Real polynomial root-finding by means of matrix and polynomial iterations

This paper presents a general analysis of all the quartic equations with real coefficients and multiple roots; this analysis revealed some unknown formulae to solve each kind of these equations and some precisions about the relation between these ones and the Resolvent Cubic; for example, it is well-known that any quartic equation has multiple roots whenever its Resolvent Cubic also has multiple roots; however, this analysis reveals that any non-biquadratic quartic equation and its Resolvent Cubic always have the same number of multiple roots; additionally, the four roots of any quartic equation with multiple roots are real whenever some specific forms of its Resolvent Cubic have three non-negative real roots. This analysis also proves that any method to solve third-degree equations is unnecessary to solve quartic equations with multiple roots, despite the existence of the Resolvent Cubic; finally, here is developed a generalized variation of the Ferrari Method and the Descartes Method, which help to avoid complex arithmetic operations during the resolution of any quartic equation with real coefficients, even though this equation has non-real roots; and a new, more simplified form of the discriminant of the quartic equations is also featured here.

Two algorithms of polynomial factorization, quadratically and cubically convergent, are presented corresponding respectively to two algorithms of polynomial root finding known as Durand-Kerner method and Aberth’s method. To apply these algorithms to the problem of factorization of a polynomial with real coefficients into at most quadratic factors within real arithmetic, a default starting value is proposed.

In this paper, from the point of view reducing the system calculation cost, a digital channelized receiver is proposed. The decimation factor of the proposed digital channelized receiver with low calculation cost is equal to the number of signal channels, which is acquired by changing the input signal of the embedded instantaneous frequency measurement receiver. Due to the increase of the system decimation factor, the data to be processed are decreased and the system processing speed is improved indirectly. In this paper, the architecture and working principle of the proposed digital channelized receiver with low calculation cost is firstly given; secondly, the overlapping channel division method used in the digital channelized receiver with low calculation cost is put forward. Meanwhile, the channel arbitration method of the unmodulated or modulated signal in the case of overlapping channel division is analyzed. Then, the frequency measurement principle of the embedded instantaneous frequency measurement receiver is deduced in detail. Through the use of the overlapping channel division method, the proposed digital channelized receiver with low calculation cost is able to avoid the existence of blind spots between channels and increases the interception probabilities of the input signals. And, the proposed digital channelized receiver can distinguish two signals coming simultaneously to the same channel by using the overlapping channel division method. In addition, the channel of the input signal truly residing can be simply determined by using the channel arbitration in the case of overlapping channel division and according to the output frequency of the embedded instantaneous frequency measurement receiver. Finally, the related simulation results verify the correctness and feasibility of the theories and method involved in this paper.

Real-time static voltage stability assessment in large-scale power systems based on spectrum estimation of phasor measurement unit data

Disentangled convolution for optimizing receptive field

In order to discuss the complexity of nonmonotonicity of a given PM function, it is needed to determine the height of PM functions. But it is not easy to do even for a polynomial because there are great difficulties in determining the number of real zeros for polynomials of higher degrees. In this paper we introduce two iterative sets to determine the nonmonotonicity height for polynomials and give a numerical algorithm for computing the height of general polynomials.

Machine learning interatomic potentials (MLIPs) have emerged as powerful tools for investigating atomistic systems with high accuracy and a relatively low computational cost. However, a common and unaddressed challenge with many current neural network (NN) MLIP models is their limited ability to accurately predict the relative energies of systems containing isolated or nearly isolated atoms, which appear in various reactive processes. To address this limitation, we present a mathematical technique for modifying any existing atom-centered NN architecture to account for the energies of isolated atoms. The result produces a consistent prediction of the atomization energy (AE) of a system using minimal constraints on the model. Using this technique, we build a model architecture that we call hierarchically interacting particle neural network (HIP-NN)-AE, an AE-constrained version of the HIP-NN, as well as ANI-AE, the AE-constrained version of the accurate NN engine for molecular energies (ANI). Our results demonstrate AE consistency of AE-constrained models, which drastically improves the AE predictions for the models. We compare the AE-constrained approach to unconstrained models as well as models from the literature in other scenarios, such as bond dissociation energies, bond dissociation pathways, and extensibility tests. These results show that the constraints improve the model performance in some of these tasks and do not negatively affect the performance on any tasks. The AE constraint approach thus offers a robust solution to the challenges posed by isolated atoms in energy prediction tasks.

Thermal conductivities from temperature profiles in the polymer electrolyte fuel cell

Convolutional neural networks (CNNs) have emerged as powerful tools for image retrieval and classification. In this paper, we study binary embedding of CNN-based image descriptors for resource-constrained devices. We propose two classes of fast computable and memory-efficient dimension reduction operators using Golay-Hadamard matrices (GHMs), which are constructed by multiplying the columns of a Hadamard matrix with a Golay sequence. Simulation results on CNN-based instance image retrieval and classification show that GHM-based operators can offer competitive performance to those of full random Gaussian matrices and Gaussian circulant matrices at much lower computational cost and storage space. This implies the potential of proposed approaches on devices with low memory, bandwidth and power restrictions.

We study asymptotic distributions of large dimensional random matrices of the\nform $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using\nfree probability and combinatorics of colored labeled noncrossing partitions.\nThese matrices are taken from the set of off-diagonal blocks of the family\n$\\mathcal{Y}$ of independent Hermitian random matrices which are asymptotically\nfree, asymptotically free against the family of deterministic diagonal\nmatrices, and whose norms are uniformly bounded almost surely. This class\nincludes unitarily invariant Hermitian random matrices with limit distributions\ngiven by compactly supported probability measures $\\nu$ on the real line. We\nexpress the limit moments in terms of colored labeled noncrossing pair\npartitions, to which we assign weights depending on even free cumulants of\n$\\nu$ and on asymptotic dimensions of blocks (Gaussianization). For products of\n$p$ independent blocks, we show that the limit moments are linear combinations\nof a new family of polynomials called generalized multivariate Fuss-Narayana\npolynomials. In turn, the product of two blocks of the same matrix leads to an\nexample with rescaled Raney numbers.\n

Applications like fuel cells, batteries or food analysis are typical fields of interest for electrical impedance spectroscopy (EIS).Another field with promising results are EIS measurements in semiconductor (sc) gas sensor applications [1].Due to their high sensitivity and low cost, sc gas sensors are ideally suited for different applications, especially in the field of safety and security.The major drawbacks of sc gas sensors, i.e. poor selectivity and sensor baseline drift, can be overcome by using temperature cycled operation (TCO) with DC measurement [2] or EIS, both in combination with intelligent signal processing.Currently, EIS for sc gas sensors is only possible in laboratory environments using commercial impedance analyzers, which are characterized by high cost, poor mobility and long acquisition time.However, for efficient development of application specific sc gas sensor systems and/or a self-monitoring strategy [3] extensive field tests are necessary.Therefore, a new hardware concept for EIS of sc gas sensors based on FPGA-technology was developed.This concept also allows a combination of EIS with TCO for the first time.Additionally, state-of-the-art feature extraction methods are applied to sc gas sensor data and new features were tested [4]. MotivationDue to their high sensitivity and low cost, semiconductor gas sensors are ideally suited for safety and security applications like fire and leakage detection [5,6].The major drawbacks of this type of sensor are the poor selectivity, the sensor baseline drift and sensor poisoning.Especially for safety and security applications these drawbacks must be overcome to allow a proper function of the system.Temperature cycled operation (TCO) and electrical impedance spectroscopy (EIS) -both in combination with intelligent signal processing -are the main approaches for improving both selectivity and stability.In TCO mode the sensor temperature is changed periodically [2] and the DC resistance of the sensor is read-out continually during this cycle.For EIS the sensor temperature is usually kept constant and the sensor impedance is measured over a wide frequency range, e.g. during a frequency sweep [1,7,8].To detect a sensor malfunction due to sensor poisoning a self-monitoring strategy can be realized by combining TCO und EIS (fig. 1) [3].This strategy is based on redundant information from both modes which are evaluated separately and comparing both results.Additional information can be used for selectivity enhancement.

The authors present an application of the general idea of preconditioning in the context of Gabor frames. While most (iterative) algorithms aim at a more or less costly exact numerical calculation of the inverse Gabor frame matrix, we propose here the use of cheap methods to find an approximation for it, based on (double) preconditioning. We thereby obtain good approximations of the true dual Gabor atom at low computational costs. Since the Gabor frame matrix commutes with certain time-frequency shifts, it is natural to make use of diagonal and circulant preconditioners sharing this property. Part of the efficiency of the proposed scheme results from the fact that all the matrices involved share a well-known block matrix structure. At least, for the smooth Gabor atoms typically used, the combination of these two preconditioners leads consistently to good results. These claims are supported by numerical experiments in this paper. For numerical evaluations we introduce two new matrix norms, which can be calculated efficiently by exploiting the structure of the frame matrix

Aliasing in computer-synthesized images not only limits the realism of the images, but also affects the user's concentration. Many antialiasing methods have been proposed to solve this problem, but almost all of them are computation intensive, and some of them are also memory intensive. While these may not be limitations for high-end applications such as medical visualization and architectural design, this kind of antialiasing methods may still be far too costly for the low-cost applications. In this paper, we propose an antialiasing method that operates in the image domain. It is based on fitting curves to the discontinuity edges extracted from the aliased images to reshade those edge pixels. (Note that a curve may be considered as a general form of lines.) To improve the performance and the simplicity of the method, we propose to preprocess all possible edge patterns and fit curves in advance. During runtime, we only need to construct an index to obtain the filtering information from a lookup table. The new method is extremely simple and efficient. It provides a very good compromise between hardware cost and output image quality. In addition, because the new method has a very low computational cost, and hence low power consumption for hardware implementation, it is particularly suitable for low-cost mobile applications such as computer game consoles and palm computers, where low implementation cost and low power consumption are important design factors.

Accurate needle placement is essential for prostate biopsy. Recently, transperineal prostate biopsies are receiving renewed interest due to concern over infection from conventional transrectal biopsies. However, accurate needle placement is more challenging in the transperineal approach than in the transrectal approach due to the long insertion distance leading to a large targeting error and repeated insertion attempts. Improved procedure planning tools that can predict the deviation of the needle can potentially reduce the targeting error and number of insertion attempts. Prediction of deflection magnitude requires a model of biopsy needle deflection, which in turn requires information about tissue material properties. However, material properties of tissue in patients cannot be easily obtained. Accounting for this uncertainty in patient tissue properties requires a model capable of quantifying uncertainty in needle deflection as a function of a distribution of tissue properties. A Monte Carlo uncertainty quantification requires 1000s of samples, but it is not possible to obtain this many samples in a short enough time for intraoperative procedure planning using published needle deflection predictionmodels. This work seeks to develop a model of needle deflection fast enough for use in intraoperative procedure planning, validate this model against experimental results, and integrate it into a Monte Carlo uncertainty quantificationmodel. This work used a mechanics-based model of biopsy needle deflection to train a Fourier feature neural network (FFNN) model in order to make predictions with a low computational cost. Both models were validated against experimental data. The neural network model was used in a Monte Carlo uncertainty quantification model to quantify uncertainty in needle deflection arising from uncertain tissue mechanicalproperties. This work (1) implemented a mechanics-based model and a FFNN model. Both models were validated against previously published experiments carried out with tissue phantoms. Both models showed close agreement with the experimental data. (2) We showed that our FFNN model was more accurate than a baseline ordinary least squares model, introducing only about 0.3-mm tip deflection error compared to the mechanics-based model. We also showed that our FFNN model makes unbiased predictions with respect to the amount of deflection. (3) We demonstrated a Monte Carlo uncertainty quantification model of needle deflection with a low computational cost of about 20 CPU s. We used our uncertainty quantification model to show how the depth, stiffness, and magnitude of uncertainty in a layer of tissue affect needle deflection. In addition, we showed a simple clinical example of the use of our model. This work demonstrates a Monte Carlo uncertainty quantification model of needle deflection with a low computational cost. This method shows promise for future applications in procedure planning for prostate biopsies as well as other transperineal procedures conducted with flexible needles such as cryoablation andbrachytherapy.

High dimensional covariance matrices have attracted much attention of statisticians and econometricians during the past decades. Vast literature is devoted to the research in high dimensional covariance matrices. However, most of them are for constant covariance matrices. In many applications, constant covariance matrices are not appropriate, e.g. in portfolio allocation, dynamic covariance matrices would make much more sense. Simply assuming each entry of a covariance matrix is a function of time to introduce a dynamic structure would not work. In this paper, we are going to introduce a class of high dimensional dynamic covariance matrices in which a kind of additive structure is embedded. We will show the proposed high dimensional dynamic covariance matrices have many advantages in applications. An estimation procedure is also proposed to estimate the proposed high dimensional dynamic covariance matrices. Asymptotic properties are built to justify the proposed estimation procedure. Intensive simulation studies show the proposed estimation procedure works very well when sample size is finite. Finally, we apply the proposed high dimensional dynamic covariance matrices, together with the proposed estimation procedure, to portfolio allocation. The results look very interesting.