Natural Interpolation Research Articles

Fuzzballs and random matrices

Black holes are believed to have the fast scrambling properties of random matrices. If the fuzzball proposal is to be a viable model for quantum black holes, it should reproduce this expectation. This is considered challenging, because it is natural for the modes on a fuzzball microstate to follow Poisson statistics. In a previous paper, we noted a potential loophole here, thanks to the modes depending not just on the n-quantum number, but also on the J-quantum numbers of the compact dimensions. For a free scalar field ϕ, by imposing a Dirichlet boundary condition ϕ = 0 at the stretched horizon, we showed that this J-dependence leads to a linear ramp in the Spectral Form Factor (SFF). Despite this, the status of level repulsion remained mysterious. In this letter, motivated by the profile functions of BPS fuzzballs, we consider a generic profile ϕ = ϕ0(θ) instead of ϕ = 0 at the stretched horizon. For various notions of genericity (eg. when the Fourier coefficients of ϕ0(θ) are suitably Gaussian distributed), we find that the J-dependence of the spectrum exhibits striking evidence of level repulsion, along with the linear ramp. We also find that varying the profile leads to natural interpolations between Poisson and Wigner-Dyson(WD)-like spectra. The linear ramp in our previous work can be understood as arising via an extreme version of level repulsion in such a limiting spectrum. We also explain how the stretched horizon/fuzzball is different in these aspects from simply putting a cut-off in flat space or AdS (i.e., without a horizon).

Promoting the heritage of the city of San Cristobal de La Laguna through a temporal link with a 16th century map

There is an increasing interest in the conservation of historical cities since they provide a link to the roots of their communities and bring cultural and economic benefits to their inhabitants. In this paper, we present an approach to promote the knowledge of the UNESCO World Heritage city of San Cristóbal de La Laguna in the Canary Islands, Spain. The city was founded in 1496 and has a unique value due to its urban design. In this work, we present a web application that allows a user to locate places and addresses of the current city in its first known map authored by Leonardo Torriani in the 16th century. To build this application we have georeferenced the ancient map in the current cartography. The georeferencing process needs the identification of homologous ground control points in the coordinate systems of both the old map and the current cartography, and the definition of a transform between them. Best results were obtained with the non-parametric natural transform interpolation leading to a global mean error of 4.9 m that reduces to 3.2 m in the historical city centre. To provide a fast response to the user of the web application, a technique to precompute offline the natural transform is presented. The web application has a simple front-end where the user fills the current city address in a form. This activates a query to obtain the geographical coordinates of the address that are transformed to map coordinates using the pre-computed transformation. These map coordinates are used by a map viewer in the front end that locates the user address in the ancient map. To test the performance of the web application, the load of the system has been analysed obtaining a latency of 1.4 s in 50 concurrent users. Results show that the web application provides accurate results in the historical centre while providing satisfactory response times.Highlights:- This study investigates the issue of georeferencing the first historic map (16th century) of San Cristóbal de La Laguna and the possibility to geolocate current city addresses.- By using the natural interpolation method, the georeferencing errors were diminished below 4 m in most of the historic city.- A user-friendly web application has been designed which precisely locates current directions in the old map providing a valuable tool for the promotion of the heritage of the city.

K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

AbstractWe show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

Finite multiple zeta values, symmetric multiple zeta values and unified multiple zeta functions

We introduce entire multiple zeta functions, which are natural interpolations of symmetric multiple zeta values. Moreover we give further generalizations which interpolate these zeta functions, $\widehat{\mathcal{A}}$-finite multiple zeta values and $\widehat{\mathcal{S}}$-symmetric multiple zeta values. We also show that the correspondence which Kaneko--Zagier conjecture suggests holds on nonpositive integers for finite multiple zeta values and special values of these multiple zeta functions.

Thermoelectric materials with crystal-amorphicity duality induced by large atomic size mismatch

Summary Discovering novel materials and attaining higher performance are the eternal pursuit of thermoelectric materials research. Here, we report a material series, (Cu1−xAgx)2(Te1−ySy) (0.16 ≤ x ≤ 0.24, 0.16 ≤ y ≤ 0.24), which adopts a complex orthorhombic structure differing from any known crystal structure of (Cu/Ag)2(S/Te). This material series is featured by the crystal-amorphicity duality induced by the large anionic size mismatch: a crystalline sublattice of highly size-mismatched anions Te/S coexists with an amorphous-like sublattice of cations Cu/Ag. In the context of structure-property correlation, the crystal-amorphicity duality gave rise to not only interesting electrical properties but also exceptionally low lattice thermal conductivities from 300 to 1,000 K. A state-of-the-art figure of merit zT of 2.0 is obtained in the x = y = 0.22 sample at 1,000 K. These results give insights into crystal-amorphicity duality as a paradigm-shifting materials design approach to develop high-performance thermoelectric materials.

On the elicitability of range value at risk

Abstract The debate of which quantitative risk measure to choose in practice has mainly focused on the dichotomy between value at risk (VaR) and expected shortfall (ES). Range value at risk (RVaR) is a natural interpolation between VaR and ES, constituting a tradeoff between the sensitivity of ES and the robustness of VaR, turning it into a practically relevant risk measure on its own. Hence, there is a need to statistically assess, compare and rank the predictive performance of different RVaR models, tasks subsumed under the term “comparative backtesting” in finance. This is best done in terms of strictly consistent loss or scoring functions, i.e., functions which are minimized in expectation by the correct risk measure forecast. Much like ES, RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, we show that a triplet of RVaR with two VaR-components is elicitable. We characterize all strictly consistent scoring functions for this triplet. Additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares regression.

Elements of Bi-cubic Polynomial Natural Spline Interpolation for Scattered Data: Boundary Conditions Meet Partition of Unity Technique

This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.

A Generalization of Parking Functions Allowing Backward Movement

Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the $n$-tuple containing the cars' parking preferences a parking function. 
 In this paper, we introduce a generalization of the parking rule allowing cars whose preferred space is taken to first proceed up to $k$ spaces west of their preferred spot to park before proceeding east if all of those $k$ spaces are occupied. We call parking preferences which allow all cars to park under this new parking rule $k$-Naples parking functions of length $n$. This generalization gives a natural interpolation between classical parking functions, the case when $k=0$, and all $n$-tuples of positive integers $1$ to $n$, the case when $k\geq n-1$. Our main result provides a recursive formula for counting $k$-Naples parking functions of length $n$. We also give a characterization for the $k=1$ case by introducing a new function that maps $1$-Naples parking functions to classical parking functions, i.e. $0$-Naples parking functions. Lastly, we present a bijection between $k$-Naples parking functions of length $n$ whose entries are in weakly decreasing order and a family of signature Dyck paths.

An improved path integration method for nonlinear systems under Poisson white noise excitation

In order to overcome some unsatisfactory trends and limitations of the traditional path integration (PI) method for Poisson white noise, a novel PI method is proposed in this paper, which includes two improved schemes. The first one is a new Transition Probability Density Function (TPDF) approximation which considers the randomness of the impulse happening time during each time intervals. The second one is a transformation of Chapman–Kolmogorov (CK) equation by a variable substitution instead of directly using it, whose numerical calculation is based on the back stepping Runge–Kutta scheme and the triangulation-based interpolation. Monte Carlo Simulations (MCS) are utilized to measure the accuracy of the improved algorithm with three illustrative nonlinear systems. The results show that compared with the traditional PI method, the improved PI method can give a more accurate description of the TPDF values, and provide more precise stationary Probability Density Function (PDF) results whenever the mean arrival rate is large or small. The improved algorithm has a wider range of choices in time interval values to maintain the accuracy of stationary PDF results. Besides, it is discovered that cubic interpolation deserves to be applied in the improved PI method more than linear and natural interpolations.

Numerical Solution of Functional Volterra–Hammerstein Integro-differential Equations of Order Two

In this paper, the functional Volterra–Hammerstein integro-differential equations (FVHIDEs) of order two with variable coefficients are studied. First, the existence and uniqueness of the solution to the mentioned problem are proved. Then a numerical method is proposed to solve the FVHIDEs. The proposed method is a combination of the well-known finite difference method with successive approximations method, trapezoidal quadrature rule and natural cubic spline interpolation method. An error bound is obtained, and the convergence of the method is proved. The numerical stability of the method is also proved by a suitable way. Finally, some examples are given to show the accuracy and efficiency of the proposed method.

ON SOME CLASS OF INTERPOLATION FUNCTORS

As it is well known, the Gustavsson Peetre construction, using the concept of unconditional convergence in Banach spaces, provides an important class of interpolation functors. In this paper, we define a new close construction, based on the use of the so-called random unconditional convergence. We find necessary and sufficient conditions, which being imposed on a generating function give us an interpolation functor defined on the category of Banach couples. It is shown that calculating the latter functor for a couple of Orlicz spaces results in the natural interpolation theorem. Moreover, we obtain conditions that guarantee the coincidence of this functor with the corresponding Gustavsson Peetre functor, as well as with the Calderon Lozanovskii method.

NUMERICAL ANALYSIS FOR (H-Q) CURVE OF CENTRIFUGAL PUMPS USING NATURAL AND CONSTRAINED CUBIC SPLINE INTERPOLATIONS

This study is founded by numerical analysis to (H-Q) curve for centrifugal device pump with radial and axial flow fitted firstly with natural spline interpolation and secondly using reserved cubic interpolation spline. Two case studies are used, first one analyzed the (H-Q) curve for axial flow with high specific speed, NS, when the H-Q curve having undesirable dip due to incorrect design of the impeller. The second case study is the parallel operation for two devices (pumps) for different (H-Q) curves, the joint the curves of combination which having important point, until the suitable and working point which reached, the first one doesn’t transfer the flow. The second pump can begin to fund to the flow, it is for this reason that in such a case, the second pump should be started first and then the first pump, locate this point provides us with energy and maintenance costs. The results shown that the normal cubic interpolation which will useful to interpolate these types of curves to obtain the undesirable overshooting points than constrained cubic spline.

XFEM formulation with sub‐interpolation, and equivalence to zero‐thickness interface elements

SummaryThis paper describes a particular formulation of the extended finite element method (XFEM) specifically conceived for application to existing discontinuities of fixed location, for instance, in geological media. The formulation is based on two nonstandard assumptions: (1) the use of sub‐interpolation functions for each subdomain and (2) the use of fictitious displacement variables on the nodes across the discontinuity (instead of the more traditional jump variables). Thanks to the first of those assumptions, the proposed XFEM formulation may be shown to be equivalent to the standard finite element method with zero‐thickness interface elements for the discontinuities (FEM+z). The said equivalence is theoretically proven for the case of quadrangular elements cut in two quadrangles by the discontinuity, and only approximate for other types of intersections of quadrangular or triangular elements, in which the XFEM formulation corresponds to a kinematically restricted version of the corresponding interface plus continuum scheme. The proposed XFEM formulation with sub‐interpolation, also helps improving spurious oscillations of the results obtained with natural interpolation functions when the discontinuity runs skew to the mesh. A possible explanation for these oscillations is provided, which also explains the improvement observed with sub‐interpolation. The paper also discusses the oscillations observed in the numerical results when some nodes are too close to the discontinuity and proposes the remedy of moving those nodes onto the discontinuity itself. All the aspects discussed are illustrated with some examples of application, the results of which are compared with closed‐form analytical solutions or to existing XFEM results from the literature.

Interpolation of matrices and matrix-valued densities: The unbalanced case

We propose unbalanced versions of the quantum mechanical version of optimal mass transport that is based on the Lindblad equation describing open quantum systems. One of them is a natural interpolation framework between matrices and matrix-valued measures via a quantum mechanical formulation of Fisher-Rao information and the matricial Wasserstein distance, and the second is an interpolation between Wasserstein distance and Frobenius norm. We also give analogous results for the matrix-valued density measures, i.e., we add a spatial dependency on the density matrices. This might extend the applications of the framework to interpolating matrix-valued densities/images with unequal masses.

AQUA Satellite Data and Imputation of Geopotential Height: A Case Study for Pakistan

In current study an attempt is carried out by filling missing data of geopotiential height over Pakistan and identifying the optimum method for interpolation. In last thirteen years geopotential height values over were missing over Pakistan. These gaps are tried to be filled by interpolation Techniques. The techniques for interpolations included Bilinear interpolations [BI], Nearest Neighbor [NN], Natural [NI] and Inverse distance weighting [IDW]. These imputations were judged on the basis of performance parameters which include Root Mean Square Error [RMSE], Mean Absolute Error [MAE], Correlation Coefficient [Corr] and Coefficient of Determination [R2]. The NN and IDW interpolation Imputations were not precise and accurate. The Natural Neighbors and Bilinear interpolations immaculately fitted to the data set. A good correlation was found for Natural Neighbor interpolation imputations and perfectly fit to the surface of geopotential height. The root mean square error [maximum and minimum] values were ranges from ±5.10 to ±2.28 m respectively. However mean absolute error was near to 1. The validation of imputation revealed that NN interpolation produced more accurate results than BI. It can be concluded that Natural Interpolation was the best suited interpolation technique for filling missing data sets from AQUA satellite for geopotential height.

Accurate Imputation for Relative Humidity over Pakistan Gathered from AQUA Satellite

The relative humidity in the atmosphere captured by AQUA satellite contains missing matrices. In order to fill such missing values four very popular imputation techniques: Bilinear, Inverse Distance Weighting, Natural Neighbor and Nearest Interpolations were tested. Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Coefficient of Determination (R2) and Correlation Coefficient (Corr), were used to check the accuracy of these interpolations. It was found that the Inverse Distance Weighting and Nearest Interpolation were proved not to be suited. Natural interpolation gave accurate results than the aforementioned two interpolations. Missing values of relative humidity were accurately refilled with Bilinear Interpolation. This interpolation produced RMSE of ±0.543 for relative humidity over 100, 150, 200, 250, 300, 400, 500 hPa while for 600, 700, 850 and 925 hPa RMSE remainnear to 1. A perfect fit to the surface and very strong correlation (value near to 0.99) was found between actual and imputed relative humidity data through Bilinear Interpolation. Therefore it was concluded that the Bilinear Interpolation is the most accurate and best imputation for missing values of relative humidity form 100 to 1000 hPa levels.

Calculation of the derivative of interpolation shape function for three-dimensional natural element method

The natural element method (NEM) is a special meshless method. Its shape functions are constructed using natural neighbor node interpolations based on the concepts of Voronoi tessellation. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. However, for a three-dimensional problem, the computation of shape function derivative of NEM is still very complicated even with the non-Sibson interpolation function, which makes the NEM an unpopular numerical method. In this paper, we adopt the direct mathematical derivative technique, and after some rigorous deduction, finally obtain the shape function derivative expression of three-dimensional NEM. Compared with the Lasserre algorithm, this algorithm is more intuitionistic and can be conveniently programmed. The NEM numerical results for cantilever beams verify the correctness of the shape function derivative expression of NEM derived in this paper.

Missing Data Imputations for Upper Air Temperature at 24 Standard Pressure Levels over Pakistan Collected from Aqua Satellite

This research was an effort to select best imputation method for missing upper air temperature data over 24 standard pressure levels. We have implemented four imputation techniques like inverse distance weighting, Bilinear, Natural and Nearest interpolation for missing data imputations. Performance indicators for these techniques were the root mean square error (RMSE), absolute mean error (AME), correlation coefficient and coefficient of determination ( R2 ) adopted in this research. We randomly make 30% of total samples (total samples was 324) predictable from 70% remaining data. Although four interpolation methods seem good (producing 0.99) found between actual and predicted air temperature data through this method. The high value of the coefficient of determination (0.99) through bilinear interpolation method, tells us best fit to the surface. We have also found similar results for imputation with natural interpolation method in this research, but after investigating scatter plots over each month, imputations with this method seem to little obtuse in certain months than bilinear method.

Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles

We compute the cone of effective divisors on the Hilbert scheme of n n points in the projective plane. We show that the sections of many stable vector bundles satisfy a natural interpolation condition and that these bundles always give rise to the edge of the effective cone of the Hilbert scheme. To do this, we give a generalization of Gaeta’s theorem on the resolution of the ideal sheaf of a general collection of n n points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that ideal sheaves of collections of points are destabilized by exceptional bundles. By studying the Bridgeland stability of exceptional bundles, we also show that our computation of the effective cone of the Hilbert scheme is consistent with a conjecture in a 2013 work by Arcara, Bertram, Coskun and the author which predicts a correspondence between Mori and Bridgeland walls for the Hilbert scheme.

Uniform bounds for convolution and restricted X-ray transforms along degenerate curves

We establish endpoint Lebesgue space bounds for convolution and restricted X-ray transforms along curves satisfying fairly minimal differentiability hypotheses, with affine and Euclidean arclengths. We also explore the behavior of certain natural interpolants and extrapolants of the affine and Euclidean versions of these operators.