Complementary Modulus Research Articles

Quaternionic extended local binary pattern with adaptive structural pyramid pooling for color image representation

This paper proposes a novel feature representation method for color images, namely quaternionic extended local binary pattern (QxLBP) with adaptive structural pyramid pooling (ASPP). First, we propose a QxLBP operator to encode local neighboring information and complementary modulus and phase information in the quaternion domain of color images. In QxLBP, an extended quaternionic representation (EQR) is proposed which introduces an information term into the real part of a quaternion. The resulting EQR enables us to flexibly encode discriminative features and handle multichannel image data. Second, we propose ASPP as a multiresolution pooling way to aggregate local features. Unlike the traditional spatial pyramid pooling which is sensitive to image rotation and spatial changes, ASPP is structure-oriented pooling which can adaptively aggregate the encoded features into multiresolution histogram representations. Experiments on four benchmark image datasets demonstrate that the proposed method achieves the state-of-the-art performance for color texture classification and scene categorization.

Definite integrals of generalized certain class of incomplete elliptic integrals

Elliptic- integral have their importance and potential in certain problems in radiation physics and nuclear technology, studies of crystallographic minimal surfaces, the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk, studies of elliptical crack problems in fracture mechanics. A number of earlier works on the subject contains remarkably large number of general families of elliptic- integrals and indeed also many definite integrals of such families with respect to their modulus (or complementary modulus) are known to arise naturally. Motivated essentially by these and many other potential avenues of their applications, our aim here is to give a systematic account of the theory of a certain family of generalized incomplete elliptic integrals in a unique and generalized manner. The results established in this paper are of manifold generality and basic in nature. By making use of the familiar Riemann-Liouville fractional differ integral operators, we establish many explicit hypergeometric representations and apply these representation in deriving several definite integrals pertaining to their, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of generalized incomplete elliptic integrals involved therein.

On Andrica’s Conjecture, Cramér’s Conjecture, Gaps between Primes and Jacobi Theta Functions III: A Simple Proof for Andrica’s Conjecture

We will use the notation of Armitage and Eberlein, [see 1, p. 103]: k is a real number such that 0< k < 1; k' = (1 - k2)1/2 is the complementary modulus, K = K(k) =∫2π0 (dψ)/(1 - k2 sin2ψ), K' = K(k'). τ = iK' / K, q=exp(πiτ).

A certain class of incomplete elliptic integrals and associated definite integrals

In many seemingly diverse physical contexts (including, for example, certain radiation field problems, studies of crystallographic minimal surfaces, the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk, studies of elliptical crack problems in fracture mechanics, and so on), a remarkably large number of general families of elliptic-type integrals, and indeed also many definite integrals of such families with respect to their modulus (or complementary modulus), are known to arise naturally. Motivated essentially by these and many other potential avenues of their applications, we present here a systematic account of the theory of a certain family of incomplete elliptic integrals in a unified and generalized manner. By means of the familiar Riemann–Liouville fractional differintegral operators, we obtain several explicit hypergeometric representations and apply these representations with a view to deriving various associated definite integrals, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of the incomplete elliptic integrals involved therein.

Calculation of potential flows

Methods for calculating potential flows based on the theory of functions of a complex variable have been widely used as fundamental investigation methods in many fields of engineering, including seepage theory, elasticity theory, continuum mechanics, heat dynamics, aero- and hydromechanics, electromagnetism, electroand radio engineering, etc. [1‐3]. In most cases, the application of these methods involves conformal mapping of the rectangle 1 — 2 — 3 — 4 of the complex domain W = ϕ + i ψ (Fig. 1a) onto a complex half-plane ζ = ξ + i η (Fig. 1b). It is well known [1‐5] that this mapping is performed by means of Jacobi elliptic functions, making use of the complete elliptic integrals of the first kind K and K ’ (Fig. 1) with the modulus λ and the complementary modulus λ ’ = , respectively. This generates considerable difficulties due to the necessity of series expansion of elliptic functions, interpolation of special nonograms and tables, solution of inverse table problems, etc., particularly when it comes to determining the current values of Jacobi functions for the rectangle interior [1‐4, 6, 7]. Moreover, the difficulty of expressing the elliptic functions in terms of elementary functions restricts the possibility of analytically representing the relationship between the physical parameters of the problem under consideration and the given boundary conditions, as well as the use of complicated calculation techniques. The above circumstances considerably constrain the further development of analytical methods for investigating engineering problems in the above-listed lines of inquiry. In this study, we present a new method for solving this problem based on the conformal mapping of a rectangle, one side of which has a vanishingly small convexity, onto a half-plane by means of elementary functions. For this purpose, in the complex half-band W = ϕ + i ψ of width H (Fig. 2), we introduce the function

Ramanujan's explicit values for the classical theta‐function

MathematikaVolume 42, Issue 2 p. 278-294 Research Article Ramanujan's explicit values for the classical theta-function Bruce C. Berndt, Bruce C. Berndt Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL, 61801 U. S. A.Search for more papers by this authorHeng Huat Chan, Heng Huat Chan School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540 U.S.A.Search for more papers by this author Bruce C. Berndt, Bruce C. Berndt Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL, 61801 U. S. A.Search for more papers by this authorHeng Huat Chan, Heng Huat Chan School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540 U.S.A.Search for more papers by this author First published: 01 December 1995 https://doi.org/10.1112/S0025579300014595Citations: 11AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume42, Issue2December 1995Pages 278-294 RelatedInformation

On a generalization of Barton's integral and related integrals of complete elliptic integrals

Let K(k) and E(k) denote respectively the complete elliptic integrals of the first and second kind with modulus k, as defined by Byrd and Friedman ([5] 110·06 and 110·07), and let k′ = √(1 − k2), the complementary modulus.

A 20-D Table of Jacobi's Nome and its Inverse

Abstract : Jacobi's Nome q is given to twenty decimals as a function of the modulus-squared, k squared, the modular angle arc sin k, and the complementary modulus k' for k squared : 0(.001) 999; arc sin k : 0(.10) 89 degrees (.01) 89.99 degrees (.0002) 90 degrees; k' : 0001(.0001).02. The latter table also gives values of an approximation valid in this range to the order of (k') squared, as well as the ratio of the true and approximate values and second central differences of the latter ratio. The fourth table gives k and k' as functions of q for q = 0(.001).5. (Author)

LONG, HORIZONTAL CYLINDRICAL ORE BODY AT ARBITRARY DEPTH IN THE FIELD OF TWO LINEAR CURRENT ELECTRODES*

AbstractThe electric potential distribution when an infinitely long, horizontal cylinder is embedded in an otherwise homogeneous earth with a flat surface, and the current is supplied by two infinitely long electrodes on the surface, parallel to the axis of the cylinder, can be readily found by solving Laplace's equation in bipolar coordinates. The expression for the apparent resistivity (in the sense of Schlumberger) can be obtained after differentiating the expression for the surface potential. This, like the expression for the potentials themselves, involves infinite series in the general case. For the particular case of a cylinder with zero resistivity the apparent resistivity (ρa) can, however, be expressed in terms of the Jacobian elliptic functions cs (2Kμ/π) and the trigonometrical functions cot u where K is the complete elliptic integral of the first kind. The value of K depends upon the radius‐to‐depth ratio of the cylinder. For a cylinder of infinite resistivity the logarithmic derivative of the theta function replaces cs (2Kμ/π). Excellent tabulations exist from which the cs and the cotangent functions can be easily computed.A set of 36 apparent resistivity curves have been computed for the particular case of a perfectly conducting horizontal cylinder midway between the electrodes (but at arbitrary depth), from which a number of interesting conclusions emerge.It is found that the resistivity anomaly (departure of apparent resistivity from its normalized value 1) vertically above the cylinder axis increases with the electrode separation (2L) but attains a maximum value only when the electrodes are at infinity, that is, when the normal electric field is homogeneous. The optimum electrode separation for detecting a cylinder at any depth whatever would therefore appear to be infinity.However, the resistivity anomaly at infinity, which will be called the asymptotic anomaly Δρa(o), in contrast to the central anomaly Δρa(o) directly above the cylinder, is zero when the electrodes are at infinity and increases as they approach each other. The asymptotic anomaly attains a maximum value when 2L is vanishingly small, that is, when the current source is a (linear) electric dipole. Its value is then exactly the same as that of the central anomaly for infinite electrode separation. The optimum electrode separation if the asymptotic anomaly were to be measured would be zero.It is interesting to note that a cylinder of finite radius‐to‐depth ratio (not equal to zero) alters the electric field even at infinity by a finite amount.In general for any 2L, with the single exception of , where h is the depth to the cylinder axis and a is the cylinder radius. In this case the two are equal.Between the central and the asymptotic resistivities for any electrode separation, there exists the relation where m1is the complementary modulus of the elliptic functions. Since K and m1are not independent, the product is a function of the ratio a/h only. The value of the central resistivity for infinite electrode separation is also given by and this fact could be used for finding the a/h ‐ratio of a horizontal cylinder by Schlumberger electric drilling above it.It is also found that the variation of the resistivity anomaly of a horizontal cylinder, in a homogeneous field, with change in depth or radius, follows very nearly an inverse 2nd power law, as for the magnetic anomaly of a cylinder in a homogeneous magnetic field.The paper discusses computations involving elliptic functions in some detail because such computations do not seem to be reported in geophysical literature.