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Boolean Algebra - Maple Programming Help

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Boolean Algebra

Background

Boolean algebra is a form of algebra where the variables only have the values true and false; 1 and 0, respectively.

The operations of Boolean algebra are:

 • AND operation (denoted $A\cdot B$),
 • OR operation (denoted $\mathrm{A}+\mathrm{B}$),
 • NOT operation (denoted A')

These operations are defined as follows:

 NOT AND OR 0' = 1 1' = 0 0 $\cdot$ 0 = 0 0 $\cdot$ 1 = 0 1 $\cdot$ 0 = 0 1 $\cdot$ 1 = 1 0 $+$ 0 = 0 0 $+$ 1 = 1 1 $+$ 0 = 1 1 $+$ 1 = 1

You will notice that these are the basic logic gates described in the Logic Gates Math App. Boolean algebra is fundamental for digital logic.

 Laws of Boolean Algebra Note: Every law in Boolean algebra has two forms that are obtained by exchanging all the ANDs to ORs and 1s to 0s and vice versa. This is known as the Boolean algebra duality principle.  The order of operations for Boolean algebra, from highest to lowest priority is NOT, then AND, then OR. Expressions inside brackets are always evaluated first.   1. Commutative Law (a) A + B = B + A (b) A $\cdot$ B = B $\cdot$ A   2. Associative Law (a) (A + B) + C = A + (B + C) (b) (A $\cdot$ B) $\cdot$ C = A $\cdot$ (B $\cdot$ C)   3. Distributive Law (a) A $\cdot$ (B + C) = A $\cdot$ B + A $\cdot$ C (b) A + (B $\cdot$ C) = (A + B) (A + C)   4. Null Law (a) 1 + A = 1    (b) 0 $\cdot$ A = 0   5. Identity law (a) 0 + A = A (b) 1 $\cdot$ A = A   6. Idempotent Law (a) A + A = A (b) A $\cdot$ A = A   7. Complementarity Law (a) A' + A = 1 (b) A' $\cdot$ A = 0   8. Uniting Theorem (a) A $\cdot$ B + A $\cdot$ B' = A (b) (A + B) $\cdot$ (A + B') = A   9. Absorption Theorem (a) A + A $\cdot$ B = A (b) A $\cdot$ (A + B) = A   10. Adsorption Theorem (a)  A + A' B = A + B (b) A $\cdot$ (A' + B) = A $\cdot$ B   11. De Morgan's Theorem (a) (A + B)' = A' $\cdot$ B' (b) (A $\cdot$ B)' = A' + B'

The following activity allows you to practice remembering the basic laws of Boolean algebra and simplifying Boolean expressions. The objective of the game is to complete the equations below in a timely manner.

To play, use the laws (above) to solve for the missing variable(s) and/or operator(s) in the equation. The orange square indicates a blank in the equation. Before the mouse reaches the gap, your objective is to use the options (variables and operators) below to correctly complete the equation. With the correct response in place, the mouse is able to run over the gap and obtain the cheese. If the response is incorrect, the mouse falls through. The difficulty (easy, medium, hard) allows you to adjust the speed of the mouse.

 CHEESE HUNT A Boolean Algebra Game

Choose Difficulty:

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What are the commands to setup SharePoint Framework?

To setup SPFX environment there are few commands that needs to be executed.

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24.11.2021 (Wednesday)

How well do we know the SMEFT, and how to learn more

Regular Seminar Veronica Sanz (University of Valencia)

 at: 14:00 QMWroom zoom abstract: [there will be a pre-seminar for students at 1.30PM; for zoom link please email s.nagy@qmul.ac.uk] The characterization of measurements using the EFT framework is becoming prevalent, not just in its traditional realm of low-energy physics, but now also with LHC high-energy probes. At the LHC, the EFT approach is viewed as a way to transcend models, to exploit the huge range of LHC topologies, and even as a form of data preservation. In this talk we will review this state-of-affairs, point out challenges with this approach and also discuss some new opportunities that more data will bring.


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# 7 The Logic of Ordinary Least Squares Estimation

This chapter begins the discussion of ordinary least squares (OLS) regression. OLS is the “workhorse” of empirical social science and is a critical tool in hypothesis testing and theory building. This chapter builds on the discussion in Chapter 6 by showing how OLS regression is used to estimate relationships between and among variables.

## 7.1 Theoretical Models

Models, as discussed earlier, are an essential component in theory building. They simplify theoretical concepts, provide a precise way to evaluate relationships between variables, and serve as a vehicle for hypothesis testing. As discussed in Chapter 1, one of the central features of a theoretical model is the presumption of causality, and causality is based on three factors: time ordering (observational or theoretical), co-variation, and non-spuriousness. Of these three assumptions, co-variation is the one analyzed using OLS. The often repeated adage, correlation is not causation’’ is key. Causation is driven by theory, but co-variation is the critical part of empirical hypothesis testing.

When describing relationships, it is important to distinguish between those that are deterministic versus stochastic. Deterministic relationships are “fully determined” such that, knowing the values of the independent variable, you can perfectly explain (or predict) the value of the dependent variable. Philosophers of Old (like Kant) imagined the universe to be like a massive and complex clock which, once wound up and set ticking, would permit perfect prediction of the future if you had all the information on the starting conditions. There is no “error” in the prediction. Stochastic relationships, on the other hand, include an irreducible random component, such that the independent variables permit only a partial prediction of the dependent variable. But that stochastic (or random) component of the variation in the dependent variable has a probability distribution that can be analyzed statistically.

### 7.1.1 Deterministic Linear Model

The deterministic linear model serves as the basis for evaluating theoretical models. It is expressed as:

$$$Y_{i} = \alpha + \beta X_{i} \tag{7.1}$$$

A deterministic model is systematic and contains no error, therefore $$Y$$ is perfectly predicted by $$X$$. This is illustrated in Figure 7.1. $$\alpha$$ and $$\beta$$ are the model parameters, and are constant terms. $$\beta$$ is the slope, or the change in $$Y$$ over the change in $$X$$. $$\alpha$$ is the intercept, or the value of $$Y$$ when $$X$$ is zero.

Given that in social science we rarely work with deterministic models, nearly all models contain a stochastic, or random, component.

### 7.1.2 Stochastic Linear Model

The stochastic, or statistical, linear model contains a systematic component, $$Y = \alpha+\beta$$, and a stochastic component called the error term. The error term is the difference between the expected value of $$Y_i$$ and the observed value of $$Y_i$$; $$Y_i-\mu$$. This model is expressed as:

$$$Y_{i} = \alpha + \beta X_{i} + \epsilon_i \tag{7.2}$$$

where $$\epsilon_i$$ is the error term. In the deterministic model, each value of $$Y$$ fits along the regression line, however in a stochastic model the expected value of $$Y$$ is conditioned by the values of $$X$$. This is illustrated in Figure 7.2.

Figure 7.2 shows the conditional population distributions of $$Y$$ for several values of $$X, p(Y|X)$$. The conditional means of $$Y$$ given $$X$$ are denoted $$\mu$$.

$$$\mu_{i} \equiv E(Y_{i}) \equiv E(Y|X_{i})=\alpha+\beta X_{i} \tag{7.3}$$$

where - $$\alpha = E(Y) \equiv \mu$$ when $$X=0$$ - Each 1 unit increase in $$X$$ increases $$E(Y)$$ by $$\beta$$

However, in the stochastic linear model variation in $$Y$$ is caused by more than $$X$$, it is also caused by the error term $$\epsilon$$. The error term is expressed as:

\begin{align*} \epsilon_i &= Y_{i}-E(Y_{i}) \\ &= Y_{i}-(\alpha+\beta X_{i}) \\ &= Y_{i}-\alpha-\beta X_{i} \end{align*} Therefore; \begin{align*} Y_{i} &= E(Y_{i})+\epsilon \\ &= \alpha+\beta X_{i}+\epsilon_{i} \end{align*}

We make several important assumptions about the error term that are discussed in the next section.

### 7.1.3 Assumptions about the Error Term

There are three key assumptions about the error term; a) errors have identical distributions, b) errors are independent, and c) errors are normally distributed.14

#### Error Assumptions

• Errors have identical distributions

$$E(\epsilon^{2}_{i}) = \sigma^2_{\epsilon}$$

• Errors are independent of $$X$$ and other $$\epsilon_{i}$$

$$E(\epsilon_{i}) \equiv E(\epsilon|x_{i}) = 0$$

and

$$E(\epsilon_{i}) \neq E(\epsilon_{j})$$ for $$i \neq j$$

• Errors are normally distributed

$$\epsilon_{i} \sim N(0,\sigma^2_{\epsilon})$$

Taken together these assumption mean that the error term has a normal, independent, and identical distribution (normal i.i.d.). However, we don’t know if, in any particular case, these assumptions are met. Therefore we must estimate a linear model.

## 7.2 Estimating Linear Models

With stochastic models we don’t know if the error assumptions are met, nor do we know the values of $$\alpha$$ and $$\beta$$; therefore we must estimate them, as denoted by a hat (e.g., $$\hat{\alpha}$$ is the estimate for $$\alpha$$). The stochastic model as shown in Equation (7.4) is estimated as:

$$$Y_{i} = \hat{\alpha} + \hat{\beta} X_{i}+ \epsilon_{i} \tag{7.4}$$$

where $$\epsilon_i$$ is the residual term, or the estimated error term. Since no line can perfectly pass through all the data points, we introduce a residual, $$\epsilon$$, into the regression equation. Note that the predicted value of $$Y$$ is denoted $$\hat{Y}$$ ($$y$$-hat).

\begin{align*} Y_{i} &= \hat{\alpha}+\hat{\beta}X_{i}+\epsilon_{i} \\ &= \hat{Y_{i}} + \epsilon_{i} \\ \epsilon_{i} &= Y_i-\hat{Y_{i}} \\ &= Y_i-\hat{\alpha}-\hat{\beta}X_i \end{align*}

### 7.2.1 Residuals

Residuals measure prediction errors of how far observation $$Y_{i}$$ is from predicted $$\hat{Y_{i}}$$. This is shown in Figure 7.3.

The residual term contains the accumulation (sum) of errors that can result from measurement issues, modeling problems, and irreducible randomness. Ideally, the residual term contains lots of small and independent influences that result in an overall random quality of the distribution of the errors. When that distribution is not random – that is, when the distribution of error has some systematic quality – the estimates of $$\hat{\alpha}$$ and $$\hat{\beta}$$ may be biased. Thus, when we evaluate our models we will focus on the shape of the distribution of our errors.

What’s in $$\epsilon$$?

Measurement Error

• Imperfect operationalizations
• Imperfect measure application

Modeling Error

• Modeling error/mis-specification
• Missing model explanation

Stochastic “noise”

• Unpredictable variability in the dependent variable

The goal of regression analysis is to minimize the error associated with the model estimates. As noted, the residual term is the estimated error, or overall miss" (e.g., $$Y_{i}-\hat{Y_{i}}$$). Specifically the goal is to minimize the sum of the squared errors, $$\sum \epsilon^{2}$$. Therefore, we need to find the values of $$\hat{\alpha}$$ and $$\hat{\beta}$$ that minimize $$\sum \epsilon^{2}$$.

Note that for a fixed set of data {$$\hat{\alpha}$$,$$\hat{\alpha}$$}, each possible choice of values for $$\hat{\alpha}$$ and $$\hat{\beta}$$ corresponds to a specific residual sum of squares, $$\sum \epsilon^{2}$$. This can be expressed by the following functional form:

$$$S(\hat{\alpha},\hat{\beta})=\sum_{i=1}^{n} \epsilon^{2}_{i}=\sum (Y_{i}-\hat{Y_{i}})^{2}=\sum (Y_{i}-\hat{\alpha}-\hat{\beta}X_{i})^{2} \tag{7.5}$$$

Minimizing this function requires specifying estimators for $$\hat{\alpha}$$ and $$\hat{\beta}$$ such that $$S(\hat{\alpha},\hat{\beta})=\sum \epsilon^{2}$$ is at the lowest possible value. Finding this minimum value requires the use of calculus, which will be discussed in the next chapter. Before that we walk through a quick example of simple regression.

## 7.3 An Example of Simple Regression

The following example uses a measure of peoples’ political ideology to predict their perceptions of the risks posed by global climate change. OLS regression can be done using the lm function in R. For this example, we are again using the class data set.

ols1 <- lm(ds$glbcc_risk~ds$ideol)
summary(ols1)
##
## Call:
## lm(formula = ds$glbcc_risk ~ ds$ideol)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -8.726 -1.633  0.274  1.459  6.506
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.81866    0.14189   76.25   <2e-16 ***
## ds$ideol -1.04635 0.02856 -36.63 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 2.479 on 2511 degrees of freedom ## (34 observations deleted due to missingness) ## Multiple R-squared: 0.3483, Adjusted R-squared: 0.348 ## F-statistic: 1342 on 1 and 2511 DF, p-value: < 2.2e-16 The output in R provides a quite a lot of information about the relationship between the measures of ideology and perceived risks of climate change. It provides an overview of the distribution of the residuals; the estimated coefficients for $$\hat{\alpha}$$ and $$\hat{\beta}$$; the results of hypothesis tests; and overall measures of model fit" – all of which we will discuss in detail in later chapters. For now, note that the estimated $$B$$ for ideology is negative, which indicates that as the value for ideology increases—in our data this means more conservative—the perceived risk of climate change decreases. Specifically, for each one unit increase in the ideology scale, perceived climate change risk decreases by -1.0463463. We can also examine the distribution of the residuals, using a histogram and a density curve. This is shown in Figure 7.4 and Figure 7.5. Note that we will discuss residual diagnostics in detail in future chapters. data.frame(ols1$residuals) %>%
ggplot(aes(ols1$residuals)) + geom_histogram(bins = 16) data.frame(ols1$residuals) %>%
ggplot(aes(ols1\$residuals)) +
geom_density(adjust = 1.5)

For purposes of this Chapter, be sure that you can run the basic bivariate OLS regression model in R. If you can – congratulations! If not, try again. And again. And again…

1. Actually, we assume only that the means of the errors drawn from repeated samples of observations will be normally distributed – but we will deal with that wrinkle later on.


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# Building a Computerized Adaptive Version of Psychological Scales

Computerized adaptive testing (CAT) is a sophisticated methodology to create measurement instruments that are highly accurate and efficient. In this post, I explain how to evaluate the feasibility of creating a computerized adaptive version of a psychological instrument.

Okan Bulut http://www.okanbulut.com/ (University of Alberta)https://www.ualberta.ca
02-20-2021

## Introduction

Psychologists, counselors, educators, and other practitioners often rely on educational and psychological instruments–such as tests, questionnaires, and inventories–to make informed decisions about individuals. Most educational and psychological instruments in use today follow the traditional approach of asking as many items as possible via a paper-and-pencil assessment. Although this approach is likely to increase the internal consistency of the instrument, it may lead to some unintended consequences, such as having low-quality responses due to test fatigue and test-taking disengagement. Therefore, many researchers have proposed systematic ways to shorten educational and psychological instruments .

In my previous posts, I demonstrated how to shorten measurement instruments using psychometric methods such as automated test assembly and data science methods such as the ant colony optimization. These methods can help researchers and practitioners build a shorter version of an instrument and thereby increasing measurement efficiency. However, as pointed out, conventional assessments with fixed items (i.e., the same items being used for everyone) tend to yield accurate results for individuals whose trait levels are around the mean of the target population but yield poor measurement results for those whose latent trait levels deviate from the mean.

A promising solution to creating assessments with high measurement accuracy for all individuals is the use of adaptive testing. Adaptive testing follows the idea of adapting an assessment to each individual’s latent trait level by administering a customized set of items, instead of administering the same set of items to all individuals. In this post, I will briefly explain how adaptive testing works and then demonstrate how to create a computerized adaptive version of an existing psychological instrument.

Computerized adaptive testing (CAT) is a sophisticated method of delivering computerized assessments with high measurement precision and efficiency . The primary goal of CAT is to customize the assessment for each individual by selecting the most suitable items based on their responses to the previously administered questions. To design and implement a CAT, the following five components are necessary :

• Item bank: A large pool of items calibrated with a particular item response theory (IRT) model
• Starting point: The point where the test begins (e.g., administering a particular item to everyone or assuming $$\theta = 0$$ for all individuals)
• Item selection algorithm: How CAT will determine which items should be administered (e.g., Maximum Fisher Information for selecting the most informative item)
• Scoring algorithm: How latent trait levels will be estimated after each response (e.g., Maximum Likelihood or Expected a Posteriori)
• Test termination criteria: How CAT will terminate the test (e.g., answering the maximum number of items or reaching a particular level of measurement precision)

In a typical CAT, each individual begins to receive items at a particular level (e.g., $$\theta = 0$$). Then, depending on the answer, the next item becomes less or more difficult. For example, if the individual answers an item with moderate difficulty correctly, then the CAT assumes that this individual’s latent trait level is above the difficulty level of the question and thus it presents a more difficult question in the next round. If, however, the individual is not able to answer the item correctly, then she/he is administered an easier question in the next round. this iterative process continues until a test termination criterion is met (e.g., answering the maximum number of questions).

Before implementing a CAT, it is important to obtain enough evidence to support its use in operational settings. For example, if researchers aim to redesign a conventional, non-adaptive instrument as a CAT, then a series of post-hoc simulations can be conducted using previously collected data. A post-hoc simulation is a psychometric procedure to evaluate the effect of different CAT algorithms under specific conditions . Using real data from a conventional instrument taken by a large group of individuals, one can create a hypothetical CAT scenario and evaluate the impact of various CAT elements (e.g., different algorithms for item selection or scoring) on measurement accuracy and efficiency. The results of these simulations can help researchers and practitioners determine the feasibility and applicability of the CAT approach for an existing instrument.

## Example

In this example, we will use real data from a sample of respondents ($$n = 4474$$) who responded to the items in the Taylor Manifest Anxiety Scale and build a CAT version of this instrument through post-hoc simulations. The Taylor Manifest Anxiety Scale is typically used for measuring anxiety as a personality trait. The scale consists of fifty statements about different indicators of anxiety. For each item, individuals select either true or false. The higher the score, the higher the anxiety level. Because some items on the scale are negatively phrased, they must be reverse-coded1. A clean version of the data (including reverse-coding) is available here. Now let’s import the data and check out its content.

# Import the TMA data into R


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## Analysis of spatial distribution of marker expression in cells using boundary distance plots.(English)Zbl 1202.62149

Summary: Boundary distance (BD) plotting is a technique for making orientation invariant comparisons of the spatial distribution of biochemical markers within and across cells/nuclei. Marker expression is aggregated over points with the same distance from the boundary. We present a suite of tools for improved data analysis and statistical inference using BD plotting. BD is computed using the Euclidean distance transform after presmoothing and oversampling of nuclear boundaries. Marker distribution profiles are averaged using smoothing with linearly decreasing bandwidth. Average expression curves are scaled and registered by $$x$$-axis dilation to compensate for uneven lighting and errors in nuclear boundary marking. Penalized discriminant analysis is used to characterize the quality of separation between average marker distributions. An adaptive piecewise linear model is used to compare expression gradients in intra, peri and extra nuclear zones. The techniques are illustrated by the following: (a) a two sample problem involving a pair of voltage gated calcium channels (Cav1.2 and AB70) marked in different cells; (b) a paired sample problem of calcium channels (Y1F4 and RyR1) marked in the same cell.

### MSC:

 62P10 Applications of statistics to biology and medical sciences; meta analysis 62A09 Graphical methods in statistics 92C40 Biochemistry, molecular biology 65C60 Computational problems in statistics (MSC2010) 62H30 Classification and discrimination; cluster analysis (statistical aspects) 92C37 Cell biology

fda (R)
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### References:

 [1] Ambler, S. K., Poenie, M., Tsien, R. Y. and Taylor, P. (1988). Agonist-stimulated oscillations and cycling of intracellular free calcium in individual cultured muscle cells. J. Biol. Chem. 263 1952-1959. [2] Anderson, T. (2003). An Introduction to Multivariate Statistical Analysis , 3rd ed. Wiley, Hoboken, NJ. · Zbl 1039.62044 [3] Bewersdorf, J., Bennett, B. and Knight, K. (2006). H2AX chromatin structures and their response to DNA damage revealed by 4Pi microscopy. Proc. Natl. Acad. Sci. USA 103 18137-18142. [4] Callinan, L., McCarthy, T., Maulet, Y. and Mackrill, J. (2005). Atypical L-type channels are down-regulated in hypoxia. Biochemical Society Transactions 33 1137-1139. [5] Fabbri, R., Costa, L., Torelli, J. and Bruno, O. (2008). 2D Euclidean distance transform algorithms: A comparative survey. ACM Computing Surveys 40 2:1-2:44. [6] Fernandez-Gonzalez, R., Munoz-Barrutia, A., Barcellos-Hoff, M. and Ortiz-De-Solorzano, C. (2006). Quantitative in vivo microscopy: The return from the ‘omics.’ Current Opinion in Biotechnology 17 501-510. [7] Friedman, J. (1989). Regularized discriminant analysis. J. Amer. Statist. Assoc. 84 165-175. [8] Gomez-Ospina, N., Barreto-Chang, O., Hu, L. and Dolmetsch, R. (2006). The C terminus of the L-type voltage-gated calcium channel Ca(V)1.2 encodes a transcription factor. Cell 127 591-606. [9] Hastie, T., Tibshirani, R. and Friedman, J. (2001). The Elements of Statistical Learning . Springer, New York. · Zbl 0973.62007 [10] Jahne, B. (2005). Digital Image Processing , 6th ed. Springer. [11] Knowles, D., Sudar, D., Bator-Kelly, C., Bissell, M. and Lelièvre, S. (2006). Automated local bright feature image analysis of nuclear protein distribution identifies changes in tissue phenotype. Proc. Natl. Acad. Sci. USA 103 4445-4450. [12] Mackrill, J. J. (1999). Protein-protein interactions in intracellular Ca2+-release channel function. Biochem. J. 337 345-361. [13] Ramsay, J. and Silverman, B. (2002). Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1011.62002 [14] Silverman, B. (1984). Spline smoothing: The equivalent variable kernel method. Ann. Statist. 12 898-916. · Zbl 0547.62024 [15] Silverman, B. (1985). Some aspects of the spline smoothing approach to non-parametric regression curve fitting. J. Roy. Stat. Soc. Ser. B 47 1-52. · Zbl 0606.62038 [16] Wahba, G. (1975). Optimal convergence properties of variable knot, kernel, and orthogonal series methods for density estimation. Ann. Statist. 3 15-29. · Zbl 0305.62021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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## File: l2hconf.pin

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latex2html 2008-debian1-7
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use vars qw(%used_icons); # Setting this variable to where your perl executable resides can lead to # better performance on some platforms. # # It is advisable to do this on an Intel system; e.g. # $PERL='g:/usr/bin/perl_.exe'; # # On a unix system it may be best left empty, or set as in: #$PERL='/usr/local/bin/perl'; # $PERL = '@PERL@'; # ############### THESE VARIABLES ARE DETERMINED BY CONFIGURE ################ # Give the paths to latex and dvips on your system: #$LATEX = '@LATEX@'; # LaTeX $DVIPS = '@DVIPS@'; # dvips # give these too, to use the -ldump feature #$TEX = "@TEX@"; # TeX $INILATEX = "@INITEX@ \"&latex\""; # initex+latex # These affect whether images are made on a white or gray background. # They are ignored when the document preamble contains similar commands. # Use these defaults for dark solid (anti-aliased) characters in text and math. #$LOAD_LATEX_COLOR = "\\usepackage[dvips]{color}"; $LATEX_COLOR = "\\pagecolor[gray]{.7}"; # -white # this overrides the above gray-scale for figures that don't need anti-aliasing #$WHITE_BACKGROUND = 0; # -image_type # This specifies the type of images produced by latex2html when processing # unknown environments and/or e.g. math formulae. # @IMAGE_TYPES = qw(@IMAGE_TYPES@); $IMAGE_TYPE =$IMAGE_TYPES[0]; # -tmp # Specify a tmp directory for image-generation (optional) # #if @texlive@ $TMP =$ENV{TMP} || $ENV{TEMP} || '.'; #else$TMP = '@TMPSPACE@'; #fi ############# HTML validation ############### # # set $HTML_VALIDATOR to the command needed to run a validator to check # the HTML pages produced; # use the -validate switch to run the validator, or set$HTML_VALIDATE # $HTML_VALIDATOR = '@HTML_VALIDATOR@'; # -validate # when$HTML_VALIDATE is 1, the validator will run as default # provided $HTML_VALIDATOR is also set; -novalidate suppresses this #$HTML_VALIDATE = 0; ########## ICONSERVER --- !! IMPORTANT !! ############################ # # LaTeX2HTML uses many small graphics as icons within the navigation # panels, and for other purposes. # You *must* specify where these graphics are to be found, for use within # your documents. # # IMPORTANT: This location must not only be accessible to you, but also # to the people who are to read your HTML documents. # # One option is to always use the -local_icons switch (see below) # to have a copy of the icons together with your HTML document. # (This is safe, but wasteful if you have a large number of documents.) # # A better option is to set $ICONSERVER to point to a location that # is known to always (at least in principle) be available publicly. # # The default value for$ICONSERVER given below is within the local # $LATEX2HTMLDIR. If this is *not* to be publicly available then you # **should change the value of$ICONSERVER to a site that is**. # # You probably have to talk to your Webmaster to provide access to the # icon directory. Some hints: # a) It's ok to set $ICONSERVER just to "/path/to/icons.$IMAGE_TYPE" (without # the internet address) if /path/to is valid for your file system, *and* # if www_root/path/to points to the same directory. Normally the www area # is located in some subsidary directory, which is pointed to by www_root # (the entry 'Document Root' of the http daemon's srm.conf file). # Make the Webmaster add appropriate links in that directory that help # locate the icon directory, or set up an Alias in srm.conf. # b) To check if the icons can be displayed properly, invoke the browser # with the *http URL to your site*, and click down to your document. # c) If you have $LOCAL_ICONS set, your document will accumulate a pretty # amount of redundant icons if you make use of segmentation. # In this case, customize &img_tag to use a central directory, say, # "../icons". # #if @texlive@ use L2hos;$ICONSERVER = L2hos->path2URL("$LATEX2HTMLDIR/icons"); #else$ICONSERVER = '@ICONSERVER@'||'@ICONPATH@'; #fi $ALTERNATIVE_ICONS = 0; # ####### YOU *MAY* WANT/NEED TO CHANGE SOME OF THESE VARIABLES ############## # -djgpp # On DOS/DJGPP systems one can easily run out of file handles. To # avoid that, set this to 1. However this affects performance. #$DJGPP = 0; # if you are having difficulties with inputs not being found, # and your system is Web2C then setting this to 1 may help. # $Web2C = @WEB2C@; # Options for dvips as determined by configure #$DVIPSOPT = '@DVIPSOPT@'; # If you already have the fonts, you may add -M to suppress font # generation # # $DVIPSOPT .= ' -M'; # If you have dvips 5.62 or higher, you can turn on generation of EPS files # by uncommenting the following line. Warning: dvips does not support # included EPS figures very well. However if you don't make use of # complicated image include commands like \includegraphics, this option # will speed up image generation a *lot*. #$DVIPSOPT .= " -E"; # (Note: this here is old, don't worry unless you really run into trouble.) # # Some dvips programs generate postscript images in the reverse order by # default. If your inlined images are all screwed up try uncommenting # the following line: # # $DVIPSOPT .= " -r0"; # Modern TeX installations have PostScript Type 1 fonts which can be # used instead of bitmaps. Use of these can give better quality images # as Ghostscript can use hinting' as well as having accurate outlines # which help with anti-aliasing. Possible options here depend upon the # TeX installation; e.g. " -Pcmz -Pams" or " -Ppdf" # #$DVIPSOPT .= " -Ppdf"; # For efficient use of font resources, minimising disk-space use, # allow GhostScript to find the fonts it needs for images. # # DO: # EITHER: edit Ghostscript's $GS_LIB/Fontmap file # OR: set the GS_FONTPATH environment variable; # e.g. #$ENV{'GS_FONTPATH'} = join(':/usr/local/texmf/fonts/type1/', '', # 'adobe','ams','bh','bsr','lucida','mt','public/cm','public/xypic'); # # AND # setup a virtual printer configuration file 'config.gs' # and listing 'psfonts.gs' of PostScript fonts to exclude from .ps files # # AND # tell dvips to use this 'virtual printer' : # (the previous item for Type 1 fonts becomes redundant) # # $DVIPSOPT .= " -Pgs"; # Local initialization files are usually named .latex2html-init # this name is hard-coded as the default with the latex2html script # It can be changed here, if desired: # #$INIT_FILE_NAME = '.latex2html-init'; # Location of texexpand, supplied with the translator # #if @texlive@ $TEXEXPAND = "$PERL $LATEX2HTMLDIR/texexpand.pl"; #else$TEXEXPAND = "$PERL @scriptdir@${dd}texexpand@scriptext@"; #fi # Location of pstoimg, supplied with the translator # #if @texlive@ $PSTOIMG = "$PERL $LATEX2HTMLDIR/pstoimg.pl"; #else$PSTOIMG = "$PERL @scriptdir@${dd}pstoimg@scriptext@"; #fi # This is used to "autoload" perl code to deal with specific style files # $LATEX2HTMLSTYLES = "$LATEX2HTMLDIR${dd}styles"; # This is used to support upcoming versions of html - directory where perl # files to handle those are #$LATEX2HTMLVERSIONS = "$LATEX2HTMLDIR${dd}versions"; # The following variable sets the default search list of directories for # latex style files that latex2html should process. It also defines a # a list of directories (: separated) which possibly contain TeX and dvips # inputs. This variable is overriden by the environment variable # TEXINPUTS, if it is specified. Internally, the directory your document # resides in, and ".", are appended to this list of directories. # # IMPORTANT: In some installations, latex and dvips are really # shell scripts which set environment TEXINPUTS (and other variables) # to predefined values, then call the real latex and dvips. If this # is true for your installation, then the $TEXINPUTS that latex2html # sees will only affect the processing of \input and \include's # by latex2html, not the operation of latex and dvips when called # by latex2html. In this case, make sure that the predefined # values of TEXINPUTS within the latex and dvips scripts at least # contains "." and ".." (".." works in the most cases). # Otherwise, latex and dvips will not find inputs from the original source # directory when called from a subdirectory contained therein. # # The single colon tells LaTeX to look on the standard places only. # If you add entries, do it colon-separated. # If you don't know where LaTeX takes its standard files from, leave the # single colon in front or at the end, or have an empty entry "::" # at some place among the other entries. # if ($Web2C) { chomp ($TEXINPUTS = # kpsewhich -progname=latex -expand-braces \\\$TEXINPUTS); #kpsewhich -v -n latex \\\$TEXINPUTS);$envkey); } else { $TEXINPUTS =$envkey; } # This line helps LaTeX2HTML to recognize your adaption everywhere. # $ENV{'TEXINPUTS'} =$TEXINPUTS unless defined $ENV{'TEXINPUTS'}; # -no_fork # If defined this will prevent the translator to crash if your operating # system does not support forking; e.g. DOS. # #if @texlive@ # works only on UNIX, or...?$CAN_FORK = L2hos->plat() =~ /unix/; #else #if @plat@ =~ /dos|win|os2/i $CAN_FORK = 0; #else$CAN_FORK = 1; #fi #fi # ############################################################################ # THERE IS NO NEED TO CHANGE ANY OF THE VARIABLES BELOW EXCEPT FOR CUSTOMISING # THE OPERATION OF LATEX2HTML. # ############################################################################ ### Command Line Argument Defaults ####################################### # -ldump # Change this to 1 if you want to speed up image processing during the 2nd # and more runs of LaTeX2HTML on the same document. # This will cause LaTeX2HTML to produce a LaTeX dump of images.tex which # is read in on subsequent runs and speeds up startup time of LaTeX on the # images.tex translation. # This actually consumes additional time on the first run, but pays off on # subsequent runs. The dump file will need about 1 Meg of disk space. # $LATEX_DUMP = 0; # -numbered_footnotes # If defined to 1 you will get every footnote applied with a subsequent # number, else with a hyperlink icon. #$NUMBERED_FOOTNOTES = 0; # -local_icons # Change this to 1 if you want to copy the navigation icons to each # document directory so that the document directory is self-contained # and can be dropped into another server tree. Note that you can also # use the command line option -local_icons # $LOCAL_ICONS = 0; # -split #$MAX_SPLIT_DEPTH = 8; # Stop making separate files at this depth # -link # $MAX_LINK_DEPTH = 4; # Stop showing child nodes at this depth # -short_extn # If this is set all HTML file will have extension ".htm" instead of # ".html". This is helpful when shipping the document to PC systems. #$SHORTEXTN = 0; # -nolatex # $NOLATEX = 0; # 1 = do not pass unknown environments to Latex # -external_images #$EXTERNAL_IMAGES = 0; # 1 = leave the images outside the document # -ps_images # 1 = use links to external postscript images rather than inlined GIF's. # $PS_IMAGES = 0; # ANTI-ALIASING within generated images # # -antialias # 1 = use anti-aliasing in the generation of images of figures . #$ANTI_ALIAS = 0; # -antialias_text # 1 = use anti-aliasing in the generation of images of typeset material; # e.g. mathematics and text, e.g. in tables and {makeimage} environments. # $ANTI_ALIAS_TEXT = 1; # -font_size # To set the point size of LaTeX-generated GIF files, uncomment the following # variable, and set it to its desired value (i.e, 10pt, 11pt, 12pt, etc.) # The default is to use the point size of the original LaTeX document. # This value will be magnified by$FIGURE_SCALE_FACTOR and # $MATH_SCALE_FACTOR (below). # #$FONT_SIZE = "12pt"; # -no_tex_defs # To suppress the interpretation of raw TeX commands, set $TEXDEFS = 0; # Note: There are many variations of \def that latex2html cannot process # correctly! #$TEXDEFS = 1; # -ascii_mode # This is different from -no_images. # If this is set, LaTeX2HTML will show textual tags rather than # images, both in navigation panel and text (Eg. [Up] instead the up # icon). # You could use this feature to create simple text from your # document, eg. with 'Save as... Text' from Netscape or with # lynx -dump. # $ASCII_MODE = 0; # 1 = do not use any icons or internal images # -t, The document title. #$default_title = '$FILE'; # -dir$DESTDIR = ''; # Put the result in this directory # -no_subdir # When this is set, the generated HTML files will be placed in the # current directory. If set to 0 the default behaviour is to create (or reuse) # another file directory. # $NO_SUBDIR = 0; # -address # Supply your own string if you don't like the default #$ADDRESS = "$address_data[0]\n$address_data[1]"; # -no_navigation # 1 = do not put a navigation panel at the top of each page # $NO_NAVIGATION = 0; # -top_navigation # Determines whether to navigation links should be at the top or the bottom # of each page. The default is at the top. #$TOP_NAVIGATION = 1; # -bottom_navigation # Determines whether to navigation links should be at the top or the bottom # of each page. The default is at the top. # $BOTTOM_NAVIGATION = 0; # -auto_navigation # Put navigation links at the top of each page. If the page exceeds #$WORDS_IN_PAGE number of words then put one at the bottom of the page. # $AUTO_NAVIGATION = 1; # -index_in_navigation # Put a link to the index page in the navigation panel #$INDEX_IN_NAVIGATION = 1; # -contents_in_navigation # Put a link to the table of contents in the navigation panel # $CONTENTS_IN_NAVIGATION = 1; # -next_page_in_navigation # Put a link to the next logical page in the navigation panel #$NEXT_PAGE_IN_NAVIGATION = 1; # -previous_page_in_navigation # Put a link to the previous logical page in the navigation panel # $PREVIOUS_PAGE_IN_NAVIGATION = 1; # -prefix # Set the output file prefix, prepended to all .html, .gif and .pl files. # See also$AUTO_PREFIX. # $PREFIX = ''; # -auto_prefix # To automatically insert the equivalent of "-prefix basename-", where # "basename" is the base name of the file being translated, set this to 1. #$AUTO_PREFIX = 0; # -up_url, -up_title, -down_url, -down_title, -prev_url, -prev_title: # If both of the following two variables are set then the "Up" button # of the navigation panel in the first node/page of a converted document # will point to $EXTERNAL_UP_LINK.$EXTERNAL_UP_TITLE should be set # to some text which describes this external link. # $EXTERNAL_UP_LINK = '';$EXTERNAL_UP_TITLE = ''; # Similarly you might set these variables to link external documents # to your navigation panel. # $EXTERNAL_DOWN_LINK = "";$EXTERNAL_DOWN_TITLE = ""; $EXTERNAL_PREV_LINK = "";$EXTERNAL_PREV_TITLE = ""; $EXTERNAL_INDEX = "";$EXTERNAL_CONTENTS = ""; # -info # 0 = do not make an "About this document..." section # $INFO = 1; # -reuse, -no_reuse, Image recycling: # If 0, do not reuse or recycle identical images. If the html subdirectory # already exists, start the interactive session. # If nonzero, do recycle them and switch off the interactive session. # If 1, only recycle images generated from previous runs. # If 2, recycle images from the current and previous runs. #$REUSE = 2; # -no_images # When $NO_IMAGES is set LaTeX2HTML will not attempt to produce any inlined images # The missing images can be generated "off-line" by restarting LaTeX2HTML # after setting$IMAGES_ONLY (see below); #if @have_images@ $NO_IMAGES = 0; #else$NO_IMAGES = 1; #fi # -images_only # When $IMAGES_ONLY is set, LaTeX2HTML will only try to convert the inlined images # in the file "images.tex" which should have been generated automatically during # previous runs. This is very useful for correcting "bad LaTeX" in this file. #$IMAGES_ONLY = 0; # -discard # When $DISCARD_PS is set, the PostScript file created for each generated image # is discarded immediately after its image has been rendered and saved in the # required graphics format. This can lead to significant savings in disk-space, # when there are a lot of images, since otherwise these files are not discarded # until the end of all processing. #$DISCARD_PS = 1; # -show_section_numbers # When this is 1, the section numbers are shown. The section numbers should # then match those that would have bee produced by LaTeX. # The correct section numbers are obtained from the $FILE.aux file generated # by LaTeX. # Hiding the seciton numbers encourages use of particular sections # as standalone documents. In this case the cross reference to a section # is shown using the default symbol rather than the section number. #$SHOW_SECTION_NUMBERS = 0; # -short_index # If this is set then makeidx.perl will construct codified names # for the text of index references. # $SHORT_INDEX = 0; # -debug # If this is set then intermediate files are left for later inspection. # This includes $$_images.tex and$$_images.log created during image # conversion. # Caution: Intermediate files can be *enormous*. #$DEBUG = 0; # -html_version # The default HTML version to be produced # $HTML_VERSION = '4.0'; # -no_math # By default the special MATH extensions are not used # since they do not conform with the HTML 3.2 standard. #$NO_SIMPLE_MATH = 1; # -unsegment # Use this to translate a segmented document as if it were not # segmented. # $UNSEGMENT = 0; ### Other global variables ############################################### # If this is set then the HTML will look better if viewed with Netscape. #$NETSCAPE_HTML = 0; # Set this to 1 if you want interlaced images, 0 otherwise. Interlaced # images build up gradually while downloading so one can get a first # impression of what the final image will look like very quickly. # $INTERLACE = 1; # Set this to 1 if you like the old LaTeX2HTML style to have a # border around the navigation links. # Values > 1 are also possible. #$NAV_BORDER = 0; ### Colors ### # # If this is set you may set colors in your document (see the LaTeX # package color.dvi and the color.perl/colordvi.perl files). # Note that HTML generated herefrom cannot be viewed by all browsers # (at least Netscape or Mosaic 2.7 should do). # $COLOR_HTML = 0; # Specify the path to your systems color database if you do not agree on # the databases provided with the translator. Eg.: /usr/lib/X11/rgb.txt # the RGB colors database #$RGBCOLORFILE = '@RGBCOLORFILE@'; # the CMYK colors database # $CRAYOLAFILE = '@CRAYOLAFILE@'; ### End Colors ### # Do not try to translate these input files, and do not # complain about a missing Perl module. # Complex LaTeX inputs, styles, or classes may cause the translator # to hang. If this occurs add the input file here. # You may also specify filename extensions here, e.g. if you do # not want to include input files matching "*.myfig", add # :.myfig''. #$DONT_INCLUDE = "2up:psfig:epsf:texinfo:pictex:" . ".ps:.eps:.fig:.pstex_t:.epsf:.epic:.eepic:.xy:.xya:.xyc:" . "titlepage:openbib:\\d+pt:twoside:twocolumn:" . "memo:dvipsfig:times:margins:aaii2:a4:art\\d+:doublespace:" . "alltt:amstex:anysize:array:article:bm:book:bookman:" . "boxedminipage:cite:comment:courier:dcolumn:doc:eepic:" . "enumerate:epic:fleqn:float:floatflt:fullpage:index:" . "inputenc:isolatin1:leqno:letter:llncs:makeidx:" . "multicol:psfig:report:shadow:shapepar:showidx:" . "slides:syntonly:sz:tabls:times:twoside:umlaut:umlaute"; # Latex2html usually does not include style files provided by # \documentstyle, \documentclass, \usepackage but tries to use the # corresponding *.perl files provided in the styles/ subdirectory. # Now if you use home-brew style files with new environments/commands # you may want to include them. E.g. if you want to include "mystyle.sty", # say $DO_INCLUDE = "mystyle" here. Separate styles with colons. This # setting overrides the settings in$DONT_INCLUDE. You may specify # filename extensions here as well. # # $DO_INCLUDE = ""; # If you have equations in your text, and must use bitmap'd fonts with # an old (pre 4.02) version of Ghostscript... # (RRM: ignore all the following with later versions) # # ... you'll get the best results with PK_GENERATION=1. # With this option switched on, DVIPS will be told to generate # all of the images for a specific screen resolution, # eliminating "blurring" of small letters and subscripts. # If any of it causes you grief, simply set PK_GENERATION=0, # and your default printer's resolution will be used. # # Sidik Isani, added this. #$PK_GENERATION = 0; # @PK_GENERATION@ # by configure # ... and set the following variable ONLY if your version of dvips # understands the "-mode" command line switch. It is a more reliable # way of setting the METAfont mode than the .dvipsrc file on versions that # support this switch. If you do this, you do not need to modify .dvipsrc # as described below. Herb Swan added this. # $DVIPS_MODE = '@METAMODE@'; # Only if you have PK_GENERATION set to 1: # A file 'modes.mf' is probably installed somewhere in your tex # tree. e.g /usr/local/tex/texmf/mf/modes.mf or something similar. # If it has and entry for 'toshiba', then you can ignore the rest. # Otherwise, you'll need to try *one* of the following things: # # o Set PK_GENERATION=0 and generate images for your default printer # o Download the latest modes.mf from the TeX archive and re-run inimf. # o Choose another LOW RESOLUTION screen or printer entry from modes.mf, # (preferably with "|blacker|" set to 0.0 and a 1:1 aspect ratio-- # If you don't have "toshiba", try "epsonlq", "lqlores", "nec", or "NEC") # If you try some other mode, remember to update two other things as well: # # 1) Change the 'toshiba' in the .dvipsrc file that came with latex2html. # 2) Put whatever the |pixels_per_inch| value is into$METAFONT_DPI # and the .dvipsrc file # # Note: The maximum usable SCALE_FACTOR is determined by this resolution. # A higher resolution will take more memory during processing (regardless # of SCALE_FACTOR) but will allow higher SCALE_FACTORs. Do NOT just # change this value though. It *must* match the |pixels_per_inch| of # the metafont mode (e.g. toshiba'). # $METAFONT_DPI = @METADPI@; # Controls which markup shows up between page and its foot.$CHILDLINE = "
\n"; # If this is set as below, LaTeX2HTML produces a directory index link to # the html document, ie. you may use # instead of . # Set it to eg. 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## Why not formulate QM in terms of |ψ| squared?

 Quote by Tosh5457 So why is that formalism only used in quantum statistical mechanics, and not on the general theory of quantum mechanics?
Because outside statistical mechanics, one can pretend that a mixed state is a mixture of pure states (though not uniquely, and the decomposition has therefore no physical meaning), and the pure state computations look slightly more familiar. Whereas in statistical mechanics you really _need- the mixed states, as every equilibrium state is mixed.

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 Quote by DrDu What I wondered, probably because I don't understand qft too well: Wouldn't it be possible to consider the charge density rho as some field, something related to its time derivative as momentum and then doing second quantization with it?
Actually, in QFT, charge density is a field, the zero component of the corresponding current.

But charge density is not the same as |psi|^2, except in the single particle case.

 Quote by A. Neumaier as every equilibrium state is mixed.
Sorry, this is off topic, but where can I find more info on this. It came up in a paper I was reading, but I don't understand why. I don't think the standard QM books I have cover this. Any advice where to look would be appreciated. Sorry to disrupt the thread.

 ... outside statistical mechanics, one can pretend that a mixed state is a mixture of pure states ...
But doesn't quantum statistical physics pretend the same? Take, for example, the density matrix for harmonic oscillator $\rho_{mn} = \frac{e^{- n\hbar \omega/ (k_B T)}}{Z} \delta_{mn}$ (in the basis of its Hamiltonian). Is not this a good example of a mixture of pure states - eigenstates of the Hamiltonian?

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 Quote by A. Neumaier Actually, in QFT, charge density is a field, the zero component of the corresponding current. But charge density is not the same as |psi|^2, except in the single particle case.
I was assuming a non-relativistic setting. There, I think, charge density and psi square are identical.

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 Quote by DrDu I was assuming a non-relativistic setting. There, I think, charge density and psi square are identical.
My statement holds both in the relativistic and the nonrelativistic case, where things look identical in the second-quantized formulation.

For a single particle, charge density and absolute psi square are identical.

For two particles, the wave function had already two position arguments, while the charge density has only one. So they cannot be the same.

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 Quote by Jano L. But doesn't quantum statistical physics pretend the same? Take, for example, the density matrix for harmonic oscillator $\rho_{mn} = \frac{e^{- n\hbar \omega/ (k_B T)}}{Z} \delta_{mn}$ (in the basis of its Hamiltonian). Is not this a good example of a mixture of pure states - eigenstates of the Hamiltonian?
This is by far not the only casae you can write this density matrix as a mixture, and none of the many mixture representations has any physical relevance as they are observationally undistinguishable.

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 Quote by DrewD Sorry, this is off topic, but where can I find more info on this. It came up in a paper I was reading, but I don't understand why. I don't think the standard QM books I have cover this. Any advice where to look would be appreciated. Sorry to disrupt the thread.
Look an any introduction to statistical mechanics, which doens't treat only the classical case.
e.g., Part II of http://lanl.arxiv.org/abs/0810.1019

 This is by far not the only casae you can write this density matrix as a mixture,
That is true. There are potentially many different ways to express that density matrix. And I agree that preferring one mixture above the others seems artificial and ungrounded.

However, the fact is that no other representation is used. Statistical physics relies on the properties of Hamiltonian, not some other operator, and really treats energy representation with distinction.

 and none of the many mixture representations has any physical relevance as they are observationally undistinguishable.
I am not so sure. In general, different mixtures are in principle distinguishable, if we know something about the preparation of the ensemble. Of course, simple counting of frequencies of results of measurement will not reveal that; but there is more to the ensemble than just results of measurements on it. This is more easily shown for spins than for energy, but why assume that canonical density matrix is different in this respect?

I think the relevance of H eigenstates is in that we need to assume ensemble of them in order to use Boltzmann's probability distribution to find the canonical density matrix in the first place. For other mixtures, how would you arrive at density matrix that is diagonal in H representation? I do not see how that is possible, but perhaps you know.

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 Quote by Jano L. However, the fact is that no other representation is used. Statistical physics relies on the properties of Hamiltonian, not some other operator, and really treats energy representation with distinction.
Statistical mechanics doesn't care at all about the basis in which things are represented, except for computational reasons. One sees it clearly once one leaves the equilibrium situation.
 Quote by Jano L. In general, different mixtures are in principle distinguishable, if we know something about the preparation of the ensemble.
One can distinguish ensembles only if they are prepared artificially, by mixing observations made at different times. But the ensembles of statistical mechanics are of a different nature; they are instantaneous, a single time already defines the ensemble. Then nothing can distinguish a decomposition into subensembles.
 Quote by Jano L. I think the relevance of H eigenstates is in that we need to assume ensemble of them in order to use Boltzmann's probability distribution to find the canonical density matrix in the first place. For other mixtures, how would you arrive at density matrix that is diagonal in H representation? I do not see how that is possible, but perhaps you know.
All that is needed is that e^{beta H} commutes with H, which is a triviality, independent of the basis used to express it. No diagonalization is needed or assumed.

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 Quote by A. Neumaier For two particles, the wave function had already two position arguments, while the charge density has only one. So they cannot be the same.
You are obviously right, here. Nevertheless my question remains: can we quantize charge density, viewed as a field, directly?
 Blog Entries: 9 Recognitions: Homework Help Science Advisor Which charge ? Electric charge ? We surely can.

 Quote by dextercioby Which charge ? Electric charge ? We surely can.
using dirac quantization condition of monopoles or reverse argument?

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# Topic: Dirichlet

###### In the News (Fri 24 May 13)

 Johann Peter Gustav Lejeune Dirichlet - Wikipedia, the free encyclopedia His family hailed from the town of Richelet in Belgium, from which his surname "Lejeune Dirichlet" ("le jeune de Richelet" = "the young chap from Richelet") was derived, and that was where his grandfather lived. Dirichlet was born in Düren, where his father was the postmaster. After his death, Dirichlet's lectures and other results in number theory were collected, edited and published by his friend and fellow mathematician Richard Dedekind under the title Vorlesungen über Zahlentheorie (Lectures on Number Theory). en.wikipedia.org /wiki/Johann_Peter_Gustav_Lejeune_Dirichlet   (305 words)

 Dirichlet series - Wikipedia, the free encyclopedia In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex line, such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes. en.wikipedia.org /wiki/Dirichlet_series   (288 words)

 Dirichlet character - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-07) Dirichlet L-series are straightforward generalizations of the Riemann zeta function and appear prominently in the generalized Riemann hypothesis. A Dirichlet L-series can be expressed as a linear combination of the Hurwitz zeta function, and thus the study of L-series can be unified through a study of the Hurwitz zeta. Dirichlet characters and their L-series were introduced by Dirichlet, in 1831, in order to prove Dirichlet's theorem about the infinitude of primes in arithmetic progressions. www.hartselle.us /project/wikipedia/index.php/Dirichlet_character   (388 words)

 Dirichlet   (Site not responding. Last check: 2007-11-07) Dirichlet did not remain in Rome for the whole period, but visited Sicily and then spent the winter of 1844/45 in Florence before returning to Berlin in the spring of 1845. Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties. Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Dirichlet.html   (2043 words)

 Johann Peter Gustav Lejeune Dirichlet - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-07) His family hailed from the town of in Belgium, from which his surname "Lejeune Dirichlet" ("le jeune de Richelet" = "the young chap from Richelet") was derived, and that was where his grandfather lived. He married, who came from a distinguished Jewish family, being a granddaughter of the philosopher Moses Mendelssohn, and a sister of the composer Felix Mendelssohn. Dirichlet, Johann Peter Gustav Lejeune, Vorlesungen uber Zahlentheorie. www.hartselle.us /project/wikipedia/index.php/Peter_Gustav_Dirichlet   (332 words)

 Dirichlet tessellation of bark beetle spatial attack points This tessellation was proposed in 1850 by Dirichlet (Upton and Fingleton 1985) and a formal mathematical definition is given by Green and Sibson (1978). Dirichlet tessellations can be thought of as representing the areas of territorial animals, allelochemic-producing plants, or the packing of cells in a tissue. Dirichlet (Voronoi) cells have been delineated by algorithms that use Delaunay triangles and circumscribing circles on a HITAC M-180 computer (Tanemura and Hasegawa 1980) or elimination of intersecting circles according to a set of rules (Honda 1978). vinsonlab.tamu.edu /former/john/papers/jae92a.htm   (6012 words)

 Correct calculation of Dirichlet polygon areas   (Site not responding. Last check: 2007-11-07) The Dirichlet cell, first proposed in 1850, has been useful in many scientific disciplines and thus is known under a variety of names including Voronoi, 1909, Thiessen, 1911, Wigner-Seitz, 1933, cell model, 1953, and S-mosaic, 1977 (Rogers 1964; Mead 1971; Rhynsburger 1973; Upton & Fingleton 1985; David 1988). More recently, a Dirichlet tessellation algorithm was developed to define colonization territories of bark beetles (Coleoptera: Scolytidae) under the bark of host trees (Byers 1992). However, calculations of the area of a Dirichlet cell are undervalued by about 10 to 30 percent. www.wcrl.ars.usda.gov /cec/z65-abs.htm   (974 words)

 Dirichlet mixtures In Section 6, we'll see that Dirichlet mixtures are superior to all the other regularizers examined, and that there is not much room for improvement to better regularizers. One way to view the posterior counts of Dirichlet mixtures is as a linear combination of pseudocount regularizers, where the weights on the combination vary from one sample to another, but the underlying regularizers are fixed. Thus a 9-component Dirichlet mixture for the amino acids will have 168 degrees of freedom: 9 pseudocount vectors with 20-components each and 9 mixture coefficients (whose sum can be normalized to 1.0). cse.ucsc.edu /research/compbio/html_format_papers/tr-95-11/node14.html   (343 words)

 Dirichlet conditions Dirichlet gave the necessary conditions for existence of a fourier spectra for any signal Dirichlet's condition is to have finite number of "jump discontinuities" for piecewise smooth function. From my understanding, Dirichlet's convergence depends on the property that you can cut the function into small monotonic(either increase or decrease) section, so that the average value at any point [f(x+)+f(x-)]/2 can be calculated. www.edaboard.com /ftopic93912.html   (418 words)

 Dirichlet Forms And Markov Processes: A Generalized Framework Including Both Elliptic And Parabolic Cases - Stannat ...   (Site not responding. Last check: 2007-11-07) We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dirichlet forms (cf. Dirichlet forms and Markov processes: a generalized framework including both elliptic and parabolic cases. citeseer.ist.psu.edu /370433.html   (547 words)

 Dirichlet characters   (Site not responding. Last check: 2007-11-07) The first zeros on the critical line of some Dirichlet L-series... Johan Bosman, Lenny Taelman On sums of sums of values of Dirichlet characters NA... Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet charac... www.scienceoxygen.com /math/324.html   (140 words)

 Dirichlet, Peter Gustav Lejeune --  Encyclopædia Britannica   (Site not responding. Last check: 2007-11-07) He taught at the universities of Breslau (1827) and Berlin (1828–55) and in 1855 succeeded Carl Friedrich Gauss at the University of Göttingen. More results on "Dirichlet, Peter Gustav Lejeune" when you join. German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic (unique factorization of every integer into a product of primes) to complex number fields. www.britannica.com /eb/article-9030603?tocId=9030603   (735 words)

 GETWGT - Dirichlet Mixture Estimation   (Site not responding. Last check: 2007-11-07) GETWGT is a set of routines for Dirichlet Mixture estimation. The main requirement in the design of this library was that there be a single routine with a simple interface through which the user interacts. This routine, called getwgt, accepts a set of nucleic acid counts, updates an internal Dirichlet mixture model, and returns its current estimate for the pseudocounts. orion.math.iastate.edu /burkardt/b_src/getwgt/getwgt.html   (297 words)

 Dirichlet Mixtures and other Regularizers   (Site not responding. Last check: 2007-11-07) The optimization problem for Dirichlet mixtures is quite difficult, since there are many local minima of similar quality, and it takes a lot of optimization to get the value of the local minimum determined well enough to distinguish it from the others. The component that 9-component mixtures have for "highly conserved residues" is also split into two components (one mainly for C and W, the other mainly for P and G---these should probably both be split further, to get down to components dominated by a single residue). The data used for training the Dirichlet mixtures described above is provided (in gzipped files of count vectors). www.soe.ucsc.edu /research/compbio/dirichlets   (1699 words)

 BDMLIB - Estimate Dirichlet Mixture Weights   (Site not responding. Last check: 2007-11-07) BDMLIB addresses the problem of determining the weights W that define a Dirichlet mixture. To generate "events", first one of the component Dirichlet PDF's is chosen based on the weights. Then the Dirichlet PDF, whose parameters ALPHA are known, is used to generate a probability vector P. This probability vector is then used with the multinomial PDF to generate an integer vector X. Sound confusing? orion.math.iastate.edu /burkardt/b_src/bdmlib/bdmlib.html   (280 words)

 LEJEUNE-DIRICHLET, PETER GUSTAV (1805 - 1859)   (Site not responding. Last check: 2007-11-07) Dirichlet proved that any arithmetic progression without trivial obstructions contains infinitely many primes. Dedekind edited Dirichlet's lectures on number theory and published these as Vorlesungen uber Zahlentheorie in 1863. It is noted by H. Edwards that: "Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." www.scs.uiuc.edu /~mainzv/exhibitmath/bkgd/dirichlet-bkgd.htm   (150 words)

 Bretton Woods Workshop   (Site not responding. Last check: 2007-11-07) The aim of this workshop is to explore the modern theory of Dirichlet series in several complex variables (multiple Dirichlet series) and to bring together experts in this and allied fields. During this period we will not have regularly scheduled talks, but conference participants will be encouraged to continue their mathematical interactions and take advantage of the proximity of so many researchers interested in the topic of multiple Dirichlet series. Function field example of a multiple Dirichlet Series, notes to a lecture by Gautam Chinta. sporadic.stanford.edu /bump/mds_workshop.html   (320 words)

 Abstract for Markov chain sampling methods for Dirichlet process mixture models''   (Site not responding. Last check: 2007-11-07) Markov chain methods for sampling from the posterior distribution of a Dirichlet process mixture model are reviewed, and two new classes of methods are presented. One new approach is to make Metropolis-Hastings updates of the indicators specifying which mixture component is associated with each observation, perhaps supplemented with a partial form of Gibbs sampling. These methods are simple to implement and are more efficient than previous ways of handling general Dirichlet process mixture models with non-conjugate priors. www.cs.utoronto.ca /~radford/mixmc.abstract.html   (270 words)

 Amazon.ca: Books: New Directions in Dirichlet Forms   (Site not responding. Last check: 2007-11-07) The theory of Dirichlet forms brings together methods and insights from the calculus of variations, sotchastic analysis, partial differential and difference equations, potential theory, Riemannian geometry and more. This book features contributions by leading experts and provides up-to-date, authoritative accounts on exciting developments in the field and on new research perspectives. the topics are featured, and it is demonstarted that the theory of Dirichlet forms provides the proper framework for exploring these connections. www.amazon.ca /exec/obidos/ASIN/0821810618   (208 words)

 Random Dirichlet functions   (Site not responding. Last check: 2007-11-07) We show that if one inserts random plus and minus signs before the coefficients of the power series of a function in the Dirichlet space, then the resulting series is almost surely a multiplier of the Dirichlet space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip_{1/2} of functions obeying a Lipschitz condition with exponent 1/2. Versions of our results hold for weighted Dirichlet spaces, and much of our work is carried out in this more general setting. www.mth.msu.edu /~shapiro/Pubvit/Downloads/RanDir/RanDir.html   (140 words)

 Amazon.ca: Books: The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics   (Site not responding. Last check: 2007-11-07) Amazon.ca: Books: The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics Top of Page : The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics www.amazon.ca /exec/obidos/ASIN/0821813846   (152 words)

 Solutions of stochastic partial differential equations considered as Dirichlet processes, Denis Laurent   (Site not responding. Last check: 2007-11-07) We consider the parabolic stochastic partial differential equation $$u(t,x)=\Phi(x)+\int_0^t Lu(s,x)+f(s,x,u(s,x),Du(s,x))\,\d s$$ $$+\int_0^t g_i(s,x,u(s,x),Du(s,x))\,\d B^i_s,$$/ \noindent where f and g are supposed to be Lipschitzian and L is a self-adjoint operator associated with a Dirichlet form defined on a finite- or infinite-dimensional space. We prove that it admits a unique solution which is a Dirichlet process and, thanks to Itô's formula for Dirichlet processes, we prove a comparison theorem. [5] Bouleau, N. and Hirsch, F. (1991) Dirichlet Forms and Analysis on Wiener Space, de Gruyter Stud. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.bj/1099579156   (467 words)

 Weighted Extended B-Spline Approximation of Dirichlet Problems Weighted Extended B-Spline Approximation of Dirichlet Problems: SIAM Journal on Numerical Analysis Vol. We describe a new finite element method which uses weighted extended B-splines on a regular grid as basis functions for solving Dirichlet problems on bounded domains in arbitrary dimensions. This web-method does not require any grid generation and can be implemented very efficiently. epubs.siam.org /sam-bin/dbq/article/37320   (116 words)

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# PCA's principle and available algorithms

If I understand correctly, PCA's principle is very simple:

1. Calculate data vectors' covariance matrix C.
2. Solve det(C - e*I) = 0, to find matrix C's eigenvalues e.
3. Calculate matrix C's eigenfunctions (from those eigenvalues).

FIRST: Is this description correct?

SECOND: Any algorithm for machine-solving of the polynomial equation det(C - e*I) = 0 ? I understand that this is a general math question (finding roots of a polynomial of degree n).

THIRD: Are there any simple implementations of PCA in C/C++?

## 1 Answer

This is not done by solving the polynomial equation, but rather by iterative operations on matrices. Start by having a look on

• the Nipals algorithm to compute only the first few PC this seems very simple and clear ;

• the QR decomposition if you want all PC (although it has never been very clear to me that it is or should be faster).

Of course there are plenty of codes for this, QR decomposition is in LAPACK, and with LAPACK you won’t have much problems to implement Nipals if you can’t google anything satisfying.

• Thanks.Is it possible that different decomposition methods give different sets of eigenvectors for the same matrix? Secondly, regarding QR decomposition: Are the eigenvectors the columns of the matrix Q? How do I find their eigenvalues? Jan 5 '12 at 20:47
• No, it is not as simple, you need to iterate. Start with $X_1$ (symmetric), compute $Q_1, R_1$ such that $X_1 = Q_1 R_1$, and let $X_2 = R_1 Q_1$ ; iterate (pass from $X_n = Q_n R_n$ to $X_{n+1} = R_n Q_n$), $X_n$ will converge to a diagonal matrix with the eigenvalues on the diagonal. The eigenvectors are the columns of $\prod_i Q_i$ (you don’t store all the $Q_i$, just compute the new product each time). Jan 5 '12 at 21:24
• Thanks. Just to make sure I understand and make it well: I'm using the Gram-Schmidt algorithm to perform QR decomposition. I should decompose the original matrix X1 (which is a covariance matrix and hence symmetric) to Q1*R1, then calculate a matrix X2 = R1*Q1, QR-decompose X2 to Q2*R2, calculate a matrix X3 = R2*Q2 and so on... until a diagonal matrix Xn is received? Thanks again. Jan 6 '12 at 10:26
• @YanRaf yes! I'll add an example in the answer in a moment. Jan 6 '12 at 12:38
• Finally I answered to your other question... Jan 6 '12 at 13:57


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## Section: New Results

### Broadcast Speedup in Vehicular Networks via Information Teleportation

In [32] presented this year at LCN our goal is to increase our understanding of the fundamental communication properties in urban vehicle-to-vehicle mobile networks by exploiting the self-similarity and hierarchical organization of modern cities. We use an innovative model called "hyperfractal" that captures the self-similarities of both the traffic and vehicle locations, and yet avoids the extremes of regularity and randomness. We use analytical tools to derive matching theoretical upper and lower bounds for the information propagation speed in an urban delay tolerant network (i.e., a network that is disconnected at all time, and thus uses a store-carry-and-forward routing model). We prove that the average broadcast time behaves as $n\left(1-\delta \right)$ (times a slowly varying function), where $\delta$ depends on the precise fractal dimension. Furthermore, we show that the broadcast speedup is due in part to an interesting self-similar phenomenon, that we denote as information teleportation. This phenomenon arises as a consequence of the topology of the vehicle traffic, and triggers an acceleration of the broadcast time. We show that our model fits real cities where open traffic data sets are available. The study presents simulations that confirm the validity of the bounds in multiple realistic settings, including scenarios with variable speed.


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## Example 8.5

$\Delta U=q+w$

Tiffany Cao 1D
Posts: 94
Joined: Fri Sep 29, 2017 7:03 am
Been upvoted: 1 time

### Example 8.5

For practice example 8.5 on page 274, why did we assume that delta U was 0 in the second step of part B?

Cam Bear 2F
Posts: 60
Joined: Thu Jul 27, 2017 3:01 am

### Re: Example 8.5

Delta U=0 for the isothermal expansion of an ideal gas.

Cam Bear 2F
Posts: 60
Joined: Thu Jul 27, 2017 3:01 am

### Re: Example 8.5

Path A is a reversible expansion and Part B is an irreversible expansion but both still take place isothermally.

Pooja Nair 1C
Posts: 55
Joined: Thu Jul 13, 2017 3:00 am

### Re: Example 8.5

Even as an ideal gas expands isothermally, its molecules continue to move at the same speed, so the kinetic energy doesn't change. Also the total potential energy of these molecules is the same because there aren't any intramolecular forces. Since neither the total kinetic energy nor the total potential energy changes, the internal energy of the gas doesn't change, making delta U = 0 for isothermal expansions of ideal gas.


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The set of all positive integers is the union of two disjoint subsets $\{f(1), f(2),\dots ,f(n),\dots\}$ and $\{g(1), g(2),\dots,g(n),\dots\},$ where $f(1) < f(2) <\dots < f(n) < \dots,$ and $g(1) < g(2) <\dots< g(n) < \dots,$ and $g(n) = f(f(n)) + 1$ for all $n \geq 1.$ What is the value of $g(1)?$

1. Zero
2. Two
3. One
4. Cannot be determined

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Math Help - About a basic property of the Laplace Transform...

1. About a basic property of the Laplace Transform...

I suppose is well known to you the so called 'Final value theorem' , connected to the Laplace Trasform:

If...

$f(s) = \mathcal{L} \{\varphi (t)\}$

... then is...

$\lim _{t \rightarrow \infty} \varphi (t) = \lim_{s \rightarrow 0} s\cdot f(s)$

... provided that both limits exist.

Let's take an example and consider the function...

$\varphi (t) = \sum_{n=0}^{\infty} \mathcal {U} (t-n)$ (1)

... where $\mathcal {U}$ is the so called 'Haeviside step function'. This function is represented here...

In this case...

$f(s) = \frac{1}{s\cdot (1-e^{-s})}$ (2)

... so that is...

$\lim_{t \rightarrow \infty} \varphi(t) = \lim_{s \rightarrow 0} \frac{1}{1-e^{-s}} = + \infty$ (3)

... and it seems to be all right. Let' try now with another function...

$\varphi (t) = \sum_{n=0}^{\infty} \mathcal {U} (t-n) - t$ (4)

... that is represented here...

It is quite obvious that such type of function doesn't have limit for $t \rightarrow \infty$. In this case is...

$f(s)= \frac{s -1 + e^{-s}}{s^{2}\cdot (1-e^{-s})}$ (5)

... and if we try to apply the 'Final value theorem', using three times the L'Hopital rule, we obtain...

$\lim_{s \rightarrow 0} s\cdot f(s) = \frac{1}{2}$ (6)

... and that's is a little surprising! ...

At this point the questions for You are...

a) did I make some mistrake?...
b) if not how to interpret all that?...

Any help will be gratefully accepted!...

Kind regards

$\chi$ $\sigma$

2. A carefull search has pemitted me to discover the mistake in my procedure: the correct diction of the 'Final value theorem' is...

If...

$f(s) = \mathcal{L} \{\varphi (t)\}$

... $\varphi(t)$ and $\varphi^{'}(t)$ are both Laplace transformable, and $s\cdot f(s)$ has no singularities of the $i\omega$ axis and in the right half plane, then ...

$\lim_{ t \rightarrow \infty} \varphi(t) = \lim_{ s \rightarrow 0} s\cdot f(s)$

.... where the limit on the right is taken along the positive $\sigma$ axis.

In both my examples $s\cdot f(s)$ did have infinite poles on the $i\omega$ axis so that ...

Kind regards

$\chi$ $\sigma$


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DESY Theory Seminar

# Some four- and five-loop results in QCD

## by Andreas Vogt

Europe/Berlin
SR2

### SR2

Description
We report on calculations in perturbative Quantum Chromodynamics (QCD)
and its generalization to Yang-Mills theories with fermions. We summarize
the present status of four-loop calculations of the splitting functions for the
scale dependence (evolution) of the quark and gluon distributions of hadrons.
Five-loop results are presented for the beta function, i.e., the running of
the coupling constant, for hadronic decays of the Higgs boson, and for the
lowest moments of the non-singlet splitting functions.