Piecewise Smooth Paths Research Articles

Toric geometry of path signature varieties

In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. The study of these signature varieties builds a bridge between algebraic geometry and stochastics, and allows a fruitful exchange of techniques, ideas, conjectures and solutions.In this paper we study the signature varieties of two very different classes of paths. The class of rough paths is a natural extension of the class of piecewise smooth paths. It plays a central role in stochastics, and its signature variety is toric. The class of axis-parallel paths has a peculiar combinatoric flavor, and we prove that it is toric in many cases.

A refinement of Grüss inequality for the complex integral

Abstract Assume that f and g are continuous on γ, γ ⊂ 𝔺 is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u and the complex Čebyšev functional is defined by 𝒟 γ ( f , g ) : = 1 w - u ∫ γ f ( z ) g ( z ) d z - 1 w - u ∫ γ f ( z ) d z 1 w - u ∫ γ g ( z ) d z . {{\cal D}_\gamma}\left({f,g} \right): = {1 \over {w - u}}\int_\gamma {f\left(z \right)} g\left(z \right)dz - {1 \over {w - u}}\int_\gamma {f\left(z \right)} dz{1 \over {w - u}}\int_\gamma {g\left(z \right)} dz. In this paper we establish some Grüss type inequalities for 𝒟 (f, g) under some complex boundedness conditions for the functions f and g.

On some Chebyshev type inequalities for thecomplex integral

Assume thatfandgare continuous onγ,γ⊂Cis a piecewisesmooth path parametrized byz(t), t∈[a, b]fromz(a) =utoz(b) =wwithw6=u, and thecomplex Chebyshev functionalis defined byDγ(f, g) :=1w−u∫γf(z)g(z)dz−1w−u∫γf(z)dz1w−u∫γg(z)dz.In this paper we establish some bounds for the magnitude of the functionalDγ(f, g)under Lipschitzian assumptions for the functionsfandg,and pro-vide a complex version for the well known Chebyshev inequality.

Rough path recursions and diffusion approximations

In this article, we consider diffusion approximations for a general class of stochastic recursions. Such recursions arise as models for population growth, genetics, financial securities, multiplicative time series, numerical schemes and MCMC algorithms. We make no particular probabilistic assumptions on the type of noise appearing in these recursions. Thus, our technique is well suited to recursions where the noise sequence is not a semi-martingale, even though the limiting noise may be. Our main theorem assumes a weak limit theorem on the noise process appearing in the random recursions and lifts it to diffusion approximation for the recursion itself. To achieve this, we approximate the recursion (pathwise) by the solution to a stochastic equation driven by piecewise smooth paths; this can be thought of as a pathwise version of backward error analysis for SDEs. We then identify the limit of this stochastic equation, and hence the original recursion, using tools from rough path theory. We provide several examples of diffusion approximations, both new and old, to illustrate this technique.

Control of mobile robot following a piecewise-smooth path

The paper describes an approach to the development of motion control system for a mobile robot following a piecewise smooth path set by the waypoints and parameterized by linear and circular segments. Control is synthesized using the differential geometrical method through nonlinear transformation of initial dynamic model. The main results presented are the model of spatial motion and relevant nonlinear control algorithms.

Dissimilarity cumulation theory in smoothly connected spaces

This is the third paper in the series introducing the Dissimilarity Cumulation theory and its main psychological application, Universal Fechnerian Scaling. The previously developed dissimilarity-based theory of path length is used to construct the notion of a smooth path, defined by the property that the ratio of the dissimilarity between its points to the length of the subtended fragment of the path tends to unity as the points get closer to each other. We consider a class of stimulus spaces in which for every path there is a series of piecewise smooth paths converging to it pointwise and in length; and a subclass of such spaces where any two sufficiently close points can be connected by a smooth “geodesic in the small”. These notions are used to construct a broadly understood Finslerian geometry of stimulus spaces representable by regions of Euclidean n -spaces. With an additional assumption of comeasurability in the small between the canonical psychometric increments of the first and second kind, this establishes a link between Universal Fechnerian Scaling and Multidimensional Fechnerian Scaling in Euclidean n -spaces. The latter was a starting point for our theoretical program generalizing Fechner’s idea that sensation magnitudes can be computed by integration of a local discriminability measure.

Gravitational Aharonov-Bohm Effect

We study a purely gravitational Aharonov-Bohm effect. The space-time curvature is concentrated in the quasiregular singularity of a cosmic string, outside of which space-time is (locally) flat. The symmetries of this field configuration are described by the groupoid symmetries rather than by the usual group symmetries. The groupoid in question is formed by homotopy classes of piecewise smooth paths in the cosmic string region. A gravitational counterpart of the Aharonov-Bohm effect occurs if the symmetry of the system, with respect to the groupoid action, is broken down.

Stochastic differential equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem

We show in this article how the theory of rough paths allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.

Unbounded Fredholm Modules and Spectral Flow

AbstractAn odd unbounded (respectively, p-summable) Fredholm module for a unital Banach *-algebra, A, is a pair (H,D) where A is represented on the Hilbert space, H, and D is an unbounded self-adjoint operator on H satisfying:(1) (1 + D2)-1 is compact (respectively, Trace_(1 + D2)-(p/2)_∞), and(2) ﹛a ∈ A | [D, a] is bounded﹜ is a dense *- subalgebra of A.If u is a unitary in the dense *-subalgebra mentioned in (2) thenuDu* = D + u[D, u*] = D + Bwhere B is a bounded self-adjoint operator. The pathis a “continuous” path of unbounded self-adjoint “Fredholm” operators. More precisely, we show thatis a norm-continuous path of (bounded) self-adjoint Fredholm operators. The spectral flow of this path is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as t runs from 0 to 1. This integer,recovers the pairing of the K-homology class [D] with the K-theory class [u].We use I.M. Singer's idea (as did E. Getzler in the θ-summable case) to consider the operator B as a parameter in the Banach manifold, Bsa(H), so that spectral flow can be exhibited as the integral of a closed 1-formon this manifold. Now, for B in ourmanifold, any X ∈ TB_Bsa(H)_ is given by an X in Bsa(H) as the derivative at B along the curve t→ B + tX in the manifold. Then we show that for m a sufficiently large half-integer:is a closed 1-form. For any piecewise smooth path {Dt = D + Bt} with D0 and D1 unitarily equivalent we show thatthe integral of the 1-form ã. If D0 and D1 are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form:for an integer. The unbounded case is proved by reducing to the bounded case via the map . We prove simultaneously a type II version of our results.

Almost sure approximation of Wong-Zakai type for stochastic partial differential equations

A solution to a stochastic partial differential equation (in the Stratonovitch form) is an almost sure limit of solutions to a sequence of approximated equations (with Brownian path w( t) being replaced by a piecewise smooth path w n ( t) approximating w( t)). This is achieved by employing a generalized Feynman-Kac formula of Pardoux and Rozovskii and proving the corresponding result for ordinary stochastic differential equations. Parabolic and hyperbolic (degenerate parabolic) evolution equations are studied.

Simplicial Algorithm to Find Zero Points of a Function with Special Structure on a Simplotope

In this paper we introduce a variable dimension simplicial algorithm on a cartesian product of unit simplices to find a zero point of a continuous function with special structure. The special structure of the function allows us to perform the linear programming pivot steps of the algorithm in a small system of equations. Moreover, a specific simplicial subdivision of the simplotope underlies the algorithm. The path of points generated by the algorithm approximately follows a piecewise smooth path in the simplotope. The latter path can be interpreted as being generated by an adjustment process. We discuss two applications, an international trade economy and an economy with increasing returns to scale. In both applications the zero points of the function induce equilibria in the economies.

Iterated Integrals of Differential Forms and Loop Space Homology

We continue to define iterated integrals 5w ... w, where wl, *., w, are 1forms. Such iterated integrals will be defined for forms wi, *-*, w, of arbitrary degrees on a manifold or, more generally, a suitably defined differentiable The space P(X) of piecewise smooth paths on a manifold X turns out to be a space. If wi, *.., w, are forms of respective degrees P1, ..., Pr on X, then the iterated integral Wi .. Wr is a form of degree Pi + * -1+ Prr on P(X). Let P(X; x0, xl) denote the subspace of P(X) consisting of paths from a given point x0 to another given point xl. We are interested in those iterated integrals or linear combinations of iterated integrals, which give rise to de Rham cohomology classes of QSX = P(X; x0, x0) and shall prove an iterated integral version of a loop space de Rham theorem, which includes the following