Noncentral Case Research Articles

Noniterative Generalized Camera Model for Near-Central Camera System.

This paper proposes a near-central camera model and its solution approach. 'Near-central' refers to cases in which the rays do not converge to a point and do not have severely arbitrary directions (non-central cases). Conventional calibration methods are difficult to apply in such cases. Although the generalized camera model can be applied, dense observation points are required for accurate calibration. Moreover, this approach is computationally expensive in the iterative projection framework. We developed a noniterative ray correction method based on sparse observation points to address this problem. First, we established a smoothed three-dimensional (3D) residual framework using a backbone to avoid using the iterative framework. Second, we interpolated the residual by applying local inverse distance weighting on the nearest neighbor of a given point. Specifically, we prevented excessive computation and the deterioration in accuracy that may occur in inverse projection through the 3D smoothed residual vectors. Moreover, the 3D vectors can represent the ray directions more accurately than the 2D entities. Synthetic experiments show that the proposed method can achieve prompt and accurate calibration. The depth error is reduced by approximately 63% in the bumpy shield dataset, and the proposed approach is noted to be two digits faster than the iterative methods.

Period Collapse in Characteristic Quasi-Polynomials of Hyperplane Arrangements

Abstract Given an integral hyperplane arrangement, Kamiya–Takemura–Terao [11, 12] introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of the arrangement modulo a positive integer. The most popular candidate for periods of the characteristic quasi-polynomials is the lcm period. In this paper, we initiate a study of period collapse in characteristic quasi-polynomials stemming from the concept of period collapse in the theory of Ehrhart quasi-polynomials. We say that period collapse occurs in a characteristic quasi-polynomial when the minimum period is strictly less than the lcm period. Our first main result is that in the non-central case, with regard to period collapse anything is possible: period collapse occurs in any dimension $\ge 1$, occurs for any value $\ge 2$ of the lcm period, and the minimum period when it is not the lcm period can be any proper divisor of the lcm period. Our second main result states that in the central case, however, no period collapse is possible in any dimension, that is, the lcm period is always the minimum period.

Is Farming a Technē?

Plato and Aristotle both grant and deny the status of techne to farming. How can we resolve this apparent inconsistency? The answer is by construing techne as a folk concept and farming as a marginal member of this category: farming is a techne in the sense of a specialized, rational practice, but it is a non-central case of techne because it is tied to the land and involves work outdoors.

Moments of a Wishart Matrix

The paper discusses the moments of Wishart matrices, in both the central and noncentral cases. The first part of the paper shows that the expectation map has certain homogeneity and equivariance properties which impose considerable structure on the moments, hitherto unrecognised. The second part of the paper explains how the moments may be computed efficiently. The two parts of the paper are completely independent, but the computations produce precisely the algebraic structure predicted in the first part, as well as reproducing all previously known formulae. A number of examples are given for the more manageable cases.

Moments of a Wishart Matrix

The paper discusses the moments of Wishart matrices, in both the central and noncentral cases. The first part of the paper shows that the expectation map has certain homogeneity and equivariance properties which impose considerable structure on the moments, hitherto unrecognised. The second part of the paper explains how the moments may be computed efficiently. The two parts of the paper are completely independent, but the computations produce precisely the algebraic structure predicted in the first part, as well as reproducing all previously known formulae. A number of examples are given for the more manageable cases.

Legislated Rights in the real world

This essay, written for a symposium in the Jerusalem Review of Legal Studies on Legislated Rights: Securing Human Rights Through Legislation (Cambridge University Press 2018, pb 2019), engages with the careful and challenging contributions by Ittai Bar-Siman-Tov, Lech Garlicki, Vicki Jackson, James Kelly, and Yaniv Roznai. Organised around the theme of the relevance of Legislated Rights in the real world of fallible legislatures, the essay explores seven themes: (1) the neglect of the legislature from court-centred modes of human rights discourse; (2) the central case method; (3) the relevance of real world legislatures to the central case method; (4) populism and non-central cases; (5) the good legislator; (6) the specification of rights and the general welfare; and (7) constitutional rights and judicial review.

Legislated Rights in the Real World

This essay, written for a symposium in the Jerusalem Review of Legal Studies on Legislated Rights: Securing Human Rights Through Legislation (Cambridge University Press 2018, pb 2019), engages with the careful and challenging contributions by Ittai Bar-Siman-Tov, Lech Garlicki, Vicki Jackson, James Kelly, and Yaniv Roznai. Organised around the theme of the relevance of Legislated Rights in the real world of fallible legislatures, the essay explores seven themes: (1) the neglect of the legislature from court-centred modes of human rights discourse; (2) the central case method; (3) the relevance of real world legislatures to the central case method; (4) populism and non-central cases; (5) the good legislator; (6) the specification of rights and the general welfare; and (7) constitutional rights and judicial review.

What’s the Point of Silent Blame?

In this article I thoroughly examine Miranda Fricker’s account of blame based on Communicative Blame as a paradigm explanation of the practice by focusing my attention on a non-central case: silent blame. I argue that this kind of instance has not been successfully connected with her selected paradigm case so I propose a different “focal meaning” or rationale to better accommodate it as part of the entire range of cases of the practice of blaming. I begin my argument drawing upon Herbert L.A. Hart’s methodology on central and borderline cases as a more accurate way to address the complex structure of the concept of blame, since this approach is sensitive to borderline cases as well. I also argue that, although marginal, silent blame fulfils a purpose which must too be vindicated. Thus, following Hart’s suggestion on the value of studying borderline cases, I propose a more accurate reframing of the general point of the practice of blame as a mechanism to secure respect for ourselves.

Exact distributions of two non-central generalized Wilks's statistics

ABSTRACTWe derive the exact expressions of the probability density function (pdf) and the cumulative distribution function (cdf) of Wilks's likelihood ratio criterion Λ and Wilks-Lawley's statistic U in the non-central linear and the non-central planar cases. Those expressions are given in rapidly converging infinite series and can be used for numerical computation. For applications, we compute the exact power of these statistics in a multivariate analysis of variance exercise, and show by simulation the precision of our analytic formulae.

Almost sure limit theorems on Wiener chaos: the non-central case

In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Ito integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].

Monotherapy laser photocoagulation for diabetic macular oedema.

Monotherapy laser photocoagulation for diabetic macular oedema.

Convergence of long-memory discrete [formula omitted]th order Volterra processes

We obtain limit theorems for a class of nonlinear discrete-time processes X(n) called the kth order Volterra processes of order k. These are moving average kth order polynomial forms: X(n)=∑0<i1,…,ik<∞a(i1,…,ik)ϵn−i1…ϵn−ik, where {ϵi} is i.i.d. with Eϵi=0, Eϵi2=1, where a(⋅) is a nonrandom coefficient, and where the diagonals are included in the summation. We specify conditions for X(n) to be well-defined in L2(Ω), and focus on central and non-central limit theorems. We show that normalized partial sums of centered X(n) obey the central limit theorem if a(⋅) decays fast enough so that X(n) has short memory. We prove a non-central limit theorem if, on the other hand, a(⋅) is asymptotically some slowly decaying homogeneous function so that X(n) has long memory. In the non-central case the limit is a linear combination of Hermite-type processes of different orders. This linear combination can be expressed as a centered multiple Wiener–Stratonovich integral.

Bound states in two spatial dimensions in the noncentral case

We derive a bound on the total number of negative energy bound states in a potential in two spatial dimensions by using an adaptation of the Schwinger method to derive the Birman–Schwinger bound in three dimensions. Specifically, counting the number of bound states in a potential gV for g=1 is replaced by counting the number of gi’s for which zero energy bound states exist, and then the kernel of the integral equation for the zero-energy wave functon is symmetrized. One of the keys of the solution is the replacement of an inhomogeneous integral equation by a homogeneous integral equation.

Largest eigenvalue of complex wishart matrices and performance analysis of MIMO MRC systems

This paper extends Khatri (1964, 1969) distribution of the largest eigenvalue of central complex Wishart matrices to the noncentral case. It then applies the resulting new statistical results to obtain closed-form expressions for the outage probability of multiple-input-multiple-output (MIMO) systems employing maximal ratio combining (known also as "beamforming" systems) and operating over Rician-fading channels. When applicable these expressions are compared with special cases previously reported in the literature dealing with the performance of (1) MIMO systems over Rayleigh-fading channels and (2) single-input-multiple-output (SIMO) systems over Rician-fading channels. As a double check these analytical results are validated by Monte Carlo simulations and as an illustration of the mathematical formalism some numerical examples for particular cases of interest are plotted and discussed. These results show that, given a fixed number of total antenna elements and under the same scattering condition (1) SIMO systems are equivalent to multiple-input-single-output systems and (2) it is preferable to distribute the number of antenna elements evenly between the transmitter and the receiver for a minimum outage probability performance.

Bound states in one and two spatial dimensions

In this article we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the well-known fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of bound states. These examples have potentials which decrease at infinity faster than expected. Using somewhat stronger conditions, we derive explicit bounds on the number of bound states in one dimension, using known results for the three-dimensional zero angular momentum. A change of variables which allows us to go from the one-dimensional case to that of two dimensions results in a bound for the zero angular momentum case. Finally, we obtain a bound on the total number of bound states in two dimensions, first for the radial case and then, under stronger conditions, for the noncentral case.

Elliptically contoured model and factorization of Wilks' $ \Lambda$: noncentral case

Kshirsagar in a series of papers, see e.g., Kshirsagar (1964, 1971), McHenry and Kshirsagar (1977), factorizes Wilks' $ \Lambda$ into a number of factors and finds the independent central multivariate beta densities of these factors. These factors are the Wilks' likelihood ratio test criteria for testing goodness of fit of certain canonical variables. Essentially the factors of Wilks' $ \Lambda$ are the factors of the determinants of certain multivariate beta distributed matrices. The Bartlett decompositions of the underlying multivariate beta distribution into independent factors, determine the distributions of these factors. The present paper generalizes Kshirsagar's (1971) normal central distribution theory to elliptically contoured model noncentral distribution theory, showing that Kshirsagar's (1971) nonnull normal theory is nonnull robust for elliptically contoured model.

A necessary condition for a quadratic form to have a chi-squared distribution: an accessible proof

The proof of a well-known necessary condition for a quadratic form, x′Ax, to have a chi-squared distribution in the noncentral case (that is, when the mean of x is a non-null vector) requires the use of advanced mathematical theory. An alternative proof is presented in this article using only simple tools of calculus and some basic matrix results.

Generalized noncentral hermite and laguerre polynomials in multiple matrices

We are concerned with the generalized “noncentral” Hermite and Laguerre polynomials in multiple matrices. We define the generalized Hermite polynomials H ( mn) k[ r]; ø ( Y [ q] ; A [ r] ) in q m × n rectangular matrix arguments Y 1,…, Y q (= Y [ q] ) and r m× m constant matrices A 1,…, A r (= A [ r] ) ( q ⩽ r), associated with the joint distribution of q independent m× n rectangular matrix-variate standard normal distributions, and present various properties of theirs. The discussion of the generalized Hermite polynomials H ( m) k[ r]; ø ( X [ q] ; A [ r] ) in q m × m symmetric matrix arguments X 1,…, X q (= X [ q] ), associated with the joint distribution of q independent m × m symmetric matrix-variate normal distributions, introduced by Chikuse, is further facilitated. The generalized noncentral Hermite polynomials are defined as H ( mn) k[ r]; ø (( Y − N) [ q] ; A [ r] ) and H ( m) k[ r]; ø (( X − M) [ q] ; A [ r] ) for q m × n rectangular and m × m symmetric matrices, N [ q] and M [ q] , respectively. Extending results of Chikuse and Davis to the noncentral case, we define and discuss the generalized noncentral Laguerre polynomials L u[q] k[r];ø(X [q]; A [r]; Ω [q]) in q m × m noncentrality matrices Ω 1,…,Ω q (= Ω [q]) , associated with the joint distribution of q independent noncentral Wishart distributions. Many unsolved problems can now be resolved by introducing these noncentral polynomials in multiple matrices. Specifically, both the differential and integral versions of the generalized multivariate Rodrigues formulae are obtained for all of these polynomials; they are of great use in multivariate distribution theory. A simple example of applications is presented.

Noncentral and central chi-square, f and beta distribution functions as special cases of the distribution function of an indefinite quadratic form

The distribution functions of central and noncentral chi-square, F and β random variables are expressed as special cases of the distribution function of an indefinite quadratic form. Some of these distribution functions, particularly the doubly noncentral cases, have traditionally involved fairly complicated expressions. The form presented in this paper is computationally straightforward and is attractive because it presents these related distributions as special cases of a single equation.

On wilks' a factorization noncentral distribution theory

Kshirsagar in a series of papers, see e.g., Kshirsagar (1964), (1971), (1972), McHenry and Kshirsagar (1977), factorizes Wilks A into a number of factors and finds the independent central multivariate beta densities of these factors. These factors are the likelihood ratio test criteria for testing the goodness of fit of certain assigned discriminant functions or canonical variables either in the space of dependent or independent variables. Essentially the factors of Wilks A are the factors of a certain multivariate beta distributed matrix or its determinant. The Bartlett decomposition of the underlying multivariate beta distribution into independent factors determines the distributions of these factors. In the noncentral case the distributions of these factors are not independent. However, the marginal noncentral distribution of each factor can be obtained. The present paper studies a typical paper of Kshirsagar's (1971), and generalizes Kshirsagar's Wilks A factorization central distribution theory to th...