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    Why the sum of cosines between "v`" and any vector =1?

    Exactly, ##1 = \sum_{k =1}^N (\frac{ \overrightarrow{v_k} \cdot \overrightarrow{q_j}}{ |\overrightarrow{v_k}||\overrightarrow{q_j}|})^2## I discovered this fact by coincidence but it turns out that it may have a nice link to the quantum mechanics. For example, if the cosine of the angle...
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    A solute added to a container with a small hole

    But I do not see where to start from your post #9. You gave the variable in units and a boundary condition when t approaches infinity.
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    A solute added to a container with a small hole

    Thank you, I think this should take me to my proposed solution under the assumption that the rate of the salt leaving the bowel is proportionated to the total salt in the bowel. (The proportion constant should be negative sign because the leaving salt reduces the total amount of salt in the...
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    A solute added to a container with a small hole

    Correct, this is the one I am seeking to solve.
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    A solute added to a container with a small hole

    Sorry that I was not clear in the description of the problem. I like the analogy of salt and water, so I will consider it here. Let`s consider a tank has water and salt. The hole near the bottom leaks water and salt. To keep the volume of the fluid constant, we add from the top a volume of water...
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    A solute added to a container with a small hole

    Yes, it leaks fluid and solute. I also assume that the added solute is instantaneously solved in the fluid with no sediments.
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    A solute added to a container with a small hole

    Homework Statement Suppose there is solute ##s## in a bowel containing fluid. There is a tiny hole near the bottom which leaks a small fixed volume of solute ##\lambda## per unit time ##dt##. In addition, there is a small added solute to the fluid in a constant rate ##\alpha## so as the volume...
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    Calculating the area of equilateral triangle using calculus

    Homework Statement Calculating the area of equilateral triangle using calculus. Homework Equations The Attempt at a Solution The area of the triangle is the area of the circle minus 3 times the area of the sector shown in (light blue). So, the target is to calculate the pink area first...
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    Velocity transformation using the chain rule

    From some help, I found a way out. First the velocity should be represented by the total derivatives not partial derivatives. ##\frac{dx}{dt}=\frac{dx}{dx`}\frac{dx`}{dt`}\frac{d t`}{dt}## Now ##\frac{dx}{dx`}## and ##\frac{d t`}{dt}## are expressed in term of partial derivatives...
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    Velocity transformation using the chain rule

    Here is another trial; ##\frac{dx}{dt}=\frac{\partial x}{\partial x`}\frac{\partial x`}{\partial t}+\frac{\partial x}{\partial t`}\frac{\partial t`}{\partial t}## But ##\frac{\partial x`}{\partial t}=\frac{\partial x`}{\partial t`}\frac{\partial t`}{\partial t}## So...
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    Velocity transformation using the chain rule

    Homework Statement How to obtain the famous formula of velocity transformation using a chain rule. I know that there is a straightforward way by dividing ##dx## as a function of ##dx`## and ##dt`## on ##dt## which is also a function of them. But I would rather try using the chain rule. Homework...
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    Why the sum of cosines between "v`" and any vector =1?

    Homework Statement Given that matrix, A can be decomposed using SVD (Singular Value Decomposition) into ##A=USV^T##`, why does always the sum of the square of cosines between v` vectors and any other column vector q representation of arbitrarily column vector Q vector sum up to 1? Homework...
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    How to calculate the state vector after n-transitions?

    The entries in column sums to 1. The matrix is 5x5 for example.
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    How to calculate the state vector after n-transitions?

    In this post, I adopted the convention of column vector instead of row because it is more conventional for me. Yes, I am looking for a closed analytic expression for ##x_n##. This is important especially if n is larger.
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    How to calculate the state vector after n-transitions?

    Homework Statement Given an initial distribution state vector that represents the probability of the system to be in one of its states. Also given a Markov transition matrix. How to calculate the state vector of the system after n-transition? Homework Equations Assuming the initial state...
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    Integration of part of a radius gives a complex number...

    d is still fixed and represents half length of the needle. I followed the same reasoning of the classical Buffon needle. In classical version, the needle crosses the vertical line when the projection of the needle on the horizontal axis is not larger than $$2dcos\theta$$ . In other words, the...
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    Integration of part of a radius gives a complex number...

    So if we taking the area under the curve, p(theta), it should represent the desired integration, right!. Remember Buffon needle problem, a needle with a length 2d and the distance between the vertical lines is L crosses those lines with a probability = $$ 2d/ \pi L$$, One of smart solution to...
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    Integration of part of a radius gives a complex number...

    I am not familiar with elliptic integrals so I don not know what z and k in this solution are related to my problem. For example, $$ \int_0^{\pi/2} r + d\cos\theta - \sqrt{d^2\cos^2\theta + r^2 - d^2 } \ d\theta = {\pi/2} r + d - f(d, r)$$. So how to represent f(d,r)? If it is not elementary...
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    Integration of part of a radius gives a complex number...

    But still I don`t know how to evaluate $$\int_0^{\pi/2} \sqrt{r^2 - d^2\sin^2\theta } \ d\theta$$ WolframAlfa didn`t give me a solution.
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    Integration of part of a radius gives a complex number...

    This is nice step that: ##d^2\cos^2\theta + r^2 - d^2 = r^2 - d^2\sin^2\theta## , I overlooked it at all. Nothing wrong with $$ \int_0^{\pi/2} r + d\cos\theta - \sqrt{d^2\cos^2\theta + r^2 - d^2 } \ d\theta$$ because I know how to evaluate the the first two terms but I stuck at the third one...
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    Integration of part of a radius gives a complex number...

    I drew another circle with more clear lines. z is always perpendicular to r. I didn`t get you at the second sentence. d is always ≤r
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    Integration of part of a radius gives a complex number...

    Yes ##\theta## varies from ##0## to ##\pi/2##. d is a constant.
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    Integration of part of a radius gives a complex number...

    True, I am using this too later in the calculation of p. Please see the word file I just attached.
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    Integration of part of a radius gives a complex number...

    Consider the triangle with sides r, d and s, r2= d2 + s2 + 2 ds cosθ This is a quadratic equation of s, given d and r.
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    Integration of part of a radius gives a complex number...

    I am interested to find the length shown in red in the attached figure. I want this length as a function of d (shown in blue) and the angle θ. Then I will integrate this length to dθ from 0 to π/2. Firstly, I used the law of the triangle to determine the length s which when subtracted from the...
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