text:
# how much heat is required to melt an ice cube that is 4.2 cm x 2.1 cm x 2.9 cm? The density of ice is 0.92 g/cm3

22,259 questions, page 95
1. ## chemistry

The value of ΔH° for the reaction below is - 6535 kJ. __________ kJ of heat are released in the combustion of 16.0 g of C6H6(l)? 2C6H6(l) + 15O2(g) → 12CO2(g) + 6H2O (l)

asked by Anonymous on November 2, 2014
2. ## Chemistry

A certain substance has a heat of vaporization of 26.49 kJ/mol. At what Kelvin temperature will the vapor pressure be 5.00 times higher than it was at 285 K?

asked by Nick Tanner on January 15, 2014
3. ## Urgent Chem Help

A certain substance has a heat of vaporization of 55.30 kJ/mol. At what Kelvin temperature will the vapor pressure be 3.50 times higher than it was at 361 K?

asked by Carl on February 10, 2013
4. ## chemistry

How much H2 gas at STP can be produced by the reaction 2 Na(s) + 2 H2O(ℓ)heat−→ H2(g) + 2 NaOH(aq) of 3.90 g of Na and excess water? Answer in units of L.

asked by Abby on February 25, 2015
5. ## explian why please its physics

expliain y the very high specific heat capacity of water leads to formation of land and sea breezes???

asked by kimberly on December 7, 2008
6. ## Physics

(30)(Cm)Δ+(335.5)(4.18)Δ(21)=0 I need to find the specific heat of the masses, but i don't know how to, i'm currently doing a lab and these are the mass and temp. numbers that i plugged in...

asked by Me and I on March 17, 2014
7. ## CHEMISTRY> HELP QUICK! pls

A system loses 14.2 kJ of heat while performing 7.2kJ of work on the surroundings. If the initial internal energy, E, is 77.6 kJ, what is the final value of E? 56.2 kJ 70.6 kJ 99.0 kJ 84.6 kJ

asked by Lucinda on August 20, 2010
8. ## Chemistry

A certain substance has a heat of vaporation 56.21 kj/mol. At what kelvin temperature will the vapor pressure be 6.50 times higher than it was at 361k? Thank you

asked by Josh on July 9, 2011
9. ## Chemistry

What is the change in internal energy (in J) of a system that absorbs 0.677 kJ of heat from its surroundings and has 0.438 kcal of work done on it?

asked by Ashley on October 20, 2010
10. ## Physics

If 500 grams of a metal requires 70 calories of energy to raise its temperature by 10 Kelvin, what is its specific heat?

asked by John on October 28, 2010
11. ## physics

How much heat is absorbed by a 30g iron skillet when its temperature rises from 24oC to 93oC? Joules do i addition everything?

asked by jake on January 24, 2017
12. ## chemistry

When 34.1 g of lead reacts with 6.81 L of oxygen gas, measured at 1.00 atm and 25.0°C, 36.1 kJ of heat is released at constant pressure. What is DH° for this reaction?

asked by george on October 5, 2014
13. ## chemistry

if heat was added at a constant rate to the solid subsance, why did the temperature remain at 16 degree C for approximately 6 minutes?

asked by amanda on October 4, 2010
14. ## Chemistry 2 ap

How much heat energy is produced when 31.6 grams of propane are burned? I asked my teacher and she said it would be a stochiometry problem but im still confuded:/

asked by Peanut on October 28, 2010
15. ## Physics

An electric filament can heat 200 cm^3 of water from 5°C to 95°C in 6 minutes. If the voltage source is a 12 V battery, how much current does it draw?

asked by Jake on July 10, 2017
16. ## Science!-meterology

What are the effects of the transfer of heat into the inside of the hydroshphere and atmosphere on the development, intensity and circulation of meterological systmems?

asked by Sara on January 17, 2007
17. ## chemistry

How long would it take for 1.50 moles of water at 100.0 to be converted completely into steam if heat were added at a constant rate of 19.0 j/s ?

asked by Anonymous on October 11, 2008
18. ## chemistry

determine the specific heat of a 150.0 gram object that requires 62.0 cal of energy to raise its temperature 12.0 C.

i am sorry to bother you Ms. Sue but i need to present your proposal of a device that minimizes heat transfer. can you please give a list of insulators

asked by banna on April 30, 2017
20. ## physical science

How much heat is absorbed by a 30g iron skillet when its temperature rises from 24oC to 93oC? Joules do i addition everything?

asked by jake on January 24, 2017
21. ## chem

If you used the 8900 kilojoules you expend in energy in one day to heat 46000 {\rm g} of water at 25^\circ {\rm C}, what would be the rise in temperature?

asked by Anonymous on September 14, 2011
22. ## Chemistry

What is final temperature of a 88 gram sample symbol of water at 295 k that absorbs 1250 Joules of heat ?

asked by Kiki on November 14, 2016
23. ## Chem

H2SO4(aq) + Ba(NO3)2(aq) = BaSO4(s) + 2HNO3(aq) overall heat gen by reaction answer is supposed to be -659.3 but I keep coming up with something else.

asked by Drew on December 14, 2010
24. ## Chemistry

Calculate the heat released when 454 g of steam at 100.0C condenses and then cools to 30.0C. State your answer in calories.

asked by Ray on April 3, 2013
25. ## chemistry

How much heat (in kilocalories) is evolved or absorbed in the reaction of 1.60 g of Na with H2O? 2Na(s)+2H2O(l)→2NaOH(aq)+H2(g), ΔH = -88.0 kcal (-368 kJ)

asked by Anonymous on October 24, 2016
26. ## Chemistry

A certain substance has a heat of vaporization of 34.65 kj/mol. At what kelvin temperature will the vapor pressure be 3.50 times higher than it was at 331 k?

asked by Josh on January 25, 2012
27. ## chemistry

Determine how much heat (in kilojoules) is evolved or absorbed in the reaction of 1.05 g of Na with H2O: 2Na(s)+2H2O(l) →2NaOH(aq)+H2(g),ΔH∘=−368.4 kJ

asked by Jay on February 11, 2016
28. ## Physics

A 0.2 kg glass cup at 20 C is filled with 0.4 kg of hot water at 70 C. Neglecting any heat losses to the environment, what is the equilibrium temperature?

asked by Bob on May 23, 2012
29. ## francais freshmen

Translate into french please: Return to heat and cook until set and edge is dry,(takes approximately 30 seconds).

asked by Gabrielle on February 20, 2011
30. ## physics

A police offer may warn that “speed kills.” In video games, high-speed crashes are common. Suppose a particular scene in a game required the following elements: • a high performance motorcycle must crash into a barrier and be stopped • a small

asked by alejandro on April 5, 2013

Some turtles live in the sea. Some turtles live in the land. Some turtles are pets and live in houses. The words below relate to different places where turtles live. Read the words. Then write the words in the groups where they belong. tank-beach-soil-heat

32. ## Mathematics

a cuboid with length 13cm width 4cm and height h State in terms of h the shaded face of the cuboid. write an expression in terms of h for the volume of the cuboid. if the volume of the cuboid is 286cm cube calculate the height h of the cuboid.

asked by SAM on May 2, 2012
33. ## mathFormula

Formula for calculating cubic feet. length(in feet) x width(in feet) x height(in feet) = volume in cubic feet. Thank you DrBob That's for a cube or four sided regular figures. For a cylinder, it's pi*(radius2)*height and there are other formulas for other

asked by Richard on August 20, 2007
34. ## chemistry

Calculate the current required to transfer 0.015 mols of electrons in a time of 20 min

asked by Priya on April 22, 2008
35. ## physics

what potential difference delta V is required to accelerate protons from the rest to 10% of the speed of light?

asked by ike on April 13, 2011
36. ## chemistry

what mass of potassium chloride is required to prepare 5.00L of a 2.00M solution of KCL

asked by TROUBLED on October 24, 2011

Determine the weight of Na2C2O4 which is required to react completely with 20.0ml of 0.02M KMnO4.

asked by Sophie on May 5, 2013
38. ## chemistry

You need to make a solution containing 150 g of potassium chloride in 300 g of water. What temperature is required?

asked by Anonymous on April 19, 2012
39. ## Chemistry 20

What volume of concentrated phosphoric acid is required to prepare 250ml of a 0.375mol/L solutions?

asked by Cheryl on June 18, 2011
40. ## Physics

Calculate the work required to be done to stop a car of 1500kg moving with a velocity of 60km/hr?

asked by Abhishek on April 8, 2013
41. ## chemistry

how many liters of 0.100 M HCL would be required to react completely with 5.00 grams of calcium hydroxide?

asked by mac on March 6, 2008
42. ## chem

What volume of 0.0658 M calcium hydroxide is required to neutralize 45.10 mL of 0.0242 M nitric acid?

asked by Anonymous on January 19, 2015
43. ## physics

two electric charges +10 microC and -20microC are located .35 m apart how much work is required to double the distance between them?

asked by Grane on November 4, 2012
44. ## Physics

What push parallel to a 42 incline is required to just start a 340 N block sliding up the plane? us = 0.6

asked by MA on November 2, 2012
45. ## Chemistry help

How many moles of Fe3O4 are required to supply enough iron to prepare 0.752 mol Fe2O3?

asked by Tina on September 20, 2015
46. ## Chemistry

What mass of Pb(NO3)2,would be required to yeild 9g of PbCl2,on d addition of excex NaCl?Pb=207,N=14,O=16,Na=23,Cl=35.5

asked by Cally on April 10, 2019
47. ## Chemistry

If 30.3 liters of O2 at 273K and 1 atm are produced by the electrolysis of water, how many coulombs of charge were required?

asked by Dana on November 19, 2010
48. ## Math

A 21% oil gas mixture is to be formed by mixing 80% and 20% oil_gas mixtures. If 35L are required l, how much of each should be used?

asked by Abdul on February 11, 2019
49. ## Chemistry

If 30.3 liters of O2 at 273K and 1 atm are produced by the electrolysis of water, how many coulombs of charge were required?

asked by Dana on November 19, 2010
50. ## Physics

What net external force is required to give a 35.7 kg suitcase an acceleration of 2.8 m/s2 to the right? Answer in units of N.

asked by Brett on September 19, 2011
51. ## Chemistry

What volume of 0.5 N nitric acid would be required to completely neutralize 300 ml of 1.0 N sodium hydroxide?

asked by Jeff on October 6, 2010
52. ## chemistry

calculate the weight of oxalic acid required to neutralise 250 mL , 0.1 N KOH sollution?

asked by Anonymous on September 8, 2013
53. ## Economics/Money

Suppose that banks today were required to keep 100 per cent reserves against their deposits. What would be the result?

asked by Labz on April 21, 2012
54. ## chem

What mass of NaCl is required to precipitate all the silver ions from 19.2 mL of 0.100 M AgNO3 solution?

asked by Sara on March 19, 2011
55. ## Physics

What net external force is required to give a 30 kg suitcase an acceleration of 2.5 m/s2 to the right? Answer in units of N

asked by Sam on September 18, 2013
56. ## Chem

What pressure is required to reduce 70 mL of a gas at standard conditions to 16 mL at a temperature of 22◦C? Answer in atm.

asked by Thomas on May 7, 2019
57. ## Biochemistry

Calculate [H+] or [OH-]as required for each of the following solutions at 25degrees and state whether the solution is neutral, acidic or basic

asked by Mervis on June 10, 2017
58. ## math

How much time required to fill the 1000litre water tank with 2 inch pipe How it calculate

asked by vimalraj on March 19, 2017
59. ## chemistry

how ,amy grams of aspirin are required to react exactly with 29.40ml of a 0.6M solution of sodium hydroxide

asked by ujay on January 17, 2010
60. ## chemistry

What volume of 1.00 M NaOH would be required to neutralize 77.4 mL of 5.4 M HCl? Round to two places after the decimal point and keep it in mL

asked by Corso on February 20, 2014
61. ## chemistry

what weight of na2c03 of 95% purity would be required to neutralize 45.6ml of 0.235N acid? Who Helps

asked by Fai on January 27, 2013
62. ## chemistry

calculate the energy required to raise the temp of 72g of water from 50c tp 150

asked by dd on December 11, 2014
63. ## chemistry

what mass of potassium hydroxide is required to produce 2.00L of a solution mthat has a pOH of 4.5?

asked by judi on May 8, 2010
64. ## CHM 152

How many gram of Fe+3 are required to make 1.0 L of an aqueous solution of 10 ppm Fe+3? Assume density of water is 1.00 g/mL.

asked by Car on June 8, 2010
65. ## chemistry

Calculate the energy required for the ionization of an electron from the ground state of the hydrogen atom.

asked by masre man on November 26, 2018
66. ## AE

steps required for a multicultural education to be effective. would that be treating each student eaqually and fairly?

asked by scooby91320002 on March 15, 2009
67. ## geometry

How much foil is required to cover a rectangular box that is 4in. by 8in. by 5in.? use 3.14 as an aproximation for pi. where appropriate.

asked by DJ on July 17, 2010
68. ## physics

the force required to maintain an object at a constant velocity in free space is eaqual to?

asked by john on February 20, 2013
69. ## math

Find the derivative algebraically. g(t) = t^2 + t at t = −1 I can't seem to start this problem correctly. What is the process required for this question?

asked by Natalie on September 16, 2015
70. ## general chemistry 2

How many grams of urea [CO(NH2)2] would be required to prepare 200 mL of a 5.0 x 10-3 molar solution? Answer

asked by Anonymous on September 26, 2012
71. ## Chemistry

What volume in liters will be required to dissolve 60.9 grams of Mg(OH)2? Ksp = 1.8×10-11 After doing the math, i got 1.58e-4, but its not right, so i have no idea what to do now.

asked by Brianna on March 25, 2014
72. ## Algebra

Show that two WRONG's can make a RIGHT. Replace each different letter in the addition problem with a different digit. It is required that O=zero.

asked by Rachel on November 21, 2010
73. ## chemistry

What amount of current, in amperes, would be required to deposit 19.7 g of Au metal in 1.00 hour from a solution of Au(NO3)3?

asked by zak on May 2, 2014
74. ## Chemistry

How many grams of sodium sulphate (mol. wt. 142) are required to make 1.2 liters of 0.5N solution

asked by Frank on September 19, 2013
75. ## physics

A capacitor has a capacitance of 3 pF. What potential difference would be required to store a charge of 21 pC? Choose one answer. a. 3 V b. 21 V c. 7 V d. 0.14 V

76. ## College Physics

how much work is required to completely separate two charges (each -1.4 microcoulombs) and leave them at rest if they were initially 8 mm apart?

asked by CourtneyW on February 3, 2011

List the steps required to be able to view each font to the selected type? I cant find the answers for that.

asked by sami on September 24, 2017
78. ## chemistry

Calculate the energy required for ionization of electron from the ground state of hydrogen atom

asked by donald on January 9, 2017
79. ## math

what is the total amount of edging that would be required to cover all edges of a rectangular prism measuring 10 cm by 30 cm by 15 cm?

asked by sharon on May 7, 2015
80. ## chemistry

What volume of ammonia gas at STP is required to neutralise 40cm of 2m sulphuric acid

asked by Anonymous on April 28, 2018
81. ## chemistry

How many moles of Fe3O4 are required to supply enough iron to prepare 0.785 mol Fe2O3?

asked by Anonymous on September 22, 2012
82. ## Chemistry

What volume of Cl2 gas, measured at 687torr and 34∘C, is required to form 27g of NaCl?

asked by Amy on April 6, 2014
83. ## chemistry

The mass of FeSO4.7H2O required to prepare 500mL of 280ppm Fe2+solutin is

asked by selvi on April 21, 2010
84. ## Health Information Management

At what point during the course of treatment of a new patient should notice of use and disclosure of health information, required under HIPAA, be given?

asked by Ella on May 21, 2009
85. ## chem 130

what mass of sodium hydroxide will be required to neutralize completely 1.60x10^23 molecules of HCl?

asked by Heather on November 17, 2015
86. ## physics

Calculate the net force required to give a 3000 kg truck an acceleration of 2.5m per squared

asked by Natasha on October 15, 2018
87. ## health information

At what point during the course of treatment of a new patient should notice of use and disclosure of health information, required under HIPPA, be given and why?

asked by Sindi on August 2, 2008
88. ## Chemistry

How many milligrams of NaHCO3 are in a 500mg tablet if 40.0 mL of 0.120M HCL is required to neutralize the sample?

asked by prince on December 9, 2010
89. ## Math

How would you do this problem. What is the growth rate of the stock with a $3.00 expected dividend and a$20.60 price with 15%required return?

asked by Robin on June 21, 2013
90. ## Maths, physics, chemistry,

The weight of H2C2O4.2H2O required to prepare 500 ml of 0.2N solution is a)126g b)12.6g c)63g d)6.3g

asked by Deepak on April 29, 2017
91. ## health

Dietary guidelines are _________________ for Americans to follow. A recommendations B required C reinforced D repeated

92. ## Physics

What is the work required for a 1000kg truck to increase in speed from 30 m/s to 40m/s? Is the answer 350 kJ?

asked by Harold on March 4, 2009
93. ## algebra

(a) A graph of is shown in the figure, where y _ L(x). As the weight of the plane increases, what can be said about the length of the runway required?

asked by logan on June 1, 2011
94. ## Math

It is required to sit 4 males and 5 females in arrow so that the male occupy the even place. How many such arrangements are possible?

asked by Honey on October 7, 2015
95. ## chemistry

What mass of NaOH is required to precipitate all the Fe2+ ions from 50.0 mL of 0.200 M Fe(NO3)2 solution?

asked by jennifer on January 11, 2010
96. ## Chemistry

How many mL of 0.024M KMn04 (Potassium Permanganate) is required to react with 20mL of 0.112M Fe2+?

asked by Joe on May 29, 2011
97. ## chemistry

The time required for the completion of 99.9% of a first order reaction is equal to how many times that of its half life? (A)2 (B)3 (C)5 (D)10

asked by Poonam on March 12, 2012
98. ## Biochemistry

Calculate the mass (in g) of solid potassium hydroxide required to prepare: a) 1.75 litres of a solution with pH 11.5

asked by Sarah on August 3, 2016
99. ## Chemistry

How many moles of Fe3O4 are required to supply enough iron to prepare 0.442 mol Fe2O3?


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# Algebraic geometry by Shafarevich I.R.

By Shafarevich I.R.

Similar algebraic geometry books

Algebraic Functions And Projective Curves

This publication provides an advent to algebraic services and projective curves. It covers a variety of fabric through shelling out with the equipment of algebraic geometry and continuing at once through valuation conception to the most effects on functionality fields. It additionally develops the idea of singular curves via learning maps to projective area, together with issues comparable to Weierstrass issues in attribute p, and the Gorenstein family for singularities of aircraft curves.

Complex Manifolds and Deformation of Complex Structures

Kodaira is a Fields Medal Prize Winner.  (In the absence of a Nobel prize in arithmetic, they're considered as the top expert honour a mathematician can reach. ) Kodaira is an honorary member of the London Mathematical Society. cheap softcover variation of 1986 vintage

Moduli of Double Epw-sextics (Memoirs of the American Mathematical Society)

The writer reviews the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of $\bigwedge^3{\mathbb C}^6$ modulo the ordinary motion of $\mathrm{SL}_6$, name it $\mathfrak{M}$. it is a compactification of the moduli house of gentle double EPW-sextics and as a result birational to the moduli house of HK $4$-folds of style $K3^{[2]}$ polarized through a divisor of sq. $2$ for the Beauville-Bogomolov quadratic shape.

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# Axiomatic definition of exponent

Here's something I'm thinking about for a while, and would like to get feedback and some relevant references.

Say I want to define by axioms an operation that will act like exponent, without using addition, multiplication, limits etc.

Let's look at set K with binary operation { , } : KxK --> K that provides for each a,x,y,z,w in K :

1. {{a,x},y}={{a,y},x}

2. if {{a,x},y} = {a,w} then {{a,{x,z}},{y,z}} = {a,{w,z}}

The reason for the first axiom is clear, the second will be clarified later.

For a,x in K, if exists a unique w in K so that {a,w} = x, then write: [a,x] := w

(I want to express log operation)

We immediately get (If the following is defined): [a,{a,x}] = {a,[a,x]} = x

In addition, it can be shown that for each a,b in K (again, if defined): [a,a] = [b,b]

(so from now on I'll write just [,] to represent such expressions)

It follows that if exists [a,x] then: {x,[,]} = x

More identities that can be proven:

{[,],x} = [,]

{x,[a,[,]]} = [,]

(when translating it to exponents and logarithms, we get the known identities.)

From now on I won't pay attention to weather a specific log is defined, when I write an expression with log, the meaning is to the case it is defined.

Now I'd like to express the arithmetic operations using those operations. For that reason let's take some element e from K, and in order to get more compact expressions, I'll write them without e, for example:

[,x] := [e,x]

{y,} := {y,e}

The multiplication can be defined by: x*y := [{{x},y}]

The addition by: x+y := [[{{{x}},{y}}]]

Provides: x+y = [{x}*{y}] as required.

The first axiom gives commutativity and associativity to multiplication and addition. The second axiom gives distributivity between them.

The expressions [,] and [,[,]] act as identity element regarding to multiplication and addition.

The expressions [{,x},] and [,[{,{,x}},]] act as inverse (to x) elements regarding to multiplication and addition.

This definition pushes us to define a whole series of operations (*n) .

Define recursively (although it can be defined explicitly): x(*n+1)y := [,{,x}(*n){,y}]

and respectively: x(*n-1)y := {,[,x](*n)[,y]}

Now if we define the (*0) operation to be: x(*0)y = {x,[,y]} then we'll get the multiplication and addition as the (*-1) and the (*-2) operations, as part of a series of commutative, associative operations with distributions between each two consecutive operations, and with identity elements of the form [,[,[,...[,]...]]] or {,{,{,...{,}...}}} or e, and inverse elements.

Back to classic arithmetic, we have expressions for identity elements (0 and 1), and for multiplication and addition, which means we can express every rational number, for example:

0 = [,[,]]

1 = [,[,{,}]] = [,]

2 = [,[,{{,},}]]

3 = [,[,{{{,},},}]]

-1 = [,[{,},]]

1/2 = [[,{{,},}],]

...

It reminds Zermelo's construction of natural numbers...

I can continue, but I think the idea is clear. I still have a big gap related to the question when log is defined. I guess it can be defined in different ways in order to get integers, rationals, or other fields. obviously we can get roots as well... I'm curious weather I can allow existence of, for example, log(0) without getting contradiction... And what else can I get? I'm also not sure if the choosing of the second axiom is the best for this purpose.

Generally, I see this construction very natural and elegant but I don't know if it has a real value, means, if it can be connected to other mathematical fields. I'll be happy to hear any idea.

yy

• There have been a number of axiomatizations of the theory of ordered fields (or real-closed fields) with exponentiation. Names I can think of casually are Wilkie, Macintyre, Ressayre. There are quite a few others. All the work I know of is model-theoretic, so not directly related to your work. – André Nicolas Feb 28 '16 at 7:49


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## Elementary Automata (Generating Random Sequences X)

30/01/2018

I was rather discontented with last week’s post results. Most automata seemed to produce self-correcting patterns, even when seeded randomly—one could argue that rand() isn’t the strongest random generator, but that wasn’t the problem. No, indeed, most automata exhibit self-correcting behavior, forming the same self-similar pattern, or worse, the same periodic pattern.

So I made a few more experiments with random seeds and larger images. The code isn’t very complicated and isn’t of interest in itself, but it reveals a couple of interesting things.

## Elementary Automata (Generating Random Sequences IX)

23/01/2018

A while ago, Wolfram (re)introduced elementary cellular automata, a special case of cellular automata that lives in one dimension. One cutesy aspect of these automata is that the rules are easy to describe and number. As any automata, it is given a context, a region of interest, and gives an output depending on the context. For example, we can have the rules:

 Context output 000 1 001 0 010 1 011 0 100 1 101 0 110 1 111 1

The output column can be viewed as a number, in this case 11010101, 0xd5, or 213. This is rule 213.

It may be tempting to use those to generate chaos, noise, and random numbers. But is it that a good idea?

16/01/2018

## #include <the_usual>

09/01/2018

Recently on Freenode channel ##cpp, I saw some code using an include-all-you-can header. The idea was to help beginners to the language, help them start programming without having to remember which header was which, and which headers are needed.

Is that really a good idea?

## Fast Recurrences

02/01/2018

Quite a while ago, I wrote a post on how to compute an arbitrarily large Fibonacci number $F_n$ in essentially $O(\log n)$ steps. Turns out, that technique generalizes to other recurrence relations, and of any order.


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# Area enclosed by a semi-circular fence

A contractor has installed a silt fence around an area that is semi-circular and level to prevent soil from the construction site entering nearby streams. The diameter of the semi-circle is 900 feet. How many linear feet of fence does the contractor need to use to enclose the area? How many acres are within the enclosed by the fence? I need an answer for part two of the question.

-
Are you trying to solve by yourself or only pasting the problem here? –  Sigur Feb 5 at 0:41
I tried to solve it but i am having problems with the second part. thanks –  Sami Alshehry Feb 5 at 0:42
So you can write what you did ans we can give you some hints. –  Sigur Feb 5 at 0:43
Area of a whole circle =$\pi r^2$, and according to Google 43560 square ft to an acre. –  Maesumi Feb 5 at 0:45
Welcome to MSE! I'd like to tell you a few things that will likely improve the reception of your question. Your title does not indicate the content of the problem. Nor does your tag. This means that people have no way of knowing that they're interested in your problem (unless they click on it, but why would they when there are other, better indicated problems?). You have also just copy-pasted the question. Many people here expect you to put in some effort before they put in the effort to help. –  mixedmath Feb 5 at 0:47
Area of the whole circle is $\pi r^2=\pi 450^2$. For a semi circle it is $\pi 450^2/2$. In acres this is $\pi 450^2/(2*43560)=7.30$


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# If $f$ is uniformly differentiable then $f '$ is uniformly continuous?

The following theorem is true?

Theorem. Let $U\subset \mathbb{R}^m$ (open set) and $f:U\longrightarrow \mathbb{R}^n$ a differentiable function.

If $f$ is uniformly differentiable $\Longrightarrow$ $f':U\longrightarrow \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ is uniformly continuous.

Note that $f$ is uniformly differentiable if

$\forall \epsilon>0\,,\exists \delta>0:|\!|h|\!|<\delta,\color{blue}{[x,x+h]\subset U} \Longrightarrow |\!|f(x+h)-f(x)-f'(x)(h)|\!|<\epsilon |\!|h|\!|$ (edited)

$\forall \epsilon>0\,,\exists \delta>0:|\!|h|\!|<\delta,\color{blue}{x,x+h\in U} \Longrightarrow |\!|f(x+h)-f(x)-f'(x)(h)|\!|<\epsilon |\!|h|\!|\qquad \checkmark$

Any hints would be appreciated.

• What is your definition for uniformly differentiable? – Tomás Sep 5 '13 at 19:49
• @Tomás $\forall \epsilon>0\,,\exists \delta>0:|\!|h|\!|<\delta,[x,x+h]\subset U \Longrightarrow |\!|f(x+h)-f(x)-f'(x)(h)|\!|<\epsilon |\!|h|\!|$ – felipeuni Sep 5 '13 at 20:02
• What did you try ? (I think you know the proof in the case $\Bbb R \rightarrow \Bbb R$). – Tony Piccolo Sep 6 '13 at 10:26
• Do you know the proof in this case @TonyPiccolo? – Tomás Sep 6 '13 at 12:22
• See my new attempt. – Tony Piccolo Sep 24 '13 at 9:29

Let's build off of Tomas' last remark, slightly modified:

Let $t>0$ be small. Then \begin{eqnarray} \|f'(x)-f'(y)\| &=& \frac{1}{t}\sup_{\|w\|=1}\|\langle f'(x)-f'(y),tw\rangle\| \nonumber \\ &\leq& \frac{1}{t}\sup_{\|w\|=1}\|f(x+tw)-f(x)-[f(y+tw)-f(y)]\| + 2\epsilon \nonumber \end{eqnarray}

It suffices to show that this weighted combination of four close points on a parallelogram can be bounded by $C\epsilon t$.

Let us bound $\|f(x+h) - f(x) + f(x+k) - f(x+h+k)\|_2 \leq C\epsilon(\|h\|+\|k\|)$, and then in this case $\|h\|=t$ and $\|k\|\leq \delta$, so if $t=\delta$ the whole expression is bounded by a constant times $\epsilon$.

Note applying uniform differentiability three times in directions $h,k,$ and $h+k$, for small $\|h\|,\|k\|$ we have

\begin{eqnarray*} \|f(x+h) - f(x) + f(x+k) - f(x+h+k)\| &\leq& \|f'(x)h + f'(x)k - f'(x)(h+k)\|_2 + 3\epsilon(\|h\|+\|k\|)\\ &=& 3\epsilon(\|h\|+\|k\|) \end{eqnarray*}

• Shouldn't the first occurrence of $2 \epsilon$ actually be $2\epsilon/t$? I'm not sure you can magick away the magnitude of $w$ like that. – Anthony Carapetis Sep 10 '13 at 1:17
• @AnthonyCarapetis The original equation had $\epsilon |w|$, but since I replaced $w$ with $tw$, that's where the extra $t$ comes from. – Evan Sep 10 '13 at 1:21
• oh right. It works. – Anthony Carapetis Sep 10 '13 at 1:43
• @Evan only a question, the following equivalence is true ? $\forall \epsilon>0\,,\exists \delta>0:|\!|h|\!|<\delta,\color{blue}{[x,x+h]\subset U} \Longrightarrow |\!|f(x+h)-f(x)-f'(x)(h)|\!\!|<\epsilon |\!|h|\!|\Longleftrightarrow \forall \epsilon>0\,,\exists \delta>0:|\!|h|\!|<\delta,\color{blue}{x,x+h \in U} \Longrightarrow |\!|f(x+h)-f(x)-f'(x)(h)|\!\!|<\epsilon |\!|h|\!|$ – felipeuni Sep 10 '13 at 5:35
• @felipeuni Yes that is true, because the property only depends on $x$ and $x+h$, so there's no distinction here. – Evan Sep 10 '13 at 5:46

This answer only the case $f:U\subset\mathbb{R}\to \mathbb{R}^n$. By hypothesis we have that: given $\epsilon>0$, there exist $\delta>0$ such that if $|x-y|<\delta$ $$\left|\frac{f(x)-f(y)}{x-y}-f'(y)\right|<\frac{\epsilon}{2}\tag{1}$$

and

$$\left|\frac{f(y)-f(x)}{y-x}-f'(x)\right|<\frac{\epsilon}{2}\tag{2}$$

From $(2)$ we get that $$\left|\frac{f(x)-f(y)}{x-y}-f'(x)\right|<\frac{\epsilon}{2}\tag{3}$$

From $(1)$ and $(3)$ $$|f'(x)-f'(y)|<\epsilon$$

I dont know how to tackle the case $U\subset \mathbb{R}^m$.

Update: I would like to add here some thoughts, maybe it can help someone to give a full answer. First note that

\begin{eqnarray} \|f'(x)-f'(y)\| &=& \sup_{\|w\|=1}\|\langle f'(x)-f'(y),w\rangle\| \nonumber \\ &=& \sup_{\|w\|=1}\|f(x+w)-f(x)-[f(y+w)-f(y)]+o(x,w)-o(y,w)\| \nonumber \end{eqnarray}

By hypothesis, $\|o(x,w)-o(y,w)\|\leq \epsilon\|w\|+\epsilon\|w\|$, hence for $\|w\|=1$ we have that $\sup_{\|w\|=1}\|o(x,w)-o(y,w)\|$ is small, independetly of $x,y$. Therefore, it remains to prove that $f$ is uniformly continuous. One way to prove it is for example showing that $f'$ is bounded, which I think is true when $U$ is bounded. When $U$ is unbounded, I think we need a direct argument to show that $f$ is uniformly continuous.

• [Tomás] Note that $[x,y]\subset \mathbb{U}$ then $U$ should be an interval, definition of uniformly differentiable. – felipeuni Sep 9 '13 at 6:36
• @felipeuni you need to specify where are $x,y$. If $x,y$ are any points on $U$, then the only thing you can conclude is that $U$ is convex. – Tomás Sep 9 '13 at 12:32
• For bounded regions, take the closure to get a compact set, and then it's clear that $f$ is uniformly continuous. For unbounded regions, as you say we need a direct argument, as $f$ is not uniformly continuous in general, with a 1-d counterexample $f(x) = x^2$... – Evan Sep 9 '13 at 23:20

The difficulty here is that the obvious estimate (as in Tomás' answer) only gives a result along lines - i.e. it can't say anything about how $f'(x)h$ changes as you move in any direction but $h$. Thus I think we need some kind of "polarization argument" to get an isotropic result. When $f$ is $C^2$, this is literally polarizing the Hessian matrix $H(x) = f''(x)$ to show it is bounded from the fact that it is bounded on the diagonal. While we cannot use this literally for general $f$, hopefully we can use a finite-difference analogy and the analysis will go through. Here's the proof for the $C^2$ case and an optimistic starting point for the general case.

For any $\epsilon>0$ there is a $\delta>0$ such that for all $x,h$ with $|h|<\delta$, we have (cf. Tomás' answer)

\begin{align} |f(x+h) - f(x) - f'(x)h| &< \epsilon |h| \\ |f(x) - f(x+h) + f'(x+h)h| &< \epsilon |h| \end{align}

which combine to give $$t|(f'(x+t v)-f'(x))v| < 2 \epsilon$$ for $v$ a unit vector, $t<\delta$.

In the $C^2$ case the left side is $t^2 |H(x)(v,v)| + o(t^2)$, so we have a uniform bound $$|H(x)(v,v)| < M \tag{1}.$$ The polarization formula is $$H(x)(v,w) = \frac12 \left( H(x)(v+w,v+w) - H(x)(v,v) - H(x)(w,w) \right). \tag{2}$$

We want to show uniform continuity of $f'$. A line integral estimate gives $$|f'(x)w-f'(y)w| \le |x-y| \sup_{z \in U} |H(z)\left(v,w\right)|$$ where $|x-y|v = x-y$; so it suffices to show that $\sup_{z \in U, |v| = 1} H(z)(v,w)$ is finite. Applying $(1)$ and $(2)$ we estimate

\begin{align} |H(z)(v,w)| & \le \frac12 M \left( |v+w|^2 + |v|^2 + |w|^2 \right)\\ & \le M \left( 1 + |w| + |w|^2 \right) \end{align}

so we are done.

Now to try generalising this. Attempting to write down the polarization formula using finite difference terms gives

\begin{align} &(f'(x+v+w) - f'(x+v))w + (f'(x+w+v)-f'(x+w))v \\ = &(f'(x+v+w) - f'(x))(v+w) - (f'(x+v)-f'(x))v-(f'(x+w) - f'(x))w \\ \end{align}

which (since $f$ is uniformly differentiable) is bounded by $2 \epsilon (|v+w| + |v| + |w|)$.

Now we would like to equate the two terms on the LHS, which is the reason polarization works for symmetric forms. Our expression is of course not symmetric but we can hope it is close enough for small $u,v$ - I have no idea whether or not this can be done without more assumptions, however. If anyone has any ideas (or has spotted any mistakes) please let me know.

• Maybe we could approximate $f$ by $C^2$ functions? – Tomás Sep 9 '13 at 14:20
• @AnthonyCarapetis@Tomás I think the following article Theorem $3.4$ could help. physics.umanitoba.ca/~khodr/Publications/RSImplicitFT-Final.pdf – felipeuni Sep 9 '13 at 22:43
• @felipeuni: that theorem is for non-Archimedian fields, however at first glance the proof seems to work for the Euclidean case too! In fact it may even directly prove what you're looking for - I haven't been through it in detail but the estimates are probably all uniform in $x_0$. I think it's very similar to Evan's answer, which seems to be a complete proof. – Anthony Carapetis Sep 10 '13 at 2:00
• @AnthonyCarapetis the link is broken. The name of the article is "The implicit function theorem in a non-Archimedean setting". – Guillaume F. Jul 18 '18 at 17:07

Okay, let's get rid of the $R'$ and try the same thing this way. f is differentiable at $X_0$ means that $\exists$ a g which is linear throughout some neighborhood U of $X_0$, g(X) = $f(X_0)$ + (M,$(X-X_0)$ [ where the second term is the inner product of $(X-X_0)$ with a constant vector M] and for which $\lim_{X \rightarrow X_0}\frac{f(X) -g(X)}{X - X_0}$ = 0. This means that f is practically linear throughout U.

Now if f were precisely linear we would have no problem, since $f(X) - f(X_0)$ = (M,$X-X_0)$ and this is uniformly continuous throughout U. The problem is that f is not exactly linear. I am using R(X) to represent the difference between f, the not quite linear, and g the exactly linear.

So the question of uniform continuity of f in the neighborhood U comes down to the behavior of R(X) in U. What do we know about R? We know that $\lim_{X \rightarrow X_0} \frac{R(X)}{X-X_0}$ = 0 and further that the convergence is uniform in U. That means that if you pick an $\epsilon$, there is a $\delta$ independent of X such that in $V \subset U$ a neighborhood of $X_0$ of radius $\delta$, if $X_1$ and $X_2$ are in V then $\frac {|R(X_1) - R(X_2)|}{|X_1-X_2|}$ < $\epsilon$. Substituting back the definition of R we have

$R(X_1)$ = $f(X_1)-g(X_1)$ = $f(X_1) - f(X_0)- (M,(X_1-X_0))$ and similarly for $X_2$. So

$\frac {|R(X_1) - R(X_2)|}{|X1-X_2|}$ = $\frac{f(X_1) -f(X_2) - [(M,(X_1-X_0))-(M,(X_2-X_0))]}{X_1 - X_2}$ < $\epsilon$ when $X_1, X_2$ are in V.

We get $|f(X_1)-f(X_2)|$ < $\epsilon|(X_1-X_2)|$ + $[(M,(X_1-X_0))-(M,(X_2-X_0))]$. The terms in square brackets go to 0 uniformly as $X_1 \rightarrow X_2$ because they are the difference of a linear function at 2 points in V, and the $\epsilon|(X_1-X_2)|$ likewise .

The proof of theorem $3.4$ in your reference contains the idea to solve the problem: one has to prove the continuity of partial derivatives.
The reasoning is valid apart from the non-Archimedean and local context of the writing.

In the $\varepsilon-\delta \,$ game let $\delta \,$ be a reply to $\varepsilon /4$ as for the differentiability of $\mathbf f$ and let $\delta_1$ be a reply to $\varepsilon \delta/4$ as for the continuity of $\mathbf f$.

In the following $f_j^i$ is the derivative of $f^i$ respect to $x_j \,$.

Now choose $\mathbf x,\mathbf y \in U$ such that $||\mathbf x-\mathbf y||<\min \{\delta,\delta_1\}$.
Then $$|f_j^i(\mathbf x)\delta-f_j^i(\mathbf y)\delta| \le$$$$|f^i(\mathbf x+\delta\hat {\mathbf e}_j)-f^i(\mathbf x)-f_j^i(\mathbf x)\delta|$$$$+ \,|f^i(\mathbf y+\delta\hat {\mathbf e}_j)-f^i(\mathbf y)-f_j^i(\mathbf y)\delta|$$$$+ \,|f^i(\mathbf x)-f^i(\mathbf y)|+|f^i(\mathbf x+\delta\hat {\mathbf e}_j)-f^i(\mathbf y+\delta\hat {\mathbf e}_j)|$$$$< \varepsilon \delta$$from which the continuity of $f_j^i \,$.
The continuity of the (total) derivative follows from $$||\mathbf f'(\mathbf x)-\mathbf f'(\mathbf y)|| \le \left \{\sum_{i,j} \;[f_j^i(\mathbf x)-f_j^i(\mathbf y)]^2 \right \} ^\frac 12$$by the Schwarz inequality.

Edit:$\;$ OP advises that the proof doesn't work: in fact the uniform continuity of $\mathbf f$ is not assured because $U$ is open.

new attempt

Let $U$ be simply open: I want to prove that $f'$ is uniformly continuous on the set$$U_{\Delta}=\{x\in U:\ {\rm dist}\,(x,\Bbb R^m\backslash U)>\Delta\}$$with $\Delta>0$ chosen sufficiently small.

Avoiding some detail, let $x \in U_{\Delta}$ and let $\delta>0$ corresponding to $\varepsilon>0$ and such that $\delta<\Delta$.

We have $$(f'(x+h)-f'(x))(k)=$$$$-[f(x+k)-f(x+h)-f'(x+h)(k-h)]+$$$$f(x+k)-f(x)-f'(x)(k)+$$$$f(x)-f(x+h)-f'(x+h)(-h)$$Take $h$ and $k$ so that $\|h\|<\frac \delta 2$ and $\|k\|=\|h\|\,$: hence $x+h,x+k \in U$.

Then $$\|(f'(x+h)-f'(x))(k)\|<4\varepsilon \|k\|$$

Now let $u \in \mathbb R^m$ such that $\|u\|=1$ and let $k=\|h\|\,u\,$: we have $$\|(f'(x+h)-f'(x))(u)\|<4\varepsilon$$Then $$\|f'(x+h)-f'(x)\|\le4\varepsilon$$

• [@TonyPiccolo] Then $\displaystyle \delta_1=\delta_1(\frac{\epsilon \delta}{4},x)$ as $f$ is continuous only, $\delta_2=\min \{\delta,\delta_1\}$ should not depend on the points $x,y.$ – felipeuni Sep 13 '13 at 16:43
• The point is that $\mathbf f$ is uniformly continuous because it is uniformly differentiable: modify slightly the statement and the proof of proposition $3.3$ in your reference and you have the result. – Tony Piccolo Sep 13 '13 at 17:17
• [@TonyPiccolo] Note that $f:\mathbb{R}\longrightarrow \mathbb{R},f(x)=x^2$ is uniformly differentiable but $f$ is non-uniformly continuous. – felipeuni Sep 13 '13 at 18:03
• Sorry, but I had forgotten that $U$ is open. – Tony Piccolo Sep 13 '13 at 22:37
• However do you think that proving the uniform continuity of partial derivatives is possible ? For me it should be the most natural way to solve your question. – Tony Piccolo Sep 15 '13 at 6:49

From the uniform differentiabilty, we can find a $t>0$ such that, for every ${\bf v}$ with $\|{\bf v}\| \le 2t$ and ${\bf x} \in U$, ${\bf x} + {\bf v} \in U$,

$$t^{-1}\| f({\bf x}) - f({\bf x}+{\bf v}) + f'({\bf x}){\bf v}\| < \frac{\epsilon}{5}$$

Now, let take any ${\bf x}$, ${\bf y}$ such that $\|{\bf x} - {\bf y}\| < t$, and ${\bf x}, {\bf y} \in U$. Define ${\bf k}$ = ${\bf y} - {\bf x}$.

Note that for any $t{\bf w}$ with $\| {\bf w} \|=1$, we have $\|t{\bf w} + {\bf k}\| < 2t$.

We then have, for every ${\bf x}$ such that an open ball of radius $2t$ is contained in $U$,

\begin{align} \left\| f'({\bf x}) - f'({\bf y})\right\| &= t^{-1}\max_{\|{\bf w}\|=1}\left\| f'({\bf x}){t\bf w} - f'({\bf x} +{\bf k})t{\bf w}\right\| \\ &\le t^{-1}\max_{|{\bf w}|=1} \left\| ( f({\bf x}+t{\bf w}) - f({\bf x}) + f({\bf x} + {\bf k}) - f({\bf x} + {\bf k} + t{\bf w}) \right\| + \frac{2}{5}\epsilon \\ &\le t^{-1}\max_{|{\bf w}|=1}\left\| f'({\bf x})({t\bf w} + {\bf h} -t{\bf w} -{\bf h})\right\| + \epsilon\\ &= \epsilon \end{align}

However, I do not know how to prove it for ${\bf x}$ less than $2t$ away from a point outside $U$.

Maybe some sort of extension theorem would help?

Otherwise, we can strengthen the definition of uniform differentiability to the following:

$\forall \epsilon>0\,,\exists \delta>0:|\!|h|\!|<\delta,\color{blue}{x \in U} \Longrightarrow |\!|f(x+h)-f(x)-f'(x)(h)|\!|<\epsilon |\!|h|\!|\qquad$

in which case it does work.

Let us start with the simplest case f:$R^1 \rightarrow R^1$. The discussion below will generalize nicely to more dimensions.

Then f is differentiable at $x_0$ iff $\lim_{x \rightarrow x_0} {\frac{f(x)-f(x_0)}{x-x_0}}$ exists and is finite. We call that limit $f'(x_0)$ . So if f is differentiable at $x_0$ then we may write

f(x) = $f(x_0) + f'(x_0)(x-x_0) + R(x)$ where $\lim_{x \rightarrow x_0} R(x)/(x-x_0)$ = 0.     (1)

If f is uniformly differentiable that means

$\lim_{x \rightarrow x_0} R(x)/(x-x_0)$ converges uniformly to zero.     (2)

From (1) we have $f'(x)$ = $f'(x_0)$ + R'(x). [We know R is differentiable because it is the difference between f which is differentiable and a linear polynomial.]

So $\lim_{x \rightarrow x_0} (f'(x)-f'(x_0))$ = $\lim_{x \rightarrow x_0} R'(x)$ = $\lim_{x \rightarrow x_0} R(x)/(x-x_0)$ = 0. And this convergence is uniform as per our hypothesis as stated in (2).

Moving on to the case where $f:R^n \rightarrow R^m$. For this part of the discussion capital letters will be used to represent vectors (in whatever dimension).

According to many sources (and I will state specifically Robert Sealey "Calculus of Several Variables") f is differentiable at $X_0$ iff there exists a linear function g (X) = $f(X_0)$ + $(M,X-X_0)$ [where M is a constant vector and (a,b) means the inner product], such that

$\lim_{X \rightarrow X_0} {\frac{f(X)-g(X)}{|X-X_0|}}$ = 0.   (3)

If f is uniformly differentiable that means simply that this limit converges uniformly.We could have stated differentiability this way in the $R^1 \rightarrow R^1$ case, but I stated it as I did in that case to throw some light on what g actually is.

Let R(X) = f(X) - g(X) $\Rightarrow$ f(X) = g(X) + R(X). Since R = f - g then $R/|x-x_0| \rightarrow 0$ uniformly. Notice also that $R(X_0)$ = 0, since it is $f(X_0) - g(X_0)$ and by (3) this must be 0. Since g is linear and f is differentiable R is also differentiable and from the above $R'(X_0)$ = 0.

Looking at $f'$ we have $f'(X) = g'(X) + R'(X) = M + R'(X)$;
$f'(X_0) = M + R'(X_0)$ and thus $f'(X_0) = M$.

Finally $\lim_{X \rightarrow X_0} (f'(X)-f'(X_0))$ = $\lim_{X \rightarrow X_0} R'(X)$ = $\lim_{X \rightarrow X_0}{\frac{R(X)-R(X_0)}{X-X_0}}$ = $\lim_{X \rightarrow X_0}{\frac{R(X)}{X-X_0}}$ and this goes to 0 uniformly because f is uniformly differentiable, R = f - g, and $R(X_0)$ is 0.

• Haven't you assumed $R'$ is continuous? Can we deduce continuously differentiable from uniformly differentiable? – Anthony Carapetis Sep 9 '13 at 6:09
• @BettyMock Note that $f$ is uniformly differentiable if $\forall \epsilon>0\,,\exists \delta>0:|\!|h|\!|<\delta,\color{blue}{[x,x+h]\subset U} \Longrightarrow |\!|f(x+h)-f(x)-f'(x)(h)|\!\!|<\epsilon |\!|h|\!|$ – felipeuni Sep 9 '13 at 6:15
• In fact I think that for your convergence to be uniform you have used that $R'$ is uniformly continuous when you replaced $\lim_{X \to X_0} R'(X)$ with $R'(X_0)$, so the argument seems circular. – Anthony Carapetis Sep 9 '13 at 6:28
• @AnthonyCarapetis -- The R(x)/$(X-X_0) \rightarrow$ 0 is exactly equation (3), which is the definition of differentiable. f is uniformly differentiable is equivalent to (3) converges uniformly. Re $R'(X_0)$ = 0 we have $(R(X)-R(X_0))/|X-X_0|$ = $R(X)/|X-X_0|$; this converges to 0 uniformly because it is equation (3) where convergence is uniform. And of course this limit is $R'(X_0)$. So I don't think I'm going in a circle. I relied strictly on the definition of differentiable and the given uniform convergence. I don't think I am assuming anywhere that R' is continuous. – Betty Mock Sep 10 '13 at 0:46
• @felipeuni. That is a correct definition of uniformly differentiable. However, every definition can be stated in different ways. I am stating it as equation (3) converges uniformly. If you check it out, you will see they are equivalent. – Betty Mock Sep 10 '13 at 0:49


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<meta http-equiv="refresh" content="1; url=/nojavascript/"> Graphs of Square Root Functions | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: CK-12 Algebra - Basic Go to the latest version.

# 11.1: Graphs of Square Root Functions

Created by: CK-12

You have used squared roots many times in this text: to simplify, to evaluate, and to solve. This lesson will focus on the graph of the square root function.

The square root function is defined by $f(x)=\sqrt{x-h}+k$, where $x-h\ge0$ and $(h,k)$ represents the origin of the curve.

The graph of the parent function $f(x)=\sqrt{x}$ is shown below. The function is not defined for negative values of $x$; you cannot take the square root of a negative number and get a real value.

By shifting the square root function around the coordinate plane, you will change the origin of the curve.

Example: Graph $f(x)=\sqrt{x}+4$ and compare it to the parent function.

Solution: This graph has been shifted vertically upward four units from the parent function $f(x)=\sqrt{x}$. The graph is shown below.

## Graphing Square Root Functions Using a Calculator

Graphing square root functions is similar to graphing linear, quadratic, or exponential functions. Use the following steps:

These figures should be side by side. Due to the captions, they have moved in a vertical alignment.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both.

1. In the definition of a square root function, why must $(x-h) \ge 0$?
2. What is the domain and range of the parent function $f(x)=\sqrt{x}$?

Identify the ordered pair of the origin of each square root function.

1. $f(x)=\sqrt{x-2}$
2. $g(x)=\sqrt{x+4}+6$
3. $h(x)=\sqrt{x-1}-1$
4. $y=\sqrt{x}+3$
5. $f(x)=\sqrt{2x}+4$

Graph the following functions on the same coordinate axes.

1. $y=\sqrt{x}, \ y=2.5 \sqrt{x}$, and $y=-2.5 \sqrt{x}$
2. $y=\sqrt{x}, \ y = 0.3 \sqrt{x}$, and $y=0.6 \sqrt{x}$
3. $y=\sqrt{x}, \ y=\sqrt{x-5}$, and $y=\sqrt{x+5}$
4. $y =\sqrt{x}, \ y = \sqrt{x} + 8$, and $y=\sqrt{x}-8$

In 12-20, graph the function.

1. $y = \sqrt{2x-1}$
2. $y = \sqrt{4x+4}$
3. $y = \sqrt{5-x}$
4. $y = 2\sqrt{x}+5$
5. $y = 3-\sqrt{x}$
6. $y = 4 + 2\sqrt{x}$
7. $y = 2\sqrt{2x+3}+1$
8. $y = 4 + 2\sqrt{2-x}$
9. $y = \sqrt{x+1}-\sqrt{4x-5}$
10. The length between any two consecutive bases of a baseball diamond is 90 feet. How much shorter is it for the catcher to walk along the diagonal from home plate to second base than the runner running from second to home?
11. The units of acceleration of gravity are given in feet per second squared. It is $g=32 \ ft/s^2$ at sea level. Graph the period of a pendulum with respect to its length in feet. For what length in feet will the period of a pendulum be two seconds?
12. The acceleration of gravity on the Moon is $1.6 \ m/s^2$. Graph the period of a pendulum on the Moon with respect to its length in meters. For what length, in meters, will the period of a pendulum be 10 seconds?
13. The acceleration of gravity on Mars is $3.69 \ m/s^2$. Graph the period of a pendulum on Mars with respect to its length in meters. For what length, in meters, will the period of a pendulum be three seconds?
14. The acceleration of gravity on the Earth depends on the latitude and altitude of a place. The value of $g$ is slightly smaller for places closer to the Equator than places closer to the Poles, and the value of $g$ is slightly smaller for places at higher altitudes that it is for places at lower altitudes. In Helsinki, the value of $g=9.819 \ m/s^2$, in Los Angeles the value of $g=9.796 \ m/s^2$, and in Mexico City the value of $g=9.779 \ m/s^2$. Graph the period of a pendulum with respect to its length for all three cities on the same graph. Use the formula to find the length (in meters) of a pendulum with a period of 8 seconds for each of these cities.
15. The aspect ratio of a wide-screen TV is 2.39:1. Graph the length of the diagonal of a screen as a function of the area of the screen. What is the diagonal of a screen with area $150 \ in^2$?

Graph the following functions using a graphing calculator.

1. $y=\sqrt{3x-2}$
2. $y=4+\sqrt{2-x}$
3. $y = \sqrt{x^2-9}$
4. $y = \sqrt{x} - \sqrt{x+2}$

Mixed Review

1. Solve $16=2x^2-3x+4$.
2. Write an equation for a line with slope of 0.2 containing the point (1, 10).
3. Are these lines parallel, perpendicular, or neither: $x+5y=16$ and $y=5x-3$?
4. Which of the following vertices minimizes the expression $20x+32y$?
1. (50, 0)
2. (0, 60)
3. (15, 30)
5. Is the following graph a function? Explain your reasoning.
6. Between which two consecutive integers is $\sqrt{205}$?

8 , 9

Feb 22, 2012

Aug 21, 2014


text:
The Community for Technology Leaders
Issue No. 04 - April (2008 vol. 41)
ISSN: 0018-9162
pp: 5-6
Space Pens and Pencils
Editor's note: The following is representative of several letters we received regarding the urban myth referred to in the The Profession column in Computer's February 2008 issue.
I was disappointed to see that Computer printed an urban myth as a fact in the The Profession column in its February issue (R. Natarajan, "On Attending Conferences," pp. 108, 107). It is ironic that this inaccurate statement is printed right after the author states that "there is a great danger of becoming shallow if we abstain from research." Although it might not have been the author's intent, this unfortunate occurrence serves only to fuel a false perception of NASA's incompetence as well as an inflammatory indication of the agency's carelessness at US taxpayer's expense.
Here are the facts:

• During the first NASA space missions, the astronauts used pencils.

• In 1965, NASA ordered 34 pencils from Tycam Engineering Manufacturing at $128.80 per unit. • Paul C. Fisher developed the Fisher Space Pen and offered it to NASA. • Mr. Fisher developed the pen without NASA funding. • NASA purchased approximately 400 pens from Fisher in 1967 at$6 per unit for Project Apollo.

• The Soviet Union also purchased 100 of the Fisher pens.

• Both American astronauts and Soviet/Russian cosmonauts have continued to use these pens.

The problem with pencils is that they are hazardous items in weightless conditions and pure oxygen atmosphere. It took only a few minutes of "research" to obtain reliable facts. They are available from http://history.nasa.gov/spacepen.html as well as from www.spacepen.com/Public/History/index.cfm. It took about the same amount of time to track down the myth sources ( www.snopes.com/business/genius/spacepen.asp).
Aleksandar Fabijanic
alex.fabijanic@computer.org
The author responds:
Thanks to Aleksandar Fabijanic and the other correspondents who pointed out that the NASA pen and pencil story is not to be taken as a real historical episode. This could be helpful to readers who might have failed to see that it was just an anecdote narrated in a lighter vein with the purpose of setting the mood before describing more true-to-life examples of innovations brought about by outsiders to various disciplines. In fact, a search for the keyword phrase "space pencil" on Google immediately reveals that this is a well-known joke, and the top hit http://www.thespacereview.com/article/613/1 provides a clear explanation of the real facts.
Raja Natarajan
raja@tifr.res.in
E-Voting
I read "Secure and Easy Internet Voting" by Giampiero E.G. Beroggi (Feb. 2008, pp. 52–56) with interest. It's great to see a system that encourages greater participation through convenience while ensuring the integrity of the election. The only concern that I did not see fully addressed was the point that "no one can intercept, change, or reroute electronically cast votes." This requirement might not be met for votes cast on a computer with malware present at time of voting.
The "Silent Banker" trojan is an example of this danger; it places itself between the browser and the SSL layer and does not need to break an SSL session to steal money from the victim. Similar trojan malware would have full control over a voting session by modifying the vote results as they are submitted and faking confirmation response screens to make the voter believe the vote was cast as intended. Even if the voting system uses a secure applet, a trojan having full control over the screen, mouse, and keyboard could tamper with the vote.
Antivirus researchers commonly place the number of infected computers worldwide at around 11 percent. If a foreign government or organization were to control a large percentage of these computers, the corrupted votes would have a significant effect on the election results.
Michael Nice
niceman@att.net
I read with interest the February article on Swiss "Secure and Easy Internet Voting." I was hoping that someone had solved the problems, but sadly, that is not the case. Easy, perhaps, but secure?
Nothing in this system prevents man-in-the-middle attacks, and recent discussions of malware that modifies a cracked system' DNS server settings shows this is critical. Nothing prevents one person from being registered multiple times or one person from voting for all the registered voters in one household. The sale of votes is not prevented. It is trivial to monitor the voter's display to make sure he's voting the way he is supposed to, and he can't change his vote later. These all seem to violate at least three of the four Federal Chancellery rules.
Further, recounts are meaningless since the same software will do the recount that did the first, and a modified database of votes will count the same twice, thrice, or more no matter whose software counts things.
The author claims that the "e-voting hardware itself is in a steel cage," but surely none of the home systems that connect to it and are part of the system as a whole are. The system's security relies on tens or eventually hundreds of thousands of home computers.
The most serious statement comes just prior to that, referring to the decision to keep the code from the public. Attackers "with such access could modify voting and auditingrecords." Security through obscurity is a dangerous way to operate, since clearly someone does have access to the code and thus could find the means to modify voting records.
And yet, this system has received an award for its "remarkably high security standard." If this is true, then I have an encryption routine I'd like to sell you. You can't see the code for it, but I promise it's very secure. In fact, I've used it to encrypt this e-mail, not just once, but twice. I call it "ROT13."
Mark Kramer
c28f62@theworld.com
"Secure and Easy Internet Voting" contains a glaring contradiction. The author concedes that the Swiss e-voting system he describes falls short of the ACM's recommendations in that it "does not lend itself to a reproducible recording of each voter's actions." He purports to excuse this shortcoming on the grounds that "a paper trail is … dangerous in that it provides a visible receipt [which] could subject voters to bribery from those seeking to buy and sell votes." Yet he overlooks the obvious fact that when a Swiss e-voter uses a computer or a mobile phone, there's no way to know who might be looking over the e-voter's shoulder, buying or coercing his vote.
What Mr. Beroggi doesn't understand about the ACM's recommendations is that the "visible receipt" can't be taken away from the polling place, and it is in fact treated by the voting authorities like a paper ballot. It thus poses far less risk of vote buying or coercion than the Swiss e-voting system, which might be easy but is far from secure.
Hamilton Richards
ham@cs.utexas.edu
Typographical Poverty
Catching up on my reading, I found "The Profession as a Culture Killer" (Sept. 2007, pp. 112, 110–111) of particular interest in its remarks on typographical poverty in personal computing and its development.
This poverty was not universal in the early days of personal computing. The Amstrad PCW 8256 word processing computer of two decades ago ( www.old-computers.com/museum/computer.asp?c=189) used a special keyboard with a program called LocoScript that allowed applying all accents to any character by simple and obvious keying. It was very popular at the time.
What a pity we can no longer do this.
Piotr Karocki
pkar@ieee.org
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text:
# U-sub integral: Not sure where i went wrong.

1. Jan 21, 2013

### dwdoyle8854

1. The problem statement, all variables and given/known data

Im not sure what i did wrong, every operation i took to evaluate the integral seems valid. I took pictures of the integral evaluated two different ways, producing two different answers. What bugs me is even if i evaluate (as i did) from say 0 to 1 the numbers are different, so they are not equivalent. I dont see any errors in the first one, but the text book lists the latter as a correct answer.

The incorrect way:

Correct:

2. Relevant equations

3. The attempt at a solution
The incorrect way:
[/PLAIN]

Correct:
[/PLAIN]

Last edited by a moderator: May 6, 2017
2. Jan 21, 2013

### Dick

Saying u/(2+u^2-2u) equals u/2+u^2/u-2u/u is seriously bad algebra. (a+b)/c=a/c+b/c. c/(a+b) IS NOT equal to c/a+c/b. Try some numbers.

3. Jan 21, 2013

### SammyS

Staff Emeritus
Your error is in doing basic algebra with a rational expression.

$\displaystyle \frac{u}{u^2-2u+2}\ne\frac{u}{u^2}-\frac{u}{2u}+\frac{u}{2}$

After all, is $\displaystyle \ \ \frac{1}{1+2+3}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}\ ? \ \$

Last edited by a moderator: May 6, 2017
4. Jan 21, 2013

### dwdoyle8854

oh christ. im dumb.

Apologies.


text:
# Sql-server – Nonclustered Index Rebuild Online on Heap Table

enterprise-editionheapnonclustered-indexsql serversql-server-2012

Recently read that rebuilding a heap online in SQL Server Enterprise is single threaded and does not go parallel.

My question is: If you try to rebuild online just non-clustered indexes on a heap table with those also be single threaded or can those go parallel?

### Background

I am trying to find the best place to actually test it, but right now I have limited hardware available to me. I am new to the company and I am not advocating a heap, I am trying to steer them towards clustered indexes, but they have random GUIDs everywhere and lots of the developers think clustered indexes will hurt performance. I am trying to educate them on the advantages of using clustered indexes rather than heaps and one of those components is index maintenance.

We had one stored procedure that was doing over 27 million logical reads. I went in and looked at fragmentation on one of the key table scans, rebuilt the heap and logical reads went down to 560,000. Just looking at forwarded records and fragmentation I would not have even rebuilt that table through a scripted job. Trying to make them understand if we are gonna keep heaps we need to perform regular maintenance on these things.

The cost of this maintenance is higher for rebuilding heaps than having clustered indexes. That is really where my question about single threaded is coming from. Sorry to be long winded but thought would help explain question.

My question is: If you try to rebuild online just non-clustered indexes on a heap table with those also be single threaded or can those go parallel?

It can be multi threaded, the restriction is ONLY to alter table rebuild command when done with ONLINE option. I would like to quote from Enabling Compression on Heap MSDN Blog

(1) If you use ONLINE option then HEAP rebuild is single threaded. Note, this restriction is not there when you are rebuilding an index which supports multi-threaded ONLINE build. However, in both cases, OFFLINE build does support multi-threading.

A non-clustered index rebuild can be multi threaded. Please note for index rebuild to use multiple threads the edition should be Enterprise edition or above( if there is such a edition). This is still true from SQL Server 2005 to SQL Server 2019.

For example, rebuilding a heap with three nonclustered indexes online:

ALTER TABLE dbo.Fish REBUILD WITH (ONLINE = ON);

The heap rebuild is serial:

...while the associated nonclustered index rebuilds are parallel:


text:
# What is the cause for the inclusion of 'thermal equilibrium' in the statement of Ergodic hypothesis?

This is the fundamental assumption of statistical mechanics:

In an isolated system in thermal equilibrium $^1$, all accessible microstates are equally probable.

But why does it mention the condition thermal equilibrium?

As Daniel V. Schroeder mentions:

Any large system in the equilibrium will be found in the macrostate with the greatest entropy.

Equilibrium is the state where there can only be that macrostate which has the greatest multiplicity that is the greatest entropy.

Also, check this pic taken from here; it clearly mentions thermal equilibrium at $q_A =4$; there is only one macrostate corresponding to $q_A= 4$ in thermal equilibrium and not more than one.

But the ergodic hypothesis says otherwise; it says in thermal equilibrium, all microstates are equally probable. Now, among those accessible microstates, there are also many which correspond to $q_A\ne 4$; these microstates define such macrostates which have lesser entropy than the macrostate to that of $q_A=4$ in the example. So, how can it be possible for those microstates of lesser-entropic macrostates $q_A\ne 4$ to exist at thermal equilibrium? For, there can only be one macrostate i.e. $q_A= 4$ at thermal equilibrium.

The famous line goes as:

Entropy is maximum at equilibrium.

You can't expect microstates of lesser-entropic macrostates in thermal equilibrium, isn't it? Equilibrium exists only at the macrostate having the greatest number of microstates; ie. $q_A= 4\;.$

So, how can I say, all microstates are equally likely in thermal equilibrium if microstates other than that of the macrostate having greatest entropy $q_A= 4$ can't exist in thermal equilibrium?

• A system in thermodynamical equilibrium still has fluctuations that move it out of equilibrium. But the probability of this happening is very low by Poincare's recursion theorem, fluctuations occur at time scales that are exponentially long on the size of the entropy fluctuation . – user83548 Nov 11 '15 at 19:39
• @bruce smitherson: So, the state defined at equilibrium is only the most entropic state, isn't it? But the microstates of the other macrostates are also equally likely in thermal equilibrium 7 that implies other macrostste an still occur under thermal equilibrium. This is what I am cofused about why those macrostates which has lesser entropy can be considered in thermal equilibrium. How? Are temperatures same ? – user36790 Nov 11 '15 at 19:47
• Yes, if the system moves to a state in which the occupied microstates result in an unlikely macrostate, then you are no longer in thermal equilibrium. However, in most cases you never observe this situation, as it is highly unlikely. I do not know why you claim that "it seems any state, no matter how less the probability be, can be in thermal equilibrium". That is seems to be incorrect. Do you have any reference that state that? – user83548 Nov 11 '15 at 20:36
• @bruce smitherson: Yes, I also deem it it incorrect. But see since all the microstates are equally likely, maybe that the system now is in such microstate which is not of the most entropic state, some other state. And as said, the microstates are equally likely, when the system is in thermal equilibrium. That's why I asked how if in thermal equilibrium, some other microstate appeared, how could it be in thermal equilibrium then. Probably that is fluctuation which becomes negligible when the numbers are very large. – user36790 Nov 12 '15 at 2:53
• An isolated system is said to be in thermal equilibrium if you can define a constant temperature across it. The fact that the system can explore all microstates with equal probability does not mean that it is in thermal equilibrium. Thermal equilibrium is actually an average. If the fluctuation is so large (in either space or time) that you can measure it, then the system is momentarily out of thermal equilibrium. – user83548 Nov 13 '15 at 18:47

A macrostate is a set of microstates. Some microstates are thermal, others are not.

Without the assumption of being in thermal equilibrium you can't assume anything since any possible microstate is possible. And lots of possibilities macrostates could be picked.

Usually you want to group your macrostates according to a state variables such as pressure, volume, total energy or something like that.

And when you break your 30 microstates into three groups: A, B, and C you can ask yourself if each group is classified according to a state variable such as pressure, volume, total energy or something like that.

And even if it is, then all you might know is the state variable, and even that maybe not precisely. For instance the volume isn't precisely known since the exact locations of all the many parts is not known.

Now even when the microstate is a particular microstate, and that microstate is assigned specifically to a particular macrostate, that doesn't tell you how that macrostate assigns probabilities to the microstates in it

And you can assume that each microstate in the macrostate is equally likely. But that is just an assumption. If energy is conserved, then the dynamics will always keep the total system at a configuration with that fixed initial energy, so it doesn't change from any state to just any state.

Doesn't thermal equilibrium mean the macrostate having the greatest multiplicity?

It means so much more. Firstly, it requires that a macrostate is a probability distribution on microstates. Secondly, the macrostate is specified by some (macro) state variables. Thirdly, the particular distribution specified by the hypothesis requires that the space of all microstates be partitioned (partitioned by different values of the state variables) and each partition has an equal probability assigned to every microstate in that part of the partition. I always imagine different floors of a mansion, where your variables constrain you to a different floor and each room in a floor is equally likely.

Now, thermal equilibrium doesn't mean having the greatest multiplicity. You could have $N$ particles of some gas at a certain pressure $P_0$ and volume $V$ and that could be one macrostate, and there might be macrostates with a larger multiplicity with the same $N$ and $V$ and larger $P_+\gt P_0$ but there isn't enough energy for those $N$ particles to have that pressure, there just isn't enough kinetic energy to spread around to get them to the $P_+\gt P_0$ macrostate variable.

If you want to go to the mansion example. Imagine that you have two mansions and one person can go up a floor if (and only if) the other person goes down a floor. When they are both by some stairs then one can go up and the other can go down. But if that places one of them into a floor with trillions more rooms available than the other one had, then they are way way more likely to stay in the configuration with the one stuck in the floor with way more rooms.

So energy can be exchanged between the two people, but the additional energy spends most of it's time with one of them having more energy if they can exchange it. Eventually they could get to a level where one gains as many rooms as the other one loses. And that joint collection roughly is the macrostate of the combined system.

And when that happens we say they are thermal equilibrium. And the thing they have in common, temperature, is how many additional rooms they gain per bit of energy. Maybe one has stairs that are longer, so going up/down one flight for it is going up/down 5 flights for the other. But maybe the rate at which the floor have more rooms changes with floors at a different rate. There could still be a $\textrm dS/\textrm dE$ in common.

Can thermal equilibrium have fluctuation?

The macrostate could, in principle change from thermal equilibrium and more to have to two subsystems be at different temperature instead of the same temperature. But that would require bouncing around until you are by some stairs going down instead of towards any of the many more options on staying on the same floor, and then continuing to do the improbable floor after floor until the temperature of the two subsystems are very out of line.

And even if it happened, it could just go back to equilibrium. The idea is that for a large enough system, the time to wait for such a thing to happen is just really really long.

• Edited my question. Please check that. – user36790 Feb 3 '16 at 5:33

I think there is some confusion of terminology here..

A system is not at thermal equilibrium if you lock it into the most probable state. A single state is not an equilibrium, it is the complete evolution of the system which has to be observed (or known) to say if it's in equilibrium.

If we take your A,B,C example, the system will be in thermal equilibrium if (over time) there is a passage between the A,B and C macrostates as related to their respective differences in free energy (which is directly related to their phase-space size of microstates).

So if you keep observing your system, it will jump between A, B and C, with C being most probable. This is indeed what the definition says as well (all microstates being equally probable) and needs the ergodic hypothesis to work.

A system out of thermal equilibrium is for example if you have some external force that keeps the system in C or similar.

• Firstly, thanks for the answer. So, could you tell what should be the definition of thermal equilibrium then? – user36790 Nov 12 '15 at 17:47
• The definition you started your post with is good. You just have to realize that equilibrium is a temporal process and not a statement of which single state is "in most equilibrium".. – BjornW Nov 12 '15 at 17:52
• No, I just wanted to get a little more elaboration of A system is not at thermal equilibrium if you lock it into the most probable state. What does that mean? It implies just by remaining at $A$ is not thermal equilibrium. Does it switch to $B,C$ even a thermal equilibrium? I would be grateful if you, please, elaborate the second para. – user36790 Nov 12 '15 at 18:42
• Yes - the system can explore the possible configurations of the system, without bias. Thermal equilibrium really just means that the temperature has become homogenous throughout the system, not that the system has stopped moving. Of course at zero temperature, the state won't change at all, but at any non-zero temperature, there is a chance that the system will move between states (more easily if more thermal energy is available). This "state-hopping" can be quantified by statistical mechanics. Maybe have a look at this: en.wikipedia.org/wiki/Thermal_fluctuations – BjornW Nov 12 '15 at 23:15

In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature, pressure, volume and density. Treatments on statistical mechanics, define a macrostate as follows. A particular set of values of energy, number of particles and volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular macrostate.

Now lets come to your example:

Suppose there are three macrostates A,B,C ; A contains 3 microstates; B contains 6 microstates & C contains 10 microstates.

To be in thermal equilibrium, they have to belong to one and the same isolated system. Three isolated systems from each other do not define one macrostate, but three, in contrast to what you assume . These cannot come into thermodynamic equilibrium because by your construction they have to be isolated.

You could say "take an isolated system and cut the volume in three, A, B, C" and continue with your example. The subsets do not define a macrostate of an isolated system, only a particular microstate of a single isolated system ,with a very small probability of ever manifesting, similar as an example to "what is the probability for all the molecules in the room to be up at the ceiling" .

The inclusion of "thermal equilibrium" is in place so that a fixed temperature can be defined for the system. Temperature is a measure of the macrostate and statistical mechanics needs to connect to thermodynamic quantities in a one to one way with statistical quantities in the state. Otherwise the identification of temperature with average kinetic energy would not be one to one.

• I thought equilibrium is a macrostate which has the maximum multiplicities. C has the maximum multiplicity; that must be the macrostate which is in equilibrium; why then would anyone say entropy is maximum in equilibrium? Equilibrium is a specific macrostate; the fluctuations comprise of another macrostate. All the molecules confined in a corner is a macrostate comprising very small multiplicity- if any of these microstates happen to occur, you can't say it happened in equilibrium which is simply another macrostate. – user36790 Jan 30 '16 at 11:26
• You are using the definition of "macrostate " wrongly. A macrostate is an "isolated system", one.. It is not comprised of systems. Thermal equilibrium means the temperature is uniform throughout. This ensures that the average kinetic energy is uniform throughout – anna v Jan 30 '16 at 12:37
• So, what should you say when it is said equilibrium has the greatest entropy? – user36790 Jan 30 '16 at 12:45
• It is a comparison in time for one macrostate. When in equilibrium it is in the greatest entropy for the system. It supposes no more microstates are created , which of course is true in a classical neutral gas but wrong when one considers black body radiation. ( there is no isolation when radiation is considered, unless one counts the leaving photons) – anna v Jan 30 '16 at 14:23
• So, does that mean thermal equilibrium is not a macrostate? – user36790 Jan 31 '16 at 20:46

Your question does not make a lot of sense, which is due to the words "microstate" and "macrostate" floating around without being given a precise meaning.

A microstate is a point in the phase space of the system - it refers to a single, unambiguous configuration of the system. For a system of $N$ freely moving particles (an ideal gas) the phase space is typically $\mathbb{R}^{6N}$.

A macrostate is a probability distribution $\rho(x,p)$ on the phase space - it is not only a "set of microstates", but it also assigns a probability to every subset of microstates, which is the probability with which the system is found in that particular microstate. In the case of finitely many microstates, it assigns a non-zero probability to each single microstate.

A macrostate (or mixed state, as this concept is called outside the history-laden field of thermodynamics) is in equilibrium if it does not change in time.

The ergodic hypothesis now asserts that the macrostates with $\partial_t \rho = 0$ are states where $\rho$ is uniform on the set where it is non-zero. This is compatible with

There is often a notion that a macrostate is given by specifying thermodynamic properties like pressure or temperature, or even defined to do so. With the definitions in this post, different macrostates can realize the same sets of such state variables, that is, such a thermodynamic state does not correspond to a unique macrostate. If you start from the notions of thermodynamic state and microstate, then the ergodic hypothesis says that a thermodynamic state that is in equilibrium is uniquely realized by the uniform macrostate, i.e. no other macrostate produces the same state variables and is in equilibrium.

• Thanks for answering; I've edited my question and tried to present my argument through an example. Please check that. Meanwhile, I have to go to bed for in India, it is 4am; I was waiting for you the whole night to talk with you at the chat : P Good night : ) – user36790 Feb 2 '16 at 22:28
• @user36790: The uniform distribution on a set is the maximum entropy distribution among all continuous distributions. I do not understand where you see the contradiction between equilibrium being the maximally entropic macrostate and it being the macrostate in which all microstates with non-zero probability are equiprobabe. – ACuriousMind Feb 2 '16 at 22:31
• the contradiction between equilibrium being the maximally entropic macrostate and it being the macrostate in which all microstates with non-zero probability are equiprobable- let me go step by step: at equilibrium, the macrostate prevailing corresponds to $q_A= 4$ as seen in the pic, right? – user36790 Feb 3 '16 at 14:37
• At equilibrium, there are microstates corresponding to $q_A= 4$ and not any other microstates corresponding to $q_A\ne 4$, right? – user36790 Feb 3 '16 at 14:39
• @user36790: "Accessible" means "compatible with the assigned state variables". Since the state variable $q_A$ has been assigned the value 4, no microstate with $q_A\neq 4$ is "accessible" - realizing such a microstate would change the state variable $q_A$, hence the thermodynamic state, so it is not a microstate compatible or "accessible" from that thermodynamic state. – ACuriousMind Feb 3 '16 at 15:00


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Can someone PLEASE test my game?

This topic is 1999 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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Hi guys,

It's not a real game yet, so just run around and explore, comment on controls and usability. Try to break it, and it probably will - I'm a long way from the end:)

Much appreciated!

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Still installing....

Firstly, I am not a fan of anything that has to download half a gig of pre-requises.

Once that it done though, I'll give it a try and let you know.

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Really? I had no idea you would need ANYTHING extra.
I published it using these instructions and it says the requirements are:  Must be running, at minimum, Windows XP Service Pack 2. Must have a video card that supports, at minimum, Shader Model 1.1.

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Not too bad for a start.

Camera seems a bit low to the ground (feels like about knee height). Think about the height of the door for instance. A normal door would be around 2 metres high - so the camera should be around the 1.7 metre mark for eye level of an average/tall person.

The field of view might be a bit narrow too. How many degrees FOV do you have set for the camera? I normally use 60 degrees. The current field of view feels very straining on the eyes. I know I wouldn't be able to play it for long this way as motion sickness would set in pretty quick (for me anyway).

Also, all of the doors are pivoting at the centre (you probably know this allready - if you don't, it may be a bug).

So, there is my input. Please reagard this as completely constructive as you have gotten your project to a point futher than I have ever persisted with mine.

Keep up the great work!

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Hi Ole_92,

It was the Client Framework for the C# component. You would not have been affected as you have the prerequisites on your dev machine.

It installed as part of your installer. It wasn't in anyway intrusive, but added substantially to the install time.

--edit--

actually the above link you posted even mentions the pre-requisites

The .NET Framework 4.0 Redistributable.
The XNA Framework Redistributable 4.0.[/quote]

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Hi Lonewolff,

First of all, thank you so much for your comments! Especially about the field of view - I've changed it to 60 degrees and it's SO MUCH better now!
I will have a menu where a user can set up their own graphics settings.

Everything else - yes, I'm aware of, it will all get fixed before the next update.

Did you have to get both of the ones you have listed?

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It did it automatically so it wasn't that bad.

I would be interested in seeing how it now looks with 60 degree FOV.

BTW what was the FOV previously?

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Well maybe it's not that much better, but it still lets you see more things!
I also tried Pi/2, which is 90 degrees. and it looks really cool! Would be good for scary scenes (I'm making a horror game).

The one I'm attaching here is 60 degrees FOV (was 45 before).

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Good work... what engine are you using for this? Sunburn, or your own thing?

It was very hard to get the key under the bed (either it was obscured by the bed or I was too far away to grab it - there was a very narrow section where I could grab it)

I can't figure out how to use the key.

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Oh, I still haven't done that yet - it will most likely unlock a locked door from inside.
Yep, I will move the key closer to the edge of the bed. Thanks for pointing that out.

I'm using Bepu physics engine, and I'm quite happy about it!

Even though I run into the problems very often, I wasn't stuck on anything longer than a couple of days..

Their forums are very helpful, there's an administrator there, "Norbo", he always replies quickly and informatively, without doing any work for you (which is just what I need!)
I highly recommend it if you're after a physics engine.

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The FOV looks much better now

I would possibly also reduce mouse sensitivity a little bit and toggling between windows and full screen mode throws out the direction atht you are looking at.

It is the finer points that makes the difference between something that is thrown together and something that is awesomely polished. It is often a fine line, but every tweak counts.

I personally don't move past a problem until I am completely happy with the result.

I look forward to seeing how this progresses!

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Ok, I've updated my game, if anyone is interested, please follow this link. Any comments or criticism will be regarded and attempted to implement

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it is still installing for me. it shouldnt be too long but i will let you know what I think of it once i play it! ill prolly also look for some bugs and things like that.

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Unfortunately I can't play the game because my graphics card isn't good enough.

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I would recommend targetting the reach profile instead of HiDef, really it doesn't make any noticable difference but allows lower end cards to be used such as the integrated chipset on my laptop, I will try it on my desktop tommorow its just right now its 11pm-ish and I can't be bothered to fire it up (I know, I'm lazy).

Easy to fix the target profile. Just click project> [projectname] properties and there will be a radio button for you to click.

And yeah, the user will need .net4 and xna, not the full SDK's thankfully (irritatingly terraria downloaded both .net 4 and xna redistributables on my lappy even though I have visual studio 2010 on here with both set-up????), the autoinstaller deals with that nicely.

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Four-month-old thread. Though I'll give the game a shot. The required distributables total roughly 50MB, but you might already have the 40MB .NET one already installed.

My opinion: the game isn't bad, I like the style even though it's not epic art. There are a few graphics glitches, for instance the door design is... well... let's just say doors don't turn like that in real life. Also the second floor doors bleed through the first floor ceiling when they're open. And the physics engine is a bit glitchy at times, I think I lost a red ball through the stairs at some point.

The controls were a bit rigid and awkward, I am used to "E" to jump and "space" to use, but that can be fixed by implementing key bindings.

Gravity is a bit weird, if feels like I'm on the moon when jumping - are you sure it's been implemented properly? Also the lack of shadows makes it quite difficult to judge distances, I think adding some shading to distinguish different surfaces a bit more would help.

Gameplay-wise, there are some odd things I had trouble with. Finding the key was easy, using it took a bit of work (I tried to move it into the lock, but then I realized it needed to be in the inventory and I had to "use" the door). In the next room, the purpose of the red balls, book and gray tablets (dunno what those were meant to be) was - and remains - unclear to me, did they actually have any use? The "forcefield" splitting the room in half, that required jumping over it to cross was not at all obvious (or was it a glitch? I'm not sure - I couldn't walk through it unless I jumped). If there was anything beyond this room, I didn't reach it - all the other rooms were empty.

All in all I don't regret downloading it, it's a good start. Would play the final version.

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Edit: Opps, forgot to check the date. Edited by Labouts


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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions

## F Distribution

Probability Density Function The F distribution is the ratio of two chi-square distributions with degrees of freedom ν1 and ν2, respectively, where each chi-square has first been divided by its degrees of freedom. The formula for the probability density function of the F distribution is
$$f(x) = \frac{\Gamma(\frac{\nu_{1} + \nu_{2}} {2}) (\frac{\nu_{1}} {\nu_{2}})^{\frac{\nu_{1}} {2}} x^{\frac{\nu_{1}} {2} - 1 }} {\Gamma(\frac{\nu_{1}} {2}) \Gamma(\frac{\nu_{2}} {2}) (1 + \frac{\nu_{1}x} {\nu_{2}})^{\frac{\nu_{1} + \nu_{2}} {2}} }$$
where ν1 and ν2 are the shape parameters and Γ is the gamma function. The formula for the gamma function is
$$\Gamma(a) = \int_{0}^{\infty} {t^{a-1}e^{-t}dt}$$
In a testing context, the F distribution is treated as a "standardized distribution" (i.e., no location or scale parameters). However, in a distributional modeling context (as with other probability distributions), the F distribution itself can be transformed with a location parameter, μ, and a scale parameter, σ.

The following is the plot of the F probability density function for 4 different values of the shape parameters.

Cumulative Distribution Function The formula for the Cumulative distribution function of the F distribution is
$$F(x) = 1 - I_{k}(\frac{\nu_{2}} {2},\frac{\nu_{1}} {2} )$$ where k = $$\nu_2/(\nu_2 + \nu_1 \cdot x)$$ and Ik is the incomplete beta function. The formula for the incomplete beta function is

$$I_{k}(x,\alpha,\beta) = \frac{\int_{0}^{x} {t^{\alpha-1}(1-t)^{\beta-1}dt}} {B(\alpha,\beta) }$$

where B is the beta function

$$B(\alpha,\beta) = \int_{0}^{1} {t^{\alpha-1}(1-t)^{\beta-1}dt}$$

The following is the plot of the F cumulative distribution function with the same values of ν1 and ν2 as the pdf plots above.

Percent Point Function The formula for the percent point function of the F distribution does not exist in a simple closed form. It is computed numerically.

The following is the plot of the F percent point function with the same values of ν1 and ν2 as the pdf plots above.

Other Probability Functions Since the F distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.
Common Statistics The formulas below are for the case where the location parameter is zero and the scale parameter is one.
 Mean $$\frac {\nu_{2}} {(\nu_{2} - 2)} \;\;\;\;\;\;\; \nu_{2} > 2$$ Mode $$\frac {\nu_{2}(\nu_{1} - 2)} {\nu_{1}(\nu_{2} + 2)} \;\;\;\;\;\;\; \nu_{1} > 2$$ Range 0 to $$\infty$$ Standard Deviation $$\sqrt{\frac {2\nu_{2}^{2}(\nu_{1} + \nu_{2} - 2)} {\nu_{1}(\nu_{2} - 2)^{2}(\nu_{2} - 4)}} \;\;\;\;\;\;\; \nu_{2} > 4$$ Coefficient of Variation $$\sqrt{\frac{2(\nu_{1} + \nu_{2} - 2)} {\nu_{1}(\nu_{2} - 4)}} \;\;\;\;\;\;\; \nu_{2} > 4$$ Skewness $$\frac {(2\nu_{1} + \nu_{2} - 2)\sqrt{8(\nu_{2} - 4)}} {\sqrt{\nu_{1}}(\nu_{2} - 6)\sqrt{(\nu_{1} + \nu_{2} - 2)}} \;\;\;\;\;\;\; \nu_{2} > 6$$
Parameter Estimation Since the F distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.
Comments The F distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. Two common examples are the analysis of variance and the F test to determine if the variances of two populations are equal.
Software Most general purpose statistical software programs support at least some of the probability functions for the F distribution.


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# Thread: convergence of series with x^n in denominsator

1. ## convergence of series with x^n in denominsator

Test for convergence of the series $\displaystyle \sum_{n=1}^{\infty} \frac{\sqrt{n}}{x^n\sqrt{n^2+1}}$ when $\displaystyle x>0$

I am confused, since 'x' is in the denominator. Otherwise it could be treated as a general power series

2. Use the quotient test, You'll obtain that for $\displaystyle x>1$ the series is convergent if $\displaystyle x>1$ and divergent if $\displaystyle 0<x<1$ . For $\displaystyle x=1$ is divergent (compare for example with $\displaystyle \sum_{n\geq 1} 1/n^{1/2}$ ).

Fernando Revilla

3. If you were to let z= 1/x, it would be a power series:
$\displaystyle \displaytype \sum_{n=1}^\infty \frac{\sqrt{n}}{\sqrt{n^2+1}}z^n$
determine its interval of convergence and use that to determine for what values of x the original series converges (it son't be an "interval").

4. For z=1/x, the series diverges for (-1,1), so for x, the series will converge for x<-1(ignore this, since x>0) and x>1. Thanks to HallsofIvy. Also at x=1, the convergence can be tested using comparison test. Thanks to FernandoRevilla.

5. Of course, the ratio test can be used here...

$\displaystyle \displaystyle \lim_{n \to \infty}\frac{a_{n+1}}{a_n} = \lim_{n \to \infty}\frac{\frac{\sqrt{n+1}}{x^{n+1}\sqrt{(n+1)^ 2+1}}}{\frac{\sqrt{n}}{x^n\sqrt{n^2+1}}}$

$\displaystyle \displaystyle = \lim_{n \to \infty}\frac{x^n\sqrt{n^2+1}\sqrt{n+1}}{x^{n+1}\sq rt{n^2 + 2n + 2}\sqrt{n}}$

$\displaystyle \displaystyle = \frac{1}{x}\lim_{n \to \infty}\sqrt{\frac{(n^2+1)(n+1)}{n(n^2+2n+2)}}$

$\displaystyle \displaystyle = \frac{1}{x}\lim_{n \to \infty}\sqrt{\frac{n^3+n^2+n+1}{n^3 + 2n^2 +2n}}$

$\displaystyle \displaystyle = \frac{1}{x}\lim_{n \to \infty}\sqrt{\frac{1 + \frac{1}{n} + \frac{1}{n^2} + \frac{1}{n^3}}{1 + \frac{2}{n} + \frac{2}{n^2}}}$

$\displaystyle \displaystyle = \frac{1}{x}\cdot 1$

$\displaystyle \displaystyle = \frac{1}{x}$.

The series converges when this limit is $\displaystyle \displaystyle <1$. So we need to evaluate the values of $\displaystyle \displaystyle x$ for which this ratio is $\displaystyle \displaystyle <1$, and since $\displaystyle \displaystyle x > 0$...

$\displaystyle \displaystyle \frac{1}{x} < 1$

$\displaystyle \displaystyle 1 < x$.

So the series is convergent when $\displaystyle \displaystyle x > 1$.

Since the ratio test fails when the limit is $\displaystyle \displaystyle 1$, you need to substitute $\displaystyle \displaystyle x=1$ into the original series and test the convergence of the series for that value.


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STD_LOGIC_VECTOR to INTEGER VHDL

The problem is to find a modulo when a three digit number is divided by its last two digits. But the three digit number is received in a binary form. So I first want to change the binary form into integer form then use the mod operator.

• Maybe your signal definitions for divident_int (and others) should be defined as unsigned? If you convert a negative int (that starts with a MSB=1), it may be truncated to a value of 0? – Nazar May 27 '17 at 13:48
• Your process is only called once (no sensitivity list) and the value of "dividend_int" is only updated after reaching the WAIT statement. modcalc looks like a combinatorial entity (no clock). You don't need any process. Alternatively, you could use a variable for dividend_int, modulo_int and modulo_1 – TEMLIB May 27 '17 at 14:15

Why do you paste your code as an image?! This makes it harder for us to help you. Please don't do that next time.

The answer is given by TEMLIB. You have two options:

1) Make a triggered process, with the internal value as variables.

begin
process(divident)
variable divident_int : integer;
variable modulo_1_int : integer;
variable modulo_int : integer;
begin
divident_int := to_integer(unsigned(divident));
modulo_1_int := divident_int mod divis_1;
modulo_int := divident_int mod modulo_1_int;
modulo <= std_logic_vector(to_unsigned(modulo_int, modulo'length));
end process;
end architecture;

2) leave out the process (and keep the signals)

begin
divident_int <= to_integer(unsigned(divident));
modulo_1 <= divident_int mod divis_1;
modulo_int <= divident_int mod modulo_1;
modulo <= std_logic_vector(to_unsigned(modulo_int, modulo'length));
end architecture;

• Thank you so much for your help! I accept your suggestion as well. I won't do so again. – Hintsa Fisseha May 30 '17 at 8:25


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# How to prove the following Whittaker formula

I am a theoretical physicist and I need help in proving the alternate Whittaker formula

$$W _ { k , m } ( z ) = \frac { \Gamma ( - 2 m ) } { \Gamma \left( \frac { 1 } { 2 } - m - k \right) } M _ { k , m } ( z ) + \frac { \Gamma ( 2 m ) } { \Gamma \left( \frac { 1 } { 2 } + m - k \right) } M _ { k , - m } ( z )$$

in case where $$| \arg z | < \frac { 3 } { 2 } \pi$$ and 2$$m$$ is not an integer. Taking into account the appropriate definitions

$$M _ { k . u } ( z ) = z ^ { \mu + 1 / 2 } e ^ { - z / 2 } \Phi \left( \frac { 1 } { 2 } - k + \mu , 2 \mu + 1 ; z \right) , \quad | \arg z | < \pi$$ $$W _ { k , \mu } ( z ) = z ^ { \mu + 1 / 2 } e ^ { - z / 2 }\Psi \left( \frac { 1 } { 2 } - k + \mu , 2 \mu + 1 ; z \right) , \quad | \arg z | < \pi$$

Interestingly, I found this function as I was working my thesis related to the article "Creation of scalar and Dirac particles in the presence of a time varying electric field in an anisotropic Bianchi type I universe Victor M. Villalba* and Walter Greiner" where they used this formula and I tried to find a way to referred references and I found the author himself who discovered these functions in his book of "A course in modern analysis by Whittaker and Watson" p.346, chap.16.

Any help is appreciated.

If you start from the expression for $$W_{k,m}(z)$$ given here,
$$W_{k,m}\left(z\right) = e^{-\tfrac{z}{2}}z^{m+\tfrac{1}{2}}U\left(m-k+\frac{1}{2}, 1+2m, z\right)$$
and express the Tricomi function $$U\left(m-k+\frac{1}{2}, 1+2m, z\right)$$ in terms of the Kummer functions $$M$$ as here $$U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a+1-b)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a+1-b,2-b,z),$$ with $$a:=m-k+\frac{1}{2}$$ and $$b:=1+2m$$, and also use the formula for $$M_{k,m}$$ in the first link, $$M_{k,m}\left(z\right) = e^{-\tfrac{z}{2}}z^{m+\tfrac{1}{2}}M\left(m-k+\frac{1}{2}, 1+2m, z\right),$$ you end up with $$W_{k,m}$$ as the linear combination of $$M_{k,m}$$ and $$W_{k,-m}$$ as you wrote.


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# Math Help - log problem

1. ## log problem

The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n+1) = 2^{T(n)}$ for $n\geq1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ such that

$\underbrace{\log_{2}\log_{2}\log_{2}...\log_{2}} B$
$k$ times

is defined?

2. Wasn't this a problem on the AMC 12A this year?


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# What is the difference between density and distribution?

1. Apr 12, 2006

### loom91

Hi,

What is the difference between a density and a distribution? As in probability distribution and probability density, or energy distribution and energy density. Thanks.

Molu

2. Apr 12, 2006

### Hurkyl

Staff Emeritus
A distribution is a gadget that tells you how much there is over any region of space.

A density is a gadget you have to integrate in order to find out how much there is over any region of space.

Incidentally, in the probability case, we will often define the distribution in terms of a "cumulative distribution function": we use p(x) to tell how much there is over the interval (-infinity, x).

(Or maybe it's (-infinity, x]. I can't remember)

3. Apr 13, 2006

### loom91

You mean the density is the exterior drivative of the distribution? Can you give an example?