Quadratic Forms Research Articles

On a divisor problem associated to Dedekind zeta function over certain quadratic forms

On a divisor problem associated to Dedekind zeta function over certain quadratic forms

Failures of integral Springer’s Theorem

We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove that this phenomenon happens infinitely often, and, conversely, establish finiteness results about the situation when the quadratic form is fixed.

A Machine Learning Toolkit Assisted Approach for IMRT Fluence Map Optimization: Feasibility and Advantages.

Purpose
Traditional machine learning (ML) and deep learning (DL) applications in treatment planning rely on complex model architectures and large, high-quality training datasets. However, they cannot fully replace the conventional optimization process. This study presents a novel application of ML in treatment planning where established ML/DL toolkits are directly applied to treatment plan optimization.
Materials and Methods
A one-layer network was designed based on the dose deposition matrix and implemented in PyTorch's L-BFGS optimizer with GPU acceleration. The classical steepest descent optimizer was selected as a reference for comparison. Both optimizers utilized identical inputs and objective functions to ensure a fair comparison. DVH- and gEUD-based objectives were implemented in standard quadratic forms. Standard uniform and 1,000 random initializations were used to test optimizer's search ability under different starting conditions for prostate and head-and-neck cases. 
Results
The MLT-assisted framework demonstrated comparable or superior plan quality to classical optimization by achieving lower objective values, improved DVHs and capturing finer modulation details in fluence maps. For gEUD-based optimization, it effectively explored beam weight elevations that classical optimization could only reach with stricter convergence criteria and many more iterations. The quality differences primarily stemmed from convergence speed. The MLT-assisted framework required significantly fewer evaluations and iterations to achieve similar or better results. Optimization on random initial maps further demonstrated that it was more robust and less likely to be trapped. It does not require stricter convergence criteria or extended runs to reach high-quality optima, making it more efficient and reliable.
Conclusion
This framework leverages ML toolkits in a novel way, enabling faster convergence, greater robustness and handling of complex constraints. As the first study of its kind, it establishes MLT-assisted optimization as a viable and effective alternative to classical methods.&#xD.

Primitive two-dimensional prime-universality of quadratic forms

Primitive two-dimensional prime-universality of quadratic forms

On the variance of Schatten p-norm estimation with Gaussian sketching matrices

Abstract Monte Carlo matrix trace estimation is a popular randomized technique to estimate the trace of implicitly-defined matrices via averaging quadratic forms across several observations of a random vector. The most common approach to analyze the quality of such estimators is to consider the variance over the total number of observations. In this paper we present a procedure to compute the variance of the estimator proposed in [W. Kong and G. Valiant, Spectrum estimation from samples, Ann. Statist. 45 2017, 5, 2218–2247] for the case of Gaussian random vectors and provide a sharper bound than previously available.

A simple construction of the Anderson operator via its quadratic form in dimensions two and three

A simple construction of the Anderson operator via its quadratic form in dimensions two and three

Symmetrizations of quadratic and Hermitian forms

The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and Hermitian forms over fields of good characteristic. Explicit results are given for the partitions (n), (1n), (2,1n-2) and (3,1n-3) as well as for all partitions of n≤7. For orthogonal groups, these symmetrizations are not irreducible, and we continue to find the determinants of their irreducible constituents, the refined symmetrizations, over fields of characteristic 0.

Weak Dynkin type and the universality of non-negative Coxeter-regular integral quadratic forms

Weak Dynkin type and the universality of non-negative Coxeter-regular integral quadratic forms

MCMC-CE: A Novel and Efficient Algorithm for Estimating Small Right-Tail Probabilities of Quadratic Forms with Applications in Genomics.

Quadratic forms of multivariate normal variables play a critical role in statistical applications, particularly in genomics and bioinformatics. However, accurately computing small right-tail probabilities ( p -values) for large-scale quadratic forms is computationally challenging due to the intractability of their probability distributions, as well as significant numerical constraints and computational burdens. To address these problems, we propose MCMC-CE, an innovative algorithm that integrates Markov Chain Monte Carlo (MCMC) sampling with the cross-entropy (CE) method, coupled with leading-eigenvalue extraction and Satterthwaite-type approximation techniques. Our approach efficiently estimates small p -values for quadratic forms with their ranks exceeding 10,000. Through extensive simulation studies and real-world applications in genomics, including genome-wide association studies and pathway enrichment analyses, our method demonstrates advantageous numerical accuracy and computational reliability compared to existing approaches such as Davies', Imhof's, Farebrother's, Liu-Tang-Zhang's and saddlepoint approximation methods. MCMC-CE provides a robust and scalable solution for accurately computing small p -values for quadratic forms, facilitating more precise statistical inference in large-scale genomic studies.

The classification of the refined Humbert invariant for curves of genus 2

The refined Humbert invariant is a positive definite quadratic form intrinsically attached to a curve [Formula: see text] of genus 2. This invariant is an algebraic generalization of the (usual) Humbert invariant. This invariant is useful because many geometric properties of [Formula: see text] are reflected in the arithmetic properties of this invariant. The purpose of this paper is to complete the classification of this invariant when the Jacobian [Formula: see text] of [Formula: see text] is isogenous to a product of an elliptic curve with complex multiplication.

An inductive formula for the Gross-Keating invariant of a quadratic form

An inductive formula for the Gross-Keating invariant of a quadratic form

Integral and rational representations of forms

Criteria are given for the integral representation of a binary quartic form by certain integral ternary quadratic forms and its rational representation by some discriminant forms. These conditions depend on integral or rational solutions of a Weierstrass equation associated to the forms.

The key challenges and best alternatives to environmental sustainability: a comprehensive study

In the modern world, the globe has been trapped by severe challenges, and an abrupt increase in environmental deterioration (ED) is one of these. The practitioners have recently tried to suggest several green initiatives to combat rising environmental concerns, but the problem remains intact. Most economies have priorities to meet their socio-economic target, and practitioners are unaware of true challenges and best alternatives. This study makes an effort to introduce the true challenges (income, urbanization, and natural resources) and best alternative (renewable energy, information & communication technology (ICT), and circular economy) for carbon footprint in the top 28 waste re-cycled economies (WRE) throughout 2000–2021. In order to obtain robust outcomes, the present study uses the most reliable estimators, and the Q-GMM is the most prominent. The outcomes describe a positive role of income, urbanization, and natural resources in carbon footprint. On the other hand, ICT, renewable energy, and the circular economy decline ED. The renewable energy and circular economy mediating role on urbanization shows supportive behavior in sustainability but finds only a significant role of circular economy in sustainable urbanization. Using a quadratic income form, this study validates the EKC & LCC hypotheses for specified economies. On behalf of outcomes, this study suggests imperative implications to become clean & green shortly.

Commutative Algebras with One-Dimensional Square

In this paper, we classify and study commutative algebras having a one-dimensional square. In finite dimension (see Theorem 3.9) besides some cases (which are all associative and nilpotent with nilpotency index 3), the algebras with zero annihilator are either of symplectic type (appearing only in characteristic 2), or evolution algebras. In infinite dimension, ruling out the associative case, we prove that our algebras are either of symplectic type or evolution algebras provided some technical conditions are satisfied (see Theorem 3.12). Our main tool is the theory of inner product spaces and quadratic forms. More precisely, if A denotes an evolution algebra with dim(A2)=1 and a a generator of A2, then A admits an inner product ⟨·,·⟩ such that the product of A is given by xy=⟨x,y⟩a. There are three classes to consider: a∈Ann(A);a∉Ann(A) and a is isotropic relative to ⟨·,·⟩;a∉Ann(A) and a is anisotropic relative to ⟨·,·⟩. The isomorphism problem among these objects is investigated. For some of these algebras, we have also determined the existence of faithful associative representations in certain Clifford algebras.

A Canonical Infinitesimally Hilbertian Structure on Locally Minkowski Spaces

Abstract The aim of this paper is to show the existence of a canonical distance $$\mathsf d'$$ d ′ defined on a locally Minkowski metric measure space $$(\textsf{X},\mathsf d,\mathfrak m)$$ ( X , d , m ) such that: $$\mathsf d'$$ d ′ is equivalent to $$\mathsf d$$ d , $$(\textsf{X}, \mathsf d', \mathfrak m)$$ ( X , d ′ , m ) is infinitesimally Hilbertian. This new regularity assumption on $$(\textsf{X}, \mathsf d,\mathfrak m)$$ ( X , d , m ) essentially forces the structure to be locally similar to a Minkowski space and defines a class of metric measure structures which includes all the reversible Finsler manifolds and it is actually strictly larger. The required distance $$\mathsf d'$$ d ′ will be the intrinsic distance $$\mathsf d_\textsf{KS}$$ d KS associated to the so-called Korevaar-Schoen energy, which is proven to be a quadratic form. In particular, we show that the Cheeger energy associated to the metric measure space $$(\textsf{X}, \mathsf d_\textsf{KS}, \mathfrak m)$$ ( X , d KS , m ) is in fact the Korevaar-Schoen energy.

Sails for universal quadratic forms

We establish a new connection between sails, a key notion in the geometric theory of generalised continued fractions, and arithmetic of totally real number fields, specifically, universal quadratic forms and additively indecomposable integers. Our main application is to biquadratic fields, for which we show that if their signature rank is at least 3, then ranks of universal forms and numbers of indecomposables grow as a power of the discriminant. We also construct a family in which these numbers grow only logarithmically.

Integral matrices as diagonal quadratic forms II

For an integer n ≥ 2 , let f ( n ) be the smallest positive integer m such that for every pairwise coprime integers a 1 , … , a m , every n × n integral matrix can be represented by the diagonal quadratic form ∑ i = 1 m a i X i 2 . In this paper, we prove that f ( 3 ) ≤ 5 and f ( n ) ≤ 6 for every integer n ≥ 4 .

CONTROL OF THE BENDING OF THE TORSO BY CHANGING THE DEPENDENCE OF THE ANGLE OF RISING OF ITS BACK RIB

A spatial curve can be specified by two natural equations: the dependence of the curvature and the dependence of the twist on the length of its arc. If such a curve is taken as the edge of the torso turn, then its bending can be controlled by changing the curvature of the curve, since the curvature does not change. However, in practice, such bending is difficult to implement, since there is no simple transition from the natural equations of the spatial curve to the parametric ones. Such a transition requires numerical methods for solving a system of differential equations. It is possible to approach the solution of this question in a different way, namely to replace the dependence of the twist on the length of the arc of the curve with the dependence of the angle of elevation also on the length of the arc of the curve. In this case, the formulas for the transition from natural equations to parametric ones are greatly simplified and in some cases do not require numerical integration. The article uses just such an approach for constructing torsos. The parametric equations of the torso in its general form are presented, in which the turning edge is a spatial curve, given by the dependence of the curvature and the angle of elevation on the length of its arc. It is shown that by changing the regularity of the angle of elevation, the process of bending the torso can be controlled. At the same time, the expression of the first quadratic form was found, which does not include the dependence of the angle of elevation of the turning edge, but only the dependence of its curvature. This indicates that it is possible to transform the back edge in different ways, that is, to bend the torso, and at the same time the first square shape remains unchanged. A well-known example of the bending of a helicoid torso by a discrete change in the elevation angle of its reverse rib is given. Torso-helicoid bending by changing the elevation angle of the turning rib according to the linear law is also demonstrated. Numerical methods were used for this. The resulting torso is no longer a torso of the same inclination of the generators. An example of bending of the torso is also given, in which the curvature and the angle of elevation of the turning rib are variable dependences on the length of the arc. They are selected in such a way that they allow obtaining intermediate positions of the torso during its bending with the help of equations in the final form. Examples are given, visualization of the obtained results is carried out.

On prime numbers and quadratic forms represented by positive-definite, primitive quadratic forms

On prime numbers and quadratic forms represented by positive-definite, primitive quadratic forms

Feedback stability analysis via dissipativity with dynamic supply rates

Feedback stability analysis via dissipativity with dynamic supply rates