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Feng Tian

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Published work

45 published item(s)

preprint2026arXiv

Collective Communication for 100k+ GPUs

The increasing scale of large language models (LLMs) necessitates highly efficient collective communication frameworks, particularly as training workloads extend to hundreds of thousands of GPUs. Traditional communication methods face significant throughput and latency limitations at this scale, hindering both the development and deployment of state-of-the-art models. This paper presents the NCCLX collective communication framework, developed at Meta, engineered to optimize performance across the full LLM lifecycle, from the synchronous demands of large-scale training to the low-latency requirements of inference. The framework is designed to support complex workloads on clusters exceeding 100,000 GPUs, ensuring reliable, high-throughput, and low-latency data exchange. Empirical evaluation on the Llama4 model demonstrates substantial improvements in communication efficiency. This research contributes a robust solution for enabling the next generation of LLMs to operate at unprecedented scales.

preprint2026arXiv

NeuroPlastic: A Plasticity-Modulated Optimizer for Biologically Inspired Learning Dynamics

Optimization algorithms are fundamental to modern deep learning, yet most widely used methods rely on update rules based primarily on local gradient statistics. We introduce NeuroPlastic, a plasticity-modulated optimizer that augments gradient-based updates with an adaptive multi-signal modulation mechanism inspired by multi-factor synaptic plasticity, a concept from neurobiology. NeuroPlastic dynamically scales gradient updates using interacting components that capture gradient, activity-like, and memory-like statistics, forming a lightweight modulation layer compatible with standard deep learning training pipelines. Across image classification benchmarks, NeuroPlastic consistently improves over a controlled gradient-only ablation, with more pronounced gains on the Fashion-MNIST benchmark and in reduced-data regimes. In transfer experiments on CIFAR-10 with ResNet-18, the method remains stable and competitive without retuning. These results suggest that multi-signal plasticity-inspired modulation can provide a useful extension to conventional gradient-driven optimization, particularly when learning signals are limited or noisy, and offer a promising direction for gradient-based methods in deep learning.

preprint2026arXiv

RSATalker: Realistic Socially-Aware Talking Head Generation for Multi-Turn Conversation

Talking head generation is increasingly important in virtual reality (VR), especially for social scenarios involving multi-turn conversation. Existing approaches face notable limitations: mesh-based 3D methods can model dual-person dialogue but lack realistic textures, while large-model-based 2D methods produce natural appearances but incur prohibitive computational costs. Recently, 3D Gaussian Splatting (3DGS) based methods achieve efficient and realistic rendering but remain speaker-only and ignore social relationships. We introduce RSATalker, the first framework that leverages 3DGS for realistic and socially-aware talking head generation with support for multi-turn conversation. Our method first drives mesh-based 3D facial motion from speech, then binds 3D Gaussians to mesh facets to render high-fidelity 2D avatar videos. To capture interpersonal dynamics, we propose a socially-aware module that encodes social relationships, including blood and non-blood as well as equal and unequal, into high-level embeddings through a learnable query mechanism. We design a three-stage training paradigm and construct the RSATalker dataset with speech-mesh-image triplets annotated with social relationships. Extensive experiments demonstrate that RSATalker achieves state-of-the-art performance in both realism and social awareness. The code and dataset will be released.

preprint2026arXiv

Vibrational resonance in a frequency-adaptive learning Duffing system

Vibrational resonance focuses on the resonance behavior of a nonlinear system when it is subjected to both a weak low-frequency characteristic signal and a high-frequency auxiliary signal. A traditional Duffing system has a fixed natural frequency and lacks adaptability to the excitation frequency, resulting in vibrational resonance occurring only in a lower frequency range, which affects the application of vibrational resonance. We propose a frequency-adaptive learning Duffing system to overcome the above problem through a learning rule of the natural frequency. The optimal vibrational resonance performance is demonstrated by examining the influence of auxiliary signal parameters, nonlinear stiffness coefficient and the learning rule on the response. The appearance of vibrational resonance is verified by numerical simulation, approximated theoretical predication and circuit simulation. In addition, the advantages of the proposed frequency-adaptive learning rule are highlighted in vibrational resonance performance by comparing with that of two other commonly used alternatives called Hebbian learning rules. The proposed learning rule makes the system more stable and have a stronger resonance degree. The results provide a useful reference for optimizing nonlinear system response and also for processing a weak characteristic signal through nonlinear resonance methods. These achievements provide a groundbreaking foundation for future applied studies especially in the field of weak and complex signal processing.

preprint2022arXiv

An Infinite Dimensional Analysis of Kernel Principal Components

We study non-linear data-dimension reduction. We are motivated by the classical linear framework of Principal Component Analysis. In nonlinear case, we introduce instead a new kernel-Principal Component Analysis, manifold and feature space transforms. Our results extend earlier work for probabilistic Karhunen-Loève transforms on compression of wavelet images. Our object is algorithms for optimization, selection of efficient bases, or components, which serve to minimize entropy and error; and hence to improve digital representation of images, and hence of optimal storage, and transmission. We prove several new theorems for data-dimension reduction. Moreover, with the use of frames in Hilbert space, and a new Hilbert-Schmidt analysis, we identify when a choice of Gaussian kernel is optimal.

preprint2022arXiv

Relevance between Information scrambling and quantum Darwinism

Quantum system interacting with environment can induce redundant encoding of the information of system into a multipartite environment, which is the essence of quantum Darwinism. At the same time, environment may scramble the initially localized information about the system. We mainly investigate the relevance between information scrambling in environment and the emergence of quantum Darwinism. First, we generally identify that when the system shows a Darwinistic behavior system information that is initially localized in the environment is not scrambled, while when Darwinism disappears scrambling occurs.We then verify our result through a collision model where the system, consisting of one or two qubits, interacts with an ensemble of environmental ancillas.Moreover, dependent on the nature of system-environment interactions, our results also shows that the single qubit and two-qubit systems behave differently for the emergence of QD and the scrambling, but the above relevance between them remains valid.

preprint2021arXiv

"Brilliant AI Doctor" in Rural China: Tensions and Challenges in AI-Powered CDSS Deployment

Artificial intelligence (AI) technology has been increasingly used in the implementation of advanced Clinical Decision Support Systems (CDSS). Research demonstrated the potential usefulness of AI-powered CDSS (AI-CDSS) in clinical decision making scenarios. However, post-adoption user perception and experience remain understudied, especially in developing countries. Through observations and interviews with 22 clinicians from 6 rural clinics in China, this paper reports the various tensions between the design of an AI-CDSS system ("Brilliant Doctor") and the rural clinical context, such as the misalignment with local context and workflow, the technical limitations and usability barriers, as well as issues related to transparency and trustworthiness of AI-CDSS. Despite these tensions, all participants expressed positive attitudes toward the future of AI-CDSS, especially acting as "a doctor's AI assistant" to realize a Human-AI Collaboration future in clinical settings. Finally we draw on our findings to discuss implications for designing AI-CDSS interventions for rural clinical contexts in developing countries.

preprint2021arXiv

CASS: Towards Building a Social-Support Chatbot for Online Health Community

Chatbots systems, despite their popularity in today's HCI and CSCW research, fall short for one of the two reasons: 1) many of the systems use a rule-based dialog flow, thus they can only respond to a limited number of pre-defined inputs with pre-scripted responses; or 2) they are designed with a focus on single-user scenarios, thus it is unclear how these systems may affect other users or the community. In this paper, we develop a generalizable chatbot architecture (CASS) to provide social support for community members in an online health community. The CASS architecture is based on advanced neural network algorithms, thus it can handle new inputs from users and generate a variety of responses to them. CASS is also generalizable as it can be easily migrate to other online communities. With a follow-up field experiment, CASS is proven useful in supporting individual members who seek emotional support. Our work also contributes to fill the research gap on how a chatbot may influence the whole community's engagement.

preprint2021arXiv

Possible Atmospheric Diversity of Low Mass Exoplanets, some Central Aspects

Exoplanetary science continues to excite and surprise with its rich diversity. We discuss here some key aspects potentially influencing the range of exoplanetary terrestrial-type atmospheres which could exist in nature. We are motivated by newly emerging observations, refined approaches to address data degeneracies, improved theories for key processes affecting atmospheric evolution and a new generation of atmospheric models which couple physical processes from the deep interior through to the exosphere and consider the planetary-star system as a whole. Using the Solar System as our guide we first summarize the main processes which sculpt atmospheric evolution then discuss their potential interactions in the context of exoplanetary environments. We summarize key uncertainties and consider a diverse range of atmospheric compositions discussing their potential occurrence in an exoplanetary context.

preprint2021arXiv

TeethTap: Recognizing Discrete Teeth Gestures Using Motion and Acoustic Sensing on an Earpiece

Teeth gestures become an alternative input modality for different situations and accessibility purposes. In this paper, we present TeethTap, a novel eyes-free and hands-free input technique, which can recognize up to 13 discrete teeth tapping gestures. TeethTap adopts a wearable 3D printed earpiece with an IMU sensor and a contact microphone behind both ears, which works in tandem to detect jaw movement and sound data, respectively. TeethTap uses a support vector machine to classify gestures from noise by fusing acoustic and motion data, and implements K-Nearest-Neighbor (KNN) with a Dynamic Time Warping (DTW) distance measurement using motion data for gesture classification. A user study with 11 participants demonstrated that TeethTap could recognize 13 gestures with a real-time classification accuracy of 90.9% in a laboratory environment. We further uncovered the accuracy differences on different teeth gestures when having sensors on single vs. both sides. Moreover, we explored the activation gesture under real-world environments, including eating, speaking, walking and jumping. Based on our findings, we further discussed potential applications and practical challenges of integrating TeethTap into future devices.

preprint2020arXiv

A Novel Software-based Multi-path RDMA Solutionfor Data Center Networks

In this paper we propose Virtuoso, a purely software-based multi-path RDMA solution for data center networks (DCNs) to effectively utilize the rich multi-path topology for load balancing and reliability. As a "middleware" library operating at the user space, Virtuoso employs three innovative mechanisms to achieve its goal. In contrast to existing hardware-based MP-RDMA solution, Virtuoso can be readily deployed in DCNs with existing RDMA NICs. It also decouples path selection and load balancing mechanisms from hardware features, allowing DCN operators and applications to make flexible decisions by employing the best mechanisms (as "plug-in" software library modules) as needed. Our experiments show that Virtuoso is capable of fully utilizing multiple paths with negligible CPU overheads

preprint2020arXiv

Feature Super-Resolution Based Facial Expression Recognition for Multi-scale Low-Resolution Faces

Facial Expressions Recognition(FER) on low-resolution images is necessary for applications like group expression recognition in crowd scenarios(station, classroom etc.). Classifying a small size facial image into the right expression category is still a challenging task. The main cause of this problem is the loss of discriminative feature due to reduced resolution. Super-resolution method is often used to enhance low-resolution images, but the performance on FER task is limited when on images of very low resolution. In this work, inspired by feature super-resolution methods for object detection, we proposed a novel generative adversary network-based feature level super-resolution method for robust facial expression recognition(FSR-FER). In particular, a pre-trained FER model was employed as feature extractor, and a generator network G and a discriminator network D are trained with features extracted from images of low resolution and original high resolution. Generator network G tries to transform features of low-resolution images to more discriminative ones by making them closer to the ones of corresponding high-resolution images. For better classification performance, we also proposed an effective classification-aware loss re-weighting strategy based on the classification probability calculated by a fixed FER model to make our model focus more on samples that are easily misclassified. Experiment results on Real-World Affective Faces (RAF) Database demonstrate that our method achieves satisfying results on various down-sample factors with a single model and has better performance on low-resolution images compared with methods using image super-resolution and expression recognition separately.

preprint2020arXiv

FISHING Net: Future Inference of Semantic Heatmaps In Grids

For autonomous robots to navigate a complex environment, it is crucial to understand the surrounding scene both geometrically and semantically. Modern autonomous robots employ multiple sets of sensors, including lidars, radars, and cameras. Managing the different reference frames and characteristics of the sensors, and merging their observations into a single representation complicates perception. Choosing a single unified representation for all sensors simplifies the task of perception and fusion. In this work, we present an end-to-end pipeline that performs semantic segmentation and short term prediction using a top-down representation. Our approach consists of an ensemble of neural networks which take in sensor data from different sensor modalities and transform them into a single common top-down semantic grid representation. We find this representation favorable as it is agnostic to sensor-specific reference frames and captures both the semantic and geometric information for the surrounding scene. Because the modalities share a single output representation, they can be easily aggregated to produce a fused output. In this work we predict short-term semantic grids but the framework can be extended to other tasks. This approach offers a simple, extensible, end-to-end approach for multi-modal perception and prediction.

preprint2016arXiv

Capture cross sections of 15N(n, γ)16N at astrophysical energies

We have reanalyzed reaction cross sections of 16N on 12C target. The nucleon density distribution of 16N, especially surface density distribution, was extracted using the modified Glauber model. On the basis of dilute surface densities, the discussion of 15N(n, γ)16N reaction was performed within the framework of the direct capture reaction mechanism. The calculations agreed quite well with the experimental data.

preprint2016arXiv

Dynamical properties of endomorphisms, multiresolutions, similarity-and orthogonality relations

We study positive transfer operators $R$ in the setting of general measure spaces $\left(X,\mathscr{B}\right)$. For each $R$, we compute associated path-space probability spaces $\left(Ω,\mathbb{P}\right)$. When the transfer operator $R$ is compatible with an endomorphism in $\left(X,\mathscr{B}\right)$, we get associated multiresolutions for the Hilbert spaces $L^{2}\left(Ω,\mathbb{P}\right)$ where the path-space $Ω$ may then be taken to be a solenoid. Our multiresolutions include both orthogonality relations and self-similarity algorithms for standard wavelets and for generalized wavelet-resolutions. Applications are given to topological dynamics, ergodic theory, and spectral theory, in general; to iterated function systems (IFSs), and to Markov chains in particular.

preprint2016arXiv

Nonuniform sampling, reproducing kernels, and the associated Hilbert spaces

In a general context of positive definite kernels $k$, we develop tools and algorithms for sampling in reproducing kernel Hilbert space $\mathscr{H}$ (RKHS). With reference to these RKHSs, our results allow inference from samples; more precisely, reconstruction of an "entire" (or global) signal, a function $f$ from $\mathscr{H}$, via generalized interpolation of $f$ from partial information obtained from carefully chosen distributions of sample points. We give necessary and sufficient conditions for configurations of point-masses $δ_{x}$ of sample-points $x$ to have finite norm relative to the particular RKHS $\mathscr{H}$ considered. When this is the case, and the kernel $k$ is given, we obtain an induced positive definite kernel $\left\langle δ_{x},δ_{y}\right\rangle _{\mathscr{H}}$. We perform a comparison, and we study when this induced positive definite kernel has $l^{2}$ rows and columns. The latter task is accomplished with the use of certain symmetric pairs of operators in the two Hilbert spaces, $l^{2}$ on one side, and the RKHS $\mathscr{H}$ on the other. A number of applications are given, including to infinite network systems, to graph Laplacians, to resistance metrics, and to sampling of Gaussian fields.

preprint2016arXiv

Positive definite kernels and boundary spaces

We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivated by) results and notions from classical harmonic analysis on the disk. Our positive definite kernels include those defined on infinite discrete sets, for example sets of vertices in electrical networks, or discrete sets which arise from sampling operations performed on positive definite kernels in a continuous setting. Below we give a summary of main conclusions in the paper: Starting with a given positive definite kernel $K$ we make precise generalized boundaries for $K$. They are measure theoretic "boundaries." Using the theory of Gaussian processes, we show that there is always such a generalized boundary for any positive definite kernel.

preprint2016arXiv

Representations of the canonical commutation relations--algebra and the operators of stochastic calculus

We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. And we thus get the operators of Malliavin's calculus of variation from a more algebraic approach than is common. And we obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.

preprint2016arXiv

The MUSCLES Treasury Survey I: Motivation and Overview

Ground- and space-based planet searches employing radial velocity techniques and transit photometry have detected thousands of planet-hosting stars in the Milky Way. The chemistry of these atmospheres is controlled by the shape and absolute flux of the stellar spectral energy distribution, however, flux distributions of relatively inactive low-mass stars are poorly known at present. To better understand exoplanets orbiting low-mass stars, we have executed a panchromatic (X-ray to mid-IR) study of the spectral energy distributions of 11 nearby planet hosting stars, the {\it Measurements of the Ultraviolet Spectral Characteristics of Low-mass Exoplanetary Systems} (MUSCLES) Treasury Survey. The MUSCLES program consists of contemporaneous observations at X-ray, UV, and optical wavelengths. We show that energetic radiation (X-ray and ultraviolet) is present from magnetically active stellar atmospheres at all times for stars as late as M5. Emission line luminosities of \ion{C}{4} and \ion{Mg}{2} are strongly correlated with band-integrated luminosities. We find that while the slope of the spectral energy distribution, FUV/NUV, increases by approximately two orders of magnitude form early K to late M dwarfs ($\approx$~0.01~to~1), the absolute FUV and XUV flux levels at their corresponding habitable zone distances are constant to within factors of a few, spanning the range 10~--~70 erg cm$^{-2}$ s$^{-1}$ in the habitable zone. Despite the lack of strong stellar activity indicators in their optical spectra, several of the M dwarfs in our sample show spectacular flare emission in their UV light curves. Finally, we interpret enhanced $L(line)$/$L_{Bol}$ ratios for \ion{C}{4} and \ion{N}{5} as tentative observational evidence for the interaction of planets with large planetary mass-to-orbital distance ratios ($M_{plan}$/$a_{plan}$) with the transition regions of their host stars.

preprint2016arXiv

The MUSCLES Treasury Survey III: X-ray to Infrared Spectra of 11 M and K Stars Hosting Planets

We present a catalog of panchromatic spectral energy distributions (SEDs) for 7 M and 4 K dwarf stars that span X-ray to infrared wavelengths (5 Å - 5.5 μm). These SEDs are composites of Chandra or XMM-Newton data from 5 - ~50 Å, a plasma emission model from ~50 - 100 Å, broadband empirical estimates from 100 - 1170 Å, HST data from 1170 - 5700 Å, including a reconstruction of stellar Lyα emission at 1215.67 Å, and a PHOENIX model spectrum from 5700 - 55000 Å. Using these SEDs, we computed the photodissociation rates of several molecules prevalent in planetary atmospheres when exposed to each star's unattenuated flux ("unshielded" photodissociation rates) and found that rates differ among stars by over an order of magnitude for most molecules. In general, the same spectral regions drive unshielded photodissociations both for the minimally and maximally FUV active stars. However, for ozone visible flux drives dissociation for the M stars whereas NUV flux drives dissociation for the K stars. We also searched for an FUV continuum in the assembled SEDs and detected it in 5/11 stars, where it contributes around 10% of the flux in the range spanned by the continuum bands. An ultraviolet continuum shape is resolved for the star ε Eri that shows an edge likely attributable to Si II recombination. The 11 SEDs presented in this paper, available online through the Mikulski Archive for Space Telescopes, will be valuable for vetting stellar upper-atmosphere emission models and simulating photochemistry in exoplanet atmospheres.

preprint2015arXiv

Characterizing the Habitable Zones of Exoplanetary Systems with a Large Ultraviolet/Visible/Near-IR Space Observatory

Understanding the surface and atmospheric conditions of Earth-size, rocky planets in the habitable zones (HZs) of low-mass stars is currently one of the greatest astronomical endeavors. Knowledge of the planetary effective surface temperature alone is insufficient to accurately interpret biosignature gases when they are observed in the coming decades. The UV stellar spectrum drives and regulates the upper atmospheric heating and chemistry on Earth-like planets, is critical to the definition and interpretation of biosignature gases, and may even produce false-positives in our search for biologic activity. This white paper briefly describes the scientific motivation for panchromatic observations of exoplanetary systems as a whole (star and planet), argues that a future NASA UV/Vis/near-IR space observatory is well-suited to carry out this work, and describes technology development goals that can be achieved in the next decade to support the development of a UV/Vis/near-IR flagship mission in the 2020s.

preprint2015arXiv

Discrete reproducing kernel Hilbert spaces: Sampling and distribution of Dirac-masses

We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs) $\mathscr{H}$ over infinite, discrete and countable sets $V$. In this setting we analyze in detail the distributions of the corresponding Dirac point-masses of $V$. Illustrations include certain models from neural networks: An Extreme Learning Machine (ELM) is a neural network-configuration in which a hidden layer of weights are randomly sampled, and where the object is then to compute resulting output. For RKHSs $\mathscr{H}$ of functions defined on a prescribed countable infinite discrete set $V$, we characterize those which contain the Dirac masses $δ_{x}$ for all points $x$ in $V$. Further examples and applications where this question plays an important role are: (i) discrete Brownian motion-Hilbert spaces, i.e., discrete versions of the Cameron-Martin Hilbert space; (ii) energy-Hilbert spaces corresponding to graph-Laplacians where the set $V$ of vertices is then equipped with a resistance metric; and finally (iii) the study of Gaussian free fields.

preprint2015arXiv

Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space (RKHS) $\mathscr{H}_{F}$. Why extensions? In science, experimentalists frequently gather spectral data in cases when the observed data is limited, for example limited by the precision of instruments; or on account of a variety of other limiting external factors. Given this fact of life, it is both an art and a science to still produce solid conclusions from restricted or limited data. In a general sense, our monograph deals with the mathematics of extending some such given partial data-sets obtained from experiments. More specifically, we are concerned with the problems of extending available partial information, obtained, for example, from sampling. In our case, the limited information is a restriction, and the extension in turn is the full positive definite function (in a dual variable); so an extension if available will be an everywhere defined generating function for the exact probability distribution which reflects the data; if it were fully available. Such extensions of local information (in the form of positive definite functions) will in turn furnish us with spectral information. In this form, the problem becomes an operator extension problem, referring to operators in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we have stressed hands-on-examples. Extensions are almost never unique, and so we deal with both the question of existence, and if there are extensions, how they relate back to the initial completion problem.

preprint2015arXiv

Generalized Gramians: Creating frame vectors in maximal subspaces

A frame is a system of vectors $S$ in Hilbert space $\mathscr{H}$ with properties which allow one to write algorithms for the two operations, analysis and synthesis, relative to $S$, for all vectors in $\mathscr{H}$; expressed in norm-convergent series. Traditionally, frame properties are expressed in terms of an $S$-Gramian, $G_{S}$ (an infinite matrix with entries equal to the inner product of pairs of vectors in $S$); but still with strong restrictions on the given system of vectors in $S$, in order to guarantee frame-bounds. In this paper we remove these restrictions on $G_{S}$, and we obtain instead direct-integral analysis/synthesis formulas. We show that, in spectral subspaces of every finite interval $J$ in the positive half-line, there are associated standard frames, with frame-bounds equal the endpoints of $J$. Applications are given to reproducing kernel Hilbert spaces, and to random fields.

preprint2015arXiv

Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions

We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding approximate solutions to boundary value problems; using multiresolution-subdivision schemes in continuous domains. In this paper, we turn the tables: our object of study is realistic infinite discrete models in their own right; and we then use an analysis of suitable continuous counterpart problems, but now serving as a tool for obtaining solutions in the discrete world.

preprint2015arXiv

Induced representations arising from a character with finite orbit in a semidirect product

Making use of a unified approach to certain classes of induced representations, we establish here a number of detailed spectral theoretic decomposition results. They apply to specific problems from non-commutative harmonic analysis, ergodic theory, and dynamical systems. Our analysis is in the setting of semidirect products, discrete subgroups, and solenoids. Our applications include analysis and ergodic theory of Bratteli diagrams and their compact duals; of wavelet sets, and wavelet representations.

preprint2015arXiv

Infinite networks and variation of conductance functions in discrete Laplacians

For a given infinite connected graph $G=(V,E)$ and an arbitrary but fixed conductance function $c$, we study an associated graph Laplacian $Δ_{c}$; it is a generalized difference operator where the differences are measured across the edges $E$ in $G$; and the conductance function $c$ represents the corresponding coefficients. The graph Laplacian (a key tool in the study of infinite networks) acts in an energy Hilbert space $\mathscr{H}_{E}$ computed from $c$. Using a certain Parseval frame, we study the spectral theoretic properties of graph Laplacians. In fact, for fixed $c$, there are two versions of the graph Laplacian, one defined naturally in the $l^{2}$ space of $V$, and the other in $\mathscr{H}_{E}$. The first is automatically selfadjoint, but the second involves a Krein extension. We prove that, as sets, the two spectra are the same, aside from the point 0. The point zero may be in the spectrum of the second, but not the first. We further study the fine structure of the respective spectra as the conductance function varies; showing now how the spectrum changes subject to variations in the function $c$.

preprint2015arXiv

Infinite weighted graphs with bounded resistance metric

We consider infinite weighted graphs $G$, i.e., sets of vertices $V$, and edges $E$ assumed countable infinite. An assignment of weights is a positive symmetric function $c$ on $E$ (the edge-set), conductance. From this, one naturally defines a reversible Markov process, and a corresponding Laplace operator acting on functions on $V$, voltage distributions. The harmonic functions are of special importance. We establish explicit boundary representations for the harmonic functions on $G$ of finite energy. We compute a resistance metric $d$ from a given conductance function. (The resistance distance $d(x,y)$ between two vertices $x$ and $y$ is the voltage drop from $x$ to $y$, which is induced by the given assignment of resistors when 1 amp is inserted at the vertex $x$, and then extracted again at $y$.) We study the class of models where this resistance metric is bounded. We show that then the finite-energy functions form an algebra of $\frac{1}{2}$-Lipschitz-continuous and bounded functions on $V$, relative to the metric $d$. We further show that, in this case, the metric completion $M$ of $(V,d)$ is automatically compact, and that the vertex-set $V$ is open in $M$. We obtain a Poisson boundary-representation for the harmonic functions of finite energy, and an interpolation formula for every function on $V$ of finite energy. We further compare $M$ to other compactifications; e.g., to certain path-space models.

preprint2015arXiv

Noncommutative analysis, Multivariable spectral theory for operators in Hilbert space, Probability, and Unitary Representations

Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods, discretization), PDEs (diffusion equations, scattering theory), representation theory; iterated function systems (fractals, Julia sets, chaotic dynamical systems), ergodic theory, operator algebras, and many more. And neighboring areas, probability/statistics (for example stochastic processes, Ito and Malliavin calculus), physics (representation of Lie groups, quantum field theory), and spectral theory for Schrödinger operators. We have strived for a more accessible book, and yet aimed squarely at applications; -- we have been serious about motivation: Rather than beginning with the four big theorems in Functional Analysis, our point of departure is an initial choice of topics from applications. And we have aimed for flexibility of use; acknowledging that students and instructors will invariably have a host of diverse goals in teaching beginning analysis courses. And students come to the course with a varied background. Indeed, over the years we found that students have come to the Functional Analysis sequence from other and different areas of math, and even from other departments; and so we have presented the material in a way that minimizes the need for prerequisites. We also found that well motivated students are easily able to fill in what is needed from measure theory, or from a facility with the four big theorems of Functional Analysis. And we found that the approach "learn-by-using" has a comparative advantage.

preprint2015arXiv

Photochemical Escape of Oxygen from Early Mars

Photochemical escape is an important process for oxygen escape from present Mars. In this work, a 1-D Monte-Carlo Model is developed to calculate escape rates of energetic oxygen atoms produced from O2+ dissociative recombination reactions (DR) under 1, 3, 10, and 20 times present solar XUV fluxes. We found that although the overall DR rates increase with solar XUV flux almost linearly, oxygen escape rate increases from 1 to 10 times present solar XUV conditions but decreases when increasing solar XUV flux further. Analysis shows that atomic species in the upper thermosphere of early Mars increases more rapidly than O2+ when increasing XUV fluxes. While the latter is the source of energetic O atoms, the former increases the collision probability and thus decreases the escape probability of energetic O. Our results suggest that photochemical escape be a less important escape mechanism than previously thought for the loss of water and/or CO2 from early Mars.

preprint2015arXiv

Transfer Operators, Induced Probability Spaces, and Random Walk Models

We study a family of discrete-time random-walk models. The starting point is a fixed generalized transfer operator $R$ subject to a set of axioms, and a given endomorphism in a compact Hausdorff space $X$. Our setup includes a host of models from applied dynamical systems, and it leads to general path-space probability realizations of the initial transfer operator. The analytic data in our construction is a pair $\left(h,λ\right)$, where $h$ is an $R$-harmonic function on $X$, and $λ$ is a given positive measure on $X$ subject to a certain invariance condition defined from $R$. With this we show that there are then discrete-time random-walk realizations in explicit path-space models; each associated to a probability measures $\mathbb{P}$ on path-space, in such a way that the initial data allows for spectral characterization: The initial endomorphism in $X$ lifts to an automorphism in path-space with the probability measure $\mathbb{P}$ quasi-invariant with respect to a shift automorphism. The latter takes the form of explicit multi-resolutions in $L^{2}$ of $\mathbb{P}$ in the sense of Lax-Phillips scattering theory.

preprint2014arXiv

Frames and Factorization of Graph Laplacians

Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $\mathscr{H}_{E}$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $\mathscr{H}_{E}$ we characterize the Friedrichs extension of the $\mathscr{H}_{E}$-graph Laplacian. We consider infinite connected network-graphs $G=\left(V,E\right)$, $V$ for vertices, and \emph{E} for edges. To every conductance function $c$ on the edges $E$ of $G$, there is an associated pair $\left(\mathscr{H}_{E},Δ\right)$ where $\mathscr{H}_{E}$ in an energy Hilbert space, and $Δ\left(=Δ_{c}\right)$ is the $c$-Graph Laplacian; both depending on the choice of conductance function $c$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $\mathscr{H}_{E}$ consisting of dipoles. Now $Δ$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}\left(V\right)$ and $\mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}\left(V\right)$-operator, but generally not as an $\mathscr{H}_{E}$ operator. Hence as an $\mathscr{H}_{E}$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^{2}\left(V\right)$.

preprint2014arXiv

Harmonic analysis of a class of reproducing kernel Hilbert spaces arising from groups

We study two extension problems, and their interconnections: (i) extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; and (ii) (in case of Lie groups $G$) representations of the associated Lie algebras $La\left(G\right)$, i.e., representations of $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space $\mathscr{H}_{F}$ (RKHS). Our analysis is non-trivial even if $G=\mathbb{R}^{n}$, and even if $n=1$. If $G=\mathbb{R}^{n}$, (ii), we are concerned with finding systems of strongly commuting selfadjoint operators $\left\{ T_{i}\right\} $ extending a system of commuting Hermitian operators with common dense domain in $\mathscr{H}_{F}$. Specifically, we consider partially defined positive definite (p.d.) continuous functions $F$ on a fixed group. From $F$ we then build a reproducing kernel Hilbert space $\mathscr{H}_{F}$, and the operator extension problem is concerned with operators acting in $\mathscr{H}_{F}$, and with unitary representations of $G$ acting on $\mathscr{H}_{F}$. Our emphasis is on the interplay between the two problems, and on the harmonic analysis of our RKHSs $\mathscr{H}_{F}$.

preprint2014arXiv

Unbounded operators, Lie algebras, and local representations

We prove a number of results on integrability and extendability of Lie algebras of unbounded skew-symmetric operators with common dense domain in Hilbert space. By integrability for a Lie algebra $\mathfrak{g}$, we mean that there is an associated unitary representation $\mathcal{U}$ of the corresponding simply connected Lie group such that $\mathfrak{g}$ is the differential of $\mathcal{U}$. Our results extend earlier integrability results in the literature; and are new even in the case of a single operator. Our applications include a new invariant for certain Riemann surfaces.

preprint2014arXiv

von Neumann indices and classes of positive definite functions

With view to applications, we establish a correspondence between two problems: (i) the problem of finding continuous positive definite extensions of functions $F$ which are defined on open bounded domains $Ω$ in $\mathbb{R}$, on the one hand; and (ii) spectral theory for elliptic differential operators acting on $Ω$, (constant coefficients.) A novelty in our approach is the use of a reproducing kernel Hilbert space $\mathscr{H}_{F}$ computed directly from $\left(Ω,F\right)$, as well as algorithms for computing relevant orthonormal bases in $\mathscr{H}_{F}$.

preprint2013arXiv

High Stellar FUV/NUV Ratio and Oxygen Contents in the Atmospheres of Potentially Habitable Planets

Recent observations of several planet-hosting M dwarfs show that most have FUV/NUV flux ratios 1000 times greater than that of the Sun. Here we show that the atmospheric oxygen contents (O2 and O3) of potentially habitable planets in this type of UV environment could be 2~3 orders of magnitude greater than those of their counterparts around Sun-like stars as a result of decreased photolysis of O3, H2O2, and HO2. Thus detectable levels of atmospheric oxygen, in combination with the existence of H2O and CO2, may not be the most promising biosignatures on planets around stars with high FUV/NUV ratios such as the observed M dwarfs.

preprint2013arXiv

Momentum Operators in The Unit Square

We investigate the skew-adjoint extensions of a partial derivative operator acting in the direction of one of the sides a unit square. We investigate the unitary equivalence of such extensions and the spectra of such extensions. It follows from our results, that such extensions need not have discete spectrum. We apply our techniques to the problem of finding commuting skew-adjoint extensions of the partial derivative operators acting in the directions of the sides of the unit square. While our results are most easily stated for the unit square, they are established for a larger class of domains, including certain fractal domains.

preprint2012arXiv

Design and Implementation of Flight Visual Simulation System

The design requirement for flight visual simulation system is studied and the overall structure and development process are proposed in this paper. Through the construction of 3D scene model library and aircraft model, the rendering and interaction of visual scene are implemented. The changes of aircraft flight attitude in visual system are controlled by real-time calculation of aircraft aerodynamic and dynamic equations and flight simulation effect is enhanced by this kind of control. Several key techniques for optimizing 3D model and relative methods for large terrain modeling are explored for improving loading ability and rendering speed of the system. Experiment shows that, with specific function and performance guaranteed as a premise, the system achieves expected results, that is, precise real-time calculation of flight attitude and smooth realistic screen effect.

preprint2012arXiv

Restrictions and extensions of semibounded operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.

preprint2012arXiv

Spectral Theory of Multiple Intervals

We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space.) The models allow for a scale of complexity, indexed by the natural numbers $\mathbb{N}$. For each $n\in\mathbb{N}$, we get families of selfadjoint operators indexed by: (i) the unitary matrix group U(n), and by (ii) a prescribed set of $n$ non-overlapping intervals. Take $Ω$ to be the complement in $\mathbb{R}$ of $n$ fixed closed finite and disjoint intervals, and let $L^{2}(Ω)$ be the corresponding Hilbert space. Moreover, given $B\in U(n)$, then both the lengths of the respective intervals, and the gaps between them, show up as spectral parameters in our corresponding spectral resolutions within $L^{2}(Ω)$. Our models have two advantages: One, they encompass realistic features from quantum theory, from acoustic wave equations and their obstacle scattering; as well as from harmonic analysis. Secondly, each choice of the parameters in our models, $n\in\mathbb{N}$, $B\in U(n)$, and interval configuration, allows for explicit computations, and even for closed-form formulas: Computation of spectral resolutions, of generalized eigenfunctions in $L^{2}(Ω)$ for the continuous part of spectrum, and for scattering coefficients. Our models further allow us to identify embedded point-spectrum (in the continuum), corresponding, for example, to bound-states in scattering, to trapped states, and to barriers in quantum scattering. The possibilities for the discrete atomic part of spectrum includes both periodic and non-periodic distributions.

preprint2012arXiv

The Ultraviolet Radiation Environment Around M dwarf Exoplanet Host Stars

The spectral and temporal behavior of exoplanet host stars is a critical input to models of the chemistry and evolution of planetary atmospheres. At present, little observational or theoretical basis exists for understanding the ultraviolet spectra of M dwarfs, despite their critical importance to predicting and interpreting the spectra of potentially habitable planets as they are obtained in the coming decades. Using observations from the Hubble Space Telescope, we present a study of the UV radiation fields around nearby M dwarf planet hosts that covers both FUV and NUV wavelengths. The combined FUV+NUV spectra are publically available in machine-readable format. We find that all six exoplanet host stars in our sample (GJ 581, GJ 876, GJ 436, GJ 832, GJ 667C, and GJ 1214) exhibit some level of chromospheric and transition region UV emission. No "UV quiet" M dwarfs are observed. The bright stellar Ly-alpha emission lines are reconstructed, and we find that the Ly-alpha line fluxes comprise ~37-75% of the total 1150-3100A flux from most M dwarfs; > 10^{3} times the solar value. The F(FUV)/F(NUV) flux ratio, a driver for abiotic production of the suggested biomarkers O2 and O3, is shown to be ~0.5-3 for all M dwarfs in our sample, > 10^{3} times the solar ratio. For the four stars with moderate signal-to-noise COS time-resolved spectra, we find UV emission line variability with amplitudes of 50-500% on 10^{2} - 10^{3} s timescales. Finally, we observe relatively bright H2 fluorescent emission from four of the M dwarf exoplanetary systems (GJ 581, GJ 876, GJ 436, and GJ 832). Additional modeling work is needed to differentiate between a stellar photospheric or possible exoplanetary origin for the hot (T(H2) \approx 2000-4000 K) molecular gas observed in these objects.

preprint2012arXiv

Time-resolved Ultraviolet Spectroscopy of the M-dwarf GJ 876 Exoplanetary System

Extrasolar planets orbiting M-stars may represent our best chance to discover habitable worlds in the coming decade. The ultraviolet spectrum incident upon both Earth-like and Jovian planets is critically important for proper modeling of their atmospheric heating and chemistry. In order to provide more realistic inputs for atmospheric models of planets orbiting low-mass stars, we present new near- and far-ultraviolet (NUV and FUV) spectroscopy of the M-dwarf exoplanet host GJ 876 (M4V). Using the COS and STIS spectrographs aboard the Hubble Space Telescope, we have measured the 1150-3140A spectrum of GJ 876. We have reconstructed the stellar HI LyA emission line profile, and find that the integrated LyA flux is roughly equal to the rest of the integrated flux (1150-1210A + 1220-3140A) in the entire ultraviolet bandpass (F(LyA)/F(FUV+NUV) ~0.7). This ratio is ~ 2500x greater than the solar value. We describe the ultraviolet line spectrum and report surprisingly strong fluorescent emission from hot H2 (T(H2) > 2000 K). We show the light-curve of a chromospheric + transition region flare observed in several far-UV emission lines, with flare/quiescent flux ratios >= 10. The strong FUV radiation field of an M-star (and specifically LyA) is important for determining the abundance of O2 -- and the formation of biomarkers -- in the lower atmospheres of Earth-like planets in the habitable zones of low-mass stars.

preprint2012arXiv

Translation Representations and Scattering By Two Intervals

Studying unitary one-parameter groups in Hilbert space (U(t),H), we show that a model for obstacle scattering can be built, up to unitary equivalence, with the use of translation representations for L2-functions in the complement of two finite and disjoint intervals. The model encompasses a family of systems (U (t), H). For each, we obtain a detailed spectral representation, and we compute the scattering operator, and scattering matrix. We illustrate our results in the Lax-Phillips model where (U (t), H) represents an acoustic wave equation in an exterior domain; and in quantum tunneling for dynamics of quantum states.

preprint2011arXiv

Momentum Operators in Two Intervals: Spectra and Phase Transition

We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs. Compared to earlier studies, it yields a more direct link between spectrum and geometry.