Searcharxiv⌕ Search

SEARCH · Searcharxiv

Results for “math.MP”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 recordsLinked to original sources
arXiv2009

Pair and Impar, Even and Odd Form Fields and Electromagnetism

In this paper after reviewing the Schouten and de Rham definition of impair and pair differential form fields (not to be confused with differential form fields of even and odd grades) we prove that in a relativistic spacetime it is possible (despite claims in contrary) to coherently formulate electromagnetism (and we believe any other physical theory) using only pair form fields or, if one wishes, using pair and impair form fields together, in an appropriate way. Those two distinct descriptions involve only a mathematical choice and do not seem to lead to any observable physical consequence if due care is taken. Moreover, we show in details that a formulation of electromagnetic theory in the Clifford bundle formalism of differential forms where the two Maxwell equations of the so called free metric approach becomes a single equation is compatible with both formulations of electromagnetism just mentioned above. Moreover we derive directly from Maxwell equation the density of force (coupling of the electromagnetic field with the charge current) that is a postulate in the free metric approach to electromagnetism. We recall also a formulation of the engineering version of Maxwell equations using electric and magnetic fields as objects of the same nature, i.e., without using polar and axial vectors.

math-ph↗
arXiv2009

Reflection positivity and phase transitions in lattice spin models

Reflection positivity (RP) is a property of Gibbs measures exhibited by a class of lattice spin systems that include the Ising, Potts and Heisenberg models. The RP property is useful because of its two basic consequences: infrared bound and chessboard estimates. These are one of basic (and rather efficient) tools for proving phase transitions in many models of physical interest. The notes presented hereby summarize the lectures on reflection positivity and its consequences that the author delivered at the Prague Summer School on Mathematical Statistical Mechanics in September 2006. The text features both the classical material on the subject from the late 1970s as well as some of the more recent developments.

math-ph↗
arXiv2013

The completeness of quantum theory for predicting measurement outcomes

The predictions that quantum theory makes about the outcomes of measurements are generally probabilistic. This has raised the question whether quantum theory can be considered complete, or whether there could exist alternative theories that provide improved predictions. Here we review recent work that considers arbitrary alternative theories, constrained only by the requirement that they are compatible with a notion of "free choice" (defined with respect to a natural causal order). It is shown that quantum theory is "maximally informative", i.e., there is no other compatible theory that gives improved predictions. Furthermore, any alternative maximally informative theory is necessarily equivalent to quantum theory. This means that the state a system has in such a theory is in one-to-one correspondence with its quantum-mechanical state (the wave function). In this sense, quantum theory is complete.

quant-ph↗
arXiv2021

The Universe from a Single Particle II

We continue to explore, in the context of a toy model, the hypothesis that the interacting universe we see around us could result from single particle (undergraduate) quantum mechanics via a novel spontaneous symmetry breaking (SSB) acting at the level of probability distributions on Hamiltonians (rather than on states as is familiar from both Ginzburg-Landau superconductivity and the Higgs mechanism). In an earlier paper [7] we saw qubit structure emerge spontaneously on $\mathbb{C}^4$ and $\mathbb{C}^8$, and in this work we see $\mathbb{C}^6$ spontaneously decomposing as $\mathbb{C}^2\otimes \mathbb{C}^3$ and very curiously $\mathbb{C}^5$ (and $\mathbb{C}^7$) splitting off one (one or three) directions and then factoring. This evidence provides additional support for the broad hypothesis: Nature will seek out tensor decompositions where none are present. We consider how this finding may form a basis for the origins of interaction and ask if it can be related to established foundational discussions such as string theory.

hep-th↗
arXiv2012

Global Classical Solutions of the Relativistic Vlasov-Darwin System with Small Cauchy Data: the Generalized Variables Approach

We show that a smooth, small enough Cauchy datum launches a unique classical solution of the relativistic Vlasov-Darwin (RVD) system globally in time. A similar result is claimed in Comm. Math. Sci. 6, 749-764 (2008) following the work in Int. Mat. Res. Not. 57191, 1-31 (2006). Our proof does not require estimates derived from the conservation of the total energy, nor those previously given on the transverse component of the electric field. These estimates are crucial in the references cited above. Instead, we exploit the formulation of the RVD system in terms of the generalized space and momentum variables. By doing so, we produce a simple a-priori estimate on the transverse component of the electric field. We widen the functional space required for the Cauchy datum to extend the solution globally in time, and we improve decay estimates given in Comm. Math. Sci. 6, 749-764 (2008) on the electromagnetic field and its space derivatives. Our method extends the constructive proof presented in "Collisionless kinetic equations from astrophysics: The Vlasov-Poisson system. Handbook of Differential Equations: Evolutionary Equations, Vol.3, Elsevier, 2007" to solve the Cauchy problem for the Vlasov-Poisson system with a small initial datum.

math-ph↗
arXiv2014

Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system I. The $2$-D and $2\frac 12$-D cases

Consider the relativistic Vlasov-Maxwell system with initial data of unrestricted size. In the two dimensional and the two and a half dimensional cases, Glassey-Schaeffer (1997, 1998, 1998) proved that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^\infty_x$ norms of the electromagnetic fields compared to Glassey-Schaeffer.

math.AP↗
arXiv2004

Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids

Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold (M,g). In this paper we study the restrictions on the topology and geometry of the fibres (the level sets) of the solutions f to (P1). We give a technique based on certain remarkable property of the fibres (the analytic representation property) for going from the initial PDE to a global analytical characterization of the fibres (the equilibrium partition condition). We study this analytical characterization and obtain several topological and geometrical properties that the fibres of the solutions must possess, depending on the topology of M and the metric tensor g. We apply these results to the classical problem in physics of classifying the equilibrium shapes of both Newtonian and relativistic static self-gravitating fluids. We also suggest a relationship with the isometries of a Riemannian manifold.

math-ph↗
arXiv2012

Continuum statistics of the Airy2 process

We develop an exact determinantal formula for the probability that the Airy$_2$ process is bounded by a function $g$ on a finite interval. As an application, we provide a direct proof that $\sup(\aip(x)-x^2)$ is distributed as a GOE random variable. Both the continuum formula and the GOE result have applications in the study of the end point of an unconstrained directed polymer in a disordered environment. We explain Johansson's [Joh03] observation that the GOE result follows from this polymer interpretation and exact results within that field. In a companion paper [MQR11] these continuum statistics are used to compute the distribution of the endpoint of directed polymers.

math.PR↗
arXiv2014

Orbit Approach to Separation of Variables in sl(3)-Related Integrable Systems

Using the orbit method we attempt to reveal geometric and algebraic meaning of separation of variables for the integrable systems on coadjoint orbits in an $\mathfrak{sl}(3)$ loop algebra. We consider two types of generic orbits embedded into a common manifold, endowed with two nonsingular Lie-Poisson brackets. We prove that separation of variables on orbits of both types is realized by the same variables of separation. We also construct the integrable systems on these orbits: a coupled 3-component nonlinear Schrödinger equation and an isotropic SU(3) Landau-Lifshitz equation.

nlin.SI↗
arXiv2021

Representations of the orthosymplectic Yangian

We give a complete description of the finite-dimensional irreducible representations of the Yangian associated with the orthosymplectic Lie superalgebra $\frak{osp}_{1|2}$. The representations are parameterized by monic polynomials in one variable, they are classified in terms of highest weights. We give explicit constructions of a family of elementary modules of the Yangian and show that a wide class of irreducible representations can be produced by taking tensor products of the elementary modules.

math.RT↗
arXiv2019

On J-equation

In this paper, we prove that for any Kähler metrics $ω_0$ and $χ$ on $M$, there exists $ω_φ=ω_0+\sqrt{-1}\partial\bar\partialφ>0$ satisfying the J-equation $\mathrm{tr}_{ω_φ}χ=c$ if and only if $(M,[ω_0],[χ])$ is uniformly J-stable. As a corollary, we can find many constant scalar curvature Kähler metrics with $c_1<0$. Using the same method, we also prove a similar result for the deformed Hermitian-Yang-Mills equation when the angle is in $(\frac{nπ}{2}-\fracπ{4},\frac{nπ}{2})$.

math.DG↗
arXiv2009

Complete characterization and synthesis of the response function of elastodynamic networks

The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function $W(ω)$, depending on the frequency $ω$, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and sufficient conditions for a given function $W(ω)$ to be the response function of an elastodynamic network, assuming there is no damping. In particular we construct an elastodynamic network that can mimic a suitable response in the frequency or time domain. Our characterization is valid for networks in three dimensions and also for planar networks, which are networks where all the elements, displacements and forces are in a plane. The network we design can fit within an arbitrarily small neighborhood of the convex hull of the terminal nodes, provided the springs and masses occupy an arbitrarily small volume. Additionally, we prove stability of the network response to small changes in the spring constants and/or addition of springs with small spring constants.

math-ph↗
arXiv2019

Two Groups in a Curie-Weiss Model with Heterogeneous Coupling

We discuss a Curie-Weiss model with two groups with different coupling constants within and between groups. For the total magnetisations in each group, we show bivariate laws of large numbers and a central limit theorem which is valid in the high temperature regime. In the critical regime, the total magnetisation normalised by $N^{3/4}$ converges to a non-trivial distribution which is not Gaussian, just as in the single-group Curie-Weiss model. Finally, we prove a kind of a `law of large numbers' in the low temperature regime, more precisely we prove that the empirical magnetisation converges in distribution to a mixture of two Dirac measures.

math.PR↗
arXiv2013

HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4

HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: one, pfq, is relevant for manipulations of hypergeometric functions_{p+1}F_p, and the second one, AppellF1F4, for manipulations with Appell hypergeometric functions F_1,F_2,F_3,F_4 of two variables.

math-ph↗
arXiv2006

Topology and Phase Transitions I. Preliminary Results

In this first paper, we demonstrate a theorem that establishes a first step toward proving a necessary topological condition for the occurrence of first or second order phase transitions: we prove that the topology of certain submanifolds of configuration space must necessarily change at the phase transition point. The theorem applies to smooth, finite-range and confining potentials V bounded below, describing systems confined in finite regions of space with continuously varying coordinates. The relevant configuration space submanifolds are both the level sets {Σ_v := V_N^{-1} (v)}_{v \in R} of the potential function V_N and the configuration space submanifolds enclosed by the Σ_{v} defined by {M_v := V_N^{-1} ((-\infty,v])}_{v \in R}, which are labeled by the potential energy value v, and where N is the number of degrees of freedom. The proof of the theorem proceeds by showing that, under the assumption of diffeomorphicity of the equipotential hypersurfaces {Σ_v}_{v \in R}, as well as of the ${M_v}_{v \in R}, in an arbitrary interval of values for \vb=v/N, the Helmoltz free energy is uniformly convergent in N to its thermodynamic limit, at least within the class of twice differentiable functions, in the corresponding interval of temperature. This preliminary theorem is essential to prove another theorem - in paper II - which makes a stronger statement about the relevance of topology for phase transitions.

math-ph↗