Density and some other function to find

In summary, the conversation involves a person seeking help with a research problem and another person suggesting that they post their question concisely in LaTex. The problem involves finding constants, a random variable, and a function that will satisfy certain conditions for a given function. The person providing help also mentions that the problem may be easier to solve by changing variables.
  • #1
Anna Kaladze
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0
Hi All,

I have been trying to unsuccessfully crack a certain problem for my research, but I get stuck. I found it is easy to describe the problem in a separate document. Can you please have a look at the attached file and give me some help? Thanks!

Anna.
 

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  • #2
You'd probably get more responses if you posted in LaTex and stated the question concisely. As I interpret the question, a change of variables would reduce it to the following:

Given the following:
The function [itex] S(t) [/itex] satisfying [itex] S(0) = 1 ; S(1)= 0 [/itex] and [itex] \frac{dS}{dt} < 0 [/itex] for [itex] 0 < t < 1 [/itex]

Find the following:
Constants [itex] X_{min} < X_{max} [/itex]
A random variable [itex] X [/itex] that has support [itex] [X_{min}, X_{max}] [/itex].
and a function [itex] F(S,X) [/itex]
that will satisfy these conditions:
[itex] F(S(0),x) = 1 [/itex] for all [itex] X_{min} \leq x \leq X_{max} [/itex]
[itex] F(S(1),x) = 0 [/itex] for all [itex] X_{min} \leq x \leq X_{max} [/itex]
[itex] \frac{\partial F(S,x)}{\partial x } < 0 [/itex] for [itex] 0 < t < 1 [/itex]
For each [itex] 0 < t < 1 [/itex], [itex] \int_{X_{min}}^{X_{max}} F(S(t),X) dX = S(t) [/itex]


You didn't give a specific [itex] S [/itex] so I assume an answer must be expressed in terms of [itex] S [/itex]. You used the constant [itex] t_0 [/itex] in your statement of the problem and you wrote [itex] S [/itex] as [itex] S(t_0, t) [/itex] suggesting that S is a function of an interval. This might give some reader a hint about the solution, but I omitted [itex] t_0 [/itex] it since it isn't necessary.
 
Last edited:

What is density and how is it calculated?

Density is a measure of the amount of mass in a given volume of a substance. It is calculated by dividing the mass of the substance by its volume.

How is density different from specific gravity?

Density and specific gravity are both measures of the mass per unit volume of a substance. However, specific gravity compares the density of a substance to the density of water, whereas density is an absolute measure.

What is the formula for finding the density of a regular shaped object?

The formula for finding the density of a regular shaped object is density = mass / volume. The mass can be measured in grams and the volume in cubic centimeters.

How can density be used to identify substances?

Density is a unique characteristic of a substance and can be used to identify it. By comparing the density of an unknown substance to the densities of known substances, it is possible to determine what the unknown substance is.

What other function can be used to find density?

Another function that can be used to find density is the buoyancy equation, which states that the density of an object is equal to the density of the fluid it is submerged in multiplied by the ratio of its volume to the displaced fluid's volume.

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