2
votes
2answers
250 views

Is Im(T) + Ker(T) the same as Im(T) union Ker(T)

If i know Im(T) and Ker(T), is Im(T) + Ker(T) the union of the two vector space? If not, how do i find the addition of the two vector space. It is best if examples can be given. Thanks.
1
vote
0answers
53 views

Cross-talk filter with known source

Hello fellow Stackers, This question was also posted on StackOverflow, but perhaps this is a more suitable location for this question. I currently work in an experimental rock mechanics lab, and when ...
3
votes
1answer
56 views

Calc 2: Integration by Parts w/ trig identities

$$\int e^{3\theta}\sec^4(e^{3\theta})\tan^{11}(e^{3\theta})d\theta$$ I just want to make sure that I'm doing this correctly so that I can understand the material. I would also appreciate any tips or ...
2
votes
3answers
108 views

Integral $\int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x}$

Integrate: $$ \int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x} $$
1
vote
0answers
58 views

Evaluating integral involving product of cosine inverse

I am trying to evaluate the below mentioned integral which involves product of two cosine inverses and two variables $x$ and $y$. I need to evaluate the integral or get an approximate value of this ...
1
vote
1answer
99 views

Showing that two functions are orthogonal on a rectangle

I was given the following question, and I think I'm nearly there, I just wanted to ask for some clarification in the last step. Derive the eigenvalues and functions of the SL problem $\phi_{xx} ...
3
votes
1answer
143 views

Uniform convergence on an interval

Let $a<c<b$. Let {$f_n$} be a sequence of functions converging uniformly on $[a,c]$ and $[c,b]$. Prove that {$f_n$} converges uniformly on $[a,b]$ My attempt: Intuitively, I see that {$f_n$} ...
2
votes
3answers
185 views

Why doesn't the definition of the interior of a set depend on the dimension of the set

I have just started with a course on convex optimization and have been introduced to the concept of the interior of a set. I have a fairly basic question. I am still trying to understand this topic, ...
0
votes
2answers
34 views

Is the following text correct about the interior of a given set? (excerpt from Stephen Boyd's convex optimization text)

How is the interior of set C empty in this example? There is definitely more than one $x \in C$ such that $B(x,\epsilon) \subset C$.
0
votes
0answers
75 views

For all $\xi \in \mathbb{C}$ we have $e^{-\pi\xi^2}=\int_{-\infty}^\infty \! e^{-\pi x^2}e^{2\pi ix\xi}\ \mathrm{d}x.$

This is Exercise 2.4 in Stein & Shakarchi's Complex Analysis. Prove that for all $\xi \in \mathbb{C}$ $$e^{-\pi\xi^2}=\int_{-\infty}^\infty \! e^{-\pi x^2}e^{2\pi ix\xi}\ \mathrm{d}x.$$ They prove ...
1
vote
1answer
47 views

Relationship between $\int_a^b f(x) dx$ and $\sum_{i= \lceil a\rceil}^{\lfloor b\rfloor} f(i)$

Let we have a continuous function $f(x)$ in the interval $ [ a,b ] $ Does there exist any relationship between its integral and summation of function-values defined at the integers between $a$ and ...
0
votes
1answer
54 views

card drawing with replacement

If 6 cards are drawn from a deck of cards with replacement. What is the probability that the outcome is a club at least once. I thought it would be 13/52. But I was wrong.
2
votes
2answers
692 views

Prove that if a and b are positive integers, then there exists integers x and y such that 1/lcm(a,b)=x/a+y/b

My professor has not taught us the technique of writing proofs, he just continues to do them for us in class. So I am really stumped on this proof. Any help is greatly appreciated!
19
votes
6answers
106k views

What is the difference between independent and mutually exclusive events?

Two events are mutually exclusive if they can't both happen. Independent events are events where knowledge of the probability of one doesn't change the probability of the other. Are these ...
2
votes
1answer
68 views

When orthogonal polynomials form an Hilbert basis?

Let $\mu$ be a probability measure on $\mathbb R$, and consider the sequence of orthonormal polynomials in $L^2(\mu)$. These polynomials are constructed by applying Gram-Schmidt to the sequence ...
1
vote
1answer
38 views

Prove this by the principle of mathematical induction.

If $S_r(n)=1^r+2^r+\cdots+n^r$, then prove that $S_r(n) \geq \int_0^nx^r\,dx$. Please help me to solve this problem. I am not able to prove that $P(k+1)$ is true using $P(k)$.
2
votes
0answers
33 views

Generalized change of variables in integral

When I read the following (http://www.math.helsinki.fi/~analysis/GraduateSchool/maly/gs.pdf ), it is hard to understand it. In particular, what does it mean by the last equation? Why does it make ...
2
votes
1answer
57 views

Why is $\dim(W)=3$?

I am teaching myself upper-division linear algebra for the moment, and I currently do not understand this example in my textbook. This is from page 50 of Linear Algebra by Friedberg, Insel, and ...
1
vote
2answers
46 views

Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
2
votes
1answer
60 views

Number of possible 3x3 matrices with 0,1 entries [closed]

I have a $3\times 3$ matrix whose entries can be $0$ or $1$. How many patterns can I make with this? I know it has something to do with the binomial coefficient, but I haven't studied it in Yeats.
0
votes
1answer
162 views

Discriminant of a ternary quadratic form

What is the discriminant of a ternary quadratic form $x^2-y^2+z^2-2xy+4yz-6xz$? The answer says, first make it $a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{23}yz+2a_{13}xz$, and then the discriminant ...
6
votes
5answers
1k views

Why consider square-integrable functions?

Why are $L^2$ functions important? From reading around I have three hypotheses: they show up in QM (but, why?) they form an inner product space (but, is that a "tight bound" or is the class easily ...
2
votes
5answers
2k views

Is it true that $n^2+3n+13$ is prime for all $n\in\mathbb ℤ^+$?

Prove or disprove the statement: If $n\in\mathbb ℤ^+$, then $n^2+3n+13$ is prime. I am lost here. All I know is that $n$ is greater than or equal to one, since it is a positive integer.
0
votes
2answers
76 views

How should I simplify this expression using the Laws of Logic?

I have this expression here that I have attempted to solve, but as of now I have no success in solving. My problem is probably the distributing part because I don't know how to continue after ...
1
vote
0answers
45 views

Proof for the number of leaves for any Binary Search Tree

A property for binary trees is that the number of leaves is the number of full nodes plus 1, in other words, $L = F + 1$ where $L$ is the number of leaves and $F$ is the number of full nodes. What ...
0
votes
2answers
32 views

Neutral element of an Algebraic structure

Consider $(\epsilon,*)$ an algebraic structure. If the neutral element of $(\epsilon,*)$ is $e$ then it is unique.
0
votes
3answers
131 views

How would I show that Tn=3^n + 2 is a solution to the recurrence?

Would anyone be able to help me or give me some advice on the following problem: Consider the recurrence with $T_0 = 3$ and $T_{n+1} = 3T_n - 4$ for all $n \in \mathbb{N}$. How would I show that ...
-1
votes
1answer
995 views

Is there any shortcut to find if a number is a perfect cube?

Is there any shortcut to find if a number is a perfect cube? I am taking for instance finding if a number is a perfect square. So , if a number ends with 2,3,7,8. It cannot be a square. But if it ...
0
votes
2answers
121 views

Are these proposed rules for the canonical factorization of algebraic integers complete?

In $\mathbb{Z}$, the rules are fairly well established, a few minor quibbles notwithstanding. But in, say, $\mathbb{Z}[\sqrt{7}]$, there are, as far as I can tell, no established rules. What I've seen ...
0
votes
0answers
23 views

Combinations: 6 numbers selected, chance of 3rd largest being 15? [duplicate]

Q: 6 numbers are selected without replacement from the list {2,3,7,8,12,15,17,21,28}. Find the probability that the third largest number is 15. My attempt: There are 9 numbers and 6 can be ...
0
votes
2answers
148 views

Applications of infinite cardinalities in real analysis

What are some topics in real analysis that make use of infinite cardinalities larger than that of the real numbers themselves, preferably [edit: but not necessarily] topics that are widely applied in ...
1
vote
1answer
146 views

Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
2
votes
1answer
258 views

Proving the general formula [nx] where [.] is the floor function.

I've been trying to solve a exercise that asks me to prove the following generalization for the floor function: $$\lfloor nx\rfloor = \sum_{k=0}^{n-1} {\lfloor x + \frac kn \rfloor}$$ I've already ...
1
vote
1answer
112 views

Prove there exist a $p$ so that the inequality holds

I am stuck with the following problem. Given the Gaussian mixture distribution $f(\cdot)$ $$ f(x) = ...
2
votes
0answers
25 views

Converting a derivative constraint into an orthogonality constraint

Let's say I'm trying to generate a quadratic curve in 3 dimensions, given two points it passes through, $\vec a$ and $\vec b$ in $\mathbb{R}^3$, and normals to the curve at those points, $\vec n_1$ ...
0
votes
3answers
102 views

Proof that the greatest common divisor of $(a, a+2)$ is $2$ if $a$ is even and $1$ if $a$ is odd

Some help would be great on this, my teacher hasn't explained how to construct proofs to us, he just keeps doing them for us in class. I have at the beginning: Let a be even. Since the sum of two ...
0
votes
2answers
318 views

Finding the probability of choosing six numbers

I just started to learn probability and came across this question: Six numbers are selected without replacement from the list {2, 3, 7, 8, 12, 15, 17, 21, 28}. Find the probability that the third ...
0
votes
1answer
697 views

Overlaying scatter points over world map in MATLAB

I am facing difficulty overlaying lightning location data over a region of the world map. I have lightning location data where the first column contains longitude and the second column latitude. I ...
0
votes
2answers
81 views

Need help ordering a list of functions

List the functions below from lowest order to highest order. If any two or more are of the same order, indicate which. $n$, $n^3$, $2^n$, $\ln n$, $n^2$, $\ln^2 n$, $\sqrt n$, $2^{n−1}$, $\ln n$, ...
1
vote
2answers
178 views

Which of the following are reduced modulo residue systems modulo 18?

Question: Which of the following are reduced modulo residue systems modulo 18? $a. 1,5,25,125,625,3125$ $b. 5, 11, 17, 23, 29, 35$ $c. 1, 25, 49, 121, 169, 289$ $d. 1, 5, 7, 11, 13, 17$ Attempt: ...
1
vote
3answers
54 views

A proof that if $S \cap T = S$, then $S \subseteq T$

I have to prove the following: If $S \cap T = S$, then $S \subseteq T$. I have no idea where to start. Here's what I have done so far: Suppose $S$ and $T$ are two sets and assume the fact that ...
1
vote
0answers
26 views

Shortest route with a requirements set

Suppose you have a weighted connected graph, $G(V, E)$, with $n$ nodes such that every node has a edge to every other node (a large clique). You are also given a set of sets, $\{l_1, l_2, ... l_n\}$ ...
7
votes
3answers
265 views

Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups

I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in this Math Overflow thread, it is mentioned ...
0
votes
1answer
214 views

Jacobian and PDE

I am wondering how to compute the Jacobian in order to know if a given PDE satisfying an initial condition has a unique solution or not. If I consider the PDE, $u_x=1$, satisfying the initial ...
-4
votes
4answers
3k views

Is ∅ equivalent to {∅}?

Is ∅ equivalent to {∅}? I think they are, but I am not sure? If anyone could clarify, that would be great. Thank you!
0
votes
2answers
47 views

Inequalities and floors.

I've been presented with a question that I actually don't understand. The question is: Given $$\lfloor a\rfloor\leq a<\lfloor a\rfloor+1$$ Write an inequality for $\lfloor a\rfloor$ I'm fine ...
1
vote
1answer
77 views

Mean value theorem on a triangle in 2D

Given three points in $\mathbb{R}^2$, $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$, and a $C^2$ surface through them $f(x,y)$, is there always a point in the interior of the triangle in formed by the ...
2
votes
1answer
54 views

Proof of the uncountability of reals using the diagonal argument—problem?

Consider a common proof of the uncountability of $(0,1]$, as presented here for example: We assume that the reals can be arranged in a sequence $x_k$, represent every number in $x_k$ by its ...
1
vote
1answer
52 views

Calculus I - Simple Difference Quotient Question

The problem is to calculate the difference quotient of $f(x) = \sqrt{x^2 +2x+1}$. But $\sqrt{x^2+2x+1}= \sqrt{x+1}^2 = x+1$ so can I just take the difference quotient of $x+1$? If not, how do I ...
0
votes
1answer
47 views

Demonstrate that (p → q) → ((q → r) → (p → r)) is a tautology.

I'm struggling to demonstrate that (p → q) → ((q → r) → (p → r)) is a tautology. I know that : ...

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