Real Functions Research Articles

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

AbstractWe show that a function $$f: X \rightarrow {\mathbb {R}}$$ f : X → R defined on a closed uniformly polynomially cuspidal set X in $${\mathbb {R}}^n$$ R n 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 $$f: X \rightarrow {\mathbb {R}}$$ f : X → R 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.

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.

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.

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<s1<s2<1<p<q<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>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: $$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$

Spectrum of all multiplicative functions with application to powerfull numbers

Roughly speaking, the spectrum of multiplicative functions is the set of all possible mean values. In this paper, we are interested in the spectra of multiplicative functions supported over powerfull numbers. We prove that its real logarithmic spectrum takes values from −2/(4+2)=−0.26160... to 1 while it is known that the logarithmic spectrum of real multiplicative functions over all natural numbers takes values from 0 to 1. In the course of this study, we correct and complete the proof of Granville and Soundararajan on the spectrum of all multiplicative functions.

Efficient quantum amplitude encoding of polynomial functions

Loading functions into quantum computers represents an essential step in several quantum algorithms, such as quantum partial differential equation solvers. Therefore, the inefficiency of this process leads to a major bottleneck for the application of these algorithms. Here, we present and compare two efficient methods for the amplitude encoding of real polynomial functions on n qubits. This case holds special relevance, as any continuous function on a closed interval can be uniformly approximated with arbitrary precision by a polynomial function. The first approach relies on the matrix product state representation (MPS). We study and benchmark the approximations of the target state when the bond dimension is assumed to be small. The second algorithm combines two subroutines. Initially we encode the linear function into the quantum registers either via its MPS or with a shallow sequence of multi-controlled gates that loads the linear function's Hadamard-Walsh series, and we explore how truncating the Hadamard-Walsh series of the linear function affects the final fidelity. Applying the inverse discrete Hadamard-Walsh transform converts the state encoding the series coefficients into an amplitude encoding of the linear function. Thus, we use this construction as a building block to achieve an exact block encoding of the amplitudes corresponding to the linear function on k0 qubits and apply the quantum singular value transformation that implements a polynomial transformation to the block encoding of the amplitudes. This unitary together with the Amplitude Amplification algorithm will enable us to prepare the quantum state that encodes the polynomial function on k0 qubits. Finally we pad n−k0 qubits to generate an approximated encoding of the polynomial on n qubits, analyzing the error depending on k0. In this regard, our methodology proposes a method to improve the state-of-the-art complexity by introducing controllable errors.

Strong law of large numbers for double sequences of pairwise M-dependent real and multivalued random variables

Strong law of large numbers for double sequences of pairwise M-dependent real and multivalued random variables

Nearly computable real numbers

We call a sequence ( a n ) n of elements of a metric space nearly computably Cauchy if for every increasing computable function r : N → N the sequence ( d ( a r ( n + 1 ) , a r ( n ) ) ) n converges computably to 0. We show that there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence ( a n ) n of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (Theory of Computing Systems 60 (2017) 396–420) and by Stephan and Wu (In New computational paradigms. First conference on computability in Europe, CiE 2005, Proceedings (2005) 461–469 Springer) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.

COVID-19 uncertainty index in Japan: Newspaper-based measures and economic activities

This paper measures economic uncertainty induced by the COVID-19 using a newspaper-based approach in Japan and examines its economic impact in the VAR model. We specify two types of uncertainty indices and structural shocks: epidemiological and policy-related uncertainties. The constructed indices can trace well the events related to COVID-19. We then find that stock market variables respond significantly to a policy-related uncertainty shock, while real variables such as mobility and consumption are significantly affected by an epidemiological one. These findings highlight the importance of considering different types of uncertainty to assess the impact of COVID-19 induced uncertainty on economic activity.

On discrete models of Boltzmann-type kinetic equations

The known nonlinear kinetic equations, in particular, the wave kinetic equation and the quantum Nordheim–Uehling–Uhlenbeck equations are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables \(F(x,y;
 v,w)\). The function \(F\) is assumed to satisfy certain simple relations. The main properties of this kinetic equation are studied. It is shown that the above mentioned specific kinetic equations correspond to different polynomial forms of the function \(F\). Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas similar to those used for construction of discrete velocity models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similarly to the Boltzmann \(H\)-function. The theorem of existence, uniqueness and convergence to equilibrium of solutions to the Cauchy problem with any positive initial conditions is formulated and discussed. The differences in long time behaviour between solutions of the wave kinetic equation and solutions of its discrete models are also briefly discussed.

Growth optimization, optical, and dielectric properties of heteroepitaxially grown ultrawide-bandgap ZnGa2O4 (111) thin film

Ultrawide bandgap ZnGa2O4 (ZGO) thin films were grown on sapphire (0001) substrates at various growth temperatures with a perspective to investigate the electrical and optical characteristics required for high-power electronic applications. Due to the variation in the vapor pressure of Zn and Ga, severe loss of Zn was observed during pulsed laser deposition, which was solved by using a zinc-rich Zn0.98Ga0.02O target. A pure phase single-crystalline ZGO thin film was obtained at a deposition temperature of 750 °C and an oxygen pressure of 1 × 10−2 Torr. The out-of-plane epitaxial relationship between the sapphire and ZGO thin film was obtained from φ-scan. The x-ray rocking curve of the ZGO thin film grown at 750 °C exhibits a full width at half maximum of ∼0.098°, which indicates a good crystalline phase and quality of the thin film. Core-level x-ray photoelectron spectroscopy of ZGO grown at 750 °C indicated that Zn and Ga were in the 2+ and 3+ oxidation states, respectively, and the atomic ratio of Zn/Ga was estimated to be ∼0.48 from the fitted values of Zn-2p3/2 and Ga-2p3/2. The high-resolution transmission electron microscopy images revealed a sharp interface with the thickness of the ZGO film of ∼265 nm, and the signature of minor secondary phases was observed. The bandgap of the ZGO film at different growth temperatures was calculated from the ultraviolet-diffuse reflectance spectroscopy spectra, and its value was obtained to be ∼5.08 eV for the 750 °C grown sample. The refractive index (n) and the extinction coefficient (k) were determined to be ∼1.94 and 0.023 from the ellipsometric data, respectively, and the real dielectric function (ɛr) was estimated to be ∼6.8 at energy 5 eV. The ultrawide bandgap and dielectric function of ZGO recommend its possible potential applications in deep-ultraviolet optoelectronic devices and high-power electronics.

Factors influencing investment into PropTech and FinTech – only new rules or a new game?

PurposeMany studies have analysed the impact of various variables on the ability of companies to raise capital. While most of these studies are sector-agnostic, literature on the effects of macroeconomic variables on sectors that established over the last 20 years like property technology and financial technology, is scarce. This study aims to identify macroeconomic factors that influence the ability of both sectors and is extended by real estate variables.Design/methodology/approachThe impact of macroeconomic and real estate related factors is analysed using multiple linear regression and quantile regression. The sample covers 338 observations for PropTech and 595 for FinTech across 18 European countries and 5 deal types between 2000–2001 with each observation representing the capital invested per year for each deal type and country.FindingsBesides confirming a significant impact of macroeconomic variables on the amount of capital invested, this study finds that additionally the real estate transaction volume positively impacts PropTech while the real estate yield-bond-gap negatively impacts FinTech.Practical implicationsFor PropTech and FinTech companies and their investors it is critical to understand the dynamic with mac-ro variables and also the real estate industry. The direct connection identified in this paper is critical for a holistic understanding of the effects of measurable real estate variables on capital investments into both sectors.Originality/valueThe analysis fills the gap in the literature between variables affecting investment into firms and effects of the real estate industry on the investment activity into PropTech and FinTech.

Ellis enveloping semigroups in real closed fields

We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgebraic functions representing Boolean combinations of d-semialgebraic sets.

Review of: "Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective"

Review of: "Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective"

The quasi-Gramian solution of a non-commutative extension of the higher-order nonlinear Schrödinger equation

The nonlinear Schrödinger (NLS) equation, which incorporates higher-order dispersive terms, is widely employed in the theoretical analysis of various physical phenomena. In this study, we explore the non-commutative extension of the higher-order NLS equation. We treat real or complex-valued functions, such as g 1 = g 1(x, t) and g 2 = g 2(x, t) as non-commutative, and employ the Lax pair associated with the evolution equation, as in the commutation case. We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation. The soliton solutions are presented explicitly within the framework of quasideterminants. To visually understand the dynamics and solutions in the given example, we also provide simulations illustrating the associated profiles. Moreover, the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns within the context of modulational instability.

부울경 아파트 전세가율 결정모형에 관한 연구

1. CONTENTS
 (1) RESEARCH OBJECTIVES
 This study is a research on the differences in apartment rental rates in Busan, Ulsan, and Gyeongnam by government, and aims to examine the differences in the effects of real estate variables and macroeconomic variables on apartment rental rates by government.
 (2) RESEARCH METHOD
 This study employed fixed effect model to achieve the major objectives of this study.
 (3) RESEARCH FINDINGS
 This study examines three issues: 1. Real estate characteristics were analyzed based on three independent variables: apartment sales price change rate, apartment rental price change rate, and apartment unsold. In the case of apartment sales price change rate, both the Moon Jae-in and Yoon Seok-yeol administrations had a negative effect, meaning that an increase in apartment sales price change rate decreased the apartment subletting rate. In the case of apartment charter price change rate, the effect was positive, meaning that an increase in apartment charter price change rate increased the apartment subletting rate. Since the change in apartment rental rate is caused by the change in apartment sales price and apartment rental price, it should be interpreted differently from previous studies. Second, this study is possible because it combines time series data and cross-sectional data to build a panel data that has the characteristics of both data, so we tried to create a variable of apartment subletting rate and identify the relationship between real estate variables and macroeconomic variables. Third, it is meaningful to identify the difference in apartment rental rates between the Moon Jae-in and Yoon Seok-yeol administrations in light of recent research on reverse subletting and charter fraud.

Understanding the implications of successive waves of currency devaluation and associated government reforms for real estate market

ABSTRACT In response to economic hurdles, several countries turn to devaluing their national currencies as a way to cope with economic difficulties. Such acts have a variety of effects on many industries, including real estate, which has led to multiple institutional and regulatory reformations aiming at minimising these effects. This paper uses a system thinking methodology that emphasises underpinning relationships among all entangled components to examine how such reforms affect minimising risks associated with currency devaluation on the real estate market. This paper uses this methodology beyond linear cause-effect analyses to consider interactions and interconnectedness in order to give some insights to gain: The impact of the currency devaluation on the real estate market based on Egyptian recent situation. The unintended consequences of government reforms on real estate market variables Solution leverage points where to intervene in the system. Holistic understanding which enables decision makers to see a wider picture and factors. Egypt has faced severe fluctuations in the economic climate, which has entailed a sharp devaluation of the currency three times between 2016 and 2023 (which could be considered a special aggressive case study between countries due to repetitiveness and effects). The effects of this devaluation have extended to all sectors, and the most important of these effects was severe challenges that faced the real estate market performance. The Real Estate market (REM) recently in Egypt has faced huge recession of transactions of the market. This research focuses on understanding the relations of reforms of the regulations which is designed to give a comprehensive overview of the government reforms of RE business, and how it has been interacting with the ramifications of those currency devaluation impacts.

Demand for Money in the United States: Stability and Forward-Looking Tests

This study shows that demand for money is a function not only of interest rate, real exchange rate, and personal consumption but also of fiscal variables like deficit, debt, and foreign-financed debt. It is stable over the short and long run. This study also covers the investigation of policy invariance of money demand, an important issue ignored so far in the existing literature on the demand for money in the United States. It was found that the behavior of agents in the money market changes as the real exchange rate, consumption, and interest rate change. Namely, agents in the money market are forward-looking, and their expectations are formed rationally. This also means that even though the parameters of money demand are stable, according to the stability test results, they can be unstable, as agents adapt their behavior based on any change in the exchange rate, consumption, and/or interest rate. In other words, the contemporaneous real exchange and interest rate variables are not superexogenous in demand for M1, and the contemporaneous consumption is not superexogenous in demand for M2.