Terms Of Functions Research Articles

Powers of Catalan generating functions for bounded operators

Let be the Catalan sequence and a linear and bounded operator on a Banach space such is a power‐bounded operator. The Catalan generating function is defined by the following Taylor series: Note that the operator is a solution of the quadratic equation . In this paper, we define powers of the Catalan generating function in terms of the Catalan triangle numbers. We obtain new formulae that involve Catalan triangle numbers: the spectrum of and the expression of for in terms of Catalan polynomials ( is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future.

A STUDY ON EXAMINATION OF PERCEPTIONS ON CRITICAL PEDAGOGY AMONG SCHOOL ADMINISTRATORS IN TERMS OF FUNCTIONS OF SCHOOLS AND ON SCHOOL PRACTICES: A CASE STUDY

A STUDY ON EXAMINATION OF PERCEPTIONS ON CRITICAL PEDAGOGY AMONG SCHOOL ADMINISTRATORS IN TERMS OF FUNCTIONS OF SCHOOLS AND ON SCHOOL PRACTICES: A CASE STUDY

Spectral problems of dissipative singular q-Sturm-Liouville operators in limit-circle case

We consider the dissipative singular q-Sturm-Liouville operators acting in the Hilbert space L2 w,q(R+), that the extensions of a minimal symmetric operator with deficiency indices (2, 2) (in limit-circle case).We construct a self-adjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation in terms of the Weyl-Titchmarsh function of a self-adjoint q-Sturm-Liouville operator. We also construct a functional model of the dissipative operator and determine its characteristic function in terms of the scattering matrix of the dilation (or of the Weyl-Titchmarsh function). Theorems on the completeness of the system of or root functions of the dissipative and accumulative q-Sturm-Liouville operators are proved.

Completeness conditions of systems of Bessel functions in weighted L2-spaces interms of entire functions

Let $J_{\nu}$ be the Bessel function of the first kind of index $\nu\ge 1/2$, $p\in\mathbb R$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Sufficient conditions for the completeness of the system $\big\{x^{-p-1}\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\mathbb N\big\}$ in the weighted space $L^2((0;1);x^{2p} dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$.

Axiomatization, Computability and Stability for Discrete Event Time Algorithms

This work proposes a formalization of discrete event time algorithms. A Church type thesis, its proof, and the notion of stability for discrete event time algorithms are presented. The Church thesis for discrete algorithms motivates us to consider the Church thesis for the case when we are dealing with discrete event time algorithms. The notions of discrete event time algorithm and discrete event time dynamical system are postulated to be equivalent. The stability concept for discrete event time algorithms is defined. The stability analysis presentation starts concentrating in discrete event time algorithms, i.e., discrete event time dynamical systems, whose Petri net model is described by difference equations, and continues considering Lyapunov energy function sin terms of the logical structures of the vocabulary. A stability analysis based on the reachability tree of the Petri net model is also discussed.

Genelleştirilmiş Lojistik Büyüme Eğrisinin Birinci Türevinin Fourier Dönüşümü

The “generalized logistic growth curve” or the “5-point sigmoid” is a typical example for sigmoidal curves without symmetry and it is commonly used for non-linear regression. The “critical point” of a sigmoidal curve is defined as the limit, if it exists, of the points where its derivatives reach their absolute extreme values. The existence and the location of the critical point of a sigmoidal curve is expressed in terms of its Fourier transform. In this work, we obtain the Fourier transform of the first derivative of the generalized logistic growth curve in terms of Gamma functions and we discuss special cases.

BOPcat software package for the construction and testing of tight-binding models and bond-order potentials

Atomistic models like tight-binding (TB), bond-order potentials (BOP) and classical potentials describe the interatomic interaction in terms of mathematical functions with parameters that need to be adjusted for a particular material. The procedures for constructing TB/BOP models differ from the ones for classical potentials. We developed the BOPcat software package as a modular python code for the construction and testing of TB/BOP parameterizations. It makes use of atomic energies, forces and stresses obtained by TB/BOP calculations with the BOPfox software package. It provides a graphical user interface and flexible control of raw reference data, of derived reference data like defect energies, of automated construction and testing protocols, and of parallel execution in queuing systems. We demonstrate the concepts and usage of the BOPcat software and illustrate its key capabilities by exemplary constructing and testing a parameterization of a magnetic BOP for Fe. We provide a parameterization protocol with a successively increasing set of reference data that leads to good transferability to a variety of properties of the ferromagnetic bcc groundstate and to crystal structures which were not part of the training set.

BMO-type estimates of Riesz transforms associated with generalized Schrödinger operators

Abstract In this paper we show that the dual Riesz transform associated with the generalized Schrödinger operator ℒ {\mathcal{L}} is bounded from BMO {\mathrm{BMO}} into BMO ℒ {\mathrm{BMO}_{\mathcal{L}}} and give the Fefferman–Stein-type decomposition of BMO ℒ {\mathrm{BMO}_{\mathcal{L}}} functions in terms of Riesz transforms.

New class of weak open sets via R*-interior and a decomposition of continuity

In this paper, some new class of sets R*I-S(X), R*I-P(X) and R*I-α(X) which are the collection of all ‘R*I-semi –open’, ‘R*I -pre-open’ and ‘R*I-α-open set’s are introduced and we analysed how the sets are related with them and examine the relationship the other similar collection of weak open sets in ideal and general topological spaces. Also we obtained an equivalent wayof representing continuous function interms of funcions which are defined using weak open sets in R*-topological spaces.

Structure learning in coupled dynamical systems and dynamic causal modelling.

Identifying a coupled dynamical system out of many plausible candidates, each of which could serve as the underlying generator of some observed measurements, is a profoundly ill-posed problem that commonly arises when modelling real-world phenomena. In this review, we detail a set of statistical procedures for inferring the structure of nonlinear coupled dynamical systems (structure learning), which has proved useful in neuroscience research. A key focus here is the comparison of competing models of network architectures-and implicit coupling functions-in terms of their Bayesian model evidence. These methods are collectively referred to as dynamic causal modelling. We focus on a relatively new approach that is proving remarkably useful, namely Bayesian model reduction, which enables rapid evaluation and comparison of models that differ in their network architecture. We illustrate the usefulness of these techniques through modelling neurovascular coupling (cellular pathways linking neuronal and vascular systems), whose function is an active focus of research in neurobiology and the imaging of coupled neuronal systems. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Efficient Bag-of-Features using Improved Whale Optimization Algorithm for Histopathological Image Classification

Background: The whale optimization algorithm is one of the popular meta-heuristic algorithms which has successfully been applied in various application areas such as image analysis and data clustering. However, the slow convergence rate and chances of sticking into the local optima due to improper balance of its exploration and exploitation phases are some of its pitfalls. Therefore, in this paper, a new improved whale optimization algorithm has been proposed. Moreover, the proposed method has been used in bag-of-features method for histopathological image classification. Methods: The new algorithm, improved whale optimization algorithm, modifies the encircling phase of original whale optimization algorithm. The proposed algorithm has been used to cluster the extracted features for finding the relevant codewords to be used in the bag-of-features method for histopathological image classification. Results: The efficiency of proposed algorithm has been analyzed on 23 benchmark functions in terms of mean fitness, standard deviation values, and convergence behavior. The performance of the improved whale optimization algorithm based histopathological image classification method has been analyzed on blue histology image dataset and compared with other meta-heuristic based bagof- features methods in terms of recall, precision, F-measure, and accuracy. The experimental results validate that the proposed method outperforms the considered state-of-the-art methods and attains 12% increase in the histopathological image classification accuracy. Conclusion: In this paper, a new improved whale optimization algorithm has been proposed and applied in bag-of-features method for histopathological image classification. The results of proposed method outperform the other existing meta-heuristic methods over standard benchmark functions and histopathological image dataset.

Sharp Approximations for the Ramanujan Constant

In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan R-function) R(a), by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of R(a), $$\sin (\pi a)$$ and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too.

SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.

Wave-optics uncertainty propagation and regression-based bias model in GNSS radio occultation bending angle retrievals

Abstract. A new reference occultation processing system (rOPS) will include a Global Navigation Satellite System (GNSS) radio occultation (RO) retrieval chain with integrated uncertainty propagation. In this paper, we focus on wave-optics bending angle (BA) retrieval in the lower troposphere and introduce (1) an empirically estimated boundary layer bias (BLB) model then employed to reduce the systematic uncertainty of excess phases and bending angles in about the lowest 2 km of the troposphere and (2) the estimation of (residual) systematic uncertainties and their propagation together with random uncertainties from excess phase to bending angle profiles. Our BLB model describes the estimated bias of the excess phase transferred from the estimated bias of the bending angle, for which the model is built, informed by analyzing refractivity fluctuation statistics shown to induce such biases. The model is derived from regression analysis using a large ensemble of Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) RO observations and concurrent European Centre for Medium-Range Weather Forecasts (ECMWF) analysis fields. It is formulated in terms of predictors and adaptive functions (powers and cross products of predictors), where we use six main predictors derived from observations: impact altitude, latitude, bending angle and its standard deviation, canonical transform (CT) amplitude, and its fluctuation index. Based on an ensemble of test days, independent of the days of data used for the regression analysis to establish the BLB model, we find the model very effective for bias reduction and capable of reducing bending angle and corresponding refractivity biases by about a factor of 5. The estimated residual systematic uncertainty, after the BLB profile subtraction, is lower bounded by the uncertainty from the (indirect) use of ECMWF analysis fields but is significantly lower than the systematic uncertainty without BLB correction. The systematic and random uncertainties are propagated from excess phase to bending angle profiles, using a perturbation approach and the wave-optical method recently introduced by Gorbunov and Kirchengast (2015), starting with estimated excess phase uncertainties. The results are encouraging and this uncertainty propagation approach combined with BLB correction enables a robust reduction and quantification of the uncertainties of excess phases and bending angles in the lower troposphere.

Modeling and Pose Control of Robotic Manipulators and Legs using Conformal Geometric Algebra

Controlling the pose of a manipulator involvesfinding the correct configuration of the robot’s elementsto move the end effector to a desired position and orientation.In order to find the geometric relationshipsbetween the elements of a robot manipulator, it is necessaryto define the kinematics of the robot. We presenta synthesis of the kinematical model of the pose forthis type of robot using the conformal geometric algebraframework. In addition, two controllers are developed,one for the position tracking problem and another for theorientation tracking problem, both using an error feedbackcontroller. The stability analysis is carried out forboth controllers, and their application to a 6-DOF serialmanipulator and the legs of a biped robot are presented.By proposing the error feedback and Lyapunov functionsin terms of geometric algebra, we are opening a newvenue of research in control of manipulators and robotlegs that involves the use of geometric primitives, suchas lines, circles, planes, spheres.

Formal pseudodifferential operators and Witten’s r-spin numbers

Abstract We derive an effective recursion for Witten’s r-spin intersection numbers, using Witten’s conjecture relating r-spin numbers to the Gel’fand–Dikii hierarchy. Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions). We use these closed-form descriptions to prove Harer–Zagier’s formula for the Euler characteristic of ℳ g , 1 {\mathcal{M}_{g,1}} . Finally, we extend Witten’s series expansion formula for the Landau–Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brézin and Hikami.

Nonlinear Closure Relations for Electron Transport in Hydrodynamical Models

Closure relations problem of hydrodynamical models in semiconductors is considered by expressing third- and fourth-order closure relations for the moments of the distribution function in terms of second-order Lagrange multipliers using a generalized Maxwell-Boltzmann distribution function within information theory. Calculation results are commented and compared with others to justify the accuracy of the approach developed in this paper. The comparison involves, in the first part with good agreements, the closure relations results obtained within extended thermodynamics which were checked by means of Monte Carlo simulations, in the second part, the results obtained by Grad's method which expands the distribution function up to fourth-order in Hermite polynomials. It is seen that the latter method cannot give any restriction on closure relations for higher-order moments, within the same conditions proposed in our approach. The important role of Lagrange multipliers for the determination of all closure relations is asserted.

Modelling the calibration of the industrial platinum-resistance thermometers according to GUM

According to the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100: 2008), the calibration process and its uncertainty evaluation should be expressed in terms of mathematical function(s) of input quantities. However, in practice, expressing measurement or calibration in a way that is fully compliant with GUM might be unrealistic and require a clear definition of the calibration process itself. Depending on the applied cali- bration process, different modelling equations with various complexities can be written. In this paper, four different approaches are given to model the calibration process of industrial platinum-resistance thermometers.

Using emission functions in modeling environmentally sustainable traffic assignment policies

Transport systems play a crucial role for sustainable development, and hence, sustainable urban transportation has recently become a major research area. Most of the existing studies propose evaluation methods that use simulation tools to assess the sustainability of different transportation policies. Although there are some recent studies, considering the sustainability dimension and the resulting policies through mathematical programming models is still an open research area. In this study, we focus on controlling the gas emissions for the environmental sustainability and propose several mathematical programming models that incorporate the measurements of gas emissions over a traffic network. We define emission functions in terms of the traffic flow so that the accumulated emission amounts can be modeled accurately, particularly in case of congestion. Using these emission functions, we introduce alternate objective functions and develop optimization models under various policies which are based on the well-known toll pricing and capacity enhancement. The proposed models both reflect the route choice decisions of the network users and the decisions of the transportation managers that aim at making the transport systems more sustainable through the policies of interest. We conduct a computational study on a well-known testing network and present numerical results to evaluate the proposed alternate models. We conclude that simultaneously applying the toll pricing and capacity enhancement policies is in general more effective in serving the travel demand and reducing the emission amounts compared to implementing these policies individually.

Semi-continuous function versions of Urysohn Lemma

Some characterizations of semi-stratifiable and k-semi-stratifiable spaces in terms of semi-continuous functions, analogous to Urysohn Lemma, were established in this paper. Some applications of them in the insertion of functions are given as well.