scholarly journals On random quadratic forms: supports of potential local maxima

2018 ◽  
Vol 55 (4) ◽  
pp. 1113-1130 ◽  
Author(s):  
Boris Pittel
Keyword(s):  
Population Genetics ◽  
Quadratic Forms ◽  
Hazard Rate ◽  
High Probability ◽  
Stable Equilibrium ◽  
Broad Class ◽  
Probability Distributions ◽  
Local Equilibrium ◽  
Local Maximum ◽  
Local Maxima

Abstract The selection model in population genetics is a dynamic system on the set of the probability distributions 𝒑=(p1,…,pn) of the alleles A1…,An, with pi(t+1) proportional to pi(t) multiplied by ∑jfi,jpj(t), and fi,j=fj,i interpreted as a fitness of the gene pair (Ai,Aj). It is known that 𝒑̂ is a locally stable equilibrium if and only if 𝒑̂ is a strict local maximum of the quadratic form 𝒑T𝒇𝒑. Usually, there are multiple local maxima and lim𝒑(t) depends on 𝒑(0). To address the question of a typical behavior of {𝒑(t)}, John Kingman considered the case when the fi,j are independent and [0,1]-uniform. He proved that with high probability (w.h.p.) no local maximum may have more than 2.49n1∕2 positive components, and reduced 2.49 to 2.14 for a nonbiological case of exponentials on [0,∞). We show that the constant 2.14 serves a broad class of smooth densities on [0,1] with the increasing hazard rate. As for a lower bound, we prove that w.h.p. for all k≤2n1∕3, there are many k-element subsets of [n] that pass a partial test to be a support of a local maximum. Still, it may well be that w.h.p. the actual supports are much smaller. In that direction, we prove that w.h.p. a support of a local maximum, which does not contain a support of a local equilibrium, is very unlikely to have size exceeding ⅔log2n and, for the uniform fitnesses, there are super-polynomially many potential supports free of local equilibriums of size close to ½log2n.

Energy Harvesting From the Vibrations of Rotating Systems

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Christopher G. Cooley ◽  
Tan Chai
Keyword(s):  
Rotation Speed ◽  
Excitation Frequency ◽  
Average Power ◽  
Local Maximum ◽  
Operating Conditions ◽  
Centripetal Acceleration ◽  
Local Maxima ◽  
Closed Form Solutions ◽  
Energy Harvesters ◽  
Wide Range

This study investigates the vibration of and power harvested by typical electromagnetic and piezoelectric vibration energy harvesters when applied to vibrating host systems that rotate at constant speed. The governing equations for these electromechanically coupled devices are derived using Newtonian mechanics and Kirchhoff's voltage law. The natural frequency for these devices is speed-dependent due to the centripetal acceleration from their constant rotation. Resonance diagrams are used to identify excitation frequencies and speeds where these energy harvesters have large amplitude vibration and power harvested. Closed-form solutions are derived for the steady-state response and power harvested. These devices have multifrequency dynamic response due to the combined vibration and rotation of the host system. Multiple resonances are possible. The average power harvested over one oscillation cycle is calculated for a wide range of operating conditions. Electromagnetic devices have a local maximum in average harvested power that occurs near a specific excitation frequency and rotation speed. Piezoelectric devices, depending on their mechanical damping, can have two local maxima of average power harvested. Although these maxima are sensitive to small changes in the excitation frequency, they are much less sensitive to small changes in rotation speed.

scholarly journals Memory Dynamics in Attractor Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Guoqi Li ◽  
Kiruthika Ramanathan ◽  
Ning Ning ◽  
Luping Shi ◽  
Changyun Wen
Keyword(s):  
Energy Function ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Memory Systems ◽  
Initial Stimulus ◽  
Attractor Network ◽  
Local Maxima ◽  
Attractor Networks ◽  
New Energy

As can be represented by neurons and their synaptic connections, attractor networks are widely believed to underlie biological memory systems and have been used extensively in recent years to model the storage and retrieval process of memory. In this paper, we propose a new energy function, which is nonnegative and attains zero values only at the desired memory patterns. An attractor network is designed based on the proposed energy function. It is shown that the desired memory patterns are stored as the stable equilibrium points of the attractor network. To retrieve a memory pattern, an initial stimulus input is presented to the network, and its states converge to one of stable equilibrium points. Consequently, the existence of the spurious points, that is, local maxima, saddle points, or other local minima which are undesired memory patterns, can be avoided. The simulation results show the effectiveness of the proposed method.

DBN Models for Visual Tracking and Prediction

2007 ◽  
pp. 176-193
Author(s):  
Qian Diao ◽  
Jianye Lu ◽  
Wei Hu ◽  
Yimin Zhang ◽  
Gary Bradski
Keyword(s):  
Visual Tracking ◽  
General Framework ◽  
Broad Class ◽  
Probability Distributions ◽  
Prediction Method ◽  
Dynamic Bayesian Networks ◽  
Time Series Models ◽  
Complex Environments ◽  
Inference Algorithms ◽  
Background Clutter

In a visual tracking task, the object may exhibit rich dynamic behavior in complex environments that can corrupt target observations via background clutter and occlusion. Such dynamics and background induce nonlinear, nonGaussian and multimodal observation densities. These densities are difficult to model with traditional methods such as Kalman filter models (KFMs) due to their Gaussian assumptions. Dynamic Bayesian networks (DBNs) provide a more general framework in which to solve these problems. DBNs generalize KFMs by allowing arbitrary probability distributions, not just (unimodal) linear-Gaussian. Under the DBN umbrella, a broad class of learning and inference algorithms for time-series models can be used in visual tracking. Furthermore, DBNs provide a natural way to combine multiple vision cues. In this chapter, we describe some DBN models for tracking in nonlinear, nonGaussian and multimodal situations, and present a prediction method to assist feature extraction part by making a hypothesis for the new observations.

scholarly journals Noise-Robust Modes of the Retinal Population Code Have the Geometry of “Ridges” and Correspond to Neuronal Communities

2017 ◽  
Vol 29 (12) ◽  
pp. 3119-3180 ◽  
Author(s):  
Adrianna Loback ◽  
Jason Prentice ◽  
Mark Ioffe ◽  
Michael Berry II
Keyword(s):  
Neural Activity ◽  
Ganglion Cells ◽  
Activity Patterns ◽  
Local Maximum ◽  
Spike Count ◽  
Neural Population ◽  
Local Maxima ◽  
Local Probability ◽  
Multiple Data Sets ◽  
Noise Robust

An appealing new principle for neural population codes is that correlations among neurons organize neural activity patterns into a discrete set of clusters, which can each be viewed as a noise-robust population codeword. Previous studies assumed that these codewords corresponded geometrically with local peaks in the probability landscape of neural population responses. Here, we analyze multiple data sets of the responses of approximately 150 retinal ganglion cells and show that local probability peaks are absent under broad, nonrepeated stimulus ensembles, which are characteristic of natural behavior. However, we find that neural activity still forms noise-robust clusters in this regime, albeit clusters with a different geometry. We start by defining a soft local maximum, which is a local probability maximum when constrained to a fixed spike count. Next, we show that soft local maxima are robustly present and can, moreover, be linked across different spike count levels in the probability landscape to form a ridge. We found that these ridges comprise combinations of spiking and silence in the neural population such that all of the spiking neurons are members of the same neuronal community, a notion from network theory. We argue that a neuronal community shares many of the properties of Donald Hebb's classic cell assembly and show that a simple, biologically plausible decoding algorithm can recognize the presence of a specific neuronal community.

scholarly journals TIR-MS: Thermal Infrared Mean-Shift for Robust Pedestrian Head Tracking in Dynamic Target and Background Variations

2019 ◽  
Vol 9 (15) ◽  
pp. 3015 ◽  
Author(s):  
Sungmin Yun ◽  
Sungho Kim
Keyword(s):  
Probability Distributions ◽  
Mean Shift ◽  
Thermal Infrared ◽  
Temperature Data ◽  
Seasonal Effect ◽  
Head Tracking ◽  
Pedestrian Tracking ◽  
Local Maxima ◽  
Dynamic Target ◽  
Feature Values

Thermal infrared (TIR) pedestrian tracking is one of the major issues in computer vision. Mean-shift is a powerful and versatile non-parametric iterative algorithm for finding local maxima in probability distributions. In existing infrared data, and mean-shift-based tracking is generally based on the brightness feature values. Unfortunately, the brightness is distorted by the target and background variations. This paper proposes a novel pedestrian tracking algorithm, thermal infrared mean-shift (TIR-MS), by introducing radiometric temperature data in mean-shift tracking. The thermal brightness image (eight-bits) was distorted by the automatic contrast enhancement of the scene such as hot objects in the background. On the other hand, the temperature data was unaffected directly by the background change, except for variations by the seasonal effect, which is more stable than the brightness. The experimental results showed that the TIR-MS outperformed the original mean-shift-based brightness when tracking a pedestrian head with successive background variations.

Nonlinear Spatial Equilibria and Stability of Cables Under Uni-axial Torque and Thrust

1994 ◽  
Vol 61 (4) ◽  
pp. 879-886 ◽  
Author(s):  
C.-L. Lu ◽  
N. C. Perkins
Keyword(s):  
Closed Form ◽  
Elliptic Integral ◽  
Broad Class ◽  
Local Equilibrium ◽  
Three Dimensional ◽  
Equilibrium States ◽  
Form Complex ◽  
Closed Form Solutions ◽  
The Stability ◽  
Low Tension

Low tension cables subject to torque may form complex three-dimensional (spatial) equilibria. The resulting nonlinear static deformations, which are dominated by cable flexure and torsion, may produce interior loops or kinks that can seriously degrade the performance of the cable. Using Kirchhoffrod assumptions, a theoretical model governing cable flexure and torsion is derived herein and used to analyze (1) globally large equilibrium states, and (2) local equilibrium stability. For the broad class of problems described by pure boundary loading, the equilibrium boundary value problem is integrable and admits closed-form elliptic integral solutions. Attention is focused on the example problem of a cable subject to uni-axial torque and thrust. Closed-form solutions are presented for the complex three-dimensional equilibrium states which, heretofore, were analyzed using purely numerical methods. Moreover, the stability of these equilibrium states is assessed and new and important stability conclusions are drawn.

The Hastings algorithm at fifty

Biometrika ◽  
2019 ◽  
Vol 107 (1) ◽  
pp. 1-23 ◽  
Author(s):  
D B Dunson ◽  
J E Johndrow
Keyword(s):  
Markov Chain ◽  
Likelihood Function ◽  
Marginal Likelihood ◽  
Broad Class ◽  
Probability Distributions ◽  
50Th Anniversary ◽  
Monte Carlo Sampling ◽  
Proposal Distribution ◽  
Target Distribution ◽  
Sampling Algorithms

Summary In a 1970 Biometrika paper, W. K. Hastings developed a broad class of Markov chain algorithms for sampling from probability distributions that are difficult to sample from directly. The algorithm draws a candidate value from a proposal distribution and accepts the candidate with a probability that can be computed using only the unnormalized density of the target distribution, allowing one to sample from distributions known only up to a constant of proportionality. The stationary distribution of the corresponding Markov chain is the target distribution one is attempting to sample from. The Hastings algorithm generalizes the Metropolis algorithm to allow a much broader class of proposal distributions instead of just symmetric cases. An important class of applications for the Hastings algorithm corresponds to sampling from Bayesian posterior distributions, which have densities given by a prior density multiplied by a likelihood function and divided by a normalizing constant equal to the marginal likelihood. The marginal likelihood is typically intractable, presenting a fundamental barrier to implementation in Bayesian statistics. This barrier can be overcome by Markov chain Monte Carlo sampling algorithms. Amazingly, even after 50 years, the majority of algorithms used in practice today involve the Hastings algorithm. This article provides a brief celebration of the continuing impact of this ingenious algorithm on the 50th anniversary of its publication.

scholarly journals Nucleation and growth of sub-3 nm particles in the polluted urban atmosphere of a megacity in China

2016 ◽  
Vol 16 (4) ◽  
pp. 2641-2657 ◽  
Author(s):  
Huan Yu ◽  
Luyu Zhou ◽  
Liang Dai ◽  
Wenchao Shen ◽  
Wei Dai ◽  
...  
Keyword(s):  
Growth Rate ◽  
Particle Size ◽  
Cloud Condensation Nuclei ◽  
Local Maximum ◽  
Particle Growth ◽  
Meteorological Conditions ◽  
Control Measures ◽  
Urban Atmosphere ◽  
Limiting Factors ◽  
Local Maxima

Abstract. Particle size distribution down to 1.4 nm was measured in the urban atmosphere of Nanjing, China, in spring, summer, and winter during 2014–2015. Sub-3 nm particle event, which is equivalent to nucleation event, occurred on 42 out of total 90 observation days, but new particles could grow to cloud condensation nuclei (CCN)-active sizes on only 9 days. In summer, infrequent nucleation was limited by both unfavorable meteorological conditions (high temperature and relative humidity – RH) and reduced anthropogenic precursor availability due to strict emission control measures during the 2014 Youth Olympic Games in Nanjing. The limiting factors for nucleation in winter and spring were meteorological conditions (radiation, temperature, and RH) and condensation sink, but for the further growth of sub-3 nm particles to CCN-active sizes, anthropogenic precursors again became limiting factors. Nucleation events were strong in the polluted urban atmosphere. Initial J1.4 at the onset and peak J1.4 at the noontime could be up to 2.1 × 102 and 2.5 × 103 cm−3 s−1, respectively, during the eight nucleation events selected from different seasons. Time-dependent J1.4 usually showed good linear correlations with a sulfuric acid proxy for every single event (R2 = 0.56–0.86, excluding a day with significant nocturnal nucleation), but the correlation among all eight events deteriorated (R2 =  0.17) due to temperature or season change. We observed that new particle growth rate (GR) did not increase monotonically with particle size, but had a local maximum up to 25 nm h−1 between 1 and 3 nm. The existence of local maxima GR in sub-3 nm size range, though sensitive to measurement uncertainties, gives new insight into cluster dynamics in polluted environments. In this study such growth rate behavior was interpreted as the solvation effect of organic activating vapor in newly formed inorganic nuclei.

Efficient Deal Identification by Constraint Relaxation for Collaborative Decision Making Using Negotiation

2014 ◽  
Vol 18 (4) ◽  
pp. 608-615
Author(s):  
Raiye Hailu ◽  
◽  
Takayuki Ito
Keyword(s):  
Decision Making ◽  
Execution Time ◽  
High Probability ◽  
Current Issue ◽  
Hill Climbing ◽  
Time Cost ◽  
Collaborative Decision Making ◽  
Local Maxima ◽  
Constraint Relaxation ◽  
Collaborative Decision

Negotiation is one means for making decision collaboratively. We propose efficient protocols for identifying deals in such negotiations. Specifically we focus on situations in which the negotiators must agree upon one option from among many. This work proposes solutions to problems faced when automating negotiations over multiple and interdependent issues. When negotiations are over issues that are interdependent, previous and future decisions concerning other issues affect how one decides the current issue. Therefore generally we must deal with all of the issues at the same time. To identify deals for negotiations over multiple and interdependent issues previous work has proposed a bidding based protocol that works well only when there is a high probability that agents in the negotiation have local maxima at similar positions in the contract space. This happens only when the contract space is small and the number of agents in the negotiation is low. Otherwise the protocol fails to identify deals. We propose a multi round bidding approach in which agents submit supersets of their bids from earlier rounds. A superset of a bid is created by relaxing the constraints that it satisfies. We will use the same concept of negotiation using relaxed constraints to extend a Hill Climbing (HC) protocol. HC has a linear execution time cost. Ordinarily it can not be used for complex negotiations. But we modify it so that it is used optimally and efficiently for such negotiations.

Local maxima of stationary processes

1966 ◽  
Vol 62 (2) ◽  
pp. 263-268
Author(s):  
M. R. Leadbetter
Keyword(s):  
Distribution Function ◽  
Stationary Process ◽  
Local Maximum ◽  
Stationary Processes ◽  
Normal Case ◽  
Local Maxima ◽  
Ergodic Process ◽  
Frequency Interpretation

AbstractTwo natural definitions for the distribution function of the height of an ‘arbitrary local maximum’ of a stationary process are given and shown to be equivalent. It is further shown that the distribution function so defined has the correct frequency interpretation, for an ergodic process. Explicit results are obtained in the normal case.

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