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# IB Notes Explaining Thermal Physics For IB Physics

22 APR 2020

## Temperature And Heat Flow

• The direction of the natural flow of thermal energy between two objects is determined by the 'hotness' of each object.

• Thermal energy naturally flows from hot to cold. The temperature of an object is a measure of how hot it is. In other words, if two objects are placed in thermal contact, then the temperature difference between the two objects will determine the direction of the natural transfer of thermal energy. Thermal energy is naturally transferred 'down' the temperature difference - from high temperature to low temperature.

• Eventually, the two objects would be expected to reach the same temperature. When this happens, they are said to be in thermal equilibrium.

• Heat is not a substance that flows from one object to another. What has happened is that thermal energy has been transferred. Thermal energy (heat) refers to the non-mechanical transfer of energy between a system and its surroundings.

## Kelvin And Celsius

• In order to use Kelvin and Celsius, you do not need to understand the details of how either of these scales has been defined, but you do need to know the relation between them. Most everyday thermometers are marked with the celsius scale and temperature Is quoted in degrees Celsius (°C).

• There is an easy relationship between a temperature 'Tas' measured on the Kelvin scale and the corresponding temperature 't' as measured on the Celsius scale. The approximate relationship is T (K) = (°C) + 273. This means that the 'size' of the units used on each scale is identical, but they have different zero points

• The Kelvin scale is an absolute thermodynamic temperature scale and a measurement on this scale is also called the absolute temperature. Zero Kelvin is called absolute zero.

## Gases

• For a given sample of gas, the pressure, the volume and the temperature are all related to one another.

• The SI units of pressure are $N{m}^{-2}$ or Pa (Pascals). 1*Pa = 1*N

• The temperature, t, of the gas is measured in °C or K

• In order to investigate how these quantities are interrelated, we choose:

•  one quantity to be the independent variable (the thing we alter and measure)

• another quantity to be the dependent variable (the second thing we measure).

• The third quantity needs to be controlled (i.e. kept constant ).

• The specific values that will be recorded also depend on the mass of gas being investigated and the type of gas being used so these need to be controlled as well.

## Internal Energy Of Molecules

• The molecules have kinetic energy. They are moving. To be absolutely precise. a molecule has either translational kinetic energy
• The molecule also has potential energy because of the intermolecular forces.
• Molecules with large mass move with a lower average speed
• Molecules with a small mass, move with a higher average speed

## Specific Heat Capacity

• Two different blocks with the same mass and same energy input will have a different temperature change. We define the thermal capacity C of an object as the energy required to raise its temperature by 1 K.
• Different objects (even different samples of the same substance) will have different values of heat capacity.
• Specific heat capacity is the energy required to raise a unit mass of a substance by 1 K. 'Specific' here just means 'per unit mass'.
• A particular gas can have many different values of specific heat capacity—it depends on the conditions.
• It generally takes the same amount of energy to raise the temperature of an object from 25 °C to 35 °C as it does for the same object to go from 402 °C to 412 °C. This is only true so long as energy is not lost from the object.
• If an object is raised above room temperature, it starts to lose energy. The hotter it becomes, the greater the rate at which it loses energy.

• ## Electrical Method

• In this method, the following are the sources of experimental error come from the loss of thermal energy from the apparatus.
• The container for the substance and the heater will also be warmed up.
• It will take some time for the energy to be shared uniformly through the substance.
• ## Method of Mixtures

• The known specific heat capacity of one substance can be used to find the specific heat capacity of another substance.
• Procedure:
•  Measure the masses of the liquids
•  Measure the two starting temperatures
•  Mix the two liquids together.
• Record the maximum temperature of the mixture
• If no energy is lost from the system then, energy lost by hot substance cooling down = energy gained by cold substance heating up
• The main source of experimental error is
• The loss of thermal energy from the apparatus; particularly while the liquids are being transferred.
• The changes in the temperature of the container also need to be taken into consideration for a more accurate result.

## The States Of Matter And Latent Heat

• When a substance changes phase, the temperature remains constant even though thermal energy is still being transferred.

• Cooling Curve For Molten Lead (Idealized)

• The amount of energy associated with the phase change is called the latent heat. The technical term for the change of phase from solid to liquid is fusion and the term for the change from liquid to gas is vaporization.

• The energy given to the molecules does not increase their kinetic energy so it must be increasing their potential energy. Intermolecular bonds are being broken and this takes energy. When the substance freezes bonds are created and this process releases energy.

• It is a very common mistake to think that the molecules must speed up during a phase change. The molecules in water vapour at 100 °C must be moving with the same average speed as the molecules in liquid water at 100 °C.

• The specific latent heat of a substance is defined as the amount of energy per unit mass absorbed or released during a change of phase. In symbols,

• Specific latent heat

• In the idealized situation of no energy loss, a constant rate of energy transfer into a solid substance would result in a constant rate of increase in temperature until the melting point is reached:

• The specific heat capacity of the solid as the gradient of the line that corresponds to the liquid phase is greater than the gradient of the line that corresponds to the solid phase. A given amount of energy will cause a greater increase in temperature for the liquid when compared with the solid.

## Methods Of Measuring Latent Heats

• The two possible methods for measuring latent heats are very similar in principle to the methods for measuring specific heat capacities. A method for measuring the

• Method 1: Electrical Circuit.

• The amount of thermal energy provided to water at its boiling point is calculated using electrical energy = I t V. The mass vaporized needs to be recorded.

• The specific latent heat

• Sources of experimental error

• Loss of thermal energy from the apparatus.

• Some water vapour will be lost before and after timing.

• Method 2: Fusion Of Water

• Providing we know the specific heat capacity of water, we can calculate the specific latent heat of fusion for water. For example, ice (at 0 °C) is added to warm water and the temperature of the resulting mix is measured.

• If no energy is lost from the system then, energy lost by water cooling down = energy gained by the ice

• Sources of experimental error

• Loss (or gain) of thermal energy from the apparatus.

• If the ice had not started at exactly zero, then there would be an additional term in the equation in order to account for the energy needed to warm the ice up to 0 °C.

•  The water clinging to the ice before the transfer.

## Gas Laws 1

• Although pressure and volume both vary linearly with Celsius temperature, neither pressure nor volume is proportional to Celsius temperature.
•  A different sample of gas would produce a different straight-line variation for pressure (or volume) against temperature but both graphs would extrapolate back to the same low temperature, -273 °C. This temperature is known as absolute zero.
• As pressure increases, the volume decreases. In fact, they are inversely proportional.
• At constant  constant (the pressure law)
• At constant  constant (Charles's law)
• At constant constant (Boyle's law)
• These relationships are known as Ideal gas laws.
• The temperature is always expressed in Kelvin
• . These laws do not always apply to experiments done with real gases.
• A real gas is said to 'deviate' from ideal behaviour under certain conditions (e.g. high pressure).

## Gas Laws 2

• The three ideal gas laws can be combined together to produce one mathematical relationship: constant

• This constant will depend on the mass and type of gas.

• If we compare the value of this constant for different masses of different gases, it turns out to depend on the number of molecules that are in the gas - not their type.

• In this case, we use the definition of the mole to state that for n moles of an ideal gas: a universal constant

• The universal constant is called the molar gas constant R. The SI unit for R is J

•  R = 8.314 J $mo{l}^{-1}$ ${K}^{-1}$

## Definitions

The concepts of the mole, molar mass and the Avogadro constant are all introduced so as to be able to relate the mass of gas (an easily measurable quantity) to the number of molecules that are present in the gas.

• Ideal gas
• An ideal gas Is one that follows them, laws for all values of P, Vand T
• ## Mole

• It is the basic SI unit for 'amount of substance'. One mole of any substance is equal to the amount of that substance that contains the same number of particles as 0.012 kg of carbon-12. When writing the unit it is (slightly) shortened to the mol.

• ## Molar mass

• The mass of one mole of a substance is called the molar mass. A simple rule applies. If an element has a certain mass number, A, then the molar mass will be A grams.

• This is the number of atoms in 0.012 kg of carbon-12. It is 6.02 x ${10}^{23}$.

• ## Ideal Gases And Real Gases

• An ideal gas is a one that follows the gas laws for all values of p, V and T and thus ideal gases cannot be liquefied. Real gases, however, can approximate to ideal behaviour providing that the intermolecular forces are small enough to be ignored. For this to apply, the pressure/density of the gas must be low and the temperature must be moderate.

## Molecular Model Of An Ideal Gas

• Kinetic Model Of An Ideal Gas
• Assumptions:
• Newton's laws apply to molecular behaviour
• There are no intermolecular forces except dining a collision
• The molecules are treated as points
• The molecules are in random motion
• The collisions between the molecules arc elastic (no energy is lost)
• There is no time spent in these collisions.

• The pressure of a gas is a result of collisions between the molecules and the walls of the container in which that gas has been stored.
• When a molecule bounces off the walls of a container its momentum changes (due to the change in direction - momentum is a vector).
• There must have been a force on the molecule from the wall (Newton II).
• There must have been an equal and opposite force on the wall from the molecule (Newton III).
• Each time there is a collision between a molecule and the wall, a force is exerted on the wall.
• The average of all the microscopic forces on the wall over a period of time means that there is effectively a constant force on the wall from the gas.
• This force per unit area of the wall is what we call pressure. $P=\frac{F}{A}$.
•  Since the temperature of a gas is a measure of the average kinetic energy of the molecules, as we lower the temperature of a gas the molecules will move slower.
• At absolute zero, we imagine the molecules to have zero kinetic energy. We cannot go any lower because we cannot reduce their kinetic energy any further!

## Pressure Law

• Macroscopically, at a constant volume, the pressure of a gas is proportional to its temperature in kelvin.
• Microscopically this can be analysed as follows
• If the temperature of gas goes up, the molecules have more average kinetic energy - they are moving faster on average.
• Fast-moving molecules will have a greater chance of momentum when they hit the walls of the container. Thus the microscopic force from each molecule will be greater.
• The molecules are moving faster so they hit the walls more often.
•  For both these reasons, the total force on the wall goes up.
• Thus the pressure goes up. low-temperature high temperature

## Charles' Law

• Macroscopically, at constant pressure, the volume of a gas is proportional to its temperature in kelvin.

• Microscopically this can be analysed as follows

• A higher temperature means faster moving molecules.

• Faster moving molecules hit the walls with a greater microscopic force

• If the volume of the gas increases, then the rate at which these collisions take place on a unit area of the wall must go down.

• The average force on a unit area of the wall can thus be the same. Thus the pressure remains the same.

## Boyle's Law

• Macroscopically, at a constant temperature, the pressure of a gas is inversely proportional to its volume.
• Microscopically this can be seen to be correct.
• The constant temperature of gas means that the molecules have a constant average speed.
• The microscopic force that each molecule exerts on the wall will remain constant.
• Increasing the volume of the container decreases the rate with which the molecules hit the wall -the average total force decreases.
• If the average total force decreases the pressure decreases.

## Sample Questions

• When the volume of a gas is isothermally compressed to a smaller volume, the pressure exerted by the gas on the container walls increases. The best microscopic explanation for this pressure increase is that at the smaller volume
• A. the individual gas molecules are compressed
• B. the gas molecules repel each other more strongly
• C. the average velocity of gas molecules hitting the wall is greater
• D. the frequency of collisions with gas molecules with the walls is greater

•  A lead bullet is fired into an iron plate, where it deforms and stops. M a result, the temperature of the lead increases by an amount AT. For an identical bullet hitting the plate with twice the speed, what is the best estimate of the temperature increase?
• A. $∆$T
• B. 2 $∆$T
• C. 4 $∆$T

• In winter, in some countries, the water in a swimming pool needs to be heated.
• Estimate the cost of heating the water in a typical swimming pool from 5 °C to a suitable temperature for swimming. You may choose to consider any reasonable size of the pool. Clearly show any estimated values. The following information will be useful:
• Specific heat capacity of water 4186 J
• The density of water 1000 kg ${m}^{-3}$
• Cost per kW h of electrical energy \$0.10
• (i) Estimated values
• (ii) Calculations

• This question is about determining the specific latent heat of fusion of ice. A student determines the specific latent heat of fusion of ice at home. She takes some ice from the freezer, measures its mass and mixes it with a known mass of water in an insulating jug. She stirs until all the ice has melted and measures the final temperature of the mixture. She also measured the temperature in the freezer and the initial temperature of the water. She records her measurements as follows:

• Mass of ice used: ${m}_{1}$ = 0.12 kg
• The initial temperature of ice: Ti —12 °C
• The initial mass of water: ${m}_{w}$ =  0.40 kg
• The initial temperature of water: ${T}_{w}$ = 22 °C
• The final temperature of the mixture: ${T}_{f}$= 15 °C
• The specific heat capacities of water and ice are
• ${c}_{w}$= 4.2 kJ $k{g}^{-1}$ °${C}^{-1}$ and ${c}_{i}$ =  2.1 kJ $k{g}^{-1}$°${C}^{-1}$

•  a) Set up the appropriate equation, representing energy transfers during the process of coming to thermal equilibrium, that will enable her to solve for the specific latent heat Li of ice. Insert values into the equation from the data above, but do not solve the equation.
• b) Explain the physical meaning of each energy transfer term in your equation (but not each symbol).
•  State an assumption you have made about the experiment, in setting up your equation in (a).
• d) Why should she take the temperature of the mixture immediately after all the ice has melted?
• e) Explain from the microscopic point of view, in terms of molecular behaviour, why the temperature of the ice does not increase while it is melting.


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Cy� ��*����xM���)>���)���C����3ŭ3YIqCo �173\hn�>#|�]n.��. Thus, if the random variable X is log-normally distributed, then Y = ln (X) has a normal distribution. Find the area under the standard normal curve to the left of 0.87. Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. $$P(Z<3)$$ and $$P(Z<2)$$ can be found in the table by looking up 2.0 and 3.0. It assumes that the observations are closely clustered around the mean, μ, and this amount is decaying quickly as we go farther away from the mean. To find the area between 2.0 and 3.0 we can use the calculation method in the previous examples to find the cumulative probabilities for 2.0 and 3.0 and then subtract. This is a special case when $${\displaystyle \mu =0}$$ and $${\displaystyle \sigma =1}$$, and it is described by this probability density function: The&normal&distribution&with¶meter&values µ=0&and σ=&1&iscalled&the&standard$normal$distribution. $$P(2 < Z < 3)= P(Z < 3) - P(Z \le 2)= 0.9987 - 0.9772= 0.0215$$. The Normal distribution is a continuous theoretical probability distribution. 624 0 obj<>stream You may see the notation $$N(\mu, \sigma^2$$) where N signifies that the distribution is normal, $$\mu$$ is the mean, and $$\sigma^2$$ is the variance. Fortunately, as N becomes large, the binomial distribution becomes more and more symmetric, and begins to converge to a normal distribution. N- set of population size. 0000010595 00000 n 0000001596 00000 n Go down the left-hand column, label z to "0.8.". a dignissimos. x�bbcec�Z� �� Q�F&F��YlYZk9O�130��g�谜9�TbW��@��8Ǧ^+�@��ٙ�e'�|&�ЭaxP25���'&� n�/��p\���cѵ��q����+6M�|�� O�j�M�@���ټۡK��C�h$P�#Ǧf�UO{.O�)�zh� �Zg�S�rWJ^o �CP�8��L&ec�0�Q��-,f�+d�0�e�(0��D�QPf ��)��l��6��H+�9�>6.�]���s�(7H8�s[����@���I�Ám����K���?x,qym�V��Y΀Á� ;�C���Z����D�#��8r6���f(��݀�OA>cP:� ��[ The normal distribution (N) arises from the central limit theorem, which states that if a sequence of random variables Xi are independently and identically distributed, then the distribution of the sum of n such random variables tends toward the normal distribution as n becomes large. The Anderson-Darling test is available in some statistical software. Cumulative distribution function: Notation ... Normal distribution is without exception the most widely used distribution. The probability to the left of z = 0.87 is 0.8078 and it can be found by reading the table: You should find the value, 0.8078. 0000024222 00000 n It also goes under the name Gaussian distribution. 0000002461 00000 n 0000003670 00000 n P- population proportion. The distribution is parametrized by a real number μ and a positive real number σ, where μ is the mean of the distribution, σ is known as the standard deviation, and σ 2 is known as the variance. For the standard normal distribution, this is usually denoted by F (z). Since z = 0.87 is positive, use the table for POSITIVE z-values. Hence, the normal distribution … 0000005852 00000 n 0000006875 00000 n And Problem 3 is looking for p(16 < X < 24). For example, 1. Next, translate each problem into probability notation. In general, capital letters refer to population attributes (i.e., parameters); and lower-case letters refer to sample attributes (i.e., statistics). Problem 1 is really asking you to find p(X < 8). NormalDistribution [μ, σ] represents the so-called "normal" statistical distribution that is defined over the real numbers. This is also known as a z distribution. Note in the expression for the probability density that the exponential function involves . Then we can find the probabilities using the standard normal tables. This is also known as the z distribution. The distribution plot below is a standard normal distribution. A Z distribution may be described as $$N(0,1)$$. Scientific website about: forecasting, econometrics, statistics, and online applications. Based on the definition of the probability density function, we know the area under the whole curve is one. 0000009248 00000 n 0000023958 00000 n Since the entries in the Standard Normal Cumulative Probability Table represent the probabilities and they are four-decimal-place numbers, we shall write 0.1 as 0.1000 to remind ourselves that it corresponds to the inside entry of the table. Excepturi aliquam in iure, repellat, fugiat illum Click on the tabs below to see how to answer using a table and using technology. 0000005340 00000 n 0000006448 00000 n A random variable X whose distribution has the shape of a normal curve is called a normal random variable.This random variable X is said to be normally distributed with mean μ and standard deviation σ if its probability distribution is given by The intersection of the columns and rows in the table gives the probability. 0000034070 00000 n $$P(2 < Z < 3)= P(Z < 3) - P(Z \le 2)= 0.9987 - 0.9772= 0.0215$$, You can also use the probability distribution plots in Minitab to find the "between.". The function $\Phi(t)$ (note that that is a capital Phi) is used to denote the cumulative distribution function of the normal distribution. In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions.One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. A Z distribution may be described as N (0, 1). One of the most popular application of cumulative distribution function is standard normal table, also called the unit normal table or Z table, is the value of cumulative distribution function of … To find the 10th percentile of the standard normal distribution in Minitab... You should see a value very close to -1.28. When finding probabilities for a normal distribution (less than, greater than, or in between), you need to be able to write probability notations. P refers to a population proportion; and p, to a sample proportion. where $$\Phi$$ is the cumulative distribution function of the normal distribution. ��(�"X){�2�8��Y��~t����[�f�K��nO݌5�߹*�c�0����:&�w���J��%V��C��)'&S�y�=Iݴ�M�7��B?4u��\��]#��K��]=m�v�U����R�X�Y�] c�ضU���?cۯ��M7�P��kF0C��a8h�! norm.pdf returns a PDF value. 0000001097 00000 n P (Z < z) is known as the cumulative distribution function of the random variable Z. From Wikipedia, the free encyclopedia In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Therefore, the 10th percentile of the standard normal distribution is -1.28. voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos Practice these skills by writing probability notations for the following problems. Note that since the standard deviation is the square root of the variance then the standard deviation of the standard normal distribution is 1. In the case of a continuous distribution (like the normal distribution) it is the area under the probability density function (the 'bell curve') from The shaded area of the curve represents the probability that Xis less or equal than x. normal distribution unknown notation. 1. 622 0 obj <> endobj 4. x- set of sample elements. Arcu felis bibendum ut tristique et egestas quis: A special case of the normal distribution has mean $$\mu = 0$$ and a variance of $$\sigma^2 = 1$$. The corresponding z-value is -1.28. For example, if $$Z$$ is a standard normal random variable, the tables provide \(P(Z\le a)=P(Z ��� ) �173\hn�. Which has an s … this Figure shows a picture of X ‘ s for! Left of 0.87 the 10th percentile of the columns and rows in the,... # |� ] n.�� a normal distribution notation normal distribution the  greater than.  Minitab find... 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# Code Documentation¶

class base.Foundation[source]

This class contains common features of all modules.

DetSymEnv()[source]

Returns a list. The list consists of all symbolic tools present among ‘sympy’,’sage’ and ‘symengine’.

class measure.Measure(dom, w=None)[source]
An instance of this class is a measure on a given set supp. The support is either
• a python variable of type set, or
• a list of tuples which represents a box in euclidean space.
Initializes a measure object according to the inputs:
• dom must be either
• a list of 2-tuples
• a non-empty dictionary
• w must be a
• a function if dom defines a region
• left blank (None) if dom is a dictionary
boxCheck(B)[source]

Checks the structure of the box B. Returns True id B is a list of 2-tuples, otherwise it returns False.

check(dom, w)[source]

Checks the input types and their consistency, according to the __init__ arguments.

integral(f)[source]

Returns the integral of f with respect to the currwnt measure over the support.

measure(S)[source]

Returns the measure of the set S. S must be a list of 2-tuples.

norm(p, f)[source]

Computes the norm-p of the f with respect to the current measure.

sample(num)[source]

Samples from the support according to the measure.

class orthsys.OrthSystem(variables, var_range, env='sympy')[source]

OrthogonalSystem class produces an orthogonal system of functions according to a suggested basis of functions and a given measure supported on a given region.

This basically performs a ‘Gram-Schmidt’ method to extract the orthogonal basis. The inner product is obtained by integration of the product of functions with respect to the given measure (more accurately, the distribution).

To initiate an instance of this class one should provide a list of symbolic variables variables and the range of each variable as a list of lists var_range.

To initiate an orthogonal system of functions, one should provide a list of symbolic variables variables and the range of each these variables as a list of lists var_range.
Basis(base_set)[source]

To specify a particular family of function as a basis, one should call this method with a list base_set of linearly independent functions.

FormBasis()[source]

Call this method to generate the orthogonal basis corresponding to the given basis via Basis method. The result will be stored in a property called OrthBase which is a list of function that are orthogonal to each other with respect to the measure measure over the given range Domain.

FourierBasis(n)[source]

Generates a Fourier basis from variables consisting of all $$sin$$ & $$cos$$ functions with coefficients at most n.

PolyBasis(n)[source]

Generates a polynomial basis from variables consisting of all monomials of degree at most n.

Series(f)[source]

Given a function f, this method finds and returns the coefficients of the series that approximates f as a linear combination of the elements of the orthogonal basis.

SetMeasure(M)[source]

To set the measure which the orthogonal system will be computed, simply call this method with the corresponding distribution as its parameter dm; i.e, the parameter is d(m) where m is the original measure.

SetOrthBase(base)[source]

Sets the orthonormal basis to be the given base.

TensorPrd(Bs)[source]

Takses a list of symbolic bases, each one a list of symbolic expressions and returns the tensor product of them as a list.

inner(f, g)[source]

Computes the inner product of the two parameters with respect to the measure measure.

project(f, g)[source]

Finds the projection of f on g with respect to the inner product induced by the measure measure.

class interpolation.Interpolation(var, env='sympy')[source]

The Interpolation class provides polynomial interpolation routines in multi variate case.

var is the list of symbolic variables and env is the the symbolic tool.

Delta(idx=-1)[source]

Construct the matrix corresponding to idx‘th point, if idx>0 Otherwise returns the discriminant.

Interpolate(points, vals)[source]

Takes a list of points points and corresponding list of values vals and return the interpolant.

Since in multivariate case, there is a constraint on the number of points, it checks for the valididty of the input. In case of failure, describes the type of error occured according to the inputs.

MinNumPoints()[source]

Returns the minimum number of points still required.

Monomials()[source]

Generates the minimal set of monomials for interpolation.

class collocation.Collocation(variables, ufunc, env='sympy')[source]

The Collocation class tries to approximate the solutions of a system of partial differential equations with respect to an orthogonal system of functions.

To initiate an instance of this class one needs to provide two set of parameters:
1. List of independent symbolic variables variables;
2. List of unknown functions to be found that depend on the independent variables ufunc.
CollPoints(pnts)[source]

Accepts alist of collocation point pnts, to form the algebraic system of equations and find the coefficients of the orthogonal functions from OrthSystem.OrthBase. Each point must be either a list or a tuple.

Condition(eq, val)[source]

List of initial and boundary conditions.

Equation(eq)[source]

To enter the system of equations, use this meyhod with a list of equations as input.

FindDomain()[source]

Finds the region that all variables are defined.

PlugPoints()[source]

Internal use: plug in collocation points to elliminate independent variables and keep the coefficients.

SetOrthSys(obj, func)[source]

To approximate the solutions of the system of pdes, the class requires an orthogonal system of functions OrthSystem. This method accepts such a system.

Solve()[source]

Solves the collocation equations and keep a dictionary of coefficients in self.Coeffs and returns a list of functions in the span of orthoginal system.

collocate()[source]

Internal use: generates the system of equations for coefficients to be used via collocation points.

setSampleMeasure(meas)[source]

Sets the measure over the domain for sampling collocation points in case of necessity.

setSolver(solver, optn='lm')[source]

Currently only two solvers are supported:

1. the sages defult solver for rather simple system of algebraic equations.

2. the scipys fsolves to handel more complex and larger systems. It also supports the following solvers from scipy:

• hybr
• lm (defauls)
• broyden1
• broyden2
• anderson
• krylov
• df-sane
class subregion.SubRegion(collsys, num_parts=[])[source]

The SubRegion class partitions the region into sub-regions, solve the system of Integro-differential equations on each and glue them together.

It takes:
1. a collocation instance collsys;
2. an optional list of positive integers num_parts which shows the number of equal length partitions for each variable.
AnalyseConditions()[source]

Associates boundary conditions to each sub-collocation system. Starting from one corner, associates boundary conditions to all adjacent subregions based on the solution of the current subregion.

ClosedForm(func)[source]

Compiles a piece-wise defined symbolic function from the solution for func which is an unknown from the original collocation system.

InitMiniSys()[source]

Initializes a set of collocation systems for all subregions.

Solve()[source]

Solves the collocation system for each region and generates boundary conditions for adjacent regions.

corners(tpl)[source]

Generates end corner points of the region represented tpl as collocation points (used internally).

class graphics.Graphics(env='numeric', numpoints=50)[source]

This class tends to provide basic graphic tools based on matplotlib and mayavi.

Accepts one optional argument env which determines the types of the function to be visualized:

• numeric: is a numerical function (regular python functions)
• sympy: Sympy symbolic function
• sage: Sage symbolic function
ParamPlot2D(funcs, rng, color='blue', legend='', thickness=1)[source]

Appends a parametric curve to the Graphics object. The parameters are as follows:

• funcs: the tupleof functions to be plotted,
• rng: a triple of the form (t, a, b), where t is the funcs‘s independents variable, over the range [a, b],
• color: the color of the current curve,
• legend: the text for the legend of the current crve.
Plot2D(func, xrng, color='blue', legend='', thickness=1)[source]

Appends a curve to the Graphics object. The parameters are as follows:

• func: the function to be plotted,
• xrng: a triple of the form (x, a, b), where x is the func‘s independents variable, over the range [a, b],
• color: the color of the current curve,
• legend: the text for the legend of the current crve.
Plot3D(func, xrng, yrng)[source]

Sets a surface to the Graphics object. The parameters are as follows:

• func: the function to be plotted,
• xrng: a triple of the form (x, a, b), where x is the first funcs independents variable, over the range [a, b],
• yrng: a triple of the form (y, c, d), where x is the second func‘s independents variable, over the range [c, d].
Point(pnts, color='blue', marker='o', legend='')[source]

Adds a list of points to the plot.

SetLabelX(lbl)[source]

Sets the label for X axis

SetLabelY(lbl)[source]

Sets the label for Y axis

SetLabelZ(lbl)[source]

Sets the label for Z axis

SetTitle(ttl)[source]

Sets the title of the graph.

interact()[source]

Shows an interavtive demo of the 3D surface, using mayavi, so it requires mayavi for python to be installed.

save(fname='fig.png')[source]

Saves the outpu of the Graphics object to the file fname.


text:
# Questions tagged [algebra]

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206 views

### Kleene Algebra for star-free regular expressions

TLDR: Is there a notion of Kleene Algebra for star-free regular expressions? Kleene Algebras are algebraic structures that are equivalent to regular expressions. A Kleene Algebra is an idempotent ...
43 views

### Relation between automorphism group of a linear code and its dual code

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322 views

### Technical lemma about curves used in original proof of PCP theorem

I am reading the proof from here and found a technical lemma that seems to be incorrect (its proof is short and very vague). I know this is rather specific and the context is problematic, but I couldn'...
280 views

### What category are Tagless Final Algebras final In?

The Haskell and Scala community have been very enamored recently with what they call tagless final 'pattern' of programming. These are referenced as dual to initial free algebras, so I was wondering ...
407 views

### Applications of algebraic geometry in type theory/programming language theory

Lately, I have become interested in algebraic geometry and have started reading on it. I still know very little about this field, but I do want to know if it has any connection with my main field, ...
259 views

### Turing Machines as Coalgebras

I'm looking to write a survey on the method of representing the dynamics of state-based computation within the framework of coalgebras. So far I've managed to find papers on coalgebra representations ...
20 views

### Worst case polynomial in elimination theory under rank conditions?

Given $n$ polynomials $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})\in\mathbb Z[x_1,\dots,x_{2n}]$ where each of $h_1(x_1,\dots,x_{2n}),\dots,h_{2n}(x_1,\dots,x_{2n})$ is homogeneous of degree ...
111 views

### Are there cascade decompositions of machines that are more general than finite automata?

The idea of decomposing automata and their associated semi-groups into irreducible sub-components is due to Krohn & Rhodes and has been explored relatively thoroughly. Krohn & Rhodes gave an ...
56 views

68 views

### Does an initial algebra for a class have to belong to the class itself?

In the context of algebraic data types, a concept of initial algebras is usually defined, e.g., in the following way: An algebra $S$ is initial in a class $C$ of algebras iff for every $A\in C$ ...
57 views

### Relation between OSAs and grammars

Are there any relation between order-sorted algebra (OSA) and grammars (context-free grammar in particular)? If I'm not mistaken, according to [1], there is an equivalence between order-sorted and ...
295 views

### If a root||nonce Proof-of-Work certificate is prime, can it be used in any other interesting proofs?

Because Bitcoin and many other cryptocurrency mining certificates are "rare" in that their respective hash is less than a very small number, can we leverage their rarity in probabilistic proofs of ...
137 views

### Connection between algebraic logic and computational complexity of logics?

I'm learning a bit about algebraic logic and I was wondering how knowing the algebraic semantics of a given logic might help the study of the logic itself from a computational point of view. In ...
227 views

### Terminology about computation and Finite algebra

I am looking for the name of something that may have one. A finite algebra $\mathcal{A} = (E, \{f_1, f_2, \ldots, f_k\})$ is a non-empty set $E$ together with some functions $f_i$ from $E^{r_i} \to E$...
105 views

### On $\Sigma \Pi \Sigma \Pi(2,r)$-circuits

As I understand from the survey "Progress on Polynomial Identity Testing - II" a polynomial-time algorithm for solving PIT for $\Sigma \Pi \Sigma \Pi (2, r)$ is unknown. However, there exists paper ...
139 views

### Is the relation decidable?

Given an ideal $I$ over $\mathbb{C}$ and P, a polynomial, is it decidable whether $P\in I$?
290 views

### What does a private coin $\mathsf{IP}$ protocol for Hilbert's Nullstellensatz look like?

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1k views

### What kind of theoretical object corresponds to a C++ concept?

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### Standard basis for recurrence relations

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### Are There Highly Symmetric NP- or P-complete Languages?

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91 views

### the number of rational points of a curve modulo 2

Consider the language $L=\{f, q\}$ - the number of solutions of equations $f(x,y)=0$ in $\mathbb{F}_q^2$ is equal to zero modulo $2$, where $q = 2^m$. Does $L$ belong to $P$? to $NP$? ($f$ is written ...
136 views

### What automorphisms on a Markov Chain imply a uniform limiting distribution?

Consider an irreducible aperiodic Markov chain $M$, modeled as a connected directed graph with weighted edges. The existence of certain (graph) automorphisms on this Markov chain imply various ...
76 views


text:
gzcon.Rd 2.51 KB
Newer Older
 Radford Neal committed May 18, 2013 1 2 % File src/library/base/man/gzcon.Rd % Part of the R package, http://www.R-project.org  Radford Neal committed Aug 26, 2016 3 % Copyright 1995-2010 R Core Team  Radford Neal committed May 18, 2013 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 % Distributed under GPL 2 or later \name{gzcon} \alias{gzcon} \title{ (De)compress I/O Through Connections } \description{ \code{gzcon} provides a modified connection that wraps an existing connection, and decompresses reads or compresses writes through that connection. Standard \code{gzip} headers are assumed. } \usage{ gzcon(con, level = 6, allowNonCompressed = TRUE) } \arguments{ \item{con}{a connection.} \item{level}{integer between 0 and 9, the compression level when writing.} \item{allowNonCompressed}{logical. When reading, should non-compressed input be allowed?} } \details{ If \code{con} is open then the modified connection is opened. Closing the wrapper connection will also close the underlying connection. Reading from a connection which does not supply a \code{gzip} magic header is equivalent to reading from the original connection if \code{allowNonCompressed} is true, otherwise an error. Compressed output will contain embedded NUL bytes, and so \code{con} is not permitted to be a \code{\link{textConnection}} opened with \code{open="w"}. Use a writable \code{\link{rawConnection}} to compress data into a variable. The original connection becomes unusable: any object pointing to it will  Radford Neal committed May 18, 2013 39 40  now refer to the modified connection. For this reason, the new connection needs to be closed explicitly.  Radford Neal committed May 18, 2013 41 42 43 44 } \value{ An object inheriting from class \code{"connection"}. This is the same connection \emph{number} as supplied, but with a modified internal  Radford Neal committed May 18, 2013 45  structure. It has binary mode.  Radford Neal committed May 18, 2013 46 47 48 49 50 51 52 53 54 55 56 57 58 59 } \seealso{\code{\link{gzfile}}} \examples{ \donttest{## Uncompress a data file from a URL z <- gzcon(url("http://www.stats.ox.ac.uk/pub/datasets/csb/ch12.dat.gz")) # read.table can only read from a text-mode connection. raw <- textConnection(readLines(z)) close(z) dat <- read.table(raw) close(raw) dat[1:4, ] } ## gzfile and gzcon can inter-work.  Radford Neal committed May 18, 2013 60 ## Of course here one would use gzfile, but file() can be replaced by  Radford Neal committed May 18, 2013 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 ## any other connection generator. zz <- gzfile("ex.gz", "w") cat("TITLE extra line", "2 3 5 7", "", "11 13 17", file = zz, sep = "\n") close(zz) readLines(zz <- gzcon(file("ex.gz", "rb"))) close(zz) unlink("ex.gz") % Cygwin seems to require a different name zz <- gzcon(file("ex2.gz", "wb")) cat("TITLE extra line", "2 3 5 7", "", "11 13 17", file = zz, sep = "\n") close(zz) readLines(zz <- gzfile("ex2.gz")) close(zz) unlink("ex2.gz") } \keyword{file} \keyword{connection}


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# Technical Fridays

Friday, September 1, 2017

In 1973, the University of California, Berkeley was sued for gender bias against women who had applied to graduate schools. The data for fall 1973 showed that men applying were far more likely to get admits than the women.

But, after examining the individual departments, it appeared that no department was significantly biased against women.

It was a case of Simpson’s paradox1, a phenomenon in which a trend appears in different groups of data but disappears or reverses when these groups are combined.

ucb is a three-dimensional table (like a matrix): Admit Status x Gender x Dept, with counts for each category as the matrix’s values. Here’s the data of six departments:

Let’s check the acceptance rate of UC Berkeley in 1973 for graduate applicants.

Overall, women have an admission rate of 30.35%, which is much lower than that of men, 44.52%.
It prompted a lawsuit against UC Berkeley which prompted the study that collected this data.

Let’s draw mosaic plot3, which provides a way to visualize contingency tables.

It seems to indicate a gender bias. However, there is a lurking variable: Dept. Here is what happens if we stratify on department:

The Admit row in our table of contents corresponds to the width of columns in the mosaic plot. More people were rejected than admitted because Rejected column is wider. Of the people admitted, a much higher proportion were Male because of the height of the rectangles. Of the people rejected, it appears to be pretty even.

A higher proportion of admitted Males were for Dept A and B compared to the proportion of admitted Females for the same Dept. On the other hand, a higher proportion of admitted Females were for Dept C – F. But, higher proportion of the Males were rejected for Dept A and B than Females as the widths of the Male rectangles are wider than their Female counterparts. Likewise for Dept C – F, a higher proportion of the Females were rejected for Dept C – F than Males.

It appears that most departments have no gender bias, and those departments that are biased favor women. First, note that Dept A and B have very few female applicants (the columns are narrow). It is also relatively easy to get into those departments—the proportion rejected is lower than other departments, especially F. So one explanation is that more males get in because they are applying to the hungrier, perhaps fastest-growing, departments.

The research paper by Bickel et al.4 concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as in the English Department), whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants (such as in engineering and chemistry).

If you express matters algebraically, the appearance of the Paradox is no surprise.
For, suppose
No one would dream of deducing that

would they?


text:
ACC 280 Week 5 DQ 3 (Aspect of This Class did you Find the Most Helpful)

# ACC 280 Week 5 DQ 3 (Aspect of This Class did you Find the Most Helpful)

C
0 points

ACC 280 Week 5

DQ-3

What aspect of this class did you find the most helpful? What would you change if you were given the opportunity?

ACC 280
Cramster

C
0 points

#### Oh Snap! This Answer is Locked

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text:
# Definition:Intersection

Definition: Intersection
If and are sets, then the intersection of and (written ) is given by .


text:
## Essential University Physics: Volume 1 (3rd Edition)

As $|A.B|=2|A\times B|$ $|A||B|cos{\theta}=2|A||B|sin{\theta}$ $\frac{sin(\theta)}{cos({\theta})}=\frac{1}{2}$ $tan(\theta)=\frac{1}{2}=27^{\circ}$


text:
# Could you please give an intuitive definition of chemical potential?

Could you please give an intuitive definition of chemical potential? It seems that it is an extremely important notion of physics but definitions are really vague.

You say that the definitions are vague, but $\mu_i=\left(\frac{\partial U}{\partial N_i}\right)_{S,V,N_{j\neq i}}=\left(\frac{\partial G}{\partial N_i}\right)_{P,T,N_{j\neq i}}$ is precise.

However, it may be helpful to use an analogy to obtain an intuitive definition. I'm sure you're familiar with the fact that systems tend to evolve to reduce gradients. Any such spontaneous change involves two conjugate parameters: a generalized force (corresponding to a gradient in some field, such as a pressure difference) and a generalized displacement (corresponding to the flow, such as a change in volume). The product of the two conjugate variables has units of energy.

In heat transfer, for example, a temperature gradient causes a spontaneous flow of energy. The "stuff" that is transferred is entropy. Thus, we obtain the differential term $dU = T\,dS$.

A pressure gradient drives a change in volume: $dU = -P\,dV$.

What would cause spontaneous movement of matter? In this case, the driving force is a gradient in the chemical potential of a material $i$: $dU = \mu_i\,dN_i$.

I'm sure you're also familiar with the concept that changes in concentration drive material transport or diffusion. This is only an approximation. It fails to explain why oil and water separate, for example. (Or why any two mixed materials would separate.) The chemical potential is like an augmented concentration that also incorporates bonding between materials (as well as concentration). It is the true arbiter of how matter will move.

You can say that its the energy needed to remove a particle from a many-body system.

More formally $$F(N+1) -F(N)=\mu$$ where $F$ is the free energy of the system and $N$ is the particle number.

• would that not be the energy needed to add a particle to a system? – Zarathustra Mar 10 '18 at 0:39

The chemical potential can also be viewed as a Lagrange multiplier related to the constraint that the number of particles of a closed system is fixed.

To understand this, consider the example of a gas made of $N$ electrons. The probability of an eigenstate of energy $E$ be occupied is given by the Fermi-Dirac distribution $$n_F(E,\beta)=\frac{1}{e^{\beta(E-\mu)}+1},\tag1$$ where $\beta$ is the inverse of temperature. This equation can very well be understood as the definition of the chemical potential. If we sum over all momentum eingestate $\vec k$ and multiply the result by two to account for the two spin states we obtain the total number of particles, $$N=2\sum_{\vec k}n_F(E(\vec k),\beta).\tag2$$ Since in general $n_F$ changes with temperature, as in Eq. (1), and yet $N$ has to be kept fixed, this intuition about the chemical potential is that it is whatever parameter $\mu$ defined by (1) which satisfies the constraint (2).

The physical interpretation of chemical potential that works best for me is the "private" Gibbs free energy per mole of each individual chemical species in a mixture, so that the total free energy of the mixture is just the number of moles of each species times the chemical potential of that species, summed over all the species.

• I think I like this. Is your curious use of the word "private" meant to be analogous to the access control declarator in C++ (and other "encapsulating" languages)? – WetSavannaAnimal Oct 20 '17 at 9:39

While each of the earlier answers gives insight into the meaning of chemical potential, personally the Wikiperdia entry https://en.wikipedia.org/wiki/Chemical_potential gives me the best sense of how to read chemical potential: as well as distinguishing "electrochemical potential" from chemical potential, it also interprets its meaning in a wider variety of physical systems.

## protected by Qmechanic♦Oct 19 '17 at 14:30

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).


text:
Question

# (i) In what ratio is the line segment joining the points (-2, -3) and (3, 7) divided by the y-axis ? Also, find the coordinates of the point of division.  (ii) In what ratio is the line segment joining (-3, -1) and (-8, -9) divided at the point (−5,−215)?

Solution

## i) Let the ratio in which the line is divided be a:1 and the point of division be (0,y) So, 0=3a−2a+1 ⇒a=23 Now, y=7×23−323+1=1 So, the point is (0,1). ii) Let the ratio in which the line is divided be k:1. So, −5=−8k−3k+1 ⇒−5k−5=−8k−3 ∴k=23 MathematicsRD SharmaStandard X

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text:
## CryptoDB

### Wonhee Cho

#### Publications

Year
Venue
Title
2021
PKC
A weak pseudorandom function (weak PRF) is one of the most important cryptographic primitives for its efficiency although it has lower security than a standard PRF. Recently, Boneh et al. (TCC'18) introduced two types of new weak PRF candidates, which are called a basic Mod-2/Mod-3 and alternative Mod-2/Mod-3 weak PRF. Both use the mixture of linear computations defined on different small moduli to satisfy conceptual simplicity, low complexity (depth-2 ${\sf ACC^0}$) and MPC friendliness. In fact, the new candidates are conjectured to be exponentially secure against any adversary that allows exponentially many samples, and a basic Mod-2/Mod-3 weak PRF is the only candidate that satisfies all features above. However, none of the direct attacks which focus on basic and alternative Mod-2/Mod-3 weak PRFs use their own structures. In this paper, we investigate weak PRFs from two perspectives; attacks, fixes. We first propose direct attacks for an alternative Mod-2/Mod-3 weak PRF and a basic Mod-2/Mod-3 weak PRF when a circulant matrix is used as a secret key. For an alternative Mod-2/Mod-3 weak PRF, we prove that the adversary's advantage is at least $2^{-0.105n}$, where $n$ is the size of the input space of the weak PRF. Similarly, we show that the advantage of our heuristic attack to the weak PRF with a circulant matrix key is larger than $2^{-0.21n}$, which is contrary to the previous expectation that `structured secret key' does not affect the security of a weak PRF. Thus, for an optimistic parameter choice $n = 2\lambda$ for the security parameter $\lambda$, parameters should be increased to preserve $\lambda$-bit security when an adversary obtains exponentially many samples. Next, we suggest a simple method for repairing two weak PRFs affected by our attack while preserving the parameters.
2019
CRYPTO
We present a new cryptanalytic algorithm on obfuscations based on GGH15 multilinear map. Our algorithm, statistical zeroizing attack, directly distinguishes two distributions from obfuscation while it follows the zeroizing attack paradigm, that is, it uses evaluations of zeros of obfuscated programs.Our attack breaks the recent indistinguishability obfuscation candidate suggested by Chen et al. (CRYPTO’18) for the optimal parameter settings. More precisely, we show that there are two functionally equivalent branching programs whose CVW obfuscations can be efficiently distinguished by computing the sample variance of evaluations.This statistical attack gives a new perspective on the security of the indistinguishability obfuscations: we should consider the shape of the distributions of evaluation of obfuscation to ensure security.In other words, while most of the previous (weak) security proofs have been studied with respect to algebraic attack model or ideal model, our attack shows that this algebraic security is not enough to achieve indistinguishability obfuscation. In particular, we show that the obfuscation scheme suggested by Bartusek et al. (TCC’18) does not achieve the desired security in a certain parameter regime, in which their algebraic security proof still holds.The correctness of statistical zeroizing attacks holds under a mild assumption on the preimage sampling algorithm with a lattice trapdoor. We experimentally verify this assumption for implemented obfuscation by Halevi et al. (ACM CCS’17).

#### Coauthors

Jung Hee Cheon (2)
Minki Hhan (1)
Jiseung Kim (2)
Jeong Han Kim (1)
Changmin Lee (1)


text:
# High School College Essays Jou. vh B 7 Steps Analysis sezo000000000... Chemistry B 2019-2020 CR OSCAR TERRAZ of the aqueous solution

###### Question:

High School
College Essays Jou.
vh
B
7 Steps Analysis
sezo000000000...
Chemistry B 2019-2020 CR
OSCAR TERRAZ
of the aqueous solution increases to 29.20°C. If the mass of the solution is 120.0 g and the specific heat of the calorimeter and
solution is 4.184 J/g • °C, how much heat is given off in the reaction? (Ignore the mass of the calorimeter in the calculation.)
Use 4-MCAT
4.62 kJ
10.0 ky
10.0 kJ
14.7 k)
o ON
38.5 kJ
Save and Exit
Next
Submit
Mark this and return

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### What is the speed of a car that traveled 325 kilometers in 6.5 hours?

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### Write an expression that best represents the statement belowA number, x, multiplied by 4 is subtracted from the difference of the same number and 65.

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text:
1. Factorisation

Factorise:

$(xy+1)^4 - 4xy(xy+1)^2 - (x^2 - y^2)^2$

Thank you.

Originally Posted by muks
Factorise:

$(xy+1)^4 - 4xy(xy+1)^2 - (x^2 - y^2)^2$

Thank you.
$(xy+1)^4 - 4xy(xy+1)^2 - (x^2 - y^2)^2$

As you see, $(xy+1)^2$ comes twice, so, Complete the square

$=[(xy+1)^2]^2-2(2xy)(xy+1)^2+(2xy)^2-(2xy)^2-(x^2-y^2)^2$

$=[(xy+1)^2-(2xy)]^2-(2xy)^2-(x^2-y^2)^2$

$=[x^2y^2+1+2xy-2xy]^2-4x^2y^2-[(x^2)^2+(y^2)^2-2x^2y^2]$

$=(x^2y^2+1)^2-[4x^2y^2+(x^2)^2+(y^2)^2-2x^2y^2]$

$=(x^2y^2+1)^2-[(x^2)^2+(y^2)^2+2x^2y^2]$

$=(x^2y^2+1)^2-(x^2+y^2)^2$

$=(x^2y^2+1+x^2+y^2)(x^2y^2+1-x^2-y^2)$

3. I agree with the answer provided, but the answer in the book is $(x+1)(x-1)(y+1)(y-1)(x^2+1)(y^2+1)$

How did they obtain?


text:
# IBPS Simplification Questions with Answers Practice online test 21

Description: free IBPS Simplification Questions with Answers Practice test 21 for IBPS PO Preliminary and Main online test Prepare bank PO banking mock exams adda

1 . What value should come in place of question mark (?) in the following questions?
77.8 $\times$ 0.8 $\times$ ? = 964.72
 13.5 14.5 15.5 16.5 17.5
2 . What value should come in place of question mark (?) in the following questions?
$\sqrt {17.64}$ $\times$ $14.0625$ = $\sqrt {0.0225}$ $\times$ ?
 1.5 115 125 135 145
3 . What value should come in place of question mark (?) in the following questions?
$7 \over 15$ of $5 \over 27$ of 45 % of 1593 = 2.1 $\times$ ?
 29.5 28.5 27.5 26.5 25.5
4 . What value should come in place of question mark (?) in the following questions?
$(357.911)^{2 \over 3}$ $\times$ $(50.41)^{3 \over 2}$ = $(7.1)^?$
 5 4 3 2 1
5 . What approximate value should come in place of question mark (?) in the following questions?
$\sqrt {6890}$ + $\sqrt [3]{50650}$ = ?
 112 114 116 118 120
6 . What approximate value should come in place of question mark (?) in the following questions?
(669.76 + 29.96 $\times$ 35.05) $\div$ 6.04 = ?
 150 290 370 420 460
7 . What approximate value should come in place of question mark (?) in the following questions?
$(44.99)^2$ $\div$ 7.538 = ?
 90 160 270 320 375
8 . What approximate value should come in place of question mark (?) in the following questions?
228% of 450 + 84% of 844.98 -1116 = ?
 360 630 625 530 620
361 $\times$ 5.96 $\times$ ? = 15227
4003 $\times$ 77 - 21015 = ? $\times$ 116


text:
# R package site incorporation into my personal website

## Overview

I’ve been developing the R package qtl2pleio. It’s now at a stage where others can use it. I recently found the R package pkgdown, which allows developers to build, in an automated fashion, websites for their packages. I posted an entry on pkgdown earlier this year. I now wish to communicate my strategy for incorporating the package site into my personal website, https://fboehm.us. I write this primarily as a reminder to myself.

## The solution: Specify destination in _pkgdown.yml

My initial strategy was to use a Makefile to automate copying of my pkgdown site from qtl2pleio/docs to a subdirectory in my website repository, which, in my case, has the path website-mediumish/docs/static/software/qtl2pleio. We know that pkgdown, by default, builds the site in the (package) subdirectory docs. What I learned only yesterday is that the field destination in _pkgdown.yml can be used to specify an alternative path at which to build the site. I thus added to my _pkgdown.yml the text:

destination: ~/Documents/website-mediumish/docs/static/software/qtl2pleio

You can view my initial _pkgdown.yml for qtl2pleio here.

Now, when I type

pkgdown::build_site()

from within the directory for the qtl2pleio R package, I see that the site is built at the specified destination, i.e., in my personal website subdirectory.

2 reticulate R package & Travis | All posts | World-wide Web and Social Media 1


text:
# How do you calculate the atomic mass of element X if there are two isotopes of this element (x-24 and x-28) and 75% of all x atoms are x-28 isotopes?

(24 xx 25%) + (28 xx 75%) = 27
75% of all X atoms are X-28, which means that 25% of all X atoms are X-24 as there are only 2 isotopes.


text:
## Exercises6.5Exercises

###### 1.

Suppose that $$G$$ is a finite group with an element $$g$$ of order $$5$$ and an element $$h$$ of order $$7\text{.}$$ Why must $$|G| \geq 35\text{?}$$

###### 2.

Suppose that $$G$$ is a finite group with $$60$$ elements. What are the orders of possible subgroups of $$G\text{?}$$

###### 3.

Prove or disprove: Every subgroup of the integers has finite index.

###### 4.

Prove or disprove: Every subgroup of the integers has finite order.

###### 5.

List the left and right cosets of the subgroups in each of the following.

1. $$\langle 8 \rangle$$ in $${\mathbb Z}_{24}$$

2. $$\langle 3 \rangle$$ in $$U(8)$$

3. $$3 {\mathbb Z}$$ in $${\mathbb Z}$$

4. $$A_4$$ in $$S_4$$

5. $$A_n$$ in $$S_n$$

6. $$D_4$$ in $$S_4$$

7. $${\mathbb T}$$ in $${\mathbb C}^\ast$$

8. $$H = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}$$ in $$S_4$$

###### 6.

Describe the left cosets of $$SL_2( {\mathbb R} )$$ in $$GL_2( {\mathbb R})\text{.}$$ What is the index of $$SL_2( {\mathbb R} )$$ in $$GL_2( {\mathbb R})\text{?}$$

###### 7.

Verify Euler's Theorem for $$n = 15$$ and $$a = 4\text{.}$$

###### 8.

Use Fermat's Little Theorem to show that if $$p = 4n + 3$$ is prime, there is no solution to the equation $$x^2 \equiv -1 \pmod{p}\text{.}$$

###### 9.

Show that the integers have infinite index in the additive group of rational numbers.

###### 10.

Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.

###### 11.

Let $$H$$ be a subgroup of a group $$G$$ and suppose that $$g_1, g_2 \in G\text{.}$$ Prove that the following conditions are equivalent.

1. $$\displaystyle g_1 H = g_2 H$$

2. $$\displaystyle H g_1^{-1} = H g_2^{-1}$$

3. $$\displaystyle g_1 H \subset g_2 H$$

4. $$\displaystyle g_2 \in g_1 H$$

5. $$\displaystyle g_1^{-1} g_2 \in H$$

###### 12.

If $$ghg^{-1} \in H$$ for all $$g \in G$$ and $$h \in H\text{,}$$ show that right cosets are identical to left cosets. That is, show that $$gH = Hg$$ for all $$g \in G\text{.}$$

###### 13.

What fails in the proof of Theorem 6.8 if $$\phi : {\mathcal L}_H \rightarrow {\mathcal R}_H$$ is defined by $$\phi( gH ) = Hg\text{?}$$

###### 14.

Suppose that $$g^n = e\text{.}$$ Show that the order of $$g$$ divides $$n\text{.}$$

###### 15.

The cycle structure of a permutation $$\sigma$$ is defined as the unordered list of the sizes of the cycles in the cycle decomposition $$\sigma\text{.}$$ For example, the permutation $$\sigma = (1 \, 2)(3 \, 4 \, 5)(7 \, 8)(9)$$ has cycle structure $$(2,3,2,1)$$ which can also be written as $$(1, 2, 2, 3)\text{.}$$

Show that any two permutations $$\alpha, \beta \in S_n$$ have the same cycle structure if and only if there exists a permutation $$\gamma$$ such that $$\beta = \gamma \alpha \gamma^{-1}\text{.}$$ If $$\beta = \gamma \alpha \gamma^{-1}$$ for some $$\gamma \in S_n\text{,}$$ then $$\alpha$$ and $$\beta$$ are conjugate.

###### 16.

If $$|G| = 2n\text{,}$$ prove that the number of elements of order $$2$$ is odd. Use this result to show that $$G$$ must contain a subgroup of order 2.

###### 17.

Suppose that $$[G : H] = 2\text{.}$$ If $$a$$ and $$b$$ are not in $$H\text{,}$$ show that $$ab \in H\text{.}$$

###### 18.

If $$[G : H] = 2\text{,}$$ prove that $$gH = Hg\text{.}$$

###### 19.

Let $$H$$ and $$K$$ be subgroups of a group $$G\text{.}$$ Prove that $$gH \cap gK$$ is a coset of $$H \cap K$$ in $$G\text{.}$$

###### 20.

Let $$H$$ and $$K$$ be subgroups of a group $$G\text{.}$$ Define a relation $$\sim$$ on $$G$$ by $$a \sim b$$ if there exists an $$h \in H$$ and a $$k \in K$$ such that $$hak = b\text{.}$$ Show that this relation is an equivalence relation. The corresponding equivalence classes are called double cosets. Compute the double cosets of $$H = \{ (1),(1 \, 2 \, 3), (1 \, 3 \, 2) \}$$ in $$A_4\text{.}$$

###### 21.

Let $$G$$ be a cyclic group of order $$n\text{.}$$ Show that there are exactly $$\phi(n)$$ generators for $$G\text{.}$$

###### 22.

Let $$n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}\text{,}$$ where $$p_1, p_2, \ldots, p_k$$ are distinct primes. Prove that

\begin{equation*} \phi(n) = n \left( 1 - \frac{1}{p_1} \right) \left( 1 - \frac{1}{p_2} \right)\cdots \left( 1 - \frac{1}{p_k} \right)\text{.} \end{equation*}
###### 23.

Show that

\begin{equation*} n = \sum_{d \mid n} \phi(d) \end{equation*}

for all positive integers $$n\text{.}$$


text:
# Surface area of a rectangular prism made of unit cubes Online Quiz

#### Surface Areas and Volumes

117 Lectures 6.5 hours

#### Class 10th - Surface Areas and Volumes

20 Lectures 1 hours

#### Class 10th - Surface Areas and Volumes (Hindi)

10 Lectures 47 mins

Following quiz provides Multiple Choice Questions (MCQs) related to Surface area of a rectangular prism made of unit cubes. You will have to read all the given answers and click over the correct answer. If you are not sure about the answer then you can check the answer using Show Answer button. You can use Next Quiz button to check new set of questions in the quiz.

Q 1 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 2 ; w = 2; h = 3

Step 2:

The surface area = 2(lw + wh + lh)

= 2(2×2 + 2×3 + 2×3)

= 2(4 + 6 + 6)

= 32 square units

Q 2 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 4 ; w = 3; h = 2

Step 2:

The surface area = 2(lw + wh + lh)

= 2(4×3 + 4×2 + 2×3)

= 2(12 + 8 + 6)

= 52 square units

Q 3 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 5 ; w = 5; h = 3

Step 2:

The surface area = 2(lw + wh + lh)

= 2(5×5 + 5×3 + 5×3)

= 2(25 + 15 + 15)

= 110 square units

Q 4 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 3 ; w = 5; h = 5

Step 2:

The surface area = 2(lw + wh + lh)

= 2(3×5 + 3×5 + 5×5)

= 2(15 + 15 + 25)

= 110 square units

Q 5 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 3 ; w = 3; h = 2

Step 2:

The surface area = 2(lw + wh + lh)

= 2(3×3 + 3×2 + 2×3)

= 2(9 + 6 + 6)

= 42 square units

Q 6 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 4 ; w = 2; h = 5

Step 2:

The surface area = 2(lw + wh + lh)

= 2(4×2 + 4×5 + 2×5)

= 2(8 + 20 + 10)

= 76 square units

Q 7 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 2 ; w = 5; h = 5

Step 2:

The surface area = 2(lw + wh + lh)

= 2(2×5 + 2×5 + 5×5)

= 2(10 + 10 + 25)

= 90 square units

Q 8 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 2 ; w = 3; h = 4

Step 2:

The surface area = 2(lw + wh + lh)

= 2(2×3 + 2×4 + 4×3)

= 2(6 + 8 + 12)

= 52 square units

Q 9 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 5 ; w = 3; h = 2

Step 2:

The surface area = 2(lw + wh + lh)

= 2(5×3 + 5×2 + 2×3)

= 2(15 + 10 + 6)

= 62 square units

Q 10 - Find the length, width and height of the following rectangular prism. Then find its surface area.

### Explanation

Step 1:

The length , width and height are l = 3 ; w = 5; h = 4

Step 2:

The surface area = 2(lw + wh + lh)

= 2(3×5 + 4×3 + 5×4)

= 2(15 + 12 + 20)

= 94 square units


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# Thermodynamics - know process, work and change in entropy. Final temperature?

So, the problem is:
A 200-gram sample of dry air is heated isobarically. Its entropy increases by 19.2 J/K and the work done by expansion is 1.61x103 J. Solve for the final temperature of the air.

Okay, since p=const => dq=CpdT thus instead of dq=du+dw we can write
dq=dq(Cv/Cp)+dw
can find dq=460.46
So i know dq,du,dw,d$$\phi$$, but how do I get final temperature? All I can find is dT but it's useless sine I don't know initial temperature.
And where do I put mass and how to use the d$$\phi$$=dq/T formula? Looks like it will be needed here...


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# Thread: find the second derivative of...

1. ## find the second derivative of...

Find the second derivative of f(x)= sq rt(x^2 + 1)

I think the first derivative is x(x^2 + 1)^-1/2

But, I don't know how to find the derivative of that?

A) x/sq rt (x^2 + 1)

B) sq rt(x^2 + 1) - 1/2(x^2 + 1)^-1/2

C) 2x(sq rt(x^2 + 1) + 1/2(x^2 + 1)^-1/2

D) 1/sq rt((x^2 + 1)^3)

2. Originally Posted by obsmith08
Find the second derivative of f(x)= sq rt(x^2 + 1)

I think the first derivative is x(x^2 + 1)^-1/2

But, I don't know how to find the derivative of that?

A) x/sq rt (x^2 + 1)

B) sq rt(x^2 + 1) - 1/2(x^2 + 1)^-1/2

C) 2x(sq rt(x^2 + 1) + 1/2(x^2 + 1)^-1/2

D) 1/sq rt((x^2 + 1)^3)
Now apply the product rule, which states $\frac{\,d}{\,dx}\left[f\!\left(x\right)g\!\left(x\right)\right]=f\!\left(x\right)\frac{\,dg}{\,dx}+g\!\left(x\rig ht)\frac{\,df}{\,dx}$. Treat $x$ as $f\!\left(x\right)$ and $\left(x^2+1\right)^{-1}$ as $g\!\left(x\right)$.


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# Recurring Decimal To Fraction Calculator

Use this calculator to convert a repeating decimal to a fraction. [Fraction to recurring decimal calculator] ( /show/calculator/fraction-to-recurring-decimal ) is also available.

#### Calculator

It answers queries like: * Convert 0.(3) to a fraction * Convert 0.33333... to a fraction * What is 0.(1) as a fraction? * Represent 0.(5) as a fraction Some numbers cannot be expressed exactly as decimals with a finite number of digits. For example, since 2/3 = 0.666666666..., to express the fraction 2/3 in the decimal system, we require an infinity of 6s. Such decimals are referred to as __recurring (or repeating) decimals__. ##Recurring decimal to fraction## Every recurring decimal has a representation as a fraction. To see that, consider a recurring fraction of the form: \$$2.5(34) = 2.534343434343434...\$$ Let's convert the recurring part of the decimal to an infinite geometric series: \$$2.5 + 0.0(34) = 2.5 + 0.034 \cdot 10^{0} + 0.034 \cdot 10^{-2} + 0.034 \cdot 10^{-4}... = 2.5 + 0.034 \cdot {\sum^{\infty}_{i=0} (10^{-2i)})} \$$ And from the formula for the sum of a geometric series we get: \$$2.5 + { {34\over 1000} \over 1 - 10^{-2} } = {25 \over 10} + {34 \over 990} \$$ which means the whole expression is a fraction. ##General Formula## We can rewrite the formula above with variables to get something more general: \$${n + r \cdot 10^{-p} \cdot \sum^{\infty}_{i=0} (10^{-i \cdot j})} = n + {r \cdot 10^{-p} \over 1 - 10^{-j} }\$$ where: \$$n \$$ is the non-recurring part \$$r \$$ r is the recurring part \$$j \$$ is the length of \$$r \$$ \$$p \$$ is the number digits preceding the recurring part and the decimal point \$$+ 1 \$$ ##Method for Human-beings## There are better methods of finding the desired fractions than using the above formula. Let's use it on an example. What fraction is \$$0.(7) \$$ equal to? Let \$$x = 0.(7) \$$. Then \$$10x = 7.777777... = 7 + 0.(7) = 7 + x \$$. So, \$$9x = 7 \$$ and lastly, \$$x = {7 \over 9} \$$.