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Bin Shang

Publications and source records attributed to Bin Shang.

8 recordsLinked to original sources
arXiv2025

Weak Harnack estimates for a doubly nonlinear nonlocal p-Laplace equation

We establish a new type of weak Harnack estimates with optimal parabolic tail for the weak supersolutions to a doubly nonlinear nonlocal $p$-Laplace equation, which is modeled on the nonlocal Trudinger equation. Our results are achieved by employing the expansion of positivity and measure theoretical techniques. In particular, the weak Harnack estimates highlight the nonlocal feature, as we only require the local positivity of weak supersolutions instead of the global one.

math.AP↗
arXiv2024

Harnack inequality for doubly nonlinear mixed local and nonlocal parabolic equations

In this paper, we establish the Harnack inequality of nonnegative weak solutions to the doubly nonlinear mixed local and nonlocal parabolic equations. This result is obtained by combining a related comparison principle, a local boundedness estimate, and an integral Harnack-type inequality. Our proof is based on the expansion of positivity together with a comparison argument.

math.AP↗
arXiv2023

Regularity of weak solutions for mixed local and nonlocal double phase parabolic equations

We study the mixed local and nonlocal double phase parabolic equation \begin{align*} \partial_t u(x,t)-\mathrm{div}(a(x,t)|\nabla u|^{q-2}\nabla u) +\mathcal{L}u(x,t)=0 \end{align*} in $Q_T=Ω\times(0,T)$, where $\mathcal{L}$ is the nonlocal $p$-Laplace type operator. The local boundedness of weak solutions is proved by means of the De Giorgi-Nash-Moser iteration with the nonnegative coefficient function $a(x,t)$ being bounded. In addition, when $a(x,t)$ is Hölder continuous, we discuss the pointwise behavior and lower semicontinuity of weak supersolutions based on energy estimates and De Giorgi type lemma. Analogously, the corresponding results are also valid for weak subsolutions.

math.AP↗
arXiv2021

Regularity theory for mixed local and nonlocal parabolic p-Laplace equations

We investigate the mixed local and nonlocal parabolic $p$-Laplace equation \begin{align*} \partial_t u(x,t)-Δ_p u(x,t)+\mathcal{L}u(x,t)=0, \end{align*} where $Δ_p$ is the local $p$-Laplace operator and $\mathcal{L}$ is the nonlocal $p$-Laplace operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi-Nash-Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.

math.AP↗
arXiv2006

Prime Factorization in the Duality Computer

We give algorithms to factorize large integers in the duality computer. We provide three duality algorithms for factorization based on a naive factorization method, the Shor algorithm in quantum computing, and the Fermat's method in classical computing. All these algorithms are polynomial in the input size.

quant-ph↗
arXiv2006

Query complexity for searching multiple marked states from an unsorted database

An important and usual problem is to search all states we want from a database with a large number of states. In such, recall is vital. Grover's original quantum search algorithm has been generalized to the case of multiple solutions, but no one has calculated the query complexity in this case. We will use a generalized algorithm with higher precision to solve such a search problem that we should find all marked states and show that the practical query complexity increases with the number of marked states. In the end we will introduce an algorithm for the problem on a ``duality computer'' and show its advantage over other algorithms.

quant-ph↗
arXiv2006

A summary on two new algorithms for Grover's unsorted database search problem

In this summary we discuss two new algorithms for Grover's unsorted database search problem that claimed to have reached exponential speedup over Grover's original algorithm. One is in the quantum setting with "power queries" that allow for exponential reduction in the number of queries over Grover's original algorithm with "bit queries". The other is to use "dubit queries" on a duality computer - a new computing model uses a quantum system's wave-particle duality, which is able to achieve even greater computing power and better capability than existent quantum computers we have been discussing. We discuss the shortages and difficulties of both schemes as well.

quant-ph↗