only the Clifford Algebraic approach. In this manner we obtain a full exposition of standard quantum mechanics using only the basic axioms of Clifford Algebra.

  • on the logical origins of quantum mechanics demonstrated by using Clifford Algebra
    Neuroquantology, 2011
    Co-Authors: E Conte
    Abstract:

    Recently we have given proof of two theorems characterizing the Clifford Algebra. By using such two theorems we have reformulated the well known von Neumann postulate on quantum measurements giving evidence of the Algebraic manner in which quantum wave function collapse of quantum mechanics happens. In the present paper we introduce logic in Clifford Algebra interpreting its idempotents as logical statements. Using the previously mentioned theorems we demonstrate that the two basic foundations of quantum mechanics, as the indeterminism and the quantum interference, do not arise from physics itself but from logic. We advance the principles that there are levels of our reality in which we lose our possibility of unconditionally define the truth. At this level of reality we cannot separate matter per se from the basic foundations of the logic that we use to describe it. This logical relativism does not characterize classical mechanics but quantum physics. According to Y. F. Orlov, at quantum level the truths of logical statements about dynamic variables become dynamic variables themselves.

  • on the logical origins of quantum mechanics demonstrated by using Clifford Algebra a proof that quantum interference arises in a Clifford Algebraic formulation of quantum mechanics
    Electronic Journal of Theoretical Physics, 2011
    Co-Authors: E Conte
    Abstract:

    We review a rough scheme of quantum mechanics using the Clifford Algebra. Following the steps previously published in a paper by another author (31), we demonstrate that quantum interference arises in a Clifford Algebraic formulation of quantum mechanics. In 1932 J. von Neumann showed that projection operators and, in particular, quantum density matrices can be interpreted as logical statements. In accord with a previously obtained result by V. F Orlov , in this paper we invert von Neumann's result. Instead of constructing logic from quantum mechanics , we construct quantum mechanics from an extended classical logic. It follows that the origins of the two most fundamental quantum phenomena , the indeterminism and the interference of probabilities, lie not in the traditional physics by itself but in the logical structure as realized here by the Clifford Algebra. c