Boundedness of planar jump discontinuities for homogeneous hyperbolic systems

Suppose that [Formula: see text] is a homogeneous constant coefficient strongly hyperbolic partial differential operator on [Formula: see text] and that [Formula: see text] is a characteristic hyperplane. Suppose that in a conic neighborhood of the conormal variety of [Formula: see text], the characteristic variety of [Formula: see text] is the graph of a real analytic function [Formula: see text] with [Formula: see text] identically equal to zero or the maximal possible value [Formula: see text]. Suppose that the source function [Formula: see text] is compactly supported in [Formula: see text] and piecewise smooth with singularities only on [Formula: see text]. Then the solution of [Formula: see text] with [Formula: see text] for [Formula: see text] is uniformly bounded on [Formula: see text]. Typically when [Formula: see text] on the conormal variety, the sup norm of the jump in the gradient of [Formula: see text] across [Formula: see text] grows linearly with [Formula: see text].

On finite sums of periodic functions

It is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

The eigenvalues’ function of the family of Sturm-Liouville operators and the inverse problems

We study the direct and inverse problems for the family of Sturm-Liouville operators, generated by fixed potential q and the family of separated boundary conditions. We prove that the union of the spectra of all these operators can be represented as a smooth surface (as the values of a real analytic function of two variables), which has specific properties. We call this function ”the eigenvalues function of the family of Sturm-Liouville operators (EVF)”. From the properties of this function we select those, which are sufficient for a function of two variables be the EVF a family of Sturm-Liouville operators.

Nearly Optimal Bernoulli Factories for Linear Functions

Suppose that X1, X2, . . . are independent identically distributed Bernoulli random variables with mean p. A Bernoulli factory for a function f takes as input X1, X2, . . . and outputs a random variable that is Bernoulli with mean f(p). A fast algorithm is a function that only depends on the values of X1, . . ., XT, where T is a stopping time with small mean. When f(p) is a real analytic function the problem reduces to being able to draw from linear functions Cp for a constant C > 1. Also it is necessary that Cp ⩽ 1 − ε for known ε > 0. Previous methods for this problem required extensive modification of the algorithm for every value of C and ε. These methods did not have explicit bounds on $\mathbb{E}[T]$ as a function of C and ε. This paper presents the first Bernoulli factory for f(p) = Cp with bounds on $\mathbb{E}[T]$ as a function of the input parameters. In fact, supp∈[0,(1−ε)/C]$\mathbb{E}[T]$ ≤ 9.5ε−1C. In addition, this method is very simple to implement. Furthermore, a lower bound on the average running time of any Cp Bernoulli factory is shown. For ε ⩽ 1/2, supp∈[0,(1−ε)/C]$\mathbb{E}[T]$≥0.004Cε−1, so the new method is optimal up to a constant in the running time.

On Hessian Limit Directions along Gradient Trajectories

Given a non-oscillating gradient trajectory |γ|of a real analytic function f, we show that the limit v of the secants at the limit point 0of |γ|along the trajectory |γ| is an eigenvector of the limit of the direction of the Hessian matrix Hess(f) at 0along |γ|. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting.