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## Deterministic criteria for the absence of arbitrage in one-dimensional diffusion models

Mijatović, Aleksandar; Urusov, Mikhail
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.14%
We obtain a deterministic characterisation of the \emph{no free lunch with vanishing risk}, the \emph{no generalised arbitrage} and the \emph{no relative arbitrage} conditions in the one-dimensional diffusion setting and examine how these notions of no-arbitrage relate to each other.; Comment: 20 pages; most results in this paper were contained in the first version of submission 0905.3701; to appear in Finance & Stochastics

## Asymptotic arbitrage and num\'eraire portfolios in large financial markets

Rokhlin, Dmitry B.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.14%
This paper deals with the notion of a large financial market and the concepts of asymptotic arbitrage and strong asymptotic arbitrage (both of the first kind), introduced by Yu.M. Kabanov and D.O. Kramkov. We show that the arbitrage properties of a large market are completely determined by the asymptotic behavior of the sequence of the num\'eraire portfolios, related to the small markets. The obtained criteria can be expressed in terms of contiguity, entire separation and Hellinger integrals, provided these notions are extended to sub-probability measures. As examples we consider market models on finite probability spaces, semimartingale and diffusion models. Also a discrete-time infinite horizon market model with one log-normal stock is examined.; Comment: 18 pages

## Generalised arbitrage-free SVI volatility surfaces

Guo, Gaoyue; Jacquier, Antoine; Martini, Claude; Neufcourt, Leo
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.14%
In this article we propose a generalisation of the recent work of Gatheral and Jacquier on explicit arbitrage-free parameterisations of implied volatility surfaces. We also discuss extensively the notion of arbitrage freeness and Roger Lee's moment formula using the recent analysis by Roper. We further exhibit an arbitrage-free volatility surface different from Gatheral's SVI parameterisation.; Comment: 20 pages, 4 figures. Section 2 more precise. Added a section on non-smooth implied volatilities

## Arbitrage Opportunities in Misspecified Stochastic volatility Models

Jena, Rudra P.; Tankov, Peter
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.14%
There is vast empirical evidence that given a set of assumptions on the real-world dynamics of an asset, the European options on this asset are not efficiently priced in options markets, giving rise to arbitrage opportunities. We study these opportunities in a generic stochastic volatility model and exhibit the strategies which maximize the arbitrage profit. In the case when the misspecified dynamics is a classical Black-Scholes one, we give a new interpretation of the classical butterfly and risk reversal contracts in terms of their (near) optimality for arbitrage strategies. Our results are illustrated by a numerical example including transaction costs.; Comment: Several typos in section 5 have been corrected in this new version (with thanks to Amy Y. Zhou from MIT)

## Derivative pricing with virtual arbitrage

Ilinski, Kirill; Stepanenko, Alexander
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.14%
In this paper we derive an effective equation for derivative pricing which accounts for the presence of virtual arbitrage opportunities and their elimination by the market. We model the arbitrage return by a stochastic process and find an equation for the average derivative price. This is an integro-differential equation which, in the absence of the virtual arbitrage or for an infinitely fast market reaction, reduces to the Black-Scholes equation. Explicit formulas are obtained for European call and put vanilla options.; Comment: Latex, 10 pages

## A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing

Fontana, Claudio
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.14%
We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper "Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing" by B. Wong and C.C. Heyde (Stochastics, 2010) in the context of incomplete It\^o-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counterexample.; Comment: 10 pages

## Credit Risk in a Geometric Arbitrage Perspective

Farinelli, Simone
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.14%
Geometric Arbitrage Theory, where a generic market is modelled with a principal fibre bundle and arbitrage corresponds to its curvature, is applied to credit markets to model default risk and recovery, leading to closed form no arbitrage characterizations for corporate bonds.; Comment: arXiv admin note: substantial text overlap with arXiv:0910.1671

## Static versus Dynamic Arbitrage Bounds on Multivariate Option Prices

d'Aspremont, Alexandre
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.14%
We compare static arbitrage price bounds on basket calls, i.e. bounds that only involve buy-and-hold trading strategies, with the price range obtained within a multi-variate generalization of the Black-Scholes model. While there is no gap between these two sets of prices in the univariate case, we observe here that contrary to our intuition about model risk for at-the-money calls, there is a somewhat large gap between model prices and static arbitrage prices, hence a similarly large set of prices on which a multivariate Black-Scholes model cannot be calibrated but where no conclusion can be drawn on the presence or not of a static arbitrage opportunity.; Comment: Submitted to IMA series

## Non-Arbitrage up to Random Horizon for Semimartingale Models

Aksamit, Anna; Choulli, Tahir; Deng, Jun; Jeanblanc, Monique
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.14%
This paper addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, which is also known in the literature as the first kind of non-arbitrage. For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and random time for which the resulting stopped model fulfills NUPBR condition. The second main result characterises the random time models that preserve the NUPBR property after stopping for any market model. These results are elaborated in a very general market model, and we also pay attention to some particular and practical models. The analysis that drives these results is based on new stochastic developments in semimartingale theory with progressive enlargement. Furthermore, we construct explicit martingale densities (deflators) for some classes of local martingales when stopped at random time.; Comment: 40 pages. This version develops in details the ideas and the results of the previous version and fixes a glitch in the quasi-left-continuous case

## No-arbitrage conditions and absolutely continuous changes of measure

Fontana, Claudio
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.14%
We study the stability of several no-arbitrage conditions with respect to absolutely continuous, but not necessarily equivalent, changes of measure. We first consider models based on continuous semimartingales and show that no-arbitrage conditions weaker than NA and NFLVR are always stable. Then, in the context of general semimartingale models, we show that an absolutely continuous change of measure does never introduce arbitrages of the first kind as long as the change of measure density process can reach zero only continuously.; Comment: 14 pages. Arbitrage, Credit and Informational Risks (C. Hillairet, M. Jeanblanc and Y. Jiao, eds.), Peking University Series in Mathematics, Vol. 6, World Scientific, 2014

## Arbitrage-free prediction of the implied volatility smile

Dellaportas, Petros; Mijatović, Aleksandar
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.14%
This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to $n$ strikes (with $n$ large, e.g. $n\geq 40$) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any $n$-dimensional time series model also has to satisfy these implicit restrictions at each time step, a condition that is impossible to meet since the model innovations can take arbitrary values. We solve this problem for any $n\in\NN$ in the context of Foreign Exchange (FX) by first encoding the option prices at each time step in terms of the parameters of the corresponding risk-neutral measure and then performing the time series analysis in the parameter space. The option price predictions are obtained from the predicted risk-neutral measure by effectively integrating it against the corresponding option payoffs. The non-linear transformation between option prices and the risk-neutral parameters applied here is \textit{not} arbitrary: it is the standard mapping used by market makers in the FX option markets (the SABR parameterisation) and is given explicitly in closed form. Our method is not restricted to the FX asset class nor does it depend on the type of parameterisation used. Statistical analysis of FX market data illustrates that our arbitrage-free predictions outperform the naive random walk forecasts...

## Periodic Sequences of Arbitrage: A Tale of Four Currencies

Cross, Rod; Kozyakin, Victor; O'Callaghan, Brian; Pokrovskii, Alexei; Pokrovskiy, Alexey
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.14%
This paper investigates arbitrage chains involving four currencies and four foreign exchange trader-arbitrageurs. In contrast with the three-currency case, we find that arbitrage operations when four currencies are present may appear periodic in nature, and not involve smooth convergence to a "balanced" ensemble of exchange rates in which the law of one price holds. The goal of this article is to understand some interesting features of sequences of arbitrage operations, features which might well be relevant in other contexts in finance and economics.; Comment: 35 pages, 48 bibliography references, submitted to Metroeconomica

## An Hilbert space approach for a class of arbitrage free implied volatilities models

Brace, A.; Fabbri, G.; Goldys, B.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.14%
We present an Hilbert space formulation for a set of implied volatility models introduced in \cite{BraceGoldys01} in which the authors studied conditions for a family of European call options, varying the maturing time and the strike price $T$ an $K$, to be arbitrage free. The arbitrage free conditions give a system of stochastic PDEs for the evolution of the implied volatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family obtained fixing a strike $K$ and varying $T$. In order to give conditions to prove an existence-and-uniqueness result for the solution of the system it is here expressed in terms of the square root of the forward implied volatility and rewritten in an Hilbert space setting. The existence and the uniqueness for the (arbitrage free) evolution of the forward implied volatility, and then of the the implied volatility, among a class of models, are proved. Specific examples are also given.; Comment: 21 pages

## Black-Scholes equation from Gauge Theory of Arbitrage

Ilinski, Kirill; Kalinin, Gleb


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# A finite set is closed

Question:

Prove that a finite subset in a metric space is closed.

My proof-sketch: Let $A$ be finite set. Then $A=\{x_1, x_2,\dots, x_n\}.$ We know that $A$ has no limits points. What's next?

Definition: Set $E$ is called closed set if $E$ contains all his limits points.

• Know we use the fact that $\emptyset \subset E$ (the set of limit points is the empty set). – Noah Olander Jun 8 '15 at 21:52
• What is the context? Subsets of $\mathbb R$ with the usual topology/metric? Without context it is impossible to answer your question as in fact in an arbitrary topological space not all finite sets need be closed. – Ittay Weiss Jun 8 '15 at 21:53
• I mean a metric space. – ZFR Jun 8 '15 at 21:55
• @IttayWeiss You are right. Although since the OP tag this as real analysis I assume he might be talking of any finite dimensional metric space, in which case the finite sets are always closed. – Alonso Delfín Jun 8 '15 at 21:56
• @Pacman do you happen to know then that in a metric space every singleton is a closed set? and that the union of finitely many closed sets is closed? combine... – Ittay Weiss Jun 8 '15 at 21:57

Thinking $A$ as a subset of a metric space $M$. An easy approach will be to use that the single points $\{x_j\}\subset M$ are closed (you know why?), then of course $$A=\bigcup_{j=1}^n \{ x_j \}$$ Since $A$ is a finite union of closed sets, it is itself closed.

• I don't know that single point is closed – ZFR Jun 8 '15 at 22:03
• Yes, in metric spaces they are. And it is very easy to prove PROOF: Take any convergent sequence in $\{ x \}$, then of course it converges to $x \in \{ x \}$, thus $\{x\}$ is closed !! – Alonso Delfín Jun 8 '15 at 22:04
• You know in Rudin's book the convergent series is after that closed sets. – ZFR Jun 8 '15 at 22:09
• @Pacman I see. Here is a proof without using the sequence criteria for closed sets. PROOF Take any $y \in M \setminus \{ x \}$ and fix $r=d(x,y)$, then clearly $B_{r/2}(y) \subset M \setminus \{ x \}$, thus $M \setminus \{ x \}$ is open and hence $\{ x \}$ must be closed $\blacksquare$ . Here $B_{r/2}(y)=\{z \in M: d(z,y)<r/2\}$ – Alonso Delfín Jun 8 '15 at 22:21
• @AlonsoDelfín Actually you could use $r$ instead of $r/2$ in that proof – Mario Carneiro Jun 8 '15 at 23:15

If $M$ is a metric space then every subset $A =\{x_1, \ldots, x_n\} \subseteq M$ is closed. In fact, if $a \notin A$ then $d(a,A)$ is the least of the numbers $d(a,x_1) ,\ldots, d(a,x_n)$ thus, $d(a,A) > 0$.

• In particular, $\{x\}$ is closed. – Aaron Maroja Jun 8 '15 at 22:12
• Why you take $a\notin A$? – ZFR Jun 8 '15 at 22:18
• $a \in \overline X \iff d(a, X) = 0$. – Aaron Maroja Jun 8 '15 at 22:20

If we show that the compliment of $A$ is open, then $A$ is closed.

Let $B=A^c$, and suppose that $y \in B$. We need to show that there is an $r >0$ for which $B_r(y) \cap A = \emptyset$, that is $B_r(y) \subset B$.

If we let $r = \min \{ d(y,x_i) : i = 1,...,n\}$ then it must be true that $B_r(y) \cap A = \emptyset$, since for every $z \in B_r(y)$ we have $d(z,y) < d(y,x_i)$ for each $i=1,...,n$.

• Can you do this without a metric? – steven gregory Jun 9 '15 at 1:58
• The metric defined your topology and therefore your open sets. In a general topology, you would not expect singletons or finite sets to be closed. – Joel Jun 9 '15 at 15:27
• @StevenGregory, in any case, the explicit use of a metric can be avoided, but it will be implicit through the topology. – Joel Jun 9 '15 at 15:29

I think the simplest answer to this, following Baby Rudin (Principles of mathematical analysis 3rd ed.) terminology, is:

The definition of a closed set is in 2.18.d: a set $$A$$ is closed if every limit point of $$A$$ is an element of it. However, none of the points $$x_i \in A$$ are limit points of $$A$$: for each $$x_i$$, pick $$r = min(d(x_{i-1} x_i), d(x_i, x_{i+1}))$$ i.e. the minimum distance between itself and its neighbours (to take care of corner cases, define $$d(x_0, \_) = \infty$$ , $$d(\_, x_{n+1}) = \infty$$). Now, any neighbourhood $$N_{r'}(x_i)$$ for $$r' < r$$ doesn't contain any other point of $$A$$ apart from $$x_i$$ so $$x_i$$ can't be a limit point. Therefore, $$A$$ is closed "vacuously" i.e. trivially, because the condition for it to be closed (all limit points of $$A$$ are elements of $$A$$) is trivially satisfied because the set of all limit points of $$A$$ is empty (a set of 0 points has any property).

If some open set $G_i$ containing $x_1$ does not contain $x_i$, for $i=1,\ldots,n$, then $\bigcap_{i=1}^n G_i$ is an open set containing $x_1$ and not containing any of $x_2,\ldots,x_n$. Therefore $x_1$ is not a limit point of the set. The same argument applies to show that $x_2,\ldots,x_n$ are not limit points.

If the space is a Hausdorff space, then certainly some open set $G_i$ containing $x_1$ does not contain $x_i$ if $i\ne1$.

If it's a metric space, you can take $G_i$ to be the open ball of radius $d(x_1,x_i)$ centered at $x_1$.

Let $(X, d)$ be a metric space and $A = \{ x_1, \ldots, x_n \} \subset X$ be finite. Then the following holds: A sequence $(a_n)_{n \in \Bbb N}$ in $A$ is convergent, if and only if there exists a $N \in \Bbb N$ and a $i \in \{ 1, \ldots, n\}$, such that $$a_n = x_i \quad \text{for all } n \geq N \; ,$$ i.e. the sequence $(a_n)_{n \in \Bbb N}$ will be constant up to a index $N$. This means, that the limit points of $A$ are $x_1, \ldots, x_n$, so $A$ contains all its limit points, thus $A$ is closed.

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# Model description for glca In glca: An R Package for Multiple-Group Latent Class Analysis

Suppose that there are $G$ groups, and the $g$th group consists of $n_g$ observations for $g=1,\ldots,G$, and there are $M$ categorical manifest items, where the $m$th item has $r_m$ categories for $m=1,\ldots,M$. Let $\mathbf{Y}{ig}=(Y{ig1}, \dots, Y_{igM})^\top$ and ${\bf y}{ig} = (y{ig1}, \ldots, y_{igM})^\top$ denote a set of item variables and their responses given by the $i$th individual within the $g$th group, respectively. The number of possible response patterns of $\mathbf{Y}{ig}$ is $\prod{m = 1}^{M}r_m$, and it is likely that most of these response patterns are sparse. The multiple-group LCA assumes that associations among manifest items can be explained by the latent classifier $L_{ig}$, where $L_{ig}$ is the latent class variable having $C$ categories for the $i$th individual within the $g$th group. To reflect multiple-group data structures, we discuss two different LCA approaches, namely fixed-effect and random-effect LCA.

## Models

### Fixed-effect latent class analysis

The fixed-effect LCA can reflect group differences in latent structure by specifying an LCR model for a given subgroup. We extend the simultaneous LCA (Clogg & Goodman, 1985) by incorporating logistic regression in the class prevalence and refer it to multiple-group latent class regression (mgLCR). Let ${\bf x}{ig}=(x{ig1}, \ldots, x_{igp})^\top$ be a subject-specific $p\times 1$ vector of covariates for the $i$th individual within the $g$th group, either discrete or continuous. Then, the observed-data likelihood of mgLCR for the $i$th individual within the $g$th group can be specified as \begin{eqnarray} \mathcal{L}{ig} &=& \sum{c=1}^C P(\mathbf{Y}{ig} = \mathbf{y}{ig}, L_{ig} = c \mid {\bf x}{ig}) \nonumber \ &=& \sum{c=1}^C \left[ P(L_{ig} = c \mid {\bf x}{ig}) \prod{m = 1}^{M} P(y_{igm} = k \mid L_{ig} = c) \right] \nonumber \ &=& \sum_{c=1}^C \left[ \gamma_{c \mid g}({\bf x}{ig}) \prod{m = 1}^{M}\prod_{k = 1}^{r_m} \rho_{mk \mid cg}^{I(y_{igm} = k)} \right], (#eq:likeli) \end{eqnarray} where $I(y_{igm}=k)$ is an indicator function that is equal to 1 when the response to the $m$th item from the $i$th individual within the $g$th group is $k$ and is otherwise equal to 0. The likelihood given in \@ref(eq:likeli) contains two types of parameters:

1. $\rho_{mk \mid cg}$ represents the probability of an individual within the $g$th group responding $k$ to the $m$th item given his or her latent class as $c$.
2. $\gamma_{c \mid g}({\bf x}{ig})$ is the probability of the $i$th individual belonging to the latent class $c$ within the $g$th group, which could be influenced by the subject-specific covariates ${\bf x}{ig}$.

The $\rho$-parameter is the measurement parameter in mgLCR (i.e., item-response probability), describing a tendency of individuals in a latent class $c$ to respond to the $m$th item for $m = 1,\ldots,M$. Comparison of estimated item-response probabilities across groups is a valuable strategy for quantifying measurement invariance because they solely determine the meaning of the latent class. By comparing the model fit with the parameter held constant across groups (i.e., $\rho_{mk \mid c} = \rho_{mk \mid c1} = \dots = \rho_{mk \mid cG}$ for $k=1,\ldots,r_m$, $m=1,\ldots,M$, and $c=1,\ldots,C$) against an alternative model with freely varying parameters, we obtain evidence on whether measurement invariance across groups can be assumed. As given in \@ref(eq:likeli), the subject-specific covariates ${\bf x}{ig}$ may influence the probability of the individual belonging to a specific class in the form of logistic regression as \begin{eqnarray} \gamma{c \mid g}(\mathbf{x}{ig}) = P(L{ig} = c \mid {\bf x}{ig}) = \frac{\exp(\alpha{c \mid g}+{\bf x}{ig}^\top \boldsymbol{\beta}{c \mid g})}{\sum_{c'=1}^C\exp(\alpha_{c' \mid g}+{\bf x}{ig}^\top \boldsymbol{\beta}{c' \mid g})}, (#eq:mgLCA-reg) \end{eqnarray} where the coefficient vector $\boldsymbol{\beta}{c \mid g} = (\beta{1c \mid g}, \ldots, \beta_{pc \mid g})^\top$ can be interpreted as the expected change in the log odds of belonging to a class $c$ versus belonging to the referent class $C$ (i.e., $\alpha_{C \mid g}=0$ and $\boldsymbol{\beta}{C \mid g} = {\bf 0}$ for $g=1, \ldots, G$). Then, the observed log-likelihood function for the mgLCR model can be specified as \ell{mgLCR} = \sum_{g = 1}^{G}\sum_{i = 1}^{n_{g}} \log \mathcal{L}{ig}. (#eq:loglik-mgLCA) It should be noted that, similar to item-response probabilities, coefficients of logistic regression can be constrained to be equal across subgroups (i.e., $\boldsymbol{\beta}{c} = \boldsymbol{\beta}{c \mid 1}= \cdots = \boldsymbol{\beta}{c \mid G}$ for $c=1, \ldots,C$) to test whether the effects of covariates are identical across groups.

### Random-effect latent class analysis

The random-effect LCA considers the group variation in the latent class prevalence for each group using random coefficients, for example, [ P(L_{ig} = c) = \frac{\exp(\alpha_{c} + \sigma_{c} \lambda_{g})}{\sum_{c' = 1}^{C}\exp(\alpha_{c'} + \sigma_{c'} \lambda_{g})}, ] where $\boldsymbol{\lambda}=(\lambda_1, \ldots, \lambda_G)^\top$ represents group variation in the class prevalence. In the parametric random-effect LCA, the random coefficients are assumed to be derived from parametric distributions such as standard normal distribution. However, the nonparametric approach assumes no specific distribution; rather, it only assumes that random coefficients follow a specific probability mass function with some mass points. In other words, the nonparametric approach employs categorical level-2 latent variable (i.e., latent cluster) $U_{g}$ whose probability mass function is $P(U_g=w) = \delta_w$ for $w=1,\ldots,W$. Using the classification mechanics of LCA, the latent cluster membership of level-2 units can be identified by the small number of representative patterns of class prevalences in multiple groups. Therefore, the meaning of the $w$th level of latent cluster variable is determined by the prevalence of latent classes $P(L_{ig} = c \mid U_{g} = w)$ for $c=1,\ldots,C$. Considering latent cluster variable as a group variable, the nonparametric approach provides more meaningful interpretations in group comparison than parametric approach; we can examine whether the latent class structure differs across latent cluster memberships. Therefore, we focus on the nonparametric random-effect LCA, hereafter referred to as nonparametric latent class regression (npLCR).

The observed-data likelihood of npLCR for the $g$th group can be expressed by \begin{eqnarray} \mathcal{L}{g} &=& \sum{w = 1}^{W} P(U_g = w) \prod_{i=1}^{n_g} \left{\sum_{c = 1}^{C} P(Y_{ig}=y_{ig}, L_{ig} = c \mid U_{g} = w, \mathbf{x}{ig}, \mathbf{z}{g})\right} \nonumber \ &=& \sum_{w = 1}^{W} P(U_g = w) \prod_{i=1}^{n_g} \left{\sum_{c = 1}^{C} P(L_{ig} = c \mid U_{g} = w, \mathbf{x}{ig}, \mathbf{z}{g}) \prod_{m=1}^M P(Y_{igm}=k \mid L_{ig} = c ) \right} \nonumber \ &=& \sum_{w = 1}^{W} \delta_{w} \prod_{i=1}^{n_g} \left{ \sum_{c = 1}^{C} \gamma_{c \mid w}(\mathbf{x}{ig}, \mathbf{z}{g}) \prod_{m = 1}^{M}\prod_{k = 1}^{r_m}\rho_{mk \mid c}^{I(y_{igm} = k)}\right}, (#eq:grouploglik-mLCA) \end{eqnarray} where ${\bf x}{ig} = (x{ig1}, \ldots, x_{igp})^\top$ and ${\bf z}g=(z{g1}, \ldots, z_{gq})^\top$ denote vectors of subject-specific (i.e., level-1) and group-specific (i.e., level-2) covariates for $i=1, \ldots, n_g$ and $g=1,\ldots,G$, respectively. The likelihood given in \@ref(eq:grouploglik-mLCA) contains three types of parameters:

1. $\rho_{mk \mid c}$ represents the probability of an individual responding $k$ to the $m$th item given his or her latent class as $c$.
2. $\gamma_{c \mid w}({\bf x}{ig}, {\bf z}_g)$ is the probability of the $i$th individual within the $g$th group belonging to the latent class $c$ given the latent cluster $w$, which could be influenced by the subject-specific covariates ${\bf x}{ig}$ and the group-specific covariates ${\bf z}_g$.
3. $\delta_w$ represents the latent cluster prevalence for $w=1,\ldots,W$.

The class prevalence can be modeled using the logistic regression as \begin{eqnarray} (#eq:MLCA-reg) \gamma_{c \mid w}(\mathbf{x}{ig}, \mathbf{z}{g}) = P(L_{ig} = c \mid U_{g} = w, \mathbf{x}{ig}, \mathbf{z}{g}) = \frac{\exp(\alpha_{c \mid w} + \mathbf{x}^{\top}{ig}\boldsymbol{\beta}{1c \mid w} + \mathbf{z}^{\top}{g}\boldsymbol{\beta}{2c})} {\sum_{c' = 1}^{C} \exp(\alpha_{c' \mid w} + \mathbf{x}^{\top}{ig}\boldsymbol{\beta}{1c' \mid w} + \mathbf{z}^{\top}{g}\boldsymbol{\beta}{2c'})}, \end{eqnarray} where vectors $\boldsymbol{\beta}{1c \mid w} = (\beta{11c \mid w}, \ldots, \beta_{1pc \mid w})^\top$ and $\boldsymbol{\beta}{2c} = (\beta{21c}, \ldots, \beta_{2qc})^\top$ are logistic regression coefficients for level-1 and level-2 covariates, respectively. Then, the observed log-likelihood of npLCR is specified as \begin{eqnarray} (#eq:loglik-mLCA) \ell_{npLCR} = \sum_{g = 1}^{G} \log \mathcal{L}_{g}. \end{eqnarray}

Note that coefficients for level-1 covariates depend on both latent classes and clusters, while coefficients for level-2 covariates depend only on latent class membership. We may refer the model \@ref(eq:MLCA-reg) to the random slope model as coefficients for level-1 covariates are different across latent clusters. The coefficients for level-1 covariates can be constrained to be equal across clusters (i.e., $\boldsymbol{\beta}{1c} = \boldsymbol{\beta}{1c \mid 1} = \cdots = \boldsymbol{\beta}{1c \mid W}$ for $c=1, \ldots,C$) to test whether the effects of level-1 covariates are identical across all latent cluster memberships. It should also be noted that the measurement invariance is assumed across latent cluster memberships in npLCR (i.e., $\rho{mk \mid c} = \rho_{mk \mid c1} = \dots = \rho_{mk \mid cW}$ for $k=1,\ldots,r_m$, $m=1,\ldots,M$, and $c=1,\ldots,C$). If not, the item response probability may vary across latent cluster memberships, suggesting that the latent class structure itself is different between latent clusters. Thus, it no longer makes sense to use latent class prevalences as identifiers for the latent cluster membership.

## Estimation

### Fixed-effect latent class analysis

The package glca finds the maximum-likelihood (ML) estimates for mgLCR and npLCR using expectation-maximization (EM) algorithm (Dempster et al., 1977). The EM algorithm iterates two steps: expectation step (E-step) and maximization step (M-step) in order to find the solution maximizing the log-likelihood functions given in \@ref(eq:loglik-mgLCA) and \@ref(eq:loglik-mLCA).

For mgLCR, E-step computes the posterior probabilities \begin{eqnarray} (#eq:post) \theta_{ig(c)} = P(L_{ig} = c \mid \mathbf{Y}{ig} = \mathbf{y}{ig}, \mathbf{x}{ig}) = \frac{\gamma{c \mid g}(\mathbf{x}{ig}) \prod{m = 1}^{M}\prod_{k = 1}^{r_m} \rho_{mk \mid cg}^{I(y_{igm} = k)}}{\sum_{c' = 1}^{C} \gamma_{c' \mid g}(\mathbf{x}{ig}) \prod{m = 1}^{M}\prod_{k = 1}^{r_m} \rho_{mk \mid c'g}^{I(y_{igm} = k)}}\ \end{eqnarray} with current estimates for $i=1, \ldots, n_g$, $g=1,\ldots,G$, and $c=1,\ldots,C$. M-step maximizes the complete-data likelihood (i.e., the likelihood for the cross-classification by $L_{ig}$ and $\mathbf{y}{ig}$) with respect to $\beta$- and $\rho$-parameters. In particular, when all values of $\theta{ig(c)}$ are known, updated estimates for $\beta$-parameters can be calculated by the Newton-Raphson algorithm for multinomial logistic regression given in \@ref(eq:mgLCA-reg), provided that the computational routine allows fractional responses rather than integer counts (Bandeen-Roche et al., 1997). Therefore, the package glca conducts one-cycle of Newton-Raphson algorithm to update $\beta$-parameters at every iteration in M-step. If there is no covariate in the model, the class prevalence can be updated directly without estimating $\beta$-parameters as $\hat{\gamma}{c \mid g} = P(L{ig}=c) = \sum_{i=1}^{n_g} \theta_{ig(c)}/n_g$ for $c=1,\ldots,C$ and $g=1,\ldots,G$. The item-response probabilities, $\rho_{mk \mid cg}$ can be interpreted as parameters in a multinomial distribution when $\theta_{ig(c)}$ is known, so we have [ \hat{\rho}{mk \mid cg} = \frac{\sum{i=1}^{n_g} \theta_{ig(c)}I(y_{igm} = k)}{\sum_{i=1}^{n_g} \theta_{ig(c)}} ] for $k=1, \ldots, r_m$, $m=1,\ldots,M$, $c=1,\ldots,C$, and $g=1,\ldots,G$. Under the measurement invariance assumption (i.e., $\rho_{mk \mid c} = \rho_{mk \mid c1} = \dots = \rho_{mk \mid cG}$), the $\rho$-parameter will be updated as [ \hat{\rho}{mk \mid c} = \frac{\sum{g = 1}^{G} \sum_{i = 1}^{n_g} \theta_{ig(c)}I(y_{igm} = k)}{\sum_{g = 1}^{G} \sum_{i = 1}^{n_g} \theta_{ig(c)}} ] for $k=1, \ldots, r_m$, $m=1,\ldots,M$, and $c=1,\ldots,C$.

## Try the glca package in your browser

Any scripts or data that you put into this service are public.

glca documentation built on Nov. 3, 2021, 1:09 a.m.


text:
JEE Advanced is just 5 days away. $$\textbf{Best of luck!}$$ to all those who are appearing. Also, after 25th May, please tell me about your experience, i.e ,how was the paper and the expected score!

Note by Avineil Jain
5 years ago

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All the best to everyone!

- 5 years ago

All the best to you as well and all others who are taking the exam!!!

- 5 years ago

Best of luck to you too! Feeling sooooo nervous!

- 5 years ago

Best of luck to all!!!

- 5 years ago

well I have 370 days to go . lucky me ... :-)

- 4 years, 12 months ago

All the best Avineil :D !!

- 5 years ago

Best of Luck to you too!

- 5 years ago

all the best bro! :)

- 5 years ago

Thnx

- 5 years ago

All the best to everybody !! :D

- 5 years ago

maintain the confidence and hit the end just the way you do on brilliant

- 5 years ago

ALL the best To u as well others too.. i will be giving nxt year would learn frm u all

- 5 years ago

all the very best to you too!! and to all those who are giving it....

- 5 years ago

Good luck Everyone be optimistic and give your best .. ;)..i ll be appearing next year and I hope to crack it

- 5 years ago

All the best to everyone!! :D

- 4 years, 12 months ago

@Avineil Jain Best luck to you all who are going to take JEE Advanced

- 4 years, 12 months ago

How was the paper?!

- 4 years, 11 months ago

avineil jaiswal hv u tooo cleared JEE advanced??

- 4 years, 11 months ago

all the best to everybody who are appearing from me ......

- 5 years ago

dude what the heck is JEE? where is it found?

- 5 years ago

You got 7 downvoted !!! .... I think that's a record .! :P

- 4 years, 12 months ago

Its Joint Entrance Examination.... and its found in India .... u cud have searched in Google...

- 4 years, 12 months ago


text:
# Slope deflection method

The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by George A. Maney.[1] The slope deflection method was widely used for more than a decade until the moment distribution method was developed. In the book, "The Theory and Practice of Modern Framed Structures", written by J.B Johnson, C.W. Bryan and F.E. Turneaure, it is stated that this method was first developed,"by Professor Otto Mohr in Germany, and later developed independently by Professor G.A. Maney". According to this book, professor Otto Mohr introduced this method for the first time in his book,"Evaluation of Trusses with Rigid Node Connections" or "Die Berechnung der Fachwerke mit Starren Knotenverbindungen".

## Introduction

By forming slope deflection equations and applying joint and shear equilibrium conditions, the rotation angles (or the slope angles) are calculated. Substituting them back into the slope deflection equations, member end moments are readily determined. Deformation of member is due to the bending moment.

## Slope deflection equations

The slope deflection equations can also be written using the stiffness factor ${\displaystyle K={\frac {I_{ab}}{L_{ab}}}}$ and the chord rotation ${\displaystyle \psi ={\frac {\Delta }{L_{ab}}}}$:

### Derivation of slope deflection equations

When a simple beam of length ${\displaystyle L_{ab}}$ and flexural rigidity ${\displaystyle E_{ab}I_{ab}}$ is loaded at each end with clockwise moments ${\displaystyle M_{ab}}$ and ${\displaystyle M_{ba}}$, member end rotations occur in the same direction. These rotation angles can be calculated using the unit dummy force into the cal method or Darcy's Law.

${\displaystyle \theta _{a}-{\frac {\Delta }{L_{ab}}}={\frac {L_{ab}}{3E_{ab}I_{ab}}}M_{ab}-{\frac {L_{ab}}{6E_{ab}I_{ab}}}M_{ba}}$
${\displaystyle \theta _{b}-{\frac {\Delta }{L_{ab}}}=-{\frac {L_{ab}}{6E_{ab}I_{ab}}}M_{ab}+{\frac {L_{ab}}{3E_{ab}I_{ab}}}M_{ba}}$

Rearranging these equations, the slope deflection equations are derived.

## Equilibrium conditions

### Joint equilibrium

Joint equilibrium conditions imply that each joint with a degree of freedom should have no unbalanced moments i.e. be in equilibrium. Therefore,

${\displaystyle \Sigma \left(M^{f}+M_{member}\right)=\Sigma M_{joint}}$

Here, ${\displaystyle M_{member}}$ are the member end moments, ${\displaystyle M^{f}}$ are the fixed end moments, and ${\displaystyle M_{joint}}$ are the external moments directly applied at the joint.

### Shear equilibrium

When there are chord rotations in a frame, additional equilibrium conditions, namely the shear equilibrium conditions need to be taken into account.

## Example

Example

The statically indeterminate beam shown in the figure is to be analysed.

• Members AB, BC, CD have the same length ${\displaystyle L=10\ m}$.
• Flexural rigidities are EI, 2EI, EI respectively.
• Concentrated load of magnitude ${\displaystyle P=10\ kN}$ acts at a distance ${\displaystyle a=3\ m}$ from the support A.
• Uniform load of intensity ${\displaystyle q=1\ kN/m}$ acts on BC.
• Member CD is loaded at its midspan with a concentrated load of magnitude ${\displaystyle P=10\ kN}$.

In the following calculations, clockwise moments and rotations are positive.

### Degrees of freedom

Rotation angles ${\displaystyle \theta _{A}}$, ${\displaystyle \theta _{B}}$, ${\displaystyle \theta _{C}}$, of joints A, B, C, respectively are taken as the unknowns. There are no chord rotations due to other causes including support settlement.

### Fixed end moments

Fixed end moments are:

${\displaystyle M_{AB}^{f}=-{\frac {Pab^{2}}{L^{2}}}=-{\frac {10\times 3\times 7^{2}}{10^{2}}}=-14.7\mathrm {\,kN\,m} }$
${\displaystyle M_{BA}^{f}={\frac {Pa^{2}b}{L^{2}}}={\frac {10\times 3^{2}\times 7}{10^{2}}}=6.3\mathrm {\,kN\,m} }$
${\displaystyle M_{BC}^{f}=-{\frac {qL^{2}}{12}}=-{\frac {1\times 10^{2}}{12}}=-8.333\mathrm {\,kN\,m} }$
${\displaystyle M_{CB}^{f}={\frac {qL^{2}}{12}}={\frac {1\times 10^{2}}{12}}=8.333\mathrm {\,kN\,m} }$
${\displaystyle M_{CD}^{f}=-{\frac {PL}{8}}=-{\frac {10\times 10}{8}}=-12.5\mathrm {\,kN\,m} }$
${\displaystyle M_{DC}^{f}={\frac {PL}{8}}={\frac {10\times 10}{8}}=12.5\mathrm {\,kN\,m} }$

### Slope deflection equations

The slope deflection equations are constructed as follows:

${\displaystyle M_{AB}=2{\frac {EI}{L}}\left(2\theta _{A}+1\theta _{B}\right)={\frac {4EI\theta _{A}+2EI\theta _{B}}{L}}}$
${\displaystyle M_{BA}={\frac {EI}{L}}\left(2\theta _{A}+4\theta _{B}\right)={\frac {2EI\theta _{A}+4EI\theta _{B}}{L}}}$
${\displaystyle M_{BC}={\frac {2EI}{L}}\left(4\theta _{B}+2\theta _{C}\right)={\frac {8EI\theta _{B}+4EI\theta _{C}}{L}}}$
${\displaystyle M_{CB}={\frac {2EI}{L}}\left(2\theta _{B}+4\theta _{C}\right)={\frac {4EI\theta _{B}+8EI\theta _{C}}{L}}}$
${\displaystyle M_{CD}={\frac {EI}{L}}\left(4\theta _{C}\right)={\frac {4EI\theta _{C}}{L}}}$
${\displaystyle M_{DC}={\frac {EI}{L}}\left(2\theta _{C}\right)={\frac {2EI\theta _{C}}{L}}}$

### Joint equilibrium equations

Joints A, B, C should suffice the equilibrium condition. Therefore

${\displaystyle \Sigma M_{A}=M_{AB}+M_{AB}^{f}=0.4EI\theta _{A}+0.2EI\theta _{B}-14.7=0}$
${\displaystyle \Sigma M_{B}=M_{BA}+M_{BA}^{f}+M_{BC}+M_{BC}^{f}=0.2EI\theta _{A}+1.2EI\theta _{B}+0.4EI\theta _{C}-2.033=0}$
${\displaystyle \Sigma M_{C}=M_{CB}+M_{CB}^{f}+M_{CD}+M_{CD}^{f}=0.4EI\theta _{B}+1.2EI\theta _{C}-4.167=0}$

### Rotation angles

The rotation angles are calculated from simultaneous equations above.

${\displaystyle \theta _{A}={\frac {40.219}{EI}}}$
${\displaystyle \theta _{B}={\frac {-6.937}{EI}}}$
${\displaystyle \theta _{C}={\frac {5.785}{EI}}}$

### Member end moments

Substitution of these values back into the slope deflection equations yields the member end moments (in kNm):

${\displaystyle M_{AB}=0.4\times 40.219+0.2\times \left(-6.937\right)-14.7=0}$
${\displaystyle M_{BA}=0.2\times 40.219+0.4\times \left(-6.937\right)+6.3=11.57}$
${\displaystyle M_{BC}=0.8\times \left(-6.937\right)+0.4\times 5.785-8.333=-11.57}$
${\displaystyle M_{CB}=0.4\times \left(-6.937\right)+0.8\times 5.785+8.333=10.19}$
${\displaystyle M_{CD}=0.4\times -5.785-12.5=-10.19}$
${\displaystyle M_{DC}=0.2\times -5.785+12.5=13.66}$

## Notes

1. ^ Maney, George A. (1915). "Studies in Engineering". Minneapolis: University of Minnesota.

## References

• Norris, Charles Head; John Benson Wilbur; Senol Utku (1976). Elementary Structural Analysis (3rd ed.). McGraw-Hill. pp. 313–326. ISBN 0-07-047256-4.
• McCormac, Jack C.; Nelson, James K. Jr. (1997). Structural Analysis: A Classical and Matrix Approach (2nd ed.). Addison-Wesley. pp. 430–451. ISBN 0-673-99753-7.
• Yang, Chang-hyeon (2001-01-10). Structural Analysis (in Korean) (4th ed.). Seoul: Cheong Moon Gak Publishers. pp. 357–389. ISBN 89-7088-709-1.


text:
Suppose S and T are mutually exclusive events. Find P(S or T) if P(S) = 1/3 and P(T) = 5/12?

3/4 7/18 5/36

Apr 20, 2017

In an OR situation you may ADD

Explanation:

$P \left(S \mathmr{and} T\right) = P \left(S\right) + P \left(T\right) = \frac{1}{3} + \frac{5}{12} = \frac{4}{12} + \frac{5}{12} = \frac{9}{12} = \frac{3}{4}$

Apr 24, 2017

$P \left(S \cup T\right) = \frac{3}{4}$

Explanation:

We use the following fundamental definition from Probability and Set Theory:

$P \left(A \cup B\right) = P \left(A\right) + P \left(B\right) - P \left(A \cap B\right)$

And if we apply this to our problem we have:

$P \left(S \cup T\right) = P \left(S\right) + P \left(T\right) - P \left(S \cap T\right)$
$\text{ } = \frac{1}{3} + \frac{5}{12} - P \left(S \cap T\right)$
$\text{ } = \frac{3}{4} - P \left(S \cap T\right)$

Now we are told that $S$ and $T$ are mutually exclusive, and so:

$P \left(S \cap T\right) = 0$

Hence,

$P \left(S \cup T\right) = \frac{3}{4}$


text:
# scipy.io.netcdf.netcdf_file¶

class scipy.io.netcdf.netcdf_file(filename, mode='r', mmap=None, version=1)[source]

A file object for NetCDF data.

A netcdf_file object has two standard attributes: dimensions and variables. The values of both are dictionaries, mapping dimension names to their associated lengths and variable names to variables, respectively. Application programs should never modify these dictionaries.

All other attributes correspond to global attributes defined in the NetCDF file. Global file attributes are created by assigning to an attribute of the netcdf_file object.

Parameters: filename : string or file-like string -> filename mode : {‘r’, ‘w’, ‘a’}, optional read-write-append mode, default is ‘r’ mmap : None or bool, optional Whether to mmap filename when reading. Default is True when filename is a file name, False when filename is a file-like object. Note that when mmap is in use, data arrays returned refer directly to the mmapped data on disk, and the file cannot be closed as long as references to it exist. version : {1, 2}, optional version of netcdf to read / write, where 1 means Classic format and 2 means 64-bit offset format. Default is 1. See here for more info.

Notes

The major advantage of this module over other modules is that it doesn’t require the code to be linked to the NetCDF libraries. This module is derived from pupynere.

NetCDF files are a self-describing binary data format. The file contains metadata that describes the dimensions and variables in the file. More details about NetCDF files can be found here. There are three main sections to a NetCDF data structure:

1. Dimensions
2. Variables
3. Attributes

The dimensions section records the name and length of each dimension used by the variables. The variables would then indicate which dimensions it uses and any attributes such as data units, along with containing the data values for the variable. It is good practice to include a variable that is the same name as a dimension to provide the values for that axes. Lastly, the attributes section would contain additional information such as the name of the file creator or the instrument used to collect the data.

When writing data to a NetCDF file, there is often the need to indicate the ‘record dimension’. A record dimension is the unbounded dimension for a variable. For example, a temperature variable may have dimensions of latitude, longitude and time. If one wants to add more temperature data to the NetCDF file as time progresses, then the temperature variable should have the time dimension flagged as the record dimension.

In addition, the NetCDF file header contains the position of the data in the file, so access can be done in an efficient manner without loading unnecessary data into memory. It uses the mmap module to create Numpy arrays mapped to the data on disk, for the same purpose.

Note that when netcdf_file is used to open a file with mmap=True (default for read-only), arrays returned by it refer to data directly on the disk. The file should not be closed, and cannot be cleanly closed when asked, if such arrays are alive. You may want to copy data arrays obtained from mmapped Netcdf file if they are to be processed after the file is closed, see the example below.

Examples

To create a NetCDF file:

>>> from scipy.io import netcdf
>>> f = netcdf.netcdf_file('simple.nc', 'w')
>>> f.history = 'Created for a test'
>>> f.createDimension('time', 10)
>>> time = f.createVariable('time', 'i', ('time',))
>>> time[:] = np.arange(10)
>>> time.units = 'days since 2008-01-01'
>>> f.close()

Note the assignment of range(10) to time[:]. Exposing the slice of the time variable allows for the data to be set in the object, rather than letting range(10) overwrite the time variable.

To read the NetCDF file we just created:

>>> from scipy.io import netcdf
>>> f = netcdf.netcdf_file('simple.nc', 'r')
>>> print(f.history)
Created for a test
>>> time = f.variables['time']
>>> print(time.units)
days since 2008-01-01
>>> print(time.shape)
(10,)
>>> print(time[-1])
9

NetCDF files, when opened read-only, return arrays that refer directly to memory-mapped data on disk:

>>> data = time[:]
>>> data.base.base
<mmap.mmap object at 0x7fe753763180>

If the data is to be processed after the file is closed, it needs to be copied to main memory:

>>> data = time[:].copy()
>>> f.close()
>>> data.mean()

A NetCDF file can also be used as context manager:

>>> from scipy.io import netcdf
>>> with netcdf.netcdf_file('simple.nc', 'r') as f:
...     print(f.history)
Created for a test

Methods

 close() Closes the NetCDF file. createDimension(name, length) Adds a dimension to the Dimension section of the NetCDF data structure. createVariable(name, type, dimensions) Create an empty variable for the netcdf_file object, specifying its data type and the dimensions it uses. flush() Perform a sync-to-disk flush if the netcdf_file object is in write mode. sync() Perform a sync-to-disk flush if the netcdf_file object is in write mode.


text:
# Matrix Multiplication Calculator

Enter the elements of the matrix 1

Enter the elements of the matrix 2

Result

The Matrix Multiplication Calculator an online tool which shows Matrix Multiplication for the given input. Byju's Matrix Multiplication Calculator is a tool
which makes calculations very simple and interesting. If an input is given then it can easily show the result for the given number.

#### Practise This Question

If the sum of a pair of opposite angles of a quadrilateral is 180, then the quadrilateral is


text:
Data for Exercise 5.68

Coins

## Format

A data frame/tibble with 12 observations on one variable

return

yearly returns on each of 12 possible investments

## References

Kitchens, L. J. (2003) Basic Statistics and Data Analysis. Pacific Grove, CA: Brooks/Cole, a division of Thomson Learning.

## Examples

qqnorm(Coins$return) qqline(Coins$return)


text:
# Recent questions and answers in 2004

Questions from: 2004

### The hormone that helps in the conversion of glucose to glycogen is :-

To see more, click for all the questions in this category.


text:
# Cox Ingersoll Ross (1985) Model [closed]

How can I convert the following process to a standard Brownian Motion?

$$\mathrm{d}r_t=(a-br_t)\mathrm{d}t+\sigma\sqrt{r_t}\mathrm{d}W_t$$

I am not quite sure I get your question. You cannot solve the model in closed-form. What you get is that \begin{align*} r_t=r_0e^{-a t}+\frac{b}{a}(1- e^{-a t})+\sigma e^{-at}\int_0^te^{a u} \sqrt{r_u}\mathrm{d}W_u. \end{align*} Furthermore, you can get that $$r_t$$ follows a (non-central) chi-squared distribution and can compute the (conditional) moments of $$r_t$$ and obtain closed-form solutions for the prices of zero-coupon bonds and European-style zero-coupon bond options.
• Thank you for your response. Yes, I am trying to obtain closed-form for zero-coupon bonds but to do that I need to work under P*, probability that converts that given $W_u$ to standard Brownian motion, say, $W*$, I am not sure how to get $r_t$ in the form of $W*$. Thank you for the reference btw. – Stochastic learner Feb 12 at 12:14
• @Stochasticlearner You cannot get $r_t$ in explicit form regardless what measure you use. If you want to price bonds, a common approach is to solve the corresponding bond price PDE. Guessing that bond price is affine-linear in $r_t$, the PDE reduces to a system of ODEs. The solution is given in the book I mentioned above. – KeSchn Feb 12 at 12:22


text:
##### -15 POintsSerpsEiy 'S.P,04z_NOTESASk TOUR EACHERvery light rigid rod with length 268 m extends straight out from one end meter stick. The combination susjended from pivot at the upper end of the rod as shown in the following figure The combination then pulled out by small angle and released_Determine the period of oscillation the svstem.(b) By what percentage docs the perlod differ from the period o slmple pendulum 768 m long?
-15 POints SerpsEiy 'S.P,04z_ NOTES ASk TOUR EACHER very light rigid rod with length 268 m extends straight out from one end meter stick. The combination susjended from pivot at the upper end of the rod as shown in the following figure The combination then pulled out by small angle and released...
##### Queston 10Express the confidence interval 757.6 40 1037.8 in the form of z + ME.T+MEQuestlon Help: DPostto lorumSubmit Queston
Queston 10 Express the confidence interval 757.6 40 1037.8 in the form of z + ME. T+ME Questlon Help: DPostto lorum Submit Queston...
##### If the print is too small I have provided them here as well; Jose and Maria...
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##### Q1 (4 points)Let13 E =and let A be a 3 X 3 matrix such that A can be reduced to E by the following elementary row operations:Rz + R1 - Rz Ra 7 Ra _5 Rz < R3 A B11Rz + Rz 2 Ra + 2R2 4 Ra Ra + 9 R1 = R: D E:Find det(A) . It is not necessary to simplify your answer:Drag and drop an image or PDF file or click to browse_
Q1 (4 points) Let 13 E = and let A be a 3 X 3 matrix such that A can be reduced to E by the following elementary row operations: Rz + R1 - Rz Ra 7 Ra _5 Rz < R3 A B 11Rz + Rz 2 Ra + 2R2 4 Ra Ra + 9 R1 = R: D E: Find det(A) . It is not necessary to simplify your answer: Drag and drop an image or P...
##### MinEnonuon nUromtrE enicai compcrcncpordoreLtctMncicoSuhaunnn nineraleahat4-9.80 mryquodratic equaton Icr [;spccdstort ust betorc ( strkcsnrcuno?pencntJEonDYFunttionotrainthecumDonenirelocty GCNiE ralnd=Yone Jus{Subattie numenValeMs' Kt}Lsc tnls comporcnt witn componcni %horconml7+vspeedMlnetnnTeatanabttnaanrEatrnneng GerealteEEJenattyrearanaae ErattatIinal{Occolargrr (an cncinicle apcrdJ6-J m/7Masuea ItHIHTS;stxg STARTEDsnicaticlloaira Ouastan KencsLcrnanaur ucarendin?Suppose arother ston
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##### Compute the two missing amounts for each case. Assume that it is the end of the...
Compute the two missing amounts for each case. Assume that it is the end of the first full year of operations for the company. (Loss amounts should be indicated with a minus sign.) Independent Cases Net Income (Loss) Total Assets Total Liabilities Stockholders' Equity A Total Total Revenues Expe...
##### To what extent do you think the EPA might impact public health disparities at the microsystem...
to what extent do you think the EPA might impact public health disparities at the microsystem level, and why?...
##### A gourmet jelly bean company can produce $100,000$ pounds of jelly beans a day. One ounce of these jelly beans contains 100 Calories. If there are 800 jelly beans in a pound, how many jelly beans can be produced in a day?
A gourmet jelly bean company can produce $100,000$ pounds of jelly beans a day. One ounce of these jelly beans contains 100 Calories. If there are 800 jelly beans in a pound, how many jelly beans can be produced in a day?...
##### Which of the following is the general solution of1 1 16 I 2y" + cy' + 22 ~ 9 1)y = 0 9 9 for â‚¬ > 0, near â‚¬ = 0 ?Oa)y = C1J(32) + c2Y(32)Ob)y = C1Ja(4r) | C2J_3(4.) Ocy = C1J(32) + 02J_4(3.) Ody = CJ,(4c/3) + c2Y (4c/3) Oel y = 01J1(32/4) + c2Yi(32/4) Ofy = CJs(4z) +CYa(4z)
Which of the following is the general solution of 1 1 16 I 2y" + cy' + 22 ~ 9 1)y = 0 9 9 for â‚¬ > 0, near â‚¬ = 0 ? Oa)y = C1J(32) + c2Y(32) Ob)y = C1Ja(4r) | C2J_3(4.) Ocy = C1J(32) + 02J_4(3.) Ody = CJ,(4c/3) + c2Y (4c/3) Oel y = 01J1(32/4) + c2Yi(32/4) Ofy = CJs(4z) +CYa(4z...
##### 891 1 1 2  9 8 1 2 h 4 1 1 1 W 38 4 8 8
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##### (a) verify that the series converges. (b) Use a graphing utility to find the indicated partial sum $S_{n}$ and complete the table. (c) Use a graphing utility to graph the first 10 terms of the sequence of partial sums. (d) Use the table to estimate the sum of the series. (e) Explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series.  egin{array}{|l|l|l|l|l|l|} hline oldsymbol{n} & 5 &
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text:
# 2.4 Distribution of molecular speeds  (Page 5/9)

 Page 5 / 9

(a) Use the ideal gas equation to estimate the temperature at which 1.00 kg of steam (molar mass $M=18.0\phantom{\rule{0.2em}{0ex}}\text{g/mol}$ ) at a pressure of $1.50\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ occupies a volume of $0.220\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}$ . (b) The van der Waals constants for water are $a=0.5537\phantom{\rule{0.2em}{0ex}}\text{Pa}·{\text{m}}^{6}\text{/}{\text{mol}}^{2}$ and $b=3.049\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}\text{/}\text{mol}$ . Use the Van der Waals equation of state to estimate the temperature under the same conditions. (c) The actual temperature is 779 K. Which estimate is better?

One process for decaffeinating coffee uses carbon dioxide $\left(M=44.0\phantom{\rule{0.2em}{0ex}}\text{g/mol}\right)$ at a molar density of about $14,600\phantom{\rule{0.2em}{0ex}}{\text{mol/m}}^{3}$ and a temperature of about $60\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ . (a) Is CO 2 a solid, liquid, gas, or supercritical fluid under those conditions? (b) The van der Waals constants for carbon dioxide are $a=0.3658\phantom{\rule{0.2em}{0ex}}\text{Pa}·{\text{m}}^{6}\text{/}{\text{mol}}^{2}$ and $b=4.286\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}\text{/}\text{mol}\text{.}$ Using the van der Waals equation, estimate the pressure of ${\text{CO}}_{2}$ at that temperature and density.

a. supercritical fluid; b. $3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{Pa}$

On a winter day when the air temperature is $0\phantom{\rule{0.2em}{0ex}}\text{°C},$ the relative humidity is $50%$ . Outside air comes inside and is heated to a room temperature of $20\phantom{\rule{0.2em}{0ex}}\text{°C}$ . What is the relative humidity of the air inside the room. (Does this problem show why inside air is so dry in winter?)

On a warm day when the air temperature is $30\phantom{\rule{0.2em}{0ex}}\text{°C}$ , a metal can is slowly cooled by adding bits of ice to liquid water in it. Condensation first appears when the can reaches $15\phantom{\rule{0.2em}{0ex}}\text{°C}$ . What is the relative humidity of the air?

$40.18%$

(a) People often think of humid air as “heavy.” Compare the densities of air with $0%$ relative humidity and $100%$ relative humidity when both are at 1 atm and $30\phantom{\rule{0.2em}{0ex}}\text{°C}$ . Assume that the dry air is an ideal gas composed of molecules with a molar mass of 29.0 g/mol and the moist air is the same gas mixed with water vapor. (b) As discussed in the chapter on the applications of Newton’s laws, the air resistance felt by projectiles such as baseballs and golf balls is approximately ${F}_{\text{D}}=C\rho A{v}^{2}\text{/}2$ , where $\rho$ is the mass density of the air, A is the cross-sectional area of the projectile, and C is the projectile’s drag coefficient. For a fixed air pressure, describe qualitatively how the range of a projectile changes with the relative humidity. (c) When a thunderstorm is coming, usually the humidity is high and the air pressure is low. Do those conditions give an advantage or disadvantage to home-run hitters?

The mean free path for helium at a certain temperature and pressure is $2.10\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$ The radius of a helium atom can be taken as $1.10\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}\phantom{\rule{0.2em}{0ex}}\text{m}$ . What is the measure of the density of helium under those conditions (a) in molecules per cubic meter and (b) in moles per cubic meter?

a. $2.21\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{27}\phantom{\rule{0.2em}{0ex}}{\text{molecules/m}}^{3};$ b. $3.67\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}{\text{mol/m}}^{3}$

The mean free path for methane at a temperature of 269 K and a pressure of $1.11\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ is $4.81\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$ Find the effective radius r of the methane molecule.

In the chapter on fluid mechanics, Bernoulli’s equation for the flow of incompressible fluids was explained in terms of changes affecting a small volume dV of fluid. Such volumes are a fundamental idea in the study of the flow of compressible fluids such as gases as well. For the equations of hydrodynamics to apply, the mean free path must be much less than the linear size of such a volume, $a\approx d{V}^{1\text{/}3}.$ For air in the stratosphere at a temperature of 220 K and a pressure of 5.8 kPa, how big should a be for it to be 100 times the mean free path? Take the effective radius of air molecules to be $1.88\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}\phantom{\rule{0.2em}{0ex}}\text{m},$ which is roughly correct for ${\text{N}}_{2}$ .

8.2 mm

determining dimensional correctness
determine dimensional correctness of,T=2π√L/g
PATRICK
somebody help me answer the question above
PATRICK
calculate the heat flow per square meter through a mineral roll insulation 5cm thick if the temperature on the two surfaces are 30degree Celsius and 20 degree Celsius respectively. thermal conduction of mineral roll is 0.04
what are the elementary compositions of a cell?
when a current pass through a material does the velocity varies
what is spin entropy ?and disorder in ferromagnetic material
diagram of an hall effect sensor
if a magnetised wire having dipole moment M is bent in the form of arc subtending angle of 45°at centre,new magnetic moment is
is this book for preparing IIT or neet?
is it possible to increase the temperature of a gas without adding heat to it?
I'm not sure about it, but I think it's possible. If you add some form of energy to the system, it's a possibility. Also, if you change the pression or the volume of the system, you'll increase the kinetic energy of the system, increasing the gas temperature. I don't know if I'm correct.
playdoh
For example, if you get a syringe and close the tip(sealing the air inside), and start pumping the plunger, you'll notice that it starts getting hot. Again, I'm not sure if I am correct.
playdoh
you are right for example an adiabatic process changes all variables without external energy to yield a temperature change. (Search Otto cycle)
when a current pass through a material does the velocity varies
lovet
yes at adiabatic compression temperature increase
Nepal
how to draw a diagram of a triode
whate is fckg diagrame?
Arzoodan
why do we use integration?
To know surfaces below graphs.
Jan
To find a Primitive function. Primitive function: a function that is the origin of another
playdoh
yes
Dharmdev
what is laps rate
Dharmdev
Г=-dT/dZ that is simply defination
Arzoodan
what is z
Dharmdev
to find the area under a graph or to accumulate .e.g. sum of momentum over time is no etic energy.
Naod
Z is alt.,dZ altv difference
Arzoodan
what is the Elasticty
it is the property of the by virtue of it regains it's original shape after the removal of applied force (deforming force).
Prema
property of the material
Prema
which type of cable is suitable for patrol station wiring
what is calorimeter
heat measuring device
Suvransu
What is mean electric potential


text:
# Floor function and 2016

Algebra Level 5

$\large \left\lfloor \frac { { 1 }^{ 2 } }{ 2016 } \right\rfloor ,\left\lfloor \frac { { 2 }^{ 2 } }{ 2016 } \right\rfloor ,\left\lfloor \frac { { 3 }^{ 2 } }{ 2016 } \right\rfloor ,\left\lfloor \frac { { 4 }^{ 2 } }{ 2016 } \right\rfloor , \ldots ,\left\lfloor \frac { { 2016 }^{ 2 } }{ 2016 } \right\rfloor$

How many distinct integers are there in the sequence above?

Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

×

Problem Loading...

Note Loading...

Set Loading...


text:
# commuting matrices are simultaneously triangularizable

###### Theorem 1.

All matrices in the below are complex $n\times n$ matrices.
Let $A$,$B$ be matrices and $AB=BA$. Then there exists a unitary matrix $Q$ such that

$Q^{H}AQ=T_{1}$ , $Q^{H}BQ=T_{2}$

where ${}^{H}$ is the conjugate transpose and $T_{1},T_{2},$ are upper triangular matrices.

Title commuting matrices are simultaneously triangularizable CommutingMatricesAreSimultaneouslyTriangularizable 2013-03-22 15:26:48 2013-03-22 15:26:48 georgiosl (7242) georgiosl (7242) 12 georgiosl (7242) Theorem msc 15A23 SimultaneousUpperTriangularBlockDiagonalizationOfCommutingMatrices


text:
# CTAN Update: glossaries

Date: July 24, 2018 2:38:27 PM CEST
Nicola Talbot submitted an update to the glossaries package. Version: 4.41 2018-07-23 License: lppl1.3 Summary description: Create glossaries and lists of acronyms Announcement text:
- added package option kernelglossredefs (Warnings are now generated if the kernel commands \makeglossary and \glossary are used. These commands are deprecated with the glossaries package. If the originals are required, by a package or class that must be used with glossaries.sty for some reason, then use kernelglossredefs=false, but beware of other possible conflicts.) - renamed some internal commands to avoid conflict
The package’s Catalogue entry can be viewed at https://ctan.org/pkg/glossaries The package’s files themselves can be inspected at http://mirror.ctan.org/macros/latex/contrib/glossaries/
Thanks for the upload. For the CTAN Team Petra Rübe-Pugliese
We are supported by the TeX users groups. Please join a users group; see https://www.tug.org/usergroups.html .

## glossaries – Create glossaries and lists of acronyms

The glossaries package supports acronyms and multiple glossaries, and has provision for operation in several languages (using the facilities of either babel or polyglossia). New entries are defined to have a name and description (and optionally an associated symbol). Support for multiple languages is offered, and plural forms of terms may be specified. An additional package, glossaries-accsupp, can make use of the accsupp package mechanisms for accessibility support for PDF files containing glossaries.

The user may define new glossary styles, and preambles and postambles can be specified. There is provision for loading a database of terms, but only terms used in the text will be added to the relevant glossary.

The package uses an indexing program to provide the actual glossary; either makeindex or xindy may serve this purpose, and a Perl script is provided to serve as interface.

This package requires the mfirstuc package.

The package supersedes the author’s glossary package (which is now obsolete).

 Package glossaries Version 4.43 2019-09-28 Copyright 2007–2019 Nicola Talbot Maintainer Nicola Talbot

more


text:
# Drawing a Circuit from Voltage-Time Graph

## Homework Statement

The attached graph shows the relationship between time and voltage for three resistors. A capacitor is discharging current through three "separate" resistors. How resistors might be positioned in a circuit to give rise to the graphs.

## Homework Equations

V = IR
CR = $$\uptau$$

## The Attempt at a Solution

Separate cannot mean in parallel, or else the graphs would all begin at the same voltage.

My attempt to prove that
"Separate cannot mean completely in series either" ended in squash:

The resistance order can be shown to be: R2>R3>R1

RC = Tau gives the time constant for a capacitor. The greater tau is, the longer it takes a capacitor discharge or acquire charge. The resistors with the greater resistances, consequently, are the ones that takes the longer to reach V = 0.

What approach can I use?

phinds
Gold Member
I take it we are supposed to guess what the graphs look like?

Sorry for forgetting to attach the graph:

nah, I wanted to determine what the circuit woud look like.

#### Attachments

• Screen Shot 2015-01-11 at 10.34.07 AM.png
2.6 KB · Views: 648
Sorry, my original post wasn't clear

1. Homework Statement

The attached graph shows the relationship between time and voltage for three resistors. A capacitor is discharging current through three "separate" resistors. How can resistors be positioned in a circuit so that the graph describes them?

## Homework Equations

V = IR
CR = $$\uptau$$

## The Attempt at a Solution

Separate cannot mean in parallel, or else the graphs would all begin at the same voltage.

My attempt to prove that
"Separate cannot mean completely in series either" ended in squash:

The resistance order can be shown to be: R2>R3>R1

RC = Tau gives the time constant for a capacitor. The greater tau is, the longer it takes a capacitor discharge or acquire charge. The resistors with the greater resistances, consequently, are the ones that takes the longer to reach V = 0.

What approach can I use?

phinds
Gold Member
Yes, but the point is that there are obviously three SEPARATE circuit layouts to get the three different graphs. What they are, based on the condition that each use exactly 3 resistors, is obvious. What do you think they might be?

three SEPARATE circuit layouts to get the three different graphs

Ah so I misunderstood the problem? I need to identfy three circuit layouts that each use three resistors.

If I understand the problem, then the approach should be easy:

We already concluded that R2>R3>R1, where R2,R3 and R1 are presumably the net resistances that combinations of the three resistors create.

The component resistors that comprise these net resistors can be assumed to be the same, without loss of generality:

so that R2 has: three Rx resistors in series
R3 has: 2 Rx in series and one in parallel
R1 has: three in parallel

Other layouts would work.

phinds
Gold Member
Yep. That's what I came up with.

NascentOxygen
Staff Emeritus


text:
Documentation

This is machine translation

Translated by
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fsolve

Solve system of nonlinear equations

Nonlinear system solver

Solves a problem specified by

F(x) = 0

for x, where F(x) is a function that returns a vector value.

x is a vector or a matrix; see Matrix Arguments.

Syntax

``x = fsolve(fun,x0)``
``x = fsolve(fun,x0,options)``
``x = fsolve(problem)``
``````[x,fval] = fsolve(___)``````
``````[x,fval,exitflag,output] = fsolve(___)``````
``````[x,fval,exitflag,output,jacobian] = fsolve(___)``````

Description

example

````x = fsolve(fun,x0)` starts at `x0` and tries to solve the equations `fun(x) = `0, an array of zeros.```

example

````x = fsolve(fun,x0,options)` solves the equations with the optimization options specified in `options`. Use `optimoptions` to set these options.```

example

````x = fsolve(problem)` solves `problem`, where `problem` is a structure described in Input Arguments. Create the `problem` structure by exporting a problem from Optimization app, as described in Exporting Your Work.```

example

``````[x,fval] = fsolve(___)```, for any syntax, returns the value of the objective function `fun` at the solution `x`.```

example

``````[x,fval,exitflag,output] = fsolve(___)``` additionally returns a value `exitflag` that describes the exit condition of `fsolve`, and a structure `output` with information about the optimization process.```
``````[x,fval,exitflag,output,jacobian] = fsolve(___)``` returns the Jacobian of `fun` at the solution `x`.```

Examples

collapse all

This example shows how to solve two nonlinear equations in two variables. The equations are

Convert the equations to the form .

Write a function that computes the left-hand side of these two equations.

```function F = root2d(x) F(1) = exp(-exp(-(x(1)+x(2)))) - x(2)*(1+x(1)^2); F(2) = x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5; ```

Save this code as a file named `root2d.m` on your MATLAB® path.

Solve the system of equations starting at the point `[0,0]`.

```fun = @root2d; x0 = [0,0]; x = fsolve(fun,x0) ```
```Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient. x = 0.3532 0.6061 ```

Examine the solution process for a nonlinear system.

Set options to have no display and a plot function that displays the first-order optimality, which should converge to 0 as the algorithm iterates.

```options = optimoptions('fsolve','Display','none','PlotFcn',@optimplotfirstorderopt); ```

The equations in the nonlinear system are

Convert the equations to the form .

Write a function that computes the left-hand side of these two equations.

```function F = root2d(x) F(1) = exp(-exp(-(x(1)+x(2)))) - x(2)*(1+x(1)^2); F(2) = x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5; ```

Save this code as a file named `root2d.m` on your MATLAB® path.

Solve the nonlinear system starting from the point `[0,0]` and observe the solution process.

```fun = @root2d; x0 = [0,0]; x = fsolve(fun,x0,options) ```
```x = 0.3532 0.6061 ```

Create a problem structure for `fsolve` and solve the problem.

Solve the same problem as in Solution with Nondefault Options, but formulate the problem using a problem structure.

Set options for the problem to have no display and a plot function that displays the first-order optimality, which should converge to 0 as the algorithm iterates.

```problem.options = optimoptions('fsolve','Display','none','PlotFcn',@optimplotfirstorderopt); ```

The equations in the nonlinear system are

Convert the equations to the form .

Write a function that computes the left-hand side of these two equations.

```function F = root2d(x) F(1) = exp(-exp(-(x(1)+x(2)))) - x(2)*(1+x(1)^2); F(2) = x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5; ```

Save this code as a file named `root2d.m` on your MATLAB® path.

Create the remaining fields in the problem structure.

```problem.objective = @root2d; problem.x0 = [0,0]; problem.solver = 'fsolve'; ```

Solve the problem.

```x = fsolve(problem) ```
```x = 0.3532 0.6061 ```

This example returns the iterative display showing the solution process for the system of two equations and two unknowns

`$\begin{array}{c}2{x}_{1}-{x}_{2}={e}^{-{x}_{1}}\\ -{x}_{1}+2{x}_{2}={e}^{-{x}_{2}}.\end{array}$`

Rewrite the equations in the form F(x) = 0:

`$\begin{array}{c}2{x}_{1}-{x}_{2}-{e}^{-{x}_{1}}=0\\ -{x}_{1}+2{x}_{2}-{e}^{-{x}_{2}}=0.\end{array}$`

Start your search for a solution at `x0 = [-5 -5]`.

First, write a file that computes `F`, the values of the equations at `x`.

```function F = myfun(x) F = [2*x(1) - x(2) - exp(-x(1)); -x(1) + 2*x(2) - exp(-x(2))];```

Save this function file as `myfun.m` on your MATLAB® path.

Set up the initial point. Set options to return iterative display.

```x0 = [-5;-5]; options = optimoptions('fsolve','Display','iter');```

Call `fsolve`.

`[x,fval] = fsolve(@myfun,x0,options)`
``` Norm of First-order Trust-region Iteration Func-count f(x) step optimality radius 0 3 47071.2 2.29e+04 1 1 6 12003.4 1 5.75e+03 1 2 9 3147.02 1 1.47e+03 1 3 12 854.452 1 388 1 4 15 239.527 1 107 1 5 18 67.0412 1 30.8 1 6 21 16.7042 1 9.05 1 7 24 2.42788 1 2.26 1 8 27 0.032658 0.759511 0.206 2.5 9 30 7.03149e-06 0.111927 0.00294 2.5 10 33 3.29525e-13 0.00169132 6.36e-07 2.5 Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient. x = 0.5671 0.5671 fval = 1.0e-006 * -0.4059 -0.4059```

Find a matrix `X` that satisfies

`$X*X*X=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],$`

starting at the point `x= [1,1;1,1]`. Examine the `fsolve` outputs to see the solution quality and process.

Create an anonymous function that calculates the matrix equation.

`fun = @(x)x*x*x - [1,2;3,4];`

Set options to turn off the display. Set the initial point `x0`.

```options = optimoptions('fsolve','Display','off'); x0 = ones(2);```

Call `fsolve` and obtain information about the solution process.

```[x,fval,exitflag,output] = fsolve(fun,x0,options); x,fval,exitflag```
```x = -0.1291 0.8602 1.2903 1.1612 fval = 1.0e-09 * -0.1618 0.0778 0.1160 -0.0474 exitflag = 1```

The exit flag value `1` indicates that the solution is reliable. To verify this manually, calculate the residual (sum of squares of `fval`) to see how close it is to zero.

`sum(sum(fval.*fval))`
```ans = 4.7957e-20```

This small residual confirms that `x` is a solution.

`fsolve` performed 35 function evaluations to find the solution, as you can see in the `output` structure.

`output.funcCount`
```ans = 35```

Input Arguments

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Nonlinear equations to solve, specified as a function handle or function name. `fun` is a function that accepts a vector `x` and returns a vector `F`, the nonlinear equations evaluated at `x`. The equations to solve are `F` = 0 for all components of `F`. The function `fun` can be specified as a function handle for a file

`x = fsolve(@myfun,x0)`

where `myfun` is a MATLAB function such as

```function F = myfun(x) F = ... % Compute function values at x```

`fun` can also be a function handle for an anonymous function.

`x = fsolve(@(x)sin(x.*x),x0);`

If the user-defined values for `x` and `F` are arrays, they are converted to vectors using linear indexing (see Array Indexing (MATLAB)).

If the Jacobian can also be computed and the Jacobian option is `'on'`, set by

`options = optimoptions('fsolve','SpecifyObjectiveGradient','on')`

the function `fun` must return, in a second output argument, the Jacobian value `J`, a matrix, at `x`.

If `fun` returns a vector (matrix) of `m` components and `x` has length `n`, where `n` is the length of `x0`, the Jacobian `J` is an `m`-by-`n` matrix where `J(i,j)` is the partial derivative of `F(i)` with respect to `x(j)`. (The Jacobian `J` is the transpose of the gradient of `F`.)

Example: `fun = @(x)x*x*x-[1,2;3,4]`

Data Types: `char` | `function_handle` | `string`

Initial point, specified as a real vector or real array. `fsolve` uses the number of elements in and size of `x0` to determine the number and size of variables that `fun` accepts.

Example: `x0 = [1,2,3,4]`

Data Types: `double`

Optimization options, specified as the output of `optimoptions` or a structure such as `optimset` returns.

Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.

Some options are absent from the `optimoptions` display. These options appear in italics in the following table. For details, see View Options.

All Algorithms
`Algorithm`

Choose between `'trust-region-dogleg'` (default), `'trust-region'`, and `'levenberg-marquardt'`.

The `Algorithm` option specifies a preference for which algorithm to use. It is only a preference because for the trust-region algorithm, the nonlinear system of equations cannot be underdetermined; that is, the number of equations (the number of elements of `F` returned by `fun`) must be at least as many as the length of `x`. Similarly, for the trust-region-dogleg algorithm, the number of equations must be the same as the length of `x`. `fsolve` uses the Levenberg-Marquardt algorithm when the selected algorithm is unavailable. For more information on choosing the algorithm, see Choosing the Algorithm.

To set some algorithm options using `optimset` instead of `optimoptions`:

• `Algorithm` — Set the algorithm to `'trust-region-reflective'` instead of `'trust-region'`.

• InitDamping — Set the initial Levenberg-Marquardt parameter λ by setting `Algorithm` to a cell array such as `{'levenberg-marquardt',.005}`.

`CheckGradients`

Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. The choices are `true` or the default `false`.

For `optimset`, the name is `DerivativeCheck` and the values are `'on'` or `'off'`. See Current and Legacy Option Name Tables.

Diagnostics

Display diagnostic information about the function to be minimized or solved. The choices are `'on'` or the default `'off'`.

DiffMaxChange

Maximum change in variables for finite-difference gradients (a positive scalar). The default is `Inf`.

DiffMinChange

Minimum change in variables for finite-difference gradients (a positive scalar). The default is `0`.

`Display`

Level of display (see Iterative Display):

• `'off'` or `'none'` displays no output.

• `'iter'` displays output at each iteration, and gives the default exit message.

• `'iter-detailed'` displays output at each iteration, and gives the technical exit message.

• `'final'` (default) displays just the final output, and gives the default exit message.

• `'final-detailed'` displays just the final output, and gives the technical exit message.

`FiniteDifferenceStepSize`

Scalar or vector step size factor for finite differences. When you set `FiniteDifferenceStepSize` to a vector `v`, the forward finite differences `delta` are

`delta = v.*sign′(x).*max(abs(x),TypicalX);`

where `sign′(x) = sign(x)` except `sign′(0) = 1`. Central finite differences are

`delta = v.*max(abs(x),TypicalX);`

Scalar `FiniteDifferenceStepSize` expands to a vector. The default is `sqrt(eps)` for forward finite differences, and `eps^(1/3)` for central finite differences.

For `optimset`, the name is `FinDiffRelStep`. See Current and Legacy Option Name Tables.

`FiniteDifferenceType`

Finite differences, used to estimate gradients, are either `'forward'` (default), or `'central'` (centered). `'central'` takes twice as many function evaluations, but should be more accurate.

The algorithm is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds.

For `optimset`, the name is `FinDiffType`. See Current and Legacy Option Name Tables.

`FunctionTolerance`

Termination tolerance on the function value, a positive scalar. The default is `1e-6`. See Tolerances and Stopping Criteria.

For `optimset`, the name is `TolFun`. See Current and Legacy Option Name Tables.

FunValCheck

Check whether objective function values are valid. `'on'` displays an error when the objective function returns a value that is `complex`, `Inf`, or `NaN`. The default, `'off'`, displays no error.

`MaxFunctionEvaluations`

Maximum number of function evaluations allowed, a positive integer. The default is `100*numberOfVariables`. See Tolerances and Stopping Criteria and Iterations and Function Counts.

For `optimset`, the name is `MaxFunEvals`. See Current and Legacy Option Name Tables.

`MaxIterations`

Maximum number of iterations allowed, a positive integer. The default is `400`. See Tolerances and Stopping Criteria and Iterations and Function Counts.

For `optimset`, the name is `MaxIter`. See Current and Legacy Option Name Tables.

`OptimalityTolerance`

Termination tolerance on the first-order optimality (a positive scalar). The default is `1e-6`. See First-Order Optimality Measure.

Internally, the `'levenberg-marquardt'` algorithm uses an optimality tolerance (stopping criterion) of `1e-4` times `FunctionTolerance` and does not use `OptimalityTolerance`.

`OutputFcn`

Specify one or more user-defined functions that an optimization function calls at each iteration. Pass a function handle or a cell array of function handles. The default is none (`[]`). See Output Function Syntax.

`PlotFcn`

Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass a built-in plot function name, a function handle, or a cell array of built-in plot function names or function handles. For custom plot functions, pass function handles. The default is none (`[]`):

• `'optimplotx'` plots the current point.

• `'optimplotfunccount'` plots the function count.

• `'optimplotfval'` plots the function value.

• `'optimplotstepsize'` plots the step size.

• `'optimplotfirstorderopt'` plots the first-order optimality measure.

For information on writing a custom plot function, see Plot Function Syntax.

For `optimset`, the name is `PlotFcns`. See Current and Legacy Option Name Tables.

`SpecifyObjectiveGradient`

If `true`, `fsolve` uses a user-defined Jacobian (defined in `fun`), or Jacobian information (when using `JacobianMultiplyFcn`), for the objective function. If `false` (default), `fsolve` approximates the Jacobian using finite differences.

For `optimset`, the name is `Jacobian` and the values are `'on'` or `'off'`. See Current and Legacy Option Name Tables.

`StepTolerance`

Termination tolerance on `x`, a positive scalar. The default is `1e-6`. See Tolerances and Stopping Criteria.

For `optimset`, the name is `TolX`. See Current and Legacy Option Name Tables.

`TypicalX`

Typical `x` values. The number of elements in `TypicalX` is equal to the number of elements in `x0`, the starting point. The default value is `ones(numberofvariables,1)`. `fsolve` uses `TypicalX` for scaling finite differences for gradient estimation.

The `trust-region-dogleg` algorithm uses `TypicalX` as the diagonal terms of a scaling matrix.

`UseParallel`

When `true`, `fsolve` estimates gradients in parallel. Disable by setting to the default, `false`. See Parallel Computing.

trust-region Algorithm
`JacobianMultiplyFcn`

Jacobian multiply function, specified as a function handle. For large-scale structured problems, this function computes the Jacobian matrix product `J*Y`, `J'*Y`, or `J'*(J*Y)` without actually forming `J`. The function is of the form

`W = jmfun(Jinfo,Y,flag)`

where `Jinfo` contains a matrix used to compute `J*Y` (or `J'*Y`, or `J'*(J*Y)`). The first argument `Jinfo` must be the same as the second argument returned by the objective function `fun`, for example, in

`[F,Jinfo] = fun(x)`

`Y` is a matrix that has the same number of rows as there are dimensions in the problem. `flag` determines which product to compute:

• If `flag == 0`, `W = J'*(J*Y)`.

• If `flag > 0`, ```W = J*Y```.

• If `flag < 0`, ```W = J'*Y```.

In each case, `J` is not formed explicitly. `fsolve` uses `Jinfo` to compute the preconditioner. See Passing Extra Parameters for information on how to supply values for any additional parameters `jmfun` needs.

Note

`'SpecifyObjectiveGradient'` must be set to `true` for `fsolve` to pass `Jinfo` from `fun` to `jmfun`.

See Minimization with Dense Structured Hessian, Linear Equalities for a similar example.

For `optimset`, the name is `JacobMult`. See Current and Legacy Option Name Tables.

JacobPattern

Sparsity pattern of the Jacobian for finite differencing. Set `JacobPattern(i,j) = 1` when `fun(i)` depends on `x(j)`. Otherwise, set ```JacobPattern(i,j) = 0```. In other words, `JacobPattern(i,j) = 1` when you can have ∂`fun(i)`/∂`x(j)` ≠ 0.

Use `JacobPattern` when it is inconvenient to compute the Jacobian matrix `J` in `fun`, though you can determine (say, by inspection) when `fun(i)` depends on `x(j)`. `fsolve` can approximate `J` via sparse finite differences when you give `JacobPattern`.

In the worst case, if the structure is unknown, do not set `JacobPattern`. The default behavior is as if `JacobPattern` is a dense matrix of ones. Then `fsolve` computes a full finite-difference approximation in each iteration. This can be very expensive for large problems, so it is usually better to determine the sparsity structure.

MaxPCGIter

Maximum number of PCG (preconditioned conjugate gradient) iterations, a positive scalar. The default is `max(1,floor(numberOfVariables/2))`. For more information, see Equation Solving Algorithms.

PrecondBandWidth

Upper bandwidth of preconditioner for PCG, a nonnegative integer. The default `PrecondBandWidth` is `Inf`, which means a direct factorization (Cholesky) is used rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution. Set `PrecondBandWidth` to `0` for diagonal preconditioning (upper bandwidth of 0). For some problems, an intermediate bandwidth reduces the number of PCG iterations.

`SubproblemAlgorithm`

Determines how the iteration step is calculated. The default, `'factorization'`, takes a slower but more accurate step than `'cg'`. See Trust-Region fsolve Algorithm.

TolPCG

Termination tolerance on the PCG iteration, a positive scalar. The default is `0.1`.

Levenberg-Marquardt Algorithm
InitDamping

Initial value of the Levenberg-Marquardt parameter, a positive scalar. Default is `1e-2`. For details, see Levenberg-Marquardt Method.

ScaleProblem

`'jacobian'` can sometimes improve the convergence of a poorly scaled problem. The default is `'none'`.

Example: `options = optimoptions('fsolve','FiniteDifferenceType','central')`

Problem structure, specified as a structure with the following fields:

Field NameEntry

`objective`

Objective function

`x0`

Initial point for `x`

`solver`

`'fsolve'`

`options`

Options created with `optimoptions`

The simplest way of obtaining a `problem` structure is to export the problem from the Optimization app.

Data Types: `struct`

Output Arguments

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Solution, returned as a real vector or real array. The size of `x` is the same as the size of `x0`. Typically, `x` is a local solution to the problem when `exitflag` is positive. For information on the quality of the solution, see When the Solver Succeeds.

Objective function value at the solution, returned as a real vector. Generally, `fval` = `fun(x)`.

Reason `fsolve` stopped, returned as an integer.

 `1` Equation solved. First-order optimality is small. `2` Equation solved. Change in `x` smaller than the specified tolerance. `3` Equation solved. Change in residual smaller than the specified tolerance. `4` Equation solved. Magnitude of search direction smaller than specified tolerance. `0` Number of iterations exceeded `options.MaxIterations` or number of function evaluations exceeded `options.MaxFunctionEvaluations`. `-1` Output function or plot function stopped the algorithm. `-2` Equation not solved. The exit message can have more information. `-3` Equation not solved. Trust region radius became too small (`trust-region-dogleg` algorithm).

Information about the optimization process, returned as a structure with fields:

 `iterations` Number of iterations taken `funcCount` Number of function evaluations `algorithm` Optimization algorithm used `cgiterations` Total number of PCG iterations (`'trust-region'` algorithm only) `stepsize` Final displacement in `x` (not in `'trust-region-dogleg'`) `firstorderopt` Measure of first-order optimality `message` Exit message

Jacobian at the solution, returned as a real matrix. `jacobian(i,j)` is the partial derivative of `fun(i)` with respect to `x(j)` at the solution `x`.

Limitations

• The function to be solved must be continuous.

• When successful, `fsolve` only gives one root.

• The default trust-region dogleg method can only be used when the system of equations is square, i.e., the number of equations equals the number of unknowns. For the Levenberg-Marquardt method, the system of equations need not be square.

Tips

• For large problems, meaning those with thousands of variables or more, save memory (and possibly save time) by setting the `Algorithm` option to `'trust-region'` and the `SubproblemAlgorithm` option to `'cg'`.

Algorithms

The Levenberg-Marquardt and trust-region methods are based on the nonlinear least-squares algorithms also used in `lsqnonlin`. Use one of these methods if the system may not have a zero. The algorithm still returns a point where the residual is small. However, if the Jacobian of the system is singular, the algorithm might converge to a point that is not a solution of the system of equations (see Limitations).

• By default `fsolve` chooses the trust-region dogleg algorithm. The algorithm is a variant of the Powell dogleg method described in [8]. It is similar in nature to the algorithm implemented in [7]. See Trust-Region Dogleg Method.

• The trust-region algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [1] and [2]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See Trust-Region fsolve Algorithm.

• The Levenberg-Marquardt method is described in references [4], [5], and [6]. See Levenberg-Marquardt Method.

References

[1] Coleman, T.F. and Y. Li, “An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds,” SIAM Journal on Optimization, Vol. 6, pp. 418-445, 1996.

[2] Coleman, T.F. and Y. Li, “On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds,” Mathematical Programming, Vol. 67, Number 2, pp. 189-224, 1994.

[3] Dennis, J. E. Jr., “Nonlinear Least-Squares,” State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269-312.

[4] Levenberg, K., “A Method for the Solution of Certain Problems in Least-Squares,” Quarterly Applied Mathematics 2, pp. 164-168, 1944.

[5] Marquardt, D., “An Algorithm for Least-squares Estimation of Nonlinear Parameters,” SIAM Journal Applied Mathematics, Vol. 11, pp. 431-441, 1963.

[6] Moré, J. J., “The Levenberg-Marquardt Algorithm: Implementation and Theory,” Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977.

[7] Moré, J. J., B. S. Garbow, and K. E. Hillstrom, User Guide for MINPACK 1, Argonne National Laboratory, Rept. ANL-80-74, 1980.

[8] Powell, M. J. D., “A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations,” Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Ch.7, 1970.

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text:
# The Photon: Particle Nature of Light

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Photoelectric effect thus gave evidence to the strange fact that light in interaction with matter behaved as if it was made of quanta or packets of energy, each of energy h ν.

We can summaries the photon picture of electromagnetic radiation as follows:

1. In the interaction of radiation with matter, radiation behaves as if it is made up of particles called photons.
2. Each photon has energy E (=hν) and momentum p (= h ν/c), and speed c, the speed of light.
3. All photons of light of a particular frequency ν, or wavelength λ, have the same energy E (=hν = hc/λ) and momentum p (= hν/c = h/λ), whatever the intensity of radiation may be. By increasing the intensity of light of given wavelength, there is only an increase in the number of photons per second crossing a given area, with each photon having the same energy. Thus, photon energy is independent of intensity of radiation.
4. Photons are electrically neutral and are not deflected by electric and magnetic fields.
5. In a photon-particle collision (such as photon-electron collision), the total energy and total momentum are conserved. However, the number of photons may not be conserved in a collision. The photon may be absorbed or a new photon may be created.