Homographic Research Articles

A Parabolic Transform and Averaging Methods for General Partial Differential Equations

Averaging method of the fractional general partial differential equations and a special case of these equations are studied, without any restrictions on the characteristic forms of the partial differential operators. We use the parabolic transform, existence and stability results can be obtained.

A Novel Hybrid PAPR Reduction Technique for NOMA and FBMC System and its Impact in Power Amplifiers

Non-orthogonal multiple access (NOMA) and filter bank multi carrier (FBMC) based on multi-carrier techniques are two best transmission schemes candidates for modern mobile communication systems. High peak to average power ratio (PAPR) is regarded as an important problem in implementing NOMA and FBMC systems. Several PAPR reduction techniques for FBMC have been presented in the literature review, but PAPR reduction techniques in NOMA are not explored so far. In that regard, there are several drawbacks of reduction methods presented in the literature. The disadvantages include a decrease in performance of bit error rate (BER), design complexity, minimum bit rate. In the present study, we have examined and compared the performance of the reduction techniques and proposed a selective mapping circular transformation hybrid (SLMCT) method. It is observed that SLMCT is the best method to enhance the PAPR and BER performance and minimizes the complexity of PTS technique. Further BER, PAPR, net gain, power reduction and CCDF parameters are examined. Finally, it is concluded that the performance of the proposed hybrid method is better than the existed peak power minimize techniques and plays an important role, when the objective is to maximize the performance of both PAPR and BER.

Arithmetic summable sequence space over non-Newtonian field

In this article, we introduce the sequence spaces AS(G) and AC(G) of arithmetic summable and arithmetic convergent sequences, respectively, suggested by the geometric sum as k ranges over the divisors of m. We further obtain an analogous of Mobius inversion formula in the sense of geometric calculus and give interesting results in the geometric field.

A New Concept of Compton Scattering Tomography and the Development of the Corresponding Circular Radon Transform

A new modality of Compton scattering tomography (CST) is presented. This model is composed of a fixed source and detectors placed on a fixed ring passing through the source. This new modality, called circular CST (CCST), allows to register a complete set of data and, regarding constructive features like compactness or the absence of moving components, outperforms other existing CST designs. A bi-imaging system combining this new system and fan-beam computed tomography can also be considered, and such system will provide both the attenuation map and the distribution on electrons of the object under study. The modeling of the CCST leads to the Radon transform (RT) on circles passing through a fixed point. A reconstruction algorithm based on an exact inversion formula of the RT on circles is proposed and numerical simulations for image formation and reconstruction are carried out. The results show the viability of this new image system which is particularly suitable for small objects, such as biomedical objects.

Prescribing inner parts of derivatives of inner functions

Let ℐ be the set of inner functions whose derivative lies in the Nevanlinna class. We show that up to a post-composition with a Mobius transformation, an inner function F ∈ ℐ is uniquely determined by the inner part of its derivative. We also characterize inner functions which can be represented as Inn F′ for some F ∈ ℐ in terms of the associated singular measure, namely, it must live on a countable union of Beurling–Carleson sets. This answers a question raised by K. Dyakonov.

On the dynamics of Riccati foliations with nonparabolic monodromy representations

In this paper, we study the dynamics of Riccati foliations over noncompact finite volume Riemann surfaces. More precisely, we are interested in two closely related questions: the asymptotic behaviour of the holonomy map H o l t ( ω ) Hol_t(\omega ) defined for every time t t over a generic Brownian path ω \omega in the base; and the analytic continuation of holonomy germs of the foliation along Brownian paths in transversal complex curves. When the monodromy representation is parabolic (i.e., the monodromy around any puncture is a parabolic element in P S L 2 ( C ) PSL_2(\mathbb {C}) ), these two questions have been solved, respectively, in [Comm. Math. Phys. 340 (2015), pp. 433–469] and [Ergodic Theory Dynam. Systems 37 (2017), pp. 1887–1914]. Here, we study the more general case where at least one puncture has hyperbolic monodromy. We characterise the lower-upper, upper-upper, and upper-lower classes of the map H o l t ( ω ) Hol_t(\omega ) for almost every Brownian path ω \omega . We prove that the main result of [Ergodic Theory Dynam. Systems 37 (2017), pp. 1887–1914] still holds in this case: when the monodromy group of the foliation is “big enough”, the holonomy germs can be analytically continued along a generic Brownian path.

The circular nature of recurrent life cycle events: a test comparing tropical and temperate phenology

Abstract The high diversity of plant species in the tropics has revealed complex phenological patterns and reproductive strategies occurring throughout the year. Describing and analysing tropical plant phenology, and detecting triggers, demands to consider the circular nature of recurrent life cycle events and the use of appropriated statistical metrics. Here, we explore analytical pitfalls potentially affecting results of studies that do not consider the circular nature of phenology data when comparing resting and non‐resting systems, especially when accounting for phylogeny. We discuss definitions of the widely used first flowering date and revisit the literature on phylogenetic signal in plant phenology. We compare statistical analyses for tropical and temperate phenology by simulating communities with known phenological and phylogenetic structures. We demonstrate that ignoring the circular nature of phenological data underestimates the phylogenetic signal in plant phenology. Using the proposed circular transformation for non‐resting tropical ecosystems and resting temperate systems prevented errors, yielding precise comparisons. Synthesis. The analysis of both non‐resting and resting systems must consider the circularity of phenological events. Circular statistics is the appropriate approach to calculate phenological parameters, identify phylogenetic signal and assess drivers, allowing accurate cross‐comparisons of phenology across environments at large spatial scales.

Comparison and Analysis of Evaluation Indexes regarding China’s Eco-industrial Parks

The industrial park plays a significant role in fueling the economic growth, expanding the foreign trade and promoting the industry cluster. However, due to the blind investment attraction, and the poor perspectiveness and systematicness of the planning, the industrial park also becomes the centralized pollutant emission area. Chinese government has implemented many meaningful explorations in promoting the ecological development of the industrial parks, successively introduced the demonstration eco-industrial park, circular transformation of industrial park, and green park construction, and separately established the evaluation index system and policy guarantee system. This paper reviews the ecological greening development course of China’s industrial parks, thoroughly introduces the evaluation index system of demonstration eco-industrial park, circular transformation index system of the industrial park, and the green park evaluation index system, and compares and analyzes the commonalities and differences of such three sets of index systems from the theoretical basis, target positioning, index requirements, index universality, evaluation methods and many other dimensions, based on which this paper proposes the policy suggestions regarding the ecological greening development of China’s industrial parks.

Elasticities and the Inverse Hyperbolic Sine Transformation

AbstractApplied econometricians frequently apply the inverse hyperbolic sine (or arcsinh) transformation to a variable because it approximates the natural logarithm of that variable and allows retaining zero‐valued observations. We provide derivations of elasticities in common applications of the inverse hyperbolic sine transformation and show empirically that the difference in elasticities driven by ad hoc transformations can be substantial. We conclude by offering practical guidance for applied researchers.

LDPC codes based on Mobius transformations

Recently, a class of low-density parity-check (LDPC) codes from affine permutation matrices, called APM-LDPC codes, have attracted because of some advantages rather than QC-LDPC codes in minimum-distance, girth, cycle distribution and error-rate performance. In this study, a new class of LDPC codes based on Mobius transformations, called MT-LDPC codes, are presented as a generalisation of APM-LDPC codes which have some new achievements rather than QC and APM LDPC codes in the terms of length, cycle distribution and error-rate performance. Moreover, each Mobius transformation is represented by a square matrix which is helpful to pursuing the cycles in the Tanner graph of an MT-LDPC code by the product of some square matrices. In continue, for a given base matrix, the authors propose a deterministic algorithm which efficiently produces MT-LDPC codes with the desired girth. Simulation results show that the binary and non-binary constructed MT-LDPC codes outperform APM, QC, PEG, random-like and some algebraic LDPC codes with the same rates and lengths.

Low-Computational Cost Stitching Method in a Three-Eyed Endoscope.

Aortic valve replacement is the only definitive treatment for aortic stenosis, a highly prevalent condition in elderly population. Minimally invasive surgery brought numerous benefits to this intervention, and robotics recently provided additional improvements in terms of telemanipulation, motion scaling, and smaller incisions. Difficulties in obtaining a clear and wide field of vision is a major challenge in minimally invasive aortic valve surgery: surgeon orientates with difficulty because of lack of direct view and limited spaces. This work focuses on the development of a computer vision methodology, for a three-eyed endoscopic vision system, to ease minimally invasive instrument guidance during aortic valve surgery. Specifically, it presents an efficient image stitching method to improve spatial awareness and overcome the orientation problems which arise when cameras are decentralized with respect to the main axis of the aorta and are nonparallel oriented. The proposed approach was tested for the navigation of an innovative robotic system for minimally invasive valve surgery. Based on the specific geometry of the setup and the intrinsic parameters of the three cameras, we estimate the proper plane-induced homographic transformation that merges the views of the operatory site plane into a single stitched image. To evaluate the deviation from the image correct alignment, we performed quantitative tests by stitching a chessboard pattern. The tests showed a minimum error with respect to the image size of 0.46 ± 0.15% measured at the homography distance of 40 mm and a maximum error of 6.09 ± 0.23% at the maximum offset of 10 mm. Three experienced surgeons in aortic valve replacement by mini-sternotomy and mini-thoracotomy performed experimental tests based on the comparison of navigation and orientation capabilities in a silicone aorta with and without stitched image. The tests showed that the stitched image allows for good orientation and navigation within the aorta, and furthermore, it provides more safety while releasing the valve than driving from the three separate views. The average processing time for the stitching of three views into one image is 12.6 ms, proving that the method is not computationally expensive, thus leaving space for further real-time processing.

Fractional Fourier Analysis Using the Möbius Inversion Formula

Compared with the Fourier analysis, the fractional Fourier analysis is more suitable to process linear frequency modulation type non-stationary functions. To the best of our knowledge, the theoretical framework of the fractional Fourier analysis has not well established yet, especially for the fractional Fourier series (FrFS) and the discrete fractional Fourier transform (DFrFT) algorithms. To tackle with these problems, the efficient FrFS and DFrFT algorithms based on the Mobius function are proposed. First, the existence and applicability of the FrFS are analyzed basing on the Mobius inversion formula. Second, two kinds of fast algorithms for the infinite/finite FrFS are proposed. Then, based on the amplitude scaling relationship between the FrFS and samples of the FrFT, two efficient DFrFT algorithms are obtained, which are noted as the arithmetic discrete fractional Fourier transform (ADFrFT)-I and the ADFrFT-II. Importantly, the multiplication complexity of the two proposed ADFrFT algorithms is reduced to O(M), which is less than that of the state-of-the-art DFrFT algorithms. The parallel butterfly structure of the ADFrFT-II algorithm is suitable for the very large scale integration implementation. Finally, the simulations justify the efficiency of the ADFrFT algorithms in filtering and parameter evaluation of radar and optical signals.

Multidomain spectral method for the Gauss hypergeometric function

We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius' method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line $\mathbb{R}\cup {\infty}$, except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier--ultraspherical spectral method.

Restoring discrete Painlevé equations from an E8(1)-associated one

We present a systematic method for the construction of discrete Painlev\'e equations. The method, dubbed `restoration', allows one to obtain all discrete Painlev\'e equations that share a common autonomous limit, up to homographic transformations, starting from any one of those limits. As the restoration process crucially depends on the classification of canonical forms for the mappings in the QRT family, it can in principle only be applied to mappings that belong to that family. However, as we show in this paper, it is still possible to obtain the results of the restoration even when the initial mapping is not of QRT type (at least for the system at hand, but we believe our approach to be of much wider applicability). For the equations derived in this paper we also show how, starting from a form where the independent variable advances one step at a time, one can obtain versions corresponding to multistep evolutions.

2S2P1D Model Calibration Error from User Panel for One Bitumen and One Bituminous Mixture

The objective of this study is to analyse the differences between experimental LVE properties of both a straight-run bitumen and a bituminous mixture and simulations with analogical 2S2P1D (2 Springs, 2 Parabolic elements, and 1 Dashpot) model fitted by 14 different users. Data for the bitumen consisted of isotherms of G∗ and φ obtained from DSR complex modulus tests at 12 different temperatures ranging from −29.9°C to 60.0°C and frequencies ranging from 6.3 to 40 Hz, for a total of 60 data points. Data for the bituminous mixture consisted of isotherms of E∗ and φ obtained from strain-controlled traction/compression complex modulus tests at 8 different temperatures ranging from −29.7°C to 38.8°C and frequencies ranging from 0.01 to 10 Hz, for a total of 55 data points. All users worked independently and for the same time duration of one hour to fit the 2S2P1D model on both sets of data. Successful simulations of experimental data of both bitumen and mixture were generally obtained by all the users over the whole range of frequencies and temperatures, regardless of their familiarity and experience with the model. The accuracy of the model to fit experimental data is all the more evident if the great spans of complex modulus (G∗ of the bitumen between 10−2 and 103 MPa, E∗ of the mixture between 10 and 40000 MPa) are considered. The obtained results highlight the convenience of 2S2P1D model to perform multiscale modelling of LVE behaviour of bituminous materials, from bitumens to mixtures.

Factorization of Darboux—Laplace Transformations for Discrete Hyperbolic Operators

We classify elementary Darboux—Laplace transformations for semidiscrete and discrete second-order hyperbolic operators. We prove that there are two types of elementary Darboux—Laplace transformations in the (semi) discrete case as in the continuous case: Darboux transformations constructed from a particular element in the kernel of the initial hyperbolic operator and classical Laplace transformations that are defined by the operator itself and are independent of the choice of an element in the kernel. We prove that on the level of equivalence classes in the discrete case, any Darboux—Laplace transformation is a composition of elementary transformations.

Investigation on the combined effect of aging temperatures and cooling medium on rheological properties of asphalt binder based on DSR and BBR

In this paper, the combined effect of aging temperatures and cooling medium on the rheological properties of asphalt binder was experimentally investigated and evaluated. Two 70/100 penetration grade binders, which were part of the core materials used by the active RILEM committee CMB-252, were selected for this purpose. The materials were artificially aged to six different conditions, first with the Rolling Thin Film Oven Test (RTFOT) at three different temperatures (123°C, 143°C, and 163°C), and then, with the standard Pressure Aging Vessel (PAV). Next, temperature and frequency sweep (T-f-sweep) tests were performed on the entire set of materials by using the Dynamic Shear Rheometer (DSR) from −30°C to 80°C while low-temperature creep tests were conducted on the long-term aged asphalt binders with the Bending Beam Rheometer (BBR) devices in air and in ethanol, respectively. Then, the 2 Spring 2 Parabolic element 1 Dashpot model (2S2P1D) and Huet model were used to evaluate the rheological properties of asphalt binders based on the DSR and BBR results, respectively. Moreover, two parameters, crossover temperature, Tδ=45°, and the difference in critical temperature, ΔTc, were used to evaluate the aging properties of the materials. Finally, the relationship between the time domain complex modulus based on DSR and the creep stiffness obtained with the BBR was established to investigate the effect of the cooling medium at low-temperatures. The results indicate that a temperature reduced process may lead to a mitigation of the short-term aging of asphalt binders, while a temperature reduction of only 40°C can result in a consistently milder long-term aging. In the BBR test, higher creep stiffness and smaller m-values can be observed in the air compared to ethanol. Huet and 2S2P1D models show that complex modulus and creep stiffness share the same core parameters only in the case of air. Based on the finding in this research, the temperature reduced mixing process and the use of air for BBR creep tests are recommended.

Evaporation-Induced Capillary Siphoning Through Hydraulically Connected Porous Domains: The Vedernikov–Bouwer Model Revisited

Evaporation-driven wicking of soil water through porous domains with contrasting hydraulic properties is studied analytically by conformal mappings and compared to numerical solutions. Initially, the connected rectangular domains are fully saturated. The first rectangle, Gp, is comprised of a coarse-textured porous medium with large permeability and low capillary forces. Evaporation-induced capillary flow pulls water horizontally across the domains to the surface of fine-textured rectangular domain, Gz, through an interfacial hydraulic exchange region that shrinks with time. The flow field in Gz is 2-D and is analytically expressed by the Vedernikov–Bouwer model that assumes a constant hydraulic conductivity for pressure heads higher than the air-entry value. The rate of 1-D drop of the phreatic surface in Gp is proportional to the evaporation rate (decreasing with time) from the Gz surface. The complex potential domain Gw “shrinks” with time, and at any time instance, it is conformally mapped onto Gz via two auxiliary planes using the Schwarz–Christoffel and Mobius transformations. The resulting Cauchy problem for an integro-differential equation with respect to an affix of the conformal mapping is solved using numerical algebra routines. A similar capillary coupled flow problem was numerically simulated using HYDRUS2D considering 2-D flow in both Gp and Gz. New insights into process dynamics are gained from a solution of an auxiliary optimization for a vertical imbibition in a column brought in contact with a water table where particle size (linking capillarity and permeability) is used as a control variable and counter-gravity front propagation dynamics as criteria.

Eisenstein Series and Equidistribution of Lebesgue Probability Measures on Compact Leaves of the Horocycle Foliations of Bianchi 3-Orbifolds

Inspired by the works of Zagier (Eisenstein series and the Riemann zeta function, Automorphic forms, representation theory and arithmetic, Tata Institute of Fundamental Research, Bombay 1979, Springer, New York, 1981) and Sarnak (Commun Pure Appl Math, 34:719–739, 1981), we study the probability measures $$\nu (t)$$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle $$\mathcal {F}({M}_D)$$ of 3-dimensional hyperbolic Bianchi orbifolds $${M}_D=\mathbb {H}^3/\widetilde{\Gamma }_D$$, of finite volume and with only one cusp. Here $$\widetilde{\Gamma }_D\subset \mathbf {PSL}(2,\mathbb {C})$$ is the Bianchi group corresponding to the imaginary quadratic field $$\mathbb {Q}(\sqrt{-D})$$. Thus $$\widetilde{\Gamma }_D$$ consist of Mobius transformations with coefficients in the ring of integers of $$\mathbb {Q}(\sqrt{-D})$$. If $$l \in \mathbb {N}$$, $$k,m \in \mathbb {Z}$$ are such that $$k,m \in [-l,l]$$, the appropriate Eisenstein series $$\widehat{E}_{km}^l ( g,s)$$ (which are defined and analytic for $$\mathrm {Re}(s) \ge 1$$) admit an analytic continuation to all of $$\mathbb {C}$$, except when $$l=k=m=0$$ in which case there is a pole for $$s=1$$. Using this fact we show that the elliptic curves which are the compact orbits of the complex horocycle flow $$h_T:\mathcal {F}({M}_D)\longrightarrow \mathcal {F}({M}_D)$$ ($$T \in \mathbb {C}$$) are expanded by the real geodesic flow $$g_t, t\in \mathbb {R}$$, and they become equidistributed in $$\mathcal {F}({M}_D)$$ with respect to the normalized Haar measure as $$t \longrightarrow \infty $$. This follows from the equidistribution of Lebesgue probability measures on compact leaves of the horocycle foliations in the orthonormal frame bundle of $$M_D$$, which is equal to the quotient $$\mathbf {PSL}(2,\mathbb {C})/\widetilde{\Gamma }_D$$. The same equidistribution property occurs for the spin bundle of $$M_D$$ which is the homogeneous space $$\mathbf {SL}(2,\mathbb {C})/\Gamma _D$$, where $$\Gamma _D$$ is the Bianchi subgroup in $$\mathbf {SL}(2,\mathbb {C})$$ consisting of matrices with entries in the ring of integers of $$\mathbb {Q}(\sqrt{-D})$$. Our method uses the theory of spherical harmonics in the unit tangent bundle orbifold $$T_1(M_D)={\mathbf {SO}}(2){\backslash }\mathbf {PSL}(2,\mathbb {C})/\widetilde{\Gamma }_D=T_1(\mathbb {H}^3)/{\Gamma ^*_D}$$, where $${\Gamma ^*_D}$$ is the action of $${\widetilde{\Gamma }}_D$$ on the unit tangent bundle of $$\mathbb {H}^3$$ via the differential of the elements of $$\widetilde{\Gamma }_D$$.

On Ramond Decorations

We impose constraints on the odd coordinates of super Teichm\"uller space in the uniformization picture for the monodromies around Ramond punctures, thus reducing the overall odd dimension to be compatible with that of the moduli spaces of super Riemann surfaces. Namely, the monodromy of a puncture must be a true parabolic element of the canonical subgroup $PSL(2,\mathbb{R})$.