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# American Institute of Mathematical Sciences

March  2017, 7(1): 53-72. doi: 10.3934/mcrf.2017004

## Control and stabilization of 2 × 2 hyperbolic systems on graphs

 Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, F-59313 -Valenciennes Cedex 9, France

* Corresponding author

Received  June 2015 Revised  February 2016 Published  December 2016

We consider 2× 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at all except one external vertices. This result is obtained with the help of a perturbation argument and an observability estimate for an associated wave type equation. An exact controllability result is also deduced.

Citation: Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004
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Prediction Measures in Beta Regression Models

# Prediction Measures in Beta Regression Models

Patrícia L. Espinheira Luana Cecília Meireles da Silva Alisson de Oliveira Silva Departamento de Estatística, Universidade Federal de Pernambuco, Cidade Universitária, Recife/PE, 50740–540, Brazil
###### Abstract

We consider the issue of constructing PRESS statistics and coefficients of prediction for a class of beta regression models. We aim at displaying measures of predictive power of the model regardless goodness-of-fit. Monte Carlo simulation results on the finite sample behavior of such measures are provided. We also present an application that relates to the distribution of natural gas for home usage in São Paulo, Brazil. Faced with the economic risk of to overestimate or to underestimate the distribution of gas was necessary to construct prediction limits using beta regression models (Espinheira et al., 2014). Thus, it arises the aim of this work, the selection of best predictive model to construct best prediction limits.

###### keywords:
Beta distribution, beta regression, PRESS, prediction coefficient.

, Corresponding author. ,

## 1 Introduction

The beta distribution is commonly used to model random variables that assume values in , such as percentages, rates and proportions. The beta density can display quite different shapes depending on the parameter values. Oftentimes the variable of interest is related to a set of independent (explanatory) variables. Ferrari and Cribari-Neto (2004) introduced a regression model in which the response is beta-distributed, its mean being related to a linear predictor through a link function. The linear predictor includes independent variables and regression parameters. Their model also includes a precision parameter whose reciprocal can be viewed as a dispersion measure. In the standard formulation of the beta regression model it is assumed that the precision is constant across observations. However, in many practical situations this assumption does not hold. Smithson and Verkuilen (2006) consider a beta regression specification in which dispersion is not constant, but is a function of covariates and unknown parameters. Parameter estimation is carried out by maximum likelihood (ML) and standard asymptotic hypothesis testing can be easily performed. Practitioners can use the betareg package, which is available for the R statistical software (http://www.r-project.org), for fitting beta regressions. Cribari-Neto and Zeileis (2010) provide an overview of varying dispersion beta regression modeling using the betareg package.

Recently Espinheira et al. (2014) built and evaluated bootstrap-based prediction intervals for the class of beta regression models with varying dispersion. However, a prior approach it is necessary, namely: the selection of the model with the best predictive ability, regardless of the goodness-of-fit. Indeed, the model selection is a crucial step in data analysis, since all inferential performance is based on the selected model. Bayer and Cribari-Neto (2014) evaluated the performance of different selection criteria models in samples of finite size in beta regression model, such as Akaike Information Criterion (AIC) (Akaike, 1973), Schwarz Bayesian Criterion (SBC) (Schwarz, 1978), residual sum of squares (RSS), and various functions of RSS such as the coefficient of determination, and the adjusted . However, these methods do not offer any insight about the quality of the predictive values. In this context, Allen (1974), proposed the PRESS (Predictive Residual Sum of Squares) criterion, that can be used as an indication of the predictive power of a model. The PRESS statistic is independent from the goodness-of-fit of the model, since that its calculation is made by leaving out the observations that the model is trying to predict. The PRESS statistics can be viewed as a sum of squares of external residuals. Thus, similarly of the approach of Mediavilla et al. (2008) proposed a coefficient of prediction based on PRESS namely . The statistic can be used to select models from a predictive perspective adding important information about the predictive ability of the model in various scenarios.

## 2 On beta regression residuals

Let be independent random variables such that each , for , is beta distributed, i.e., each has density function given by

 f(yt;μt,ϕt)=Γ(ϕt)Γ(μtϕt)Γ((1−μt)ϕt)yμtϕt−1t(1−yt)(1−μt)ϕt−1,0

where and . Here, and , where . In the beta regression model introduced by Ferrari and Cribari-Neto (2004) the mean of can be written as

 g(μt)=x⊤tβ=ηt. (2)

In addition to the relation given in (2), it is possible to assume that the precision parameter is not constant and write

 h(ϕt)=z⊤tγ=ϑt. (3)

In (2) and (3), and are linear predictors, and are unknown parameter vectors (; ), and are fixed covariates () and and are link functions, which are strictly increasing and twice-differentiable.

The PRESS statistic is based on sum of external residuals obtained from exclusion of observations. For beta regression models Ferrari et al. (2011) present a standardized residual obtained using Fisher’s scoring iterative algorithm for under varying dispersion. Here, we propose a new residual based on a combination of ordinary residuals obtained using the algorithms for and under varying dispersion. At the outset, consider the Fisher’s scoring iterative algorithm for estimating (see the Appendix A). From (21) it follows that the th step of the scoring scheme is

 β(m+1)=β(m)+(X⊤Φ(m)W(m)X)−1Φ(m)X⊤T(m)(y∗−μ∗(m)), (4)

where the th elements of the vectors and are given, respectively, by

 y∗t=log{yt/(1−yt)}andμ∗t=ψ(μtϕt)−ψ((1−μt)ϕt), (5)

denoting the digamma function, i.e., for . The matrices and are given in (17) and (19), respectively, is an matrix whose th row is and . Note that (see (22); Appendix A). Similarly, from (21) it follows that the th step of the scoring scheme for is given by

 γ(m+1)=γ(m)+(Z⊤D(m)Z)−1Z⊤H(m)a(m), (6)

where the th element of is give by

 at=μt(y∗t−μ∗t)+log(1−yt)−ψ((1−μt)ϕt)+ψ(ϕt) (7)

and the matrices and are given in (18) and (20), respectively, and is an matrix that th row is . It is possible to write the iterative schemes in (4) and (6) in terms of weighted least squares regressions, respectivelly as and . Where , with , , with and given in (7). Upon convergence,

 ˆβ=(X⊤ˆΦˆWX)−1ˆΦX⊤ˆWu1%andˆγ=(Z⊤ˆDZ)−1Z⊤ˆDu2,whereu1=ˆη+ˆW−1ˆT(y∗−ˆμ∗)andu2=ˆϑ+ˆD−1ˆHˆa. (8)

Here, , , and are the matrices , , and respectively, evaluated at the maximum likelihood estimator. We note that and in (8) can be viewed as the least squares estimates of and obtained by regressing and on and , respectively. The residuals ordinary obtained of interactive process of and are given by and , respectively. Hence, using the definitions of the matrices given from (17) to (21), we can rewrite the residuals obtained from the iterative process of and respectively, as

 rβt=y∗t−ˆμ∗t√ˆvtandrγt=ˆat√ˆςt, (9)

where and are given in (19) and (20), respectively. Thus, we propose a new residual based on and , which we shall refer to as the combined residual where and are given in (5). Assuming that and are known and from (22) to (26) it follows that , with

 ζt=(1+μt)2ψ′(μtϕt)+μ2tψ′((1−μt)ϕt)−ψ′(ϕt). (10)

Then, we can define the following standardized combined residual:

 rβγp,t=(y∗t−ˆμ∗t)+ˆat√ˆζt (11)

Here, is in (10) evaluated at e . It is important to note that when is constant it is only necessary replace by at all elements of (11). We should emphasize that here we are just interested in evaluating the in the composition of the PRESS statistic.

## 3 P2 Statistics

Consider the linear model, where is a vector of responses, is a known matrix of covariates of dimension , is the parameter vector of dimension and is a vector of errors distributed as . Let , , and let be the estimate of without the i observation and be the case deleted predicted value of the response when the independent variable has value . Thus, for multiple regression which can be rewritten as , where is the t diagonal element of the matrix .

In the beta regression model in (8) can be viewed as the least squares estimate of obtained by regressing

 ˇy=ˆΦ1/2ˆW1/2u1onˇX=ˆΦ1/2tˆW1/2X. (12)

Thus, the prediction error is . Using the ideas proposed by (Pregibon, 1981) and fact that

 ˆβ(t)=ˆβ−(X⊤ˆΦˆWX)−1xtˆϕ1/2tˆw1/2trβt(1−h∗tt),

where is given in (9) and is the th diagonal element of

 H∗=(ˆWˆΦ)1/2X(XˆΦˆWX)−1X⊤(ˆΦˆW)1/2

it then follows that Finally, for the beta regression model the PRESS statistic is given by

 PRESS=n∑t=1(ˇyt−^ˇy(t))2=n∑t=1(rβt1−h∗tt)2. (13)

In (13) the th observation is not used in fitting the regression model to predict , then both the external predicted values and the external residuals are independent of . This fact enables the PRESS statistic to be a true assessment of the prediction capabilities of the regression model regardless of the overall quality of the fit of the model.

Considering the same approach of the coefficient of determination , we can think in a prediction coefficient based on PRESS, namely

 P2=1−PRESSSST(t), (14)

wherein and is the arithmetic average of the . It can be shown that , wherein is the number of model parameters. In the beta regression model with varying dispersion, , is the is the arithmetic average of the given in (12) and .

Cook and Weisberg (1982) suggest other versions of PRESS statistics based on different residuals. Thus, we present another version of PRESS statistics and associated considering a new residual presented in (11), such that

 PRESSβγ=n∑t=1⎛⎝rβγp,t1−h∗tt⎞⎠2andP2βγ=1−PRESSβγSST(t), (15)

respectively. It is noteworthy that the measures and are distinct, since that the propose to measure the quality of fit of the model and the and measure the predictive power. Additionally, and are not positive measure. In fact, the is a positive quantity, thus the and the associated given in (14) and (15), respectively, take values in . The closer to one the better is the predictive power of the model. In order to check the goodness-of-fit of the estimated model, we used the approach suggested by Bayer and Cribari-Neto (2014) for beta regression models with varying dispersion, a version of based on likelihood ratio, given by: wherein is the maximum likelihood achievable (saturated model) and is the achieved by the model under investigation.

### 3.1 Monte Carlo results

The Monte Carlo experiments were carried out using using both fixed and varying dispersion beta regressions as data generating processes. All results are based on 10,000 Monte Carlo replications. Table 1 contains numerical results for the fixed dispersion beta regression model as data generating processe, given by

 log(μt1−μt)=β1+β2xt2+β3xt3+β4xt4+β5xt5,t=1,…,n,

The covariate values were independently obtained as random draws of the following distributions: , and were kept fixed throughout the experiment. The precisions, the sample sizes and the mean response are, respectively, , , , and . To investigate the performances of statistics in the omission of covariates, we considered the Scenarios 1, 2 and 3, in which are omitted, three, two and one covariate, respectively. In the fourth scenario the estimated model is correctly specified. Additionally we calculate the for the same scenarios. The results in Table 1 show that the values of all statistics increase as important covariates are included in the model. Statistics behave similarly as the sample size and the precisions values indicating that the most important factor is the correct specification of the model. Considering the three ranges for the it should be noted that the statistic values are considerably larger when and the values approaching one when the estimated model is closest to the true model. For instance, in Scenario 4 for , the values of and are, respectively, (0.8354, 0.9357, 0.9748) and (0.8349, 0.9376, 0.9758).

The statistics finite sample behavior substantially change when . It is noteworthy the reduction of the statistic values, revealing the difficulty in to fit the model and make prediction when . Indeed, in this range of is more difficult to make prediction that to fit the model. For example, in Scenario 1, when three covariates are omitted from the model, when and the values equals, 0.0580, 0.0636 and 0.0972 whereas the the values are 0.1553, 0.1999 and 0.2496, respectively. Similar results were obtained for . Even when for the correctly specified four covariate model (Scenario 4) the predictive power of the model is more affected than the quality of fit of the model by the fact of . In this situation, it is noteworthy that the finite sample performances predictive power model improve when the value of the precision parameter increases. For instance, when and we have and , respectively. Here it is possible see that the statistic always shows larger values than the statistic when the mean responses are close to of the upper limit of the standard unit interval. However, the two measures behave similarly when used to investigate model misspecification.

The same difficulty in obtaining predictions and in fitting the regression model occurs when . Once again the greatest difficulty lies on the predictive power of the model. It is also noteworthy that when the point prediction becomes even less reliable than when , since the and values decreased substantially and become considerably distant from the values. When the mean responses are close to of the lower limit of the standard unit interval, the seems to be more able in identify poor predictions. For instance, in Scenario 4 (model correctly specified; four covariates) when and , we have and , respectively.

We have also carried out Monte Carlo simulations using a varying dispersion beta regression model, in which we increased the number of covariates, used different covariates in the mean and precision submodels. In this case the data generating process and the postulated model is the same . We report results for , , , and . Here,

 λ=ϕmaxϕmin=maxt=1,…,n{ϕt}mint=1,…,n{ϕt}, (16)

is the measure the intensity of nonconstant dispersion. The covariate values in the mean submodel and in the precision submodel were obtained as random draws from the and distributions, respectively, such that the covariate values in the two submodels are not the same. At the end, we also considered a covariate values generated from (Student’s t-distribution with 3 degrees of freedom). The results are presented in Table 2. We should emphasize that were generated only covariates values and the covariates values are replications of original set. In this sense, the intensity of nonconstant dispersion remains the same over the sample size.

When the mean responses are scattered on the standard unit interval () the three statistics display similar values. It seems that neither the degree of intensity of nonconstant dispersion nor the simultaneous increase in the number of covariates in the two submodels noticeably affect the predictive power and fit of the model when the sample size is fixed. However, it is noteworthy a reduction of statistic values when the response values are close to one or close to zero, making clear the difficulty in fitting the regression model and obtaining good predictions when or and the precision is modelled. The minor values of statistic reveals the problem in to make good predictions when , whereas when this problem is singled out by smaller values of statistic. Here, the model fit is more affect when the number of covariates increases simultanealy in the two submodels. For instance consider , and . At the Scenario 5 (one covariate in both submodels), we have = 0.8117, = 0.3677 and = 0.8228. Whereas in Scenario 8 (four covariate in both submodels) we have = 0.8627, = 0.4863 and = 0.6447;

We also displayed in Table 2 the statistic values when the model is correctly specified, but we introduced leverage points in the data. To that end, only the values were obtained as random draws of the distribution and concerned ourselves with , which yielded one point which has leverage measure ten times greater than the average value when , two high leverage points when and three, . Here, we used as measure of leverage the leverage generalized (Espinheira et al., 2008). Notice that in Scenarios 5, 6 and 7 the measure seems more able to identify correctly that the leverage points affect the goodness of prediction than the measure. On the order hand, the outperforms the in Scenario 8. It is interesting to notice that in Scenario 5, which represents one covariate in both submodels, with the only one covariate of mean submodel had values generated from the occurs the smaller values of the three statistics. Thus, the statistics correctly lead to the conclusion that as greatest is the influence of leverage point in the data, worst are the predictions and the model fit.

Finally, were carried out Monte Carlo simulations to assess the performance of statistics when the dispersion modelling is neglected. To that end, the true data generating process considers varying dispersion but a fixed dispersion beta regression is estimated; see Table 3. In this case we have misspecification. Thus, we hope that the statistics display smaller values in comparison with Table 3. In this sense, when

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Cartan–Hadamard theorem In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898. Élie Cartan generalized the theorem to Riemannian manifolds in 1928 (Helgason 1978; do Carmo 1992; Kobayashi & Nomizu 1969). The theorem was further generalized to a wide class of metric spaces by Mikhail Gromov in 1987; detailed proofs were published by Ballmann (1990) for metric spaces of non-positive curvature and by Alexander & Bishop (1990) for general locally convex metric spaces.

Conteúdo 1 Geometria Riemanniana 2 Metric geometry 2.1 Generalization to locally convex spaces 3 Significado 4 Veja também 5 References Riemannian geometry The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected complete Riemannian manifold of non-positive sectional curvature is diffeomorphic to Rn. Na verdade, for complete manifolds of non-positive curvature, the exponential map based at any point of the manifold is a covering map.

The theorem holds also for Hilbert manifolds in the sense that the exponential map of a non-positively curved geodesically complete connected manifold is a covering map (McAlpin 1965; Lang 1991, IX, §3). Completeness here is understood in the sense that the exponential map is defined on the whole tangent space of a point.

Metric geometry In metric geometry, the Cartan–Hadamard theorem is the statement that the universal cover of a connected non-positively curved complete metric space X is a Hadamard space. Em particular, if X is simply connected then it is a geodesic space in the sense that any two points are connected by a unique minimizing geodesic, and hence contractible.

A metric space X is said to be non-positively curved if every point p has a neighborhood U in which any two points are joined by a geodesic, and for any point z in U and constant speed geodesic γ in U, um tem {estilo de exibição d(z,gama (1/2))^{2}leq {fratura {1}{2}}d(z,gama (0))^{2}+{fratura {1}{2}}d(z,gama (1))^{2}-{fratura {1}{4}}d(gama (0),gama (1))^{2}.} This inequality may be usefully thought of in terms of a geodesic triangle Δ = zγ(0)c(1). The left-hand side is the square distance from the vertex z to the midpoint of the opposite side. The right-hand side represents the square distance from the vertex to the midpoint of the opposite side in a Euclidean triangle having the same side lengths as Δ. This condition, called the CAT(0) condition is an abstract form of Toponogov's triangle comparison theorem.

Generalization to locally convex spaces The assumption of non-positive curvature can be weakened (Alexander & Bishop 1990), although with a correspondingly weaker conclusion. Call a metric space X convex if, for any two constant speed minimizing geodesics a(t) and b(t), a função {displaystyle tmapsto d(uma(t),b(t))} is a convex function of t. A metric space is then locally convex if every point has a neighborhood that is convex in this sense. The Cartan–Hadamard theorem for locally convex spaces states: If X is a locally convex complete connected metric space, then the universal cover of X is a convex geodesic space with respect to the induced length metric d.

Em particular, the universal covering of such a space is contractible. The convexity of the distance function along a pair of geodesics is a well-known consequence of non-positive curvature of a metric space, but it is not equivalent (Ballmann 1990).

Significance The Cartan–Hadamard theorem provides an example of a local-to-global correspondence in Riemannian and metric geometry: nomeadamente, a local condition (non-positive curvature) and a global condition (simple-connectedness) together imply a strong global property (contractibility); or in the Riemannian case, diffeomorphism with Rn.

The metric form of the theorem demonstrates that a non-positively curved polyhedral cell complex is aspherical. This fact is of crucial importance for modern geometric group theory.

See also Glossary of Riemannian and metric geometry Cartan–Hadamard manifold Cartan–Hadamard conjecture References McAlpin, John (1965), "Infinite dimensional manifolds and Morse theory", Tese, Columbia University. Alexandre, Stephanie B.; Bishop, Richard L. (1990), "The Hadamard-Cartan theorem in locally convex metric spaces", Enseign. Matemática., Series 2, 36 (3-4): 309-320. Ballmann, Werner (1995), Lectures on spaces of nonpositive curvature, DMV Seminar 25, Basileia: Birkhäuser Verlag, pp. viii+112, ISBN 3-7643-5242-6, MR 1377265. Bridson, Martin R.; Haefliger, André (1999), Metric spaces of non-positive curvature, fundamentos de ciencias matematicas 319, Berlim: Springer-Verlag, pp. xxii+643, ISBN 3-540-64324-9, MR 1744486. do Carmo, Manfredo Perdigão (1992), Geometria Riemanniana, Matemática: theory and applications, Boston: Birkhauser, pp. xvi+300, ISBN 0-8176-3490-8. Kobayashi, Shoshichi; Nomizu, Katsumi (1969), Foundations of Differential Geometry, Volume. II, Tratados de Matemática 15, Nova york: Wiley Interscience, pp. xvi+470, ISBN 0-470-49648-7. Helgason, Sigurdur (1978), Geometria diferencial, Lie groups and symmetric spaces, Matemática Pura e Aplicada 80, Nova york: Imprensa Acadêmica, pp. xvi+628, ISBN 0-12-338460-5. Lang, Sarja (1999), Fundamentals of differential geometry, Textos de Graduação em Matemática, volume. 191, Berlim, Nova york: Springer-Verlag, ISBN 978-0-387-98593-0, MR 1666820. Categorias: Metric geometryTheorems in Riemannian geometry

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# Integer Cech cohomology of icosahedral projection tilings

Gähler F, Hunton J, Kellendonk J (2008)
ZEITSCHRIFT FUR KRISTALLOGRAPHIE 223(11-12): 801-804.

Journal Article | Published | English

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Abstract
The integer Cech cohomology of canonical projection tilings of dimension three and codimension three is derived. These formulae are then evaluated for several icosahedral tilings known from the literature. Rather surprisingly, the cohomologies of all these tilings turn out to have torsion. This is the case even for the Danzer tiling, which is, in some sense, the simplest of all icosahedral tilings. This result is in contrast to the case of two-dimensional canonical projection tilings, where many examples without torsion are known.
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### Cite this

Gähler F, Hunton J, Kellendonk J. Integer Cech cohomology of icosahedral projection tilings. ZEITSCHRIFT FUR KRISTALLOGRAPHIE. 2008;223(11-12):801-804.
Gähler, F., Hunton, J., & Kellendonk, J. (2008). Integer Cech cohomology of icosahedral projection tilings. ZEITSCHRIFT FUR KRISTALLOGRAPHIE, 223(11-12), 801-804.
Gähler, F., Hunton, J., and Kellendonk, J. (2008). Integer Cech cohomology of icosahedral projection tilings. ZEITSCHRIFT FUR KRISTALLOGRAPHIE 223, 801-804.
Gähler, F., Hunton, J., & Kellendonk, J., 2008. Integer Cech cohomology of icosahedral projection tilings. ZEITSCHRIFT FUR KRISTALLOGRAPHIE, 223(11-12), p 801-804.
F. Gähler, J. Hunton, and J. Kellendonk, “Integer Cech cohomology of icosahedral projection tilings”, ZEITSCHRIFT FUR KRISTALLOGRAPHIE, vol. 223, 2008, pp. 801-804.
Gähler, F., Hunton, J., Kellendonk, J.: Integer Cech cohomology of icosahedral projection tilings. ZEITSCHRIFT FUR KRISTALLOGRAPHIE. 223, 801-804 (2008).
Gähler, Franz, Hunton, John, and Kellendonk, Johannes. “Integer Cech cohomology of icosahedral projection tilings”. ZEITSCHRIFT FUR KRISTALLOGRAPHIE 223.11-12 (2008): 801-804.
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arXiv 0809.4442


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dc.contributor.author Nixon, MC dc.contributor.author Nikku, Madhusudhan dc.date.accessioned 2022-01-11T00:32:20Z dc.date.available 2022-01-11T00:32:20Z dc.date.issued 2021-08 dc.identifier.issn 0035-8711 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/332606 dc.description.abstract Understanding the internal structures of planets with a large H$_2$O component is important for the characterisation of sub-Neptune planets. The finding that the mini-Neptune K2-18b could host a liquid water ocean beneath a mostly hydrogen envelope motivates a detailed examination of the phase structures of water-rich planets. To this end, we present new internal structure models for super-Earths and mini-Neptunes that enable detailed characterisation of a planet's water component. We use our models to explore the possible phase structures of water worlds and find that a diverse range of interiors are possible, from oceans sandwiched between two layers of ice to supercritical interiors beneath steam atmospheres. We determine how the bulk properties and surface conditions of a water world affect its ocean depth, finding that oceans can be up to hundreds of times deeper than on Earth. For example, a planet with a 300 K surface can possess H$_2$O oceans with depths from 30-500 km, depending on its mass and composition. We also constrain the region of mass-radius space in which planets with H/He envelopes could host liquid H$_2$O, noting that the liquid phase can persist at temperatures up to 647 K at high pressures of $218$-$7\times10^4$ bar. Such H/He envelopes could contribute significantly to the planet radius while retaining liquid water at the surface, depending on the planet mass and temperature profile. Our findings highlight the exciting possibility that habitable conditions may be present on planets much larger than Earth. dc.publisher Oxford University Press (OUP) dc.rights Attribution 4.0 International dc.rights.uri https://creativecommons.org/licenses/by/4.0/ dc.subject planets and satellites: composition dc.subject planets and satellites: general dc.subject planets and satellites: interiors dc.subject planets and satellites: oceans dc.subject planets and satellites: surfaces dc.title How deep is the ocean? Exploring the phase structure of water-rich sub-Neptunes dc.type Article dc.publisher.department Institute of Astronomy dc.date.updated 2022-01-09T07:33:32Z prism.endingPage 3432 prism.issueIdentifier 3 prism.publicationDate 2021 prism.publicationName Monthly Notices of the Royal Astronomical Society prism.startingPage 3414 prism.volume 505 dc.identifier.doi 10.17863/CAM.80053 rioxxterms.versionofrecord 10.1093/mnras/stab1500 rioxxterms.version VoR dc.contributor.orcid Nixon, MC [0000-0001-8236-5553] dc.contributor.orcid Nikku, Madhusudhan [0000-0002-4869-000X] dc.identifier.eissn 1365-2966 dc.publisher.url http://dx.doi.org/10.1093/mnras/stab1500 rioxxterms.type Journal Article/Review cam.issuedOnline 2021-06-17 cam.depositDate 2022-01-09 pubs.licence-identifier apollo-deposit-licence-2-1 pubs.licence-display-name Apollo Repository Deposit Licence Agreement

### This item appears in the following Collection(s)

Except where otherwise noted, this item's licence is described as Attribution 4.0 International


text:
# diophantus

Hello, this is beta version of diophantus. If you want to report about a mistake, please, write to hello@diophantus.org

#### Sufficiency Criterion for the Validity of the Adiabatic Approximation

09 Apr 2007 quant-ph arxiv.org/abs/0704.1184
Abstract. We examine the quantitative condition which has been widely used as a criterion for the adiabatic approximation but was recently found insufficient. Our results indicate that the usual quantitative condition is sufficient for a special class of quantum mechanical systems. For general systems, it may not be sufficient, but it along with additional conditions is sufficient. The usual quantitative condition and the additional conditions constitute a general criterion for the validity of the adiabatic approximation, which is applicable to all $N-$dimensional quantum systems. Moreover, we illustrate the use of the general quantitative criterion in some physical models.

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Feb 17, 2014

# Life cycle of silk moth

Describe various stages in the life cycle of a silk moth?

Syeda

Member since Jan 25, 2017

 Answer. Sericulture, or silk farming, is the rearing of silkworms for the production of raw silk. Life cycle of a silk moth: 1. Egg →2.First stage of larvae→3.Second stage of larvae→4.final stage of larvae→ 5.Pupa →6.Adult The female silk moth lays eggs, from which hatch larvae which are called caterpillars or silkworms. They grow in size and when the caterpillar is ready to enter the next stage of its life history called pupa, it first weaves a net to hold itself. Then it swings its head from the side to side and secrets fibre made of a protein which hardens on exposure to air and becomes silk fibre. Soon the caterpillar completely covers itself by silk fibres and turns into pupa. This covering is known as cocoon. The further development of the pupa into moth continues inside the cocoon. SME Approved

Member since

The total Life-Cycle of a Silkworm ranges from 6-8 weeks. Generally, the warmer the weather the quicker the Silkworm will complete its life-cycle, however, other factors such as humidity and exposure to sunlight are also very important. Ideally, Silkworms will experience 12 hours of sunlight, and 12 hours of darkness per day, a temperature of 23-28 degrees Celsius, as well as humidity levels of 85-90%. Under these conditions, your eggs should hatch in a period of 7-10 days.� Eggs: In Australia, a Silkworm will naturally hatch in late July to early August – depending on the weather where you live – this time amazingly coincides with the Mulberry Tree regaining its leaves after the tree loses them in Winter. It is recommended that you store all of your eggs in a a cold space (a fridge is ideal) because if there are fluctuations in the weather, your eggs may hatch earlier than expected. When placed in a cold area, you can control when your eggs hatch, therefore ensuring they will always have a food source. If your eggs have been placed in the fridge, they will begin developing and hatch after 10-25 days, depending on the warmth of the weather, with warmer weather causing quicker hatching. Eggs 10-25 days Lava and worms 20-33 days Pupa 10-14 days Moth 5-10 days Lava/Cocoon: After 20-33 days of constantly munching away at Mulberry leaves or chow, your Silkworm will feel the urge to cocoon. This process can take up to 48 hours for the Silkworm to fully complete, and the result will be a perfectly ovate cocoon, with one continuous strand of silk up to 1 kilometre long! Inside the cocoon, the Silkworm will moult moults for the last time, as it approaches the last stage of its life-cycle to become a moth. Sometimes Silkworms do funny things, and this is evident when a Silkworm decides to spin a cocoon with a friend! When this happens, the cocoon will be noticeably larger, however if the two are members of the opposite sex, sometimes the cocoon does not hatch – and you open it to find two dead Silk-Moths and a bunch of eggs inside the cocoon! es_whiteseductressleaf3 Moth: After 10-14 days of developing into a moth in its cocoon, the Silk-Moth will appear from the cocoon and will excrete a brownish fluid upon emergence – don’t worry, this is completely normal, and is a sign the the Silk Moth is cleaning out its system. Silk-Moths cannot fly, as a result of thousands of years of domestication, however male Silk-Moths may leave their container/shoebox as they search for a female mate. Some males may be able to almost fly if you drop the from a small height, however it is uncontrolled, ad they hit the ground after a few moments. It is quite easy to separate the male and female Silk-Moths, with the females having larger abdomens and males possessing smaller abdomens. Males also tend to be more active, as they are constantly searching for a mate. The moth lives a very brief life of 5-10 days, with males generally living longer than Females. Silk-Moths will begin to search for a mate almost immediately after emerging from their cocoon, with some seen mating with a moth of the opposite sex inside their cocoon should two moths share one! After mating, the Female Moth will lay between 300-500 eggs, and die, whereas the male Silk-Moth will search for another mate, should it not be too old. es_mothcloseupweb At Everything Silkworms, we follow the practice of producing “Ahimsa” Silk, also known as “Peaceful” Silk – meaning that we allow every moth to emerge from its cocoon.

Deepak

Member since

Basically a life cycle of silk worm is very same as butterfly. First stage it changes into caterpillar from eggs layed by female silk worm Then caterpillar produces Larva and in few days it is covered with larva and is called cocoon There is pupa in cocoon which develops and becomes silk worm and the life cycle continuous

Member since Apr 24, 2013

Ans: Sericulture, or silk farming, is the rearing of silkworms for the production of raw silk. Life cycle of a silk moth: 1. Egg →2.First stage of larvae→3.Second stage of larvae→4.final stage of larvae→ 5.Pupa →6.Adult The female silk moth lays eggs, from which hatch larvae which are called caterpillars or silkworms. They grow in size and when the caterpillar is ready to enter the next stage of its life history called pupa, it first weaves a net to hold itself. Then it swings its head from the side to side and secrets fibre made of a protein which hardens on exposure to air and becomes silk fibre. Soon the caterpillar completely covers itself by silk fibres and turns into pupa. This covering is known as cocoon. The further development of the pupa into moth continues inside the cocoon.
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# Inconsolata italics?

Please see https://stackoverflow.com/questions/23975828/how-to-specify-non-standard-font-in-fancyvrb-verbatim/ for the start of this discussion.

I am using fancyvrb to create a fancy Verbatim environment. I have

\usepackage{inconsolata}

to load the Inconsolata font, then

\DefineVerbatimEnvironment{VerbOut}
{Verbatim}{fontfamily=zi4,fontsize=\relsize{-1}}

for the environment itself. I would like it to display in italics, but it appears the inconsolata package doesn't load an italicized version of that font. Is there some other package that does, or some other way I can get it?

• Welcome to TeX.SX! There's no italic version of Inconsolata, I'm afraid. – egreg Jun 4 '14 at 22:44
• Reference: Inconsolata in the LaTeX Font Catalogue only shows \normalfont. – Werner Jun 4 '14 at 23:12

The font Inconsolata does not have an italic or slanted variant (June 2014).

Some TeX engines can fake a slanted font, based on a font transformation feature of the PDF format (therefore PDF mode only).

## LuaTeX/XeTeX

\documentclass{article}
\usepackage{fontspec}
\setmonofont[
AutoFakeSlant,
BoldItalicFeatures={FakeSlant},
]{Inconsolatazi4}

\begin{document}
\ttfamily
Regular
\textbf{Bold}
\textsl{Slanted}
\textit{Italics}
\textbf{\textsl{BoldSlanted}}
\textbf{\textit{BoldItalics}}
\end{document}

The slanted/italic versions are only faked. Also the italics correction is not set.

## pdfTeX

A SlantFont operator can be used in the .map file.

The following Perl script zi4-sl.pl reads zi4.map (found via kpsewhich) and modifies the entries:

• -sl is added to the font (TFM) name,
• .167 SlantFont is added.

Perl script:

#!/usr/bin/env perl
use strict;
$^W=1; my$outfile = 'zi4-sl.map';
my $infile = kpsewhich zi4.map; chomp$infile;

open(IN, '<', $infile) or die "!!! Error: Cannot open zi4.map':$!";
open(OUT, '>', $outfile) or die "!!! Error: Cannot write $outfile': $!"; while (<IN>) { next if /^\s*$/;
s/^([^- ]+)-([^- ]+)/$1-$2-sl/ or die "!!! Error: Cannot parse TFM name!\n";
s/ " / " .167 SlantFont / or die "!!! Error: Cannot insert SlantFont!\n";
} continue {
print OUT;
}

close(IN);
close(OUT);
__END__

The following example:

• includes the map file via \pdfmapfile,
• defines the font shapes (would go into .fd files normally).
\pdfmapfile{+zi4-sl.map}

\documentclass{article}
\usepackage{inconsolata}

\makeatletter
\expandafter\ifx\csname zifour@scaled\endcsname\relax
\let\zifour@scaled\@empty
\fi
\expandafter\ifx\csname zifour@opt\endcsname\relax
\def\zifour@opt{\z@}\def\zifour@altopt{\tw@}
\fi
\DeclareFontShape{\f@encoding}{zi4}{m}{sl}{%
<-> \zifour@scaled ot1-zi4r-sl-\zifour@opt}{}%
\DeclareFontShape{\f@encoding}{zi4}{m}{it}{%
<-> ssub * zi4/m/sl}{}
\DeclareFontShape{\f@encoding}{zi4}{b}{sl}{%
<-> \zifour@scaled ot1-zi4b-sl-\zifour@opt}{}
\DeclareFontShape{\f@encoding}{zi4}{bx}{sl}{%
<-> ssub * zi4/b/sl}{}
\DeclareFontShape{\f@encoding}{zi4}{b}{it}{%
<-> ssub * zi4/b/sl}{}
\DeclareFontShape{\f@encoding}{zi4}{bx}{it}{%
<-> ssub * zi4/b/sl}{}
\makeatother

\begin{document}
\ttfamily
Regular
\textbf{Bold}
\textsl{Slanted}
\textit{Italics}
\textbf{\textsl{BoldSlanted}}
\textbf{\textit{BoldItalics}}
\end{document}

Finally the TFM files are needed. A poor man's solution is to use the TFM files for the original fonts. The example would need (bash syntax):

cp $(kpsewhich ot1-zi4r-0.tfm) ot1-zi4r-sl-0.tfm cp$(kpsewhich ot1-zi4b-0.tfm) ot1-zi4b-sl-0.tfm

Again, the slanted and italic variants are only faked. Also the italics correction is missing, but it could be added by editing the .tfm files.

• While I admire the technical wizardry in this answer (+1), I cannot refrain from disparaging the purpose to which it is turned. ;) – cfr Jun 5 '14 at 1:32
• Hmm, looks like fontspec should also slant the bold font with AutoFakeSlant. Bug added. – Will Robertson Jun 5 '14 at 7:01
• @cfr It should be noted that it's quite common for font designers to simply slant the glyphs for an italic monospaced font. I'd consider this less of a travesty than, say, slanting a text font. – Will Robertson Jun 5 '14 at 7:03
• This one might give even another option to slant a font. But haven't tried it, so it might give problems. – Manuel Jun 5 '14 at 10:18
• @Manuel: There are problems with line and page breaks inside the argument. Then the coordinate systems of TeX and the PDF output device get out of sync. Safely it can be used inside \mbox, which forbids line breaks. – Heiko Oberdiek Jun 5 '14 at 10:48

There are now two variants of Inconsolata that have italics (as well as bold, bold italic and Cyrillic and Greek support for all of the weights and styles).

These are Inconsolata LGC and Inconsolata LGC Markup. The latter also tries to distinguish various kinds of quote and apostrophe-like characters, en/em dashes and hyphens,... It's all listed on the link.

It seems there is no italic version for your font (at least not that I'm aware of), but you can always search for a monospaced font you like in the LaTeX Font Catalogue


text:
# Is pollution the reason why rainbows are so rare nowadays?

Today I saw a rainbow after almost a decade. I live in Noida (a city) in India. Most people I know believe that: we don't see rainbows nowadays because of pollution and now after nearly 3 months since the COVID lock-down, the pollution must have come down and the rainbow was visible. Is this the reason why rainbows have become so rare or is it a false belief?

Here is news about the rainbow I saw here(so you can see the photos)

According to this site https://aqicn.org/map/delhi/ PM2.5 in Delhi(very close to Noida) today is $$74$$ and last year in 2019 on this day was $$151$$.

If the answer is yes, how does pollution impact rainbows?

• for this to be true there has to be less sunlight(it is not) or it has to be less rainfall(in some areas this is true and in others it is more rainfall).i vote to close this as opinion based. – trond hansen May 31 at 16:48
• I Googled rainbows pollution and found some analysis: Scientific Attitude on Understanding Disappearance of Rainbow. – Keith McClary May 31 at 17:05
• Is the supposed scarcity of rainbows a local phenomenon? They don't seem to be any less common where I live (western USA), but then we don't experience a high level of air pollution. (Except during major fires.) But if your local air pollution is so bad that it makes the sky look like a brown haze, I could understand not seeing the rainbows due to a lot of the light being scattered. – jamesqf Jun 1 at 4:38
• Refraction in water or ice drops causes rainbows, (Mie-)scattering happens through pollution. So there are two different principles here. If there are less rainbows in an area, than there's probably less water in the air. One could check a past climate trend for the area, but it may also be just a felt differencce. I am not into believing :-) – a_donda Jun 1 at 8:27
• @a_donda: What I'm suggesting is that the rainbow happens where the rain is falling, but that pollution scatters the light on the way from the rain to the observer, so the rainbow isn't visible. Basically the same reason you can see the distant mountains on a clear day, but not on a hazy one. – jamesqf Jun 1 at 17:47


text:
# IEEE Transactions on Circuits and Systems I: Regular Papers

## Filter Results

Displaying Results 1 - 25 of 31

Publication Year: 2014, Page(s):C1 - C4
| PDF (150 KB)
• ### IEEE Transactions on Circuits and Systems—I: Regular Papers publication information

Publication Year: 2014, Page(s): C2
| PDF (133 KB)
• ### Single Transistor Active Filters: What is Possible and What is Not

Publication Year: 2014, Page(s):2517 - 2524
Cited by:  Papers (5)
| | PDF (2424 KB) | HTML

This paper systematically investigates the design of single transistor second-order active filters out-lining all possible architectures and possible impedance settings using an exhaustive MAPLE search code for all stable cases with the minimum number of passive elements (two resistors and two energy storage elements). The search is performed on six general voltage-input voltage-output transfer fu... View full abstract»

• ### Quantization Noise Reduction Techniques for Digital Pulsed RF Signal Generation Based on Quadrature Noise Shaped Encoding

Publication Year: 2014, Page(s):2525 - 2536
Cited by:  Papers (3)
| | PDF (2774 KB) | HTML

The digital generation of pulsed radio frequency (RF) signals based on noise shaped encoding in complex baseband is a popular concept in digital RF transmitters. However, as the pulsed complex-baseband signals are up-converted to an RF frequency, a conjugate image overlaps with the signal band, which complicates the center frequency tuning for the transmitters in question. In this paper novel ways... View full abstract»

• ### A Harmonic Class-C CMOS VCO-Based on Low Frequency Feedback Loop: Theoretical Analysis and Experimental Results

Publication Year: 2014, Page(s):2537 - 2549
Cited by:  Papers (7)
| | PDF (2740 KB) | HTML

A novel harmonic Class-C CMOS VCO architecture with improved phase noise performance and power efficiency is presented in this paper. The VCO is based on the widely adopted topology consisting in a crossed pair of NMOS devices refilling a symmetric resonator with a center tapered inductor and biased by a top PMOS current generator. The Class-C operation mode is obtained through a low frequency fee... View full abstract»

• ### A Low-Power Multifunctional CMOS Sensor Node for an Electronic Facade

Publication Year: 2014, Page(s):2550 - 2559
Cited by:  Papers (4)
| | PDF (2721 KB) | HTML

In this paper, a low-power, multifunctional CMOS smart sensor node is designed for an electronic facade, which provides an alternative solution to the concept of energy-efficient responsive buildings. Various sensing capabilities, including light intensity sensing, temperature sensing, motion tracking, and compressive image acquisition, are implemented on the sensor node. An 80 × 80 image p... View full abstract»

• ### Analysis and Modeling of a Gain-Boosted N-Path Switched-Capacitor Bandpass Filter

Publication Year: 2014, Page(s):2560 - 2568
Cited by:  Papers (14)
| | PDF (2710 KB) | HTML

It has been studied that, an N-path switched-capacitor (SC) branch driven by an N-phase non-overlapped local oscillator (LO), is equivalent to a tunable parallel-RLC tank suitable for radio-frequency (RF) filtering. This paper proposes a gain-boosted N-path SC bandpass filter (GB-BPF) with a number of sought features. It is based on a transconductance amplifier (Gm) with an N-path SC branch as its... View full abstract»

• ### Stability Analysis of a Charge Pump Phase-Locked Loop Using Autonomous Difference Equations

Publication Year: 2014, Page(s):2569 - 2577
| | PDF (2445 KB) | HTML

The CP-PLL is a common component in modern communication circuits. It is used among others in frequency synthesis, synchronization and clock and data recovery. Since the CP-PLL constitutes a mixed-signal architecture and a nonlinear behavior, it is difficult to use general theories to characterize the dynamic behavior of this feedback system. Therefore, linear continuous-time and linear discrete-t... View full abstract»

• ### 40 nm Bit-Interleaving 12T Subthreshold SRAM With Data-Aware Write-Assist

Publication Year: 2014, Page(s):2578 - 2585
Cited by:  Papers (18)
| | PDF (2014 KB) | HTML

This paper presents a new bit-interleaving 12T subthreshold SRAM cell with Data-Aware Power-Cutoff (DAPC) Write-assist to improve the Write-ability to mitigate increased device variations at low supply voltage under deep sub-100 nm processes. The disturb-free feature facilitates the bit-interleaving architecture that can reduce multiple-bit errors in a single word and enhance soft error immunity b... View full abstract»

• ### Tunable CMOS Delay Gate With Improved Matching Properties

Publication Year: 2014, Page(s):2586 - 2595
Cited by:  Papers (5)
| | PDF (2124 KB) | HTML

This paper presents the analysis and design of a tunable CMOS delay gate with improved matching properties as compared with the commonly used “current starved inverter” (CSI). The main difference between these structures lies in the location of the current limiting transistor on the inverter's output rather than on the side of the power rail. This improves the dynamic performance of ... View full abstract»

• ### Modeling of Retention Time for High-Speed Embedded Dynamic Random Access Memories

Publication Year: 2014, Page(s):2596 - 2604
Cited by:  Papers (3)
| | PDF (1608 KB) | HTML

Embedded dynamic random access memory (eDRAM) is becoming a popular choice for large cache applications due to its density, speed, and power benefits. One of the crucial challenges in eDRAM design is meeting the retention time specification. Due to implementation in logic process, usually eDRAM suffers from poor retention time compared to commodity DRAM. The retention time of eDRAM designed in sca... View full abstract»

• ### A Low Active Leakage and High Reliability Phase Change Memory (PCM) Based Non-Volatile FPGA Storage Element

Publication Year: 2014, Page(s):2605 - 2613
Cited by:  Papers (26)
| | PDF (1827 KB) | HTML

The high leakage current has been one of the critical issues in SRAM-based Field Programmable Gate Arrays (FPGAs). In recent works, resistive non-volatile memories (NVMs) have been utilized to tackle the issue with their superior energy efficiency and fast power-on speed. Phase Change Memory (PCM) is one of the most promising resistive NVMs with the advantages of low cost, high density and high re... View full abstract»

• ### A Low Power Localized 2T1R STT-MRAM Array With Pipelined Quad-Phase Saving Scheme for Zero Sleep Power Systems

Publication Year: 2014, Page(s):2614 - 2623
Cited by:  Papers (3)
| | PDF (2136 KB) | HTML

The high leakage power due to process nodes scaling down has been one of the critical issues in CMOS circuits, especially the sleep power critical systems. The conventional retention CMOS register based approaches cannot fully address the high standby energy issue in long time standby systems. The recent non-volatile Flip-Flop (nvFF) based approaches may achieve zero sleep power consumption, but s... View full abstract»

• ### Performance Analysis of the CS-DCSK/BPSK Communication System

Publication Year: 2014, Page(s):2624 - 2633
Cited by:  Papers (17)
| | PDF (2792 KB) | HTML

As a spread spectrum communication system, code-shifted differential chaos shift keying (CS-DCSK) is required to provide reasonable bit error performance in a coexisting scenario where the conventional communication system is an interferer. Meantime, the bit error performance of the conventional communication system should not deteriorate dramatically in the presence of a CS-DCSK signal, which is ... View full abstract»

• ### A Survey on Circuit Modeling of Spin-Transfer-Torque Magnetic Tunnel Junctions

Publication Year: 2014, Page(s):2634 - 2643
Cited by:  Papers (9)
| | PDF (1415 KB) | HTML

Accurate modeling of magnetic tunnel junction (MTJ) is critical for design of memories such as spin-transfer-torque magnetoresistive random access memory (STT-MRAM) and spin logic circuits such as spin flip flops. This paper reviews several static and dynamic models for the MTJ and compares them for their capabilities and limitations. Furthermore, a Verilog-A model is developed to predict dynamic ... View full abstract»

• ### A Krein Space Approach to $H_{\infty}$ Filtering of Discrete-Time Nonlinear Systems

Publication Year: 2014, Page(s):2644 - 2652
Cited by:  Papers (4)
| | PDF (2455 KB) | HTML

In this paper, a Krein space approach to finite horizon H∞ filtering is proposed for a class of affine nonlinear discrete-time systems. It is shown that the problem of H∞ nonlinear filtering can be converted into a minimum of an indefinite quadratic form. Hence, a relationship between H∞ nonlinear filter in Hilbert space and nonlinear estimatio... View full abstract»

• ### Multimode Oscillations in Coupled Oscillators With High-Order Nonlinear Characteristics

Publication Year: 2014, Page(s):2653 - 2662
Cited by:  Papers (2)
| | PDF (3434 KB) | HTML

We propose two van der Pol oscillators with high-order nonlinearities coupled with an inductor and investigate various types of multimode oscillations. We confirm that these oscillations are stably excited by numerical calculations, theoretical analyses and circuit experiments. In particular, we aim to show relations between the multimode oscillations and the number of order nonlinear characterist... View full abstract»

• ### Bias-Compensated Least Squares Identification of Distributed Thermal Models for Many-Core Systems-on-Chip

Publication Year: 2014, Page(s):2663 - 2676
Cited by:  Papers (9)
| | PDF (2344 KB) | HTML

The thermal wall for many-core systems on-chip calls for advanced management techniques to maximize performance, while capping temperatures. Distributed and compact thermal models are a cornerstone for such techniques. System identification methodologies allow to extract models directly from the target device thermal response. Unfortunately, standard Auto-Regressive eXogenous models and Least Squa... View full abstract»

• ### Synchronization of Nonlinear Circuits in Dynamic Electrical Networks With General Topologies

Publication Year: 2014, Page(s):2677 - 2690
Cited by:  Papers (19)
| | PDF (3129 KB) | HTML

Sufficient conditions are derived for global asymptotic synchronization in a system of identical nonlinear electrical circuits coupled through linear time-invariant (LTI) electrical networks. In particular, the conditions we derive apply to settings where: i) the nonlinear circuits are composed of a parallel combination of passive LTI circuit elements and a nonlinear voltage-dependent current sour... View full abstract»

• ### A SAW-Less Receiver Front-End Employing Body-Effect Control IIP2 Calibration

Publication Year: 2014, Page(s):2691 - 2698
Cited by:  Papers (2)
| | PDF (1927 KB) | HTML

An IM2 product cancellation technique that employs a body-effect control of transistors is presented by developing a design for low-cost and high-linearity receiver front-end in LTE/WCDMA applications. To accomplish a high IIP2 for strong out-of-band blockers, the body terminal voltage of the mixer switches is properly adjusted to provide intentional asymmetry and suppress the IM2 at the mixer out... View full abstract»

• ### High-Throughput Turbo Decoder With Parallel Architecture for LTE Wireless Communication Standards

Publication Year: 2014, Page(s):2699 - 2710
Cited by:  Papers (10)
| | PDF (3081 KB) | HTML

This work focuses on the VLSI design aspect of high- speed maximum a posteriori (MAP) probability decoders which are intrinsic building-blocks of parallel turbo decoders. For the logarithmic-Bahl-Cocke-Jelinek-Raviv (LBCJR) algorithm used in MAP decoders, we have presented an ungrouped backward recursion technique for the computation of backward state metrics. Unlike the conventional decoder archi... View full abstract»

• ### Tail-Overlapped SISO Decoding for High-Throughput LTE-Advanced Turbo Decoders

Publication Year: 2014, Page(s):2711 - 2720
Cited by:  Papers (1)
| | PDF (2232 KB) | HTML

This paper proposes a novel decoding algorithm to enhance the throughput of turbo decoding in LTE-Advanced systems and its hardware realization. The proposed method completely removes the undesired phase-switching latency by partially overlapping in-ordered and interleaved decoding phases, and as a result, achieves a significant increase of decoding throughput. Moreover, the algorithm does not deg... View full abstract»

• ### A 5.4 $\mu{\rm W}$ Soft-Decision BCH Decoder for Wireless Body Area Networks

Publication Year: 2014, Page(s):2721 - 2729
Cited by:  Papers (4)
| | PDF (1885 KB) | HTML

This paper presents an IEEE 802.15.6 compliant soft-decision BCH decoder for energy-constrained wireless body area networks. The proposed soft-decision decoder (SDD) provides a 1 dB coding gain compared to the hard-decision decoder (HDD). The improvement in BER performance can translate into power savings at the transmitter. The energy dissipation and area of the soft-decision BCH decoder is minim... View full abstract»

• ### Implementation of a High-Throughput Modified Merge Sort in MIMO Detection Systems

Publication Year: 2014, Page(s):2730 - 2737
Cited by:  Papers (3)
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Wireless communication technology continues to advance at a rapid pace, and researchers have made tremendous progress in extending single-input single-output (SISO) systems to multiple-input multiple-output (MIMO) systems such as IEEE 802.11n, WiMAX, and LTE. However, a MIMO system requires a detector circuit to separate received data. To reduce the number of comparators required, the method of mo... View full abstract»

• ### A Fully Parallel LDPC Decoder Architecture Using Probabilistic Min-Sum Algorithm for High-Throughput Applications

Publication Year: 2014, Page(s):2738 - 2746
Cited by:  Papers (11)
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This paper presents a normalized probabilistic min-sum algorithm for low-density parity-check (LDPC) codes, where a probabilistic second minimum value, instead of the true second minimum value, is used to facilitate fully parallel decoder realization. The comparators in each check-node unit (CNU) are connected through an interconnect network based on a mix of tree and butterfly networks such that ... View full abstract»

## Aims & Scope

The theory, analysis, design, and practical implementations of circuits, and the application of circuit techniques to systems and to signal processing.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief
Andreas Demosthenous
Dept. Electronic & Electrical Engineering
University College London
London WC1E 7JE, UK


text:
# Uniform power-saving estimate for an exponential sum

Let $N$ be a large natural number. Define an expoential sum $$I_m=\sum_{x,y=1}^N e^{2\pi i\frac{x^2-y^2}{N}m}, \,\, m=1,2,...,N-1.$$

The trivial bound for $I_m$ is $N^2$, as there are $N^2$ terms. My question is can we do better? Is it possible to have $$|I_m|<C N^{2-\delta}$$ for some positive $\delta$ and $C$ (indepdent of $m$)? And what about the case when x^2-y^2 is replaced with P(x)-P(y) for a general nonlinear polynomial?

We set $$e_N(\theta) := e^{2 \pi I \theta/N}.$$ Your sum which is equal to $$\left|\sum_x e_N(x^2 m)\right|^2$$ and is known as a Gauss sum. Gauss himself studied these extensively. Following him, we use the substitution $x = y+h$, $$\sum_{x,y} e_N((x^2 - y^2)m) = \sum_{y,h} e_N((2hy + h^2)m) = N \sum_h e_N(mh^2) 1_{N | 2hm}$$ Now $N | 2hm$ for $h = 1, \ldots , N$ if and only if $N / \gcd(2m , N)$ divides $h$. Letting $h = k N / \gcd(2m , N)$ we get the above is $$= N \sum_{k=1}^{\gcd(N , 2m)} e_N(\left(\frac{N }{ gcd(2m, N)}\right)^2 k^2m) \leq N\gcd(N , 2m).$$ Note that the above is an inequality for $N$ odd, so we cannot expect a uniform bound without knowing further information about $\gcd(N , 2m)$. For instance if $m = 0$ there is no cancellation. Note if $N$ is prime and $m \neq 0$, then we have square root cancellation ($\delta = C = 1$ in your notation).
For general polynomials, you should look at the Weil conjectures. This handles the case when $N$ is prime, getting square root cancellation just like in the Gauss sums above.


text:
2021-12-19

All the real zeros of the given polynomial are integers. Find the zeros, and write the polynomial in factored form.
$P\left(x\right)={x}^{3}-4{x}^{2}-19x-14$

Durst37

Expert

Step 1
Given polynomial is, $P\left(x\right)={x}^{3}-4{x}^{2}-19x-14$
In factored form, it is written as follows:
${x}^{3}-4{x}^{2}-19x-14={x}^{3}+{x}^{2}-5{x}^{2}-5x-14x-14$
$={x}^{2}\left(x+1\right)-5x\left(x+1\right)-14\left(x+1\right)$
$=\left(x+1\right)\left({x}^{2}-5x-14\right)$
=(x+1)(x(x+2)-7(x+2))
=(x+1)(x+2)(x-7)
Step 2
It zeroes are find by putting it equal to zeros.
${x}^{3}-4{x}^{2}-19x-14=0$
(x+1)(x+2)(x-7)=0
x+1=0 or x+2=0 or x-7=0
x=-1, x=-2, x=7

Ella Williams

Expert

$P\left(x\right)={x}^{3}-4{x}^{2}-19x-14$
rational roots of the function are
x=+-{1,2,7,14}
actual roots occur at
x = -1 , -2 , 7
hence zeros are
x = -1 , -2 , 7
polynomial in factored form is
P(x) = (x+1) ( x+2) (x-7)

Do you have a similar question?


text:
### Thousand Words

Here the diagram says it all. Can you find the diagram?

### Complex Squares

What happens when we square complex numbers? Can the square of a complex number be real?

### A Brief Introduction to Complex Numbers

In this problem, we define complex numbers and invite you to explore what happens when you add and multiply them.

# A Brief Introduction to the Argand Diagram

##### Stage: 4 and 5 Challenge Level:

Pick a point for $z_1$ such as $2+i$. Move $z_2$ around, and keep a record of the points where $z_3$ is on the x-axis. What do you notice about the coordinates of $z_2$ and $z_3$?

Repeat for some other points.


text:
# What is the kinetic energy of an object with a mass of 3 kg that has been in freefall for 28 s?

Dec 21, 2015

$112.9 \text{kJ}$

#### Explanation:

We need to get the velocity of the object after it has fallen under gravity for 28s.

We can use $v = u + a t$

This becomes:

$v = 0 + 9.8 \times 28$

$\therefore v = 274.4 \text{m/s}$

The kinetic energy is given by:

KE = $\frac{1}{2} m {v}^{2}$

$\therefore \text{KE"=1/2xx3xx274.4^(2)=112,943"J}$

$= 112.9 \text{kJ}$


text:
## Introduction

The unique properties of terahertz waves (0.1–10 THz)1,2, such as the nonionizing photon energy, spectral fingerprint, and transparency for most nonpolar materials, have attracted much research interest and enabled many applications, such as nondestructive testing3, biodetection4,5,6, and cancer imaging7. However, the long wavelength of THz waves (1 THz ~ 300 μm) typically limits the resultant imaging resolution to greater than the millimetre scale in conventional far-field imaging methods due to the well-known Rayleigh diffraction limit, therefore restricting their use in many emergent applications such as cellular imaging. On the other hand, near-field imaging by mapping the object-modulated evanescent waves paves the way towards deep subwavelength resolution and is particularly desirable at long wavelengths, for example, at THz frequencies. AFM- or STM-tip-enhanced THz probes8,9 or microantenna/aperture THz probes10 have achieved micrometre- or even atomic-scale resolution9. Nevertheless, these techniques require mechanical raster scanning of the surface of an object pixel by pixel, with a relatively poor signal-to-noise ratio (SNR). Recently, near-field ghost imaging techniques11,12,13,14 have been experimentally demonstrated to increase the SNR by $$\sqrt N$$ times (where N denotes the pixel number of a digital picture) over that in raster scanning15. In these approaches, THz images are spatially encoded in the near field with deterministic patterns (e.g., the Walsh-Hadamard matrix11,12,13,14,15,16,17). Then, the total intensities (or electric field amplitudes) of encoded THz images are collected and detected with a single-pixel detector in the far field. After postprocessing via computational algorithms to correlate the detected THz intensities (or electric field amplitudes) with the deterministic patterns18,19, near-field images can be reconstructed. In this scheme, the subwavelength spatial information “hidden” in the diffracted far-field distribution can be recovered from the intensities (or amplitude fields) recorded by a mere single-pixel detector.

To encode THz images in a single pixel, the conventional method is to use photogenerated spatial THz wave modulators as reconfigurable masks11,12,13. However, the THz wave amplitude passing through the subwavelength apertures of modulators follows the scaling rule 1/a3 (where a denotes the diameter of the apertures on the mask)20, which fails to image deep subwavelength structures.

Alternatively, directly detecting THz images14 or generating patterned THz waves16,17 by encoding femtosecond (fs) laser pulses in nonlinear electrooptic (EO) crystals (such as ZnTe) has been proposed to bypass the scaling rule. However, subwavelength structures are rapidly blurred upon propagation in hundreds of micrometre-thick EO crystals14 (Supplementary Fig. S1). Recently, the time-resolved single-pixel detection of THz pulses has been theoretically proposed16 and experimentally demonstrated17 to recover higher-resolution images. However, suffering from the milimeter-scale-thick nonlinear crystal, THz-wave generation along with diffraction occurs across the entire volume of the crystal, leading to a limited spatial resolution of 50–100 μm. Although the spatial resolution can be further improved by the inverse propagation algorithm, this requires full-wave measurements of THz pulses in the time domain with extra cost in terms of detection resources and involves an inverse problem. Furthermore, it only applies to recovering 2D-structured images. In addition, constrained by the electromagnetic boundary conditions (Supplementary Fig. S2), the distribution of the polarized THz field in subwavelength structures presents a “distorted” image11,12,13,14 that might lead to misrecognition of the object’s morphology.

In this work, we utilize a spintronic THz emitter (STE) to illuminate an object in the near field. STEs are a novel type of THz emitter based on the spin-related effects21,22 in ferromagnetic/nonmagnetic (FM/NM) heterostructures23,24,25,26, which are only a few nanometres thick but offer generation efficiency comparable to conventional milimetre-thick EO crystals. In principle, an STE can fully cover the 0.1–30 THz frequency range24 without phonon absorption, which is superior to all the current solid emitters. Limited by the 1-mm-thick ZnTe used as the detector (3-THz detection bandwidth) and the pulse duration of the pump laser (~90 fs), the STE in our work provided a bandwidth of up to 2 THz. To date, all applications of STEs have only focused on their far-field properties, whereas the highly efficient few-nanometre-thick STE is capable of illuminating an object at an extremely near field, which naturally breaks the diffraction limit. The key challenge for using STEs in near-field imaging is how to map the object-modulated THz field without near-field scanning probes. In view of this, we developed a near-field illuminating “array” (STEA) whose “elements” are coherent and individually programmable in binary states of either “on” or “off” by photoexcitation. Combining the programmable near-field illuminating STEA and far-field single-pixel detection, we designed and demonstrated a novel ghost spintronic THz-emitter-array microscope (GHOSTEAM) for THz wave imaging with deep subdiffraction resolution. A minimum resolution of 6.5 μm at a single pixel was demonstrated with a contrast of more than 57 ± 21% (>20% required by the Rayleigh criterion14) in a 6-µm metal gap. In addition, polarization effects on the subdiffraction-limited image were eliminated via postprocessing of two images with mutually orthogonal polarizations. In addition, TOF microscopic topography was demonstrated with a 3D silica structure.

## Results

### Concept design

In our design, as shown in Fig. 1a, the STEA under external magnetic field B generates spatially structured THz pulses upon fs-laser-pulse spatial photoexcitation. The STEA, consisting of a W(2 nm)/Fe(2 nm)/Pt(2 nm) trilayer heterostructure on a transparent MgO substrate (Fig. 1b), provides great THz conversion efficiency in terms of output amplitude, comparable to that of 1-mm ZnTe24. The STEA is capped with a 150-nm SiO2 layer (n = 1.97) to protect it from being damaged by the fs laser. The output THz electric field E(t) is linearly polarized perpendicular to the applied magnetic field B, as described by E(t)  (Jc = Js × B)21,22,23,24,25,26, where Js represents the spin current induced by the fs laser and Jc represents the charge current converted in the NM metals under B. To perform ghost imaging (see “Materials and methods” for details), the Walsh–Hadamard matrix27 was used to code the STEA due to its unrivalled noise suppression performance among various measurement matrices11,12,13,14,15,16,17. The patterns, programmed in the order of the Walsh–Hadamard masks, were spatially encoded on the excitation fs laser beam by a digital micromirror device (DMD) with a switching time of 5 μs. On the exit surface of the STEA, the profile of the output THz pulse is as accurate as that of the excitation fs laser because the 150 nm propagation distance in the SiO2 protective layer is too thin for the THz wave to be diffracted (150 nm ≈ 5 × 10−4λ0/n, where λ0 = 600 μm and n = 1.97; see Supplementary Fig. S1 for the theoretical calculation of the near-field evanescent wave). The spatial near-field profile of the THz pulse from the STEA consists of individually real-time programmable “elements” up to 128 × 128 (see Supplementary Fig. S3) in either “on” or “off” binary emission states, as shown in Fig. 1b. The pixel size of each “element” is 6.5 μm × 6.5 μm in our experiments, which can be zoomed in or out by using an optical projection imaging system depending on the practical requirements. After being transmitted through an object placed in the near-field region (z = 150 nm), the structured THz pulses were collected and focused onto a 1-mm-thick (110) ZnTe crystal for single-pixel coherent detection by EO sampling. The peak electric amplitudes were recorded for reconstructing THz subdiffraction ghost images.

After reflection upon a DMD, the fs laser pulse suffers from wavefront tilt in the DMD deflection direction (horizontal) since the DMD is essentially a reflective blazed grating28. The resultant fs laser pulse with a tilted wavefront will badly lose its temporal coherence among individual micromirrors along the deflection direction. To correct the wavefront tilt, another DMD (DMD2) was imaged onto the encoding DMD (DMD1) by a 4f system with a 1:1 imaging ratio, as shown in Fig. 1a. Due to the fixed time delay of ghost imaging (corresponding to the peak of the THz pulses), the DMD-induced temporal smearing should be minimized. Otherwise, decoherence (large time delay in the wavefront) would result in severely distorted images (see Supplementary Fig. S6). The temporal delay in the horizontal direction |ΔtH| was ultimately corrected to less than 69 ± 13 fs (Fig. 1c) but could not be completely eliminated due to the manufacturing tolerance of the tilt angles of the two DMDs (see Supplementary Fig. S6). Meanwhile, the temporal delay in the vertical direction |ΔtV| was measured to be less than 8 fs (Fig. 1d), which is expected because the two DMDs only deflect in the horizontal direction. The spatiotemporal THz waveforms (Fig. 1c, d) were measured by ghost imaging using 64-order Walsh–Hadamard coding 1-D masks (see “Materials and methods” and Supplementary Fig. S5 for detailed information and Supplementary GIFs. 1 and 2 for raw data).

An object was illuminated by the coherent programmable STEA using a sequence of prearranged Walsh–Hadamard masks, and the computational ghost imaging algorithm was applied to correlate the peak amplitude measured by a single-pixel detector with the sequence of predetermined incident THz wave patterns (Fig. 1e, with the detailed procedure shown later). The reconfigurable area on the STEA, defined as the first imaging field of view FOV1, was measured as 834 μm × 834 μm, with up to 128 × 128 coding pixels. The SNRs of reconstructed images encoded by Walsh–Hadamard masks were numerically determined as a function of the pixel number (or pixel size in FOV1; see “Materials and methods”), which gave a reasonable value of ~47.02 when FOV1 was coded into 16 × 16 pixels under our experimental conditions (pump fluence ~2.88 mJ cm−2, dynamic range of peak field ~1043, and pulse fluctuation ~0.7%), as shown in Fig. 1f. It is worth noting that although the detected THz band was less than 2 THz due to the 90-fs pulse and 3-THz detection bandwidth (1-mm ZnTe), the STE has demonstrated complete coverage of the entire THz region, i.e., 0.1–30 THz, when a shorter pump pulse (10 fs) and a wide-band detector are used24,26. More experimental details are given in “Materials and methods” and Supplementary section 2.

### Subdiffraction ghost imaging

The STEA was experimentally reconfigured in a 64 × 64-order Walsh–Hadamard matrix sequence to acquire THz subdiffraction-limited images of an object (see Fig. 2a) positioned z = 150 nm from the STEA (thickness of the protective SiO2 layer on top of the STEA). The field amplitudes of spatially coded THz waves that passed through the object were measured at a fixed time delay of 0 ps (as indicated by the black dotted lines in Fig. 1c, d). The reconstructed 64 × 64 ghost images in FOV1 = 834 μm × 834 μm with mutually orthogonal polarizations are shown in Fig. 2b, c, whose pixel sizes are both 13.0 μm × 13.0 μm. The reconstructed 64 × 64 ghost image in a smaller area FOV2 = 417 μm × 417 μm (indicated by the black dashed box in Fig. 2a) with a smaller pixel size of 6.5 μm × 6.5 μm is shown in Fig. 2d. The y-dependent THz field distribution across the slit region (indicated by the black dashed arrow in Fig. 2b) was extracted from the associated pixels, as shown in Fig. 2e. Note that every pixel value was averaged from the identical-row pixels within the slit region, with prior knowledge about the object’s horizontal homogeneity in the slit region. A contrast ratio of 57 ± 21% (>20% required by the Rayleigh criterion14 for distinguishing slits) was observed in the narrowest 6-μm metal slit within FOV2 (see Supplementary section 5 for the quantification process and Supplementary GIFs. 35 for the raw data), which clearly proves that our GHOSTEAM system deeply breaks the diffraction barrier of ~366 μm for the 600-μm THz wave (Rayleigh resolution of 0.61λ0/NA). The spatial resolution of the GHOSTEAM system using this experimental setup is limited to the minimum available pixel size of 6.5 μm, which depends on the accuracy of the mask patterns on the STEA projected by the DMD. As the micromirrors of the DMD are arranged in the diamond orientation, the accuracy of the coding profile on the STEA is decreased since a mask pixel is formed by fewer micromirrors (see Supplementary Fig. S3). Regardless of the diffraction of the coding optical light, the resolution of GHOSTEAM is limited by the propagation distance between the emission surface of the STEA and the object (the thickness of the protective 150-nm-thick SiO2 layer in this experiment), which is theoretically estimated as submicrometer (see Supplementary Fig. S1).

### Polarization-free image

Polarization impacts the subwavelength imaging (see the dark region where the slits are along the wave polarization in Fig. 2b, c), which is expected due to the electrical field boundary conditions at the subwavelength scale11,13,14 (see Supplementary Fig. S2) and hinders accurate resolution of the object’s morphology. With advancement of the in-line rotating THz wave polarization by an external magnetic field, we secured two images with mutually orthogonal polarizations. This feature allows us to achieve a higher-contrast polarization-free image through postfusion of the two images. To do this, 2D Fourier transform was applied to Fig. 2b, c individually to quantify the distribution in the spatial frequency domain |F(ux,uy)| along the x and y directions. The amplitude |F(ux,uy)| as functions of spatial frequencies ux and uy is shown in Fig. 3a, b, which indicates loss of the high spatial frequency (>10/λ) components parallel to ETHz in the polarization images. To acquire polarization-free THz image XF, the THz ghost images with horizontal polarization XH (Fig. 2b) and vertical polarization XV (Fig. 2c) were fused by the weighted average method, namely, XF = rXH + (1 − r)XV, where r represents the weight fraction and lies within [0, 1]. The total variation (TV), which is a common assessment parameter for image clearness, was chosen to guide the optimization of the fusion process. The TV of XF was calculated by

$${\mathrm{TV}} = \left( {\mathop {\sum }\limits_{i = 2}^{64} \mathop {\sum}\limits_{j = 2}^{64} {\left[ {\left( {\nabla _{\mathrm{H}}{\mathbf{X}}_{\mathrm{F}}\left( {i,j} \right)} \right)^2 + \left( {\nabla _{\mathrm{V}}{\mathbf{X}}_{\mathrm{F}}\left( {i,j} \right)} \right)^2} \right]} } \right)^{1/2}$$
(1)

where H and V are the discretized gradient operators along the horizontal and vertical directions, respectively. At the ratio of r = 0.53 in Fig. 3c, the minimal TV was achieved, and the corresponding optimal fused image is shown in Fig. 3d. In addition, the SNR (see Supplementary Fig. S13) and directional spatial gradients29 of the fused image are also given in Fig. 3c. The directional spatial gradients of XF were used to quantify the spatial information of the fused image and were calculated by

$$G_{{\mathrm{row}}} = \left( {\mathop {\sum }\limits_i^{64} \mathop {\sum}\limits_{j = 2}^{64} {\left[ {\left( {\nabla _{\mathrm{H}}{\mathbf{X}}_{\mathrm{F}}(i,j)} \right)^2} \right]} } \right)^{1/2}$$
(2)
$$G_{{\mathrm{col}}} = \left( {\mathop {\sum }\limits_{i = 2}^{64} \mathop {\sum}\limits_j^{64} {\left[ {\left( {\nabla _{\mathrm{V}}{\mathbf{X}}_{\mathrm{F}}(i,j)} \right)^2} \right]} } \right)^{1/2}$$
(3)

As shown in Fig. 3c, the fused image at r = 0.53 shows more homogeneous spatial information in the vertical and horizontal directions, as visually presented in Fig. 3d, which proves that the polarization impacts are effectively overcome and that the original object’s morphology is better reproduced.

### Time-of-fight topography

Since the THz emission from the STEA used in our GHOSTEAM system is coherent, TOF measurements30 of THz pulses from the STEA can be adopted to enable microscopic topography of objects with depth resolution. TOF topography utilizes the optical path differences among different media to acquire the depth (z axis) distribution of a 3D object. For the sake of illustration, a 3D object made of silica by fs laser manufacturing (see “Materials and methods”) with three SiO2/air interfaces with air depths of 0, 100 and 200 μm (illustrated in Fig. 4a) was used to demonstrate TOF microscopic topography using the GHOSTEAM system. The structured silica was attached to the external surface of the STEA. Transient THz waveforms from the STEA that passed through the object within and outside the region of the trilayer structure were measured and are shown in Fig. 4b. Three THz ghost images relevant to the three interfaces, under the field of view FOV1, were measured at fixed time delays of 0, −0.33 and −0.66 ps (indicated by the three solid circles in Fig. 4b), as shown in Fig. 4c–e. The reconstructed images agree with the simulated electrical field distributions (Fig. 4f–h; see “Materials and methods” for simulation details). The subwavelength structures (scale of ~λ0/6) of the three interfaces are well resolved both experimentally and theoretically (note that diffraction leads to distortion in reconstruction of the middle interface).

## Discussion

In conclusion, we have presented a novel GHOSTEAM system for THz wave near-field microscopic topography. Compared with other existing THz wave near-field imaging systems, the GHOSTEAM system utilizes a real-time, reconfigurable, coherent STEA for structured near-field illumination. It was numerically and experimentally demonstrated that such a system can feature both micrometre-scale-resolved microscopy (≤6.5 μm) and depth-resolved topography. Further optimized optical projection imaging systems could enable submicrometre-scale resolution. Polarization-free subdiffraction-limited imaging was also achieved owing to the flexible tunability of the STEA.

The capacity of an STE to provide efficient THz pulses would further enable label-free cellular imaging or ultraprecise topography, a feature lacking in the current THz near-field imaging systems. STEs fabricated on flexible substrates31 would extend our GHOSTEAM system to imaging of objects with curved surfaces. Oscillator fs lasers have been demonstrated to drive STEs with a dynamic range above 60 dB (1000:1)26. Taking advantage of the MHz repetition rate and stable pulse energy of oscillators, the GHOSTEAM system is endowed with a much faster acquisition speed (in principle, an 18-fold improvement can be expected; see “Materials and methods” and Supplementary Fig. S9 for the estimation) than all other existing THz wave subdiffraction ghost imaging systems11,12,13,14,16,17, which require an amplified fs laser with a few-kilohertz repetition rate, leading to a slow acquisition speed and a complex system. In addition, using a shorter fs pulse and a wide-band detector, the ultrabroadband property of STEs, fully covering 0.1–30 THz frequencies24, has great potential for broadband THz applications. In addition, the GHOSTEAM system is compatible with compressive sensing (see Supplementary GIFs. 15), adaptive sampling12,32, parallel acquisition in k-space33 and coherent time-resolved full-wave detection in the time domain16,17.

## Materials and methods

### Experimental details

THz pulses (0.2–1.7 THz with a central frequency of 0.5 THz) were generated in W(2 nm)/Fe(2 nm)/Pt(2 nm) trilayer heterostructures [placed in a rotatable dc field (|B| = 80 mT)] by an 800-nm laser (duration of 90 fs, repetition rate of 1 kHz, and pump pulse energy of 20 μJ). The THz signals were electrooptically sampled by 1-mm-thick ZnTe (110) in combination with a balanced detector and were eventually recorded by a lock-in amplifier with an integration time of 100 ms (see Supplementary Fig. S4 for details). All the THz ghost images were multiplexed using the Walsh–Hadamard matrix and operated at respective fixed time delays (e.g., the peak time delays for Fig. 2b–d and appropriate time delays for Fig. 4c–e indicated in Fig. 4b). The acquisition time for each mask was 2 s, and the imaging time for each 64 × 64 ghost image (Fig. 2b-d) was ~4.5 h.

To optimize the wavefront of fs pulses, three steps were adopted to determine the positions of DMD1 and DMD2 (Wintech DMD4500, which contains 1140 × 912 diamond-arrayed micromirrors with a tilt angle of 12 ± 1° and a micromirror pitch of 7.6 μm). First, we acquired the one-to-one image of DMD2 by placing a CCD (see Supplementary Fig. S4c) at the presupposed position of DMD1 to adjust and determine the positions of DMD2 and two lenses (focal lengths of 100 mm). Second, by adjusting the position of DMD1, it could be determined when the images (at the presupposed position of the STE) shaped by DMD1 and DMD2 individually were simultaneously sharp. Third, their positions were accurately adjusted by measuring the waveforms of the Walsh–Hadamard masks of #2p and #2n (see Supplementary section 3) until reaching the minimal peak time delay difference between the two waveforms. The tilt angle difference between DMD1 and DMD2 was measured as 0.40° by comparing the zero-order diffraction angles of the two DMDs in the case of normal incidence.

### Computational ghost imaging

Let O represent the pixelated imaging target, consisting of N unknown elements O(i) at pixel i. O is a vector reshaped from the initial L × L image matrix Om, where L × L = N. The DMD is used to display the Walsh–Hadamard masks ϕ1, ϕ2, …, ϕN in sequence. ϕi with mask number i (1 ≤ i ≤ N) is an L × L matrix reshaped from the ith row of the N-order Walsh–Hadamard matrix Φ. Then, the correlation between the mask and object, which is recorded by a single-pixel detector, can be mathematically described by their inner product

$$y_i = < {\mathbf{\varphi }} _i,{\mathbf{O}} >$$
(4)

where φi is an N-length vector reshaped from ϕi. The complete measurement vector is then given by

$${\mathbf{Y}} = {\mathbf{\Phi }} {\mathbf{O}}$$
(5)

In the experiments, the Walsh-Hadamard matrix consisting of “+1” and “−1” elements was realized by

$${\mathbf{\Phi}} = {\mathbf{\Phi}} ^{\left( {{p}} \right)} - {\mathbf{\Phi}} ^{\left( {{n}} \right)}$$
(6)

since the DMD can only modulate the amplitude of the incident light. In Equation (6), Φ(p) is constructed by substituting all “−1” elements in Φ with “0”. Φ(n) is acquired by Φ(p)Φ.

Ultimately, the ghost image can be reconstructed by

$${\mathbf{X}} = {\mathbf{\Phi}} ^{ - 1}{\mathbf{Y}} = {\mathbf{\Phi }} ^{ - 1}\left( {{\mathbf{\Phi}} {\mathbf{O}}} \right) = {\mathbf{O}}$$
(7)

### Spatiotemporal THz waveform mapping

Let E(ξ, t) represent the THz spatiotemporal waveform to be measured, where ξ and t represent the spatial and temporal coordinates, respectively. E(ξ, t) consists of N time-dependent vectors

$${\boldsymbol{E}}\left( {\xi ,t} \right) = \left| {\begin{array}{*{20}{c}} {{\boldsymbol{E}}_1\left( t \right)} \\ {{\boldsymbol{E}}_2\left( t \right)} \\ \vdots \\ {{\boldsymbol{E}}_N\left( t \right)} \end{array}} \right|$$
(8)

The sequentially recorded signals S(ξ,t) can be written as

$${\boldsymbol{S}}\left( {\xi ,t} \right) = \left| {\begin{array}{*{20}{c}} {{\boldsymbol{s}}_1\left( t \right)} \\ {{\boldsymbol{s}}_2\left( t \right)} \\ \vdots \\ {{\boldsymbol{s}}_N\left( t \right)} \end{array}} \right| = {\mathbf{\Phi }}{\boldsymbol{E}}\left( {\xi ,t} \right)$$
(9)

Once the complete measurements are obtained, E(ξ, t) can be calculated as

$${\boldsymbol{E}}\left( {\xi ,t} \right) = {\mathbf{\Phi }}^{ - 1}{\boldsymbol{S}}\left( {\xi ,t} \right)$$
(10)

In the experiments, the ith mask had N = 64 identical rows (columns) for spatiotemporal waveform mapping in the vertical (horizontal) direction, and every row (column) of the mask was the ith row of Φ. The spatial and temporal resolutions were 13 μm and 33 fs, respectively, for both Fig. 1c, d. The wavefront was then regarded as a linear fit of the peak time delay in Ei(t) to the corresponding spatial coordinate ξ (see Supplementary Fig. S5).

### Estimation of the SNR and potential frame rate of ghost images

The SNRs of ghost images obtained via Hadamard multiplexing (Fig. 1f) were estimated by the following equation (see Supplementary section 4 for the detailed mathematical derivation):

$${\mathrm{SNR}}_{\mathrm{H}} = \sqrt {\frac{{\it{k}}}{{2{\boldsymbol{N}}\left[ {{\it{\upgamma }}_{\mathbf{d}}^2 + \left( {{\it{\upgamma }}_{\mathbf{s}}/2} \right)^2} \right]}}}$$
(11)

where γd denotes the ratio of the dark noise to the THz peak, γs denotes the ratio of the peak root mean square error to the THz peak, N denotes the pixel number of the ghost image and k denotes the number of measurements for each mask. In our experimental setup, γd was measured as 1 × 10−3, and γs was measured as ~7 × 10−3, as indicated in Fig. 1f. Each mask value was averaged for k = 15. With these parameters, the SNRH was estimated as 11.76 in the case of N = 64 × 64, according to Eq. (11).

The frame rate equals FPS = 1/(2Nrctmask), where tmask represents the acquisition time for each mask and rc represents the compressive ratio. tmask can be expressed as

$$t_{{\mathrm{mask}}} = 2N\left( {\gamma _{{\mathrm{d}}0}^2 + \gamma _{{\mathrm{s}}0}^2/4} \right)t_0{\mathrm{SNR}}_{\mathrm{H}}^2$$
(12)

where t0 denotes the pulse period and γd0 and γs0 denote the ratio of the dark noise to a single THz peak and the pulse fluctuation ratio within the “integration time” of t0, respectively. For an 80-MHz-oscillator-driven GHOSTEAM system, reasonable values are t0 = 12.5 ns, γd0 = 2.83 (corresponding to a dark-noise-to-peak ratio of γd = 1 × 10−3)26, and γs0 = 3.8 × 10−3 (see Supplementary Fig. S8). With these parameters and SNRH = 11.76, the acquisition time 2Ntmask for a 64 × 64 ghost image was calculated as ~15 min, yielding an imaging speed improvement by a factor of ~18 (4.5 h/15 min).

### Fs laser manufacturing

The silica sample with three air/silica interfaces for topography was manufactured by the femtosecond laser ablation method. In this method, a Ti-sapphire laser beam (800-nm central wavelength, 30-fs pulse width, 1-kHz repetition rate and 100-mW average power) was focused onto a silica substrate (1-mm thickness) using an objective lens (10×, numerical aperture NA = 0.25). The silica substrate travelled along multicircular trajectories at a speed of 100 μm/s (original radius r0 was 10 μm, interval between two adjacent circular trajectories Δr was 10 μm, travel number nr was 10, and the radius of the inner circular was r = r0 + Δr(nr − 1) = 100 μm). The laser ablation depth under the above parameters was ~50 μm, and the same ablation process was repeated to obtain an ablation depth of 100 μm, while the silica substrate was moved 50 μm up from the laser spot. Using the same laser fabrication method, the outer circular was produced with R0 = 250 μm, ΔR = 10 μm, nr = 15, and R = R0 + ΔR(nr − 1) = 400 μm. The process was repeated to obtain a deeper interface.

### Denoising and simulating near-field ghost topography

The results of near-field ghost topography (Fig. 4c–e) were denoised using the MATLAB toolbox Stationary Wavelet Transform Denoising 2-D. A five-level Haar wavelet was used to decompose the images. The selected threshold method was penalized low soft thresholding.

The electromagnetic field distributions as the THz pulse propagates in the near field (Fig. 4f–h) were simulated using the Wave Optics module of the commercial software COMSOL Multiphysics. Certain THz waves acquired in the experiment were set as the incident source, and the time-dependent solver was used to resolve the electromagnetic field distributions at discrete times (the time step was 33 fs). The study domain was a rectangle that was divided into silica (nsilica = 1.97) and air (nair = 1), as illustrated in Fig. 4a.