Singular Quadratic Forms Research Articles

The behavior of singular quadratic forms under purely inseparable extensions

Let F be a field of characteristic 2 and K a purely inseparable extension of F. When K is modular, we give a complete classification of anisotropic semisingular F-quadratic forms that have over K a maximal Witt index and a defect index at least equal to the half of the dimension of the quasilinear part. The case of totally singular quadratic forms will be also treated. Moreover, without the modularity hypothesis, we give necessary and sufficient conditions under which an anisotropic semisingular F-quadratic form has a given Witt index over K. The quasi-hyperbolicity of semisingular F-quadratic forms over function fields of certain irreducible polynomials will be treated, extending to such forms many results established by the first author in [11].

An ultimate proof of Hoffmann–Totaro’s conjecture

We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic \ne2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the author in 2002. In characteristic 2 , the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.

Isotropy of orthogonal involutions in characteristic two

A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only if its corresponding quadratic form is isotropic.

Arason's filtration of the Witt group of dyadic valued fields

Generalizing a theorem of Springer, we construct an extended Arason filtration by subgroups for the Witt group of quadratic forms of a general valued field, relating these subgroups with Witt-like groups of the residue field, in arbitrary characteristic. These Witt-like groups involve totally singular quadratic forms. In the case of a discretely valued field, we recover the original Arason filtration.

Orthogonal involutions and totally singular quadratic forms in characteristic two

We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can be used to classify totally decomposable algebras with orthogonal involution. Also, using this form, a criterion is obtained for an orthogonal involution on a split algebra to be conjugated to the transpose involution.

Hoffmann’s conjecture for totally singular forms of prime degree

One of the most significant discrete invariants of a quadratic form $\phi$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behaviour of $\phi$ under scalar extension to arbitrary overfields of $k$. A similarly important, but more accessible variant of this notion is that of the Knebusch splitting pattern of $\phi$, which captures the isotropy behaviour of $\phi$ as one passes over a certain prescribed tower of $k$-overfields. In this paper, we determine all possible values of this latter invariant in the case where $\phi$ is totally singular. This includes an extension of Karpenko's theorem (formerly Hoffmann's conjecture) on the possible values of the first Witt index to the totally singular case. Contrary to the existing approaches to this problem (in the nonsingular case), our results are achieved by means of a new structural result on the higher anisotropic kernels of totally singular quadratic forms. Moreover, the methods used here readily generalise to give analogous results for arbitrary Fermat-type forms of degree $p$ over fields of characteristic $p>0$. Related problems concerning the Knebusch splitting of symmetric bilinear forms over fields of characteristic 2 and the full splitting of totally singular quadratic forms are also considered.

Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator

We design and analyze variational and non-variational multigrid algorithms for the Laplace-Beltrami operator on a smooth and closed surface. In both cases, a uniform convergence for the V -cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid. The main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface. In addition, the vanishing mean value constraint is imposed on each level, thereby avoiding singular quadratic forms without adding additional computational cost. Numerical results supporting our analysis are reported. In particular, the algorithms perform well even when applied to surfaces with a large aspect ratio.

On splitting of totally singular quadratic forms

In this note we completely study the standard splitting of quasi-Pfister forms and their neighbors, and we include some general results on standard splitting towers of totally singular quadratic forms.

Quadratic forms and Pfister neighbors in characteristic 2

We study Pfister neighbors and their characterization over fields of characteristic 2 2 , where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form q q is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form φ \varphi which is dominated by q q . From this, we derive an analogue in characteristic 2 2 of a result by Knebusch saying that, in characteristic ≠ 2 \neq 2 , a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch’s result generally fails in characteristic 2 2 for singular forms. As an application, we characterize certain forms of height 1 1 in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.

On Field Theory Methods in Singular Perturbation Theory

Singular and supersingular finite rank perturbations of self-adjoint operators are studied using methods from renormalization theory for quantum fields. It is shown that the ideas from dimensional and Pauli-Villars regulatizations can be applied to determine uniquely certain finite rank supersingular perturbations. Approach is based on the regularization of homogeneous singular quadratic forms.

Generalized quadratic minimization and blind multichannel deconvolution

This paper deals with the problem of quadratic minimization subject to linear equality constraints. Unlike the standard formulation, we assume the most general case of a possibly singular quadratic form. We explain that the existing formal solution to this problem has several drawbacks. Our new approach is free from most of these drawbacks. In particular, it has a simple interpretation and is relatively easy to implement. The practical importance of this result lies in its numerous applications: filter design, spectral analysis, direction finding, and blind deconvolution of multiple FIR channels. Here, we focus on the blind deconvolution application for which we present a novel solution with enhanced performance.

Probability of resolution of the MUSIC algorithm

The MUSIC algorithm is well known for its high-resolution capability, and various aspects of its statistical performance have been investigated. However, rigorous asymptotic analysis of one of its most important performance measures, the probability of resolution, is not available yet. We analyze the probability of the MUSIC algorithm resolving two spatially separated signal sources in the context of array processing. By formulating the resolution problem in the framework of statistical decision theory and directly determining the probability density function (PDF) of the indefinite and singular quadratic form that defines the resolution event, we arrive at an exact asymptotic formula for the probability of resolution. This is accomplished by a multistep procedure. Computer simulations have been performed to confirm the validity of the theory. >

Problems of distance geometry and convex properties of quadratic maps

A weighted graph is calledd-realizable if its vertices can be chosen ind-dimensional Euclidean space so that the Euclidean distance between every pair of adjacent vertices is equal to the prescribed weight. We prove that if a weighted graph withk edges isd-realizable for somed, then it isd-realizable for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaaceiGaa8hzaiabg2da9maadmaabaWaaeWaaeaadaGc% aaqaaiabiIda4iaa-TgacqGHRaWkcqaIXaqmaSqabaGccqGHsislcq% aIXaqmaiaawIcacaGLPaaacqGGVaWlcqaIYaGmaiaawUfacaGLDbaa% aaa!47D5! $$d = \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]$$ (this bound is sharp in the worst case). We prove that for a graphG withn vertices andk edges and for a dimensiond the image of the so-called rigidity map ℝ dn →ℝ k is a convex set in ℝ k provided % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaaceiGaa8hzaiabgwMiZoaadmaabaWaaeWaaeaadaGc% aaqaaiabiIda4iaa-TgacqGHRaWkcqaIXaqmaSqabaGccqGHsislcq% aIXaqmaiaawIcacaGLPaaacqGGVaWlcqaIYaGmaiaawUfacaGLDbaa% aaa!4895! $$d \geqslant \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]$$ . These results are obtained as corollaries of a general convexity theorem for quadratic maps which also extends the Toeplitz-Hausdorff theorem. The main ingredients of the proof are the duality for linear programming in the space of quadratic forms and the “corank formula” for the strata of singular quadratic forms.

Regular restrictions of singular quadratic forms

Regular restrictions of singular quadratic forms

Hydromechanics of swimming propulsion. Part 2. Some optimum shape problems

The optimum shape problems considered in this part are for those profiles of a two-dimensional flexible plate in time-harmonic motion that will minimize the energy loss under the condition of fixed thrust and possibly also under other isoperimetric constraints. First, the optimum movement of a rigid plate is completely determined; it is necessary first to reduce the original singular quadratic form representing the energy loss to a regular one of a lower order, which is then tractable by usual variational methods. A favourable range of the reduced frequency is found in which the thrust contribution coming from the leading-edge suction is as small as possible under the prescribed conditions, outside of which this contribution becomes so large as to be hard to realize in practice without stalling. This optimum solution is compared with the recent theory of Lighthill (1970); these independently arrived-at conclusions are found to be virtually in agreement.The present theory is further applied to predict the movement of a porpoise tail of large aspect-ratio and is found in satisfactory agreement with the experimental measurements. A qualitative discussion of the wing movement in flapping flight of birds is also given on the basis of optimum efficiency.The optimum shape of a flexible plate is analysed for the most general case of infinite degrees of freedom. It is shown that the solution can be determined to a certain extent, but the exact shape is not always uniquely determinate.