Smooth Vector Research Articles

On the generalized moment separability theorem for type 1 solvable Lie groups

Abstract Let G be a type 1 connected and simply connected solvable Lie group. The generalized moment map for π in {\widehat{G}} , the unitary dual of G, sends smooth vectors of the representation space of π to {{\mathcal{U}(\mathfrak{g})}^{*}} , the dual vector space of {\mathcal{U}(\mathfrak{g})} . The convex hull of the image of the generalized moment map for π is called its generalized moment set, denoted by {J(\pi)} . We say that {\widehat{G}} is generalized moment separable when the generalized moment sets differ for any pair of distinct irreducible unitary representations. Our main result in this paper provides a second proof of the generalized moment separability theorem for G.

Strange Attractors Generated by Multiple-Valued Static Memory Cell with Polynomial Approximation of Resonant Tunneling Diodes.

This paper brings analysis of the multiple-valued memory system (MVMS) composed by a pair of the resonant tunneling diodes (RTD). Ampere-voltage characteristic (AVC) of both diodes is approximated in operational voltage range as common in practice: by polynomial scalar function. Mathematical model of MVMS represents autonomous deterministic dynamical system with three degrees of freedom and smooth vector field. Based on the very recent results achieved for piecewise-linear MVMS numerical values of the parameters are calculated such that funnel and double spiral chaotic attractor is observed. Existence of such types of strange attractors is proved both numerically by using concept of the largest Lyapunov exponents (LLE) and experimentally by computer-aided simulation of designed lumped circuit using only commercially available active elements.

Sensor-based reactive navigation in unknown convex sphere worlds

We construct a sensor-based feedback law that provably solves the real-time collision-free robot navigation problem in a compact convex Euclidean subset cluttered with unknown but sufficiently separated and strongly convex obstacles. Our algorithm introduces a novel use of separating hyperplanes for identifying the robot’s local obstacle-free convex neighborhood, affording a reactive (online-computed) continuous and piecewise smooth closed-loop vector field whose smooth flow brings almost all configurations in the robot’s free space to a designated goal location, with the guarantee of no collisions along the way. Specialized attention to planar navigable environments yields a necessary and sufficient condition on convex obstacles for almost-global navigation towards any goal location in the environment. We further extend these provable properties of the planar setting to practically motivated limited range, isotropic and anisotropic sensing models, and the non-holonomically constrained kinematics of the standard differential-drive vehicle. We conclude with numerical and experimental evidence demonstrating the effectiveness of the proposed sensory feedback motion planner.

Transference for Banach space representations of nilpotent Lie groups. Part 1. Irreducible representations

We establish a general CCR (liminarity) property for uniformly bounded irreducible representations of nilpotent Lie groups on reflexive Banach spaces, extending the well-known property of unitary irreducible representations of these groups on Hilbert spaces. We also prove that this conclusion fails for many representations on non-reflexive Banach spaces. Our approach to these results blends the method of transference from abstract harmonic analysis and a systematic use of spaces of smooth vectors with respect to Lie group representations.

Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor

In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.

Smooth cuspidal automorphic forms and integrable discrete series

In this paper we construct smooth cuspidal automorphic forms related to integrable discrete series of a connected semisimple Lie group with finite center for classical and adelic situation as an application of the theory of Schwartz spaces for automorphic forms developed by Casselman. In the classical situation, smooth cuspidal automorphic forms are constructed via an explicit continuous map from the Frechet space of smooth vectors of a Banach realization inside $$L^1(G)$$ of an integrable discrete series into the space of smooth vectors of a strong topological dual of an appropriate Schwartz space.

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

Abstract In this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

Symmetries in 4-dimensional manifolds with metric of neutral signature

This paper studies some general principles of metric (Killing) symmetry on a 4-dimensional, connected manifold M admitting a metric of neutral signature. First a general survey of neutral geometry is presented and this is followed by a discussion of the generalised distribution, integral manifolds and orbits which arise from a general Lie algebra of global smooth vector fields on M . Next, this theory is applied to the finite-dimensional Lie algebra of global Killing vector fields on M and gives a decomposition of M into its various Killing orbits. The zeros of Killing vector fields and the consequent isotropies are then introduced, the latter being described in terms of subalgebras of o ( 2 , 2 ) . The concepts of stability and dimensional stability for orbits are described and theorems are proved which relate the dimension of the Killing algebra to the orbit and isotropy dimensions. Certain special subalgebras are then introduced and related to isotropies in dimensionally stable orbits. Several examples of metrics with symmetries are then presented and which reveal many of the above features. Finally, a theorem describing the consequences for the Ricci and Weyl tensors under Killing isotropy is given in terms of algebraic classifications of these tensors. A table is provided which gives full details of the above results.

Strong comparison principle for 𝑝-harmonic functions in Carnot-Caratheodory spaces

We extend Bony’s propagation of support argument to C 1 C^1 solutions of the nonhomogeneous subelliptic p p -Laplacian associated to a system of smooth vector fields satisfying Hörmander’s finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that generalize results of Tolksdorf.

Computational anatomy and diffeomorphometry: A dynamical systems model of neuroanatomy in the soft condensed matter continuum.

The nonlinear systems models of computational anatomy that have emerged over the past several decades are a synthesis of three significant areas of computational science and biological modeling. First is the algebraic model of biological shape as a Riemannian orbit, a set of objects under diffeomorphic action. Second is the embedding of anatomical shapes into the soft condensed matter physics continuum via the extension of the Euler equations to geodesic, smooth flows with inverses, encoding divergence for the compressibility of atrophy and expansion of growth. Third, is making human shape and form a metrizable space via geodesic connections of coordinate systems. These three themes place our formalism into the modern data science world of personalized medicine supporting inference of high-dimensional anatomical phenotypes for studying neurodegeneration and neurodevelopment. The dynamical systems model of growth and atrophy that emerges is one which is organized in terms of forces, accelerations, velocities, and displacements, with the associated Hamiltonian momentum and the diffeomorphic flow acting as the state, and the smooth vector field the control. The forces that enter the model derive from external measurements through which the dynamical system must flow, and the internal potential energies of structures making up the soft condensed matter. We examine numerous examples on growth and atrophy. This article is categorized under: Analytical and Computational Methods > Computational Methods Laboratory Methods and Technologies > Imaging Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Engineering of a rough auxotrophic mutant Salmonella Typhimurium for effective delivery.

Live Salmonella vaccine vectors offer a remarkable platform for delivering immunogens and therapeutic molecules by mimicking natural intracellular infections; however, pre-existing anti-vector immunity can impede effective deployment. Measures to alleviate pre-existing immunity include the use of heterologous vectors, development of highly attenuated strain enabling greater payload, removal of major immunoreactive components from the vector, and/or augmentation of delivered antigens via increased presentation in antigen presenting cells. Here we report a Salmonella Typhimurium (ST) vector-JOL1800 that embodies these requisite properties. JOL1800 is a highly attenuated, auxotrophic, and O-antigen deficient rough-mutant strain. Heterologous bacterial and viral antigens were expressed and delivered using JOL1800 in mice, irrespective of the inoculation route successful inductions of the mucosal and systemic humoral responses were observed. Compared to smooth LPS vector delivery, we observed an increased fraction of delivered-antigen presenting dendritic cells and a higher frequency of delivered-antigen displayed per macrophage. Upon post-priming with JOL1800 delivery, efficacy of the delivery was minimally affected as indicated by insignificant decrease in colonization, humoral and cellular responses. Our results show that the generated vector is capable of remote antigen delivery, manifests higher antigen presentation, is Differentiating Infected from Vaccinated Animals (DIVA) capable, evades normal pre-existing immunity, and can be deployed for effective delivery.

Fast Diffeomorphic Image Registration via Fourier-Approximated Lie Algebras

This paper introduces Fourier-approximated Lie algebras for shooting (FLASH), a fast geodesic shooting algorithm for diffeomorphic image registration. We approximate the infinite-dimensional Lie algebra of smooth vector fields, i.e., the tangent space at the identity of the diffeomorphism group, with a low-dimensional, bandlimited space. We show that most of the computations for geodesic shooting can be carried out entirely in this low-dimensional space. Our algorithm results in dramatic savings in time and memory over traditional large-deformation diffeomorphic metric mapping algorithms, which require dense spatial discretizations of vector fields. To validate the effectiveness of FLASH, we run pairwise image registration on both 2D synthetic data and real 3D brain images and compare with the state-of-the-art geodesic shooting methods. Experimental results show that our algorithm dramatically reduces the computational cost and memory footprint of diffemorphic image registration with little or no loss of accuracy.

N-D SVPWM With DC Voltage Balancing and Vector Smooth Transition Algorithm for a Cascaded Multilevel Converter

Direct current voltages unbalance of respective power cells is an inherent and critical problem in a cascaded multilevel converter, which is widely used in the solid-state transformer. In this paper, an n -dimension space vector pulse width modulation is presented to address this issue in a three-level H-bridge-based cascaded multilevel converter. This algorithm utilizes the redundant switching pairs of the inner cell and the redundant vectors of mutual cells to balance capacitors’ and cells’ voltages. Meanwhile, a smooth vector transition strategy is included in this algorithm to avoid the level-skip phenomenon. The calculation processes are analyzed in one-cell, two-cells, and three-cells modes, respectively. It is simple, calculable, and flexible to extend in n -cells mode. Finally, the voltage balancing capacity of the inner cell and mutual cells with smooth level transition are verified by simulations and experiments not only at startup but also with unbalanced loads.

Ensembling Downscaling Techniques and Multiple GCMs to Improve Climate Change Predictions in Cryosphere Scarcely-Gauged Catchment

Future projections of climate variables are the key for the development of mitigation and adaptation strategy to changing climate. However, such projections are often subjected to large uncertainties which make implementation of climate change strategies on water resources system a challenging job. Major uncertainty sources are General Circulation models (GCMs), post-processing and climate heterogeneity based on catchment characteristics (e.g. scares data and high-altitude). Here we presents the comparisons between different GCMs, statistical downscaling and bias correction approaches and finally climate projections, with the integration of gridded and converted (monthly to daily) data for a high-altitude, scarcely-gauged Jhelum River basin, Pakistan. Current study relies on climate projections obtained from factorial combination of 5-GCMs, 2 statistical downscaling and 2 bias correction methods. In addition, we applied bias corrected APHRODITE, converted daily data using MODAWEC model and observed data. Further, five GCMs (CGCM3, HadCM3, CCSM3, ECHAM5 and CSIRO-MK3.5) were tested to scrutinize two suitable GCMs integrated with Statistical Downscaling Model (SDSM) and Smooth Support Vector Machine (SSVM). Results illustrate that the CGCM3 and HadCM3 were suitable GCMs for selected study basin. Both downscaling techniques are able to simulate precipitation, however, SSVM performed slightly better than SDSM. We found that the integration of CGCM3 with SSVM (SSVM-CGCM3) generates precipitation and temperature better than the CGCM3 (SDSM-CGCM3) and HadCM3 (SDSM-HadCM3) with SDSM. Furthermore, the low elevation stations were influenced by monsoon, significantly prone to rise in precipitation and temperature, while high-altitude stations were influenced by westerlies circulations, less prone to climate change. The projections indicated rise in basin-wide annual precipitation by 25.51, 36.76 and 45.52 mm and temperature by 0.64, 1.47 and 2.79 °C, during 2030s, 2060s and 2090s, respectively. The methods and results of this study can be adopted to evaluate climate change implications in the catchments of characteristics similar to Jhelum River basin.

A note on second-order Karush–Kuhn–Tucker necessary optimality conditions for smooth vector optimization problems

The aim of this note is to present some second-order Karush–Kuhn–Tucker necessary optimality conditions for vector optimization problems, which modify the incorrect result in ((10), Thm. 3.2).

Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori

Let $P$ be a Laplace type operator acting on a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold given locally by $P= - [g^{\mu\nu} u(x)\partial_\mu\partial_\nu + v^\nu(x)\partial_\nu + w(x)]$ where $u,\,v^\nu,\,w$ are $M_N(\mathbb{C})$-valued functions with $u(x)$ positive and invertible. For any $a \in \Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr} (a e^{-tP}) \underset{t \downarrow 0^+}{\sim} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the coefficients $a_r(a, P)$ can be written locally as $a_r(a, P)(x) = \text{tr}[a(x) \mathcal{R}_r(x)]$. The computation of $\mathcal{R}_2$ is performed opening the opportunity to calculate the modular scalar curvature for noncommutative tori.

A Newton-Like Method for Variable Order Vector Optimization Problems

A Newton approach is proposed for solving variable order smooth constrained vector optimization problems. The concept of strong convexity is presented, and its properties are analyzed. It is thus obtained that the Newton direction is well defined and that the algorithm converges. Moreover, the rate of convergence is obtained under ordering structures satisfying a mild hypothesis.

Smooth invariant densities for random switching on the torus

We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal to each other at all points of the torus and that each of them allows for a smooth invariant density and no periodic orbits, we prove that the switched system also has a smooth invariant density, for every switching rate. Our approach is based on an integration by parts formula inspired by techniques from Malliavin calculus.

Flows of measures generated by vector fields

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

On a non-smooth potential analysis for Hörmander-type operators

We develop a potential theory approach for some degenerate parabolic operators in non-divergence form and with non-smooth coefficients, which are modeled on smooth Hormander vector fields. We prove necessary and sufficient Wiener-type tests for the regularity of boundary points. As a consequence we obtain, in particular, a cone-type criterion. We also investigate the related boundary value problem and the Holder regularity at the boundary.