Affine Subspace Research Articles

An upper bound of the Hausdorff dimension of singular vectors on affine subspaces

In Diophantine approximation, the notion of singular vectors was introduced by Khintchine in the 1920’s. We study the set of singular vectors on an affine subspace of R n \mathbb {R}^n . We give an upper bound of its Hausdorff dimension in terms of the Diophantine exponent of the parameter of the affine subspace.

Algebraic hierarchical locally recoverable codes with nested affine subspace recovery

AbstractCodes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed–Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed–Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.

On affine spaces of rectangular matrices with constant rank

Let F be a field, and n≥p≥r>0 be integers. In a recent article, Rubei has determined, when F is the field of real numbers, the greatest possible dimension for an affine subspace of n–by–p matrices with entries in F in which all the elements have rank r. In this note, we generalize her result to an arbitrary field with more than r+1 elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of Ms(F) with dimension (s2). The latter is known to be connected to the classification of nonisotropic quadratic forms over F up to congruence.

How to project onto the intersection of a closed affine subspace and a hyperplane

How to project onto the intersection of a closed affine subspace and a hyperplane

Affine subspaces of matrices with rank in a range

Affine subspaces of matrices with rank in a range

Good functions, measures, and the Kleinbock–Tomanov conjecture

Abstract In this paper, we prove a conjecture of Kleinbock and Tomanov (2007) on Diophantine properties of a large class of fractal measures on Q p n \mathbb{Q}_{p}^{n} . More generally, we establish the 𝑝-adic analogues of the influential results of Kleinbock, Lindenstrauss and Weiss (2004) on Diophantine properties of friendly measures. We further prove the 𝑝-adic analogue of one of the main results of Kleinbock (2008) concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of Kleinbock, Lindenstrauss and Weiss is a result on ( C , α ) (C,\alpha) -good functions whose proof crucially uses the Mean Value Theorem. Our main technical innovation is an alternative approach to establishing that certain functions are ( C , α ) (C,\alpha) -good in the 𝑝-adic setting. We believe this result will be of independent interest.

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

Closed-loop identification of stabilized models using dual input–output parameterization

This paper introduces a dual input–output parameterization (dual IOP) for the identification of linear time-invariant systems from closed-loop data. It draws inspiration from the recent input–output parameterization developed to synthesize a stabilizing controller. The controller is parameterized in terms of closed-loop transfer functions, from the external disturbances to the input and output of the system, constrained to lie in a given subspace. Analogously, the dual IOP method parameterizes the unknown plant with analogous closed-loop transfer functions, also referred to as dual parameters. In this case, these closed-loop transfer functions are constrained to lie in an affine subspace guaranteeing that the identified plant is stabilized by the known controller. Compared with existing closed-loop identification techniques guaranteeing closed-loop stability, such as the dual Youla parameterization, the dual IOP requires neither a doubly-coprime factorization of the controller nor a nominal plant that is stabilized by the controller. The dual IOP does not depend on the order and the state-space realization of the controller either, as in the dual system-level parameterization. Simulation shows that the dual IOP outperforms the existing benchmark methods.

Operators with a non-trivial closed invariant affine subspace

We are concerned with the question of the existence of an invariant proper affine subspace for an operator A on a complex Banach space. It turns out that the presence of the number 1 in the spectrum of A or in the spectrum of its adjoint operator A∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*$$\\end{document} is crucial. For instance, an algebraic operator has an invariant proper affine subspace if and only if 1 is its eigenvalue. For an arbitrary operator A, we show that it has an invariant proper hyperplane if and only if 1 is an eigenvalue of A∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*$$\\end{document}. If A is a power bounded operator, then every invariant proper affine subspace is contained in an invariant proper hyperplane, moreover, A has a non-trivial invariant cone.

On affine spaces of alternating matrices with constant rank

Let F be a field, and n ≥ r > 0 be integers, with r even. Denote by A n ⁡ ( F ) the space of all n-by-n alternating matrices with entries in F . We consider the problem of determining the greatest possible dimension for an affine subspace of A n ⁡ ( F ) in which every matrix has rank equal to r (or rank at least r). Recently Rubei [Affine subspaces of skewsymmetric matrices with constant rank. Linear Multilinear Algebra. 2023. in press. doi: 10.1080/03081087.2023.2198759.] has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n ≥ r + 3 and | F | ≥ min ( r − 1 , r 2 + 2 ) , we also determine the affine subspaces of rank r matrices in A n ⁡ ( F ) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case n ≤ r + 2 .

Proximity Testing with Logarithmic Randomness

A fundamental result dating to Ligero (Des. Codes Cryptogr. '23) establishes that each fixed linear block code exhibits proximity gaps with respect to the collection of affine subspaces, in the sense that each given subspace either resides entirely close to the code, or else contains only a small portion which resides close to the code. In particular, any given subspace's failure to reside entirely close to the code is necessarily witnessed, with high probability, by a uniformly randomly sampled element of that subspace. We investigate a variant of this phenomenon in which the witness is not sampled uniformly from the subspace, but rather from a much smaller subset of it. We show that a logarithmic number of random field elements (in the dimension of the subspace) suffice to effect an analogous proximity test, with moreover only a logarithmic (multiplicative) loss in the possible prevalence of false witnesses. We discuss applications to recent noninteractive proofs based on linear codes, including Brakedown (CRYPTO '23).

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.

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} $.

On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property

On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property

Correction of the article "On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property"

Correction of the article "On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property"

Chebyshev unions of planes, and their approximative and geometric properties

We study approximative and geometric properties of Chebyshev sets composed of at most countably many planes (i.e., closed affine subspaces). We will assume that the union of planes is irreducible, i.e., no plane in this union contains another plane from the union. We show, in particular, that if a Chebyshev subset M of a Banach space X consists of at least two planes, then it is not B-connected (i.e., its intersection with some closed ball is disconnected) and is not B̊-complete. We also verify that, in reflexive (CLUR)-spaces (and, in particularly, in complete uniformly convex spaces), a set composed of countably many planes is not a Chebyshev set. For finite unions, we show that any finite union of planes (involving at least two planes) is not a Chebyshev set for any norm on the space. Several applications of our results in the spaces C(Q), L1 and L∞ are also given.

On the image of an affine subspace under the inverse function within a finite field

On the image of an affine subspace under the inverse function within a finite field

Geometric integral formulas of cylinders within slabs

We present new expressions for the integrals of mean curvature of domains in Rn by means of sections with cylinders. Then, we combine these expressions with the corresponding version of the invariant density of affine subspaces in Rn, in order to obtain pseudo-rotational formulae for all the integrals of mean curvature of ∂K. As particular cases, we present pseudo-rotational integral formulas for the volume, area, integral of mean curvature, and Euler-Poincaré characteristic of a connected domain of R3, whose boundary is a surface, considering slabs in R3 whose central plane passes through a fixed point, and cylinders contained in these slabs.

A Fubini-type theorem for Hausdorff dimension

It is well known that a classical Fubini theorem for Hausdorff dimension cannot hold; that is, the dimension of the intersections of a fixed set with a parallel family of planes do not determine the dimension of the set. Here we prove that a Fubini theorem for Hausdorff dimension does hold modulo sets that are small on all Lipschitz graphs.We say that \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G \\subset {\\mathbb{R}^k} \ imes {\\mathbb{R}^n}$$\\end{document} is Γk-null if for every Lipschitz function \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:{\\mathbb{R}^k} \ o {\\mathbb{R}^n}$$\\end{document} the set \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{t \\in {\\mathbb{R}^k}:(t,f(t)) \\in G\\} $$\\end{document} has measure zero. We show that for every Borel set \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\subset {\\mathbb{R}^k} \ imes {\\mathbb{R}^n}$$\\end{document} with dim \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\rm{pro}}{{\\rm{j}}_{{\\mathbb{R}^k}}}E) = k$$\\end{document} there is a Γk-null subset G ⊂ E such that \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dim (E\\backslash G) = k + {\\rm{ess}} - \\sup (\\dim {E_t})$$\\end{document} where ess- sup(dim Et) is the essential supremum of the Hausdorff dimension of the vertical sections \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\{{E_t}\\} _{t \\in {\\mathbb{R}^k}}}$$\\end{document} of E.In addition, we show that, provided that E is not Γk-null, there is a Γk-null subset G ⊂ E such that for F = E G, the Fubini property holds, that is, dim (F) = k + ess-sup(dim Ft).We also obtain more general results by replacing ℝk by an Ahlfors–David regular set. Applications of our results include Fubini-type results for unions of affine subspaces, connection to the Kakeya conjecture and projection theorems.

Irreducible Subcube Partitions

A subcube partition is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates.&#x0D; We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of $\{0,\dots,q-1\}^n$, and partitions of $\mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal size.&#x0D; Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.