The monotonic Quartic Spline Method (QSM) for conservative transport problems

A quartic spline based remapping algorithm is developed and illustrative tests of it are presented herein. To ensure mass conservation, the scheme solves an integral form of the transport equation rather than the differential form. The integrals are computed from reconstructed quartic splines with mass conservation constraints. For higher dimensions, this remapping can be used within a standard directional splitting methodology or within the flow-dependent cascade splitting approach. A high-order grid and sub-grid based monotonic filter is also incorporated into the overall scheme. This filter is independent of the underlying spline representation adopted here, and is of more general application.     

High spatial resolution multi-site EPR oximetry. The use of convolution-based fitting method

We describe a new method to enhance the spatial resolution of multi-site electron paramagnetic resonance (EPR) oximetry. The method is suitable for any shape (density distribution function) of a solid paramagnetic material implanted in tissue. It corrects distortions of lineshapes caused by the gradient and thus overcomes limitations of previous multi-site EPR oximetry methods that restricted the ratio of the particle size to the distance between sites. The new method is based on consecutive applications of magnetic field gradients with the same direction but with a different magnitude and uses a convolution-based fitting algorithm to derive Lorentzian EPR linewidths of each individual peak of the EPR spectrum. The method is applicable for any particulate EPR oxygen sensitive materials whose EPR spectra can be approximated by a Lorentzian function or a superposition of Lorentzian functions. By incorporating this model of the lineshape in the data processing, we are able to decrease significantly the number of parameters needed for the calculations and to recover the oxygen concentration, even from quite noisy spectra. We (i) describe our method and the data-processing algorithm, (ii) demonstrate our approach in model and in vivo experiments, and (iii) discuss the limitations.     

High Spatial Resolution Multi-Site EPR Oximetry: The Use of a Convolution-Based Fitting Method

We describe a new method to enhance the spatial resolution of multi-site electron paramagnetic resonance (EPR) oximetry. The method is suitable for any shape (density distribution function) of a solid paramagnetic material implanted in tissue. It corrects distortions of lineshapes caused by the gradient and thus overcomes limitations of previous multi-site EPR oximetry methods that restricted the ratio of the particle size to the distance between sites. The new method is based on consecutive applications of magnetic field gradients with the same direction but with a different magnitude and uses a convolution-based fitting algorithm to derive Lorentzian EPR linewidths of each individual peak of the EPR spectrum. The method is applicable for any particulate EPR oxygen sensitive materials whose EPR spectra can be approximated by a Lorentzian function or a superposition of Lorentzian functions. By incorporating this model of the lineshape in the data processing, we are able to decrease significantly the number of parameters needed for the calculations and to recover the oxygen concentration, even from quite noisy spectra. We (i) describe our method and the data-processing algorithm, (ii) demonstrate our approach in model and in vivo experiments, and (iii) discuss the limitations.     

The Largest Eigenvalue of Real Symmetric,Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source,Part I

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “one-band” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0.     

A robust edge detection method with sub-pixel accuracy

In this paper, a new edge detection approach combining gray-moments operator with smoothing spline algorithm is proposed, which is invariant to additive and multiplicative noises in the image. This approach consists of two steps: firstly, a continuous blurred edge model is obtained using the smoothing spline algorithm in the edge region detected by Sobel operator; then a gray-moment solution is derived for both the one- and two-dimensional situations using the blurred edge model. Testing of this new detection approach demonstrates more robustness against the white Gaussian noise and speckle noise, and run time very close to the gray-moment and space-moment operators. The above advantages indicate this approach is very suitable for on-line accurate detection.     

Finite-Dimensional PT-Symmetric Hamiltonians

This paper investigates finite-dimensional PT-symmetric Hamiltonians. It is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) The usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the first approach the spectrum remains real, while in the second approach the spectrum remains real if the PT symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D>2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional PT-symmetric matrix Hamiltonian.     

Optical geometries and related structures

Two natural optical geometries on the space of all null directions over a four-dimensional Lorentzian manifold are defined and studied. One of this geometries is never integrable and the other is integrable iff the metric of is conformally flat. Sections of forming a zero set of integrability conditions for the latter optical geometry are interpreted as principal null directions on .

Certain well-defined conditions on are shown to be equivalent to the vanishing of the traceless part of the Ricci tensor of . Sections of forming a zero set for these new conditions correspond to the eigendirections of the Ricci tensor of .

An analogy between optical and Hermitian geometries is discussed. Existing (or possible to exist) mutual counterparts between facts from optical and Hermitian geometries are listed. In this analogy, construction of the optical geometries on constitutes a Lorentzian counterpart of the Atiyah-Hitchin-Singer construction of two natural almost Hermitian structures on the twistor space of four-dimensional Euclidean manifold.     

Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit.     

Strain profile reconstruction of fiber Bragg grating with gradient using chaos genetic algorithm and modified transfer matrix formulation

In this paper, a new heuristic approach based on chaos genetic algorithm and modified transfer matrix method is presented to demodulate the fiber Bragg grating strain sensors. To facilitate accurate calculation of the grating reflected intensity spectrum, the modified transfer matrix approach is applied. The motivation for using the spline smooth function scheme is to provide nice approximation of strain profiles from a scattered data set and overcome the difficulty of calculating the strain gradients in local period function of the modified T-matrix formulation for the piecewise constant strain field assumption. The chaos genetic algorithm is developed to improve the performance of GA and optimize the data of control nodes of the spline smooth function, and more valuably, the reconstructed strain profile is continuous with discretionary spatial resolution. The proposed method is verified through numerical example reconstructions of Bragg grating sensor simulated strain profile cases.     

An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher–Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach.     

A dual image approach for bias field correction in magnetic resonance imaging

In this paper, we propose a dual image approach to correcting intensity inhomogeneities for MR images acquired using surface coils. Previous methods are usually not satisfactory due to restricted application domains, considerable human interactions, or some undesirable artifacts. The proposed algorithm provides nice correction results for a variety of surface-coil MR images. It is accomplished by using an additional body-coil MR image of a smaller size captured at the same position as that of the surface-coil image to facilitate the estimation of the bias field function. The correction algorithm consists of aligning the surface-coil image with the body-coil image and fitting a spline surface from a sparse set of data points for the associated bias field function. Experiments on some real images show satisfactory correction results by using the proposed algorithm.     

Inference and Learning in a Latent Variable Model for Beta Distributed Interval Data

Latent Variable Models (LVMs) are well established tools to accomplish a range of different data processing tasks. Applications exploit the ability of LVMs to identify latent data structure in order to improve data (e.g., through denoising) or to estimate the relation between latent causes and measurements in medical data. In the latter case, LVMs in the form of noisy-OR Bayes nets represent the standard approach to relate binary latents (which represent diseases) to binary observables (which represent symptoms). Bayes nets with binary representation for symptoms may be perceived as a coarse approximation, however. In practice, real disease symptoms can range from absent over mild and intermediate to very severe. Therefore, using diseases/symptoms relations as motivation, we here ask how standard noisy-OR Bayes nets can be generalized to incorporate continuous observables, e.g., variables that model symptom severity in an interval from healthy to pathological. This transition from binary to interval data poses a number of challenges including a transition from a Bernoulli to a Beta distribution to model symptom statistics. While noisy-OR-like approaches are constrained to model how causes determine the observables’ mean values, the use of Beta distributions additionally provides (and also requires) that the causes determine the observables’ variances. To meet the challenges emerging when generalizing from Bernoulli to Beta distributed observables, we investigate a novel LVM that uses a maximum non-linearity to model how the latents determine means and variances of the observables. Given the model and the goal of likelihood maximization, we then leverage recent theoretical results to derive an Expectation Maximization (EM) algorithm for the suggested LVM. We further show how variational EM can be used to efficiently scale the approach to large networks. Experimental results finally illustrate the efficacy of the proposed model using both synthetic and real data sets. Importantly, we show that the model produces reliable results in estimating causes using proofs of concepts and first tests based on real medical data and on images.     

阿贝尔逆变换数据处理算法在电弧诊断中的应用

提出了一种由等离子体辐射投影值重建发射系数的阿贝尔逆变换数据处理算法。采用计算简单、变换精度高的Bockasten插值方法实现阿贝尔逆变换积分方程的离散化。在阿贝尔逆变换前,运用傅里叶变换频域低通滤波去除实验数据中的噪声,通过多项式插值弥补数据采样率过低、提高空间分辨力,并对投影数据进行对称化处理以消除数据偏离对称对结果的影响。运用此算法对实验数据进行处理,得到电流200 A、弧长5 mm电弧温度在阴极下方0.5 mm处最高,超过22000 K,与文献中结果一致。该算法能够有效地克服噪声对变换结果的影响,运算速度快、计算精度高、稳定性好,处理大量数据时具有明显的优势。     

Some consequences of exchangeability in random-matrix theory

Properties of infinite sequences of exchangeable random variables result directly in explicit expressions for calculating asymptotic densities of eigenvalues rho(infinity)(lambda) of any ensemble of random matrices H whose distribution depends only on tr(H+H), where H+ is the Hermitian conjugate of H. For real symmetric matrices and for Hermitian matrices, the densities rho(infinity)(lambda) are constructed by summing up Wigner semicircles with varying radii and weights as confirmed by Monte Carlo simulations. Extensions to more general matrix ensembles are also considered.     

Proof of the Symmetry of the Off-Diagonal¶Hadamard/Seeley–deWitt's Coefficients in¶C ∞ Lorentzian Manifolds by a “Local Wick Rotation”

Completing the results achieved in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth D-dimensional Lorentzian manifolds. This result is relevant because it plays a central rôle in Physics, in particular in the theory of the stress-energy tensor renormalization procedure in quantum field theory in curved spacetime. To this end, it is shown that, in any Lorentzian manifold, a sort of "local Wick rotation" of the metric can be performed provided the metric is a (locally) analytic function of the coordinates and the coordinate are appropriate. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point (more generally, in a neighborhood of a space-like (Cauchy) hypersurface) into a Riemannian metric. The continuation locally preserves geodesically convex neighborhoods. In order to make rigorous the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or Kählerian) manifold is introduced and some features are analyzed. Using these tools, the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients is proven in Lorentzian analytical manifolds by analytical continuation of the (symmetric) Riemannian heat-kernel coefficients. This continuation is performed in geodesically convex neighborhoods in common with both the metrics. Then, the symmetry is generalized to C^X non analytic Lorentzian manifolds by approximating Lorentzian C^X metrics by analytic metrics in common geodesically convex neighborhoods.     

Restoration of the signal and its spectral characteristics with an irregular set of measurements

An extension of the algorithm for singular value decomposition (SVD)-based restoration of the signal is suggested that makes it possible to gain information on the dynamics of spectral lines (response of the medium) recorded in a single series of irregular measurements of the signal’s integral characteristics with the help of a sliding window. The size and displacement of the window fluctuate in time. SVD-based restoration is accomplished both directly from irregular measurement data and after interpolation of the data to the nodes of a regular grid. An algorithm of signal restoration that is based on spline interpolation is also proposed. When combined with SVD-based restoration, this algorithm allows almost complete elimination of spurious frequencies. In the case of integral measurements using an irregular-size window with its center displaced, the signal restoration quality and determination of the spectral line dynamics are shown to be no worse than in the case of fluctuation-free measurements. Analysis is performed both for a model signal and for the real response considered earlier in the physical experiment.     

Standard quantum mechanics featuring probabilities instead of wave functions

A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.     

The use of cubic splines in vibration-rotation problems of diatomic molecules

Vibration-rotation energies of diatomic molecules are computed in a basis of cubic splines using a series expansion of the centrifugal potential instead of its fitting representation. The vibrational continuum wavefunctions can be evaluated in a mixed basis set, as the spline fitting technique previously used for the electronic potential makes the analytical integration of the new functions possible. The same fitting algorithm suggests a fast method for defining approximately the best locations of the knots to be used in the least squares spline representation of the potential.     

Quantum Brachistochrone problem and the geometry of the state space in pseudo-Hermitian quantum mechanics

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.