Piecewise polynomials, Minkowski weights, and localization on toric varieties

We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the scalar curvature of a toric metric on a toric variety in terms of polytope data.

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K ˆ S1† to be a torus. Invent. math. 131, 25±83 (1998)

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch‐Riemann‐Roch theorem. As important special cases, we obtain new (or recover well‐known) formulae for the Baum‐Fulton‐MacPherson Todd (or MacPherson's Chern) classes of toric varieties, as well as for the Thom‐Milnor L‐classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus‐orbit decomposition and the motivic properties of the motivic Chern and respectively homology Hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the (dual) Todd classes of closures of orbits. This method even applies to torus‐invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice‐point counting in lattice polytopes and their subcomplexes, yielding generalized Pick‐type formulae. Second, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz‐Riemann‐Roch theorem of Edidin‐Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd, and mock L‐classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.© 2015 Wiley Periodicals, Inc.

Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus Tk. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus Φ, which was introduced independently by Krichever and Löffler in 1974, albeit from radically different viewpoints. In fact, Φ is a version of tom Dieck’s bundling transformation of 1970, defined on Tk-equivariant complex cobordism classes and taking values in the complex cobordism algebra of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera and refer to the index-theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework and to introduce parametrized versions that apply to bundles equipped with a stably complex structure on the tangents along their fibers. In the presence of isolated fixed points, we obtain universal localization formulae, whose applications include the identification of Krichever’s generalized elliptic genus as universal among the genera that are rigid on SU-manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites, we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s.

We first investigate torus‐equivariant motivic characteristic classes of toric varieties, and then apply them via the equivariant Riemann–Roch formalism to prove very general Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes. We consider ‐equivariant versions and of the motivic Chern and, resp., Hirzebruch characteristic classes of a toric variety (with corresponding torus ), and extend many known results from the non‐equivariant context to the equivariant setting. For example, the equivariant motivic Chern class is computed as the sum of the equivariant Grothendieck classes of the ‐equivariant sheaves of Zariski ‐forms weighted by . Using the motivic, as well as the characteristic class nature of , the corresponding generalized equivariant Hirzebruch ‐genus of a ‐invariant Cartier divisor on is also calculated. Further global formulae for are obtained in the simplicial context based on the Cox construction and the equivariant Lefschetz–Riemann–Roch theorem of Edidin–Graham. Alternative proofs of all these results are given via localization techniques at the torus fixed points in ‐equivariant ‐ and, resp., homology theories of toric varieties, due to Brion–Vergne and, resp., Brylinski–Zhang. These localization results apply to any toric variety with a torus fixed point. In localized ‐equivariant ‐theory, we extend a classical formula of Brion for a full‐dimensional lattice polytope to a weighted version. We also generalize the Molien formula of Brion–Vergne for the localized class of the structure sheaf of a simplicial toric variety to the context of . Similarly, we calculate the localized Hirzebruch class in localized ‐equivariant homology, extending the corresponding results of Brylinski–Zhang for the localized Todd class (fitting with the equivariant Hirzebruch class for ). As main applications of our equivariant characteristic class formulae, we provide a geometric perspective on several weighted Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes (corresponding to simplicial toric varieties), coming from the equivariant toric geometry via the equivariant Hirzebruch–Riemann–Roch (for an ample torus invariant Cartier divisor). Our main results even provide generalizations to arbitrary equivariant coherent sheaf coefficients , including algebraic geometric proofs of (weighted versions of) the Euler–Maclaurin formulae of Cappell–Shaneson, Brion–Vergne, Guillemin, and so forth (all of which correspond to the choice of the structure sheaf), via the equivariant Hirzebruch–Riemann–Roch formalism. In particular, we give a first complete proof of the Euler–Maclaurin formula of Cappell–Shaneson. Our approach, based on motivic characteristic classes , allows us to obtain such Euler–Maclaurin formulae also for (the interior of) a face, as well as for the polytope with several facets (i.e., codimension one faces) removed, for example, for the interior of the polytope (as well as for equivariant characteristic class formulae for locally closed ‐invariant subsets of a toric variety). Moreover, we prove such results also in the weighted context, as well as for ‐Minkowski summands of the given full‐dimensional lattice polytope (corresponding to globally generated torus invariant Cartier divisors in the toric context). Some of these results are extended to local Euler–Maclaurin formulae for the tangent cones at the vertices of the given full‐dimensional lattice polytope (fitting with the localization at the torus fixed points in equivariant ‐theory and equivariant (co)homology). Finally, we also give an application of our abstract Euler–Maclaurin formula to generalized reciprocity for Dedekind sums.

We classify smooth toric Fano varieties of dimension $n\\geq 3$ containing a toric divisor isomorphic to the $(n-1)$-dimensional projective space. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\\geq 3$ with a fixed point $x\\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to the $n$-dimensional projective space or to the blow-up of the $n$-dimensional projective space along an invariant linear codimension two subspace. As expected, such results are proved\nusing toric Mori theory due to Reid.

Solving parametric piecewise polynomial systems

The meeting brought together people with different mathematical background, who all use cohomological methods to study symmetries of manifolds. The main aim was the exchange of ideas, recent results, and the discussion of open problems and questions from diverse viewpoints. Altogether there were 27 talks (including an evening talk on computer programs), and a Hausmusik evening with an artistic juggling intermission. All talks reflected the central theme of the workshop, namely the use of (equivariant) cohomology in studying Lie group actions on manifolds. Contribution to the following subjects were given: Especially the informal discussions among participants both with similar and with diverse mathematical background were a very important aspect of the workshop. We believe that the meeting has stimulated further cooperation in the study of actions of Lie groups between mathematicians from different areas.

In smooth support vector machine (SSVM), the plus function must be approximated by some smooth function, and the approximate error will affect the classification ability. This paper studies the smooth approximation to the plus function by piecewise polynomials. First, some standard piecewise polynomial smooth approximation problems are formulated. Then, the existence and uniqueness of solution for these problems are proved and the analytic solutions are achieved. The comparison between the results in this paper and the previous ones shows that the piecewise polynomial functions in this paper achieve better approximation to the plus function

T-duality and homological mirror symmetry for toric varieties

Abelianization for hyperkähler quotients

The aim of this thesis is to study discrete structures associated with group- based models. Group-based models are statistical models on phylogenetic trees that can be parametrized by polynomial maps. The algebraic varieties given by these polynomial maps are toric. By the well-known correspondence between toric varieties and polyhedral fans, we can associate discrete structures, such as lattice polytopes and affine semigroups, with group-based models. We follow three main lines in this dissertation. In Chapters 2 and 3, we study the Hilbert polynomials of group-based models. This is motivated by a result of Buczyńska and Wiśniewski stating that the Hilbert polynomial of the Jukes- Cantor binary model on a trivalent tree does not depend on the shape of the tree. In Chapter 2, we give a simple combinatorial proof to this statement, and in Chapter 3, we show that the analogous statement does not hold for the Kimura 3-parameter model. In Chapters 4 and 5, we study the phylogenetic semigroups on graphs that generalize the Jukes-Cantor binary model on trees. In Chapter 4, we study the maximal degrees of the minimal generators of these semigroups. In Chapter 5, we investigate the minimal generators of the phylogenetic semigroups on graphs with a few holes, extending the work of Buczyńska. Finally, in Chapter 6, we establish a connection between Berenstein-Zelevinsky triangles from representation theory and group-based models. This is motivated by the recent work of Sturmfels, Xu, and Manon related to conformal block algebras.

Over the last decades the finite element method, which was introduced by engineers in the 1960s, has become the perhaps most important numerical method for partial differential equations, particularly for equations of elliptic and parabolic types. This method is based on the variational form of the boundary value problem and approximates the exact solution by a piecewise polynomial function. It is more easily adapted to the geometry of the underlying domain than the finite difference method, and for symmetric positive definite elliptic problems it reduces to a finite linear system with a symmetric positive definite matrix. We first introduce this method in Sect. 5.1 for the case of a two-point boundary value problem and show a number of error estimates. In Sect. 5.2 we then formulate the method for a two-dimensional model problem. Here the piecewise polynomial approximations are defined on triangulations of the spatial domain, and in the following Sect. 5.3 we study such approximation in more detail. In Sect. 5.4 we show basic error estimates for the finite element method for the model problem, using piecewise linear approximating functions. All error bounds derived up to this point contain a norm of the unknown exact solution and are therefore often referred to as a priori error estimates. In Sect. 5.5 we show a so-called a posteriori error estimate in which the error bound is expressed in terms of the data of the problem and the computed solution. In Sect. 5.6 we analyze the effect of numerical integration, which is often used when the finite element equations are assembled in a computer program. In Sect. 5.7 we briefly describe a so-called mixed finite element method.KeywordsFinite Element MethodElliptic EquationQuadrature FormulaInterpolation ErrorPosteriori Error EstimateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents an event-driven simulation methodology for injection-locked oscillators. The proposed method adopts a phase-domain macromodel based on a perturbation projection vector (PPV), which expresses the oscillator's phase response using a nonlinear, time-varying differential equation. By expressing the PPV using a piecewise polynomial function, the PPV model can be simulated in an event-driven fashion on a digital logic simulator like SystemVerilog. While this piecewise polynomial approximation introduces a trade-off between the simulation accuracy and speed, our results show that the simulation accuracy quickly reaches its optimal value as the piecewise interval number increases even with first-order polynomials. The experimental results with an LC oscillator, ring oscillator, and burst-mode CDR models operating with both periodic and aperiodic inputs demonstrate that the proposed models achieve speeds up to 2 ~ 30× that of transistor-level Spectre simulations with the same accuracy.

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In practice, we are interested in approximating a function f(x) near x = a by a polynomial of degree M. Other issues involved in approximation are how good the approximation is and over what interval can we expect the approximation to be good. In this paper we present approximation that is realized as piecewise polynomial function where in each interval ƒ(x) is approximated by a Taylor polynomial of low order. The piecewise polynomial approximation can be effectively implemented using Farrow structure.