Multi-Dimensional Sigma-Functions

In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by K.Weierstrass and F.Klein was aimed to extend to higher genera Weierstrass theory of elliptic functions based on the Weierstrass $\sigma$-functions. Our development was motivated by the recent achievements of mathematical physics and theory of integrable systems that were based of the results of classical theory of multi-dimensional theta functions. Both theta and sigma-functions are integer and quasi-periodic functions, but worth to remark the fundamental difference between them. While theta-function are defined in the terms of the Riemann period matrix, the sigma-function can be constructed by coefficients of polynomial defining the curve. Note that the relation between periods and coefficients of polynomials defining the curve is transcendental. Since the publication of our 1997-review a lot of new results in this area appeared (see below the list of Recent References), that promoted us to submit this draft to ArXiv without waiting publication a well-prepared book. We complemented the review by the list of articles that were published after 1997 year to develop the theory of $\sigma$-functions presented here. Although the main body of this review is devoted to hyperelliptic functions the method can be extended to an arbitrary algebraic curve and new material that we added in the cases when the opposite is not stated does not suppose hyperellipticity of the curve considered.Comment: 267 pages, 4 figure

Identities for hyperelliptic P-functions of genus one, two and three in covariant form

We give a covariant treatment of the quadratic differential identities satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of genera 1, 2 and 3

Abelian functions associated with a cyclic tetragonal curve of genus six

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y^4 = x^5 + λ[4]x^4 + λ[3]x^3 + λ[2]x^2 + λ[1]x + λ[0]. We construct Abelian functions using the multivariate sigma-function associated with the curve, generalizing the theory of theWeierstrass℘-function. We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi inversion problem, a power series expansion for σ(u) and a new addition formula

Quantum discrete Dubrovin equations

The discrete equations of motion for the quantum mappings of KdV type are given in terms of the Sklyanin variables (which are also known as quantum separated variables). Both temporal (discrete-time) evolutions and spatial (along the lattice at a constant time-level) evolutions are considered. In the classical limit, the temporal equations reduce to the (classical) discrete Dubrovin equations as given in a previous publication. The reconstruction of the original dynamical variables in terms of the Sklyanin variables is also achieved.Comment: 25 page

A constructive study of the module structure of rings of partial differential operators

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. © 2014 Springer Science+Business Media

Numerical Nonlinear Algebra

Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications.Comment: 40 pages, many figure

The personalized advantage index: Translating research on prediction into individualized treatment recommendations. A demonstration

Background: Advances in personalized medicine require the identification of variables that predict differential response to treatments as well as the development and refinement of methods to transform predictive information into actionable recommendations. Objective: To illustrate and test a new method for integrating predictive information to aid in treatment selection, using data from a randomized treatment comparison. Method: Data from a trial of antidepressant medications (N = 104) versus cognitive behavioral therapy (N = 50) for Major Depressive Disorder were used to produce predictions of post-treatment scores on the Hamilton Rating Scale for Depression (HRSD) in each of the two treatments for each of the 154 patients. The patient's own data were not used in the models that yielded these predictions. Five pre-randomization variables that predicted differential response (marital status, employment status, life events, comorbid personality disorder, and prior medication trials) were included in regression models, permitting the calculation of each patient's Personalized Advantage Index (PAI), in HRSD units. Results: For 60% of the sample a clinically meaningful advantage (PAI≥3) was predicted for one of the treatments, relative to the other. When these patients were divided into those randomly assigned to their "Optimal" treatment versus those assigned to their "Non-optimal" treatment, outcomes in the former group were superior (d = 0.58, 95% CI .17-1.01). Conclusions: This approach to treatment selection, implemented in the context of two equally effective treatments, yielded effects that, if obtained prospectively, would rival those routinely observed in comparisons of active versus control treatments. © 2014 DeRubeis et al

Asymmetry of wind waves studied in a laboratory tank

International audienceAsymmetry of wind waves was studied in laboratory tank tinder varied wind and fetch conditions using both bispectral analysis of wave records and third-order statistics of the surface elevation. It is found skewness S (the normalized third-order moment of surface elevation describing the horizontal asymmetry waves) varies only slightly with the inverse wave u*/Cm (where u* is the air friction velocity and Cm is phase speed of the dominant waves). At the same time asymmetry A, which is determined from the Hilbert transform of the wave record and characterizes the skewness of the rate of change of surface elevation, increase consistently in magnitude with the ratio u*/Cm. This suggests that nonlinear distortion of the wave profile determined by the degree of wind forcing and is a sensitive indicator of wind-wave interaction processes. It is shown that the asymmetric profile of waves can described within the frameworks of the nonlinear nonspectral concept (Plate, 1972; Lake and Yuen, 197 according to which the wind-wave field can be represented as a coherent bound-wave system consisting mainly of dominant component w. and its harmonics propagating with the same speed C. , as observed by Ramamonjiaris and Coantic (1976). The phase shift between o). harmonics is found and shown to increase with the asymmetry of the waves