All Questions

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sum of series [n/p]+[n/$p^2$]+[n/$p^3$]..

Is there a general formula sum of series [n/p]+[n/$p^2$]+[n/$p^3$]... where [] denotes greatest integer function and p is prime. If yes please explain the formula
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Continuous function of the Hyperspace of Compact Sets

Let $X$ be separable and completely metrizable. Let the hyperspace of compact sets be denoted $H(X)$. I want to show that $H(X)^2 \rightarrow H(X)$ defined by $(H,G)\rightarrow H \cup G$ is ...
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Improper Integral in four dimensions

Considering the integral $I = \prod_{i=1}^4 (\int_\infty^\infty dk_i sinc (k_ia_i + b_i)sinc (k_iA_i + B_i) )\frac {k_1^n}{k_1^2+k_2^2+k_3^2+k_4^2}$ with parameters $n,a,b,A,B$ I tried to use ...
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How many triangles can be formed from the 12 non-collinear points?

There are 12 distinct non-collinear points in a same plane, they are points A,B,....L. How many different triangle can be formed, with criteria one of its vertice must be contain point A? My ...
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Example of smooth function of a smooth submanifold cannot be obtained by restriction of a smooth function in the manifold

Anyone can give the example of smooth function of a smooth submanifold N of M cannot be obtained by restriction of a smooth function in the smooth manifold M?
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Conditional Distribution and Marginal Distribution

What is the formula for Conditional and Marginal distribution? I have this exercise: ...
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Extension of Dedekind domains with the same quotient field

Let $K$ be a field. Let $A \subseteq B$ be an extension of Dedekind domains that are both finitely generated $K$-algebras. Assume that $A$ and $B$ have the same quotient field. Can we find an element ...
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Rotating people at a dinner table - 19 people, 4 tables, 3 rotations

I am organising a dinner where some members need to rotate between tables, while others remain stationary. We have 8 seats per table, with 4 tables and the following stationary people Table 1: 3 ...
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Real Analysis II: Application of Inverse Function Theorem

I missed two weeks' worth of classes in my Real Analysis II course die to personal issues, and while going over past exam questions for midterm revision, I came across some problems that I had trouble ...
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Differential of a quadratic map over a finite field

I have some trouble with the definition of differential of a map over a finite field $\mathbb{F}_q$, where $q$ is a power of $2$. I have found that, given $f$ a quadratic map over $\mathbb{F}_q$, the ...
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Where is the Hausdorff condition used?

Here https://en.m.wikipedia.org/wiki/Continuous_functions_on_a_compact_Hausdorff_space It says that the space $C(\Omega)$ is a normed space if $\Omega$ is a compact hausdorff space. But why do we ...
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Offset switch partitions

Say we have 6 tiles numbered: $$\begin{array}{|c|c|c|c|c|c|} \hline 6&5&4&3&2&1\\ 0&1&2&3&4&5\\ \hline \end{array}$$ You must pick one of the numbers from ...
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Continuity of multiplication of maps in the strong topology

Prove that multiplication of maps is a continuous operation in the strong topology on the unit balls of L(X,U) and L(U,W),where X,U,W are all Banach spaces. The strong topology is the weakest topology ...
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Finding an angle in a quadrilateral

Let ABCD be a convex quadrilateral such that angle BAD = 90 degree, angle BAC = 2*angle BDC and angle DBA + angle DCB = 180 degree. Find angle DBA.
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Little Graph Theory Problem

Let $G$ be a finite graph and let $H_1,\ldots, H_n$ be some distinct subgraphs with the same number of vertices, and with the property that each edge of $G$ belongs to the same number of the $H_i$. ...
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Seeking a combinatorial proof $\sum _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$

I would appreciate if somebody could help me with the following problem Q: Seeking a combinatorial proof $(\binom{n}{k}=\frac{n!}{k! (n-k)!} )$ $$\sum _{k=0}^n (n-2 k)^2 \binom{n}{k}=n\times 2^n$$
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Rational points of conics over $\mathbb{Q}$

I am starting to read lecture notes on basics of arithmetic geometry by A. V. Sutherland. In the second lecture, there is a procedure how to decide whether a conic over $\mathbb{Q}$ has a rational ...
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When do the solutions to the linear system $Ax=b$ form a vector subspace?

When do the solutions to the linear system $Ax=b$ form a vector subspace? (Full text below) Is the answer b? As homogeneous solution are closed under addition and multiplication.
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How many invertible 3x3 matrices?

How many invertible 3x3 matrices exist over 2-element field? Obviously if some field has only 2 elements, those elements must be $0$ and $1$. A matrix is invertible if and only if its determinant ...
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The time at which the tank would be full given the following conditions?

A rectangular tank has dimensions 5 m x 3 m x 2 m. There are three inlet pipes P. Q. R, which have filling rates of 2 m3 / hr. 3 m3 / hr and 5 m3 / hr respectively. At 9:00 am. when the tank was ...
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Abel transform of $R_z(\Delta)$

The spectrum of the Laplace operator $\Delta$, as self-adjoint unbounded operator on $L^2$ is equal $]-\infty ,a]$. The resolvent $R_z$ is defined for $z \in \mathbb C \setminus ]-\infty ,a]$. Using ...
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Prove that normals at the points where line intersects a parabola meet at point on normal at a given point on the parabola.(Loney XXX 18)

Prove that the normals at the points, where the straight line $lx+my=1$ meets the parabola, meet on the normal at the point $(\frac{4am^2}{l^2},\frac{4am}{l})$ of the parabola. As the line intersects ...
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SweSAT Data Sufficiency

A product has undergone two price increases with the same percentage, ie by x percent each time. How big was the increase in percent each time? ( 1 ) The total price increase was 450 dollars (2) ...
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conditional probability on Truth and lies

Mike tells truth with probability 1/3 and lies with probability 2/3. Independently, David tells truth with probability 3/4 and lies with probability 1/4. Both watch a soccer match. David tells you ...
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ind the time in which the 15th tap alone can fill the empty cistern given the following conditions?.

There are t taps numbered 1,2 and so on till t, each of which can fill a cistern. the rate of filling of the nth ta is such that it is equal to twice that of all the taps from 1 to (n-1) put ...
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Find and describe all relative extrema

I've encountered this question: Find and describe all local extrema of $f(x) = x^{5 /3} − 5x^{2/3}$. Also indicate on which regions the function is increasing and decreasing. Using the derivative, I ...
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Real Analysis II: Chain Rule Application

I missed two weeks' worth of classes in my Real Analysis II course die to personal issues, and while going over past exam questions for midterm revision, I came across some problems that I had trouble ...
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The number of points of inflection on the curve $y=\arccos(\frac{2x}{1+x^2})$ is/are

The number of points of inflection on the curve $y=\arccos(\frac{2x}{1+x^2})$ is/are $(A)1\hspace{1cm}(B)2\hspace{1cm}(C)3\hspace{1cm}(D)0$ How to find the number of points of inflection of a ...
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Does this rule have a name?

I call this the Ateekster Rule (my nickname lol) The Sum of All Numbers Equals The Sum of Their Digits: 1 21 3 1+21+3=25 Add the two numbers 2+5 = 7 Now add the total digits: ...
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How does scaling rows to sum to 1, of a positive matrix change the perron vector?

Let $A$ be a $N\times N$ positive matrix such that $A_{ij}>0$. By perron-frobenius theorem, there is a unique positive unique eigenvector called perron vector $x$ corresponding to the largest ...
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what are the current research trends and areas in coding theory?

I want to know the fertile research areas in error correction coding especially topics that use error correcting methods and techniques to computer science.
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how to understand this comparison of screw theory to other methods?

there is a paragraph in the second page of chapter 2 in the book "A mathematical introduction to robotic manipulation" as follows: "There are two main advantages to using screws, twists, and wrenches ...
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What is an “empirical distribution”?

Let $\lambda_{n1} \leq \lambda_{n2} .. \leq \lambda_{nn}$ be the $n$ real eigenvalues of a ransom symmetric matrix, $X$ of dimension $n$. Then in this case one seems to define the "empirical ...
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How to find the number of solutions to an equation in wolfram alpha

Given the equation: $$sin(x)=\frac{x}{218}$$How do I use Wolfram Alpha to find the number of positive solutions?
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Expected value of distinct balls

My friend posed me the following question: We have a bowl with 70 balls, 7 colors, and 10 balls in each color. You draw 20 balls simultaneously from the bowl. What is the expected number of ...
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Expected number of person not to get shot - reloaded

Inspired from this question I came up with a seemingly simpler problem that I could not solve either. There are $n$ people sit on a round table. At noon, each person shots and kills one of their ...
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Prove that for any fixed $n>1$ there exists an infinite amount of such prime $p$, that $p\equiv 1 (n)$

Prove that for any fixed $n>1$ there exists an infinite amount of such prime $p$, that $p\equiv 1 (n)$ Essintially, what's in the title, how do I that?
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Proving Fibonacci sequence with mathematical induction

Okay, so I have the following thing: $$\sum_{i=1}^a F_{2i}=F_{2a+1}-1$$ It's to do with Fibonacci sequence. I can do the basis step of MI fine (proving for $a = 1$) However the inductive step has ...