url
stringlengths 14
5.47k
| tag
stringclasses 1
value | text
stringlengths 60
624k
| file_path
stringlengths 110
155
| dump
stringclasses 96
values | file_size_in_byte
int64 60
631k
| line_count
int64 1
6.84k
|
|---|---|---|---|---|---|---|
http://math.stackexchange.com/questions/134782/how-to-express-cosh4x-as-a-polynomial-in-coshx
|
math
|
My friend needs help with this question and I don't know it.
If you use the definition of the hyperbolic cosine then, dropping the factor 2, you have:
$\exp(4x)+\exp(-4x) = (\exp(2x)-\exp(-2x))^2+2$
$\exp(2x)-\exp(-2x) = (\exp(x)+\exp(-x))(\exp(x)-\exp(-x))$
if you do it you should get something like (if I'm not mistaken)
$\cosh(4x) = 8(\cosh(x))^2(\sinh(x))^2+1$
then, if you want to replace the $\sinh$ you can use the fact that $(\cosh(x))^2-(\sinh(x))^2=1$.
For a mechanical way to translate the more familiar trigonometric identities to hyperbolic function identities, use $$\cosh x=\cos(ix),\qquad \sinh x=-i\sin(ix).$$
But perhaps an identity for $\cos(4w)$ does not qualify as familiar. However, it is easy to derive from standard facts: $$\cos(4w)=2\cos^2 (2w) -1=2(2\cos^2 w -1)^2-1.$$ Putting $w=ix$, we obtain in a mechanical way $$\cosh(4x)=2(2\cosh^2 x -1)^2-1.$$
|
s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257828010.15/warc/CC-MAIN-20160723071028-00164-ip-10-185-27-174.ec2.internal.warc.gz
|
CC-MAIN-2016-30
| 879 | 9 |
http://www.purplemath.com/learning/viewtopic.php?f=14&t=304&p=838
|
math
|
how to locate and classify the maximum and minimum values of the function y= 4x^3 + 3x^2 - 60x - 12.
as far as I know that at a stationery point dy/dx = 12x² + 6x – 60 =0
You can start the solution of the first derivative by noting the common factor
and dividing it out:
. . . . .
Then you solve the quadratic by factoring
or else by the Quadratic Formula
I don't know what tools you have at that point. You'll either need to find the sign of the derivative between the stationary (or "critical") points, or else apply the Second Derivative Test. (Or you can work straight from your knowledge of what positive cubics look like
to classify the two points in question.)
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865830.35/warc/CC-MAIN-20180523215608-20180523235608-00184.warc.gz
|
CC-MAIN-2018-22
| 670 | 9 |
https://www.maths.ox.ac.uk/node/14848
|
math
|
If $R = F_q[t]$ is the polynomial ring over a finite field
then the group $SL_2(R)$ is not finitely generated. The group $SL_3(R)$ is
finitely generated but not finitely presented, while $SL_4(R)$ is
finitely presented. These examples are facets of a larger picture that
I will talk about.
- Algebra Seminar
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585916.29/warc/CC-MAIN-20211024081003-20211024111003-00436.warc.gz
|
CC-MAIN-2021-43
| 307 | 6 |
https://community.qlik.com/thread/121704
|
math
|
This content has been marked as final. Show 3 replies
Hi Folks ,
I'm facing on small issue while calculating average in Straight table and in Bar charts. Please see below:
In this Straight table below I'm calculating Act. Tanks Scrapped % as expression = avg(Scrap) which is giving me result 3.4% which is right number. (Here I have set expression total as Average so I'm getting 3,4% which is correct)
Now the same thing I'm calling in a Bar chart calling the same expression =Avg(Scrap) , but here I'm getting average as 3,5% which is not the exact average . As we don't have option for setting Expression Total as Average for Bar chart. I want to show here exactly 3.4% instead of 3.5% .
Kindly help me here in bar chart , how to get exactly average numbers.
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591596.64/warc/CC-MAIN-20180720115631-20180720135631-00294.warc.gz
|
CC-MAIN-2018-30
| 761 | 6 |
http://www.eduessays.com/Essays-y84110.htm
|
math
|
|Leibniz And Spinoza As Applied To Baseball
First we will consider the assigned baseball scenario under Leibniz’s system of metaphysics. In the baseball scenario, the aggregate of the player, bat, pitch, swing and all the other substances in the universe are one and all contingent. There are other possible things, to be sure; but there are also other possible universes that could have existed but did not. The totality of contingent things, the bat, the player, etc., themselves do not explain themselves. Here Leibniz involves the principle of reason; “there can be found no fact that is true or existent, or any true proposition, without there being a sufficient reason for its being so and not otherwise.” There must be, Leibniz insists, something outside the totality of contingent things (baseball games) which explains them, something which is itself necessary and therefore requires no explanation other than itself.
This forms Leibniz’s proof for the exis...
|
s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00397-ip-10-147-4-33.ec2.internal.warc.gz
|
CC-MAIN-2014-15
| 977 | 3 |
https://support.jumpdesktop.com/hc/en-us/community/posts/360077999432-Feature-request-Webcam-redirection
|
math
|
During these times of remote work, a lot of applications for remote meetings have soared in usage (WebEx, Microsoft Teams, etc.).
It would be amazing to introduce a feature that allows a webcam redirection so that we could remotely allow the webcam to function.
This is what I mean.
Say that I have a Mac#1 and a Mac#2.
I use Mac#2 to establish a connection to Mac#1 and attend a meeting on Mac#1.
It would be amazing if, on Mac#1, I was allowed to share the camera from Mac#2 in software such as Teams, WebEx, etc.
Please sign in to leave a comment.
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662675072.99/warc/CC-MAIN-20220527174336-20220527204336-00537.warc.gz
|
CC-MAIN-2022-21
| 550 | 7 |
http://www.pvv.org/~leifmk/english/gurps/gurpsme.html
|
math
|
Character: Leif Kjønnøy
Occupation: Computer nerd (software/database repair guy for the national university libraries).
Hobbies: Gaming, martial arts, taking college classes for fun (history and stuff like that).
Description: 6'4", 220 lbs, wider shoulders than waist. Age 30. Fair skin (*does* tan somewhat if given a chance, but this is the wrong part of the world for that to happen on its own and I can't be bothered), blonde hair (usually somewhere between shaved off (in summer) and just long enough to look messy), blue/gray eyes, usually wears glasses but sometimes contacts. Usually wearing deceptively normal clothes -- black or blue jeans in relatively good repair, unexceptional T-shirts (and sweaters, jackets, etc as demanded by local weather); occasionally featuring clothes or accessories with weird designs or slogans (Miskatonic University T-shirt, Illuminati pin, etc).
Point total: 19 pts, or 88 pts if a 40-point disadvantage limit is enforced.
Attributes (+40 pts balance)
ST 12 (have in fact been working out lately)
DX 10 (kind of average there, I guess)
IQ 12 (fairly bright -- hey, I made it to a Master's in math with little effort)
HT 10 (fair shape all things considered)
Advantages (total +33 pts)
Attractive (supposedly; +5 pts).
3 Extra Hit Points (13 in all; I am sort of big and rugged; +15 pts).
Illuminated (no points -- see below).
Light Hangover (only rarely get drunk anyway; +3 pts).
Mathematical Ability (always had an easy time with math; +10 pts).
(total -109 pts worth, guess I'm having a bad day)
Bad Back (-15 pts).
Bad Sight (Seriously nearsighted, but correctible; -10 pts).
Compulsive Behavior (roleplaying games; -5 pts).
Cowardice (could claim Pacifism, but I'm in a cynical mood today; -10 pts).
Delusion: Think I'm Illuminated (mild version; -5 pts).
Honesty (-10 pts).
Incompetence: Singing (to spare my surroundings I usually don't even try; -1 pt)
Intolerance vs. Intolerant People (-5 pts)
Klutz (I really have this disadvantage; -5 pts)
Laziness (-10 pts)
Low Self-Image (-10 pts)
Oblivious (-3 pts)
(-5 pts worth at least, these are some of the more significant ones):
Reads a lot, prefers good SF but will read nearly anything.
Enjoys trashy movies (as well as good ones).
Pedestrian, by birth and by choice.
Usenet junkie (I could stop if I wanted to, really).
Prefers a nocturnal lifestyle (sadly, circumstances intrude).
Lesbian trapped in a man's body.
Skills (+60 pts total) Pts Level Area Knowledge: Where I Live (M/E) 12 Astronomy (M/H) 10 Bardic Lore, spec. SF and gaming anecdotes (M/H) 11 Bicycling (P/E) 10 Board Games (M/E) 12 Boating (P/A) [0.5] 8 Cartography (M/A) 11 Chemistry (M/H) 10 Computer Operation (M/E) 12 Computer Programming (M/H) 16(*) Conspiracy Theory (M/H) 11 Driving: Stock Car (P/A) 9 First Aid (M/E) 12 Hiking (P/A) 9 History (M/H) 11 Judo (P/H) 10 Karate (P/H) 11 Language: Norwegian (native speaker) 15 Language: English (M/A) 15 Language: German (M/A) [0.5] 10 Literature (M/H) 10 Mathematics (M/H) 16(*) Motorcycle (P/E) [0.5] 9 Mythos Lore (M/VH) 10 Nuclear Physics (M/VH) 13(*) Occultism (M/A) 11 Performance (M/A) 11 Philosophy (M/H) 10 Physics (M/H) 15(*) Powerboat (P/A) [0.5] 8 Research (M/A) 11 Skating (P/H) [0.5] 7 Skiing (P/H) [0.5] 7 Strategy (M/H) 10 Writing (M/A) 12 (*): Includes a +3 bonus for Mathematical Ability.
Back to my GURPS page.
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358591.95/warc/CC-MAIN-20211128194436-20211128224436-00015.warc.gz
|
CC-MAIN-2021-49
| 3,380 | 38 |
https://brainmass.com/business/finance/lock-boxes-120151
|
math
|
Anne Teak, the financial manager of a furniture manufacturer, is considering operating a lock-box system. She forecasts that 400 payments a day will be made to lock boxes with an average payment size of $2,000. The bank's charge for operating the lock boxes is $.40 a check. The interest rate is .015 percent per day.
a. If the lock box saves 2 days in collection float, is it worthwhile to adopt the system?
b. What minimum reduction in the time to collect and process each check is needed to justify use of the lock-box system?
a. We need to compare the cost and benefit of operating the lock box on a per check basis. The cost is the lock box cost and the benefit is the savings in ...
The solution explains how to evaluate a lock box system.
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585382.32/warc/CC-MAIN-20211021071407-20211021101407-00370.warc.gz
|
CC-MAIN-2021-43
| 745 | 5 |
http://mathhelpforum.com/geometry/208381-prove-similar-triangles-parallelogram.html
|
math
|
Given a parallelogram ABCD and 2 segments FH and EG intersecting the diagonal AC at point P and terminating on 2 opposite parallelogram sides respectively, plus segments HG and EF.
Pprove that triangles EFP and HGP are similar.
Could I get a hint on how to solve this problem?
If it could be proven that HG and EF are parallel or by the same token, that angle AFE=GHC, the proof would follow. I have been unable to make any headway.
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189686.56/warc/CC-MAIN-20170322212949-00644-ip-10-233-31-227.ec2.internal.warc.gz
|
CC-MAIN-2017-13
| 432 | 4 |
https://www.stephenwolfram.com/questions/2019/12/06/what-time-do-you-get-up/
|
math
|
Stephen Wolfram Q&ASubmit a question
Some collected questions and answers by Stephen Wolfram
Questions may be edited for brevity; see links for full questions.
December 6, 2019
From: Interview by Jeff D’Alessio, The News-Gazette (unpublished)
What time do you get up?
I start my day around 10 am…
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224649348.41/warc/CC-MAIN-20230603233121-20230604023121-00120.warc.gz
|
CC-MAIN-2023-23
| 300 | 7 |
https://socialcar-project.eu/en/what-is-the-difference-between-mass-and-weight-there-is-no-difference-weight-is-the-amount-of-ma.1567460.html
|
math
|
what is the difference between mass and weight. * there is no difference. weight is the amount of matter, and mass is the force due to gravity. mass is the amount of matter, and weight is the force due to gravity.
There is a basic difference, because mass is the actual amount of material contained in a body and is measured in kg, gm, etc. Whereas weight is the force exerted by the gravity on that object mg. Note that mass is independent of everything but weight is different on the earth, moon,
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104597905.85/warc/CC-MAIN-20220705174927-20220705204927-00308.warc.gz
|
CC-MAIN-2022-27
| 498 | 2 |
https://www.physicsforums.com/threads/proton-and-electric-field.152531/
|
math
|
The direction of an electric field at a point is the same as
a.the force on a neutron placed at that point.
b.the force on a proton placed at that point.
c.the force on an electron placed at that point.
d.the force on a hydrogen molecule placed at that point.
This is really a conceptual question.
The Attempt at a Solution
It would be choice B. because a positively charged particle's force is in the same direction of the electric field, right?
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335058.80/warc/CC-MAIN-20220927194248-20220927224248-00243.warc.gz
|
CC-MAIN-2022-40
| 446 | 8 |
https://tropicalessays.com/solution-ra495200765/
|
math
|
(b)lf the tax rate is 35 percent, what is the after-tax cost of debt as an APR? (Do not round your intermediate calculations.) (Click to select)
v Sixx AM Manufacturing has a target debt-equity ratio of 0.67. Its cost of equity is 18 percent, and its cost of debt is 11 percent. If the tax rate is 33 percent, what is the company’s WACC?
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337631.84/warc/CC-MAIN-20221005140739-20221005170739-00556.warc.gz
|
CC-MAIN-2022-40
| 339 | 2 |
https://progress.lawlessfrench.com/questions/view/inverted-questions-1
|
math
|
The question was "Fais-tu la vaisselle" (no question mark or full stop). Because there was no question mark, I put "Do the dishes." The answer was "Are you doing the dishes?". I know it was supposed to be a question, but was not asked as a question. So, I think I should be marked correct.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224645089.3/warc/CC-MAIN-20230530032334-20230530062334-00201.warc.gz
|
CC-MAIN-2023-23
| 289 | 1 |
https://newyorkessays.com/essay-language-teaching-methodology/
|
math
|
Language Teaching Methodology
One would expect that students who consistently perform well in the classroom (tests, quizes, etc. would also perform well on a standardized achievement test (0 – 100 with 100 indicating high achievement). A teacher decides to examine this hypothesis. At the end of the academic year, she computes a correlation between the students achievement test scores (she purposefully did not look at this data until after she submitted students grades) and the overall g. p. a. for each student computed over the entire year.
The data for her class are provided below. What does this statistic mean concerning the relationship between achievement test prformance and g. p. a.? 3. What percent of the variability is accounted for by the relationship between the two variables and what does this statistic mean? 4. What would be the slope and y-intercept for a regression line based on this data? 5. If a student scored a 93 on the achievement test, what would be their predicted G. P. A.? If they scored a 74? A 88?
Language Teaching Methodology Essay Example
A professor in the psychology department would like to determine whether there has been a significant change in grading practices over the years. It is known that the overall grade distribution for the department in 1985 had 14% As, 26% Bs, 31% Cs, 19% Ds and 10% Fs. A sample of n=200 psychology students from last semester produced the following grade distribution:for independent- online calculator
Research has demonstrated strong gender differences in teenagers approaches to dealing with mental health issues (Chandra & Minkovitz, 2006). In a typical study, eight-graders are asked to report their willingness to use mental health services in the event they were experiencing emotional or other mental health problems. Typical data for a sample of n=150 students are shown in the table. Do the data show a significant relationship between gender and willingness to seek mental health assistance?
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588257.34/warc/CC-MAIN-20211028034828-20211028064828-00238.warc.gz
|
CC-MAIN-2021-43
| 1,983 | 6 |
https://www.expertsmind.com/library/determine-whether-interest-parity-rate-exists-5710125.aspx
|
math
|
Already have an account? Get multiple benefits of using own account!
Login in your account..!
Don't have an account? Create your account in less than a minutes,
Forgot password? how can I recover my password now!
Enter right registered email to receive password!
Assume zero transaction costs. As of now, the Japanese one-year interest rate is 3%, and the US one-year interest rate is 9%. The spot rate of the Japanese yen is $.0090 and the one-year forward rate of Japanese yen is $.0097.
A) Determine whether interest parity rate exists, or whether the quoted forward rate is quoted too high or too low.
B) Based on the information provided in (A), is covered interest arbitrage feasible for US investors, for Japanese investors, for both types or for neither types of investors
4. Which of the following accurately defines the term compound growth? a. A fee paid for the use of another person's money
Assume your instructor has two bonds in his portfolio. Both have face values of $1,000 and pay a 10% annual coupon rate. Bond L (longer maturity) matures in 15 years and Bond S (shorter maturity) matures in 1 year
- What were the probable misrepresentations and/or omissions of facts in the amended S-1 filing? - Did the lead underwriters violate Fair Dealing with regards to the conveyance of updated revenue forecasts? If yes, how so?
Define life safety as related to fire safety programs. Organizations provide comprehensive fire safety programs to mitigate hazards involving fire in the workplace.
2a. What is the present value of $7,900 in 10 years at 11%? 5a. If you invest $9,000 today, how much will you have in 2 years at 9%? 5d. In 25 years at 14% compounded semiannually?
Explain Dalton's law of partial pressure. Does water vapor conform to Dalton's law? What are some other sources of pressure hazards.
Twin City Printing is considering two financial alternatives for financing a major expansion program. Under either alternative EBIT is expected to be $15.6 million. Currently the firm's capital structure consists of 4 million shares of common stoc..
1. Determine if the implied interest rate can be uniquely determined if you know volatility; consider the derivative dC/dr 3. Assume that the volatility is 10%/year
(a) Calculate the firm's operating return on assets. Assume that the firm's year-end total assets balance for the prior year was $6 million. (b) Calculate the firm's net working capital(c) What is the number of "accounts receivable days" for Abbee ..
Explain why a comprehensive confined space management policy is important in any organization. What areas should a confined space management policy cover.
Objective type questions on current assets and liabilities and Which of the following statements is CORRECT
determine the present values if 5000 is received in the future i.e.at the end of each indicated time period in each of
calculate the value of a bond with a face value of 1000 a coupon interest rate of 8 percent paid semiannually and a
Consider the following information for a mutual fund, the market index,
in 1971 president nixon unilaterally suspended the fixed rate between dollar and gold and effectively moved
a company has a bond issue outstanding that pays 150 annual interest plus 1000 at maturity. the bond has a maturity of
gamma medicals stock trades at 90 a share. the company is contemplating a 3-for-2 stock split. assuming the stock split
Use the Black-Scholes model to find the price for a call option with the following inputs: (1) current stock price is $21, (2) strike price is $24, (3) time to expiration is 5 months.
Bernie and Pam Britten are a young married couple starting careers and establishing a household. They will every make about $50,000 next year and will have accumulated about $40,000 to invest.
Discuss how managements' discretion in applying accounting rules can mislead investors. Provide three examples and how the discresion can distort results.
What are the components of WACC? Which component has the most significance in the total? Over which component does management have the greatest influence?
Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!
All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949678.39/warc/CC-MAIN-20230331175950-20230331205950-00420.warc.gz
|
CC-MAIN-2023-14
| 4,335 | 31 |
https://bestengineeringprojects.com/forced-vibration-of-a-mass-spring-system-with-damping/
|
math
|
In foundation soil system damping is always present in one form or another. For this case fig. 1 (a), the equation of motion is:
The solution of equation (2) is done by applying the concept of rotating vector. In the figure 1 the exciting force vector Q0 is placed with a phase angle ahead of the displacement vector A. The equation of displacement may be expressed as:
In figure 1 (b) the position of motion vector is shown. In figure 1 (c) the position of force vector is shown. The force vectors act opposite to that of motion vectors. The force vector Q0 is placed with a phase angle ahead of the displacement vector A Resolving the force in the vertical and horizontal direction, we have,
Solving for A and we have,
The equations are plotted for various values D as shown in figure 2(a) and (b). These curves are referred to here as response curves for Constant-force-amplitude-excitation. In the figure it is seen that maximum amplitude occurs at a frequency slightly less than the un-damped natural circular frequency, where . The frequency at maximum amplitude will be referred to as resonant frequency fm for constant force Q0.
Resonant frequency at maximum amplitude
Put and , then
Taking positive sign, we get,
Magnification factor at resonant frequency is given by,
When , fmax = 0 which means that the maximum response is the static response.
The maximum amplitude at resonance will be,
When damping in the system is neglected, then
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257646178.24/warc/CC-MAIN-20180318224057-20180319004057-00723.warc.gz
|
CC-MAIN-2018-13
| 1,445 | 12 |
https://lab591.org/and-pdf/319-vector-differentiation-and-integration-pdf-782-903.php
|
math
|
Vector Differentiation And Integration PdfBy Tilly C. In and pdf 12.05.2021 at 09:26 5 min read
File Name: vector differentiation and integration .zip
- 4.1: Differentiation and Integration of Vector Valued Functions
- MULTIVARIABLE AND VECTOR ANALYSIS
- Vector Calculus
- Differentiation and integration of vectors
To browse Academia.
In mathematics , an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation , integration is a fundamental operation of calculus, [a] and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
4.1: Differentiation and Integration of Vector Valued Functions
Class Central is learner-supported. Johns Hopkins University. Korea Advanced Institute of Science and Technology. Start your review of Vector Calculus for Engineers. Anonymous completed this course. Syed Murtaza Jaffar completed this course. Get personalized course recommendations, track subjects and courses with reminders, and more.
Vector Calculus for Engineers covers both basic theory and applications. In the first week we learn about scalar and vector fields, in the second week about differentiating fields, in the third week about multidimensional integration and curvilinear coordinate systems.
These theorems are needed in core engineering subjects such as Electromagnetism and Fluid Mechanics. Instead of Vector Calculus, some universities might call this course Multivariable or Multivariate Calculus or Calculus 3.
Two semesters of single variable calculus differentiation and integration are a prerequisite. The course is organized into 53 short lecture videos, with a few problems to solve following each video.
And after each substantial topic, there is a short practice quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. There are a total of five weeks to the course, and at the end of each week there is an assessed quiz. Vectors A vector is a mathematical construct that has both length and direction. We will define vectors and learn how to add and subtract them, and how to multiply them using the scalar and vector products dot and cross products.
We will use vectors to learn some analytical geometry of lines and planes, and learn about the Kronecker delta and the Levi-Civita symbol to prove vector identities. The important concepts of scalar and vector fields will be introduced.
Differentiation Scalar and vector fields can be differentiated. We define the partial derivative and derive the method of least squares as a minimization problem. We learn how to use the chain rule for a function of several variables, and derive the triple product rule used in chemical engineering.
We define the gradient, divergence, curl and Laplacian. We learn some useful vector calculus identities and how to derive them using the Kronecker delta and Levi-Civita symbol. Vector identities are then used to derive the electromagnetic wave equation from Maxwell's equation in free space.
Electromagnetic waves form the basis of all modern communication technologies. Integration and Curvilinear Coordinates Integration can be extended to functions of several variables. We learn how to perform double and triple integrals. Curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, are used to simplify problems with circular, cylindrical or spherical symmetry.
We learn how to write differential operators in curvilinear coordinates and how to change variables in multidimensional integrals using the Jacobian of the transformation. Line and Surface Integrals Scalar or vector fields can be integrated on curves or surfaces.
We learn how to take the line integral of a scalar field and use line integrals to compute arc lengths. We then learn how to take line integrals of vector fields by taking the dot product of the vector field with tangent unit vectors to the curve. Consideration of the line integral of a force field results in the work-energy theorem.
Next, we learn how to take the surface integral of a scalar field and compute surface areas. We then learn how to take the surface integral of a vector field by taking the dot product of the vector field with the normal unit vector to the surface.
The surface integral of a velocity field is used to define the mass flux of a fluid through the surface. Fundamental Theorems The fundamental theorem of calculus links integration with differentiation. Here, we learn the related fundamental theorems of vector calculus. These include the gradient theorem, the divergence theorem, and Stokes' theorem.
We show how these theorems are used to derive continuity equations, derive the law of conservation of energy, define the divergence and curl in coordinate-free form, and convert the integral version of Maxwell's equations into their more aesthetically pleasing differential form. Taught by Jeffrey R. Select a rating. I can only deliver a mixed review. The course presents a generous amount of material, and all the basics are covered, but the presentation, especially in the final week, is perfunctory at best, grinding through derivations and leaving many steps for the The course presents a generous amount of material, and all the basics are covered, but the presentation, especially in the final week, is perfunctory at best, grinding through derivations and leaving many steps for the student simply to "look up".
Therefore, I recommend the course only as a review for anyone who already knows the material; trying to learn the details for the first time from this rushed and compressed presentation is likely to be frustrating, if not discouraging. This is a likely related to Coursera's pressure to cram course contents into 4-week lumps as much as of anything else.
That said, the lectures are well-organized trips through the standard derivations of results in Cartesian and spherical coordinate systems, but the motivation of the utility of scalar and vector products as projections and volumes is left behind once it has been given that perfunctory treatment in the early lectures.
The course makes no attempt to get beyond dimension that can treated with analytic geometry of planes and 3-space; we do see the fluxes on cube faces and on the boundary of spheres; sadly, these treatments are too rushed. Vector calculus is a rich and beautiful subject, but don't come here to try to learn it for the first time. Helpful 2. Professor Chasnov is highly organized and presents the contents in a clear manner.
I have become fond of his excellent teaching style. Over and above, all engineers must take this course. This is terrific effort from him. I wish the best comes his way as a reward for his dedication. God bless. Week three is the pivotal week for learning that I struggled with. Line and Surface integrals just did not come easy to me.
A tutorial on the line and surface integrals in greater depth would have helped me since it is difficult to visualize what these always mean. The instruction was excellent, but I feel I needed extra help.
Would love to take a course in just line and surface integrals. An extremely valuable course for anyone in physics or engineering. Take it as soon as you can. In short duration it could cover all areas of vector calculus I request sir to include more no.
Finished all the course in about 2 weeks. It is very good if you want to refresh your memory on vector calculus in my case. If you want a solid foundation, then you should supplement it with lots of more examples from some textbook s. Otherwise, things are explained very well, and the examples are not too difficult to scare you away!
Great course. Helpful 1. A great refresher course if you already know vector calculus and would like to take a cursory glance to brush up the concepts. I didn't have the in-depth knowledge of the topic but tackling it on your own can at first seem daunting. It had been something It had been something of a personal challenge for me. This course seemed to offer a practical grasp of the topics in four weeks.
I figured if I could manage this, I will be able to gather enough courage to independently study the topic in more detail. Thanks to the incredible instructor, Professor Chasnov, the material didn't seem too hard.
But I should mention that I am a physics major and I was already comfortable with working with vectors and had a good enough grasp on Calculus 2. My only complaint was with problems in lecture 41 which I personally thought could have done with an additional video on how to apply the theorems to Navier Stokes equation. I think I was looking a bit more information for the physical meaning behind the problem and how the theorems exactly help us.
Something you can find out online I certainly had to work a lot more than what the course suggests is the time required to complete a given week.
All in all, I would definitely recommend this course to anyone who wants to get a working understanding of using multivariable calculus. My review here isn't so much about this particular course. Instead, it is about the instructor Jeff Chasnov. I enjoyed all of them. I'm excited about his new course Numerical Methods for Engineers, beginning Jan, which I will not miss. Heck, I wish that I could work for him!
He enthusiastically engages with students in the discussion forums, and responds well to constructive criticism, a rare quality. There are practice quizzes as well as weekly graded quizzes. Other instructors should take note of him. He really cares. Love his use of the lightboard. I would say that his courses target advanced high school students through undergrad school, but refreshers for graduate students or engineers are appropriate as well.
The only negatives I can mention is that he keeps his courses too short IMO, usually 4 weeks, but he crams a lot of topics into them, and that PDF handouts were not available. But it is easy to snapshot the lightboard.
MULTIVARIABLE AND VECTOR ANALYSIS
Exercice de Physique Chimie 6eme Two integrals of the same function may differ by a constant. The derivative of any function is unique but on the other hand, the integral of every function is not unique. Time can play an important role in the difference between differentiation and integration. Both differentiation and integration, as discussed are inverse processes of each other. Exercice de Physique Chimie 5eme Python Mini Projects for Beginners.
Vector calculus , or vector analysis , is concerned with differentiation and integration of vector fields , primarily in 3-dimensional Euclidean space R 3. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering , especially in the description of electromagnetic fields , gravitational fields , and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their book, Vector Analysis. A scalar field associates a scalar value to every point in a space.
Calculus Notes Pdf Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. Yusuf and Prof. Bernoulli in Consider a bead sliding under gravity. Faculty of Science at Bilkent University. However, I will use linear algebra. Maths Study For Student. Underline all numbers and functions 2.
In mathematics , matrix calculus is a specialized notation for doing multivariable calculus , especially over spaces of matrices. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering , while the tensor index notation is preferred in physics. Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector.
Class Central is learner-supported. Johns Hopkins University. Korea Advanced Institute of Science and Technology. Start your review of Vector Calculus for Engineers.
Collapse menu 1 Analytic Geometry 1. Lines 2. Distance Between Two Points; Circles 3. Functions 4. The slope of a function 2.
Differentiation and integration of vectors
Let us generalize these concepts by assigning n-squared numbers to a single point or n-cubed numbers to a single. Curves in R3. We will also use X denote the space of input values, and Y the space of output values. Further necessary conditions 57 3.
Search this site. According to Mark PDF. Advances in Cardiac Surgery: v. Analysis PDF. Art Therapy PDF. Atlantis, the Daughter PDF. Bergisel PDF.
Freeman Vector Calculus Website W. Exercises Book: Multivariable Calculus Clark Bray. Find Answer. Do not enter an equivalent expression when your answer is marked wrong.
Подумайте о юридических последствиях. Звонивший выдержал зловещую паузу. - А что, если мистер Танкадо перестанет быть фактором, который следует принимать во внимание. Нуматака чуть не расхохотался, но в голосе звонившего слышалась подозрительная решимость. - Если Танкадо перестанет быть фактором? - вслух размышлял Нуматака.
Ему на руку была даже конструкция башни: лестница выходила на видовую площадку с юго-западной стороны, и Халохот мог стрелять напрямую с любой точки, не оставляя Беккеру возможности оказаться у него за спиной, В довершение всего Халохот двигался от темноты к свету. Расстрельная камера, мысленно усмехнулся. Халохот оценил расстояние до входа. Семь ступеней.
- Почему же вся переписка Северной Дакоты оказалась в твоем компьютере. - Я ведь тебе уже говорил! - взмолился Хейл, не обращая внимания на вой сирены. - Я шпионил за Стратмором. Эти письма в моем компьютере скопированы с терминала Стратмора - это сообщения, которые КОМИНТ выкрал у Танкадо.
Насколько. Сьюзан не понимала, к чему клонит Стратмор.
Все еще темно? - спросила Мидж. Но Бринкерхофф не ответил, лишившись дара речи. То, что он увидел, невозможно было себе представить. Стеклянный купол словно наполнился то и дело вспыхивающими огнями и бурлящими клубами пара. Бринкерхофф стоял точно завороженный и, не в силах унять дрожь, стукался лбом о стекло.
Сквозь строй - лучший антивирусный фильтр из всех, что я придумал. Через эту сеть ни один комар не пролетит. Выдержав долгую паузу, Мидж шумно вздохнула. - Возможны ли другие варианты.
Я, университетский профессор, - подумал он, - выполняю секретную миссию.
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662558015.52/warc/CC-MAIN-20220523101705-20220523131705-00220.warc.gz
|
CC-MAIN-2022-21
| 16,269 | 55 |
http://fx-fx.info/index.php/books/bilinear-control-systems-matrices-in-action-169-applied-mathematical
|
math
|
By David Elliott
Read Online or Download Bilinear Control Systems: Matrices in Action: 169 (Applied Mathematical Sciences) PDF
Best group theory books
Groupoids, Inverse Semigroups, and their Operator Algebras (Progress in Mathematics)
In recent times, it has turn into more and more transparent that there are very important connections pertaining to 3 strategies -- groupoids, inverse semigroups, and operator algebras. there was loads of development during this sector over the past 20 years, and this e-book provides a cautious, updated and fairly large account of the subject material.
Asymptotic Geometric Analysis: Proceedings of the Fall 2010 Fields Institute Thematic Program: 68 (Fields Institute Communications)
Asymptotic Geometric research is anxious with the geometric and linear homes of finite dimensional gadgets, normed areas, and convex our bodies, specifically with the asymptotics in their numerous quantitative parameters because the size has a tendency to infinity. The deep geometric, probabilistic, and combinatorial tools built listed here are used outdoor the sphere in lots of parts of arithmetic and mathematical sciences.
The food of the Alpha and Beta 1: Mathematics is your food
The e-book “Mathematics is your foodstuff” is all approximately creating a easy basic subject your energy and effort to extend your wisdom for the time being learning arithmetic. The publication is a strong selective booklet intended to inculcate arithmetic wisdom into you and boost the educational lifetime of a scholar.
Alternative Pseudodifferential Analysis: With an Application to Modular Forms (Lecture Notes in Mathematics)
This quantity introduces a completely new pseudodifferential research at the line, the competition of which to the standard (Weyl-type) research could be stated to mirror that, in illustration idea, among the representations from the discrete and from the (full, non-unitary) sequence, or that among modular sorts of the holomorphic and replacement for the standard Moyal-type brackets.
- A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences: With Complete Bibliography
- Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics: 2
- Random Walks on Reductive Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)
- Groups St Andrews 2013 (London Mathematical Society Lecture Note Series)
Extra resources for Bilinear Control Systems: Matrices in Action: 169 (Applied Mathematical Sciences)
- Mechanical Design by Peter R. N. Childs BSc.(Hons),D. Phil C.Eng F.I.Mech.E.
- 75 Remarkable Fruits For Your Garden by Jack Staub,Ellen Buchert
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943562.70/warc/CC-MAIN-20230320211022-20230321001022-00704.warc.gz
|
CC-MAIN-2023-14
| 2,719 | 18 |
https://practice.geeksforgeeks.org/problems/smallest-number-in-one-swap/0
|
math
|
Given a non-negative number N. The task is to apply at most one swap operation on the number N so that the resultant is the smallest possible number.
Note: Input and Output should not contain leading zeros.
The first line of input contains an integer T denoting the number of test cases. Then T test cases follow. Each test case contains a number N as input.
For each test case, print the smallest number possible. if it is not possible to make smallest number than N then print N as it is.
2<=|Number length |<=105
If you have purchased any course from GeeksforGeeks then please ask your doubt on course discussion forum. You will get quick replies from GFG Moderators there.
|
s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738950.61/warc/CC-MAIN-20200813014639-20200813044639-00145.warc.gz
|
CC-MAIN-2020-34
| 676 | 6 |
https://tmeastafrica.org/and-pdf/879-probability-questions-and-answers-examples-pdf-738-369.php
|
math
|
File Name: probability questions and answers examples .zip
- Probability | Theory, solved examples and practice questions
- Probability Questions with Solutions
- Probability Practice Problems
Read the lesson on probability problems for more information and examples. Try the free Mathway calculator and problem solver below to practice various math topics.
Probability | Theory, solved examples and practice questions
To have no repeated digits, all four digits would have to be different, which is selecting without replacement. The probability of no repeated digits is the number of 4 digit PINs with no repeated digits divided by the total number of 4 digit PINs. This probability is. Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket. Compute the probability that you win the second prize if you purchase a single lottery ticket.
Probability Questions with Solutions
Solved probability problems and solutions are given here for a concept with clear understanding. Students can get a fair idea on the probability questions which are provided with the detailed step-by-step answers to every question. Solved probability problems with solutions :. The graphic above shows a container with 4 blue triangles, 5 green squares and 7 red circles. A single object is drawn at random from the container. A single card is drawn at random from a standard deck of 52 playing cards.
divided by the cardinality of the sample space S. • Permutations: P(n, k) value is recorded. Create a probability table (pdf) for this probability experiment, and.
Probability Practice Problems
The probability of throwing a 3 or a 4 is double that, or 2 in 6. This can be simplified by dividing both 2 and 6 by 2. Note that H represents heads and T represents tails:.
Example: At a car park there are vehicles, 60 of which are cars, 30 are vans and the remainder are lorries. If every vehicle is equally likely to leave, find the probability of: a a van leaving first. Solution: a Let S be the sample space and A be the event of a van leaving first. Probability of a van leaving first:.
Understanding the basic rules and formulas of probability will help you score high in the entrance exams. In mathematics too, probability indicates the same — the likelihood of the occurrence of an event. Either an event will occur for sure, or not occur at all.
Random Experiment : An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment. A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the number that appears on its upper face.
Students and other people can get benefit from this Quiz. On the basis of this information, the professor states that the average age of all the students in the university is 21 years. A distribution formed by all possible values of a statistics is called a Binomial distribution b Hypergeometric distribution c Normal distribution d Sampling distribution MCQ The solved questions answers in this Probability And Statistics - MCQ Test 1 quiz give you a good mix of easy questions and tough questions.
Из тени на авенида дель Сид появилась фигура человека. Поправив очки в железной оправе, человек посмотрел вслед удаляющемуся автобусу. Дэвид Беккер исчез, но это ненадолго.
- Эксклюзивные права у вас. Это я гарантирую. Как только найдется недостающая копия ключа, Цифровая крепость - ваша. - Но с ключа могут снять копию. - Каждый, кто к нему прикоснется, будет уничтожен.
Введя несколько модифицированных команд на языке Паскаль, он нажал команду ВОЗВРАТ. Окно местоположения Следопыта откликнулось именно так, как он рассчитывал. ОТОЗВАТЬ СЛЕДОПЫТА. Он быстро нажал Да.
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303747.41/warc/CC-MAIN-20220122043216-20220122073216-00501.warc.gz
|
CC-MAIN-2022-05
| 4,232 | 19 |
https://www.onlinemathlearning.com/understand-fractions.html
|
math
|
Plans and Worksheets for Grade 3
Plans and Worksheets for all Grades
Lessons for Grade 3
Common Core For Grade 3
Videos, examples, solutions, and lessons to help Grade 3 students learn to understand a fraction 1/b
as the quantity formed by 1 part when a whole is partitioned into b
equal parts; understand a fraction a
as the quantity formed by a
parts of size 1/b
Common Core: 3.NF.1
Suggested Learning Target
Understanding fraction 1/b as part of a whole partitioned into b equal parts
- I can explain any fraction (1/b) as one part of a whole.
- I can explain any fraction (a/b) as "a" (numerator) being the numbers of parts and "b" (denominator) as the total number of equal parts in the whole.
In this lesson you will learn that fractions are equal size parts of a whole by looking at fair and unfair situations.
Understand any fraction (a/b) as "a" (numerator) being the numbers of parts and "b" (denominator) as the total number of equal parts in the whole
As the number of equal pieces in the whole increases the size of the fractional pieces decreases.
The denominator represents the number of equal parts.
The numerator of a fraction is the count of the number of equal parts.
Fair sharing means equal sharing. In word problems it means fractions.
1. Six children share four brownies equally so that each child receives a fair share. What portion of each brownie will each child receive?
2. What fraction of the rectangle is shaded? How might you draw the rectangle in another way but with the same fraction shaded?
This lesson will help teachers and parents to understand and explain fractions.
Partitioning Shapes Using Models: 3.NF.1
How to express a shaded amount as a fraction?
Also learn about partitioned shapes using models with more than one whole.
Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233506399.24/warc/CC-MAIN-20230922102329-20230922132329-00635.warc.gz
|
CC-MAIN-2023-40
| 2,105 | 30 |
https://quizlet.com/297379715/g4-ccss-math-vocabulary-flash-cards/
|
math
|
G4 CCSS Math Vocabulary
Terms in this set (143)
an angle with a measure less than 90 degrees
to combine, to put together two or more quantities
any number being added
problems that ask how much more or less one amount is than another
a step-by-step method for computing
two rays that share an endpoint
The measure of the size of an angle. It tells how far one
side is turned from the other side. A one degree angle turns through 1/360 of a full circle.
Part of a circle between any two of its points
The measure, in square units, of the inside of a plane figure
A model of multiplication that shows each place value product
An arrangement of objects in equal rows
Associative Property of Addition
Changing the grouping of three or more addends does not change the sum
Associative Property of Multiplication
Changing the grouping of three or more factors does not change the product
A characteristic of an object, such as color, shape, size, etc
Fractions that are commonly used for estimation
Capacity refers to the amount of liquid a container can hold
A metric unit of length equal to 0.01 of a meter
A plane figure with all points the same distance from a fixed point called
To sort into categories or to arrange into groups by attributes
For two or more fractions, a common denominator is a common multiple of the denominators
Commutative Property of Addition
Changing the order of the addends does not change the sum
Commutative Property of Multiplication
Changing the order of the factors does not
change the product
To decide if one number is greater than, less than, or equal to
Used to represent larger and smaller amounts in a comparison situation. Can be used to represent all four operations. Different lengths of bars are drawn to represent each number
To put together components or basic elements
A number greater than 0 that has more than two different
Having exactly the same size and shape
A customary unit of capacity. 1 cup = 8 fluid ounces
A system of measurement used in the U.S. The system includes
units for measuring length, capacity, and weight
A collection of information gathered for a purpose. Data may be in the form of either words or numbers
A number with one or more digits to the right of a decimal point
A fractional number with a denominator of 10 or a power of 10. Usually written with a decimal point
A number containing a decimal point
A dot (.) separating the whole number from the fraction in decimal notation
To separate into components or basic elements
degree (angle measure)
A unit for measuring angles. Based on dividing one complete circle into 360 equal parts
The quantity below the line in a fraction. It tells how many equal parts are in the whole
A tool used to measure and draw angles.
A customary unit of capacity. 1 ______ = 2 pints or 1 _____ = 4 cups
The answer to a division problem.
The difference between the greatest number and the least number in a set of data.
A part of a line that has one endpoint and goes on forever in one direction.
An answer that is based on good number sense.
________ addition and subtraction facts or _______ multiplication and division facts. Also called fact family.
The amount left over when one number is divided by another.
An angle that measures exactly 90º.
A triangle that has one 90º angle.
round a whole number
To find the nearest ten, hundred, thousand, (and so on).
One sixtieth of a minute. There are 60 _______ in a minute.
A set of numbers arranged in a special order or pattern.
When a fraction is expressed with the fewest possible pieces, it is in ______ ______. (Also called lowest terms.)
To express a fraction in simplest form.
A unit, such as square centimeter or square inch, used to measure area.
A common or usual way of writing a number using digits.
An operation that gives the difference between two numbers.
The answer to an addition problem.
One of the equal parts when a whole is divided into 10 equal parts.
A duration of a segment of time.
Having length and width. Having area, but not volume. Also called a plane figure.
A fraction that has 1 as its numerator.
_________ that are not equal.
A letter or symbol that represents a number.
The point at which two line segments, lines, or rays meet to form an angle.
The number of cubic units it takes to fill a figure.
The measure of how heavy something is.
____ ______ are zero and the counting numbers 1, 2, 3, 4, 5, 6, and so on.
A way of using words to write a number.
A customary unit of length. 1 _____ = 3 feet or 36 inches.
Zero Property of Multiplication
The product of any number and zero is zero.
_________ in two or more fractions that
are the same.
A set of connected points continuing without end
in both directions.
line of symmetry
A line that divides a figure into two congruent halves
that are mirror images of each other.
A diagram showing frequency of data on a number line.
A part of a line with two endpoints.
line symmetric figures
Figures that can be folded in half and its two parts match
The basic unit of capacity in the metric system.
1 ____ = 1,000 milliliters
When a fraction is expressed with the fewest possible pieces, it is in _____ ____. (Also called simplest form.)
The amount of matter in an object.
A standard unit of length in the metric system
A system of measurement based on tens. The basic unit of
capacity is the liter. The basic unit of length is the meter. The basic unit of mass is the gram.
A customary unit of length.
1 ____ = 5,280 feet
A metric unit of capacity.
1,000 _______ = 1 liter
A metric unit of length.
1,000 __________ = 1 meter
One sixtieth of an hour or 60 seconds.
A number that has a whole number (not 0) and a fraction.
A product of a given whole number and any other whole number.
Compare by asking or telling how many times more one
amount is as another. For example, 4 times greater than.
The operation of repeated addition of the same number.
A diagram that represents numbers as points on a line.
The number written above the line in a fraction. It tells how many equal parts are described in the fraction.
An angle with a measure greater than 90º but less than 180º.
Order of Operations
A set of rules that tells the order in which to compute.
A customary unit of weight equal to one sixteenth of a pound. 16 _________ = 1 pound
Lines that are always the same distance apart. They do not intersect.
Used in mathematics as grouping symbols for operations.
A repeating or growing sequence or design. An ordered set of numbers or shapes arranged according to a rule.
The distance around the outside of a figure.
In a large number, periods are groups of 3 digits separated by commas or by spaces.
Two intersecting lines that form right angles.
A customary unit of capacity. 1 ______ = 2 cups
The value of the place of a digit in a number.
A two-dimensional figure.
The exact location in space represented by a dot.
A customary unit of weight. 1 _______ = 16 ounces.
A whole number greater than 0 that has exactly two different factors, 1 and itself.
The answer to a multiplication problem.
Any of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
The amount that remains after one quantity is subtracted
When one of the factors of a product is a sum, multiplying
each addend before adding does not change the product.
To separate into equal groups and find the number in each group or the number of groups.
A number that is divided by another number.
The number by which another number is divided.
A point at either end of a line segment, or a point at one end of a ray.
Having the same value.
A mathematical sentence with an equals sign. The amount on one side of the equals sign has the same value as the amount on the other side.
Fractions that have the same value.
To find a number close to an exact amount, an
estimate tells about how much or about how many.
To find the value of a mathematical expression.
A way to write numbers that shows the place value of each digit.
A mathematical phrase without an equal sign.
A group of related facts that use the same numbers.
Also called related facts
The whole numbers that are multiplied to get a product.
A set of two whole numbers when multiplied, will result in a
A customary unit of length. 1 ____ = 12 inches
A rule that is written as an equation.
A way to describe a part of a whole or a part of a group by
using equal parts.
A table that lists pairs of numbers that follow a rule.
A customary unit of capacity. 1 ____ = 4 quarts
The standard unit of mass in the metric system.
1,000 ______ = 1 kilogram
________ ____ is used to compare two numbers when the first number is larger than the second number.
A unit of time. 1 ____ = 60 minutes. 24 _____ = 1 day
One of the equal parts when a whole is divided into 100
In the decimal numeration system, __________ is the name
of the next place to the right of tenths.
Identity Property of Addition
If you add zero to a number, the sum is the same as that
Identity Property of Multiplication
If you multiply a number by one, the product is the same
as that number.
A term for a fraction whose numerator is greater than or
equal to its denominator.
A customary unit of length.
12 _______ = 1 foot
Lines that cross at one point.
Operations that undo each other.
A metric unit of mass equal to 1000 grams.
A metric unit of length equal to 1000 meters.
How long something is. The distance from one
point to another.
____ ____ is used to compare two numbers when the first number is smaller than the second number.
YOU MIGHT ALSO LIKE...
Numbers & Animals Vocabulary | Everyday Traditional Mandarin Chinese
Math 4th Grade Vocabulary
3rd Grade Math Vocabulary
OTHER SETS BY THIS CREATOR
ccss ela grade 4
Alipne Lakes Grade 4 Math All
Lesson 4 tsunami Earth Science
Proper Fractions / Improper Grade 4
THIS SET IS OFTEN IN FOLDERS WITH...
Engage NY Grade 4 - Module 2
Grade 4 Math Review
OSB Gr4 geometry List 1
Grade 4 Math Vocabulary
|
s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628001089.83/warc/CC-MAIN-20190627095649-20190627121649-00183.warc.gz
|
CC-MAIN-2019-26
| 9,864 | 193 |
https://www.essaymill.net/2021/investments-analysis-13815/
|
math
|
Answer Question 1 (this question is compulsory)
Estimate the price, duration and convexity for all 3 bonds in the table below. You may assume the current date is 31st December 2019 and that the bonds mature on the last day of the years shown below. (35 Marks)
Company Moody’s Credit rating Maturity Date Coupon
[% p.a.] Payment
[in years] Yield
[% p.a.] Convexity
[in years] Bond Price
Motor Co ltd B++ 2025 6.5% annual ??? 4.5%
Mining Co ltd A1 2027 3% annual ??? 2.5%
Water Co ltd A3 2023 4.5% annual ??? 1.5% ??? ???
The longer dated bonds above have higher levels of convexity. Holding all other bond characteristics equal, it is normally better to hold a bond with higher convexity in volatile markets conditions. Explain why this is the case.
(c) You buy an annual coupon £100 bond with a 5% p.a. coupon when interest rates are 3% p.a. At this point in time the bond has 15 years until maturity. Interest rates rise to 4% p.a. immediately after your purchase and remain constant until you sell it in 8 years-time. What is your Realised Compound Yield on this bond?
(d) Explain the differences between the following bond management strategies: (i) buy and hold (ii) laddering (iii) matched funds. Identify the advantages and disadvantages of each.
Answer EITHER Question 2 OR Question 3. DO NOT answer both.
Question 2: The Dividend Discount Model
LoBank plc has just paid an annual dividend of £0.50 per share. The required rate of return is 8% p.a.
If that level of dividend payment is expected to be constant into the future, what is the intrinsic value (or fair price) of the shares?
If the next dividend payment of LoBank plc is expected to be 5% higher than the last, and if this rate of dividend growth is expected to be maintained over time, what is the intrinsic value of the shares?
HiBank plc has a new product and is enjoying rapid growth. It is estimated that dividends will grow at an annual rate of 15% over the next five years. After that, the growth rate will fall to 7% and remain at that rate. The directors have just paid an annual dividend of £2.50 per share. Calculate the intrinsic value of the share if the required rate of return is 10% p.a.
MedBank plc has seen dividends per share grow at 10% p.a. recently but expects the growth rate to decline linearly over the next 6 years (period 2H) to 3% p.a. The beta of the stock is 0.75, the risk-free rate is 2% p.a. and the market risk premium is 4% p.a. The last reported dividend was £0.50. Calculate the intrinsic value of the share.
Explain how the 3-stage DDM takes into consideration: (i) changes in dividend growth rates (ii) changes in the cost of capital and (iii) changes in the pay-out ratio. Discuss the extent to which the additional parameters incorporated in this model make it more realistic than the 2-stage and H models. (45 marks)
Note. For the 2-rate growth model the intrinsic value may be estimated as:
P_0=∑_(t=1)^(t=n)▒(〖DPS〗_0 (1+g_a )^t)/〖(1+k_e)〗^t +((〖DPS〗_n (1+g_n ))/(k_e-g_n )*1/(1+k_e )^n )
For the H-model:
P_0=(〖DPS〗_0*(1+g_n ))/((k_e-g_n ) )+(〖DPS〗_0*H*(g_a-g_n ))/((k_e-g_n ) )
For the 3-stage model:
P_0=∑_(t=1)^(t=n1)▒(〖EPS〗_0 (1+g_a )^t*⊓_a)/((1+k_(e,hg) ) )+∑_(t=n1+1)^(t=n2)▒〖DPS〗_t/((1+k_(e,t) )^t )+((〖EPS〗_n2 (1+g_n)*⊓_n ))/(〖(k〗_(e,st)-g_n)(1+r)^n ))
Question 3: Technical Analysis
Sketch two diagrams to show the following technical chart patterns:
head and shoulders.
Explain the process through which EITHER a continuation pattern, like a flag, OR a trend reversal pattern, like a head and shoulders, may develop.
Identify the nature of the technical relationships found in the in the London silver fix and Halliburton share price charts shown below. Explain the reasons why such technical patterns may develop.
London Silver price fix September 2001 – November 2007
Halliburton share price: August 1999 – April 2000
With the aid of examples, describe how technical analysts use moving averages to identify buy and sell signals. Explain why moving averages identify changes in the momentum of price movements.
Present Value Factors:
where n is the number of periods and r is the interest (discount) rate as a decimal
Present Value of an Annuity:
Compound Sum factor:
Compound Sum of Annuity Factor:
Table 1a: Compound (Future) Value Factors for £1 Compounded at R Percent for N periods
Table 1b: Compound (Future) Value Factors for £1 Compounded at R Percent for N periods
Table 2a: Present Value Factors (at R per cent) for £1 received at end of N periods
Table 2b: Present Value Factors (at R per cent) for £1 received at end of N periods
Table 3a: Compound Sum Annuity for £1 Compounded at R percent for N periods
Table 3b: Compound Sum Annuity for £1 Compounded at R percent for N periods
Table 4a: Present Value Annuity Factors (at R percent period) for £1 received per period for each of N periods
Table 4b: Present Value Annuity Factors (at R percent period) for £1 received per period for each of N period
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991685.16/warc/CC-MAIN-20210512070028-20210512100028-00405.warc.gz
|
CC-MAIN-2021-21
| 5,008 | 48 |
https://optionsbistro.com/tag/coveredputs/
|
math
|
#CoveredPuts Sold $CAT Mar 3 87 puts @ 1.06, against short stock I was assigned early overnight.
$NFLX #CoveredPuts – Sold NFLX Nov 11 2016 119.0 Puts @ 0.60.
#Earnings #ShortStock #CoveredPuts #ironbutterfly
I am still short some stock so I’ve sold several puts today and over the past week… Nov 4th, every strike from 120 to 126.
I also STO FB 120/130/140 iron butterfly for 6.15, at @thomberg1201‘s suggestion.
#ShortStock #CoveredPuts #Earnings #Rolling some $FB positions…
BTC FB Oct 14th 125 put for .07. Sold for 1.10 on Sept 16th.
BTC FB Dec 90 put for .11. Sold for 2.68 on June 15th.
STO FB Oct 28th 127 put for 1.11
#Earnings #CoveredPuts #ShortStock
And my short stock position continues. Someday my cost basis will equal the stock price…. some day.
STO $FB Oct 21st 126 covered put for 1.25.
#CoveredPuts STO $FB Oct 14th 125 put for 1.45
#Earnings #CoveredPuts STO $FB Oct 7th 125 covered put for 1.10
#VXXGame BTC $UVXY Oct 21st 55 call for .84. Sold for 4.25 on Aug 2nd.
#ContangoETFs BTC $NUGT Sept 16th 23 puts for 4.45 and 5.20. Sold for 1.45 on Aug 26th.
Rolled to: Sept 30th 16 puts for 1.00, and waiting for fills on others.
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320695.49/warc/CC-MAIN-20170626083037-20170626103037-00147.warc.gz
|
CC-MAIN-2017-26
| 1,158 | 17 |
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?reload=true&arnumber=1182435
|
math
|
Skip to Main Content
In OFDM systems both time synchronization error and channel estimation error contribute to a post-FFT phase error in the received symbols. The combined effect of this phase error and the channel noise results in bit errors. A numerical technique for evaluating the probability of error due to the post-FFT phase error and channel noise in the context of OFDM (where small constellations such as BPSK and QPSK are used) has not been reported in the literature so far. In this paper, a Fourier series based numerical technique of evaluating the probability of error in OFDM systems is presented. Formulae are derived for probability of error in terms of the phase error variance and channel noise variance for an OFDM system incorporating BPSK and QPSK constellations. A comparison between the numerical results for the proposed technique and the average symbol/bit error rate (SER/BER) estimates obtained through simulation demonstrates the accuracy of the technique.
|
s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398465089.29/warc/CC-MAIN-20151124205425-00239-ip-10-71-132-137.ec2.internal.warc.gz
|
CC-MAIN-2015-48
| 987 | 2 |
https://www.allassignmenthelp.com/questions/san-diego-state-university-operations-and-supply-chain-management-assignment-help-regression-2553499249
|
math
|
San Diego State University Operations And Supply Chain Management Assignment Help - Regression
Question - In 2009, the New York Yankees won 103 baseball games during the regular season. The table on the
next page lists the number of victories (W), the earnedrun average (ERA), and the batting average
(AVG) of each team in the American League. The ERA is one measure of the effectiveness of the
pitching staff, and a lower number is better. The batting average is one measure of effectiveness of
the hitters, and a higher number is better.
(a) Develop a regression model that could be used to predict the number of victories based on the
(b) Develop a regression model that could be used to predict the number of victories based on the
(c) Which of the two models is better for predicting the number of victories?
(d) D ...Read More
Solution Preview - No Solution Preview Available
|
s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890566.2/warc/CC-MAIN-20200706222442-20200707012442-00496.warc.gz
|
CC-MAIN-2020-29
| 881 | 11 |
http://coveredcallsadvisor.blogspot.com/2015/12/established-new-position-in-enterprise.html
|
math
|
Today, the Covered Calls Advisor established a new position in Enterprise Products Partners LP (ticker symbol EPD) by selling three Dec2015 Put options at the $21.00 strike price. This position is a conservative one since it was established with 3.0% downside protection to the strike price.
As detailed below, the Enterprise Products Partners investment will yield a +3.2% absolute
return in 12 days (which is equivalent to a +96.5% annualized
return-on-investment) if EPD closes above the $21.00 strike price on the Dec 18th options expiration date.
This potential return is very nice given the downside protection (from the $21.64 stock price to the $21.00 strike price) when the position was established. Because of the recent rapid decline in Master Limited Partnerships (MLPs), including another 5% today when this position was established, the implied volatility in the options had ballooned to 55; so the $.70 per share price when the Puts were sold is a very attractive premium received.
1. Enterprise Products Partners LP (EPD) -- New 100% Cash-Secured Puts Position
12/07/2015 Sold 3 EPD Dec2015 $21.00 100% cash-secured Put options @ $.70
Note: the price of EPD was $21.64 today when this transaction was executed.
The Covered Calls Advisor does not use margin, so the detailed
information on this position and a potential result shown below
reflect the fact that this position was established using 100% cash
securitization for the Put options sold.
A possible overall performance result (including commissions) would be as follows:
100% Cash-Secured Cost Basis: $6,300.00
Note: the price of EPD was $21.64 when these options were sold
(a) Options Income: +$199.80
= ($.70*300 shares) - $10.20 commissions
(b) Dividend Income: +$0.00
(c) Capital Appreciation (If EPD is above $21.00 strike price at Dec2015 expiration): +$0.00
= ($21.00-$21.00)*300 shares
Total Net Profit (If EPD is above $21.00 strike price at Dec2015 options expiration): +$199.80
= (+$199.80 options income +$0.00 dividend income +$0.00 capital appreciation)
Absolute Return (If EPD is above $21.00 strike price at Dec2015 options expiration): +3.2%
Annualized Return: +96.5%
= (+$199.80/$6,300.00)*(365/12 days)
downside 'breakeven price' at expiration is at $20.30 ($21.00 - $.70),
which is 6.2% below the current market price of $21.64.
Using the Black-Scholes Options Pricing Model in the Schwab
Hypothetical Options Pricing Calculator, the probability of
making a profit (if held until the Dec 18th, 2015 options expiration) for
this EPD short Puts position is 65%. This compares with a
profit of 50.4% for a buy-and-hold of EPD stock over the same
Using this probability of profit of 65%, the expected value annualized return-on-investment (if held until expiration) is +62.7% (+96.5% *
65%), an extraordinarily high value for this investment.
'crossover price' at expiration is $22.34 ($21.64 + $.70). This is the
price above which it would have been more profitable to simply
buy-and-hold Enterprise Products Partners until the Dec2015 options expiration date
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591719.4/warc/CC-MAIN-20180720174340-20180720194340-00151.warc.gz
|
CC-MAIN-2018-30
| 3,051 | 37 |
https://slideplayer.com/slide/7860057/
|
math
|
Presentation on theme: "Similar figures have exactly the same shape but not necessarily the same size. Corresponding sides of two figures are in the same relative position, and."— Presentation transcript:
1 Similar figures have exactly the same shape but not necessarily the same size. Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if and only if the lengths of corresponding sides are proportional and all pairs of corresponding angles have equal measures.
2 When stating that two figures are similar, use the symbol ~ When stating that two figures are similar, use the symbol ~. For the triangles above, you can write ∆ABC ~ ∆DEF. Make sure corresponding vertices are in the same order. It would be incorrect to write ∆ABC ~ ∆EFD.You can use proportions to find missing lengths in similar figures.
3 Example 1A: Finding Missing Measures in Similar Figures Find the value of x the diagram.∆MNP ~ ∆STU
4 Example 1B: Finding Missing Measures in Similar Figures Find the value of x the diagram.ABCDE ~ FGHJK
5 Reading MathAB means segment AB. AB means the length of AB.A means angle A mA the measure of angle A.
6 Check It Out! Example 1Find the value of x in the diagram if ABCD ~ WXYZ.ABCD ~ WXYZ
7 You can solve a proportion involving similar triangles to find a length that is not easily measured. This method of measurement is called indirect measurement. If two objects form right angles with the ground, you can apply indirect measurement using their shadows.
8 Example 2: Measurement Application A flagpole casts a shadow that is 75 ft long at the same time a 6-foot-tall man casts a shadow that is 9 ft long. Write and solve a proportion to find the height of the flag pole.
9 Helpful HintA height of 50 ft seems reasonable for a flag pole. If you got 500 or 5000 ft, that would not be reasonable, and you should check your work.
10 Check It Out! Example 2aA forest ranger who is 150 cm tall casts a shadow 45 cm long. At the same time, a nearby tree casts a shadow 195 cm long. Write and solve a proportion to find the height of the tree.
11 Check It Out! Example 2bA woman who is 5.5 feet tall casts a shadow 3.5 feet long. At the same time, a building casts a shadow 28 feet long. Write and solve a proportion to find the height of the building.
12 If every dimension of a figure is multiplied by the same number, the result is a similar figure. The multiplier is called a scale factor.
13 Example 3A: Changing Dimensions The radius of a circle with radius 8 in. is multiplied by 1.75 to get a circle with radius 14 in. How is the ratio of the circumferences related to the ratio of the radii? How is the ratio of the areas related to the ratio of the radii?Circle ACircle B
14 Example 3B: Changing Dimensions Every dimension of a rectangular prism with length 12 cm, width 3 cm, and height 9 cm is multiplied by to get a similar rectangular prism. How is the ratio of the volumes related to the ratio of the corresponding dimensions?Prism APrism B
15 Helpful HintA scale factor between 0 and 1 reduces a figure. A scale factor greater than 1 enlarges it.
16 Check It Out! Example 3A rectangle has width 12 inches and length 3 inches. Every dimension of the rectangle is multiplied by to form a similar rectangle. How is the ratio of the perimeters related to the ratio of the corresponding sides?Rectangle ARectangle B
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585424.97/warc/CC-MAIN-20211021133500-20211021163500-00080.warc.gz
|
CC-MAIN-2021-43
| 3,451 | 17 |
https://www.arxiv-vanity.com/papers/gr-qc/0605106/
|
math
|
A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
A key result in the proof of black hole uniqueness in -dimensions is that a stationary black hole that is “rotating”—i.e., is such that the stationary Killing field is not everywhere normal to the horizon—must be axisymmetric. The proof of this result in -dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, . This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than , it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the -dimensional proof, that the spacetime is analytic.
Consider an -dimensional stationary spacetime containing a black hole. Since the event horizon of the black hole must be mapped into itself by the action of any isometry, the asymptotically timelike Killing field must be tangent to the horizon. Therefore, we have two cases to consider: (i) is normal to the horizon, i.e., tangent to the null geodesic generators of the horizon; (ii) is not normal to the horizon. In -dimensions it is known that in case (i), for suitably regular non-extremal vacuum or Einstein-Maxwell black holes, the black hole must be static [42, 5]. Furthermore, in -dimensions it is known that in case (ii), under fairly general assumptions about the nature of the matter content but assuming analyticity of the spacetime and non-extremality of the black hole, there must exist an additional Killing field that is normal to the horizon. It can then be shown that the black hole must be axisymmetric111 In this paper, by “axisymmetric” we mean that spacetime possesses one-parameter group of isometries isomorphic to whose orbits are spacelike. We do not require that the Killing field vanishes on an “axis.” as well as stationary [18, 19]. This latter result is often referred to as a “rigidity theorem,” since it implies that the horizon generators of a “rotating” black hole (i.e., a black hole for which is not normal to the horizon) must rotate rigidly with respect to infinity. A proof of the rigidity theorem in -dimensions which partially eliminates the analyticity assumption was given by Friedrich, Racz, and Wald [9, 32], based upon an argument of Isenberg and Moncrief [27, 20] concerning the properties of spacetimes with a compact null surface with closed generators. The above results for both cases (i) and (ii) are critical steps in the proofs of black hole uniqueness in -dimensions, since they allow one to apply Israel’s theorems [23, 24] in case (i) and the Carter-Robinson-Mazur-Bunting theorems [2, 36, 25, 1] in case (ii).
Many attempts to unify the forces and equations of nature involve the consideration of spacetimes with dimensions. Therefore, it is of considerable interest to consider a generalization of the rigidity theorem to higher dimensions, especially in view of the fact that there seems to be a larger variety of black hole solutions (see e.g., [7, 12, 15]), the classification of which has not been achieved yet. 222 There have recently appeared several works on general properties of a class of stationary, axisymmetric vacuum solutions, including an -dimensional generalization of the Weyl solutions for the static case (see e.g., [3, 6, 16, 17], and see also [26, 43] and references therein for some techniques of generating such solutions in -dimensions). The purpose of this paper is to present a proof of the rigidity theorem in higher dimensions for non-extremal black holes.
The dimensionality of the spacetime enters the proof of the rigidity theorem in -dimensions in the following key way: The expansion and shear of the null geodesic generators of the horizon of a stationary black hole can be shown to vanish (see below). The induced (degenerate) metric on the -dimensional horizon gives rise to a Riemannian metric, , on an arbitrary -dimensional cross-section, , of the horizon. On account of the vanishing shear and expansion, all cross-sections of the horizon are isometric, and the projection of the stationary Killing field onto gives rise to a Killing field, , of on . In case (ii), does not vanish identically. Now, when , it is known that must have the topology of a -sphere, . Since the Euler characteristic of is nonzero, it follows that must vanish at some point . However, since is -dimensional, it then follows that the isometries generated by simply rotate the tangent space at . It then follows that all of the orbits of are periodic with a fixed period , from which it follows that, after period , the orbits of on the horizon must return to the same generator. Consequently, if we identify points in spacetime that differ by the action of the stationary isometry of parameter , the horizon becomes a compact null surface with closed null geodesic generators. The theorem of Isenberg and Moncrief [27, 20] then provides the desired additional Killing field normal to this null surface.
In dimensions, the Euler characteristic of may vanish, and, even if it is non-vanishing, if there is no reason that the isometries generated by need have closed orbits even when vanishes at some point . Thus, for example, even in the -dimensional Myers-Perry black hole solution with cross section topology , one can choose the rotational parameters of the solution so that the orbits of the stationary Killing field do not map horizon generators into themselves.
One possible approach to generalizing the rigidity theorem to higher dimensions would be to choose an arbitrary and identify points in the spacetime that differ by the action of the stationary isometry of parameter . Under this identification, the horizon would again become a compact null surface, but now its null geodesic generators would no longer be closed. The rigidity theorem would follow if the results of [27, 20] could be generalized to the case of compact null surfaces that are ruled by non-closed generators. We have learned that Isenberg and Moncrief are presently working on such a generalization , so it is possible that the rigidity theorem can be proven in this way.
However, we shall not proceed in this manner, but rather will parallel the steps of [27, 20], replacing arguments that rely on the presence of closed null generators with arguments that rely on the presence of stationary isometries. Since on the horizon we may write
where is tangent to the null geodesic generators and is tangent to cross-sections of the horizon, the stationarity in essence allows us to replace Lie derivatives with respect to by Lie derivatives with respect to . Thus, equations in [27, 20] that can be solved by integrating quantities along the orbits of the closed null geodesics correspond here to equations that can be solved if one can suitably integrate these equations along the orbits of in . Although the orbits of are not closed in general, we can appeal to basic results of ergodic theory together with the fact that generates isometries of to solve these equations.
For simplicity, we will focus attention on the vacuum Einstein’s equation, but we will indicate in section 4 how our proofs can be extended to models with a cosmological constant and a Maxwell field. As in [18, 19] and in [27, 20], we will assume analyticity, but we shall indicate how this assumption can be partially removed (to prove existence of a Killing field inside the black hole) by arguments similar to those given in [9, 32]. The non-extremality condition is used for certain constructions in the proof (as well as in the arguments partially removing the analyticity condition), and it would not appear to be straightforward to generalize our arguments to remove this restriction when the orbits of are not closed.
Our signature convention for is . We define the Riemann tensor by and the Ricci tensor by . We also set .
2 Proof of existence of a horizon Killing field
Let be an -dimensional, smooth, asymptotically flat, stationary solution to the vacuum Einstein equation containing a black hole. Thus, we assume the existence in the spacetime of a Killing field with complete orbits which are timelike near infinity. Let denote a connected component of the portion of the event horizon of the black hole that lies to the future of . We assume that has topology , where is compact. Following Isenberg and Moncrief [27, 20], our aim in this section is to prove that there exists a vector field defined in a neighborhood of which is normal to and on satisfies
where is an arbitrary vector field transverse to . As we shall show at the end of this section, if we assume analyticity of and of it follows that is a Killing field. We also will explain at the end of this section how to partially remove the assumption of analyticity of and .
We shall proceed by constructing a candidate Killing field, , and then proving that eq. (2) holds for . This candidate Killing field is expected to satisfy the following properties: (i) should be normal to . (ii) If we define by
then, on , should be tangent to cross-sections333Note that as already mentioned above, since is mapped into itself by the time translation isometries, must be tangent to , so is automatically tangent to . Condition (iii) requires that there exist a foliation of by cross-sections such that each orbit of is contained in a single cross-section. of . (iii) should commute with . (iv) should have constant surface gravity on , i.e., on we should have with constant on , since, by the zeroth law of black hole mechanics, this property is known to hold on any Killing horizon in any vacuum solution of Einstein’s equation.
We begin by choosing a cross-section , of . By arguments similar to those given in the proof of proposition 4.1 of , we may assume without loss of generality that has been chosen so that each orbit of on intersects at precisely one point, so that is everywhere transverse to . We extend to a foliation, , of by the action of the time translation isometries, i.e., we define , where denotes the one-parameter group of isometries generated by . Note that the function on that labels the cross-sections in this foliation automatically satisfies
Next, we define and on by
where is normal to and is tangent to . It follows from the transversality of that is everywhere nonvanishing and future-directed. Note also that on . Our strategy is to extend this definition of to a neighborhood of via Gaussian null coordinates. This construction of obviously satisfies conditions (i) and (ii) above, and it also will be shown below that it satisfies condition (iii). However, it will, in general, fail to satisfy (iv). We shall then modify our foliation so as to produce a new foliation so that (iv) holds as well. We will then show that the corresponding satisfies eq. (2).
Given our choice of and the corresponding choice of on , we can uniquely define a past-directed null vector field on by the requirements that , and that is orthogonal to each . Let denote the affine parameter on the null geodesics determined by , with on . Let be local coordinates on an open subset of . Of course, it will take more than one coordinate patch to cover , but there is no problem in patching together local results, so no harm is done in pretending that covers . We extend the coordinates from to by demanding that they be constant along the orbits of . We then extend and to a neighborhood of by requiring these quantities to be constant along the orbits of . It is easily seen that the quantities define coordinates covering a neighborhood of . Coordinates that are constructed in this manner are known as Gaussian null coordinates and are unique up to the choice of and the choice of coordinates on . It follows immediately that on we have
and we extend and to a neighborhood of by these formulas. Clearly, and commute, since they are coordinate vector fields.
Note that we have
so everywhere, not just on . Similarly, we have everywhere. It follows that in Gaussian null coordinates, the metric in a neighborhood of takes the form
where, again, is a labeling index that runs from to . We write
Note that , , and are independent of the choice of coordinates, , and thus are globally defined in an open neighborhood of . From the form of the metric, we clearly have and . It then follows that is the orthogonal projector onto the subspace of the tangent space perpendicular to and , where here and elsewhere, all indices are raised and lowered with the spacetime metric . Note that when , i.e., off of the horizon, differs from the metric , on the -dimensional submanifolds, , of constant , since fails to be perpendicular to these surfaces. Here, is defined by the condition that is the orthogonal projector onto the subspace of the tangent space that is tangent to ; the relationship between and is given by
However, since on (where ), we have , we will refer to as the metric on the cross-sections of .
Thus, we see that in Gaussian null coordinates the spacetime metric, , is characterized by the quantities , , and . In terms of these quantities, if we choose , then the condition (2) will hold if and only if the conditions
hold on .
Since the vector fields and are uniquely determined by the foliation and since (i.e., the time translations leave the foliation invariant), it follows immediately that and are invariant under . Hence, we have , so, in particular, condition (iii) holds, as claimed above. Similarly, we have and throughout the region where the Gaussian null coordinates are defined. Since , we obtain from eq. (8)
Contraction of this equation with yields
Contraction with then yields
and we then also immediately obtain
The next step in the analysis is to use the Einstein equation on , in a manner completely in parallel with the 4-dimensional case . This equation is precisely the Raychaudhuri equation for the congruence of null curves defined by on . Since that congruence is twist-free on , we obtain on
where denotes the expansion of the null geodesic generators of , denotes their shear, and is the affine parameter along null geodesic generators of with tangent . Now, by the same arguments as used to prove the area theorem , we cannot have on . On the other hand, the rate of change of the area, , of (defined with respect to the metric ) is given by
However, since is related to by the isometry , the left side of this equation must vanish. Since on , this shows that on . It then follows immediately that on . Now on , the shear is equal to the trace free part of while the expansion is equal to the trace of this quantity. So we have shown that on . Thus, the first equation in eq. (2) holds with .
However, in general fails to satisfy condition (iv) above. Indeed, from the form, eq. (8), of the metric, we see that the surface gravity, , associated with is simply , and there is no reason why need be constant on . Since on , the Einstein equation on yields
(see eq. (79) of Appendix A) where denotes the derivative operator on , i.e., . Thus, if is not constant on , then the last equation in eq. (2) fails to hold even when . As previously indicated, our strategy is repair this problem by choosing a new cross-section so that the corresponding arising from the Gaussian null coordinate construction will have constant surface gravity on . The determination of this requires some intermediate constructions, to which we now turn.
First, since we already know that everywhere and that on , it follows immediately from the fact that on that
on (for any choice ). Thus, is a Killing vector field for the Riemannian metric on . Therefore the flow, of yields a one-parameter group of isometries of , which coincides with the projection of the flow of the original Killing field to .
We define to be the mean value of on ,
where is the area of with respect to the metric . In the following we will assume that , i.e., that we are in the “non-degenerate case.” Given that , we may assume without loss of generality, that .
We seek a new Gaussian null coordinate system satisfying all of the above properties of together with the additional requirement that , i.e., constancy of the surface gravity. We now determine the conditions that these new coordinates would have to satisfy. Since clearly must be proportional to , we have
for some positive function . Since , we must have . Since on we have and is given by
we find that must satisfy
The last equality provides an equation that must be satisfied by on . In order to establish that a solution to this equation exists, we first prove the following lemma:
For any , we have
Furthermore, the convergence of the limit is uniform in . Similarly, -derivatives of converge to uniformly in as .
Proof: The von Neumann ergodic theorem (see e.g.,) states that if is an function for on a measure space with finite measure, and if is a continuous one-parameter group of measure preserving transformations on , then
converges in the sense of (and in particular almost everywhere). We apply this theorem to , , , and , to conclude that there is an function on to which the limit in the lemma converges. We would like to prove that is constant. To prove this, we note that eq. (18) together with the facts that and yields
where are constants independent of and , and where is finite because is compact. Consequently, is uniformly bounded in and in . Thus, for all , we have
Let be such that converges as . (As already noted above, existence of such a is guaranteed by the von Neumann ergodic theorem.) The above equation then shows that, in fact, must converge for all as and that, furthermore, the limit is independent of , as we desired to show. Thus, is constant, and hence equal to its spatial average, . The estimate (30) also shows that the limit (24) is uniform in . Similar estimates can easily be obtained for the norm with respect to of , for any . These estimates show that -derivatives of converge to uniformly in .
We now are in a position to prove the existence of a positive function on satisfying the last equality in eq. (23) on . Let
where is the function on defined by
The function is well defined for almost all because for any and sufficiently large , by Lemma 1. It also follows from the uniformity statement in Lemma 1 that is smooth on . By a direct calculation, using Lemma 1, we find that satisfies
as we desired to show.
We now can deduce how to choose the desired new Gaussian null coordinates. The new coordinate must satisfy
as before. However, in view of eq. (21), it also must satisfy
Since , we find that on , must satisfy
Substituting from eq. (34), we obtain
Thus, if our new Gaussian null coordinates exist, there must exist a smooth solution to this equation. That this is the case is proven in the following lemma.
There exists a smooth solution to the following differential equation on :
Proof: First note that the orbit average of any function of the form where is smooth must vanish, so there could not possibly exist a smooth solution to the above equation unless the average of over any orbit is equal to . However, this was proven to hold in Lemma 1. In order to get a solution to the above equation, choose , and consider the regulated expression defined by
Due to the exponential damping, this quantity is smooth, and satisfies the differential equation
We would now like to take the limit as to get a solution to the desired equation. However, it is not possible to straightforwardly take the limit as of , for there is no reason why this should converge without using additional properties of . In fact, we will not be able to show that the limit as of exists, but we will nevertheless construct a smooth solution to eq. (39).
To proceed, we rewrite eq. (40) as
where denotes the pull-back map on tensor fields associated with . Taking the gradient of this equation and using eq. (27), we obtain
where here and in the following we use differential forms notation and omit tensor indices. Since clearly commutes with and since is just the derivative along the orbit over which we are integrating, we can integrate by parts to obtain
It follows from the von Neumann ergodic theorem444 Here, the theorem is applied to the case of a tensor field of type on a compact Riemannian manifold , rather than a scalar function, and where the measure preserving map is a smooth one-parameter family of isometries acting on via the pull back. To prove this generalization, we note that a tensor field of type on a manifold may be viewed as a function on the fiber bundle, , of all tensors of type over that satisfies the additional property that this function is linear on each fiber. Equivalently, we may view as a function, , on the bundle, , of unit norm tensors of type that satisfies a corresponding linearity property. A Riemannian metric on naturally gives rise to a Riemannian metric (and, in particular, a volume element) on , and is compact provided that is compact. Since the isometry flow on naturally induces a volume preserving flow on , we may apply the von Neumann ergodic theorem to to obtain the orbit averaged function . Since will satisfy the appropriate linearity property on each fiber, we thereby obtain the desired orbit averaged tensor field . (see eq. (25)) that the limit
exists in the sense of . Furthermore, the limit in the sense of also exists of all -derivatives of the left side. Indeed, because is an isometry commuting with the derivative operator of the metric , we have
The expression on the right side converges in , as by the von Neumann ergodic theorem, meaning that
for all , where denotes the Sobolev space of order . By the Sobolev embedding theorem,
where the embedding is continuous with respect to the sup norm on the all derivatives in the space , i.e., for all . Thus, convergence of the limit (45) actually occurs in the sup norms on . Thus, in particular, .
Now pick an arbitrary , and define by
where the integral is over any smooth path connecting and . This integral manifestly does not depend upon the choice of , independently of the topology of . By what we have said above, the function is smooth, with a smooth limit
which is independent of the choice of . Furthermore, the convergence of and its derivatives to and its derivatives is uniform. Now, by inspection, is a solution to the differential equation
Furthermore, the limit
exists by the ergodic theorem, and vanishes by Lemma 1. Thus, the smooth, limiting quantity satisfies the desired differential equation (39).
We now define a new set of Gaussian null coordinates as follows. Define on to be a smooth solution to eq. (38), whose existence is guaranteed by Lemma 2. Extend to by eq. (35). It is not difficult to verify that is given explicitly by
is the map projecting any point in to the point on the cross section on the null generator through . Let denote the surface on . Then our desired Gaussian null coordinates are the Gaussian null coordinates associated with . The corresponding fields satisfy all of the properties derived above for and, in addition, satisfy the condition that is constant on .
Now let . We have previously shown that on , since this relation holds for any choice of Gaussian null coordinates. However, since our new coordinates have the property that is constant on , we clearly have that on . Furthermore, for our new coordinates, eq. (18) immediately yields on . Thus, we have proven that all of the relations in eq. (2) hold for .
on . Since , with tangent to , and since all quantities appearing in eq. (55) are Lie derived by , we may replace in this equation all Lie derivatives by . Hence, we obtain
Integration of this equation yields
where is a tensor at that is independent of . Integrating this equation (and absorbing constant factors into ), we obtain
However, since is a Riemannian isometry, each orthonormal frame component of at is uniformly bounded in by the Riemannian norm of , i.e., . Consequently, the limit of eq. (59) as immediately yields
from which it then immediately follows that
Thus, we have , and therefore on , as we desired to show.
Thus, we now have shown that the first equation in (2) holds for , and that the other equations hold for , for the tensor fields associated with the “tilde” Gaussian null coordinate system, and . In order to prove that eq. (2) holds for all , we proceed inductively. Let , and assume inductively that the first of equations (2) holds for all , and that the remaining equations hold for all . Our task is to prove that these statements then also hold when is replaced by . To show this, we apply the operator to the Einstein equation (see eq. (78)) and restrict to . Using the inductive hypothesis, one sees that on , thus establishes the second equation in (2) for . Next, we apply the operator to the Einstein equation (see eq. (81)), and restrict to . Using the inductive hypothesis, one sees that on , thus establishes the third equation in (2) for . Next, we apply the operator to the Einstein equation (see eq. (82)), and restrict to . Using the inductive hypothesis and the above results and , one sees that the tensor field satisfies a differential equation of the form
on . By the same argument as given above for , it follows that . This establishes the first equation in (2) for , and closes the induction loop.
Thus far, we have assumed only that the spacetime metric is smooth (). However, if we now assume that the spacetime is real analytic, and that is an analytic submanifold, then it can be shown that the vector field that we have defined above is, in fact, analytic. To see this, first note that if the cross section of is chosen to be analytic, then our Gaussian null coordinates are analytic, and, consequently, so is any quantity defined in terms of them, such as and . Above, was defined in terms of a certain special Gaussian normal coordinate system that was obtained from a geometrically special cross section. That cross section was obtained by a change (53) of the coordinate . Thus, to show that is analytic, we must show that this change of coordinates is analytic. By eq. (53), this will be the case provided that and are analytic. We prove this in Appendix C.
Since and are analytic, so is . It follows immediately from the fact that this quantity and all of its derivatives vanish at any point of that where defined, i.e., within the region where the Gaussian null coordinates are defined. This proves existence of a Killing field in a neighborhood of the horizon. We may then extend by analytic continuation. Now, analytic continuation need not, in general, give rise to a single-valued extension, so we cannot conclude that there exists a Killing field on the entire spacetime. However, by a theorem of Nomizu (see also ), if the underlying domain is simply connected, then analytic continuation does give rise to a single-valued extension. By the topological censorship theorem [10, 11], the domain of outer communication has this property. Consequently, there exists a unique, single valued extension of to the domain of outer communication, i.e., the exterior of the black hole (with respect to a given end of infinity). Thus, in the analytic case, we have proven the following theorem:
Let be an analytic, asymptotically flat -dimensional solution of the vacuum Einstein equations containing a black hole and possessing a Killing field with complete orbits which are timelike near infinity. Assume that a connected component, , of the event horizon of the black hole is analytic and is topologically , with compact and that (where is defined eq. (20) above). Then there exists a Killing field , defined in a region that covers and the entire domain of outer communication, such that is normal to the horizon and commutes with .
The assumption of analyticity in this theorem can be partially removed in the following manner, using an argument similar to that given in . Since , the arguments of show that the spacetime can be extended, if necessary, so that is a proper subset of a regular bifurcate null surface in some enlarged spacetime
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304835.96/warc/CC-MAIN-20220125130117-20220125160117-00293.warc.gz
|
CC-MAIN-2022-05
| 28,542 | 92 |
https://writemia.com/testbank/suppose-x-is-a-normally-distributed-random-variabl
|
math
|
01 Nov Suppose x is a normally distributed random variabl
Suppose x is a normally distributed random variable with muμequals=1212and sigmaσequals=22. Find each of the following probabilities.a. P(xgreater than or equals≥13.513.5) b. P(xless than or equals≤8.58.5) c. P(13.3813.38less than or equals≤xless than or equals≤17.5417.54) d. P(7.427.42less than or equals≤xless than or equals≤14.914.9)One method of purification of molten salt involves oxidation. An important aspect of the purification process is the rising velocity of oxygen bubbles in the molten salt. An experiment was conducted in which oxygen was inserted (at a designated sparging rate) into molten salt and photographic images of the bubbles taken. A random sample of 2525 images yielded the data on bubble velocity (measured in meters per second) shown in the accompanying table. Complete parts a and b below.LOADING… Click the icon to view the table of bubble velocity data.a. Use technology to find a 9595% confidence interval for the mean bubble rising velocity of the population. Interpret the result.The confidence interval is left parenthesis nothing comma nothing right parenthesis,.(Type integers or decimals rounded to three decimal places as needed.)Interpret the result. Select the correct choice below and fill in any answer boxes to complete your choice.(Round to three decimal places as needed. Use ascending order.)A.With nothing% confidence, the true mean bubble rising velocity is between nothing and nothing.B.With nothing% confidence, the true variance of the bubble rising velocity is between nothing and nothing.C.With nothing% confidence, the true mean bubble rising velocity is exactly nothing.b. The researchers discovered that the mean bubble rising velocity is muμequals=0.3410.341 when the sparging rate of oxygen is 3.363.36times×10 Superscript negative 610−6. Do you believe that the data in the table were generated at this sparging rate? Explain. Choose the correct answer below.A.Yes. The discovered mean velocity is close enough to the confidence interval calculated in part a that it is reasonable for these data to have been generated using this sparging rate.B.No. The discovered mean velocity is equal to one of the bounds of the confidence interval. Since the bounds are not included in the interval, it is not reasonable that these data could have been generated using this sparging rate.C.Yes. The discovered mean velocity lies near the center of the confidence interval calculated in part a. Thus, it is very likely that these data were generated using this sparging rate.D.No. The discovered mean velocity lies well outside the confidence interval calculated in part a. Thus, it is very unlikely that these data were generated using this sparging rate.Click to select your answer(s).
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487655418.58/warc/CC-MAIN-20210620024206-20210620054206-00113.warc.gz
|
CC-MAIN-2021-25
| 2,815 | 2 |
https://www.researchwithrutgers.com/en/projects/rui-spectral-theory-and-geometric-analysis-in-several-complex-var-2
|
math
|
Many physical and social phenomena can be modeled mathematically using partial differential operators. The Laplace operator is a differential operator that has long played an important role in mechanics, physics, and mathematics. The complex Laplace operator is a natural outgrowth of the classical Laplace operator in complex analysis of several variables, a branch of mathematics where algebra, analysis, and geometry intertwine. This research project investigates analytic and geometric properties of the complex Laplace operator, in particular, its spectrum. Spectral analysis is a major tool in scientific research, and spectral properties of the complex Laplace operator are known to be closely related to certain quantum phenomena in physics. The goal of this project is to understand how algebraic, analytic, and geometric structures of the underlying complex space interact with each other. The project combines ideas and methods from several branches of mathematics, and the techniques under development could potentially have applications in other areas of mathematics and physical sciences. This project involves undergraduate students in research activities and broadens participation of underrepresented groups in mathematics. The complex Neumann Laplace operator is a prototype of an elliptic operator with non-coercive boundary conditions. Since the work of Kohn and Hörmander in the 1960's, there have been extensive studies on regularity theory of the complex Neumann Laplace operator that led to important discoveries in both partial differential equations and several complex variables. The main thrust of this proposal is to study spectral theory of the complex Neumann Laplace operator, with emphasis on the interplay between the spectral behavior of the operator and the underlying geometric structures. Among the problems studied in this project are stability of the spectrum as the underlying structures deform and characterization of complex manifolds whose complex Laplace operator has discrete spectrum. Also investigated are regularity theory of the Cauchy-Riemann operator on complex manifolds, reproducing kernels, invariant metrics, and their applications to problems in complex algebraic geometry. This project supports research activities of undergraduate and graduate students, facilitates the development of new courses that attract students into mathematics, and fosters interdisciplinary research.
|Effective start/end date||6/1/15 → 5/31/18|
- National Science Foundation (National Science Foundation (NSF))
Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500035.14/warc/CC-MAIN-20230202165041-20230202195041-00476.warc.gz
|
CC-MAIN-2023-06
| 2,712 | 4 |
https://electronics.stackexchange.com/questions/246157/calculating-the-leaking-current-for-a-tantalum-capacitor
|
math
|
I've been looking at these tantalum caps for a coin cell operated BLE device, but I'm a little confused by specified leakage current. The datasheet lists the leakage current at <= 0.003CV uA. With a 470uA cap at 6.0V, my calculated leakage current is 0.003 * 0.000470 * 6.0, which is 0.00000846. The datasheet says that current is in uA.
Is the leakage current really 0.00000846uA? Or is the C (capacitive) value in the equation supposed to be in uF, not F? That would make more sense if the leakage current was 8.46uA, although that seems high for a "low leakage" cap.
And yes, I am well aware of the potential dangers of tantalums. Operating voltage is going to be in the 3.0V - 2.0V range. We have a few other (highly experienced) EEs working on this project, and I'm trying to get up the learning curve on some aspects of datasheet specs.
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662636717.74/warc/CC-MAIN-20220527050925-20220527080925-00761.warc.gz
|
CC-MAIN-2022-21
| 842 | 3 |
http://0674x7e3.ga/fede/permutations-and-combinations-homework-help-cop.php
|
math
|
Permutations and combinations homework help
Algebra I, Algebra II, Geometry: homework help by free math tutors,.Our homework help online can be right solution for your problem that we offer with best efficiency,.Simple counting problems allow one to list each possible way that an event can occur.Most tutoring sessions...REDDIT and the ALIEN Logo are registered trademarks of reddit inc.
Minitab Homework Help - Tutors On NetProfessional Help Writing Website, Do My Physics Homework High. see Permutations and Combinations.
Probability and Statistics Factorials and PermutationsBe sure your doc is accessible to those who will read your essay.
Clearly, our carpenter is still not getting it, but besides that.what do we know.One permutation is The Odyssey, The Iliad and the other permutation is The Iliad, The Odyssey.Blog features how Shodor is transforming STEM education Shodor works with middle school students at Purdue University Shodor returns to the Bridge Downeast.There are ways to order 3 books when we have 4 books to choose from.Calculate 9C 2:First calculate it using paper and pencil only,. and all the permutations had the little.
Permutations & Combinations? - Tutorhelpdesk.com Free
Homeworkhelp.com - The Best Place to Find Live HomeworkDefinition: A permutation is an ordered combination of a set of items.
Permutations & Combinations Worksheets
Both permutations and combinations are two numbers: the first is the total number of objects you have to pick.Then we permute (abbreviated by the letter P ) using r places.
Tutorvista.com - Online Tutoring, Homework Help in Math
Homework Help Online, Do My Homework - Homework1In probabilities where multiple events must occur together, you multiply the individual probabilities so you can find the above three probabilities and find the product.If you think this is a server error, please contact the webmaster.Probability is the number ways to get the desired outcome divided by the total possible outcomes.
Permutation and Combinations help!? - Weknowtheanswer
Mathopolis Question DatabaseHave some respect for people who take time to answer your question and follow the posting rules.View Homework Help - Permutations and Combinations Homework from MATH 2610 at UGA. Permutations and Combinations Homework. (PERMUTATIONS AND COMBINATIONS).
Finding Number of Ways using Permutation and CombinationStudypool is a marketplace that helps students get efficient academic help.
Give a real-world example of how permutations andFor example, ABC is a distinct permutation from ACB because they are ordered differently.Walls-of-text are almost impossible to edit with any effectiveness.
Get Help with Permutations from Experts - Tutorvista
Free Math Tutoring - Get Help from Online Math TutorHow many distinguishable permutations are possible using the letters of the.
Hi Please check the attached file for details, let me know if you have any question. James.Khan Academy is a nonprofit with the mission of providing a free,.If you entered the URL manually please check your spelling and try again.When we put an object in a space, we have fewer objects left to choose from for the next space.
Business Statistics Tutoring and Homework HelpWe could look at The Odyssey, put it back, and then look at The Odyssey again.Determine whether permutation or combination is involved and explain.You may need to add four spaces before or put backticks around math fragments.
Permutations and Combinations, Permutations and Combinations in GMAT,. free homework help forum.Combinations. Tags. Permutations (math. all of the permutations-- and permutations are when you think about all the.
Algebra II Lesson on Combinations & Permutations TutorialDetermine the range and mean for the following set of numbers: -10, 33, 46, 12,20, -8, homework help.To summarize, if we have n objects and we choose r of them, with repetition allowed, there are n r.
Rather than choosing a new object, you may be doing a double-take.
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948521188.19/warc/CC-MAIN-20171213030444-20171213050444-00101.warc.gz
|
CC-MAIN-2017-51
| 3,967 | 22 |
https://answeringexams.com/category/dock-and-harbor-engineering/
|
math
|
Assertion A : Depth and width required at the entrance to a harbour are more than those required in the channel. Reason R : The entrance to a harbour is usually more exposed to waves as compared to the harbour itself__________________?…
In basins subjected to strong winds and tide, the length of the berthing area should not be less than____________________?
A. the length of design vessel
B. the length of design vessel + 10% clearance between adjacent vesselsC. the…
As per Berlin’s formula, the length of wave in metres is given by_______________?
A. 1.3412B. 1.5612
where ‘t’ is the period in seconds for two successive waves to pass the same section…
In a two lane channel, bottom width of channel is given by_______________?
A. Manoeuvring lane + 2 x Bank clearance lane
B. 2 x Manoeuvring lane + 2 x Bank clearance laneC. 2 x Manoeuvring lane + 2 x Bank clearance lane + ship…
Select the incorrect statement ?
A. The progress of work in low level method of mound construction is very slow.B. Barge method of mound construction is economical.
C. In low level method of mound construction, the area of working…
If H is the height of the wave expected, then the height of the breakwater is generally taken as______________________?
A. 1.2 H to 1.25 H above the datum
B. 1.2 H to 1.25 H above the low water levelC. 1.2 H to 1.25 H above the…
By increasing the rise of lockgates_______________?
(i) the length of the lock gate will increase
(ii) transverse stress due to water pressure on the gate will increase
(iii) compressive force on the gate will increase
The significant wave height is defined is the average height of the_________________?
A. one – third highest waves
B. one – fourth highest waves
C. one – fifth highest waves
D. one – tenth highest waves…
When a wave strikes a vertical breakwater in deep water, it is reflected back and on meeting another advancing wave of similar amplitude merges and rises vertically in a wall of water. This phenomenon is called_________________?
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780055684.76/warc/CC-MAIN-20210917151054-20210917181054-00034.warc.gz
|
CC-MAIN-2021-39
| 2,025 | 26 |
http://www.fact-index.com/t/to/torus.html
|
math
|
In geometry, a torus is a solid of revolution generated by revolving a circle about an axis coplanar with the circle. The sphere is a special case of the torus obtained when the axis of rotation is a diameter of the circle. If the axis of rotation does not intersect the circle, the torus has a hole in the middle and resembles a ring doughnut, a hula hoop or an inflated tyre (U.S. tire). The other case, when the axis of rotation is a chord of the circle, produces a sort of squashed sphere resembling a round cushion. Torus was the Latin word for a cushion of this shape.
According to the broadest definition, the generator of a torus need not be a circle but could also be an ellipse or any other conic section.
picture of torus
with one homology class
In topology torus
means the product
of a number of circles, the surface of a doughnut shape being the product of two. (In proper mathematical usage, a solid as described above would be spoken of as generated from a disk, i.e., a filled-in circle.) An algebraic torus in the theory of algebraic groups is so-called not directly for topological reasons, but more on account of the analogy with the way actual tori play a role in the theory of compact Lie groups
(which begins from the idea of maximal torus
In nuclear physics a torus is a large fusion reactor which is very roughly the shape of an elliptical torus. Examples are JET in the UK, JT-60 in Japan, TFTR in the USA and the proposed ITER.
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337322.29/warc/CC-MAIN-20221002115028-20221002145028-00378.warc.gz
|
CC-MAIN-2022-40
| 1,453 | 9 |
http://www.physics.udel.edu/~watson/phys208/97f/exercises/hints/29-62P.html
|
math
|
The torque created will be out of the page (screen) tending to rotate the spool up the incline.
The additional forces acting on the spool are the force of gravity mg, acting downward from the center of mass, the normal force of the incline N, acting perpendicularly to the incline through the center of mass, and the force of friction Ffric, acting up the incline at the point of contact.
Draw a free body diagram!
Set up Newton's second law for acceleration down the incline and Newton's second law for rotation about the center of the spool. Find the least current (least dipole moment) that will be required for both the linear acceleration and the angular acceleration to be zero.
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948615810.90/warc/CC-MAIN-20171218102808-20171218124808-00731.warc.gz
|
CC-MAIN-2017-51
| 684 | 4 |
http://pmzilla.com/4-types-eac-formulas-when-use-what
|
math
|
4 Types of EAC formulas, when to use what
I see there are 4 types of EAC forecast calculation formulas available, but confused when to use each type.
EAC = BAC / CPI
EAC = ETC + AC
EAC = (BAC-EV) + AC
EAC = [BAC-EV / spi * cpi] + AC
Please need a brief justification / explanation.
Thank you all for the helpful info.
|
s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315321.52/warc/CC-MAIN-20190820092326-20190820114326-00456.warc.gz
|
CC-MAIN-2019-35
| 317 | 8 |
https://scripbox.com/mf/absolute-return-vs-cagr/
|
math
|
Returns from an investment can be estimated using both absolute returns and CAGR. On the one hand, absolute returns are a measure of the total return from an investment, irrespective of the time period. CAGR, on the other hand, is the return from an investment during a specific period. Both absolute returns and CAGR are used for determining the return from an investment. However, both use different ways to calculate the return. This article covers absolute return and CAGR in detail and elaborates on absolute return vs CAGR.
What Are Absolute Returns in a Mutual Fund?
Absolute returns in mutual funds refer to the return from a fund over a certain period of time. It is the total return from a mutual fund from the date of investment. Absolute returns are expressed as a percentage and show how much the investment has grown or depreciated in value.
Absolute returns are pure returns from the investment and don’t compare to any other benchmark. Also, absolute returns can be positive or negative. The fund managers of mutual funds seek a positive return by using multiple strategies like short selling or derivatives.
While calculating absolute returns, the tenure of the investment is the least important. Only actual investment and the current value of the investment are considered while estimating the absolute return.
The formula for Absolute return:
((Current value of the investment/ Actual investment) – 1) * 100.
Let’s understand absolute returns with an example. An investor invests INR 1,00,00 in a mutual fund. Over a certain period of time, the investment grows to INR 3,00,000. The absolute return from this investment can be calculated using the above formula.
Absolute return = (300000/100000 – 1) * 100
The absolute return of the above investment is 200%. The tenure of the investment is not considered while calculating the return from the investment. The 200% return could’ve been earned over a period of months, years or decades. Using absolute return alone, one cannot determine whether the investment is good or not as the tenure of the investment isn’t known.
Therefore, absolute returns only tell how much the investment depreciated or appreciated. It doesn’t tell how fast the investment grew or fell. Hence, absolute returns cannot be used for comparison of two different investments.
What is Compound Annual Growth Rate (CAGR) in a Mutual Fund?
CAGR (Compounded annual growth rate) is the rate of return from a mutual fund during a specific period of time, assuming the profits are reinvested. In other words, CAGR shows how much the investment has grown from the beginning to ending value over a period of time.
CAGR shows the rate at which the investment grows each year to reach the investment’s final value. It smoothes out the performance of a fund so that it can be easily understood and becomes comparable to other investments. One can use CAGR to compare two investments and determine which has performed better during a specific period of time.
The formula for Compounded Annual Growth Rate CAGR
CAGR = ((Ending value/ Beginning value) ^ (1/n)) – 1
Where, n is tenure of the investment
Let’s understand CAGR better with the help of an example. An investor invested INR 2,50,000 lump sum in a mutual fund. And the investment grew to INR 4,00,000 in 3 years. The CAGR of this investment can be calculated using the above formula.
CAGR = ((400000/250000) ^ (1/3)) – 1
CAGR = 16.96%
This means the investment grew 16.96% every year for three years for it to grow to INR 4,00,000. In other words, the average return from this mutual fund is 16.96%. The absolute return from this investment is 60%. But the CAGR is 16.96%.
CAGR enables investors to compare multiple investments and help them plan their financial future. Let’s say an investor has an opportunity to invest in stocks and bonds that have a CAGR of 18% and 15%, respectively. The investor will choose stocks as it has a higher CAGR when compared to bonds.
Moreover, CAGR can be used to estimate the average growth of an investment. Due to market volatility, the investment might grow by 10% one year and grow only by 2% the other year. CAGR helps smoothen out the returns and gives a better picture of an investment’s overall growth.
Calculating CAGR can be a time taking process. Hence one can use a CAGR calculator to estimate returns from a mutual fund investment. Scripbox’s CAGR calculator is a simple online tool that helps in calculating the CAGR of investment to analyze an investment opportunity.
All one has to do is enter the initial value, final value, and investment tenure. The calculator will estimate the CAGR within seconds.
Difference Between Absolute Return Vs CAGR
Investments are made to earn profits. There are different ways to represent returns from an investment. Absolute returns is a simple method that helps in determining the return from an investment, irrespective of the period or tenure of the investment. Absolute returns simply take the initial investment amount and the maturity amount. On the other hand, compounded annual growth rate takes into account the investment duration or tenure. Hence, it gives a more accurate and comparable earnings percentage.
The formula for Absolute return
((Current value of the investment/ Initial investment) – 1) * 100.
The formula to calculate CAGR
CAGR = ((Ending value/ Beginning value) ^ (1/n)) – 1
Where, n is tenure of the investment
Example for Absolute Return Vs CAGR
To understand the difference between absolute return and CAGR better, let’s take an example of Mr Krishna who invested INR 5,00,000 lump sum in a mutual fund in 2010. He withdrew the investment in 2020, and the value of the investment is 8,00,000.
The absolute return for Mr. Krishna is ((8,00,000/5,00,000) – 1) * 100
Absolute Return = 60%
While, the CAGR is ((8,00,000/5,00,000)^(1/10)) – 1
CAGR = 4.81%
While the above absolute return looks promising, the investment has actually grown only 4.81% every year for ten years.
Absolute returns tell how much an investment depreciated or appreciated. And, it doesn’t tell how fast the investment grew or fell. Therefore, absolute returns are not ideal for comparing two different investments.
On the other hand, CAGR can be used to determine an investment’s average growth. With markets being volatile, the returns are never the same over the years. For example, an investment might grow by 12% one year and grow only by 5% the other year. Therefore, CAGR addresses the volatility and smoothens out the returns. Hence, it gives a clear picture of an investment’s overall growth. Also, it is a good measure to compare different investments.
In short, if absolute return is the distance your investment has travelled, then CAGR is the rate at which your investment has travelled or grown.
CAGR Vs Absolute Return – Which is Better?
Both absolute returns and compounded annual growth rate are useful in determining the returns from an investment. However, the difference between the two lies in the aspect of time consideration. For investments with longer durations, the CAGR value is a better measure. CAGR determines an investment’s annual growth rate, whose value usually fluctuates over the investment tenure. While on the other hand, absolute returns consider only the purchase value and sale value of an investment to calculate returns.
For investments with a duration of less than a year, one can consider the absolute return. While, for investments with tenure greater than a year, CGAR gives a better picture. Also, with CAGR, one can compare two or more investments held for different periods. When it comes to tenure less than one year, CAGR may inflate or shrink the returns therefore, not giving the actual return.
Frequently Asked Questions
Annualized return is the measure of an investment’s performance during a specific period. In other words, annualized return shows how much your investment has grown from the beginning to ending value over a certain period of time. It is the same as CAGR.
To convert absolute returns to CAGR, one should take the nth root of (Current value of the investment/ Actual investment) and subtract 1 from it. In other words,
((Current value of the investment/ Actual investment)^(1/n)) – 1 will give the CAGR value.
CAGR considers the tenure of an investment and helps in determining the annual growth rate. On the other hand, absolute return considers only the investment value and the maturity value. Therefore, the absolute return cannot be used for comparison of different investments. Hence, CAGR is a good metric that helps investors compare the performance of different investments. It also gives a complete picture of the gains made from your investments. Furthermore, for investments with a tenure of more than one year, CAGR gives a better picture of the returns. Finally, CAGR determines the return from an investment over a specific period of time while taking into consideration the market volatility.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100146.5/warc/CC-MAIN-20231129204528-20231129234528-00009.warc.gz
|
CC-MAIN-2023-50
| 9,007 | 50 |
https://www.phrases.org.uk/bulletin_board/60/messages/689.html
|
math
|
In Reply to: The Whole Nine Yards posted by David FG on August 10, 2009 at 18:41:
: Maybe there is something deeply flawed in my maths (which is very possible) but by my reckoning 17.5 x 1.5 isn't 27.47.
You're right even a slide rule would have done better. 26.25 feet, but still less than a foot from 9 yards. Actually if you used the exact rule for l = PI (r/3) then l = (3.14159) * (17.5/3) = 18.3259 feet; the 1.5l = 27.4889 feet. You get my first length if you use PI to two decimals. Thought I had used l ~ r when I posted, but you pick em:
PI ~ 3 l = 17.5 feet and 1.5l = 26.25 feet
PI ~ 3.14 l = 18.3167 feet and 1.5l = 27.47 feet
PI ~ 3.14159 l = 18.3259 feet and 1.5l = 27.4889126 or 27.49 feet
BUT BACK TO THE original question: is this close enough to the Whole Nine Yards? May not have been the first use but sure kept it going in the 60's.
|
s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347388012.14/warc/CC-MAIN-20200525063708-20200525093708-00037.warc.gz
|
CC-MAIN-2020-24
| 854 | 7 |
https://www.teacherspayteachers.com/Product/Monopoly-and-Line-of-Best-Fit-470354
|
math
|
A good data set to explore linear regression is analyzing the relationship between the number of spaces a property is from GO and the cost of that property (on a classic Monopoly board). This is actually a TI-Nspire activity which can be found on their website.
I don’t like their set up for three reasons. First, their worksheet gives students all the data in a pre-filled table. I think students are more invested in a problem if they do the “work” themselves to gather the data, even if it is just counting spaces from go.
Second, it ignores the utilities and the railroads which are good outliers.
Finally, I think this could be done by hand instead of on the calculator. Students could always compare their manual calculations with the calculator answer at the end if using the calculators is one of the objectives of this activity.
So I created this handout which is, as always, on a single page.
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221212910.25/warc/CC-MAIN-20180817202237-20180817222237-00148.warc.gz
|
CC-MAIN-2018-34
| 908 | 5 |
http://www.glassmessages.com/index.php/topic,4015.msg31615.html
|
math
|
Hi again Everyone!
Does anyone know if Barovier also made mermaid figures? I have some Barbini and Seguso mermaids, but this one is not like the others. Specially the tail part looks interesting with some stretched bubbles in it, and the shell itself is iridato? or mirrored? It doesnt really reflect colors as an irridescent piece, but is very reflective.
Any ideas would be great. Thanks!
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917119995.14/warc/CC-MAIN-20170423031159-00419-ip-10-145-167-34.ec2.internal.warc.gz
|
CC-MAIN-2017-17
| 390 | 3 |
https://digitalcommons.mtu.edu/michigantech-p/5321/
|
math
|
On the thermodynamic theory of fluid interfaces: Infinite intervals, equilibrium solutions, and minimizers
We outline a thermodynamic theory for one-dimensional fluid interfaces and compare our findings with the classical results of the variational van der Waals-Cahn-Hilliard approach. After establishing necessary and sufficient conditions for their equivalence, we list all types of possible solutions giving the structure of the density profile in an infinite interval. Then we examine the stability of these solutions, strictly within a variational thermodynamic context and prove that transitions are minimizers, but reversals and oscillations are not. To the best of our knowledge, this is the first proof available for this old problem. It substantiates previous intuitive statements and makes rigorous certain mathematical assertions existing in the physical literature. © 1986.
Journal of Colloid And Interface Science
On the thermodynamic theory of fluid interfaces: Infinite intervals, equilibrium solutions, and minimizers.
Journal of Colloid And Interface Science,
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/5321
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077843.17/warc/CC-MAIN-20210414155517-20210414185517-00047.warc.gz
|
CC-MAIN-2021-17
| 1,146 | 6 |
https://www.jstor.org/stable/2243747
|
math
|
You are not currently logged in.
Access JSTOR through your library or other institution:
If You Use a Screen ReaderThis content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Further Monotonicity Properties for Specialized Renewal Processes
The Annals of Probability
Vol. 9, No. 5 (Oct., 1981), pp. 891-895
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2243747
Page Count: 5
You can always find the topics here!Topics: Mathematical monotonicity, Mathematical functions, Probabilities, Distributivity, Atoms, Laplace transformation
Were these topics helpful?See something inaccurate? Let us know!
Select the topics that are inaccurate.
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Preview not available
Define Z(t) to be the forward recurrence time at t for a renewal process with interarrival time distribution, F, which is assumed to be IMRL (increasing mean residual life). It is shown that Eφ(Z(t)) is increasing in t ≥ 0 for all increasing convex φ. An example demonstrates that Z(t) is not necessarily stochastically increasing nor is the renewal function necessarily concave. Both of these properties are known to hold for F DFR (decreasing failure rate).
The Annals of Probability © 1981 Institute of Mathematical Statistics
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189802.18/warc/CC-MAIN-20170322212949-00200-ip-10-233-31-227.ec2.internal.warc.gz
|
CC-MAIN-2017-13
| 1,574 | 16 |
https://demerarawaves.com/2020/11/03/farmer-who-allegedly-stole-police-force-handgun-on-bail/
|
math
|
Last Updated on Tuesday, 3 November 2020, 10:18 by Denis Chabrol
A farmer of Springlands, Corriverton, Berbice, who allegedly stole a handgun and ammunition belonging to the Guyana Police Force, has been granted bail.
He is 26-year old Symeon “Burn Hand” Lynch of Hazel Street, Springlands.
Police say that on October 28 at Number 76 Village, Springlands, Corriverton, Berbice, he robbed Shibike Calder and Anette Mckenzie of: one Samsung A20 cellphone valued GYD$55,000 Guyana Currency and GYD$45,000 cash a total value of GYD$100,000 and one .38 Taurus Revolver with five .38 rounds of ammunition property of the force.
Mr. Lynch is also accused of stealing one Samsung J3 Prime valued GYD$38,000, one Haversack valued GYD$10,000 and GYD$30,000 cash, a total value of GYD$40,000 on October 28 #78 Springlands, Corriverton, Berbice. Those items were the property of Pamela Trotman and Anett Mckenzie.
He was not required to plea and was granted GYD$70,000 bail each and GYD$70,000 on the two counts of simple larceny by Magistrate Alex Moore. The accused returns to court on January 7,2021.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224652116.60/warc/CC-MAIN-20230605121635-20230605151635-00726.warc.gz
|
CC-MAIN-2023-23
| 1,095 | 6 |
https://dr.lib.iastate.edu/handle/20.500.12876/qzoDMpmw
|
math
|
Polychromatic X-ray CT Image Reconstruction and Mass-Attenuation Spectrum Estimation
Is Version Of
Electrical and Computer Engineering
We develop a method for sparse image reconstruction from polychromatic computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident-energy spectrum are unknown. We obtain a parsimonious measurement-model parameterization by changing the integral variable from photon energy to mass attenuation, which allows us to combine the variations brought by the unknown incident spectrum and mass attenuation into a single unknown mass-attenuation spectrum function; the resulting measurement equation has the Laplace integral form. The mass-attenuation spectrum is then expanded into first order B-spline basis functions. We derive a block coordinate-descent algorithm for constrained minimization of a penalized negative log-likelihood (NLL) cost function, where penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density map. The image sparsity is imposed using total-variation (TV) and ℓ1 norms, applied to the density-map image and its discrete wavelet transform (DWT) coefficients, respectively. This algorithm alternates between Nesterov's proximal-gradient (NPG) and limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) steps for updating the image and mass-attenuation spectrum parameters. To accelerate convergence of the density-map NPG step, we apply a step-size selection scheme that accounts for varying local Lipschitz constant of the NLL. We consider lognormal and Poisson noise models and establish conditions for biconvexity of the corresponding NLLs. We also prove the Kurdyka-Łojasiewicz property of the objective function, which is important for establishing local convergence of the algorithm. Numerical experiments with simulated and real X-ray CT data demonstrate the performance of the proposed scheme.
This is a pre-print of the article Gu, Renliang, and Aleksandar Dogandžić. "Polychromatic X-ray CT image reconstruction and mass-attenuation spectrum estimation." arXiv preprint arXiv:1509.02193 (2015). DOI: 10.48550/arXiv.1509.02193. Copyright 2015 The Authors. Posted with permission.
|
s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296818293.64/warc/CC-MAIN-20240422113340-20240422143340-00373.warc.gz
|
CC-MAIN-2024-18
| 2,282 | 5 |
https://1hotelsouthbeachforsale.com/qa/question-what-does-an-r2-value-of-0-1-mean.html
|
math
|
- What does an r2 value of 0.5 mean?
- What is a good R value for correlation?
- What does an R squared value of 0.4 mean?
- What is a good R squared value?
- What does an R 2 value mean?
- What is a good R value in statistics?
- Is 0.6 A strong correlation?
- How do you know if a correlation is significant?
- What is a good r2 value for regression?
- How do you tell if a regression model is a good fit?
- How do you calculate r2 value?
- What does a low R squared value mean?
- What does an R squared value of 0.3 mean?
- What does an R squared value of 0.2 mean?
- What does an R squared value of 0.6 mean?
- How do you interpret an R?
- Is a low R Squared good?
- Can R Squared be above 1?
What does an r2 value of 0.5 mean?
Key properties of R-squared Finally, a value of 0.5 means that half of the variance in the outcome variable is explained by the model.
Sometimes the R² is presented as a percentage (e.g., 50%)..
What is a good R value for correlation?
The relationship between two variables is generally considered strong when their r value is larger than 0.7. The correlation r measures the strength of the linear relationship between two quantitative variables.
What does an R squared value of 0.4 mean?
R-squared = Explained variation / Total variation. R-squared is always between 0 and 100%: 0% indicates that the model explains none of the variability of the response data around its mean. 100% indicates that the model explains all the variability of the response data around its mean.
What is a good R squared value?
Any study that attempts to predict human behavior will tend to have R-squared values less than 50%. However, if you analyze a physical process and have very good measurements, you might expect R-squared values over 90%.
What does an R 2 value mean?
R-squared (R2) is a statistical measure that represents the proportion of the variance for a dependent variable that’s explained by an independent variable or variables in a regression model. … It may also be known as the coefficient of determination.
What is a good R value in statistics?
For a natural/social/economics science student, a correlation coefficient higher than 0.6 is enough. Correlation coefficient values below 0.3 are considered to be weak; 0.3-0.7 are moderate; >0.7 are strong. You also have to compute the statistical significance of the correlation.
Is 0.6 A strong correlation?
Correlation Coefficient = 0.8: A fairly strong positive relationship. Correlation Coefficient = 0.6: A moderate positive relationship. … Correlation Coefficient = -0.8: A fairly strong negative relationship. Correlation Coefficient = -0.6: A moderate negative relationship.
How do you know if a correlation is significant?
To determine whether the correlation between variables is significant, compare the p-value to your significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. An α of 0.05 indicates that the risk of concluding that a correlation exists—when, actually, no correlation exists—is 5%.
What is a good r2 value for regression?
25 values indicate medium, . 26 or above and above values indicate high effect size. In this respect, your models are low and medium effect sizes. However, when you used regression analysis always higher r-square is better to explain changes in your outcome variable.
How do you tell if a regression model is a good fit?
The best fit line is the one that minimises sum of squared differences between actual and estimated results. Taking average of minimum sum of squared difference is known as Mean Squared Error (MSE). Smaller the value, better the regression model.
How do you calculate r2 value?
The R-squared formula is calculated by dividing the sum of the first errors by the sum of the second errors and subtracting the derivation from 1.
What does a low R squared value mean?
A low R-squared value indicates that your independent variable is not explaining much in the variation of your dependent variable – regardless of the variable significance, this is letting you know that the identified independent variable, even though significant, is not accounting for much of the mean of your …
What does an R squared value of 0.3 mean?
– if R-squared value < 0.3 this value is generally considered a None or Very weak effect size, ... - if R-squared value 0.5 < r < 0.7 this value is generally considered a Moderate effect size, - if R-squared value r > 0.7 this value is generally considered strong effect size, Ref: Source: Moore, D. S., Notz, W.
What does an R squared value of 0.2 mean?
R^2 of 0.2 is actually quite high for real-world data. It means that a full 20% of the variation of one variable is completely explained by the other. It’s a big deal to be able to account for a fifth of what you’re examining. GeneralMayhem on [–] R-squared isn’t what makes it significant.
What does an R squared value of 0.6 mean?
An R-squared of approximately 0.6 might be a tremendous amount of explained variation, or an unusually low amount of explained variation, depending upon the variables used as predictors (IVs) and the outcome variable (DV).
How do you interpret an R?
To interpret its value, see which of the following values your correlation r is closest to:Exactly –1. A perfect downhill (negative) linear relationship.–0.70. A strong downhill (negative) linear relationship.–0.50. A moderate downhill (negative) relationship.–0.30. … No linear relationship.+0.30. … +0.50. … +0.70.More items…
Is a low R Squared good?
Regression models with low R-squared values can be perfectly good models for several reasons. … Fortunately, if you have a low R-squared value but the independent variables are statistically significant, you can still draw important conclusions about the relationships between the variables.
Can R Squared be above 1?
some of the measured items and dependent constructs have got R-squared value of more than one 1. As I know R-squared value indicate the percentage of variations in the measured item or dependent construct explained by the structural model, it must be between 0 to 1.
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153899.14/warc/CC-MAIN-20210729234313-20210730024313-00182.warc.gz
|
CC-MAIN-2021-31
| 6,127 | 55 |
https://www.coursehero.com/file/p2h3cu/In-the-first-experiment-reaction-for-the-black-prime-was-quicker-for-the-gun/
|
math
|
PRACTICE QUESTION: WEEK 38Chapter 314.On a standard measure of hearing ability, the mean is 300 and the standard deviation is 20. Give the Zscores for persons who score (a) 340, (b) 310, and (c) 260. Give the raw scores for persons whose Zscores on this test are (d) 2.4, (e) 1.5, (f) 0, and (g) –4.5.
15.A person scores 81 on a test of verbal ability and 6.4 on a test of quantitative ability. For the verbal ability test, the mean for people in general is 50 and the standard deviation is 20. For the quantitative ability test, the mean for people in general is 0 and the standard deviation is 5. Which is this person’s stronger ability: verbal or quantitative? Explain your answer to a person who has never had a course in statistics.
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057158.19/warc/CC-MAIN-20210921041059-20210921071059-00090.warc.gz
|
CC-MAIN-2021-39
| 741 | 2 |
http://cogsci.stackexchange.com/questions/tagged/perception+devices
|
math
|
Cognitive Sciences Meta
to customize your list.
more stack exchange communities
Start here for a quick overview of the site
Detailed answers to any questions you might have
Discuss the workings and policies of this site
Is Apple's iPhone Retina Display really accurate to human eye resolution?
Apple based their Retina Display on the following claim, as cited by Wikipedia: The display has a contrast ratio of 800:1. The screen is marketed by Apple as the "Retina Display", based on the ...
Mar 2 '12 at 18:53
newest perception devices questions feed
Hot Network Questions
Did Obama say the USA is "no longer a Christian Nation"?
Draw the South Korean flag
Is there anything preventing the NSA from becoming a root CA?
In the films, why does Sauron choose Azog instead of the ringwraiths to lead his legions?
what does the '-1' superscript mean in units?
bivariate normal density function
Word/phrase for "the one that brings bad luck" (e.g. to a group)
What is it called in linguistics when you change a word from one part of speech to another?
Why is it called ISO "speed"?
Can you cancel out a term if equal to zero?
Can you jump in combat?
How can native English speakers read an unknown word correctly?
Short Deadfish Numbers
Contacted by a Company's Client to do a project - should I inform my company?
I'm disagreeing with my chemistry teacher over this enthalpy of formation problem, so can someone tell me if I am right or why I am wrong?
Recommended books for undergraduate electrodynamics
Fixing instruments via a swift punch
Replace a string containing newline characters
Why didn't Walter White consume his own product?
Is there a fastest way to shutdown the system?
How to solve this equation? quadratic equation?
Can a satellite orbit Earth so that it always has the Moon in line of sight?
Do hard drives really have open cases now?
Cola Machine #1
more hot questions
Life / Arts
Culture / Recreation
TeX - LaTeX
Unix & Linux
Ask Different (Apple)
Geographic Information Systems
Science Fiction & Fantasy
Seasoned Advice (cooking)
Personal Finance & Money
English Language & Usage
Mi Yodeya (Judaism)
Cross Validated (stats)
Theoretical Computer Science
Meta Stack Exchange
Stack Overflow Careers
site design / logo © 2014 stack exchange inc; user contributions licensed under
cc by-sa 3.0
|
s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507446231.28/warc/CC-MAIN-20141017005726-00360-ip-10-16-133-185.ec2.internal.warc.gz
|
CC-MAIN-2014-42
| 2,305 | 53 |
https://www.thestudentroom.co.uk/showthread.php?t=4538342
|
math
|
I've been stuck on this for a while now (as you can see from my crossing out) would be great if someone could help me
enthalpy change question Watch
- Thread Starter
Last edited by noor.m; 01-02-2017 at 11:06.
- 01-02-2017 11:05
- Official TSR Representative
- 03-02-2017 12:23
Sorry you've not had any responses about this. Are you sure you've posted in the right place? Here's a link to our subject forum which should help get you more responses if you post there.
Just quoting in Amusing Elk so she can move the thread if neededSpoiler:Show(Original post by Amusing Elk)
- 03-02-2017 22:50
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934804125.49/warc/CC-MAIN-20171118002717-20171118022717-00237.warc.gz
|
CC-MAIN-2017-47
| 592 | 10 |
https://besttutorshelp.com/solution-9433/
|
math
|
The Dewitt Corporation has determined the following discrete probability distributions for net cash flows generated by a contemplated project:
a. Assume that probability distributions of cash flows for future periods are independent. Also, assume that the risk-free rate is 7 percent. If the proposal will require an initial outlay of 55,000, determine the mean net present value.
Save your time - order a paper!
Get your paper written from scratch within the tight deadline. Our service is a reliable solution to all your troubles. Place an order on any task and we will take care of it. You wonât have to worry about the quality and deadlinesOrder Paper Now
b. Determine the standard deviation about the mean.
c. If the total distribution is approximately normal and assumed continuous, what is the probability of the net present value being zero or less?
d. What is the probability that the net present value will be greater than zero?
e. What is the probability that the profitability index will be 1.00 or less?
f. What is the probability that the profitability index will be greater than 2.00?
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949958.54/warc/CC-MAIN-20230401094611-20230401124611-00650.warc.gz
|
CC-MAIN-2023-14
| 1,100 | 9 |
https://www.simszoo.de/forum/thread/367-weckermaus-egyptian-stuff/?postID=6561
|
math
|
Nice forum here but why are most stuff not on this site, there a lot of other websites that refer to this site
for the downloads but there is nothing on this site, the best stuff I found out is from
some awesome person called 'Weckermauss', made a lot of cool Egyptian stuff (like the rugs for example)
but also nowhere to be found on this site, I read something about a crash on the main page...
So does this mean a lot of stuff went gone on this site?
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585380.70/warc/CC-MAIN-20211021005314-20211021035314-00331.warc.gz
|
CC-MAIN-2021-43
| 453 | 5 |
http://www.ask.com/web?qsrc=6&o=102140&oo=102140&l=dir&gc=1&q=A+Location+the+Place+Where+Two+Lines+Cross+or+Intersect
|
math
|
In Euclidean geometry, the intersection of a line and a line can be the empty set,
a point, or a line. Distinguishing these cases and finding the intersection point
have use, for example, in compute...
The place where two lines cross or intersect? The place where two lines ... What
are two lines that are not coplanar and do not intersect? Parallel lines. 2 people ...
What is the intersection of two rays with a common endpoint? ... A Location the
Place Where Two Lines Cross or Intersect · The Total Area of All the Surfaces of ...
An intersection is a single point where two lines meet or cross each other. In the
figure above we would say that "point K is the intersection of line segments PQ ...
In figure, there are two lines AB and CD which intersect each other at O ... Step 1:
Intersecting lines cross at a common point. Step 2: Figure 2 has two lines, ...
Jun 27, 2006 ... Point – an exact location in space represented by a dot. Line segment ...
Intersecting lines – lines that meet or cross at one point. Parallel lines .... Place
two dots three to five inches apart in the center of the paper. Lightly label ...
Here we will cover a method for finding the point of intersection for two linear ...
That is, we will find the (x, y) coordinate pair for the point were two lines cross.
Oct 5, 1997 ... Could you help me prove that parallel lines meet at infinity or that infinity begins
... In this context, there is a single "infinity" location where all lines meet. ... two
non-parellel lines do not intersect at infinity but intersect only at the ...
Dec 31, 2013 ... What's the path from home to office that best skirts identified zones of location
based spam? .... A LineString may cross itself (i.e. be complex and not simple ).
.... For example, two LineStrings may intersect along a line and at a point. .....
equal to the other at all points to specified decimal place precision.
Jun 27, 2012 ... One of the main purposes of maps is to identify the location of places or ... This is
a grid which is based on two sets of lines which intersect each other at ... of the
Prime Meridian it is important to place an "E" or "W" after the degrees. ... In other
words, two lines of latitude will never inter...
|
s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396222.11/warc/CC-MAIN-20160624154956-00004-ip-10-164-35-72.ec2.internal.warc.gz
|
CC-MAIN-2016-26
| 2,225 | 27 |
http://dunawi.blogspot.com/2011/10/floret-pentagonal-quilt.html
|
math
|
Here's another one:
This one's a Floret Pentagonal tiling. It's the dual of the snub hex tiling I posted yesterday. A dual is basically when you convert the vertices of a tiling into faces and vice versa. One of the nifty properties of the snub hex tiling (and all the other uniform tilings including the rhombi tri hex) is that every vertex is the same as all the others. In the case of the snub hex every corner is made up of a hexagon and 4 triangles. This means that a nifty property of its dual is that every face is the same as all the others (i.e. a one patch quilt; it would be a good pattern for a charm quilt).
Each tiling would also be a good quilting pattern (by this I mean the stitching on top) for the piecing pattern of its dual.
I think it would be cool to make these projects in pairs (gifts for couples, perhaps) where one quilt's quilting is the other quilt's piecing.
Is your story F**CKING great?
2 years ago
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866276.61/warc/CC-MAIN-20180524112244-20180524132244-00201.warc.gz
|
CC-MAIN-2018-22
| 930 | 6 |
https://math.arizona.edu/events/9333
|
math
|
Analysis of the Discrete Painlevé Equations and Their Degenerations.
The discrete Painlevé equations have been of great interest in recent years, due to their applications in many combinatorial and physical settings. In the 1990's, Ramani, Grammaticos, and Hietarinta showed that autonomous forms of these equations were instances of QRT mappings. A decade later, Sakai studied these equations from a different perspective, using rational surfaces. We will examine concrete examples using both viewpoints, and discuss how certain techniques could translate to degenerations of the discrete Painlevé equations.
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829568.86/warc/CC-MAIN-20181218184418-20181218210418-00467.warc.gz
|
CC-MAIN-2018-51
| 612 | 2 |
https://www.jiskha.com/questions/1149041/there-are-no-examples-of-this-type-of-problem-in-my-book-so-if-you-could-help-walk
|
math
|
there are no examples of this type of problem in my book so if you could help walk me through it - that would be extremely helpful. thanks ahead of time.
Find the extreme values of the function on the interval and where they occur.
4) F(x)=³√(x); -3</=x</=64
A. Maximum at (64, 4), and minimum at (-3, ³√-3)
B. Maximum at (-64, 4), and minimum at (0,0)
C. Maximum at (0,0), and minimum at (64,4)
D. Maximum at (64,4), and minimum at (-64,-4).
Find an equation for the hyperbola described. Foci at (-4,0) and (4,0); asymptote the line y= -x. Please explain the steps of this problem. My book does not get an example of this type of problem and I'm not having any luck
i'm reading a book called "1984" i have an assignment that got to do with propaganda in 1984. i have to find 4 propaganda examples so far i got 2/4 examples i still need 2 more examples i don't know what other propaganda there are
Book 1 255 book 2 235 book 3 178 book 4 299 book 5 150 book 6 67 book 7 82 book 8 267 book 9 188 book 10 142 book 11 281 book 12 138 book 13 326 book 14 264 book 15 103 ^ number of pages What would be the best graph or display to
How much energy is evolved when 2.65 mg of Cl(g) atoms adds electrons to give Cl1-(g) ions. No clue how to go about solving this problem as it doesn't really relate to any of my examples in my text book or notes. All I can find
from our text book "Bittenger, M. L. & Beecher, J. A. (2007). Introductory and intermediate algebra (3rd ed.). Boston: Pearson-Addison Wesley" can someone please help me with "Review examples 2, 3, and 4 in section 8.4 of the
We has to read a book called alas babylon and we has to write an essay about how did fort repose adapt to the changes after the day. and I ran into two examples in the book that i didn't quite comprehend It says people
Q:Describe, with specific examples, what kind of art the Hiberno-Saxons are best known for. Include a basic progression of this art-type’s development ? A: The kind of art that the Hiberno-Saxons are best known for is their
I noticed someone else is having problems with the same problem below that I am. Since I have a Matt in my class he is having the same problem I am and could use some help. Fib. numbers are not in my math book and I cannot find
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160620.68/warc/CC-MAIN-20180924165426-20180924185826-00051.warc.gz
|
CC-MAIN-2018-39
| 2,260 | 15 |
https://hammadalitv.com/mth643-midterm-past-papers/
|
math
|
Most students need mth643 solved midterm papers. We will provide virtual university mth643 past papers which cover important topics and MCQs in this midterm exam.
Also, we recommend that you practice the maths questions given in the handout and solve them on MathType especially the maths subject papers should be practiced within a given time frame.
How to Download VU Past Papers
Now students can check below midterm past papers you can click to download easily according to your desired file.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100287.49/warc/CC-MAIN-20231201120231-20231201150231-00583.warc.gz
|
CC-MAIN-2023-50
| 495 | 4 |
https://ru-facts.com/how-do-you-calculate-monthly-amortization-in-the-philippines/
|
math
|
How do you calculate monthly amortization in the Philippines?
How to Calculate Monthly Payment on a Loan?
- a: Loan amount (PHP 100,000)
- r: Annual interest rate divided by 12 monthly payments per year (0.10 ÷ 12 = 0.0083)
- n: Total number of monthly payments (24)
How do you calculate amortization on a balance sheet?
The company should subtract the residual value from the recorded cost, and then divide that difference by the useful life of the asset. Each year, that value will be netted from the recorded cost on the balance sheet in an account called “accumulated amortization,” reducing the value of the asset each year.
How do I use Ipmt in Excel?
The formula to be used will be =IPMT( 5%/12, 1, 60, 50000). In the example above: As the payments are made monthly, it was necessary to convert the annual interest rate of 5% into a monthly rate (=5%/12), and the number of periods from years to months (=5*12).
How do I create a loan amortization table?
Creating an amortization table is a 3 step process:
- Use the =PMT function to calculate the monthly payment.
- Create the first two lines of your table using formulas with the correct relative and absolute references.
- Use the Fill Down feature of Excel to create the rest of the table.
How do banks compute monthly amortization?
Divide the interest rate (expressed as a decimal) by the number of repayments you’ll make throughout the loan term. For example, if your loan term is two years and you’ll make monthly payments, divide the interest rate by 24. Multiply the result by the balance of the loan.
How do you calculate principal amortization?
The amount of principal due in a given month is the total monthly payment (a flat amount) minus the interest payment for that month. For the next month, the outstanding loan balance is calculated as the previous month’s outstanding balance minus the most recent principal payment.
How do I calculate an amortization schedule?
Amortized payments are calculated by dividing the principal — the balance of the amount loaned after down payment — by the number of months allotted for repayment. Next, interest is added. Interest is calculated at the current rate according to the length of the loan, usually 15, 20, or 30 years.
How do I create a custom amortization schedule?
Method 1 of 2: Creating an Amortization Schedule Manually Open a new spreadsheet in Microsoft Excel. Create labels in column A. Create labels for your data in the first column to keep things organized. Enter the information pertaining to your loan in column B. Calculate your payment in cell B4. Create column headers in row 7. Populate the Period column. Fill out the other entries in cells B8 through H8.
How is an amortization schedule calculated?
Calculations in an Amortization Schedule. The Interest portion of the payment is calculated as the rate ( r) times the previous balance, and is usually rounded to the nearest cent. The Principal portion of the payment is calculated as Amount – Interest. The new Balance is calculated by subtracting the Principal from the previous balance.
How to calculate investment amortization schedules?
How to Calculate Investment Amortization Schedules Before the Loan Starts. To set up an amortization schedule, create a chart with columns for the period, the payment, the payment interest portion, the payment principal portion and the The First Payment. In the first period you can begin to calculate the composition of the payment and the remaining principal. Changes in Principal. Cost Evaluation.
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337524.47/warc/CC-MAIN-20221004184523-20221004214523-00111.warc.gz
|
CC-MAIN-2022-40
| 3,545 | 26 |
https://corescholar.libraries.wright.edu/mme/423/
|
math
|
Molecular dynamics simulations are performed to investigate the plastic response of a model glass to a local shear transformation in a quiescent system. The deformation of the material is induced by a spherical inclusion that is gradually strained into an ellipsoid of the same volume and then reverted back into the sphere. We show that the number of cage-breaking events increases with increasing strain amplitude of the shear transformation. The results of numerical simulations indicate that the density of cage jumps is larger in the cases of weak damping or slow shear transformation. Remarkably, we also found that, for a given strain amplitude, the peak value of the density profiles is a function of the ratio of the damping coefficient and the time scale of the shear transformation.
Priezjev, N. V.
(2015). The Effect of a Reversible Shear Transformation on Plastic Deformation of an Amorphous Solid. Journal of Physics: Condensed Matter, 27, 435002.
Additional FilesNVP_APS_15.pdf (1276 kB)
|
s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947473598.4/warc/CC-MAIN-20240221234056-20240222024056-00242.warc.gz
|
CC-MAIN-2024-10
| 1,002 | 4 |
https://www.math.harvard.edu/event/cmsa-condensed-matter-math-seminar-topological-qauntum-field-theory-in-31d-and-a-potential-origin-of-dark-matter-2-2-3-2-3-2-2-2-2-2-3-2-2-2-2-2-2-3-2-2-2-2-2-2-3-2-3-2-2-2-2-2-2-3-3/
|
math
|
CMSA Quantum Matter in Mathematics and Physics: Construction of Lattice Chiral Gauge Theory
Herbert Neuberger - Rutgers University
The continuum formal path integral over Euclidean fermions in the background of a Euclidean gauge field is replaced by the quantum mechanics of an auxiliary system of non-self-interacting fermions. No-go "theorems" are avoided.
The main features of chiral fermions arrived at by formal continuum arguments are preserved on the lattice.
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710771.39/warc/CC-MAIN-20221130192708-20221130222708-00317.warc.gz
|
CC-MAIN-2022-49
| 466 | 4 |
https://www.physicsforums.com/threads/piston-question-have-solution-dont-understand-rationale.780193/
|
math
|
1. The problem statement, all variables and given/known data The drawing below shows a hydraulic chamber in which a spring (spring constant = 1580 N/m) is attached to the input piston (A1 = 15.3 cm2), and a rock of mass 41.9 kg rests on the output plunger (A2 = 66.6 cm2). The piston and plunger are nearly at the same height, and each has a negligible mass. By how much is the spring compressed from its unstrained position? 2. Relevant equations F1/A1=F2/A2 F=kx 3. The attempt at a solution F1/15.3 cm^2= 410.62 N/66.6 cm^2 F1=94.33 N 94.33 N= 1580 N/m 0.0597 m I'm pretty sure this is the correct solution, I just don't understand why I should use F=kx rather than F=kx^2.
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589172.41/warc/CC-MAIN-20180716021858-20180716041858-00631.warc.gz
|
CC-MAIN-2018-30
| 676 | 1 |
http://www.faqs.org/patents/app/20120227500
|
math
|
Patent application title: Apparatus and Method for Determining Formation Anisotropy
Jennifer Market (Rosehill, TX, US)
Paul F. Rodney (Spring, TX, US)
HALLIBURTON ENERGY SERVICES, INC.
IPC8 Class: AG01N2904FI
Class name: Measuring or testing system having scanning means by reflected wave having separate sonic transmitter and receiver
Publication date: 2012-09-13
Patent application number: 20120227500
A method of generating an axial shear wave in a formation surrounding a
wellbore comprising urging a clamp pad into contact with a wall of the
wellbore, and applying an axial force to the clamp pad to impart a shear
force into the wall of the wellbore to generate a shear wave in the
1. A method for determining at least one characteristic of an anisotropic
earth formation, comprising: transmitting dipole acoustic energy into the
earth formation at a first location in a wellbore where the acoustic
energy propagates as a fast polarized shear wave and a slow polarized
shear wave in a plane of the formation orthogonal to a first longitudinal
axis of the wellbore at the first location; receiving at the first
location composite waveforms comprising components of both a fast
polarized shear wave and a slow polarized shear wave from the plane of
the formation orthogonal to a first longitudinal axis of the wellbore at
the first location; transmitting dipole acoustic energy into the earth
formation at a second location in a wellbore where the second location is
axially displaced from the first location and a second longitudinal axis
of the wellbore at the second location is substantially orthogonal to the
first longitudinal axis of the wellbore at the first location and where
the acoustic energy propagates as a fast polarized shear wave and a slow
polarized shear wave in a plane of the formation orthogonal to the second
longitudinal axis of the wellbore at the second location; receiving at
the second location composite waveforms comprising components of both a
fast polarized shear wave and a slow polarized shear wave from a plane of
the formation orthogonal to the second longitudinal axis of the wellbore
at the second location; and combining the received signals at the first
location and the second location to determine the least one
characteristic of the anisotropic formation.
2. The method of claim 1 further comprising transmitting the at least one determined characteristic of the anisotropic earth formation to a surface location.
3. The method of claim 1 wherein the at least one characteristic of the anisotropic earth formation comprises at least one of: a three dimensional stress field of the formation and a three dimensional velocity field of the formation.
4. The method of claim 1 wherein transmitting acoustic dipole energy into the earth formation further comprises firing a first dipole transmitter in a first direction, then firing a second dipole transmitter in a second direction substantially azimuthally perpendicular to the first direction.
5. The method of claim 1 further comprising adjusting the direction of the wellbore based at least in part on the determined characteristic of the anisotropic formation.
6. A method for determining at least one characteristic of an anisotropic earth formation, comprising: transmitting dipole acoustic energy into the earth formation at a first location in a first wellbore where the acoustic energy propagates as a fast polarized shear wave and a slow polarized shear wave in a plane of the formation orthogonal to a first longitudinal axis of the wellbore at the first location; receiving at the first location in the first wellbore composite waveforms comprising components of both a fast polarized shear wave and a slow polarized shear wave from the plane of the formation orthogonal to a first longitudinal axis of the first wellbore at the first location; transmitting dipole acoustic energy into the earth formation at a second location in an offset wellbore where a second longitudinal axis of the offset wellbore at the second location is inclined to the first longitudinal axis of the first wellbore at the first location and where the acoustic energy propagates as a fast polarized shear wave and a slow polarized shear wave in a plane of the formation orthogonal to the second longitudinal axis of the offset wellbore at the second location; receiving at the second location composite waveforms comprising components of both a fast polarized shear wave and a slow polarized shear wave from a plane of the formation orthogonal to a second longitudinal axis of the offset wellbore at the second location; and combining the received signals at the first location and the second location to determine the least one characteristic of the formation.
7. The method of claim 6 further comprising transmitting the at least one determined characteristic of the anisotropic earth formation to a surface location.
8. The method of claim 6 wherein the at least one characteristic of the anisotropic earth formation comprises at least one of: a three dimensional stress field of the formation and a three dimensional velocity field of the formation.
9. The method of claim 6 wherein measurements in the first wellbore and measurements in the offset wellbore occur at different times.
10. An apparatus comprising: an extendable member controllably extendable from a housing in a wellbore, the extendable member urging a clamp pad into engagement with a wall of the wellbore; and an axial force assembly to cooperatively act with the extendable member and the clamp pad to move the clamp pad in an axial direction to impart an axial shear force into the formation.
11. The apparatus of claim 10 wherein the axial force assembly comprises at least one of a piezoelectric member and a magnetostrictive member to impart axial force to move the clamp pad.
12. The apparatus of claim 10 wherein the clamp pad comprises a plurality of clamp pads distributed at locations around the circumference of the wellbore.
13. The apparatus of claim 12 wherein the extendable member comprises a plurality of extendable members distributed at locations around the circumference of the housing.
14. The apparatus of claim 10 further comprising a controller to control the motion of the clamp pad.
15. The apparatus of claim 10 wherein the extendable member comprises a telescoping cylinder.
16. A method of generating an axial shear wave in a formation surrounding a wellbore comprising: urging a clamp pad into contact with a wall of the wellbore; and applying an axial force to the clamp pad to impart a shear force into the wall of the wellbore to generate a shear wave in the formation.
17. The method of claim 16 wherein urging a clamp pad into contact with a wall of the wellbore comprises extending an extendable member coupled to the clamp pad from a housing the wall of the wellbore.
18. The method of claim 16 wherein applying an axial force to the clamp pad to impart a shear force into the wall of the wellbore to generate a shear wave in the formation comprises actuating at least one of a piezoelectric member and a magnetostrictive member.
The present disclosure relates generally to the field of acoustic logging.
Certain earth formations exhibit a property called "anisotropy", wherein the velocity of acoustic waves polarized in one direction may be somewhat different than the velocity of acoustic waves polarized in a different direction within the same earth formation. Anisotropy may arise from intrinsic structural properties, such as grain alignment, crystallization, aligned fractures, or from unequal stresses within the formation. Anisotropy is particularly of interest in the measurement of the velocity of shear/flexural waves propagating in the earth formations. Shear or S waves are often called transverse waves because the particle motion is in a direction "transverse", or perpendicular, to the direction that the wave is traveling.
Acoustic waves travel fastest when the direction of particle motion polarization direction is aligned with the material's stiffest direction. If the formation is anisotropic, meaning that there is one direction that is stiffer than another, then the component of particle motion aligned in the stiff direction travels faster than the wave component aligned in the other, more compliant, direction in the same plane. In the case of 2-dimensional anisotropy, a shear wave induced into an anisotropic formation splits into two components, one polarized along the formation's stiff (or fast) direction, and the other polarized along the formation's compliant (or slow) direction. Generally, the orientation of these two polarizations is substantially orthogonal (components which are at a 90° angle relative to each other). The fast wave is polarized along the direction parallel to the fracture strike and a slow wave in the direction perpendicular to it.
A significant number of hydrocarbon reservoirs comprise fractured rocks wherein the fracture porosity makes up a large portion of the fluid-filled space. In addition, the fractures also contribute significantly to the permeability of the reservoir. Identification of the direction and extent of fracturing is important in reservoir development for at least two reasons.
One reason for identification of fracture direction is that such a knowledge makes it possible to drill deviated or horizontal boreholes with an axis that is preferably normal to the plane of the fractures. In a rock that otherwise has low permeability and porosity, a well drilled in the preferred direction will intersect a large number of fractures and thus have a higher flow rate than a well that is drilled parallel to the fractures. Knowledge of the extent of fracturing also helps in making estimates of the potential recovery rates in a reservoir and in enhancing the production from the reservoir.
BRIEF DESCRIPTION OF THE DRAWINGS
A better understanding of the present invention can be obtained when the following detailed description of example embodiments are considered in conjunction with the following drawings, in which:
FIG. 1A shows an example of a drilling system traversing a downhole formation;
FIG. 1B shows an example of a drilling system traversing a dipping downhole formation;
FIG. 2 shows an example of an acoustic tool;
FIG. 3 shows an example set of decomposed received signals;
FIG. 4 shows an example of logging in two wells, inclined to each other, in the same formation;
FIG. 5 shows an example of an acoustic tool having an axial shear wave generator; and
FIG. 6 shows an example of an axial shear wave generator in a wellbore.
FIG. 1A shows a schematic diagram of a drilling system 110 having a downhole assembly according to one embodiment of the present invention. As shown, the system 110 includes a conventional derrick 111 erected on a derrick floor 112 which supports a rotary table 114 that is rotated by a prime mover (not shown) at a desired rotational speed. A drill string 120 comprising a drill pipe section 122 extends downward from rotary table 114 into a directional borehole, also called a wellbore, 126, through subsurface formations A and B. Borehole 126 may travel in a two-dimensional and/or three-dimensional path. A drill bit 150 is attached to the downhole end of drill string 120 and disintegrates the geological formation 123 when drill bit 150 is rotated. The drill string 120 is coupled to a drawworks 130 via a kelly joint 121, swivel 128 and line 129 through a system of pulleys (not shown). During the drilling operations, drawworks 130 may be operated to raise and lower drill string 120 to control the weight on bit 150 and the rate of penetration of drill string 120 into borehole 126. The operation of drawworks 130 is well known in the art and is thus not described in detail herein.
During drilling operations a suitable drilling fluid (also called "mud") 131 from a mud pit 132 is circulated under pressure through drill string 120 by a mud pump 134. Drilling fluid 131 passes from mud pump 134 into drill string 120 via fluid line 138 and kelly joint 121. Drilling fluid 131 is discharged at the borehole bottom 151 through an opening in drill bit 150. Drilling fluid 131 circulates uphole through the annular space 127 between drill string 120 and borehole 126 and is discharged into mud pit 132 via a return line 135. A variety of sensors (not shown) may be appropriately deployed on the surface according to known methods in the art to provide information about various drilling-related parameters, such as fluid flow rate, weight on bit, hook load, etc.
In one example, a surface control unit 140 may receive signals from downhole sensors (discussed below) via a telemetry system and processes such signals according to programmed instructions provided to surface control unit 140. Surface control unit 140 may display desired drilling parameters and other information on a display/monitor 142 which may be used by an operator to control the drilling operations. Surface control unit 140 may contain a computer, memory for storing data and program instructions, a data recorder, and other peripherals. Surface control unit 140 may also include drilling models and may process data according to programmed instructions, and respond to user commands entered through a suitable input device, such as a keyboard (not shown).
In one example embodiment of the present invention, bottom hole assembly (BHA) 159 is attached to drill string 120, and may comprise a measurement while drilling (MWD) assembly 158, an acoustic tool 190, a drilling motor 180, a steering apparatus 161, and drill bit 150. MWD assembly 158 may comprise a sensor section 164 and a telemetry transmitter 133. Sensor section 164 may comprise various sensors to provide information about the formation 123 and downhole drilling parameters.
MWD sensors in sensor section 164 may comprise a device to measure the formation resistivity, a gamma ray device for measuring the formation gamma ray intensity, directional sensors, for example inclinometers and magnetometers, to determine the inclination, azimuth, and high side of at least a portion of BHA 159, and pressure sensors for measuring drilling fluid pressure downhole. The above-noted devices may transmit data to a telemetry transmitter 133, which in turn transmits the data uphole to the surface control unit 140. In one embodiment a mud pulse telemetry technique may be used to generate encoded pressure pulses, also called pressure signals, that communicate data from downhole sensors and devices to the surface during drilling and/or logging operations. A transducer 143 may be placed in the mud supply line 138 to detect the encoded pressure signals responsive to the data transmitted by the downhole transmitter 133. Transducer 143 generates electrical signals in response to the mud pressure variations and transmits such signals to surface control unit 140. Alternatively, other telemetry techniques such as electromagnetic and/or acoustic techniques or any other suitable telemetry technique known in the art may be utilized for the purposes of this invention. In one embodiment, drill pipe sections 122 may comprise hard-wired drill pipe which may be used to communicate between the surface and downhole devices. Hard wired drill pipe may comprise segmented wired drill pipe sections with mating communication and/or power couplers in the tool joint area. Such hard-wired drill pipe sections are commercially available and will not be described here in more detail. In one example, combinations of the techniques described may be used. In one embodiment, a surface transmitter/receiver 180 communicates with downhole tools using any of the transmission techniques described, for example a mud pulse telemetry technique. This may enable two-way communication between surface control unit 140 and the downhole tools described below.
FIG. 2 shows an example of acoustic tool 190. FIG. 2 shows the tool 190 disposed in BHA 159 within a fluid filled borehole 126. Alternatively, the tool 190 may be suspended within the borehole by a multi-conductor armored cable known in the art.
The tool 190 comprises a set of dipole transmitters: a first dipole transmitter 20, and a second dipole transmitter 22. In the perspective view of FIG. 2, only one face of each of the dipole transmitters 20, 22 may be seen. However, one of ordinary skill in the art understands that a complimentary face of each dipole transmitter 20 and 22 is present on a back surface of the tool 10. The dipole transmitters may be individual transmitters fired in such a way as to act in a dipole fashion. The transmitter 20 induces its acoustic energy along an axis, which for convenience of discussion is labeled X in the FIG. 2. Transmitter 22 induces energy along its axis labeled Y in FIG. 2, where the X and Y axes (and therefore transmitters 20, 22) may be, in one example, orthogonal. The orthogonal relationship of the transmitters 20, 22 need not necessarily be the case, but a deviation from an orthogonal relationship complicates the decomposition of the waveforms. The mathematics of such a non-orthogonal decomposition are within the skill of one skilled in the art without undue experimentation.
Tool 190 may also comprise a plurality of receiver pairs 24 and 26 at elevations spaced apart from the transmitters 20, 22. In one embodiment tool 190 comprises four pairs of dipole receivers 24 A-D and 26 A-D. However, any number of receiver pairs may be used without departing from the spirit and scope of the invention. In the example shown in FIG. 2, the receivers are labeled 24A-D and 26A-D. In one example, each set of dipole receivers at a particular elevation has one receiver whose axis is coplanar with the axis of transmitter 20 (in the X direction) and one receiver whose axis is coplanar with the axis of transmitter 22 (in the Y direction). For example, one set of dipole receivers could be receivers 24A and 26A. Thus, the dipole receivers whose axes are coplanar with the axis of transmitter 20 are the transmitters 24A-D Likewise the dipole receivers whose axes are coplanar with the axis of transmitter 22 are receivers 26 A-D. It is not necessary that the axes of the receivers be coplanar with the axes of one of the transmitters. However, azimuthally rotating any of the receiver pairs complicates the trigonometric relationships and, therefore, the data processing. The mathematics of such a non-orthogonal decomposition are within the skill of one skilled in the art without undue experimentation.
Anisotropic earth formations tend to break an induced shear wave into two components: one of those components traveling along the faster polarization direction, and the second component traveling along the slower polarization direction, where those two directions are substantially orthogonal. The relationship of the fast and slow polarizations within the formation, however, rarely lines up with the orthogonal relationship of the dipole transmitters 20, 22. For convenience of the following discussion and mathematical formulas, a strike angle Θ is defined to be the angle between the X direction orientation (the axis of dipole transmitter 20) and the faster of the two shear wave polarizations (see FIG. 2). Further, it must be understood that the shear wave of interest does not propagate in the X or Y direction, but instead propagates in the Z direction where the Z direction is parallel to the axial direction.
Operation of the tool 190 involves alternative firings of the transmitters 20, 22. Each of the receivers 24A-D and 26A-D create received waveforms designated R, starting at the firing of a particular transmitter. Each of the received waveforms or signals has the following notation: R.sub.[receiver][source]. Thus, for the firing of transmitter 20 in the X direction, and receipt by one of the receivers having an axis coplanar to the axis of transmitter 20 (receivers 24A-D), the time series received signal is designated as RXX. Likewise, the cross-component signal, the signal received by the dipole receiver whose axis is substantially perpendicular to the axis of the firing transmitter, is designated RYX in this situation. In similar fashion, firing of the transmitter whose axis is oriented in the Y direction, transmitter 22, results in a plurality of received signals designated as RYY for the axially parallel receivers, and RXY for the cross-components. Thus, each transmitter firing creates two received signals, one for each receiver of the dipole receiver pair. It follows that for a set of dipole transmitter firings, four signals are received at each receiver pair indicative of the acoustic signals propagated through the formation. The acoustic signals may be processed using transform techniques known in the art to indicate formation anisotropy.
In one example, a processing method comprises calculating, or estimating, source signals or source wavelets that created each set of received signals by assuming a transfer function of the formation. Estimating source wavelets can be described mathematically as follows:
where SESTi is the estimated source signal calculated for the ith set of receivers, [TF] is the assumed transfer function of the formation in the source to receiver propagation, and Ri is the decomposed waveforms (described below) for the ith receiver set. Thus, for each set of received signals Ri, an estimate of the source signal SESTi is created. The estimated source signals are compared using an objective function. Minimas of a graph of the objective function are indicative of the angle of the anisotropy, and the slowness of the acoustic waves through the formation. Further, depending on the type objective function used, one or both of the value of the objection function at the minimas, and the curvature of the of the objective function plot near the minimas, are indicative of the error of the slowness determination.
Thus, a primary component of the source signal estimation is the assumed transfer function [TF]. The transfer function may be relatively simple, taking into account only the finite speed at which the acoustic signals propagate and the strike angle, or may be very complex, to include estimations of attenuation of the transmitted signal in the formation, paths of travel of the acoustic signals, the many different propagation modes within the formation (e.g. compressional waves, shear waves, Stonely waves), and if desired even the effects of the acoustic waves crossing boundaries between different layers of earth formations. For reasons of simplicity of the calculation, the preferred estimated transfer functions take into account only the propagation speed (slowness) of the acoustic energy in the formation and the strike angle of the anisotropy.
Each of the received signals in the case described above contains components of both the fast and slow shear waves, and hence can be considered to be composite signals. That is, for example, an RXX receiver signal contains information regarding both the fast and slow polarized signals. These composite signals may be decomposed into their fast and slow components using equations as follows:
FP(t)=cos2(θ)RXX(t)+sin(θ)cos(θ)[RXY(t)- +RYX(t)]+sin2(θ)RYY(t) (2)
SP(t)=sin2(θ)RXX(t)-cos(θ)sin(θ)[RXY(t)- +RYX(t)]+cos2(θ)RYY(t) (3)
sin(2θ)[RXX(t)-RYY(t)]-cos(2θ)[RXY(t)+RY- X(t)]=0 (4)
where FP(t) is the fast polarization time series, SP(t) is the slow polarization time series, and θ is the strike angle as defined above. The prior art technique for decomposing the multiple received composite signals involved determining the strike angle θ by solving equation (4) above, and using that strike angle in equations (2) and (3) to decompose the composite signals into the fast and slow time series.
In another example for decomposing the composite signals into the fast and slow time series, a close inspection of equations (2) and (3) above for the fast and slow polarization time series respectively shows two very symmetric equations. Taking into account the trigonometric relationships:
sin θ=cos(90°-θ) (5)
cos θ=sin(90°-θ) (6)
it may be recognized that either the fast polarization equation (2) or the slow polarization equation (3) may be used to obtain either the fast or slow polarization signals by appropriately adjusting the angle θ used in the calculation. Stated otherwise, either the fast or slow polarization equations (2) or (3) may be used to decompose a received signal having both fast and slow components into individual components if the strike angle θ is appropriately adjusted.
Rather than using a single strike angle in both equations (2) and (3) above, each assumed transfer function comprises a strike angle. A plurality of transfer functions are assumed over the course of the slowness determination, and thus a plurality of strike angles are used, preferably spanning possible strike angles from -90° to)+90° (180°. For each assumed transfer function (and thus strike angle), the four received signals generated by a set of receivers at each elevation are decomposed using the following equation:
DS(t)=cos2(θ)RXX(t)+sin(θ)cos(θ)(RXY(t)- +RYX(t))+sin2(θ)RYY(t) (7)
where DS(t) is simply the decomposed signal for the particular strike angle used. This process is preferably repeated for each set of received signals at each level for each assumed transfer function. Equation (7) is equation (2) above; however, equation (3) may be equivalently used if the assumed strike angle is appropriately adjusted.
Consider a set of four decomposed signals, see FIG. 3, that are created using equation (7) above for a particular transfer function (strike angle). In the exemplary set of decomposed signals, R1 could be the decomposed signal created using the strike angle from the assumed transfer function and the composite signals received by the set of receivers 24A, 26A. Likewise, decomposed signal R2 could be the decomposed signal created again using the strike angle from the assumed transfer function and the composite signals created by the set of receivers 24B, 26B. In this example, the amplitude of the decomposed signal of the set of receivers closest to the transmitters, decomposed signal R1, is greater than the decomposed signals of the more distant receivers, for example R4. The waveforms may shift out in time from the closest to the more distant receivers, which is indicative of the finite speed of the acoustic waves within the formation.
For a particular starting time within the decomposed signals, for example starting time T1, and for a first assumed transfer function having an assumed strike angle and slowness, portions of each decomposed signal are identified as being related based on the transfer function. Rectangular time slice 50 of FIG. 3 is representative of a slowness in an assumed transfer function (with the assumed strike angle used to create the decomposed signals exemplified in FIG. 3). In particular, the slope of the rectangular time slice is indicative of the slowness of the assumed transfer function. Stated another way, the portions of the decomposed signals within the rectangular time slice 50 should correspond based on the assumed slowness of the formation of the transfer function. The time width of the samples taken from each of the received signals may be at least as long as each of the source signals in a firing set. In this way, an entire source waveform or source wavelet may be estimated. However, the time width of the samples taken from the decomposed signals need not necessarily be this width, as shorter and longer times would be operational.
Thus, the portions of the decomposed signals in the rectangular time slice 50 are each used to create an estimated source signal. These estimated source signals are compared to create an objective function that is indicative of their similarity. In one example, the estimated source signals may be compared using cross correlation techniques known in the art. In another example, cross correlation of the frequency spectra of the received signals may be compared using techniques known in the art. The process of assuming a transfer function, estimating source wavelets based on decomposed signals and creating an objective function may be repeated a plurality of times. The rectangular time slices 50 through 54 are exemplary of multiple assumed transfer functions used in association with starting time T1 (and the a strike angle used to create the decomposed signals). Estimating source wavelets in this fashion (including multiple assumed transfer functions) may also be repeated at multiple starting times within the decomposed signals.
The value of the objective function may be calculated for each assumed transfer function and starting time. Calculating the objective function of the first example technique comprises comparing estimated source signals to determine a variance between them. This slowness determination comprises calculating an average of the estimated source signals within each time slice, and then calculating a variance against the average source signal. In more mathematical terms, for each assumed transfer function, a series of estimated source waveforms or signals SESTi are calculated using equation (1) above.
From these estimated source signals, an average estimated source signal may be calculated as follows:
S EST AVG ( t ) = 1 N i = 1 N S EST i ( t ) ( 8 ) ##EQU00001##
where SESTiAVG is the average estimated source signal, N is the number of decomposed received signals, SESTi is the source wavelet estimated for each decomposed received signal within the time slice, and t is time within the various time series.
The average estimated source signal is used to calculate a value representing the variance of the estimated source signals from the average estimated source signal. The variance may be calculated as follows:
δ 2 = i = 1 N ( S EST i ( t ) - S EST AVG ( t ) ) 2 ( 9 ) ##EQU00002##
where δ2 is the variance. In one embodiment, the variance value is determined as a function of slowness, starting time, and strike angle. Large values of the variance indicate that the assumed transfer function (assumed strike angle and/or assumed slowness) did not significantly match the actual formation properties. Likewise, small values of the variance indicate that the assumed transfer function closely matched the actual formation properties. Thus, the minimas of the objective function described above indicate the slowness of the fast and slow polarized waves as well as the actual strike angle. The value of the variance objective function at the minimas is indicative of the error of the determination of the acoustic velocity and strike angle. The curvature of the variance objective function at the minima is indicative of the error of the calculation.
A second embodiment for calculating an objective function is based on determining a difference between each estimated source signal. As discussed above, using the assumed transfer function, an estimated source signal is created using the portions of the decomposed signal within a time slice. Differences or differentials are calculated between each estimated source signal, for example between the source signal estimated from a portion of the R1 signal and the source signal estimated from the portion of the R2 signal. This difference is calculated between each succeeding receiver, and the objective function in this embodiment is the sum of the square of each difference calculation. The differential objective function is generated as a function of slowness, starting time, and strike angle. However, the function obtained using the differential slowness calculation has slower transitions from maximas to minimas which therefore makes determining the minimas (indicative of the actual slowness of the fast and slow polarizations) easier than in cases where the function has relatively steep slopes between minima and maxima More mathematically, the objective function of this second embodiment is calculated as follows:
ζ = i = 1 N - 1 ( S EST i + 1 - S EST i ) 2 ( 10 ) ##EQU00003##
where ζ is the objective function, and N is the number of receivers. Much like using the variance as the objective function, this differential objective function is a function of slowness versus starting time versus strike angle. Known techniques may be used to determine minima of these functions, and the locations of the minima are indicative of formation slowness and the strike angle.
Either of the two calculational techniques may be used. Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. For example, the disclosed method for determining shear wave velocity and orientation may be implemented using any number of receiver levels and different receiver types for the acoustic logging tool. Indeed, even a single set of dipole receivers may be used relying on rotation of the tool to obtain additional composite signals for decomposition. Further, the source may be located at any arbitrary angle relative to the receivers. Moreover, processing of the data after collection at receivers can be performed downhole in real time with only the results being transferred uphole to a computer system for storage. Throughout this discussion, the various earth formation characteristics were discussed with reference to finding minimas of the objective function. However, one of ordinary skill in the art could easily invert the values used, thus making a determination a search for maximum values in the plot, and this would not deviate from the scope and spirit of the invention. While assuming the transfer functions in the embodiments described involved thus far assume a strike angle, it is possible that the transfer function need not include a strike angle estimation, and instead the composite signals could be decomposed for the range of possible strike angles independent of an assumed transfer function. It is also possible to solve for the strike angle using equation (4) above and decompose the composite waveforms using that strike angle; and thereafter, estimate and apply transfer functions to the decomposed signals, thus also removing the strike angle from the transfer function.
As discussed above, crossed-dipole acoustic tools use a pair of orthogonal acoustic sources to create acoustic surface waves on the borehole wall. These surface waves (flexural waves) are strongly influenced by the mechanical stresses in the formations surrounding the borehole as well as any intrinsic anisotropy (such as fine layering in shales). The tools measure the anisotropy in the X-Y plane that is orthogonal to the tool longitudinal axis. The tool is substantially insensitive to anisotropy in the Z axis aligned with the tool longitudinal axis. In several drilling situations, complex stress regimes in the formations of interest make it desirable to know the three-dimensional stress field surrounding the borehole.
As indicated, the acoustic tool described herein, provides information related to the anisotropy in the plane perpendicular to the local Z axis of the tool. At the L0 location in FIG. 1A, the XY plane of the tool is aligned with the XY plane of the earth system G. As the tool progresses, during drilling, along the path of borehole 126 in FIG. 1A, the local coordinate system rotates from vertical to horizontal, as indicated by the local coordinate systems L0, L1, and L2. When acoustic tool 190 is in the horizontal section of the borehole, the Z axis of the earth coordinate system falls in the tools XY measurement plane. Thus by measuring in both the substantially vertical and substantially horizontal sections of the wellbore 126, the horizontal (earth) field measurements from location L0 and the vertical (earth) field measurements from L2 may be combined using suitable techniques known in the art to provide a three dimensional stress field.
FIG. 1B shows a system similar to that described above traversing through a formation that is dipping, or tilted, with respect to the earth's coordinate system G. The properties of the dipping formation are aligned to the coordinate system F, where the XY plane is substantially parallel the bed interface 90. Acoustic measurements made at location L0 will measure components of the formation Z axis anisotropy. However, depending on the dipping angle, the sensitivity to the formation Z axis anisotropy may be weak. By again measuring in both the vertical (earth) and horizontal (earth) planes, the combined measurements may be related to the three dimensional stress field of the formation. In one example, the wellbore 126 may be drilled along a trajectory based on the three dimensional stress field. For example, the wellbore may be drilled to intersect fractures. In another example, the wellbore may be drilled along a path of minimum stresses. In one example, the calculations may be made downhole and may be used with drilling models stored in the downhole processor to adjust steering assembly 160 to drill the wellbore along a predetermined path based on the calculated anisotropic characteristics.
In one example, see FIG. 4, the formation B is not large enough in the axial direction to allow the wellbore 126' to be turned to the horizontal direction. Alternatively, the well plan may not call for an inclined or horizontal section in the particular well. It may be possible to acquire suitable acoustic anisotropy measurements in an offset wellbore 126'' that penetrates formation B at an inclination αc from vertical. Offset wellbore 126'' may have been drilled and logged prior to the drilling of wellbore 126'. In one example, the measurements from tool 190'' in well 126'' may be stored and later downloaded in memory of tool 190' before deployment of tool 190'. The stored measurements may be combined with measurements made by tool 190' and the resulting anisotropy results transmitted to the surface using known MWD telemetry techniques. Alternatively, tool 190'' may take measurements at approximately the same time as tool 190'. Measurements from both tool 190' and 190'' may alternatively be processed in a surface control unit 140, or at a remote site using techniques known in the art.
In another example, see FIG. 5, instead of taking measurements at different axially displaced, orthogonal locations to acquire 3-D anisotropy results, a 3-axis acoustic tool 400 excites shear waves in all 3 axes by including an axial shear wave generator 401. In one example, acoustic tool 400 comprises the 2-D tool 190 described previously and axial shear wave generator 401. Axial shear wave generator 401 comprises a clamping device 405 that is extendable from the axial shear wave generator body 402 to engage the borehole wall around at least a portion of the circumference of the borehole wall. Clamp 405 is forced into cyclical axial motion by a force element in generator body 402. The cyclical axial motion generates shear on the borehole wall in the axial motion direction. The resulting shear waves propagate away from the borehole wall. The shear waves produced by the clamped axial generator propagate substantially orthogonal to the shear waves generated by the dipole sources 20, 22 described above.
In an isotropic medium, the clamped axial shear wave generator 401 will produce shear waves that move out into the formation and compressional waves along the borehole axis. If there is anisotropy, the wave from the clamped dipole source may split producing wave components along the three principle axes depending on the orientation of those axes relative to the borehole. The signals propagate out into the formation and are reflected back to the receivers 24 and 26 described previously. In one example, the signals may be processed in a downhole processor, using techniques known in the art, to determine the 3-D anisotropy characteristics of the formation, and the results transmitted to the surface using known telemetry techniques. Alternatively, the raw data may be transmitted to the surface and processed at the surface. The anisotropic characteristics comprise at least one of a three dimensional stress field and a three dimensional velocity field of the formation.
FIGS. 6A and 6B show one example of an axial shear wave generator 401 comprising a housing 402 that may be in drillstring 122 (see FIGS. 1A and 1B). As used herein, the term axial is intended to mean along, or parallel to, the longitudinal axis of the wellbore. An extendable member 409 is controllably extendable outward from housing 402 toward the wall 430 of wellbore 426. In one example, a clamp pad 407 is attached to extendable member 409, and engages wall 430. As shown in FIG. 5B, each pads 407 A-D may approximate a circumferential ring attached to wall 430 when all of pads 407 A-D are extended to engage wall 430. In one embodiment, extendable member 409 may be part of a telescoping cylinder located on a movable base 410 disposed in housing 402. In one example, movable base is 410 is attached to an axial force assembly 412 that provides axial back and forth motion to movable base 410, thus providing axial motion to clamp pads 407. In one embodiment, axial force assembly 412 comprises a stack of piezoelectric disks 413 polarized to extend and contract axially when excited by a suitable electric signal. In one embodiment, a backing mass 450 is mounted between the piezoelectric disks 413 and a shoulder 403 in housing 402. In one example, backing mass 450 may comprise a tungsten material and/or a tungsten carbide material. Backing mass 451 helps to ensure that the majority of axial movement of the piezoelectric stack is directed toward the clamp bands. In one example, controller 415 comprises suitable electric circuits and processors to power the crystals and control the extension, and/or retraction, of extendable members 409. Power source 420 may comprise suitable batteries for powering the axial shear wave generator during operation. Controller 415 may be in suitable data communication with other controllers in the downhole tool. Programmed instructions in controller 415 may be used control shear wave generation, data acquisition, and calculation of the anisotropic properties of the formation. In an alternative embodiment, magnetostrictive materials may be used to power the back and forth movement of clamp members 407 to generate axial shear waves in the surrounding formation. Such magnetostrictive materials may include nickel and rare earth materials for example a terbium-dysprosium-iron material. Such materials are known in the art.
While described above with relation to an MWD/LWD system, one of ordinary skill in the art will appreciate that the apparatus and methods described herein may be used with wireline, slickline, wired drill pipe, and coiled tubing to convey the acoustic tools into the wellbore.
Patent applications by Paul F. Rodney, Spring, TX US
Patent applications by HALLIBURTON ENERGY SERVICES, INC.
|
s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345769121/warc/CC-MAIN-20131218054929-00073-ip-10-33-133-15.ec2.internal.warc.gz
|
CC-MAIN-2013-48
| 43,142 | 115 |
https://askthemufti.us/zakaah-for-forgotten-gold/
|
math
|
Fatwaa ID: 689
Salaams a friend had jewells kept by a friend fo 18 years
N over those years she forgot sbt it
N nev giv zakaath
Now da Jewellery is valued 100.000
What must she do
How muc mus giv
Salaams u wanted to know why it was kept by a friend
Maaf actually she says her brother was keeping it in his flat for safety cause her house was not safe enough
In the Name of Allaah, the Most Gracious, the Most Merciful.
As-salaamu ‘alaykum wa-rahmatullaahi wa-barakaatuh.
We understand from the details of your query that the brother kept his sister’s gold or silver jewelry in his possession with her permission for safe keeping. However, the sister forgot about this for the past 18 years. She thus did not pay zakaah on it for these past years as well. Hence, in the enquired situation, zakaah will have to be backdated for the past years for the said jewelry. She will find out the scrap value of gold/silver for the past 18 years on her zakaah date and figure out the scrap value of her jewelry. She will then pay 2.5% of its scrap value for each of the years.
And Allaah Ta’aala knows best.
Mufti Muajul I. Chowdhury
Darul Iftaa New York
08/04/1444 AH – 02/24/2023 CE | AML1-7174
وصل اللهم وسلم وبارك على سيدنا محمد وعلى ءاله وصحبه أجمعين
النتف في الفتاوى (1/172)
اﻟﻤﺎﻝ اﻟﻐﺎﺋﺐ اﻟﺬاﻫﺐ
ﻭاﻟﻮﺟﻪ اﻟﺜﺎﻟﺚ اﻟﻤﺎﻝ اﻟﻐﺎﺋﺐ اﻟﺬﻱ ﺫﻫﺐ ﻣﻨﻪ ﻭﻫﻮ ﻋﻠﻲ ﺧﻤﺴﺔ ﺃﻭﺟﻪ
اﺣﺪﻫﺎ اﻟﻤﻐﺼﻮﺏ ﻭاﻟﺜﺎﻧﻲ اﻟﻤﺴﺮﻭﻕ ﻭاﻟﺜﺎﻟﺚ اﻵﺑﻖ ﻓﻼ ﺯﻛﺎﺓ ﻋﻠﻴﻪ ﻓﻴﻬﺎ ﻭاﻟﺮاﺑﻊ اﻟﺬﻱ ﺃﺿﻠﻪ ﻭاﻟﺨﺎﻣﺲ اﻟﺬﻱ اﺧﻔﺎﻩ ﻭﻧﺴﻲﻫ ﻓﻬﻮ ﻋﻠﻰ ﻭﺟﻬﻴﻦ
اﺣﺪﻫﻤﺎ ﻳﻜﻮﻥ اﺧﻔﺎﻩ ﻓﻲ ﻣﻠﻜﻪ ﻣﺜﻞ ﺩاﺭﻩ ﻭﺻﻨﺪﻭﻗﻪ ﻭﻧﺤﻮﻫﺎ ﻓﺎﺫا ﻭﺟﺪﻩ ﻓﻌﻠﻴﻪ ﺯﻛﺎﺗﻬﻦ ﻟﻤﺎ ﻣﻀﻰ ﻭاﻟﻮﺟﻪ اﻟﺜﺎﻧﻲ اﻥ ﻳﻜﻮﻥ ﻗﺪ اﺧﻔﺎﻩ ﻓﻲ ﻏﻴﺮ ﻣﻠﻜﻪ ﻣﺜﻞ ﺧﺮﺑﺔ ﺃﻭ ﺑﺮﻳﺔ ﻭﻧﺤﻮﻫﺎ ﻓﺎﻥ ﻭﺟﺪﻩ ﻓﻼ ﺯﻛﺎﺓ ﻋﻠﻴﻪ ﻟﻤﺎ ﻣﻀﻰ
ﻭﺃﻣﺎ اﻟﺬﻱ اﺿﻠﻪ ﻓﺤﻜﻤﻪ ﻛﺤﻜﻢ اﻟﺬﻱ اﺧﻔﺎﻩ ﻭنسيه ﺑﻌﻴﻨﻪ
تحفة الفقهاء (1/296)
ﻭﺃﺟﻤﻌﻮا ﺃﻧﻪ ﺇﺫا ﺩﻓﻦ ﻓﻲ اﻟﺤﺮﺯ ﻣﻦ اﻟﺪﻭﺭ ﻭﻧﺤﻮﻫﺎ ﻭﻧﺴﻲﻫ ﺛﻢ ﺗﺬﻛﺮ ﻓﺈﻧﻪ ﺗﺠﺐ ﻋﻠﻴﻪ ﺯﻛﺎﺓ ﻣﺎ ﻣﻀﻰ
ﻭﻛﺬﻟﻚ ﺇﺫا ﺃﻭﺩﻉ ﺭﺟﻼ ﻣﻌﺮﻭﻓﺎ ﺛﻢ ﻧﺴﻲﻫ ﺳﻨﻴﻦ ﺛﻢ ﺗﺬﻛﺮ ﻓﺈﻧﻪ ﻳﺠﺐ ﺑﺎﻹﺟﻤﺎﻉ
بدائع الصنائع (2/9)
ﻭﻟﻮ ﺩﻓﻊ ﺇﻟﻰ ﺇﻧﺴﺎﻥ ﻭﺩﻳﻌﺔ ﺛﻢ ﻧﺴﻲ اﻟﻤﻮﺩﻉ ﻓﺈﻥ ﻛﺎﻥ اﻟﻤﺪﻓﻮﻉ ﺇﻟﻴﻪ ﻣﻦ ﻣﻌﺎﺭﻓﻪ ﻓﻌﻠﻴﻪ اﻝﺯﻛﺎﺓ ﻟﻤﺎ ﻣﻀﻰ ﺇﺫا ﺗﺬﻛﺮ؛ ﻷﻥ ﻧﺴﻲاﻥ اﻟﻤﻌﺮﻭﻑ ﻧﺎﺩﺭ ﻓﻜﺎﻥ ﻃﺮﻳﻖ اﻟﻮﺻﻮﻝ ﻗﺎﺋﻤﺎ؛ ﻭﺇﻥ ﻛﺎﻥ ﻣﻤﻦ ﻻ ﻳﻌﺮﻓﻪ ﻓﻼ ﺯﻛﺎﺓ ﻋﻠﻴﻪ ﻓﻴﻤﺎ ﻣﻀﻰ ﻟﺘﻌﺬﺭ اﻟﻮﺻﻮﻝ ﺇﻟﻴﻪ
Darul Iftaa New York answers questions on issues pertaining to Shari’ah. These questions and answers are placed for public view on askthemufti.us for educational purposes. The rulings given here are based on the questions posed and should be read in conjunction with the questions. Many answers are unique to a particular scenario and cannot be taken as a basis to establish a ruling in another situation.
Darul Iftaa New York bears no responsibility with regard to its answers being used out of their intended contexts, nor with regard to any loss or damage that may be caused by acting on its answers or not doing so.
References and links to other websites should not be taken as an endorsement of all contents of those websites.
Answers may not be used as evidence in any court of law without prior written consent of Darul Iftaa New York.
|
s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817144.49/warc/CC-MAIN-20240417044411-20240417074411-00369.warc.gz
|
CC-MAIN-2024-18
| 4,227 | 32 |
https://iris.uniroma3.it/handle/11590/416955
|
math
|
Spontaneous symmetry breaking (SSB) is mathematically tied to some limit, but must physically occur, approximately, before the limit. Approximate SSB has been independently understood for Schrodinger operators with double well potential in the classical limit [1, 2] and for quantum spin systems in the thermodynamic limit [3, 4]. We relate these to each other in the context of the Curie-Weiss model, establishing a remarkable relationship between this model (for finite N) and a discretized Schrodinger operator with double well potential.
van de Ven, C., Groenenboom, G.c., Reuvers, R., Landsman, N.p. (2020). Quantum spin systems versus Schrodinger operators: A case study in spontaneous symmetry breaking. SCIPOST PHYSICS, 8(2) [10.21468/SciPostPhys.8.2.022].
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337473.26/warc/CC-MAIN-20221004023206-20221004053206-00151.warc.gz
|
CC-MAIN-2022-40
| 764 | 2 |
https://www.mylespaul.com/threads/are-custom-shop-guitars-worth-the-money.406083/page-3
|
math
|
- Sep 29, 2011
- Reaction score
Yet they sell everyday for 4,5,6, 10k, so, they are WORTH it to somebody.Even using Made In US cost$
Woods, craftsmanship, labor . . .
I find it hard to believe any top-of-the-line solid body electric guitar could be WORTH more than $1500.
( I picked that amount from my arse, it could be $1300 or $1600, but you get the drift)
Every penny after that, is perceived value IMO. It becomes an I WANT factor.
Is it worth it to YOU.
If $$$$ is spare change to you. You may want, and get, lots of these if you like them enough.
|
s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202474.26/warc/CC-MAIN-20190320230554-20190321012554-00095.warc.gz
|
CC-MAIN-2019-13
| 553 | 9 |
https://byjus.com/question-answer/the-incorrect-statements-among-the-followingenthalpy-of-neutralization-of-formic-acid-vs-naoh-is-greater/
|
math
|
The correct options are
B Heat capacity of O2 is independent of temperature
C Boiling of water at 374 K is an equilibrium process
a) Since formic acid is a stronger acid than acetic acid, enthalpy of neutralization will be greater.
b) For di and polyatomic molecules, vibrational degrees of freedom contribute to heat capacity at higher temperature.
c) Boiling of water at 374 K is a spontaenous process. At this temperature ΔH>TΔS
d) P1V1=P2V2 for isothermal reversible compression
P1Vγ1=P3Vγ2 for adiabatic compression.
Since V1 is greater than V2 and γ is greater than 1, P2 is less than P3. Hence more work is done in adiabatic compression.
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362571.17/warc/CC-MAIN-20211203000401-20211203030401-00258.warc.gz
|
CC-MAIN-2021-49
| 649 | 9 |
https://www.studysmarter.us/textbooks/physics/physics-for-scientists-and-engineers-a-strategic-approach-with-modern-physics-4th/fluids-and-elasticity/q-12-the-container-shown-in-figure-ex1412-is-filled-with-oil/
|
math
|
The container shown in FIGURE EX14.12 is filled with oil. It is open to the atmosphere on
the left.a. What is the pressure at point A?b. What is the pressure difference between points A and B? Between points A and C?
a) Total pressure = 106 kPa
b) PC-PA = 4263 Pa and PB - PA = 4263 Pa
The tank and points (A, B and C) are given in figure
Calculate the pressure at point A
PA = h ρ g
= (0.5m) x (870 kg / m3) (9.8 m/s2) = 4263 Pa
PTotal = Patm + Poil
PTota = 101325+4263 Pa=106 kPa
A plastic "boat" with a square cross section floats in a liquid. One by one, you place masses inside the boat and measure how far the boat extends below the surface. Your data are as follows:
Draw an appropriate graph of the data and, from the slope and intercept of the best-fit line, determine the mass of the boat and the density of the liquid.
Glycerin is poured into an open U-shaped tube until the height in both sides is . Ethyl alcohol is then poured into one arm until the height of the alcohol column is . The two liquids do not mix. What is the difference in height between the top surface of the glycerin and the top surface of the alcohol?
94% of StudySmarter users get better grades.Sign up for free
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710534.53/warc/CC-MAIN-20221128171516-20221128201516-00613.warc.gz
|
CC-MAIN-2022-49
| 1,196 | 14 |
https://openreview.net/forum?id=J0zi7_8ZIv1
|
math
|
Keywords: Hypergraph, Diffusion Kernal, Graph Convolutional Networks, Node Classification
Abstract: Kernels on discrete structures evaluate pairwise similarities between objects which capture semantics and inherent topology information. Existing kernels on discrete structures are only developed by topology information(such as adjacency matrix of graphs), without considering original attributes of objects. This paper proposes a two-phase paradigm to aggregate comprehensive information on discrete structures leading to a Discount Markov Diffusion Learnable Kernel (DMDLK). Specifically, based on the underlying projection of DMDLK, we design a Simple Hypergraph Kernel Convolution (SHKC) for hidden representation of vertices. SHKC can adjust diffusion steps rather than stacking convolution layers to aggregate information from long-range neighborhoods which prevents over-smoothing issues of existing hypergraph convolutions. Moreover, we utilize the uniform stability bound theorem in transductive learning to analyze critical factors for the effectiveness and generalization ability of SHKC from a theoretical perspective. The experimental results on several benchmark datasets for node classification tasks verified the superior performance of SHKC over state-of-the-art methods.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100989.75/warc/CC-MAIN-20231209233632-20231210023632-00539.warc.gz
|
CC-MAIN-2023-50
| 1,288 | 2 |
http://angkormovie.com/ebooks/a-course-in-mathematical-physics-iv-quantum-mechanics-of-large-systems-volume
|
math
|
By Walter Thirring, E.M. Harrell
During this ultimate quantity i've got attempted to give the topic of statistical mechanics in keeping with the elemental ideas of the sequence. the trouble back entailed following Gustav Mahler's maxim, "Tradition = Schlamperei" (i.e., dirt) and clearing away a wide component to this tradition-laden sector. the result's a publication with little in universal with such a lot different books at the topic. the normal perturbation-theoretic calculations aren't very valuable during this box. these tools have by no means ended in propositions of a lot substance. even if perturbation sequence, which for the main half by no means converge, could be given a few asymptotic that means, it can't be made up our minds how shut the nth order approximation involves the precise end result. due to the fact that analytic options of nontrivial difficulties are past human features, for greater or worse we needs to accept sharp bounds at the amounts of curiosity, and will at such a lot try to make the measure of accuracy passable.
Read Online or Download A Course in Mathematical Physics IV. Quantum Mechanics of Large Systems: Volume 4: Quantum Mechanics of Large Systems PDF
Similar mathematics books
Written essentially for undergraduate scholars of arithmetic, technological know-how, or engineering, who regularly take a direction on differential equations in the course of their first or moment 12 months. the most prerequisite is a operating wisdom of calculus.
the surroundings within which teachers educate, and scholars examine differential equations has replaced drastically some time past few years and keeps to adapt at a quick velocity. Computing gear of a few style, even if a graphing calculator, a laptop computer, or a laptop notebook is on the market to such a lot scholars. The 7th version of this vintage textual content displays this altering setting, whereas while, it keeps its nice strengths - a modern process, versatile bankruptcy development, transparent exposition, and extraordinary difficulties. moreover many new difficulties were additional and a reorganisation of the fabric makes the suggestions even clearer and extra comprehensible.
Like its predecessors, this variation is written from the point of view of the utilized mathematician, focusing either at the conception and the sensible functions of differential equations as they follow to engineering and the sciences.
This famous paintings covers the answer of quintics by way of the rotations of a typical icosahedron round the axes of its symmetry. Its two-part presentation starts with discussions of the idea of the icosahedron itself; commonplace solids and idea of teams; introductions of (x + iy) ; a press release and exam of the elemental challenge, with a view of its algebraic personality; and common theorems and a survey of the topic.
Within the previous few many years, multiscale algorithms became a dominant development in large-scale clinical computation. Researchers have effectively utilized those ways to quite a lot of simulation and optimization difficulties. This booklet provides a normal review of multiscale algorithms; functions to normal combinatorial optimization difficulties resembling graph partitioning and the touring salesman challenge; and VLSICAD functions, together with circuit partitioning, placement, and VLSI routing.
- A Binary Images Watermarking Algorithm Based on Adaptable Matrix
- Inquiry into the Validity of a Method recently proposed by George B. Jerrard, Esq., for Transforming and Resolving Equations of Elevated Degrees: undertaken at the Request of the Association
- Combinatorial Mathematics IX, Brisbane, Australia: Proceedings, 1981
- Schaum's Outline of Trigonometry (5th Edition) (Schaum's Outlines Series)
- Mathematical Methods in Particle Transport Theory
Additional info for A Course in Mathematical Physics IV. Quantum Mechanics of Large Systems: Volume 4: Quantum Mechanics of Large Systems
Other solutions are excluded by the continuity requirement (Problem 1).
This means that no vector in the Hilbert space of a representation of type 11 or Ill corresponds to a pure state on the algebra. 4. Any operator a of an algebra of type 111 is of course bounded, so Tr pa is well defined for any p E 1(X), only p can not come from the algebra, which contains no element of a trace class (other than 0). 3. Let us end the section by recapitulating the physical significance of the new mathematical phenomena that make an appearance in infinite systems. 1. Inequivalent Representations Since vectors that differ globally are always orthogonal, globally different situations lead to inequivalent representations.
Constructed with 0.... , ® 0 (1 ® 0. is the weak closure of d, and Z = (1 + weak limits • 1}, which is a reducible factor representation. 8) 1. 5); as mentioned above, the vector C fl® ... has no counterpart in the since the corresponding functional in sr,, would earlier representations then be strongly continuous. The state defined by (10 ®... on d. a (norm) Continuous linear functional, and therefore extensible to the whole C* algebra generated by d, but it still need not be strongly continuous in a representation: For instance, in the representation using is 1 + iN -.
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948541253.29/warc/CC-MAIN-20171214055056-20171214075056-00427.warc.gz
|
CC-MAIN-2017-51
| 5,272 | 18 |
https://pdfslide.net/documents/parameterized-complexity-of-finding-small-degree-constrained-subgraphs.html
|
math
|
Journal of Discrete Algorithms 10 (2012) 7083
Contents lists available at ScienceDirect
Oa Cb Cc T
15doJournal of Discrete Algorithms
arameterized complexity of nding small degree-constrainedubgraphs,
mid Amini a, Ignasi Sau b,, Saket Saurabh c
NRS, DMA, ENS, Paris, FranceNRS, LIRMM, Montpellier, Francehe Institute of Mathematical Sciences, Chennai, India
r t i c l e i n f o a b s t r a c t
ticle history:ceived 15 March 2010ceived in revised form 22 December 2010cepted 16 May 2011ailable online 19 May 2011
ywords:rameterized complexitygree-constrained subgraphed-parameter tractable algorithm-hardnesseewidthnamic programmingcluded minors
In this article we study the parameterized complexity of problems consisting in ndingdegree-constrained subgraphs, taking as the parameter the number of vertices of thedesired subgraph. Namely, given two positive integers d and k, we study the problem ofnding a d-regular (induced or not) subgraph with at most k vertices and the problem ofnding a subgraph with at most k vertices and of minimum degree at least d. The latterproblem is a natural parameterization of the d-girth of a graph (the minimum order of aninduced subgraph of minimum degree at least d).We rst show that both problems are xed-parameter intractable in general graphs. Moreprecisely, we prove that the rst problem is W -hard using a reduction from Multi-Color Clique. The hardness of the second problem (for the non-induced case) follows froman easy extension of an already known result. We then provide explicit xed-parametertractable (FPT) algorithms to solve these problems in graphs with bounded local treewidthand graphs with excluded minors, using a dynamic programming approach. Althoughthese problems can be easily dened in rst-order logic, hence by the results of Frickand Grohe (2001) are FPT in graphs with bounded local treewidth and graphs withexcluded minors, the dependence on k of our algorithms is considerably better than theone following from Frick and Grohe (2001) .
2011 Elsevier B.V. All rights reserved.
Problems of nding subgraphs with certain degree constraints are well studied both algorithmically and combinatorially,d have a number of applications in network design (cf. for instance [1,20,25,29,35]). In this article we consider two naturalch problems: nding a small regular (induced or not) subgraph and nding a small subgraph with given minimum degree.e discuss in detail these two problems in Sections 1.1 and 1.2, respectively.
This work has been partially supported by European project IST FET AEOLUS, PACA region of France, Ministerio de Ciencia e Innovacin, Europeangional Development Fund under project MTM2008-06620-C03-01/MTM, and Catalan Research Council under project 2005SGR00256.An extended abstract of this work appeared in: Proceedings of the International Workshop on Parameterized Complexity (IWPEC), May 2008, LNCS,
l. 5018, pp. 1329.Corresponding author.E-mail addresses: [email protected] (O. Amini), [email protected] (I. Sau), [email protected] (S. Saurabh).
70-8667/$ see front matter 2011 Elsevier B.V. All rights reserved.i:10.1016/j.jda.2011.05.001
O. Amini et al. / Journal of Discrete Algorithms 10 (2012) 7083 71
in. Finding a small regular subgraph
The complexity of nding regular graphs as well as regular (induced) subgraphs has been intensively studied in theerature [68,11,24,30,31,35,36]. One of the rst problems of this kind was stated by Garey and Johnson: Cubic Subgraph,at is, the problem of deciding whether a given graph contains a 3-regular subgraph, is NP-complete . More generally,e problem of deciding whether a given graph contains a d-regular subgraph for any xed degree d 3 is NP-complete onneral graphs as well as in planar graphs (where in the latter case only d = 4 and d = 5 were considered, sincey planar graph contains a vertex of degree at most 5). For d 3, the problem remains NP-complete even in bipartiteaphs of degree at most d + 1 . Note that this problem is clearly polynomial-time solvable for d 2. If the regularbgraph is required to be induced, Cardoso et al. proved that nding a maximum cardinality d-regular induced subgraphNP-complete for any xed integer d 0 (for d = 0 and d = 1 the problem corresponds to Maximum Independent Setd Maximum Induced Matching, respectively).Concerning the parameterized complexity of nding regular subgraphs, Moser and Thilikos proved that the followingoblem is W -hard for every xed integer d 0 :
k-size d-Regular Induced SubgraphInput: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a subset S V , with |S| k, such that G[S] is d-regular?
On the other hand, the authors proved that the following problem (which can be seen as the dual of the above one) isP-complete but has a problem kernel of size O(kd(k + d)2) for d 1 :
k-Almost d-Regular GraphInput: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a subset S V , with |S| k, such that G[V \ S] is d-regular?
Mathieson and Szeider studied in variants and generalizations of the problem of nding a d-regular subgraph (for 3) in a given graph by deleting at most k vertices. In particular, they answered a question of , proving that the k-lmost d-Regular Graph problem (as well as some variants) becomes W -hard when parameterized only by k (that is, itunlikely that there exists an algorithm to solve it in time f (k) nO(1) , where n = |V (G)| and f is a function independentn and d).Given two integers d and k, it is also natural to ask for the existence of an induced d-regular graph with at most krtices. The corresponding parameterized problem is dened as follows:
k-size d-Regular Induced Subgraph (kdRIS)Input: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a subset S V , with |S| k, such that G[S] is d-regular?
Note that the hardness of k-size d-Regular Induced Subgraph does not follow directly from the hardness of k-sizeRegular Induced Subgraph as, for instance, the approximability of the problems of nding a densest subgraph on at leastvertices or on at most k vertices are signicantly different . In general, a graph may not contain an induced d-regularbgraph on at most k vertices, while containing a non-induced d-regular subgraph on at most k vertices. This observationads to the following problem:
k-size d-Regular Subgraph (kdRS)Input: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a d-regular subgraph H G , with |V (H)| k?
Observe that k-size d-Regular Subgraph could a priori be easier than its corresponding induced version, as it happensr the MaximumMatching (which is in P) and the Maximum Induced Matching (which is NP-hard) problems.The two parameterized problems dened above have not been considered in the literature. We prove in Section 2 thatth problems are W -hard for every xed d 3, by reduction from Multi-Color Clique.2. Finding a small subgraph with given minimum degree
For a nite, simple, and undirected graph G = (V , E) and d N, the d-girth gd(G) of G is the minimum order of anduced subgraph of G of minimum degree at least d. The notion of d-girth was proposed and studied by Erdos et al. [18,19]
72 O. Amini et al. / Journal of Discrete Algorithms 10 (2012) 7083
wd Bollobs and Brightwell . It generalizes the usual girth, the length of a shortest cycle, which coincides with the 2-rth. (This is indeed true because every induced subgraph of minimum degree at least two contains a cycle.) Combinatorialunds on the d-girth can also be found in [4,27]. The corresponding optimization problem has been recently studied in ,here it has been proved that for any xed d 3, the d-girth of a graph cannot be approximated within any constant factor,less P = NP . From the parameterized complexity point of view, it is natural to introduce a parameter k N and askr the existence of a subgraph with at most k vertices and with minimum degree at least d. The problem can be formallyned as follows:
k-size Subgraph of Minimum Degree d (kSMDd)Input: A graph G = (V , E) and a positive integer k.Parameter: k.Question: Does there exist a subset S V , with |S| k, such that G[S] has minimum
degree at least d?
te that the case d = 2 in P, as discussed above. The special case of d = 4 appears in the book of Downey and Fellows [15,457], where it is announced that H.T. Wareham proved that kSMD4 is W -hard. (However, we were not able to nd aoof.) From this result, it is easy to prove that kSMDd is W -hard for every xed d 4 (see Section 2). The complexitythe case d = 3 remains open (see Section 4). Note that in the kSMDd problem we can assume without loss of generalityat we are looking for the existence of an induced subgraph, since we only require the vertices to have degree at least d.Besides the above discussion, another motivation for studying the kSMDd problem is its close relation to the well studiednse k-Subgraph problem [3,14,20,28], which we proceed to explain. The density (G) of a graph G = (V , E) is dened as
(G) := |E||V | . More generally, for any subset S V , we denote its density by (S), and dene it to be (S) := (G[S]). Thense k-Subgraph problem is formulated as follows:
Dense k-Subgraph (DkS)
Input: A graph G = (V , E).Output: A subset S V , with |S| = k, such that (S) is maximized.
derstanding the complexity of DkS remains widely open, as the gap between the best hardness result (Apx-hardness )d the best approximation algorithm (with ratio O(n1/3) ) is huge. Suppose we are looking for an induced subgraph[S] of size at most k and with density at least . In addition, assume that S is minimal, i.e., no subset of S has densityeater than (S). This implies that every vertex of S has degree at least /2 in G[S]. To see this, observe that if there is artex v with degree strictly smaller than /2, then removing v from S results in a subgraph of density greater than (S)d of smaller size, contradicting the minimality of S . Secondly, if we have an induced subgraph G[S] of minimum degreeleast , then S is a subset of density at least /2. These two observations together show that, modulo a constant factor,oking for a densest subgraph of G of size at most k is equivalent to looking for the largest possible value of d for whichMDd returns Yes. As the degree conditions are more rigid than the global density of a subgraph, a better understandingthe kSMDd problem could provide an alternative way to approach the DkS problem.Finally, we would like to point out that the kSMDd problem has practical applications to trac grooming in opticaltworks. Trac grooming refers to packing small trac ows into larger units then can then be processed as singletities. For example, in a network using both time-division and wavelength-division multiplexing, ows destined to ammon node can be aggregated into the same wavelength, allowing them to be dropped by a single optical Add-Dropultiplexer. The main objective of grooming is to minimize the equipment cost of the network, which is mainly given inavelength-Division Multiplexing optical networks by the number of electronic terminations. (We refer, for instance, to r a general survey on grooming.) It has been recently proved by Amini, Prennes and Sau that the Trac Groomingoblem in optical networks can be reduced (modulo polylogarithmic factors) to DkS, or equivalently to kSMDd. Indeed, inaph theoretic terms, the problem can be translated into partitioning the edges of a given request graph into subgraphsith a constraint on their number of edges. The objective is then to minimize the total number of vertices of the subgraphsthe partition. Hence, in this context of partitioning a given set of edges while minimizing the total number of vertices,e problems of DkS and kSMDd come into play. More details can be found in .
. Presentation of the results
We do a thorough study of the kdRS, the kdRIS, and the kSMDd problems in the realm of parameterized complexity,hich is a recent approach to deal with intractable computational problems having some parameters that can be relativelyall with respect to the input size. This area has been developed extensively during the last decade (the monograph ofwney and Fellows provides a good introduction, and for more recent developments see the books by Flum andohe and by Niedermeier ).For decision problems with input size n and parameter k, the goal is to design an algorithm with running time f (k)nO (1) ,
here f depends only on k. Problems having such an algorithm are said to be xed-parameter tractable (FPT). There is
O. Amini et al. / Journal of Discrete Algorithms 10 (2012) 7083 73
kSso a theory of parameterized intractability to identify parameterized problems that are unlikely to admit xed-parameteractable algorithms. There is a hierarchy of intractable parameterized problem classes above FPT, the important ones being:
FPT M W M W W [P ] X P .The principal analogue of the classical intractability class NP is W , which is a strong analogue, because a fundamentaloblem complete for W is the k-Step Halting Problem for Nondeterministic Turing Machines (with unlimited non-terminism and alphabet size); this completeness result provides an analogue of Cooks theorem in classical complexity.convenient source of W -hardness reductions is provided by the result stating that k-Clique is complete for W . Theincipal working algorithmic way of showing that a parameterized problem is unlikely to be xed-parameter tractable, isprove its W -hardness using a parameterized reduction (dened in Section 2).Our results can be classied into two categories:
eneral graphs: We show in Section 2 that kdRS is not xed-parameter tractable by showing it to be W -hard for any 3 in general graphs. We will see that the graph constructed in our reduction implies also the W -hardness of kdRIS.general, parameterized reductions are quite stringent because of parameter-preserving requirements of the reduction, andquire some technical care. Our reduction is based on a new methodology emerging in parameterized complexity, calledulti-color clique edge representation. This has proved to be useful in showing various problems to be W -hard recently .e rst spell out step-by-step the procedure to use this methodology, which can be used as a template for future purposes.en we adapt this methodology to the reduction for the kSMDd problem. The hardness of kSMDd for d 4 follows fromeasy extension of a result of H.T. Wareham [15, p. 457].
raphs with bounded local treewidth and graphs with exc...
|
s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027323328.16/warc/CC-MAIN-20190825105643-20190825131643-00431.warc.gz
|
CC-MAIN-2019-35
| 14,528 | 45 |
https://www.wyzant.com/Kemah_physics_tutors.aspx
|
math
|
Math, Physics, and Computer Science: All Ages, All Abilities
...am a computer scientist, engineer, and Texas-certified teacher who is highly qualified to teach mathematics and computer programming. Many seek my help with Algebra, Geometry, Calculus, Statistics, and Physics. It is a joy to teach computer...
I am an engineer who has taken advanced courses in physics including mechanics, fluid dynamics and thermodynamics. I can help you with all aspects of your high school and college level physics, including theoretical concepts, mathematical...
Replies in 5 hours
In-person + online
| 10+ other subjects
|
s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443738004493.88/warc/CC-MAIN-20151001222004-00046-ip-10-137-6-227.ec2.internal.warc.gz
|
CC-MAIN-2015-40
| 608 | 6 |
https://puzzling.stackexchange.com/questions/109448/numbers-with-minimal-sum-at-the-vertices-of-a-cube
|
math
|
The eight vertices of a cube are marked with numbers from 1 to 8 such that the sum of any three numbers on any face is not less than 10.
What is the minimum sum of the four numbers on a face?
Puzzling Stack Exchange is a question and answer site for those who create, solve, and study puzzles. It only takes a minute to sign up.Sign up to join this community
Old school proof of optimality:
Let n be the largest number on the face with the smallest sum. The remaining three numbers on that face must sum to at least 10 so cannot be all less than 5 (2+3+4=9) the largest must therefore be at least 6. + 10 = 16.
2---3 /| /| 5-+-6 | | 8-+-7 |/ |/ 4---1
There is no unique way to label the vertices. So does the question mean “minimum” for a specific consistent cube? Or “minimum” over all consistent cubes? I choose the former because it’s more interesting, and it subsumes the other question
If the vertices 1&2 are diagonally opposite one another, then the labels 1...5 are fixed, up to rotation and reflection. But the other 3 vertices can be labelled 6...8 in any order, without risk of any triple being too light.
The vertex-sums of the faces are: 10+a, 26-a, 11+b, 25-b where a&b are distinct and drawn from 6...8. Some values appear twice. The minimum vertex-sum is 16 iff a is 6. The minimum is 18 (actually achieved by every face) iff a=8 and b=7. Otherwise the minimum is 17.
On the other hand if the 1&2 are on the same face (but not adjacent) then so are 7&8. The other 4 vertices can be labelled in 4 ways. Either all the sums are 18, or there is one 17/19 pair.
I didn’t know one could label the corners of a cube so that the sum of the 4 values on any face is the same: 18. Interesting
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499695.59/warc/CC-MAIN-20230128220716-20230129010716-00610.warc.gz
|
CC-MAIN-2023-06
| 1,710 | 11 |
https://brainmass.com/math/linear-programming/quantitative-method-mcq-91493
|
math
|
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) In linear programming, sensitivity analysis is associated with
(1) objective function coefficients
(2) right hand side values of constraints
(3) constraint coefficients
A) 1 and 2 B) 1 and 3 C) 2 and 3 D) 1, 2, and 3
2) Sensitivity analysis is the analysis of the effect of ________ changes on the ________.
A) price, company B) parameter, optimal solution
C) cost, production D) none of the above
3) For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lb, and the range of feasibility (sensitivity range) for this constraint is from 3 lb to 6 lb, increasing the amount of this resource by 1 lb will result in the
A) same product mix, different total profit.
B) different product mix, different total profit.
C) same product mix, same total profit.
D) different product mix, same total profit as before.
4) The production manager for Beer etc. produces two kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He can get, at most, 4800 oz. of malt per week and 3200 oz. of wheat per week. Each bottle of light beer requires 12 oz. of malt and 4 oz. of wheat, while a bottle of dark beer uses 8 oz. of malt and 8 oz. of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. What is the objective function?
A) Z = $12L + $8D
B) Z = $1L + $2D
C) Z = 12 lb + 4 lb
D) Z = $4L + $8D
E) Z = $2L + $1D
5) The production manager for the Whoppy soft drink company is considering the production of two kinds of soft drinks: regular and diet. The company operates one 8 hour shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup, is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case, and profits for diet soft drink are $2.00 per case. What is the time constraint?
A) 2 R + 4 D ≥ 480
B) 2 D + 4 R ≤ 480
C) 2 R + 3 D ≤ 480
D) 2 R + 4 D ≤ 480
E) 3 R + 2 D ≤ 480
TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.
6) Decision variables must be clearly defined after constraints are written.
7) The sensitivity range for an objective coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal.
8) If the original amount of a resource is 15, and the range of feasibility for it can increase by 5, then the amount of the resource can increase to 20.
9) The reduced cost (shadow price) for a positive decision variable is 0.
10) A maximization problem may be characterized by all greater than or equal to constraints.© BrainMass Inc. brainmass.com April 3, 2020, 3:54 pm ad1c9bdddf
Solution contains answers of multiple choice questions.
|
s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371876625.96/warc/CC-MAIN-20200409185507-20200409220007-00165.warc.gz
|
CC-MAIN-2020-16
| 3,047 | 33 |
https://asp-eurasipjournals.springeropen.com/articles/10.1186/1687-6180-2014-56
|
math
|
An Erratum to this article was published on 18 August 2014
This paper investigates security-oriented beamforming designs in a relay network composed of a source-destination pair, multiple relays, and a passive eavesdropper. Unlike most of the earlier works, we assume that only statistical information of the relay-eavesdropper channels is known to the relays. We propose beamforming solutions for amplify-and-forward (AF) and decode-and-forward (DF) relay networks to improve secrecy capacity. In an AF network, the beamforming design is obtained by approximating a product of two correlated Rayleigh quotients to a single Rayleigh quotient using the Taylor series expansion. Our study reveals that in an AF network, the secrecy capacity does not always grow as the eavesdropper moves away from the relays or as total relay transmit power increases. Moreover, if the destination is nearer to the relays than the eavesdropper is, a suboptimal power is derived in closed form through monotonicity analysis of secrecy capacity. While in a DF network, secrecy capacity is a single Rayleigh quotient problem which can be easily solved. We also found that if the relay-eavesdropper distances are about the same, it is unnecessary to consider the eavesdropper in a DF network. Numerical results show that for either AF or DF relaying protocol, the proposed beamforming scheme provides higher secrecy capacity than traditional approaches.
Cooperative communications, in which multiple nodes help each other transmit messages, has been widely acknowledged as an effective way to improve system performance [1–3]. However, due to the broadcast property of radio transmission, wireless communication is vulnerable to eavesdropping which consequently makes security schemes of great importance as a promising approach to communicate confidential messages.
The traditional secure communication schemes rely on encryption techniques where secret keys are used. However, as the high-layer secure protocols have attracted growing attacks in recent years, the implementation of security schemes at physical layer becomes a hotspot. It was first proved by Wyner that it is possible to communicate perfectly at a non-zero rate without a secret key if the eavesdropper has a worse channel than the destination . This work was extended to Gaussian channels in and to fading channels in . Recently, there has been considerable work on secure communication in wireless relay networks (WRNs) [7–15]. A widely acknowledged measurement of system security in WRNs is the maximal rate of secret information exchange between source and destination which is defined as secrecy capacity. A decode-and-forward (DF)-based cooperative beamforming scheme which completely nulls out source signal at eavesdropper(s) was proposed in , and this work was extended to the amplify-and-forward (AF) protocol and cooperative jamming in . Hybrid beamforming and jamming was investigated in where one relay was selected to cooperate and the other to make intentional interference in a DF network. Combined relay selection and cooperative beamforming schemes for DF networks were proposed in where two best relays were selected to cooperate. The authors of [11, 12] considered the scenario where the relay(s) could not be trusted in cooperative MIMO networks. Additionally, a new metric of system security is brought up in as intercept probability and optimal relay selection schemes for AF and DF protocols based on the minimization of intercept probability were proposed.
In earlier works, it is widely assumed that the relays have access to instantaneous channel state information (CSI) of relay-eavesdropper (RE) channels [7, 8, 13–15]. This assumption is ideal but unpractical in a real-life wiretap attack since the malicious eavesdropper would not be willing to share its instantaneous CSI. Thus, security schemes using instantaneous CSI of the eavesdropper cannot be adopted anymore. However, the instantaneous CSI of relay-destination (RD) channels is available since the destination is positive. The statistical information of the RE channels is also available through long-term supervision of the eavesdropper's transmission . It is worth mentioning that even if the relays do not have access to the perfect CSI of RD channels, they can still estimate these channels by training sequences and perform beamforming based on the estimated CSI .
Our focus is on secrecy capacity, and we are interested in maximizing it with appropriate weight designs of relays. The remainder of this paper is organized as follows. Section 2 introduces system model under AF and DF protocols using relay beamforming. The optimization problem in an AF network is addressed and solved in Section 3 along with some analyses of secrecy capacity. Section 4 provides the optimal beamforming design for a DF network along with a surprising finding that considering the eavesdropper sometimes may not be necessary. Numerical results are given in Section 5 to compare the performances of different designs, and Section 6 provides some concluding remarks.
2. System model
Consider a cooperative wireless network consisting of a source node S, a legitimate destination D, an eavesdropper E, and M relays Ri, i = 1,…, M as shown in Figure 1. Each node is equipped with single antenna working in half-duplex mode. Assume that there is no direct link between the source and the destination/eavesdropper, i.e., neither the destination nor the eavesdropper is in the coverage area of the source. For notational convenience, we denote the source-relay (SR) channels as fi, the RD channels as gi, and the RE channels as hi. All the channels are modeled as independent and identically distributed (i.i.d.) Rayleigh fading channels, i.e., , , and . Considering the path loss effect and setting the path loss exponent to 4 (for an urban environment), we have , , and , where dAB is the distance between nodes A and B. We assume the relays to know instantaneous CSI of SR channels and RD channels, but only statistical information of RE channels. Without loss of generality, we also assume the additive noises to be i.i.d. and follow a distribution.
In an AF protocol, the source broadcasts in the first hop where the information symbol s is selected from a codebook and is normalized as E|s|2 = 1, and Ps is the transmit power. The received signal at Ri is
where vi is the additive noise at Ri.
In the second hop, each relay forwards a weighted version of the noisy signal it just received. More specifically, Ri normalizes ri with a scaling factor and then transmits a weighted signal ti = wiρiri. The transmit power of Ri is Pi = |wi|2. The received signal at the destination is
where w = (w1, …, wM)T, ρfg = (ρ1f1g1, …, ρMfMgM)T, ρg = (ρ1g1, …, ρMgM)T, v = (v1, …, vM)T, and vD represents additive white Gaussian noise (AWGN) at the destination. The total relay transmit power is wHw = P.
Meanwhile, the eavesdropper also gets a copy of s:
where ρfh = (ρ1f1h1, …, ρMfMhM)T, ρh = (ρ1h1, …, ρMhM)T, and vE represents AWGN at the eavesdropper.
In a DF protocol, the first hop is the same as in an AF protocol. While in the second hop, instead of simply amplifying the received signal, Ri decodes the message s and multiplies it with a weighted factor wi to generate the transmit signal ti = wis. The transmit power of Ri is still Pi = |wi|2. The received signals at the destination and the eavesdropper can be expressed, respectively, as
where g = (g1, …, gM)T and h = (h1, …, hM)T.
3. Distributed beamforming design for AF
In the following sections, we consider the security issue of the above relay network. The metric of interest is secrecy capacity which is defined as
where , , and γD and γE are received signal-to-noise ratios (SNRs) at the destination and the eavesdropper, respectively. We aim to improve CS by exploiting appropriate beamforming designs. The following subsection describes the proposed beamforming design for an AF network.
3.1 Proposed design for AF (P-AF)
In distributed beamforming schemes, the relays compute the received SNRs at the destination and the eavesdropper from Equations 2 and 3, respectively, as
where Γg = diag(ρ12|g1|2, …, ρM2|gM|2), , and Now we discuss how to design w to maximize CS, and the proposed solution is denoted by . It is obvious that maximizing CS is equivalent to maximizing . Hence, in what follows, the objective function will be .
where . This is a product of two correlated Rayleigh quotients which is generally difficult to maximize. However, it would be much easier to get a suboptimal solution if we approximate the objective function to a single Rayleigh quotient.
Rewrite the optimization problem as
Denote the matrices Dh + P- 1I, Γg + P- 1I, and Γh + P- 1I as A, B, and C, respectively. For simplicity, we also let ai, bi, and ci represent the i th diagonal entry of A, B, and C, respectively, and define p = (P1, …, PM)T.
Since Pi = |wi|2, the denominator can be rewritten as . According to the Taylor series expansion ,
if we expand f(p) at . Since
where and , we have . Substituting this partial derivative into (11), we further have where . It can be proved that K is negligible either with small P or large P if we make a commonly used assumption that the SR distances are about the same (see Appendix for details). Thus, we omit this part and rewrite f(p) approximately as
So the optimization problem in (9) can be approximated to
This is a single Rayleigh quotient problem. It has been reported in that if U is Hermitian and V is positive definite Hermitian, for any non-zero column vector x, we have where λmax(V-1U) is the largest eigenvalue of V-1U. The equality holds if x = cumax(V-1U) where c can be any non-zero constant and umax(V-1U) is the unit-norm eigenvector of V-1U corresponding to λmax(V-1U). As a result, the optimal solution to (14) is
where Φ = A- 1B- 1C(PsρfgρfgH + Γg + P- 1I).
To show the agreement of the approximated denominator and the exact denominator, we calculated them numerically, and the results are shown in Figure 2. The channel information we used are listed in Table 1 where f = (f1, …, fM)T, g = (g1, …, gM)T, and . f and g are generated randomly.
For comparison purpose, we present two other beamforming designs. First, for the optimization of a product of two correlated Rayleigh quotient problems, a method was proposed recently in to maximize the upper and lower bounds. Note that where and . is bounded as
As a result, the bounds maximization design for AF (B-AF) should be
We also address the traditional design for AF (T-AF) where the eavesdropper is ignored and the goal is to maximize CD. It can be easily proved that the optimal solution is where cAF is a constant chosen to satisfy .
3.2 Discussion about secrecy capacity in AF networks
It is natural to conjecture that secrecy capacity would grow as the eavesdropper moved away or as the total relay transmit power increased. However, we find that this conjecture is not always right.
For simplicity, we assume the distances between relays are much smaller than those between the relays and the source, so the path losses of the SR channels are almost the same. The same assumption is also made to the destination/eavesdropper. Denote the SR, RD, and RE distances as dSR, dRD, and dRE, respectively, and the corresponding channel variances as , , and , respectively.
Proposition 1.If the destination is much nearer to the relays than the eavesdropper is in an AF network, CSdoes not always grow as the total relay transmit power increases, and a suboptimal value of the total relay transmit power is found as
Proof. Recall that . No matter how we design the beamforming vector w, is bounded as
Due to the difficulty of calculating the eigenvalues of Φ, we replace the non-diagonal elements in Φ with their mean value 0 and the i th diagonal element with where . Thus, Φ becomes λ(P)I after replacement.
Define . Now we investigate the monotonicity of CS(P). The first-order derivatives of CS(P) and λ(P) can be computed, respectively, as
By setting , we obtain the positive stationary point of CS(P) as described in (18).
If dRE > dRD (), ∀P∈(0, Psubopt), we have ; ∀P∈(Psubopt, + ∞), we have . Hence, if the destination is much nearer than the eavesdropper is, CS(P) is an increasing function over (0, Psubopt) and a decreasing function over (Psubopt, + ∞), which means that CS(Psubopt) is the maximum of CS(P).
This monotonicity of CS and the accuracy of Psubopt under the case of dRE > dRD will be verified in the next section. It needs to be pointed out that the above analysis is not for any certain design, so the optimal value of P for a certain design would be different from but around Psubopt. It also needs to be pointed out that the replacement of the channel coefficients in Φ with their mean values may result in the loss of the security benefit that is supposed to be achieved by exploiting the perfect CSI of SR and RD channels. This loss does not affect the monotonicity of CS greatly under the case of dRE > dRD because the destination is much nearer and therefore much more advantageous in communication than the eavesdropper is. However, when dRE < dRD (or dRE = dRD), such replacement becomes inappropriate, since the instantaneous CSI of fi and gi improves the system security significantly.
We can further compute the second-order derivatives of CS(P) and λ(P), respectively, as
It can be observed from (22) that the positivity of depends on the value of P. Thus, CS(P) is neither convex nor concave.
Remark 1. In an AF network, if the total relay transmit power is large, the AWGNs in the second hop are negligible compared to the forwarded versions of the AWGNs in the first hop. Thus, can be approximately written as
This equation does not involve , which implies CS is a constant in this case wherever the eavesdropper is.
4. Distributed beamforming design for DF
This section focuses on the security-oriented beamforming design for DF protocol. Similar to the design for AF protocol, the mission is to find the optimal design under a total relay transmit power constraint to maximize secrecy capacity.
4.1 Proposed design for DF (P-DF)
From Equations 4 and 5, the received SNRs at the destination and the eavesdropper are obtained, respectively, as follows:
Let be the optimal solution of the proposed design, then
For comparison purpose, we also address the traditional design for DF (T-DF) and denote the solution by . The optimization problem is formulated as , and the optimal solution is obviously where cDF is a constant chosen to satisfy the power constraint .
4.2 Discussion about secrecy capacity in DF networks
It is a natural thought that no matter under what channel assumption, secrecy capacity achieved by security-oriented designs would be higher than that achieved by traditional designs. However, the fact is that these designs may have the same performance which means that sometimes we can just ignore the eavesdropper.
Remark 2. In a DF network, if the RE distances are about the same (which is widely assumed), it is unnecessary to consider the eavesdropper as the security-oriented design and the traditional design are indeed the same.
Noticing that in this scenario, we can write Dh as . Thus, one can rewrite (27) as . Since
we have umax(I + PggH) = umax(PggH). Thus, which is the same as .
5. Numerical results
In this section, we investigate the performance of the above beamforming designs numerically. The simulation environment follows the model of Section 2. We perform Monte Carlo experiments consisting of 10,000 independent trials to obtain the average results.
Assume the number of relays is M = 6, and the source transmit power is Ps = 10 dB. In order to show the influence of the RE distance in AF protocol, we fix the source at (0,0), the destination at (2,0), and the relays at (1,0) and move the eavesdropper from (1.25,0) to (5,0). We assume that the distances between relays are much smaller than SR/RD/RE distances. Therefore, the SR channels and RD channels follow a distribution, and the RE distance dRE varies from 0.25 to 4.
Figure 3 shows the relationship between average secrecy capacity and total relay transmit power with the eavesdropper in different locations using P-AF design. We can see that if the eavesdropper is nearer to the relays than the destination is, the relays should use the maximal power to transmit. However, if the destination is much nearer, there is an optimal value of total relay transmit power which is about 12 dB under the case of dRE = 2, while the theoretical value in (18) is which is not very accurate but close. The reason is that Psubopt satisfies while the optimal power for P-AF design should satisfy . However, it is difficult to express λmax(Φ) in terms of the total relay transmit power and the channel coefficients, not to mention to solve the latter equation analytically. It can also be seen that as the total relay transmit power increases, the secrecy capacity tends to keep constant no matter where the eavesdropper is.
Figure 4 compares different AF beamforming designs. It can be seen that the B-AF design shows a slight advantage over the T-AF design only when the total relay transmit power is small in the dRE = 1/2 case, while our proposed design always performs the best.
The relationship between average secrecy capacity and total relay transmit power with the eavesdropper in different locations using P-DF design is demonstrated in Figure 5. We still assume the RE distances to be the same. Results show that the secrecy capacity of a DF network grows as the total relay transmit power increases or as the eavesdropper moves away.
To verify Remark 2, we now examine the P-DF design and T-DF design under different variance assumptions of the RE channels.
The average secrecy capacities of P-DF and T-DF designs under different RE channel assumptions are demonstrated in Figure 6. Our design outperforms the traditional design in case 1 and case 2. While in case 3, the two designs have the same performance. This indicates that the greater the RE channels differ from each other, the more superior the P-DF design is. If the RE distances are almost the same, the eavesdropper can be ignored.
In this paper, we focused on security-oriented distributed beamforming designs for relay networks in the presence of a passive eavesdropper. We provided two beamforming designs under a total relay transmit power constraint, one of which is for AF and the other is for DF. Each design is to maximize secrecy capacity by exploiting information of SR, RD, and RE channels. To derive the beamforming solution for AF requires approximating the optimization objective by using the Taylor series expansion, while the solution for DF is obtained much more easily. We also found that secrecy capacity does not always grow if the relays use more power to transmit or if the eavesdropper gets farther from the relays, and that taking the eavesdropper into consideration is not always necessary. Moreover, for AF, we derived a suboptimal value of the total relay transmit power if the destination is nearer than the eavesdropper is. Numerical results showed the efficiency of the proposed designs.
we have with small P, and
with large P. If we make the assumption that the SR distances are all about the same, i.e., the 's are about the same (which is also assumed in ), and replace |fi|2 in the expression of with its mean value , we have
Thus, K ≈ 0 with large P.
Sendonaris A, Erkip E, Aazhang B: User cooperative diversity-part I: system description. IEEE Trans. Commun. 2003, 51(11):1927-1938. 10.1109/TCOMM.2003.818096
Dong L, Zhu H, Petropulu AP, Poor HV: Secure wireless communication via cooperation. In The 46-th Annual Allerton Conference on Communication, Control and Computing. Urbana-Champaign, IL, USA; 23–26 September 2008:1132-1138.
This work is supported by the Natural Science Foundation of China under Grants 61372126 and 61302101, and the open research fund of National Mobile Communications Research Laboratory in Southeast University under Grant 2012D11.
Authors and Affiliations
The Key Lab of Broadband Wireless Communication and Sensor Network Technology (Ministry of Education), Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu, 210003, China
Mujun Qian, Chen Liu & Youhua Fu
National Mobile Communications Research Laboratory, Southeast University, Nanjing, Jiangsu, 210096, China
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Qian, M., Liu, C. & Fu, Y. Distributed beamforming designs to improve physical layer security in wireless relay networks.
EURASIP J. Adv. Signal Process.2014, 56 (2014). https://doi.org/10.1186/1687-6180-2014-56
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500392.45/warc/CC-MAIN-20230207071302-20230207101302-00602.warc.gz
|
CC-MAIN-2023-06
| 21,514 | 83 |
https://cms.math.ca/cjm/msc/55P45
|
math
|
1. CJM 2009 (vol 62 pp. 284)
||Self-Maps of Low Rank Lie Groups at Odd Primes|
Let G be a simple, compact, simply-connected Lie group localized at an odd prime~p. We study the group of homotopy classes of self-maps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.
Keywords:Lie group, self-map, H-map
Categories:55P45, 55Q05, 57T20
2. CJM 2007 (vol 59 pp. 1154)
||$k(n)$-Torsion-Free $H$-Spaces and $P(n)$-Cohomology |
The $H$-space that represents Brown--Peterson cohomology
$\BP^k (-)$ was split by the second author into indecomposable
factors, which all have torsion-free homotopy and homology.
Here, we do the same for the related spectrum $P(n)$, by constructing
idempotent operations in $P(n)$-cohomology $P(n)^k ($--$)$ in the style
of Boardman--Johnson--Wilson; this relies heavily on the
Ravenel--Wilson determination of the relevant Hopf ring.
The resulting $(i- 1)$-connected $H$-spaces $Y_i$ have
free connective Morava $\K$-homology $k(n)_* (Y_i)$, and may be
built from the spaces in the $\Omega$-spectrum for $k(n)$
using only $v_n$-torsion invariants.
We also extend Quillen's theorem on complex cobordism to show that
for any space $X$, the \linebeak$P(n)_*$-module $P(n)^* (X)$ is generated
by elements of $P(n)^i (X)$ for $i \ge 0$. This result is essential
for the work of Ravenel--Wilson--Yagita, which in many cases allows
one to compute $\BP$-cohomology from Morava $\K$-theory.
3. CJM 2003 (vol 55 pp. 181)
||Homotopy Decompositions Involving the Loops of Coassociative Co-$H$ Spaces |
James gave an integral homotopy decomposition of $\Sigma\Omega\Sigma X$,
Hilton-Milnor one for $\Omega (\Sigma X\vee\Sigma Y)$, and Cohen-Wu gave
$p$-local decompositions of $\Omega\Sigma X$ if $X$ is a suspension. All
are natural. Using idempotents and telescopes we show that the James and
Hilton-Milnor decompositions have analogues when the suspensions are
replaced by coassociative co-$H$ spaces, and the Cohen-Wu decomposition
has an analogue when the (double) suspension is replaced by a coassociative,
cocommutative co-$H$ space.
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891485.97/warc/CC-MAIN-20180122153557-20180122173557-00566.warc.gz
|
CC-MAIN-2018-05
| 2,270 | 33 |
https://brainly.com/question/134269
|
math
|
So it l;ooks as though you have a large number minus a number. the equation answer is 36 minus 13 equals 23. 23 plus 5 equals 28 = 47 - 19. i hope i helped you out. remember that it takes time. i had tried three other combination before i got this answer. study, work hard. and focus. that will get you a lond way in school.
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560284411.66/warc/CC-MAIN-20170116095124-00423-ip-10-171-10-70.ec2.internal.warc.gz
|
CC-MAIN-2017-04
| 324 | 1 |
https://nrich.maths.org/public/leg.php?code=72&cl=2&cldcmpid=8283
|
math
|
It would be nice to have a strategy for disentangling any tangled ropes...
Can you tangle yourself up and reach any fraction?
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
Can all unit fractions be written as the sum of two unit fractions?
Find out what a "fault-free" rectangle is and try to make some of your own.
This challenge asks you to imagine a snake coiling on itself.
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Can you find sets of sloping lines that enclose a square?
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
A collection of games on the NIM theme
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
It starts quite simple but great opportunities for number discoveries and patterns!
Can you explain the strategy for winning this game with any target?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823067.51/warc/CC-MAIN-20171018180631-20171018200631-00690.warc.gz
|
CC-MAIN-2017-43
| 6,531 | 50 |
https://www.savouidakis.gr/en/recipies/lachanika/manitaria-gemista-me-lachanika/
|
math
|
• 1 kg fresh white mushrooms (large)
• "Lof" Extra Virgin Olive Oil
• 2 carrots
• 1 yellow pepper
• 1 red pepper
• 1 bunch of spring onions
• 1 large tomato, not too ripe
• 1/2 bunch of parsley
• Freshly ground pepper
• Clean the mushrooms with a very lightly wet cloth and cut their stalks.
• Sprinkle them with the juice of a lemon, so that they do not turn black.
• Grease a baking dish and put the mushrooms inside.
• Sprinkle them with some "Lof" Extra Virgin Olive Oil and season with salt and pepper.
• In a pan, add some "Lof" Extra Virgin Olive Oil and sauté all the vegetables and the tomato, after cutting them in small pieces.
• Sauté them for 5 mins and stuff the mushrooms.
• Cook for 20 mins in a preheated oven at 180°C using the upper & lower heating elements and the convection function.
• Garnish with finely chopped fresh parsley.
|
s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103036176.7/warc/CC-MAIN-20220625220543-20220626010543-00058.warc.gz
|
CC-MAIN-2022-27
| 887 | 17 |
https://boundaryvalueproblems.springeropen.com/articles/10.1186/1687-2770-2013-223
|
math
|
- Open Access
Multiple solutions for the -Laplacian problem involving critical growth with aparameter
Boundary Value Problems volume 2013, Article number: 223 (2013)
By energy estimates and establishing a local condition, existence of solutions for the-Laplacian problem involving critical growth in abounded domain is obtained via the variational method under the presence ofsymmetry.
MSC: 35J20, 35J62.
In recent years, the study of problems in differential equations involving variableexponents has been a topic of interest. This is due to their applications in imagerestoration, mathematical biology, dielectric breakdown, electrical resistivity,polycrystal plasticity, the growth of heterogeneous sand piles and fluid dynamics,etc. We refer readers to [1–7] for more information. Furthermore, new applications are continuing to appear,see, for example, and the references therein.
With the variational techniques, the -Laplacian problems with subcritical nonlinearities havebeen investigated, see [9–13]etc. However, the existence of solutions for -Laplacian problems with critical growth is relatively new.In 2010, Bonder and Silva extended the concentration-compactness principle of Lions to the variableexponent spaces, and a similar result can be found in . After that, there have been many publications for this case, see [16–19]etc.
In this paper, we study the existence and multiplicity of solutions for the quasilinearelliptic problem
where , () is a bounded domain with smooth boundary, is a real parameter, , are continuous functions on with
Related to f, we assume that is a Carathéodory function satisfying for every , and the subcritical growth condition:
(f1) for all , where is a continuous function in satisfying , .
For , we suppose that f satisfies the following:
(f2) there are constants and such that for every , a.e. in Ω,
(f3) there are constants and a continuous function , , with , such that for every , a.e. in Ω,
(f4) there are , and with such that
Now we state our result.
Theorem 1.1 Assume that (1.2), (1.3) and(f1)-(f4) are satisfied with, is odd in s. Then,given, there existssuch that problem (1.1) possesses atleast k pairs of nontrivial solutions forall.
(g1) there is such that
(g2) , odd with respect to t and
(g3) for all and a.e. in Ω, where .
Moreover, they assumed that
and the result is the following theorem.
Theorem 1.2 Assume that (1.2), (1.3), (1.4) and(g1)-(g3) are satisfied with. Then there exists asequencewithsuch that for, problem (1.1) has atleast k pairs of nontrivial solutions.
Note that (f2) is a weaker version of (g3). This conditioncombined with (f1) and the concentration-compactness principle in will allow us to verify that the associated functional satisfies the condition below a fixed level for sufficiently small. Conditions (f3) and(f4) provide the geometry required by the symmetric mountain pass theorem . Compared with (g2), there is no condition imposed on fnear zero in Theorem 1.1. Furthermore, we should mention that our Theorem 1.1 improvesthe main result found in . In that paper, the authors considered only the case where is constant, while in our present paper, we have showedthat the main result found in is still true for a large class of functions.
The paper is organized as follows. In Section 2, we introduce some necessarypreliminary knowledge. Section 3 contains the proof of our main result.
We recall some definitions and basic properties of the generalized Lebesgue-Sobolevspaces and , where is a bounded domain with smooth boundary. And Cwill denote generic positive constants which may vary from line to line.
For any , we define the variable exponent Lebesgue space
with the norm
where is the set of all measurable real functions defined onΩ.
Define the space
with the norm
By , we denote the subspace of which is the closure of with respect to the norm . Further, we have
There is a constantsuch that for all,
So, and are equivalent norms in . Hence we will use the norm for all .
Set. For, we have:
Ifwitha.e. in Ω, then thereexists the continuous embedding.
Ifandfor any, the embeddingis compact.
The conjugate space ofis, where. For anyand,
The energy functional corresponding to problem (1.1) is defined on as follows:
Then and ,
We say that is a weak solution of problem (1.1) in the weak sense iffor any ,
So, the weak solution of problem (1.1) coincides with the critical point of. Next, we need only to consider the existence of criticalpoints of .
We say that satisfies the condition if any sequence , such that and as , possesses a convergent subsequence. In this article, weshall be using the following version of the symmetric mountain pass theorem .
Let, where E is a real Banach spaceand V is finite dimensional. Supposethatis an even functionalsatisfyingand
there is a constantsuch that;
there is a subspace W of E withand there issuch that;
consideringgiven by (ii), I satisfiesfor.
Then I possesses at leastpairs of nontrivial critical points.
Next we would use the concentration-compactness principle for variable exponent spaces.This will be the keystone that enables us to verify that satisfies the condition.
Let and be two continuous functions such that
Letbe a weakly convergent sequenceinwith weak limit u such that:
weakly in the sense of measures;
weakly in the sense of measures.
Also assume thatis nonempty. Then, for some countableindex set K, we have:
whereand S is the best constant in theGagliardo-Nirenberg-Sobolev inequality for variable exponents,namely
3 Proof of main results
Lemma 3.1 Assume that f satisfies (f1)and (f2) with. Then, given, there existssuch thatsatisfies thecondition for all, provided.
Proof (1) The boundedness of the sequence.
Let be a sequence, i.e., satisfies , and as . If , we have done. So we only need to consider the case that with . We know that
From (f2), we get
Notice that , , then from Lemmas 2.3, 2.4, , so . Let , then , and from the Hölder inequality,
In addition, from Lemma 2.2(2), we can also obtain that
So we have
From (3.1), (3.3) and (f1), we have
Noting that , we have that is bounded.
Up to a subsequence, in .
By Lemma 2.7, we can assume that there exist two measures μ,ν and a function such that
Choose a function such that , on and on . For any , and , let . It is clear that is bounded in . From , we can obtain , as , i.e.,
From (f1), by Lemma 2.7, we have
By the Hölder inequality, it is easy to check that
From (3.5), as , we obtain . From Lemma 2.7, we conclude that
Given , set
where S is given by (2.5). Considering , we have
We claim that . Indeed, if , this follows by (3.7). Otherwise, taking in (3.2), we obtain
Therefore, by (3.8), the claim is proved. As a consequence of this fact, we concludethat for all . Therefore, in . Then, with the similar step in , we can get that in . □
Next we prove Theorem 1.1 by verifying that the functional satisfies the hypotheses of Lemma 2.6. First, we recallthat each basis for a real Banach space E is a Schauder basis forE, i.e., given , the functional defined by
Lemma 3.2 Givenfor alland, there issuch that for all, .
Proof We prove the lemma by contradiction. Suppose that there exist and for every such that . Taking , we have for every and . Hence is a bounded sequence, and we may suppose, without loss ofgenerality, that in . Furthermore, for every since for all . This shows that . On the other hand, by the compactness of the embedding, we conclude that . This proves the lemma. □
Lemma 3.3 Suppose that f satisfies (f3),then there existandsuch thatfor all.
Proof Now suppose that , with , . From (f3), we know that
Consequently, considering to be chosen posteriorly by Lemma 3.2, we have, for all and j sufficiently large,
Now taking such that and noting that , so , if . We can choose such that . Next, we take such that for ,
for every , , the proof is complete. □
Lemma 3.4 Suppose that f satisfies (f4),then, given, there exist asubspace W ofand a constantsuch thatand.
Proof Let and be such that , and . First, we take with . Considering , we have . Let and such that , and . Next, we take with . After a finite number of steps, we get such that , , and for all . Let , by construction, , and for every ,
consider the case that , then . Now it suffices to verify that
From condition (f4), given , there is such that for every , a.e. x in ,
Consequently, for and ,
where and . Observing that W is finite dimensional, we have, , and the inequality is obtained by taking. The proof is complete. □
Proof of Theorem 1.1 First, we recall that , where and are defined in (3.9). Invoking Lemma 3.3, we find, and satisfies (i) with . Now, by Lemma 3.4, there is a subspace W of with such that satisfies (ii). By Lemma 3.1, satisfies (iii). Since and is even, we may apply Lemma 2.6 to conclude that possesses at least k pairs of nontrivial criticalpoints. The proof is complete. □
Bocea M, Mihăilescu M: Γ-convergence of power-law functionals with variable exponents. Nonlinear Anal. 2010, 73: 110-121. 10.1016/j.na.2010.03.004
Bocea M, Mihăilescu M, Popovici M: On the asymptotic behavior of variable exponent power-law functionals andapplications. Ric. Mat. 2010, 59: 207-238. 10.1007/s11587-010-0081-x
Bocea M, Mihăilescu M, Pérez-Llanos M, Rossi JD: Models for growth of heterogeneous sandpiles via Mosco convergence. Asymptot. Anal. 2012, 78: 11-36.
Chen Y, Levine S, Rao R: Variable exponent, linear growth functionals in image processing. SIAM J. Appl. Math. 2006, 66: 1383-1406. 10.1137/050624522
Fragnelli G: Positive periodic solutions for a system of anisotropic parabolic equations. J. Math. Anal. Appl. 2010, 367: 204-228. 10.1016/j.jmaa.2009.12.039
Halsey TC: Electrorheological fluids. Science 1992, 258: 761-766. 10.1126/science.258.5083.761
Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 1987, 9: 33-66.
Boureanu MM, Udrea DN:Existence and multiplicity result for elliptic problems with-growth conditions. Nonlinear Anal., Real World Appl. 2013, 14: 1829-1844. 10.1016/j.nonrwa.2012.12.001
Boureanu MM, Preda F: Infinitely many solutions for elliptic problems with variable exponent andnonlinear boundary conditions. Nonlinear Differ. Equ. Appl. 2012, 19(2):235-251. 10.1007/s00030-011-0126-1
Chabrowski J, Fu Y:Existence of solutions for -Laplacian problems on a bounded domain. J. Math. Anal. Appl. 2005, 306: 604-618. 10.1016/j.jmaa.2004.10.028
Dai GW, Liu DH:Infinitely many positive solutions for a -Kirchhoff-type equation involving the-Laplacian. J. Math. Anal. Appl. 2009, 359: 704-710. 10.1016/j.jmaa.2009.06.012
Fan XL, Zhang QH:Existence of solutions for -Laplacian Dirichlet problem. Nonlinear Anal. 2003, 52: 1843-1852. 10.1016/S0362-546X(02)00150-5
Mihăilescu M:On a class of nonlinear problems involving a -Laplace type operator. Czechoslov. Math. J. 2008, 58(133):155-172.
Bonder JF, Silva A: The concentration compactness principle for variable exponent spaces andapplications. Electron. J. Differ. Equ. 2010., 2010: Article ID 141
Fu YQ:The principle of concentration compactness in spaces and its application. Nonlinear Anal. 2009, 71: 1876-1892. 10.1016/j.na.2009.01.023
Silva, A: Multiple solutions for the -Laplace operator with critical growth. Preprint
Alves CO, Barrwiro JLP:Existence and multiplicity of solutions for a -Laplacian equation with critical growth. J. Math. Anal. Appl. 2013, 403: 143-154. 10.1016/j.jmaa.2013.02.025
Fu YQ, Zhang X:Multiple solutions for a class of -Laplacian equations in involving the critical exponent. Proc. R. Soc., Math. Phys. Eng. Sci. 2010, 466(2118):1667-1686. 10.1098/rspa.2009.0463
Bonder, JF, Saintier, N, Silva, A: On the Sobolev trace theorem for variableexponent spaces in the critical range. Preprint
Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7
Silva EAB, Xavier MS: Multiplicity of solutions for quasilinear elliptic problems involving criticalSobolev exponents. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2003, 20(2):341-358. 10.1016/S0294-1449(02)00013-6
Fan X, Zhao D:On the space and . J. Math. Anal. Appl. 2001, 263: 424-446. 10.1006/jmaa.2000.7617
Kovacik O, Rakosnik J:On spaces and . Czechoslov. Math. J. 1991, 41: 592-618.
Lindenstrauss J, Tzafriri L: Classical Banach Spaces, I. Springer, Berlin; 1977.
Marti JT: Introduction to the Theory of Bases. Springer, New York; 1969.
The authors would like to express their gratitude to the anonymous referees forvaluable comments and suggestions which improved our original manuscript greatly. Thefirst author is supported by NSFC-Tian Yuan Special Foundation (No. 11226116),Natural Science Foundation of Jiangsu Province of China for Young Scholar (No.BK2012109), the China Scholarship Council (No. 201208320435), the FundamentalResearch Funds for the Central Universities (No. JUSRP11118, JUSRP211A22). The secondauthor is supported by NSFC (No. 10871096). The third author is supported by GraduateEducation Innovation of Jiangsu Province (No. CXZZ13-0389).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
About this article
Cite this article
Yang, Y., Zhang, J. & Shang, X. Multiple solutions for the -Laplacian problem involving critical growth with aparameter. Bound Value Probl 2013, 223 (2013). https://doi.org/10.1186/1687-2770-2013-223
- -Laplacian problem
- critical Sobolev exponents concentration-compactness principle
|
s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145500.90/warc/CC-MAIN-20200221080411-20200221110411-00475.warc.gz
|
CC-MAIN-2020-10
| 13,495 | 123 |
https://nctbsolution.com/nctb-class-6-science-chapter-1-scientific-process-and-measurement-solution/
|
math
|
NCTB Class 6 Science Chapter 1 Scientific Process and Measurement Solution
Bangladesh Board Class 6 Science Solution Chapter 1 Scientific Process and Measurement Solution Exercises Question and Answer by Experienced Teacher.
NCTB Solution Class 6 Chapter 1 Scientific Process and Measurement:
|NCTB Bangladesh Board|
Scientific Process and Measurement
Fill in the gaps:-
(1) There are two types of unit of measurements —— and——–.
Ans:-Fundamental and Derived.
(2) As per the international system the number of funder mental S.I units is——.
Ans :- 7
(3) In all the systems of measurement the unit of time is —-.
Multiple choice questions:-
(1)The unit of mass is— (b) kilogram
(2) The space deceived by a brick is called — (b)Volume.
(3) The volume of the stone in fig B—- (a)5cc [15cc-10cc]
(4) To determine the volume of the substance in fig A, multiplication of– (c) There units is required.
Short answer questions:-
(1) Measurement is necessary in every events of our life like, purchasing rice or dal from the grocers, making clothes or starting and finishing classes in time. Without measurement it is difficult to decide how many rooms can be built within the certain area of a house or which room will be of what size. Moreover, While cooking, it is essential to add exact amounts. In fact a correct measurement is necessary in every field.
(2) The fundamental units are –
(a) The unit of length is meter.
(b) The unit of mass is kilogram.
(c) The unit of time is second.
(d) The unit of temperature is Kelvin.
(e) The unit of electric current is ampere.
(f) The unit of luminous intensity is candela.
(g) The unit of substance is mole.
(3) The multiples and sub-multiples of units are necessary because. to measure the length of pencil and the thickness of acorn we use centimetre and millimetre respectively, to measure long time we use, minute, hour, days, month, year, to measure smaller mass we use gram etc.
(4) The main difference between area and volume in case of measurement is,
In case of, measurement of area, we have to multiply 2 units. [Because, area = length x breadth]
But in case of measurement of volume, we have to multiply 3 units.
[Because, volume = length x breadth x height]
(1) A surface – Length = 5meters
breath = 4meter
There fore, the area of the surface is, Area = length x breadth
= 20 square meter.
(2) The area of a room= 20 square meters
room’s length = 5meters
There fore, the breadth of the room is, breadth = Area/length
= 20/5 = 4meter.
(3) A box is long = 20cm
broad = 10cm
high = 5cm.
There fore, the volume of the box is,= long x broad x high
= 1000 cube cm
There fore, the volume of 100 similar boxes is, 1000×100 = 100000 cube cm
(1) (a) The unit of length is meter.
(b) Fareham’s mother want to put another table of same size as the reading table. So we have measure the size, We have to measure the length.
(1)(a) The area of recoding room is 40 square meters length of the recoding room is 10 meters.
There fore, the breadth = Area/length = 40/10 = 4 meters.
(d) The length of the recoding table = 1meter
breadth is 50 cm = 100cm
There fore, the area of recoding table is,
Length x breadth = 100×50= 5000 square cm
= 5000/100×10 square cm
= 0.5 square cm meter.
There fore, the total area covered by two table is,
(0.5+0.5) = 1 square metres.
The area of recoding room is – 40 square meters.
There fore, after keeping two table. (40-1)=39 square meters area will remain unoccupied in the room.
(2) (a) Candela is the S.I unit. Like, to measure area of a class room, we have to multiply two units (area= length x breadth) and to measure volume, we have to multiply three units (volume = length x breadth x height). These combined units are known as derived units.
(2) (c) 5 metric tons = (5×1000) kilograms
= 5000 kilograms
There fore, 5000 kilograms of jute was exported this year by Hossain Uddin Sarkar.
(d) M.k.S system is advantageous between the two system mentioned in the above because all countries dose not use F.P.S system. If we use F.P.S system then problem may occurs. But according to International system of units, M.K.S system was introduced for all countries. So if we use M.K.S system then we do not face any problem.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100164.15/warc/CC-MAIN-20231130000127-20231130030127-00120.warc.gz
|
CC-MAIN-2023-50
| 4,226 | 64 |
https://books.google.ca/books?id=aG_vAAAAMAAJ&q=Wheeler+Peak&dq=related:ISBN0120790696&lr=&source=gbs_word_cloud_r&hl=en
|
math
|
An Eye for Fractals: A Graphic & Photographic Essay
Fractional geometry posits that a natural visual complexity can arise from iteration of simple rules and simple shapes. An Eye for Fractals is a fascinating study of the converse premise: that nature’s complexity implies an underlying simplicity that can be traced back to fractal geometry.The book effectively integrates art with science, illustrating the natural occurrence of mathematics and geometry in lava flows, kelp beds, cloud formations and aspen groves. The book is enhanced with more than 150 photographs and drawings, including some color illustrations. An Eye for Fractals is a beautiful introduction to fractal geometry, a graphic, visual approach that should appeal to all who feel the fascination of this artful mathematics.
6 pages matching Wheeler Peak in this book
Results 1-3 of 6
What people are saying - Write a review
We haven't found any reviews in the usual places.
Ansel Adams Anza Borrego Desert Arastradero Preserve artistic Aspens attracted to infinity attractor point Barnsley behavior Benoit Mandelbrot Bifurcation Diagram California chaos circle collage theorem color complex number complex plane computer graphics Curie Curves cycles dimension 2.9 dimensional dust equation EYE FOR FRACTALS fern formations frac fractal dimension fractal geometry fractal structure grid growth rate H. O. Peitgen imaginary number itera iteration of simple Iterations 50 Julia sets Kelp Pool Koch snowflake landscape linear Lobos log(n log(number of pieces M. F. Barnsley Mandelbrot set mathematics matter Maui ment rule midpoint natural world nonlinear North Cascades number of iterations outline Pahoehoe painting phase transition photograph physics population R-square random range of scale real number replacement rule result Road to Hana Rock Form self-similar sense shape Sierpinski triangle Sierra Nevada simple rules sion square starting point three transformations tions trans tree Utah visual cortex Wheeler Peak zero zoom sequence
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886416.17/warc/CC-MAIN-20180116105522-20180116125522-00115.warc.gz
|
CC-MAIN-2018-05
| 2,011 | 7 |
http://garmasoftware.com/sample-size/sample-size-calculation-using-margin-of-error.php
|
math
|
Reply Nida Madiha says: March 6, 2015 at 3:30 am Thanks a lot for the fast answer. What confidence level do you need? A SurveyMonkey product. What is the population size? click site
Distribution, on the other hand, reflects how skewed the respondents are on a topic. This is a constant value needed for this equation. Reply RickPenwarden says: August 1, 2014 at 1:32 pm Thanks Matt! drenniemath 37.779 προβολές 11:04 17. https://www.sophia.org/tutorials/finding-sample-size-with-predetermined-margin-of-e--2
A 95% degree confidence corresponds to = 0.05. So in short, the 10 times formula is total nonsense. Hope this helps! Reply RickPenwarden says: March 3, 2015 at 10:17 am Hi Nida, Need help with your homework?
The region to the left of and to the right of = 0 is 0.5 - 0.025, or 0.475. One way to answer this question focuses on the population standard deviation. Leave this as 50% % For each question, what do you expect the results will be? Find Sample Size Given Margin Of Error Calculator The sample size calculator computes the critical value for the normal distribution.
A larger sample can yield more accurate results — but excessive responses can be pricey. Sample Size Equation Erison Valdez 10.909 προβολές 5:54 95% Confidence Interval - Διάρκεια: 9:03. There is a powerpoint of definitions and examples, as well as examples for you to do on your own. recommended you read ME = Critical value x Standard error = 1.96 * 0.013 = 0.025 This means we can be 95% confident that the mean grade point average in the population is 2.7
Now all you have to do is choose whether getting that lower margin of error is worth the resources it will take to sample the extra people. Sample Size Calculator Online Your example fits the bill. ProfessorSerna 39.483 προβολές 12:39 Φόρτωση περισσότερων προτάσεων… Εμφάνιση περισσότερων Φόρτωση... Σε λειτουργία... Γλώσσα: Ελληνικά Τοποθεσία περιεχομένου: Ελλάδα Λειτουργία περιορισμένης πρόσβασης: Ανενεργή Ιστορικό Βοήθεια Φόρτωση... Φόρτωση... Φόρτωση... Σχετικά με Τύπος Πνευματικά δικαιώματα Menu Search Create Account Sign In Don't lose your points!
For the purpose of this example, let’s say we asked our respondents to rate their satisfaction with our magazine on a scale from 0-10 and it resulted in a final average
View Mobile Version Υπενθύμιση αργότερα Έλεγχος Υπενθύμιση απορρήτου από το YouTube, εταιρεία της Google Παράβλεψη περιήγησης GRΜεταφόρτωσηΣύνδεσηΑναζήτηση Φόρτωση... Επιλέξτε τη γλώσσα σας. Κλείσιμο Μάθετε περισσότερα View this message in English Το Find Sample Size Given Margin Of Error And Confidence Level Calculator Unfortunately, it is sometimes much more expensive to incentivize or convince your target audience to take part. Sample Size Table If the population is N, then the corrected sample size should be = (186N)/( N+185).
Simply click here or go through the FluidSurveys website’s resources to enter our Survey Sample Size Calculator. get redirected here Many statisticians do not recommend calculating power post hoc. You designed your study to have a certain margin of error, based on certain assumptions. Afterwards, you can empirically confirm your margin The short answer to your question is that your confidence levels and margin of error should not change based on descriptive differences within your sample and population. Suppose you chose the 95% confidence level – which is pretty much the standard in quantitative research1 – then in 95% of the time between 85% and 95% of the population Margin Of Error Calculator Statistics
This means that you are 100% certainty that the information you collected is representative of your population. Thanks Reply RickPenwarden says: May 25, 2015 at 2:10 pm Hello Panos! Or, following on our previous example, it tells you how sure you can be that between 85% and 95% of the population likes the ‘Fall 2013’ campaign. navigate to this website T-Score vs.
Alternate scenarios With a sample size of With a confidence level of Your margin of error would be 9.78% 6.89% 5.62% Your sample size would need to be 267 377 643 How To Find Sample Size With Margin Of Error On Ti 84 Get Started *The FluidSurveys Sample Size Calculator uses a normal distribution (50%) to calculate your optimum sample size. Here are the z-scores for the most common confidence levels: 90% - Z Score = 1.645 95% - Z Score = 1.96 99% - Z Score = 2.576 If you choose
in what occasion should we use a particular number of confidence level? Reply Nida Madiha says: March 6, 2015 at 9:40 pm Thanks a lot Rick! If the entire population responds to your survey, you have a census survey. Margin Of Error Calculator Without Population Size Learn more You're viewing YouTube in Greek.
Join for free An error occurred while rendering template. Recently Added Descriptive Research: Defining Your Respondents and Drawing Conclusions Posted by FluidSurveys Team on July 18, 2014 Causal Research: Identifying Relationships and Making Business Decisions through Experimentation Posted by FluidSurveys The choice of t statistic versus z-score does not make much practical difference when the sample size is very large. my review here Hope this information helps!
If you send all 100 staff a survey invite, they are all in your potential sample. This section describes how to find the critical value, when the sampling distribution of the statistic is normal or nearly normal. If the population standard deviation is known, use the z-score. Sign In Sign In New to Sophia?
Reply RickPenwarden says: March 6, 2015 at 11:44 am Hi Nida, 95% is an industry standard in most research studies.
|
s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583660070.15/warc/CC-MAIN-20190118110804-20190118132804-00512.warc.gz
|
CC-MAIN-2019-04
| 6,000 | 15 |
http://www.yourarticlelibrary.com/tag/articles-on-input-output-analysis/
|
math
|
Input-Output Analysis: Features, Static and Dynamic Model! Input-output is a novel technique invented by Professor Wassily W. Leontief in 1951. It is used to analyse inter-industry relationship in order to understand the inter-dependencies and complexities of the economy and thus the conditions for maintaining equilibrium between supply and demand.
Tag Archives | Input-Output Analysis
Some of the importance of input-output analysis are as follows: (i) A producer can know from the input-output table, the varieties and quantities of goods which he and the other firms buy and sell to each other. In this way, he can make the necessary adjustments and thus improve his position vis-a-vis other producers.
Major limitations faced by input-output analysis are as follows: 1. Its framework rests on Leontiefs basic assumption of constancy of input co-efficient of production which was split up above as constant returns of scale and technique of production. The assumption of constant returns to scale holds good in a stationary economy, while that of constant […]
|
s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886107065.72/warc/CC-MAIN-20170821003037-20170821023037-00057.warc.gz
|
CC-MAIN-2017-34
| 1,066 | 4 |
http://www.amazon.co.uk/review/R5KWCGKOJ2QOI
|
math
|
19 of 21 people found the following review helpful
It's over my head,
This review is from: Taming the Infinite: The Story of Mathematics (Paperback)
This is university level pure mathematics so the Waterstone reviewer who wrote "guaranteed to illuminate even the most number-shy" could not have read it. I don't know where this paperback is supposed to fit in: it is not a text book and nor is it a layman's paperback but requires a good level of mathematical knowledge and a high intellect to get anywhere near grasping the concepts. A number of tantalising concepts could have made this book more interesting if they had been explained eg. what is 196,884 dimensional algebra and although it is good to know that the Greeks solved cubic equations using conic sections how did they do it? The index is not very good. I don't know who this book could be recommended to.
Tracked by 2 customers
Sort: Oldest first | Newest first
Showing 1-3 of 3 posts in this discussion
Initial post: 8 Dec 2010 22:16:10 GMT
S R Scott says:
Above you head is it ? There are many many readers who have run through first year University level mathematics course. Most science and engineering courses as well as economics and some of the humanities include this. So there is a very wide numerate readership out there. Professor Steward is a great teacher and communicator - don't knock this book because you cannot handle it.
In reply to an earlier post on 1 Sep 2011 19:43:33 BDT
I agree with the reviewer about the index. It should be more comprehensive. I agree with nothing else. The book doesn't require any particular mathematical skills but it does require some minimal intelligence. My maths-phobic cousin (educated in the humanities) stumbled upon the book, browsed it rather thoroughly, and she decided to buy it as a present for me, intending to borrow it back when I was finished. If the book is over anyone's head, that person simply can't think logically.
Posted on 8 May 2012 11:40:45 BDT
J. Peters says:
I agree with the original poster. I am educated to A-level in mathematics and physics, and I program computers for a living (in very obscure low-level languages and boolean logic). I am no Steven Hawking, but I am no thickie either. I have read a number of popular science/maths books over the years which include many difficult concepts yet I understand most. By contrast I found reading this book disappointing and bewildering as it lost me several times. By the way, I am surprised by some of the snide comments in here. I would have thought anyone clever enough to buy, read and understand this book must be intelligent enough to realise we all learn differently - some people will respond well to the author's style, others won't. We're not school kids any more, so play nicely :-)
‹ Previous 1 Next ›
|
s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500834494.74/warc/CC-MAIN-20140820021354-00400-ip-10-180-136-8.ec2.internal.warc.gz
|
CC-MAIN-2014-35
| 2,809 | 16 |
https://scholarship.claremont.edu/jhm/vol9/iss1/4/
|
math
|
Abstract / Synopsis
In this article I describe Adelphi University's Mathematics Orientation Seminar, a new course that was introduced into the mathematics major to help students find their passion in mathematics and to strengthen the educational community within our department. I discuss quantitative and qualitative results of surveys among students in the Mathematics Orientation Seminar in Fall 2016 and Fall 2017, which suggest that this might be a useful course for other institutions to utilize within any major. Finally, I explore faculty perspectives and describe what I believe to be the final version of this course.
Salvatore J. Petrilli, "The Mathematics Orientation Seminar: A Tool for Diversity and Retention in the First Year of College," Journal of Humanistic Mathematics, Volume 9 Issue 1 (January 2019), pages 24-48. DOI: 10.5642/jhummath.201901.04. Available at: https://scholarship.claremont.edu/jhm/vol9/iss1/4
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100942.92/warc/CC-MAIN-20231209170619-20231209200619-00337.warc.gz
|
CC-MAIN-2023-50
| 932 | 3 |
https://www.homeschoolmath.net/reviews/adventures_penrose_mathematical_cat.php
|
math
|
Review of the book The Adventures of Penrose The Mathematical Cat
Some of us are math enthusiastics: we love math, think it is full of beauty, logic, and patterns; we think it can be fun, exciting, amusing even. (I include myself in this category.)
And then some of us don't care for math, didn't like it in school, didn't do well in it, don't want anything to do with it anymore. And I'm sure there are people in between these extremes, too.
As a teacher, you would like your students to belong to the first category, of course. Now, I sincerely think that the biggest factor in whether a student ends up liking or hating math is what kind of math teacher and teaching he/she gets. Parents' attitudes play a role too, but the teacher's attitude to math and the way math is taught are the weightiest factors, I feel.
But books such as The Adventures of Penrose The Mathematical Cat by Theoni Pappas can help too. It is intended to be a fun book where you explore exciting mathematical topics in an easy-to-understand way.
The storyline includes Penrose the mathematical cat who seems pretty lazy but often gets interested in her mistress' math papers, and thus ends up learning about this and that.
But this isn't a book about adding, fractions, ratios, algebra, or any other "regular" math topic. It is about fascinating and fun mathematical ideas you seldom encounter in school. Via Penrose, they are presented in an easy way for kids to understand.
For example, you get to encounter 0s and 1s (binary numbers), fractals, infinity, Mobious strip, Pascal triangle, golden rectangles, paperfolding, tessellations, abacus, magic squares, tangrams, nanoworld... For math geeks, these are familiar topics. They show some of the beautiful, fun, and intriguing aspects of math.
Each chapter spans about four pages, has a short storyline, and often a problem or challenge in the end. Answers are included in the end of the book. The mathematical content is suitable for about grades 3-9, but even younger kids can enjoy some of it.
I liked the layout a lot; it is very clear and spacey with generous amount of pictures to hold the little reader's interest.
You could say this book is about "Math appreciation". We are told to teach our children to appreciate fine arts, paintings, sculptures, and music. In my opinion, learning about fascinating math topics is important also. It helps children to see an ARTISTIC or CREATIVE side to mathematics. Art and math are connected anyway – you'll find out about that in this book also.
Math is so much more than your basic facts, fractions, and algebra. Via The Adventures of Penrose, you can give your child a glimpse of all that, encourage her to study math, and let her mathematical understanding grow a few leaps.
Penrose meets Fibonacci rabbit and of course learns about Fibonacci numbers.
The Adventures of Penrose The Mathematical Cat, about $9 new. The author Theoni Pappas has written several other math books, too.
Review by Maria Miller
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224656788.77/warc/CC-MAIN-20230609164851-20230609194851-00714.warc.gz
|
CC-MAIN-2023-23
| 2,986 | 15 |
https://essayking.icu/2021/04/18/assignment-problem-in-operations-research_z8/
|
math
|
150 per copy. assignment topics for literature review problems operational research level 4 assignment problem in operations research 1 assignment problems operational research- level 4. how assignment problems in solving problems math operation research it works. 105 exact algorithm for the weapon target assignment and fire scheduling problem journal of society of korea industrial and systems engineering, vol. assignment problem (one's method) presentation 1. the hungarian method how to do a cover page for a research paper can critical reading and thinking also how to write a conclusions solve such assignment problems , as it is easy to obtain argumentative philosophy essay topics an equivalent minimization problem by converting every number in the matrix to an opportunity loss some assignment problems entail maximizing the profit, effectiveness, or assignment problem in operations research layoff of an assignment of persons to tasks or of jobs to machines. this paper presents a new algorithm for solving the assignment problem. all that you should know about example of case study research paper writing assignments. 50 per copy. it purchases the diaries at rs. operation research calculations is made easier here reduction theorem in operations research || assignment problem in operations research reduction theorem harry potter essays in assignment problem in hindies video me maine assignment problem se ek important theorem redu. assignment problems operational research level 4 1 assignment problems operational research- level 4. 123 help essay.
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989526.42/warc/CC-MAIN-20210514121902-20210514151902-00483.warc.gz
|
CC-MAIN-2021-21
| 1,569 | 1 |
https://math.answers.com/Q/How_do_you_multiply_decimals_and_fractions
|
math
|
You EITHER, change them BOTH to decimals, OR you change them BOTH to vulgar (common) FRACTIONS EXAMPLE 2.3 X1&2/3 equals 2&3/10 X 1&2/3 equals 23/10 X 5/3 etc OR 2.3 X 1.66 equals etc.
Convert one to the other.
to order fractions you can cross multiply two fractions at a time or you can convert all the fractions into decimals.
no, that's only if it's fractions.:}
turn both fractions into decimals and then multiply!
Change the decimal into a fraction or the easier way is to turn the fration into a decimal, then multiply.
It's because decimals are really fractions and all numbers get smaller when you multiply them by fractions.
Yes providing you change the fractions into decimals or change the decimals into fractions
Multiply them by a number greater than 100.
Decimals and fractions are PART of a whole
I Think Decimals Are Better Than Fractions
All numbers can be changed from fractions to decimals.
A strong working understanding of fractions and decimals is essential for nurses. They must be familiar enough with fractions and decimals to quickly and accurately divide, multiply, add and subtract dosages as well as convert fractions to decimals and vice versa. Conceptual understating of fractions and decimals is essential since half doses, extra doses and time-delayed dosages must be calculated correctly. Nurses also need to know how to convert fractions and decimals to percentages in order to explain medication instructions accurately and easily to their patients. Read more about how math is related to nursing at the link I provided below.
All terminating decimals can be written as fractions.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679099514.72/warc/CC-MAIN-20231128115347-20231128145347-00761.warc.gz
|
CC-MAIN-2023-50
| 1,616 | 14 |
https://www.fr.freelancer.com/projects/sales/all-sale/?ngsw-bypass=&w=f
|
math
|
Budget ₹400-750 INR / heure
I need some help with selling something.
Compétences : Ventes
en voir plus :
need help selling product, need help selling ebay, need help selling service, need help selling, electronics need help selling, need help selling amazon, need help selling items, need help selling beats, i need help advertising my website, i need help designing a form, i need help designing a shirt, i need help designing clothes, i need help designing my van wrap, i need help find worker, i need help finding a manufacturer, i need help finding a shoes designer, i need help from an engineer, i need help from scientist, i need help getting my book edited not much money, i need help in designing a powerpoint presentation for a conference
Nº du projet : #27450679
Hello, My name is Syra am interested in this project for you. I have been working as a Customer service for five years. I have learned how to listen and show empathy whenever a customer has an issue. I have been workin
L'adresse e-mail est déjà associé à un compte Freelancer.
Entrez votre mot de passe ci-dessous pour lier les comptes :
Liez votre compte à un nouveau compte Freelancer
Liez à votre compte Freelancer existant
|
s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176256.21/warc/CC-MAIN-20201124111924-20201124141924-00156.warc.gz
|
CC-MAIN-2020-50
| 1,209 | 11 |
http://www.cfd-online.com/Forums/openfoam-solving/58859-no-slip-boundary-conditions-print.html
|
math
|
Hi People, I am trying to j
I am trying to justify my choice of no slip boundary conditions for a wall shear stress study:
(1) it simply means the fluid is not moving at the wall surface
(2) slip can be used if the shear components are specified, but will not be appropriate in this case since i am interested to find the wall shear stress
(3) the no slip boundary condition should account for highest wall shear stress compared to cases with slip, thus conservative results
Please correct me ( especially (3)) if i am wrong. any comments and helps will be very much appreciated.
All 3 of your statements seem
All 3 of your statements seem accurate to me.
By turning on the no slip boun
By turning on the no slip boundary condition you are essentially saying that the surface has roughness and so you will get a proper boundary layer profile. Without it, you won't get any surface shear stresses or any viscous drag.
(1) A boundary layer has zero velocity at the wall and tends to 0.99*Ufreestream at its edge. So, yes.
(2) I suppose you could, but why since it is a bit of a fudge? You need no-slip if you want to predict what these shear stresses are.
(3) Not sure what you mean here.
I find it strange you have to justify applying the no-slip condition, especially for a shear stress study. I thought it'd be more the case if it was the other way around.
Hi Edward and Adriano, Than
Hi Edward and Adriano,
Thanks for your replies and help. Seems like no-slip Boundary condition is the right way to go for this case.
I am asking for help here because someone questioned my choice of such condition but now i will be much more confident when explaining the choice.
|All times are GMT -4. The time now is 00:43.|
|
s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541187.54/warc/CC-MAIN-20161202170901-00044-ip-10-31-129-80.ec2.internal.warc.gz
|
CC-MAIN-2016-50
| 1,712 | 19 |
https://www.alpineschool.org/Page/576
|
math
|
Alpine School offers a challenging and comprehensive math curriculum which is aligned with the newly adopted New Jersey Student Learning Standards. The curriculum stresses the acquisition of mathematical fundamentals and important mathematical practices in grades K-5 using our new enVisions 2.0 Math Program by Pearson. In grades 6-8 the math program builds on the foundations set in the previous grades with a sequence of courses that lead students to study in grades 6 & 7: Ratios & Proportional Relationships, The Number System, Expressions & Equations, Geometry, and Statistics & Probability. In grade 8 the students study: The Number System, Expressions & Equations, Functions, Geometry, and Statistics & Probability. In addition the students move through pre-algebra and algebra, with the most skilled students studying advanced algebra. This 6-8 curriculum is based on two research based programs, CMP3 and the University of Chicago’s Algebra program.
|
s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578841544.98/warc/CC-MAIN-20190426153423-20190426175423-00423.warc.gz
|
CC-MAIN-2019-18
| 961 | 1 |
http://lipaperjlde.supervillaino.us/profit-maximisation-model.html
|
math
|
Main propositions of the profit-maximization model profit maximization has always been considered the primary goal of firmsthe firm's owner is the manager of. Managerial economics august 15, 2007 the key points underpinning the economics of a profit maximizing firm neoclassical model of the firm states that organization will have the main objective of maximizing its profit within a given period of time.
Production maximization and cost minimization recall that in consumer choice we take budget constraint as fixed and move indifference curves to find the optimal point. Profit maximization • a profit-maximizing firm chooses both its inputs and its outputs with the goal of model • firm has inputs (z 1,z 2) prices (r 1,r 2). Baumol's sales revenue maximization model highlights that the primary objective of a firm is to maximize its sales rather than profit maximization. Chapter 9 profit maximization economic theory normally uses the profit maximization assumption in studying the firm just as it uses the utility.
When a firm applies profit maximization, it is basically saying that its primary focus is on profits, and it will use its resources solely to get the biggest profits possible, regardless of the consequences or the risk involved. This study focused on an area of emerging research: managing a multi-product and multi-echelon supply chain which produces and sells deteriorating goods in the marketplace. What is profit maximization why would we want to maximize our profits, rather than revenues or sales in this lesson we'll discuss what profit.
Profit maximizers the aim of profit maximizing companies is to create as much net income, or profit, as possible with the resources and market share currently at their disposal. Economics for dummies firm a determines the profit-maximizing quantity of in the stackelberg duopoly model, one firm determines its profit-maximizing.
Value maximization and stakeholder theory. C g h sum rhs amount 1 1 1 revenue 4 6 10 20 cost -3 -4 -8 -15 = 5 profit hours 05 1 2 35 = 0 g non neg 1 1 = 0.
The theory of consumer behavior uses the law of diminishing marginal utility to explain how consumers allocate their incomes the utility maximization model is built based on the following assumptions:. Econ 600 lecture 3: profit maximization i the concept of profit maximization profit is defined as total revenue minus total cost π = tr – tc. Video created by university of california, irvine for the course strategic business management - microeconomics 2000+ courses from schools like stanford and yale - no application required. Risks associated with the theory of the firm's profit maximization goal porter's 5 forces is a model that identifies and analyzes the competitive forces that.Download
|
s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221215176.72/warc/CC-MAIN-20180819125734-20180819145734-00334.warc.gz
|
CC-MAIN-2018-34
| 2,760 | 6 |
http://www.ptsim.com/forum/viewtopic.php?p=4108
|
math
|
Here is an explanation about VTP hooks.
If, in the Australia folder, you open the file HL986450.BGL with the recent TmfViewer (the one dated 2004) and go to:
you can get the following picture. Look to the 3 views of the same line junction. They use different settings of the viewer.
On the top only the points are drawn. You will see the hooks. On the centre I asked TmfViewer to draw wide lines as polygons. On the bottom I asked TmfViewer to fill polygons.
You see that there is a starting hook and a ending hook. Points 1 and 2 and points N-1 and N are used to make the side of a triangle. Note that VTP lines are in fact polygons. Here is what I wrote recently on the AVSIM Scenery forum:
I think that any VTP line should have at least 4 points. Points 1 and 2 will generate the "true first" point and points N-1 and N will generate the "true last point". Based on TMFViewer display I assume that each midle point is internally translated to 2 points. I am using VTP Lines in SBuilder as follows:
1) I do not allow the width of the line to be greater than 255
2) I start the line with 2 points. These 2 points are perpendicular to the first line segment and their distance is the starting width of the line. The "mean position" of these 2 points is the position of the starting point of the source line.
3) I end lines with 2 points which are perpendicular to the last segment of the line. Again their distance is the width of the last point in the source line. The "mean position" of these 2 points is the position of the ending point of the source line.
4) I created an option in the VTP file import which I call "Remove Hooks". If this option (a check box) is set, Sbuilder imports 4 points lines as 2 points. If I do not set the option, Sbuilder import the points as if they all are different (as for example you see them in LWMViewer).
I suggest that you allways set this box.
General discussion about Scenery Design. Questions about SBuilder for Flight Simulator FS2004.
1 post • Page 1 of 1
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988831.77/warc/CC-MAIN-20210508001259-20210508031259-00572.warc.gz
|
CC-MAIN-2021-21
| 2,003 | 13 |
https://www.arxiv-vanity.com/papers/1701.00771/
|
math
|
Local index theorem for orbifold Riemann surfaces
We derive a local index theorem in Quillen’s form for families of Cauchy-Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.
Key words and phrases:Fuchsian groups, determinant line bindles, Quillen’s metric, local index theorems
1991 Mathematics Subject Classification:14H10, 58J20, 58J52
Quillen’s local index theorem for families of Cauchy-Riemann operators explicitly computes the first Chern form of the corresponding determinant line bundles equipped with Quillen’s metric. The advantage of local formulas becomes apparent when the families parameter spaces are non-compact. In the language of algebraic geometry, Quillen’s local index theorem is a manifestation of the “strong” Grothendieck-Riemann-Roch theorem that claims an isomorphism between metrized holomorphic line bundles.
The literature on Quillen’s local index theorem is abundant, but mostly deals with families of smooth compact varieties. In this paper we derive a general local index theorem for families of Cauchy-Riemann operators on Riemann orbisurfaces, both compact and with punctures, that appear as quotients of the hyperbolic plane by the action of finitely generated cofinite Fuchsian groups . The main result (cf. Theorem 2) is the following formula on the moduli space associated with the group :
Here is the first Chern form of the determinant line bundle of the vector bundle of square integrable meromorphic -differentials on equipped with the Quillen’s metric, is a symplectic form of the Weil-Petersson metric on the moduli space, is a symplectic form of the cuspidal metric (also known as Takhtajan-Zograf metric), is the symplectic form of a Kähler metric associated with elliptic fixpoints, is the 2nd Bernoulli polynomial, and is the fractional part of . We refer the reader to Sections 2.1–2.3 and 3.2 for the definitions and precise statements. Note that the above formula is equivalent to formula (2) for because the Hermitian line bundles and on the moduli space are isometrically isomorphic, see Remark 3.
Note that the case of smooth punctured Riemann surfaces was treated by us much earlier in , and now we add conical points into consideration. The motivation to study families of Riemann orbisurfaces comes from various areas of mathematics and theoretical physics – from Arakelov geometry to the theory of quantum Hall effect . In particular, the paper that establishes the Riemann-Roch type isometry for non-compact orbisurfaces as Deligne isomorphism of metrized -line bundles stimulated us to extend the results of to the orbisurface setting.
The paper is organized as follows. Section 2 contains the necessary background material. In Section 3 we prove the local index theorem for families of -operators on Riemann orbisurfaces that are factors of the hyperbolic plane by the action of finitely generated cofinite Fuchsian groups. Specifically, we show that the contribution to the local index formula from elliptic elements of Fuchsian groups is given by the symplectic form of a Kähler metric on the moduli space of orbisurfaces. Since the cases of smooth (both compact and punctured) Riemann surfaces have been well understood by us quite a while ago [14, 10], in Section 3.2 we mainly emphasize the computation of the contribution from conical points corresponding to elliptic elements. In Section 4.1 we find a simple formula for a local Kähler potential of the elliptic metric, and in Section 4.2 we show that in the limit when the order of the elliptic element tends to the elliptic metric coincides with the corresponding cusp metric. Finally, in Section 4.3 we give a simple example of a relation between the elliptic metric and special values of Selberg zeta function for Fuchsian groups of signature (0;1;2,2,2).
We thank G. Freixas i Montplet for showing to us a preliminary version of and for stimulating discussions. Our special thanks are to Lee-Peng Teo for carefully reading the manuscript and pointing out to us a number of misprints.
2.1. Hyperbolic plane and Fuchsian groups
We will use two models of the Lobachevsky (hyperbolic) plane: the upper half-plane with the metric , and the Poincaré unit disk with the metric . The biholomorphic isometry between the two models is given by the linear fractional transformation for any .
A Fuchsian group of the first kind is a finitely generated cofinite discrete subgroup of acting on (it can also be considered as a subgroup of acting on ). Such has a standard presentation with hyperbolic generators , parabolic generators and elliptic generators of orders satisfying the relations
where is the identity element. The set , where , is called the signature of , and we will always assume that
We will be interested in orbifolds (or , if we treat as acting on ) for Fuchsian groups of the first kind. Such an orbifold is a Riemann surface of genus with punctures and conical points of angles . By a -differential on the orbifold Riemann surface we understand a smooth function on that transforms according to the rule . The space of harmonic -differentials, square integrable with respect to the hyperbolic metric on , we denote by . The dimension of the space of square integrable meromorphic (with poles at punctures and conical points) -differentials on , or cusp forms of weight for , is given by Riemann-Roch formula for orbifolds:
where denotes the integer part of (see [9, Theorem 2.24]). In particular,
The elements of the space are called harmonic Beltrami differentials and play an important role in the deformation theory of Fuchsian groups, see Sect. 2.3. To study the behavior of harmonic Beltrami differentials at the elliptic fixpoints we use the unit disk model. Take and let be an elliptic element of order with fixpoint . The pushforward of to by means of the map is just the multiplication by , the -th primitive root of unity. The pushforward of to (that, slightly abusing notation, we will denote by the same symbol) develops into a power series of the form
Moreover, since we have unless , so that
In particular, for and for .
As in , for we put , where
is the Laplace operator (or rather of the Laplacian) in the hyperbolic metric acting on . The function is regular on and satisfies
The following result is analogues to Lemma 2 in and describes the behavior of at . We will use polar coordinates on such that .
be the Fourier series of the function on . Then
as , where
For the constant term we have
where is the integral kernel of on , and .
Since is a regular solution of the equation at , we have in polar coordinates
where we used (2.1) for and the analogous expansion
for . Then for the term of the Fourier series (2.2) we have the differential equation
From here we get that as , where
For the coefficients with we have
so that as . This proves parts (i) and (ii) of the lemma, from where it follows that . To prove part (iii) it is sufficient to observe that
2.2. Laplacians on Riemann orbisurfaces
Let us now switch to the properties of the Laplace operators on the hyperbolic orbifold , where is a Fuchsian group of the first kind. Here we give only a brief sketch, and the details can be found in , . Denote by the Hilbert space of -differentials on , and let be the Cauchy-Riemann operator acting on -differentials (in terms of the coordinate on we have ). Denote by the formal adjoint to and define the Laplace operator acting on -differentials on by the formula .
We denote by the integral kernel of on the entire upper half-plane (where is the identity operator in the Hilbert space of -differentials on ). The kernel is smooth for and has an important property that for any . For and we have the explicit formula
Furthermore, denote by the integral kernel of the resolvent of on (where is the identity operator in the Hilbert space ). For and the Green’s function is a smooth function on away from the diagonal (i. e. for ). For we have the following Laurent expansion near :
as , where is the hyperbolic area of . Then for any integer we have
This series converges absolutely and uniformly on compact sets for .
We now recall the definition of the Selberg zeta function. Let be a Fuchsian group of the first kind, and let be a unitary character. Put
where runs over the set of classes of conjugate hyperbolic elements of , and is the norm of defined by the conditions (in other words, is the length of the closed geodesic in the free homotopy class associated with ). The product (2.8) converges absolutely for and admits a meromorphic continuation to the complex -plane.
Except for the last section, in what follows we will always assume that and will denote simply by . The Selberg trace formula relates to the spectrum of the Laplacians on , and it is natural (cf. ) to define the regularized determinants of the operators by the formula
(note that has a simple zero at ).
2.3. Deformation theory
We proceed with the basics of the deformation theory of Fuchsian groups. Let be a Fuchsian group of the first kind of signature . Consider the space of quasiconformal mappings of the upper half-plane that fix 0, 1 and . Two quasiconformal mappings are equivalent if they coincide on the real axis. A mapping is compatible with if for all . The space of equivalence classes of -compatible mappings is called the Teichmüller space of and is denoted by . The space is isomorphic to a bounded complex domain in . The Teichmüller modular group acts on by complex isomorphisms. Denote by the subgroup of consisting of pure mapping classes (i. e. those fixing the punctures and elliptic points on pointwise). The factor is isomorphic to the moduli space of smooth complex algebraic curves of genus with labeled points.
Note that , as well as the quotient space , actually depends not on the signature of , but rather on its signature type, the unordered set , where and is the number of elliptic points of order (see ).
The holomorphic tangent and cotangent spaces to at the origin are isomorphic to and respectively (where, as before, ). Let be the unit ball in with respect to the norm and let be the Bers map. It defines complex coordinates in the neighborhood of the origin in by the assignment
where , is a basis for , and is a quasiconformal mapping of that fixes 0, 1, and satisfies the Beltrami equation
For denote by and the partial derivatives along the holomorphic curve in , where is a small parameter.
The Cauchy-Riemann operators form a holomorphic -invariant family of operators on . The determinant bundle associated with is a holomorphic -invariant line bundle on whose fibers are given by the determinant lines . Since the kernel and cokernel of are the spaces of harmonic differentials and respectively, the line bundle is Hermitian with the metric induced by the Hodge scalar products in the spaces (note that each orbifold Riemann surface inherits a natural metric of constant negative curvature ). The corresponding norm in we will denote by . Note that by duality between and the determinant line bundles and are isometrically isomorphic.
The Quillen norm in is defined by the formula
for and is extended for all by the isometry . The determinant defined via the Selberg zeta fuction is a smooth -invariant function on .
3. Main results
Our objective is to compute the canonical connection and the curvature (or the first Chern form) of the Hermitian holomorphic line bundle on . By Remark 1 can be thought of as holomorphic -line bundle on the moduli space .
3.1. Connection form on the determinant bundle
We start with computing the connection form on the determinant line bundle for relative to the Quillen metric. The following result generalizes Lemma 3 in :
For any integer and we have
where , and is the Euclidean area form on .
The integral in (3.1) is absolutely convergent if for all . If for some , then this integral should be understood in the principal value sense as follows. Let be the fixpoint of the elliptic generator of order 2, and consider the mapping . Denote by the disk of radius in with center at 0. Since is discrete, for small enough we have unless and is either or . The subset
is -invariant, where denotes the cyclic group of order generated by . The factor is an orbifold Riemann surface with holes centered at the conical points of angle . The integral in the right hand side of (3.1) we then define as
The integrand in the right hand side is smooth and the integral is absolutely convergent, cf. (2.7). We need to show that
(if there is we understand this integral as the principle value, see Remark 2).
Without loss of generality we may assume that and has one elliptic generator of order with fixpoint . Then by (2.5) we have
where is the cyclic group generated by (the stabilizer of in ), and
Since , it is easy to check that the last expression in the above formula is a (meromorphic) quadratic differential on . Using the standard substitution we get
that proves the theorem for (in the last line we used polar coordinates on ).
We have to be more careful in the case , since the contribution from elliptic elements is no longer absolutely convergent and should be considered as the principal value, see Remark 2. From now on we assume that acts on the unit disk , so that is generated by . Since is discrete, there exists . Therefore, we can choose a small such that unless . The set is -invariant, and the factor is a Riemann surface with a small hole centered at the conical point. In this case we have
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585768.3/warc/CC-MAIN-20211023193319-20211023223319-00510.warc.gz
|
CC-MAIN-2021-43
| 14,258 | 66 |
https://www.wyzant.com/resources/answers/radical_expressions
|
math
|
Simplify radical expression
√ 12 + √ 1/3 = x can you help me find the answer asap it is on an algebra1 worksheet operations with radical expressions adding...
Can some explain to me how to simplify this radical expression √2+√7/√8+√7
Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. ^4√9ab^3 X √3a^4b
Determine whether √a+√a=2a is sometimes, always, or never true. My Answer: never A playground is shaped like a rectangle with a width 5 times its length, l. What is a simplified...
Strontium-90 is a radioactive isotope used in some cancer treatments. It has a half life of 28.8 days. If you receive .45 grams of strontium-90 in a radiation session, how much will you receive in...
I need help solving this: 3√2 (√8 - 8√6)
The total wages W in a metropolitan area compared to it's population p can be approximated by a power function of the form W=a*p^(9/8) where a is a constant. About how many times as great does the...
learning operations with radicals. no logarithms please.
simplify radical expression square root of 8 plus square root of 2
USE DISCRIMININT TO FIND THE NUMBER OF REAL SOLUTIONS OF 2X(SQUARED) +9X+7
|
s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645281115.59/warc/CC-MAIN-20150827031441-00338-ip-10-171-96-226.ec2.internal.warc.gz
|
CC-MAIN-2015-35
| 1,202 | 11 |
http://japanese.stackexchange.com/questions/3845/what-is-the-difference-between-%E8%A6%B3%E5%AF%9F-and-%E8%A6%B3%E6%B8%AC/3877
|
math
|
According to JDIC, both mean "observation". Judging from the Kanji used, I'm guessing the difference is 観測 has a nuance of observing something and measuring some aspect of it or taking data, whereas 観察 is just to observe and monitor what's happening.
Japanese Language Stack Exchange is a question and answer site for students, teachers, and linguists wanting to discuss the finer points of the Japanese language. It's 100% free, no registration required.
Here's how it works:
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
You guess is right. And 観測 can also mean prediction based on the data observed (e.g. 希望的観測).
|
s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783397842.93/warc/CC-MAIN-20160624154957-00133-ip-10-164-35-72.ec2.internal.warc.gz
|
CC-MAIN-2016-26
| 693 | 7 |
https://brahma.tcs.tifr.res.in/events/balanced-information-inequalities?page=1
|
math
|
- A-212 (STCS Seminar Room)
In this talk we will discuss about linear information inequalities, both discrete and continuous ones. We will prove that every discrete information inequality is associated with a “balanced” information inequality and a set of “residual weights.” To prove the inequality, it is necessary and sufficient to prove that its “balanced” version is valid and all its residual weights are nonnegative. For a continuous information inequality, we will prove that it is valid if and only if its discrete counterpart is balanced and valid.
This work is due to Terence H. Chan.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945168.36/warc/CC-MAIN-20230323132026-20230323162026-00585.warc.gz
|
CC-MAIN-2023-14
| 607 | 3 |
https://back40-sweetpea.blogspot.com/2012/10/math-one-understood.html
|
math
|
I used to love math. But over the years the educators have been changing the "rules" of mathematics. No wonder our children are confused. Old math, new math, and back to old math again.
Have you solved the problem above? I personally refer to this as one understood. -(-36) is like -1*-36=36
So -19-17=-36+36=0-12=-12. Have I lost you yet? When I first looked at the problem I was lost. I had to walk away and come back to it before I could solve it.
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039476006.77/warc/CC-MAIN-20210420152755-20210420182755-00322.warc.gz
|
CC-MAIN-2021-17
| 450 | 3 |
https://www.heldermann.de/JCA/JCA13/JCA133/jca13057.htm
|
math
|
Journal of Convex Analysis 13 (2006), No. 3, 759--772
Copyright Heldermann Verlag 2006
Three Theorems on Subdifferentiation of Convex Integral Functionals
Dept. of Mathematics, Technion - Israel Inst. of Technology, Haifa 32000, Israel
The paper is devoted to calculation of subdifferential of the integral functional $\int_Tf(t,x)d\mu(t)$ with $f$ being a normal convex integrand. The first theorem gives a precise formula for the case when $x$ is finite dimensional and some qualification condition, which reduce to the general qualification condition for the sum rule of convex analysis, is satisfied. The two other theorems deal with infinite dimensional case and the absence of any qualification conditions and offer some approximations, one for the case when the space of $x$ is reflexive and separable and the other, when it is just separable.
Keywords: Convex function, normal convex integrand, subdifferential, normal cone, fuzzy calculus.
MSC: 90C25, 52A41, 26E15
[ Fulltext-pdf (367 KB)] for subscribers only.
|
s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038064898.14/warc/CC-MAIN-20210411174053-20210411204053-00536.warc.gz
|
CC-MAIN-2021-17
| 1,020 | 8 |
https://epublications.marquette.edu/mscs_fac/255/
|
math
|
Format of Original
Cambridge University Press
Proceedings of the Edinburgh Mathematical Society
Original Item ID
A semigroup B in which every element is an idempotent can be embedded into such a semigroup B′, where all the local submonoids are isomorphic, and in such a way that B and B′ satisfy the same equational identities. In view of the properties preserved under this embedding, a corresponding embedding theorem is obtained for regular semigroups whose idempotents form a subsemigroup.
Albert, Justin and Pastijn, Francis, "Uniform Bands" (2014). Mathematics, Statistics and Computer Science Faculty Research and Publications. 255.
|
s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100258.29/warc/CC-MAIN-20231130225634-20231201015634-00554.warc.gz
|
CC-MAIN-2023-50
| 643 | 6 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.