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3f978da85b2740be5b99dc1d020e468a7b671d90 | subsection | 63 | 64 | Colimits | Now the preimages of \varphi and
\varphi ^{\prime } will be sent to the same compatible family for R \cap R^{\prime }
and thus \varphi = \varphi ^{\prime }.Proposition 50
The category has all (small) colimits.Given a diagram of spaces F A ,
let F A Setop be the underlying
diagram of presheaves. We can compute
the coli... | {
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36ac6dbaebc5de34957c47d3ba817687c68f974d | abstract | 0 | 39 | Abstract | The continuous-discrete filtering problem requires the solution of a partial
differential equation known as the Fokker-Planck-Kolmogorov forward equation
(FPKfe). In this paper, it is pointed out that for a state model with an
affine, linear drift and state-independent diffusion matrix the fundamental
solution can be o... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
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] | [
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a2fe2c10880b698fc3104d97bb3d54ec2c8cdb9b | subsection | 1 | 39 | Introduction | The problem of continuous-discrete (continuous-continuous) filtering is to estimate the state that is described by a continuous-time stochastic process from the observations of a related discrete-time (continuous-time) stochastic process called the measurement process. The complete solution of the filtering problem, in... | {
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State Model with State-Independent Diffusion Matrix | [
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e4de18283b3bc98bec06150d11d83992de43160e | subsection | 2 | 39 | Introduction | Note that the measurement model in the continuous-discrete Yau case can be arbitrary, and so a more general filtering problem is solved than for the continuous-continuous Yau case. In a following paper, it is shown that the fundamental solution of the FPKfe for another special case of the Yau filter, namely one with a ... | {
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a42611274eebc61bc34ff7a6e0ee7074ff2536e3 | subsection | 3 | 39 | Langevin Equation, the FPKfe and its Fundamental Solution | The general continuous-time state model is described by the following stochastic differential equation (SDE):d{\mathsf {x}}(t)=f({\mathsf {x}}(t),t)dt+e({\mathsf {x}}(t),t)d{\mathsf {v}}(t),\qquad {\mathsf {x}}(t_0)=x_0.Here {\mathsf {x}}(t) and f({\mathsf {x}}(t), t) are n-dimensional column vectors, the diffusion vie... | {
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State Model with State-Independent Diffusion Matrix | [
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a72872b06db4f26557f2371eb56425f318873859 | subsection | 4 | 39 | Langevin Equation, the FPKfe and its Fundamental Solution | In this instance, the physical interpretation is that it is the transition probability density.From the fundamental solution one can compute the probability at a later time for an arbitrary initial condition as followsIn this paper, all integrals are assumed to be from -\infty to +\infty , unless otherwise specified.:p... | {
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State Model with State-Independent Diffusion Matrix | [
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134fbfbfcab17f6c9707b8bcc0d47a6a03fb908e | subsection | 5 | 39 | Continuous-Discrete Filtering | In ths paper, it is assumed that the measurement model is described by the following discrete-time stochastic process{\mathsf {y}}(t_k)=h({\mathsf {x}}(t_k),t_k,{\mathsf {w}}(t_k)),\qquad k=1,2,\ldots ,\qquad t_k>t_0,where y(t)\in \mathbb {R}^{m\times 1}, h\in \mathbb {R}^{m\times 1}, and the noise process {\mathsf {w}... | {
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ba6c21f8c57a48bca5e70f0eb8beab2433f20400 | subsection | 6 | 39 | Fundamental Solution I: Additive Noise | In this section, the following general affine, linear state model with additive noise is considered:d{\mathsf {x}}(t)=( F(t){\mathsf {x}}(t)+ l(t))dt+e(t)d{\mathsf {v}}_i(t),\qquad i=1,\ldots , n.It is assumed that F(t) commutes at different times, i.e.,\left[ F(t),F(t^{\prime }) \right]&\equiv F(t)F(t^{\prime })-F(t^{... | {
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68c628ded61d1320c17a3f38bc41d5230f8009cd | subsection | 7 | 39 | Fundamental Solution I: Additive Noise | From the Leibniz rule it therefore follows that\frac{d{\mathsf {x}}}{dx}(t)&=F(t)U(t,t_0){\mathsf {x}}(t_0)+\int _{t_0}^tF(t)U(t,\tau )\left[ l(\tau )+e(\tau ){\mathsf {\nu }}(\tau ) \right]d\tau ++U(t,t)(l(t)+e(t){\mathsf {\nu }}(t)),\\ &=F(t)\left[ U(t,t_0){\mathsf {x}}(t_0)+\int _{t_0}^tU(t,\tau )\left[ l(\tau )+e(\... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
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d177abf9e52bdeb8c0bed2e2b53ecdeddc1a423e | subsection | 8 | 39 | Fundamental Solution I: Additive Noise | Therefore, it is completely characterized by the mean vector and covariance matrix.The mean vector is\mu (t,t_0)&=\left\langle {\mathsf {x}}(t)\right\rangle ,\\ &=U(t,t_0)x(t_0)+\tilde{l}(t),\qquad \tilde{l}(t)\equiv \int _{t_0}^tU(t,t_0)l(t)dt,and the covariance matrix is\Sigma (t,t_0)&=\left\langle \left[ {\mathsf {x... | {
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State Model with State-Independent Diffusion Matrix | [
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02c1c92ec9652d88775d369d820af569aa25a9c0 | subsection | 9 | 39 | Fundamental Solution I: Additive Noise | This result will be used in the following section. | {
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State Model with State-Independent Diffusion Matrix | [
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fd56cbb58272842e55824c2022322f73540907cc | subsection | 10 | 39 | Fundamental Solution II: State-Independent Diffusion Matrix | In Section , the fundamental solution was derived for the time-independent affine, linear state model with additive noise. In this section, it is pointed out that a similar result follows if the noise is multiplicative but with state dependent diffusion matrix. This is a straightforward generalization of the result der... | {
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1bfdaa60e2feddd3d8a63c31416310d6b3370807 | subsection | 11 | 39 | Fundamental Solution II: State-Independent Diffusion Matrix | This model is also more general than the Yau filter case where no explicit time dependence is assumed.The FPKfe for the transition probability density P(t,x|t^{\prime },x^{\prime }) is{\left\lbrace \begin{aligned}\frac{\partial P}{\partial t}(t,x|t_0,x_0)&=-\sum _{i,j=1}^n\frac{\partial }{\partial x_i}\left(\left( F_{i... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
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a69bbd1b1188065b85fada67ab87076e803c4785 | subsection | 12 | 39 | Fundamental Solution II: State-Independent Diffusion Matrix | Note that if there is no explicit time dependence, the argument of \mu and \Sigma is the difference between the times, i.e., t-t_0.Therefore\left( -i\sum _{i=1}^nk_i\dot{\mu }_i-\frac{1}{2}\sum _{i,j=1}^nk_ik_j\dot{\Sigma }_{ij}+i\sum _{i,j=1}^nF_{ij}(t)k_i\mu _j+\sum _{i=1}^nl_ik_i+\sum _{i,j,l=1}^nF_{ij}k_i\Sigma _{j... | {
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State Model with State-Independent Diffusion Matrix | [
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36c9877ae3a4a4a0acd5b52325038a41fc60f990 | subsection | 13 | 39 | Practical Implementation | In this section, some practical implementational aspects are discussed. Specifically, a computationally efficient filtering algorithm based the results derived in the previous sections is presented in Section REF . Some additional aspects are discussed in Section REF .It is important to note that the transition probabi... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
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0fb06c5a155cce8273345fd0a12cc3b3f72b73ee | subsection | 14 | 39 | Sparse Kernel Grid Filtering Algorithm | The implementation of the continous-discrete filtering solution exploiting the sparsity property of the transtition probability density, or kernel, is thus straightforward.Precompute the transition probability density is given by (t^{\prime \prime }>t^{\prime })
&P(t^{\prime \prime },x^{\prime \prime }|t^{\prime },x^{... | {
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State Model with State-Independent Diffusion Matrix | [
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faf9170c73f622966150ea9a489814e03c2d145a | subsection | 15 | 39 | Sparse Kernel Grid Filtering Algorithm | Then, at observation time t_k, the conditional density is given byp(t_k,x|Y(t_k))=\frac{p(y(t_k)|x)p(t_k,x|Y(t_{k-1}))}{\int p(y(t_k)|\xi )p(t_k,\xi |Y(t_{k-1}))\left\lbrace d^n\xi \right\rbrace },where p(y(t_k)|x) is given byp(y(t_k)|x)=\frac{1}{\left( (2\pi )^m\det R(t_k) \right)^{1/2}}\exp \left\lbrace -\frac{1}{2}(... | {
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6fb86147c0b469109b1b3a7fa9f707cde0940360 | subsection | 16 | 39 | Additional Remarks | Observe also that the fundamental solution has a simple and clear physical interpretation. Specifically, when the signal model noise is small the transition probability is significant only near trajectories satisfying the noiseless equation. The noise variance quantifies the extent to which the state may deviate from t... | {
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89dba2c48af561aafb6ec6c53d86d588066d4232 | subsection | 17 | 39 | Additional Remarks | Note that in the case of the PDEs, even when the exact solution is known, it often does not yield accurate numerical solutions. | {
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8239677ff37fa9ad6407155f9275cc955fa175c1 | subsection | 18 | 39 | Discrete-Discrete Filtering | In Section , the continuous-time state process has been converted to a discrete-time state sequence. This result is simple and exact because the model is linear. This is not possible for a general nonlinear state model, i.e., there is no such general formula relating the continuous-time process with its equivalent disc... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.05338045582175255,
0.0026552884373813868,
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0.05936248600482941,
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-0.02211214415729046,
-0.0038265150506049395,
... | |
ed4e35b4ddd90edbd4de0abaa0aafafd66e15bba | subsection | 19 | 39 | Discrete-Discrete Filtering | Thus, if p(x_{k-1}|y_{1:k-1}) is known (note p(x_0|y_0)=p(x_0) is the prior, or initial pdf), then{\left\lbrace \begin{aligned}p(x_k|y_{1:k-1})&=\int p(x_k|x_{k-1})p(x_{k-1}|y_{1:k-1})dx_{1:k-1},\qquad \text{(Prediction)}\\
p(x_k|y_{1:k})&=\frac{p(y_k|x_k)p(x_k|y_{1:k-1})}{\int p(y_k|x_k)p(x_k|y_{1:k-1})dx_k },\qquad \... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.030048539862036705,
-0.010140810161828995,
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0.01791619509458542,
0.06482793390750885,
0.049933381378650665,
0.021075183525681496,
0.02705742046236992,
-0.05677022412419319,
-0.03482517600059509,
0.009019140154123306,
-... | |
9d70ac3ea7445c2c0ee75e03f60593dfd1b6c0cb | subsection | 20 | 39 | Sparse Kernel Grid Filtering | It is usually stated that grid-based methods are computationally prohibitive when dealing with high-dimensional spaces. However, the transition probability density tensor is sparse, with the sparsity determined by the grid spacing, grid size and signal model noise. Likewise, the correction due to measurements is going ... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.08442673087120056,
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0.009691065177321434,
0.019107423722743988,
0.007283560466021299,
0.041358720511198044,
-0.07191228121519089,
-0.004631871357560158,
0.002874888712540269... | |
abdfabf64ecc5018aecb59822e838c7fd726ebcb | subsection | 21 | 39 | Some Remarks on Particle Filtering | An alternative to grid based techniques is particle filtering (see, for instance, ). A recursive Bayesian filter is obtained by Monte Carlo simulations. The idea is to represent the required conditional probability density by a set of random samples with associated weights and then to compute estimates based on these s... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 84,
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"raw": "M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking,” Signal Processing, IEEE Transactions on [see also Acoustics,... | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.013135870918631554,
-0.00047438268666155636,
-0.03725644201040268,
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0.0048553901724517345,
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0.01111438125371933,
0.039972104132175446,
-0.04711215943098068,
0.00017783390649128705,
-0.01087790448218584,... | |
741111a7a72d1372ba7b79b10974894acd5153d4 | subsection | 22 | 39 | Example | In the examples, use is made of the Tensor toolbox in MATLAB developed by Bader and Kolda . It has the multininear sparse tensor class, essential for SGF.
[Figure: A sample of a the {\mathsf {x}}_1 state process, its conditional mean \langle x_1\rangle and standard deviation \sigma _1.]Consider the 2D coordinated turn ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 91,
"openalex_id": "",
"raw": "B. W. Bader and T. G. Kolda, “Efficient MATLAB computations with sparse and factored tensors,” Tech. Rep. SAND2006-7592, Dec. 2006, see also http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox.",
... | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.06256760656833649,
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0.008835767395794392,
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0.026110531762242317,
0.012078600935637951,
0.... | |
8cfae1fd5633810e5d91535beb41e4c2026152cd | subsection | 23 | 39 | Conclusion | In this paper it has been shown that the continuous-discrete filtering problem with an affine, linear state model and with state independent diffusion matrix can be solved accurately using the exact fundamental solution of the corresponding FPKfe valid for an arbitrary time step size. Unlike the continuous-continuous c... | {
"cite_spans": [
{
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"doi": "",
"end": 996,
"openalex_id": "",
"raw": "B. Balaji, “Universal nonlinear filtering using path integrals I: The continuous-discrete model with additive noise,” submitted to IEEE Transactions on Aerospace and Electronic Systems, 2006. [Onlin... | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.047440305352211,
-0.04252848029136658,
0.00031437783036381006,
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0.024482859298586845,
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0.045030154287815094,
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0.02237779088318348,
0.07206045091152191,
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-0.014499038457870483,
-0.009564331732690334,
-... | |
3432e089c50a9a7668b713ddff7ae307ebd1a22e | subsection | 24 | 39 | General Time Dependent Case | The state transition matrix is the solution of the following equation:{\left\lbrace \begin{aligned}\frac{dU}{dt}(t,t_0)&=F(t)U(t,t_0),\\
U(t_0,t_0)&=I.
\end{aligned}
\right.}The results derived in Section and are not valid when the time-dependent matrix, F(t), does not commute at different times (note that [F(t),F(t)]=... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.0005815804470330477,
-0.005728090647608042,
-0.0655946433544159,
-0.0017313935095444322,
0.007806525565683842,
0.02808365970849991,
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0.056777503341436386,
0.03859405964612961,
0.02242421545088291,
-0.024209000170230865,
-0.014850322157144547,
-0.002158521441742778,
... | |
b0b68fd34c24bcf362a610b0e14e2c6c1d7c6e89 | subsection | 25 | 39 | General Time Dependent Case | Also, uniqueness of the solution follows because it is the solution of a first-order differential equation with a given initial value.An alternative form is the following:U(t,t_0)=1+\int _{t_0}^tdt_1F(t_1)+\int _{t_0}^tdt_1\int _{t_0}^{t_1}dt_2F(t_1)F(t_2)+\int _{t_0}^tdt_1\int _{t_0}^{t_1}dt_2\int _{t_0}^{t_2}dt_3F(t_... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
0.005907432176172733,
0.007638855371624231,
-0.0317300446331501,
-0.008794408291578293,
0.006731193047016859,
0.0014358607586473227,
0.023294122889637947,
0.044544097036123276,
0.02620779536664486,
0.042194854468107224,
-0.024041609838604927,
0.024209411814808846,
-0.008229980245232582,
0.... | |
7721c33b46db69fb42f047628a46b1cc955216b5 | subsection | 26 | 39 | Application: Maneuvering Target Tracking Signal Models | In this section, we summarize the results for many of the linear models that arise in maneuvering target tracking problems. For a nice, up-to-date review, the reader is referred to . Since the models are not time-independent, we can express results in terms of the time difference, or equivalently, set the initial time ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 182,
"openalex_id": "",
"raw": "X. R. Li and V. P. Jilkov, “A survey of maneuvering target tracking i: dynamic models,” IEEE Transactions on Aerospace and Electronic Systems, vol. 39, pp. 1333–1364, 2003.",
"source_ref_id": ... | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.009572800248861313,
-0.015163926407694817,
-0.039969302713871,
-0.0009830236667767167,
-0.026239389553666115,
-0.023447640240192413,
0.01076272688806057,
0.038901422172784805,
0.02053384855389595,
0.01548429112881422,
-0.017391223460435867,
-0.02219669334590435,
0.012166229076683521,
0.... | |
1042bffaa32286f25d47aaf800394efffb4b9a79 | subsection | 27 | 39 | Nilpotent or Orthogonal Matrix | The matrix exponential function can be explicitly written in a closed form using the power series method for the following two cases:Nilpotent F, i.e., F^r=0 for some positive integer r;
Skew-symmetric F, i.e., F^T=-F.The simplest example is the white noise acceleration model is\begin{bmatrix}
\dot{{\mathsf {x}}}_1(t)... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.0318528451025486,
0.021936967968940735,
-0.004893103614449501,
-0.0019698268733918667,
0.007303424179553986,
-0.03609379008412361,
0.06590243428945541,
0.0790829211473465,
0.0544610433280468,
0.04411802068352699,
-0.007978466339409351,
0.009175999090075493,
0.0030262491200119257,
-0.029... | |
adb89d0beb881f9c504145a951d334305802a167 | subsection | 28 | 39 | Nilpotent or Orthogonal Matrix | In this case,U(t,0)=
\begin{bmatrix}
1&t&0&0&0\\
0&1&0&0&0\\
0&0&1&t&0\\
0&0&0&1&0\\
0&0&0&0&1
\end{bmatrix},\qquad \Sigma (t)-\Sigma (0)=
\begin{bmatrix}
t^3/3&t^2/2&0&0&0\\
t^2/2&t&0&0&0\\
0&0&t^3/2&0&0\\
0&0&0&t&0\\
0&0&0&0&t
\end{bmatrix}.All these examples correspond to the nilpotent case.The coordinated turn stat... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.03830438479781151,
0.045080140233039856,
-0.003496610326692462,
-0.012971604242920876,
0.017168300226330757,
-0.007496824488043785,
0.015855878591537476,
0.040837664157152176,
0.021273430436849594,
0.029102176427841187,
-0.006894026417285204,
0.02711828425526619,
-0.004448497202247381,
... | |
f3a6587ccd13e372973071f0341cce95e9ccf3b7 | subsection | 29 | 39 | Nilpotent or Orthogonal Matrix | The characteristic equation for \Omega is\lambda ^3+\omega ^2\lambda =0,\qquad \omega ^2=\omega _1^2+\omega _2^2+\omega _3^2.This implies that\Omega ^3=-\omega ^2\Omega ,\qquad \text{or }\quad \widehat{\Omega }^3=-\widehat{\Omega },\qquad \text{where }\qquad \widehat{\Omega }=\frac{\Omega }{\omega }.Therefore, all powe... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.025659529492259026,
0.05064782127737999,
-0.01707075722515583,
-0.057848356664180756,
-0.0033542742021381855,
-0.059465426951646805,
-0.013203521259129047,
0.0335007905960083,
0.04155562445521355,
-0.019481105729937553,
-0.01888614520430565,
0.005339378956705332,
0.019633658230304718,
-... | |
11d38c4dc2b585733d50b49f69f6df3522da2a4d | subsection | 30 | 39 | Nilpotent or Orthogonal Matrix | Since\Omega ^2=
\begin{bmatrix}
\omega _1^2-\omega ^2&\omega _1\omega _2&\omega _1\omega _3\\
\omega _2\omega _1&\omega _2^2-\omega ^2&\omega _2\omega _3\\
\omega _1\omega _3&\omega _2\omega _3&\omega _3^2-\omega ^2
\end{bmatrix},we finally obtainU(t,0)= \begin{bmatrix}
\cos \omega t+\widehat{\omega }_1^2(1-\cos \omega... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.002809837693348527,
0.019411344081163406,
-0.041172802448272705,
-0.014459313824772835,
0.010636561550199986,
-0.018144723027944565,
0.0037273745983839035,
0.04108123853802681,
-0.014795045368373394,
0.01532916259020567,
-0.0020372753497213125,
0.003462223568931222,
-0.0020372753497213125... | |
4ed93be32a86b31c79351e0530e44cc85a2a1920 | subsection | 31 | 39 | Other Cases | For the general case, a systematic and elegant method for computing the exponential of an arbitrary matrix is presented in . A particularly attractive feature is that explicit formulas can be systematically and elegantly derived for an (up to) arbitrary 4\times 4 matrix F in terms of F and its eigenvalues. It requires ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 124,
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"raw": "H. W. Cheng and S. S.-T. Yau, “More explicit formulas for the matrix exponential,” Linear Algebra and its Applications, vol. 262, pp. 131–163(33), 1 September 1997. [Online]. Available: http://www.i... | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.02218540571630001,
0.04180743917822838,
-0.009154913015663624,
-0.015616755932569504,
-0.0021075373515486717,
-0.05648581683635712,
0.02517601102590561,
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0.019820386543869972,
-0.046384893357753754,
0.02148352935910225,
-0.020110292360186577,
... | |
152a384f0e4babb923f1990b4672b79383568b82 | subsection | 32 | 39 | Other Cases | Of course, this technique could also have been applied to the orthogonal and nilpotent F cases studied in the previous section.The constant turn model with known turn rate is defined as\begin{bmatrix}
\dot{{\mathsf {x}}}_1(t)\\
\dot{{\mathsf {x}}}_2(t)\\
\dot{{\mathsf {x}}}_3(t)\\
\dot{{\mathsf {x}}}_4(t)
\end{bmatrix}... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.027174659073352814,
0.007804513443261385,
-0.04031187668442726,
0.027144143357872963,
-0.00005930390761932358,
-0.029997406527400017,
0.05016860365867615,
0.07055341452360153,
0.01000167801976204,
0.04714750126004219,
-0.014960558153688908,
-0.005233525298535824,
-0.012351424433290958,
... | |
5efecfd4126be40fbf5e291ce0a5cd5475db3011 | subsection | 33 | 39 | Other Cases | Note that the resulting model is nonlinear:\begin{bmatrix}
\dot{{\mathsf {x}}}_1(t)\\
\dot{{\mathsf {x}}}_2(t)\\
\dot{{\mathsf {x}}}_3(t)\\
\dot{{\mathsf {x}}}_4(t)\\
\dot{{\mathsf {x}}}_5(t)
\end{bmatrix}=
\begin{bmatrix}
0&1&0&0&0\\
0&0&0&-{\mathsf {x}}_5(t)&0\\
0&0&0&1&0\\
0&{\mathsf {x}}_5(t)&0&0&0\\
0&0&0&0&0
\end... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.0336865596473217,
0.013105413876473904,
-0.030360622331500053,
-0.005816576536744833,
-0.0014989603077992797,
0.0522080659866333,
0.025142867118120193,
0.05666298791766167,
-0.011434816755354404,
0.023693490773439407,
0.004542650654911995,
-0.002259882865473628,
-0.0029921995010226965,
... | |
913760f6ea804ff366104e55761d07216c5458f8 | subsection | 34 | 39 | Other Cases | The exponential isFor simplicity, in the remainder of the section, only the expression for U(t,0) will be written down; the expressions for \mu and \Sigma follow from it straightforwardly.U(t,0)&=A_2(t)F^2+A_1(t)F+I_3,\\ &=
\begin{bmatrix}
1&A_1(t)&A_2(t)\\
0&1&A_1(t)-\alpha A_2(t)\\
0&0&\alpha ^2A_2(t)-\alpha A_1(t)+1... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.029472591355443,
0.04053816944360733,
-0.031410977244377136,
-0.0016302646836265922,
0.018193334341049194,
-0.03797400742769241,
0.019078580662608147,
0.05009272322058678,
0.01903279311954975,
0.039133984595537186,
0.004678068216890097,
0.01578180119395256,
-0.016163373365998268,
0.0045... | |
2f362b6fe4672608697a68c52131a11c3cc306ed | subsection | 35 | 39 | Other Cases | In this caseU(t,0)&=\frac{F^2}{\omega ^2}(1-\cos \omega t)+tF\sin \omega t+I_3,\\ &=\begin{bmatrix}
1&t\sin \omega t&\frac{1}{\omega ^2}(1-\cos \omega t)\\
0&\cos \omega t&t\sin \omega t\\
0&-\omega ^2t\sin \omega t&\cos \omega t
\end{bmatrix}.The state model\begin{bmatrix}
\dot{{\mathsf {x}}}_1(t)\\
\dot{{\mathsf {x}}... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
-0.030150678008794785,
0.011886324733495712,
-0.04580583795905113,
-0.02969292365014553,
0.016570664942264557,
-0.0021495327819138765,
0.03497234359383583,
0.053435057401657104,
-0.025832539424300194,
0.010284188203513622,
-0.03573526442050934,
0.0314018689095974,
0.00319473585113883,
-0.0... | |
d2eb2b7eaa8203f6b6d65704009649e994766363 | subsection | 36 | 39 | Other Cases | The exponential of F is given byU(t,0)&=A_2(t)F^2+A_1(t)F+I_3,\\ &=
\begin{bmatrix}
1&A_1(t)-\beta A_2(t)&A_2(t)\\
0&1+\beta ^2A_2(t)-\beta A_1(t)&A_1(t)-(\alpha +\beta )A_2(t)\\
0&0&1-\alpha A_1(t)+\alpha ^2A_2(t)
\end{bmatrix},whereA_1(t)&=\frac{1}{\alpha \beta (\beta -\alpha )}\left[ (\alpha ^2-\beta ^2)+\beta ^2e^{... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
"physics.data-an"
] | 2,008 | en | Physics | [
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13accf408fb96b96a22fe5ee257c82b2809754e3 | subsection | 37 | 39 | Other Cases | Since the eigenvalues are distinct, the characteristic polynomial is the minimal polynomial and one obtainsU(t,0)&=A_2(t)F^2+A_1(t)F+I_3,\\ &=
\begin{bmatrix}
1&A_1(t)&A_2(t)\\
0&1-\beta A_2(t)&-\alpha A_2(t)+A_1(t)\\
0&\alpha \beta A_2(t)-\beta A_1(t)&(\alpha ^2-\beta )A_2(t)-\alpha A_1(t)+1
\end{bmatrix},whereA_1(t)&... | {
"cite_spans": []
} | 0807.1705 | Universal, Continuous-Discrete Nonlinear Yau Filtering I: Affine, Linear
State Model with State-Independent Diffusion Matrix | [
"Bhashyam Balaji"
] | [
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] | 2,008 | en | Physics | [
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9a6f7ccbb5cd01fe02b11a95aee776ffb22054d7 | subsection | 38 | 39 | Other Cases | The expression for U(t,0) is\frac{1}{(\lambda _1-\lambda _2)(\lambda _2-\lambda _3)(\lambda _3-\lambda _1)}&\Big [ (\lambda _2-\lambda _3)(F-\lambda _2I_3)(F-\lambda _3I_3)\\ &\qquad +(\lambda _3-\lambda _1)(F-\lambda _3I_3)(F-\lambda _1I_3)+(\lambda _1-\lambda _2)(F-\lambda _1I_3)(F-\lambda _1I_3) \Big ].The Markov ac... | {
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State Model with State-Independent Diffusion Matrix | [
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1874d7486f9092ca3ce4b8eeaff38ee884f85c8f | abstract | 0 | 12 | Abstract | Prompted by high resolution observations, I propose an explanation for the
40+ year old problem of structure and energy balance in the solar transition
region. The ingredients are simply cross-field diffusion of neutral atoms from
cool threads extending into the corona, and the subsequent excitation,
radiation and ioni... | {
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} | 10.1086/591470 | 0807.1706 | An explanation of the solar transition region | [
"Philip G. Judge"
] | [
"astro-ph"
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65afc3e468c4d6c50da2143348eaf10d58f17ae5 | subsection | 1 | 12 | Introduction | The upper transition region (henceforth, “TR”) - plasma with
electron temperatures in the range 2\times 10^5 T_e 10^6 K,
is adequately described by field-aligned thermal conduction down from
the corona. The lower TR (10^4 K < T_e < 2\times 10^5 K) however,
is not so easily understood , . Models
dominated by field-align... | {
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... | 10.1086/591470 | 0807.1706 | An explanation of the solar transition region | [
"Philip G. Judge"
] | [
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ec54d39708ad6e142e8f346a40b4512266b0a427 | subsection | 2 | 12 | A new scenario | L\alpha network emission, at 03
resolution appears mostly as
threads of relatively uniform
intensity, of 5-10 Mm length and 1Mm diameter (PGV). PGV argued
that “the different appearance the TR has in the quiet Sun
[i.e. network] is suggesting that the bulk of its emission comes from
structures other than the footpoints... | {
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a5630e51c503b062a68141273f3ab8991174c156 | subsection | 3 | 12 | Initial diffusion, relaxation, radiation | Imagine an injection of dense neutral material into the tube footpoint by
some
chromospheric process.
The tube surface acts as a
semi-permeable membrane. Neutral particles travel freely between
collisions, but ions gyrate about magnetic field lines with gyro radii
orders of magnitude smaller than mean free paths
(“mfp”... | {
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... | 10.1086/591470 | 0807.1706 | An explanation of the solar transition region | [
"Philip G. Judge"
] | [
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5bd27335eb67b09393b778b31313bc9c38706d98 | subsection | 4 | 12 | Initial diffusion, relaxation, radiation | The warm
neutrals relax via collisions with
the cool neutrals.The initial electron evolution is largely determined by inelastic
collisions with hydrogen:
each hot electron typically has sufficient energy to excite and ionize
5 neutral hydrogen atoms, which takes
\sim 7\tau _{1\kappa }\sim 0.6s. (Electron-electron colli... | {
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"Philip G. Judge"
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8812f7aef71dfcae4f4afcb785c4ab802c55d396 | subsection | 5 | 12 | A multi fluid calculation | To examine the evolution at later times,
multi-fluid equations for conservation of mass, momentum and energy
were solved as functions of time and distance
x across the field lines following and HLH.
Just electrons, protons and
neutral hydrogen atoms were treated.
Cartesian geometry is used because the diffusion region ... | {
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ddb7d3bb5e3ccd1fb39d99744d3996297f29edac | subsection | 6 | 12 | A multi fluid calculation | Thus, a simple recipe for calculating
cross-field transport via the fluid equations
is to set both F and \frac{\partial p_p}{\partial x}
terms to zero in the proton momentum equation, and modify the d_p
terms to account for the reduced displacements. Field-free
diffusion is described by equations (4.41),
(4.46) and (4.... | {
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"Philip G. Judge"
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025b718f8a9e5b3c1e3dfa1d1bcf9c8902769efc | subsection | 7 | 12 | A multi fluid calculation | I is computed to be just a factor of 10-30 below
observed active network thread intensities, and 3 below
average network intensities.A calculation with twice the coronal density, more
appropriate for active network, yields smaller diffusion speeds and
L\alpha fluxes which are just 1.7 times higher. EUV/X-ray
coronal in... | {
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} | 10.1086/591470 | 0807.1706 | An explanation of the solar transition region | [
"Philip G. Judge"
] | [
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84f5b1d54818db124cfec004cd146e91c1bedc4a | subsection | 8 | 12 | Discussion, speculations | Based upon observations of spicules and other fine, thread-like
structures on the solar disk, it is clear that non-planar thermal interfaces
exist at the base of the corona, and that the morphology of the TR emission
from such interfaces cannot be explained by field-aligned particle transport at
the base of coronal loo... | {
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... | 10.1086/591470 | 0807.1706 | An explanation of the solar transition region | [
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6792bdf709a1cf8a22b5d63e9142c2bca49eff69 | subsection | 9 | 12 | Discussion, speculations | Such
calculations will also show if the emission lines of trace species
(ions of carbon, oxygen etc. in the TR) can be explained.Cool threads are observed in different coronal environments (PGV)-
their intensities appear to vary relatively little compared with the
embedding coronal intensities. This fact is part of Fel... | {
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} | 10.1086/591470 | 0807.1706 | An explanation of the solar transition region | [
"Philip G. Judge"
] | [
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ab4a395de24b9a3105f88128500a65bcea4a2525 | subsection | 10 | 12 | Discussion, speculations | The scenario might
therefore explain most of the observed puzzling facets noted by
Feldman and colleagues, yet still maintain a strong energetic link
between the corona and TR, and thereby resolve a long-standing debate
(see the different perspectives of
and
, for example).To see if the scenario survives scrutiny,
mor... | {
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5ecbe024e692d508c5f29e3e884c6c89c8c2bc7f | subsection | 11 | 12 | Discussion, speculations | Theabscissa is distance x across the field lines, the initial cool fluxtube extends from zero to the dashed line.] | {
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9b063da39bb16b5aeaa7943d3526de09615069dc | abstract | 0 | 6 | Abstract | Backward waves and negative refraction are shown to exist in plasmonic
crystals whose lattice cell size is a very small fraction of the vacuum
wavelength (less than 1/40th in an illustrative example). Such
``quasi-homogeneity'' is important, in particular, for high-resolution imaging.
Real and complex Bloch bands are c... | {
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} | 10.1088/1464-4258/11/11/114025 | 0807.1707 | Quasi-Homogeneous Backward-Wave Plasmonic Structures: Theory and
Accurate Simulation | [
"Igor Tsukerman"
] | [
"physics.optics",
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fcde0b6da6a9d084c5f01df03d49884406ea3bda | subsection | 1 | 6 | Body | Quasi-Homogeneous Backward-Wave Plasmonic Structures:Theory and Accurate SimulationIgor Tsukerman
Department of Electrical and Computer Engineering, The
University of Akron, OH 44325-3904, USAigor@uakron.eduBackward waves and negative refraction are shown to exist in
plasmonic crystals whose lattice cell size is a very... | {
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76bd11f0bd44ddbd54e91ac700a432765b6aaed0 | subsection | 2 | 6 | Body | Quadratic eigenproblems with
respect to the Bloch number usually arise and can be converted to
linear ones by introducing auxiliary unknowns either on the
continuous level (e.g. solving for both fields \mathbf {E},
\mathbf {H} instead of just one) or, alternatively, on the linear
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bd7ea17bb5e09df613fb937930aa2b784bb76460 | subsection | 3 | 6 | Body | The boundary conditions for the
full H-field are “scaled-periodic” due to the Bloch exponential
\exp (\mathrm {i}\mathbf {K} \cdot \mathbf {r}):with similar conditions at the boundaries y
= \pm a/2.Accurate local analytical approximations that FLAME relies on are
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"raw": "I. Tsukerman. Electromagnetic applications of a new finite-difference calculus. IEEE Trans. Magn., 41(7):2206–2225, 2005.",
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"... | 10.1088/1464-4258/11/11/114025 | 0807.1707 | Quasi-Homogeneous Backward-Wave Plasmonic Structures: Theory and
Accurate Simulation | [
"Igor Tsukerman"
] | [
"physics.optics",
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] | 2,008 | en | Physics | [
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be9cdc2c7a3ce10bfe5397e33ea35db614350647 | subsection | 4 | 6 | Body | Note
that this example is not computationally favorable because
the filling factor is high and the gaps between the rods are narrow.
[Figure: A backward-wave mode (TE3, circles, dotted line) in a plasmonic crystal with thelattice cell size as small as \sim \lambda _0 / 40.Parameters: \tilde{\omega }_p = 0.25, r_\mathrm... | {
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"raw": "J. Yuan and Y. Lu. Photonic bandgap calculations with Dirichlet-to-Neumann maps. JOSA A, 23(12):3217–3222, 2006.",
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Accurate Simulation | [
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a068a71a322bb1f902d4ae083f4476432a1a0864 | subsection | 5 | 6 | Body | For this “proof-of-concept” example, losses are
neglected and the Drude-like dielectric function is set as
\epsilon _\mathrm {rod} = 1 - \tilde{\omega }_p^2 / \tilde{\omega }^2,
with \tilde{\omega }_p = \omega _p a / c.
A fragment of the band diagram in the \Gamma X direction for
\tilde{\omega }_p = 0.25 is shown in
Fi... | {
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{... | 10.1088/1464-4258/11/11/114025 | 0807.1707 | Quasi-Homogeneous Backward-Wave Plasmonic Structures: Theory and
Accurate Simulation | [
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6a68e2b6a696e8137330b91fe9a777a0bf46776e | abstract | 0 | 8 | Abstract | Phase separation under directional quenching has been studied in a
Cahn-Hilliard model. In distinct contrast to the disordered patterns which
develop under a homogeneous quench periodic stripe patterns are generated
behind the quench front. Their wavelength is uniquely defined by the velocity
of the quench interface in... | {
"cite_spans": []
} | 10.1103/PhysRevE.79.035302 | 0807.1708 | Formation of regular structures in the process of phase separation | [
"Alexei Krekhov"
] | [
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6633c45938c99efdc3825f0c6292a79aa454640a | subsection | 1 | 8 | Body | Formation of regular structures in the process of phase separationAlexei Krekhov
Physikalisches Institut, Universität Bayreuth,
D-95440 Bayreuth, GermanyPhase separation under directional quenching has been studied in a
Cahn-Hilliard model.
In distinct contrast to the disordered patterns which develop under a
homogeneo... | {
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{
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"doi": "",
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"raw": "J. D. Gunton, M. San Miguel, and P. S. Sahni, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic Press, London, 1983), Vol. 8.",
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4e11353c91a80d9c8c0088585ba9e15a24a15c2e | subsection | 2 | 8 | Body | The homogeneous solution u=0 becomes unstable for \epsilon > 0 against
linear perturbations \sim e^{\sigma t + i q x} with wavenumber
q \in (0, \sqrt{\epsilon }) and growth rate \sigma = q^2(\epsilon - q^2).
The most unstable (fastest growing) mode is characterized by
q_m = \sqrt{\epsilon /2} with \sigma _m = \epsilon ... | {
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33627e56b1351703c281a8222f7a7a645c61c4c5 | subsection | 3 | 8 | Body | The quench interface (referred to as QI in the sequel) at
x_q is moving in the laboratory frame with a velocity v, i.e.,Numerical simulations of the 1d CH model (REF ) with the directional
quenching (REF ) demonstrate that a periodic solution develops
behind the QI in the unstable region.
Typical examples for large and... | {
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c1a41236892e3afcb040a95d2d3bb281026efe28 | subsection | 4 | 8 | Body | (REF ) for
v > v^\star , which do not depend on v, agree perfectly with v^\star
and q^\star given by Eq. (REF ) (Fig. REF ).In the opposite limit v \rightarrow 0 our starting point is a particular
stationary solution of Eq. (REF ) for v = 0 interpolating between
u=0 at x < 0 and u=\sqrt{\epsilon } at x > 0 which is ch... | {
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a0ffcd1bc971d72a31403afb60a36b934943a586 | subsection | 5 | 8 | Body | This gives for u_0=u_{in}(x_0) the following expressionStarting from u=u_0 the outer solution u_{out} will grow until at
x=\lambda _0 (second kink) the maximal possible amplitude
u_{max}=\sqrt{\epsilon } is reached.
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} | 10.1103/PhysRevE.79.035302 | 0807.1708 | Formation of regular structures in the process of phase separation | [
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9b538903163ce2e373fcb8c9502ba637d1553fe1 | subsection | 6 | 8 | Body | Thus the well-known Ginzburg criterion, necessary for the validity of a
mean-field description of a phase separation process , is
satisfied: in fact the dynamics does not depend on the particular choice of
\delta u.For the off-critical quench \langle u \rangle \ne 0 when v < v^\star
always regular stripe patterns with... | {
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4f8226dea77e6bbc6c6315aff67e4f413b2de148 | subsection | 7 | 8 | Body | REF (d)] at intermediate QI velocities,
in analogy to the transition from parallel to perpendicular stripes for
\langle u \rangle = 0.
[Figure: Snapshots of the phase separation in 2d at the time when the QIalmost reaches the left boundary.Straight QI () for \epsilon =1,\langle u \rangle = 0.1, v=0.05 (a).Modulated QI ... | {
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d850d20ed3bbe1599dd40bd0ca2f4a2f1dc1b008 | abstract | 0 | 23 | Abstract | A spin metal-oxide-semiconductor field-effect-transistor (spin MOSFET), which
combines a Schottky-barrier MOSFET with ferromagnetic source and drain
contacts, is a promising device for spintronic logic. Previous simulation
studies predict that this device should display a very high magnetoresistance
(MR) ratio (between... | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
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0b00b7ec823f0b5ce604af3395a07dfeda67853d | subsection | 1 | 23 | INTRODUCTION | In recent years, a vigorous research effort to demonstrate spintronic devices , has been pursued. One of the motivations has been that spin-based devices are identified as one of the most promising alternatives to traditional, charge-based logic devices by the International Technology Roadmap for Semiconductors . Simul... | {
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a1222449b22e4636e7905b0ceb6a3fd0d1f5b6f7 | subsection | 2 | 23 | INTRODUCTION | Together the directions of magnetization and the gate bias determine the current through the transistor.In all types of spin transistors, the key device metric is the magnetoresistance (MR) ratio, i.e., the ratio of currents for parallel and anti-parallel magnetizations. It is a measure of the control of carrier transp... | {
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"Tony Low",
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23c45cc718fe5adc1a35e6111c5c1511766283a4 | subsection | 3 | 23 | INTRODUCTION | The influence of the `magnetically dead layer' is incorporated via a scattering self-energy of interaction of spin of carriers and localized electrons, derived within the self-consistent Born approximation , . We quantify the effects of spin relaxation on the MR ratio of the spin MOSFET.
The rest of this paper is organ... | {
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3843cf4bee4ab25fbaabbce3ef8c686e0ee751f5 | subsection | 4 | 23 | MODEL DESCRIPTION | A schematic drawing of a spin MOSFET is illustrated in Fig.REF . blackWe employed here a double-gate structure with a thin film semiconductor for which the gate control of electrostatics is optimal. The channel is a semiconductor and the source/drain contacts are half-metal ferromagnets (HMF) with magnetization of \mat... | {
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ad77b6c1fdb6f6e78f5c1415ac0948716f7bc605 | subsection | 5 | 23 | MODEL DESCRIPTION | Assuming that the electrostatic potential is separable, i.e., can be represented as V(\mathbf {r})\approx V_{1}(x)+V_{2}(z), we can further reduce our system Eq. (REF ) to a one-dimensional problem It is also theoretically possible to further decouple the Hamiltonian H into two 1D problem when V(\mathbf {r}) can be wri... | {
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1725d0042f7d443c60dad2bb38121a637106d049 | subsection | 6 | 23 | MODEL DESCRIPTION | We will designate the majority spin as `spin up' and the minority spins as `spin down'. k_{i}^{\uparrow }=[2m_{x}(\epsilon -\epsilon _{y}-\epsilon _{j}-E_{i}^{\uparrow })]^{1/2}/\hbar is the wave-vector in the contact i, and the energy of the majority band edge is E_{i}^{\uparrow }. A similar identity holds for k_{i}^{... | {
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} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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4a729d8242ec15eeadd0cdd18b7707a2c346cd8a | subsection | 7 | 23 | MODEL DESCRIPTION | The transverse modes can be summed over and we obtain the aggregated electron correlation function \tilde{G}^{n},\tilde{G}^{n}(\epsilon _{x})=G(\epsilon _{x})\tilde{\Sigma }_{c}^{in}(\epsilon _{x})G(\epsilon _{x})^{\dagger }with the aggregated filling function defined as\tilde{\Sigma }_{c}^{in}(\epsilon _{x})=\sum _{j}... | {
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} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
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] | 2,008 | en | Physics | [
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d1e54e88c293999d8044ff656921b1810f85fd03 | subsection | 8 | 23 | MODEL DESCRIPTION | In the self-consistent Born approximation , one can express the in- and out-scattering self-energy as a function of the electron and hole correlation functions , ,\Sigma ^{in}_{s,ij}(\epsilon )&=& \gamma (\epsilon ) \Phi ^{n}_{ijkl} G^{n}_{kl}(\epsilon ), \\
\Sigma ^{out}_{s,ij}(\epsilon )&=& \gamma (\epsilon ) \Phi ^{... | {
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... | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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b9bd4514f1909b17d1f8d2b872ea232aa3a33bc7 | subsection | 9 | 23 | MODEL DESCRIPTION | Their specific form and derivations are provided in Appendix .In the case of spin=1/2 reservoir and diagonal density matrix
(\Delta =0) the explicit form can be derived\Sigma _{s}^{in}(\epsilon _{x})|_{\epsilon _{y},\epsilon _{j}}&=&
{\gamma (\epsilon _{x})}
\left[
\begin{array}{cc}
F_{u}G^{n}_{\downarrow \downarrow }(... | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
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] | 2,008 | en | Physics | [
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cc2ed22d125c1d30b2149313672c54b2ea7270fc | subsection | 10 | 23 | MODEL DESCRIPTION | This iterative process makes it numerically prohibitive to solving realistic transport problems.blackIn this work, we shall introduce some simplifications to make the numerics more tractable. Firstly, we assumed that the relaxation rate \gamma is energy independent. Under the condition where the impurity spin state is ... | {
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} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
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] | 2,008 | en | Physics | [
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f4e793c0dc686636408e67b72ab61fe5f7e2c86b | subsection | 11 | 23 | MODEL DESCRIPTION | The aggregated electron correlation function can then be computed self-consistently from,\tilde{G}^{n}(\epsilon _{x})=G(\epsilon _{x})\left[\tilde{\Sigma }_{s}^{in}(\epsilon _{x})+\tilde{\Sigma }_{c}^{in}(\epsilon _{x})\right]G(\epsilon _{x})^{\dagger }where the aggregated in-scattering self-energy for a spin \frac{1}{... | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
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] | 2,008 | en | Physics | [
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a1201270ed20e55600b369b8f1db26a8f8d15b47 | subsection | 12 | 23 | Spin-MOSFET: Coherent Regime | We consider a spin MOSFET with a double-gated structure as depicted in Fig.REF with a 3nm thin film Si channel. The channel length is 12nm with gate oxide thickness of 1nm. Due to the strong body confinement, electrons predominantly occupy the doubly degenerate valleys along \mathbf {k}=(0,0,1). Its energy dispersion c... | {
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"do... | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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1ae1bdad2eab48dd27f8a968aa1515897de54eed | subsection | 13 | 23 | Spin-MOSFET: Coherent Regime | These resonance states results in the oscillatory behavior in the derivatives of the potential profile in the anti-parallel configuration. blackIt is also numerically challenging to resolve these states in the energy domain due to the relatively fine linewidth in these strongly localised resonance levels i.e. a numeric... | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
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] | 2,008 | en | Physics | [
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a6e7afbb7d003640be4f41dcdc1984aa08eb69f1 | subsection | 14 | 23 | Spin-MOSFET: Incoherent Regime | Spin exchange scattering processes between the tunneling electron and the localized spin impurities at the HMF/Si interfaces are responsible for the incoherent nature of the electron transport. These localized spin impurities lead to decoherence of the electronic spins states. In our model, we assume that there are ext... | {
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} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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1a5f240fe87612d5c6e76f7b45d46df66b1087c1 | subsection | 15 | 23 | Spin-MOSFET: Incoherent Regime | The spin polarization in the channel decreases with the increase of spin relaxation strength in an approximately linear fashion.Fig.REF b and REF c plots the drain current vs. drain voltages at V_{G}=0.6V for both the parallel and anti-parallel configuration, in the presence of spin relaxation at the HMF/semiconductor ... | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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f1ed4b5d952d57ce6f12bf803350834b59041262 | subsection | 16 | 23 | Influence of Spin relaxation on MR Ratio | In this section, we discuss MR ratio in the presence of spin relaxation. Fig.REF (a) shows the MR ratio versus the drain voltage bias at V_{G}=0.6V in the presence of spin relaxation at the HMF/semiconductor interfaces on both the detector and injector sides, characterized by spin relaxation strength of \gamma a = 0-1e... | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
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] | 2,008 | en | Physics | [
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fac48a7a78d823f705f2b751b57901cfa1117ef9 | subsection | 17 | 23 | Influence of Spin relaxation on MR Ratio | A plausible explanation to why spin relaxation at the detector side produces more minority spin current in anti-parallel case could be understood from Fig.REF (d). When V_{D}<\Delta _{s}-E_{w}, majority carrier undergoing spin relaxation at the detector interface will either be admitted into the drain as a minority spi... | {
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{
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... | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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9c4c9570fd3e2e5a7bcf449078362049028d47c1 | subsection | 18 | 23 | Summary | In this paper, we have simulated the operation of a spin MOSFET using the non-equilibrium Green's function approach. A large spin-splitting energy \Delta _{s} for the HMF contacts is beneficial for achieving a large MR ratio. This is because the spin-splitting energy presents a potential barrier at the drain side in th... | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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a16b5d0c19065878b0176010b25f7c64340af47e | subsection | 19 | 23 | Derivation of Scattering Spin Tensors | In this appendix we provide a simplified derivation and the
explicit form in a particular case of spin=1/2 of the scattering tensors in
Eqs. (REF ).
It follows earlier papers , ,
but is presented here for completeness sake.
In general, the scattering tensor is determined by the Hamiltonian
of interaction with the reser... | {
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Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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b07efa5f870d444db4ebd64e6e6a4525f7cb46d6 | subsection | 20 | 23 | Derivation of Scattering Spin Tensors | The scattering tensor becomes\Phi ^{n}_{ijkl} =
(F_u + F_d) s^z_{lj} s^z_{ik}
+ \Delta s^+_{lj} s^z_{ik}
- \Delta s^z_{lj} s^+_{ik} \\
+ \Delta ^* s^-_{lj} s^z_{ik}
- \Delta ^* s^z_{lj} s^-_{ik}
+ F_d s^+_{lj} s^-_{ik}
+ F_u s^-_{lj} s^+_{ik} ,and similarly for \Phi ^{p}. | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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0f9a3d58f34f44d1299d8bc900d7389a065b9d8c | subsection | 21 | 23 | Electrical Analysis of HMF/FM Contacts in Ohmic Regime | blackThe magnetocurrent ratio is an important device metric which provides a measure of the degree of distinguishability between the parallel and anti-parallel states of the spin transistor through their measured current. The Julliere's description of the magnetocurrent ratio in terms of the available effective tunneli... | {
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Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
] | [
"cond-mat.mes-hall"
] | 2,008 | en | Physics | [
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f6f4252e768ddcb434743c4f89c7021ac82fd30e | subsection | 22 | 23 | Electrical Analysis of HMF/FM Contacts in Ohmic Regime | We can also derive similar expressions for normal ferromagnetic contacts;r_{P}&=&\frac{2r_{\alpha }r_{\beta }}{r_{\tau }}+\frac{(2r_{\gamma }r_{\alpha }+r_{\tau } c)(2r_{\gamma }r_{\beta }+r_{\tau } r_{c})}{r_{\tau }(2r_{\gamma }r_{\beta }+r_{\tau }r_{c})+r_{\tau }(2r_{\gamma }r_{\alpha }+r_{\tau } r_{c})}\\
r_{AP}&=&\... | {
"cite_spans": []
} | 10.1063/1.3013438 | 0807.1709 | Modeling of Spin Metal-Oxide-Semiconductor Field-Effect-Transistor: A
Non-Equilibrium Green's Function Approach with Spin Relaxation | [
"Tony Low",
"Mark S. Lundstrom",
"Dmitri E. Nikonov"
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72f419c885422c44b76680236c4fbd9bb9cd1e01 | abstract | 0 | 8 | Abstract | Jin et al reported that axisymmetric simulations of NS-like objects with
polytropic EOS undergo critical gravitational collapse. As the critical
collapse observed via fine-tuning of the adiabatic index $\Gamma$, they
conjecture that critical phenomena may occur in realistic astrophysical
scenarios. To clarify the impli... | {
"cite_spans": []
} | 0807.1710 | Dynamical Analysis of the Structure of Neutron Star Critical Collapses | [
"M. -B. Wan",
"K. -J. Jin",
"W. -M. Suen"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
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5d89395937610a5a7e2f0e81cac91f414c4e3394 | subsection | 1 | 8 | Body | Dynamical Analysis of the Structure of Neutron Star Critical CollapsesM.-B. Wan^{2}, K.-J. Jin^{2}, W.-M. Suen^{1,2}^1Department of Physics, The University of
Hong Kong, Hong Kong^2McDonnell Center for the Space Sciences,
Department of Physics, Washington University, St. Louis,
Missouri 63130Jin et al reported that axi... | {
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... | 0807.1710 | Dynamical Analysis of the Structure of Neutron Star Critical Collapses | [
"M. -B. Wan",
"K. -J. Jin",
"W. -M. Suen"
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799cf772896746bfe11a7fe7064a5c1dbe8dad50 | subsection | 2 | 8 | Body | Since exact critical
solutions in principle stay on the critical point as time proceeds to infinity, we perform a perturbative mode analysis and
compare the mode frequencies of the critical solution with frequencies of specifically non-radiating modes for TOV configurations
with the same baryonic mass.We next determine... | {
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"M. -B. Wan",
"K. -J. Jin",
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] | [
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b970aa135d2aa5e5bc76d3942c982f5e4479e92c | subsection | 3 | 8 | Body | We note that the exact critical solution is non-radiative in
principle, and therefore, we focus on the l=0,1 modes. Using the following perturbation metric on a TOV background :with a gauge choice that leaves the exterior solution invariantly spherically symmetric, namely G=K\equiv 0, and h_{0}=h_{1}\equiv 0, we obtain... | {
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{
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... | 0807.1710 | Dynamical Analysis of the Structure of Neutron Star Critical Collapses | [
"M. -B. Wan",
"K. -J. Jin",
"W. -M. Suen"
] | [
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47128c8d8b220210c1103b82e49f22a0b0e732f7 | subsection | 4 | 8 | Body | The universality of the critical
solution with respect to a 1-parameter family of initial data is thus considered unclear . Therefore, in this section, we
present a completely different family of initial data where the matter field consists of packets of matter whose densities are
characterized by Gaussian distribution... | {
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{
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"raw": "C. Gundlach, J.M. Martín-García, http://www.livingreviews.org/lrr-2007-5.",
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... | 0807.1710 | Dynamical Analysis of the Structure of Neutron Star Critical Collapses | [
"M. -B. Wan",
"K. -J. Jin",
"W. -M. Suen"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
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b7bc55b15fc55a08d7dbd4e1fbf8d964ce4b045d | subsection | 5 | 8 | Body | This indicates the universality of the NS critical solution
with respect to a 1-parameter family of initial data, and that the NS attraction basin constitutes a universal attractor.
[Figure: NO_CAPTION][Figure: Left legend shows contour densities at which measures are performed.Polar and equatorial proper radii shown o... | {
"cite_spans": []
} | 0807.1710 | Dynamical Analysis of the Structure of Neutron Star Critical Collapses | [
"M. -B. Wan",
"K. -J. Jin",
"W. -M. Suen"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
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a00253048dd70fc219b08ff7b63b7890a36b91d3 | subsection | 6 | 8 | Body | Further, the oscillation
phases of certain evolution variables (as those mentioned in Sec. 3) correlate with each other such that they form closed periodic
orbits in the phase diagrams, similar to the circle in the phase diagram of a simple pendulum. In infinite-dimensional
dynamical systems theory , we note that phase... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 455,
"openalex_id": "",
"raw": "J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).",
"source_ref_id": "d53fa15bedcffe29a41b358a8628fa208b641675",... | 0807.1710 | Dynamical Analysis of the Structure of Neutron Star Critical Collapses | [
"M. -B. Wan",
"K. -J. Jin",
"W. -M. Suen"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
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b3962cc36b5e9b9f4773db6844519bbdcf0d3835 | subsection | 7 | 8 | Body | Young for helpful discussions. Numerical computations are performed on WUGRAV machines and NCSA clusters.
The research is supported by the McDonnell Center for the Space Sciences, Washington University, and the NSF Grant MCA93S025. | {
"cite_spans": []
} | 0807.1710 | Dynamical Analysis of the Structure of Neutron Star Critical Collapses | [
"M. -B. Wan",
"K. -J. Jin",
"W. -M. Suen"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
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12ef29b4fd47112fa56e55ec3e5770e295c1d1d0 | abstract | 0 | 18 | Abstract | In previous work the authors considered the asymmetric simple exclusion
process on the integer lattice in the case of step initial condition, particles
beginning at the positive integers. There it was shown that the probability
distribution for the position of an individual particle is given by an integral
whose integr... | {
"cite_spans": []
} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
"Craig A. Tracy",
"Harold Widom"
] | [
"math.PR",
"math-ph",
"math.MP"
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182d448dbd2da4aa6047a13e7343e720ef06cfaa | subsection | 1 | 18 | Body | August 9, 2008Asymptotics in ASEP with Step Initial ConditionCraig A. TracyDepartment of MathematicsUniversity of CaliforniaDavis, CA 95616, USAemail: tracy@math.ucdavis.eduHarold WidomDepartment of MathematicsUniversity of CaliforniaSanta Cruz, CA 95064, USAemail: widom@ucsc.eduIn previous work the authors considered ... | {
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{
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"doi": "",
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... | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
"Craig A. Tracy",
"Harold Widom"
] | [
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337a53010c3fb08aea6042084ad9f955042205af | subsection | 2 | 18 | Body | For fixed m and fixed x<m we have, as t\rightarrow \infty ,It is clear probabilistically that \mathbb {P}\left(x_m(t)>x\right)=0 for all t when x\ge m: for a particle to be to the right of its initial position all particles to its right would have to move simultaneously to the right, which surely has probability zero. ... | {
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"Harold Widom"
] | [
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