Microlocal Analysis of $d$-Plane Transform on the Euclidean Space

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k > 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

Consider an n-point metric space M = (V,δ), and a transmission range assignment r: V → ℝ + that maps each point v ∈ V to the disk of radius r(v) around it. The symmetric disk graph (henceforth, SDG) that corresponds to M and r is the undirected graph over V whose edge set includes an edge (u,v) if both r(u) and r(v) are no smaller than δ(u,v). SDGs are often used to model wireless communication networks.Abu-Affash, Aschner, Carmi and Katz (SWAT 2010, [1]) showed that for any n-point 2-dimensional Euclidean space M, the weight of the MST of every connected SDG for M is O(logn) ·w(MST(M)), and that this bound is tight. However, the upper bound proof of [1] relies heavily on basic geometric properties of constant-dimensional Euclidean spaces, and does not extend to Euclidean spaces of super-constant dimension. A natural question that arises is whether this surprising upper bound of [1] can be generalized for wider families of metric spaces, such as high-dimensional Euclidean spaces.In this paper we generalize the upper bound of Abu-Affash et al. [1] for Euclidean spaces of any dimension. Furthermore, our upper bound extends to arbitrary metric spaces and, in particular, it applies to any of the normed spaces ℓ p . Specifically, we demonstrate that for any n-point metric space M, the weight of the MST of every connected SDG for M is O(logn) ·w(MST(M)).KeywordsSpan TreeTransmission RangeMinimum Span TreeHamiltonian PathUnit Disk GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The solution of many applied problems is to find a solution of nonlinear equations systems in finite- dimensional Euclidean spaces. The problem of finding the solution of a nonlinear system is divided into two problems: 1. The existence of a solution of a nonlinear equations system; in the case of nonunique of the solution, it is necessary to find the number of these solutions and their surroundings. 2. Finding the solution of a system of nonlinear equations with a given accuracy. Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence and estimates of the solution accuracy. In contrast to problem 2, for problem 1 there is no general algorithm for solving this task, there are no constructive conditions for the existence of a solution of a nonlinear equations system in Euclidean spaces. In this article, in finite-dimensional Euclidean spaces, the constructive conditions for the existence of a solution of nonlinear systems of polynomial form are found. The connection of these conditions with the linear polynomial interpolant of the minimum norm, generated by a scalar product with Gaussian measure and the conditions of its existence, is given.

To suppress the streak artifacts in images reconstructed from sparse-view projections in computed tomography (CT), a residual, attention-based, dense UNet (RAD-UNet) deep network is proposed to achieve accurate sparse reconstruction. The filtered back projection (FBP) algorithm is used to reconstruct the CT image with streak artifacts from sparse-view projections. Then, the image is processed by the RAD-UNet to suppress streak artifacts and obtain high-quality CT image. Those images with streak artifacts are used as the input of the RAD-UNet, and the output-label images are the corresponding high-quality images. Through training via the large-scale training data, the RAD-UNet can obtain the capability of suppressing streak artifacts. This network combines residual connection, attention mechanism, dense connection and perceptual loss. This network can improve the nonlinear fitting capability and the performance of suppressing streak artifacts. The experimental results show that the RAD-UNet can improve the reconstruction accuracy compared with three existing representative deep networks. It may not only suppress streak artifacts but also better preserve image details. The proposed networks may be readily applied to other image processing tasks including image denoising, image deblurring, and image super-resolution.

In this article we present a comparison of the performance between a metaheuristic optimization method, Game of Patterns (GofP), so-called by the author, and the well-known genetic algorithms (GAs), through two implementations, namely: the GA of Scilab (SGA); and the GA of the R Project for Statistical Computing (RGA). For this purpose, we have selected a set of multimodal objective functions in the n-dimensional Euclidean space \(\mathbb {R}^{n}\) with a unique global minimum. For comparing both metaheuristic optimization approaches, a performance indicator of quality, denoted Q(p, n, s), was defined, which allows us to measure the quality of the obtained global optimal solution for each pth problem, in the n-dimensional space, when it is solved by each metaheuristic optimization method \(s\in \{\texttt {GofP},\texttt {SGA},\texttt {RGA}\}\). The indicator Q(p, n, s) then depends on: the number of evaluations of the pth optimization problem in the Euclidean space \(\mathbb {R}^{n}\), which has required the s metaheuristic optimization method for identifying the global minimum; and the distance between the location of its respective unique global minimum and the location of the minimum that has been identified by the s metaheuristic optimization method. The paper also offers a brief explanation of the GofP method, which has been developed for solving unconstrained mixed integer problems in the \(n\times m\)-dimensional Euclidean space \(\mathbb {R}^{n}\times \mathbb {Z}^{m}\).

The purpose of this paper is to provide necessary and sufficient conditions on a continuous and matrix valued radial kernel on a Euclidean space in order that it be conditionally positive definite of a fixed order. Except for the one dimensional Euclidean space, the strict conditional positive definiteness of the kernel is fully characterized.

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in m m -dimensional Euclidean space, or a set of n n -by- m m systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in m m -dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of n n -by- m m systems of linear forms. Consequently, a class of m m -dimensional results in Diophantine approximation can be bootstrapped to corresponding n n -by- m m -dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.

PurposeAs computed tomography (CT) imaging is the most commonly used modality for treatment planning in radiation therapy, metal artifacts in the planning CT images may complicate the target delineation and reduce the dose calculation accuracy. Although current CT scanners do provide certain correction steps, it is a common understanding that there is not a universal solution yet to the metal artifact reduction (MAR) in general. Particularly noting the importance of MAR for radiation treatment planning, we propose a novel MAR method in this work that recruits an additional tilted CT scan and synthesizes nearly metal‐artifact‐free CT images.MethodsThe proposed method is based on the facts that the most pronounced metal artifacts in CT images show up along the x‐ray beam direction traversing multiple metallic objects and that a tilted CT scan can provide complementary information free of such metal artifacts in the earlier scan. Although the tilted CT scan would contain its own metal artifacts in the images, the artifacts may manifest in a different fashion leaving a chance to concatenate the two CT images with the metal artifacts much suppressed. We developed an image processing technique that uses the structural similarity (SSIM) for suppressing the metal artifacts. On top of the additional scan, we proposed to use an existing MAR method for each scan if necessary to further suppress the metal artifacts.ResultsThe proposed method was validated by a simulation study using the pelvic region of an XCAT numerical phantom and also by an experimental study using the head part of the Rando phantom. The proposed method was found to effectively reduce the metal artifacts. Quantitative analyses revealed that the proposed method reduced the mean absolute percentages of the error by up to 86% and 89% in the simulation and experimental studies, respectively.ConclusionsIt was confirmed that the proposed method, using complementary information acquired from an additional tilted CT scan, can provide nearly metal‐artifact‐free images for the treatment planning.

We consider a finite subgroup Θ n of the group O(N) of orthogonal matrices, where N = 2n, n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2n-dimensional Euclidean space RN. We prove that representations of the group Θ n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1–3] imply that the orbit Θ n,2 x t of any initial point x on the sphere S N − 1 is a 7-design in the Euclidean space of dimension 2n.

Sparse-view computed tomography (CT) has been attracting attention for its reduced radiation dose and scanning time. However, analytical image reconstruction methods suffer from streak artifacts due to insufficient projection views. Recently, various deep learning-based methods have been developed to solve this ill-posed inverse problem. Despite their promising results, they are easily overfitted to the training data, showing limited generalizability to unseen systems and patients. In this work, we propose a novel streak artifact reduction algorithm that provides a system- and patient-specificsolution. Motivated by the fact that streak artifacts are deterministic errors, we regenerate the same artifacts from a prior CT image under the same system geometry. This prior image need not be perfect but should contain patient-specific information and be consistent with full-view projection data for accurate regeneration of the artifacts. To this end, we use a coordinate-based neural representation that often causes image blur but can greatly suppress the streak artifacts while having multiview consistency. By employing techniques in neural radiance fields originally proposed for scene representations, the neural representation is optimized to the measured sparse-view projection data via self-supervised learning. Then, we subtract the regenerated artifacts from the analytically reconstructed original image to obtain the final correctedimage. To validate the proposed method, we used simulated data of extended cardiac-torso phantoms and the 2016 NIH-AAPM-Mayo Clinic Low-Dose CT Grand Challenge and experimental data of physical pediatric and head phantoms. The performance of the proposed method was compared with a total variation-based iterative reconstruction method, naive application of the neural representation, and a convolutional neural network-based method. In visual inspection, it was observed that the small anatomical features were best preserved by the proposed method. The proposed method also achieved the best scores in the visual information fidelity, modulation transfer function, and lung nodulesegmentation. The results on both simulated and experimental data suggest that the proposed method can effectively reduce the streak artifacts while preserving small anatomical structures that are easily blurred or replaced with misleading features by the existing methods. Since the proposed method does not require any additional training datasets, it would be useful in clinical practice where the large datasets cannot becollected.

In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve. We obtain the Frenet apparatus of these spherical indicatrices and the characterizations of being general helix and slant helix. Moreover we give some properties between the spherical indicatrices and their associated curves.

Helices and constant procession curves are special examples of slant curves. However, there is no example of a $k$-slant curve for a positive integer $k\geq 2$ in three dimensional Euclidean spaces. Furthermore, the position vector of a $k$-slant curve for a positive integer $k\geq 2$ has not been known thus far. In this paper, we propose a method for constructing $k$-slant curves in three dimensional Euclidean spaces. We then show that spherical $k$-slant curves and $N_{k}$-constant procession curves can be derived from circles, for $k \in \mathbb{N}$, the set of all nonnegative integers. In addition, we provide a new proof of the spherical curve characterization and define a curve in the sphere called a spherical prime curve. Afterward, we apply $k$-slant curves to magnetic curves. Finally, we discuss the need for further research.

Let the cardinal invariant s n {\mathfrak s}_{n} denote the least number of continuously smooth n n -dimensional surfaces into which ( n + 1 ) (n+1) -dimensional Euclidean space can be decomposed. It will be shown to be consistent that s n {\mathfrak s}_{n} is greater than s n + 1 {\mathfrak s}_{n+1} . These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.

Abstract. In this paper, we consider and study a new class of general-ized vector quasi-variational type inequalities and obtain some existencetheorems for both under compact and noncompact assumptions in topo-logical vector spaces without using monotonicity. For the noncompactcase, we use the concept of escaping sequences. 1. IntroductionThe vector variational inequality problem was initiated by Giannessi [11] in nite dimensional Euclidean spaces with applications. Later on many authors[6, 7, 12, 15, 24] generalized vector variational inequalities in abstract spacesin several ways. The vector variational-like inequality is one of the generalizedforms of vector variational inequalities [11]. In 1989, Parida et al [19] stud-ied the existence of solutions for variational-like inequalities in R n space andhave shown a relationship between variational like inequalities problems andconvex programming as well as with complementarity problems. The vectorvariational like inequality and generalized vector variational like inequality arepowerful tools to study non-convex vector optimization problems and convexand nondi erentiable vector optimization problems respectively, see [10, 18, 21].In 1973, Bensoussan and Lions [3] introduced the quasi-variational inequalityproblem (QVIP). Since then, many generalizations of the QVIP have beenappeared in the literature (see, for example, [1, 2]). In the last decade, becauseof applications in the optimization problems, mathematical programming andequilibrium problems, the QVIP has been intensively studied by many authors[5, 8, 16, 20, 22, 23, 25].In this paper, we consider the generalized vector quasivariational type in-equality problem (GVQVTIP) and obtain some existence theorems for solutions