Real Valued Research Articles

Existence Results of Some Nonlinear Minimization Problems with Application to a System of PDE

A new class of non-self mappings, called proximal condensing operators, is introduced and used to investigate the existence of best proximity points for non-self mappings in reflexive Banach spaces by applying an appropriate measure of noncompactness. In this way, we present generalizations of well-known Schauder and Darbo fixed point problems. We then define the notion of best proximity points for multiplication of two non-self mappings in the setting of Banach algebras and present some sufficient conditions to ensure the existence of such points. Finally, we renorm a Banach space consists of all real valued and continuous functions with two variables to obtain the strict convexity condition and then we survey the existence of an optimal solution for a system of partial differential equations.

Inversion of the attenuated momenta ray transform of planar symmetric tensors

We present a reconstruction method that stably recovers the real valued, symmetric tensors compactly supported in the Euclidean plane, from knowledge of their attenuated momenta ray transform. The problem is recast as an inverse boundary value problem for a system of transport equations, which we solve by an extension of Bukhgeim’s A-analytic theory. The method of proof is constructive. To illustrate the reconstruction method, we present results obtained in the numerical implementation for the non-attenuated case of one-tensors.

Regularity of the Optimal Sets for a Class of Integral Shape Functionals

We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}. The main feature of these functionals is that the minimality of a domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions j(u,x)=-g(x)u+Q(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j(u,x)=-g(x)u+Q(x)$$\\end{document}, where u is the solution of the PDE -Δu=f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\Delta u=f$$\\end{document} with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal u from the inwards/outwards optimality of Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} and then we use the stability of Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} with respect to variations with smooth vector fields in order to study the blow-up limits of the state function u. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove C∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^\\infty $$\\end{document} regularity of the regular part of the free boundary when the data are smooth.

Optimized adaptation algorithms for low-complexity adaptive equalizers.

In order to address the high-power consummation issue of conventional multi-input and multi-output (MIMO) adaptive equalizer (AEQ) for short-reach coherent transmissions, several state-of-the-art low-complexity AEQs have been proposed. In our work, optimized adaptation algorithms for low-complexity real-valued (RV) AEQs with different structures are analyzed. Moreover, an approach to avoid introducing additional computational complexity due to the optimized adaptation process is presented here. The advantages of proposed optimized adaptation algorithms are experimentally demonstrated in a 25 Gbaud dual-polarization 16-quadrature-amplitude-modulation (DP-16QAM) back-to-back (BtB) intradyne system with an overall bandwidth of 14 GHz. Experimental results show that a similar performance as the conventional AEQ could be achieved by using proposed adaptation algorithms and reducing the number of multiplications with up to ∼65%.

MONETARY UTILITY FUNCTIONS ON Cb(X) SPACES

We will characterize robust monetary utility functions defined on the space of real valued (bounded) continuous functions on a Polish space.

RGA: a unified measure of predictive accuracy

AbstractA key point to assess statistical forecasts is the evaluation of their predictive accuracy. Recently, a new measure, called Rank Graduation Accuracy (RGA), based on the concordance between the ranks of the predicted values and the ranks of the actual values of a series of observations to be forecast, was proposed for the assessment of the quality of the predictions. In this paper, we demonstrate that, in a classification perspective, when the response to be predicted is binary, the RGA coincides both with the AUROC and the Wilcoxon-Mann–Whitney statistic, and can be employed to evaluate the accuracy of probability forecasts. When the response to be predicted is real valued, the RGA can still be applied, differently from the AUROC, and similarly to measures such as the RMSE. Differently from the RMSE, the RGA measure evaluates point predictions in terms of their ranks, rather than in terms of their values, improving robustness.

Invariance of the microsupport of some evolution problems

This paper is devoted to prove the invariance of the microsupport under the influence of the evolution of the solution in of the following Cauchy problem: where is a semiclassical parameter, and ( ) is a pseudo‐differential operator of symbol where as and is real valued, that is,

Clutter Covariance Matrix Estimation for Radar Adaptive Detection Based on a Complex-Valued Convolutional Neural Network

In this paper, we address the problem of covariance matrix estimation for radar adaptive detection under non-Gaussian clutter. Traditional model-based estimators may suffer from performance loss due to the mismatch between real data and assumed models. Therefore, we resort to a data-driven deep-learning method and propose a covariance matrix estimation method based on a complex-valued convolutional neural network (CV-CNN). Moreover, a real-valued (RV) network with the same framework as the proposed CV network is also constructed to serve as a natural competitor. The obtained clutter covariance matrix estimation based on the network is applied to the adaptive normalized matched filter (ANMF) detector for performance assessment. The detection results via both simulated and real sea clutter illustrate that the estimator based on CV-CNN outperforms other traditional model-based estimators as well as its RV competitor in terms of probability of detection (PD).

Low complexity joint clock recovery and adaptive equalization based on a baud-rate timing phase error detector for short-reach digital coherent transmission.

Digital coherent receivers adopting joint clock recovery and adaptive equalization (JCA) can avoid failures of the adaptive equalizer (AEQ) or clock recovery algorithm (CRA) due to clock asynchrony and chromatic dispersion (CD). But in the previous JCA scheme, the AEQ has a heavy computational load as it has to generate two samples per symbol (SPS) for the subsequent timing phase error detector (TPED) which is the core of the CRA. Furthermore, the previous JCA scheme cannot compensate for receiver skew or accommodate Nyquist signals with small roll-off factors (ROFs). These shortcomings hinder its practical applications in ultrahigh-speed short-reach coherent transmission requiring low power consumption, high spectral efficiency, whilst being sensitive to receiver skew. To solve this problem, we propose a new JCA scheme by integrating a two-section real-valued (RV) AEQ with an all-digital feedback CRA based on a baud-rate TPED versatile for different ROFs. Experiments for 61-GBaud dual-polarization (DP) Nyquist 16QAM signals with an ROF of 0.01 show that, compared with the previous JCA scheme, the proposed scheme can reduce the AEQ computational load by about 70% for 10-km transmission, whilst improving the receiver sensitivity by more than 1.7 dB for a receiver skew of 1.5 ps. As far as we know, the proposed JCA scheme is the most comprehensive and efficient solution for ultrahigh-speed short-reach coherent transmission where CD, receiver skew, clock asynchrony, and Nyquist signals with small ROFs have to be dealt with.

A 2D-LUT Scheme Design for Complex-Valued Spline Adaptive Filter

This brief presents a two dimensional Look-Up-Table (2D-LUT) scheme for complex-valued (CV) spline adaptive filter. The key idea of this structure is to model a complex-valued nonlinear static function by implementing the Carstesian product of two univariate cubic uniform B-spline curves. Therefore, the real and imaginary components of the system complex-valued output can be readily achieved hinging on two different sets of spline control point respectively, each of which is stored in a 2D-LUT separately. In addition, as an extension of the real-valued (RV) sign spline adaptive algorithm, the CV counterpart is also derived to update both the weight coefficients of the linear subsystem and spline control points associated with 2D-LUTs. Simulations on the identification of RF nonlinear power amplifier (PA) show that the proposed scheme outperforms the existing methods in terms of estimation accuracy against impulsive noises, as well as lower computational complexity.

Alternative Forward Models for Imaging Thick Specimens in Transmission Electron Microscopy.

I have investigated two different forward models for image formation in transmission electron microscopy of thick specimens, the 3DCtf model, which introduces a defocus gradient in the linear approximation, and the multislice model. An important result is that the 3DCtf model does not seem to be compatible with the multislice image formation model. A second very useful finding is that the exit wave in the multislice model has an imaginary part, which, in first-order approximation, is a pure projection of the specimen and is not affected by the defocus gradient. The defocus gradient only comes into play in real valued and higher-order imaginary terms. If the multislice model is closer to reality than the 3DCtf-model, then the best way to retrieve the specimen projection for thicker specimens should be a procedure for retrieving the exit wave's imaginary term, for example using images recorded at different defocus values.

Allowable complex scalars from Kaluza-Klein compactifications and metric rescalings

Recently there have been discussions about which complex metrics should be allowable in quantum gravity. These discussions assumed that the matter fields were real valued. We make the observation that for compactified solutions it makes sense to demand convergence of the theory's path integral in the higher-dimensional parent theory. Upon compactification this allows for more general matter configurations in the lower-dimensional theory, in particular it allows for complex scalar fields, with a bound on their imaginary parts. Similar considerations apply to metric rescalings in the presence of higher curvature corrections. We illustrate this effect with the example of the no--boundary proposal, in which scalar fields are typically required to take complex values. We find that complex no-boundary solutions exist, and satisfy the derived bound, if the potential is sufficiently flat. For instance, for a compactification from $D$ dimensions, the bound on the imaginary part $\textrm{Im}\,(\phi)$ of the internal volume modulus reads $V_{,\phi}/V < \sqrt{\frac{D-4}{D-2}}/3\sqrt{2}.$ This leads to a mild tension with swampland conjectures.

Complex-Valued End-to-End Deep Network With Coherency Preservation for Complex-Valued SAR Data Reconstruction and Classification

Deep learning models have achieved remarkable success in many different fields and attracted many interests. Several researchers attempted to apply deep learning models to Synthetic Aperture Radar (SAR) data processing, but it did not have the same breakthrough as the other fields, including optical remote sensing. SAR data are in complex domain by nature and processing them with Real-Valued (RV) networks neglects the phase component which conveys important and distinctive information. A Complex-Valued (CV) end-to-end deep network is developed in this study for the reconstruction and classification of CV-SAR data. Azimuth subaperture decomposition is utilized to incorporate physics-aware attributes of the CV-SAR into the deep model. Moreover, the correlation coefficient amplitude (Coherence) of the CV-SAR images depends on the SAR system characteristics and physical properties of the target. This coherency should be considered and preserved in the processing chain of the CV-SAR data. The coherency preservation of the CV deep networks for CV-SAR images, which is mostly neglected in the literature, is evaluated in this study. Furthermore, a large-scale CV-SAR annotated dataset for the evaluation of the CV deep networks is lacking. A semantically annotated CV-SAR dataset from Sentinel-1 Single Look Complex StripMap mode data (S1SLC_CVDL dataset) is developed and introduced in this study. The experimental analysis demonstrated the better performance of the developed CV deep network for CV-SAR data classification and reconstruction in comparison to the equivalent RV model and more complicated RV architectures, as well as its coherency preservation and physics-aware capability.

Uniform boundedness for the optimal controls of a discontinuous, non-convex Bolza problem,

We consider a Bolza type optimal control problem of the form [see formula in PDF] Subject to: [see formula in PDF] where Λ(s, y, u) is locally Lipschitz in s, just Borel in (y, u), b has at most a linear growth and both the Lagrangian Λ and the end-point cost function g may take the value +∞. If b ≡ 1, g ≡ 0, (Pt, x) is the classical problem of the Calculus of Variations. We suppose the validity of a slow growth condition in u, introduced by Clarke in 1993, including Lagrangians of the type [see formula in PDF] and [see formula in PDF] and the superlinear case. We show that, if Λ is real valued, any family of optimal pairs (y*, u*) for (Pt,x) whose energy Jt(y*, u*) is equi-boundcd as (t, x) vary in a compact set, has L∞ – equibounded controls. Moreover, if Λ is extended valued, the same conclusion holds under an additional lower semicontinuity assumption on (s, u) ↦ Λ(s, y, u) and requiring a condition on the structure of the effective domain. No convexity, nor local Lipschitzianity is assumed on the variables (y, u). As an application we obtain the local Lipschitz continuity of the value function under slow growth assumptions.

Generalised Kuramoto models with time-delayed phase-resetting for [formula omitted]-dimensional clocks

We consider a class of Kuramoto models, with an array of N individual k-dimensional clocks (k>1), coupled within a directed, range dependent, network. For each directed connection, a signal triggered at the sending clock incurs a (real valued) time delay before arriving at the receiving clock, where it induces an instantaneous phase reset affecting all k-phases. Instantaneous phase resetting maps (PRMs) for k-dimensional clocks have received little attention. The system may be treated as open and subject to periodic, or other types of, PRM forcing at any individual clock, as a result of external forcing stimuli. We show how the full system, with Nk phase variables, responds to such stimuli, as the impact spreads across the entire network. Beyond simulations, we employ methods to reverse engineer the dynamical behaviour of the whole: estimating the intrinsic dimensions of the responses to different experiments; and by analysing pairwise comparisons between those responses. This shows that the system’s responses are governed by a hierarchy of internal dynamical modes, existing across both the Nk phases and over time.We argue that this Kuramoto system is a model for the human cortex, where each k-dimensional clock models the dynamics of a single neural column, which contains 10,000 densely inter-connected neurons. The Kuramoto model exploits the natural network of networks architecture of the human cortex. An array of N=1M such columns/clocks is at the 10B neuron scale of the human cortex. However its simulation is far more accessible than very large scale (VLS) simulations of neuron-to-neuron systems on supercomputers. The latent modes may have important implications for cognition (information processing) and for consciousness (the existence of internal phenomenological experiences). We argue that the existence of the latter plays a key role in preconditioning the former, reducing the decision sets and the cognitive load, and thus enabling a fast-thinking evolutionary advantage.This is the first time that systems of k-dimensional clocks (k> 1), coupled via time-lagged PRMs, within range dependent networks, have been deployed to demonstrate the basic internal phenomenological elements (of consciousness) and their potential role within immediate cognition.Statement of Significance: We argue that this Kuramoto system is a model for the human cortex, where each of 1M k-dimensional clocks models the dynamics of a single neural column (containing 10,000 densely inter-connected neurons). This Kuramoto model exploits the natural network of networks architecture of the human cortex. Large scale human cortex simulations, with 10B neutrons, usually require a super computer. We show that similar results, using this model, can be obtained on a laptop. In particular we show that such dynamical can support internal phenomenological elements (of conscious experience) and we discuss their potential role in preconditioning immediate cognition, furnishing a “fast thinking” evolutionary advantage to the human brain.

Normalized ground states for a Hardy–Littlewood–Sobolev upper critical Schrödinger equation with double Choquard type nonlinear terms

In this paper, we study the existence of normalized ground states for a Hardy–Littlewood–Sobolev upper critical Schrödinger equation with double Choquard type nonlinear terms −Δu=λu+μ(Iα∗|u|p)|u|p−2u+(Iα∗|u|2α∗)|u|2α∗−2uinRN,∫RN|u|2=a,where N≥3, μ>0, N+αN<p<N+α+2N, α∈(0,N), Iα is the Riesz potential. We show that the associated energy functional admits a normalized ground state, which is a real valued, positive, radially symmetric, radially non-increasing function.

Novel complex-valued deep learning applied to automatic classification of heart sounds

Abstract Introduction Cardiovascular disease (CVD) is a problem facing governments around the world. Early detection saves the healthcare system thousands of dollars by allowing prevention and avoiding costly treatments when the disease has progressed to advanced stages. According to the World Health Organization (WHO), cardiovascular diseases represent the leading cause of mortality worldwide (WHO, 2019). That is why, we have decided to develop novel artificial intelligence models for the detection of heart disease, with accuracies that allow us to put the obtained algorithms into production. The revolution of artificial intelligence allows the application Deep Learning techniques of two-dimensional images in the domain of both real numbers and complex value numbers, as classifiers of heart sounds, for the detection of normality or abnormality in the functioning of the heart. Purpose In the present work, we propose the comparison of a novel 2D convolutional neural network (CNN) algorithm in the domain of complex value numbers with its counterpart in the domain of real numbers for the automatic classification of heart sounds into normal or abnormal. Material The database we decided to use for our research is Pascal, which has audio files of cardiac activity, distributed in 351 normal sounds and 129 pathological sounds. Methods The following steps were applied to get the objectives of our work: 1) automatic segmentation of a single heartbeat, 2) conversion of the segmented sound into its associated image scalogram using the Hilbert transform, 3) classification of the sounds into normal and abnormal using the proposed algorithms, and 4) measurement and comparison of the results obtained by performing a two-tailed t-student hypothesis test and cross-validation. Results We present a comparative table between the two proposed models, finding that Accuracy, F1 Score, Precision and Recall metrics obtained using complex-valued convolution networks present significant improvements compared with the real valued one. The following table show us the obtained numbers. For all cases, the t-student test shows us p-values less than 0.05%, giving statistical evidence that the means are significantly different between the two proposed models. Besides, in all cases, the performance of the Complex-valued model is better compared with the Real-valued one. Conclusion Complex-valued neural networks propose a significant advance around Deep learning, since they present a better performance than the traditional counterparts based on real numbers. This proposes an experimental basis for the construction of a new Deep learning paradigm, where information in another numerical domain, is better exploited with the help of mathematical transforms. The latter is a significant advance in health sciences, where the demand is higher, in terms of performance of the proposed models. Funding Acknowledgement Type of funding sources: Other. Main funding source(s): eVIDA research Group

The Barbero–Immirzi Parameter: An Enigmatic Parameter of Loop Quantum Gravity

The Barbero–Immirzi parameter, (γ), is introduced in loop quantum gravity (LQG), whose physical significance is still the biggest open question because of its profound traits. In some cases, it is real valued, while it is complex valued in other cases. This parameter emerges in the process of denoting a Lorentz connection with a non-compact group SO(3,1) in the form of a complex connection with values in a compact group of rotations, either SO(3) or SU(2). Initially, it appeared in the Ashtekar variables. Fernando Barbero proposed its possibility for inclusion within formalism. Its present value is fixed by counting micro states in loop quantum gravity and matching with the semi-classical black hole entropy computed by Stephen Hawking. This parameter is used to count the size of the quantum of area in Planck units. Until the discovery of the spectrum of the area operator in LQG, its significance remained unknown. However, its complete physical significance is yet to be explored. In the present paper, an introduction to the Barbero–Immirzi parameter in LQG, a timeline of this research area, and various proposals regarding its physical significance are given.

Examination of Generalized Statistical Convergence of Order α on Time Scales

In this paper, we introduce the concepts of deferred statistical convergence of order α and strongly deferred Cesàro summable functions (real valued) of order α on time scales and give some relationships between deferred statistical convergence of order α and strongly deferred Cesàro summable functions (real valued) of order α on time scales.

Markovian imprecise jump processes: Extension to measurable variables, convergence theorems and algorithms

The existing framework of Markovian imprecise jump processes, also known as imprecise continuous-time Markov chains, is limited to bounded real variables that depend on the state of the system at a finite number of (future) time points. This is an issue in many applications, because typically the variables of interest depend on the state of the system at all time points in some – possibly unbounded – (future) interval, and they can be unbounded or even extended real valued; examples of such variables are temporal averages, the number of (selected) jumps in some interval and hitting times. To eliminate this shortcoming, we assume that the sample paths are càdlàg and use measure theory to extend the domain of Markovian imprecise jump processes to extended real-valued variables that may depend on the state of the system at all (future) time points – that is, the extended real variables that are bounded below or above and are measurable with respect to the σ-algebra generated by the cylinder events. We investigate the continuity properties of the extended lower and upper expectations with respect to point-wise convergent sequences, and this yields generalisations of the Monotone Convergence Theorem and Lebesgue's Dominated Convergence Theorem. For two particular classes of variables, we strengthen these convergence theorems and present an iterative scheme to approximate their lower and upper expectations. The first class is the number of selected jumps in some interval, and the second class are real variables that take the form of a Riemann integral over some interval; this second class includes temporal averages and occupancy times.