Decay rate of iterated integrals of branched rough paths

Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals.     

D'Alembert's proof of the fundamental theorem of algebra

D'Alembert's proof of the fundamental theorem of algebra (FTA), the first published, is still widely misunderstood. Typical of d'Alembert, his work is bold and imaginative but in need of significant repair. The proof is examined in detail, in both the 1746 and 1754 versions, along with commentary over 250 years and recent efforts to revive d'Alembert's reputation. A particular challenge is to work with algebraic equations while avoiding dependence on the FTA itself. A repaired version is offered.     

H-convex standard fundamental domain of a subgroup of a modular group

The two major ways of obtaining fundamental domains for discrete subgroups of SL(2,?) are the Dirichlet Polygon construction (see Lehner in Discontinuous Groups and Automorphic Functions, American Mathematical Society, Providence, 1964) and Ford’s construction (see Ford in Automorphic Functions, McGraw–Hill, New York, 1929). Each of these two methods yield a hyperbolically convex fundamental domain for any discrete subgroup of SL(2,?).However, the Dirichlet polygon construction and Ford’s construction are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction and their reliance on knowing almost all elements of the group under discussion.

A third-and most important and practical-method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. Let Γ₂ be a subgroup of Γ₁ and

$\Gamma_{1}=\Gamma_{2}\cdot \{L_{1},L_{2},\ldots,L_{m}\}.$

If \(\mathbb{F}\) is a fundamental domain of the bigger group Γ₁, then the set

$\mathcal{R}_{\Gamma}=\Biggl(\overline{\bigcup_{k=1}^{m}L_{k}(\mathbb{F})}\,\Biggr)^{o}$

(1)

is a fundamental domain of Γ₂. One can ask at this juncture, is it possible to choose the right cosets suitably so that the set ?_Γ is hyperbolically convex? We will answer this question affirmatively for

$\Gamma_{1}=\Gamma(1)\quad \mbox{and}\quad \mathbb{F}=\biggl\{\tau \in \mathbb{H}:|\tau|>1\ \&;\ |\mathrm{Re}(\tau)|<\frac{1}{2}\biggr\}.$

     

Lie group symmetries as integral transforms of fundamental solutions

We obtain fundamental solutions for PDEs of the form ut=σxγuxx+f(x)ux−μxru by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be reduced to inverting a Laplace transform or some other classical transform.     

On rational invariants of the group

We prove rationality of the field of invariants in several variables of a minimal irreducible representation of a simple algebraic group of type over an algebraically closed field of characteristic zero.

     

Exponential Hilbert series and the Stirling numbers of the second kind

We consider the exponential generating function whose coefficients encode the dimensions of irreducible highest weight representations which lie on a given ray in the dominant chamber of the weight lattice. This formal power series can be considered as an exponential version of the Hilbert series of a flag variety. In this context, we compute a simple closed form for the exponential generating function in terms of finitely many differential operators and the Stirling polynomials. We prove that this series converges to a product of a rational polynomial and an exponential, and that, by summing the constant term and linear coefficient of this polynomial, we recover the dimension of the representation.     

Flat orbits and kernels of irreducible representations of the group algebra of a completely solvable Lie group

We show that the kernel of an irreducible unitary representation π of the group algebra L¹(G) of a completely solvable Lie group G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit Oπ of π if and only if this orbit Oπ is flat. This is a generalization of a result obtained before for nilpotent Lie groups.     

Asymptotics of matrix integrals and tensor invariants of compact Lie groups

In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant-theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of permutation operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces.

     

Free and non-free subgroups of the fundamental group of the Hawaiian Earrings

The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called “The Hawaiian Earrings”. The fundamental group of this space is known to be a subgroup of the inverse limit of the finitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space usually serves as the simplest example in this context. This paper contributes to understanding this group and corresponding phenomena by pointing out that several subgroups that are constructed according to similar schemes partially turn out to be free and not to be free. Amongst them is a countable non-free subgroup, and an uncountable free subgroup that is not contained in two other free subgroups that have recently been found. This group, although free, contains infinitely huge “virtual powers”, i.e. elements of the fundamental group of that kind that are usually used in proofs that this fundamental group is not free, and, although this group contains all homotopy classes of paths that are associated with a single loop of the Hawaiian Earrings, this system of ‘natural generators’ can be proven to be not contained in any free basis of this free group.     

The fundamental group of a locally finite graph with ends

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the free groups π₁(G^′) with G^′⊆G finite.     

On a relative uniform completion of an archimedean lattice ordered group

The notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups. Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) of G is dealt with. It is known that exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in . Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

A note on the fundamental group of Finsler manifolds with nonnegative flag curvature

In this note we prove that the fundamental group of any forward complete Finsler manifold with nonnegative flag curvature is finitely generated provided the line integral of T-curvature is small. In particular, the fundamental group of any forward complete Berwald manifold with nonnegative flag curvature is finitely generated.     

Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds

A substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.     

Deviation inequalities and the law of iterated logarithm on the path space over a loop group

A law of iterated logarithm (LIL) in small time and an asymptotic estimate of modulus of continuity are proved for Brownian motion on the loop group ?(G) over a compact connected Lie group G. Upper bounds are obtained via infinite-dimensional deviation inequalities for functionals on the path space ?(?(G)) on ?(G), such as the supremum of Brownian motion on ?(G), which are proved from the Clark–Ocone formula on ?(?(G)). The lower bounds rely on analog finite-dimensional results that are proved separately on Riemannian path space.     

First integrals and bifurcations of a Lane-Emden equation of the second kind

First integrals admitted by an approximate Lane-Emden equation modelling a thermal explosion in a rectangular slab and cylindrical vessel are investigated. By imposing the boundary conditions on the first integrals we obtain a nonlinear relationship between the temperature at the center of the vessel and the temperature gradient at the wall of the vessel. For a rectangular slab the presence of a bifurcation indicates multivalued solutions for the temperature at the center of the vessel when the temperature gradient at the wall is fixed. For a cylindrical vessel we find a bifurcation indicating multivalued solutions for the temperature gradient at the walls of the vessel when the temperature at the center of the vessel is fixed.     

Generalized multiple stochastic integrals and the representation of wiener functionals

In this paper we study a generalized multiple stochastic integral for non-adapted integrands following Skorohod's approach. The main properties of this integral are derived. In particular, we prove a Fubini type result and discuss the relation of this multiple integral to the Malliavin calculus. It turns out that this integral includes other kinds of multiple stochastic integrals like those of Hajek and Wong. Finally, we apply these results to the representation of functionals of the multiparameter Wiener process, obtaining explicit formulas for the kernels of the representation in terms of conditional expectations of Malliavin derivatives     

Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

We consider the highly oscillatory integral $F(w):={\int }_{-\infty }^{\infty }{e}^{iw(t^{K+2}+e^{i\theta }{t}^p)}g(t)dt$ for large positive values of w, , K and p positive integers with $1\le p\le K$, and $g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when $w\to +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. For $p\ne 1$, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p; the special case $p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals ${\mathrm{\Psi }}_K(x_1,x_2,\text{…},x_K)$ for large values of one of its variables, say $x_p$, and bounded values of the remaining ones. This family of integrals may be written in the form $F(w)$ for appropriate values of the parameters w, θ and the function $g(t)$. Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large $|x_p|$. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.