Related papers: The real part of the complementary error function

Arithmetic progressions of length $3$ may be found in compact subsets of the reals that satisfy certain Fourier -- as well as Hausdorff -- dimensional requirements. It has been shown that a very similar result holds in the integers under…

We prove fractional boundary Hardy's inequality in dimension one for the critical case $sp =1$. Optimality of the inequality is obtained for any $p$. The extra logarithmic correction term appears in usual fashion. We also provide a concrete…

The fundamental gap of a domain is the difference between the first two eigenvalues of the Laplace operator. In a series of recent and celebrated works, it was shown that for convex domains in $\mathbb R^n$ and $\mathbb S^n$ with Dirichlet…

We consider the problem of approximation of a continuous function $f$ defined on a compact metric space $X$ by elements from a sum of two algebras. We prove a de la Vall\'{e}e Poussin type theorem, which estimates the approximation error…

Fault tolerant protocol assumes the application of error correction after every quantum gate. However, correcting errors is costly in terms of time and number of qubits. Here we demonstrate that quantum error correction can be applied…

The spectral deferred correction method is a variant of the deferred correction method for solving ordinary differential equations. A benefit of this method is that is uses low order schemes iteratively to produce a high order…

The set of all error--correcting block codes over a fixed alphabet with $q$ letters determines a recursively enumerable set of rational points in the unit square with coordinates $(R,\delta)$:= (relative transmission rate, relative minimal…

In a weak measurement the real and imaginary parts of a weak value participate in the shifts of the complementary variables of a pointer. While the real part represents the value of an observable in the limit of zero measurement strength,…

Random Fourier Features (RFF) is among the most popular and broadly applicable approaches for scaling up kernel methods. In essence, RFF allows the user to avoid costly computations on a large kernel matrix via a fast randomized…

We present a method of concatenated quantum error correction in which improved classical processing is used with existing quantum codes and fault-tolerant circuits to more reliably correct errors. Rather than correcting each level of a…

This article is an exposition of recent results on self-similar sets, asserting that if the dimension is smaller than the trivial upper bound then there are almost overlaps between cylinders. We give a heuristic derivation of the theorem…

For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local…

Elementary function operations such as sin and exp cannot in general be computed exactly on today's digital computers, and thus have to be approximated. The standard approximations in library functions typically provide only a limited set…

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors…

In this work we proof the following theorem which is, in addition to someother lemmas, our main result:\noindent \textbf{theorem}. Let$\ X=\{ ( x\_{1}\text{, }%t\_{1}) \text{, }( x\_{2}\text{, }t\_{2}) \text{, ..., }(x\_{n}\text{,…

The Riemann Hypothesis, originally proposed by the eminent mathematician Bernard Riemann in 1859, remains one of the most profound challenges in number theory. It posits that all non-trivial zeros of the Riemann zeta function {\zeta}(s) are…

State-of-the-art static analysis tools for verifying finite-precision code compute worst-case absolute error bounds on numerical errors. These are, however, often not a good estimate of accuracy as they do not take into account the…

Elementary function calls are a common feature in numerical programs. While their implementions in library functions are highly optimized, their computation is nonetheless very expensive compared to plain arithmetic. Full accuracy is,…

This paper is a corrigendum to the article 'On the ideal theorem for number fields`. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.

Khabibullin's conjecture for integral inequalities has two numeric parameters $n$ and $\alpha$ in its statement, $n$ being a positive integer and $\alpha$ being a positive real number. This conjecture is already proved in the case where…