An Optimization Problem Related to Minkowski’s Successive Minima

The purpose of this paper is to establish an inequality connecting the lattice point enumerator of a 0-symmetric convex body with its successive minima. To this end, we introduce an optimization problem whose solution refines former methods, thus producing a better upper bound. In particular, we show that an analogue of Minkowski’s second theorem on successive minima with the volume replaced by lattice point enumerator is true up to an exponential factor, whose base is approximately 1.64.     

Hadamard’s Formula Inside and Out

Our goal is to explore boundary variations of spectral problems from the calculus of moving surfaces point of view. Hadamard’s famous formula for simple Laplace eigenvalues under Dirichlet boundary conditions is generalized in a number of significant ways, including Neumann and mixed boundary conditions, multiple eigenvalues, and second order variations. Some of these formulas appear for the first time here. Furthermore, we present an analytical framework for deriving general formulas of the Hadamard type.     

Plateau’s Problem and Riemann’s Mapping Theorem

First the connection between conformal mappings and Plateau’s problem is pointed out. Then the relation between minimizers of area and energy under Plateau boundary conditions is discussed (joint work with F. Sauvigny). Finally, generalizations of the mapping theorems of Riemann and Koebe for Riemannian metrics are presented (joint work with H. von der Mosel).     

A Note on Erdös and Kac’s Identity: Boundary Crossing Probabilities of Brownian Motion Over Constant Boundaries

Methodology and Computing in Applied Probability - The finite Markov chain imbedding technique is an emerging approach for calculating boundary crossing probabilities for high-dimensional Brownian...     

Fractional Brownian Motion and Sheet as White Noise Functionals

In this short note, we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises, and the fractional white noise calculus developed by Hu and Фksendal follows directly from the classical white noise functional calculus. As examples we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts. An extension to the fractional Brownian sheet is also briefly discussed.     

Stratonovich’s signatures of Brownian motion determine Brownian sample paths

The signature of Brownian motion in $\mathbb R ^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\ge 2$ , almost all Brownian motion sample paths (running up to time $T$ ) are determined by their signature over $[0,T]$ .     

On the Maximum of a Brownian Motion and Its Location

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Power law Pólya’s urn and fractional Brownian motion

We introduce a natural family of random walks $S_n$ on $\mathbb{Z }$ that scale to fractional Brownian motion. The increments $X_n := S_n - S_{n-1} \in \{\pm 1\}$ have the property that given $\{ X_k : k < n \}$ , the conditional law of $X_n$ is that of $X_{n - k_n}$ , where $k_n$ is sampled independently from a fixed law $\mu $ on the positive integers. When $\mu $ has a roughly power law decay (precisely, when $\mu $ lies in the domain of attraction of an $\alpha $ -stable subordinator, for $0<\alpha <1/2$ ) the walks scale to fractional Brownian motion with Hurst parameter $\alpha + 1/2$ . The walks are easy to simulate and their increments satisfy an FKG inequality. In a sense we describe, they are the natural “fractional” analogues of simple random walk on $\mathbb{Z }$ .     

On Shallit’s Minimization Problem

Mathematical Notes - In Shallit’s problem (SIAM Review, 1994), it was proposed to justify a two-term asymptotics of the minimum of a rational function of $$n$$ variables defined as the sum of...     

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

We provide a partial result on Taylor’s modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analog for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence n-permutability for a fixed n, and satisfying a non-trivial congruence identity.     

On the Geometrical and Physical Meaning of Newton’s Solution to Kepler’s Problem

In the short treatise De Motu (1684),which serves as a precursor to the Principia Mathematica (1687),Newton essentially deals with the following two problems.     

Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem

Let K be a totally real number field, π an irreducible cuspidal representation of ${{\rm GL}_{2}(K){\backslash}{\rm GL}_{2}(\mathbb{A}K)}$ with unitary central character, and χ a Hecke character of conductor ${\mathfrak{q}}$ . Then ${L(1/2, \pi\oplus\chi) \ll (\mathcal{N}\mathfrak{q})^{\frac{1}{2}-\frac{1}{8}(1-2\theta)+\epsilon}}$ , where 0 ≤ θ ≤ 1/2 is any exponent towards the Ramanujan–Petersson conjecture (θ =  1/9 is admissible). The proof is based on a spectral decomposition of shifted convolution sums and a generalized Kuznetsov formula.     

Stationarity and Self-Similarity Characterization of the Set-Indexed Fractional Brownian Motion

The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin–Merzbach (J. Theor. Probab. 19(2):337–364, 2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to satisfy a strengthened definition of increment stationarity. This new definition for stationarity property allows us to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0<H<1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0<H<1.     

Laplace’s method for iterated complex Brownian integrals

A kind of Laplace’s method is developped for iterated stochastic integrals where integrators are complex standard Brownian motions. Then it is used to extend properties of Bougerol and Jeulin’s path transform in the random case when simple representations of complex semisimple Lie algebras are not supposed to be minuscule.     

A Note on Lewin’s Problem

The parameter l(G) for a primitive digraph G introduced by Lewin is the minimum positive integer k for which there are walks of both lengths k and k + 1 from some vertex u to some vertex v. We obtain upper bounds on l(G) if G is primitive ministrong, or G is just primitive and not necessarily ministrong, or G is primitive symmetric. We also discuss the numbers attainable as l(G).AMS Subject Classification (2000): 05C20, 15A48Partially supported by the National Natural Science Foundation of China (19771040) and the Guangdong Provincial Natural Science Foundation of China (990447).     

On Heilbronn’s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]², there must be three points which form a triangle of area at most O(1/n²). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]² where all triangles have area at least (log n/n²). Considering a generalization of this problem to dimensions d3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]^d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least (1/n^d). We improve on this lower bound by a logarithmic factor (log n).     

Further results on Hilbert’s Tenth Problem

Hilbert's Tenth Problem(HTP) asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring Z of integers.This was finally solved negatively by Matiyasevich in 1970.In this paper we obtain some further results on HTP over Z.We prove that there is no algorithm to determine for any P(z_1,...,z_9) ∈ Z[z_1,...,z_9] whether the equation P(z_1,...,z_9)=0 has integral solutions with z_9≥0.Consequently,there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z_1,...,z_(11))=0(with integer coefficients) in 11 unknowns has integral solutions,which provides the best record on the original HTP over Z.We also prove that there is no algorithm to test for any P(z_1,...,z_(17))∈Z[z_1,...,z_(17)] whether P(z_1,...,z_(17))=0 has integral solutions,and that there is a polynomial Q(z_1,...,z_(20))∈Z[z_1,...,z_(20)] such that {Q(z_1~2,...,z_(20)~2):z_1,...,z_(20)∈Z}∩ {0,1,2,...} coincides with the set of all primes.     

Hidden Oscillations in Dynamical Systems. 16 Hilbert’s Problem,Aizerman’s and Kalman’s Conjectures,Hidden Attractors in Chua’s Circuits

The present survey is devoted to efficient methods for localization of hidden oscillations in dynamical systems. Their application to Hilbert’s sixteenth problem for quadratic systems, Aizerman’s problem, and Kalman’s problem on absolute stability of control systems, and to the localization of chaotic hidden attractors (the basin of attraction of which does not contain neighborhoods of equilibria) is considered. The synthesis of the describing function method with the applied bifurcation theory and numerical methods for computing hidden oscillations is described.