Non-constant Weight Research Articles

Exponential Stability and Hypoelliptic Regularization for the Kinetic Fokker–Planck Equation with Confining Potential

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker–Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-distance between a Fokker–Planck-solution and the steady state has always a sharp decay estimate of the order O((1+t)e-tν/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal O\\big ( (1+t)e^{-t\ u /2}\\big )$$\\end{document}, with ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker–Planck equations (from a weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-space to a weighted H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-space).

A Compressive Sensing CMOS Image Sensor With Partition Sampling Technique

A CMOS image sensor (CIS) that performs compressive sensing (CS) image encoding without compromising the operating speed and hardware complexity is presented in this article. The conversion rate and the frame rate of the sensor are increased by using a high-order sigma-delta (ΣΔ) analog-to-digital converter (ADC) to obtain linear measurements of selected pixel values for CS encoding. Image distortion caused by the nonconstant weight function of high-order ΣΔ ADCs is eliminated by using the proposed sampling technique. The sampling technique makes the effective weights of the inputs to the ADC equal by applying a solution for the set partition problem. In addition, unlike the existing block-based CS encoding scheme that requires complicated analog multiplexers, the proposed column-based CS encoding scheme can be implemented in a hardware-efficient column-parallel fashion, as in conventional CISs. The CIS is fabricated in a 0.11-μm 1P4M CIS process, and the sampling technique and encoding scheme are successfully demonstrated. It achieves a readout noise of 2.63 e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rms</sub> and a dynamic range of 67.96 dB with a power consumption of 56.38 mW. The resulting figure of merit is 0.375 e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sup> ·nJ, which is the lowest among those of recently reported state-of-the-art CS-CISs.

A practical method to evaluate major statistical errors in isothermal titration calorimetry

Continuous efforts have been devoted to improve the accuracy and reliability of isothermal titration calorimetry (ITC) in terms of instrumental design, modeling and data processing. Among various factors, background noise, proportional error and injection volume error lead to heteroscedasticity (non-constant variance) of the titration heat. The variance function method provides an efficient way to estimate the non-constant variance and weight of the titration heat. Following this approach, in this work, the heat residual recorded by calorimeter model of Nano ITC Standard Volume was examined in detail. Based on the CaCl2-EDTA saturated titration and water blank titration, 9 heat variances, each from 60 injections, have been analyzed, which provide reliable information about the variance function. The major statistical errors for general titrations were estimated and the non-constant variance distribution of typical titration (BaCl2 to 18-Crown-6) was successfully predicted.

Nonresonance conditions on the potential for a nonlinear nonautonomous Neumann problem

The aim of this paper is to establish the existence of the principal eigencurve of the p-Laplacian operator with the nonconstant weight subject to Neumann boundary conditions. We then study the nonresonce phenomena under the first eigenvalue and under the principal eigencurve, thus we obtain existence results for some nonautonomous Neumann elliptic problems involving the p-Laplacian operator.

A note on the Kakeya maximal operator and radial weights on the plane

We obtain an estimate of the operator norm of the weighted Kakeya (Nikodým) maximal operator without dilation on $L^2(w)$. Here we assume that a radial weight $w$ satisfies the doubling and supremum condition. Recall that, in the definition of the Kakeya maximal operator, the rectangle in the supremum ranges over all rectangles in the plane pointed in all possible directions and having side lengths $a$ and $aN$ with $N$ fixed. We are interested in its eccentricity $N$ with $a$ fixed. We give an example of a non-constant weight showing that $\sqrt{\log N}$ cannot be removed.

Weighted tridiagonal matrix enhanced multivariance products representation (WTMEMPR) for decomposition of multiway arrays: applications on certain chemical system data sets

This work focuses on the utilization of a very recently developed decomposition method, weighted tridiagonal matrix enhanced multivariance products representation (WTMEMPR) which can be equivalently used on continuous functions, and, multiway arrays after appropriate unfoldings. This recursive method has been constructed on the Bivariate EMPR and the remainder term of each step therein has been expanded into EMPR from step to step until no remainder term appears in one of the consecutive steps. The resulting expansion can also be expressed in a three factor product representation whose core factor is a tridiagonal matrix. The basic difference and novelty here is the non-constant weight utilization and the applications on certain chemical system data sets to show the efficiency of the WTMEMPR truncation approximants.

Note on the tight relative 2-designs on [formula omitted

Bannai–Bannai–Bannai studied intensively the tight relative 2-designs on two shells in H(n,2) and gained many important results. Their results indicate the relationship among relative t-designs in binary Hamming association schemes, combinatorial t-designs and Hadamard matrices. In this note we study the further relationship between tight relative 2-designs on two shells in H(n,2) and them. We first prove that the existence of a special tight relative 2-design of H(n,2) on two shells is equivalent to that of a Hadamard matrix of size m+1, where m≡−1(mod4) and n=2m; we also show how tight relative 2-designs of H(n,2) on two shells and BIBD or λ-linked designs are related. Next, for each integer n, where 30<n≤50, we determine whether there exists such a tight relative 2-design or not using the above results. Finally, we give a possible list of parameters for the existence of tight relative 2-designs on two shells in H(n,2) with non-constant weight.

On the existence of tight relative 2-designs on binary Hamming association schemes

It is known that there is a close analogy between “Euclidean t-designs vs. spherical t-designs” and “Relative t-designs in binary Hamming association schemes vs. combinatorial t-designs”. In this paper, we want to prove how much we can develop a similar theory in the latter situation, imitating the theory in the former one. We first prove that the weight function is constant on each shell for tight relative t-designs on p shells on a wide class of Q-polynomial association schemes, including Hamming association schemes. In the theory of Euclidean t-designs on 2 concentric spheres (shells), it is known that the structure of coherent configurations is naturally attached. However, it seems difficult to prove this claim in a general context. In the case of tight 2-designs in combinatorial 2-designs, there are great many tight 2-designs, i.e., symmetric 2-designs, while there are very few tight 2e-designs for e≥2. So, as a starting point, we concentrate our study to the existence problem of tight relative 2-designs, in particular on 2 shells, in binary Hamming association schemes H(n,2). We prove that every tight relative 2-design on 2 shells in H(n,2) has the structure of coherent configuration. We determine all the possible parameters of coherent configurations attached to such tight relative 2-designs for n≤30. Moreover for each of them we determine whether there exists such a tight relative 2-design or not, either by constructing them from symmetric 2-designs or Hadamard matrices, or theoretically showing the non-existence. In particular, we show that for n≡6(mod8), there exist such tight relative 2-designs whose weight functions are not constant. These are the first examples of those with non-constant weight.

New technique for construction of a zero cross correlation code

This paper proposes a new technique for constructing zero cross correlation codes (ZCC). In the existing techniques, especially for weight w more than one, the weight is always fixed to the maximum number of users (i.e. the code size). To overcome this difficulty, an efficient design is presented. The code design procedure can be easily used to construct variable length and non-constant weight w. A comparison has been established between existing ZCC optical codes and the novel ZCC classes.

Asymptotic expansions for polynomials orthogonal with respect to a complex non-constant weight function

Asymptotic expansions for polynomials orthogonal with respect to a complex non-constant weight function

On Euclidean tight 4-designs

A spherical t -design is a finite subset X in the unit sphere S n - 1 ⊂ R n which replaces the value of the integral on the sphere of any polynomial of degree at most t by the average of the values of the polynomial on the finite subset X . Generalizing the concept of spherical designs, Neumaier and Seidel (1988) defined the concept of Euclidean t -design in R n as a finite set X in R n for which ∑ i = 1 p ( w ( X i ) / ( | S i | ) ) ∫ S i f ( x ) d σ i ( x ) = ∑ x ∈ X w ( x ) f ( x ) holds for any polynomial f ( x ) of deg ( f ) ≤ t , where { S i , 1 ≤ i ≤ p } is the set of all the concentric spheres centered at the origin and intersect with X , X i = X ∩ S i , and w : X → R > 0 is a weight function of X . (The case of X ⊂ S n - 1 and with a constant weight corresponds to a spherical t -design.) Neumaier and Seidel (1988), Delsarte and Seidel (1989) proved the (Fisher type) lower bound for the cardinality of a Euclidean 2 e -design. Let Y be a subset of R n and let 𝒫 e ( Y ) be the vector space consisting of all the polynomials restricted to Y whose degrees are at most e . Then from the arguments given by Neumaier-Seidel and Delsarte-Seidel, it is easy to see that | X | ≥ dim ( 𝒫 e ( S ) ) holds, where S = ∪ i = 1 p S i . The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as designs on S , the bound dim ( 𝒫 e ( S ) ) is natural and universal. In this point of view, we call a Euclidean 2 e -design X with | X | = dim ( 𝒫 e ( S ) ) a tight 2 e -design on p concentric spheres. Moreover if dim ( 𝒫 e ( S ) ) = dim ( 𝒫 e ( R n ) ) ( = n + e e ) holds, then we call X a Euclidean tight 2 e -design. We study the properties of tight Euclidean 2 e -designs by applying the addition formula on the Euclidean space. Furthermore, we give the classification of Euclidean tight 4-designs with constant weight. It is possible to regard our main result as giving the classification of rotatable designs of degree 2 in R n in the sense of Box and Hunter (1957) with the possible minimum size n + 2 2 . We also give examples of nontrivial Euclidean tight 4-designs in R 2 with nonconstant weight,which give a counterexample to the conjecture of Neumaier and Seidel (1988) that there are no nontrivial Euclidean tight 2 e -designs even for the nonconstant weight case for 2 e ≥ 4 .

Exact maximum likelihood time delay estimation for short observation intervals

An exact solution is presented to the problem of maximum likelihood time delay estimation for a Gaussian source signal observed at two different locations in the presence of additive, spatially uncorrelated Gaussian white noise. The solution is valid for arbitrarily small observation intervals; that is, the assumption T>> tau /sub c/, mod d mod made in the derivation of the conventional asymptotic maximum likelihood (AML) time delay estimator (where tau /sub c/ is the correlation time of the various random processes involved and d is the differential time delay) is relaxed. The resulting exact maximum likelihood (EML) instrumentation is shown to consist of a finite-time delay-and-sum beamformer, followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one-dimensional integral equation with nonconstant weight. The solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities. Finally, the performance of the EML and AML estimators is compared by means of computer simulations. >

Synthesis of Nonconstant Weight Binary Codes for Minimum Diode Counters

Synthesis of Nonconstant Weight Binary Codes for Minimum Diode Counters