736 research outputs found
The Rough Veronese variety
We study signature tensors of paths from a geometric viewpoint. The
signatures of a given class of paths parametrize an algebraic variety inside
the space of tensors, and these signature varieties provide both new tools to
investigate paths and new challenging questions about their behavior. This
paper focuses on signatures of rough paths. Their signature variety shows
surprising analogies with the Veronese variety, and our aim is to prove that
this so-called Rough Veronese is toric. The same holds for the universal
variety. Answering a question of Amendola, Friz and Sturmfels, we show that the
ideal of the universal variety does not need to be generated by quadrics
Toric geometry of path signature varieties
In stochastic analysis, a standard method to study a path is to work with its
signature. This is a sequence of tensors of different order that encode
information of the path in a compact form. When the path varies, such
signatures parametrize an algebraic variety in the tensor space. The study of
these signature varieties builds a bridge between algebraic geometry and
stochastics, and allows a fruitful exchange of techniques, ideas, conjectures
and solutions. In this paper we study the signature varieties of two very
different classes of paths. The class of rough paths is a natural extension of
the class of piecewise smooth paths. It plays a central role in stochastics,
and its signature variety is toric. The class of axis-parallel paths has a
peculiar combinatoric flavour, and we prove that it is toric in many cases.Comment: Code for the computations is available at
https://sites.google.com/view/l-colmenarejo/publications/cod
Collisions of fat points and applications to interpolation theory
We address the problem to determine the limit of the collision of fat points
in $\mathbb{P}^n. We give a description of the limit scheme in many cases, in
particular in low dimension and multiplicities. The problem turns out to be
closely related with interpolation theory, and as an application we exploit
collisions to prove some new cases of Laface-Ugaglia Conjecture.Comment: 19 page
Generic identifiability of pairs of ternary forms
We prove that two general ternary forms are simultaneously identifiable only
in the classical cases of two quadratic and a cubic and a quadratic form. We
translate the problem into the study of a certain linear system on a projective
bundle on the plane, and we apply techniques from projective and birational
geometry to prove that the associated map is not birational
Secant non-defectivity via collisions of fat points
Secant defectivity of projective varieties is classically approached via
dimensions of linear systems with multiple base points in general position. The
latter can be studied via degenerations. We exploit a technique that allows
some of the base points to collapse together. We deduce a general result which
we apply to prove a conjecture by Abo and Brambilla: for and , the Segre-Veronese embedding of in
bidegree is non-defective.Comment: 36 pages, 4 pages, all comments are welcome
Enhanced Effective Thickness of Multi-Layered Laminated Glass
The stiffness and strength of laminated glass, a composite of glass layers bonded together by polymeric interlayers, depend upon shear coupling between the glass plies through the polymer. In the design practice, this effect is commonly considered by defining the effective thickness, i.e., the thickness of a monolith with equivalent bending properties. Traditional formulations have been proposed for a package of two layers of glass and one polymeric interlayer, but their extrapolation to a higher number of layers gives in general inaccurate results. Here, the recently-proposed Enhanced Effective Thickness method is extended to the case of laminated glass beams composed i) by three layers of glass of arbitrary thickness, or ii) by an arbitrary number of equally-thick glass layers. Comparison with numerical experiments confirms the accuracy of the proposed approach
Composite beams with viscoelastic interaction. An application to laminated glass.
A practical way to calculate the response of laminated glass is to consider both glass and polymeric interlayer as linear elastic materials; the viscoelastic behavior of the polymer is evaluated assuming equivalent elastic moduli, that is, the relaxed moduli under constant strain after a time equal to the duration of the design action. Here, we analytically solve the time-dependent problem of simply-supported laminated-glass beams, modeling the response of the polymer by a Prony’s series of Maxwell elements. The obtained results, in agreement with a full 3-D viscoelastic finite-element numerical analysis, emphasize that there is a noteworthy difference between the state of strain and stress calculated in the full-viscoelastic case or in the aforementioned “equivalent” elastic problem. The second approach gives in general results that are on the side of safeness, but the design may be too conservative for short-time actions, whose duration
depends upon the polymer type
Effective Thickness of Laminated Glass Beams.New expression via a variational approach.
The performance of laminated glass, which consists of two or more glass plies bonded together by polymeric interlayers, depends upon shear coupling between the plies through the polymer. This is commonly considered by defining the effective thickness, i.e., the thickness of a monolithic beam with equivalent bending properties in terms of stress and deflection. General expressions have been proposed on the basis of simplified models by Newmark and Wölfel-Bennison, but they are either diffcult to apply or inaccurate. Here, a variational approach to the problem is presented. By choosing appropriate shape functions for the laminated-beam deformation, minimization of the strain energy functional gives new expressions for the effective thickness under any constraint- and load-conditions, embracing the classical formulations as particular cases. Comparisons with numerical experiments confirm the better accuracy of the proposed approach with respect to the previous ones
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