On the difference of inverse coefficients of convex functions

Let f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${\mathcal {S}}$$ S be the subclass of normalised univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . Let F be the inverse function of f defined in some set $$|\omega |\le r_{0}(f)$$ | ω | ≤ r 0 ( f ) , and be given by $$F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$$ F ( ω ) = ω + ∑ n = 2 ∞ A n ω n . We prove the sharp inequalities $$-1/3 \le |A_4|-|A_3| \le 1/4$$ - 1 / 3 ≤ | A 4 | - | A 3 | ≤ 1 / 4 for the class $${\mathcal {K}}\subset {\mathcal {S}}$$ K ⊂ S of convex functions, thus providing an analogue to the known sharp inequalities $$-1/3 \le |a_4|-|a_3| \le 1/4$$ - 1 / 3 ≤ | a 4 | - | a 3 | ≤ 1 / 4 , and giving another example of an invariance property amongst coefficient functionals of convex functions.

Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.

Joint estimation of Robin coefficient and domain boundary for the Poisson problem

We consider the problem of simultaneously inferring the heterogeneous coefficient field for a Robin boundary condition on an inaccessible part of the boundary along with the shape of the boundary for the Poisson problem. Such a problem arises in, for example, corrosion detection, and thermal parameter estimation. We carry out both linearised uncertainty quantification, based on a local Gaussian approximation, and full exploration of the joint posterior using Markov chain Monte Carlo (MCMC) sampling. By exploiting a known invariance property of the Poisson problem, we are able to circumvent the need to re-mesh as the shape of the boundary changes. The linearised uncertainty analysis presented here relies on a local linearisation of the parameter-to-observable map, with respect to both the Robin coefficient and the boundary shape, evaluated at the maximum a posteriori (MAP) estimates. Computation of the MAP estimate is carried out using the Gauss-Newton method. On the other hand, to explore the full joint posterior we use the Metropolis-adjusted Langevin algorithm (MALA), which requires the gradient of the log-posterior. We thus derive both the Fréchet derivative of the solution to the Poisson problem with respect to the Robin coefficient and the boundary shape, and the gradient of the log-posterior, which is efficiently computed using the so-called adjoint approach. The performance of the approach is demonstrated via several numerical experiments with simulated data.

Invariance Properties of the Entropy Production, and the Entropic Pairing of Inertial Frames of Reference by Shear-Flow Systems

This study examines the invariance properties of the thermodynamic entropy production in its global (integral), local (differential), bilinear, and macroscopic formulations, including dimensional scaling, invariance to fixed displacements, rotations or reflections of the coordinates, time antisymmetry, Galilean invariance, and Lie point symmetry. The Lie invariance is shown to be the most general, encompassing the other invariances. In a shear-flow system involving fluid flow relative to a solid boundary at steady state, the Galilean invariance property is then shown to preference a unique pair of inertial frames of reference—here termed an entropic pair—respectively moving with the solid or the mean fluid flow. This challenges the Newtonian viewpoint that all inertial frames of reference are equivalent. Furthermore, the existence of a shear flow subsystem with an entropic pair different to that of the surrounding system, or a subsystem with one or more changing entropic pair(s), requires a source of negentropy—a power source scaled by an absolute temperature—to drive the subsystem. Through the analysis of different shear flow subsystems, we present a series of governing principles to describe their entropic pairing properties and sources of negentropy. These are unaffected by Galilean transformations, and so can be understood to “lie above” the Galilean inertial framework of Newtonian mechanics. The analyses provide a new perspective into the field of entropic mechanics, the study of the relative motions of objects with friction.

Optimal 3D Angle of Arrival Sensor Placement with Gaussian Priors

Sensor placement is an important factor that may significantly affect the localization performance of a sensor network. This paper investigates the sensor placement optimization problem in three-dimensional (3D) space for angle of arrival (AOA) target localization with Gaussian priors. We first show that under the A-optimality criterion, the optimization problem can be transferred to be a diagonalizing process on the AOA-based Fisher information matrix (FIM). Secondly, we prove that the FIM follows the invariance property of the 3D rotation, and the Gaussian covariance matrix of the FIM can be diagonalized via 3D rotation. Based on this finding, an optimal sensor placement method using 3D rotation was created for when prior information exists as to the target location. Finally, several simulations were carried out to demonstrate the effectiveness of the proposed method. Compared with the existing methods, the mean squared error (MSE) of the maximum a posteriori (MAP) estimation using the proposed method is lower by at least 25% when the number of sensors is between 3 and 6, while the estimation bias remains very close to zero (smaller than 0.15 m).

Formation of Hybrid Computational Structure for the Analysis of Critical Control Scenarios in Biocybernetics

We have considered methods of computational modeling of rapid processes in ecosystems and events of changes with extreme amplitude of abundance. For biocybernetics, the phenomena of degradation of the commercial population in the form of a sudden collapse for specialists and an explosive increase in the number of a new species after invasion — outbreaks are of the type predicted with difficulties. We have developed and ecologically substantiated a method for modeling a group of rapid phenomena, including the calculation of threshold states and transient modes. Algorithmically implemented computational structure, which describes spontaneous modes of rapid transformations in ecodynamics based on the internal properties of biosystems. New model is based on the formalization of threshold effects in regulation of reproduction, included as additional functionals in the basic hybrid structure for research in scenario experiments. Computational scenario is obtained for a generalized description of the extreme population process. We have considered the situation of the collapse of the commercial population with a quota-regulated harvest on the example of the king crab Paralithodes camtschaticus near the coast of Alaska. In the simulation scenario of the crab collapse, we took into account the logic of expert management of the level of exploitation of biological resources. The resulting control scenarios using the iterative model use bifurcations and the loss of the invariance property by the attractor. Modeling with the expert logic of fishery management revealed the characteristic signs of the dynamics of crab collapse and predicted important stages in the process of degradation of exploited biological resources.

Combination of Texture and Shape Features Using Machine and Deep Learning Algorithms for Breast Cancer Diagnosis

Worldwide, breast cancer is a commonly occurring disease in women. Automatic diagnosis of the lesions based on mammographic images is playing an essential role to assist experts. A novel Computer-Aided Diagnosis (CADx) scheme of breast lesion classification is proposed in this paper based on an optimized combination of texture and shape features using machine and deep learning algorithms for mass classification as benign-malignant namely C(M-ZMs)*. The main advantage of using Zernike moments for shape feature extraction is their scale, translation, and rotation invariance property, this allows omitting some of the preprocessing stages in our case. We implemented for texture feature extraction the Monogenic-Local Binary Pattern taking the advantage of lower time and space complexity because monogenic signal analysis needs fewer convolutions and generates more compact feature vectors. Therefore, we used Zernike moments for shape feature extraction due to their scale, translation, and rotation invariance property, this allows omitting some of the preprocessing stages in our proposed system. The proposed system proves its performance on some challenging breast cancer cases where the lesions exist in dense breast tissues. Validation has been undertaken on 520 mammograms from the Digital Database for Screening Mammography Database (DDSM), yielding an accuracy rate of 99.5\%.

Fourier-Bessel beams of finite energy

In this paper, we consider a new type of Bessel beams having Fourier-invariance property and, therefore, called Fourier-Bessel beams. In contrast to the known Bessel beams, these beams have weak side lobes. Analytical expressions for the complex amplitude of the proposed field in the initial plane of the source and in the far field region have been obtained. It is shown that the proposed Fourier-Bessel beams have a finite energy, although they do not have a Gaussian envelope. Their complex amplitude is proportional to a fractional-order Bessel function (an odd integer divided by 6) in the initial plane and in the Fraunhofer zone. The Fourier-Bessel modes have a smaller internal dark spot compared to the Laguerre-Gauss modes with a zero radial index. The proposed beams can be generated with a spatial light modulator and may find uses in telecommunications, interferometry, and the capture of metal microparticles.

Random-effect based test for multinomial logistic regression: choice of the reference level and its impact on the testing

ABSTRACTRandom-effect score test has become an important tool for studying the association between a set of genetic variants and a disease outcome. While a number of random-effect score test approaches have been proposed in the literature, similar approaches for multinomial logistic regression have received less attention. In a recent effort to develop random-effect score test for multinomial logistic regression, we made the observation that such a test is not invariant to the choice of the reference level. This is intriguing because binary logistic regression is well-known to possess the invariance property with respect to the reference level. Here, we investigate why the multinomial logistic regression is not invariant to the reference level, and derive analytic forms to study how the choice of the reference level influences the power. Then we consider several potential procedures that are invariant to the reference level, and compare their performance through numerical studies. Our work provides valuable insights into the properties of multinomial logistic regression with respect to random-effect score test, and adds a useful tool for studying the genetic heterogeneity of complex diseases.