Search results for "Linear"

From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture

2020

Abstract In this paper we study the joint convexity/concavity of the trace functions Ψ p , q , s ( A , B ) = Tr ( B q 2 K ⁎ A p K B q 2 ) s , p , q , s ∈ R , where A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of ( p , q , s ) ∈ R 3 for Ψ p , q , s to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of ( α , z ) for α-z Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψ p…

Chebyshev’s Method on Projective Fluids

2020

We demonstrate the acceleration potential of the Chebyshev semi-iterative approach for fluid simulations in Projective Dynamics. The Chebyshev approach has been successfully tested for deformable bodies, where the dynamical system behaves relatively linearly, even though Projective Dynamics, in general, is fundamentally nonlinear. The results for more complex constraints, like fluids, with a particular nonlinear dynamical system, remained unknown so far. We follow a method describing particle-based fluids in Projective Dynamics while replacing the Conjugate Gradient solver with Chebyshev’s method. Our results show that Chebyshev’s method can be successfully applied to fluids and potentially…

A remark on weakly convex continuous mappings in topological linear spaces

2009

Abstract Let C be a compact convex subset of a Hausdorff topological linear space and T : C → C a continuous mapping. We characterize those mappings T for which T ( C ) is convexly totally bounded.

TOPOLOGICAL PARTIAL *-ALGEBRAS: BASIC PROPERTIES AND EXAMPLES

1999

Let [Formula: see text] be a partial *-algebra endowed with a topology τ that makes it into a locally convex topological vector space [Formula: see text]. Then [Formula: see text] is called a topological partial *-algebra if it satisfies a number of conditions, which all amount to require that the topology τ fits with the multiplier structure of [Formula: see text]. Besides the obvious cases of topological quasi *-algebras and CQ*-algebras, we examine several classes of potential topological partial *-algebras, either function spaces (lattices of Lp spaces on [0, 1] or on ℝ, amalgam spaces), or partial *-algebras of operators (operators on a partial inner product space, O*-algebras).

Fibre Bundle for Spin and Charge in General Relativity

2000

The Lorentzian and spin structures of general relativity are shown to allow a natural extension, by means of which the set of possible electromagnetic bundles is linked to the topology and geometry of the underlying causal structure. Further, both the Dirac operator and the electromagnetic potential are obtainable from a single linear connection 1-form.

2014

This paper investigates the finite-time distributedL2–L∞consensus control problem of multiagent systems with parameter uncertainties. The relative states of neighboring agents are used to construct the control law and some agents know their own states. By substituting the control input into multiagent systems, an augmented closed-loop system is obtained. Then, we analyze its finite-time boundedness (FTB) and finite-timeL2–L∞performance. A sufficient condition for the existence of the designed controller is given with the form of linear matrix inequalities (LMIs). Finally, simulation results are described.

Spectral WENO schemes with Adaptive Mesh Refinement for models of polydisperse sedimentation

2012

The sedimentation of a polydisperse suspension with particles belonging to N size classes (species) can be described by a system of N nonlinear, strongly coupled scalar first-order conservation laws. Its solutions usually exhibit kinematic shocks separating areas of different composition. Based on the so-called secular equation [J. Anderson, Lin. Alg. Appl. 246, 49–70 (1996)], which provides access to the spectral decomposition of the Jacobian of the flux vector for this class of models, Burger et al. [J. Comput. Phys. 230, 2322–2344 (2011)] proposed a spectral weighted essentially non-oscillatory (WENO) scheme for the numerical solution of the model. It is demonstrated that the efficiency …

Conservation Laws and Asymptotic Behavior of a Model of Social Dynamics

2008

Abstract A conservative social dynamics model is developed within a discrete kinetic framework for active particles, which has been proposed in [M.L. Bertotti, L. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Mod. Meth. Appl. Sci. 14 (2004) 1061–1084]. The model concerns a society in which individuals, distinguished by a scalar variable (the activity) which expresses their social state, undergo competitive and/or cooperative interactions. The evolution of the discrete probability distribution over the social state is described by a system of nonlinear ordinary differential equations. The asymptotic trend of their solutions…

Computation of travelling wave solutions of scalar conservation laws with a stiff source term

2003

Abstract In this paper we propose a nonoscillatory numerical technique to compute the travelling wave solution of scalar conservation laws with a stiff source term. This procedure is based on the dynamical behavior described by the associated stationary ODE and it reduces/avoids numerical errors usually encountered with these problems, i.e., spurious oscillations and incorrect wave propagation speed. We combine this treatment with either the first order Lax–Friedrichs scheme or the second order Nessyahu–Tadmor scheme. We have tested several model problems by LeVeque and Yee for which the stiffness coefficient can be increased. We have also tested a problem with a nonlinear flux and a discon…

On the hyperbolicity of certain models of polydisperse sedimentation

2012

The sedimentation of a polydisperse suspension of small spherical particles dispersed in a viscous fluid, where particles belong to N species differing in size, can be described by a strongly coupled system of N scalar, nonlinear first-order conservation laws for the evolution of the volume fractions. The hyperbolicity of this system is a property of theoretical importance because it limits the range of validity of the model and is of practical interest for the implementation of numerical methods. The present work, which extends the results of R. Burger, R. Donat, P. Mulet, and C.A. Vega (SIAM Journal on Applied Mathematics 2010; 70:2186–2213), is focused on the fluxes corresponding to the …