More Like this

1. The quasi-periods of the Weierstrass zeta-function

   https://doi.org/10.4171/LEM/1085  

   Bonk, Mario (June 2025, L’Enseignement Mathématique)

   We study the ratiop=\eta_{1}/\eta_{2}of the quasi-periods of the Weierstrass\zeta-function in dependence of the ratio\tau=\omega_{1}/\omega_{2}of the generators of the underlying rank-2 lattice. We will give an explicit geometric description of the map\tau\mapsto p(\tau). As a consequence, we obtain an explanation of a theorem by Heins who showed thatpattains every value in the Riemann sphere infinitely often. Our main result is implicit in the classical literature, but it seems not to be very well known.Essentially, this is an expository paper. We hope that it is easily accessible and may serve as an introduction to these classical themes. 

   more » « less
2. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan; Zhang, Yanzhi (January 2022, Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$$ (- \Delta)^\frac{{ \alpha}}{{2}} $$\end{document} for \begin{document}$$ \alpha \in (0, 2) $$\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$$ {\mathcal O}(h^2) $$\end{document}, while \begin{document}$$ {\mathcal O}(h^4) $$\end{document} for quadratic basis functions with \begin{document}$ h $$\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$$ \alpha \in (0, 2) $$\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$$ u \in C^{m, l}(\bar{ \Omega}) $$\end{document} for \begin{document}$$ m \in {\mathbb N} $$\end{document} and \begin{document}$$ 0 < l < 1 $$\end{document}, our method has an accuracy of \begin{document}$$ {\mathcal O}(h^{\min\{m+l, \, 2\}}) $$\end{document} for constant and linear basis functions, while \begin{document}$$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $$\end{document}$ for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency. 

   more » « less
3. A motivic analogue of the $K(1)$-local sphere spectrum

   https://doi.org/10.4171/JEMS/1641  

   Balderrama, William; Ormsby, Kyle; Quigley, James D (April 2025, Journal of the European Mathematical Society)

   We identify the motivicKGL/2-local sphere as the fiber of\psi^{3}-1on(2,\eta)-completed HermitianK-theory, over any base scheme containing1/2. This is a motivic analogue of the classical resolution of theK(1)-local sphere, and extends to a description of theKGL/2-localization of an arbitrary motivic spectrum. Our proof relies on a novel conservativity argument that should be of broad utility in stable motivic homotopy theory. 

   more » « less
4. An Efficient ZK Compiler from SIMD Circuits to General Circuits

   https://doi.org/10.1007/s00145-024-09531-4  

   Bui, Dung; Chu, Haotian; Couteau, Geoffroy; Wang, Xiao; Weng, Chenkai; Yang, Kang; Yu, Yu (January 2025, Journal of Cryptology)

   Abstract We propose a generic compiler that can convert any zero-knowledge (ZK) proof for SIMD circuits to general circuits efficiently, and an extension that can preserve the space complexity of the proof systems. Our compiler can immediately produce new results improving upon state of the art.By plugging in our compiler to Antman, an interactive sublinear-communication protocol, we improve the overall communication complexity for general circuits from$$\mathcal {O}(C^{3/4})$$ $O\left(C^{3/4}\right)$to$$\mathcal {O}(C^{1/2})$$ $O\left(C^{1/2}\right)$. Our implementation shows that for a circuit of size$$2^{27}$$ $2^{27}$, it achieves up to$$83.6\times $$ $83.6\times$improvement on communication compared to the state-of-the-art implementation. Its end-to-end running time is at least$$70\%$$ $70\%$faster in a 10Mbps network.Using the recent results on compressed$$\varSigma $$ $\Sigma$-protocol theory, we obtain a discrete-log-based constant-round zero-knowledge argument with$$\mathcal {O}(C^{1/2})$$ $O\left(C^{1/2}\right)$communication and common random string length, improving over the state of the art that has linear-size common random string and requires heavier computation.We improve the communication of a designatedn-verifier zero-knowledge proof from$$\mathcal {O}(nC/B+n^2B^2)$$ $O(nC/B+n^2{B}^2)$to$$\mathcal {O}(nC/B+n^2)$$ $O(nC/B+n^2)$.To demonstrate the scalability of our compilers, we were able to extract a commit-and-prove SIMD ZK from Ligero and cast it in our framework. We also give one instantiation derived from LegoSNARK, demonstrating that the idea of CP-SNARK also fits in our methodology. 

   more » « less
5. Quantum adiabatic theorem for unbounded Hamiltonians with a cutoff and its application to superconducting circuits

   https://doi.org/10.1098/rsta.2021.0407  

   Mozgunov, Evgeny; Lidar, Daniel A. (January 2023, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We present a new quantum adiabatic theorem that allows one to rigorously bound the adiabatic timescale for a variety of systems, including those described by originally unbounded Hamiltonians that are made finite-dimensional by a cutoff. Our bound is geared towards the qubit approximation of superconducting circuits and presents a sufficient condition for remaining within the $2^n$-dimensional qubit subspace of a circuit model of $n$qubits. The novelty of this adiabatic theorem is that, unlike previous rigorous results, it does not contain $2^n$as a factor in the adiabatic timescale, and it allows one to obtain an expression for the adiabatic timescale independent of the cutoff of the infinite-dimensional Hilbert space of the circuit Hamiltonian. As an application, we present an explicit dependence of this timescale on circuit parameters for a superconducting flux qubit and demonstrate that leakage out of the qubit subspace is inevitable as the tunnelling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a $2^n\times 2^n$effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters. This article is part of the theme issue ‘Quantum annealing and computation: challenges and perspectives’. 

   more » « less