In algebraic geometry, enumerating or finding superspecial curves in positive characteristic $p$ is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus $4$ with automorphism group properly containing the Klein $4$-group. Executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for every $p$ with $19 \leq p < 500$, and in finding a single one for every $p$ with $19 \leq p < 7000$.
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While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.
A new area of application of methods of algebra of logic and to valued logic, which has emerged recently, is the problem of recognizing a variety of objects and phenomena, medical or technical diagnostics, constructing modern machines, checking test problems, etc., which can be reduced to constructing an optimal extension of the logical function to the entire feature space. For example, in logical recognition systems, logical methods based on discrete analysis and propositional calculus based on it are used to build their own recognition algorithms. In the general case, the use of a logical recognition method provides for the presence of logical connections expressed by the optimal continuation of a k-valued function over the entire feature space, in which the variables are the logical features of the objects or phenomena being recognized. The goal of this work is to develop a logical method for object recognition consisting of a reference table with logical features and classes of non-intersecting objects, which are specified as vectors from a given feature space. The method consists of considering the reference table as a logical function that is not defined everywhere and constructing an optimal continuation of the logical function to the entire feature space, which determines the extension of classes to the entire space.
We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solut ion over a more restricted monoid exists. As a corollary, we obtain a complexity classification for the same problem over groups.
We study the notion of $k$-stabilizer universal quantum state, that is, an $n$-qubit quantum state, such that it is possible to induce any stabilizer state on any $k$ qubits, by using only local operations and classical communications. These states generalize the notion of $k$-pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and $k$-vertex-minor universal graphs. First, we demonstrate the existence of $k$-stabilizer universal graph states that are optimal in size with $n=\Theta(k^2)$ qubits. We also provide parameters for which a random graph state on $\Theta(k^2)$ qubits is $k$-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of $k$-stabilizer universal graph states on $n = O(k^4)$ qubits. Both rely upon the incidence graph of the projective plane over a finite field $\mathbb{F}_q$. This provides a major improvement over the previously known explicit construction of $k$-pairable graph states with $n = O(2^{3k})$, bringing forth a new and potentially powerful family of multipartite quantum resources.
We investigate various forms of (model-theoretic) stability for hypergraphs and their corresponding strengthenings of the hypergraph regularity lemma with respect to partitions of vertices. On the one hand, we provide a complete classification of the various possibilities in the ternary case. On the other hand, we provide an example of a family of slice-wise stable 3-hypergraphs so that for no partition of the vertices, any triple of parts has density close to 0 or 1. In particular, this addresses some questions and conjectures of Terry and Wolf. We work in the general measure theoretic context of graded probability spaces, so all our results apply both to measures in ultraproducts of finite graphs, leading to the aforementioned combinatorial applications, and to commuting definable Keisler measures, leading to applications in model theory.
We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.
The Weisfeiler-Leman algorithm ($1$-WL) is a well-studied heuristic for the graph isomorphism problem. Recently, the algorithm has played a prominent role in understanding the expressive power of message-passing graph neural networks (MPNNs) and being effective as a graph kernel. Despite its success, $1$-WL faces challenges in distinguishing non-isomorphic graphs, leading to the development of more expressive MPNN and kernel architectures. However, the relationship between enhanced expressivity and improved generalization performance remains unclear. Here, we show that an architecture's expressivity offers limited insights into its generalization performance when viewed through graph isomorphism. Moreover, we focus on augmenting $1$-WL and MPNNs with subgraph information and employ classical margin theory to investigate the conditions under which an architecture's increased expressivity aligns with improved generalization performance. In addition, we show that gradient flow pushes the MPNN's weights toward the maximum margin solution. Further, we introduce variations of expressive $1$-WL-based kernel and MPNN architectures with provable generalization properties. Our empirical study confirms the validity of our theoretical findings.
Krylov methods rely on iterated matrix-vector products $A^k u_j$ for an $n\times n$ matrix $A$ and vectors $u_1,\ldots,u_m$. The space spanned by all iterates $A^k u_j$ admits a particular basis -- the \emph{maximal Krylov basis} -- which consists of iterates of the first vector $u_1, Au_1, A^2u_1,\ldots$, until reaching linear dependency, then iterating similarly the subsequent vectors until a basis is obtained. Finding minimal polynomials and Frobenius normal forms is closely related to computing maximal Krylov bases. The fastest way to produce these bases was, until this paper, Keller-Gehrig's 1985 algorithm whose complexity bound $O(n^\omega \log(n))$ comes from repeated squarings of $A$ and logarithmically many Gaussian eliminations. Here $\omega>2$ is a feasible exponent for matrix multiplication over the base field. We present an algorithm computing the maximal Krylov basis in $O(n^\omega\log\log(n))$ field operations when $m \in O(n)$, and even $O(n^\omega)$ as soon as $m\in O(n/\log(n)^c)$ for some fixed real $c>0$. As a consequence, we show that the Frobenius normal form together with a transformation matrix can be computed deterministically in $O(n^\omega \log\log(n)^2)$, and therefore matrix exponentiation~$A^k$ can be performed in the latter complexity if $\log(k) \in O(n^{\omega-1-\varepsilon})$, for $\varepsilon>0$. A key idea for these improvements is to rely on fast algorithms for $m\times m$ polynomial matrices of average degree $n/m$, involving high-order lifting and minimal kernel bases.
The Crank-Nicolson (CN) method is a well-known time integrator for evolutionary partial differential equations (PDEs) arising in many real-world applications. Since the solution at any time depends on the solution at previous time steps, the CN method is inherently difficult to parallelize. In this paper, we consider a parallel method for the solution of evolutionary PDEs with the CN scheme. Using an all-at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can prove that most eigenvalues of preconditioned matrices are equal to 1 and the others lie in the set: $\left\{z\in\mathbb{C}: 1/(1 + \alpha) < |z| < 1/(1 - \alpha)~{\rm and}~\Re{\rm e}(z) > 0\right\}$, where $0 < \alpha < 1$ is a free parameter. Besides, the efficient implementation of the proposed preconditioner is described. Given certain conditions, we prove that the preconditioned GMRES method exhibits a mesh-independent convergence rate. Finally, we will verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs.
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