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... which can be seen in Eq. (22). Since the 2 nd order tensor B in the diffusion profile is only of rank-1, it is rank deficient, whereas in general the 2 nd order tensor C, in the quadratic form would include both full rank, and rank deficient tensors. ...
... In the case of ternary quartics, it can be shown that the entire space of non-negative polynomials over entire R 3 (and not only over the unit sphere), can be described by the sum of squares of quadratic polynomials. Examples in [22] of non-negative polynomials of degree k > 4 that cannot be written as sums of squares of lower order polynomials indicate that not all non-negative polynomials of arbitrary degree k can be decomposed into sums of squares of lower order polynomials. Hilbert's theorem, which identifies all the classes of non-negative multi-variate polynomials that can be always decomposed as sums of squares of lower order polynomials is also presented in [22]. ...
... Examples in [22] of non-negative polynomials of degree k > 4 that cannot be written as sums of squares of lower order polynomials indicate that not all non-negative polynomials of arbitrary degree k can be decomposed into sums of squares of lower order polynomials. Hilbert's theorem, which identifies all the classes of non-negative multi-variate polynomials that can be always decomposed as sums of squares of lower order polynomials is also presented in [22]. ...
High Order Cartesian Tensors (HOTs) were introduced in Generalized DTI (GDTI) to overcome the limitations of DTI. HOTs can model the apparent diffusion coefficient (ADC) with greater accuracy than DTI in regions with fiber heterogeneity. Although GDTI HOTs were designed to model positive diffusion, the straightforward least square (LS) estimation of HOTs doesn’t guarantee positivity. In this chapter we address the problem of estimating 4th order tensors with positive diffusion profiles.Two known methods exist that broach this problem, namely a Riemannian approach based on the algebra of 4th order tensors, and a polynomial approach based on Hilbert’s theorem on non-negative ternary quartics. In this chapter, we review the technicalities of these two approaches, compare them theoretically to show their pros and cons, and compare them against the Euclidean LS estimation on synthetic, phantom and real data to motivate the relevance of the positive diffusion profile constraint.
... His idea is to start with a ternary quartic that is obviously a sum of three squares of polynomials, such as f 0 = 1 + X 4 1 + X 4 2 , to carefully choose a path from f 0 to f in the space of positive semidefinite ternary quartics and to deform the representation of f 0 as a sum of three squares to one for f . We refer to [26] or [31] for modern accounts of Hilbert's proof and to [28] for recent developments. ...
... This bound is sharp: a general nonnegative ternary quartic is not a sum of two squares. This result has attracted attention over the years; see [18,20,21,22]. The most recent elementary proof is due to Pfister and Scheiderer [15]. ...
A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real projective variety $X$ of minimal degree is a sum of $\dim(X)+1$ squares of linear forms. This strengthens one direction of a recent result due to Blekherman, Smith, and Velasco. Our upper bound is the best possible, and it implies the existence of low-rank factorizations of positive semidefinite bivariate matrix polynomials and representations of biforms as sums of few squares. We determine the number of equivalence classes of sum-of-squares representations of general quadratic forms on surfaces of minimal degree, generalizing the count for ternary quartics by Powers, Reznick, Scheiderer, and Sottile.
... Proof: Let ¯ τ be the optimal solution for (26). Since f (u) − ¯ τ is a quadratic function of v and is non-negative for all v, it can always be written as a sum of squares of polynomials[30]. In other words, the solution of (26) and ...
In this work, we investigate an efficient numerical approach for solving higher order statistical methods for blind and semi-blind signal recovery from non-ideal channels. We develop numerical algorithms based on convex optimization relaxation for minimization of higher order statistical cost functions. The new formulation through convex relaxation overcomes the local convergence problem of existing gradient descent based algorithms and applies to several well-known cost functions for effective blind signal recovery including blind equalization and blind source separation in both single-input-single-output (SISO) and multi-input-multi-output (MIMO) systems. We also propose a fourth order pilot based cost function that benefits from this approach. The simulation results demonstrate that our approach is suitable for short-length packet data transmission using only a few pilot symbols.
... This question is related to the 17th of Hilbert's 23 problems (see [7]). Recently Dritschel proved that a positive Laurent polynomial in multivariate settings can be written as a sum of square magnitudes of Laurent polynomials(see [3]). ...
We study the number of Laurent polynomials in a sum of square magnitudes of a nonnegative Laurent polynomial associated with bivariate box splines on the three-and the four-direction meshes. In addition, we study the same problem associated with trivariate box splines. The number of Laurent polynomials in a sum of squares magnitude of a nonnegative Laurent polynomial determines the number of multivariate box spline tight wavelet frame generators under the simple construction method in [5]. As a result, the numbers of bivariate box spline tight wavelet frame generators on the three-and the four-direction meshes under certain condition and the one for trivariate box splines are much smaller than the number of multivariate box spline tight wavelet frame generators constructed via Kronecker poduct method in [2].
... [21]). This is related to the 17th of Hilbert's famous 23 [33]). It poses the question if any real positive polynomial can be written as a finite sum of squares of rational functions. ...
Two simple constructive methods are presented to compute compactly supported tight wavelet frames for any given refinable function whose mask satisfies the QMF or sub-QMF conditions in the multivariate setting. We use one of our constructive methods in order to find tight wavelet frames associated with multivariate box splines, e.g., bivariate box splines on a three or four directional mesh. Moreover, a construction of tight wavelet frames with maximum vanishing moments is given, based on rational masks for the generators. For compactly supported bi-frame pairs, another simple constructive method is presented.
... Any positive-definite polynomial can be written as a sum of squares of lower order polynomials [10,11]. In our particular case, any positive-definite K th -order homogeneous polynomial in 3 variables can be written as a sum of squares of K/2 th -order homogeneous polynomials p(g 1 , g 2 , g 3 ; c), where c is a vector that contains the polynomial coefficients. ...
Cartesian tensors of various orders have been employed for either modeling the diffusivity or the orientation distribution function in Diffusion-Weighted MRI datasets. In both cases, the estimated tensors have to be positive-definite since they model positive-valued functions. In this paper we present a novel unified framework for estimating positive-definite tensors of any order, in contrast to the existing methods in literature, which are either order-specific or fail to handle the positive-definite property. The proposed framework employs a homogeneous polynomial parametrization that covers the full space of any order positive-definite tensors and explicitly imposes the positive-definite constraint on the estimated tensors. We show that this parametrization leads to a linear system that is solved using the non-negative least squares technique. The framework is demonstrated using synthetic and real data from an excised rat hippocampus.
... Several authors have given fully detailed accounts of Hilbert's proof in recent years. We mention the approach due to Cassels, published in Rajwade's book ([8] chapter 7), and the two articles by Rudin [9] and Swan [11]. These approaches also show some characteristic differences. ...
In 1888, Hilbert proved that every non-negative quartic form f=f(x,y,z) with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. Up to now, no elementary proof is known. Here we present a completely new approach. Although our proof is not easy, it uses only elementary techniques. As a by-product, it gives information on the number of representations f=p_1^2+p_2^2+p_3^2 of f up to orthogonal equivalence. We show that this number is 8 for generically chosen f, and that it is 4 when f is chosen generically with a real zero. Although these facts were known, there was no elementary approach to them so far. Comment: 26 pages
... Hence, here we are concerned with the positive definiteness of homogenous polynomials of degree 4 in 3 variables, or the so called ternary quartics. Hilbert in 1888 proved the following theorem on ternary quartics (see[34,32]for a modern exposition) that we will employ in this work: ...
In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing, a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. From this tensor approximation, one can compute useful scalar quantities (e.g. anisotropy, mean diffusivity) which have been clinically used for monitoring encephalopathy, sclerosis, ischemia and other brain disorders. It is now well known that this 2nd-order tensor approximation fails to capture complex local tissue structures, e.g. crossing fibers, and as a result, the scalar quantities derived from these tensors are grossly inaccurate at such locations. In this paper we employ a 4th order symmetric positive-definite (SPD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the SPD property. Several articles have been reported in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positivity of the estimates, which is a fundamental constraint since negative values of the diffusivity are not meaningful. In this paper we represent the 4th-order tensors as ternary quartics and then apply Hilbert's theorem on ternary quartics along with the Iwasawa parametrization to guarantee an SPD 4th-order tensor approximation from the DW-MRI data. The performance of this model is depicted on synthetic data as well as real DW-MRIs from a set of excised control and injured rat spinal cords, showing accurate estimation of scalar quantities such as generalized anisotropy and trace as well as fiber orientations.
... Hilbert [5] showed that every non-negative real ternary quartic form is a sum of three squares of quadratic forms. His proof (see [8], [9] for modern expositions) was non-constructive: The map ...
David Hilbert proved that a non-negative real quartic form f(x,y,z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the complex plane curve Q defined by f is smooth, then f has exactly 8 such representations, up to equivalence. They correspond to those real 2-torsion points of the Jacobian of Q which are not represented by a conjugation-invariant divisor on Q. Comment: 4 pages
... In 1888, Hilbert proved [8] that Σ 3,4 = P 3,4 ; more specifically, every p ∈ P 3,4 can be written as the sum of three squares of quadratic forms. (An elementary proof, with " five " squares is in [2, pp.16-17]; for modern expositions of Hilbert's proof, see [24] and [21].) Hilbert also proved in [8] that the preceding are the only cases for which ∆ n,m = ∅. ...
Hilbert showed that for most $(n,m)$ there exist psd forms $p(x_1,...,x_n)$ of degree $m$ which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form $h$ so that $h^2p$ is a sum of squares of forms; that is, $p$ is a sum of squares of rational functions with denominator $h$. We show that, for every such $(n,m)$ there does not exist a single form $h$ which serves in this way as a denominator for {\it every} psd $p(x_1,...,x_n)$ of degree $m$.
... Three years later, in a single remarkable paper, Hilbert [11] resolved the question. He first showed that F ∈ P 3,4 is a sum of three squares of quadratic forms; see [23] and [26] for recent expositions and [17] [18] for another approach. Hilbert then described a construction of forms in ∆ 3,6 and ∆ 4,4 ; after multiplying these by powers of linear forms if necessary, it follows that ∆ n,m = ∅ if n ≥ 3 and m ≥ 6 or n ≥ 4 and m ≥ 4. The goal of this paper is to isolate the underlying mechanism of Hilbert's construction , show that it applies to situations more general than those in [11], and use it to produce many new examples. ...
In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert's construction and present many such polynomials.
Hilbert showed that for most ( n , m ) (n,m) there exist positive semidefinite forms p ( x 1 , … , x n ) p(x_1,\dots ,x_n) of degree m m which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form h h so that h 2 p h^2p is a sum of squares of forms; that is, p p is a sum of squares of rational functions with denominator h h . We show that, for every such ( n , m ) (n,m) there does not exist a single form h h which serves in this way as a denominator for every positive semidefinite p ( x 1 , … , x n ) p(x_1,\dots ,x_n) of degree m m .
Hilbert’s ternary quartic theorem states that every nonnegative degree 4 homogeneous polynomial in three variables can be written as a sum of three squares of homogeneous quadratic polynomials. We give a linear-algebraic approach to Hilbert’s theorem by showing that a structured cone of positive semidefinite matrices is generated by rank 1 elements.
In diffusion-weighted magnetic resonance imaging (DW-MRI) of white matter fiber microstructure modeling, higher order tensor (HOT) is a model to describe complex white matter fiber structures that the commonly used diffusion tensor imaging (DTI) model fails to describe. However some obstacles still exisit in HOT such as complexity in extraction and large amount of calculation is required. Based on the HOT model, this paper presented a novel method of eigenvector extraction using subdivision theory and iterative search. In the first step, grid subdivision was used to roughly determine direction of the regional characteristics; next, accurate of high order tensor model features direction was obtained according to the area segmentation and iterative search. The method provided a solution to the problems including (1) symbols calculation method is likely to fall into the local extreme value point, (2) the search errors caused by no convergence and computational efficiency. In order to examine the efficacy of the proposed method, simulated data were used to analyze the recognition ability and the precision of eigenvectors in different numbers and different point of fiber cross cases. After that, evaluation was conducted using a real DW-MRI dataset. Compared with the existing symbols calculation method, stable fiber characteristic direction can be obtained by the proposed algorithm under 6 th and higher order model.
Fiber tracking is a basic task in analyzing data obtained by diffusion tensor magnetic resonance imaging (DT-MRI). In order to get a better tracking result for crossing fibers with noise, an improved fiber tracking method is proposed in this paper. The method is based on the framework of Bayesian fiber tracking, but improves its ability to deal with crossing fibers, by introducing the high order tensor (HOT) model as well as a new fiber direction selection strategy. In this method, orientation distribution function is first obtained from HOT model, and then used as the likelihood probability to control fiber tracing. On this basis, the direction in candidates that has the smallest change relative to current two previous directions is selected as the next tracing direction. By this means, our method achieves better performance in processing crossing fibers.
Diffusion MRI is a new medical imaging modality of great interest in neuroimaging research. This modality enables the characterization in vivo of local micro-structures. Tensors have commonly been used to model the diffusivity profile at each voxel. This multivariate data set requires the design of new dedicated image processing techniques. The context of this thesis is the automatic analysis of intra-patient longitudinal changes with application to the follow-up of neuro-degenerative pathologies. Our research focused on the development of new models and statistical tests for the automatic detection of changes in temporal sequences of diffusion images. Thereby, this thesis led to a better modeling of second order tensors (statistical tests on positive definite matrices), to an extension to higher-order models, and to the definition of refined neighborhoods on which tests are conducted, in particular the design of statistical tests on fiber bundles.
The positive definiteness of a diffusion tensor is important in magnetic resonance imaging because it reflects the phenomenon of water molecular diffusion in complicated biological tissue environments. To preserve this property, we represent it as an explicit positive semidefinite (PSD) matrix constraint and some linear matrix equalities. The objective function is the regularized linear least squares fitting for the log-linearized Stejskal-Tanner equation. The regularization term is the heuristic nuclear norm of the PSD matrix, since we expect it to be of low rank. In this way, we establish a convex quadratic semidefinite programming (SDP) model, whose global solution exists. The optimal solution could be solved by three efficient methods. While there are two state-of-the-art solvers-SDPT3 and QSDP- for the primal problem, we design a new augmented Lagrangian based alternating direction method (ADM) for the dual problem. Sensitivity analyses on the coefficients of the optimal diffusion tensor and the optimal objective function value with respect to noise-corrupted signals are presented. Experiments on synthetic data with multiple fibers show that the new method is robust to the Rician noise and outperforms several existing methods. Furthermore, when the fiber orientation distribution function is considered, the new method is competitive with the Q-ball imaging. Using the human brain data, we illustrate that the new method could capture the crossing of three nervous fiber bundles. Additionally, the new method generates positive definite generalized diffusion tensors in all voxels, while the unconstrained least squares fitting fails. Finally, we confirm that the ADM solver is more efficient than SDPT3 and QSDP for this special problem.
Let X be an integral plane quartic curve over a field k, let f be an equation for X. We first consider representations (*) cf = p1p2 -P 20 (where c ∈ k* and the pi are quadratic forms), up to a natural no-tion of equivalence. Using the general theory of determinantal varieties we show that equivalence classes of such representations correspond to nontrivial globally generated torsion-free rank one sheaves on X with a self-duality which are not exceptional, and that the exceptional sheaves are in bijection with the k-rational singular points of X. For k = ℂ, the number of representations (*) (up to equivalence) depends only on the singularities of X, and is determined explicitly in each case. In the second part we focus on the case where k = ℝ and f is nonnegative. By a famous theorem of Hubert, such f is a sum of three squares of quadratic forms. We use the Brauer group and Galois cohomology to relate identities (**) f = P02 + P21 +P22 to (*), and we determine the number of equivalence classes of representations (**) for each f. Both in the complex and in the real definite case, our results are considerably more precise since they give the number of representations with any prescribed base locus.
Positive definite forms f∈R[x1,…,xn]f∈R[x1,…,xn] which are sums of squares of forms of R[x1,…,xn]R[x1,…,xn] are constructed to have the additional property that the members of any collection of forms whose squares sum to ff must share a nontrivial complex root in CnCn.
We have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask which integers can be represented in the form x
2 + 2y
2 or, more generally, in the form ax
2 + 2bxy + cy
2, where a, b, c are given integers. The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange’s theorem that every positive integer is a sum of four squares, was continued during the 19th century by Dirichlet, Hermite, H.J.S. Smith, Minkowski and others. In the 20th century Hasse and Siegel made notable contributions. With Hasse’s work especially it became apparent that the theory is more perspicuous if one allows the variables to be rational numbers, rather than integers. This opened the way to the study of quadratic forms over arbitrary fields, with pioneering contributions by Witt (1937) and Pfister (1965–67).
Michael Steele describes the fundamental topics in mathematical inequalities and their uses. Using the Cauchy-Schwarz inequality as a guide, Steele presents a fascinating collection of problems related to inequalities and coaches readers through solutions, in a style reminiscent of George Polya, by teaching basic concepts and sharpening problem solving skills at the same time. Undergraduate and beginning graduate students in mathematics, theoretical computer science, statistics, engineering, and economics will find the book appropriate for self-study.
Hilbert proved that a non-negative real quartic form f(x,y,z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the plane curve Q defined by f is smooth, then f has exactly 8 such representations, up to equivalence. They correspond to those real 2-torsion points of the Jacobian of Q which are not represented by a conjugation-invariant divisor on Q. To cite this article: V. Powers et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).
On a smooth curve a theta-characteristic is a line bundle L, the square of which is the canonical line bundle ω. The equivalent condition ℋom(L, ω) ≅ L generalizes well to singular curves, as applications show. More precisely, a theta-characteristic is a torsion-free sheaf
ℱ of rank 1 with ℋom(ℱ, ω) ≅ ℱ. If the curve has non-ADE singularities, then there are infinitely many theta-characteristics. Therefore, theta-characteristics
are distinguished by their local type. The main purpose of this article is to compute the number of even and odd theta-characteristics
(that is ℱ with h0(C, ℱ)≡ 0 and h0(C, ℱ)≡ 1 modulo 2, respectively) in terms of the geometric genus of the curve and certain discrete invariants of a fixed local
type.
Two new parameter-dependent bounded real lemmas (BRLs) are
proposed for continuous-time stochastic systems with polytopic
uncertainties. One is based on the decoupling of product terms
between a Lyapunov matrix and system matrices, which is realized
by transforming the original system to its equivalent descriptor
form; and the other is derived by defining a multiple Lyapunov
function combined with a new bounding technique. These two
improved BRLs are expressed as linear matrix inequality
conditions, which can be easily tested by using standard
numerical software. Numerical examples show the reduced
conservativeness of the proposed BRLs.
In this paper, we discuss extremal extensions of entanglement witnesses based
on Choi's map. The constructions are based on a generalization of the Choi map
due to Osaka, from which we construct entanglement witnesses. These extremal
extensions are powerful in terms of their capacity to detect entanglement of
positive under partial transpose (PPT) entangled states and lead to unearthing
of entanglement of new PPT states. We also use the Cholesky-like decomposition
to construct entangled states which are revealed by these extremal entanglement
witnesses.
We investigate whether a second order Wick polynomial T of a free scalar
field, including derivatives, is essentially self-adjoint on the natural
(Wightman) domain in a quasi-free (i.e. Fock space) Hadamard representation. We
restrict our attention to the case where T is smeared with a test-function from
a particular class S, namely the class of sums of squares of testfunctions.
This class of smearing functions is smaller than the class of all non-negative
testfunctions -- a fact which follows from Hilbert's Theorem. Exploiting the
microlocal spectrum condition we prove that T is essentially self-adjoint if it
is a Wick square (without derivatives). More generally we show that T is
essentially self-adjoint if its compression to the one-particle Hilbert space
is essentially self-adjoint. For the latter result we use W\"ust's Theorem and
an application of Konrady's trick in Fock space. In this more general case we
also prove that one has some control over the spectral projections of T, by
describing it as the strong graph limit of a sequence of essentially
self-adjoint operators.
In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing a 2nd
order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. It is now well known that this 2nd
-order approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4th
order symmetric positive semi-definite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semi-definite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4th
order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert’s theorem on ternary quartics. This parametric representation is then used in a nonlinear-least squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higher-order tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DW-MRI from an isolated rat hippocampus.
These informal notes deal with Fourier series in one or more variables, Fourier transforms in one variable, and related matters.
Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.
1 Carleson measures
2 Lipschitz maps
3 Hausdorff dimension and Hausdorff measures
4 Density properties of Hausdorff measures
5 Rectifiable and purely unrectifiable sets