text:
# Visualization of data and simulations¶

In this notebook, we illustrate the visualization functions of petab.

[1]:

from petab.visualize import plot_data_and_simulation
import matplotlib.pyplot as plt

[2]:

folder = "example_Isensee/"

data_file_path = folder + "Isensee_measurementData.tsv"
condition_file_path = folder + "Isensee_experimentalCondition.tsv"
visualization_file_path = folder + "Isensee_visualizationSpecification.tsv"
simulation_file_path = folder + "Isensee_simulationData.tsv"

# function to call, to plot data and simulations
ax = plot_data_and_simulation(data_file_path,
condition_file_path,
visualization_file_path,
simulation_file_path)
plt.show()

Now, we want to call the plotting routines without using the simulated data, only the visualization specification file.

[3]:

ax_without_sim = plot_data_and_simulation(
data_file_path,
condition_file_path,
visualization_file_path)
plt.show()

We can also call the plotting routine without the visualization specification file, but by passing a list of lists as dataset_id_list. Each sublist corresponds to a plot, and contains the datasetIds which should be plotted. In this simply structured plotting routine, the independent variable will always be time.

[4]:

ax_without_sim = plot_data_and_simulation(
data_file_path,
condition_file_path,
dataset_id_list = [['JI09_150302_Drg345_343_CycNuc__4_ABnOH_and_ctrl',
'JI09_150302_Drg345_343_CycNuc__4_ABnOH_and_Fsk'],
['JI09_160201_Drg453-452_CycNuc__ctrl',
'JI09_160201_Drg453-452_CycNuc__Fsk',
'JI09_160201_Drg453-452_CycNuc__Sp8_Br_cAMPS_AM']])
plt.show()

Let’s look more closely at the plotting routines, if no visualization specification file is provided. If such a file is missing, PEtab needs to know how to group the data points. For this, five options can be used: * dataset_id_list * sim_cond_id_lis * sim_cond_num_list * observable_id_list * observable_num_list

Each of them is a list of lists. Again, each sublist is a plot and its content are either simulation condition IDs or observable IDs (or their corresponding number when being enumerated) or the dataset IDs.

We want to illustrate this functionality by using a simpler example, a model published in 2010 by Fujita et al.

[5]:

data_file_path = "example_Fujita/Fujita_measurementData.tsv"
condition_file_path = "example_Fujita/Fujita_experimentalCondition.tsv"

# Plot 4 axes objects, plotting
# - in the first window all observables of the  1st, 2nd, and 3rd simulation condition
# - in the second window all observables of the 1st, 3rd, and 4th simulation condition
# - in the third window all observables of the  1st, 4th, and 5th simulation condition
# - in the fourth window all observables of the 1st, 5th, and 6th simulation condition
plot_data_and_simulation(data_file_path, condition_file_path,
sim_cond_num_list = [[0, 1, 2], [0, 2, 3], [0, 3, 4], [0, 4, 5]])
plt.show()

/home/polina/Documents/Development/PEtab/petab/visualize/helper_functions.py:157: UserWarning: DatasetIds would have been available, but other grouping was requested. Consider using datasetId.
warnings.warn("DatasetIds would have been available, but other "

[6]:

# Plot 4 axes objects, plotting
# - in the first window  all observables of the simulation condition 'model1_data1'
# - in the second window all observables of the simulation conditions 'model1_data2', 'model1_data3'
# - in the third window  all observables of the simulation conditions 'model1_data4', 'model1_data5'
# - in the fourth window all observables of the simulation condition 'model1_data6'
plot_data_and_simulation(
data_file_path, condition_file_path,
sim_cond_id_list = [['model1_data1'], ['model1_data2', 'model1_data3'],
['model1_data4', 'model1_data5'], ['model1_data6']])
plt.show()

[7]:

# Plot 5 axes objects, plotting
# - in the first window  the 1st observable for all simulation conditions
# - in the second window the 2nd observable for all simulation conditions
# - in the third window  the 3rd observable for all simulation conditions
# - in the fourth window the 1st and 3rd observable for all simulation conditions
# - in the fifth window  the 2nd and 3rd observable for all simulation conditions
plot_data_and_simulation(
data_file_path, condition_file_path,
observable_num_list = [[0], [1], [2], [0, 2], [1, 2]])
plt.show()

[8]:

# Plot 3 axes objects, plotting
# - in the first window  the observable 'pS6_tot' for all simulation conditions
# - in the second window the observable 'pEGFR_tot' for all simulation conditions
# - in the third window  the observable 'pAkt_tot' for all simulation conditions

plot_data_and_simulation(
data_file_path, condition_file_path,
observable_id_list = [['pS6_tot'], ['pEGFR_tot'], ['pAkt_tot']])
plt.show()

[9]:

# Plot 2 axes objects, plotting
# - in the first window  the observable 'pS6_tot' for all simulation conditions
# - in the second window the observable 'pEGFR_tot' for all simulation conditions
# - in the third window  the observable 'pAkt_tot' for all simulation conditions
# while using the noise values which are saved in the PEtab files

plot_data_and_simulation(
data_file_path, condition_file_path,
observable_id_list = [['pS6_tot'], ['pEGFR_tot']],
plotted_noise='provided')
plt.show()


text:
Linear representation theory of projective general linear group of degree two over a finite field

Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: projective general linear group of degree two.
View linear representation theory of group families | View other specific information about projective general linear group of degree two

This article describes the linear representation theory of the projective general linear group of degree two over a finite field. The order (size) of the field is $q$, and the characteristic prime is $p$. $q$ is a power of $p$. The group is denoted $PGL(2,q)$ or $PGL_2(q)$.

See also the linear representation theory for: special linear group, projective special linear group, and general linear group.

Summary

Item Value
degrees of irreducible representations over a splitting field Case $q$ odd: 1 (2 times), $q - 1$ ($(q - 1)/2$ times), $q$ (2 times), $q + 1$ ($(q - 3)/2$ times)
Case $q$ even: 1 (1 time), $q - 1$ ($q/2$ times), $q$ (1 time), $q + 1$ ($(q - 2)/2$ times)
number of irreducible representations Case $q$ odd: $q + 2$, case $q$ even: $q + 1$
See number of irreducible representations equals number of conjugacy classes, element structure of projective general linear group of degree two over a finite field#Conjugacy class structure
quasirandom degree (minimum degree of nontrivial ireducible representation) 1
maximum degree of irreducible representation $q + 1$
lcm of degrees of irreducible representations Case $q$ odd: $q(q + 1)(q - 1)/2 = (q^3 - q)/2$; Case $q$ even: $q(q+1)(q-1) = q^3 - q$
sum of squares of degrees of irreducible representations $q(q + 1)(q - 1) = q^3 - q$, equal to the group order; see sum of squares of degrees of irreducible representations equals group order

Particular cases

$q$ (field size) $p$ (underlying prime, field characteristic) Case for $q$ Group $PGL(2,q)$ Order of the group ($= q^3 - q$) Degrees of irreducible representations (ascending order) Number of irreducible representations ($= q + 1$ if $q$ even, $q + 2$ if $q$ odd) Linear representation theory page
2 2 even symmetric group:S3 6 1,1,2 3 linear representation theory of symmetric group:S3
3 3 odd symmetric group:S4 24 1,1,2,3,3 5 linear representation theory of symmetric group:S4
4 2 even alternating group:A5 60 1,3,3,4,5 5 linear representation theory of alternating group:A5
5 5 odd symmetric group:S5 120 1,1,4,4,5,5,6 7 linear representation theory of symmetric group:S5
7 7 odd projective general linear group:PGL(2,7) 336 1,1,6,6,6,7,7,8,8 9 linear representation theory of projective general linear group:PGL(2,7)
8 2 even projective special linear group:PSL(2,8) 504 1,7,7,7,7,8,9,9,9 9 linear representation theory of projective special linear group:PSL(2,8)
9 3 odd projective general linear group:PGL(2,9) 720 1,1,8,8,8,8,9,9,10,10,10 11 linear representation theory of projective general linear group:PGL(2,9)

Irreducible representations

Case $p \ne 2$, $q$ odd

Description of collection of representations Parameter for describing each representation How the representation is described Degree of each representation Number of representations Sum of squares of degrees
Trivial -- $x \mapsto 1$ 1 1 1
Sign representation -- Kernel is projective special linear group of degree two, image is $\{ \pm 1 \}$ 1 1 1
Unclear a nontrivial homomorphism $\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast$, with the property that $\varphi(x)^{q + 1} = 1$ for all $x$, and $\varphi$ takes values other than $\pm 1$. Identify $\varphi$ and $\varphi^q$. unclear $q - 1$ $(q - 1)/2$ $(q - 1)^3/2 = (q^3 - 3q^2 + 3q - 1)/2$
Nontrivial component of permutation representation of $PGL_2$ on the projective line over $\mathbb{F}_q$ -- -- $q$ 1 $q^2$
Tensor product of sign representation and nontrivial component of permutation representation on projective line -- -- $q$ 1 $q^2$
Induced from one-dimensional representation of Borel subgroup $\alpha$ homomorphism $\mathbb{F}_q^\ast \to \mathbb{C}^\ast$, with $\alpha$ taking values other than $\pm 1$, up to inverses. Induced from the following representation of the image of the Borel subgroup: $\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\alpha(d)^{-1}$ $q + 1$ $(q - 3)/2$ $(q + 1)^2(q - 3)/2 = (q^3 - q^2 -5q - 3)/2$
Total NA NA NA $q + 2$ $q^3 - q$

format: plain


text:
Documentation

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doppler.rounded

Construct rounded Doppler spectrum object

Syntax

`dop = doppler.roundeddop = doppler.rounded(coeffrounded)`

Description

The `doppler.rounded` function creates a rounded Doppler spectrum object that is used for the `DopplerSpectrum` property of a channel object (created with either the `rayleighchan` or the `ricianchan` function).

`dop = doppler.rounded` creates a rounded Doppler spectrum object with default polynomial coefficients ${a}_{0}=1$, ${a}_{2}=-1.72$, ${a}_{4}=0.785$ (see Theory and Applications for the meaning of these coefficients). The maximum Doppler shift ${f}_{d}$ (in Hertz) is specified by the `MaxDopplerShift` property of the channel object.

`dop = doppler.rounded(coeffrounded)`, where `coeffrounded` is a row vector of three finite real numbers, creates a rounded Doppler spectrum object with polynomial coefficients, , given by `coeffrounded(1)`, `coeffrounded(2)`, and `coeffrounded(3)`, respectively.

Properties

The rounded Doppler spectrum object contains the following properties.

PropertyDescription
`SpectrumType`Fixed value, `'Rounded'`
`CoeffRounded`Vector of three polynomial coefficients (real finite numbers)

Theory and Applications

A rounded spectrum is proposed as an approximation to the measured Doppler spectrum of the scatter component of fixed wireless channels at 2.5 GHz [1]. However, the shape of the spectrum is influenced by the center carrier frequency.

The normalized rounded Doppler spectrum is given analytically by a polynomial in f of order four, where only the even powers of f are retained:

where

`${C}_{r}=\frac{1}{2{f}_{d}\left[{a}_{0}+\frac{{a}_{2}}{3}+\frac{{a}_{4}}{5}\right]}$`

${f}_{d}$ is the maximum Doppler shift, and are real finite coefficients. The fixed wireless channel model of IEEE 802.16 [1] uses the following parameters: ${a}_{0}=1$, ${a}_{2}=-1.72$, and ${a}_{4}=0.785$. Because the channel is modeled as Rician fading with a fixed line-of-sight (LOS) component, a Dirac delta is also present in the Doppler spectrum at $f=0$.

Examples

The following code creates a Rician channel object with a maximum Doppler shift of ${f}_{d}=10$. It then creates a rounded Doppler spectrum object with polynomial coefficients ${a}_{0}=1.0$, ${a}_{2}=-0.5$, ${a}_{4}=1.5$, and assigns it to the `DopplerSpectrum` property of the channel object.

```chan = ricianchan(1/1000,10,1); dop_rounded = doppler.rounded([1.0 -0.5 1.5]); chan.DopplerSpectrum = dop_rounded;```

References

[1] IEEE 802.16 Broadband Wireless Access Working Group, "Channel models for fixed wireless applications," IEEE 802.16a-03/01, 2003-06-27.

Introduced in R2007a

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is a display equation.

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text:
# Definition:Decimal Notation

Jump to navigation Jump to search

## Definition

Decimal notation is the quotidian technique of expressing numbers in base $10$.

Every number $x \in \R$ is expressed in the form:

$\ds x = \sum_{j \mathop \in \Z} r_j 10^j$

where:

$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$

## Examples

### Decimal Number $209$

In decimal notation $209$ means:

 $\ds 209$ $=$ $\ds 2 \times 10^2 + 0 \times 10^1 + 9 \times 10^0$ $\ds$ $=$ $\ds 200 + 0 + 9$

### Decimal Number $4129$

In decimal notation $4129$ means:

 $\ds 4129$ $=$ $\ds 4 \times 10^3 + 1 \times 10^2 + 2 \times 10^1 + 9 \times 10^0$ $\ds$ $=$ $\ds 4000 + 100 + 20 + 9$

## Historical Note

The earliest known discussion of the system of decimal notation is found in a work by Bhaskara II Acharya from the $12$th century.

However, there is evidence to suggest that some form of decimal notation was known about for some $500$ years before that.


text:
My watch list
my.chemeurope.com

# Micro heat exchanger

Micro heat exchangers, Micro-scale heat exchangers, or microstructured heat exchangers are heat exchangers in which (at least one) fluid flows in lateral confinements with typical dimensions below 1 mm. The most typical such confinement are microchannels, channels with a hydraulic diameter below 1 mm.

## Background

Investigation of microscale thermal devices is motivated by the single phase internal flow correlation for convective heat transfer:

$h=Nu_c \frac{k}{d}$

Where h is the heat transfer coefficient, Nuc is the Nusselt number, k is the thermal conductivity of the fluid and d is the hydraulic diameter of the channel or duct. In internal laminar flows, the Nusselt number becomes a constant. This is a result which can be arrived at analytically: For the case of a constant wall temperature, Nuc = 3.657 and for the case of constant heat flux Nuc = 4.364[1]. As Reynolds number is proportional to hydraulic diameter, fluid flow in channels of small hydraulic diameter will predominantly be laminar in character. This corellation therefore indicates that the heat transfer coeficient increases as channel diameter decreases. Should the hydraulic diameter in forced convection be on the order of tens or hundreds of micrometres, an extremely high heat transfer coefficient should result.

This hypothesis was initially investigated by Tuckerman and Pease [2]. Their positive results have since fueled a hot field of research ranging from classical investigations of single channel heat transfer [3] to more applied investigations in parallel micro-channel and micro scale plate fin heat exchangers. Recent work in the field has focused on the potential of two-phase flows at the micro-scale. [4][5][6]

## Classification of micro heat exchangers

Just like "conventional" or "macro scale" heat exchangers, micro heat exchangers have either one or two fluidic passages. In the case of one passage, heat is transferred to the fluid [7] from electrically powered heater cartridges, or removed from the fluid by electrically powered elements like Peltier chillers. In the case of two fluidic passages[7], micro heat exchangers are usually classified by the orientation of the fluid passages to another as "cross flow" or "counter flow" devices. If a chemical reaction is conducted inside a micro heat exchanger, the latter is also called a microreactor.

## Notes and references

1. ^ Incropera & Dewitt text - more complete reference later
2. ^ Tuckerman D.B., and Pease, R.F., High Performance Heat Sinking for VLSI, IEEE Electron Device Letters, vol. EDL-2, pp126-129, 1981
3. ^ Santiago, Kenny, Goodson, Zhang - more complete reference later
4. ^ Yen, Tzu-Hsiang; Kasagi, Nobuhide; Suzuki, Yuji; Forced convective boiling heat transfer in microtubes at low mass and heat fluxes; Int'l J Multiphase Flow; 29; 2003; pp.1771-1792
5. ^ Steinke, M. E., and Kandlikar, S. G., “An Experimental Investigation of Flow Boiling Characteristics of Water in Parallel Microchannels,” Journal of Heat Transfer, 126, no. 4, pp. 518-526, 2004.
6. ^ Mudawar - more complete reference later
7. ^ a b each of the fluids can be a gas, a liquid, or a multiphase flow


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Question: Evaluate $\int \frac{z^2-2z+4}{z^2-1} dz$ over the curve C where C is the circle |z-1|=1
0

Subject: Applied Mathematics 4

Topic: Complex Integration

Difficulty: Medium

m4e(34) • 122 views
 modified 10 weeks ago  • written 5 months ago by
0

$f(z) = \frac{z^2 - 2z + 4}{z^2 - 1}$

For poles z$^2$ - 1 = 0; (z-1)(z+1) = 0

Therefore, z = 1,-1 are the two poles of which z = 1 is the only pole which lies inside.

Residue of f(z) at z = 1 is:

$\lim_{z \to 1} (z-1) \frac{z^2 - 2z + 4}{(z-1)(z+1)} = \frac{1-2+4}{1+1} = \frac{3}{2} \\ \therefore \int_c \frac{z^2 - 2z + 4}{z^2 - 1} \,\, dz = 2 \pi i \frac{3}{2} = 3 \pi i$


text:
## 30 May 2013

### Talk: 'Hydrodynamical limits for the fractional diffusion/Stokes equation'

What do the trajectories of pollen particles suspended in water and the evolution of heat in a room have in common? These observable (macroscopic) physical phenomena are related through their microscopic dynamics. This link and its generalisations are based on standard mathematical objects from PDE and Probability theory: (fractional) diffusion equation, random walks, Brownian motion and Levy processes.

I will present a recent result concerning fractional hydrodynamical limits. Starting from a linear kinetic equation (which describes the microscopic dynamics), we derive a fractional Stokes equation governing the associated macroscopic quantities (mass, flux and temperature).

This is joint work with Sabine Hittmeir from Technische Universitat of Vienna.

Keywords: fractional laplacian, super-diffusion, hydrodynamical limit, non-locality, Levy processes, heavy-tailed distribution, anomalous transport, scale-invariance, Stokes equation.

## Kinetic Description of Multiscale Phenomena 17th-28th June 2013

#### Heraklion, Crete

The meeting intends to address questions relating to multi–scale modelling, kinetic modelling and the interactions between microscopic structure on the one hand and effective equations for its description at a macroscopic scale on the other.

http://www.acmac.uoc.gr/KDM2013/index.php

## Conference on "Mathematical topics in Kinetic Theory" 17th-21st June 2013

#### Cambridge, UK

There have been many recent progresses in the last decade in the mathematics of kinetic theory. The field is developping rapidly and many more are to come. One of the notable feature of these progresses is the interplay between different communities. This workshop aims at bringing together experts in theoretical PDE's, numerical PDE's, modelling and probabilistic aspects of kinetic theory to share and foster these advances. Cambridge in the middle of june will provide a nice and sunny environnement for this workshop.

This conference aims therefore at presenting some challenging developments and perspectives in these fields. It aims also at introducing students and young researchers to the fascinating questions open by these topics.

http://math.univ-lyon1.fr/homes-www/filbet/nusikimo/CBG2013/

## 25 August 2012

### What do I do? (and why is it important?)

Note: I will add corrections and improvements depending on feedback.

Mathematicians, those adorable and nerdy creatures... Not many people know what they actually do... or even if what they do is useful, but almost everybody has a mental picture of what they look like. Some people imagine bearded men walking aimlessly in circles while muttering words to themselves; others picture men with thick glasses making sums and multiplications all day long with a powerful mental skill; the most generous ones think of 'beautiful minds'.

I am a mathematician myself, my name is Sara Merino and currently I carry out a PhD in Mathematics. Just to answer some of the questions that people usually ask me: no, I do not work with numbers all day long, actually I 'see' numbers very rarely... basically when I pay my bills; I work with equations... not of this kind...

$(x + 1)^2 = x^2 + 2x + 1\,$

but more of this kind...

$\frac{\partial f_i}{\partial t} + \frac{\mathbf{p}_i}{m_i}\cdot\nabla f_i + \mathbf{F}\cdot\frac{\partial f_i}{\partial \mathbf{p}_i} = \left(\frac{\partial f_i}{\partial t} \right)_\mathrm{coll}$

and, no, my ultimate goal is not to obtain a number. No, I do not do a PhD because I want to become a high school teacher, but because I want to do research. No, not everything has been discovered in Mathematics. Actually there is still a lot to be discoreved. I hope I do not need to tell you that I am not a man. And yes,... I wear thick glasses.

#### INTRODUCTION

• In this entry I want to give you a flavour of the kind of problems I try to solve and the mathematical tools I use, namely differential equations and Probability. Don't worry, I will not get technical, I promise not to show any symbol... except for stetical purposes... Let's get to it!
• I work in a field of Mathematical Physics called Statistical Mechanics. In this document we will see how Statistical Mechanics was born to solve problems that Classical Mechanics could not solve and how Mathematics played a fundamental role.

#### MATHEMATICAL MODELING: Patterns in Physics

Physicists recognise patterns in nature and describe them mathematically to make predictions.
• For example, in Classical Mechanics, through Newton's equations, we can predict the trajectories of the planets around the sun.

The mathematical tool used to describe the physical law governing the orbits is called differential equation. Newton's equations are just a particular instance of a differential equation.
• What is a differential equation? How does a differential equation works? A differential equation is a special type of equation. To explain how a differential equation works lets take as an example Newton's one applied to the movements of the planets.
• A differential equation requires certain information, in our example, the position and velocities of the planets at a given time. With this information, if the differential equation can be “solved1, it provides the positions and velocities of the planets in the future, i.e., it predicts their trajectories.
• Lets consider now another example. With Newton's equations, it can also be modelled the behaviour of a gas. However, it is impossible to make predictions from them. Why is this so?

• There is a problem of lack of information when we study the evolution of a gas; we need to know the position and velocity of the particles at a given time to make the prediction, but this measurement is technically impossible. Moreover, even if we could make the measurements, the differential equations are so complex (due to the large amount of particles) that cannot be studied mathematically.
Summarizing, Newton's equations work well to predict the planetary movement but it becomes intractable when studying a gas. It does not mean that Newton's model is wrong, simply that it is not practical for the study of a gas.
• Since we are lacking information, we have to work with guesses. This is how a new mathematical field, Probability, entered the study of Physics. We can predict general features of physical phenomena despite lacking information.
• For example, we do not know the outcome of tossing a coin, but we know that if we toss it a lot of times, roughly half of the time we will get tails and the other half, heads.
Ludgwig Boltzmann founded Statistical Mechanics by studying the dynamics of gases using this new approach based on Probability and random (or stochastic) models.
 Ludwig Boltzmann

• Thanks to Probability and stochastic models, specially one called Brownian motion, we can not only study a gas but also other physical phenomena like the erratic trajectories of nano particles in water
or the following sound which is known as white noise.
• What is a stochastic model or process? In mathematics, which is the difference between a deterministic process and a stochastic one?
• A deterministic process is, for example, when we know exactly the trajectory of a particle; so Newton's laws state that a particle will move in a straight line at a constat velocity if there are no other interactions with the particle. A stochastic or random process would be one in which we cannot know exactly how the particle will move but we know some properties of its behaviour.
• An example of stochastic process is the so-called random walk, which is the following: imagine that you want to take a walk. You allow yourself to only move to the left or to the right, one step at a time, and to determine in which direction to go, you toss a coin; if it is head, you turn right; if it is tails, you turn left. At the beginning of your walk, you do not know which is the path that you are going to take, but, roughly, half of the times you will turn right, and the other half you will turn left. Your trajectory is the stochastic process called random walk, and mathematicians study this kind of processes and are able to prove properties about them. We will come back to stochastic processes when we talk about a very special one that we have mentioned before, Brownian motion, which is a generalisation of a random walk.
• Thanks to this new mathematical tools and the ideas behind them, Boltzmann entered a new conception in Physics with which he was able to explain, among other things, why the world is irreversible, namely, why we move from the past to the future without the time never going backwards. We will see this later, after explaining the new model that Boltzmann proposed for the study of gases, called Boltzmann equation.

THE BOLTZMANN EQUATION AND THE DIFFERENT SCALES OF DESCRIPTION
What is the Boltzmann equation? What is this model different from Newton's equations?
• To understand what the Boltzmann equation is, we need to put it into a context. The Boltzmann equation is a point of view. Let me explain this. If we observe each particle of a gas with its exactly position and velocity, then we use Newton's equations. However, not always we want to have so much detail; sometimes we just want to know the general behaviour of the gas, namely, what can be observe by the naked eye. For that, we have hydrodynamical equations. The difference between the two models is the point of view of description; Newton's equations have all the detailed information of the microscopic system, while hydrodynamical equations is a rough description of what we observe. Nevertheless, keep in mind that the physical phenomena is the same; the dynamics of a gas. And here is where the Boltzmann equation comes in; it is a model between these two levels of description. Instead of knowing exactly what which particle does, we know the proportion of particles that does it; so this model gives less information than the Newton's model but more information than the hydrodynamical ones, you could think of it is as a blurry image of Newton's model.

• Remember that all this started because in Classical Mechanics we have lack of information, the Boltzmann equation deals with less information by working with proportions (or sets) of particles instead of dealing with the exact particles.

WHY IS IT IMPORTANT? Applications
The Boltzmann equation has important practical and theoretical applications.
• Some of the practical applications are in aeronautics at high altitude or interactions in dilute plasmas. Also, it allows to make predictions in specific situations in which the ones provided by hydrodynamical equations are not accurate enough. For more information on the practical applications look at the book of Cercignani 'Rarefied Gas Dynamics'.
• The theoretical applications of the Boltzmann equation help us to understand better the world. Here is an example.
Thanks to his probabilistic approach, Boltzmann was able to give an explanation for the irreversibility in physical phenomena. Irreversibility is associated with the fact that time goes in one direction, hence we cannot go back to the past.
• For example, a manifestation of irreversibility in the physical world is the box with two types of sugar [reference here]. Imagine that you have a box with the lower part filled with white sugar and the upper part filled with brown sugar. If we shake the box for a while, we expect the two types of sugar to mix uniformly. We will not expect that, if we keep shaking, at some point we will have the initial configuration of brown sugar on top, white sugar at the bottom, i.e., the process will not reverse to its initial state2.
• In the same way, irreversibility appears when observing a gas. For example, in this video we have a box divided in two. In each side there are gas particles at different temperature (and color). When the wall disappears between the two compartments, we expect the blue and red particles to start mixing, becoming in the end, homogeneously distributed in the room, reaching an equilibrium and making the temperature of the box uniform. We do not expect to have again, in the future, the blue particles on the left-hand side and the red ones on the right-hand side, i.e., we do not expect reversibility. However, Newton's laws tell us that that is possible.

• Newton's equations are reversible, meaning that if we invert the velocities of the gas particles at a given time, then they will go back to its initial position; it will look like time runs backwards. However, this does not happens with the Boltzmann equation; it is not reversible.
• Newton's equation and the Boltzmann equation are models for the same physical phenomena, but the first is reversible and the second not. How can this apparent contradiction be explained?
• Boltzmann explained it using, as we said, Probability. In Classical Mechanics everything is deterministic and a particular phenomena is possible or impossible to happen. In Statistical Mechanics, since we work with uncertainties, the concepts of possible and impossible are transformed into probable and highly improbable. In this way, to observe reversibility becomes highly improbable but not impossible.
• How did Boltzmann use this difference of concept to explain the irreversibility that we observe around us? He said that the number of configurations, i.e., the number of possible positions and velocities of the particles that make us observe, to the naked eye, uniformity of particles or equilibrium, is infinitely bigger than the number of microscopic configurations that will make us observe reversibility. Hence, it is much more probable that the configuration of the particles 'fall' into one that will make us observe equilibrium than one that make us observe reversibility.
To make an analogy, imagine that we toss a coin and let it fell to the floor. We always consider the outcome to be heads or tails, however, there is another possibility: that it stands on its edge. The probability of that is so low that we do not consider it; we do not expect to experience it. In the same sense, expecting to observe reversibility is like expecting to get the coin on its edge; not impossible, but highly improbable.
• On one hand, Newton's equations, since they have all the possible information of a gas, consider all the microscopic configurations (in the analogy, it considers also the possibility of getting the coin on its edge). On the other hand, Boltzmann's equation does not have all the information and, hence, gathers together different microscopic states that give the same macroscopic picture and consider only the macroscopic pictures that are highly probable to happen, i.e., the ones that reach an equilibrium (in the analogy, in discards the possibility of getting the coin on its edge); this makes his equation non reversible.
• Thanks to the introduction of Probability, Boltzmann was able to explain physical phenomena that could not be explained in Classical Mechanics, like irreversibility, existence of equilibrium and entropy3.

Here you have an excellent clip in which Brian Cox explains the concept of entropy.

GOOD MATHEMATICAL MODELS? THE PROBLEM OF COHERENCE (Hilbert's 6th problem)
Which is the kind of questions I am trying to solve? Now that I have explained you all this. Let me explain you which are the kind of questions I am trying to give an answer to.
• Remember that physicists recognise patterns in nature and find mathematical models to describe them and make predictions. Afterwards, mathematicians have to analyse these models to check their coherence, validity and information that can be obtained from them.
• Let's go back to the gas dynamics and the different mathematical models that we have for it. We have different mathematical models at different levels of description, namely, Newton's, Boltzmann's and hydrodynamical equations. Each model, though, was derived independently from each other using physical intuition. However, if the models are correct, we expect some coherence between them since the physical phenomena that they model is the same; the dynamics of a gas.
This coherence between the models means that we expect to be able to derive, mathematically, the models at a larger scale from the ones at a lower scale; the behaviour of atoms determines what we observe by the naked eye. This is called Hilbert's 6th problem, proposed by Hilbert, one of the greatest mathematicians of the XX century in the International Congress of Mathematics in 1900.
• Partial answers to Hilbert's 6th problem have been given and I am currently working in this direction; I am trying to derive hydrodynamical models from the models in Statistical Mechanics. For example, it has been proven that a simplified version of the Boltzmann equation derives at macroscopic level into a Heat Equation4, which is the equation that models how the temperature in a room evolves over time.
• The tools to prove this link are differential equations and Probability.
• Allow me to give you a small flavour of how this link between the models was proven. As we saw before, in Probability, we use random processes, like the Brownian motion and Stochastic differential equations, which are the analog of differential equations for random processes instead of deterministic processes.
• Brownian motion is a generalization of the random walk that we saw before. In the plane (two dimensions), it will look as follows. Imagine that, instead of walking only to right or left, we also move forwards of backwards, one step at a time, and we decide which direction to take randomly, having each direction the same probability to happen. The video here shows one possible trajectory that such random walk could produce. This is approximately, a Brownian motion. It has been seen that the trajectory of a particle which follows the Heat equation corresponds to a Brownian motion.
• The derivation of the Heat equation from the simplified Boltzmann equation is done using Brownian motion. Observe the following video in which appears a gas with a singled out particle. The trajectory followed by the singled out particle seen at a larger scale and speeding up in time produces a Brownian motion, which corresponds, as we just said, to the trajectory of a particle under the Heat equation.

• This kind of problems are fundamental, among other reasons, because the models need to be validated, i.e., we need to check their correctness; that they provide a good description of the physical phenomena. For example, there was a huge controversy when Boltzmann presented his equation. An important part of the scientific community, including Poincare [add link], did not accept his model.
• Boltzmann had a hard time defending his theory. However, if Boltzmann would have obtained his equation from Newton's one, there would have been no controversy and would have been able to explain, from the very beginning, the apparent incoherences that appeared in his theory, including the irreversibility of his equation, that we have mentioned before5.

SUMMARY
• Summarizing, to describe and predict the physical world around us, physicists use mathematical models. Newton's equations, in Classical Mechanics, are a particular type of model called differential equation. It is based on deterministic processes and has proven to be very useful to describe particular physical phenomena, like the planetary movements. However, differential equations requires an initial amount of information that cannot be provided in particular physical systems, like when studying a gas. To work with this lack of information, Boltzmann proposed a new model based on random processes instead of deterministic ones where the lack of information was dealt with the use of Probability.
• The Boltzmann equation has proven to be both, practical and theoretically, useful for physicists and engineers. For example, by introducing Probability and random processes to the study of Physics, Boltzmann provides a new conception in which he can explain phenomena like the irreversibility in our world.
• How can we be sure that a model is “correct”? The mathematical derivation of models having less information from the ones having more information (Hilbert's 6th problem) is fundamental towards the understanding of these models and proving their validity. Mathematicians have been able to do so for some particular cases. The use of probabilistic tools, like Brownian motion, help us to make and understand the link between these models.

I hope you have enjoyed this because a consequence of irreversibility is that...

Cercignani. 'Lugwig Boltzmann. The man who trusted atoms.'
Wonderful article by Clement Mouhot (in French): "Les surprenantes propriétés des plasmas. Ou comment l’irréversible émerge du réversible..."

1Differential equations can be solved in few cases. In the major part of the cases, it is necessary to carry out, on one hand, a mathematical analysis to find quantitative and qualitative properties of the equations, and, on the other hand, computer simulations to approximate the solution of the equation.
2This example was given by someone else to explain the concept of entropy. I think that the author is Cercignani, though I am not sure.
3Entropy is a fundamental concept in the theory of thermodynamics. Due to lack of space, we do not deal here with it.
5The interested reader can look for 'Locksmith Paradox' and 'Poincaré recurrence theorem'.

## 21 April 2011

### Days on PDEs 2011

Description:
The aim of this international conference is to have every year an overview of the most striking advances in PDEs. Moreover, a 6h course by a first class mathematician will be given. Another important role of this conference is to promote young researchers. The organization participates in particular to the local expanses of PhD students and postdocs. Let us finally mention that the proceedings of this conference are published since 1974.

Mini course (6h):
Cédric Villani (Université de Lyon)
"Régularité du transport optimal et géométrie riemannienne lisse et non lisse"

Speakers:
Hajer Bahouri (Paris 12)
Massimiliano Berti (Naples)
Nicolas Burq (Paris XI)
Benoît Desjardins (ENS Paris)
Benjamin Dodson (Berkeley)
Rupert Frank (Princeton)
Camille Laurent (Ecole Polytechnique)
Michel Ledoux (Toulouse)
Claudio Munoz (Bilbao)
Stéphane Nonnenmacher (CEA Saclay)
Felix Otto (Institut Max Planck, Leipzig)
Igor Rodnianski (Princeton)
Frédéric Rousset (Rennes)
Benjamin Schlein (Bonn)


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# Tag Info

11

Based on the documentation cosine_similarity(X, Y=None, dense_output=True) returns an array with shape (n_samples_X, n_samples_Y). Your mistake is that you are passing [vec1, vec2] as the first input to the method. Also your vectors should be numpy arrays: from sklearn.metrics.pairwise import cosine_similarity import numpy as np vec1 = np.array([[1,1,0,1,1]]...

10

The cosine distance formula is: And the formula used by the cosine function of the spatial class of scipy is: So, the actual cosine similarity metric is: -0.9998. So, it signifies complete dissimilarity.

9

Short answer: Cosine distance is not the overall best performing distance metric out there Although similarity measures are often expressed using a distance metric, it is in fact a more flexible measure as it is not required to be symmetric or fulfill the triangle inequality. Nevertheless, it is very common to use a proper distance metric like the Euclidian ...

8

There's a related example to your problem in the Spark repo here. The strategy is to represent the documents as a RowMatrix and then use its columnSimilarities() method. That will get you a matrix of all the cosine similarities. Extract the row which corresponds to your query document and sort. That will give the indices of the most-similar documents. ...

4

Your input matrices (with 3 rows and multiple columns) are saying that there are 3 samples, with multiple attributes. So the output you will get will be a 3x3 matrix, where each value is the similarity to one other sample (there are 3 x 3 = 9 such combinations) If you were to print out the pairwise similarities in sparse format, then it might look closer to ...

3

Similarity measures are subjective and so are they ways to combine them. You should decide what is your subjective definition of similarity and then find a way to combine them that fit your definition. In general, I like to reduce similarity problems into classification problems. Given the dataset of items you have, create a new dataset of item pairs. The ...

2

If you have trained a gensim model, that object can act as a dictionary to give you the vector projection (via https://radimrehurek.com/gensim/models/word2vec.html) $model['computer'] # raw numpy vector of a word array([-0.00449447, -0.00310097, 0.02421786, ...], dtype=float32) So it is possible to manually implement any vector comparison that you ... 2 "Source credibility" of Internet articles is best calculated through the Page Rank algorithm. Algorithmically determining writing quality might be intractable. However Page Rank could be a proxy. If an article is a hub then it is the authority on the topic and can be assumed well written (or at least very useful). 2 Let us try and understand how Word2Vector actually works before looking at distances: There are 2 ways of generating vectors for a word : Continuous bag of words Skip grams The following diagram explains the difference between the two approaches. In case you want to further understand the nitty gritty of these two approaches, there are tons of blogs out ... 2 My question is, do I need to normalize each product's vector before using columnSimilarities()? No, you do not need to normalize each product's vector before using columnSimilarities() since it is performed within the operation already. I think your confusion comes from the fact that your considering dot product and cosine similarity to be the same. They ... 2 Since the time-series are annual, the data points you have for each time-series are limited and also quite distant (the values are 1 year apart). So I wouldn't use Dynamic Time Wrapping on your data. If you are interested in comparing the patterns, a very simple approach would be Pearson's correlation. Keep in mind that this will not compare the actual ... 2 As mentioned in other answers, traditionally cosine is used to measure similarity between vectors whereas Levenshtein is used as a string similarity measure, i.e. measuring the distance between sequences of characters. Nevertheless they both can be used in non-traditional settings and are indeed comparable: the vectors compared with cosine can for instance ... 2 If I understand correctly, you're trying to map abstracts to their research papers. Here is a simple starting point: Compute a TF IDF model using the entire corpus (all abstracts + research papers). Use this model to transform your abstracts and research papers into a weighted vector representation. Under the TF IDF weighting scheme, these documents will ... 2 There are libraries that are specialized in exactly that task, for instance FAISS by Facebook AI Research: Faiss is a library for efficient similarity search and clustering of dense vectors. It contains algorithms that search in sets of vectors of any size, up to ones that possibly do not fit in RAM. It also contains supporting code for evaluation and ... 1 @lsbister, You could create a pandas dataframe and use a dask function/lambda function to parellize the computation of one vs all at the same time. If you use dask, you can create partitions and map the response back. In case you use pandas, you can use the apply function and parallelize the computations to a certain extent. 1 To the extent possible you should try to evaluate based on your data rather than some ad-hoc measure. As you rightly noticed, there is a real risk that the ad-hoc measure would just confirm the predictions of the model, since it uses a somewhat similar method. I would suggest that you split your data between a training set and test set (or even better use ... 1 The first one is for computing the similarity between objects considering their representations as vectors. The second one is for computing the similarity between sequences of characters. 1 Yes, Cosine TF-IDF is quite transparent so it's usually reasonably easy to visualize the words which contribute the most to a score. Cosine is defined as the dot product divided by the product of the norms, so you can isolate the terms: dotproduct(d_1,d_2) = tfidf(w1,d1) * tfidf(w1,d2) + tfidf(w2,d1) * tfidf(w2,d2) + ... + tfidf(wN,dN) Ranking the words ... 1 If you are using Python try the fuzzywuzzy package: FuzzyWuzzy Fuzzy string matching like a boss. It uses Levenshtein Distance to calculate the differences between sequences in a simple-to-use package. (Source) 1 You can try some approximate string matching which gives a confidence score. For example, you can try out with Levenshtein distance, but adjusted with the length of the strings using a probabilistic model; or, you can try out with Jaccard similarity on 3-grams and special treatment on word boundaries, and then calibrated into probabilities. Now you have an n ... 1 Jaccard - measures similarity of assymetric, binary attributes. For example, if you have insurance claims with binary attributes ("poor driving record", "premium paid in cash") you can compare claims with those attributes. Cosine - measures similarity between vectors, like feature vectors. Could be used in a recommender system where a user asks to see items ... 1 as Lejafar mentioned cosine violates triangle inequality however maybe this repo will help you 1 A common and simple method to match "documents" is to use TF-IDF weighting, as you have described. However, as I understand your question, you want to rank each career (-document) based on a set of users skills. If you create a "query vector" from the skills, you can multiply the vector with your term-career matrix (with all the tf-idf weights as values). ... 1 Perhaps you could use word embeddings to better represent the distance between certain skills. For instance, "Python" and "R" should be closer together than "Python" and "Time management" since they are both programming languages. The whole idea is that words that appear in the same context should be closer. Once you have these embeddings, you would have a ... 1 You want to use all of the terms in the vector. In your example, where your query vector$\mathbf{q} = [0,1,0,1,1]$and your document vector$\mathbf{d} = [1,1,1,0,0]$, the cosine similarity is computed as similarity$= \frac{\mathbf{q} \cdot \mathbf{d}}{||\mathbf{q}||_2 ||\mathbf{d}||_2} = \frac{0\times1+1\times1+0\times1+1\times0+1\times0}{\sqrt{1^2+1^2+...

1

Filtering users will create bias in your training data. This may be good or bad, depending on your data and goals. The best way to find out, for your specific system, is to try and test both methods, optimizing for whatever metric you choose is best. Honestly, I think you should consider all users. Say you have a system for recommending movies. In this ...

1

As of now, I can think of two ways to formulate this problem: 1. Search problem Parse your job listings and index them in some sort of search engine like Solr or ElasticSearch. You can build capabilities like Semantic search using Word2Vec models, etc. Now write a query engine which takes a resume and queries this Search engine. It will be blazing fast ...

1

Yes, they are the same. The array [1, 0, 1, 0] represents a vector in $4$ dimensional euclidean space ($\mathbb{R}^4$) with tails at [0, 0 , 0 ,0] and head at [1, 0, 1, 0]. So your vector is "pointing" in the direction of your given set of coordinates. This might be easier to visualize in $2$ dimensions: for example, the array [1,0] corresponds to the ...

1

As you ask specifically for the Cosine Similarity technique, it has magnitude and direction, and similar to a vector which is used in Physics, as Cosine Similarity deals with vectors in an inner product space. So, the magnitude of vectors is exactly the same as the formula in Physics (summating over the squares of the vector elements.)

1

Implementation-side, there is a good reason to make 0 correspond to not rated. Since most users haven't rated most books, 0 will be the most common value and the cosine similarity function can use sparse matrices internally to speed up the computation. The sparse matrix shortcut is the main reason why people use cosine similarity in the first place. On ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# Signing a contract on the blockchain

Most if not all of us have signed a contract before, be it digitally printing it on PDFs or physically with a pen.

How does this translate to the contracts on the blockchain?

Suppose we have a contract which states various clauses which have to be satisfied.

This contract is usually several pages long. As such we can’t possibly transfer it over the blockchain, considering the costs incurred.

What we can do however, is to hash it.

This generates a number (also known as a hash as well), which we can use to verify its legitmacy.

The hash is significantly smaller than the original contract, and can be transferred over the network.

Next, we want to sign the “contract”.

To do so, we get the other party to use their private key to sign it,

returning v, r, c parameters (instead of their private key).

These parameters can then be used by the eth function ecrecover to obtain the original network address of the signatory.


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Lesson plan

Identifying Proportionality From Tables and Verbal Descriptions Lesson Plan

Overview

In this lesson, students will learn how to identify proportionality from tables and verbal descriptions through students’ discussion and problem solving. Students will use tables to help them process identifying proportionality. Then, students will practice with other tables and verbal descriptions with a partner or table group. You can expect this lesson with additional practice to take one 45-minute class period.

Rates And Proportional Relationships
7.RP.A.2.A
Step-by-step help

ByteLearn gives students targeted feedback and hints based on their specific mistakes

Preview step-by-step-help

Objective

Students will be able to identify proportionality from tables and verbal descriptions.

Materials

• Teacher Slideshow
• Online Practice

How to Teach Identifying Proportionality From Tables and Verbal Descriptions

Warm-Up

To begin the lesson, provide students with an opportunity to fill in the table of values with context on their own. If students are stuck, it may help to let them know that the table represents a proportional relationship, so they can try to find a pattern in the table.

Once students have an opportunity to answer the questions independently, they should check their work and discuss their process with a partner or table group. Once students have an opportunity to share their thoughts, ask if anyone worked with someone who solved the problem a different way but got the same answers.

Different methods to finding missing values

The goal of having students discuss their methods is for students to understand that there are multiple ways they could look at the information given to help solve the proportion. Here are some ways students may look at the information:

• They will recognize that the number of tickets sold times 5 gives them the total price.
• They will recognize that as the number of tickets sold goes up by 1, the total price goes up by 5.
• They may use one of the filled in rows to calculate the unknown rows. For example, 1 is half of 2, so the total price would be half of $10. If they doubled 2 tickets for 4 tickets, then they would double $10.

It is important that students understand there is more than one way to solve a problem like this. Once the table is completely filled in, ask students if the relationship is proportional and why. If there are any methods that students do not use but are mentioned above, it may be helpful to mention them directly to students. For some students, seeing a different method can help them understand the information, so it is generally best for students to know what options they have to help identify proportionality from tables and verbal descriptions.

Determining proportionality from descriptions

With the next example, it may be helpful to reiterate that a proportional relationship means there is a constant ratio. At this point in time, you do not need to focus on students identifying the constant of proportionality. The goal is for students to identify proportionality from tables and verbal descriptions using any mathematically correct method with the information they’re given.

Here are the different ways in which students might prove that this is a proportional relationship:

• They may find the unit rate (2.5 cups of water per tablespoon of \text{OJ})
• They may simplify the ratio of \text{OJ} to water (2:5).
• They may also see if they can multiply or divide by the same numbers to go from 10 to 24 and 25 to 60 respectively.

Determining proportionality from a table

With this example, you want to emphasize that students need to find the ratio for every row in the table.

Students may set up a ratio for each row and simplify the ratio. Some might divide one column by the other to see if they get the same number. However, some might see right away that while 12\times 2=24, 20\times 2 is not 45.

Last example as a class

By this example, students should be able to identify proportionality from the verbal description, even though this is presented more like a table. Allow students time to work and discuss if the image represents a proportional relationship.

Some students may think the relationship is proportional because the total cost goes up by the same amount every time; however, with the pattern, 0 cucumbers would cost \$1. Other students may find the ratio of the number of cucumbers to cost or vice versa and recognize that each row has a different unit rate, so it does not represent a proportional relationship.

Conversion of units

Allow students to try this example on their own, and then have students check their work with a partner.

Some students may look at the given values and say it is not proportional without recognizing that the units for the amount of fabric was different. There should be some students who recognize that the units were changed, and will convert one of the given fabric lengths to the other unit, such as inches to yards or yards to inches.

Identifying Proportionality From Tables and Verbal Descriptions Practice

After you’ve completed the examples with the whole class, it’s time for some independent practice! ByteLearn gives you access to tons of practice problems for identifying proportionality from tables and verbal descriptions. Check out the online practice and assign to your students for classwork and/or homework!

Identifying Proportionality From Tables and Verbal Descriptions Practice
Problem 1 of 5

View this practice


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# A Generic Derivation of Magnitude and Phase Plots

How Magnitude and Phase Information are Separated

## Separation of Magnitude and Phase

Start with a transfer function with an mth order numerator and and nth order denominator

$$H(s) = C{{\prod\limits_{k = 1}^m {\left( {s + {z_k}} \right)} } \over {\prod\limits_{i = 1}^n {\left( {s + {p_i}} \right)} }}$$

The first step is to rewrite the function so that the poles and zeros are of the form (1+s/ω0).

$$H(s) = C{{\prod\limits_{k = 1}^m {{z_k}} } \over {\prod\limits_{i = 1}^n {{p_i}} }}{{\prod\limits_{k = 1}^m {\left( {1 + {s \over {{z_k}}}} \right)} } \over {\prod\limits_{i = 1}^n {\left( {1 + {s \over {{p_i}}}} \right)} }} = C'{{\prod\limits_{k = 1}^m {\left( {1 + {s \over {{z_k}}}} \right)} } \over {\prod\limits_{i = 1}^n {\left( {1 + {s \over {{p_i}}}} \right)} }}{\rm{, where }}\;C' = \;C{{\prod\limits_{k = 1}^m {{z_k}} } \over {\prod\limits_{i = 1}^n {{p_i}} }}$$

The function is now a quotient of products.  For easy hand manipulation, we'd prefer to use only addition and subtraction.  To do this, let's represent the transfer function (with s=jω) as a phasor.

$$H(j\omega ) = \left| {H(j\omega )} \right|\angle H(j\omega )$$

where

$$\left| {a + jb} \right| = \sqrt {{a^2} + {b^2}}$$

and

$$\angle \left( {a + jb} \right) = \arctan \left( {{b \over a}} \right)$$

### Calculation of magnitude

Calculation of the magnitude begins with the fact that

$$\left| {H(j\omega )} \right| = \left| {C'{{\prod\limits_{k = 1}^m {\left( {1 + {{j\omega } \over {{z_k}}}} \right)} } \over {\prod\limits_{i = 1}^n {\left( {1 + {{j\omega } \over {{p_i}}}} \right)} }}} \right| = \left| {C'} \right|{{\prod\limits_{k = 1}^m {\left| {1 + {{j\omega } \over {{z_k}}}} \right|} } \over {\prod\limits_{i = 1}^n {\left| {1 + {{j\omega } \over {{p_i}}}} \right|} }}$$

This is still a quotient of products.  To simplify we will express the result in deciBels which transforms multiplication and division to addition and subtraction.

\eqalign{ 20 \cdot {\log _{10}}\left( {\left| {H(j\omega )} \right|} \right) &= 20 \cdot {\log _{10}}\left( {\left| {{C'}} \right|{{\prod\limits_{k = 1}^m {\left| {1 + {{j\omega } \over {{z_k}}}} \right|} } \over {\prod\limits_{i = 1}^n {\left| {1 + {{j\omega } \over {{p_i}}}} \right|} }}} \right) \cr &= 20 \cdot {\log _{10}}\left( {\left| {{C'}} \right|} \right) + 20 \cdot {\log _{10}}\left( {\prod\limits_{k = 1}^m {\left| {1 + {{j\omega } \over {{z_k}}}} \right|} } \right) + 20 \cdot {\log _{10}}\left( {{1 \over {\prod\limits_{i = 1}^n {\left| {1 + {{j\omega } \over {{p_i}}}} \right|} }}} \right) \cr &= 20 \cdot {\log _{10}}\left( {\left| {{C'}} \right|} \right) + 20 \cdot {\log _{10}}\left( {\prod\limits_{k = 1}^m {\left| {1 + {{j\omega } \over {{z_k}}}} \right|} } \right) - 20 \cdot {\log _{10}}\left( {\prod\limits_{i = 1}^n {\left| {1 + {{j\omega } \over {{p_i}}}} \right|} } \right) \cr &= 20 \cdot {\log _{10}}\left( {\left| {{C'}} \right|} \right) + \sum\limits_{k = 1}^m {20 \cdot {{\log }_{10}}\left( {\left| {1 + {{j\omega } \over {{z_k}}}} \right|} \right)} - \sum\limits_{i = 1}^n {20 \cdot {{\log }_{10}}\left( {\left| {1 + {{j\omega } \over {{p_i}}}} \right|} \right)} \cr}

Voilà!  We have changed the products and quotients into addition and subtraction.  As a bonus (which comes from the first step of rewriting the function), there are only two types of terms: the constant term and the simple zeros and poles (which are added and subtracted, respectively).

### Calculation of phase

The phase term is a little easier to develop, since they add and subtract naturally.  Calculation of phase begins, and ends, with the fact that

$$\angle H(j\omega ) = \angle {C'} + \sum\limits_{k = 1}^m {\angle \left( {1 + {{j\omega } \over {{z_k}}}} \right)} - \sum\limits_{i = 1}^n {\angle \left( {1 + {{j\omega } \over {{p_i}}}} \right)}$$

Again, there are only two types of terms: the constant term and the simple zeros and poles.

##### Key Concept

Starting from a transfer function it is possible to express both its magnitude and phase as a sum of simple terms.

References


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1. Jul 7, 2012

GarageDweller

So as I have read in Schutzs book on GR, and I'm finding his section on tensors and one forms very confusing.
Schutz describes gradients of functions as a one form, I cannot quite grasp why. In calculus I was taught that the gradient was a vector pointing in the direction of the fastest increase. Could someone shed some light on this?

2. Jul 7, 2012

Bill_K

They transform differently when you make a coordinate change.

Say r' = 2r. Then the circles r' = const are twice as close together as the r circles were. If v is a contravariant vector, visualized as an arrow, it now spans twice as many circles, implying that its radial component is now twice as great. And in fact, vr' = (∂r'/∂r) vr = 2 vr.

On the other hand if v is a covariant vector ("one form"), such as the gradient of a function V(r), then its radial component says how much it changes from one circle to the next, which is now clearly half as much as it was. And in fact, vr' = (∂r/∂r') vr = ½ vr.

3. Jul 7, 2012

HallsofIvy

Staff Emeritus
Roughly speaking, the set of all one forms is the "dual space" to the set of vectors. Given any vector space, it "dual space" is the set of all linear functions whose domain is the set of vectors and whose range is the set of real numbers. That is, a "one form" maps every vector to a number. We can think of that as $\int_A v\cdot d\omega$ where "A" is the manifold over which we are defining the vectors and one-forms, v is a vector and $d\omega$ is a one-form.

Some authors refer to the vectors as being in the "tangent space" at a point on the manifold and one forms as being in the "cotangent space". Strictly speaking, since we define the operation of the one-form on a "vector" as the integral over the entire manifold, we are really talkig about vector fields rather than individual vectors. But if we have a metric or "connection" defined on the manifold, we can associate a vector field over the entire manifold with a vector at a specific point in much the same we talk about "moving" vectors in classical physics.

You might get a better response if this were posted in "Mathematics- Topology and Differential Geometry".

4. Jul 7, 2012

lavinia

The gradient is a vector but it becomes a 1 form when you takes its inner product with other vectors.

the 1 form is v -> <grad f,v>

In regular calculus, the inner product is assumed to be the dot product. But on a manifold this is replaced with a Riemannian metric i.e an inner product on each tangent space. A different Riemannian metric will give you a different gradient.

Last edited: Jul 7, 2012
5. Jul 8, 2012

Rasalhague

Given a scalar field $f$, some people use the word gradient for the cotangent vector field $\mathrm{d}f$ which tells you how fast the scalar field is changing in each direction. It specifies a cotangent vector $\mathrm{d}f |_p$ at each point $p$ which maps a tangent vector $v$ at that point to a number $\mathrm{d}f |_p (v)$, namely the rate of change of f in the direction of that tangent vector, multiplied by the magnitude of the tangent vector: $||v|| \; \mathrm{d}f |_p (u)$, where $||v||$ is the magnitude of $v$, and $u$ a unit tangent vector in the same direction as $v$.

Given a metric tensor field $g$, which specifies a metric tensor $g |_p$ (i.e. an inner product) at each point $p$, we have a mapping $b$ between tangent and cotangent vectors: $b(v) := g |_p (v,\cdot )$, where the dot indicates a "slot waiting to be filled." Other people, including most introductory texts and courses on multivariable calculus and vector analysis, use the word gradient to refer to the tangent vector field $V$ whose value $V|_p$ at each point $p$ is such that $\mathrm{d}f|_p = b(V|_p) = g|_p (V|_p,\cdot )$.

When working with a fixed coordinate system, the two definitions are as good as each other, but in switching coordinate systems and their corresponding coordinate bases, the component functions of the vector field $V$ must be derived afresh from the cotangent vector field $\mathrm{d}f$, whose components transform in the usual "covariant" way. (I.e. the components of $V$ with respect to one coordinate system can't necessarily be derived from those of the other by the usual "contravariant" transformation.) This makes $\mathrm{d}f$ the more natural and fundamental entity representing the rate of change of the scalar field $f$.

6. Jul 8, 2012

Hurkyl

Staff Emeritus
By applying the metric in various ways, you can convert a geometric object into other objects.

When you learned multivariable calculus, you only knew about scalars and vectors, and the course did not intend to introduce tensors, so everything got converted into a scalar or a vector. (3 is the largest dimension in which this trick can be pulled)

So, when you learn about the gradient -- the function that tells you how fast a function varies in various directions -- you weren't taught the gradient directly. Instead, you were taught about the direction of steepest ascent and the dot-product formula that relates it to directional derivatives.

Notions such as "axial vectors", "pseudoscalars" or "densities" are other examples of this. An axial vector is really a bivector -- but one can delay learning about tensors by using the metric to convert it into a vector, so long as one remembers it's not really a vector but an "axial vector".

This works pretty well to some extent -- if you never bother with coordinate transformations, or stick to orthogonal ones only -- you might not even notice that things are weird. The first clue that something's up comes when you think about reflections, but everything gets really screwy when you start considering more general transformations. e.g. the notion of density is notoriously tricky when you rescale things.

It turns out the notion of covector is rather trivial to treat, though. I even figured it when I took multivariable calculus -- long before I had any inkling of notions like "tensor" or "dual space". Geometric vectors are "column vectors": 3x1 matrices. However the gradient is best thought of as a "row vector": a 1x3 matrix.

It turns out this is really equivalent to the distinction between vector and 1-forms: relative to a basis, the coordinate representation of a vector really is a column vector, and the coordinate representation of a 1-form really is a row vector.

Now, to make things even more annoying, some people argue that because there is are two standard notations for the 1-form version -- $\nabla f$ and $df$ -- then one should reserve $\mathop{\mathrm{grad}} f$ for the vector version.

7. Jul 8, 2012

lavinia

the differential of a function, df, is a 1 form but it is not the gradient of the function. The gradient is a vector. The equation relating the two is

df(v) = <grad f,v> where <,> is the inner product. But if you change the inner product, the gradient also changes. It is a different vector. But df is the same always.

At each point of a manifold, a 1 form is a linear map from the tangent space at that point into the base field - although this idea can be generalized to linear maps into vector bundles.

The 1 form is smooth if the values that it takes on an arbitrary smooth vector field is a smooth function.

If there is a Riemannian metric on the manifold then each vector field,v, determines a 1 form by the rule, w -> <v,w>. Conversely each 1 form may be expressed as an inner product with a vector field.

When one has a basis for a vector space, one can immediately define the dual basis for the linear maps of the vector space into the base field. These are so called co-vectors I think.
So the choice of a basis defines an isomorphic correspondence between the vector space and its dual space. An inner product also defines an isomorphic correspondence. With an inner product, no basis is required. Without a choice of basis or an inner product there is no natural way to define this isomorphism. A choice of a basis for the vector fields in a region, e,g, with a coordinate system, similarly gives you a dual basis of 1 forms.

Vector fields and 1 forms transform in opposite ways - one is called covariant, the other contravariant. The transformations can be defined without the use of a basis as follows.

Given a a function from one region to another say U into V,e.g. a change of coordinates, then df(u) is a vector in V when u is a vector in U. If one thinks of vectors as operators on functions then df(u).g = u.(g$\circ$f).

Given a 1 form on V (not on U) then f determines a 1 form on U by the transformation (df*ω)(v) = ω(df(v))

In term of bases these transformations appear as the usual covariant and contravariant transformation rules that you find in books on the theory of relativity.

In mathematics there is a general concept of covariant and contravariant functors. A functor associates objects in one category with objects in another and maps in the first category with maps in the second. depending on the way maps are associated, the functor is called covariant or contravariant. Vectors and 1 forms are an example of this general idea.

To each smooth manifold (the objects of the first category) associate its tangent bundle (the objects in the second category) To each smooth map between manifolds associate its differential, which is a map between the tangent spaces of the two manifolds. This is a covariant functor.

If on the other hand one associates, the dual tangent space to each manifold and the pull back map, df*, to each smooth map, one gets a contravariant functor.

Last edited: Jul 8, 2012
8. Jul 11, 2012

dydxforsn

Well for the gradient to be a one form it has to obey the definition of a one form, which is simply a rule that one forms must obey under a coordinate transformation. The rule is that each component of a one form transform with a weighted combination of all of the components of the one form from the previous coordinate system. Specifically for the gradient vector, we need to prove the following equation true:
$$(\nabla f)_{i'} = \sum_{i}{\frac{{\partial}{x_{i}}}{{\partial}{x_{i'}}} (\nabla f)_{i}}$$ or
$$\frac{{\partial}{f}}{{\partial}{x_{i'}}} = \sum_{i}{\frac{{\partial}{x_{i}}}{{\partial}{x_{i'}}} \frac{{\partial}{f}}{{\partial}{x_{i}}}}$$
Here the primed variables are in the new coordinate system and the un-primed variables are in the coordinate system we've transformed from.

We prove this equation is true by simply noticing that the left side of the equation can be expanded using the chain rule seeing as how each $x_{i'}$ is a function of each of the original variables in the non-primed coordinate system (all of the $x_{i}$'s..)
$$\sum_{j}{\frac{{\partial}{f}}{{\partial}{x_{j}}} \frac{{\partial}{x_{j}}}{{\partial}{x_{i'}}}} = \sum_{i}{\frac{{\partial}{x_{i}}}{{\partial}{x_{i'}}} \frac{{\partial}{f}}{{\partial}{x_{i}}}}$$
It doesn't matter what dummy index we sum over, so we'll just make them both over 'i', and swapping the order of multiplication of one side (it obviously doesn't matter), the two sides are equal:
$$\sum_{i}{\frac{{\partial}{x_{i}}}{{\partial}{x_{i'}}} \frac{{\partial}{f}}{{\partial}{x_{i}}}} = \sum_{i}{\frac{{\partial}{x_{i}}}{{\partial}{x_{i'}}} \frac{{\partial}{f}}{{\partial}{x_{i}}}}$$
So it's been proven that the components of the gradient vector transform covariantly, and thus it is a one form.

As a quick note one forms are often called "dual vectors". All dual vectors have to transform between coordinate systems via a covariant transformation (well, their components do.) Thus we started the proof from that requirement, and it is THE definition of the types of vectors we call "one forms".

So vectors in your calculus class like the gradient could have easily have been one forms because they're just special types of vectors. Note that dual vectors are indistinguishable with their counterparts (vectors who's components transform "contravariantly") in Cartesian coordinates, and so the distinction is usually only brought up when one discusses generalized curvilinear coordinates.

Last edited: Jul 11, 2012
9. Jul 11, 2012

lavinia

The gradient is not a 1 form. It is a vector. The gradient of a function,f, is the vector dual to the 1 form, df, the differential of the function. This vector is only defined when there is a Riemannian metric on the tangent space.

10. Jul 11, 2012

Muphrid

Would it not be correct to say that the gradient of a scalar field yields a cotangent vector field, or does the cotangent space only exist under, say, the more specialized case of relativity and not in general?

11. Jul 11, 2012

lavinia

The gradient is a tangent vector. The differential is a cotangent vector. There may be some terminology confusion here but in mathematics gradients are vectors always not cotangent vectors.

There is a profound point here. The differential of a function can be defined without an inner product. One does not need a metric to take a derivative. But without a metric you only have a cotangent vector. If you want to get a vector you need a metric to find the dual of the differential. This dual to the differential is the gradient.

12. Jul 11, 2012

Muphrid

Fair enough; it was precisely that subtle distinction that I suspected was being invoked and hoped to be made clear. I suspect the physics literature GarageDweller will be browsing may not be so observant of the distinction, but it seems a reasonable one to maintain for the purposes of clarity.

Overall, then, GarageDweller, you should take this away from the discussion: typical vector calculus courses use the metric of flat space to convert differentials (which are cotangent vector fields or higher ranking analogues of such) into tangent vector fields that the course had already introduced you to. In doing so, these courses avoid a lot of extra complications that aren't really relevant to the flat space case--these transformations are largely trivial. Nevertheless, the differential of a scalar field is a field in the cotangent space, and it just simplifies to gradient and such through relatively trivial manipulations when you're talking about just flat space.

13. Jul 11, 2012

lavinia

Well said

14. Jul 11, 2012

dydxforsn

Hmmm, that's interesting. I think we might have an issue of semantics here because I have a source that says otherwise:

"As in the traditional approach, vectors (which utilize contravariant components) expand using original basis vectors, while one-forms (which utilize covariant components) expand using basis one-forms, which are equivalent to dual basis vectors in the traditional approach."

Fleisch, Daniel. A Student's Guide to Vectors and Tensors. New York: Cambridge University Press, 2012. 156-157. Print.

Based upon this definition of one-forms I did indeed prove that the gradient vector is a one-form. I come from a pure physics background and have never taken a formal course in Differential Geometry, and you clearly have much more mathematical sophistication than I, could this be a mathematics versus physics terminology dispute or am I doing something wrong? I would authentically like to get to the bottom of this, the source could obviously be wrong, but wolfram seems to agree as well.

15. Jul 11, 2012

lavinia

A good application of these concepts is the definition of the Laplacian of a function.
The Laplacian is the divergence of the gradient. The Laplacian changes if the shape of the surface changes.

16. Jul 11, 2012

lavinia

Not sure. What I said is definitely right though. But I will get out a physics book and see how these transformations are defined. then get back to you.

But again, the important point is that the differential of a function is defined without a metric while the gradient requires a metric because the metric allows you to express the 1 form as an inner product with a fixed vector. This is the key idea, one needs a metric, the other does not. This means that the changes if the metric changes.

BTW: In vector calculus the differential of a function applied to a vector is described as the
dot product of the gradient with the vector. The dot product is just the inner product on flat Euclidean space. Without the dot product you can not express the value of the differential in this way. Instead you must compute it by taking the directional derivative of the function with respect to the vector.

Last edited: Jul 11, 2012


text:
# Independent Chapters and Unique References

I am trying to setup a report structure so that each author can work on their chapter on their own and once all chapters are finished I can put all together. For this to work well I need each chapter to be completely independent (I was thinking on standalone package) and have their own references. Right now I have all chapters being compiled independently, but once I try to integrated them using import on the main file it just does not work. I am using natbib package.

If anyone have an working example with natbib working together with standalone and import packages, please, share with me! :)

Each chapter should have its own reference file as well!

Thanks!

• Welcome to TeX.SX! In my opinion, \include{chapter1} should work, since the references are maintained, but I am unsure about natbib in this case. Perhaps you should give us a starter document that shows your setup – user31729 Dec 16 '15 at 18:41
• when this need came up at the american math society, the approach we took was to create an add-on package for amsbook that makes certain modifications to the processing of \chapter (to allow an author line as part of the chapter heading) and carries that information into the toc. each author is provided with a copy of the main file that is used to pull all chapters together; an author-specific version contains only one \include statement, for that author's chapter. having a single main file ensures, among other things, that all packages used are compatible. – barbara beeton Dec 16 '15 at 19:06
• Thank you for your responses. After some trials I finally came up with a 'ok' solution. I was trying to setup something like in this [website] (sharelatex.com/learn/Multi-file_LaTeX_projects) . But all worked out nicely when I removed the document class statement for the subfiles \documentclass[float=false, crop=false]{standalone} and used just normal document class statement on them like if they were separate files. As a summary, I used \usepackage{chapterbib}, \usepackage[colon, sort&compress]{natbib} and \usepackage{standalone} all together. I will add example later. – Diogo Silva Dec 17 '15 at 18:58
• I posted a solution, but it was deleted. Worked for me. – Diogo Silva Dec 22 '15 at 7:04
• @DiogoSilva: That wasn't recognizable as solution – user31729 Dec 31 '15 at 9:55

Introduction copied from author's comment and screenshot added by cfr to try to get this recognised as a solution and undeleted.

After some trials I finally came up with a 'ok' solution. I was trying to setup something like in this website.

But all worked out nicely when I removed the document class statement for the subfiles \documentclass[float=false, crop=false]{standalone} and used just normal document class statement on them like if they were separate files.

As a summary, I used \usepackage{chapterbib}, \usepackage[colon, sort&compress]{natbib} and \usepackage{standalone} all together.

# Working solution

The main file (main.tex) contains:

\documentclass[12pt,letterpaper,oneside]{report}
\usepackage{standalone}
\usepackage[colon, sort&compress]{natbib}
\setlength{\bibsep}{12pt}
\usepackage{chapterbib}
\title{Independent Chapters}
\author{Diogo Silva}
\begin{document}
\maketitle
\include{chapter1}
\include{chapter2}
\end{document}

The chapter1.tex contains:

\documentclass[12pt,letterpaper,oneside]{report}
\usepackage[colon, sort&compress]{natbib}
\setlength{\bibsep}{12pt}
\begin{document}
\chapter{Chapter1}
Reference \cite{deutsch1992}.
\bibliographystyle{apalike}
\bibliography{./ch01}
\end{document}

The chapter2.tex contains:

\documentclass[12pt,letterpaper,oneside]{report}
\usepackage[colon, sort&compress]{natbib}
\setlength{\bibsep}{12pt}
\begin{document}
\chapter{Chapter2}
Reference \cite{journel1999markov}.
\bibliographystyle{apalike}
\bibliography{./ch02}
\end{document}

The ch01.bib contains:

@article{deutsch1992,
title={Geostatistical software library and user’s guide},
author={Deutsch, Clayton V and Journel, Andr{\'e} G},
journal={New York},
year={1992}
}

The ch02.bib contains:

@article{journel1999markov,
title={Markov models for cross-covariances},
author={Journel, AG},
journal={Mathematical Geology},
volume={31},
number={8},
pages={955--964},
year={1999},
publisher={Springer}
}


text:
# Free monoidal category over a set

From nLab's article on coherence theorems, there seems to be a notion of free monoidal category over a set $S$.

I guess this corresponds to the left adjoint to the functor $Ob : MonCat \to Set$ which given a monoidal category $C$, it gives back the set of objects of $C$. The free monoidal category over $S$ would be a monoidal category $C$ together with a function $i : S \to Ob C$ such that for any other monoidal category $D$ and function $f : S \to Ob D$, I get a unique monoidal functor $\bar{f} : C \to D$ such that $Ob \bar{f} \circ i = f$.

After thinking of that definition, I thought that the objects of the free monoidal category over $\{ * \}$ would be $I$, $\{ * \}$, $\{ * \}^2$, etc.

Then, I thought that a similar presentation can be made for a free strict monoidal category over a set $S$, by replacing "monoidal" by "strict monoidal".

However, I started to wonder, wouldn't the free strict monoidal category over $\{ * \}$ satisfy the condition of being the free (non-necessarily strict) monoidal category over $\{ * \}$? If that was the case, I think the nLab page wouldn't make much sense. What's wrong here? Is my definition of free monoidal category?

The construction you describe is exactly similar to the free-strict-monoidal category but it is not the free-monoidal category: the difference being that in a strict monoidal category associators and left and right units are required to be the identities of the respective objects, a requirement that you did not add in your construction.

The free-monoidal-(biased)-category $F(S)$ over $S$ should have:

• as set of object the set of all the formal expressions you can build from the set of symbols in $S$ with the operators $\otimes$ and $1$
• as set of morphisms is generated by associators, left and right unit (and their inverses) through application of compositions and monoidal products, quotienting by the various equations needed for a monoidal categories (associativity and identities for compositions, functoriality of the monoidal product and most importantly coherence conditions).

One of the interesting things is that coherence conditions give you the ability to describe more explicitly the set of morphisms in this category: for each pair of objects in $F(S)$ you have exactly one morphism between two objects if and only if they are equivalent up to associativity and unit.

This data give you a category with the wished property: namely that for every monoidal category $\mathbf C$ and a function $f \colon S \to Ob(\mathbf C)$ there is a monoidal functor $\bar f \colon F(S) \to \mathbf C$ such that the equation $f=Ob(\bar f) \circ i$ holds, where $i \colon S \to Ob(F(S))$ is the obvious embedding sending every element of $S$ in itself seen as an element of $F(S)$.

P.s.: I realize that what I've described is not a fully formal description of what the free monoidal category over the set $S$ should be, but I hope that is enough to get the idea.

P.p.s.: above I've called $F(S)$ to distinguish it from the free-unbiased-monoidal categories described in Leinster's book Higher Operads, Higher categories.

• But doesn't the free strict monoidal category (the one I described, not all expressions as objects but only "normalized" ones) also satisfy this definition of free monoidal category? For which $\mathbb{C}$ and $f : S \to Ob(\mathbb{C})$ it doesn't hold? – user329838 Apr 8 '16 at 20:16
• Not if you are working in the $1$-category of monoidal categories and functor between them. To understand why try to answer this question: if your category were initial it should exist a functor into my category, what should be the image of an object $x_1 x_2 x_3$ through this functor? Should it be $(x_1 x_2)x_3$ or $x_1 (x_2 x_3)$? – Giorgio Mossa Apr 8 '16 at 20:22
• If I can choose any of the two, there is more than one $\bar{f}$ possible? So it is essential that $\bar{f}$ is unique? – user329838 Apr 8 '16 at 20:30
• If you work with $1$-categories I guess so. Nonetheless if you instead want to consider the $2$-category of monoidal categories and monoidal functor between them.... well I think the story could be different: I think (if not entirely sure) that in that case the two categories should be equivalent, but then you shouldn't have a unique functor $\bar f$, only one unique up to natural isomorphism. – Giorgio Mossa Apr 8 '16 at 20:35
• Just a speculation. – Giorgio Mossa Apr 8 '16 at 20:36


text:
# Caption and reference of a figure in the introduction chapter

I have an introduction chapter defined with the command:

\chapter*{Introduction}

The captions of the figures within this chapter, that doesn't have a number like the others, are named:

"Figure 1", "Figure 2", etc.

then when I reference a figure with \ref{labelname} I see a number like 1,2 etc.

I don't like this way to reference a figure, because isn't like the other chapters. The captions of the first chapter for example are:

"Figure 1.1", "Figure 1.2" etc.

I'd like that the captions of the figures in the introduction to be:

"Figure I.1", "Figure I.2" etc.

then when I reference a figure with \ref{labelname} I see a number like I.1, I.2 etc. How can I do that?

Here an example code:

\documentclass[a4paper]{book}

\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{lipsum}

\begin{document}
\chapter*{Introduction}
\lipsum[1]
\begin{figure}[h!]
\vspace{2cm}
\label{fig:intro}
\caption{Figure in the introduction}
\end{figure}

Look at the Fig.~\ref{fig:intro}.
I want it referenced as Fig.~I.1.

\chapter{A sample chapter}
\lipsum[1]

\begin{figure}[h!]
\vspace{2cm}
\label{fig:chap1}
\caption{Figure in chapter 1}
\end{figure}

Look at the Fig.~\ref{fig:chap1}.

\chapter*{Conclusion}
\lipsum[1]
\begin{figure}[h!]
\vspace{2cm}
\caption{Figure in the conclusion}
\label{fig:conc}
\end{figure}

Look at the Fig.~\ref{fig:conc}.
I want it referenced as Fig.~C.1.

\end{document}

-
Please provide a small MWE so that we don't have to construct a document for you. Easier for us to play around. – LaRiFaRi Mar 9 '14 at 10:30

This is a very simple solution. Notice! I moved the label below the caption statement.

\documentclass[a4paper]{book}

\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{lipsum}

\begin{document}
\chapter*{Introduction}
\renewcommand{\thefigure}{I.\arabic{figure}}
\lipsum[1]
\begin{figure}[h!]
\vspace{2cm}
\caption{Figure in the introduction}
\label{fig:intro}
\end{figure}

Look at the Fig.~\ref{fig:intro}.
I want it referenced as Fig.~I.1.

\chapter{A sample chapter}
\renewcommand{\thefigure}{\arabic{chapter}.\arabic{figure}}
\lipsum[1]

\begin{figure}[h!]
\vspace{2cm}
\caption{Figure in chapter 1}
\label{fig:chap1}
\end{figure}

Look at the Fig.~\ref{fig:chap1}.

\end{document}

-
Thank you! It works well! But I have another problem... if I write a conclusion chapter like the introduction, the the figure within the conclusion are referenced starting from the last figure number in the last chapter. – nunzio13n Mar 9 '14 at 13:45
What's your goal? – Oliver Mar 9 '14 at 13:49
I have edited the question, but I have solved it by adding the line \setcounter{figure}{0} before the commend \renewcommand in the conclusion. – nunzio13n Mar 9 '14 at 13:54


text:
Preparing NOJ

YAPTCHA

5000ms 32768K

Description:

The math department has been having problems lately. Due to immense amount of unsolicited automated programs which were crawling across their pages, they decided to put Yet-Another-Public-Turing-Test-to-Tell-Computers-and-Humans-Apart on their webpages. In short, to get access to their scientific papers, one have to prove yourself eligible and worthy, i.e. solve a mathematic riddle.

However, the test turned out difficult for some math PhD students and even for some professors. Therefore, the math department wants to write a helper program which solves this task (it is not irrational, as they are going to make money on selling the program).

The task that is presented to anyone visiting the start page of the math department is as follows: given a natural n, compute

where [x] denotes the largest integer not greater than x.

Input:

The first line contains the number of queries t (t <= 10^6). Each query consist of one natural number n (1 <= n <= 10^6).

Output:

For each n given in the input output the value of Sn.

Sample Input:

13
1
2
3
4
5
6
7
8
9
10
100
1000
10000

Sample Output:

0
1
1
2
2
2
2
3
3
4
28
207
1609

Info

Provider HDU

Code HDU2973

Tags

Submitted 194

Passed 64

AC Rate 32.99%

Date 06/03/2019 16:32:15

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