Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y=Φx of a k-sparse signal x∈? ^N , where Φ satisfies the restricted isometry property, then the approximate recovery x ^# via ? ₁-minimization is within O(δ) of x. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth-order ΣΔ (Sigma–Delta) quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduced approximation error that is of the order δ(k/m)^(r?1/2)α for any 0<α<1, if m? _r,α k(logN)^1/(1?α). The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x whose magnitudes are suitably bounded away from zero on their support.     

Erratum to: Risk-Sensitive Control with Near Monotone Cost

The infinite horizon risk-sensitive control problem for non-degenerate controlled diffusions is analyzed under a ‘near monotonicity’ condition on the running cost that penalizes large excursions of the process.     

Frame Fundamental Sensor Modeling and Stability of One-Sided Frame Perturbation

We demonstrate that for all linear devices and/or sensors, signal requisition and reconstruction is naturally a mathematical frame expansion and reconstruction issue, whereas the measurement is carried out via a sequence generated by the exact physical response function (PRF) of the device, termed sensory frame {h _n }. The signal reconstruction, on the other hand, will be carried out using the dual frame $\{\tilde{h}^{a}_{n}\}$ of an estimated sensory frame {h _n ^a }. This consequently results in a one-sided perturbation to a frame expansion, which resides in each and every signal and image reconstruction problem. We show that the stability of such a one-sided frame perturbation exits. Examples of image reconstructions in de-blurring are demonstrated.     

Paley-Wiener Approximations and Multiscale Approximations in Sobolev and Besov Spaces on Manifolds

An approximation theory by bandlimited functions (≡ Paley-Wiener functions) on Riemannian manifolds of bounded geometry is developed. Based on this theory multiscale approximations to smooth functions in Sobolev and Besov spaces on manifolds are obtained. The results have immediate applications to the filtering, denoising and approximation and compression of functions on manifolds. There exists applications to problems arising in data dimension reduction, image processing, computer graphics, visualization and learning theory.      

Erratum to: A Nonlinear Generalization of Singular Value Decomposition and Its Applications to Mathematical Modeling and Chaotic Cryptanalysis

Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively applied to Signal Processing, Statistical Analysis and Mathematical Modeling. We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. The method is to augment the embedding matrix with additional nonlinear columns derived from the initial embedding vectors and extract the nonlinear relationship using SVD. The paper demonstrates an application of nonlinear SVD to identify parameters when the signal is generated by a nonlinear transformation. Examples of maps (Logistic map and Henon map) and flows (Van der Pol oscillator and Duffing oscillator) are used to illustrate the method of nonlinear SVD to identify parameters. The paper presents the recovery of parameters in the following scenarios: (i) data generated by maps and flows, (ii) comparison of the method for both noisy and noise-free data, (iii) surrogate data analysis for both the noisy and noise-free cases. The paper includes two applications of the method: (i) Mathematical Modeling and (ii) Chaotic Cryptanalysis.     

Erratum to: Stability of Diffusion Coefficients in an Inverse Problem for the Lotka-Volterra Competition System

First we establish a Carleman estimate for Lotka-Volterra competition-diffusion system of three equations with variable coefficients. Then the internal observations with two measurements are allowed to obtain the stability result for the inverse problem consisting of retrieving two smooth diffusion coefficients in the given parabolic system for the dimension n≤3. The proof of the results rely on Carleman estimates and certain energy estimates for parabolic system.     

Erratum to: Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.     

Erratum to: “Subnormal, permutable, and embedded subgroups in finite groups”

The original version of the article was published in Central European Journal of Mathematics, 2011, 9(4), 915–921, DOI: 10.2478/s11533-011-0029-8. Unfortunately, the original version of this article contains a mistake: Lemma 2.1 (2) is not true. We correct Lemma 2.2 (2) and Theorem 1.1 in our paper where this lemma was used.     

Ultra-Second Quantization and “Ghosts” in Quantized Entropy

We previously constructed the theory of quantum thermodynamics, which assigns operators to dual variables (for example, pressure and volume or temperature and entropy, i.e., dual pairs of extensive and intensive variables similar to momenta and coordinates in classical mechanics) similarly to the theory of quantum mechanics. Here we show that in both the bosonic and fermionic cases, the quantized entropy introduced as an operator in special Fock spaces containing a new variable, called the statistical spin, depends on some variables that do not affect any physical results and are hence called ghosts.     

Ground States and Critical Points for Aubry–Mather Theory in Statistical Mechanics

We consider several models of networks of interacting particles and prove the existence of quasi-periodic equilibrium solutions. We assume (1) that the network and the interaction among particles are invariant under a group that satisfies some mild assumptions; (2) that the state of each particle is given by a real number; (3) that the interaction is invariant by adding an integer to the state of all the particles at the same time; (4) that the interaction is ferromagnetic and coercive (it favors local alignment and penalizes large local oscillations); and (5) some technical assumptions on the regularity speed of decay of the interaction with the distance. Note that the assumptions are mainly qualitative, so that they cover many of the models proposed in the literature. We conclude (A) that there are minimizing (ground states) quasi-periodic solutions of every frequency and that they satisfy several geometric properties; (B) if the minimizing solutions do not cover all possible values at a point, there is another equilibrium point which is not a ground state. These results generalize basic results of Aubry–Mather theory (take the network and the group to be ?). In particular, we provide with a generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls–Nabarro barrier vanishes.     

A Differential Game Related to Terrorism: Nash and Stackelberg Strategies

The question of how best to prosecute the ‘war on terror’ leads to strategic interaction in an intertemporal setting. We consider a nonzero sum differential game between a government and a terrorist organisation. Due to the state-separability of the game we are able to determine Nash and Stackelberg solutions in analytic form. Their comparison as well as the sensitivity analysis deliver interesting insight into the design of efficient measures to combat terror.     

Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups

Let G(O_S)\mathbf{G}(\mathcal{O}_{S}) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S)\mathbf{G}(\mathcal{O}_{S}) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S)\mathbf{G}(\mathcal{O}_{S}) established in an earlier paper is sharp in this case.     

ℓ-norm and its Application to Learning Theory

We investigate connections between an important parameter in the theory of Banach spaces called the -norm, and two properties of classes of functions which are essential in Learning Theory – the uniform law of large numbers and the Vapnik–Chervonenkis (VC) dimension. We show that if the -norm of a set of functions is bounded in some sense, then the set satisfies the uniform law of large numbers. Applying this result, we show that if X is a Banach space which has a nontrivial type, then the unit ball of its dual satisfies the uniform law of large numbers. Next, we estimate the -norm of a set of {0,1}-functions in terms of its VC dimension. Finally, we present a `Gelfand number' like estimate of certain classes of functions. We use this estimate to formulate a learning rule, which may be used to approximate functions from the unit balls of several Banach spaces.     

Obstruction Theory and Coincidences in Positive Codimension

Let f, g ： X→Y be two maps between closed manifolds with dim X ≥ dim Y = n ≥ 3. We study the primary obstruction on（f, g） to deforming f and g to be coincidence free on the n-th skeleton of X. We give examples for which obstructions to deforming f and g to be coincidence free are detected by on （f, g）.     

Sobolev Mappings,Degree, Homotopy Classes and Rational Homology Spheres

In this paper we investigate the degree and the homotopy theory of Orlicz–Sobolev mappings W ^1,P (M,N) between manifolds, where the Young function P satisfies a divergence condition and forms a slightly larger space than W ^1,n , n=dim M. In particular, we prove that if M and N are compact oriented manifolds without boundary and dim M=dim N=n, then the degree is well defined in W ^1,P (M,N) if and only if the universal cover of N is not a rational homology sphere, and in the case n=4, if and only if N is not homeomorphic to S ⁴.     

Toeplitz Quantization and Asymptotic Expansions: Peter–Weyl Decomposition

We obtain the Peter–Weyl decomposition of the star product and star restriction associated to the Toeplitz calculus on complex and real symmetric domains, respectively, under the action of the maximal compact subgroup. Both the Berezin and the Berezin–Toeplitz cases are covered.