0
votes
1answer
71 views

Evaluating the indefinite integral $\int\frac{dx}{qx+c}$

Evaluate the indefinite integral (remember to use $\ln |u|$ where appropriate) $$\int\frac{dx}{qx+c}\qquad (q\neq 0) $$ I have no idea how to approach this. But here's what a have so far using ...
1
vote
2answers
91 views

Algebraic variety in $\mathbb{C^{2}}$

would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry: I have to find smallest algebraic variety (irreducible algebraic set) in ...
10
votes
2answers
239 views

Computing an indefinite integral: $\int \frac{2n!\sin x + x^n }{e^x + \sin x + \cos x + P_n (x)}\, dx $

Let $\displaystyle P_n (x) = 1 + \frac{x}{1!} + \frac{x^2 }{2!} + \cdots + \frac{x^n }{n!} \ $ and $$ I(x) = \int \frac{2n!\sin x + x^n }{e^x + \sin x + \cos x + P_n (x)}\, dx $$ (where $\ n \to ...
2
votes
0answers
64 views

Differential Equation for brownian bridge?

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is ...
2
votes
0answers
22 views

Probablistic bound for $\|RR^TM\|$ for uniformly random orthonormal matrix $R$

I am stuck on a finding a probablistic bound on a nonstandard random matrix. I looked around on the internet and couldn't find any results. This could be because I don't know the key words or because ...
1
vote
1answer
21 views

Calculate the value of the integral.

See the attached image for the question. My intuition tells me to use the fundamental theorem of calculus to begin this problem. So I will have F(1) - F(0) = 9. For part A, since the value of f(t) ...
1
vote
1answer
49 views

Does the order of $a$ in $\left(\mathbb{Z}/ \Phi_n(a) \mathbb{Z} \right )^{\times}$ equal $n$?

Here, $\Phi_n(a)$ is the nth cyclotomic polynomial evaluated at a. It's obvious that the multiplicative order of $a$ modulo $\Phi_n(a)$ divides $n$, because $a^n \equiv (a^n-1)+1 \equiv P(a)\cdot ...
-1
votes
1answer
44 views

Prove $\left\lfloor n\right\rfloor = n + O(1)$

Can someone show me why $\left\lfloor n\right\rfloor = n + O(1)$. Using the same logic, can anyone derive a similar proposition for $\left\lceil n\right\rceil$?
1
vote
2answers
93 views

How find this $a$ such $(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$

Question: find all the complex $a$,such for every complex $z_{1},z_{2}(|z_{1}|,|z_{2}|<1,z_{1}\neq z_{2})$,such $$(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$$ My idea: ...
0
votes
1answer
133 views

Determining functions types. one - one , onto and bijective

Could anyone give an idea to start up with this question: Let $f$ be a function $f: \mathbb{R}^3 \to \mathbb{R}$ such that $f(x,y,z) = xyz$ . How to verify whether it's one to one or onto function?
1
vote
1answer
55 views

Prove an $n\times n$ matrix is negative definite

I wonder is there any way to prove the $n\times n$ matrix with elements below is negative definite: $$ \sigma_{ij} = \frac{a_ia_j}{\sum_k s_ka_k} \space; i \neq j \text{ (off diagonal terms)}$$ ...
4
votes
2answers
126 views

A result on sequences: $x_1\to x$ implies $\frac{x_1+\dots+x_n}n\to x$ without using Stolz-Cesaro

If $x_n \to x$, how might we prove $$\lim_{n \to \infty} \frac{\sum_{i=1}^{n} x_i}{n} = x$$ Of course, one has $\limsup x_n = \liminf x_n = x$, and thus, using the Stolz-Cesaro theorem: $$\liminf ...
1
vote
1answer
103 views

Probability: basic question and concept

I have always been struggling with the problem, in particular, I usually have great difficulty in differenting when should I multiply n! to take care of the ordering, and when should I not do so. For ...
0
votes
1answer
127 views

Operations on a set of numbers to leave the median unchanged

Set Q contains 14 distinct numbers. Which of the following operations would decrease the average of set Q while leaving the median unchanged? A. Decreasing all 14 numbers by 2 each B. Increasing the ...
6
votes
4answers
129 views

If $a+b+c=1$ and $abc>0$, then $ab+bc+ac<\frac{\sqrt{abc}}{2}+\frac{1}{4}.$

Question: For any $a,b,c\in \mathbb{R}$ such that $a+b+c=1$ and $abc>0$, show that $$ab+bc+ac<\dfrac{\sqrt{abc}}{2}+\dfrac{1}{4}.$$ My idea: let $$a+b+c=p=1, \quad ab+bc+ac=q,\quad ...
1
vote
0answers
37 views

Is the empty set a member of itself [duplicate]

I'm taking Discrete Mathematics and am a little confused on the "empty set" or "null set" defined by ∅. I understand that ∅ is not a member of {∅} but is ∅ a member of {∅, {∅}} given that there is a ...
1
vote
2answers
73 views

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$ $ $ This appears to be an easy problem, but it is consuming a lot of time, I am wondering if an easy way is possible. WHAT I DID : Wrote this as ...
0
votes
1answer
24 views

visualizing a function from the plane into the reals

If $$(x_1,y_1), (x_2,y_2),(x_3,y_3)$$ are points in the plane and if $a,b$ are fixed real numbers, how can I visualize $$f=(ax_1+b-y_1)^2+(ax_2+b-y_2)^2+(ax_3+b-y_3)^2$$ as a function from the plane ...
-2
votes
1answer
61 views

Prove or Disprove? $\log(n^n)\text{ is } \Theta(\log n)$

I need help confirming that my way of proof is alright. This is my first class in algorithms so I just wanna know if I'm on the right track. :)
2
votes
3answers
119 views

Limit of $\cfrac{\sin(\theta)}{\theta}$ in degrees

What does $\lim \limits_{\theta\to0}\cfrac{\sin(\theta)}{\theta}$ equal when $\theta$ is expressed in degrees? I know that theta in degrees is $\frac{\pi}{180}$ theta radians, but I don't get the ...
0
votes
1answer
21 views

Are irreducible polynomials and irreducibles (in an integral domain) different?

There's this theorem that you can factor polynomials (in $\mathbb{Z}$[x]) into polynomials of lower degrees r and s in $\mathbb{Q}$[x] iff you can factor that polynomial into polynomials of the same ...
1
vote
0answers
62 views

The use of Schauder fixed point in ladyzehskaya

The book Linear and Quasilinear equations of parabolic type gives the uniform parabolic pde theory in the literature. Ladyzhenskaya use Leray-Schauder rather than Schauder fixed point theorem. why? ...
3
votes
3answers
128 views

Why does $y = x\sin(\frac{180}{x})$ approach $\pi$?

A few days ago I was playing on my scientific calculator and I ran over an interesting little equation: $180\sin(1)$ is extremely close to $\pi$. At first I thought it was a coincidence, but then I ...
4
votes
2answers
85 views

Product $\sigma$ algebra

Consider $\mathbb{R}$ with the $\sigma$-algebra of Borel sets, and $\mathbb{R}^\mathbb{R}$ with the product $\sigma$-algebra(see p.22 of 'Real Analysis - Gerald B. Folland'). Does $[0,1]^\mathbb{R} ...
1
vote
1answer
35 views

sequence in an infinite series

Let $x_n \rightarrow x$ as $n \rightarrow \infty$ such that $x_n \in \mathbb{R}$. Show that $$\lim_{p \rightarrow 1 ^ {-}} (1-p) \sum_{n=0}^{\infty} x_n p^n = x.$$ I think I am getting a little ...
1
vote
1answer
49 views

$\varphi(f)$ is invertible iff $f$ is non-degenerate?

Let $E$ be the vectorial space of the bilinear functions $\varphi: \mathbb R^n\times \mathbb R^n\to \mathbb R$. Then, there is a canonical isomorphism between $E$ and the set of the real matrices ...
3
votes
1answer
36 views

Prove that $\max\{|x_i|: 1 \leq i \leq n\} \leq \|\vec{x}\| \leq \sum_{i=1}^{n} |x_i|$

If $\|\vec{x}\|$ denotes the Euclidean Norm of $\vec{x} \in R^n$, show that $$ \max\left\{|x_i|: 1 \leq i \leq n\right\} \leq \|\vec{x}\| \leq \sum_{i=1}^{n} |x_i| $$
4
votes
3answers
366 views

How to solve limits?

The above limit was solved by making a seemingly arbitrary substitution. The previous limit was solved by making a linear substitution $y=mx$. Which again seemed a bit out of the blue. For another ...
1
vote
0answers
26 views

approximation of a complex valued real rational function

all. I am now struggling with a approximation problem. Suppose we have a matrix-valued measure $\mathrm{d}\Lambda(\omega)$, with compact support $[a,b]$, then its Cauchy transform is a well-defined ...
0
votes
1answer
79 views

How to prove that the L2 norm is a non-increasing function of time for a 2nd-order PDE?

I am having a test in few days and I saw an interesting question while I was skimming through the book problems. The problem is concerned about initial-boundary value problem of 2nd order PDEs. To ...
6
votes
1answer
321 views

What is meant by “folkloric result” in category theory?

I often will see the word "folklore" used in papers on category theory, e.g. in Barr's paper on Isbell Duality, he states a result, which he proves, is "folkloric", on page 512: The following ...
0
votes
1answer
641 views

Transformation from cartesian to polar Coordinates of Vector Field

This is fairly low-level, still I would like to get a verification: I vector field $$\mathbf{F}=F_x \hat{e_x} + F_y \hat{e_y} = F_r \hat{e_r} + F_{\phi} \hat{e_\phi}$$ given in cartesian coordinates, ...
0
votes
1answer
190 views

What is the truth table for the logical expression NOT(NOT(A) OR NOT(B)).

What is the truth table for the logical expression NOT(NOT(A) OR NOT(B)). A B NOT(NOT(A) OR NOT(B)) 0 0 0 1 1 0 1 1 And also what logic ...
0
votes
1answer
90 views

Calculating running time for C code

The problem is this: How many array accesses does the following code fragment make as a function of $N$? ...
0
votes
0answers
372 views

Determine the interior, boundary, exterior and closure of the set $S= \{(x_1,…,x_n)\in\mathbb R^n\mid \forall x_i\in \mathbb Q\}$

Determine the interior, boundary, exterior and closure of the set $$S= \{(x_1,...,x_n)\in\mathbb R^n\mid \forall x_i\in \mathbb Q\}$$ I´m using the following definitions: A set is closed if it is ...
0
votes
1answer
18 views

how to show the probability of no occurrences of a single digit in a fractional expansion is a simple formula?

Say I have a binary string of length 1. What is the probability I never see a 0? My binary string can only be a 0 or 1. So obviously the probability I never see a 0 is 1/2. If instead I have a base ...
1
vote
1answer
32 views

Comparing $X^{-1}(E(X| \mathscr{G})(A))$ and $A$

Is it true that $X^{-1}(E(X| \mathscr{G})(A)) = A$ where $A \in \mathscr{F} $ and $A\notin \mathscr{G}$ where $(\Omega, \mathscr{F}, P)$ is the probability space?
2
votes
1answer
69 views

Solve integral $\int \sqrt{3x^2 - 2x}\ dx$

Find the following: $$\int \sqrt{3x^2 - 2x}\ dx$$ I've tried completing the square and doing trigonometric substitution but I think I am making a mistake somewhere. Thanks!
0
votes
1answer
4k views

Continuity and differentiability on piecewise function

Let $$f(x)=\begin{cases}x^2-3, & x<0;\\-3, & x\geq 0.\end{cases}$$ (a) Find the value of $x$ where $f$ is discontinuous (b) Find the value of $x$ where $f$ is non-differentiable ...
0
votes
0answers
84 views

differential equation with offset, characteristic polynomial equation

I have seen a lot of example problems on differential equations on forming a characteristic polynomial equation with the following diff-eq form: $\ddot{y}^2 + y = 0$ But what do you do when there is ...
3
votes
2answers
73 views

Suppose $(s_n)$ converges and that $s_n \geq a$ for all but finitely many terms, show $\lim s_n \geq a$

I've got a few questions about the problem. Prob :Suppose $(s_n)$ converges and that $s_n \geq a$ for all but finitely many terms, show $\lim s_n \geq a$ The solution here breaks this problem ...
2
votes
1answer
63 views

Are there PL-exotic $\mathbb{R}^4$s?

The title may or may not say it all. I know that there are examples of topological 4-manifolds with nonequivalent PL structures. In some lecture notes, Jacob Lurie mentions that not every PL manifold ...
5
votes
4answers
1k views

Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
4
votes
0answers
33 views

How can I compute this?(method) [duplicate]

$$\mu = \sum\limits_{k=0}^{\infty}\dfrac{2^{k}}{\displaystyle{2k+1 \choose k}}$$ The answer is $\dfrac{\pi}{2}$. But I don't know how to do it. Please show me the way! Thank you.
0
votes
1answer
34 views

Example for $(a,b,n,k) \in \mathbb{N}^4$, $r(a^n) = b^k$, where $r$ is the reverse of a number

While writing this answer I got an idea to generalize the problem. Question. Could anybody tell me such $(a,b,n,k) \in \mathbb{N}^4$ $4$-tuple, $a,b,n,k>1$, $a \neq b$, which satisfies the ...
1
vote
1answer
42 views

Is it true that the mobius function $\mu(\frac{n}{d})=(-1)^k\mu(d)$?

Is it true that the mobius function $\mu(\frac{n}{d})=(-1)^k\mu(d)$? and what does the $k$ represent? I am sorry for the simplicity of this questions, I am sure I have read this somewhere but I ...
2
votes
2answers
62 views

Terminology - variant of a hypergraph

In a hypergraph, we have vertices $V$ and hyperedges $H$, where each hyperedge is a subset of $V$. Suppose that we would like the hyperedges to be (ordered) tuples, rather than subsets. Does this ...
2
votes
2answers
36 views

Notation on partial deriviatives

If I need to find $$\frac{\partial ^{2}g}{\partial u \partial v}$$ Then do I want to perform $$ \frac{\partial} { \partial v}\ \big( \frac{\partial g}{\partial u} \big) $$ or $$ \frac{\partial g} ...
1
vote
1answer
37 views

How do I prove (scalar1 + vector1) * scalar2 is not equal to scalar1 * scalar2 + scalar2 * vector1?

I am taking a Linear Algebra course and have been stumped on a homework question for a few hours. How do I prove for two scalars, $c_1$ and $c_2$, and a vector $v$: $(c_1 + v)c_2 \neq c_1c_2 + c_2v$ ...
2
votes
3answers
170 views

Prove or Disprove n! = BigOh(2^n) via mathematical induction.

My computer science professor has us tasked with proving or disproving the statement the n! = BigOh(2^n). We are then suppose to say if it's always true, always false, or non-conclusive, ...

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