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At least 451 records · Page 25Linked to original sources
arXiv2013

Locally $C^*$ Algebras, $C^*$ Bundles and Noncommutative Spaces

This work provides a generalization of the Gelfand duality to the context of noncommutative locally $C^*$ algebras. Using a reformulation of a theorem proven by Dauns and Hofmann in the 60's we show that every locally $C^*$ algebra can be realized as the algebra of continuous sections of a $C^*$ bundle over a compactly generated topological space. This result is used then to show that on certain special cases locally $C^*$ algebras can be used to define certain sheaves of locally $C^*$ algebras that, inspired by the analogy with commutative geometry, we call noncommutative spaces. The last section provides some examples, motivated by mathematical physics, for this definition of noncommutative space. Namely we show that every local net of $C^*$ algebras defines a noncommutative space and, based on a loose generalization of the original construction by Doplicher, Fredenhagen and Roberts, construct what we propose to call a "locally covariant quantum spacetime".

math.OA↗
arXiv2013

Einstein's random walk and thermal diffusion

Thermal diffusion has been studied for over 150 years. Despite of the long history and the increasing importance of the phenomenon, the physics of thermal diffusion remains poorly understood. In this paper Ludwig's thermal diffusion is explained using Einstein's random walk. The only new structure added is the spatial heterogeneity of the random walk to reflect the temperature gradient of thermal diffusion. Hence, the walk length and the walk speed are location dependent functions in this paper. Then, a mathematical understanding of such a random walk gives the foundation of the thermal diffusion as clearly as the original homogeneous case of Einstein.

math-ph↗
arXiv2013

$N$-point locality for vertex operators: normal ordered products, operator product expansions, twisted vertex algebras

In this paper we study fields satisfying $N$-point locality and their properties. We obtain residue formulae for $N$-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants of $N$-point local fields which includes normal ordered products and coefficients of operator product expansions. We show that examples of $N$-point local fields include the vertex operators generating the boson-fermion correspondences of type B, C and D-A. We apply the normal ordered products of these vertex operators to the setting of the representation theory of the double-infinite rank Lie algebras $b_{\infty}, c_{\infty}, d_{\infty}$. Finally, we show that the field theory generated by $N$-point local fields and their descendants has a structure of a twisted vertex algebra.

math-ph↗
arXiv2013

An extension of the Dirac theory of constraints

Constructions introduced by Dirac for singular Lagrangians are extended and reinterpreted to cover cases when kernel distributions are either nonintegrable or of nonconstant rank, and constraint sets need not be closed.

math-ph↗
arXiv2013

Analytic Extension of the Birkhoff Normal Forms for the Free Rigid Body Dynamics on SO(3)

Birkhoff normal form is a power series expansion associated with the local behavior of the Hamiltonian systems near a critical point. It is known to be convergent for integrable systems under some non-degeneracy conditions. By means of an expression of the inverse of Birkhoff normal form by a period integral, analytic continuation of the Birkhoff normal forms is considered for the free rigid body dynamics on SO(3). The global behavior of the Birkhoff normal forms is clarified in relation to the monodromy of an elliptic fibration which naturally arises from the free rigid body dynamics.

math-ph↗
arXiv2013

Geometrical and physical models of abrasion

We extend the geometrical theory presented in [5] for collisional and frictional particle abrasion to include an independent physical equation for the evolution of mass and volume. We introduce volume weight functions as multipliers of the geometric equations and use these mutipliers to enforce physical volume evolution in the unified equations. The latter predict, in accordance with Sternberg's Law, exponential decay for volume evolution. We describe both the PDE versions, which are generalisations of Bloore's equations and their heuristic ODE approximations, called the box equations. The latter are suitable for tracking the collective abrasion of large particle populations. The mutual abrasion of identical particles, called the self-dual ows, play a key role in explaining geological scenarios. We give stability criteria for the self-dual ows in terms of the parameters of the physical volume evolution models and show that under reasonable assumptions these criteria can be met by physical systems. We also study a natural generalisation, the unidirectional Bloore equation, covering the case of unidirectional abrasion. We have previously shown that his equation admits travelling front solutions with circular profiles. More generally, in three dimensions, they are so-called linear or special Weingarten surfaces.

physics.geo-ph↗
arXiv2013

Asymptotic behaviour of the relativistic Boltzmann equation in the Robertson-Walker spacetime

In this paper, we study the relativistic Boltzmann equation in the spatially flat Robertson-Walker spacetime. For a certain class of scattering kernels, global existence of classical solutions is proved. We use the standard method of Illner and Shinbrot for the global existence and apply the splitting technique of Guo and Strain for the regularity of solutions. The main interest of this paper is to study the evolution of matter distribution, rather than the evolution of spacetime. We obtain the asymptotic behaviour of solutions and will understand how the expansion of the universe affects the evolution of matter distribution.

math-ph↗
arXiv2013

Partial Hamiltonian formalism, multi-time dynamics and singular theories

We formulate singular classical theories without involving constraints. Applying the action principle for the action (27) we develop a partial (in the sense that not all velocities are transformed to momenta) Hamiltonian formalism in the initially reduced phase space (with the canonical coordinates $q_{i},p_{i}$, where the number $n_{p}$ of momenta $p_{i}$, $i=1,\...,n_{p}$ (17) is arbitrary $n_{p}\leq n$, where $n$ is the dimension of the configuration space), in terms of the partial Hamiltonian $H_{0}$ (18) and $(n-n_{p})$ additional Hamiltonians $H_α$, $α=n_{p}+1,\...,n$ (20). We obtain $(n-n_{p}+1)$ Hamilton-Jacobi equations (25)-(26). The equations of motion are first order differential equations (33)-(34) with respect to $q_{i},p_{i}$ and second order differential equations (35) for $q_α$. If $H_{0}$, $H_α$ do not depend on $\dot{q}_α$ (42), then the second order differential equations (35) become algebraic equations (43) with respect to $\dot{q}_α$. We interpret $q_α$ as additional times by (45), and arrive at a multi-time dynamics. The above independence is satisfied in singular theories and $r_{W}\leq n_{p}$ (58), where $r_{W}$ is the Hessian rank. If $n_{p}=r_{W}$, then there are no constraints. A classification of the singular theories is given by analyzing system (62) in terms of $F_{αβ}$ (63). If its rank is full, then we can solve the system (62); if not, some of $\dot{q}_α$ remain arbitrary (sign of a gauge theory). We define new antisymmetric brackets (69) and (80) and present the equations of motion in the Hamilton-like form, (67)-(68) and (81)-(82) respectively. The origin of the Dirac constraints in our framework is shown: if we define extra momenta $p_α$ by (86), then we obtain the standard primary constraints (87), and the new brackets transform to the Dirac bracket. Quantization is discussed.

math-ph↗
arXiv2013

Some remarks on resonances in even-dimensional Euclidean scattering

The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions $d \geq 2$. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto $Λ$, the Riemann surface of the logarithm function. We relate the multiplicities of the poles of the continued scattering matrix to the multiplicities of the poles of the resolvent. Moreover, we show that the poles of the scattering matrix on the $m$th sheet of $Λ$ are related to the zeros of a scalar function defined on the physical sheet. This paper contains a number of results about "pure imaginary" resonances. As an example, in contrast with the odd-dimensional case, we show that in even dimensions there are no "purely imaginary" resonances on any sheet of $Λ$ for Schrödinger operators with potentials $0 \leq V \in L_0^\infty (\R^d)$.

math-ph↗
arXiv2013

An alternative description of braided monoidal categories

We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to be equivalent to the usual associativity, unit and braiding axioms. We also discuss the next dimensional version, that is, b-structures on bicategories. As an application, we show how special b-categories result in the Yang-Baxter equation, and how special b-bicategories produce Zamolodchikov's tetrahedron equation. Finally, we define a cohomology theory (the b-cohomology) which plays a role analogous to the one abelian group cohomology has for braided monoidal categories.

math.CT↗
arXiv2013

$E_8$ Bundles and Rigidity

In this paper, we establish rigidity and vanishing theorems for Dirac operators twisted by $E_8$ bundles.

math.DG↗
arXiv2013

On string density at the origin

In [V. Barcilon Explicit solution of the inverse problem for a vibrating string. J. Math. Anal. Appl. {\bf 93} (1983) 222-234] two boundary value problems were considered generated by the differential equation of a string $$ y^{\prime\prime}+λp(x)y=0, \ \ 0\leq x \leq L<+\infty \eqno{(*)} $$ with continuous real function $p(x)$ (density of the string) and the boundary conditions $y(0)=y(L)=0$ the first problem and $y^{\prime}(0)=y(L)=0$ the second one. In the above paper the following formula was stated $$ p(0)={1}{L^2μ_1}\mathop{\prod}\limits_{n=1}^{\infty}{λ_n^2}{μ_n μ_{n+1}} \eqno{(**)} $$ where $\{λ_k\}_{k=1}^{\infty}$ is the spectrum of the first boundary value problem and $\{μ_k\}_{k=1}^{\infty}$ of the second one. Rigorous proof of (**) was given in [C.-L. Shen On the Barcilon formula for the string equation with a piecewise continuous density function. Inverse Problems {\bf 21}, (2005) 635--655] under more restrictive conditions of piecewise continuity of $p^{\prime}(x)$. In this paper (**) was deduced using $$ p(0)=\lim\limits_{λ\to +\infty}({ϕ(L,-λ)}{λ^{{1}{2}}ψ(L,-λ)})^2 \eqno{(***}) $$ where $ϕ(x,λ)$ is the solution of (*) which satisfies the boundary conditions $ϕ(0)-1=ϕ^{\prime}(0)=0$ and $ψ(x,λ)$ is the solution of (*) which satisfies $ψ(0)=ψ^{\prime}(0)-1=0$. In our paper we prove that (***) is true for the so-called M.G. Krein's string which may have any nondecreasing mass distribution function $M(x)$ with finite nonzero $M^{\prime}(0)$. Also we show that (**) is true for a wide class of strings including those for which $M(x)$ is a singular function, i.e. $M^{\prime}(x)=p(x)\mathop{=}\limits^{a.e.}0$.

math-ph↗
arXiv2013

Note on the method of matched-asymptotic expansions for determining the force acting on a particle

This paper is an addendum to the article by Candelier, Mehaddi & Vauquelin (2013) where the motion of a particle in a stratified fluid is investigated theoretically, at small Reynolds and Péclet numbers. We review briefly the method of matched asymptotic expansions which is generally used in order to determine the force acting on a particle embedded in a given flow, in order to account for small, but finite, inertia effects. As part of this method, we present an alternative matching procedure, which is based on a series expansion of the far-field solution of the problem, performed in the sense of generalized functions. The way to perform such a series is presented succinctly and a simple example is provided.

physics.flu-dyn↗
arXiv2013

The resolvent algebra: Ideals and dimension

Let (X,σ) be a symplectic space admitting a complex structure and let R(X,σ) be the corresponding resolvent algebra, i.e. the C*-algebra generated by the resolvents of selfadjoint operators satisfying canonical commutation relations associated with (X,σ). In previous work this algebra was shown to provide a convenient framework for the analysis of quantum systems. In the present article its mathematical properties are elaborated with emphasis on its ideal structure. It is shown that R(X,σ) is always nuclear and, if X is finite dimensional, also of type I (postliminal). In the latter case dim(X) labels the isomorphism classes of the corresponding resolvent algebras. For X of arbitrary dimension, principal ideals are identified which are the building blocks for all other ideals. The maximal and minimal ideals of the resolvent algebra are also determined.

math.OA↗
arXiv2013

On turbulence: deciphering a renormalization flow out of an elliptic curve, II

Reaching for a better understanding of turbulence, a line of investigation was followed, its main presupposition being that each scale dependent state, in a general renormalization flow, is a state that can be modeled using a class of ninth degree polynomials. These polynomials are deduced from the Weierstrass models of a certain kind of elliptic curves. As the consequences of this presupposition unfolded, leading to the numerical study of a few samples of elliptic curves, the L functions associated with these later were considered. Their bifurcation diagrams were observed and their escape rates were determined. The consistency of such an approach was put to a statistical test, measuring the rank correlation between escape rates and values taken by these L functions on the point z=1+0i. In the most significant case, the rank correlation coefficient found, r_s, was about r_s=-0.78, with an associated p-value of an order of magnitude close to the (-69) power of 10.

math-ph↗
arXiv2013

Localisations of cobordism categories and invertible TFTs in dimension two

Cobordism categories have played an important role in classical geometry and more recently in mathematical treatments of quantum field theory. Here we will compute localisations of two-dimensional discrete cobordism categories. This allows us, up to equivalence, to determine the category of invertible two-dimensional topological field theories in the sense of Atiyah. We are able to treat the orientable, non-orientable, closed and open cases.

math-ph↗
arXiv2013

Universal scaling function for the two-dimensional Ising model in an external field: A pragmatic approach

We report an effective functional form for the spin-spin correlation function of the 2D Ising model as a function of temperature and field. Although the Ising model has been well studied, no analytical result for the spin-spin correlation function exists for arbitrary magnetic fields and temperatures. We show the validity of our form by comparison with simulations using the Wolff algorithm, and obtain remarkable precision by including analytic corrections to scaling. Given recent interest in comparing biomembrane heterogeneity to Ising criticality, our spin-spin correlation function may be used as a predictive quantitative measure for FRET or NMR membrane experiments.

cond-mat.stat-mech↗