Incomplete Beta Research Articles

Contrast enhancement for image by WNN and GA combining PSNR with information entropy

A new contrast enhancement algorithm for image is proposed combining genetic algorithm (GA) with wavelet neural network (WNN). In-complete Beta transform (IBT) is used to obtain non-linear gray transform curve so as to enhance global contrast for an image. GA determines optimal gray transform parameters. In order to avoid the expensive time for traditional contrast enhancement algorithms, which search optimal gray transform parameters in the whole parameters space, based on gray distribution of an image, a classification criterion is proposed. Contrast type for original image is determined by the new criterion. Parameters space is, respectively, determined according to different contrast types, which greatly shrink parameters space. Thus searching direction of GA is guided by the new parameter space. Considering the drawback of traditional histogram equalization that it reduces the information and enlarges noise and background blur in the processed image, a synthetic objective function is used as fitness function of GA combining peak signal-noise-ratio (PSNR) and information entropy. In order to calculate IBT in the whole image, WNN is used to approximate the IBT. In order to enhance the local contrast for image, discrete stationary wavelet transform (DSWT) is used to enhance detail in an image. Having implemented DSWT to an image, detail is enhanced by a non-linear operator in three high frequency sub-bands. The coefficients in the low frequency sub-bands are set as zero. Final enhanced image is obtained by adding the global enhanced image with the local enhanced image. Experimental results show that the new algorithm is able to well enhance the global and local contrast for image while keeping the noise and background blur from being greatly enlarged.

Does Tricuspid Regurgitation Contribute to Renal Dysfunction in Patients With Heart Failure?

Results: There was no morbidity associated with exercise testing. Baseline estimated metabolic equivalents (METs) were 6.1± 3.0 (mean±S.D.) for patients with severe LVD and 7.3± 2.3 for patients with moderate LVD (p< 0.001). Pressure-rate product (PRP;×100) was 147± 44 for severe LVD and 163± 42 for moderate LVD (p< 0.05). During the program, 18 patients were referred to other rehabilitation units. By the end of the 6-week exercisetraining program (ET), 157 patients (62%) completed cardiac rehabilitation out of the remaining 252 eligible patients. The benefits of completed ET were examined by repeat exercise testing. For the 57 patients with severe LVD, METs improved with ET from 6.5± 2.4 to 8.9± 2.9 (p< 0.001), although PRP was unchanged (155± 49 versus 166± 49). For the 100 patients with moderate LVD, METs improved from 7.3± 2.1 to 10.3± 2.6 (p< 0.001), and PRP increased at peak exercise from 156± 47 to 171± 49 (p< 0.01). Conclusions: Stable patients with moderate or severe LVD can achieve reasonable workloads at exercise testing, with a significant improvement in workloads (average 2–3 METs) in patients completing 6 weeks of ET. This in turn would suggest a potential for at least a 26% reduction in mortality based on current data. doi:10.1016/j.hlc.2008.05.336 336 111± 20). For pts with moderate LVD, resting HR did not change during CR (73± 13 vs. 70± 12), and resting systolic BP rose a little from 113± 19 to 117± 21 (p< 0.05). Conclusions: In this cohort of stable pts with moderate to severe LVD, there was a high level of BB and ACE use, especially in pts with severe LVD. The levels observed in this study are congruent with or better than recently presented Australian studies [Driscoll et. al., Bennett et al. Asia-Pacific Congress of Heart Failure; 2008]. Resting HR averaged≥70, implying incomplete beta blockade inmany pts. doi:10.1016/j.hlc.2008.05.337

Weighted integral and integro-differential inequalities

A general weighted integral inequality for two continuous functions on an interval [ a , b ] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson's integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals.

Dyson indices and Hilbert–Schmidt separability functions and probabilities

A confluence of numerical and theoretical results leads us to conjecture that the Hilbert–Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 × 4 density matrices ρ) are and , respectively. Central to our reasoning are the modifications of two ansätze, recently advanced by Slater (2007 Phys. Rev. A 75 032326), involving incomplete beta functions Bν(a, b), where . We, now, set the separability function proportional to . Then, in the complex case—conforming to a pattern we find, manifesting the Dyson indices (β = 1, 2, 4) of random matrix theory—we take proportional to . We also investigate the real and complex qubit–qutrit cases. Now, there are two variables, , but they appear to remarkably coalesce into the product , so that the real and complex separability functions are again univariate in nature.

On partial derivatives of the incomplete beta function

The incomplete beta function B x ( a , b ) is defined for a , b > 0 and 0 < x < 1 and its definition was extended to negative integer values of a and b in [E. Özçag¯, İ. Ege, H. Gürçay, An extension of the incomplete beta function for negative integers, J. Math. Anal. Appl., in press (doi:10.1016/j.jmaa.2007.05.075)]. In this work we define the partial derivatives of the incomplete beta function B x ( a , b ) for negative integer values of a and b .

Convexity of the incomplete beta functions

In this paper the author finds the condition on b and c such that the incomplete beta function φ(b, c; z) defined by φ(b, c; z):=z 2 F 1(1, b; c; z) and the normalized confluent hypergeometric function are convex in Δ. Conditions on a, b, and c for the convexity of z 2 F 1(a, b; c; z) and z 1 F 1(a; c; z) for a≠1 remains an open problem.

An extension of the incomplete beta function for negative integers

The incomplete beta function B x ( a , b ) is defined for a , b > 0 and 0 < x < 1 . Its definition can be extended, by regularization, to negative non-integer values of a and b. In this paper we define the incomplete beta function B x ( a , b ) for negative integer values of a and b. Further we prove that the function ∂ m + n ∂ a m ∂ b n B x ( a , b ) exists for m , n = 0 , 1 , 2 , … and all a and b.

Multitude of beta distributions with applications

The beta family of distributions, whose origin can be traced to 1676 in a letter from Sir Issac Newton to Henry Oldenbeg [Dutka, J., 1981, The incomplete beta function–A historic profile. Archive for History of Exact Sciences, 24, 11–29.], has been utilized extensively in statistical theory and practice for over a century. Originally defined on the unit interval [0, 1] but easily extended to any finite interval, the beta distributions can take an amazingly great variety of forms. They can be fitted practically to any data representing a phenomenon in almost any field of application. In this paper, we provide a review of the properties and the variations of beta distributions as well as their relationship to other distributions. We also illustrate two real life applications to economic and drought data sets.

Two-qubit separability probabilities and Beta functions

Due to recent important work of Zyczkowski and Sommers (quant-ph/0302197 and quant-ph/0304041), exact formulas are available (both in terms of the Hilbert-Schmidt and Bures metrics) for the (n^2-1)-dimensional and (n(n-1)/2-1)-dimensional volumes of the complex and real n x n density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n=4, that is, the two-qubit systems. Making use of the density matrix (rho) parameterization of Bloore (J. Phys. A 9, 2059 [1976]), we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, nu = (rho_{11} rho_{44})/{rho_{22} rho_{33})$. The associated integrand in each case is the product of a known (highly oscillatory near nu=1) jacobian and a certain unknown univariate function, which our extensive numerical (quasi-Monte Carlo) computations indicate is very closely proportional to an (incomplete) beta function B_{nu}(a,b), with a=1/2, b=sqrt{3}in the real case, and a=2 sqrt{6}/5, b =3/sqrt{2} in the complex case. Assuming the full applicability of these specific incomplete beta functions, we undertake separable volume calculations.

Conservative adjustment of permutation p-values when the number of permutations is limited

In this study, we show through the theory of order statistics that a considerable proportion of p-values will be under-evaluated by the permutation procedure. To solve this problem, we propose to conservatively adjust permutation p-values. The adjustment requires no parametric assumption on the distribution of test statistics. The solution can be expressed by a normalised incomplete beta function. The related normal distribution approximation is also discussed. Simulations are conducted to illustrate the proposed method and two microarray gene expression data sets are considered for applications.

Evaluation of Orbital‐ and Ground State Energies of Some Open‐ and Closed‐Shell Atoms over Integer and Noninteger Slater Type Orbitals

AbstractAb initio calculations of the orbital and the ground state energies of some open‐ and closed‐shell atoms over Slater type orbitals with quantum numbers integer and Slater type orbitals with quantum numbers noninteger have been performed. In order to increase the efficiency of these calculations the atomic two‐electron integrals were expressed in terms of incomplete beta function. Results were observed to be in good agreement with the literature.

Sums, products and ratios of generalized beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, products and ratios of independent beta random variables. In this paper, we extend professor Pham-Gia's results when X1 and X2 are independent random variables distributed according to two generalized beta distributions. For each of these distributions, we derive exact expressions for the densities of S=X1+X2, D=X1−X2, P=X1X2, and R=X2/X1. The expressions turn out to involve the incomplete beta function as well as the hypergeometric functions of one and two variables.

A Case of Type 2 Diabetes Mellitus in an Elderly Patient with Rapid Attenuation of Insulin Secretion that Resembled Fulminant Type 1 DM but with Incomplete Beta Cell Damage

We recently encountered a 66-year-old Japanese man who had suffered from acute hyperglycemia following flu-like symptoms during treatment of type 2 diabetes. Despite significantly increased plasma glucose levels, HbA1c was only slightly elevated. The possibility of autoimmune type 1 diabetes was excluded because of negative islet-related autoantibodies. Serum levels of pancreatic exocrine enzymes, amylase, lipase, and elastase-l were elevated. However, the insulin-secreting function of his islets was not severely damaged. This case is particularly notable for two reasons. First, it showed a fulminant type 1 diabetes-like clinical onset, but his beta cell function was fairly preserved. Second, it developed during the treatment of type 2 diabetes in an elderly patient.

Some properties of an integral transform

Let A be the class of normalized analytic functions in the unit disk U , and S , S ∗ ( α ) and K ( α ) denote the subclasses of A consisting of univalent functions, starlike functions of order α and convex functions of order α in U , respectively. Y.C. Kim gave the following conjecture: Let 0 ≤ α < 1 , β > 1 . If f ∈ S , or S ∗ ( α ) , or K ( α ) , then ϕ ( 3 , 3 + β ; z ) ∗ f ( z ) belongs to the same class, where ϕ ( a , c ; z ) is the incomplete beta function. In this work, we prove that Kim’s conjecture is true for f ∈ S ∗ ( α ) or f ∈ K ( α ) and improve some other results.

Some bivariate beta distributions

We introduce three new bivariate beta distributions based on a characterizing property, involving products of beta random variables. We derive various representations for their joint densities, product moments, conditional densities and conditional moments. Some of these representations involve special functions such as the incomplete beta and Gauss hypergeometric functions. We also discuss ways to construct multivariate generalizations.

Analytical results for the statistical distribution related to a memoryless deterministic walk: Dimensionality effect and mean-field models

Consider a medium characterized by N points whose coordinates are randomly generated by a uniform distribution along the edges of a unitary d-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go to the nearest point which has not been visited in the preceding mu steps (deterministic tourist walk). Each trajectory generated by this dynamics has an initial nonperiodic part of t steps (transient) and a final periodic part of p steps (attractor). The neighborhood rank probabilities are parametrized by the normalized incomplete beta function Id= I1/4 [1/2, (d+1) /2] . The joint distribution S(N) (mu,d) (t,p) is relevant, and the marginal distributions previously studied are particular cases. We show that, for the memory-less deterministic tourist walk in the euclidean space, this distribution is Sinfinity(1,d) (t,p) = [Gamma (1+ I(-1)(d)) (t+ I(-1)(d) ) /Gamma(t+p+ I(-1)(d)) ] delta(p,2), where t=0, 1,2, ... infinity, Gamma(z) is the gamma function and delta(i,j) is the Kronecker delta. The mean-field models are the random link models, which correspond to d-->infinity, and the random map model which, even for mu=0 , presents nontrivial cycle distribution [ S(N)(0,rm) (p) proportional to p(-1) ] : S(N)(0,rm) (t,p) =Gamma(N)/ {Gamma[N+1- (t+p) ] N( t+p)}. The fundamental quantities are the number of explored points n(e)=t+p and Id. Although the obtained distributions are simple, they do not follow straightforwardly and they have been validated by numerical experiments.

Incomplete beta-function expansions of the solutions to the confluent Heun equation

Several expansions of the solutions to the confluent Heun equation in terms of incomplete Beta functions are constructed. A new type of expansion involving certain combinations of the incomplete Beta functions as expansion functions is introduced. The necessary and sufficient conditions when the derived expansions are terminated, thus generating closed-form solutions, are discussed. It is shown that termination of a Beta-function series solution always leads to a solution that is necessarily an elementary function.

Powers of a hypergeometric function

A new expansion for integral powers of the hypergeometric function corresponding to a special case of the incomplete beta function is summarized and consequences, including two new sums involving digamma (psi) functions, are presented.

On classes of analytic functions defined by convolution with incomplete beta functions

Carlson and Shaffer [SIAM J. Math. Anal. 15 (1984) 737–745] defined a convolution operator L ( a , c ) on the class A of analytic functions involving an incomplete beta function ϕ ( a , c ; z ) as L ( a , c ) f = ϕ ( a , c ) ⋆ f . We use this operator to introduce certain classes of analytic functions in the unit disk and study their properties including some inclusion results, coefficient and radius problems. It is shown that these classes are closed under convolution with convex functions.

Adaptive Semi-blind Immune Algorithm for Image Enhancement

The principle and steps of semi-blind immune algorithm are studied, and an adaptive image enhancement method using semi-blind immune algorithm is proposed. The non-linear transform of gray level is an efficient way of image enhancement. In classical image enhancement methods, the specific transform function is determined according to the gray level distribution in the processed image. Tubbs proposed a normalized incomplete Beta function to represent the four kinds of non-linear transform functions most commonly used. But how to adaptively define the coefficients of the Beta function is still a problem. We adopt an adaptive semi-blind immune algorithm that explicitly searches the optimal or suboptimal coefficients more quickly. Compared with the common image adjustment approach, our method is more efficient and powerful.