scholarly journals Doubling coverings via resolution of singularities and preparation

2020 ◽  
pp. 2050018
Author(s):  
Raf Cluckers ◽  
Omer Friedland ◽  
Yosef Yomdin
Keyword(s):  
Level Sets ◽  
Arbitrary Dimension ◽  
General Setting ◽  
Upper Bounds ◽  
Algebraic Set ◽  
Dimension Formula ◽  
First Case ◽  
Polynomial Zero ◽  
Subanalytic Sets ◽  
Low Dimensional

In this paper, we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form [Formula: see text] This is done in a rather general setting, i.e. for the [Formula: see text]-complement of a polynomial zero-level hypersurface [Formula: see text] and for the regular level hypersurfaces [Formula: see text] themselves with no assumptions on the singularities of [Formula: see text]. The coefficient [Formula: see text] is the ambient dimension [Formula: see text] in the first case and [Formula: see text] in the second case. However, the question of a uniform behavior of the coefficient [Formula: see text] remains open. As a second theme, we confirm in arbitrary dimension the upper bound for the number of a-charts covering a real semi-algebraic set [Formula: see text] of dimension [Formula: see text] away from the [Formula: see text]-neighborhood of a lower dimensional set [Formula: see text], with bound of the form [Formula: see text] holding uniformly in the complexity of [Formula: see text]. We also show an analogue for level sets with parameter away from the [Formula: see text]-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

scholarly journals On a Counting Theorem for Weakly Admissible Lattices

2020 ◽  
Author(s):  
Reynold Fregoli
Keyword(s):  
Diophantine Approximation ◽  
General Setting ◽  
Upper Bounds ◽  
Lattice Points ◽  
Precise Estimate ◽  
Linear Subspaces ◽  
Minimal Structure ◽  
Partition Method

Abstract We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve “hyperbolic spikes” and occur naturally in multiplicative Diophantine approximation. We use Wilkie’s o-minimal structure $\mathbb{R}_{\exp }$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The 1st one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of the author. The 2nd application establishes new examples of linear subspaces of Khintchine type thereby refining a theorem by Huang and Liu. For the proof of our counting result, we develop a sophisticated partition method that is crucial for further upcoming work on sums of reciprocals of fractional parts over distorted boxes.

Bayesian analysis of longitudinal and multidimensional functional data

Biostatistics ◽  
2020 ◽  
Author(s):  
John Shamshoian ◽  
Damla Şentürk ◽  
Shafali Jeste ◽  
Donatello Telesca
Keyword(s):  
Functional Data ◽  
Observational Studies ◽  
Children With Autism ◽  
Autism Spectrum ◽  
Computationally Efficient ◽  
Modeling Framework ◽  
Simulated Environment ◽  
First Case ◽  
Low Dimensional

Summary Multi-dimensional functional data arises in numerous modern scientific experimental and observational studies. In this article, we focus on longitudinal functional data, a structured form of multidimensional functional data. Operating within a longitudinal functional framework we aim to capture low dimensional interpretable features. We propose a computationally efficient nonparametric Bayesian method to simultaneously smooth observed data, estimate conditional functional means and functional covariance surfaces. Statistical inference is based on Monte Carlo samples from the posterior measure through adaptive blocked Gibbs sampling. Several operative characteristics associated with the proposed modeling framework are assessed comparatively in a simulated environment. We illustrate the application of our work in two case studies. The first case study involves age-specific fertility collected over time for various countries. The second case study is an implicit learning experiment in children with autism spectrum disorder.

scholarly journals Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes

2019 ◽  
Vol 29 (06) ◽  
pp. 951-1007
Author(s):  
Nithi Rungtanapirom ◽  
Jakob Stix ◽  
Alina Vdovina
Keyword(s):  
Arbitrary Dimension ◽  
Congruence Subgroups ◽  
Quaternion Algebras ◽  
Ramanujan Graphs ◽  
Dimension Formula ◽  
Cubical Sets ◽  
Higher Dimensional ◽  
Residually Finite Groups ◽  
Doubling Construction ◽  
Effective Use

We construct vertex transitive lattices on products of trees of arbitrary dimension [Formula: see text] based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher-dimensional cubical version of Ramanujan graphs (optimal expanders).

scholarly journals Decoherence-Free Subspaces for a Quantum Register Interacting with a Spin Environment

2013 ◽  
Vol 20 (04) ◽  
pp. 1350014 ◽  
Author(s):  
Paweł Należyty ◽  
Dariusz Chruściński
Keyword(s):  
Necessary Conditions ◽  
General Setting ◽  
Quantum Spin ◽  
High Dimensional ◽  
Higher Dimension ◽  
Quantum Register ◽  
Measurement Limit ◽  
Sufficient And Necessary Conditions ◽  
Low Dimensional ◽  
Decoherence Free Subspaces

We study a model of a quantum spin register interacting with an environment of spin particles in quantum-measurement limit. In the limit of collective decoherence we obtain the form of state vectors that constitute high-dimensional decoherence-free subspaces (DFS). In a more general setting we present sufficient and necessary conditions for the existence of low-dimensional DFSs that can be used to construct subspaces of higher dimension.

Minimal homeomorphisms on low-dimensional tori

2009 ◽  
Vol 29 (5) ◽  
pp. 1515-1528
Author(s):  
N. M. DOS SANTOS ◽  
R. URZÚA-LUZ
Keyword(s):  
Affine Transformation ◽  
Homology Group ◽  
Arbitrary Dimension ◽  
Linear Part ◽  
Linear Mapping ◽  
Sufficient Condition ◽  
Skew Product ◽  
Lefschetz Fixed Point Theorem ◽  
Low Dimensional ◽  
Minimal Homeomorphisms

AbstractWe study minimal homeomorphisms (all orbits are dense) of the tori Tn, n≤4. The linear part of a homeomorphism φ of Tn is the linear mapping L induced by φ on the first homology group of Tn. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n≤4, then L is quasi-unipontent, that is, all of the eigenvalues of L are roots of unity and conversely if L∈GL(n,ℤ) is quasi-unipotent and 1 is an eigenvalue of L, then there exists a C∞ minimal skew-product diffeomorphism φ of Tn whose linear part is precisely L. We do not know whether these results are true for n≥5. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

scholarly journals TORSIONAL RIGIDITY OF MINIMAL SUBMANIFOLDS

2006 ◽  
Vol 93 (1) ◽  
pp. 253-272 ◽  
Author(s):  
STEEN MARKVORSEN ◽  
VICENTE PALMER
Keyword(s):  
Warped Product ◽  
Torsional Rigidity ◽  
Exit Time ◽  
General Setting ◽  
Minimal Submanifolds ◽  
Upper Bounds ◽  
Product Model ◽  
Product Comparison ◽  
Mean Exit Time ◽  
Model Spaces

We prove explicit upper bounds for the torsional rigidity of extrinsic domains of minimal submanifolds $P^m$ in ambient Riemannian manifolds $N^n$ with a pole $p$. The upper bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained via previously established isoperimetric inequalities, which are here extended to hold for this more general setting based on warped product comparison spaces. We also characterize the geometry of those situations in which the upper bounds for the torsional rigidity are actually attained and give conditions under which the geometric average of the stochastic mean exit time for Brownian motion at infinity is finite.

scholarly journals Finite-Time Consensus Using an Adaptive Terminal Sliding Mode Control Subjected to Input Saturation

2020 ◽  
Vol 49 (3) ◽  
pp. 412-420
Author(s):  
Ming Ren ◽  
Heyan HUANG ◽  
Esmaiel Mirabdollahi
Keyword(s):  
Sliding Mode Control ◽  
Finite Time ◽  
Sliding Mode ◽  
Input Saturation ◽  
Upper Bounds ◽  
Terminal Sliding Mode Control ◽  
Terminal Sliding Mode ◽  
Mode Control ◽  
First Case ◽  
High Dependency

In this paper, finite-time consensus of double-integrator multi-agent systems is investigated. A new adaptive-terminal sliding mode control is proposed to satisfy the goal within a finite time by considering disturbances and input saturation. The problem is solved for two cases. In the first case, the agents are subjected to disturbances with known upper bounds and input saturation parameters. For the case, the control inputs are designed based on a terminal sliding mode technique to achieve the consensus aim within the finite time as a summation of settling and reaching times. Then, a fast terminal sliding mode control is applied and the control inputs are modified to reduce the high dependency of reaching times to initial speeds. In the second case, the upper disturbance bounds are unknown. To handle this problem, the control laws are adopted by an adaptive-terminal sliding mode method. The upper bounds of disturbances are estimated in the finite time. In both cases, the maximum control efforts are adjusted to always be less than the saturation boundary by optional tuning parameters. The proposed methods efficiency is verified by numerical simulations.

The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring

2019 ◽  
Vol 18 (11) ◽  
pp. 1950212 ◽  
Author(s):  
Honglin Zou ◽  
Dijana Mosić ◽  
Jianlong Chen
Keyword(s):  
Sufficient Conditions ◽  
General Setting ◽  
Upper Bounds ◽  
Block Matrix ◽  
Drazin Inverse ◽  
Operator Matrices ◽  
Skew Field ◽  
Block Matrices ◽  
Necessary And Sufficient ◽  
Drazin Index

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of [Formula: see text] block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

scholarly journals Computing tight upper bounds for Bhattacharyya parameters of binary polar code kernels with arbitrary dimension

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tikui Zhang ◽  
Sensen Li ◽  
Bin Yu
Keyword(s):  
Arbitrary Dimension ◽  
Common Factor ◽  
Upper Bounds ◽  
Polar Codes ◽  
Construction Process ◽  
Polar Code ◽  
Tree Construction ◽  
Calculation Process ◽  
The Common ◽  
Parameter Bounds

AbstractMulti-kernel polar codes have recently received considerable attention since they can provide more flexible code lengths than do the original ones. The construction process of them can be simplified by obtaining the Bhattacharyya parameter bounds of the kernels employed. However, there has been currently no generic method for seeking such bounds. In this paper, therefore, we focus on the upper Bhattacharyya parameter bounds of the standard binary polar code kernels with an arbitrary dimension of $$l\ge 2$$ l ≥ 2 . A calculation process composing of four steps, the common column binary tree construction for the channel inputs, the common factor extraction, the calculation feasibility testing, and the upper bound calculation based on pattern matching, is formulated with a computational complexity of $$O(2^l)$$ O ( 2 l ) . It is theoretically proved that the upper bounds obtained by the proposed method are tight, which can lay the foundation to compare the reliability of the synthesized channels in polar codes.

Sign in / Sign up

Export Citation Format

Share Document