Independent Definition and 744 Threads

The Independent is a British online newspaper that was established in 1986 as a national morning printed paper. Nicknamed the Indy, it began as a broadsheet and changed to tabloid format in 2003. The last printed edition was published on Saturday 26 March 2016, leaving only the online edition.It tends to take a pro-market stance on economic issues. The newspaper was controlled by Tony O'Reilly's Irish Independent News & Media from 1997 until it was sold to the Russian oligarch and former KGB Officer Alexander Lebedev in 2010. In 2017, a Saudi Arabian investor bought a 30% stake in it.In June 2015, the newspaper had an average daily circulation of just below 58,000, 85% down from its 1990 peak, while the Sunday edition had a circulation of just over 97,000. The daily edition was named National Newspaper of the Year at the 2004 British Press Awards. The website and mobile app have a combined monthly reach of 22,939,000.

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  1. M

    Linearly independent vs dependent functions

    For part(b), My solution is, ##(c_1, c_2, c_3, c_4) = (c_1, 5c_1, -c_1, -3c_1)## They have taken the case that c_1 = -1, which gives their expression for a linear dependent function as they have shown. However, I'm confused that the functions are linearly dependent for any value of ##c_1 \in...
  2. T

    Understanding the Proper Uncertainty Formula to Use

    Good evening, I'm running into a little confusion on the second part of this problem due to finding two different formulas for calculating the uncertainty in multiplied quantities. The way that I was taught was something like this. If ##z = x \cdot y##, then: $$\dfrac{\delta z}{z} =...
  3. D

    Probability density function of Z=X+Y and Z=XY

    I assume since ##X## and ##Y## are independent and their probability density functions are functions of ##x## $$f_{Z_1}=\frac{f_X(x)+f_Y(x)}{2}$$ and $$f_{Z_2}=\frac{f_X(x)\cdot f_Y(x)}{\int_{\mathbb{R}}(f_X(x)\cdot f_Y(x))dx}.$$ The divisions occur because it must be that...
  4. MatinSAR

    How many independent components does a tensor have?

    It's a 4th-dimensional 4th-rank tensor so at first we have ##4^4=256## components. According to the book, Given that ##R_{iklm}=-R_{ikml}## 256 components reduces to 96. But I cannot see how. For one pair of i,k 16 components are dependent. We have 12 pairs of i,k(for ##i≠k## becsuse for i=k...
  5. A

    I The covariance of a sum of two random variables X and Y

    Suppose X and Y are random variables. Is it true that Cov (Z,K) = Cov(X,K)+Cov(Y,K) where Z=X+Y?
  6. vibhuav

    I Physical meaning of two independent, non-interacting parts

    I keep coming across this descriptor, "two (or three) independent, non-interacting parts," in many books on QM (for example, Penrose's Shadows of the Mind). It is usually followed by a mathematical description (for example, state vector |A>|B>). I can wrap my mind around the quantum paradox of...
  7. P

    A system of independent particles (energy levels)

    Hi guys, Can you give me some feedback on whether my calculation is correct? I applied the formula below (Boltzmann Distribution) but I didn‘t know what to use for the variable z. I don‘t even know if I used the correct equation. Can you help me further? The task is: Consider a system of...
  8. F

    I Linear Model with independent categorical variable

    Hello, I have been pondering on the following: we have data for blood pressure BP (response variable) and data about age and gender (categorical variable with two levels). We can build two linear regression models: $$BP=b_0+b_1 age+b_2 gender$$ $$BP=b_0+b_1 age$$ The first model does not take...
  9. chwala

    Test of hypothesis for independent samples

    Reference; https://www.statisticshowto.com/probability-and-statistics/t-test/ My question is, can we as well have 'subtract each ##x## score from each ##y## score?' thanks. ...t-tests after all are easy to comprehend; as long as one knows the types; i.e 1. indepedent sample tests (compares...
  10. SrivastavaHarsh

    A A game of seemingly mutually independent events

    There are 4 players numbered 1 to 4. There is a room with an entrance door to one side and exit door to opposite side. Inside the room, there are 4 boxes numbered 1 to 4. Inside each box, there is a chit containing a number (with equal probability) from 1 to 4 (inclusive). No two chits can have...
  11. H

    I Explanation of all "its linearly independent derivatives"

    I'm studying Differential Equations from Tenenbaum's, and currently going through non-homogeneous second order linear differential equations with constant coefficients. Method of Undetermined Coefficients is the concerned topic here. I will put forth my doubt through an example. Let's say we are...
  12. yucheng

    I Why are there 2s -1 independent integrals of motion?

    I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and it can be considered an additive constant of time. Hence I tried searching it up online...
  13. Ahmed1029

    I Can I always consider velocities and coordinates to be independent?

    It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.
  14. M

    I Light speed independent of source

    Please bear with me I am trying to get a grip with underlying principles. Starting to try and understand Einstein’s second postulate and distinguish that the speed of light is independent of the speed of the source – v - of objects, other than light with travel initiated independent of the...
  15. Shreya

    Maxwell's Capacitor argument for a time independent Current

    Is it because the current applied to a capacitor will never be time independent? Please help me out🙏
  16. T

    MTW Ex 21.15 - Curvature independent of Lapse and Shift functions

    Can anyone out there give me a hint as to where to start with this problem? I've been looking at it for a while and can't see a way forward. What exactly is "the curvature itself" here?BTW I think the dynamic initial value equations 21.116 and 21.117 are incorrect. MTW should have inserted to...
  17. L

    I Can configuration space be observer independent?

    We can formulate the spacetime in an observer/coordinate independent way, i.e. a particle becomes a worldline in the 4d space. Then relative to each observer, the worldline can be casted to a function in R^3. However, I haven't found any reference on formulating configuration space in a...
  18. M

    MHB For which values are the variables independent?

    Hey! :giggle: We have the table of distribution of $X$, $Y$ and their joint random variable : with $$(c,d)\in \left \{(c,d)\in \mathbb{R}^2\mid 0\leq c\leq \frac{1}{4}, \ 0\leq d\leq \frac{1}{2}, \ \frac{1}{4}\leq c+d\leq \frac{1}{2}\right \}$$ I want to calculate the values of $c$ and $d$...
  19. K

    I Independent coordinates are dependent

    (This is not about independence of ##q##, ##\dot q##) A system has some holonomic constraints. Using them we can have a set of coordinates ##{q_i}##. Since any values for these coordinates is possible we say that these are independent coordinates. However the system will trace a path in the...
  20. A

    Physics Viability of publishing as an independent researcher

    I'm as mainstream as a physicist can be, so you will not see my trying to publish things like "relativity is wrong" or "there is no proof of quantum mechanics". But I live and work in a 3rd world country with an awful policy for teaching and research, and the universities I work for only pay me...
  21. Thomas Michael

    I How did the author transform the original equation into the Legendre equation?

    I'm reading "Differential Equations with Applications and Historical Notes" by George F. Simmons and I am confused about something on pages 431-432 He has the second order ordinary differential equation $$\frac {d^2v} {d\phi^2} + \frac {\cos(\phi)} {\sin(\phi)} \frac {dv} {d\phi} + n(n+1)v = 0...
  22. R

    Circuit Analysis Theorem For the Number of Independent Equations

    I attached a screenshot of the book (sorry no pdf available for this book). Right above the somewhat central line they give the theorem that if there are m currents and n nodes, then there will be n - 1 independent equations from the current law and m - n - 1 from the voltage law. I count 4...
  23. M

    Problem with independent events in possibilities

    Summary:: problem with independent events in possibilities A forest has a fire per year with probability 30% and it has 2 fires per year with possibility 5%.The probability of 3 or more fires is 0%.The forest might have disasters from the wind which blows with probability of 100% or has...
  24. M

    I Sum independent normal variables

    (I know how to prove it). Prove that a finite sum of of independent normal random variables is normal. I suspect that independence may not be necessary.
  25. F

    B Choosing which variable is the dependent or independent variable

    Hello, I am review some basics. In the case of two variables like ##t## and ##C##, which represent the time in year and the cost in $ of a product respectively, we can graphically represent this pair of variables on a Cartesian plane. The horizonal axis is generally assigned the variable ##t##...
  26. karush

    MHB Q01 are linearly independent vectors, so are....

    Let A be invertible. Show that, if $\textbf{$v_i,v_2,v_j$}$ are linearly independent vectors, so are \textbf{$Av_1,Av_2,Av_3$} https://drive.google.com/file/d/1OuHxfUdACbpK4E5aca2oBzdaxGR0IYKv/view?usp=sharing ok I think this is the the definition we need for this practice exam question...
  27. karush

    MHB 115.C51 find a linearly independent set T so that T=S

    $\tiny{115.C51}$ find a linearly independent set T so that $\langle T\rangle =\langle S\rangle$ $S=\left\{ \left[\begin{array}{r}2\\-1\\2\end{array}\right], \left[\begin{array}{r}3\\0\\1\end{array}\right], \left[\begin{array}{r}1\\1\\-1\end{array}\right]...
  28. X

    Engineering Step response of RC circuit with independent voltage and current sources

    Dear PF, In the figure down below is Q7.47 which asks to determine the voltage v(t) across the capacitor for t > 0. Since it is given that V(0) = 0 there are two scenario's which is between time interval 0 < t < 1 and t > 1 according to the independent sources. For the scenario 0 < t < 1 the...
  29. L

    A Matrix multiplication, Orthogonal matrix, Independent parameters

    Matrix multiplication is defined by \sum_{k}a_{ik}b_{kj} where ##a_{ik}## and ##b_{kj}## are entries of the matrices ##A## and ##B##. In definition of orthogonal matrix I saw \sum_{k=1}^n a_{ki}a_{kj}=\delta_{ij} This is because ##A^TA=I##. How to know how many independent parameters we have in...
  30. A

    Engineering Circuit Analysis -- How many independent equations can be written using KCL?

    I think there are 4 essential node. So we can write 3 KCL independent equations.
  31. R

    Other How Can I Independently Conduct and Publish Research in Theoretical Physics?

    Hi everyone , I have done Bachelor's in Physics from Delhi University , India . I discontinued Master's from Indian Institute of Technology more-than-1 year before completion , it was 2 year course. I want to do research in Physics independently and publish ground-breaking...
  32. D

    Three independent random variables having Normal distribution

    Let ##X_1 X_2 X_3 ## be three independent random variables having Normal(Gaussian ) distribution, all with mean ##\mu##=20 and variance ##\sigma^2##=9. Also let ##S=X_1+ X_2 +X_3## and let ##N## be the number of the ##X_i## assuming values greater than 25. ##E\left[N\right]##=? I did not...
  33. Lucy_1192

    I Are Intensity and Magnitude independent of the radius of a star?

    For a star.. Apparent Magnitude = -2.5log10 I K And I = I0/d^2 So in terms of I0... Apparent Magnitude = -2.5log10 (I0/d^2) And the Stefan-Boltzmann law says: Energy Flux = Sigma(T^4) In my reading it says that Intensity is the energy emitted per unit of area per unit of time. It says the same...
  34. brotherbobby

    Independent parameters of the rotation tensor ##R_{ij}##

    I am afraid I had no credible attempt at solving the problem. My poor attempt was writing the matrix ##\mathbb R## as a ##3 \times 3## square matrix with elements ##a_{ij}## and use the matrix form of the orthogonality relation ##\mathbb R^T \mathbb R = \mathbb I##, where ##\mathbb I## is the...
  35. M

    MHB Show that these elements are linearly independent

    Hey! 😊 Let $1\leq n\in \mathbb{N}$, let $\mathbb{K}$ be a field, $V$ a $\mathbb{K}$-vector space with $\dim_{\mathbb{K}}V=n$ and let $\phi:V\rightarrow V$ be linear. - Let $0\neq v\in V$ and $1\leq m\in \mathbb{N}$, such that $\phi^{m-1}(v)\neq 0=\phi^m(v)$. Show that $\phi^0(v)=v, \phi (v)...
  36. M

    MHB Statements with linearly independent vectors

    Hey! 😊 Let $\mathbb{K}$ a field and let $V$ a $\mathbb{K}$-vector space. Let $1\leq m, n\in \mathbb{N}$ and $n=\dim_{\mathbb{K}}V$. Let $v_1, \ldots , v_m\in V$ be linearly independent. Let $\lambda_1, \ldots , \lambda_m, \mu_1, \ldots , \mu_m\in \mathbb{K}$ such that...
  37. Addez123

    Are these two dice rolls dependent or independent?

    Summary:: Roll a dice twice. Event A: First dice roll yield 2 or 5. Event B: Sum of the two results are atleast 7 Is A and B independent? If they are independent then $$P(A \cap B) = P(A) * P(B)$$ P(A) = 2/6 = 1/3 P(B) = 1 - 9/36 = 27/36 $$P(A \cap B) = 7/36 \neq P(A)*P(B) = 1/4$$Yet book...
  38. E

    How can angular velocity be independent of the choice of origin?

    In a previous thread it was concluded that both torque and angular momentum are taken about a chosen origin, and these quantities are generally not invariant under translations of the origin (since ##\vec{r}## changes but ##\vec{v}## does not, etc.). The moment of inertia tensor doesn't...
  39. dRic2

    I The density of states independent of Boundary Conditions

    Most undergrad textbook simply say that it is intuitive that boundary conditions should not play a role if the box is very large. Other textbooks suggest that this should be taken for granted since the number of particles at the surface are orders of magnitude smaller that the number of bulk...
  40. E

    B Is "ten" independent of the chosen number base?

    This topic is quite confusing. For instance, if I write ##14_6##, do I pronounce this as "ten", even though we'd probably just say "one-four, base six"? That is to say that we'd treat "ten" as a number of things that we could count out (i.e. corresponding to a certain number of sticks). Another...
  41. Adesh

    Confusion about the independent variable of function in magnetostaics.

    When we calculate the curl of magnetic field, that is the curl of Biot-Savart equation for magnetic field. Please consider these . The working of last equation $$ \nabla \times \mathbf {B} = \frac...
  42. brotherbobby

    Proving that the two given functions are linearly independent

    Summary:: I attach a picture of the given problem below, just before my attempt to solve it. We are required to show that ##\alpha_1 \varphi_1(t) + \alpha_2 \varphi_2(t) = 0## for some ##\alpha_1, \alpha_2 \in \mathbb{R}## is only possible when both ##\alpha_1, \alpha_2 = 0##. I don't know...
  43. P

    Show that the metric tensor is independent of coordinate choice

    I need to use some property of the relalation between the coordinate systems to prove that g_{hk} is independent of the choice of the underlying rectangular coordinate system. I will try to borrow an idea from basic linear algebra. I expect any transformation between the rectangular systems to...
  44. W

    B Why does Jupiter have a stronger gravitational pull than Earth?

    Objects fall on Earth at 9.8 m/sec² independent of mass. If gravity is independent of mass why does Jupiter pull an object more than the Earth does? Is that inconsistency within the law or in my perception?
  45. karush

    MHB 004- USMA(WP) entrance exam question independent variable

    ok not sure what forum this was supposed to go in,,so... If the independent variable of $W(\theta)=2\theta^2$ is restricted to values in the interval [2,6] What is the interval of all possible values of the dependents variable?
  46. M

    MHB Are the vectors linearly independent?

    Hey! :o We have that the vectrs $\vec{v},\vec{w}, \vec{u}$ are linearly independent. I want to check if the pairs $\vec{v}, \vec{v}+\vec{w}$ $\vec{v}+\vec{u}$, $\vec{w}+\vec{u}$ $\vec{v}+\vec{w}$, $\vec{v}-\vec{w}$ are linearly indeendent or not. Since $\vec{v}, \vec{w}, \vec{u}$...
  47. jk22

    I Integration : Are a function and it's derivative independent?

    The question is a bit confused, but it refers to if the following integration is correct : $$I=\int \frac{1}{1+f'(x)}f'(x)dx$$ $$df=f'(x)dx$$ $$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$ The last equality would come if I suppose $f,f'$ are independent variables.
  48. megf

    I Probability Density Function of the Product of Independent Variables

    How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which...
  49. P

    I Do These Probability Conditions Imply Independence of Events A, B, and C?

    Hello everyone. Let us consider 3 events A,B,C such that: $$P((A \cap B )\cup C)=P(A)*P(B)*P(C)$$ Notice that the second term is a union and not an intersection. Are they independent? And what if the assumption was: $$P(A \cap( B \cup C))=P(A)*P(B)*P(C)$$? I know that the independence condition...
  50. fight_club_alum

    Determining the voltage polarity of the independent current source

    I am so sorry if I am posting this in the wrong forum; it is just not a homework problem, and I can't find the right place - it's more of a study help question.
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