Arthur R. C. McCray, Tao Zhou, Saugat Kandel
et al.
AbstractThe manipulation and control of nanoscale magnetic spin textures are of rising interest as they are potential foundational units in next-generation computing paradigms. Achieving this requires a quantitative understanding of the spin texture behavior under external stimuli using in situ experiments. Lorentz transmission electron microscopy (LTEM) enables real-space imaging of spin textures at the nanoscale, but quantitative characterization of in situ data is extremely challenging. Here, we present an AI-enabled phase-retrieval method based on integrating a generative deep image prior with an image formation forward model for LTEM. Our approach uses a single out-of-focus image for phase retrieval and achieves significantly higher accuracy and robustness to noise compared to existing methods. Furthermore, our method is capable of isolating sample heterogeneities from magnetic contrast, as shown by application to simulated and experimental data. This approach allows quantitative phase reconstruction of in situ data and can also enable near real-time quantitative magnetic imaging.
The article provides the theoretical framework of Probabilistic Shoenfield Machines (PSMs), an extension of the classical Shoenfield Machine that models randomness in the computation process. PSMs are introduced in contexts where deterministic computation is insufficient, such as randomized algorithms. By allowing transitions to multiple possible states with certain probabilities, PSMs can solve problems and make decisions based on probabilistic outcomes, thus expanding the variety of possible computations. We provide an overview of PSMs, detailing their formal definitions, the computation mechanism, and their equivalence with Non-deterministic Shoenfield Machines (NSMs)
We introduce a novel conceptual Case Frame model that represents the content of cases involving statutory interpretation within civil law frameworks, accompanied by an associated argument scheme enriched with critical questions. By validating our approach with a modest dataset, we demonstrate its robustness and practical applicability. Our model not only provides a structured method for analyzing statutory interpretation but also highlights the distinct needs of lawyers operating under statutory law compared to those reasoning with common law precedents. The model presented here is a step towards developing a hybrid Machine Learning and Argumentation system that includes a module for constructing well-structured arguments from textual datasets.
In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured matrices and computing their kernels. The challenging problem is to reduce the size of the corresponding matrices. In this paper, we show how to convert the problem of computing recurrence relations of multi-dimensional sequences into computing the orthogonal of certain ideals as subvector spaces of the dual module of polynomials. We propose an algorithm using efficient dual module computation algorithms. We present a complexity bound for this algorithm, carry on experiments using Maple implementation, and discuss the cases when using this algorithm is much faster than the existing approaches.
Consider a sparse polynomial in several variables given explicitly as a sum of non-zero terms with coefficients in an effective field. In this paper, we present several algorithms for factoring such polynomials and related tasks (such as gcd computation, square-free factorization, content-free factorization, and root extraction). Our methods are all based on sparse interpolation, but follow two main lines of attack: iteration on the number of variables and more direct reductions to the univariate or bivariate case. We present detailed probabilistic complexity bounds in terms of the complexity of sparse interpolation and evaluation.
Identities compactly describe properties of a mathematical expression and can be leveraged into faster and more accurate function implementations. However, identities must currently be discovered manually, which requires a lot of expertise. We propose a two-phase synthesis and deduplication pipeline that discovers these identities automatically. In the synthesis step, a set of rewrite rules is composed, using an e-graph, to discover candidate identities. However, most of these candidates are duplicates, which a secondary deduplication step discards using integer linear programming and another e-graph. Applied to a set of 61 benchmarks, the synthesis phase generates 7215 candidate identities which the deduplication phase then reduces down to 125 core identities.
Let $\mathcal{C}$ be a plane curve given by an equation $f(x,y)=0$ with $f\in K[x][y]$ a monic squarefree polynomial. We study the problem of computing an integral basis of the algebraic function field $K(\mathcal{C})$ and give new complexity bounds for three known algorithms dealing with this problem. For each algorithm, we study its subroutines and, when it is possible, we modify or replace them so as to take advantage of faster primitives. Then, we combine complexity results to derive an overall complexity estimate for each algorithm. In particular, we modify an algorithm due to Böhm et al. and achieve a quasi-optimal runtime.
Alkali‐metal scandium oxoselenates(IV) ASc[SeO3]2 (A = Na – Cs) are known since a few years and a hydrothermal synthesis was used to obtain them. In our new studies we applied a flux‐supported solid‐state reaction and produced colorless single crystals as well. All representatives ASc[SeO3]2 with A = Na – Cs crystallize in the orthorhombic space group Pnma, in contrast to earlier reports for hexagonal RbSc[SeO3]2. Furthermore we have extended this field with some crystals showing a mixed occupation on the alkali‐metal site, namely (K,Na)Sc[SeO3]2, (Rb,K)Sc[SeO3]2, and (Cs,Rb)Sc[SeO3]2. Since all of them contain [ScO6]9– octahedra and [SeO3]2– ψ1‐tetrahedra the diverse connectivity of the distinct alkali‐metal centered oxygen polyhedra differentiates the compounds with the smaller alkali metals (A′ = Na and K) from those with the bigger ones (A′′ = Rb and Cs). For the mixed crystals the amount of smaller or bigger alkali metal is responsible, which design is chosen by the system. This forces the mixed crystal (Rb,K)Sc[SeO3]2 with a higher amount of potassium instead of rubidium to crystallize isotypically with KSc[SeO3]2 and NaSc[SeO3]2, whereas the pure rubidium compound RbSc[SeO3]2 adopts the CsSc[SeO3]2‐type structure. These findings are supported by single‐crystal Raman spectroscopy.
This paper extends the classical Ostrogradsky-Hermite reduction for rational functions to more general functions in primitive extensions of certain types. For an element $f$ in such an extension $K$, the extended reduction decomposes $f$ as the sum of a derivative in $K$ and another element $r$ such that $f$ has an antiderivative in $K$ if and only if $r=0$; and $f$ has an elementary antiderivative over $K$ if and only if $r$ is a linear combination of logarithmic derivatives over the constants when $K$ is a logarithmic extension. Moreover, $r$ is minimal in some sense. Additive decompositions may lead to reduction-based creative-telescoping methods for nested logarithmic functions, which are not necessarily $D$-finite.
Seung Gyu Hyun, Stephen Melczer, Catherine St-Pierre
We present an algorithm which computes the $D^{th}$ term of a sequence satisfying a linear recurrence relation of order $d$ over a field $K$ in $O( \mathsf{M}(\bar d)\log(D) + \mathsf{M}(d)\log(d))$ operations in $K$, where $\bar d \leq d$ is the degree of the squarefree part of the annihilating polynomial of the recurrence and $\mathsf{M}$ is the cost of polynomial multiplication in $K$. This is a refinement of the previously optimal result of $O( \mathsf{M}(d)\log(D) )$ operations, due to Fiduccia.
We compute the free energy of the planar monomer-dimer model. Unlike the classical planar dimer model, an exact solution is not known in this case. Even the computation of the low-density power series expansion requires heavy and nontrivial computations. Despite of the exponential computational complexity, we compute almost three times more terms than were previously known. Such an expansion provides both lower and upper bound for the free energy, and allows to obtain more accurate numerical values than previously possible. We expect that our methods can be applied to other similar problems.
Shaoshi Chen, Mark van Hoeij, Manuel Kauers
et al.
Continuing a series of articles in the past few years on creative telescoping using reductions, we adapt Trager's Hermite reduction for algebraic functions to fuchsian D-finite functions and develop a reduction-based creative telescoping algorithm for this class of functions, thereby generalizing our recent reduction-based algorithm for algebraic functions, presented at ISSAC 2016.
Sajjad Rahmany, Abdolali Basiri, Benyamin M. -Alizadeh
There are several efficient methods to solve linear interval polynomial systems in the context of interval computations, however, the general case of interval polynomial systems is not yet covered as well. In this paper we introduce a new elimination method to solve and analyse interval polynomial systems, in general case. This method is based on computational algebraic geometry concepts such as polynomial ideals and Groebner basis computation. Specially, we use the comprehensive Groebner system concept to keep the dependencies between interval coefficients. At the end of paper, we will state some applications of our method to evaluate its performance.
We design an algorithm for computing the $p$-curvature of a differential system in positive characteristic $p$. For a system of dimension $r$ with coefficients of degree at most $d$, its complexity is $\softO (p d r^ω)$ operations in the ground field (where $ω$ denotes the exponent of matrix multiplication), whereas the size of the output is about $p d r^2$. Our algorithm is then quasi-optimal assuming that matrix multiplication is (\emph{i.e.} $ω= 2$). The main theoretical input we are using is the existence of a well-suited ring of series with divided powers for which an analogue of the Cauchy--Lipschitz Theorem holds.
We present the tensor computer algebra package xTras, which provides functions and methods frequently needed when doing (classical) field theory. Amongst others, it can compute contractions, make Ansätze, and solve tensorial equations. It is built upon the tensor computer algebra system xAct, a collection of packages for Mathematica.
We present a highly scalable algorithm for multiplying sparse multivariate polynomials represented in a distributed format. This algo- rithm targets not only the shared memory multicore computers, but also computers clusters or specialized hardware attached to a host computer, such as graphics processing units or many-core coprocessors. The scal- ability on the large number of cores is ensured by the lacks of synchro- nizations, locks and false-sharing during the main parallel step.
For a regular chain $R$, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of $R$, that is, the set $\bar{W(R)} \setminus W(R)$. Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of $R$. We focus on the case where this saturated ideal has dimension one and we discuss extensions of this work in higher dimensions. We provide experimental results illustrating the benefits of our algorithms.
The library \emph{fast\_polynomial} for Sage compiles multivariate polynomials for subsequent fast evaluation. Several evaluation schemes are handled, such as Hörner, divide and conquer and new ones can be added easily. Notably, a new scheme is introduced that improves the classical divide and conquer scheme when the number of terms is not a pure power of two. Natively, the library handles polynomials over gmp big integers, boost intervals, python numeric types. And any type that supports addition and multiplication can extend the library thanks to the template design. Finally, the code is parallelized for the divide and conquer schemes, and memory allocation is localized and optimized for the different evaluation schemes. This extended abstract presents the concepts behind the \emph{fast\_polynomial} library. The sage package can be downloaded at \url{http://trac.sagemath.org/sage_trac/ticket/13358}.