Norm Of Space Research Articles

Optimal Joint Path Planning of a New Virtual-Linkage-Based Redundant Finishing Stage for Additive-Finishing Integrated Manufacturing

This paper describes the optimal path planning of a redundant finishing mechanism developed for joint space-based additive-finishing integrated manufacturing (AFM). The research motivation results from an inevitable one-sided layout of a finishing stage (FS) with regard to the additive stage (AS) in the AFM. These two stages share a 2-dof bed stage (BS), and the FS can lightly shave off the rough-surfaced 3D print on the bed. Since the FS located at the side of the AS cannot reach all the target points of the 3D print, the bed should be able to rotate the 3D print about the z-axis and translate it in the z-axis. As a result, the AS has 4-dof joints for 2P and 1P1R during the additive process with AS-BS, and FS has 4-dof and 2-dof integrated joints for 2P2R and 1P1R during the finishing process with FS-BS, respectively. For the kinematic modeling of the FS part and the BS, the virtual linkage connecting the bed frame origin and the FS’s joint frame for approaching the BS is considered to realize seamless kinematic redundancy. The minimum Euclidian norm of the joint velocity space is the objective function to find the optimal joint space solution for a given tool path. To confirm the feasibility of the developed joint path planning algorithm in the proposed FS-BS mechanism, layer-by-layer slicing of a given 3D print’s CAD model and tool path generation were performed. Then, the numerical simulations of the optimal joint path planning for some primitive 3D print geometries were conducted. As a result, we confirmed that the maximum and mean pose error in point-by-point only, with the developed optimal joint path planning algorithm, were less than 202 nm and 153 nm, respectively. Since precision and general machining accuracies in tool path generation are in the range of ±10 μm and 20 μm, the pose error in this study fully satisfies the industry requirements.

A duality approach to regularized learning problems in Banach spaces

Regularized learning problems in Banach spaces, which often minimize the sum of a data fidelity term in one Banach norm and a regularization term in another Banach norm, is challenging to solve. We construct a direct sum space based on the Banach spaces for the fidelity term and the regularization term, and recast the objective function as the norm of a quotient space of the direct sum space. We then express the original regularized problem as an optimization problem in the dual space of the direct sum space. It is to find the maximum of a linear function on a convex polytope, which may be solved by linear programming. A solution of the original problem is then obtained by using related extremal properties of norming functionals from a solution of the dual problem. Numerical experiments demonstrate that the proposed duality approach is effective for solving the regularization learning problems.

Horofunction compactifications of symmetric cones under Finsler distances

In this paper we consider symmetric cones equipped with invariant Finsler distances, namely the Thompson distance and the Hilbert distance. We give a complete characterisation of the horofunctions of the symmetric cone \(A_+^\circ\) under the Thompson distance and establish a correspondence between the horofunction compactification of \(A_+^\circ\) and the horofunction compactification of the normed space in the tangent bundle. More precisely, we show that the exponential map extends as a homeomorphism between the horofunction compactification of the normed space in the tangent bundle, which is a JB-algebra, and the horofunction compactification of \(A_+^\circ\). Analogues results are established for the Hilbert distance on the projective symmetric cone \(PA_+^\circ\). The analysis yields a concrete description of the horofunction compactifications of these spaces in terms of the facial structure of the closed unit ball of the dual norm of the norm in the tangent space.

Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product

The aim of this paper is to investigate when a linear normed space is an inner product space. Several conditions in a linear normed space are formulated with the help of inequalities. Some of them are from the literature and others are new. We prove that these conditions are equivalent with the fact that the norm is induced by an inner product. One of the new results is the following: in an inner product space, the sum of opposite edges of a tetrahedron are the sides of an acute angled triangle. The converse of this result holds also. More precisely, this property characterizes inner product spaces. Another new result is the following: in a tetrahedron, the sum of squares of opposite edges are the lengths of a triangle. We prove also that this property characterizes inner product spaces. In addition, we give simpler proofs to some theorems already known from the publications of other authors.

Интегральное уравнение Вольтерра в банаховом пространстве

Despite the significant results achieved in the study of operator equations (including Volterra equations) in normalized Banach spaces, fundamental research in this field of mathematics attracts the attention of a huge number of mathematicians around the world. The solutions of the Volterra equation describe many important processes in various fields of science and technology. Studies of various inverse problems, experimental or experimental data processing problems related to the study of spherical or axisymmetric plasma formations, numerous mathematical models of the existence of various biological systems lead to the consideration and solution of this type of integral equations. A great contribution to the development of this theory was made by N.A. Magnitsky, L.I. Panov, A.N. Tikhonov, M.M. Lavrentiev and others. Fundamental results were obtained in the study of multiple operator equations with singularities in various functional spaces. Solutions depending on many parameters were constructed for the above equations. Currently, such problems are considered in spaces of arbitrary dimension and with coefficients having a derivative of finite order. In this paper, a finite set of solutions in a certain functional space is constructed for an integral equation of the first kind. The kernel of the integral operator has a finite order and is sufficiently differentiable near zero. The integral equation under consideration is reduced to an integro-differential equation representing two terms. For the first term, it is possible to solve the corresponding inhomogeneous equation and obtain a set of solutions in some functional normalized space. For the second term, we obtain an equation with an operator whose norm in some operator space is arbitrarily small near zero. Such splitting of the integral operator makes it possible to construct a partial and general solution of the corresponding integro-differential equation in the form of convergent equations. Applying modern methods of functional analysis, it is possible, by studying two separate equations, to construct a multiparametric family of solutions with values in some Banach space with weight for the original equation under consideration.

Between norms and margins: The “Norms–Fringe–Margins” model. A dynamic model of psychosocial self-construction through an expanded notion of working

This article addresses the new forms and relationships to work currently developing, in line with the upheavals of the Anthropocene. Occupational and counseling psychology are developing models that increasingly integrate systemic and contextual dimensions to understand careers and work-related issues. We propose a theoretical model “Norms–Fringe–Margins” to approach these new forms of work as hybrid constructs, sources of both personal and social innovation. We argue that these new ways of relating to work can contribute to sustainable and just commitments to citizenship, both at the individual and social levels. The model distinguishes three interrelated spaces of psychosocial self-construction through work: a normed space (Norm), a space on the margin (Margin), and an intermediate space (Fringe). To analyze the psychosocial construction of a person, we study the dynamic interactions of these three psychosocial spaces as a presence supported by a set of tensions to overcome.

Adaptive stabilized finite elements via residual minimization onto bubble enrichments

The Adaptive Stabilized Finite Element method (AS-FEM) developed in [16] combines the idea of the residual minimization method with the inf-sup stability offered by the discontinuous Galerkin (DG) frameworks. As a result, the discretizations deliver stabilized conforming approximations and residual representatives in a DG space that can drive automatic adaptivity. In this work, we extend AS-FEM by considering continuous enriched test spaces. Thus, we propose a residual minimization method on a stable Continuous Interior Penalty (CIP) formulation, which considers a C0-conforming trial FEM space and a test space based on the bubble enrichment of that trial space. In our numerical experiments, the test space choice significantly reduces the total degrees of freedom compared to the DG test spaces of [16], while recovering the expected convergence rate for the error in the corresponding trial space norm.

Near-Field Matching and Universal Limits on Electromagnetic Energy Transfer

This article introduces the concept of near-field (NF) matching as a continuum-mode generalization of port matching in circuit theory suitable for field-theoretic electromagnetic energy transfer scenarios, with a focus on spatio-frequency processes in coupled systems. The concept is rigorously formulated using the full electromagnetic Green’s function of a generic receiving surface interacting with arbitrary illumination fields where the Riemannian structure and the electromagnetic boundary condition of the problem are encoded into the tensor structure of a Green’s function on a manifold. After a carefully selected combination of proper function spaces for the various field quantities involved, we utilize exact methods to estimate the sizes of various operator quantities using the appropriate function space norms. A field-theoretic measure of power transfer efficiency in generalized NF matching scenarios is introduced, and exact upper bounds on this efficiency are derived using Young’s inequality for integral kernel operators. This theoretical study complements and generalizes the largely empirical and problem-specific literature on wireless energy transfer by providing an exact and rigorous mathematical framework that can guide and inform future optimization and design processes.

Correction: On Banach space norms for $${\mathcal {I}}$$-bounded sequences and their quotient

Correction: On Banach space norms for $${\mathcal {I}}$$-bounded sequences and their quotient

Feedback control for fluid mixing via advection

This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain Ω⊂Rd, with d=2 or d=3. The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space (H1(Ω))′ of H1(Ω) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws.

Embeddings of Decomposition Spaces

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two? A decomposition space D ( Q , L p , Y ) \mathcal {D}(\mathcal {Q}, L^p, Y) is determined by a covering Q = ( Q i ) i ∈ I \mathcal {Q} = (Q_i)_{i \in I} of the frequency domain, an integrability exponent p p , and a sequence space Y ⊂ C I {Y \subset \mathbb {C}^I} . Given these ingredients, the decomposition space norm of a distribution g g is defined as ‖ g ‖ D ( Q , L p , Y ) = ‖ ( ‖ F − 1 ( φ i ⋅ g ^ ) ‖ L p ) i ∈ I ‖ Y , { \left \Vert g \right \Vert _{\mathcal {D}(\mathcal {Q}, L^p, Y)} = \left \Vert \left ( \left \Vert \mathcal {F}^{-1} \left ( \varphi _{i} \cdot \widehat {g} \right ) \right \Vert _{L^{p}} \right )_{i \in I} \right \Vert _{Y} } , where ( φ i ) i ∈ I (\varphi _i)_{i \in I} is a suitable partition of unity for Q \mathcal {Q} . We establish readily verifiable criteria which ensure the existence of a continuous inclusion (“an embedding”) D ( Q , L p 1 , Y ) ↪ D ( P , L p 2 , Z ) \mathcal {D}(\mathcal {Q},L^{p_1},Y) \hookrightarrow \mathcal {D}(\mathcal {P},L^{p_2},Z) , mostly concentrating on the case where Y = ℓ w q 1 ( I ) Y = \ell _{w}^{q_{1}} ( I ) and Z = ℓ v q 2 ( J ) Z = \ell _{v}^{q_{2}} (J) . Under suitable assumptions on Q , P \mathcal {Q}, \mathcal {P} , we will see that the relevant sufficient conditions are p 1 ≤ p 2 p_{1} \leq p_{2} and finiteness of a nested norm of the form \[ ‖ ( ‖ ( α i β j ⋅ v j / w i ) i ∈ I j ‖ ℓ t ) j ∈ J ‖ ℓ s , with I j = { i ∈ I : Q i ∩ P j ≠ ∅ } for j ∈ J . \left \Vert \left ( \left \Vert \left ( \alpha _{i} \beta _{j} \cdot v_{j} / w_{i} \right )_{ i \in I_{j}} \right \Vert _{\ell ^{t}} \right )_{j\in J} \right \Vert _{\ell ^{s}} \, , \quad \text {with} \quad I_{j} = \left \{ i \in I \,:\, Q_{i} \cap P_{j} \neq \varnothing \right \} \quad \text {for } j \in J \,. \] Like the sets I j I_j , the exponents t , s t, s and the weights α , β \alpha , \beta only depend on the quantities used to define the decomposition spaces. In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings. These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of p 1 , p 2 p_1, p_2 , our criteria yield a complete characterization for the existence of the embedding. The same holds for arbitrary values of p 1 , p 2 p_1, p_2 under more strict assumptions on the coverings. We also prove a rigidity result, namely that—for ( p 1 , q 1 ) ≠ ( 2 , 2 ) (p_{1}, q_{1}) \neq (2, 2) —two decomposition spaces D ( Q , L p 1 , ℓ w q 1 ) \mathcal {D}(\mathcal {Q}, L^{p_1}, \ell _w^{q_1}) and D ( P , L p 2 , ℓ v q 2 ) \mathcal {D}(\mathcal {P}, L^{p_2}, \ell _v^{q_2}) can only coincide if their “ingredients” are equivalent, that is, if p 1 = p 2 p_{1} = p_{2} and q 1 = q 2 q_{1} = q_{2} and if the coverings Q , P \mathcal {Q}, \mathcal {P} and the weights w , v w, v are equivalent in a suitable sense. The resulting embedding theory is illustrated by applications to α \alpha -modulation and Besov spaces. All known embedding results for these spaces are special cases of our approach; often, we improve considerably upon the state of the art.

On Banach space norms for $${\mathcal {I}}$$-bounded sequences and their quotient

On Banach space norms for $${\mathcal {I}}$$-bounded sequences and their quotient

Conservative semi-Lagrangian method for continuity equation with various time steps in different parts of computational domain

The system of Navier Stokes differential equations describes gas flow near rocket nozzle. To find solution of this system in general case, scientists and engineers use numerical methods. Modern numerical methods do not allow engineers to model all features of gas flow. It happened because of sophisticated physical processes and limitations of computational hardware. So, at least where are two ways to improve it: to enlarge hardware or reduce computational complicacy. The global aim of many science investigations is to develop numerical approach with reduced computational complicacy and at the same time without loss of computational accuracy. In 1959, Aksel C. Wiin-Nielsen proposed a new and effective trajectories method for problem of numerical forecasting. In 1966, K. M. Magomedov developed similar approach (method of characteristics) for numerical modelling of space (three dimensional on space) gas flow. In 1982, O. Pironneau showed a new and effective approach for two dimensional approximation of the Navier Stokes problem. It was based on method of characteristics also. Nowadays, these methods are called semi-Lagrangian or Eulerian Lagrangian methods. They use Lagrangian nature of the transport process. To apply this advantage, scientists decompose each equation of Navier Stokes system into three parts: convective part (hyperbolic type), elliptic part and part of right-hand side. Scientists use semi-Lagrangian approach to approximate the convective parts of equations. To develop and test modern algorithms from family of semi-Lagrangian methods, we use continuity equation from the system of Navier Stokes equations. Conservative versions of semi-Lagrangian approach are based on Gauss Ostrogradsky (divergence) theorem. It allows scientists to get conservation low (balance equation) for numerical solution in norm of L1 space. Aim of our investigation is to use different time steps in different parts of computation domain. It enables us to obtain at the same time three advantages: convergence of numerical solution to exact solution, reduction of computational complicacy, implementation of conservation low (balance equation) without weight coefficients. For this purpose we decompose computational domain into two parts (subdomains) and use different time steps in them. Main complication is design algorithm in the boundaries of computational subdomains. We use one dimensional (on space) problem to demonstrate ability of developing numerical method with described advantages. Generalization of considered approach for two- or even three-dimensional cases allows engineers to model gas flow more accurately and without artificial viscosity. It is essentially important in the parts of computational domain with high level of solution gradient.

Solvability for a system of Hadamard-type hybrid fractional differential inclusions

Abstract In this article, a new system of Hadamard-type hybrid fractional differential inclusions equipped with Dirichlet boundary conditions was constructed. By virtue of a fixed-point theorem due to B. C. Dhage, (Existence results for neutral functional differential inclusions in Banach algebras, Nonlinear Anal. 64 (2006), no. 6, 1290–1306, doi: https://doi.org/10.1016/j.na.2005.06.036), the existence results of solutions for the considered problem are derived in a new norm space for multivalued maps. A numerical example is provided to illustrate our main results.

The Fuzzy Width Theory in the Finite-Dimensional Space and Sobolev Space

This paper aims to fuzzify the width problem of classical approximation theory. New concepts of fuzzy Kolmogorov n-width and fuzzy linear n-width are introduced on the basis of α-fuzzy distance which is induced by the fuzzy norm. Furthermore, the relationship between the classical widths in linear normed space and the fuzzy widths in fuzzy linear normed space is discussed. Finally, the exact asymptotic orders of the fuzzy Kolmogorov n-width and fuzzy linear n-width corresponding to a given fuzzy norm in finite-dimensional space and Sobolev space are estimated.

Generalized Quasilinear Elliptic Equations in {mathbb {R}}^N

In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation -div(a(x,u,∇u))+At(x,u,∇u)+V(x)|u|p-2u=g(x,u)inRN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} - \ extrm{div} (a(x,u,\ abla u)) + A_t(x,u,\ abla u) + V(x) {\\vert u \\vert }^{p-2} u= g(x,u) \\quad \\quad \\hbox { in }{{\\mathbb {R}}}^{N} \\end{aligned}$$\\end{document}with p>1, Nge 2 and V:{mathbb {R}}^Nrightarrow {mathbb {R}} suitable measurable positive function. Here, we suppose A: {mathbb {R}}^N times {mathbb {R}}times {mathbb {R}}^N rightarrow {mathbb {R}} is a given {C}^{1}-Carathéodory function which grows as |xi |^p, with A_t(x,t,xi ) = frac{partial A}{partial t}(x,t,xi ), a(x,t,xi ) = nabla _xi A(x,t,xi ), V:{mathbb {R}}^Nrightarrow {mathbb {R}} is a suitable measurable function and g:{mathbb {R}}^N times {mathbb {R}}rightarrow {mathbb {R}} is a given Carathéodory function which grows as |xi |^q with 1<q<p. Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on A(x,t,xi ), V(x) and g(x, t), we study the interaction of two different norms in a suitable Banach space with the aim of obtaining a good variational approach. Thus, a minimization argument on bounded sets can be used to state the existence of a nontrivial weak bounded solution on an arbitrary bounded domain. Then, one nontrivial bounded solution of the given equation can be found by passing to the limit on a sequence of solutions on bounded domains. Finally, under slightly stronger hypotheses, we can able to find a positive solution of the problem.

Space Diplomacy from an Astronaut’s Viewpoint

Summary Intergovernmental space diplomacy has played an essential role in human space activities. The International Space Station of low-Earth orbit and the Artemis programme to explore the moon and beyond are essential examples. Current noteworthy developments that are influencing space diplomacy include the following: (1) the number of governments that engage in human space activities has been increasing; and (2) inter-industry and grassroots co-operation has had a significant impact. We astronauts, and I believe anybody who aims to explore space, appreciate the value of norms in space, which assure a safe environment for our activities and secure future possibilities to expand our frontier to the moon and beyond.

Metaverse electronic jurisdiction: challenges and risks of legal regulation of virtual reality

This article is a continuation of the study of the problems of legal regulation of the Metaverse presented in the work “Genesis of legal regulation web and the model of the electronic jurisdiction of the metaverse”. The article proposes the creation of an electronic jurisdiction and a model of Codes and laws of the Metaverse. Metaverse – the real world, legal disputes are often resolved through courts that operate under a certain jurisdiction. However, in the Metaverse, which is still in its early stages of development, there is no single jurisdiction that covers all of its activities. Therefore, there is an urgent need for electronic jurisdiction in the Metaverse. Electronic jurisdiction involves the application of both modern and traditional legal norms in the electronic space. This will provide for the resolution of legal disputes through an electronic court system that will operate using certain algorithms of the traditional judicial system and the latest legal norms. Electronic jurisdiction will include not only codes and laws, but also the Constitution of the Metaverse, the Report on Relict Technical Standards, Cybersecurity Rules, the Criminal Electronic Code, Military Regulations, and other regulatory legal acts.One of the main advantages of electronic jurisdiction in the Metaverse is that it will provide a reliable and consistent means of resolving disputes, thanks to the reliability of Metaverse technologies. The existence of an electronic court system will ensure fair and impartial resolution of disputes, regardless of where the parties involved are located.&#x0D; Another advantage of electronic jurisdiction is that it will help create a sense of trust and confidence in the Metaverse. Users will know that their rights are protected and that they have the means to resolve any disputes that may arise. This will create a more stable and secure environment for all participants, encouraging further growth and development of the Metaverse.

The rule of unreason: Analyzing (anti‐)democratic regression

The rule of unreason: Analyzing (anti‐)democratic regression

On the Strong Subregularity of the Optimality Mapping in an Optimal Control Problem with Pointwise Inequality Control Constraints

This paper presents sufficient conditions for strong metric subregularity (SMsR) of the optimality mapping associated with the local Pontryagin maximum principle for Mayer-type optimal control problems with pointwise control constraints given by a finite number of inequalities G_j(u)le 0. It is assumed that all data are twice smooth, and that at each feasible point the gradients G_j'(u) of the active constraints are linearly independent. The main result is that the second-order sufficient optimality condition for a weak local minimum is also sufficient for a version of the SMSR property, which involves two norms in the control space in order to deal with the so-called two-norm-discrepancy.