# All Questions

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### Proof of Equation by Well Ordering Principle

I have an assignment question I am able to solve this question using basic Induction, but not able to figure out how to do it by using Well Ordering Principle. Any Solutions or hints would be very ...
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### Finding the expectation value of a random variable counting the occurrences of certain events

Let there be $M$ (distinguishable) boxes and $N$ balls, which we uniformly distribute among the boxes. For $k \leq N$, let $g_k: \Omega \rightarrow \mathbb{Z}$ be the function counting the number of ...
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My question is very simple. I'm studying a course telling about algebraic surfaces but i think that i need some knowledge about basic algebraic geometry. Do you have some suggestions?
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### Primitive subgroups with different order

I have been studying the permutation group $S_{n}$ in combination with transitivity and blocks. Which leads to primitive groups. Now I was wondering if for $n=6$, so $S_{6}$ could have two primitive ...
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### Square root of 10=0?

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
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### showing a local minimum of a function

I have a problem that is a little bit of a struggle for me. I'm pretty sure what I'm trying to prove is a minimum at the point a, but how I go about proving it is eluding me. Any help would be greatly ...
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### Arrange the following growth rates in increasing order: $O (n (\log n)^2), O (35^n), O(35n^2 + 11), O(1), O(n \log n)$

I want to Arrange the following growth rates in increasing order This order are following : $O (n (\log n)^2), O ((35)^n), O(35n^2 + 11), O(1), O(n \log n)$ Please give me idea how to arrange growth ...
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### State the value of x after the statement

State the value of x after the statement if P(x) then x := 1 is executed, where P(x) is the statement “x > 1,” if the value of x when this statement is reached is x = 0. x = 1. x = 2. this answer ...
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### Introduction to Real Analysis Continuity Proof

Suppose $f,g: D\to R$ are both continuous on D. Define $h: D \to R$ by $h(x) =$ max{$f(x),g(x)$}. Show that $h$ is continuous on $D$. Here is what I have so far: Since $f,g$ are continuous, for ...
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### Does an operator of x commute with the differential operator with respect to x?

While solving a problem in Quantum Mechanics I got an expression $\frac{d}{dx}V(x)-V(x)\frac{d}{dx}$. The first term is just the derivative of the potential but the second one seems a bit weird. Is ...
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### calculating the density on balance scale

If i had 1531 grams of silver on one end of a balance scale, how much of 24ct gold would i have to put on the other end of the scale to balance out? Keep in mind that the density of both metals are ...
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Assume the tridiagonal matrix $T$ is in this form: $$T = \begin{bmatrix} a & c & & & \\ b & a & c & ... 1answer 23 views ### The definition of an elliptic curve? I've seen two different definitions of an elliptic curve. The first one being that it is a cubic curve of the form y^2=x^3+ax^2+bx+c, where all the (complex) roots are different. The other ... 3answers 13 views ### no. of monomials in variables w,x,y,\ldots,z of degree m The formula for no. of monomials in variables w,x,y,\ldots,z of degree m,\,\,\,\,\, (where e.g. x^iy^jz^k degree m=i+j+k) is: \,\,\,\,\,$$\binom{m+n-1}{n-1}$where$m$is degree of ... 2answers 20 views ### Can we find an$x, y : x < y$and$x, y > 0$and$\lfloor \frac{n}{x}\rfloor$<$\lfloor \frac{n}{y}\rfloor$for some integer$n > 0$? I know there are no solutions when we have just the fraction without the floor, but how do we consider solutions when the floor is there? 1answer 14 views ### 'Product rule' when$f$differentiable at$c,$and$g$not differentiable at$c$, but$f(c)=0$Take$h(x)=f(x)g(x)$mapping some subset of$\mathbf{R}^n$to$\mathbf{R}.$If$f(c)=0$and$f$is differentiable at$c,g$continuous at$c$, can we say that the derivative is still ... 0answers 10 views ### Extending a theorem true over the integers to reals and complex numbers How does one generally extend a theorem proved over the integers to the real numbers and beyond e.g. induction proofs, De Moivre's Theorem? I am aware that to extend a theorem proved over ... 1answer 9 views ### Proving property for all predicates in first order logic Let's consider language with predicate$P$and following derivation $${{{[P[a/x]]^1} \over {P[a/x] \rightarrow P[a/x]}}\rightarrow I^1 \over {\forall_x (P(x) \rightarrow P(x))}}\forall I$$ Doesn't ... 2answers 15 views ### Probability problem involving waiting time at pharmacy I'm working on the following problem: The case of Safeway is very easy as the answer is simply the mean of$f_T$, namely$1$minute. However, I'm having problems solving this problem for Target. ... 0answers 18 views ### finding presentation of a group I was working through presentation of the quaternion group (with element 8), and I let$a = i$and$b = j$. I immediately said$a^4 = b^4 = 1$, and$ab^2 a = 1$. Since I have a relation for each ... 1answer 13 views ### Probability of random assignment to form pairs So the question goes: I have 100 individuals and 100 different buses, and I randomly assigned each individual to sit on a bus (each bus has equal probability of being selected). How many buses are ... 0answers 8 views ### Bordered minor and rank of a matrix Let$M\in\mathbf{R}^{n\times n}$be a matrix. Suppose that there is a$k\times k$minor$M_k$of rank k. Now this reference (Algebra For Iit Jee 7.65) here states that if all the$k+1$th minors ... 0answers 13 views ### Morphism between two$\mathbb Q-$modules I am trying to do the following exercise: Let$V$and$W$be two$\mathbb Q-$modules and$f: V \to W$a function. Prove that$f$is a$\mathbb Q-$module morphism if and only if it is a group ... 0answers 8 views ### Questions on Strongly Differentiability. Definition: Let$U\subseteq \Bbb R^m$be an open set. Let$f: U \to \Bbb R^n$be a function and$T: \Bbb R^m \to \Bbb R^n$be a linear transformation. We say that$f$is strongly differentiable ... 3answers 28 views ### Use the law of logarithms to expand an expression $$\log\sqrt [ 3 ]{ \frac { x+2 }{ x^{ 4 }(x^{ 2 }+4) } }$$ How is this answer incorrect? $$\frac { 1 }{ 3 } [\log(x+2)-(4\log x+\log(x^ 2+4))]$$ 1answer 9 views ### Conditional Probability Problem for a Joint Distribution We have a joint probability distribution$f_{X,Y}(x,y)=\frac{1}{10}$, defined over the domain$(x,y)\in[-1,1]\times[-2,2]\cup[1,2]\times[-1,1]$. From this, I need to find the conditional PDFs ... 0answers 18 views ### Find the characteristic polynomial and the eigenvalues$\lambda_1, \lambda_2,\dots,\lambda_n$of the matrix$A$Let$\lambda_1, \lambda_2, \dots, \lambda_n$be eigenvalues of the matrix$A=(a_{ij})_{n\times n}$. Is it true that$|A|=\lambda_a\lambda_2\dots\lambda_n$, and$tr(A)=\displaystyle\sum_i^n\lambda_i$. ... 0answers 3 views ### Can the power of a test of an inconsistent estimator still go to 1 as N goes to infinity? This question is related with this one.«Can the power of a test of an inconsistent estimator still go to 1 as N goes to infinity?» However, the accepted answer doesn't explain one thing that I would ... 3answers 30 views ### 4 equations 3 unknowns If I have 4 equations and 3 unknowns, I could solve for the 3 unknowns using the first 3. How does it ensure that the 4th equation is also satisfied? In this case, what should be the usual strategy to ... 0answers 5 views ### The effect on k(G) of removing a vertex or an edge Consider an (a, b) graph G (ie a graph with a vertices and b edges). Let k (G), the minimum amount of vertices that can be removed to disconnect the graph, be n>=1. What would be the possible effects ... 5answers 34 views ### Linear independence of the functions$1,\cos(x),\cos(2x)$I want to show that the functions$1,\cos(x),\cos(2x)$are linearly independent in$C[-\pi,\pi]$. I computed the Wronskian determinat of these functions but at the points$x=0,-\pi,\pi$the obtained ... 1answer 17 views ### a problem in exact sequence First, sorry for this ugly diagram Suppose$\require{AMScd}$\begin{CD} ...@.\acute M @>f>> M @>g>>\check M @>>> O\\ @. @V \alpha V V\ @VV \beta V @VV ... 0answers 14 views ### how to prove the eigenvalues of tridiagonal matrix Assume the tridiagonal matrix$T$in this form: $$T = \begin{bmatrix} a & c & & & \\ b & a & c & ... 0answers 10 views ### What's the easiest way to approximate an answer to n places after the decimal? Say you wanted a decimal answer. How do you get one? I know evalf() can be used but it counts ALL digits, not just after the decimal. For example evalf(19/4, 2) gives 4.8 but I'm looking for a ... 2answers 20 views ### Xlosure subsets of a metric space X If A and B are subsets of a metric space X, then cl(A ∩ B)= cl(A)∩ cl(B) Why is it false?? but it is true when it is AuB. 1answer 13 views ### Talyor's Theorem in Nocedal's Numerical Optimization Please kindly refer to the figure below. I understand that (2.4) is just another formulation of Mean Value Theorem and I understand its geometrical meaning in 1-D case. However, I do not know what ... 0answers 11 views ### How to approach a minmax problem? Starting with a certain geometric problem, I have reached this function:$$R(s,t,u,v)=\max(s-u,s+u,t-v,t+v,sX+tY+u, tX-sY+v)$$where X\geq0 and Y\geq0 are parameters. I have to find the minimum ... 2answers 12 views ### Big-O Function for f(x) I'm currently taking a Discrete Mathematics course which just started the chapter on The Growth of Functions. A (very) brief overview was given in lecture that covered the Big-O definition. Let ... 0answers 3 views ### Practical convolution of poisson and log-normal distribution Hi guys, Im trying to make a Loss Distribution (in Excel), following the "Loss Distribution Approach". I do understand that the main idea is that we have a severity distribution and a frequency ... 0answers 12 views ### Problem with constructing a uniform probability measure over \mathbb{N_0} using rationals on the unit interval I've been toying with the possibility of constructing a uniform probability measure over \mathbb{N^0}. Obviously, one cannot just assign each non-negative integer a probability of 0 and call it a ... 0answers 8 views ### 3D extension of Euclidean algorithm jigsaw method - help! Recently I've been learning about how the Euclidean algorithm = jigsaw method (filling a rectangle with squares) = forming continued fractions. And today I'm wondering how a 3D version of the jigsaw ... 0answers 14 views ### Example of a special kind of infinite dimensional vector space and a linear map on it Give example of an infinite dimensional vector-space V and a linear transform T on V such that T \circ S=S\circ T , \forall S \in \mathscr L(V) , but V has a non-zero vector which is not ... 0answers 23 views ### Finishing a problem using equalities This is my problem: Let a, b and c be positive real numbers with abc=1. Prove that$$\frac{a^{n+2}}{a^n + (n-1)\,b^n} + \frac{b^{n+2}}{b^n + (n-1)\,c^n} + \frac{c^{n+2}}{c^n + ... 1answer 30 views ### Countable and uncountable sets. a) Show that$\left \{ n^{2}+m^{2}:n,m\in \mathbb{N} \right \}$is countable. b) Show that$\left \{ x\in \mathbb{R}:x(x-2)<0 \right \}$is uncountable. My answers: a) Is it possible to define ... 0answers 22 views ### statistics problem, where did I mistake? I searched interesting problem about statistic from http://www.mast.queensu.ca/~stat353/resources/pastfinals/final12sol.pdf  But at the question No.2, I have some problem the red box  ... 3answers 28 views ### Prove by induction for every integer$\; n\ge 5$,$2^n\gt n^2$. Prove by induction for every integer$ \;n\ge 5$,$2^n\gt n^2$. My try: $$p(n):\;2^n>n^2$$ verify$P(5)$$$p(5):\;2^5>5^2 = 32 > 25$$ Of course the trick is in the induction step and ... 0answers 10 views ### Bruns-Herzog, Cohen-Macaulay Rings, Exercise 10.1.16 This question is from the Bruns-Herzog, Cohen-Macaulay Rings, Exercise 10.1.16(a). Let$x_1,..., x_n, y, z$be elements of$R$such that ideals$(x_1,..., x_n,y)$and$(x_1,..., x_n,z)$are ... 2answers 21 views ### Find the maximum points of$f(x)=e^{-x}\sin^2(\pi x) \hspace{0.4cm},0<x<10$Find the maximum points of $$f(x)=e^{-x}\sin^2(\pi x) \hspace{0.4cm},0<x<10$$ My calculations:I have calculated$f'(x)=\pi e^{-x}\sin(2\pi x)-e^{-x}\sin^2(\pi x)f''(x)=e^{-x}\sin^2(\pi ...
Legendre's conjecture implies that the prime gap above any natural number $n$ is bounded by the product of a constant and the $\sqrt n$. Actually, I have strong reason to think that this factor by ...