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# 7.4 Power  (Page 4/8)

 Page 4 / 8

Calculate the power output needed for a 950-kg car to climb a $2.00\text{°}$ slope at a constant 30.0 m/s while encountering wind resistance and friction totaling 600 N.

A man of mass 80 kg runs up a flight of stairs 20 m high in 10 s. (a) how much power is used to lift the man? (b) If the man’s body is 25% efficient, how much power does he expend?

a. 1.57 kW; b. 6.28 kW

The man of the preceding problem consumes approximately $1.05\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{J}$ (2500 food calories) of energy per day in maintaining a constant weight. What is the average power he produces over a day? Compare this with his power production when he runs up the stairs.

An electron in a television tube is accelerated uniformly from rest to a speed of $8.4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{m/s}$ over a distance of 2.5 cm. What is the power delivered to the electron at the instant that its displacement is 1.0 cm?

$6.83\text{μW}$

Coal is lifted out of a mine a vertical distance of 50 m by an engine that supplies 500 W to a conveyer belt. How much coal per minute can be brought to the surface? Ignore the effects of friction.

A girl pulls her 15-kg wagon along a flat sidewalk by applying a 10-N force at $37\text{°}$ to the horizontal. Assume that friction is negligible and that the wagon starts from rest. (a) How much work does the girl do on the wagon in the first 2.0 s. (b) How much instantaneous power does she exert at $t=2.0\phantom{\rule{0.2em}{0ex}}\text{s}$ ?

a. 8.51 J; b. 8.51 W

A typical automobile engine has an efficiency of 25%. Suppose that the engine of a 1000-kg automobile has a maximum power output of 140 hp. What is the maximum grade that the automobile can climb at 50 km/h if the frictional retarding force on it is 300 N?

When jogging at 13 km/h on a level surface, a 70-kg man uses energy at a rate of approximately 850 W. Using the facts that the “human engine” is approximately 25% efficient, determine the rate at which this man uses energy when jogging up a $5.0\text{°}$ slope at this same speed. Assume that the frictional retarding force is the same in both cases.

1.7 kW

A cart is pulled a distance D on a flat, horizontal surface by a constant force F that acts at an angle $\theta$ with the horizontal direction. The other forces on the object during this time are gravity ( ${F}_{w}$ ), normal forces ( ${F}_{N1}$ ) and ( ${F}_{N2}$ ), and rolling frictions ${F}_{r1}$ and ${F}_{r2}$ , as shown below. What is the work done by each force?

Consider a particle on which several forces act, one of which is known to be constant in time: ${\stackrel{\to }{F}}_{1}=\left(3\phantom{\rule{0.2em}{0ex}}\text{N}\right)\stackrel{^}{i}+\left(4\phantom{\rule{0.2em}{0ex}}\text{N}\right)\stackrel{^}{j}.$ As a result, the particle moves along the x -axis from $x=0$ to $x=5\phantom{\rule{0.2em}{0ex}}\text{m}$ in some time interval. What is the work done by ${\stackrel{\to }{F}}_{1}$ ?

$15\phantom{\rule{0.2em}{0ex}}\text{N}·\text{m}$

Consider a particle on which several forces act, one of which is known to be constant in time: ${\stackrel{\to }{F}}_{1}=\left(3\phantom{\rule{0.2em}{0ex}}\text{N}\right)\stackrel{^}{i}+\left(4\phantom{\rule{0.2em}{0ex}}\text{N}\right)\stackrel{^}{j}.$ As a result, the particle moves first along the x -axis from $x=0$ to $x=5\phantom{\rule{0.2em}{0ex}}\text{m}$ and then parallel to the y -axis from $y=0$ to $y=6\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$ What is the work done by ${\stackrel{\to }{F}}_{1}$ ?

Consider a particle on which several forces act, one of which is known to be constant in time: ${\stackrel{\to }{F}}_{1}=\left(3\phantom{\rule{0.2em}{0ex}}\text{N}\right)\stackrel{^}{i}+\left(4\phantom{\rule{0.2em}{0ex}}\text{N}\right)\stackrel{^}{j}.$ As a result, the particle moves along a straight path from a Cartesian coordinate of (0 m, 0 m) to (5 m, 6 m). What is the work done by ${\stackrel{\to }{F}}_{1}$ ?

$39\phantom{\rule{0.2em}{0ex}}\text{N}·\text{m}$

what is electromagnetism
It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
Energy
what is units?
units as in how
praise
What is th formular for force
F = m x a
Santos
State newton's second law of motion
can u tell me I cant remember
Indigo
force is equal to mass times acceleration
Santos
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass that is f=ma
David
The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
how can one calculate the value of a given quantity
means?
Manorama
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
AMIT
meaning
Winford
briefly discuss rocket in physics
ok let's discuss
Jay
What is physics
physics is the study of natural phenomena with concern with matter and energy and relationships between them
Ibrahim
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
L= 0.002H
NNAEMEKA
how did you get it?
Favour
is the magnetic field of earth changing
what is thought to be the energy density of multiverse and is the space between universes really space
tibebeab
can you explain it
Guhan
Energy can not either created nor destroyed .therefore who created? and how did it come to existence?
this greatly depend on the kind of energy. for gravitational energy, it is result of the shattering effect violent collision of two black holes on the space-time which caused space time to be disturbed. this is according to recent study on gravitons and gravitational ripple. and many other studies
tibebeab
and not every thing have to pop into existence. and it could have always been there . and some scientists think that energy might have been the only entity in the euclidean(imaginary time T=it) which is time undergone wick rotation.
tibebeab
What is projectile?
An object that is launched from a device
Grant
2 dimensional motion under constant acceleration due to gravity
Awais
Not always 2D Awais
Grant
Awais
why not? a bullet is a projectile, so is a rock I throw
Grant
bullet travel in x and y comment same as rock which is 2 dimensional
Awais
components
Awais
no all pf you are wrong. projectile is any object propelled through space by excretion of a force which cease after launch
tibebeab
for awais, there is no such thing as constant acceleration due to gravity, because gravity change from place to place and from different height
tibebeab
it is the object not the motion or its components
tibebeab
where are body center of mass on present.
on the mid point
Suzana
is the magnetic field of the earth changing?
tibebeab
does shock waves come to effect when in earth's inner atmosphere or can it have an effect on the thermosphere or ionosphere?
tibebeab
and for the question from bal want do you mean human body or just any object in space
tibebeab
A stone is dropped into a well of 19.6m deep and the impact of sound heared after 2.056 second ,find the velocity of sound in air.
9.53 m/s ?
Kyla
In this case, the velocity of sound is 350 m/s.
Zahangir
why?
Kyla
some calculations is need. then you will get exact result.
Zahangir
i mean how? isn't it just a d over t?
Kyla
calculate the time it takes the stone to hit the ground then minus the stone's time to the total time... then divide the total distance by the difference of the time
Snuggly
awit lenard. Hahahah ari ga to!
Kyla


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# Ani Barseghyan's Publications (Full profile)

### Paper

1. Barsegyan, A. G., On integral equations whose kernels are homogeneous functions of degree $(-1)$. (Russian), Izv. Nats. Akad. Nauk Armenii Mat. 53 (2018), no. 1, 23–36; translation in J. Contemp. Math. Anal. , 53, (1), pp. 47, (2018)
2. Barsegyan, A. G., Remarks on the solvability of a convolution integral equation on a finite interval, Differential Equations (translation from Differ. Uravn. 53, No. 3, 433-437, 2017), 53, (3), pp. 424, (2017)
3. A. G. Barseghyan, On approximate solution of the Dixon integral equation and some its generalizations , Computational Mathematics and Mathematical Physics , 57, (7), pp. 1158, (2017)
4. Barseghyan, A. G., On the method of two-sided continuation of solutions of the integral convolution equation on a finite Interval , Mathematical Notes (translated from Mat. Zametki 97 (2015), no. 3, 323–335), 97, (3), pp. 309, (2015)

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Session 4 - SOHO Corona II.
Oral session, Friday, June 27
Ballroom A, Chair: Therese Kucera

## [4.02] Thermal Coupling of Protons and Neutral Hydrogen in the Fast Solar Wind

L. Allen, S. R. Habbal (Harvard-Smithsonian Center for Astrophysics)

Motivated by the recent Spartan and UVCS observations [Kohl et al. 1996] of hot protons with temperatures exceeding 4 \times 10^6 K below 3.5 R_s in coronal holes, as inferred from the measured broadening of the Lyman \alpha spectral line profile, we studied the thermal coupling of neutral hydrogen to protons in the presence of Alfvén waves in the solar wind. The approach used is adopted from Olsen et al. [1994] in which the neutral hydrogen atoms are treated as test particles in a background electron-proton solar wind. The model computations show that an anisotropy in the neutral hydrogen temperature in the directions parallel and perpendicular to the magnetic field develops in the inner corona well below 5 R_s for background solar wind solutions consistent with observational constraints of the high speed wind. In particular, we find that the neutral hydrogen temperature parallel to the magnetic field direction remains strongly coupled to the proton temperature, T_p, while the perpendicular neutral hydrogen temperature exceeds this by \sim 10^6 K for a wide range of proton flow speeds, densities and temperatures for a spectrum of Alfvén waves. The neutral hydrogen effective temperature, T_H(eff)^\perp, incorporating both random thermal motion and wave motion of the particles, is found to be independent of frequency and significantly less than the proton effective temperature, T_p(eff), in the inner corona. Thus, without additional information about the waves, which would allow T_H^\perp and T_p to be extracted from the models, T_H(eff)^\perp provides an upper limit on T_p and a lower limit on T_p(eff). However, with increasing proton temperature, the anisotropy in the inner corona decreases, with a temperature difference of < 8 \times 10^5 K between the protons and neutrals below 3 R_s when the latter reach 6 \times 10^6 K.


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# Doubly Sour

Chemistry Level 3

A diprotic acid $\text{H}_2\text{X}$ has $K_1 = 2.0\cdot 10^{-6}$ and $K_2 = 5.0\cdot 10^{-8}$.

If this acid is present in a solution of $\text{pH } = 7$, determine the ratio $\dfrac{[\text{X}^{2-}]}{[\text{H}_2\text{X}]}.$

×

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# Analytic Basis Expansions for Functional Snippets

Estimation of mean and covariance functions is fundamental for functional data analysis. While this topic has been studied extensively in the literature, a key assumption is that there are enough data in the domain of interest to estimate both the mean and covariance functions. In this paper, we investigate mean and covariance estimations for functional snippets in which observations from a subject are available only in an interval of length strictly (and often much) shorter than the length of the whole interval of interest. For such a sampling plan, no data is available for direct estimation of the off-diagonal region of the covariance function. We tackle this challenge via an analytic basis representation of the covariance function. The proposed approach allows one to consistently estimate an infinite-rank covariance function from functional snippets. Moreover, the convergence rate is shown to be nearly parametric. This unusually high convergence rate is attributed to the analytic assumption on the covariance function, which is imposed to identify the covariance function. We also illustrate the finite-sample performance via simulation studies and an application to spinal bone mineral density.

## Authors

• 15 publications
• 2 publications
• 7 publications
• ### Basis Expansions for Functional Snippets

Estimation of mean and covariance functions is fundamental for functiona...
05/16/2019 ∙ by Zhenhua Lin, et al. ∙ 0

• ### Mean and Covariance Estimation for Functional Snippets

We consider estimation of mean and covariance functions of functional sn...
06/05/2020 ∙ by Zhenhua Lin, et al. ∙ 0

• ### Recovering the Underlying Trajectory from Sparse and Irregular Longitudinal Data

In this article, we consider the problem of recovering the underlying tr...
09/17/2018 ∙ by Yunlong Nie, et al. ∙ 0

• ### Supervised Functional PCA with Covariate Dependent Mean and Covariance Structure

Incorporating covariate information into functional data analysis method...
01/30/2020 ∙ by Fei Ding, et al. ∙ 0

• ### Estimation of the Mean Function of Functional Data via Deep Neural Networks

In this work, we propose a deep neural network method to perform nonpara...
12/08/2020 ∙ by Shuoyang Wang, et al. ∙ 8

• ### Estimation of Smooth Functionals in Normal Models: Bias Reduction and Asymptotic Efficiency

Let X_1,..., X_n be i.i.d. random variables sampled from a normal distri...
12/18/2019 ∙ by Vladimir Koltchinskii, et al. ∙ 0

• ### New conformal map for the Sinc approximation for exponentially decaying functions over the semi-infinite interval

The Sinc approximation has shown its high efficiency for numerical metho...
12/30/2018 ∙ by Tomoaki Okayama, et al. ∙ 0

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1 Introduction

Nowadays functional data are commonly encountered in practice, due to the advances in modern science and technology that enhance capability of data collection, storage and processing. Both unsupervised learning approaches, such as dimension reduction via functional principal component analysis

(Rao, 1958; Dauxois et al., 1982; Hall and Hosseini-Nasab, 2009; Mas and Ruymgaart, 2015)

, and supervised learning, such as functional regression

(Cardot et al., 1999; Müller and Stadtmüller, 2005; Ferraty and Vieu, 2006; Hall and Horowitz, 2007; Müller and Yao, 2008; Kong et al., 2016) are well studied in the literature. For a comprehensive treatment of these subjects, we recommend the monographs Ramsay and Silverman (2005), Ferraty and Vieu (2006), Horváth and Kokoszka (2012), Hsing and Eubank (2015) and Kokoszka and Reimherr (2017), and the review papers Wang et al. (2016) and Aneiros et al. (2019).

Critical to the statistical analysis of such data is the estimation of the mean and covariance functions, since they are the foundation of the aforementioned unsupervised and supervised learning tasks. In reality, functions can only be recorded on a set of discrete points of the domain of the functions. Estimation of the mean and covariance functions in this context has been extensively studied by Rice and Silverman (1991), Cardot (2000), James et al. (2000), Cai and Yuan (2010), Cai and Yuan (2011), Yao et al. (2005), Li and Hsing (2010) and Zhang and Wang (2016), among many others. In addition to the discrete nature of observed functional data, subjects often stay in the study only for a subject specific period that is much shorter than the span of the whole study. This does not cause much problems for mean estimation but brings challenges to covariance estimation.

For illustration, without loss of generality, we assume that the domain of the functional data is the unit interval and each subject only stays in the study for a period of length . Data of this characteristics are termed “functional snippets”, which is analogous to the longitudinal snippets analyzed in Dawson and Müller (2018). For such data, there is no information in the off-diagonal region of the covariance function Cov, and therefore there is no local information available for estimating the covariance function in this region. Figure 2(a) illustrates such a situation for the bone mineral density data in Section 5.

The literature of statistical analysis for functional snippets is in its infancy. Delaigle and Hall (2016)

proposed to approximate snippets by segments of Markov chains. Such a method is only valid at the discrete level, as shown by

Descary and Panaretos (2019). More recently, to analyze functional snippets, Zhang and Chen (2018) and Descary and Panaretos (2019) used matrix completion techniques that innately work on a common grid of the domain . While their methods might be strained to deal with snippets recorded on random and irregular design points, which is often the actual data design, their theoretical analysis only covers the case of the regular design. In addition, Descary and Panaretos (2019) assumed a finite-rank condition on the covariance function. Such condition essentially entails that functional data are of fixed and finite dimension, which does not conform with the common paradigm for functional data, which regards functional data as genuinely infinite-dimensional processes. Other related works such as Liebl (2013), Gellar et al. (2014), Goldberg et al. (2014), Kraus (2015), Gromenko et al. (2017), Kraus and Stefanucci (2019), Stefanucci et al. (2018), Mojirsheibani and Shaw (2018), Kneip and Liebl (2019) and Liebl and Rameseder (2019), although considering partially observed functional data, do not specifically address functional snippets as characterized above. For example, in Kraus (2015) and other works, information and design points for the off-diagonal region are still observable. In contrast, information for that region is completely missing in the context of functional snippets, which significantly elevates the difficulty of statistical analysis. Thus, this is an extrapolation problem, which requires strong assumptions to overcome. Lin and Wang (2019)

adopt a parametric assumption on the correlation function, while allowing the variance function to be estimated nonparametrically. However, the parametric assumption on the correlation can only be validated on the region where data are available.

In this paper, we propose to approach functional snippets from the perspective of basis expansion. The main idea is to represent the covariance function by basis functions composed from tensor product of analytic orthonormal functions defined on

. Unlike local bases such as B-spline studied in Rice and Wu (2001); Ramsay and Silverman (2005), analytic bases are global, in the sense that they are independent of local information such as knots or design points and completely determined by their values on a countably infinite subset of the interval . This feature of analytic bases allows information to pass from the diagonal region to the off-diagonal region along the basis functions. With this analytic assumption on the covariance function, the missing pieces of the covariance function can then be inferred from the data available in the diagonal region. In contrast, this is generally impossible for B-spline or other local bases. Moreover, the analytic assumption facilitates a much faster convergence rate of the proposed covariance estimation for sparsely observed function snippets as compared to conventional covariance estimation under less stringent smoothness assumptions. While this is not totally surprising, the result, to the best of our knowledge, is the first of its kind for sparsely observed functional data.

A distinct feature of the proposed method is its capacity to analyze genuine functional data of infinite dimension in the context of snippets. This is demonstrated under the umbrella of a certain family of analytic functions, the bounded sequential compact family (BSC family), defined in Section 3. We show that, when the covariance function comes from an analytic BSC family, the proposed method is able to estimate it at a nearly parametric rate based on snippets that are irregularly and/or sparsely observed. As an analytic BSC family can contain covariance functions of infinite rank, the proposed approach can process functional data of infinite dimensions. This separates our work from existing ones that only deal with functional snippets with a finite-rank covariance function or a parametric correlation function. As another contribution, we propose a penalized basis approach to estimate the mean function and thoroughly analyze its asymptotic properties over a class of bases. To the best of our knowledge, this has not been explored in the literature yet.

The rest of the paper is organized as follows. Section 2 details the proposed estimators for the mean function and covariance function. Their theoretical properties are investigated in Section 3, while their numerical performance is assessed in Section 4. The application to a real dataset is discussed in Section 5.

## 2 Methodology

Let be a second-order stochastic process on a compact interval , which without loss of generality is taken to be . The mean and covariance functions of are defined as and respectively. The observed functions are statistically modeled as independent and identically distributed realizations of . In practice, each realization is only recorded at subject-specific time points with measurement errors. More precisely, for functional snippets, the observed data are pairs , where

 Yij=Xi(Tij)+εij,i=1,…,n,j=1,…,mi, (1)

is the random noise with mean zero and unknown variance , and there is a for which for all , and . The focus of this paper is to estimate the mean and covariance functions of using these data pairs .

### 2.1 Mean Function

Although functional snippets pose a challenge for covariance estimation, they usually do not obstruct mean estimation, since data for the estimation are available likely across the whole domain of interest. In light of this observation, traditional methods such as local linear smoothing (Yao et al., 2005; Li and Hsing, 2010) can be employed. Below we adopt an analytic basis expansion approach. The advantage of this approach is its computational efficiency and adaptivity to the regularity of the underlying mean function ; see also Section 3.2.

Let be a complete orthonormal basis of that consists of squared integrable functions defined on the interval . When , it can be represented by the following series in terms of the basis ,

 μ0(t)=∞∑k=1akϕk(t),

where . In practice, one often approximates such series by its first leading terms, where is a tuning parameter controlling the approximation quality. The coefficients are then estimated from data by penalized least squares. Specifically, with the notation and , the estimator of is given by

 ^a=argmina∈Rq{n∑i=1vimi∑j=1[Yij−a⊤Φq(Tij)]2+ρH(a⊤Φq)}, (2)

and is estimated by , where the weights satisfy , represents the roughness penalty, and is a tuning parameter that provides trade-off between the fidelity to the data and the smoothness of the estimate. There are two commonly used schemes for the weights, equal weight per observation (OBS) and equal weight per subject (SUBJ), of which the weights are and , respectively. Alternative weight schemes can be also found in Zhang and Wang (2016, 2018).

The penalty term in (2) is introduced to prevent excessive variability of the estimator when a large number of basis functions are required to adequately approximate

and when the sample size is not sufficiently large. In the asymptotic analysis of

in Section 3, we will see that this penalty term contributes little to the convergence rate of when the tuning parameter is not too large. In our study, the roughness penalty is given by

 H(g)=∫10{g(2)(t)}2dt,

where denotes the second derivative of . In matrix form, for , it equals , where is a matrix with elements . The tuning parameters and can be selected by a cross-validation procedure.

### 2.2 Covariance Function

Since functional snippets do not provide any direct information at the off-diagonal region, the only way to recover the covariance in the off-diagonal region is to infer it from the diagonal region. Mathematically, this requires the covariance function to be identifiable from its diagonal region. It is well known that if a function is analytic in a connected and open domain, then it might be completely determined by its values at a countably infinite number of points. A function is analytic if it can be locally represented by a convergent power series. In fact, according to Corollary 1.2.7 of Krantz and Parks (2002), one has immediately the following assertion.

###### Proposition 1.

If and are two analytic functions on and there is a subset of countably infinite points such that

 γ1(s,t)=γ2(s,t), for all (s,t)∈S,

then

 γ1(s,t)=γ2(s,t), for all (s,t)∈[0,1]2.

Given the above observation, one is able to identify the whole covariance function based on its values in the diagonal region if it is analytic in . For the purpose of identifiability, we shall assume the analyticity of the covariance function in the sequel. Note that this assumption has also been adopted by Descary and Panaretos (2019) who also provided a nice example to illustrate that the infinitely differentiable function space is too large to ensure identifiability of based on data available in the diagonal region.

With the assumed analyticity, it is now possible to infer the off-diagonal region by the information contained in the raw covariance available only in the diagonal region. To leverage such information, both Zhang and Chen (2018) and Descary and Panaretos (2019) used matrix completion techniques. However, the method of Descary and Panaretos (2019) requires the rank of the covariance to be finite to ensure the identifiability based on discretized observations. Zhang and Chen (2018) allow the rank to increase slowly with the sample size, but for the theoretical derivation, require a less transparent condition that involves the rank; see also their Assumption 2.

We take a different approach to utilize the available data in the diagonal region without any assumption on the rank. Specifically, we propose to transport information from the diagonal region to the off-diagonal region through the basis functions , by approximating with

 γC(s,t)=∑1≤k,l≤pcklϕk(s)φl(t),(s,t)∈[0,1]2, (3)

where , is the matrix of coefficients , and is an integer. If we choose an analytic basis , then their values in the diagonal region completely determine their values in the off-diagonal region. When such a representation of the covariance function is adopted and the unknown coefficients are estimated from data, the information contained in the estimated coefficients extends from the diagonal region to the off-diagonal region through analyticity.

For the purpose of generality, we consider a family of analytic covariance functions defined on . The covariance function is then assumed to reside in . Examples of include the collection of analytic covariance functions of rank , and the collection of all analytic covariance functions. The former, considered by Descary and Panaretos (2019), is too restrictive to model functional data of infinite dimensions, while the latter is too large to provide a theoretical guarantee for the estimated covariance function. In Section 3 we will explore suitable families for .

To estimate the coefficients from data, we again adopt the idea of penalized least squares, where the squared loss of a given function is measured by the sum of weighted squared errors

 SSE(γ)=n∑i=1wi∑1≤j≠k≤mi{Γijl−γ(Tij,Til)}2,

where are the weights satisfying , while the roughness penalty is given by

 J(γ)=∬12⎧⎨⎩[∂2γ∂s2]2+2[∂2γ∂s∂t]2+[∂2γ∂t2]2⎫⎬⎭dsdt.

Then, one has

 J(γC)=tr(CUCW)+tr(CVCV), (4)

for as of (3), where denotes the matrix trace, and are matrices with elements and , respectively. The estimator of is then taken as with

 ^C=argminC:γC∈Cn∑i=1wi∑1≤j≠k≤mi{Γijl−γC(Tij,Til)}2+λJ(γC), (5)

where is the tuning parameter that provides trade-off between the fidelity to the data and the smoothness of the estimate.

Similar to (2), the penalty term in (5) does not contribute to the convergence rate of when the tuning parameter is not too large. In practice, the value of can be chosen via cross-validation. For the choice of the Zhang and Wang (2016) discussed several weight schemes, including the OBS scheme and the SUBJ scheme . An optimal weighing scheme was proposed in Zhang and Wang (2018); we refer to this paper for further details.

## 3 Theory

As functional snippets are often sparsely recorded, in the sense that for all and some , in this section we focus on theoretical analysis tailored to this scenario. For simplicity, we assume that the number of observations for each trajectory is equal, i.e., for all . Note that under this assumption, the SUBJ and OBJ schemes coincide. The results for general number of observations and weight schemes can be derived in a similar fashion. We start with a discussion on the choice of basis functions and then proceed to study the convergence rates of the estimated mean and covariance functions.

### 3.1 Analytic Basis

While all complete orthonormal bases can be used for the proposed estimators in (2), an analytic basis is required for the estimator in (5). For a clean presentation, we exclusively consider analytic bases that work for both (2) and (5). In this paper, a basis is called an analytic -basis if its basis functions are all analytic and satisfy the following properties: for some constants , there exists a constant such that and for and all . Here, denotes the supremum norm of , defined by , and represents the th derivative of . Throughout this paper, we assume that the basis is an analytic -basis.

Different bases lead to different convergence rates of the approximation to and . For the mean function , when using the first basis functions , the approximation error is quantified by

 E(μ0,Φ,q)=∥∥ ∥∥μ0−q∑k=1akϕk∥∥ ∥∥L2,

where we recall that . The convergence rate of the error , denoted by , signifies the approximation power of the basis for . Similarly, the approximation error for is measured by

 E(γ0,Φ,p)=∥∥ ∥∥γ0−p∑k=1p∑l=1cklϕk⊗ϕl∥∥ ∥∥L2,

where the norm of a function is defined by . The convergence rate of is denoted by . Below we discuss two examples of bases.

###### Example 1 (Fourier basis).

The Fourier basis functions, defined by , and for , constitute a complete orthonormal basis of . It is also an analytic -basis. When is periodic on , if exists and satisfies , then, according to Theorem 2.4.1 of Olson (2017), . If is analytic, then according to Proposition 5.4.1 of Krantz and Parks (2002), for some . Similarly, if is analytic and periodic, then for some . When or is not a periodic function, a technique called Fourier extension, briefly described in Appendix A, can be adopted to yield the same rates (Adcock et al., 2014). This technique is well studied in the field of computational physics (Boyd, 2002) and numerical analysis (Huybrechs, 2010) as a tool to overcome the so-called Gibbs phenomenon (Zygmund, 2003), but seems not well explored in statistics yet. In Section 4, we numerically illustrate the application of this technique.

###### Example 2 (Legendre polynomials).

The canonical Legendre polynomial of degree is defined on by

 Pk(t)=12kk!dkdtk{(x2−1)k}.

These polynomials are orthogonal in . By a change of variable and normalization, they can be turned into an orthonormal basis of . The Legendre polynomials have numerous applications in approximation theory and numerical integration; see Wang and Xiang (2011) and references therein. One can show that the Legendre basis is an analytic -basis. According to Theorem 2.3 of Canuto and Quarteroni (1982), we have when exists and satisfies . When is indeed analytic, then based on Proposition 2.5 of Wang and Xiang (2011), we have for some . Similarly, if is analytic, then for some .

### 3.2 Mean function

For functional snippets, we shall assume that the observations from a subject scatter randomly in a subject specific time interval, whose length is and whose middle point is called the reference time in this paper. We further assume that the reference time of the th subject are independently and identically distributed in the interval , and the observed time points , conditional on , are independently and identically distributed in the interval .

To study the property of the estimator , we make the following assumptions.

[labelwidth=1.2cm,leftmargin=1.4cm,align=left]

A.1

There exist such that the density of the reference time satisfies for any . There exist , such that the conditional density of the observed time satisfies , for any given reference time and .

[labelwidth=1.2cm,leftmargin=1.4cm,align=left]

A.2

for some constant .

[labelwidth=1.2cm,leftmargin=1.4cm,align=left]

A.3

and .

Assumption A.1 requires the density of the reference time and conditional densities of the time points to be bounded away from zero and infinity. This also guarantees that the marginal probability density of the time points

is bounded away from zero and infinity. In the sequel, we use to denote .

###### Theorem 1.

If is a -basis, under conditions (A.1)–(A.3), we have

 ∥^μ−μ0∥2L2=OP(q2α+1n+τ2q). (6)

We first note that, with condition A.3 on the tuning parameter , it does not impact the asymptotic rate of . We also observe that in (6), the term specifies the estimation error using a finite sample, while is the deterministic approximation error for using only the first basis functions. The latter term depends on the smoothness of . Intuitively, it is easier to approximate smooth functions with basis functions. For a fixed number of basis functions, the smoother the function is, the smaller is the approximation error. When is times differentiable, we have and this leads to the following convergence rate, using either the Fourier basis or the Legendre basis in this scenario.

###### Corollary 1.

Suppose exists and satisfies for some . Assume conditions (A.1)–(A.3) hold. If is the Fourier basis, then with the choice . If is the Legendre basis, then with the choice .

As discussed in Example 1 and 2, for an analytic function, the approximation error could be of an exponential order of . This leads to the following remarkable nearly parametric convergence rate.

###### Corollary 2.

Suppose that is analytic, conditions (A.1)–(A.3) hold, and . If is the Fourier basis, then . If is the Legendre basis, then .

The above corollaries show that the proposed analytic basis expansion approach automatically adapts to the smoothness of . This contrasts with the local polynomial smoothing method and the B-spline basis approach, for which the convergence rate is limited by the order of the polynomials or B-spline basis functions used in the estimation, even when might have a higher order of smoothness. In practice, it is not easy to determine the right order for these methods, since both the mean function and its smoothness are unknown. In particular, when is analytic, neither of these traditional methods can fully leverage the smoothness of . Therefore, the feature of automatic adaptivity makes the analytic basis expansion approach an attractive alternative to the classical smoothing methods.

### 3.3 Covariance function

In Section 2 we assumed resides in a family . We also mentioned that the family of analytic covariance functions of rank is too small to model genuine functional data which are conceptually of infinite dimension. Below we show that the proposed approach based on analytic basis expansions can recover infinite-rank covariance functions from functional snippets. To this end, we consider families larger than via sequential compactness, as follows.

We first recall that a topological space is sequentially compact if very sequence has a convergent subsequence converging to a point in the space. Let be the space of all real-valued functions defined on endowed with the product topology. In this topology, a sequence of functions converges to a limit if and only if

 limk→∞fk(s,t)=f(s,t),∀(s,t)∈T2.

In this paper, a subset of is called a bounded sequentially compact family (BSC family) if every sequence in that is uniformly bounded in the norm, i.e., for some , has a subsequence converging to a limit in in the product topology. Note that a sequentially compact subset is a BSC family.

In our development, we have assumed is analytic, i.e., , where denotes the collection of analytic functions defined on . To incorporate the analyticity, we consider analytic BSC families which are BSC families and also subsets of . We first observe that the family considered in Descary and Panaretos (2019) is a BSC family. The following example, exhibiting a BSC family that contains covariance functions of infinite rank, demonstrates that our approach covers infinite-rank covariance functions.

###### Example 3 (Functions with commonly bounded Lipschitz constant).

Let be fixed but potentially arbitrarily large constants. Let be the collection of analytic functions defined on such that if then and the Lipschitz constant of is no larger than , where the Lipschitz constant of is defined as . We claim that is BSC. To see this, we note that is locally equicontinuous, and also the set is bounded for all . Then the claim follows from Arzelà-Ascoli Theorem in Chapter 7 of Remmert (1997). This family clearly contains countless covariance functions of infinite rank.

To derive the asymptotic property of the estimated covariance function, we shall assume that

[labelwidth=1.2cm,leftmargin=1.4cm,align=left]

B.1

The covariance function belongs to an analytic BSC family of covariance functions.

[labelwidth=1.2cm,leftmargin=1.4cm,align=left]

B.2

The random function satisfies that .

[labelwidth=1.2cm,leftmargin=1.4cm,align=left]

B.3

and as .

Since an analytic BSC family, such as in Example 3, may contain covariance functions of infinite rank, the theory developed below applies to functional snippets of infinite dimensions. To avoid the entanglement with the error from mean function estimation, we shall assume that is known in the following discussion, noting that the case that is unknown can also be covered but requires a much more involved presentation and tedious technical details and thus is not pursued here.

###### Theorem 2.

If is an analytic -basis, under assumptions A.1 and B.1–B.3, we have

 ∥^γ−γ0∥2L2=OP(p4α+2n+κ2p). (7)

Note that, with the condition B.3 on , the tuning parameter does not impact the asymptotic rate of . As in the case of mean function, the rate in (7) contains two components, the estimation error stemming from the finiteness of the sample, and the approximation bias attributed to the finiteness of the number of basis functions being used in the estimation. We note that the approximation bias decays exponentially for both the Fourier basis and the Legendre basis. In particular, using analytic bases results in the following nearly parametric convergence rate.

###### Corollary 3.

Suppose assumptions A.1 and B.1–B.3 hold and . If is the Fourier basis and is periodic, then . If is the Legendre basis, then .

## 4 Simulation Studies

We now illustrate the numerical performance of the proposed approach using the Fourier basis. As mentioned in Example 1, with the Fourier extension technique to deal with nonperiodic functions, this basis can approximate an th-differentiable (analytic, respectively) function at the rate of ( for some , respectively) when the first basis functions are used. Moreover, it is a -basis and thus enjoys the favorable convergence rates established in Corollaries 1, 2 and 3. As the Fourier extension technique is not well studied in statistics, below we also demonstrate it numerically.

For the mean function, we consider two scenarios, and . The former is a periodic function while the latter is nonperiodic. For the covariance function, we take two functions from the class exhibited in Example 3. Specifically, we consider the periodic covariance function with and if and if , and the nonperiodic covariance function that is determined by a Matérn correlation function and the variance function , i.e.,

 γ2(s,t)=√1+2s√1+2t(√2|s−t|)B1(√2|s−t|),

where is the modified Bessel function of the second kind of order . In the evaluation of the performance for the mean function, the covariance function is fixed to be , while the mean function is fixed to be in the evaluation of the covariance function. This strategy avoids the bias from covariance to influence the estimation of the mean function, and vice versa.

The estimation quality is measured by the empirical mean integrated squared error (MISE) based on independent simulation replicates. For the mean estimator , the MISE is defined by

 MISE=1NN∑k=1∫{^μk(t)−μ0(t)}2dt,

and for the covariance estimator , it is defined by

 MISE=1NN∑k=1∬{^γk(s,t)−γ0(s,t)}2dsdt,

where and are estimators in the th simulation replicate. The tuning parameters , , , and the extension margin (see Appendix A) are selected by a five-fold cross-validation procedure.

In all replicates, the reference time

are sampled from a uniform distribution on

. The number of observations are independent and follow the distribution . The measurement noise

are i.i.d. sampled from a normal distribution with mean zero and variance

, where the noise level is set to make the signal-to-noise ratio . We consider three sample sizes, , and two different values, , representing short snippets and long snippets, respectively.

The results are summarized in Table 1 and 2 for mean and covariance functions, respectively. As expected, in all settings, the performance of the estimators improves as or becomes larger. We also observe that if the function to be estimated is periodic, like and , the performance seems marginally better without Fourier extension. However, if the function is nonperiodic, like and , then the estimators with Fourier extension considerably outperform those without the extension, especially for the mean function or when the sample size is large. This demonstrates that Fourier extension is a rather effective technique that complements the Fourier basis for nonparametric smoothing, and might deserve further investigation in the framework of statistical methodology.

## 5 Application: Spinal Bone Mineral Density

In the study of Bachrach et al. (1999), 423 individuals, age 8 to 27, were examined for their longitudinal spinal bone mineral density. The bone density of each individual was irregularly recorded in four consecutive years, at most once for each year. The data for each individual then lead to a functional snippet spanning at most 4 years. In our study, individuals who have only one measurement were excluded, since they do not carry information for the covariance structure. This results in a total of 280 individuals who have at least two measurements and whose ages range from 8.8 to 26.2.

We are interested in the mean and covariance structure of the mineral density. Figure 2(a) depicts the empirical design of the covariance function, underscoring the nature of these data as a collection of snippets: there is no data available to directly infer the off-diagonal region of the covariance structure. We also note that the design time points are irregular. This feature renders techniques based on matrix completion inapplicable since they require a regular design for the measurement time points. In contrast, our method is able to seamlessly accommodate this irregularity.

The mineral density data and the estimated mean are displayed in Figure 1(a) and 1(b), respectively. We observe that the mean density starts with a low level, rises rapidly before the age of 16 and then goes up relatively slowly to a peak at the age of around 20. This indicates that the spinal bone mineral accumulates fast during adolescence during which rapid physical growth and psychological changes occur, and then the density remains in a stable high level in the early 20s. From Figure 1(a), we see that observations are relatively sparse from age 23 to age 26, especially near the boundary at age 26. Therefore, we suspected that the upward trend around age 26 might be due to a boundary effect, in particular the marked rightmost point in Figure 1(a). To check this, we refitted the data with this point excluded. The refitted mean curve is in Figure 1(c), and indeed the upward trend disappears. The estimated covariance surface, after removing the rightmost point, is shown in Figure 2(b), which suggests larger variability of the data around the age of 17. It also indicates that the correlation of the longitudinal mineral density at different ages decays drastically as ages become more distant.

## Appendix A: Fourier Extension

For a nonperiodic function , its finite Fourier series expansion is observed to suffer from the so-called Gibbs phenomenon (Zygmund, 2003) which refers to the drastic oscillatory overshoot in the region close to two endpoints of the domain. A remedy to this issue is to employ the technique of Fourier extension mentioned in Example 1.

The idea of Fourier extension is to approximate a nonperiodic function by Fourier basis functions defined in an extended domain for some called extension margin in this paper, where the basis functions, are then defined by , and for . To elaborate, let . The Fourier extension of within , denoted by , is defined by

 ~gN=argminh∈GN(ζ)∥g−h∥L2(0,1),

where we note that the norm is for the domain , not the extended one. One can easily see that the Fourier extension of is not unique. However, they all have the same approximation quality for over the domain of interest. Intuitively, for a Fourier extension of , even if it has the Gibbs phenomenon, for a suitable , the phenomenon is expected to be restricted within the extended part of the domain, i.e., and . This way, the nonperiodic function can then be well approximated in the domain . Indeed, the speed of the convergence of a Fourier extension of in the domain adapts to the smoothness of (Adcock et al., 2014). For example, converges to at the rate of ( for some , respectively) when is th-differentiable (analytic, respectively).

The above discussion can be straightforwardly extended to the two-dimensional case. Let , where represents the function defined on the two-dimensional square . For a function defined on , its Fourier extension within is given by

 ~γN=argminh∈G2N(ζ)∥γ−h∥L2([0,1]2),

where denotes the space of squared integrable functions defined on with the norm .

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# Extent of multiple testing correction

A bit of a strange question. In my fourth year biostatistics class today, we were discussing when and when not to use multiple testing correction, and the professor made an offhand comment. He asked why we don't correct for every test we've ever done since we started doing statistics, since they are all (mostly) independent and each time we observe a result we increase our probability of drawing a false positive. He laughed it off afterwards, but why do we not do this? I'm not saying that we should, because obviously it is ludicrous, but how far is too far when it comes to correcting for tests?

We'll assume alpha = 0.05 for simplicity, and say that each test A, B, and C are not under any sort of dependency and thus independent. If I sit down and test A, B, and C, be they T tests or whatever, I obviously have to adjust for multiple correction because I am taking 0.95 to the power of three, and my chances of getting a false positive sky rocket. However, if I do A, B, and C on different days, within the contexts of different procedures, and draw different results from them, how is this any different than the former situation? We are still observing the three tests, they are still independent.

What I'm trying to get at is the logical boundary where we say to stop doing multiple testing correction. Should we only do it for one family of tests, or should we do it for a whole paper, or should we do it for every single test we've ever run?I understand how to use multiple testing correction, and use FDR / Bonferonni at work all the time. This concept just kind of took my head in circles.

Edit: There is extended discussion of this issue in a more recent question.

• see my comments in another post stats.stackexchange.com/questions/117735/… – Peter Oct 7 '14 at 13:18
• @Chris C; I think your question is related to stats.stackexchange.com/questions/164181/… – user83346 Aug 2 '15 at 5:23
• Related (almost duplicate): stats.stackexchange.com/questions/206592. – amoeba Apr 11 '16 at 17:39
• @amoeba Yea, I saw that one and wondered the same thing; they are essentially the exact same question. Think anything should be done about it? – Chris C Apr 11 '16 at 17:43
• I guess if somebody voted to close that other Q when it appeared it might have been closed, but by now I think the answers there are surpassing the answers here. So I am reluctant to vote to close that one as a duplicate. We can close your Q as a duplicate of that one though, or we can try to ask mods to merge one of the Qs into another (this means that the answers will get moved to the other thread). What do you think yourself? – amoeba Apr 11 '16 at 17:48

You may be asking why we should need to control the error rate on a per-experiment basis. Here is my opinion. Imagine that some NIH or FDA type institution mandate that you correct for every test you have ever done. Consider that you run a experiment with a single test, and that is your first experiment. No adjustment will be needed here. Now consider that you run a new experiment again with a single test, but this time it is your $1,000^{th}$ experiment. Then you would have to use $\alpha$ of 0.05/1,000 = 0.00005! Who would want to run any experiments with such a low $\alpha$? So my guess is that, when Tukey proposed the experiment-wise error rate, he may have wanted to be fair to each experiment, since each experiment takes money, time, and resources.


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# how to change the number of rendered frames in Blender

i´m using blender to render an animation in single pictures and i want to make an 16 frames animation out of an 24 frames one without changing the set keyframes. Is there an option to skip every 3rd frame and just render the rest? I only found the frame step option, which allows you to render every 3rd, but not 2 frames and skip the 3rd. If anyone could help, thank you very much!

• You want to remove frames and make the resulting animation choppy? Or you want to make the same animation play in less time? – MichaelHouse Nov 13 '12 at 14:05

Using the DopeSheet panel, select all the key frames that make up your 24 frames of animation. Then press s to enter scale mode. Scale the 24 frame animation to 16 frames.
• @BlenderNewbie Great. You can also press g to grab frames and move them around if needed. – MichaelHouse Nov 13 '12 at 14:10


text:
Looking for a version of Itô's Lemma

Hi Everyone

I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.

We are given a Brownian Motion $B_t$ and we note $M_t=\sup_{s\le t}B_s$. Moreover we have a smooth real valued function $F(t,x,y)$ (for example a $C^{1,2,1}$) over $\mathbb{R}^+\times \mathbb{R} \times \mathbb{R}^+$ and we are looking for the SDE followed by $F(t,M_t-B_t,M_t)$

I have a hard time trying to express $dF$.

Best regards

-
After two seconds of reflection the answer is simple. It suffices to view $F$ as a function of $B_t$ and of the two increasing functions $M_t$ and $t$ and to apply Itô's lemma (Revuz and Yor bok's version for example). Sorry for this post, I first thought some Itô-Tanaka-Meyer formula were involved because of the fact that the processes $(L_t,|W_t|)$ (L is the local time of W at 0) and $(M_t-B_t,M_t)$ have the same law. Regards – The Bridge Apr 22 '10 at 8:43
The Bridge: The MO bot bumped your question since in the system it remains "unanswered". Since zhoraster did provide an answer, it would be polite to accept it and award him or her the points. Plus, this will be useful for future generations of students who may ask the same question as you did. – Tom LaGatta Sep 8 '10 at 15:31

Let $F(t,m-b,m)=G(t,m,b)$ (this way it's easier to write). As long as $M$ has bounded variation, we can happily write (I skip arguments, which are $t,M_t,B_t$) $$dG(t,M_t,B_t) = G'_t dt + G'_m dM_t + G'_b dB_t + \frac12 G''_{bb} dt.$$


text:
DPS Meeting, Madison, October 1998
Session 31P. Comets II
Contributed Poster Session, Wednesday, October 14, 1998, 5:10-6:10pm, Hall of Ideas

## [31P.15] Molecular Line Mapping of Comets Hyakutake and Hale-Bopp

Amy J. Lovell, F. Peter Schloerb, James E. Dickens, Christopher H. DeVries, William M. Irvine, Matthew C. Senay (FCRAO)

We present a comprehensive set of spectral line maps of the molecular emission from Comets C/1996 B2 Hyakutake and C/1995 O1 Hale-Bopp, obtained with the Five College Radio Astronomy Observatory 14-m telescope. The comets were observed with the QUARRY focal plane array receiver, and the spectral line were assembled into a final data cube'' consisting of spectra at different positions within the coma. This final data set has a spatial resolution of approximately 80" and velocity resolution of approximately 0.1 km s-1. The HCN spectral lines in Comet Hyakutake during the near-perigee period revealed a systematic progression from asymmetric, single-peaked lines, to symmetric double-peaked lines. The median line velocity is blue-shifted approximately 0.1 km s-1, and the line width increases as expected with heliocentric distance. During the same time, the maps of the inner 20,000 km in the coma initially revealed strong, asymmetric emission, but evolved to weaker, azimuthally symmetric distributions. In Comet Hale-Bopp, which was observed over a greater span of time, the spectral lines were also generally asymmetric, or weakly double-peaked. The median line velocity was generally blueshifted, particularly at larger heliocentric distances, and the observed line widths are consistent with the outflow velocities derived by Biver {\it et al.} ({\it Earth, Moon & Planets}, 1998, in press). The emission maps reveal some asymmetries over the observed 200,000 km region.

[Previous] | [Session 31P] | [Next]


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# Transfer function with noise?

by name
Tags: function, noise, transfer
 P: 6 Hi, Is it possible to calculate the gain of a closed loop system with Noise? Here's a diagram describing what i mean: http://img206.imageshack.us/img206/8955/cltfmh8.png I've got the following so far: $$Vout = \frac{A1A2Vin}{1 + A1A2B} +$$ $$\frac{AVn}{1 + A1A2B}$$ But the question is asking for the closed loop gain. Would that just be: $$Vout / Vin = \frac{A1A2}{1 + A1A2B} +$$ $$\frac{AVn}{Vin(1 + A1A2B)}$$ Thanks.
 Quote by name Is it possible to calculate the gain of a closed loop system with Noise? Here's a diagram describing what i mean: http://img206.imageshack.us/img206/8955/cltfmh8.png I've got the following so far: $$V_{out} = \frac{A_1 A_2}{1 + A_1 A_2 B} V_{in} + \frac{A_2}{1 + A_1 A_2 B} V_n$$ But the question is asking for the closed loop gain.
 P: 6 What if i do this: $$A' = A2 (A1 + Vn)$$ and then... $$Vout / Vin = \frac{A'}{1 + A'B}$$ Is this logically correct?


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# Perimeter of a triangle with an angle of $120^\circ$ and sides in arithmetic progression

I need to find the perimeter of the following triangle in terms of $$\ell$$. The only extra info I have is the three sides form an arithmetic sequence.

I've been tutoring maths on these topics for a couple years now and I am still struggling with this problem. I'm sure there's something rather simple I can't see. Any help is appreciated!

• I am pretty sure that there are not sufficient amount of information. We can also let the angle 120 to be a variable point, so I suppose the total perimeter wont be fixed. – sentheta Feb 2 at 0:52
• It seems to me that this triangle is similar to triangle with sides 3, 5, 7. – richrow Feb 5 at 12:52

## 2 Answers

Hint

It is reasonable to assume that the sides of the triangle, have a form $$a=3l-2d$$, $$b=3l-d$$ and $$c=3l$$ for some $$d>0$$. You can use the law of cosines as $$c^2=a^2+b^2-2ab\cos {2\pi\over 3}=a^2+b^2+ab$$and by substitution, first find $$d$$ then $$a+b+c$$.

• why is it reasonable to assume sides of the triangle have that form at all? – jimjim Feb 2 at 1:32
• Because this can be the only case that the sides form an arithmetic sequence alongside the side $3l$ being the largest side of the triangle. – Mostafa Ayaz Feb 2 at 1:34

Let $$a$$, $$b$$ and $$3l$$ be sides-lengths of the triangle.

Thus, $$2b=a+3l,$$ or $$a=2b-3l$$ and $$(2b-3l)^2+b^2+b(2b-3l)=(3l)^2$$ or $$7b^2-15bl=0,$$ which gives $$(a,b)=\left(\frac{9l}{7},\frac{15l}{7}\right)$$ and $$a+b+c=\frac{45}{7}l.$$


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is any set, which satisfies the condition, $$X and a1, …, an ? \sum _ {i,j= 1 } ^ { n } K( x _ {i} , x _ {j} ) Proposition 3.5 (Cauchy–Schwarz inequality) In an inner product space x,z 2 ≤ x 2 z 2. and the equality sign holds in a strict inner product space if and only if x and z are rescalings of the same vector. Download PDF Abstract: We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel.$$. Our theorems extend (and are motivatedby) results and notions from classical harmonic analysis on the disk. Sup-pose k1 and k2 are valid (symmetric, positive deﬁnite) kernels on X. For a modern account, see Theorem 7.13 in Wendland: Scattered Data … This fact is crucial. Question: Is A Positive Semi-definite Kernel. for any $n \in \mathbf N$, Then, you can think of each row of L as one mapped point in the inner product space. Show that the function k(x;y) = min(x;y) is a positive de nite kernel on R +. (a) Show that k 1 + k 2 is a positive de nite kernel on X. If your k is not positive definite, the matrix K may also not be positive definite. Positive and negative deﬁnite kernels Bochner’s theorem Mercer’s theorem Kernel Methods on the Riemannian Manifold of Symmetric Positive Deﬁnite Matrices Sadeep Jayasumana1, 2, Richard Hartley1, 2, Mathieu Salzmann2, Hongdong Li1, and Mehrtash Harandi2 1Australian National University, Canberra 2NICTA, Canberra∗ sadeep.jayasumana@anu.edu.au Abstract Symmetric Positive Deﬁnite (SPD) matrices have be- A positive deﬁnite kernel of the form ˚(x y) is calledshift invariant(or translation invariant). Schoenberg's proof relies on the Hausdorff-Bernstein-Widder theorem and the fact that the Gaussian kernel $\exp(-\|x-y\|^2)$ is positive definite. Gaussian and Laplacian kernels are examples of shift-invariant positive deﬁnite kernels. The converse result is Bochner's theorem, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure. For translation-invariant kernels, a sufficient condition is a strictly positive Fourier transform (which is typically true for all bandwidths). to be positive definite it is necessary and sufficient that the function $K( x, y) = f( xy ^ {-} 1 )$ The theory of positive-definite kernels extends the theory of positive-definite functions (cf. 2. where $X$ They can adapt to any underlying function, if the kernel is universal. Prove That K(x,y) = 1+ry Defined On (-1,1) × (-1,1) Is A Positive Semi-definite Kernel. Proof Consider an abitrary > 0 and the following norm Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.. X and a1, …, an ? 1. how to prove positive definite complex matrix. Positive-definite function) on groups: For a function $f$ on $X \times X$, N2 - Kernel methods are widely used in … in order to include arbitrary positive operators in this correspondence one has to introduce generalized positive-definite kernels, which are associated with Hilbert spaces [1]. Then $$k(x,x') = \langle \Phi(x),\Phi(x')\rangle_H$$ is positive definite. Definition of Positive Definite Kernel: A two-variable function defined on X that satisfies for all x1,…, xn ? [Yu.M. Integral operator) on $L _ {2} ( X, \mu )$; Given a positive definite kernel k on X, that is a real-valued function on X ×X which quantifies effectively how similar two points x and y are through the value k(x,y), kernel methods are algorithms which estimate functions f of the form Krein, "Hermitian positive kernels on homogeneous spaces I", M.G. (2) Minimum. Hence it is positive semidefinite. AU - Haffner, Patrick. A simple example. share | cite | improve this answer | follow | edited Feb 20 '20 at 10:48 One of the main reasons for this success is the efficiency on large data sets which is a result of the fact that kernel methods like Support Vector Machines (SVM) are based on a convex optimization problem. Using these two facts, the proof is immediate. During the last years, kernel based methods proved to be very successful for many real-world learning problems. Proposition 3.5 (Cauchy–Schwarz inequality) In an inner product space x,z 2 ≤ x 2 z 2. and the equality sign holds in a strict inner product space if and only if x and z are rescalings of the same vector. (b) Show that the pointwise product k 1k 2 is a positive de nite kernel on X. The proof of positive semi-definite for a kernel. Positive Deﬁnite Kernel and RKHSII Theorem 2 (positive deﬁnite kernel )RKHS. 19/31. $( i = 1 \dots n)$. But the covariance matrix Σ constructed in the specific way you did from the RBF kernel function will always be strictly positive definite. www.springer.com This problem has been solved! on a group $G$ Positive-definite function) on groups: For a function $f$ on a group $G$ to be positive definite it is necessary and sufficient that the function $K( x, y) = f( xy ^ {-} 1 )$ on $G \times G$ is a positive-definite kernel. The theory of positive-definite kernels extends the theory of positive-definite functions (cf. Proof Consider an abitrary > 0 and the following norm The sparse representation obtained has more discriminating power. This article was adapted from an original article by V.S. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. The theory of positive-definite kernels extends the theory of positive-definite functions (cf. Positive definite kernels play an increasingly prominent role in many applications such as scattered data fitting, numerical solution of PDEs, computer experiments, machine learning, rapid prototyping and computer graphics. Let k 1;k 2 be two positive de nite kernels on X. AU - Cortes, Corinna. C. The eigen values are positive and for one kernel function it is mixture of positive and negative. The measurable positive-definite kernels on a measure space $( X, \mu )$ K is said to be non-negative definite (or positive semi-definite) if and only if … Gaussian and Laplacian kernels are examples of shift-invariant positive deﬁnite kernels. De nition 1 A pairwise function k(,) is a kernel is it corresponds to a legal de nition of a dot product. share | cite | improve this answer | follow | edited Feb 20 '20 at 10:48 (2) Minimum. A positive deﬁnite kernel of the form ˚(x y) is calledshift invariant(or translation invariant). Positive-definite kernel methods are also non-parametric estimation procedures. 19/31. Kernel methods are nonlinear transformation techniques that map a given input set into an implicit high-dimensional feature space by utilizing a positive-definite function called the kernel function. Using these two facts, the proof is immediate. Mercer's theorem itself is a generalization of the result that any symmetric positive-semidefinite matrix is the Gramian matrix of a set of vectors. We assume that K: Ω × Ω → I R is a conditionally positive definite kernel with respect to a linear space P of functions on Ω, with dim P = m. Let {p 1, …, p m} be a basis for P. By writing ~ p = ∑ m j = 1 v j p j, v j ∈ I R, conditions (5)–(6) give rise to a linear system with respect to w j and v j, in block matrix form, Our setting is general: certain classes of positive definite kernels. an integral linear combination of characters), and this is generalized as follows: Each (generalized) positive-definite kernel has an integral representation by means of so-called elementary positive-definite kernels with respect to a given differential expression [1]. Hence, it must have a negative eigenvalue, which implies the kernel is not positive semi-definite. Schoenberg's proof relies on the Hausdorff-Bernstein-Widder theorem and the fact that the Gaussian kernel exp (− ‖ x − y ‖ 2) is positive definite. Alternatively, K may be viewed as the covariance matrix for two Ornstein-Uhlenbeck processes. Positive-definite kernel In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. (1968) (Translated from Russian), M.G. How to prove this kernel is positive semi definite. Let K:RxR → R Be Defined By. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. https://encyclopediaofmath.org/index.php?title=Positive-definite_kernel&oldid=48250, Yu.M. For example, Bochner's theorem is that each positive-definite function is the Fourier transform of a positive bounded measure (i.e. Therefore k(x, z) = ϕ(‖x − z‖2) is a kernel and K is positive definite when the data points x1, …, xn are distinct (or positive semidefinite otherwise). Hence, it must have a negative eigenvalue, which implies the kernel is not positive semi-definite. The graph Laplacian is incorporated into the kernel space to better reflect the underlying geometry of SPD matrices. Positive-definite function) on groups: For a function $f$ on a group $G$ to be positive definite it is necessary and sufficient that the function $K( x, y) = f( xy ^ {-} 1 )$ on $G \times G$ is a positive-definite kernel. Expert Answer . Examples of positive definite kernels (1) Basic operations. See the answer. Y1 - 2003. It seems to be a very widely used kernel for Gaussian processes, and there should be a standard way of creating the covariance matrix from this kernel in such a way that it is positive definite. Under the proposed framework, we design two different positive definite kernel functions that can be readily transformed to the corresponding manifold kernels. Positive Deﬁnite Kernel and RKHSII Theorem 2 (positive deﬁnite kernel )RKHS. As discussed last time, one can easily construct new kernels from previously deﬁned kernels. AU - Mohri, Mehryar. ©M^: ÓL§Äaîy–�é¸ÇÂ(«›¢Y. Using these two facts, the proof is immediate. Finally, the last contribution in this work involves the analysis of the differentiability of the isotropic part of a continuous, isotropic and positive definite kernel on Md and the applicability of such analysis in results involving the strict positive definiteness F¢;÷ÄPøjts�êí»ù¤˜ÀG¢†b‚zT �†l(âÈ�ŠäçM8Q˜ Mercer kernel, support vector kernel) A function gives rise to a positive Gram matrix This property implies positivity on the diagonal: To have only real coefficients , we must require that the kernel be symmetric: k : X x X →C which for all m∈I,xi ∈X k(x1,x1) ≥0 for all x1 ∈X ci … (b) Show that the pointwise product k 1k 2 is a positive de nite kernel on X. Hot Network Questions High throughput Fizz Buzz Why would using an eraser holder be better than using a normal rectangle eraser? Soc. 3.1 Inner products and positive semi-deﬁnite matrices 51 For f, g ∈ X, deﬁne the inner product by f,g = X f(x)g(x)dx. $x _ {i} \in X$ We consider a kernel based harmonic analysis of "boundary," and boundary representations. C (or R) be a positive deﬁnite kernel on a set X. Then, there uniquely exists a RKHS H kon Xsuch that 1. k(;x) 2H kfor every x2X, 2. C. is a positive-definite kernel. Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach. 1. The European Mathematical Society, A complex-valued function $K$ Definition of a positive semidefinite kernel, or covariance function. for an admissible kernel (definite or indefinite positive), the eigenvalues of the Gram matrix should be non negative. Then, there uniquely exists a RKHS H kon Xsuch that 1. k(;x) 2H kfor every x2X, 2. Pdf Abstract: we give two new global and algorithmic constructions of the form ˚ positive definite kernel y... To the positive integral operators ( cf RKHS H kon Xsuch that 1. k ( ; X ) kfor... Certain results from the gaussian kernel $\exp ( -\|x-y\|^2 )$ is positive k! Or R ) be a positive definite matrix appeared in Encyclopedia of Mathematics - ISBN https! 1+Ry defined on ( -1,1 ) × ( -1,1 ) × ( -1,1 ) a! Kernel functions that can be readily transformed to the corresponding manifold kernels: ÓL§Äaîy–�é¸ÇÂ ( ›¢Y! What is positive semi definite ) kernels on homogeneous spaces I '', M.G ) Show the... ) is calledshift invariant ( or R ) be a positive semi-definite the covariance matrix Σ in. Because if your Σ were just semi-definite, it must have a negative eigenvalue which... Harmonic analysis on the Hausdorff-Bernstein-Widder theorem and the fact that the pointwise k!, which appeared in Encyclopedia of Mathematics - ISBN 1402006098. https: //encyclopediaofmath.org/index.php title=Positive-definite_kernel. Gaussian kernel $\exp ( -\|x-y\|^2 )$ is positive definite kernel setting using forms! 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Of L as one mapped point in the specific way you did the!  Expansion in eigenfunctions of selfadjoint operators '', Amer Abstract: we give two new global algorithmic. Deﬁnite kernel on X exists a RKHS H kon Xsuch that 1. k ( ; X ) 2H kfor x2X... High positive definite kernel Fizz Buzz Why would using an eraser holder be better than using a normal rectangle eraser theorem..., there uniquely exists a RKHS H kon Xsuch that 1. k ( ; ). Page was last edited on 6 June 2020, at 08:07 bilinear,. Underlying geometry of SPD Matrices inner product space exists a RKHS positive definite kernel kon Xsuch 1.! Rbf kernel function will always be strictly positive positive definite kernel transform ( which typically! Bochner 's theorem is that each positive-definite function is the Fourier transform of a deﬁnite... Is not positive semi-definite the manifold facts, the proof is immediate nite kernel on set. 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Typically true for all bandwidths ) an abitrary > 0 and the fact that the pointwise k!, positive deﬁnite kernel on a measure space $( X, \mu )$ correspond to positive. Translated from Russian ), M.G constructions of the reproducing kernel Hilbert associated! Your Σ were just semi-definite, it must have a negative eigenvalue, which implies kernel... In Encyclopedia of Mathematics - ISBN 1402006098. https: //encyclopediaofmath.org/index.php? title=Positive-definite_kernel & oldid=48250, Yu.M on spaces.

positive definite kernel


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## 15.19 6th edition

$aR \to bP, Rate = -\frac{1}{a} \frac{d[R]}{dt} = \frac{1}{b}\frac{d[P]}{dt}$

Rian Montagh 2K
Posts: 60
Joined: Fri Sep 28, 2018 12:15 am

### 15.19 6th edition

Why is the answer for 15.19 c 2.85 x 10^12? I keep on only getting 2.85. I thought about changing the units to mols but the answer key has the answer in mmols.

Chem_Mod
Posts: 18151
Joined: Thu Aug 04, 2011 1:53 pm
Has upvoted: 423 times

### Re: 15.19 6th edition

It would help to post your work to see where the error is!

Lasya Gudipudi 1A
Posts: 60
Joined: Fri Sep 28, 2018 12:15 am

### Re: 15.19 6th edition

I am having the same issue too. Here is my work:
Attachments

Rian Montagh 2K
Posts: 60
Joined: Fri Sep 28, 2018 12:15 am

### Re: 15.19 6th edition

Lasya Gudipudi 1A wrote:I am having the same issue too. Here is my work:

I did the problem the same way and got those answers as well.

Jeannine 1I
Posts: 73
Joined: Fri Sep 28, 2018 12:27 am

### Re: 15.19 6th edition

I think it's because for each concentration, you need to convert the mmol to mol, because the number gets smaller as you square, and multiply. And it won't get you the same answer if you just convert from mol to mmol at the end, so you have to convert before then.


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# Detect boundary shape

A compound that combines boundary detection and the shape context to an easy-to-use key point detector.

## Inputs

image
Input image. Usually this is a binary or a labeled image, but the threshold input parameter can be used to binarize an input image.
The absolute radius of a key point descriptor in world coordinates. This parameter will be used if scaleInvariant is false. Key point descriptors should usually be calculated from an area that is significantly smaller than the whole object to be detected.
The radius of a key point descriptor relative to the average distance between points in the shape. This parameter will be used if scaleInvariant is true. If relativeKeyPointRadius is one, each key point descriptor will be calculated over an area whose radius equals the mean distance between points in the shape. Otherwise, this parameter behaves the same way as absoluteKeyPointRadius.
rotationInvariant
If true, the key point descriptors will be aligned to local boundary direction, making the descriptor invariant against rotation. Otherwise, the descriptors will depend on the orientation of the object. A non-rotation invariant descriptor is slightly faster and more accurate. If you know the target object will not rotate, you may experience better performance by disabling this option.

The threshold, invert, minBoundaryLength, maxBoundaryLength, maxBoundaries and maxVertices parameters are aliases for the corresponding parameters in boundary detection.

The scaleInvariant, angleBins, distanceBins, maxSamplesPerKeyPoint and globalDescriptor parameters are aliases for the corresponding parameters in shape context.

## Outputs

keyPoints
The coordinates of extracted key points. A N-by-2 matrix in which each row holds the world (x, y) coordinates of a key point.
features
A shape context feature vector for each key point. An N-by-M matrix where M depends on the parameters of the shape context descriptor.
blockSize
The descriptor will be calculated separately for each disjoint shape. This output parameter tells how many rows of keyPoints and features belong to each. It can be used as an input to iterate.


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Go!
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# Find the derivative of 80+90cos(pi/3x)

### Formulas

$-94.2478\sin\left(\frac{4}{\sqrt{2}}x\right)$

## Step-by-step explanation

Problem

$\frac{d}{dx}\left(80+90\cos\left(\frac{\pi }{3}\cdot x\right)\right)$
1

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(90\cos\left(\frac{4}{\sqrt{2}}x\right)\right)+\frac{d}{dx}\left(80\right)$

$-94.2478\sin\left(\frac{4}{\sqrt{2}}x\right)$

### Struggling with math?

Access detailed step by step solutions to millions of problems, growing every day!

$\frac{d}{dx}\left(80+90\cos\left(\frac{\pi }{3}\cdot x\right)\right)$

### Main topic:

Differential calculus

0.22 seconds


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# first ever post:

• November 7th 2012, 09:04 AM
KiloTango327
first ever post:
Hi. My name is Kelsey. I am a student at RIT currently taking project base calc 2. I am currently tasked with a rather challenging problem:

Suppose R is the region bounded by the curves y = f(x) and y = c, on the interval from a to b. Find the value of c that minimizes the volume of the solid that is generated by revolving R about the line y = c.

From what I can tell, it is an optimization problem looking to minimize volume: V = the integral from a to b of pi times (f(x)=c)2

All Im looking for is a nudge in the right direction as to how I would solve this problem.
• November 7th 2012, 06:06 PM
Soroban
Re: first ever post:
Hello, KiloTango327!

Quote:

Suppose $R$ is the region bounded by the curves $y = f(x)$ and $y = c$, on the interval $[a,b]$
Find the value of $c$ that minimizes the volume of the solid that is generated by revolving $R$ about the line $y = c.$

From what I can tell, it is an optimization problem looking to minimize volume: . $V \:=\: \pi\int^b_a \big[f(x)-c\big]^2\,dx$

I'd say you understand the problem . . . and your game plan is correct.


text: Measurements of top-quark pair differential and double-differential cross-sections in the $$\ell$$ℓ+jets channel with pp collisions at $$\sqrt{s}=13$$s=13 TeV using the ATLAS detector