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# Pure Component Helmholtz EoS¶

The Helmholtz Equation of State (EoS) classes serve as a common core for pure component property packages where very accurate and thermodynamically consistent pure component properties are required. This contains general information. Thermodynamic properties for all Helmholtz EoS packages are calculated by the core class only the parameters differ between specific component implementation. Specific implementations may also contain additional properties such as viscosity and thermal conductivity. For specific property packages details see the pages below.

The basic Helmholtz EoS is described “Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use.”. The Helmholtz EoS as used in the IAPWS-95 contains non-analytic terms to improve accuracy near the critical point. These terms, however cause a singularity at the critical point and can causes computational difficulty, so the non-analytic where omitted in the IDAES implementation.

The IDAES implementation is of the Helmholtz EoS makes use of external function for many of the properties. Solving the VLE and changing state variables require solution of non-linear equations with multiple solutions, so solving then externally provides a method of decomposition where it can be guaranteed that the nonlinear equations associated with the Helmholtz EoS are solved correctly. The external functions provide first and second derivatives, and are compatible with advanced optimization solvers. Phase change does cause problems to be non-smooth, but, as a practical matter, problem using the IDAES implementation of the Helmholtz EoS, still seem to solve well even with phase change.

## Units¶

The iapws95 property module uses SI units (m, kg, s, J, mol) for all public variables and expressions. Temperature is in K. Note that this means molecular weight is in the unusual unit of kg/mol.

A few expressions intended used internally and all external function calls use units of kg, kJ, kPa, and K. These generally are not needed by the end user.

## Phase Presentation¶

The property package wrapper can present fluid phase information to the IDAES framework in different ways. For specifics on how to set the options see a specific implementation page.

The PhaseType.MIX option causes to modeling framework to view water and steam as a single mixed liquid and vapor phase. This generally reduces model complexity. Phase equilibrium is still calculated and vapor_frac and individual phase properties are available, just are they would be with the two-phase presentation. The mixed-phase presentation can be used with most standard unit models that do not provide phase separation. If phase separation is required, either use the two-phase presentation or create a custom model.

The PhaseType.LG option appears to the IDAES framework to be two phases “Vap” and “Liq”. This option requires one of two unit model options to be set. You can use the total material balance option for unit models, to specify that only one material balance equation should be written not one per phase. The other possible option is to specify has_phase_equlibrium=True. This will write a material balance per phase, but will add a phase generation term to the model. For Helmholtz EoS packages, it is generally recommended that specifying total material balances is best because it results in a problem with fewer variables, and phase equilibrium is always calculated by the property package.

There are two single phase options PhaseType.L and PhaseType.G; these present a single phase “Liq” or “Vap” to the framework. The vapor fraction will also always return 0 or 1 as appropriate. These options can be used when the phase of a fluid is know for certain to only be liquid or only be vapor. For the temperature-pressure-vapor fraction formulation, this eliminates the complementarity constraint, but for the enthalpy-pressure formulation, where the vapor fraction is always calculated, the single phase options probably do not provide any real benefit.

## State Variables¶

There is a choice of state variables, pressure-enthalpy and temperature-pressure-vapor fraction. In general the enthalpy-pressure form is preferable. Both the pressure and enthalpy variables are smooth and sufficient to define the fluid state. For systems where two-phases may be present, it is expected that pressure-enthalpy is the best choice of state variables.

The temperature-pressure-vapor fraction form is more convenient, since temperature is directly measurable and more familiar than enthalpy. Complementarity constraints are used to deal with the vapor fraction variable, but the additional complimentary constraints may make the problem less robust. Temperature-pressure is often a good choice of state variables where there is only a single known phase.

### Pressure-Enthalpy Formulation¶

The advantage of this choice of state variables is that it is more robust when phase changes occur, and is especially useful when it is not known if a phase change will occur. The disadvantage of this choice of state variables is that for equations like heat transfer equations that are highly dependent on temperature, a model could be harder to solve near regions with phase change. Temperature is a non-smooth function with non-smoothness when transitioning from the single-phase to the two-phase region. Temperature also has a zero derivative with respect to enthalpy in the two-phase region, so near the two-phase region solving a constraint that specifies a specific temperature may be difficult.

The variables for this form are flow_mol (mol/s), pressure (Pa), and enth_mol (J/mol).

Since temperature and vapor fraction are not state variables in this formulation, they are provided by expressions, and cannot be fixed. For example, to set a temperature to a specific value, a constraint could be added which says the temperature expression equals a fixed value.

These expressions are specific to the P-H formulation:

temperature

Expression that calculates temperature by calling an ExternalFunction of enthalpy and pressure. This expression is non-smooth in the transition from single-phase to two-phase and has a zero derivative with respect to enthalpy in the two-phase region.

vapor_frac

Expression that calculates vapor fraction by calling an ExternalFunction of enthalpy and pressure. This expression is non-smooth in the transition from single-phase to two-phase and has a zero derivative with respect to enthalpy in the single-phase region, where the value is 0 (liquid) or 1 (vapor).

### Temperature-Pressure-Vapor Fraction¶

This formulation uses temperature (K), pressure (Pa), and vapor fraction as state variables. When a single phase option is given, the vapor fraction is fixed to the appropriate value and the complementarity constraint is deactivated.

A complementarity constraint is required for the T-P-x formulation when two-phases may be present. First, two expressions are defined below where $$P^-$$ is pressure under saturation pressure and $$P^+$$ is pressure over saturation pressure. The $$\max()$$ function is provided as an IDAES utility which provides a smooth max expression.

$P^- = \max(0, P_{\text{sat}} - P)$
$P^+ = \max(0, P - P_{\text{sat}})$

With the “pressure over” and “pressure under” expressions a complementarity constraint can be written. If the pressure under saturation is more than zero, only vapor exists. If the pressure over saturation is greater than zero only a liquid exists. If both are about zero two phases can exist. The saturation pressure function maxes out at the critical pressure and any temperature above the critical temperature will yield a saturation pressure that is the critical pressure, so supercritical fluids will be classified as liquids as is the convention for this property package.

$0 = xP^+ - (1 - x)P^-$

Assuming the vapor fraction ($$x$$) is positive and noting that only one of $$P^+$$ and $$P^-$$ can be nonzero (approximately), the complementarity equation above requires $$x$$ to be 0 when $$P^+$$ is not zero (liquid) or $$x$$ to be 1 when $$P^-$$ is not zero (vapor). When both $$P^+$$ and $$P^-$$ are about 0, the complementarity constraint says nothing about x, but it basically reduces another constraint, that $$P=P_{\text{sat}}$$. When two phases are present $$x$$ is found by the unit model energy balance, where the temperature will be $$T_{\text{sat}}$$ (because $$P=P_{\text{sat}}$$).

An alternative approach is sometimes useful to simplify the problem when it is certain that there are two phases. The complementarity constraint can be deactivated and a $$P=P_{\text{sat}}$$ or $$T=T_{\text{sat}}$$ constraint can be added.

Using the T-P-x formulation requires better initial guesses than the P-H form. It is not strictly necessary but it is best to try to get an initial guess that is in the correct phase region for the expected result model.

### Expressions¶

Unless otherwise noted, the property expressions are common to both the T-P-x and P-H formulations. For phase specific properties, valid phase indexes are "Liq" and "Vap". Even when using the mixed phase version of the property package, both liquid and vapor properties are available.

Expression

Description

mw

Molecular weight (kg/mol)

tau

Critical temperature divided by temperature (unitless)

temperature

Temperature (K) if PH form

temperature_red

Reduced temperature, temperature divided by critical temperature (unitless)

temperature_sat

Saturation temperature (K)

tau_sat

Critical temperature divided by saturation temperature (unitless)

pressure_sat

Saturation pressure (Pa)

dens_mass_phase[phase]

Density phase (kg/m3)

dens_phase_red[phase]

Phase reduced density ($$\delta$$), mass density divided by critical density (unitless)

dens_mass

Total mixed phase mass density (kg/m3)

dens_mol

Total mixed phase mole density (kg/m3)

flow_vol

Total volumetric flow rate (m3/s)

enth_mass

Mass enthalpy (J/kg)

enth_mol_sat_phase[phase]

Saturation enthalpy of phase, enthalpy at P and Tsat (J/mol)

enth_mol

Molar enthalpy (J/mol) if TPx form

enth_mol_phase[phase]

Molar enthalpy of phase (J/mol)

energy_internal_mol

molar internal energy (J/mol)

energy_internal_mol_phase[phase]

Molar internal energy of phase (J/mol)

entr_mol_phase

Molar entropy of phase (J/mol/K)

entr_mol

Total mixed phase entropy (J/mol/K)

cp_mol_phase[phase]

Constant pressure molar heat capacity of phase (J/mol/K)

cv_mol_phase[phase]

Constant pressure volume heat capacity of phase (J/mol/K)

cp_mol

Total mixed phase constant pressure heat capacity (J/mol/K)

cv_mol

Total mixed phase constant volume heat capacity (J/mol/K)

heat_capacity_ratio

cp_mol/cv_mol

speed_sound_phase[phase]

Speed of sound in phase (m/s)

dens_mol_phase[phase]

Mole density of phase (mol/m3)

vapor_frac

Vapor fraction, if PH form

phase_frac[phase]

Phase fraction

flow_mol_comp["H2O"]

Same as total flow since only water (mol/s)

P_under_sat

Pressure under saturation pressure (kPa)

P_over_sat

Pressure over saturation pressure (kPa)

### ExternalFunctions¶

This provides a list of ExternalFuctions available in the wrapper. These functions do not use SI units and are not usually called directly. If these functions are needed, they should be used with caution. Some of these are used in the property expressions, some are just provided to allow easier testing with a Python framework.

All of these functions provide first and second derivative and are generally suited to optimization (including the ones that return derivatives of Helmholtz free energy). Some functions may have non-smoothness at phase transitions. The delta_vap and delta_liq functions return the same values in the critical region. They will also return real values when a phase doesn’t exist, but those values do not necessarily have physical meaning.

There are a few variables that are common to a lot of these functions, so they are summarized here $$\tau$$ is the critical temperature divided by the temperature $$\frac{T_c}{T}$$, $$\delta$$ is density divided by the critical density $$\frac{\rho}{\rho_c}$$, and $$\phi$$ is Helmholtz free energy divided by the ideal gas constant and temperature $$\frac{f}{RT}$$.

Object

C Function

Returns

Arguments

func_p

p

pressure (kPa)

$$\delta, \tau$$

func_p_stau

p_stau

pressure (kPa)

s (kJ/kg/K), $$\tau$$

func_u

u

internal energy (kJ/kg)

$$\delta, \tau$$

func_s

s

entropy (kJ/K/kg)

$$\delta, \tau$$

func_h

h

enthalpy (kJ/kg)

$$\delta, \tau$$

func_hvpt

hvpt

vapor enthalpy (kJ/kg)

P (kPa), $$\tau$$

func_hlpt

hlpt

liquid enthalpy (kJ/kg)

P (kPa), $$\tau$$

func_svpt

svpt

vapor entropy (kJ/kg/K)

P (kPa), $$\tau$$

func_slpt

slpt

liquid entropy (kJ/kg/K)

P (kPa), $$\tau$$

func_uvpt

uvpt

vapor internal energy (kJ/kg)

P (kPa), $$\tau$$

func_ulpt

ulpt

liquid internal energy (kJ/kg)

P (kPa), $$\tau$$

func_tau

tau

$$\tau$$ (unitless)

h (kJ/kg), P (kPa)

func_tau_sp

tau_sp

$$\tau$$ (unitless)

s (kJ/kg/K), P (kPa)

func_tau_up

tau_up

$$\tau$$ (unitless)

u (kJ/kg), P (kPa)

func_vf

vf

vapor fraction (unitless)

h (kJ/kg), P (kPa)

func_vfs

vfs

vapor fraction (unitless)

s (kJ/kg/K), P (kPa)

func_vfu

vfu

vapor fraction (unitless)

u (kJ/kg), P (kPa)

func_g

g

Gibbs free energy (kJ/kg)

$$\delta, \tau$$

func_f

f

Helmholtz free energy (kJ/kg)

$$\delta, \tau$$

func_cv

cv

const. volume heat capacity (kJ/K/kg)

$$\delta, \tau$$

func_cp

cp

const. pressure heat capacity (kJ/K/kg)

$$\delta, \tau$$

func_w

w

speed of sound (m/s)

$$\delta, \tau$$

func_delta_liq

delta_liq

liquid $$\delta$$ (unitless)

P (kPa), $$\tau$$

func_delta_vap

delta_vap

vapor $$\delta$$ (unitless)

P (kPa), $$\tau$$

func_delta_sat_l

delta_sat_l

sat. liquid $$\delta$$ (unitless)

$$\tau$$

func_delta_sat_v

delta_sat_v

sat. vapor $$\delta$$ (unitless)

$$\tau$$

func_p_sat

p_sat

sat. pressure (kPa)

$$\tau$$

func_tau_sat

tau_sat

sat. $$\tau$$ (unitless)

P (kPa)

func_phi0

phi0

$$\phi$$ idaes gas part (unitless)

$$\delta, \tau$$

func_phi0_delta

phi0_delta

$$\frac{\partial \phi_0}{\partial \delta}$$

$$\delta$$

func_phi0_delta2

phi0_delta2

$$\frac{\partial^2 \phi_0}{\partial \delta^2}$$

$$\delta$$

func_phi0_tau

phi0_tau

$$\frac{\partial \phi_0}{\partial \tau}$$

$$\tau$$

func_phi0_tau2

phi0_tau2

$$\frac{\partial^2 \phi_0}{\partial \tau^2}$$

$$\tau$$

func_phir

phir

$$\phi$$ real gas part (unitless)

$$\delta, \tau$$

func_phir_delta

phir_delta

$$\frac{\partial \phi_r}{\partial \delta}$$

$$\delta, \tau$$

func_phir_delta2

phir_delta2

$$\frac{\partial^2 \phi_r}{\partial \delta^2}$$

$$\delta, \tau$$

func_phir_tau

phir_tau

$$\frac{\partial \phi_r}{\partial \tau}$$

$$\delta, \tau$$

func_phir_tau2

phir_tau2

$$\frac{\partial^2 \phi_r}{\partial \tau^2}$$

$$\delta, \tau$$

func_phir_delta_tau

phir_delta_tau

$$\frac{\partial^2 \phi_r}{\partial \delta \partial \tau}$$

$$\delta, \tau$$

## Initialization¶

The IAPWS-95 property functions do provide initialization functions for general compatibility with the IDAES framework, but as long as the state variables are specified to some reasonable value, initialization is not required. All required solves are handled by external functions.

## References¶

Although a general Helmholtz EoS was developed, the equations where taken from the IAPWS-95 standard. For specific parameter sources see specific implementation documentation.

International Association for the Properties of Water and Steam (2016). IAPWS R6-95 (2016), “Revised Release on the IAPWS Formulation 1995 for the Properties of Ordinary Water Substance for General Scientific Use,” URL: http://iapws.org/relguide/IAPWS95-2016.pdf

<NAME>., <NAME> (2002). “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use.” J. Phys. Chem. Ref. Data, 31, 387-535.

<NAME>. (2008). “A Reliable and Useful Method to Determine the Saturation State from Helmholtz Energy Equations of State.” Journal of Thermal Science and Technology, 3(3), 442-451.


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# How can you find the vapor pressure of a liquid?

And since vapour pressure is a function of temperature, there are usually tables that quote the vapour pressure of a given liquid at a given temperature. For water, the so-called $\text{saturated vapour pressure}$ is readily available. At $100$ ""^@C, ${P}_{\text{SVP}} = 1 \cdot a t m$. Why should this be so?

format: plain


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Get Free NCERT Solutions for Class 10 Maths Chapter 3 Ex 3.1 PDF. Pair of Linear Equations in Two Variables Class 10 Maths NCERT Solutions are extremely helpful while doing your homework. Exercise 3.1 Class 10 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of प्रश्नावली 3.1 का हल हिंदी में Class 10 Pair of Linear Equations in Two Variables Exercise 3.1 provided in NCERT TextBook.

Topics and Sub Topics in Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables:

 Section Name Topic Name 3 Pair of Linear Equations in Two Variables 3.1 Introduction 3.2 Pair Of Linear Equations In Two Variables 3.3 Graphical Method Of Solution Of A Pair Of Linear Equations 3.4 Algebraic Methods Of Solving A Pair Of Linear Equations 3.4.1 Substitution Method 3.4.2 Elimination Method 3.4.3 Cross-Multiplication Method 3.5 Equations Reducible To A Pair Of Linear Equations In Two Variables 3.6 Summary

## NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.1

NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.1 are part of NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.1

 Board CBSE Textbook NCERT Class Class 10 Subject Maths Chapter Chapter 3 Chapter Name Pair of Linear Equations in Two Variables Exercise Ex 3.1 Number of Questions Solved 3 Category NCERT Solutions

Algebra substitution calculator, writing expressions as trinomials, free print on singapore primary school worksheet, exponents calculator.

Ex 3.1 Class 10 Maths Question 1.
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be”. Isn’t this interesting? Represent this situation algebraically and graphically.
Solution:

You can also download the free PDF of  Ex 3.1 Class 10 Pair of Linear Equations in Two Variables NCERT Solutions or save the solution images and take the print out to keep it handy for your exam preparation.

Download NCERT Solutions For Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables PDF

Ex 3.1 Class 10 Maths Question 2.
The coach of a cricket team buys 3 bats and 6 balls for ₹ 3900. Later, she buys another bat and 3 more balls of the same kind for ₹1300. Represent this situation algebraically and geometrically.
Solution:

Ex 3.1 Class 10 Maths Question 3.
The cost of 2 kg of apples and 1 kg of grapes on a day was found to be ₹160.  After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹300. Represent the situation algebraically and geometrically.
Solution:

### Class 10 Maths Pair of Linear Equations in Two Variables Mind Map

#### System of a Pair of Linear Equations in Two Variables

An equations of the form Ax + By + C = 0 is called a linear equation in two variables x and y where A, B, C are real numbers.
Two linear equations in the same two variables are called a pair of linear equations in two variables. Standard form of linear equations in two variables.
a1x + b1y + c1 = 0, a2x + b2y + c1 = 0
where a1, a2, b1, b2, c1, c2 are real numbers such that

#### Representation of Linear Equation In Two Variables

Every linear equation in two variables graphically represents a line and each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.

Solve using substitution calculator is used to solve the substitution of the linear equations.

#### Ploting Linear Equation in Two Variables on the Graph

There are infinitely many solutions of each linear equation. So, we choose at least any two values of one variable & get the value of other variable by substitution, i.e; Consider; Ax + By + C = 0 We can write the above linear equation as:

Here, we can choose any values of x & can find corresponding values of y.
After getting the values of (x, y) we plot them on the graph thereby getting the line representing Ax + By + C = 0.

#### Method of Solution of a Pair of Linear Equations in Two Variables

Coordinate of the point (x, y) which satisfy the system of pair of linear equations in two variables is the required solution. This is the point where the two lines representing the two equations intersect each other.
There are two methods of finding solution of a pair of Linear equations in two variables.
(1) Graphical Method : This method is less convenient when point representing the solution has non-integral co-ordinates.
(2) Algebraic Method : This method is more convenient when point representing the solution has non-integral co-ordinates.
This method is further divided into three methods:
(i) Substitution Method,
(ii) Elimination Method and
(iii) Cross Multiplication Method.

#### Consistency and Nature of the Graphs

Consider the standard form of linear equations in two variables.
a1x + b1y + C1 = 0; a2x + b2y + c2 = 0
While solving the above system of equation following three cases arise.
(i) If $$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$$; system is called consistent, having unique solution and pair of straight lines representing the above equations intersect at one point only
(ii) If $$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$$ ; system is called dependent and have infinetly many solution. Pair of lines representing the equations coincide.
(iii) If $$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}$$; system is called inconsistent and has no solution. Pair of lines representing the equations are parallel or do not intersect at any point.

#### Algebraic Method of Solution

Consider the following system of equation
a1x + b1y + c1 =0; a2x + b2y + c1 =0
There are following three methods under Algebraic method to solve the above system.

##### (i) Substitution method

(a) Find the value of one variable, say y in terms of x or x in terms of y from one equation.
(b) Substitute this value in second equation to get equation in one variable and find solution.
(c) Now substitute the value/solution so obtained in step (b) in the equation got in step (a).

##### (ii) Elimination Method

(a) If coefficient of any one variable are not same in both the equation multiply both the equation with suitable non-zero constants to make coefficient of any one variable numerically equal.
(b) Add or subtract the equations so obtained to get equation in one variable and solve it.
(c) Now substitute the value of the variable got in the above step in either of the original equation to get value of the other variable.

#### (iii) Cross multiplication method

For the pair of Linear equations intwo variables:
a1x + b1y + C1 = 0
a2x + b2y + c2 = 0
Consider the following diagram.

Solve it to get the solution, provided a1b2 – a2b1 ≠ 0

#### Equations Reducible to a Pair of Linear Equations in Two Variables

Sometimes pair of equations are not linear (or not in standard form), then they are altered so that they reduce to a pair of linear equations in standard form.
For example;

Here we substitute $$\frac{1}{x}$$ = p & \frac{1}{y} = q, the above equations reduces to:
a1p + b1q = c1 ; a2p – b2q = c2
Now we can use any method to solve them.

### NCERT Solutions for Class 10 Maths Chapter 3 Pairs of Linear Equations in Two Variables (Hindi Medium) Ex 3.1

We hope the NCERT Solutions for Class 10 Maths Chapter Pair of Linear Equations in Two Variables Ex 3.1, help you. If you have any query regarding NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.1, drop a comment below and we will get back to you at the earliest.


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# How to use horizontal fill in math mode for LyX

I am in math mode and I would like to push text (that justifies a step in an equation) to the end of the page. Usually I would apply "Horizontal Fill"; however, this option is removed when I am in math mode. For example, I would like to push the expression (by (3)) to the margin of the page.

Here is the source for my chain of equations.

\begin{eqnarray*}
\text{Hdim}K_{0}\left(\theta\right) & \le &
\limsup_{i\to\infty}\frac{\log\#\sigma\left(\alpha\right)}{\log n_{k_{i}+1}-\log
n_{k_{i}}}\\
& = & \limsup_{i\to\infty}\frac{\log a_{k_{i}+1}}{\log n_{k_{i+1}}-\log
\end{eqnarray*}

MWE: I would like expressions used to justify steps in a chain of equations/inequalities to appear on the right margin just as (3) does in the following pdf image

Here is the Latex Source for this example.

\begin{lem}
\begin{doublespace}
\textup{Let $\alpha\in J$ and $k=\left|\alpha\right|$.}
$$\#\sigma\left(\alpha\right)\le a_{k+1}$$
\end{doublespace}
\end{lem}

• Well actually your caption don't let me see well the equations for understand better your issue. Could you bring us a Minimal Working Example of your document for understand you better? – Aradnix Apr 29 '14 at 2:17
• I've added the Latex Source from my LyX document. – The Substitute Apr 29 '14 at 2:25
• I asked for a whole minimal working example, not only a portion of the document, but I see you don't use the amsmathpackage or another that improve and ease the work with mathematical equations. I have few ideas, but without a MWE perhaps will be better wait. – Aradnix Apr 29 '14 at 23:19

Use the \tag* macro from amsmath:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\begin{align}
y &= f(x) \tag*{foo}
\end{align}

\end{document}

You can insert it inside LyX with Ctrl-L (Eval Red Text) as LaTeX code: LyX inserts the backslash and insert tag*{foo} which will then be collapsed to:

However, do not use the environment eqnarray, it has a lousy formatting. Use environment align instead.

• Where should the code be inserted, and does it start at \begin or \documentclass{article}? My document class is aricle, and I included the \usepackage{amsmath} to my latex preamble, but I am not sure where/how to implement this. I'm not sure if ERT works in math mode, and I cannot generate a pdf when I paste this code in ERT. – The Substitute Apr 30 '14 at 6:23
• Write your equation as usual and at the end of the equation – user2478 Apr 30 '14 at 6:40
• Herbert, I see how to use this code if my chain of equalities is 1 line, but where to I insert this code if I have a vertical chain of equalities and I want "foo" to appear in one of the rows? – The Substitute Apr 30 '14 at 11:06
• then at the end of that row – user2478 Apr 30 '14 at 17:22


text:
# 3 DIFFERENCES between CHINA and GREECE… .3 SIMILARITIES between CHINA and GREECE

###### Question:

3 DIFFERENCES between CHINA and GREECE… .3 SIMILARITIES between CHINA and GREECE

### Es necesario estudiar mucho. = _________ que estudiar mucho.

Es necesario estudiar mucho. = _________ que estudiar mucho....

### At the bakery ratio of cookie sales to donut sales is 4 to 3.Cookies and donuts cost the same amount.If the bakery earned 375.00 on Tuesday from the sale of cookies.Write a proportion to find how much money the bakery earned from sale of donuts on Tuesday?

At the bakery ratio of cookie sales to donut sales is 4 to 3.Cookies and donuts cost the same amount.If the bakery earned 375.00 on Tuesday from the sale of cookies.Write a proportion to find how much money the bakery earned from sale of donuts on Tuesday?...

### What are three antonyms and three synonyms for the noun of the word Market? market as in a place to buy and sell.

What are three antonyms and three synonyms for the noun of the word Market? market as in a place to buy and sell....

### A geometric progression has a third term of 20 and a sum of infinity which is 3 times the first term. Find the first term

A geometric progression has a third term of 20 and a sum of infinity which is 3 times the first term. Find the first term ...

### If it is __________ in the Northern Hemisphere, then it must be __________ in the Southern Hemisphere.

If it is __________ in the Northern Hemisphere, then it must be __________ in the Southern Hemisphere....

### Can you write a thesis statement about how maps lie or anything relating to maps for my AP Human geography question?

Can you write a thesis statement about how maps lie or anything relating to maps for my AP Human geography question?...

### HELP 50 POINTSS AND BRAINLIEST Which statement is correct about the mass of carbon dioxide used during photosynthesis? It is equal to the total mass of oxygen used. It is less than the total amount of glucose and water used. It is equal to the total mass of glucose and oxygen formed. It is less than the total mass of glucose and oxygen formed.

HELP 50 POINTSS AND BRAINLIEST Which statement is correct about the mass of carbon dioxide used during photosynthesis? It is equal to the total mass of oxygen used. It is less than the total amount of glucose and water used. It is equal to the total mass of glucose and oxygen formed. It is less ...

### BEST ANSWER GETS BRAINLIEST!!!!! Two factors of –48 have a difference of 19. The factor with a greater absolute value is positive. What is the sum of the factors? A) -19 B) -13 C) 13 D) 16

BEST ANSWER GETS BRAINLIEST!!!!! Two factors of –48 have a difference of 19. The factor with a greater absolute value is positive. What is the sum of the factors? A) -19 B) -13 C) 13 D) 16...

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Which stars have the lowest absolute brightness?...

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A fund-raising company is putting together a student kit of candy bars for a school to use in its annual support campaign. To maximize sales, the company will put more of the most popular candy bar sold in an average day in each kit. The company’s research shows that of the five most popular candi...

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text:
# Coriolis force in free fall

Does the coriolis force has any measurable effect in free fall from large heights?

Take for example the sky diving experiment by <NAME> who started from a height of about 40 km above New Mexico. Does the coriolis force has any measurable effect in this experiment?

I think there should be a deviation to the east and another one to the south (which should be smaller in size). But I don't have an idea how large the deviation would be.

How can one calculate the size of the deviations?

-
forgive my ignorance but isn't the coriolis force only a matter of changing the frame of reference? The coriolis force shouldn't er... have a physical effect on Baumgartner right? –  drN Oct 21 '12 at 23:30
@drN, the coriolis force won't have an effect on Baumgartner in an inertial frame. But it will cause him to hit the ground at a different point than directly below where he began his fall. –  <NAME> Oct 22 '12 at 1:10
@ColinMcFaul True that! –  drN Oct 22 '12 at 12:46

In a rotating frame of reference, like the earth's, when we differentiate a dynamical variable (like the position of an object -- here, Baumgartner) with respect to time we get two terms. The first is just the (infinitesimal) change in position with respect to the (infinitesimal) change in time in the rotating reference system. The second term, not present in an inertial frame (a rotating frame is not inertial (why?)) is proportional to the (instantaneous) angular velocity of the rotating frame with respect to an (instantaneous) inertial frame of the object. The second term is the Coriolis effect and it arises due to the fact that the even when the object isn't moving with respect to the non-inertial frame, it's position in the inertial frame changes due to the fact that the position vector (say, $\mathbf{r}$) changes due to the fact that we chose a moving coordinate system. The relevant equation is \begin{align} \frac{d\mathbf{r}}{dt}\Big|_I &= \frac{d\mathbf{r}}{dt}\Big|_N + \boldsymbol{\omega}\times\mathbf{r}, \end{align} in a hopefully obvious notation. Of course, if you use a reference frame at rest with respect to, say, the center of the earth, you don't have any Coriolis terms in your equations. But you do have to take into account that the target, here the earth, is moving.
If you open a decent undergraduate or graduate classical mechanics book like Davis, Symon, or (egads!) Goldstein you can find a full development. But we can do an order of magnitude calculation, quick like. Baumgartner fell for 4m, 19s. Let's call this 250s, for ease in figuring. The magnitude of the angular velocity of the earth, which we'll take (incorrectly, but this gives an upper bound) as perpendicular to a position vector from the center of the earth to Baumgartner, is 2\pi / 24 hours. Since 24h = 86,400s the magnitude of the Coriolis effect on Baumgarnter results in an additional velocity of about 10^3/10^5 ($\omega$ in $s^{-1}$) * 10 (height in miles), giving about 1/10 of a mile/second. So Baumgartner would have had to correct for this effect. And this can be done with high precision.
@PieterMüller: with your assumptions, no. The Coriolis effect implies that the landing point is to the west of the point the object would land at if $\omega = 0$. This is because the earth rotates about it's axis with a tangential velocity to the east. The angular velocity of the earth is a vector whose direction points from south to north. The radius vector (in either frame) points from the center of the earth to the object. With this information you can calculate using the relevant equation above that the second term for $d\mathbf{r}/dt\Big|_N$, the cross term, points to the west. –  MarkWayne Nov 3 '12 at 5:25


text:
# Anisotropy Parameter¶

For linearly polarized light the angular distribution of photodetached electrons from negative ions is given by

$I(\epsilon, \theta) = \frac{\sigma_\text{total}(\epsilon)}{4\pi} [ 1 + \beta(\epsilon) P_2(\cos\theta)],$

where $$\beta(\epsilon)$$ is the electron kinetic energy ($$\epsilon$$) dependent anisotropy parameter, which varies between −1 and +2, and $$P_2(\cos\theta)$$ is the 2nd-order Legendre polynomial in $$\cos\theta$$. $$\sigma_\text{total}$$ is the total photodetachment cross section. The anisotropy parameter provides phase information about the dynamics of the photon process [1].

## Methods¶

PyAbel provides several methods to determine the anisotropy parameter $$\beta$$:

Method 1: linbasex evaluates $$\beta$$ directly, available as the class attribute Beta[1].

This method fits spherical harmonic functions to the velocity-map image to directly determine the anisotropy parameter as a function of the radial coordinate. This parameter has greater uncertainty in radial regions of low intensity, and so it is commonly plotted as the product $$I \times \beta$$. See Example: Linbasex.

Method 2: using abel.tools.vmi.radial_integration().

This method determines the anisotropy parameter from the inverse Abel-transformed image, by extracting intensity vs angle for each specified radial range and then fitting the intensity formula given above. This method is best applied to the radial ranges corresponding to strong spectral intensity in the image. It has the advantage of providing the least-squares fit error estimate for the parameter(s).

Method 3: using abel.tools.vmi.Distributions.

This method, like the previous one, works on the inverse Abel-transformed image, but fits the angular intensity dependence at each radius, providing radially dependent anisotropy parameters, like in the first method. If the anisotropy parameters are known to be smooth radial functions, a moving-window averaging can be employed for noise reduction.

## Example¶

See Example: Anisotropy parameter. In this case the anisotropy parameter is determined using each method. Note:

• In method 1, the filter parameter threshold=0.2 is set to a larger value so as to exclude evaluation in regions of weak intensity.
• Method 2 evaluates the anisotropy parameter for particular radial regions of strong intensity.
• In method 3, the anisotropy parameter is calculated with 9-pixel radial averaging and plotted only in the regions with > 1 % of the maximal intensity.

A demonstration of using Distributions for incomplete images is also included in Example: rBasex beam block.

## Reference¶

 [1] <NAME>, <NAME>, “Angular Distribution of Photoelectrons”, J. Chem. Phys. 48, 942–943 (1968).


text:
# Is Induction applicable only to well-ordered sets that are not bounded above?

I think the set needs to be bounded below to avoid occurrence of infinite descending chains for induction to be applicable but couldn't decide if it can be bounded above as well.

Inductive step is P(n) implies P(n+1). But if the set on which we are applying induction contains a greatest element, say a. And if P(a) is true then what can we say about P(a+1). I think induction would still be applicable whether or not a+1 belongs to the set.

I would really appreciate if an example could be provided in which induction is being applied on a set bounded above.

• It's funny that you bring up the transfinite-induction tag, as induction could be viewed as transfinite induction with an upper bound! – <NAME> Jan 26 '18 at 13:52

That is, if $a$ is your upper bound, then with the step you could say that what you are going to show is that: for all $0 \le n < a$: if $P(n)$ then $P(n+1)$


text:
tripes3h

2022-07-05

Let X be a random variable with $E\left[X\right]=1$ and such that $X{1}_{\left\{X\le 0\right\}}=0$ a.s.
Question Does it follow that $X>0$ a.s.?
My progress I tried to write $X={X}^{+}-{X}^{-}$, where both ${X}^{+},{X}^{-}$ are nonnegative, but I did not make any progress...

<NAME>

Expert

Observe that:
$X{1}_{X\le 0}\ne 0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}X\ne 0\wedge X\le 0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}X<0$
so that
$X{1}_{X\le 0}=0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}X\ge 0$
hence:

Do you have a similar question?


text:
# Contents

## Idea

One way to make sense of the path integral used for quantization of classical field theory to quantum field theory is to define it as a resummation of the the Feynman perturbation series of Feynman diagrams.

But for a generic action functional this prescription produces in each term of this series an ill-defined diverging term. Renormalization is a process of adjusting the action functional such that the classical field theory remains the same, but while at the same time adding interactions whose effect is to counteract these divergences to yield a series that is termwise finite. The terms for these extra interactions in the action functional are therefore called counterterms.

A well-developed method for renormalization, described in more detail below, expresses each of the terms in the perturbation series as a Laurent series in a certain parameter which is divergent at the parameter value of interest. In this description the process of renormalization corresponds to discarding the poles of this series.

The precise way in which this Laurent series is constructed has recently been understood to have an elegant desciption in terms of Hopf algebra and is now known as Hopf algebraic renormalization .

## Definitions

There are different procedures of renormalization

1. Stückelberg-Bogoliubov-Epstein-Glaser renormalizationenormalization)

2. BPHZ and Hopf-Algebraic renormalization

3. Of theories in BV-CS form

(…)

### BPHZ and Hopf-algebraic renormalization

#### The phenomenon

In the study of perturbative quantum field theory one is concerned with functions – called amplitudes – that take a collection of graphs – called Feynman graphs – to Laurent polynomials in a complex variable $z$ – called the (dimensional) regularization parameter

$Amplitude : CertainGraphs \to LaurentPolynomials$

and wishes to extract a “meaningful” finite component when evaluated at vanishing regularization parameter $z = 0$.

A prescription – called renormalization scheme – for adding to a given amplitude in a certain recursive fashion further terms – called counterterms – such that the resulting modified amplitude – called the renormalized amplitude – is finite at $z=0$ was once given by physicists and is called the BPHZ-procedure .

This procedure justifies itself mainly through the remarkable fact that the numbers obtained from it match certain numbers measured in particle accelerators to fantastic accuracy.

#### Its combinatorial Hopf-algebraic interpretation

The combinatorial Hopf algebraic approach to perturbative quantum field theory, see for instance

• <NAME>, <NAME>, Combinatorial Hopf algebras in quantum field theory I (arXiv),

starts with the observation that the BPHZ-procedure can be understood

• by noticing that there is secretly a natural group structure on the collection of amplitudes;

• which is induced from the fact that there is secretly a natural Hopf algebra structure on the vector space whose basis consists of graphs;

• and with respect to which the BPHZ-procedure is simply the Birkhoff decomposition of group valued functions on the circle into a divergent and a finite part.

The Hopf algebra structure on the vector space whose basis consists of graphs can be understood most conceptually in terms of pre-Lie algebras.

#### The Connes-Kreimer theorem

A Birkhoff decomposition of a loop $\phi : S^1 \to G$ in a complex group $G$ is a continuation of the loop to

• a holomorphic function $\phi_+$ on the standard disk inside the circle;

• a holomorphic function $\phi_-$ on the complement of this disk in the projective complex plane

• such that on the unit circle the original loop is reproduced as

$\phi = \phi_+ \cdot (\phi_-)^{-1} \,,$

with the product and the inverse on the right taken in the group $G$.

Notice that by the assumption of holomorphicity $\phi_+(0)$ is a well defined element of $G$.

###### Theorem

(Connes-Kreimer)

1. If $G$ is the group of characters on any graded connected commutative Hopf algebra $H$

$G = Hom(H,\mathbb{C})$

then the Birkhoff decomposition always exists and is given by the formula

$\phi_- : (X \in H) \mapsto Counit(X) - PolePartOf( Product(\phi_- \otimes \phi) \circ (1 \otimes (1 - Counit)) \circ Coproduct (X) ) \,.$
2. There is naturally the structure of a Hopf algebra, $H = Graphs$, on the graphs considered in quantum field theory. As an algebra this is the free commutative algebra on the “1-particle irreducible graphs”. Hence QFT amplitudes can be regarded as characters on this Hopf algebra.

3. The BPHZ renormalization-procedure for amplitudes is nothing but the first item applied to the special case of the second item.

###### Proof

The proof is given in

#### The Hopf-algebra perspective on QFT

This result first of all makes Hopf algebra an organizational principle for (re-)expressing familiar operations in quantum field theory.

Computing the renormalization $\phi_+$ of an amplitude $\phi$ amounts to using the above formula to compute the counterterm $\phi_-$ and then evaluating the right hand side of

$\underbrace{\phi_+}_{renormalized amplitude} = \underbrace{\phi}_{amplitude} \underbrace{\cdot}_{convolution product} \underbrace{\phi_-}_{counterterm} \,,$

where the product is the group product on characters, hence the convolution product of characters.

Every elegant reformulation has in it the potential of going beyond mere reformulation by allowing to see structures invisible in a less natural formulation. For instance <NAME> claims that the Hopf algebra language allows him to see patterns in perturbative quantum gravity previously missed.

#### Gauge theory and BV-BRST with Hopf algebra

<NAME> is thinking about the Hopf-algebraic formulation of BRST-BV methods in nonabelian gauge theory

• <NAME>?, Renormalization of gauge fields using Hopf algebra, (arXiv)

he reviews the central idea: the BRST formulation of Yang-Mills theory manifests itself at the level of the resulting bare i.e. unnormalized amplitudes in certain relations satisfied by these, the Slavnov-Taylor identities .

Renormalization of gauge theories is consistent only if these relations are still respected by renormalized amplitudes, too. We can reformulate this in terms of Hopf algebra now:

the relations between amplitudes to be preserved under renormalization must define a Hopf ideal in the Hopf algebra of graphs.

<NAME> proves this to be the case for Slavnov-Taylor in his theorem 9 on p. 12

As a payoff, he obtains a very transparent way to prove the generalization of Dyson’s formula to nonabelian gauge theory, which expresses renormalized Green’s functions in terms of unrenormalized Green’s functions “at bare coupling”. This is his corollary 12 on p. 13.

In the context of BRST-BV quantization these statements are subsumed, he says, by the structure encoded in the Hopf ideal which corresponds to imposing the BV-master equation. See also (Suijlekom).

### Of theories in BV-CS form

In (Costello 07) a comparatively simple renormalization procedure is given that applies to theories that are given by action functionals which can be given in the form

$S(\phi) = \langle \phi , Q \phi \rangle + I(\phi)$

where

1. the fields $\phi$ are sections of a graded field bundle $E$ on which $Q$ is a differential, $\langle -,-\rangle$ a compatible antibracket pairing such that $(E,Q, \langle \rangle)$ is a free field theory (as discussed there) in BV-BRST formalism;

2. $I$ is an interaction that is at least cubic.

These are action functionals that are well adapted to BV-BRST formalism and for which there is a quantization to a factorization algebra of observables.

Most of the fundamental theories in physics are of this form, notably Yang-Mills theory. In particular also all theories of infinity-Chern-Simons theory-type coming from binary invariant polynomials are perturbatively of this form, notably ordinary 3d Chern-Simons theory.

For a discussion of just the simple special case of 3d CS see (Costello 11, chapter 5.4 and 5.14).

For comparison of the following with other renormalization schemes, see at (Costello 07, section 1.7).

#### The setup

###### Definition

A free field theory $(E, \langle, -,-\rangle. Q)$:

Write $\mathcal{E}_c \coloneqq \Gamma_{cp}(E)$ for the space of sections of the field bundle of compact support. Write

$\langle -,-\rangle \;\colon\; \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C}$

for the induced pairing on sections

$\langle \phi, \psi\rangle = \int_{x \in X} \langle \phi(x), \psi(x)\rangle_{loc} \,.$

The paring being non-degenerate means that we have an isomorphism $E \stackrel{\simeq}{\to} E^* \otimes Dens_X$ and we write

$E^! \coloneqq E^* \otimes Dens_X \,.$

dynamics

• A differential operator on sections of the field bundle

$Q \;\colon\; \mathcal{E} \to \mathcal{C}$

of degree 1 such that

1. $(\mathcal{E}, Q)$ is an elliptic complex;

2. $Q$ is self-adjoint with respect to $\langle -,-\rangle$ in that for all fields $\phi,\psi \in \mathcal{E}_c$ of homogeneous degree we have $\langle \phi , Q \psi\rangle = (-1)^{{\vert \phi\vert}} \langle Q \phi, \psi\rangle$.

From this data we obtain:

• The action functional $S \colon \mathcal{E}_c \to \mathbb{C}$ of this corresponding free field theory is

$S \;\colon\; \phi \mapsto \int_X \langle \phi, Q \phi\rangle \,.$
• The classical BV-complex is the symmetric algebra $Sym \mathcal{E}^!$ of sections of $E^!$ equipped with the induced action of the differential $Q$ and the pairing

$\{\alpha,\beta\} \coloneqq \int_{x \in X} \langle \alpha(x), \beta(x)\rangle \,.$
###### Definition

An interaction term $I \in \cdots$

(…)

###### Definition

A gauge fixing operator $Q^{GF}$ such that

$H \coloneqq [Q, Q^{GF}]$

#### Operator (heat) kernels and propagators

###### Definition

For $K = \sum K' \otimes K'' \in \mathcal{E} \otimes \mathcal{E}$ the corresponding convolution operator $K \star \colon \mathcal{E} \to \mathcal{E}$ is

$K \star e = (-1)^{\vert e\vert} \sum K' \otimes \langle K'', e\rangle \,.$
###### Definition

For $H \colon \mathcal{E} \to \mathcal{E}$ a linear operator, a heat kernel for it is a function $K_{(-)} \colon \mathbb{R}_{\gt 0} \to \mathcal{E}\otimes \mathcal{E}$ such that for each $t \in \mathbb{R}_{\gt 0}$ the convolution with $K_t$, def. 4, reproduces the exponential of $t\cdotH$:

$K_t \star = \exp(-t H) \;\;\;\;\; \forall t \in \mathbb{R}_{\gt 0} \,.$
###### Proposition

For $H$ a generalized Laplace operator such as the $[Q,Q^{GF}]$ of def. 3 there is a unique heat kernel which is moreover a smooth function of $t$.

###### Definition

For $\phi = \sum \phi' \otimes \phi'' \in Sym^2 \mathcal{E}$ write

$\partial_{\phi} \coloneqq \frac{1}{2} \sum \partial_{\phi''} \partial_{\phi'} \,.$

This is (Costello 07, p. 32).

###### Proposition
$[\partial_\phi, Q] = \partial_{Q \phi} \,.$

#### The renormalization group operator

###### Definition

For $Q^{GF}$ the gauge fixing operator of def. 3, and $K_t$ the heat kernel of the corresponding generalized Laplace operator $H = [Q, Q^{GF}]$ by prop. 1, write for $\epsilon, T \in \mathbb{R}_{\gt 0}$

$P(\epsilon, T) \coloneqq \int_{\epsilon}^T (Q^{GF} \otimes 1) K_t \; d t \;\;\;\;\; \in \mathcal{E} \otimes \mathcal{E}\,.$
###### Definition

Write

$\Gamma(P(\epsilon,T), S) \coloneqq \hbar log \left( \exp\left(\hbar \partial_{P(\epsilon,T)}\right) \exp\left(I/\hbar\right) \right) \;\; \in \mathcal{O}(\mathcal{E}, \mathbb{C}[ [ \hbar ] ]) \,,$

where $\partial_{\cdots}$ is given by def. 6.

#### The path integral

###### Proposition

If $X = *$ is the point, then the path integral over the action functional exists as an ordinary integral and is equal to

$\hbar log \int_{x \in (Im Q^{GF})_{\mathbb{R}}} \exp\left( \tfrac{1}{2} \langle x, Q x\rangle + I(x + a) d \mu \right) = \Gamma(P(0,\infty), I)(a)$

This is (Costello 07, lemma 6.6.2).

###### Remark

For $X$ of positive dimension, the limit

$\underset{\epsilon \to 0}{\lim} \Gamma(P(\epsilon,\infty), I)(a)$

does not in general exist. Renormalization is the process of adding $\hbar$-corrections to the action – the counterterms – such as to make it exist after all. In this case we may regard the limit, by prop. 3, as the definition of the path integral.

#### Renormalized action

###### Definition

Given the action functional $S \colon \phi \mapsto \frac{1}{2}\langle \phi, Q phi\rangle + I(\phi)$, a renormalization is a power series

$I^R(\hbar, \epsilon) = I(\hbar) - \underset{{i \gt 0} \atop { k \geq k}}{\sum} \hbar^i I^{CT}_{i,k}(\epsilon)$

such that the limit

$\underset{\epsilon \to 0}{\lim} \Gamma(P(\epsilon,T), I^R(\epsilon)) \,,$

by def. 8, exists.

The $I^{CT}_{i,k}$ are called the counterterms.

#### Renormalization

###### Definition

A renormalization scheme is a decomposition of functions $\mathcal{A}$ on $(0,1)$, as a vector space, into a direct sum

$\mathcal{A} \simeq \mathcal{A}_{\geq 0} \oplus \mathcal{A}_{\gt 0}$

such that the functions $f \in \mathcal{A}_{\geq 0}$ are non-singular in that $\underset{\epsilon \to 0}{\lim} f(\epsilon)$ exists.

Hence this is a choice of picking the singularities in functions that are not necessarily defined at $\epsilon = 0$.

###### Theorem

Given any choice of renormalization scheme, def. 10, there exists a unique choice of counterterms $\{I^{CT}_{k,i}\}$, def. 9 such that

• each counterterm is in the chosen $\mathcal{A}_{\gt 0}$ as a function of $\epsilon$;

• for $k \gt 0$ the germ of a counterterm at $x \in X$ depends only on the germ of the given field theory data $(E, Q, Q^{GF})$ at that point.

This is (Costello 07, theorem B, p. 38).

###### Definition

Given a renormalization $I^{R}$, write for all $T \in \mathbb{R}_{\ht 0}$

$\Gamma^R(P(0,T), I) \coloneqq \underset{\epsilon \to 0}{\to} \Gamma(P(\epsilon, R), I - I^{CT}(\epsilon)) \,.$
###### Remark

We think of $\Gamma^R(P(0,T), I)$ as the renormalized effective action of the original action at scale? $T$.

###### Proposition
$\Gamma^R(P(0,T'), I) = \Gamma(P(T,T'), \; \Gamma^R(P(0,T), I ))$

holds.

## References

Review of renormalization and specifically of the Hopf algebra description of the BPHZ method is in

Reviews include

(in the context of effective field theory)

Original articles:

BPHZ renormalization was introduced in particular in

• K. Hepp.: Théorie de la Renormalisation Lect. Notes in Phys. Springer (1969)

### In BV formalism

Discussion in the context of BV-BRST formalism is in

building on

A vaguely related approach earlier appeared in

### In AQFT

A discussion of renormalization in the context of AQFT, as well as a detailed comparison of major renormalization schemes is in

For more see at perturbation theory – In AQFT.

### Hopf algebraic BPHZ formulation

• <NAME>, <NAME>, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), no. 1, 249–273, hep-th/9912092, MR2002f:81070, doi, II. The $\beta$-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001), no. 1, 215–241; hep-th/0003188, MR2002f:81071, doi

An introduction and review to the Hopf-algebraic description of renormalization is in

A textbook treatment is

• <NAME>, Knots and Fenyman diagrams , Cambridge Lecture Notes in Physics. 13. Cambridge: Cambridge University Press.
• <NAME>, The interface of noncommutative geometry and physics, hep-th/0206007

Some heavywheight automated computations using this formalism are discussed in

• <NAME>, Representing Feynman graphs on BV-algebras , (arXiv)


text:
How to use \pdfcolorstack?

Consider this code:

``````\chardef\Color=\pdfcolorstackinit page direct{0 g}
\pdfcolorstack \Color push {1 0 0 rg 1 0 0 RG}
red
\pdfcolorstack \Color pop {}
black
\hrule height 1pt width 50pt

\bye
``````

I try to use the color stack to write "red" in red and switch back to the previous color (black) and write the word "black" in that color (works fine). But why is the `\hrule` still affected by the color switch?

Obviously I don't know how to use `\pdfcolorstack`, but what is the intended semantic? I'd like to keep `1 0 0 RG` as I also want to set the color for stroking operations.

• a bit strange. As you hint, it works fine if you just push `{1 0 0 rg }` including colouring any rules that are in that scope. so presumably rules are filled rather than stroked, I haven't checked. Does pdftex introduce stroke rendering itself or do you mean to interact with other code from pdfliteral? latex uses rg and RG I'll see what it does ... – <NAME> Sep 13 '13 at 13:10

You have already found the missing `0 G` in the initialization. But the latter is not needed for the main color stack. From `NEWS` of pdfTeX:

• \pdfcolorstackinit
A stack is initialized by \pdfcolorstackinit. It expands to the number of the new stack. The common case, the traditinal color stack is already initialized as first stack: \pdfcolorstackinit page direct {0 g 0 G} The keyword "page" means that this stack is page based. At the start of the new page, the current stack value is automatically set.

All you need is:

``````\chardef\Color=0 %
``````

BTW, the LaTeX `color` package with driver `pdftex.def` can also be used with plain TeX via the help of `miniltx.tex`:

``````\input miniltx
\makeatletter
\input color.sty
\resetatcatcode

\textcolor{red}{red} \ black
\hrule height 1pt width 50pt

\bye
``````

• Great! I was able to significantly reduce my TeX code from 17 lines to 6 lines (4 TeX commands) :) – topskip Sep 13 '13 at 17:09

Embarrassing:

It seems as if two characters fix the issue (`0 G` in `\pdfcolorstackinit`):

``````\chardef\Color=\pdfcolorstackinit page direct{0 g 0 G}
\pdfcolorstack \Color push {1 0 0 rg 1 0 0 RG}
red
\pdfcolorstack \Color pop {}
black
\hrule height 1pt width 50pt

\bye
``````

It seems to be what PDFTeX uses to reset the color. I wonder where this is documented...


text:
# Joy in Discovery: Galois theory

post by <NAME> (eigil-rischel) · 2019-09-02T19:16:46.542Z · LW · GW · 3 comments

## Contents

  Requirements/Hints
None

Inspired by: Joy in Discovery [LW · GW], Class Project [LW · GW]

• There exists a polynomial of degree five over the rational numbers, with a root , so that cannot be obtained from the rationals by field operations and iterated extraction of roots.
• There is no set formulas, given in terms of field operations and iterated extraction of roots, expressing the roots of a fifth-degree polynomial over the rationals in terms of the coefficients.

Note that the former clearly implies the latter, but you are free to prove either.

By "field operations", i mean just . By "extraction of roots", i mean operations of the form "^n\sqrt{(-)}\$. You are allowed to take even roots of negative numbers, i.e to work inside the complex numbers.

To clarify:

• In the case of the first problem, I am looking for a specific equation of the form , with rational numbers, ( nonzero) which has at least one solution , so that cannot be written in terms of rational numbers, addition, subtraction, multiplication, division, and taking roots.
• When taking roots, you can also take a non-principal root. For instance, you can take both (the positive real fifth root of ten), but also the four other roots, , for .
• The solution may be a complex number.
• In the case of the second problem, we just want to prove that there is no single formula which works for all fifth-degree polynomials, expressing the roots in terms of the coefficients.

Reading further will give you more information, which may not be in the spirit of things. I will take care not to give too many spoilers, but read carefully.

(You have already been spoiled that the solution involves something called "Galois theory", but if you know what that is, you almost certainly already know it applies to this question).

I just started taking a course on Galois theory for my master's degree in mathematics. The course is pretty far below my level; I am taking it for annoying bureaucratic reasons. Despite that fact, although I have been exposed to the some of the ideas before(which probably made it significantly easier for me), I've never really scratched the surface of Galois theory. But during the first lecture, as the professor started sketching out the direction of the course, how we would [do Galois theory] and use this to show the impossibility of solving the general quintic, it occurred to me that I could pretty much see how the argument would go. After the lecture, I wrote it down. And then I thought back to the posts above, one of which I had recently read for the first time, and I thought: this may actually be an opportunity for someone else to practice.

(It was an opportunity for me to practice, but not as much as it would be for someone who hadn't just been told what input was necessary, as I had been)

From here, anything I consider a spoiler goes in rot13

## Requirements/Hints

In the most general sense, this question is solvable with nothing more than very simple 19th-century mathematics - we know, because it was solved by those methods. Nevertheless, if you want to solve this without being as good as Galois and Abel, perhaps some sort of checklist is nice, so you can know if you have a "fair shot" at this. I think anyone with an undergraduate degree in math probably has the necessary basics, but I am not certain about that. I will give the fields of math necessary, in very broad terms, but be warned that one of them may be non-obvious, and so constitute a mild spoiler: "Onfvp nofgenpg nytroen, v.r tebhc gurbel, evat gurbel, svryqf, cbylabzvny evatf.".

More specifically, there are some background concepts that may be hard to invent de novo. But again, these may contain more information than desired, in that it's not obvious that they apply to the problem at all. Nevertheless, they are "Gur abgvba bs "fbyinoyr tebhc". Gur vqrn bs nffbpvngvat n Tnybvf tebhc gb n svryq rkgrafvba, naq gur Tnybvf pbeerfcbaqrapr.."

I am very uncertain about the difficulty of this problem. It depends a lot on your mathematical background. But I don't think it's easy.

Update: After writing out my solution in more details, thinking through some of the things one would have to invent to tackle this, I am now fairly certain that this is a very hard problem.


text:
# Where are the non-trivial PHP-questions lately?

When I look at the first page of open PHP questions, I find almost only bad questions. E.g. one person doesn't understand what a "Notice" error message is, and wants it to be fixed. Most questions have poor indentation. Posters don't seem to be motivated to proofread their own questions even once. They are asking poor write-me-the-code questions that are so trivial that you are wondering if these people will ever be able to make money by working as a programmer. (Like "how do I write a function that multiplies its input by two, but only if it is smaller than 95". These kind of "questions" are usually missing the question mark ("?"), because they really are no questions. Or this question. Or using file_get_contents on a URL on the same webserver instead of just calling a proper function to get a value.) These questions are not the exception but the norm. There are almost no good questions.

Sorry for ranting...

I have a feeling that PHP questions should be flagged by default, and only un-flagged if there is approval from someone who can prove that he is able to get basic stuff done in the language. Well... I will probably stop answering PHP questions until this changes. When I have a question, I normally use Google and I find an answer. In the future I will probably use this site more for asking than for answering, at least regarding PHP.

• I almost had to clean my monitor after reading the first sentence of the last paragraph. – Mysticial Apr 27 '14 at 21:55
• I don't agree with the question needing to be flagged by default. They should be closed by default. – PeeHaa Apr 27 '14 at 21:56
• Maybe this explains a few things: blog.codinghorror.com/the-php-singularity. In case you've missed it, the PHP Tag Wiki is a gold mine; a substantial percentage of questions asked in the PHP tag on a regular basis could probably be closed as duplicates of some question found in the "frequently asked questions" section of the Tag Wiki. – <NAME> Apr 27 '14 at 22:00
• You don't have enough answers under your belt to post this. – <NAME> Apr 27 '14 at 22:02
• I think that this problem isn't only existent in the php tag. There's too much to rant about this. Think about students that don't have an idea what they are doing, developers by chance not by choice or helpvampires that got this far by sucking blood from good developers. – HamZa Apr 27 '14 at 22:02
• @RobertHarvey Room 11 is trying hard to kill new duplicates or plainly said, remove crappy questions/answers, but unfortunately the force is weak compared to influx of aformentioned. Partially, it's Stackoverflow fault because it enables such behavior. – <NAME> Apr 27 '14 at 22:14
• @TOOTSKI: I especially like the 9-star comment made by tereško in Room 11. Is he a regular, or does he just like to shop for bicycles in a PHP store? :P – <NAME> Apr 27 '14 at 22:21
• @RobertHarvey Yes, he's room owner too and he hates you. Your argument is not valid. – <NAME> Apr 27 '14 at 22:31
• The problem isn't limited to PHP, @HamZa... But I suspect they do get a bit more than their fair share of it. – Shog9 Apr 28 '14 at 6:43
• Your highest voted answer is 3. Your average answer score is 0.6 over 37 answers... You can't say this. – user1596138 Apr 28 '14 at 14:52
• @Will I know you're joking, but there's nothing you can't do in PHP. – ajacian81 Apr 28 '14 at 15:00
• @muistooshort How do you know I didn't mean "cone shaped answer"? ;) Good catch thx. – AaronLS Apr 29 '14 at 20:22
• @AaronLS Would a hyperbolic, parabolic, elliptical, or circular answer apply? I can come up with a hyperbolic answer for anything :) – mu is too short Apr 29 '14 at 20:48
• @Jhawins He can say whatever he likes, mate. – Lightness Races in Orbit May 1 '14 at 20:10
• @Will, "When has PHP ever been used for non trivial tasks?". Sadly, often, e.g. Facebook. – <NAME> May 2 '14 at 1:43

## 7 Answers

See also: "Why is Stack Overflow so negative of late?", and this specific answer. I'm a type 1 / type 4 hybrid, and it sounds like you're getting there as well.

Some of the problems you cite are not specific to . If you want to be truly frightened, take a peek at mornings GMT. The bad questions in make me sad, but the bad questions in make me rage. I don't have enough closevotes to make it through the latest 100 questions...

Some of the problems you cite are not specific to Stack Overflow. This has all happened before ... and will happen again. The internet calls it Eternal September. We're already shoving the community guidelines down people's throats and they still don't get it. They're beyond any structured help.

Some of the problems you cite are specific to a particular breed of programmer. Again, they're beyond any structured help.

Some of the problems you cite are not specific to PHP. All languages have newbies that ask stupid questions.

Some of the problems you cite are not specific to PHP developers. All new programmers ask stupid questions and most don't know how to search and research.

It just happens that PHP has such an incredibly low barrier to entry that it attracts people both new to the language and new to programming. This can make it seem like a huge number of the questions are mind-blowingly basic. Becoming competent at PHP seems to also correspond to figuring out how to Google. Correlation is not causation, mind you.

So, where are the good questions? You can often find interesting ones under framework and library tags. Advanced, complex modern libraries. Like, not Cake or CodeIgniter. Or Laravel, but I'm a snob. Notice that sometimes they aren't tagged ! This happens a lot. Maybe they're embarrassed?

Sometimes these advanced questions aren't actually even on Stack Overflow. By the time you're building an n-tier framework-leveraging enterprise-grade PHP application with all that this entails, you aren't going to be asking PHP questions that fit on SO any longer. I have a draft in progress (that I probably won't post) that I can only fit in to Programmers because it deals with software architectural concepts that will lead to opinion-based or experience-based answers instead of concrete facts. Hell, it might even be a better fit for Code Review.

The bottom line, though, is yes, there's a lot of crap, and no, that crap is never going away. Use closevotes as needed and use downvotes with little discrimination. PeeHaa was only serious about PHP questions needing to be closed by default nowadays. You might have only a limited number every day, but question downvotes are free and should be given out like candy.

The citizens of Room 11 have a closevote recommendation system, if you'd like to come help filter the ocean of horrible questions using a hand-held water purifier...

• Downvotes are free? You've only got 40 votes per day and they cost 1 rep per answer (free for questions though), whereas you have 50 close votes per day. But I never run out of votes, because by that time I would have run out of patience already. – Antony Apr 28 '14 at 7:32
• We're not talking about answers though. – BoltClock Apr 28 '14 at 7:32
• Hmm... it's been such a long time since I've cared enough to actually downvote all the things.. there is a limit, isn't there? – Charles Apr 28 '14 at 7:33
• Same thing (or even worse) with vba, where people who never programmed ask you to write code. – CommonGuy Apr 28 '14 at 14:30
• Oh, I am running into my limit almost everyday. And half of votes are going into void, as Stack Overflow is full of kind-hearted nurses who are always ready to comfort a poor noobie. And I limit myself to a few sub-tags, mind you - almost never taking my foot onto raw [php]. – Your Common Sense Apr 28 '14 at 14:31
• @YourCommonSense: Lucky you, lol. If you were following the MySQL tag you'd be neck deep in the PHP slime. (And the Ruby on Rails and Django slimes are not much better on Postgres.) :-| – <NAME> Apr 28 '14 at 14:52
• @Denis, I am not that lucky. Some time ago I've found myself a quiet refugee in mysqli tag - keeping me apart from many mysql questions... And then all of sudden treacherous PHP team decided to deprecate the old ext! Eventually directing all the sewage into my quiet backwater. – Your Common Sense Apr 28 '14 at 15:04
• To summarize several of your points: PHP is effectively the new VB6. Many of the people that yesterday would have learned VB, today learn PHP. VB's half-hearted and broken objects system gives way to PHP's disorganized and poorly documented API. – <NAME> Apr 28 '14 at 15:15
• @YourCommonSense Your ...And then all of sudden treacherous PHP team... comment is the funniest and most spot-on thing I've read in a while. – MonkeyZeus Apr 28 '14 at 21:21
• @Denis I hear you on the Rails/PostgreSQL mess. Not only are most Rails people ignorant about databases but the culture encourages this ignorance. I tend to stay away from ruby-on-rails these days to avoid the adolescent attitude. I suspect that soon we'll see thousands of "Rails is the new PHP" garbage apps kicking around, might be a good way to make money if you don't mind wearing a hazmat suit :) – mu is too short Apr 29 '14 at 19:48
• It's just a matter of time before ElitistStackOverflow.com is up and running, where you need to have 10k rep to ask a question. Don't get me wrong, I'm all for it, once I get 10k rep. – keyser May 2 '14 at 20:50

Yes. It happens once in a while.

The most funny fact is that without a manually attached bounty a good question scarcely gets an answer at all - the regular sewage flow is so fast, that good questions almost never have a chance to meet someone who can answer.

To me, this fact is a foul shame for a site that was intended for programmers but turned into online typo-spotting / trivial-code-writing service.

As of the general flow - don't forget answers: it is not uncommon, when a trivial question gets a half-dozen of answers, ALL of which are wrong. Because of the same tolerant policy - writing total rubbish costs you nothing but occasionally can bring some points from a fellow illiterate or a kind-hearted nurse. And even being forced to delete his answer, rep-whore keeps overall balance at zero.

The problem is that Stack Overflow has changed. And for a changed site one needs changed rules. There is always a balance between scarcity and abundance - one has to moderate them wisely, as the latter is no better than the former. When visitors are scarce, they have to be lured in with all the might. Yet, when they are overflowing, there should be some moderation.

Old man Parkinson once discovered that principle in his truly brilliant article "THE SHORT LIST, OR PRINCIPLES OF SELECTION", where he clearly shows how a position has to be equally fitted with baits and spikes.

I'll take the liberty to cite

Wanted-- Acrobat capable of crossing a slack wire 200 feet above raging furnace. Twice nightly, three times on Saturday. 53 Salary offered £25 (or $70 U.S.) per week. No pension and no compensation in the event of injury. Apply in person at Wildcat Circus between the hours of 9 A.M. and 10 A.M. The wording of this may not be perfect but the aim should be so to balance the inducement in salary against the possible risks involved that only a single applicant will appear. It is needless to ask for details of qualifications and experience. No one unskilled on the slack wire would find the offer attractive. It is needless to insist that candidates should be physically fit, sober, and free from fits of dizziness. They know that. It is just as needless to stipulate that those nervous of heights need not apply. They won't. The skill of the advertiser consists in adjusting the salary to the danger. An offer of £1000 (or$3000 U.S.) per week might produce a dozen applicants. An offer of £15 (or $35 U.S.) might produce none. Somewhere between those two figures lies the exact sum to specify, the minimum figure to attract anyone actually capable of doing the job. If there is more than one applicant, the figure has been placed a trifle too high. Let us now take, for comparison, a less extreme example. Wanted-- An archaeologist with high academic qualifications willing to spend fifteen years in excavating the Inca tombs at Helsdump on the Alligator River. Knighthood or equivalent honor guaranteed. Pension payable but never yet claimed. Salary of £2000 (or$6000 U.S.) per year. Apply in triplicate to the Director of the Grubbenburrow Institute, Sickdale, Ill., U.S.A.

Here the advantages and drawbacks are neatly balanced. There is no need to insist that candidates must be patient, 54 tough, intrepid, and single. The terms of the advertisement have eliminated all who are not.

I hope you get the idea.

The same principle should be applied to the modern SO.

1. First, number of baits can be reduced. Believe me, there are many people who are willing to share their knowledge just for sake of it, not for rep points or badges.
2. Second, number and size of spikes can be enraged. Say, current ratio 1:5 is embarrassing. Especially taking into account comforting upvotes (or solidarity ones, when two illiterates, got downvoted for their wild guess answers missing the point by a mile, happily upvoting each other). A comforting upvote will reward one fivefold! I'd make ratio 1:2.
Incidentally, why not to make an upvote to cost at least a penny? It will make all the repwhores think twice before spending!

There are could be other measures. But it's no use to sit and devise them, unless some political will to the change will be shown.

Beside other benefits, it will give back value to the SO reputation, which has devaluated into soil.

• Could you explain how that principle should be applied to SO? Not sure I quite follow what you mean. – femtoRgon Apr 28 '14 at 15:35
• okay added some. – Your Common Sense Apr 29 '14 at 3:31
• +1 for "without a manually attached bounty a good question scarcely gets an answer at all." Good questions become tumbleweeds. – James King Apr 29 '14 at 22:45

Downvote them... Seriously. You've 50 votes per day; use them to downvote 50 questions.

The "demonstrate a minimal understanding" close reason is no more, and the powers that be said "downvote" instead. So downvote.

Invite other PHP devs to do the same. With enough perseverance, people who should never have begun a profession in programming to begin with might stop asking. If not, at the very least you'll see giant walls of downvoted questions which you can ignore.

• -1 for the imaginary practice. It doesn't work in real Stack Overflow for a multitude reasons – Your Common Sense Apr 28 '14 at 14:21
• Any reference for "the powers that be said "downvote" instead"? I'd have guessed that "This question appears to be off-topic because it lacks sufficient information to diagnose the problem. Describe your problem in more detail or include a minimal example in the question itself." could be used instead. – Arjan Apr 28 '14 at 15:05
• Well, it could work if we actually followed the advice you've just downvoted @YourCommonSense. So the advice itself doesn't really deserve a downvote. The fact that it's not applied in practice does ... but we can't (yet) downvote such general notions. – Bart Apr 28 '14 at 15:16
• @Arjan: see discussions and comments related to meta.stackexchange.com/q/215596 or meta.stackexchange.com/q/211080 or meta.stackexchange.com/q/216585 – <NAME> Apr 28 '14 at 15:23
• @YourCommonSense - Then again I'm not implying I believe it would work, hehe. But seeing the reactions I got when I actually cared much (e.g. meta.stackexchange.com/q/178234), I'm at a loss as to what to hope for. – <NAME> Apr 28 '14 at 15:33
• @dallin: I fail to see the point you're trying to make, then, if you don't wish them to post here either. When you post a question, you research it before showing up asking what you've missed -- and it shows. Contrast these questions with the smelly sewage spewed by random incompetents on the PHP tag and elsewhere. The part that makes me cringe is how many of those inept coders are actually working as programmers. Because I'm later tasked by clients with cleaning up the puke they left behind. – <NAME> Apr 29 '14 at 21:59
• @Denis "inept coders with an IQ no larger than a shoe-size"? My point is you do not know their IQ, their proficiency, their area of focus, their circumstances, or anything. You don't know what they are trying to achieve. You don't know how long they've tried to learn programming. I understand you're frustrated with their questions that don't belong here, but telling people they should not try to become a programmer or that they're too stupid to without even knowing them is not an attitude we should have. – dallin Apr 29 '14 at 22:10
• @dallin: I think we'll have to agree to disagree for now, and revisit this chat in a few years after you've dealt with your fair share of stinky code bases left behind by incompetent subcontractors. – <NAME> Apr 29 '14 at 22:17
• @Denis lol, I agree to disagree as well. As a side note, I have dealt with a lot of incompetent code left by learning programmers and understand why it frustrates you. – dallin Apr 30 '14 at 0:06
• @dallin; It has nothing to do with skill level. Beginners, by definition, have a low skill level. It has everything to do with willingness to learn and improve, as opposed to just dumping questions on us. – <NAME> May 1 '14 at 20:18

Lack of administration will undoubtedly lead to more stupid questions. Though it may feel wasted at the time, scorning people for bad questions will likely lead to a more refined question that abides by the criteria further down the road. You just may not be there to witness it and see the result of your effort.

That's how SE started for me. I discovered the site, asked a really dumb question, and people yelled at me for it. You could probably see my inquisitions get better and more concise over the next 2 years. Being scolded really made me feel like an idiot and drove me to want to be better.

• I think that "scorning people for bad questions" is not really the right solution. It doesn't solve the problem because there is an endless supply of people with bad questions. I think we need to find ways to do a better job at stopping people from asking the bad questions before they do. – dallin Apr 29 '14 at 21:47

Smart people who started using PHP have outgrown it, and are asking good questions about other languages and web frameworks instead. Leaving just the people who don't know any better to stumble their way around PHP.

Trivial questions get pounced upon, like a piranha feeding frenzy. Or downvoted/closed, before you can even type "Google it". If all that bothers you, stop looking at the recent questions page! Bookmark these instead:

https://stackoverflow.com/questions/tagged/php?sort=unanswered&pageSize=50

Or if those are too hard, then try these: (a question has to survive a couple of days to have a bounty, so they should be worth reading)

https://stackoverflow.com/questions/tagged/php?sort=featured&pageSize=50

I'm amazed by the fact that everyone just blindly agrees with this statement. And answers such as downvote everything, and close all by default have the majority's support. I agree with the fact that a question should show sufficient effort of research trough use of Google at least. What I don't understand is why you doubt someone would make a living just because they lack knowledge of PHP. The person might not be interested in a career as a backend programmer but just needs help with a small partion of his/her webpage that requires PHP or some other language they never completely mastered. Perhaps the person asking the question is a hybrid webdesigner/frontend developer. Do you expect them to hire a PHP developer for every trivial problem they run in to?

So, to summerize. I agree that questions that show lack of effort are to be downvoted, however when the OP takes the time to edit the question, remove your downvote accordingly.

Also, if the question shows effort but the OP does not seem to have the correct knowledge of terminoligy, try pointing them in the right direction with hints trough comment like: "Try Googling random term".

A little less negativity and a little more patience would help this community a great deal.

Just my two cents.

• "Do you expect them to hire a PHP developer for every trivial problem they run in to?" No but we expect them to put in some minimal amount of effort before they post a new (duplicate/incomprehensible) question – OGHaza Apr 28 '14 at 15:11
• If people aren't actually programmers then this site is not for them. This is a site specifically created for professional and enthusiast programmers. It is not a site for people who don't want to learn how to program to have programmers search Google for them. If people can't even be bothered to do a minimum amount of research before asking a question, such as sticking their question into a search engine, then this site is, again, not for them. The site expects question authors to have done some research. If they are unwilling to do so, their questions don't belong here. – Servy Apr 28 '14 at 15:12
• It is people like you who are encouraging these users that the site has made very clear do not belong here that is the cause of this problem here. By encouraging people asking bad questions, they ask more bad questions. That is really bad for the site. – Servy Apr 28 '14 at 15:13
• Many a SO veteran wishes, as can be seen from the recent meta post related to SO negativity, that users hire a developer instead of posting questions like stackoverflow.com/q/17403265 – <NAME> Apr 28 '14 at 15:40
• the problem with stupid questions is that they do repell the "experts". and as soon as the noobs succeed in repelling the "experts", this site will become useless, because it then will be a site where noobs give advice to noobs. and by "expert" I mean everyone but the noobs, you know, people who understand the difference between a global variable, a local variable and an instance variable. a real expert (without quotes) will probably leave much earlier. – Michael Apr 29 '14 at 14:33
• @Servy "the site has made very clear do not belong here" - and this is why a lot of experienced developers don't want to participate here anymore. Some people want SO to be some sort of exclusive club. Their motivation is something other than "helping people" - particularly those who close questions as "too specific". It's like a doctor saying too few people have this disease to be worth my while trying to find a cure. – alcalde Apr 29 '14 at 21:06
• How do you know they didn't Google it first? SO factors in the first 5 spots on Google more often than not for any programming question I ask. I would assume most/all newbies found SO by searching for their problem on Google. After getting to SO, they looked around, couldn't find their answer so they did what seemed logical... they posted a question. – davidethell Apr 29 '14 at 21:07
• @alcalde So it seems you're not familiar with the mission of this site, to which the founders have made quite clear. It's not a place where someone goes to ask a question and have someone help them, to which some useless artifact is left around. That's what the site's competition was for decades, and the SO founders were tired of it. Rather than having the same questions asked day in day out, they wanted place to create a repository of knowledge in which each question creates an artifact for future visitors. That is why this site exists. – Servy Apr 29 '14 at 21:09
• @alcalde To stick with your analogy, consider SO the equivalent of a research hospital. It's not here to just help the handful of people that can pass through its doors each day; its goal is to benefit the entire community through expanded knowledge. There are non-research clinics where people can go for individualized care. In the same vein; if someone wants personalized help with a problem that's just for them, then there are places for that. SO isn't one of them. – Servy Apr 29 '14 at 21:11
• @Servy, and because this repository has such great depth, it is the first place they'll come from a Google search. – davidethell Apr 29 '14 at 21:11
• @davidethell When I can take the title of a question, stick it into google, and get an answer to the question, I can be fairly confident that the OP did not do sufficient research before asking their question. As one moves further along between obvious cases like that, and harder cases where I can only find the question because I happened to have already known the "magic words" to find it, there is a grey area. There certainly is a degree of subjectivity as to what level of research is to be expected, and whether a given question meets that criteria. – Servy Apr 29 '14 at 21:11
• @davidethell Okay. Fine. If they can find an answer from another SO question, great, if they can't they should still look elsewhere. After all, SO is not a replacement for documentation, tutorials, etc. – Servy Apr 29 '14 at 21:12
• @Servy, why post if not for personalized help? The only questions I've posted were because I was stuck on a personal problem. I didn't necessarily care if the problem was going to help N number of programmers down the road. I posted a question, hopefully well-formed, and almost always have gotten great answers. I'm sure several of my questions are edge cases or were fairly minor once the problem came to light. – davidethell Apr 29 '14 at 21:13
• @Servy, ok, I'll grant you those last two comments. I do agree that posting on SO should be after exhausting as many other options as possible. More Google searching is usually the better solution most of the time. – davidethell Apr 29 '14 at 21:14
• @davidethell Whether someone comes here for personalized help isn't really the issue. The whole design of the site is that someone comes here for personalized help, they get it, but in the process a useful and high quality artifact is created for future googlers. If the question is one that cannot, or probably will not, generate such an artifact, then it should be downvoted/closed. The motivations of the asker aren't the issue here, but whether or not the question is of sufficient quality to be useful. Ironically, most questions people ask intending to create useful artifacts are low quality. – Servy Apr 29 '14 at 21:16


text:
# Impedance calculations. Multisim VS Hand-calculated

Discussion in 'Homework Help' started by JamesCA, Jul 13, 2015.

1. ### JamesCA Thread Starter New Member

Sep 4, 2013
2
0
Hello all,

i'm doing some impedance calculations, and what was going to be prove of concept has failed . The diagrams are pretty self-explanatory:

The circuit, simulated in multisim. What i'm interested in is the output RMS voltage at probe 1 (with no load).

My calculations:

Xc = 1/(2*pi*f*c) = 1/(2*pi*100hz*1e-6) = 1591.5ohm
R = 1k ohm

for completeness:

Z1 is just the real part of Z1 which is R = 1k
Z2 is just the imaginary part of Z2 which is Xc= 1591.5 ohm

which gives me, using voltage divider formula ==> vout = 20*(Z2/(Z1+Z2)) = 20*(1591.5/(2591.5)) = 12.28v RMS

Multisim gives however 16,9 volts. I must be doing some fundimental error here. This is not homework, but seemed the best section to fit this in.

2. ### Lool Member

May 8, 2013
117
12
I got the same answer you did.

3. ### Jony130 AAC Fanatic!

Feb 17, 2009
4,082
1,149
Your calculation are wrong. The correct one looks like this

$Vout=Vin*\frac{Xc}{\sqrt{R^2+Xc^2}}=20*\frac{1.59}{\sqrt{1^2+1.59^2}}=16.9V$

4. ### Lool Member

May 8, 2013
117
12
Yep, we both forgot the j=sqrt(-1) for the capacitive reactance. Xc=-j/(2piC) which will give a phase shift and magnitude 16.9V

5. ### WBahn Moderator

Mar 31, 2012
19,145
5,177
The sqrt(-1) is not part of the reactance.

Z = R + j X

where R is the resistance and X is the reactance.

In this case, X = -1/(2piC)

6. ### Lool Member

May 8, 2013
117
12
Yes, sorry for my miswording. I'm rusty on this topic.


text:
Enter the current value and the initial value into the Reduction Rate Calculator. The calculator will evaluate the Reduction Rate.

Reduction Rate Formula

The following two example problems outline the steps and information needed to calculate the Reduction Rate.

RR = (IV -CV) / IV * 100
• Where RR is the Reduction Rate (%)
• CV is the current value
• IV is the initial value

To calculate the reduction rate, divide the difference in values by the initial value, then multiply by 100.

How to Calculate Reduction Rate?

The following example problems outline how to calculate Reduction Rate.

Example Problem #1:

1. First, determine the current value.
• The current value is given as: 30.
2. Next, determine the initial value.
• The initial value is provided as: 50.
3. Finally, calculate the Reduction Rate using the equation above:

RR = (IV -CV) / IV * 100

The values provided above are inserted into the equation below and computed.

RR = (50 -30) / 50 * 100 = 40.00 (%)

Example Problem #2:

For this problem, the variables required are provided below:

current value = 40

initial value = 100

Test your knowledge using the equation and check your answer with the calculator..

RR = (IV -CV) / IV * 100 = ?


text:
Ask Your Question

# Revision history [back]

### How to iterate over coefficients

I am a relative newbie to Sage but not to algebraic manipulation in general.

I have a family of expressions with non-linear coefficients which I am not sure how best to represent. In the end it has to be an (infinite) ring of polynomials representing a Volterra space, but I have the same terms inside non-linear functions (which I need to evaluate to a known value) and as the polynomial variables.

Here is an example, where simple symbolic manipulation doesn't work:

f_m = a*x^b
ex = expand(f_m.subs({b : b_m + (b_s-b_m)^3})).derivative(b_m).full_simplify()
ex.subs({b_m + (b_s-b_m)^3 : b_m }).show()

Producing:

  -(3*a*b_m^2 - 6*a*b_m*b_s + 3*a*b_s^2 - a)*x^(-b_m^3 + 3*b_m^2*b_s - 3*b_m*b_s^2 + b_s^3 + b_m)*log(x)

When what I want is:

  -(3*a*b_m^2 - 6*a*b_m*b_s + 3*a*b_s^2 - a)*x^(-b_m)*log(x)

How can I iterate over non-linear elements (exponent and the arguments to any non-monomial terms) carrying out direct substitutions, while leaving monomials alone. I want the result to remain as a polynomial ring on b_s

### How to iterate over coefficients

I am a relative newbie to Sage but not to algebraic manipulation in general.

I have a family of expressions with non-linear coefficients which I am not sure how best to represent. In the end it has to be an (infinite) ring of polynomials representing a Volterra space, but I have the same terms inside non-linear functions (which I need to evaluate to a known value) and as the polynomial variables.

Here is an example, where simple symbolic manipulation doesn't work:

f_m = a*x^b
ex = expand(f_m.subs({b : b_m + (b_s-b_m)^3})).derivative(b_m).full_simplify()
ex.subs({b_m + (b_s-b_m)^3 : b_m }).show()

Producing:

  -(3*a*b_m^2 - 6*a*b_m*b_s + 3*a*b_s^2 - a)*x^(-b_m^3 + 3*b_m^2*b_s - 3*b_m*b_s^2 + b_s^3 + b_m)*log(x)

When what I want is:

  -(3*a*b_m^2 - 6*a*b_m*b_s + 3*a*b_s^2 - a)*x^(-b_m)*log(x)

How can I iterate over non-linear elements (exponent and the arguments to any non-monomial terms) carrying out direct substitutions, while leaving monomials alone. I want the result to remain as a polynomial ring on b_s

Clarification: The intent is to use the code for a generic non-linear construct. It's intended to handle any general non-linear expression in L2, but these are manipulated to belong to a class of Volterra polynomials with non-linear coefficients. like the log(x) and x^bm on the above example. Those are the coefficients of the monomials.

Idealy whatever process I follow should be able to identify the non-monomial coefficients in the expression and remove the monomial terms from them by doing the substitution b_s = b_m. Basically:

    x.subs(b_s = b_m) for x not a monomial term in expression.

Which would make the resulting expression strictly a polynomial ring on b_s with non-linear coefficients on x and b_m.

### How to iterate over coefficients

I am a relative newbie to Sage but not to algebraic manipulation in general.

I have a family of expressions with non-linear coefficients which I am not sure how best to represent. In the end it has to be an (infinite) ring of polynomials representing a Volterra space, but I have the same terms inside non-linear functions (which I need to evaluate to a known value) and as the polynomial variables.

Here is an example, where simple symbolic manipulation doesn't work:

f_m = a*x^b
ex = expand(f_m.subs({b : b_m + (b_s-b_m)^3})).derivative(b_m).full_simplify()
ex.subs({b_m + (b_s-b_m)^3 : b_m }).show()

Producing:

  -(3*a*b_m^2 - 6*a*b_m*b_s + 3*a*b_s^2 - a)*x^(-b_m^3 + 3*b_m^2*b_s - 3*b_m*b_s^2 + b_s^3 + b_m)*log(x)

When what I want is:

  -(3*a*b_m^2 - 6*a*b_m*b_s + 3*a*b_s^2 - a)*x^(-b_m)*log(x)

How can I iterate over non-linear elements (exponent and the arguments to any non-monomial terms) carrying out direct substitutions, while leaving monomials alone. I want the result to remain as a polynomial ring on b_s

Clarification: The intent is to use the code for a generic non-linear construct. It's intended to handle any general non-linear expression in L2, but these are manipulated to belong to a class of Volterra polynomials with non-linear coefficients. like the log(x) and x^bm on the above example. Those are the coefficients of the monomials.

Idealy whatever process I follow should be able to identify the non-monomial coefficients in the expression and remove the monomial terms from them by doing the substitution b_s = b_m. Basically:

    x.subs(b_s = b_m) for x not a monomial term in expression.

Which would make the resulting expression strictly a polynomial ring on b_s with non-linear coefficients on x and b_m.


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# Convergent Periodic Sequence

how do I show that every convergent periodic sequence of Real numbers is constant?I have the intuition that is true,but I don't know how to prove.thanks.

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## 1 Answer

HINT: Let $\langle a_k:k\in\mathbb{N}\rangle$ be your sequence. Suppose that it has period $p$ and that $a_n\ne a_m$. What can you say about the subsequences $\langle a_{n+kp}:k\in\mathbb{N}\rangle$ and $\langle a_{m+kp}:k\in\mathbb{N}\rangle$?

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they are constant, and have different limits, thus the sequence would have 2 accumulation points, a contradiction.Thanks. –  Jr. Jan 13 '12 at 0:47
@Jr.: Yep, you’ve got it. –  <NAME> Jan 13 '12 at 0:54
Another one: If a monotone sequence has convergent subsequence,prove that the sequence itself is convergent. –  Jr. Jan 13 '12 at 1:01
@Jr.: Let $L$ be the limit of the convergent subsequence $\langle a_{n_k}:k\in\mathbb{N}\rangle$. Assume without loss of generality that the sequence is non-decreasing. Let $\epsilon>0$; there is an $m$ such that $a_{n_k}\in(L-\epsilon,L]$ whenever $k\ge m$. What can you say about $a_k$ when $k\ge n_m$? Use monotonicity. –  <NAME> Jan 13 '12 at 1:07