Four-dimensional Function Research Articles

Hidden conformal symmetry in AdS2×S2 beyond tree level

Correlators of N=2 hypermultiplets with two-derivative interactions in AdS2×S2 exhibit a hidden four-dimensional conformal symmetry which allows one to repackage all tree-level four-point correlators into a single four-dimensional object corresponding to a contact diagram arising from a massless ϕ4 theory in AdS2×S2. This theory serves as a toy model for IIB string theory in AdS5×S5 and is interesting in its own right because AdS2×S2 describes the near-horizon limit of extremal black holes in four dimensions. We argue that, after acting with an SU(1,1)×SU(2) Casimir, all one-loop correlators can similarly be encoded by a four-dimensional function which arises from a one-loop scalar bubble diagram in AdS2×S2, explaining how the hidden conformal symmetry extends beyond tree level. Finally, we conjecture a scalar effective field theory with a two-derivative interaction in AdS2×S2 whose Witten diagrams should directly reproduce four-point correlators to all loops without acting with Casimirs. Published by the American Physical Society 2024

Effect of heat treatment on microstructure and functional properties of additively manufactured NiTi shape memory alloys

Additive manufacturing of NiTi shape memory alloys has attracted attention in recent years, due to design flexibility and feasibility to achieve four-dimensional (4D) function response. To obtain customized 4D functional responses in NiTi structures, tailorable phase transformation temperatures and stress windows as well as one-way or two-way shape memory properties are required. To achieve this goal, various heat treatments, including direct aging, annealing and annealing followed by aging, were optimized for the Ti-rich NiTi (Ni49.6Ti (at. %)) fabricated by laser powder bed fusion (L-PBF). Microstructural evolution, phase transformation, precipitation and shape memory behaviour were systematically investigated by multiscale correlative microstructural, differential scanning calorimetry analysis and thermomechanical analysis. Based on optimized heat treatments, ∼25 K phase transformation temperature windows and ∼90 MPa stress windows were achieved for the one-way shape memory effect. Solutionized annealing was found to be the most effective way to improve one-way shape memory degradation resistance, due to the reduction of defects and solid solution strengthening. One of the main findings of this study is that the heterogonous microstructures between hard intergranular Ti2NiOx and soft NiTi matrix, induced by solutionized annealing with subsequent aging, result in strain partitioning and enclosing the internal stress state, which was found to promote a pronounced two-way shape memory effect response. The results of this work provide in-depth knowledge on tailoring and designing functional shape memory characteristics via heat treatments, which contributes to expanding L-PBF NiTi application fields, such as biomedical implants, aerospace components, and other advanced engineering applications.

Space-variant Shack-Hartmann wavefront sensing based on affine transformation estimation.

The space-variant wavefront reconstruction problem inherently exists in deep tissue imaging. In this paper, we propose a framework of Shack-Hartmann wavefront space-variant sensing with extended source illumination. The space-variant wavefront is modeled as a four-dimensional function where two dimensions are in the spatial domain and two are in the Fourier domain with priors that both gently vary. Here, the affine transformation is used to characterize the wavefront space-variant function. Correspondingly, the zonal and modal methods are both escalated to adapt to four-dimensional representation and reconstruction. Experiments and simulations show double to quadruple improvements in space-variant wavefront reconstruction accuracy compared to the conventional space-invariant correlation method.

ONE CLASS OF SMOOTH BOUNDED SOLUTIONS TO THE CAUCHY PROBLEM FOR A THREE-DIMENSIONAL FILTRATION MODEL WITH DARCY'S LAW

In the filtration theory there are numerous approaches to solving three-dimensional problems of fluid motion in a porous medium. Basically, the solutions of such problems are obtained by numerical methods. The question of finding an analytical solution of precisely three-dimensional problems of fluid motion is open. The first results on the use of the apparatus of four-dimensional mathematics for solving the three-dimensional model of the Navier-Stokes equations by the analytical method were obtained by the Kazakh mathematician Professor M.M. Abenov. After the author of this article with other researchers proved the theorem on the existence of a solution to the Cauchy problem for a three-dimensional model of filtration theory. This paper is devoted to the study of a three-dimensional model of the filtration theory in one of the spaces of four-dimensional numbers. The purpose of this article is to obtain an analytical solution of the three-dimensional Cauchy problem for the mathematical model of linear filtration model by the method of four-dimensional regular functions.  In this study, a class of infinitely differentiable and bounded functions of the initial conditions of the Cauchy problem, satisfying the Cauchy-Riemann condition, with five degrees of freedom for a specific four-dimensional function is found, and also a class of infinitely differentiable and bounded solutions of this problem is found that satisfy the linear Darcy law.

Functions in one space of four-dimensional numbers

For the first time, the theory of functions of four-dimensional numbers with commutative product was described in works of Abenov M.M., in which the mathematical apparatus was defined, algebraic operations and their properties were determined, functions of four-dimensional numbers, their limits, continuity and differentiability were found. The continuation was the joint work of Abenov M.M. and Gabbasov M.B., where similar anisotropic four-dimensional spaces (withthe notation M2-M7) were defined, which are also commutative with zero divisors. This work is devoted to the study of functions of a four-dimensional variable, definitions and analysis of fourdimensional functions, their properties, as well as the regularity of functions. The purpose of this work is to analyze the definition of functions of four-dimensional variables of the space M5, as well as theorems on the continuity and existence of differentiability of functions of four-dimensionalvariables. This work is descriptive for comparing the spaces of four-dimensional numbers M5 and M3. In the article, theorems on the continuity and differentiability of functions of four-dimensional variables and their properties are proved, and the Cauchy-Riemann conditions are found. The form of trigonometric, exponential, logarithmic, exponential and power functions of four-dimensional variables is determined and the regularity of functions of M5 space is proved. Key words: four-dimensional function, continuity, differentiability, regular function, Cauchy-Riemann condition.

Multi-fidelity Bayesian neural networks: Algorithms and applications

We propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a posteriori probability (MAP) method to fit the low-fidelity data; the second is a Bayesian neural network employed to capture the cross-correlation with uncertainty quantification between the low- and high-fidelity data; and the last one is the physics-informed neural network, which encodes the physical laws described by PDEs. For the training of the last two neural networks, we first employ the mean-field variational inference (VI) to maximize the evidence lower bound (ELBO) to obtain informative prior distributions for the hyperparameters in the BNNs, and subsequently we use the Hamiltonian Monte Carlo (HMC) method to estimate accurately the posterior distributions for the corresponding hyperparameters. We demonstrate the accuracy of the present method using synthetic data as well as real measurements. Specifically, we first approximate a one- and four-dimensional function, and then infer the reaction rates in one- and two-dimensional diffusion-reaction systems. Moreover, we infer the sea surface temperature (SST) in the Massachusetts and Cape Cod Bays using satellite images and in-situ measurements. Taken together, our results demonstrate that the present method can capture both linear and nonlinear correlation between the low- and high-fidelity data adaptively, identify unknown parameters in PDEs, and quantify uncertainties in predictions, given a few scattered noisy high-fidelity data. Finally, we demonstrate that we can effectively and efficiently reduce the uncertainties and hence enhance the prediction accuracy with an active learning approach, using as examples a specific one-dimensional function approximation and an inverse PDE problem.

Exploration and Practice of Blended Teaching Reform of Advanced English

Because traditional Advanced teaching mode runs counter to the Syllabus for English Majors in Colleges and Universities, it is far from the talent training objectives of English majors, the needs of the talent market and the cognitive characteristics of college students in the self-media environment. It is necessary to innovate the teaching concept of Advanced English, advocate foreign language teaching concept of four-dimensional function(communicative, cognitive emotional and humanistic parenting functions ) and follow the teaching principle of emphasis on subject, openness, activity and process. The teaching process of self-learning before class, classroom discussion and post-class extension, and the evaluation of students' learning by the combination of formative and final evaluation are encouraged and conducted.

Implementation of a Depth from Light Field Algorithm on FPGA

A light field is a four-dimensional function that grabs the intensity of light rays traversing an empty space at each point. The light field can be captured using devices designed specifically for this purpose and it allows one to extract depth information about the scene. Most light-field algorithms require a huge amount of processing power. Fortunately, in recent years, parallel hardware has evolved and enables such volumes of data to be processed. Field programmable gate arrays are one such option. In this paper, we propose two hardware designs that share a common construction block to compute a disparity map from light-field data. The first design employs serial data input into the hardware, while the second employs view parallel input. These designs focus on performing calculations during data read-in and producing results only a few clock cycles after read-in. Several experiments were conducted. First, the influence of using fixed-point arithmetic on accuracy was tested using synthetic light-field data. Also tests on actual light field data were performed. The performance was compared to that of a CPU, as well as an embedded processor. Our designs showed similar performance to the former and outperformed the latter. For further comparison, we also discuss the performance difference between our designs and other designs described in the literature.

An Application of Sequential Variational Method without Tangent Linear and Adjoint Model Integrations

Abstract The sequential variational (SVAR) method minimizes the weakly constrained four-dimensional cost function by splitting it into a set of smaller cost functions. This study shows how it is possible to apply SVAR in practice by reducing the computational effort required by the algorithm. A major finding of the study is that, instead of using tangent linear and adjoint models, it is possible to estimate the largest eigenvalues and the corresponding eigenvectors of the evolution of the background error covariances only by applying successive nonlinear model integrations. Another major finding is that the impact of future observations on previous state estimates may be obtained in an accurate and numerically stable way by using suitably defined cost functions and control space transformations without any additional model integrations. The new method is applied in a realistic data assimilation experiment with a primitive equations ocean model.

Domain Coloring of Complex Functions: An Implementation-Oriented Introduction

This article gives a short overview of domain coloring for complex functions that have four-dimensional function graphs and therefore can't be visualized traditionally. The authors discuss several color schemes, focus on various aspects of complex functions, and provide Java-like pseudocode examples explaining the crucial ideas of the coloring algorithms to allow for easy reproduction.

Analytic phase shifts for yukawa potentials

The Yukawa or Debye-Hueckel potential, by a suitable choice of scale factor (d) and the use of the corresponding energy unit (h/sup 2//2md/sup 2/), may be written in the form V(r:Z) = -(2Z/r)e/sup -r/. For Z much less than 1 or l much greater than 1 or E = k/sup 2/ sufficiently large the partial wave phase shifts are given by the Born approximation which mathematically vanishes at E = 0. On the other hand, Levinson's theorem indicates that a phase shift, in units of ..pi.., is equal to n(Z,l), the integral number of bound states sustainable by the potential. In this work an optical model code written by Relyea for molecular potentials is adapted to be suitable for singular attractive potentials such as the Yukawa potential. This modified code is used to obtain the numerical Schroedinger phase shifts for a broad range of Z and l values. By the addition of integers to assure continuity of the phase shifts and the satisfaction of Levinson's theorem the authors obtain an array of continuous functions which, in effect, interpolate between the Levinson limit and the Born limit. Next, they find an accurate analytic characterization of the number of bound states n/submore » a/ (Z,l) suggested by the Sommerfeld model. Finally, they develop a simple two-term analytic formula which interpolates between the Levinson limit and the Born limit and fits the rectified set of Schroedinger phase shifts quite well. In effect, they arrive at a four-dimensional function delta(E,l,Z,d) that should be useful as a convenient approximation in atomic, nuclear, particle, and plasma physics. 12 references, 9 figures, 2 tables.« less

Observables in the Turaev-Viro and Crane-Yetter models

We define an invariant of graphs embedded in a 3-manifold and a partition function for 2-complexes embedded in a triangulated 4-manifold by specifying the values of variables in the Turaev-Viro and Crane-yetter state-sum models. In the case of the three-dimensional invariant, we prove a duality formula relating its Fourier transform to another invariant defined via the colored Jones polynomial. In the case of the four-dimensional partition function, we give a formula for it in terms of a regular neighborhood of the 2-complex and the signature of its complement. Some examples are computed which show that the partition function determines an invariant which can detect non locally-flat surfaces in a 4-manifold.

Regional 4-D modeling of the ionospheric electron density

The knowledge of the electron density is the key point in correcting ionospheric delays of electromagnetic measurements and in studying the ionosphere. During the last decade GNSS, in particular GPS, has become a promising tool for monitoring the total electron content (TEC), i.e., the integral of the electron density along the ray-path between the transmitting satellite and the receiver. Hence, geometry-free GNSS measurements provide informations on the electron density, which is basically a four-dimensional function depending on spatial position and time. In addition, these GNSS measurements can be combined with other available data including nadir, over-ocean TEC observations from dual-frequency radar altimetry (T/P, JASON, ENVISAT), and TECs from GPS-LEO occultation systems (e.g., FORMOSAT-3/COSMIC, CHAMP) with heterogeneous sampling and accuracy. In this paper, we present different multi-dimensional approaches for modeling spatio-temporal variations of the ionospheric electron density. To be more specific, we split the target function into a reference part, computed from the International Reference Ionosphere (IRI), and an unknown correction term. Due to the localizing feature of B-spline functions we apply tensor-product spline expansions to model the correction term in a certain multi-dimensional region either completely or partly. Furthermore, the multi-resolution representation derived from wavelet analysis allows monitoring the ionosphere at different resolutions levels. For demonstration we apply three approaches to electron density data over South America.

Wavelet modelling in support of IRI

The slant total electron content (STEC) of the ionosphere is defined as the integral of the electron density along the ray-path of the signal between the transmitter and the receiver. So-called geometry free GPS measurements provide information on the electron density, which is basically a four-dimensional function depending on spatial position and time. Since ground-based measurements are not very sensitive to the vertical structure within the atmosphere, the ionosphere is often represented by a spherical layer, where all electrons are concentrated. Then the STEC is transformed into the vertical total electron content (VTEC), which is a three-dimensional function depending on longitude, latitude and time.In our approach, we decompose an ionospheric function, i.e. the electron density or the VTEC, into a reference part computed from a given model like the International Reference Ionosphere (IRI) and an unknown correction term expanded in a multi-dimensional series in terms of localizing base functions. The corresponding series coefficients are calculable from GPS measurements applying parameter estimation procedures. Since the GPS receivers are located rather unbalanced, finer structures are modelable just in regions with a sufficient number of observation sites. Due to the localizing feature of B-spline functions we apply a tensor product spline expansion to model the correction term regionally. Furthermore, the multi-resolution representation derived from wavelet analysis allows monitoring the ionosphere at different resolutions levels. We demonstrate the advantages of this procedure by representing a simulated VTEC data set over South America.

KEKB beam instrumentation systems

For the stable high-luminosity operation and luminosity increase, the electron and positron storage rings of the KEK B-Factory (KEKB) is equipped with various beam instrumentations, which have been working well since the start of the commissioning in December, 1998. Details and performance of the beam-position monitor system based on the spectrum analysis using DSPs, the turn-by-turn BPM with four-dimensional function available for measurements of the individual bunch position, phase and intensity, the parametric beam-DCCTs designed so as to avoid the magnetic-core-selection problems for the parametric flux modulation, the bunch-by-bunch feedback system indispensable to suppress the strong multibunch instabilities in KEKB, the various optical beam diagnostic systems, such as synchrotron radiation interferometers for precise beam-size measurement, the tune meters, the bunch length monitors and the beam-loss monitors are described. Delicate machine tuning of KEKB is strongly supported by these instrumentations.

Wigner function for highly convergent three-dimensional wave fields

The angle-impact Wigner function for highly convergent three-dimensional scalar wave fields is derived directly by use of the three-dimensional generalized optical transfer function rather than from a six-dimensional Wigner function. The angle-impact Wigner function is a real four-dimensional function from which the intensity at any point in space is readily determined.

Scientific Insight Need Not Conflict With Scientific Rigor

It has been said that inside every white box are two black ones waiting to jump out (Glanville, 1982). In other words, whenever science makes a breakthrough in the understanding of an important phenomenon, twice as many new questions arise. Vallacher and Nowak provide an overview of nonlinear dynamical systems (NDS) concepts and summarize how they improved the description or explanation of social psychological phenomena beyond what was hitherto possible. They raise the question, however, of whether scientific development would, or should, proceed in a direction based on its insights and intuitive understanding of phenomena at the expense of methodological rigor. This question is based on the inherent unpredictability of many dynamical events, the paucity of techniques for direct hypothesis testing, and the tendency for new ideas generated at the outset of a new scientific paradigm to outpace the data. The main point of this commentary is to explain that no contradiction need be taking place. Vallacher and Nowak advocate this general line of reasoning as well. Nonetheless, some elaborations of newly available techniques for testing hypotheses germane to NDS phenomena in social psychology and elsewhere would be useful here. Additionally, the new classes of phenomena generate new types of hypotheses. It is true that, in its early stages of development, NDS numerics were nonstatistical in nature and required extremely large datasets; a four-dimensional function requires in excess of 10,000 datapoints (Casdagli, 1992). Commonly constructed social psychology experiments could not be expected to generate meaningful data in such quantities. Nonparametric statistical approaches were later developed that could be operational on datasets in the neighborhood of 500 datapoints (Brock, 1991; Tong, 1990). The structural equation approach, which has gained some attention recently, however, can operate on more tractable sample sizes of perhaps 50 to 100 datapoints (Guastello, 1992, 1995a). This commentary continues with the essential points of hypothesis testing with structural equations.

Illustration of second-order design methods: global merit function plots for a class of projection systems

Recently developed methods [ J. Opt. Soc. Am. A11, 3292– 3307 ( 1994)] permit definitive studies of two-mirror systems for a variety of applications. As an illustration of such a study, unobstructed, telecentric, plane-symmetric systems of two conic mirrors that are intended for distortion-free projection are investigated here. The magnification, speed, and field size in the examples are arbitrarily chosen to correspond to values that are appropriate for full-field soft-x-ray projection lithography. Low-order imaging constraints are applied to eliminate most of the 13 parameters that specify the configuration of such a system. It is shown that there are just two three-parameter families of systems and a single four-parameter family. The associated three-and four-dimensional merit function spaces are mapped, and selected systems from various regions of the spaces are discussed.

Surface Green function matching theory of magnetoelastic surface waves

The author shows that the surface Green function matching method can be easily applied to ferromagnetic media by defining a four-dimensional Green function that includes the elastic and magnetic fields. For a system with an interface this method reduces to one-half, with respect to other treatments, the order of the secular determinant. An application is made to the study of shear horizontal waves for surfaces and interfaces when the applied field is perpendicular to the propagation direction.

Study of interface and surface elastic waves in piezoelectric materials by using the surface green function matching (SGFM) method

It is shown that the surface Green function matching method can be readily applied to piezoelectric materials by defining a four-dimensional Green function which includes both the elastic and the electrostatic fields. For an interface system this reduces to one half (with respect to the usual treatments) the order of the secular determinant, with consequent saving in computation. As an illustration an application is made to the study of shear horizontal waves for surfaces and interfaces with 6mm symmetry.