Real Functions Research Articles

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.

Currency Devaluation and Economic Growth in Nigeria

This study examined currency devaluation and economic growth in Nigeria. Specifically, the study economic growth, balance and external debt on currency devaluation in Nigeria and tracked the direction of causality between currency devaluation and real economic variables in Nigeria. The study adopted the qualitative and quantitative research design. Secondary time series data covering 1990 to 2020 were obtained in the study. Estimation methods used in the study’s analysis includes descriptive statistics, correlation analysis, ARDL Co-integration analysis, parsimonious error correction model, granger causality test and other post estimation tests. Findings from the study revealed that gross domestic product exert positive significant impact on currency devaluation captured with currency devaluation both in the short and long run; balance of trade exert positive significant impact on Nigeria currency devaluation both in the short run and insignificant positive impact in the long run; external debt exerts negative significant impact on Nigeria currency devaluation in the short run; and positive significant impact in the long run; and a unidirectional relationship exists between external debt and currency devaluation, balance of trade and currency devaluation, external debt and balance of trade respectively. Premised on these findings, the study advocated that government in collaboration with monetary agencies should seek sound measures to reduce import and encourage exports towards maintaining positive trade balance and improve economic growth; government should create and introduce economic reforms or macroeconomic policies targeted at causing friendly business environment, boost economic output and improving the performance of the economy and government and the Central Bank of Nigeria should ensure debts contracted are deployed towards encouraging investment in exportable goods and services.

Enhancing Wave Function Collapse Algorithm for Procedural Map Generation Problem

In this study, the Improved Map Generation Algorithm (GHAO) method is presented to improve traditional methods in procedural map creation. Traditional procedural map generation techniques using noise generation exhibit shortcomings in the consistent composition of a real map with its uniformly distributed features. On the other hand, procedural map creation techniques that use wave function collapse require that some map pieces already exist to create a map. The observed disadvantages were eliminated by using a hybrid technique with the designed GHAO method. The developed algorithm is similar to real maps in terms of the distribution of map regions, does not need 3D model parts, and performs map creation operations without increasing the algorithm's time complexity. The evaluation of IMGA was carried out by coding the method into the Unity game engine.

Computational Study of the Effect of the Size-Dependent Dielectric Functions of Gold Nanomaterials on Optical Properties

The effect of size on the optical properties of gold nanomaterials has been studied using the theoretical Drude–Sommerfield model. The real and imaginary parts of the dielectric function of bulk as a function of wavelength due to free electron contribution and the real and imaginary parts of the dielectric function of nanogold materials as a function of wavelength due to free electron and bond electron contribution are calculated. The real and imaginary parts of the dielectric function of bulk as a function of wavelength due to the free electron contribution graph and The real and imaginary parts of the dielectric function of nanogold materials as a function of wavelength due to free electron and bond electron contributions are plotted. As we observed from the graphs, the real dielectric functions of both bulk and nanogold materials are inversely proportional to wavelength. The imaginary part of the dielectric function of bulk gold materials is independent of wavelength. At high wavelengths, the size of the gold nanomaterial is highly influenced by both real and imagined dielectric functions at high waves. As the wavelength increases, the effect of the size on the dielectric function also increases. The size-dependent dielectric function of nanomaterials is highly influenced by their optical properties and electrical structure.

A Hierarchical Heuristic Architecture for Unmanned Aerial Vehicle Coverage Search with Optical Camera in Curve-Shape Area

This paper focuses on the problem of dynamic target search in a curve-shaped area by an unmanned aerial vehicle (UAV) with an optical camera. Our objective is to generate an optimal path for UAVs to obtain the maximum detection reward by a camera in the shortest possible time, while satisfying the constraints of maneuverability and obstacle avoidance. First, based on prior qualitative information, the original target probability map for the curve-shaped area is modeled by Parzen windows with 1-dimensional Gaussian kernels, and then several high-value curve segments are extracted by density-based spatial clustering of applications with noise (DBSCAN). Then, given an example that a target floats down river at a speed conforming to beta distribution, the downstream boundary of each curve segment in the future time is expanded and predicted by the mean speed. The rolling self-organizing map (RSOM) neural network is utilized to determine the coverage sequence of curve segments dynamically. On this basis, the whole path of UAVs is a successive combination of the coverage paths and the transferring paths, which are planned by the Dubins method with modified guidance vector field (MGVF) for obstacle avoidance and communication connectivity. Finally, the good performance of our method is verified on a real river map through simulation. Compared with the full sweeping method, our method can improve the efficiency by approximately 31.5%. The feasibility is also verified through a real experiment, where our method can improve the efficiency by approximately 16.3%.

An inequality for log-concave functions and its use in the study of failure rates

Abstract We establish here an integral inequality for real log-concave functions, which can be viewed as an average monotone likelihood property. This inequality is then applied to examine the monotonicity of failure rates.

Hepatocyte-specific METTL3 ablation by Alb-iCre mice (GPT), but not by Alb-Cre mice (JAX), resulted in acute liver failure (ALF) and postnatal lethality.

In 2019, to examine the functions of METTL3 in liver and underlying mechanisms, we generated mice with hepatocyte-specific METTL3 homozygous knockout (METTL3Δhep) by simultaneously crossing METTL3fl/fl mice with Alb-iCre mice (GPT) or Alb-Cre mice (JAX), respectively. In this study, we explored the potential reasons why hepatocyte-specific METTL3 homozygous disruption by Alb-iCre mice (GPT), but not by Alb-Cre mice (JAX), resulted in acute liver failure (ALF) and then postnatal lethality. Mice with hepatocyte-specific METTL3 knockout were generated by simultaneously crossing METTL3fl/fl mice with Alb-iCre mice (GPT; Strain No. T003814) purchased from the GemPharmatech Co., Ltd., (Nanjing, China) or with Alb-Cre mice (JAX; Strain No. 003574) obtained from The Jackson Laboratory, followed by combined-phenotype analysis. The publicly available RNA-sequencing data deposited in the NCBI Gene Expression Omnibus (GEO) database under the accession No.: GSE198512 (postnatal lethality), GSE197800 (postnatal survival) and GSE176113 (postnatal survival) were mined to explore the potential reasons why hepatocyte-specific METTL3 homozygous deletion by Alb-iCre mice (GPT), but not by Alb-Cre mice (JAX), leads to ALF and then postnatal lethality. Firstly, we observed that hepatocyte-specific METTL3 homozygous deficiency by Alb-iCre mice (GPT) or by Alb-Cre mice (JAX) caused liver injury, abnormal lipid accumulation and apoptosis. Secondly, we are surprised to find that hepatocyte-specific METTL3 homozygous deletion by Alb-iCre mice (GPT), but not by Alb-Cre mice (JAX), led to ALF and then postnatal lethality. Our findings clearly demonstrated that METTL3Δhep mice (GPT), which are about to die, exhibited the severe destruction of liver histological structure, suggesting that METTL3Δhep mice (GPT) nearly lose normal liver function, which subsequently contributes to ALF, followed by postnatal lethality. Finally, we unexpectedly found that as the compensatory growth responses of hepatocytes to liver injury induced by METTL3Δhep (GPT), the proliferation of METTL3Δhep hepatocytes (GPT), unlike METTL3Δhep hepatocytes (JAX), was not evidenced by the significant increase of Ki67-positive hepatocytes, not accompanied by upregulation of cell-cycle-related genes. Moreover, GO analysis revealed that upregulated genes in METTL3Δhep livers (GPT), unlike METTL3Δhep livers (JAX), are not functionally enriched in terms associated with cell cycle, cell division, mitosis, microtubule cytoskeleton organization, spindle organization, chromatin segregation and organization, and nuclear division, consistent with the loss of compensatory proliferation of METTL3Δhep hepatocytes (GPT) observed in vivo. Thus, obviously, the loss of the compensatory growth capacity of METTL3Δhep hepatocytes (GPT) in response to liver injury might contribute to, at least partially, ALF and subsequently postnatal lethality of METTL3Δhep mice (GPT). These findings from this study and other labs provide strong evidence that these phenotypes (i.e., ALF and postnatal lethality) of METTL3Δhep mice (GPT) might be not the real functions of METTL3, and closely related with Alb-iCre mice (GPT), suggesting that we should remind researchers to use Alb-iCre mice (GPT) with caution to knockout gene in hepatocytes in vivo.

Beyond HbA1c: Identifying Gaps in Glycemic Control Among Children and Young People with Type 1 Diabetes Using Continuous Glucose Monitoring.

To describe continuous glucose monitoring (CGM) derived glycemic variables, and study their association with HbA1c and socio-economic factors in young people with Type 1 diabetes mellitus (T1DM). Ninety-two participants [age 15.7 ± 5.0 y (mean ± SD), HbA1c 8.0 ± 1.5% (mean ± SD)] wore a professional CGM sensor for 14 d. Median (IQR) time in range (TIR) was 41 (18)%. Participants spent 41 ± 20% of their day in hyperglycemia (>180mg/dl), and 14 (13)% in hypoglycemia (<70mg/dl). High glycemic variability (percent CV >36%) was seen in 92% participants. Older age at diagnosis was associated with higher TIR (β = 0.267, p = 0.01), lower time above range (TAR) (β = -0.352, p <0.001), but higher time below range (TBR) (β = 0.274, p = 0.006). The use of NPH vs. glargine basal insulin was associated with higher TBR (β = -0.262, p = 0.009) but lower TAR (β = 0.202, p = 0.041). HbA1c showed negative correlation with TIR (r= -0.449, p <0.001) and TBR (r= -0.466, p <0.001) and positive correlation with TAR (r = 0.580, p <0.001) and mean glucose (r = 0.589, p <0.001). These data demonstrate wide gaps between the recommended vs. real world glycemic variables in patients with T1DM in thisregionon multiple daily insulin injections. CGM identifies glycemic variability and complements HbA1c in improving glycemic control.

Can you hear me now? Identifying the effect of Chinese monetary policy announcements

Do Chinese monetary policy announcements matter? This study evaluates how relevant Chinese monetary policy announcements are to Chinese financial markets and the real side of the economy. Chinese monetary policy is identified by estimating a “target” factor measuring policy surprises and a “path” factor measuring future expectations of policy using price changes to Chinese financial derivatives on policy announcement dates. Local projection results show that 1) Chinese Treasury yields and interbank rates respond persistently, suggesting that policy transmission through an interest-rate channel exists; 2) Equities and exchange rates do not respond to policy announcements instantaneously, but with a lag; 3) Future expectations of Chinese monetary policy play a larger role than surprises, 4) Real variables show evidence of policy transmission, and 5) policy transmission may be sticky due to information frictions.

On axially rational regular functions and Schur analysis in the Clifford-Appell setting

In this paper we start the study of Schur analysis for Cauchy–Fueter regular quaternionic-valued functions, i.e. null solutions of the Cauchy–Fueter operator in R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^4$$\\end{document}. The novelty of the approach developed in this paper is that we consider axially regular functions, i.e. functions spanned by the so-called Clifford-Appell polynomials. This type of functions arises naturally from two well-known extension results in hypercomplex analysis: the Fueter mapping theorem and the generalized Cauchy–Kovalevskaya (GCK) extension. These results allow one to obtain axially regular functions starting from analytic functions of one real or complex variable. Precisely, in the Fueter theorem two operators play a role. The first one is the so-called slice operator, which extends holomorphic functions of one complex variable to slice hyperholomorphic functions of a quaternionic variable. The second operator is the Laplace operator in four real variables, that maps slice hyperholomorphic functions to axially regular functions. On the other hand, the generalized CK-extension gives a characterization of axially regular functions in terms of their restriction to the real line. In this paper we use these two extensions to define two notions of rational function in the regular setting. For our purposes, the notion coming from the generalized CK-extension is the most suitable. Our results allow to consider the Hardy space, Schur multipliers and their relation with realizations in the framework of Clifford-Appell polynomials. We also introduce two notions of regular Blaschke factors, through the Fueter theorem and the generalized CK-extension.

On real analytic functions on closed subanalytic domains

We show that a function f:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f: X \\rightarrow {\\mathbb {R}}$$\\end{document} defined on a closed uniformly polynomially cuspidal set X in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document} is real analytic if and only if f is smooth and all its composites with germs of polynomial curves in X are real analytic. The degree of the polynomial curves needed for this is effectively related to the regularity of the boundary of X. For instance, if the boundary of X is locally Lipschitz, then polynomial curves of degree 2 suffice. In this Lipschitz case, we also prove that a function f:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f: X \\rightarrow {\\mathbb {R}}$$\\end{document} is real analytic if and only if all its composites with germs of quadratic polynomial maps in two variables with images in X are real analytic; here it is not necessary to assume that f is smooth.

Reliability analysis of tunnels considering the rock-support interaction using a metamodel-based reliability method

The convergence-confinement method (CCM) is an important approach to analysis the rock-support interaction and to design tunnel and its support. To make an effort on efficient reliability analysis of tunnels integrated with CCM, of which performance functions are mostly highly nonlinear and implicit, a hybrid intelligent algorithm combing uniform design (UD) and Gaussian process regression (GPR) is proposed. Based on the generated UD samples, GPR-parameter metamodels to predict the important parameters of CCM and GPR-probability metamodels to approximate the real performance functions are constructed consequently. Then sensitivity analysis, determination of the optimal preliminary support timing and failure probabilities calculations can be performed in conjunction with Monte Carlo simulation (MCS) based on the constructed GPR-parameter metamodels and GPR-probability metamodels. To verify the efficiency and accuracy of the proposed method, a typical circle tunnel is analyzed and good results are achieved. Finally, a practical hydraulic tunnel is analyzed using the proposed method and some beneficial conclusions are drawn.

An Explicit Form of Ramp Function

In this paper, an analytical exact form of the ramp function is presented. This seminal function constitutes a fundamental concept of the digital signal processing theory and is also involved in many other areas of applied sciences and engineering. In particular, the ramp function is performed in a simple manner as the pointwise limit of a sequence of real and continuous functions with pointwise convergence. This limit is zero for strictly negative values of the real variable x, whereas it coincides with the independent variable x for strictly positive values of the variable x. Here, one may elucidate beforehand that the pointwise limit of a sequence of continuous functions can constitute a discontinuous function, on the condition that the convergence is not uniform. The novelty of this work, when compared to other research studies concerning analytical expressions of the ramp function, is that the proposed formula is not exhibited in terms of miscellaneous special functions, e.g., gamma function, biexponential function, or any other special functions, such as error function, hyperbolic function, orthogonal polynomials, etc. Hence, this formula may be much more practical, flexible, and useful in the computational procedures, which are inserted into digital signal processing techniques and other engineering practices.

Applications of Residue Theorem to Some Selected Integrals

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyzes the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. This paper introduces an important theorem in complex analysis, which is Cauchy’s residue theorem. A relative general expression for a complex integral along a simple closed contour is provided by Cauchy's residue theorem. Cauchy's residue theorem can be used to select an acceptable closed contour for the calculation of some unusual definite integrals that may be highly complex and challenging to solve using traditional methods. The residue at the function's isolated singularities is then established. In the formula that is subtracted in Cauchy's residue theorem, the values of the residues are extracted. The integral along a simple closed contour can then be represented in two pieces, in which one along the real axis and the other along the circle.

Positive Matrix Representations of Rational Positive Real Functions of Several Variables

Positive Matrix Representations of Rational Positive Real Functions of Several Variables

The Sign-Changing Solution for Fractional (p,q)-Laplacian Problems Involving Supercritical Exponent

In this article, we consider the following fractional (p,q)-Laplacian problem (−Δ)ps1u+(−Δ)qs2u+V(x)(|u|p−2u+|u|q−2u)=f(u)+λ|u|r−2u, where x∈RN, (−Δ)ps1 is the fractional p-Laplacian operator ((−Δ)qs2 is similar), 0&lt;s1&lt;s2&lt;1&lt;p&lt;q&lt;Ns2, qs2*=NqN−s2q, r≥qs2*, f is a C1 real function and V is a coercive function. By using variational methods, we prove that the above problem admits a sign-changing solution if λ&gt;0 is small.

Scale recurrence lemma and dimension formula for Cantor sets in the complex plane

Abstract We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2)154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Math. Z.303 (2023), 3], to prove that under the right hypothesis for the Cantor sets $K_1,\ldots ,K_n$ and the function $h:\mathbb {C}^{n}\to \mathbb {R}^{l}$ , the following formula holds: $$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$