Dimensional Affine Space Research Articles

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} $.

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.

Families of commuting automorphisms, and a characterization of the affine space

We prove that the affine space of dimension $n\geq 1$ over an uncountable algebraically closed field ${\bf k}$ is determined, among connected affine varieties, by its automorphism group (viewed as an abstract group). The proof is based on a new result concerning algebraic families of pairwise commuting automorphisms.

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.

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.

Derived categories of coherent sheaves on some zero-dimensional schemes

Let R N = k [ y 1 , … , y N ] / ( y 1 , … , y N ) 2 be the truncated polynomial algebra . Geometrically, it is the algebra of functions on the second infinitesimal neighborhood of a closed point in N -dimensional affine space . In this note we study D b ( R N − mod ) , the bounded derived category of finitely generated R N -modules. We show that for N ≥ 2 the lattice of triangulated subcategories in D b ( R N − mod ) has a rich structure (which is probably wild), in contrast to the case of zero-dimensional complete intersections. Our homological methods produce some applications to universal localizations of free associative algebras . These applications are based on a relation between triangulated subcategories in D b ( R N − mod ) and universal localizations of a free graded associative algebra in N variables.

Every centroaffine Tchebychev hyperovaloid is ellipsoid

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure admitting a canonically defined closed conformal vector field. Then, by taking the advantage of properties about Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding problem of trying to generalize the classical theorem of Blaschke and Deicke on affine hyperspheres in equiaffine differential geometry to that in centroaffine differential geometry.

On effective computations in subsemigroups of affine Cremona semigroup and implentations of new postquantum multivariate cryptosystems

Multivariate cryptography (MC) together with Latice Based, Hash based, Code based and Superelliptic curves based Cryptographies form list of the main directions of Post Quantum Cryptography.Investigations in the framework of tender of National Institute of Standardisation Technology (the USA) indicates that the potential of classical MC working with nonlinear maps of bounded degree and without the usage of compositions of nonlinear transformation is very restricted. Only special case of Rainbow like Unbalanced Oil and Vinegar digital signatures is remaining for further consideration. The remaining public keys for encryption procedure are not of multivariate. nature. The paper presents large semigroups and groups of transformations of finite affine space of dimension n with the multiple composition property. In these semigroups the composition of n transformations is computable in polynomial time. Constructions of such families are given together with effectively computed homomorphisms between members of the family. These algebraic platforms allow us to define protocols for several generators of subsemigroup of affine Cremona semigroups with several outputs. Security of these protocols rests on the complexity of the word decomposition problem, Finally presented algebraic protocols expanded to cryptosystems of El Gamal type which is not a public key system.

Hilbert schemes of nonreduced divisors in Calabi–Yau threefolds and W-algebras

A W-algebra action is constructed on the equivariant Borel-Moore homology of the Hilbert scheme of points on a nonreduced plane in three dimensional affine space, identifying it to the vacuum W-module. This is based on a generalization of the ADHM construction as well as the W-action on the equivariant Borel-Moore homology of the moduli space of instantons constructed by Schiffmann and Vasserot.

Nonlinear methods for model reduction

Typical model reduction methods for parametric partial differential equations construct a linear space Vn which approximates well the solution manifold M consisting of all solutions u(y) with y the vector of parameters. In many problems of numerical computation, nonlinear methods such as adaptive approximation, n-term approximation, and certain tree-based methods may provide improved numerical efficiency over linear methods. Nonlinear model reduction methods replace the linear space Vn by a nonlinear space Σn. Little is known in terms of their performance guarantees, and most existing numerical experiments use a parameter dimension of at most two. In this work, we make a step towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation, we give a first comparison of their performance with the performance of standard linear approximation for any compact set. We then study these methods for solution manifolds of parametrized elliptic PDEs. We study a specific example of library approximation where the parameter domain is split into a finite number N of rectangular cells, with affine spaces of dimension m assigned to each cell, and give performance guarantees with respect to accuracy of approximation versus m and N.

On hypersurfaces satisfying conditions determined by the Opozda–Verstraelen affine curvature tensor

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this tensor for locally strongly convex hypersurfaces M, n>2, with two distinct affine principal curvatures or with three distinct affine principal curvatures assuming that at least one affine principal curvature has multiplicity 1.

Translation surfaces in affine 3-space

In this paper, we study translation surfaces in three dimensional affine space. We characterize the finite type non-degenerate translation surfaces with respect to the first affine and the second affine fundamental forms.

Pathologies on the Hilbert scheme of points

We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil’s Murphy’s Law holds up to retraction for this scheme. Our main tool is a generalized version of the Bialynicki-Birula decomposition.

Classification of locally strongly convex isotropic centroaffine hypersurfaces

In this paper, we establish a complete classification of locally strongly convex isotropic centroaffine hypersurfaces in the (n+1)-dimensional affine space Rn+1.

MODELS OF TORSORS OVER AFFINE SPACES

Let $X:=\mathbb{A}^{n}_{R}$ be the $n$-dimensional affine space over a discrete valuation ring $R$ with fraction field $K$. We prove that any pointed torsor $Y$ over $\mathbb{A}^{n}_{K}$ under the action of an affine finite type group scheme can be extended to a torsor over $\mathbb{A}^{n}_{R}$ possibly after pulling $Y$ back over an automorphism of $\mathbb{A}^{n}_{K}$. The proof is effective. Other cases, including $X=\alpha_{p,R}$, will also be discussed.

A Bernstein Property for F Relative Parabolic Affine Hyperspheres

Let \(\mathbf f : M\rightarrow R^{n+1}\) be a locally strongly convex hypersurface, locally given as graph of a strongly convex function \(\xi _{n+1}=u(\xi _1,\xi _2,\ldots ,\xi _n)\) defined on a convex domain in n dimensional affine space \(R^{n}\). We study a relative affine differential geometry with the conormal field U given by \(U=F^{-1}(-u_1,-u_2,\ldots ,-u_n,1)\), where \(F>0\) is a function of \(\rho \), called F normaliztion. We derive the differential equations satisfied by the F relative affine hyperspheres and prove a Bernstein property of these hypersurfaces.

Games for eigenvalues of the Hessian and concave/convex envelopes

We study the PDE λj(D2u)=0, in Ω, with u=g, on ∂Ω. Here λ1(D2u)≤...≤λN(D2u) are the ordered eigenvalues of the Hessian D2u. First, we show a geometric interpretation of the viscosity solutions to the problem in terms of convex/concave envelopes over affine spaces of dimension j. In one of our main results, we give necessary and sufficient conditions on the domain so that the problem has a continuous solution for every continuous datum g. Next, we introduce a two-player zero-sum game whose values approximate solutions to this PDE problem.

Commuting matrices and the Hilbert scheme of points on affine spaces

Abstract We give linear algebraic and monadic descriptions of the Hilbert scheme of points on the affine space of dimension n which naturally extends Nakajima’s representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of c points on ℂ3 is irreducible for c ≤ 10.

Full Characterization of Generalized Bent Functions as (Semi)-Bent Spaces, Their Dual, and the Gray Image

A natural generalization of bent functions is a class of functions from ${\mathbb F}_{2}^{n}$ to ${\mathbb Z}_{2^{k}}$ which is known as generalized bent (gbent) functions. The construction and characterization of gbent functions are commonly described in terms of the Walsh transforms of the associated Boolean functions. Using similar approach, we first determine the dual of a gbent function when $n$ is even. Then, depending on the parity of $n$ , it is shown that the Gray image of a gbent function is $(k-1)$ or $(k-2)$ plateaued, which generalizes previous results for $k$ = 2,3, and 4. We then completely characterize gbent functions as algebraic objects. More precisely, again depending on the parity of $n$ , a gbent function is a $(k-1)$ -dimensional affine space of bent functions or semi-bent functions with certain interesting additional properties, which we completely describe. Finally, we also consider a subclass of functions from ${\mathbb F}_{2}^{n}$ to ${\mathbb Z}_{2^{k}}$ , called ${\mathbb Z}_{q}$ -bent functions (which are necessarily gbent), which essentially gives rise to relative difference sets similarly to standard bent functions. Two examples of this class of functions are provided and it is demonstrated that many gbent functions are not ${\mathbb Z}_{q}$ -bent.

R-norm bounds and metric properties for zero loci of real analytic functions

We consider the problem of deciding whether or not a zero locus, X , of multivariate real analytic functions crosses a given r -norm ball in the real n -dimensional affine space. We perform a local study of the problem, and we provide both necessary and sufficient conditions to answer the question. Our conditions derive from the analysis of differential geometric properties of X at the center of the ball. An algorithm to evaluate r -norms distances is proposed.