Real Line Research Articles

Stereo matching algorithm using deep learning and edge-preserving filter for machine vision

Machine vision research began with a single-camera system, but these systems had various limitations from having just one point-of-view of the environment and no depth information, therefore stereo cameras were invented. This paper proposes a hybrid method of a stereo matching algorithm with the goal of generating an accurate disparity map critical for applications such as 3D surface reconstruction and robot navigation to name a few. Convolutional neural network (CNN) is utilised to generate the matching cost, which is then input into cost aggregation to increase accuracy with the help of a bilateral filter (BF). Winner-take-all (WTA) is used to generate the preliminary disparity map. An edge-preserving filter (EPF) is applied to that output based on a transform that defines an isometry between curves on the 2D image manifold in 5D and the real line to eliminate these artefacts. The transform warps the input signal adaptively to allow linear 1D filtering. Due to the filter's resistance to high contrast and brightness, it is effective in refining and removing noise from the output image. Based on experimental research employing a Middlebury standard validation benchmark, this approach gives high accuracy with an average non-occluded error of 6.71% comparable to other published methods.

Asymptotics of the sticky particles evolution

We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the sticky particles dynamics is confined to a compact set.

S-asymptotically ω periodic solutions of heat equation

We mean by an S-asymptotically ω-periodic function any continuous and bounded function from the real axis to a Banach space that converges to a periodic function as t tends to infinity. We study in this paper the existence, uniqueness and differentiable dependence on the S-asymptotically ω-periodic mild solution of a heat equation on the space of continuous functions from a non-empty n-dimensional bounded domain with a Lipschitz boundary to the real axis. The dependence of the solution concerns the initial conditions, more precisely when the initial conditions is a S-asymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of heat equation. To show our main result in this work we introduce the properties of the superposition operator, or also called Nemytskii operator, in the space of S-asymptotically ω-periodic functions. The notion of derivation for the last operator will also be highlighted. We also use in this paper semi-groups which have become important tools for differential equations. In this study, our focus shifted from seeking an s-asymptotically w-periodic solution for our heat equation, which is a problem in dynamic systems, to one in functional analysis. More precisely, our strategy consists of applying the implicit function theorem on a certain operator that we constructed on our workspaces in order to achieve the objective described in our main theorem. In fact our theorem gives conditions to ensure that around a mild S-asymptotically ω-periodic solution of our heat equation with an initial value, there exists a regular (in the usual sens) mild S-asymptotically ω-periodic solution which depends on a neighboring initial condition.

The spectral determinant for second-order elliptic operators on the real line

The spectral determinant for second-order elliptic operators on the real line

Deep-water and shallow-water limits of the intermediate long wave equation

In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs -Sobolev space for s>12 , to those of BO in the deep-water limit (as δ→∞ ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required s>32 in the deep-water limit and s⩾2 in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity ∂x(uk) for k⩾2 . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity s⩾34 . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.

New neighborhoods in robustness and least favorable pairs for special capacities

We propose a new neighborhood of distributions for robust inference, which is generated by a certain special capacity with three constants. It covers not only the familiar neighborhoods defined in terms of ε-contamination, total variation and their combination, but also their intersections. It is shown that the neighborhood is also obtained from contaminating some neighborhood which consists of absolutely continuous distributions. We derive the explicit forms of Radon-Nikodym derivatives and the least favorable pairs for such special capacities and neighborhoods on the real line, which generalize the earlier results of Huber and Rieder for minimax testing, and reinforce the general results of Bednarski.

Error bounds for Gauss–Jacobi quadrature of analytic functions on an ellipse

For the Gauss–Jacobi quadrature on [ − 1 , 1 ] [-1,1] , the location is estimated where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. For the Jacobi weight ( 1 − t ) α ( 1 + t ) β (1-t)^\alpha (1+t)^\beta ( α > − 1 \alpha >-1 , β > − 1 \beta >-1 ) except for the Gegenbauer weight ( α = β \alpha =\beta ), the location is the intersection point of the ellipse with the real axis in the complex plane. For the Gegenbauer weight, it is the intersection point(s) with either the real or the imaginary axis or other axes with angle 1 4 π \tfrac {1}{4}\pi and 3 4 π \tfrac {3}{4}\pi . Our results support the empirical results provided by Gautschi and Varga [SIAM J. Numer. Anal, 20 (1983), pp. 1170–1186] for the Jacobi weight, the Gegenbauer weight and the Legendre weight. The results obtained are also illustrated numerically.

Revisiting O(N) σ model at unphysical pion masses and high temperatures

Roy-equation analysis on lattice data of ππ scattering phase shifts at mπ=391 MeV reveals that the lowest f0 meson becomes a bound state under this condition. In addition, there is a pair of complex poles below threshold generated by crossing symmetry [X.-H. Cao , ]. We use the N/D method to partially recover crossing symmetry of the O(N) σ model amplitude at leading order of 1/N expansion, and qualitatively reproduce the pole structure and pole trajectories with varying pion masses as revealed by Roy-equation analyses. The σ pole trajectory with varying temperature is also discussed and found to be similar to its properties when varying mπ. As the temperature increases, the complex σ poles firstly move from the second Riemann sheet to the real axis becoming two virtual state poles, and then one virtual state pole moves to the first sheet turning into a bound state pole and finally tends to the pion pole position at high temperature which is as expected from the chiral symmetry restoration. Our results provide further evidence that the lowest f0 state extracted from experiments and lattice data plays the role of the σ meson in the spontaneous breaking of chiral symmetry. Finally, we also briefly discuss the problems of the effective potential in the situation when mπ and the temperature become large. Published by the American Physical Society 2024

Layered Patterns in Reaction–Diffusion Models with Perona–Malik Diffusions

In this paper we deal with a reaction–diffusion equation in a bounded interval of the real line with a nonlinear diffusion of Perona–Malik’s type and a balanced bistable reaction term. Under very general assumptions, we study the persistence of layered solutions, showing that it strongly depends on the behavior of the reaction term close to the stable equilibria ±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm 1$$\\end{document}, described by a parameter θ>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta >1$$\\end{document}. If θ∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in (1,2)$$\\end{document}, we prove existence of steady states oscillating (and touching) ±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm 1$$\\end{document}, called compactons, while in the case θ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta =2$$\\end{document} we prove the presence of metastable solutions, namely solutions with a transition layer structure which is maintained for an exponentially long time. Finally, for θ>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta >2$$\\end{document}, solutions with an unstable transition layer structure persist only for an algebraically long time.

The descriptive complexity of the set of Poisson generic numbers

Let [Formula: see text] be an integer. We show that the set of real numbers that are Poisson generic in base [Formula: see text] is [Formula: see text]-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base [Formula: see text] and not Poisson generic in base [Formula: see text] is complete for the class given by the differences between [Formula: see text] sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.

Annihilating Branching Brownian Motion

Abstract We study an interacting system of competing particles on the real line. Two populations of positive and negative particles evolve according to branching Brownian motion. When opposing particles meet, their charges neutralize and the particles annihilate, as in an inert chemical reaction. We show that, with positive probability, the two populations coexist and that, on this event, the interface is asymptotically linear with a random slope. A variety of generalizations and open problems are discussed.

An Optimization Strategy for Truncating Ultralong Bus Lines Integrated with Metro Networks

For most ultralong bus lines, low punctuality rates and vehicle bunching are two main challenges. One potential strategy to solve these two problems is to split an ultralong line into two or three shorter lines according to historical passenger flow patterns as well as other presetting objectives and constraints. Thus, in this paper, a multiobjective optimization model is first established in consideration of both passenger flow distribution and coline metro networks. Then, the optimal truncated stop is determined by estimating the model results generated by enumerating each potential bus stop derived from related metro stops. The nondominated sorting genetic algorithm II (NSGA-II) is applied to solve the model efficiently. A case study conducted on a real bus line associated with metro networks in Beijing, China, demonstrates the efficiency of the optimization strategy adoption in terms of operating cost and passenger travel time, even with a slight increase in passenger cost. Besides, the multivehicle type is also referred in the experiments to verify economic efficiency to reduce operational costs.

On two consequences of CH established by Sierpiński

AbstractWe study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

Core decreasing functions

Given a measure space and a totally ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to define the down space of a Banach function space. This is done using a variant of the Köthe dual restricted to core decreasing functions. To study down spaces, the least core decreasing majorant construction and the level function construction, already known for functions on the real line, are extended to this general setting. These are used to give concrete descriptions of the duals of the down spaces and, in the case of universally rearrangement invariant (u.r.i.) spaces, of the down spaces themselves.The down spaces of L1 and L∞ are shown to form an exact Calderón couple with divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of level functions; and the down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property. The dual couple is also an exact Calderón couple with divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of least core decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property.

Smallest gaps between zeros of stationary Gaussian processes

In this paper, we study the smallest gaps between successive zeros of nondegenerate smooth stationary centered Gaussian processes on the real line with the assumption that the covariance kernel κ(x) and its derivatives decay to 0 as |x|→∞. We prove that, after rescaling, the smallest gaps converge to a Poisson point process with a specific rate. Moreover, the positions where these smallest gaps occur tend to a uniform distribution. Consequently, we can derive the limiting density for the k-th smallest gap.

Indeterminate Hamburger moment problem: Entropy convergence

The indeterminate Hamburger moment problem is considered, jointly with all its real axis supported probability density functions. As a consequence of entropy functional concavity, out of such densities there is one which has largest entropy and that plays a fundamental role: we call it fhmax. It is proved that the approximate Maximum Entropy (MaxEnt) densities constrained by an increasing number of moments converge in entropy to fhmax where the value of its entropy can be finite or −∞.

Description of the set of all possible masses at a fixed point of solutions of a truncated matricial Hamburger moment problem

We describe the set of all possible masses at a fixed point of the real axis in the context of the most general case of a truncated matricial (possibly degenerate) Hamburger moment problem. This set is a matricial interval, which we explicitly construct from the given data in the non-degenerate and degenerate cases simultaneously. In particular, we determine the maximum possible mass.

Limiting Weak-Type Behavior of the Centered Hardy–Littlewood Maximal Function of General Measures on the Positive Real Line

Limiting Weak-Type Behavior of the Centered Hardy–Littlewood Maximal Function of General Measures on the Positive Real Line

Orthonormal expansions for translation-invariant kernels

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of ℒ2(R). This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.

A Geometric Theory for Binary Classification of Finite Datasets by DNNs with Relu Activations

In this paper we investigate deep neural networks for binary classification of datasets from geometric perspective in order to understand the working mechanism of deep neural networks. First, we establish a geometrical result on injectivity of finite set under a projection from Euclidean space to the real line. Then by introducing notions of alternative points and alternative number, we propose an approach to design DNNs for binary classification of finite labeled points on the real line, thus proving existence of binary classification neural net with its hidden layers of width two and the number of hidden layers not larger than the cardinality of the finite labelled set. We also demonstrate geometrically how the dataset is transformed across every hidden layers in a narrow DNN setting for binary classification task.