Abstract Quantum walks are powerful tools for building quantum search algorithms. However, these success probabilities far below 1. Amplitude amplification is usually used to amplify probability, but the souffl'e problem follows. Only stop at right step can we achieve a maximum probability. Otherwise, as number of steps increases, probability may decrease, which will cause troubles in practical...

We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, hyperbolic equation and multiscale system with quantum algorithms (hence referred to as "quantum methods"). For equations we study impact explicit implicit discretizations on advantages over classical method. problem, find complexity both treatment scheme scales $\mathcal{O}(1/\v...

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $(k - 1)$-colorable, $L$ a list-assignment for $G$, then \textit{$L$-critical} not $L$-colorable but induced is. In 2014, Kostochka Yancey proved lower bound on the average degree an $n$-vertex $k$-critical tending to $k \frac{2}{k 1}$ large $n$ that tight infinitely many values $n$, they asked how their may be i...

In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields characterization (up to exceptional sets zero length) tangent points Jordan curve in terms finiteness associated $\varepsilon^2$-square function.

In this paper we show how ideas from spline theory can be used to construct a local basis for the space of translates general iterated Brownian Bridge kernel $k_{\beta,\varepsilon}$ $\beta\in\mathbb{N}$, $\varepsilon\geq 0$. simple case $\beta=1$, derive an explicit formula corresponding Lagrange basis, which allows us solve interpolation problems without inverting any linear system. We use pro...

We evaluate asymptotically the variance of number squarefree integers up to $x$ in short intervals length $H < x^{6/11 - \varepsilon}$ and arithmetic progressions modulo $q$ with $q > x^{5/11 + \varepsilon}$. On assumption respectively Lindel\"of Hypothesis Generalized we show that these ranges can be improved x^{2/3 x^{1/3 Furthermore obtaining a bound sharp factors $H^{\varepsilon}$ full rang...

Abstract We show that for any $\varepsilon&amp;gt;0$, $\alpha \in [0,\frac {1}{2})$, as $g\to \infty $ a generic finite-area genus $g$ hyperbolic surface with $n=O\left (g^{\alpha }\right )$ cusps, sampled probability arising from the Weil–Petersson metric on moduli space, has no non-zero eigenvalue of Laplacian below $\frac {1}{4}-\left (\frac {2\alpha +1}{4}\right )^{2}-\varepsilon $. For =0$...

Abstract In this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. particular, under some proper assumptions, prove that attractor $\{A_{\varepsilon }(t)\}_{t\in \mathbb{R}}$ { A ? <mml:m...

Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's in semi-classical regime. Here solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, sense that dynamics of states and induced observables can occur on different spatial temporal scales. Such finds many applications, including Born-O...