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# Postdoctoral Fellow Seminars

## Lecture: TBD

March 10, 2021 / 11:00am – 1:00pm
Virtual
Speaker: Wesley Hamilton, UNC CH

TBD

## Lecture: Dense Weighted Networks Featuring Communities with Augmented Degree Correction

March 3, 2021 / 11:00am – 1:00pm
Virtual
Speaker: Benjamin Leinwand, UNC CH

### Abstract

Dense networks with weighted connections often exhibit a community-like structure, where although most nodes are connected to each other, different patterns of edge weights may emerge depending on each node’s community membership. We propose a new framework for generating and estimating dense, weighted networks with potentially different connectivity patterns across different communities. The proposed model relies on a particular class of functions that map individual node characteristics to the edges connecting those nodes, allowing for flexibility while requiring a relatively small number of parameters relative to the number of edges. By leveraging the estimation techniques, we also develop a bootstrap methodology for generating new networks on the same set of vertices, which may be useful in circumstances where multiple data sets cannot be collected.

## Lecture: Sufficient Dimension Reduction in Time Series

February 24, 2021 / 11:00am – 1:00pm
(Meet and Greet with Postdocs and Grad Students 10:30am-11:00am)
Virtual
Speaker: Seyed Samadi, Southern Illinois University

### Abstract

Dimensionality reduction has always been one of the most important and challenging problems in high-dimensional data analysis. In the context of time series analysis, we are interested in estimating the conditional mean and variance functions. Using the central and central mean subspaces, that preserve sufficient information about the response, one can estimate the unknown mean and variance functions. There are different approaches in the literature to estimate the time series central mean subspace (TS-CMS). However, those methods are computationally intensive and not feasible in practice. In this talk, we describe the Fourier transformation technique that is used to estimate the TS-CMS. The proposed estimators are shown to be consistent and asymptotically normal under some mild conditions. Simulation results and a real data analysis will be presented to demonstrate the performance of our methodology and compare it with existing methods.

*Working Group Summary in the Data Science in the Social and Behavioral Sciences program at 1:15pm:

2. Austin Ferguson (Psych network and viewpoint modeling)
3. Ed Tam (Causal Networks)

## Lecture: Hypothesis Testing in Nonlinear Function on Scalar Regression with Application to Child Growth Study

February 17, 2021 / 1:15pm – 2:15pm
Virtual
Speaker: Mityl Biswas, North Carolina State University

### Abstract

We propose a kernel machine-based hypothesis testing procedure in nonlinear function-on-scalar regression model. Our research is motivated by the Newborn Epigenetic Study (NEST) where the question of interest is whether a pre-specified group of toxic metals is associated with child growth. We take the child growth trajectory as the functional response, and model the toxic metal measurements jointly using a nonlinear function. We use a kernel machine approach to model the unknown function and transform the hypothesis of no effect to an appropriate variance components test. We demonstrate our proposed methodology using a simulation study and by applying it to analyze the NEST data.

## Lecture: Theory of Deep Convolutional Neural Networks

February 10, 2021 / 11:00am – 1:00pm
(Meet and Greet with Postdocs and Grad Students 10:30am-11:00am)
Virtual
Speaker: Ding-Xuan Zhou, City University of Hong Kong and SAMSI

### Abstract

Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, natural language processing, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoretical foundation for understanding the modelling, approximation, or generalization ability of deep learning models with network architectures. Here we are interested in deep convolutional neural networks (CNNs) with convolutional structures. The convolutional architecture gives essential differences between deep CNNs and fully-connected neural networks, and the classical approximation theory for fully-connected networks developed around 30 years ago does not apply. This talk describes an approximation theory of deep CNNs associated with the rectified linear unit (ReLU) activation function. In particular, we prove the universality of deep CNNs, meaning that a deep CNN can be used to approximate any continuous function to an arbitrary accuracy when the depth of the neural network is large enough. We also show that deep CNNs perform at least as well as fully-connected neural networks for approximating general functions, and much better for approximating radial functions in high dimensions.

*Working Group Summary in the Data Science in the Social and Behavioral Sciences program at 1:15pm:

• Benjamin Leinwand (Networks of Networks working group)
• Alejandro Martinez (Simulation to Understand working group)
• Wesley Hamilton (Brain Networks working group)

## Lecture: Modeling and Parameter Subset Selection for Fibrin Matrix Polymerization Kinetics

February 7, 2021 / 1:15pm – 2:15pm
Virtual
Speaker: Katherine Pearce, SAMSI RA

### Abstract

In this talk, we present a model for fibrin matrix polymerization in a biomimetic wound healing application and an algorithm for parameter subset selection. We first contextualize the model by briefly summarizing the relevant details of hemostasis, the initial stage of wound healing in which there is rapid accumulation of fibrin matrix (our primary QoI). We then discuss the associated chemical reaction network and idealized ODE system derivation, as well as relevant conservation laws and constraints. An optimization problem is subsequently formulated for (reaction rate) parameter estimation, and we illustrate our iterative procedure that, within each iteration: (i) determines candidates for non-identifiability, (ii) fixes non-identifiable parameters at nominal values, and (iii) via this procedure, decreases the magnitude of the least squares residual objective function. Lastly, we describe how our procedure may be used for model reduction through the elimination of reactions associated with the most non-identifiable rate constants.

## Lecture: Finding Significant Communities in Cross-correlation Networks Derived from Multi-view Data

January 27, 2021 / 1:15pm – 2:15pm
Virtual

### Abstract

Multi-view data, obtained by taking multiple types of measurements of the same samples, is now common in many scientific disciplines like genomics, ecology, and climate science. An important exploratory problem in the analysis of such data is to identify interactions between features from the different measurement types. Assuming linear association, these interactions can be captured by a bipartite cross-correlation network, whose nodes are features from two distinct measurement types and edges represent feature pairs that are truly correlated. We will introduce the Bimodule Search Procedure (BSP), which performs repeated hypothesis tests on the data to find communities in the bipartite cross-correlation network and demonstrate its application to the problem of eQTL analysis in genomics. BSP works directly with the data rather than only the sample correlation matrices, and it is thus able to borrow strength and account for correlations within features of the same measurement type. We will also briefly discuss some theoretical questions motivated by community detection in correlation networks and other correlation network mining procedures.

Multi-view data, obtained by taking multiple types of measurements of the same samples, is now common in many scientific disciplines like genomics, ecology, and climate science. An important exploratory problem in the analysis of such data is to identify interactions between features from the different measurement types. Assuming linear association, these interactions can be captured by a bipartite cross-correlation network, whose nodes are features from two distinct measurement types and edges represent feature pairs that are truly correlated. We will introduce the Bimodule Search Procedure (BSP), which performs repeated hypothesis tests on the data to find communities in the bipartite cross-correlation network and demonstrate its application to the problem of eQTL analysis in genomics. BSP works directly with the data rather than only the sample correlation matrices, and it is thus able to borrow strength and account for correlations within features of the same measurement type. We will also briefly discuss some theoretical questions motivated by community detection in correlation networks and other correlation network mining procedures.

## Lecture: Advancements in Hybrid Iterative Methods for Inverse Problems

September 16, 2020, 1:15pm – 2:15pm
Virtual
Speaker: Julianne Chung, Virginia Tech University

### Abstract

In many physical systems, measurements can only be obtained on the exterior of an object (e.g., the human body or the earth’s crust), and the goal is to estimate the internal structures. In other systems, signals measured from machines (e.g., cameras) are distorted, and the aim is to recover the original input signal. These are natural examples of inverse problems that arise in fields such as medical imaging, astronomy, geophysics, and molecular biology.

Hybrid iterative methods are increasingly being used to solve large, ill-posed inverse problems, due to their desirable properties of (1) avoiding semi-convergence, whereby later reconstructions are no longer dominated by noise, and (2) enabling adaptive and automatic regularization parameter selection.  In this talk, we describe some recent advancements in hybrid iterative methods for computing solutions to large-scale inverse problems. First, we consider a hybrid approach based on the generalized Golub-Kahan bidiagonalization for computing Tikhonov regularized solutions to problems where explicit computation of the square root and inverse of the covariance kernel for the prior covariance matrix is not feasible. This is useful for large-scale problems where covariance kernels are defined on irregular grids or are only available via matrix-vector multiplication. Second, we describe flexible hybrid methods for solving $\ell_p$ regularized inverse problems, where we approximate the p-norm penalization term as a sequence of 2-norm penalization terms using adaptive regularization matrices, and we exploit flexible preconditioning techniques to efficiently incorporate the weight updates.  We introduce a flexible Golub-Kahan approach within a Krylov-Tikhonov hybrid framework, such that our approaches extend to general (non-square) l_p regularized problems. Numerical examples from dynamic photoacoustic tomography, space-time deblurring, and passive seismic tomography demonstrate the range of applicability and effectiveness of these approaches.

## Lecture: Sharp 2-norm Error Bounds for the Conjugate Gradient Method and LSQR

September 23, 2020, 1:15pm – 2:15pm
Virtual
Speaker: Eric Hallman, North Carolina State University

### Abstract

When running any iterative algorithm it is useful to know when to stop. Here we review the conjugate gradient method, an iterative method for solving Ax=b where A is symmetric positive definite, as well as estimates for the 2-norm error \|x-x_*\|_2, where x_* is the solution to the linear system. We introduce a new method for computing an upper bound on the 2-norm error, and show that given certain mild assumptions our bounds are optimal. Experimental results are discussed as well as the implications of our work for solving the least-squares problem \min_x \|Ax-b\| using the iterative algorithm LSQR.

## Lecture: Convergence of the Parameters in Mixture Models with Repeated Measurements

September 30, 2020, 1:15pm – 2:15pm
Virtual
Speaker: Yun Wei, SAMSI

### Abstract

Latent structure models with many observed variables are among the most powerful and widely used tools in statistics for learning about heterogeneity within data population(s). An important canonical example of such models is the mixture of product distributions. We consider the finite mixture of product distribution with the special structure that the product distributions in each mixture component are also  identically distributed.  In this setup, each mixture component consists of samples from repeated measurements and thus such data are exchangeable sequences. Applications of the model include psychological study and topic modeling.

We show that with sufficient repeated measurements, a model that is not originally identifiable becomes identifiable. The posterior contraction rate for the parameter estimation is also obtained and it shows that repeated measurements are beneficial for estimating parameters in each mixture component. Such results hold for general probability kernels including all regular exponential families and can be applied to hierarchical models.

Based on joint work with Xuanlong Nguyen.

## Lecture: Randomized Approaches to Accelerate MCMC Algorithms for Bayesian Inverse Problems

October 7, 2020, 1:15pm – 2:15pm
Virtual
Speaker: Arvind Saibaba, North Carolina State University

### Abstract

Markov chain Monte Carlo (MCMC) approaches are traditionally used for uncertainty quantification in inverse problems where the physics of the underlying sensor modality is described by a partial differential equation (PDE). However, the use of MCMC algorithms is prohibitively expensive in applications where each log-likelihood evaluation may require hundreds to thousands of PDE solves corresponding to multiple sensors; i.e., spatially distributed sources and receivers perhaps operating at different frequencies or wavelengths depending on the precise application. In this talk, I will show how to mitigate the computational cost of each log-likelihood evaluation by using several randomized techniques and embed these randomized approximations within MCMC algorithms. These MCMC algorithms are computationally efficient methods for quantifying the uncertainty associated with the reconstructed parameters. We demonstrate the accuracy and computational benefits of our proposed algorithms on a model application from diffuse optical tomography where we invert for the spatial distribution of optical absorption.

## Lecture: Individual Level Always Survivor, Direct, Spillover Effects with Applications

October 14, 2020, 1:15pm – 2:15pm
Virtual
Speaker: Jaffer Zaidi, SAMSI

### Abstract

We provide investigators with the ability to quantify individual level always survivor, direct, and spillover effects. The survivor average causal effect is commonly identified with more assumptions than those guaranteed by the design of a randomized clinical trial. This paper demonstrates that individual level causal effects in the `always survivor’ principal stratum can be identified with no stronger identification assumptions than randomization. We illustrate the practical utility of our methods using data from a clinical trial on patients with prostate cancer. We also provide another application on the spillover effects of randomized get out the vote campaigns. Our methodology is the first and, as of yet, only proposed procedure that enables detecting individual level causal effects in the presence of truncation by death using only the assumptions that are guaranteed by design of the clinical trial.

## Lecture: Scalable Bayesian Inference for Time Series via Divide-and-conquer

October 21, 2020 1:15pm – 2:15pm
Virtual
Speaker: Deborshee Sen, SAMSI

### Abstract

Bayesian computational algorithms tend to scale poorly as the size of data increases. This had led to the development of divide-and-conquer-based approaches for scalable inference. These divide the data into chunks, perform inference for each chunk in parallel, and then combine these inferences. While appealing theoretical properties and practical performance has been demonstrated for independent observations, scalable inference for dependent data remains challenging. In this work, we study the problem of Bayesian inference from very long time series.  The literature in this area focuses mainly on approximate approaches that lack any theoretical guarantees and may provide arbitrarily poor accuracy in practice.  We propose a simple and scalable divide-and-conquer algorithm, and provide accuracy guarantees.  Numerical simulations and real data applications demonstrate the effectiveness of our approach.

## Lecture: Probabilistic Learning on Manifolds

October 28, 2020 1:15pm – 2:15pm
Virtual
Speaker: Ruda Zhang, SAMSI

### Abstract

Probabilistic models of data sets often exhibit salient geometric structure. Such a phenomenon is summed up in the manifold distribution hypothesis, and can be exploited in probabilistic learning tasks such as density estimation and generative modeling. In this talk I present a framework for probabilistic learning on manifolds (PLoM), which uses manifold learning to discover low-dimensional structures within high-dimensional data, and exploits topological properties of the learned manifold to efficiently build probabilistic models. A joint distribution is partitioned into a marginal distribution on the manifold and conditional distributions on normal spaces of the manifold. The marginal distribution can be estimated using Riemannian kernels, and the conditional distributions can be estimated discretely by normal-bundle bootstrap or continuously using Gaussian kernels. Combining the marginal and conditional models gives a joint generative model. I will also talk about related algorithms and software development, and potential applications.

## Lecture: Competition and Spreading of Low and High-Quality Information in Online Social Networks

November 4, 2020 1:15pm – 2:15pm
Virtual
Speaker: Diego Fregolente, SAMSI

### Abstract

The advent of online social networks as major communication platforms for the exchange of information and opinions is having a significant impact on our lives by facilitating the sharing of ideas. Through networks such as Twitter and Facebook, users are exposed daily to a large number of transmissible pieces of information that compete to attain success. Such information flows have increasingly consequential implications for politics and policy, making the questions of discrimination and diversity more important in today’s online information networks than ever before.   However, while one would expect the best ideas to prevail, empirical evidence suggests that high-quality information has no competitive advantage. We investigate this puzzling lack of discriminative power through an agent-based model that incorporates behavioral limitations in managing a heavy flow of information and measures the relationship between the quality of an idea and its likelihood to become prevalent at the system level. We show that both information overload and limited attention contribute to a degradation in the system’s discriminative power. A good tradeoff between discriminative power and diversity of information is possible according to the model. However,  calibration with empirical data characterizing information load and finite attention in real social media reveals a weak correlation between quality and popularity of information. In these realistic conditions, the model provides an interpretation for the high volume of viral misinformation we observe online.

## Lecture: Retrospective Causal Inference via Matrix Completion, with an Evaluation of the Effect of European Integration on Labour Market Outcomes

November 11, 2020 1:15pm – 2:15pm
Virtual
Speaker: Jason Poulus, SAMSI

### Abstract

We propose a method of retrospective counterfactual prediction in panel data settings with units exposed to treatment after an initial time period (later-treated), and always-treated units, but no never-treated units. We invert the standard setting by using the observed post-treatment outcomes to predict the counterfactual pre-treatment potential outcomes under treatment for the later-treated units. We impute the missing outcomes via a matrix completion estimator with a propensity- and elapsed-time weighted objective function that corrects for differences in the covariate distributions and elapsed time since treatment between groups. Our methodology is motivated by evaluating the effect of two milestones of European integration on the share of cross-border workers in sending border regions. We provide evidence that opening the border increased the probability of working beyond the border in Eastern European regions.


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## Return to Question

2 added 75 characters in body

If $M$ is a combinatorial model category, it's known by the experts that there is a ''natural'' model structure on diagram categories $Hom(C,M)$, which is the projective model structure. The fibrations and weak equivalences are defined point wise.

There is also the one called injective model structure, where the cofibrations and weak equivalences are defined point wise.

I would like to know if for a given a morphism $\alpha \in Arr(C)$, the evaluation at $\alpha$ can be a right Quillen functor with the injective model structures on each side:

$Ev_\alpha: Hom(C,M) \to M^2$

Thanks !

Edit: Here $M^2= Hom([0 \to 1], M)= Arr(M)$, sorry for the confusion.

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# Evaluation functors and injective model structure on diagram categories

If $M$ is a combinatorial model category, it's known by the experts that there is a ''natural'' model structure on diagram categories $Hom(C,M)$, which is the projective model structure. The fibrations and weak equivalences are defined point wise.

There is also the one called injective model structure, where the cofibrations and weak equivalences are defined point wise.

I would like to know if for a given a morphism $\alpha \in Arr(C)$, the evaluation at $\alpha$ can be a right Quillen functor with the injective model structures on each side:

$Ev_\alpha: Hom(C,M) \to M^2$

Thanks !

format: plain


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# A 5 letter word for what sodium does to bond with chlorine to form table salt??

Mar 31, 2016

I don't think it matters so much that you get the right word, but if you get the thought process it should be enough.

Refer to the periodic table here:

A chlorine atom, with $Z = 17$, is one element to the left of the noble gas argon, so it desires (or wants, if 5 letters I guess) one electron. That way, it achieves a full octet, which is generally said to be favorable because all the valence electrons are occupying all the possible quantum states that they can.

On the other hand, a sodium atom, with $Z = 11$, is one element in front of the noble gas neon, so it wants to give (or gives, for 5 letters?) one electron away, e.g. to chlorine, to achieve its octet as well.

You should notice that these behaviors are opposite in direction yet complementary to each other; sodium can give away one electron, while chlorine wants one.

Hence, they willingly form $\text{NaCl}$, the soluble table salt we should use to flavor our food. You can think of a simple equation for this as:

$\text{Na"^(+) + "Cl"^(-) -> "NaCl}$


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# Lie point symmetry of KdV.

I'm asked to consider the 1-param. group of transformations generated by $V = \dfrac{\partial}{\partial u} + \alpha t \dfrac{\partial}{\partial x}$, which easily enough yields $g^{\epsilon}(x,t,u) = (x+\alpha t \epsilon, t, u+\epsilon)$.

Then I'm asked to find $\alpha$ s.t. $V$ generates a Lie point symmetry of $\Delta[x,t,u] = u_t + u_{xxx} - 6uu_x$, which is where I come unstuck.

I wrote $g^{\epsilon} = (\tilde{x}, \tilde{t}, \tilde{u})$ and attempted to compute $\Delta[\tilde{x}, \tilde{t}, \tilde{u}]$, but I continually obtain results such as $\tilde{u}_{\tilde{t}} = -\alpha \epsilon u_x + u_t$, which do not seem correct (given the presence of the parameter epsilon).

Any ideas on the best way to find $\alpha$?


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# Introduction

APOs are something I've been focusing on for a while now, mainly in terms of how we can write and manipulate them in ways that create functions that adhere to our specifications. Interpolation is a special case of this. When we talk about univariate interpolation in the traditional sense, we're discussing:

The lowest degree polynomial which satisfies a given number of points.

There are two things to unpack here; "polynomial" and "lowest degree". Recall that the general form of an anonymous function is:

$\phi^\star(x)=\phi(x)+[\phi(x)]r(x)$

Where $$\phi(x)$$ is a function satisfying the given points, $$[\phi(x)]$$ is zero at those given points and $$r(x)$$ is any arbitrary function. This function generates an infinite family of functions, a subset of which can be derived using our piecewise techniques.

In order to reduce of our infinite family functions to a single function, we need more information. The fact our functions are polynomials reduces this family significantly. Furthermore, the "lowest degree" requirement reduces this even more so; for $$[\phi(x)]$$ to be a polynomial, it must have as many roots as the number of points given to it; hence its degree is automatically greater than the lowest degree polynomial $$\phi(x)$$. This, in fact, refers to the 'uniqueness of the interpolating polynomial' theorem.

What we have, but seldom in depth, discussed, is how we choose to write these polynomials. Granted, we have discussed the Lagrange approach and Newton approach in the context of APOs (producing Lagrange and Newton-form polynomials respectively), but not particularly formally in the latter case.

In fact, today we'll introduce the piecewise approach to generating Newton-Form polynomials in the pure sense and then apply, and extend, a similar notion to a problem: how to write a "perfect power" ($$x^n$$) in terms of "falling powers" ($$x^{\underline{n}}$$). Don't worry if you don't know what this means just yet; we'll get to it later.

# Newton-Form Polynomials

As discussed previously, one is able to produce a Newton-form polynomial by following the naïve algorithm:

1. Write out the interpolation problem in piecewise form.
2. Subtract out a value corresponding to interpolated point(s)/piece values.
3. Factor out the root from the piecewise object.
4. Expand any brackets using the distributive property of multiplication.
5. Repeat 2-5 until one piece remains; the remaining piecewise object becomes the value of this piece.

In a piecewise sense, this algorithm becomes 'impure' at step 4; we aren't in fact generating a Newton polynomial, we're generating a nested polynomial (if this step was removed). Instead, we perform the following:

1. Write out the interpolation problem in piecewise form.
2. Take a subset of $$n-1$$ pieces (where $$n$$ is the number of pieces in the outer piecewise object) and condense these into a single piecewise object. The outer condition becomes the product of the roots of the piece conditions inside the nested piecewise object.
3. Repeat 2 until each (nested) piecewise object contains two pieces each.
4. Perform the interpolation on the deepest object, then subtract it out of the piecewise object and evaluate at relevant points. Repeat until the interpolation is complete.

Ultimately, we'll have a polynomial in the form:

$\phi^\star(x)=a_1(x-x_1)+a_2(x-x_1)(x-x_2)+\dots+a_n(x-x_1)(x-x_2)\cdots(x-x_n)$

Where the coefficients can be given by this derivation, or divided differences. Let us give an example:

\begin{align} x^3 &=\begin{cases} -1 & x=-1 \\ 0 & x=0 \\ 1 & x=1 \\ 8 & x=2 \\ \star & \star \end{cases} \\ &=\begin{cases} \begin{cases} -1 & x=-1 \\ 0 & x=0 \\ 1 & x=1 \\ \star & \star \end{cases} & x(x-1)(x+1)=0\\ 8 & x=2 \\ \star & \star \end{cases} \\ &=\begin{cases} \begin{cases} \begin{cases} -1 & x=-1 \\ 0 & x=0 \\ \star & \star \end{cases} & x(x+1)=0\\ 1 & x=1 \\ \star & \star \end{cases} & x(x-1)(x+1)=0\\ 8 & x=2 \\ \star & \star \end{cases} \\ &=\begin{cases} \begin{cases} x & x(x+1)=0\\ 1 & x=1 \\ \star & \star \end{cases} & x(x-1)(x+1)=0\\ 8 & x=2 \\ \star & \star \end{cases} \\ &=x+\begin{cases} \begin{cases} 0 & x(x+1)=0\\ 1-1 & x=1 \\ \star & \star \end{cases} & x(x-1)(x+1)=0\\ 8-2 & x=2 \\ \star & \star \end{cases} \\ &=x+\begin{cases} 0 & x(x-1)(x+1)=0\\ 6 & x=2 \\ \star & \star \end{cases} \\ &=x+x(x-1)(x+1)\\ \end{align}

While this simplified out nicely, not all problems will. This is also, as seen, fairly tedious to do. Hence the existence of the method of divided differences, which is a recursive method of generating the coefficients of all the relevant polynomial factors.

# Introducing Falling Powers (Factorials)

We define the falling powers/factorials like so (where $$n\in\mathbb{Z}^+$$):

$x^{\underline{n}}=x(x-1)(x-2)\dots(x-(n-1))$

We might as well introduce rising factorial, also known as the Pochhammer function, too, although they won't be used here:

$x^{\overline{n}}=x(x+1)(x+2)\dots(x+(n-1))$

Furthermore when $$n=0$$ we define both of these to be the empty product; $$1$$.

We notice immediate that, for integer $$x=n$$, $$n^{\underline{n}}=n!$$. There are a variety of properties associated with both the falling factorial and rising factorial, which can be found here, but for today's post we'll be focusing on a slightly more naïve approach to an interesting problem.

Just by observation, it shouldn't be (comparatively) difficult to expand on the falling factorials to get them in terms of 'perfect' powers. And while this could be its own inherently interesting problem, we'll instead focus on another: the opposite. That is: How do we write $$x^n$$ for some positive integer $$n$$ in terms of falling factorials?

One approach would be to expand the expression given by:

$x^n=a_0x^{\underline{0}}+a_1x^{\underline{1}}+\dots+a_nx^{\underline{n}}$

Noting that the highest degree falling factorial, $$x^{\underline{n}}$$ has degree $$n$$ and therefore has, by observation, a coefficient of $$1$$. Irrespective of this, trying to expand this expression out would be tedious (oh the irony -- you'll see why). Instead, we notice that:

$x^n=a_0+a_1x+a_2x(x-1)+\dots+a_nx(x-1)(x-2)\cdots(x-(n-1))$

If you have keen observation skills, you'll notice this looks a lot like a Newton polynomial. In fact, it is. And this has beautiful implications. In particular, we realise that the actual method of writing $$x^n$$ is pretty much irrelevant, despite that being our end goal.

There are some important relationships to be made here: Firstly, the coefficients of this polynomial are actually called the Stirling Numbers of the $$2^{\textrm{nd}}$$ kind. Secondly, this means that ultimately, Newton's method of divided difference are related to the Stirling numbers. Lastly, the fact that the falling factorials form a polynomial basis. It is, in fact, possible to rewrite every polynomial in terms of factorial powers, as we'll explicitly see via our derivation.

# Recursive Hell & Polynomials as Falling Powers

One important observation to make is that a polynomial $$x^n$$, in order to be written as a linear combination of falling powers, must be interpolated at points:

$x=0,1,\dots,n$

Furthermore, we must interpolate for the lowest degree polynomial. That is, we write the following:

$x^n=\begin{cases} 0 & x=0 \\ 1 & x=1 \\ 2^n & x=2 \\ \vdots & \vdots \\ n^n & x=n \\ \star & \star \end{cases}$

Okay, so this looks simple enough but probably difficult to tackle the same way we did before. We attempt to nest this problem once:

$x^n=\begin{cases} \begin{cases} 0 & x=0 \\ 1 & x=1 \\ 2^n & x=2 \\ \vdots & \vdots \\ (n-1)^n & x=n-1 \end{cases} & x^{\underline{n}}=0\\ n^n & x=n \\ \star & \star \end{cases}$

Immediately, we into a problem. That is, we don't have an expression for the inner piecewise object. We could, of course, write the nested piecewise object as $$x^n$$ also, but that would lead us nowhere. We could also try to squeeze some expression out of it. That would be difficult. Instead, let us introduce a generalised function:

$T_{n,k}(x)=\begin{cases} 0 & x=0 \\ 1 & x=1 \\ 2^n & x=2 \\ \vdots & \vdots \\ k^n & x=k \\ \star & \star \end{cases}$

Then we can do the same thing, but go further:

$T_{n,k}(x)=\begin{cases} \begin{cases} 0 & x=0 \\ 1 & x=1 \\ 2^n & x=2 \\ \vdots & \vdots \\ (k-1)^n & x=k-1 \end{cases} & x^{\underline{k}}=0\\ k^n & x=k \\ \star & \star \end{cases}=\begin{cases} T_{n,k-1}(x) & x^{\underline{k}}=0 \\ k^n & x=k \end{cases}$

And just like that, we have a recursive piecewise relation. Amazing! We also take note that $$T_{n,1}(x)=x$$. Continuing along our derivation, we reach:

\begin{align} T_{n,k}(x) &=T_{n,k-1}(x)+\begin{cases} 0 & x^\underline{k}=0 \\ k^n - T_{n,k-1}(k) & x=k \\ \star & \star \end{cases} \\ &=T_{n,k-1}(x)+x^\underline{k}\begin{cases} \frac{k^n - T_{n,k-1}(k)}{k^\underline{k}} & x=k \\ \star & \star \end{cases} \\ &=T_{n,k-1}(x)+\frac{k^n - T_{n,k-1}(k)}{k!}x^\underline{k} \end{align}

What we'll do now is expand recursively on the term $$T_{n,k-1}(x)$$:

$T_{n,k}(x)=x^\underline{1}+\sum_{i=2}^{k}{\frac{i^n-T_{n,i-1}(i)}{i!}x^\underline{i}}$

Noting that this is still a recurrence relation ($$T_{n,i-1}(i)$$). We also note that this equation is identically given by following through with our Newton algorithm as given before:

$T_{n,k}(x)=T_{n,1}(x)+\begin{cases} 0 & x^\underline{2}=0 \\ 2^n-T_{n,1}(2) & x=2 \\ \star & \star \end{cases}+\begin{cases} 0 & x^\underline{3}=0 \\ 3^n-T_{n,2}(3) & x=3\\ \star & \star \end{cases}+\dots+\begin{cases} 0 & x^\underline{k}=0 \\ k^n-T_{n,k-1}(k) & x=k\\ \star & \star \end{cases}$

It's here we take a step back; it is not trivial to solve this recurrence relation independently. At all. Actually it's a pain in the ass. Don't do it. Like actually don't even try. Instead, we consider defining:

$T_{n,i-1}(i):=C_n(i)=\begin{cases} 0 & i=0 \\ 1 & i=1 \\ 2^n & i=2 \\ \vdots & \vdots \\ (i-1)^n & i=i-1 \\ \star & \star \end{cases}$

Which gives us, noting that $$C_n(1)=0$$ by definition:

$T_{n,k}(x)=\sum_{i=1}^{k}{\frac{i^n-C_n(i)}{i!}x^\underline{i}}$

And therefore we no longer recursively define $$T_{n,k}(x)$$. The only matter is finding $$C_n(i)$$ for general $$i$$. In fact, we can just use any interpolation method available to us; for computational purposes. Symbolically, doesn't matter too much. For our formulation, we'll use the lagrange method, which I'll skip over, giving:

$C_n(i)=\sum_{p=0}^{i-1}{p^n\prod_{\substack{q=0\\q\neq p}}^{i-1}{\frac{i-q}{p-q}}}$

I prefer not to write the whole formula at once. It's not very nice. However, the relevant formulation we're looking for is:

$x^n=T_{n,n}(x)=\sum_{i=1}^{n}{\frac{i^n-C_n(i)}{i!}x^\underline{i}}$

As a limitation of our formulation, this definition works only for $$n\in\mathbb{Z}^+$$. We in fact get $$x^0=0$$ if $$n=0$$...

# Ideas

We mentioned changed of basis of polynomials earlier. While my knowledge doesn't extend to this topic at all, we know that we can write any polynomial as a linear combination of "perfect powers":

$P(x)=\sum_{n}{a_nx^n}=\sum_{n}{a_nT_{n,n}(x)}$

Therefore, we can also write this polynomial as:

$P(x)=\sum_{m}{b_mx^{\underline{m}}}$

Where this can be written when our polynomials are factored by falling power. We know we can do this as each power is interpolated at the same points, and end up being a sum of ascending/descending falling powers. This is a change of basis; the basis we use become falling factorials rather than individual powers.

Another thing we noted earlier is that the coefficients of each falling power make up the Stirling ling numbers of the second kind. That is,

$S(n,k)=\frac{k^n-C_n(k)}{k!}$

Now, as far as definitions go, ours isn't terribly far off. In fact, these are formulated elsewhere as:

$S(n,k)=\frac{1}{k!}\sum_{i=0}^{k}{(-1)^i\begin{pmatrix}n\\k\end{pmatrix}(k-i)^n}$

What I find absolutely beautiful is the term $$\frac{k^n-C_n(k)}{k!}$$ -- despite the massive size of the exponential and function $$C_n(k)$$ for large values of $$k$$ and $$n$$, this term finds itself much smaller than any of them, and always an integer.

In fact, the Stirling numbers are therefore related to Newton's divided differences; the divided differences method gives the coefficients of the falling powers. As do the Stirling numbers. It thus stands to reason that divided differences in this very specific example give the Stirling numbers also (and would also be a pretty viable way of generating them, given how simple divided differences is).

Let's not forget how combinatoric-y this is, but also both the algebraic and analytic implications. We love it.

# Other Way Round

Here's a nice challenge. Using the same principle as above, rewrite $$x^{\underline{n}}$$ in terms of perfect powers. Basically, expand the factors ;)


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# MCQs Sampling 2

Multiple Choice Questions from Statistical Inference for the preparation of exam and different statistical job tests in Government/ Semi-Government or Private Organization sectors. These tests are also helpful in getting admission in different colleges and Universities. MCQs Sampling and Smaping Distribution Online Test.

Most of the MCQs on this page are covered from Sampling and Sampling DistributionsEstimate and Estimation, Testing of Hypothesis, Parametric and Non-Parametric tests etc.

Question 1: The sampling Error is
a) Equal to the population mean
b) Population parameter
c) The difference between the sample statistic and the population parameter
d) Always Positive
e) Always Negative

Question 2: The height distribution of a few students in a school is an example of
a) Element
b) Parameter
c) Population
d) Statistic

Question 3: The unbiased point estimator of the population mean is
a) Sample proportion
b) Sample mean
c) Sample Variance
d) Sample standard deviation
e) None of these

Question 4: Bias is
a) Cumulative
b) Decumulative
c) Decreasing
d) None of these

Question 5: The sampling technique in which every element of the population has an equal, non-zero probability of being selected in a sample is called
a) Probability Sampling
b) Convenience Sampling
c) Purposive Sampling
d) Quota Sampling
e) Stratified Sampling

Question 6: If the mean of the sampling distribution is equal to the parameters then estimators will be
a) Biased
b) Consistent
c) Sufficient
d) Unbiased
e) Invariant

Question 7: Probability distribution of $\overline{X}$ is called its
a) Expected Value
b) Standard Error
c) Sampling Distribution
d) Standard Deviation
e) None of these

Question 8: Probability distribution of a statistic is called
a) Sampling Distribution
b) Standard Error
c) Sampling Error
d) Parameter
e) Bias

Question 9: A magazine conducts a survey and asks its readers to cut the questionnaire from the magazine, fill it and send it via mail. It is a type of
a) Purposive Sampling
b) Snowball Sampling
c) Sequential Sampling
d) Convenience Sampling

Question 10: Standard error of mean is the standard deviation of the
a) Population
b) Sample
c) Sampling Distribution of $\overline{X}$
d) None of these


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Sometimes when you are trying to compile a LaTeX document on Overleaf, you get a "compilation timeout" message without further information about what's causing the problem. This article provides some advice on figuring out what might be causing your project to take too long to compile. If you are trying to compile a big project with many images, go to the compiling big projects section. Contents 1 Looking for common problems 2 Debugging your project 3 Enabling additional error messages 4 Compiling big projects  Looking for common problems Below, a list of common problems that may cause compilation to take too long is presented. These are good places to start looking for errors when something goes wrong. Tables Errors in tabular, longtable and array environments. If you place invalid commands, forget or add extra & or \\ characters or forget the closing environment command, this may cause LaTeX to take too long to figure out what's wrong. TikZ pictures Missing semicolon in a tikz environment. This is a very common error, and the next snippet will cause a compilation timeout error because of the missing semicolon at the end of the first \draw command. \begin{tikzpicture} \draw (0,0)--(3,4) \draw (2,2)--(5,1); \end{tikzpicture} Adding commands before the \documentclass may cause your project to timeout. Inserting a blank line in multiple author document may cause an infinite loop in the compilation process. See the example below. \documentclass{article} \usepackage[utf8]{inputenc}   \title{Example of faulty code}   \author{John Doe   \and Hubert Harnsworth} ... If you still are unable to find something wrong in your project, you can try to debug it by hand. See the next section.  Debugging your project In absence of log messages, the only way to solve your problem is to look for the faulty chunk of code. If your document imports files to generate the final output, commenting out some lines will help to find the faulty one. Be careful, do not comment out parts that can generate more errors. It's recommended to leave uncommented the preamble and the bibliography-related commands. See the example below: \documentclass{article} \usepackage[utf8]{inputenc}   \begin{document} \input{introduction.tex}   \input{chapter1.tex}   %\input{chapter2.tex}   %\input{apendix.tex} \end{document} In this example only the chapter1 will be compiled. Once you identify the problematic file, or if your document has only one source file, comment out parts of the text and check if your project compiles with no errors. To comment out a block of text import the comment package and use the comment environment. \documentclass{article} \usepackage[utf8]{inputenc}   \usepackage{tikz} \usepackage{comment}   \begin{document} \section{Introduction} This document contains some dummy information and figures with the only purpose of taking up some space.   \begin{comment} \section{Second section} \begin{tikzpicture} \draw (0,0)--(1,1) \draw (0,1)--(1,0); \end{tikzpicture}   This text and the previous figure won't work \end{comment} \end{document} In this example, only the title of the first section, Introduction, and the text corresponding to that section will be printed in the PDF output. Everything else is commented out.  Enabling additional error messages When your project times out you usually don't see any error messages. A simple thing that you can try is to add the command \tracingall in your .tex file, right below \begin{document}. This will print some additional error messages in some cases. \documentclass{article} \usepackage[utf8]{inputenc}   \begin{document} \tracingall   \section{Introduction}   ... \end{document} The additional error messages take about three seconds to appear after the compilation project stops. However, the \tracingall command itself greatly increases the time taken to compile a project, and can even cause a project with no errors to time out. If you are working on a very large project, it's best to remove the \tracingall command altogether and to debug as described above.  Compiling big projects If you have a big document with several high resolution images, your project may take too long to compile, even when there are no errors in the code. The recommendation is to use lower resolution images where possible and to add the draft option to your \documentclass command. The draft option will disable rendering images and will show a placeholder instead, hence improving the compilation time. Helpful to debug the project before the final version. \documentclass[draft]{article} \usepackage[utf8]{inputenc}   \begin{document}   \section{Introduction} 1 Introduction Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisissem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi necante. Donec ullamcorper..   \begin{figure}[ht] \includegraphics{lion-logo} \caption{Something not important} \end{figure}   Let's cite... You can enable Draft mode in the Compile Mode menu: A final recommendation is not to let the errors to pile up just because the PDF is rendered. Ignoring errors and warnings in large documents may slow down the compilation process.


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# We can get _____________ types of option for Hyphenation in PageaMaker

1

2

3

4

Please do not use chat terms. Example: avoid using "grt" instead of "great".


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# Quark Matter 2018

May 13 – 19, 2018
Venice, Italy
Europe/Zurich timezone
The organisers warmly thank all participants for such a lively QM2018! See you in China in 2019!

## Transport properties from Charm to Bottom: $p_T$ suppression, anisotropic flow $v_n$ and their correlations to the bulk dynamics

May 15, 2018, 3:20 PM
20m
Sala Mosaici-2, 3rd Floor (Palazzo del Casinò)

### Sala Mosaici-2, 3rd Floor

#### Palazzo del Casinò

Parallel Talk Open heavy flavour

### Speaker

Salvatore Plumari (University of Catania (Italy))

### Description

We study the propagation of charm and bottom quarks in the quark-gluon plasma (QGP) by means of a relativistic Boltzmann transport approach that in the large M/T limit recovers the standard Langevin dynamics. The non-perturbative interaction between heavy quarks and light quarks is described by means of a quasi-particle approach in which light partons are dressed with thermal masses. The last tuned to lattice QCD thermodynamics naturally induce a non-perturbative interaction that entails only a weak dependence on the temperature especially around the critical temperature $T_c$, which plays a fundamental role to describe simultaneously the experimental data for the nuclear suppression factor $R_{AA}$ and the elliptic flow $v_2(p_T)$ of D mesons from RHIC to LHC energies. In the same scheme we present predictions for B mesons at 5.02 ATeV that shows a quite significant suppression and allow a determination of the space-diffusion coefficient that is practically independent on the transport scheme for HQ (Boltzmann vs Langevin). The last is seem to largely deviate from from pQCD estimate but also to be still somewhat larger than AdS/CFT and quite close to lattice QCD calculations. Finally it will be discussed the relevance of initial state fluctuations that allows to extend the analysis to high order anisotropic flows $v_3(p_T)$ and $v_4(p_T)$ as well as to investigate the role of QCD interaction in developing correlations between the light and the heavy flavor anisotropic flows. These will provide novel and powerful constraints for the transport coefficients.

[1] F. Scardina, S. K. Das, V. Minissale, S. Plumari, V. Greco, Phys.Rev. C96 (2017) no.4, 044905.
[2] S. K. Das, F. Scardina, S. Plumari, V. Greco, Phys.Lett. B747 (2015) 260.
[3] F. Scardina, D. Perricone, S. Plumari, M. Ruggieri, V. Greco, Phys.Rev. C90 (2014) no.5, 054904.
[4] S. Plumari, G. L. Guardo, F. Scardina, V. Greco, Phys.Rev. C92 (2015) no.5, 054902.

Content type Theory Presenter name already specified

### Primary authors

Salvatore Plumari (University of Catania (Italy)) Vincenzo Greco (University of Catania)

### Co-authors

Santosh Kumar Das (University of Catania, Italy) Gabriele Coci (INFN - National Institute for Nuclear Physics) francesco scardina (INFN Catania)


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Home > Error Propagation > Propagation Of Error When Taking An Average

# Propagation Of Error When Taking An Average

## Contents

Why don't browser DNS caches mitigate DDOS attacks on DNS providers? We will state the general answer for R as a general function of one or more variables below, but will first cover the specail case that R is a polynomial function The second thing I gathered is that I'm not sure if this is even a valid question since it appears as though I am comparing two different measures. is it ok that we set the SD of each rock to be 2 g despite the fact that their means are different (and thus different relative errors). click site

haruspex, May 25, 2012 May 25, 2012 #6 viraltux haruspex said: ↑ Sorry, a bit loose in terminology. Now the question is: what is the error of that average? Error propagation rules may be derived for other mathematical operations as needed. The errors are said to be independent if the error in each one is not related in any way to the others. navigate to these guys

## Propagation Of Error Division

A + ΔA A (A + ΔA) B A (B + ΔB) —————— - — ———————— — - — ———————— ΔR B + ΔB B (B + ΔB) B B (B This step should only be done after the determinate error equation, Eq. 3-6 or 3-7, has been fully derived in standard form. Everyone who loves science is here!

• I would believe $$σ_X = \sqrt{σ_Y^2 + σ_ε^2}$$ haruspex, May 27, 2012 May 28, 2012 #15 viraltux haruspex said: ↑ viraltux, there must be something wrong with that argument.
• More precise values of g are available, tabulated for any location on earth.
• Please try the request again.

A consequence of the product rule is this: Power rule. I really appreciate your help. of the dataset, whereas SDEV estimates the s.d. Multiplying Uncertainties R x x y y z z The coefficients {cx} and {Cx} etc.

That was exactly what I was looking for. Average Uncertainty The error in the sum is given by the modified sum rule: [3-21] But each of the Qs is nearly equal to their average, , so the error in the sum I think this should be a simple problem to analyze, but I have yet to find a clear description of the appropriate equations to use. http://math.stackexchange.com/questions/123276/error-propagation-on-weighted-mean in each term are extremely important because they, along with the sizes of the errors, determine how much each error affects the result.

Let's say that the mean ± SD of each rock mass is now: Rock 1: 50 ± 2 g Rock 2: 10 ± 1 g Rock 3: 5 ± 1 g Error Propagation Square Root The average values of s and t will be used to calculate g, using the rearranged equation: [3-11] 2s g = —— 2 t The experimenter used data consisting of measurements Then to get the variance and mean for this you simply take the mean and variance of the sum of all the X(i)'s and this will give you a mean and Let Δx represent the error in x, Δy the error in y, etc.

## Average Uncertainty

Hi TheBigH, You are absolutely right! https://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm Generated Mon, 24 Oct 2016 19:59:55 GMT by s_wx1157 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection Propagation Of Error Division Share a link to this question via email, Google+, Twitter, or Facebook. Error Propagation Calculator Spectral Standard Model and String Compactifications Partial Differentiation Without Tears Frames of Reference: A Skateboarder’s View Ohm’s Law Mellow So I Am Your Intro Physics Instructor Why Is Quantum Mechanics So

Do this for the indeterminate error rule and the determinate error rule. http://spamdestructor.com/error-propagation/propagate-error-through-average.php One simplification may be made in advance, by measuring s and t from the position and instant the body was at rest, just as it was released and began to fall. In other words, the error of $x + y$ is given by $\sqrt{e_1^2 + e_2^2}$, where $e_1$ and $e_2$ and the errors of $x$ and $y$, respectively. Indeterminate errors have unknown sign. Error Propagation Physics

It is therefore likely for error terms to offset each other, reducing ΔR/R. How would I then correctly estimate the error of the average? –Wojciech Morawiec Sep 29 '13 at 22:17 1 Even if you don't mind systematic errors, if you agree that But I note that the value quoted, 24.66, is as though what's wanted is the variance of weights of rocks in general. (The variance within the sample is only 20.1.) I'm navigate to this website Generated Mon, 24 Oct 2016 19:59:55 GMT by s_wx1157 (squid/3.5.20)

The uncertainty in the weighings cannot reduce the s.d. Error Propagation Chemistry Suppose n measurements are made of a quantity, Q. You will sometimes encounter calculations with trig functions, logarithms, square roots, and other operations, for which these rules are not sufficient.

## I presume a value like $6942\pm 20$ represents the mean and standard error of some heating measurements; $6959\pm 19$ are the mean and SE of some cooling measurements.

It will be hard to estimate $\mu$ because you have little information about $\delta_h$ or $\delta_c$. The relative error in R as [3-4] ΔR ΔAB + ΔBA ΔA ΔB —— ≈ ————————— = —— + —— , R AB A B this does give us a very I think this should be a simple problem to analyze, but I have yet to find a clear description of the appropriate equations to use. Error Propagation Inverse We weigh these rocks on a balance and get: Rock 1: 50 g Rock 2: 10 g Rock 3: 5 g So we would say that the mean ± SD of

What I am struggling with is the last part of your response where you calculate the population mean and variance. Yes and no. Please try the request again. my review here So which estimation is the right one?

I have looked on several error propagation webpages (e.g. Now I have two values, that differ slighty and I average them. When a quantity Q is raised to a power, P, the relative error in the result is P times the relative error in Q. You want to know how ε SD affects Y SD, right?

What is the error then?


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# AP Board 6th Class Maths Solutions Chapter 5 Fractions and Decimals Ex 5.2

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 5 Fractions and Decimals Ex 5.2 Textbook Questions and Answers.

## AP State Syllabus 6th Class Maths Solutions 5th Lesson Fractions and Decimals Ex 5.2

Question 1.
Find the product of the following.
i) 3 × $$\frac{5}{12}$$
ii) $$\frac{15}{8}$$ × 12
iii) 1$$\frac{3}{4}$$ × $$\frac{12}{21}$$
iv) $$\frac{4}{5}$$ × $$\frac{12}{7}$$

Question 2.
Which is greater?
i) $$\frac{1}{2}$$ of $$\frac{6}{7}$$ or $$\frac{2}{3}$$ of $$\frac{3}{7}$$
ii) $$\frac{2}{7}$$ of $$\frac{3}{4}$$ or $$\frac{3}{5}$$ of $$\frac{5}{8}$$
i) $$\frac{1}{2}$$ of $$\frac{6}{7}$$ or $$\frac{2}{3}$$ of $$\frac{3}{7}$$

$$\frac{3}{7}$$ or $$\frac{2}{7}$$
So, $$\frac{3}{7}$$ > $$\frac{2}{7}$$
∴ $$\frac{1}{2}$$ of $$\frac{6}{7}$$ > $$\frac{2}{3}$$ of $$\frac{3}{7}$$

ii) $$\frac{2}{7}$$ of $$\frac{3}{4}$$ or $$\frac{3}{5}$$ of $$\frac{5}{8}$$

Convert them into like fractions
LCM of denominators = 2 × 2 × 2 × 7 = 56
$$\frac{3 \times 4}{14 \times 4}$$ or $$\frac{3 \times 7}{8 \times 7}$$
$$\frac{12}{56}$$ or $$\frac{21}{56}$$
So, $$\frac{12}{56}$$ < $$\frac{21}{56}$$
∴ $$\frac{2}{7}$$ of $$\frac{3}{4}$$ < $$\frac{3}{5}$$ of $$\frac{5}{8}$$

Question 3.
Find:
i) $$\frac{7}{11}$$ of 330
ii) $$\frac{5}{9}$$ of 108
iii) $$\frac{2}{7}$$ of 16
iv) $$\frac{1}{7}$$ of $$\frac{3}{10}$$
i) $$\frac{7}{11}$$ of 330
= $$\frac{7}{11}$$ × 330
= $$\frac{7}{11}$$ × 11 × 30
= 7 × 30 = 210

ii) $$\frac{5}{9}$$ of 108
= $$\frac{5}{9}$$ × 108
= $$\frac{5}{9}$$ × 9 × 12
= 5 × 12 = 60

iii) $$\frac{2}{7}$$ of 16
= $$\frac{2}{7}$$ × 16
= $$\frac{2×16}{7}$$
= $$\frac{32}{7}$$ or 4$$\frac{4}{7}$$

iv) $$\frac{1}{7}$$ of $$\frac{3}{10}$$
= $$\frac{1}{7}$$ × $$\frac{3}{10}$$
= $$\frac{1 \times 3}{7 \times 10}$$
= $$\frac{3}{70}$$

Question 4.
If the cost of a notebook is Rs. 10$$\frac{3}{4}$$. Then, find the cost of 36 books.
Cost of each notebook = Rs. 10$$\frac{3}{4}$$ (or) $$\frac{43}{4}$$
Cost of 36 notebooks = 36 × 10$$\frac{3}{4}$$ = 36 × $$\frac{43}{4}$$ = $$\frac{9 \times 4 \times 43}{4}$$ = 9 × 43
A motor bike runs 52$$\frac{1}{2}$$ km using 1 litre of petrol. How much distance will it cover for 2$$\frac{3}{4}$$ litres of petrol ?
Distance covered by the motor bike 1 litre petrol = 52$$\frac{1}{2}$$ km (or) $$\frac{105}{2}$$ km
Distance covered by the motor bike for 2$$\frac{3}{4}$$ litres of petrol = 2$$\frac{3}{4}$$ of 52$$\frac{1}{2}$$
= $$\frac{11}{4}$$ × $$\frac{105}{2}$$
= $$\frac{11 \times 105}{4 \times 2}$$
∴ Distance covered by the motor bike for 2$$\frac{3}{4}$$ litres of petrol = $$\frac{1155}{8}$$ = 144$$\frac{3}{8}$$ km.


text:
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text:
Photons as virtual particles in High energy

1. Oct 27, 2013

da_steve

Hi,

So i can accept that photons are virtual particles for the electromagnetic force but i have a question.

Considering two stationary point charges. There are photon-like particles exchanged between them to produce the force? If so then placing a double slit between them should create some interference pattern (or even if there was just one and measuring behind it). I know it isn't from maxwells (potential is scalar field) but wouldn't the photons interfere if they were the force carriers?

Just confused :/

Thanks!

2. Oct 27, 2013

Staff Emeritus
Your mistake is in thinking virtual particles mean "little ping-pong balls moving back and forth". Virtual particles are not real (indeed, the term is used in opposition to "real particles"), so they don't have trajectories or wavelengths. Indeed, you don't even have a fixed number of them. So you cannot ascribe the behavior of real photons to them.

3. Oct 27, 2013

nikol

If I can add something to the discussion, you can thing this way: in order to see interference pattern you need a lot of photons(e.g. a beam) with the same wavelength (e.g.energy since for light $E=\frac{hc}{\lambda}$ to go trough the slits, so this is making the experiment impossible, since the virtual photons can have any energy (wavelength).

4. Oct 27, 2013

Fields

What is really making the experiment impossible is the infinitessimally short amount of time they exist. Have we even detected any virtual particles yet? Let alone virtual photons?

5. Oct 27, 2013

Staff Emeritus
Let me say it again. Virtual particles are not real. You can't see them or count them, and they don't have wavelengths.

6. Oct 27, 2013

Fields

That is your belief, and you are entitled to it. It is my belief they are real, in that they really do exist. This is not religion any more than the Higgs Boson was a religion before it's discovery.

7. Oct 27, 2013

Staff Emeritus
My belief comes from studying physics for many years, earning a PhD in it, and practicing it professionally for two decades.

Where does your belief come from?

8. Oct 28, 2013

Simon Bridge

It is not a matter of belief.
The detected Higgs boson, your example, was a real one. The virtual Higgs boson is just that: virtual. i.e. it is not real. It exists only as a step in a calculation process. It is important not to confuse the map for the territory.

It is not useful to think of charges firing virtual photons at other charges - it gives you the wrong physics. Which is what you have discovered. You should also consider how an electric field, if it comes from actual photons getting exchanged by charges, could possibly give you attractive forces.

The photon that is mediating the EM interaction does not have the same sort of reality as the photons that make up, say, a laser beam.

You are welcome to go on believing they are - but I don't think that would be an accepted/mainstream belief. Which makes it out of bounds as a topic for this forum.

Unless you can produce an accepted source saying otherwise of course.
For a list of accepted sources, please see the forum rules.

9. Oct 28, 2013

tiny-tim

(we are talking about virtual particles in the context of "mediating" a force)

Fields, do you mean on-shell virtual particles, or off-shell virtual particles exist (and "mediate" a force)?

(You presumably don't believe they both exist)
The Higgs Boson was predicted as part of a scientific theory, based on equations.

The turtles on which the universe stands are predicted as part of a religious theory, not based on equations.

There is nothing in the equations to support your idea that virtual particles "mediating" a force exist in actual space-time.

EDIT: just remembered … i like comparing this with epicycles, the imaginary circles in space whose centres rotate around larger circles, and which explained (very accurately) the motion of the planets (before we knew they were ellipses) …

we now know that the epicycles have no actual existence: they are only a mathematical trick for approximating a planet's orbit (very like a fourier series)

however, actual epicycles existing in space would at least be consistent with the equations, since the equations actually describe where they are!

it would therefore be possible to believe scientifically in their existence until observation confirmed or disproved it

however, that is not the case with virtual particles "mediating" a force … the equations do not describe where they are!

Last edited: Oct 28, 2013
10. Oct 28, 2013

Bill_K

The following quote was posted by Naty1, I don't have the original source:

11. Oct 28, 2013

Staff Emeritus
Bill, this line of argument is unhelpful. Like pornography, "I know it when I see it" - real particles and virtual particles behave differently. As there is a real qualitative distinction between real and virtual behavior, despite the fact that treating them in the same mathematical framework is sometimes convenient, it is not helpful to clear up misunderstandings like the OP's.

12. Oct 28, 2013

tiny-tim

13. Oct 28, 2013

Andy Resnick

This is a tempest in a teapot: Virtual particles are so named because their (brief) existence is limited by their violation of mass-energy conservation (Eisberg and Resnick, 'Quantum Physics', pg 634). Alternatively, intermediate states have a conserved 3-momentum but the energy is not; these are said to be virtual states (Landau and Lifshitz v.4, pg 312). Virtual particles disobey the relationship p^2 = m^2 satisfied by real particles (ibid.). L&L have a whole section (79) in that volume dedicated to the issue. An alternative name for virtual particles is 'off-mass-shell' particles (Gross, 'Relativistic Quantum Mechanics and Field Theory', p247). Similarly, virtual transitions (transitions between virtual states) occur in nonlinear optical processes (Boyd, 'Nonlinear optics', pp 6, 229 and Yariv, 'quantum electronics') such as two-photon processes. Jackson has a lengthy section dedicated to 'virtual quanta'.

14. Oct 28, 2013

tiny-tim

Hi Andy!
Aren't those three books referring to three different (and incompatible) versions of virtual particle?

The first is the "energy-borrowing" creation of particles out of nothing (except energy "borrowed" from the vacuum): these are not involved in "mediation" of forces (the subject of this thread).

The second is virtual particles of the first kind (as in Feynman diagrams of the first kind): their masses are correct ("on-shell"), they exist in position space, and energy is conserved at collisions but 3-momentum isn't.

The third is virtual particles of the second kind (as in Feynman diagrams of the second kind): their masses are all wrong ("off-shell"), they exist in momentum space (ie the maths doesn't even suggest they exist in actual space), and 4-momentum (including energy) is conserved at collisions.

15. Oct 28, 2013

Bill_K

Going from "virtual particles of the first kind" to "virtual particles of the second kind" is just a change of basis, tiny-tim, like we do all the time in quantum mechanics. A photon state in momentum space is a linear combination of photon states in position space, and vice-versa. That does not mean they are two different kinds of particles, or that they exist in different spaces! :uhh:

In the Coulomb example that the OP questioned, the field in position space is 1/r. Take the Fourier transform and you see that this can be written as a linear combination of photons of well-defined momentum k, with amplitude 1/k2. Virtual particles are typically integrated over. Does not imply a lack of reality on their part.

16. Oct 28, 2013

tiny-tim

sorry, i'm not understanding this

a "first kind" virtual particle has a 3-momentum p (and an "on-shell" energy √(p2 + m2)),

and all calculations involve triple-integrals: ∫∫∫ d3p

a "second kind" virtual particle has a 4-momentum q (and is "off-shell"),

and all calculations involve quadruple-integrals: ∫∫∫∫ d4q

that isn't a change to a different basis in the same space, it's to a different space (usually called momentum space)

where in position space is any momentum space virtual particle? (or where is it centred?) … the concept of position doesn't seem to apply in momentum space calculations

moreover (and i should have mentioned this in my last post), quantum field theory says that "first kind" virtual particles have creation and annihilation operators, but "second kind" don't … again, that's not merely a change in basis

finally, if "first kind" actually exist, then they have a 4-momentum, and it's on-shell; if "second kind" actually exist, then they have a 4-momentum, and it's off-shellhow can these be the same real particles?

17. Oct 28, 2013

Staff Emeritus
None of this is helping the OP.

18. Oct 28, 2013

Simon Bridge

Good point.
The thread has been hijacked hasn't it?

Actually we have not heard from OP (daSteve) since the start.

The original question is about whether you can get interference effects in electrostatic fields.
The argument is that perhaps you aught to because photons mediate the field, and you get interference effects with photons.

19. Oct 29, 2013

Crazymechanic

But then has anyone got an interference pattern with an static charge ? As long as I can tell no, so if they are not physically real then why should they be theoretically?

20. Oct 29, 2013

PICOGRAV

What happened to my post?


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# Thermodynamics - Energy analysis of piston-cylinder device

1. Sep 25, 2007

### eh87

Hello :)

I'm having trouble working on this homework question (please see attachment). The answers are supposed to be
a) 22.61 KJ
b) 36.79 KJ
c) 151.8*C

I solved (a) and I noticed that the answer to (c) is that saturation temperature at 500kPa, but I just cannot get (b)! In fact, I'm not really sure I even know what's given for part (b). Obviously P=500kPa, but what else is known? Would Vb = 0.6(Va) = 0.0551898 m^3 ? It seems like I need another intensive property (other than P) to identify the state as a start.

help

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2. Sep 25, 2007

### eh87

So I worked on it some more, just trying to understand the darn thing.. and yes, $$V_{b}$$ = 0.0551898 m^3. Now which formula to use for boundary work?
$$W_{b}$$ = $$\int$$$$mRT/V$$dV = mRT*ln($$V_{2}$$/$$V_{1}$$)