Real Functions Research Articles

Radial behavior of Mahler functions

Many papers have been recently devoted to the study of the radial behavior as [Formula: see text] of transcendental r-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, r-Mahler functions behave like [Formula: see text] for some [Formula: see text] and [Formula: see text] is a real analytic function of [Formula: see text] such that [Formula: see text]. They did not provide a formula for [Formula: see text] which was made explicit only in a few examples of r-Mahler functions of orders 1 and 2, and for specific values of r. In this paper, we first provide an explicit expression of [Formula: see text] as an exponential of a Fourier series in the variable [Formula: see text] for every r-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute [Formula: see text] associated to the Dilcher–Stolarsky function (a 4-Mahler function of order 2 in [Formula: see text]), we provide an explicit expression of [Formula: see text] for every r-Mahler function of order 2 under mild assumptions on the coefficients in [Formula: see text] of the underlying r-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for r-Mahler functions solutions of inhomogeneous Mahler equations of order 1.

The velocity of a quantum particle in the case of real solutions of the time independent Schrödinger equation

In the de Broglie–Bohm interpretation of quantum mechanics, the momentum of a particle in a box of perfectly reflecting walls is vanishing. On the contrary, the classical physics predicts an oscillatory motion between the walls. The resulting difference between the behavior of a body in classical and quantum regime is unsatisfactory for Einstein and many other scientists. In order to solve the problem, we show that it is possible to obtain a motion of the body using the rules of quantum mechanics. The result holds for the quantum systems described by solutions of the time independent Schrödinger equation that are real functions.

Properties of classical evanescent waves obtained by using the linear independence concept

Normally, the properties of evanescent optical waves are obtained by developing the Fresnel equations that are expressed in the complex numbers field when the incident angle exceeds the critical angle. Instead of using complex numbers, here we use real functions and the mathematical concept of linear independence to obtain all the properties of the evanescent waves.

An Exact Solution to the Inverse Problem of Steady Free-Surface Flow over Topography

A simple exact solution is presented to the inverse problem in steady, two-dimensional idealised flow over topography that seeks the bottom profile given knowledge of the free-surface data. Attention is focused on the case when a uniform stream flows over a localised obstacle, although the solution is not restricted to this case. The inverse problem is formulated as a Stieltjes integral equation which is solved exactly using a Fourier transform. The solution requires the analytic continuation of two real functions representing the surface speed and the angle between the surface velocity vector and the horizontal. Some example surface profiles and their corresponding bottom topographies are discussed. Although the solution requires the prescription of the surface as a function of the velocity potential, it is shown to closely resemble the corresponding profile in physical space, even for quite large surface displacements, while significant discrepancy occurs at the bottom. Inference of the bottom profile from discrete surface data is accomplished by way of polynomial interpolation and rational approximation in the complex plane for the sample case of a hydraulic fall.

Positioning corporate real estate to drive competitive advantage and facilitate corporate goals

Actively managing a corporate real estate (CRE) portfolio as a strategic asset may not ‘make’ the profit and loss (P&L) but failing to do so can certainly break it. A robust CRE function enhances the organisation’s competitive stance, enabling it to control costs and step up rapidly to embrace new business opportunities. This paper is directed to senior executives whose remit is to foresee and then mitigate risk to the business as well as to periodically benchmark the return on investment of company assets as marked to market. It urges senior management, especially chief financial officers (CFOs), to recognise the value and risk of CRE and to involve their CRE team in asset strategy planning. It draws on the authors’ experience and conversations with CRE leaders, as well as examples of how a strong CRE function can mitigate risk, enhance the corporation’s competitive advantage, seize opportunities, realise hidden value and staunch bleeding P&Ls. It outlines a model structure for corporations seeking to position the corporate real estate function to control risk and participate in asset strategy planning for real estate assets they own or lease in support of their core business. The paper’s premise is that CRE is a strategic asset whose cost, deployment of capital and risk demand an appropriately active management model that promotes business goals and protects against the significant risks inherent in these long-term, illiquid assets. Nonetheless, in many organisations, real estate is handled as a routine administrative function and static cost centre far removed from senior decision makers. Target audiences are CFOs, controllers and private equity operating partners typically in medium-sized companies that are planning or undergoing significant growth, leading to increasing complexity, growing portfolios and higher transaction volumes. It is equally appropriate for executives who insist on being actively involved in real estate decisions to protect their P&Ls and forestall being ‘stuck’ with either too expensive, poorly located or ill-configured real estate with few options to fix it. The concepts described also apply to companies with larger portfolios that need to exert oversight and/or cost control due to evolving market conditions.

Systems of linear equations and the theory of divisibility in the structure of mathematical competence of a chemistry teacher

This article is dedicated to the analysis of the role of linear algebraic models and related divisibility theory issues in the mathematical activity of a chemistry teacher. The work has investigated the place and informational-logical connections of linear algebraic models in the system of mathematical competence of a chemistry teacher and has justified its significance. Examples from the school chemistry course have been given, where systems of linear algebraic equations have been actively used both over the field of real numbers and over the ring of integers. Various scenarios for solving problems, in fact reduced to linear algebraic models, have been illustrated. Such tasks have included classic problems with many reagent components as well as setting coefficients in complex chemical equations. The advantages and disadvantages of synthetic methods using chemical reasoning and formal algebraic methods have been analyzed. It has been substantiated that the teacher needed to understand the mathematical essence of the corresponding models to generate any solution and guide the students' corresponding work. The possibility of applying the basics of number theory in teaching chemistry has been shown. An analysis of the possession of relevant mathematical skills of working chemistry teachers has been conducted. Approaches were developed to improve the level of chemistry teachers’ skills of solving and analyzing systems of linear equations with real and integer variables in the conditions of postgraduate pedagogical education of teachers. The results, obtained in the work, have shown the need for an analysis of mathematics curricula in pedagogical and classical universities, sections related to linear algebra, as well as the introduction of the basics of number theory into mathematics curricula. The need to introduce mathematical training into the system of advanced training of chemistry teachers has been proven. Possible further research on this topic is related to the active implementation of blocks of linear algebraic models into the courses of advanced training of chemistry teachers and further analysis of their effectiveness

Sagittal Plane Alignment for First Metatarsal Phalangeal Arthrodesis Correlated with Postoperative Function: What is the Optimal Position?

There have been many reports describing the proposed alignment of a first metatarsal phalangeal arthrodesis to obtain optimum function. Most of these recommendations are based upon historical and anecdotal evidence. Furthermore, there are few reports directly comparing alignment to patient reported function. We studied radiographic sagittal plane alignment in a group of 60 patients (80 feet) who had undergone a first metatarsal phalangeal joint arthrodesis (20 of the 60 had bilateral arthrodesis) to better understand how this component of the arthrodesis position translates to real world function. The patients in this study had completed a functional survey in 2022 at a mean of 28.4 (median 27.8; range 13.2-45.7) months with very high satisfaction for return to activities of daily living and recreational sports. We measured the sagittal plane position of the first metatarsal relative to the proximal phalanx in this cohort with known post operative activity data. We found that a mean (standard of deviation) sagittal plane angle (angle between the anatomic axis of the first metatarsal and the proximal phalanx) of 15.4 (SD 7.4) degrees and a proximal phalanx head to ground height of 12.7 (SD 3.3) mm was present in this group. Comparing the functional and positional results we conclude that this sagittal plane position provides a good recommendation for alignment.

Тайповая идентичность молодежи Чеченской Республики: 11 лет спустя (опыт социологического исследования)

The Chechen Taip Institute remains an interesting topic for scientists. The article presents the results of an applied sociological study addressing the issue of the Typic identity among Chechens. It was implemented by the interview method (n = 750) and allowed to characterize the state and specificity of transformations of the university students’ Taip identity in the region. The paper presents the results of a comparative analysis of sociological measurements in an 11-year time interval. The analysis of the survey materials allows us to draw conclusions regarding the absence of real social functions of the Taip in modern Chechen society. The position of this entity is “weakening” against the background of “strengthening” Chechen ethno-confessional and civil identities. It is emphasized that the social institution of the Taip is gradually acquiring the characteristics of a “mythical” brotherhood against the backdrop of integration and globalization processes, which are accompanied by the individualization of self-awareness among modern Chechens and their recognition of the priority of personal manifestations over collective models of behavior determined by such a social community as the Taip.

Translating Engagement Resources in English Political Texts Into Arabic

This study sheds light on the translation of 'Engagement Resources' in political texts from English into Arabic. It aims to specify and analyze the 'Engagement Resources' in English political texts and their renditions into Arabic texts by comparing and contrasting them based on Martin and White's Engagement System, which is part of Appraisal Theory (2005) to check whether the translators convey the same effect of these resources or make shifts. If shifts are found, how are those resources shifted in the translation process? The data of this study draw on two texts collected from the ‘Voice Section’ of British online newspapers ‘Independent’ two versions released in English and Arabic. This study hypothesizes that the failure to catch some Engagement resources of the source text (ST) led to violating the Engagement positioning in the target text (TT). After analyzing the two texts, four methods of translation led to a shift in Engagement: addition, omission, modulation, and couplet methods. This study has concluded that the real functions of these resources are depleted. Moreover, translators may convey the Engagement Resources while failing to convey the exact subcategories of these resources.

Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map

We study a family of one-dimensional quasi-periodically forced maps [Formula: see text], where [Formula: see text] is real, [Formula: see text] is an angle, and [Formula: see text] is an irrational frequency, such that [Formula: see text] is a real piecewise-linear map with respect to [Formula: see text] of certain kind. The family depends on two real parameters, [Formula: see text] and [Formula: see text]. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For [Formula: see text] and any [Formula: see text] there is only one continuous invariant curve. For [Formula: see text] there exists a smooth map [Formula: see text] such that: (a) For [Formula: see text], [Formula: see text] has two continuous attracting invariant curves and one continuous repelling curve; (b) For [Formula: see text] it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For [Formula: see text] it has one continuous attracting invariant curve. The case [Formula: see text] is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family [Formula: see text] for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when [Formula: see text].

Extension of unbounded uniformly continuous functions and pseudometrics

In this paper we aim to characterize uniformly continuous real functions and pseudometrics on metric spaces, having uniformly continuous extension. For functions we use a very similar approach as McShane in [7] using moduli of continuity. By doing that we obtain an explicit formula for the extension. We also show that our characterization for functions is equivalent to one proposed in [8] for uniform spaces. We then show that a similar approach can be done for uniformly continuous pseudometrics.To do so we use the notion of chainable metric spaces and intrinsic metrics defined in [9]. A somewhat similar approach has been studied in [6] for normed linear spaces.

Measuring real money demand in Cambodia: an ARDL approach to cointegration

The findings obtained through the ARDL approach to cointegration indicate that real income, consumer price index, and interest rate are cointegrated with real money demand. In terms of the estimated results derived from the long-run model, it is observed that real income has a statistically significant positive impact on real money demand, while the general price level and interest rate have a negative impact. The short-run dynamic model, known as the error correction model, demonstrates that all explanatory variables collectively account for the growth rate of real money balances. In the short-run, the growth rate of real income exhibits a positive relationship with the demand for real money, whereas the inflation rate and changes in interest rate have a significant negative effect on the demand for real money balances. The estimated slope coefficient of the short-run dynamic model, which measures the speed of adjustment, is projected to be 60.67%. This outcome suggests that the real money balance model takes no more than two quarters to adjust towards long-run equilibrium in response to short-run dynamic shocks. Stability tests, such as the CUSUM and CUSUMSQ tests, indicate that the real money demand function in Cambodia remains stable in the long-term. The findings derived from this study provide empirical evidence in favor of the quantity theory of money.

Unified approach for spectral properties of weighted adjacency matrices for graphs with degree-based edge-weights

Let G be a graph and di be the degree of a vertex vi in G. For a symmetric real function f(x,y), one can get an edge-weighted graph in such a way that for each edge vivj of G, the weight of vivj is assigned by f(di,dj). Hence, we have a weighted adjacency matrix Af(G) of G, in which the ij-entry is equal to f(di,dj) if vivj∈E(G) and 0 otherwise. In this paper, we use a unified approach to deal with the spectral properties of Af(G) for f(x,y) to be the functions of graphical or topological function-indices. Firstly, we obtain uniform interlacing inequalities for the weighted adjacency eigenvalues. For the edge-weight functions defined by almost a half of popularly used topological indices, it can be shown that our inequalities cannot be improved. Secondly, we establish a uniform equivalent condition for a connected graph G to have m distinct weighted adjacency eigenvalues. As an application, a combinatorial characterization for a graph to have two and three distinct weighted adjacency eigenvalues is presented, respectively. Moreover, bipartite graphs and unicyclic graphs with three distinct weighted adjacency eigenvalues are characterized. This paper attempts to unify the spectral study for weighted adjacency matrices of graphs with degree-based edge-weights.

Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals

Abstract We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

Intelligent Vehicle Path Planning Based on Optimized A* Algorithm.

With the rapid development of the intelligent driving technology, achieving accurate path planning for unmanned vehicles has become increasingly crucial. However, path planning algorithms face challenges when dealing with complex and ever-changing road conditions. In this paper, aiming at improving the accuracy and robustness of the generated path, a global programming algorithm based on optimization is proposed, while maintaining the efficiency of the traditional A* algorithm. Firstly, turning penalty function and obstacle raster coefficient are integrated into the search cost function to increase the adaptability and directionality of the search path to the map. Secondly, an efficient search strategy is proposed to solve the problem that trajectories will pass through sparse obstacles while reducing spatial complexity. Thirdly, a redundant node elimination strategy based on discrete smoothing optimization effectively reduces the total length of control points and paths, and greatly reduces the difficulty of subsequent trajectory optimization. Finally, the simulation results, based on real map rasterization, highlight the advanced performance of the path planning and the comparison among the baselines and the proposed strategy showcases that the optimized A* algorithm significantly enhances the security and rationality of the planned path. Notably, it reduces the number of traversed nodes by 84%, the total turning angle by 39%, and shortens the overall path length to a certain extent.

On the derived numbers of a real function

The derived numbers of a real function at a point of its domain provide useful information about the variation of the function around that point. In this note, we introduce the concept of one-sided derived numbers and we obtain specific characterizations of some functional properties. We exemplify the usefulness of our results through some interesting consequences and applications.

On semi-stratifiable frames

In this paper, we introduce the notion of a semi-stratifiable frame as an extension of classical semi-stratifiability and also as the monotonization of perfect frames. We show that stratifiable frames are precisely the monotonically normal semi-stratifiable frames. Moreover, we present an insertion theorem for semi-stratifiable frames in terms of real functions and thereby obtain an insertion theorem for stratifiable frames.

Some remarks about $$ \rho $$-regularity for real analytic maps

Some remarks about $$ \rho $$-regularity for real analytic maps

Distributed real-time pricing of smart grid considering individual differences

The utility function that characterizes customers’ satisfaction with electricity consumption plays an important and irreplaceable role in the real-time pricing mechanism based on the social welfare maximization model. Without the accurate quantification of customers’ utility, the real-time pricing will deviate from the reality. In fact, the utility functions of different types of customers in different regions should be obtained by fitting a large amount of historical data over a long period of time based on insight into the relationship between factors and utility. Based on the consideration of customers’ individual differences, a new utility function is proposed, which enriches the form of the utility function and provides a reference for fitting a real and accurate utility function. Further, based on this proposed utility function, a real-time pricing model of social welfare maximization is developed to obtain the fair electricity price between customers and the power supplier. On the basis of the separable structure of variables, we design distributed algorithms with global convergence for the pricing model and estimate a worst-case convergence rate. Numerical simulations verify the feasibility and effectiveness of our algorithms and the rationality of the new utility function, i.e., the electricity price based on the proposed utility function is more robust than the existing ones.

On the dimensional connection between a class of real number sequences and local fractal functions with a single unbounded variation point

In this paper, we investigate the connection between a class of real number sequences and local fractal functions in terms of fractal dimensions. Under certain conditions, we show that the Box dimension of the graph of a local fractal function with a single unbounded variation point is equal to that of its zero points set plus one. Several concrete examples of such functions whose Box dimension can take any numbers belonging to [1,2] have also been given. This work may provide new approaches to the construction of various local fractal functions with the required Box dimension in the future.