0
votes
0answers
2 views

Solving limit without L'Hôpital

I need to solve this limit without L'Hôpital's rule. These questions always seem to have some algebraic trick which I just can't see this time. $$ \lim_{x\to0} \frac{5-\sqrt{x+25}}{x}$$ Could ...
1
vote
1answer
38 views

single simple pole, $e^{i\theta } \frac{1-\overline{z_0}z}{z-z_0}$

$D= \big\{z:|z|\leq 1\big\}$, $|z_{0}|\in D$. A function $f(z)$ such that 1).$f(z)$ is analytic on $D\setminus \{z_0\}$; 2).$f$ has a single simple pole at $z_0$ ; 3). $f(z)\ne 0$, ...
0
votes
1answer
15 views

Logic type problem.

A school has 90 children.During the day each child attends 4 classes.Each class has 15 children and 1 teacher.During the day each teacher has 3 classes.What is the smalles number of teachers the ...
5
votes
5answers
80 views

Formula for $\sum_{k=0}^n k^d {n \choose 2k}$

If $d \geq 1$ is an integer, is there a general formula for $$\sum_{k=0}^n k^d {n \choose 2k}\,?$$ We know that $\sum_{k=0}^n k {n \choose 2k} = \frac{n2^n}{8}$ and $\sum_{k=0}^n k^2 {n \choose 2k} = ...
0
votes
1answer
17 views

sequences of six digits (0-9)

How many sequences of six digits(0-9) contain at least one 3, at least one 5 , and at least one 8? Can someone please give me a hint?
0
votes
0answers
6 views

How can I setup a triple integral for ellipsoid volume?

Find the volume of the ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1$$ where $a,b,c \in \mathbb R$. Attempt: Ok well, I figured spherical coordinates would probably be ...
0
votes
1answer
16 views

inner product of trivector and bivector in geometric algebra

Hestenes's "New Foundations for Classical Mechanics" book (page 47, 1.1c) sets a problem to show: $\begin{aligned}\left( \mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c} \right) \cdot B=\mathbf{a} ...
0
votes
0answers
9 views

The boundary area of union of balls

Let {v_i} be countable many points in an n-dimensional unit cube. Let A be the union of all of the balls around {v_i} which have radius 1/10. Is it true that the (n-1)-dimensional Hausdorff measure of ...
0
votes
1answer
14 views

Does a differentiable function map open intervals to open intervals?

I know that preimages are open for open images, since differentiability implies continuity. I suspect there is a counterexample to the above though. This is not homework, just study.
0
votes
1answer
4 views

The natural projection mapping $\pi : G -> G/N$ defined by $\pi(x) = xN$, for all $x$ in $G$, is a homomorphism, and $\ker(\pi) = N$.

The question again... The natural projection mapping $\pi : G -> G/N$ defined by $\pi(x) = xN$, for all $x$ in $G$, is a homomorphism, and $\ker(\pi) = N$. I am wanting to prove that ...
2
votes
2answers
41 views

How do I evaluate this integral by hand?

TL;DR how do I evaluate $\int_0^{2 \pi } \frac{1}{\cos ^2(\theta )+1} \, d\theta$ by hand? I'm trying to solve this problem: Find the volume of the region defined by $x^2+xy+y^2+yz+z^2\le1$. ...
0
votes
1answer
10 views

Determining if position vectors are on a line through the origin

I'm given the following $\text{true}$ or $\text{false}$ statement, where i must provide justification for my answer: $$\text{The points in the plane corresponding to} \begin{bmatrix} ...
1
vote
2answers
44 views

finite sum of powers

Can you find what $\sum_{i=1}^{n}\left(n/i\right)^{i}$ is equal to? By simulation, I know that a loose upper bound is $2^n$. I am happy with a proof of such upper bound if an exact expression is not ...
0
votes
0answers
4 views

An issue of applying Fubini's theorem in Fourier transform on Schwartz space.

Let $\hat{f}$, $\hat{g}$ be the the Fourier transform of $f$ and $g$ respectively where $f$ and $g$ are the members of the Schwartz space $\scr{S}{(\mathbb{R}^{N})}$. Then in the process of ...
0
votes
2answers
204 views

How many ways are there to divide into unordered piles…?

How many ways are there to divide five pears, five apples, five doughnuts, five lollipops, five chocolate cats, and five candy rocks into two (unordered) piles of fifteen objects each?
0
votes
0answers
7 views

Where and how a sequence of functions converge

$f_n:R \to R$ for n= 1,2, ... given by $f_n(x)= \frac{x}{n}sin(\frac{x}{n})$ Give the set of all points in $R$ where {$f_n$} converges pointwise. On what intervals does {$f_n$} converge uniformly? On ...
2
votes
0answers
14 views

Partial Differential Equations Exam help

new poster here. I have an exam in a PDE course in 72 hours which uses the second edition of Strauss. I am pretty scared because I don' exactly understand what I am doing. For instance, Chapter 6 ...
19
votes
3answers
652 views

Totally lost and frustrated [closed]

I love maths. Rather I think I love maths. I am not sure. Whenever I study maths, I like it but there are times that i get really frustrated and stop studying it. I am preparing for an entrance exam ...
0
votes
0answers
9 views

$C[G]$-module map

Suppose: $z$ is an element of the center of $G$, and let $V$ be a $C[G]$-module, and let $T_z$ be the linear transform that arises from multiplication by $z$ ($T_z(v) = zv$). Then I want to show that ...
0
votes
0answers
10 views

What is the proper name of the term which is inverted?

Is there a proper name for a term in the formula which requires to be inverted? For instance $$C(A+B)^{-1}D$$ So what is the name of the term $A+B$? Inverse term? Or just call it the term which is ...
0
votes
2answers
12 views

$T$ bounded linear, show $A_2T=TA_1$ for $A_2, A_1$ compact

Let $T:H_1 \to H_2$ be a bounded linear map between two infinite dimensional Hilbert spaces and suppose that $T$ is both surjective and injective. Let $A_2 \in K(H_2)$ (where $K(H)$ denotes the set ...
4
votes
1answer
46 views

How to find $\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}$?

find the integral of $f(x)=\frac1{(x-1)^2\sqrt{x^2+6x}}$ my attempt = $(x-1)=a$, $a=x+1$ so the integral'd be $\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}=\int\frac{da}{a^2\sqrt{a^2+8a+7}} $ lets ...
0
votes
2answers
9 views

Probability question (2 conditions)

Female smokers - 20 Male smokers - 40 Total smokers - 60 Non-smoking females - 70 Non-smoking males - 70 Total non-smokers - 140 Total females - 90 Total males - 110 Total - 200 If two people ...
0
votes
1answer
26 views

Factoring $x^4-2$ over intermediate subfields

Let $\alpha = 2^{1/4}$. Factor the polynomial $x^4-2$ into irreducible factors over each of the fields, ...
8
votes
3answers
117 views

Proof: Show there is set of $n+1$ points in $\mathbb{R}^n$ such that distance between any two distinct points is $1$?

Argh, I hate to ask a question again so soon, especially one I feel like I should know. Linear algebra is taking its toll, and I am not quite used to the theory side of mathematics. Anyways, I ...
1
vote
3answers
42 views

Are truth tables a valid method to prove an iff statement?

I recently had a homework assignment returned to me (for a Differential Geometry course, undergrad level) in which my instructor wrote "You cannot use truth tables to prove an if and only if ...
0
votes
1answer
15 views

Non-homogenous System where did I go wrong?

Solve the system $\vec{x^{'}}=\begin{pmatrix}2 & -5\\1 & -2 \end{pmatrix}\vec{x}+ \begin{pmatrix} -\cos t\\ \sin t \end{pmatrix}$ The Eigenvalues are $(2-\lambda)(-2-\lambda)+5=0 \implies ...
0
votes
1answer
22 views

What does this word represent?

What five digit number does serve represent if VCR+VCCT=SERVE? I have tried using 4,5,and 6 for C but that doesn't seem to work. Please help I do not understand.
0
votes
0answers
12 views

Eigenvalues bounded below

If we have a sequence of matrices $A_n$ such that all of the eigenvalues are positive and bounded away from $0$, is it true that $|A_nx| \geq \lambda|x|$ for some $\lambda>0$? Thank you
0
votes
0answers
12 views

Analysis of Integral of a continuous function

one more question today I've been thinking on... Prove that if $f$ is continuous on $[a,b]$, $0<a<b$, then $\int_{a}^{b} {f(t) \over t} dt$ $=$ $\int_{a}^{s} {f(t) \over a} dt$ for some $s \in ...
2
votes
1answer
76 views
+50

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
0
votes
0answers
4 views

The use of wavelets in time series modelling ( feature extraction part)

I have been working on modelling a time series using wavelets for a long time. I am quite familiar with the wavelet theory and all...However, I have a big understanding issue and really appreciate it ...
0
votes
0answers
21 views

Integration Question

If we know the integral $$\int \frac{\mathrm{d}x}{f(x)+1}$$ can we find the integral of $$\int\frac{\mathrm{d}x}{f(x)+c}$$ for arbitrary $c\in\mathbb{R}$ (where defined)? Does it make a difference if ...
0
votes
3answers
27 views

If G is a finite abelian group and $a_1,…,a_n$ are all its elements, show that $x=a_1a_2a_3…a_n$must satisfy $x^2=e$.

I have already tried with $S_3$, and indeed, the product is $(13)$, and $(13)^2=e$ But what about this: I define + in this way:$45=2$,$26=3$, $1$ is the identity. therefore, $123456=2326=233=3$, ...
-2
votes
0answers
45 views

Why are academic topics in mathematics so far away from real concrete mathematics? [on hold]

Academic mathematics seems to eat for breakfast zeta functions and Mellin transforms while in reality, the zeta function is only a heavy notation for the lexicographic order of the prime factorization ...
0
votes
1answer
32 views

Hilbert proof systems with hypothesis

This is my set of axioms: $A \rightarrow (B\rightarrow A)$ $(A\rightarrow(B\rightarrow C))\rightarrow ((A\rightarrow B) \rightarrow (A \rightarrow C))$ $(\neg A \rightarrow B)\rightarrow ((\neg A ...
1
vote
2answers
1k views

Find the equation of a parabola (in general form)

Find the equation of the parabola with axis parallel to the $y$-axis, passing through $(1/2,-5/2),(3/2,-9/4)$ and $(-7/2,3/2)$.
0
votes
0answers
38 views

find the period of a trigonometric function

I've found the period of this trigonometric function, $$y=\sin^n(x)+\cos^n(x)$$. when n ($n\neq2$)is odd, the period is $2\pi$, when n is even, the period is $\frac{\pi}{2}$. but how to proof it?
0
votes
0answers
16 views

Shoelace Formula for polygon, pre calc, help?

a.) Given polygon P with vertices (1,5), (4,8), (8,5), (6,2) and (2,1), find the following: Find the area of P. Tip: Make sure to move COUNTERCLOCKWISE from point to point to ensure you get a ...
2
votes
3answers
34 views

derivation of formula to determine determinants

Please explain the derivation of formula to determine determinant. e.g., to calculate determinant of why do we first multiply $a_{11}$ and $a_{22}$? Why not $a_{11}$ and $a_{21}$? Also why do we ...
0
votes
0answers
3 views

Asymptomatic distribution

For a series of independent binary random variables $X_i$, where $i=1,\dots,n$ each with a $\text{Binomial}(1,p)$ distribution, the logarithm transformation is defined by: ...
0
votes
0answers
20 views

Proving a series is greater than zero

I wish to prove that an equation is greater than zero. Let ...
0
votes
2answers
30 views

Evaluating $\int \frac{1}{9+4x^2} dx$

Evaluate $$\int \dfrac{1}{9+4x^2} dx$$ I let $u = 9 + 4x^2$ so $du$ would be $8x$. But I don't know any way to make the numerator $1$ become $8x$. I could multiply by $1/8$ but then I'd still ...
3
votes
1answer
26 views

When does conformal equivalence guarantee the existence of a “conformal homotopy”?

Suppose $f$ is a conformal equivalence between two domains $D_1$ and $D_2$ in $\mathbb{C}$. Does this imply the existence of a map $F_t(z): D_1 \times [0, a] \rightarrow \mathbb{C}$ such that each ...
-2
votes
0answers
14 views

Convergence of a sequence of functions on a compact set

Let {$f_n$} be a sequence of functions defined on a compact set and decreasing pointwise to 0. Show that {$f_n$} converges to 0 uniformly.
1
vote
2answers
15 views

Convergence in distribution to a standard normal

How to show $$ \lim_{n\to\infty}\left(1-\frac{t^2}{2n}-\frac{t^3}{2n^{\frac32}}\right)^{-n}=e^{\Large\frac{t^2}{2}}\,\,? $$ LHS is expanded and approximated form of a moment generating function. RHS ...
0
votes
1answer
9 views

Find radius of convergence, and then test the endpoints to determine the interval of convergence.

Consider $$\sum_{k=1}^{\infty}\left(\dfrac{x}{5}\right)^k$$ In class we had an extremely brief discussion on this topic, and so I still have many questions on how to start these problems. It ...
0
votes
0answers
8 views

Riemman sum to get surface area of cone, and cone height

I have this itch to think about math ideas in my free time. Like when I play a game or something, I kind of "work in parallel" on math ideas. Back in one of my calculus classes we worked on Riemann ...
0
votes
1answer
10 views

Is direct sum of two commutative rings is still commutative

Is direct sum of two commutative rings is still commutative?
2
votes
2answers
66 views

Constructing $\mathbb{R}$ from $\mathbb{Q}$ and showing $\mathbb{Q}$ is dense in $\mathbb{R}$

This is a very long, multi-part problem that we were told to figure out by any means possible. There are no limits on getting help or finding answers online. I haven't had much luck at all solving ...

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