Projection Theorem Research Articles

On projections of free semialgebraic sets

Abstract We examine to what extent a projection theorem is possible in the non-commutative (free) setting. We first review and extend some results that count against a full free projection theorem. We then obtain a weak version of the projection theorem: projections along linear and separated variables yield semialgebraically parametrised free semi-algebraic sets.

On the topological $K$-theory of twisted equivariant perfect complexes

We construct a comparison map from the topological K-theory of the dg-category of twisted perfect complexes on certain global quotient stacks to twisted equivariant K-theory, generalizing constructions of Halpern-Leistner-Pomerleano and Moulinos. We prove that this map is an equivalence if a version of the projective bundle theorem holds in twisted equivariant K-theory. Along the way, we give a new proof of a theorem of Moulinos that the comparison map is an equivalence in the non-equivariant case.

On the mean projection theorem for determinantal point processes

On the mean projection theorem for determinantal point processes

A nonlinear projection theorem for Assouad dimension and applications

We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to completely resolve the planar distance set problem for Assouad dimension, both dealing with the awkward `critical case' and providing sharp estimates for sets with Assouad dimension less than 1. In the higher dimensional setting we connect the problem to the dimension of the set of exceptions in a related (orthogonal) projection theorem. We also obtain results on pinned distance sets and our results still hold when the distances are taken with respect to a sufficiently curved norm. As another application we prove a radial projection theorem for Assouad dimension with sharp estimates on the Hausdorff dimension of the exceptional set.

A short proof of Host’s equidistribution theorem

This note contains a new proof of Host’s equidistribution theorem for multiplicatively independent endomorphisms of ℝ/ℤ. The method is a simplified version of our recent work on equidistribution under toral automorphisms [2] and is related to the argument in [3], but avoids the use of the scenery flow and of Marstrand’s projection theorem, using instead a direct Fourier argument to establish smoothness of the limit measure.

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes.

Tensor fields are useful for modeling the structure of biological tissues. The challenge to measure tensor fields involves acquiring sufficient data of scalar measurements that are physically achievable and reconstructing tensors from as few projections as possible for efficient applications in medical imaging. In this paper, we present a filtered back-projection algorithm for the reconstruction of a symmetric second-rank tensor field from directional X-ray projections about three axes. The tensor field is decomposed into a solenoidal and irrotational component, each of three unknowns. Using the Fourier projection theorem, a filtered back-projection algorithm is derived to reconstruct the solenoidal and irrotational components from projections acquired around three axes. A simple illustrative phantom consisting of two spherical shells and a 3D digital cardiac diffusion image obtained from diffusion tensor MRI of an excised human heart are used to simulate directional X-ray projections. The simulations validate the mathematical derivations and demonstrate reasonable noise properties of the algorithm. The decomposition of the tensor field into solenoidal and irrotational components provides insight into the development of algorithms for reconstructing tensor fields with sufficient samples in terms of the type of directional projections and the necessary orbits for the acquisition of the projections of the tensor field.

Symmetry methods for a hyperbolic model for a class of populations

We investigate a hyperbolic system that describe the dispersal dynamics of a population, introduced by Méndez and Camacho (1997)[1], in Lie symmetry classification perspective. A Lie group classification is provided for different forms of two constitutive functions: the propagation coefficient D(u) and the reaction term r(u). We establish the Lie symmetry determining equations by utilizing an equivalence generator and the projection theorem. Several extensions of principal Lie algebra are found for different forms of D(u) and r(u). By performing classical reductions we obtain several exact solutions.

Energy optimal guidance for proximity approach with obstacle avoidance

This paper presents an energy optimal guidance algorithm for the approach of a damaged spacecraft with the consideration of the arbitrary number of moving obstacles. The proposed optimal guidance laws are derived by projecting the control function in L2 Space to obtain the minimum norm controller. Unlike conventional trajectory optimization, this algorithm constructs an L2 space based on the state transfer matrix and describes the path constraints as inner products in L2 space. Following this, the proximity approach and obstacle avoidance/rendezvous are combined into a single but global optimal step by the Projection Theorem while obtaining the analytical energy consumption. Examples for multiple static and moving obstacles are presented to demonstrate the advantages of the proposed guidance over traditional trajectory optimization methods. The optimization result of the proposed method is slightly better than GPOPS and the computation time is about five-thousandths of GPOPS.

A non-linear version of Bourgain’s projection theorem

We prove a version of Bourgain’s projection theorem for parametrized families of C^2 maps, which refines the original statement even in the linear case by requiring non-concentration only at a single natural scale. As one application, we show that if A is a Borel set of Hausdorff dimension close to 1 in \mathbb{R}^2 or close to 3/2 in \mathbb{R}^3 , then for y\in A outside of a very sparse set, the pinned distance set \{|x-y|:x\in A\} has Hausdorff dimension at least 1/2+c , where c is universal. Furthermore, the same holds if the distances are taken with respect to a C^2 norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between \delta -balls and \delta -neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into “Frostman pieces” that may be of independent interest.

Integrability of orthogonal projections, and applications to Furstenberg sets

Let G(d,n) be the Grassmannian manifold of n-dimensional subspaces of Rd, and let πV:Rd→V be the orthogonal projection. We prove that if μ is a compactly supported Radon measure on Rd satisfying the s-dimensional Frostman condition μ(B(x,r))⩽Crs for all x∈Rd and r>0, then∫G(d,n)‖πVμ‖Lp(V)pdγd,n(V)<∞,1⩽p<2d−n−sd−s. The upper bound for p is sharp, at least, for d−1⩽s⩽d, and every 0<n<d.Our motivation for this question comes from finding improved lower bounds on the Hausdorff dimension of (s,t)-Furstenberg sets. For 0⩽s⩽1 and 0⩽t⩽2, a set K⊂R2 is called an (s,t)-Furstenberg set if there exists a t-dimensional family L of affine lines in R2 such that dimH⁡(K∩ℓ)⩾s for all ℓ∈L. As a consequence of our projection theorem in R2, we show that every (s,t)-Furstenberg set K⊂R2 with 1<t⩽2 satisfiesdimH⁡K⩾2s+(1−s)(t−1). This improves on previous bounds for pairs (s,t) with s>12 and t⩾1+ϵ for a small absolute constant ϵ>0. We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in Rd. As another corollary of our method, we obtain a δ-discretised sum-product estimate for (δ,s)-sets. Our bound improves on a previous estimate of Chen for every 12<s<1, and also of Guth-Katz-Zahl for s⩾0.5151.

Projection Theorems for Hewitt–Stromberg and Modified Intermediate Dimensions

Projection Theorems for Hewitt–Stromberg and Modified Intermediate Dimensions

Adaptive Current Control for Grid-Connected Inverter with Dynamic Recurrent Fuzzy-Neural-Network

The grid-connected inverter is a vital power electronic equipment connecting distributed generation (DG) systems to the utility grid. The quality of the grid-connected current is directly related to the safe and stable operation of the grid-connected system. This study successfully constructed a robust control system for a grid-connected inverter through a dynamic recurrent fuzzy-neural-network imitating sliding-mode control (DRFNNISMC) framework. Firstly, the dynamic model considering system uncertainties of the grid-connected inverter is described for the global integral sliding-mode control (GISMC) design. In order to overcome the chattering phenomena and the dependence of the dynamic information in the GISMC, a model-free dynamic recurrent fuzzy-neural-network (DRFNN) is proposed as a major controller to approximate the GISMC law without the extra compensator. In the DRFNN, a Petri net with varied threshold is incorporated to fire the rules, and only the parameters of the fired rules are adapted to alleviate the computational workload. Moreover, the network is designed with internal recurrent loops to improve the dynamic mapping capability considering the uncertainties in the control system. In addition, to assure the parameter convergence in the adaptation and the stability of the designed control system, the adaptation laws for the parameters of the DRFNN are deduced by the projection theorem and Lyapunov stability theory. Finally, the experimental comparisons with the GISMC scheme are performed in an inverter prototype to verify the superior performance of the proposed DRFNNISMC framework for the grid-connected current control.

ℓ₂-ℓ∞ Control of Discrete-Time State-Delay Interval Type-2 Fuzzy Systems via Dynamic Output Feedback.

This article investigates the design of the l2-l∞ dynamic output-feedback (DOF) controller for interval type-2 (IT2) T-S fuzzy systems with state delay. For nonlinear systems, the IT2 fuzzy model is an efficient modeling method which can better express uncertainties than the (type-1) fuzzy model. In addition, state delay is also a general factor that affects system performance. After analyzing the stability of the system, based on convex linearization and the projection theorem, this article proposes a delay-dependent output-feedback controller design method. The IT2 membership functions (MFs) of the fuzzy controller are chosen to be different from those of the model so as to increase the freedom of controller selection. A membership-function-dependent (MFD) method based on the staircase MFs is applied to relax the stability analysis results. Finally, a numerical simulation example is given to illustrate the effectiveness of the results.

Construction a Non-parametric Prediction Model for Spatial Random Field Using Projection Theorem

Identifying the best prediction of unobserved observation is one of the most critical issues in spatial statistics. In this line, various methods have been proposed, that each one has advantages and limitations in application. Although the best linear predictor is obtained according to the Kriging method, this model is applied for the Gaussian random field. The uncertainty in the distribution of random fields makes researchers use a method that makes the nongaussian prediction possible. In this paper, using the Projection theorem, a non-parametric method is presented to predict a random field. Then some models are proposed for predicting the nongaussian random field using the nearest neighbors. Then, the accuracy and precision of the predictor will be examined using a simulation study. Finally, the application of the introduced models is examined in the prediction of rainfall data in Khuzestan province.

A Measure Concentration Effect for Matrices of High, Higher, and Even Higher Dimension

Let $n>m$, and let $A$ be an $(m\times n)$-matrix of full rank. Then obviously the estimate $\|Ax\|\leq\|A\|\|x\|$ holds for the euclidean norm of $x$ and $Ax$ and the spectral norm as the assigned matrix norm. We study the sets of all $x$ for which, for fixed $\delta<1$, conversely $\|Ax\|\geq\delta\,\|A\|\|x\|$ holds. It turns out that these sets fill, in the high-dimensional case, almost the complete space once $\delta$ falls below a bound that depends on the extremal singular values of $A$ and on the ratio of the dimensions. This effect has much to do with the random projection theorem, which plays an important role in the data sciences. As a byproduct, we calculate the probabilities this theorem deals with exactly.

Kinematic Model and Redundant Space Analysis of 4-DOF Redundant Robot

The kinematics and redundant space analysis of redundant robots constitutes important research content. Currently, methods such as geometric method and iterative optimization method are relatively complicated and inconvenient for programming and practical control applications. Moreover, little research has been conducted on redundant space analysis. This paper takes the 4-DOF PowerCube redundant robot as the research object. The forward and inverse kinematics equation of the robot are established based on the D-H matrix method, projection and cosine theorem method, and vector coordinate method. Then, the redundant space of the robot is obtained and the redundant space range of the robot is analyzed. Finally, the simulation of kinematic and redundant space research is conducted. It is shown that the change of each joint angle of the robot is smooth without mutation, so it can ensure the stable motion of the robot. Additionally, the different position and redundant variable have a big influence on robotic configuration. The calculation amount is relatively small and convenient for programming application, and the path calculation data of the joints are obtained, which are conducive to the motion control of modular robots. The method used in this paper can be extended to the multi-DOF redundant robot. Finally, the motion optimization, control, and obstacle-avoidance of redundant robots are briefly studied. The results of this paper provide an important basis for these aspects of research.

A Quantification of a Besicovitch Non-linear Projection Theorem via Multiscale Analysis

The Besicovitch projection theorem states that if a subset E of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of E to a line will have zero measure. In other words, the Favard length of a purely unrectifiable 1-set vanishes. In this article, we show that when linear projections are replaced by certain non-linear projections called curve projections, this result remains true. In fact, we go further and use multiscale analysis to prove a quantitative version of this Besicovitch non-linear projection theorem. Roughly speaking, we show that if a subset of the plane has finite length in the sense of Hausdorff and is nearly purely unrectifiable, then its Favard curve length is very small. Our techniques build on those of Tao, who in (Proc Lond Math Soc 98:559–584, 2009) proves a quantification of the original Besicovitch projection theorem.

Design of Distributed Fusion Predictor and Filter without Feedback for Nonlinear System with Correlated Noises and Random Parameter Matrices

Abstract This work presents distributed predictor and filter without feedback for nonlinear stochastic uncertain system with correlated noises. Firstly, for the problem that the process noise and measurement noise are correlated, the two-step prediction theorem based on projection theorem is used to replace the one-step prediction theorem, and the two-step prediction value of a single sensor is obtained. Secondly, the two-step prediction value of each sensor state is used as the measurement information to modify the distributed fusion predictor to obtain the distributed fusion prediction value. Then, according to the projection theorem, the prediction value of distributed fusion is used as measurement information to modify the filtering value of distributed fusion. Finally, the Cubature Kalman filter (CKF) algorithm is used to implement the algorithm proposed in this paper. By comparison with existing methods, the algorithm proposed in this paper solves the problem that existing methods cannot handle state estimation and prediction problems for nonlinear multi-sensor stochastic uncertain systems with correlated noises.

Distributed Robust Kalman Filters under Model Uncertainty and Multiplicative Disturbance

This paper considers the problem of distributed robust state estimation for sensor networks in the presence of model uncertainty and multiplicative noise. More precisely, we assume that the modeling uncertainty, i.e., the actual state-space model belongs to an ambiguity set or a set of convex polytopic uncertain parameters. Several robust Kalman filters are proposed based on projection theorem, variance-constrained optimization, and robust mean square error estimation with different types of ambiguity sets. Stability analysis and simulation example verify the presented distributed robust filters.

Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings

The LA‐module is a nonassociative structure that extends modules over a nonassociative ring known as left almost rings (LA‐rings). Because of peculiar characteristics of LA‐ring and its inception into noncommutative and nonassociative theory, drew the attention of many researchers over the last decade. In this study, the ideas of projective and injective LA‐modules, LA‐vector space, as well as examples and findings, are discussed. We construct a nontrivial example in which it is proved that if the LA‐module is not free, then it cannot be a projective LA‐module. We also construct free LA‐modules, create a split sequence in LA‐modules, and show several outcomes that are connected to them. We have proved the projective basis theorem for LA‐modules. Also, split sequences in projective and injective LA‐modules are discussed with the help of various propositions and theorems.