Dimensional Affine Research Articles

Joint calibration to SPX and VIX options with signature‐based models

AbstractWe consider a stochastic volatility model where the dynamics of the volatility are described by a linear function of the (time extended) signature of a primary process which is supposed to be a polynomial diffusion. We obtain closed form expressions for the VIX squared, exploiting the fact that the truncated signature of a polynomial diffusion is again a polynomial diffusion. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log‐price and the VIX squared as linear functions of the signature of the corresponding augmented process. This feature can then be efficiently used for pricing and calibration purposes. Indeed, as the signature samples can be easily precomputed, the calibration task can be split into an offline sampling and a standard optimization. We also propose a Fourier pricing approach for both VIX and SPX options exploiting that the signature of the augmented primary process is an infinite dimensional affine process. For both the SPX and VIX options we obtain highly accurate calibration results, showing that this model class allows to solve the joint calibration problem without adding jumps or rough volatility.

On one-sided testing affine subspaces

We study the query complexity of one-sided ϵ-testing the class of Boolean functions f:Fn→{0,1} that describe affine subspaces and Boolean functions that describe axis-parallel affine subspaces, where F is any finite field. We give polynomial-time ϵ-testers that ask O˜(1/ϵ) queries. This improves the query complexity O˜(|F|/ϵ) in [14]. The almost optimality of the algorithms follows from the lower bound of Ω(1/ϵ) for the query complexity proved by Bshouty and Goldreich [3].We then show that any one-sided ϵ-tester with proximity parameter ϵ<1/|F|d for the class of Boolean functions that describe (n−d)-dimensional affine subspaces and Boolean functions that describe axis-parallel (n−d)-dimensional affine subspaces must make at least Ω(1/ϵ+|F|d−1log⁡n) and Ω(1/ϵ+|F|d−1n) queries, respectively. This improves the lower bound Ω(log⁡n/log⁡log⁡n) that is proved in [14] for F=GF(2). We also give testers for those classes with query complexity that almost match the lower bounds.1

On the asymmetric additive energy of polynomials

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy.

Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves

Let C be a Gorenstein noncomplete intersection monomial curve in the 4-dimensional affine space with defining ideal I(C). In this article, we use the minimal generating set of I(C) to give a criterion for determining whether the tangent cone of C is Cohen-Macaulay. We also show that if the tangent cone of C is Cohen-Macaulay, then the minimal number of generators of the ideal I(C)⁎ is either 5 or an even integer of the form 2d+2, for a suitable integer d. Additionally, we provide a family of Gorenstein noncomplete intersection monomial curves C whose tangent cone is Cohen-Macaulay and the minimal number of generators of I(C)⁎ is large.

Semi-classical 2-dimensional Minkowski planes

AbstractSemi-classical geometries have been investigated in the context of 2-dimensional affine planes, projective planes, Möbius planes and Laguerre planes. Here we deal with the case of 2-dimensional Minkowski planes. Semi-classical 2-dimensional Minkowski planes are obtained by pasting together two halves of the classical real Minkowski plane along two circles or parallel classes. By solving some functional equations for the functions that describe the pasting we determine all semi-classical 2-dimensional Minkowski planes. In contrast to the rich variety of other semi-classical planes there are only very few models of such Minkowski planes.

Classification of the conformally flat Tchebychev affine Kähler hypersurfaces

Tchebychev affine Kähler hypersurfaces were investigated firstly by Xu and Li, who classified the case of dimension 2. As for the case of higher dimension, Xu-Li also proved a classification for the Tchebychev affine Kähler hypersurfaces with complete Calabi metric and nonnegative Ricci curvature. In this paper, the classification of Tchebychev affine Kähler hypersurfaces is obtained under the assumption that Weyl curvature tensor vanishes. As corollaries, we completely determine 3-dimensional Tchebychev affine Kähler hypersurfaces in local sense and obtain the classification of Calabi complete Tchebychev affine Kähler hypersurfaces in R4, which partially improves the results of Xu-Li. In addition, a Bernstein-type result is got for the entire Tchebychev affine Kähler hypersurface in R4.

Generalized bent functions with large dimension

By a fundamental result by Mesnager et al. in 2018, a generalized bent function (originally defined as a class of functions from an $ n $-dimensional vector space $ \mathbb{V}_n^{(p)} $ into a cyclic group), is a bent function $ g: \mathbb{V}_n^{(p)}\rightarrow \mathbb{F}_p $ with a partition $ \mathcal{P} $ of $ \mathbb{V}_n^{(p)} $, such that for every function $ C $ which is constant on the sets of $ \mathcal{P} $, the function $ g + C $ is bent. The set of these bent functions forms then an affine space of dimension $ |\mathcal{P}| \le p^{n/2} $. This characterization of generalized bent functions is much more comprehensive than any earlier description.In this article, we analyse some classes of bent functions under this perspective. As shown earlier, Maiorana-McFarland bent functions permit the largest possible partitions, giving rise to $ p^{n/2} $-dimensional affine bent function spaces. The reason behind is their characterization as the bent functions, which are affine restricted to the $ n/2 $-dimensional affine subspaces of a trivial cover of $ \mathbb{V}_n^{(p)} $. We will show that maximal possible partitions can also be obtained for other classes of (regular) bent functions. Most notably, these classes are described as bent functions which are affine restricted to non-trivial covers of $ \mathbb{V}_n^{(p)} $. We investigate (largest) partitions for other types of bent functions, including weakly regular and non-weakly respectively non-dual bent functions. We round off the article giving partitions for bent functions obtained from bent partitions, which includes the partial spread class, and partitions for Carlet's class $ \mathcal{C} $ and $ \mathcal{D} $.

О подстановках, разрушающих структуру подпространств определённых размерностей

We consider the sets Pnkconsisting of invertible functions F : F2n → F2n such that any U ⊆ F2n and its image F(U) are not simultaneously k-dimensional affine subspaces of F2n, where 3 ⩽ k ⩽ n - 1. We present lower bounds for the cardinalities of all such Pkn ∩...∩ Pn-1n that improve the result of W. E. Clark, X. Hou, and A. Mihailovs, 2007, providing that these sets are not empty. We prove that almost all permutations of F2n belong to P4n∩...∩ Pn-1n. Asymptotic lower and upper bounds of |P3n| up to o(2n!) are obtained. They are correct for |P3n ∩...∩ Pn-1n| as well. The number of functions from P4n∩...∩ Pn-1n that map exactly one 3-dimensional affine subspace of F2n to an affine subspace is estimated. The connection between the restrictions of component functions of F and the case when both U and F(U) are affine subspaces of is obtained. The characterization of differentially 4-uniform permutations in the mentioned terms is provided.

Free Metaplectic Wigner Distribution: Definition and Heisenberg’s Uncertainty Principles

Inspired by a definition of the closed-form instantaneous cross-correlation Wigner distribution (Zhang in IEEE Trans Signal Process 67(21):5527–5542, 2019), we generalize the notion of Wigner distribution to the so-called free metaplectic Wigner distribution (FMWD) through three free metaplectic transforms, in order to tackle a challenge in high-dimensional complex features information processing. We provide some representative special cases for this general form, including the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -dimensional nonseparable affine characteristic Wigner distribution, kernel function Wigner distribution, convolution representation Wigner distribution and instantaneous cross-correlation Wigner distribution. We establish the standard Heisenberg’s uncertainty principles (HUPs) of the real-valued function for the FMWD. We also establish the standard HUPs of the complex-valued function for some specific (i.e., the orthogonal, the orthonormal, the minimum eigenvalue commutative and the maximum eigenvalue commutative) FMWDs. In view of the one-dimensional case of our results, we solve a burning question regarding the limit of time-frequency superresolution triggered by the linear canonical transform free parameters.

On the dimension of planar self-affine sets with non-invertible maps

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition the dimension equals to the affinity dimension for a typical choice of the linear-parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.

Surjective morphisms from affine space to its Zariski open subsets

We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set [Formula: see text], we construct an endomorphism of [Formula: see text] with [Formula: see text] as its image. By Noether’s normalization lemma, these results extend to give surjective maps from any [Formula: see text]-dimensional affine variety [Formula: see text] to [Formula: see text].

Cameron–Liebler sets for maximal totally isotropic flats in classical affine spaces

AbstractLet be the ‐dimensional classical affine space with parameter over a ‐element finite field , and be the set of all maximal totally isotropic flats in . In this paper, we discuss Cameron–Liebler sets in , obtain several equivalent definitions and present some classification results.

Bicycling geodesics are Kirchhoff rods

A bicycle path is a pair of trajectories in , the ‘front’ and ‘back’ tracks, traced out by the endpoints of a moving line segment of fixed length (the ‘bicycle frame’) and tangent to the back track. Bicycle geodesics are bicycle paths whose front track’s length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily of Kirchhoff rods, a class of curves introduced in 1859 by Kirchhoff, generalizing the planar elastic curves of Bernoulli and Euler.

On the dimension of the set of minimal projections

Let X be a finite-dimensional normed space and let Y⊆X be its proper linear subspace. The set of all minimal projections from X to Y is a convex subset of the space all linear operators from X to X and we can consider its affine dimension. We establish several results on the possible values of this dimension. We prove optimal upper bounds in terms of the dimensions of X and Y. Moreover, we improve these estimates in the polyhedral normed spaces for an open and dense subset of subspaces of the given dimension. As a consequence, in the polyhedral normed spaces a minimal projection is unique for an open and dense subset of hyperplanes. To prove this, we establish certain new properties of the Chalmers-Metcalf operator. Another consequence is the fact, that for every subspace of a polyhedral normed space, there exists a minimal projection with many norming pairs.

Deformations of arcs and comparison of formal neighborhoods for a curve singularity

Abstract Let $ {\mathcal C} $ be an algebraic curve and c be an analytically irreducible singular point of ${\mathcal C}$ . The set ${\mathscr {L}_{\infty }}({\mathcal C})^c$ of arcs with origin c is an irreducible closed subset of the space of arcs on ${\mathcal C}$ . We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along ${\mathscr {L}_{\infty }}({\mathcal C})^c$ the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space.

Bisections of Mass Assignments Using Flags of Affine Spaces

We use recent extensions of the Borsuk–Ulam theorem for Stiefel manifolds to generalize the ham sandwich theorem to mass assignments. A k-dimensional mass assignment continuously imposes a measure on each k-dimensional affine subspace of $${\mathbb {R}}^d$$ . Given a finite collection of mass assignments of different dimensions, one may ask if there is some sequence of affine subspaces $$S_{k-1} \subset S_k \subset \ldots \subset S_{d-1} \subset {\mathbb {R}}^d$$ such that $$S_i$$ bisects all the mass assignments on $$S_{i+1}$$ for every i. We show it is possible to do so whenever the number of mass assignments of dimensions $$(k,\ldots ,d)$$ is a permutation of $$(k,\ldots ,d)$$ . We extend previous work on mass assignments and the central transversal theorem. We also study the problem of halving several families of $$(d-k)$$ -dimensional affine spaces of $${\mathbb {R}}^d$$ using a $$(k-1)$$ -dimensional affine subspace contained in some translate of a fixed k-dimensional affine space. For $$k=d-1$$ , there results can be interpreted as dynamic ham sandwich theorems for families of moving points.

A converse statement to Hutchinson’s theorem and a dimension gap for self-affine measures

A well-known theorem of J. E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the affinity dimension, then the system necessarily consists of similarity transformations. We obtain this result by showing that the equilibrium measures of an affine iterated function system are never Bernoulli measures unless the system either is reducible or consists of similarity transformations. The proof builds on earlier results in the thermodynamic formalism of affine iterated function systems due to Bochi, Feng, Käenmäki, Shmerkin and the first named author and also relies on the work of Benoist on the spectral properties of Zariski-dense subsemigroups of reductive linear groups.

Heteroclinic Cycles Generated by Piecewise Affine Systems in ℝn

It is essential yet challenging to investigate heteroclinic trajectories in chaotic dynamics of high-dimensional systems. This paper presents an efficient approach for precise prediction of heteroclinic cycles in [Formula: see text]-dimensional ([Formula: see text]) piecewise affine systems. Moreover, it systemically investigates different types of heteroclinic cycles in the considered systems, and proposes the corresponding existence conditions with rigorous proofs by analyzing the dynamics of stable and unstable manifolds. In addition, three numerical examples with both a heteroclinic cycle and a chaotic set are provided to verify the established results.

A note on relative Vaserstein symbol

In an unpublished work of Fasel–Rao–Swan the notion of the relative Witt group [Formula: see text] is defined. In this paper, we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol [Formula: see text]. We establish injectivity of this symbol if [Formula: see text] is a non-singular affine algebra of dimension 3 over a perfect [Formula: see text]-field and [Formula: see text] is a local complete intersection ideal of [Formula: see text]. It is natural to ask whether the Vaserstein symbol is injective for a singular 3-dimensional affine algebra. At the end of the paper, we will give an example of a singular 3-dimensional algebra over a perfect [Formula: see text]-field for which the Vaserstein symbol is injective.

Homological mirror symmetry for Milnor fibers via moduli of A∞$A_\infty$‐structures

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of A ∞ $A_\infty$ -structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an n $n$ -dimensional affine hypersurface of degree n + 2 $n+2$ and the double cover of the n $n$ -dimensional affine space branched along a degree 2 n + 2 $2n+2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel (Proceedings of the International Congress of Mathematicians, 2002) which may be of independent interest.