Related papers: Functions consistent with real numbers, and global…

We exhibit a class of "relatively curved" $\vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t))$, so that the pertaining multi-linear maximal function satisfies the sharp range of H\"{o}lder exponents, \[ \left\| \sup_{r > 0} \ \frac{1}{r}…

For functions in the Sobolev space $H^s$ and decreasing sequences $t_n\to 0$ we examine convergence almost everywhere of the generalized Schr\"odinger means on the real line, given by \[S^af(x,t_n)=\exp( it_n (-\partial_{xx})^{a/2})f(x);\]…

For an integer b>=2, let s_b(n) be the sum of the digits of the integer n when written in base b, and let S_b(N) be the sum of s_b(n) over n=0,...,N-1, so that S_b(N) is the sum of all b-ary digits needed to write the numbers 0,1,...,N-1.…

For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number $k\geq 2$, there is a function in the sequence with $k$ slopes. This…

Temporal difference learning with linear function approximation is a popular method to obtain a low-dimensional approximation of the value function of a policy in a Markov Decision Process. We give a new interpretation of this method in…

We address the decision problem for a fragment of real analysis involving differentiable functions with continuous first derivatives. The proposed theory, besides the operators of Tarski's theory of reals, includes predicates for…

We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of…

Let L_a(x) be Lebesgue's singular function with a real parameter a (0<a<1, a not equal to 1/2). As is well known, L_a(x) is strictly increasing and has a derivative equal to zero almost everywhere. However, what sets of x in [0,1] actually…

TD(0) is one of the most commonly used algorithms in reinforcement learning. Despite this, there is no existing finite sample analysis for TD(0) with function approximation, even for the linear case. Our work is the first to provide such…

We consider a probabilistic approach to compute the Wiener--Young $\Phi$-variation of fractal functions in the Takagi class. Here, the $\Phi$-variation is understood as a generalization of the quadratic variation or, more generally, the…

A real valued function $f$ defined on a real open interval $I$ is called $\Phi$-convex if, for all $x,y\in I$, $t\in[0,1]$ it satisfies $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+t\Phi\big((1-t)|x-y|\big)+(1-t)\Phi\big(t|x-y|\big), $$ where…

We consider transcendental entire functions of finite order for which the zeros and $1$-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that…

The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial…

In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The $n$-dimensional…

A time-varying empirical spectral process indexed by classes of functions is defined for locally stationary time series. We derive weak convergence in a function space, and prove a maximal exponential inequality and a…

We deal with the problem of asymptotic distribution of first return times to shrinking balls under iteration generated by a large general class of dynamical systems called weakly Markov. Our ultimate main result is that these distributions…

Purpose of writing this paper is to solve a transcendental function containing a product of a variable and its double exponential by a unique method of approximation. If the value of the said product is given, then its inverse function is…

We classify transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. We focus on the case where the escape is degenerate in the sense that points from…

In this paper we characterize the set of points where the lateral derivatives of the Takagi-Van der Waerden functions are infinite. We also prove that the set of points with infinite derivative has Hausdorff dimension one and Lebesgue…

A fast approximation to the Boys functions (related to the lower incomplete gamma function of half-integer parameter) by a single closed-form analytical expression for all argument values have been developed and tested. Besides the…