All Questions
1,604,797
questions
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How to prove $G_{A^*}\supseteq (V G_A)^\perp$?
If $A:D\subset H \rightarrow H$ on a Hilbert space $H$ is a unbounded densely defined operator. Then its adjoint exists and
$$\left\langle \begin{pmatrix}x \\ Ax \end{pmatrix},\begin{pmatrix}0 & -...
- 622
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0
answers
5
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Proof for basic properties of proximal operators
I am reading the paper "Proximal Algorithms" by N. Parikh and S. Boyd, and I found interesting the basic properties of proximal operators. However, I can't prove the equivalence for the ...
0
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0
answers
3
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Helmholtz Proof, Trouble with Integral
I am on the last part trying to understand the vector potential of Helmholtz's Theorem.
Where $F = -\nabla A + \nabla \times B$. I simply don't see how this integral equals zero
$$ \int_v c(r_2)\cdot \...
- 1
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0
answers
6
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Notation for choosing the minimum of a set after applying the norm
I have a function $x: \mathbb{R^+} \mapsto \mathbb{R}^n$ defined as $x(k) = \text{min}\{p \in S: \lVert p \rVert > k\}$ (where $S \subseteq \mathbb{R}^n$). I want to choose the point with the ...
-1
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0
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20
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Totally bounded
Consider the following definition: Let d be a metric on X, we say that $d$ is fully bounded on $X$ if for each $\epsilon>0$, there exists a finite set $X_0 \subseteq X$ such that $X = \cup_{x \in ...
- 9
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9
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Determine whether $\{A_{\alpha} \cap B: \alpha \in I\}$ is a partition of $B \subseteq X$, where $\{A_{\alpha}: \alpha \in I\}$ partitions $X$.
Let $X$ be a nonempty set, and $\{A_{\alpha}: \alpha \in I\}$ be a partition of $X$. Further, let $B \subseteq X$, such that $A_{\alpha} \cap B \neq \emptyset$ for all $\alpha \in I$. Is $\{A_{\alpha} ...
- 275
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3
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Lower boudns on the maximum eigenvalue of the adjacency matrix of an edge weighted graph
If $G$ is a simple graph with adjacency matrix $A$ then the following inequalities are known to hold:
$$\sqrt{\Delta(G)}, d_{\text{avg}}(G) \leq \lambda_{\text{max}}(A) \leq \Delta(G)$$
Where $\Delta(...
- 1,484
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4
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Class of graphs with superpolynomial / subexponential number of maximal cliques
It is a classical result of Moon & Moser that the maximum number of maximal cliques in a graph with n vertices is $3^{n/3}$. There are several similar examples with exponentially many maximal ...
- 2,092
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27
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Let $p$ and $q$ be prime numbers. What is the largest possible value of the greatest common divisor of $(p+q)^4$ and $p-q?$
Anyone knows how to solve this? Largest I got so far is 8 but the answer is 16.
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19
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Analytic subset of complex manifold
I am reading the book Introduction to complex analytic geometry by Stanislaw Lojasiewicz. In Chapter II, we have that a subset $Z$ of a complex manifold $M$ is called an analytic subset of $M$ if ...
- 710
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9
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Constructing a basis for a matroid with a circuit in it.
Here is the question I am trying to solve (Matroid Theory, second edition by Martin Oxley, Chapter 1, section 2):
Prove that if $C$ is a circuit of a matroid $M$ and $e \in C,$ then $M$ has a basis $B$...
- 1,829
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0
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6
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Expectation problem of postman
A postman brought N letters to a house with two letter-boxes. Since the two boxes were empty, he puts 1 mail in each of the two mail boxes. Then he chooses one of boxes with probability proportional ...
- 73
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19
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Evaluating a value from symmetric polynomials
Some time ago I was given the following system of equations:
$$a^2 + b^2 + ab = 25$$
$$b^2 + c^2 + bc = 49$$
$$c^2 + a^2 + ac = 64$$
and tasked with finding $(a+b+c)^2$.
Clearly, the expressions here ...
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answers
16
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Sequence ,series and summation
Let {x{n}} be a sequence of non-negative real numbers such that x{n + 1}^2 =6x{n} +7 for all n >= 2
Which one of the following is true? (A) If x{2} > x{1} > 7 then sum n = 1 to ∞ x {n} is ...
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9
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Counting the Ways to Place Two Bishops on a Chessboard without Attacking Each Other
How many ways are there to place two bishops on a chessboard so that
they do not attack each other?
Attempt: Note that for a chessboard with dimensions $2n \times 2n$, where $n>1$, we will have $n-...
- 4,205
-3
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answers
24
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Discrete Math Proof of Logics
In a question, given that $P(n)$ is a statement for integers $n \geq 1$ such that for all $n : P(n) \implies P(n+4)$ . How can I determine if
$$ P(1) \wedge \neg P(4n + 1) $$ is true or false.
- 1
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1
answer
18
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The group of all bijective functions on S with composition as its binary operation is finitely-generated iff S is a finite set
Here is the problem, I thought it the I've been thinking for a long time, but I still don't have any ideas.
Problem: Let A(S) be the group of all bijective functions on S with composition as its ...
- 9
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0
answers
8
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Lebesgue-Stieltjes Measure on $\Bbb R^2$
If $(\Bbb R,B(\Bbb R),\mu)$ is measure space such that $\mu$ is finite on all bounded set in $B(\Bbb R)$.Then it correspondence to family $\mathcal A$ of non-decreasing and right continuous real ...
- 80
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18
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universe elementay properties in set theory
A universe $\Bbb U$ is a set satisfing the following properties:
1.$\emptyset \in \Bbb U$
2.$u\in \Bbb U$ implies $u\subset \Bbb U$
3.$u\in \Bbb U$ implies ${u}\in \Bbb U$
4.$u\in \Bbb U$ implies $P(...
- 307
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answers
7
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Multiples of Bernoulli numbers
I read somewhere on wikipedia that $2^{m+1}(2^{m+1}-1)\frac{B_{m+1}}{m+1}$ is an integer, where $B_m$ is the $m$-th Bernoulli number. Why this true? I could not find the proof nor a reference in ...
- 425
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36
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if $f''\geq0$, then $f$ is greater than some linear function
Suppose that for all reals $f''(x)\geq0$. Does that imply that there exists $a,b$ such that $f(x)\geq ax+b$ for all $x\in\mathbb{R}$?
- 400
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0
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20
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Say set S ={a+b/2^n : a,b,n ∈ Z}, what is wrong with defining f: S → S as f(a + b/2^n=a-b/2^n for a,b,n in Z?
Trying to get into more abstract maths. One of my first times properly getting into sets, and the textbook I have doesn't have answers or hints or anything.
The question is, if
Set S = $$\{{a+\frac{b}{...
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0
answers
25
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Convergence of $\sqrt{3 + \sqrt{3 + \sqrt{3 + ...}}}$
I need proof that the sequence defined by $a_1$ = 3 and $a_{n - 1} = \sqrt{3 + \sqrt{a_{n - 1}}$ converges.
The proof:
Let $S = \sqrt{3 + \sqrt{3 + \sqrt{3 + ...}}}$. Then $S^2 = 3 + \sqrt{3 + \sqrt{3 ...
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0
answers
27
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How to prove that $IJ\subseteq P\implies I\subseteq R \vee J \subseteq R$
I'm having some trouble with the following exercise:
Let $R$ be a non trivial ring with unity, and let $P\neq 0$ be an ideal of $R$. Prove that if $R/P$ is a prime ring, then for all right Ideals $I,...
- 4,195
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answers
11
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Proving that support of a measure is a carrier
I am reading Introduction to Model Spaces and their operators by Garcia, Mashreghi and Ross. In first page, the authors define what a support and carrier of a measure. The measures considered here are ...
- 3,811
-1
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answers
11
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Given a unique set of values, what is the probability of duplicate values when sampling with replacement
Suppose I have a set of numbers from 1 to 𝑁: {1, 2, 3, ..., 𝑁}. I draw a sample size of S from this set with replacement each time I draw a number. What is the probability of getting at least a pair ...
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votes
1
answer
35
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Matrix of $T$ for the linear transformation $T:\mathcal{M}_{2\times 3}\longrightarrow\mathcal{M}_{3\times 2}$ where $T(A)=A^{T}$
I'm trying to figure out a matrix for $T$ with respect to the standard basis of $\mathcal{M}_{2\times 3}$ (let's call this basis $B$) for the linear transformation
$T: \mathcal{M}_{2\times 3}\...
- 47
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answers
23
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Proving the quotient map is well defined, linear, and isomorphic
Reading Hoffman and Kunze, and there's a proof they write about quotient mapping that I have a few questions about:
Some points I'd like any explication on, if possible:
a) Why does it follow that if ...
-1
votes
0
answers
20
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Solve $(A∩X)^C$=(X\B)∪A
$(A∩X)^C$=(X\B)∪A Please help me solve the equation. I've been trying to solve it for an hour, but all the time I come to the contradictions. The following solution is presented in the answers: $X = A^...
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votes
2
answers
35
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Law of sines: How to choose unknown angle
Say we know an angle $\alpha$ and two sides $a$ and $b$.
Let $b$ be the opposite side of the unknown angle $\beta$.
By the law of sines we have that
$\sin(\beta)=\frac{b \sin{\alpha}}{a}$.
When we ...
-2
votes
0
answers
33
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How much is $a_{2010}$?
If $2/7 =0.a_1a_2...a_n....$, find out $a_{2010}$, i.e. $a_{2010}$=?
I tried to do some form of Gauss Sum and other formulas who are pretty much a like with Gauss Sum, but it didn't work and gave me ...
- 1
0
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0
answers
6
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Dimension of a meshing space for finite-element methods
In my Finite-Element Method class, we defined a meshing $\tau_{h}$ for an interval $I = [0,10]$ with $h$ being the size of a cell (unit in the partition of $I$). Then, internal approximation requires ...
- 15
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0
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7
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Conditional Mean simulation
The joint p.d.f of $f(x,y)=2 $ if $0<x<y<1$ and zero else where, I get the conditional expectation $\mathbb{E}(Y|X=x)=\dfrac{x+1}{2}, ~0<x<1$, I verified this with my Matlab code
...
- 2,627
1
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0
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20
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Prove that $n^b \ll a^n$ as $n \to \infty$ Where $b > 0$ and $a > 1$
I got a little stuck on this problem. My attempt was to try and prove it using $\log(n) \ll n^a$ as $n \to \infty$ because I have proven this before. I am not sure what is the best way to go about it. ...
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0
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24
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Probability Task | Need help figuring out what to do next.
Here is a task that I am trying to un-cracK.
We have the deck of 60 playing cards in which there are 25 different red cards, 20 different
green cards and 15 different white cards. We draw 7 cards. ...
- 21
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0
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20
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Monty Hall problem with no guarantee
There 's an 80% chance a ball is in a chest of drawers with 4 drawers.
If we open 3 and find they're empty ,what is the probability it s in the 4th (all drawers have an identical chance of having the ...
- 305
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0
answers
22
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Generating the topology of a base in a proper subset of a set?
Let $S$ be a set, $\tau_0$ be a subset of $2^S.$ We want to think of $\tau_0$ as a base for a topology on S. Suppose we have the situation where
$$S_0=\bigcup_{s\in\tau_0}s\subset S.$$ That is, it is ...
- 389
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1
answer
29
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$\exists X, Y, Z \in \mathbb{Z}[\omega]$ such that $X^3 + Y^3 + Z^3 = \omega$?
I am considering the following problem: Denote by $\mathbb{Z}[\omega]$ the set of Eisenstein integers. Let $X, Y, Z \in \mathbb{Z}[\omega]$ be non-zero integers coprime to $1-\omega$. Is it possible ...
- 121
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0
answers
31
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Is there an easy way to calculate this infinite summation
Is there an easy way to calculate this summation of integral:
$\Sigma_{n=0}^\infty \int_{r=0}^1 \frac {(r-.5)cos(c.ln(r+n))} {(r+n)^{1-b}} dr $
The most obvious approach is to calculate the integral ...
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votes
0
answers
10
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Smith Normal Form with only one basis change
Is there a nice normal form for matrices with entries in a PID that only uses a change of basis of the source, i.e., a pleasant $N$ such that $M= NT$ where $M$ is our original matrix and $T$ is an ...
- 546
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0
answers
19
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Neuromorphic Imaging: Modeling, Bilevel Optimization, and Generalized Nash Equilibrium
please I'm looking for the complete full text of this thesis
https://www.proquest.com/openview/d2ed29ee78d4f0082d42272034330776/1.pdf?pq-origsite=gscholar&cbl=18750&diss=y
1
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0
answers
10
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is the orbit space of a proper (but non-open) topological groupoid always Hausdorff?
I feel this ought to be a very simple question, but I seem to be asking it at a greater level of generality than people usually do:
Is the orbit space $|X|$ of every proper topological groupoid $G \...
- 131
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0
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11
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Question on definition of inverse number theoretic transformation
in a paper I found the following statement on the second page regarding the Inverse Number Theoretic Transformation (INTT), there the INTT is defined as follows:
$$h_l = N^{-1} \sum_{k=0}^{N-1} H_k g^{...
- 41
1
vote
1
answer
36
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Find Bezout's identity coefficients for $18 + \sqrt{-19}$, $18 - \sqrt{-19}$ in $\mathbb{Z}[\sqrt{-19}]$
I want to find coefficients $a$ and $b$ in $\mathbb{Z}[\sqrt{-19}]$ such that $a(18 + \sqrt{-19}) + b(18 - \sqrt{-19}) = 1$ (I know that these elements are coprime in $\mathbb{Z}[\sqrt{-19}]$).
At ...
- 1,509
2
votes
1
answer
16
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Finding the dimension and kernel of a linear map
I am working on solving this below problem:
Let $F$ be a field, $n$ a natural number, and $T: F^n \to F^n$ a linear map defined by
$$
T((v_1, \ldots, v_n)) = (v_1 + v_2, v_2 + v_3, \ldots, v_{n-1} + ...
-1
votes
2
answers
26
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Linear system of 2 ODEs with respect to 2 functions. Stuck at solving.
The system is
\begin{align}
y'+x'&=\sin(t) \\
y+4x'+3x&=\cos(t).
\end{align}
Subtracting the equations (after multiplying the second one with the $\tfrac{d}{dt}$ operator, a lengthy proof ...
- 131
0
votes
0
answers
12
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Defining the Fourier projection operator on $H^{-1}$ so that it moves freely on the dual pairing?
Let us consider the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and for any $f \in L^2(\mathbb{T},\mathbb{C})$, define $P_N : L^2(\mathbb{T},\mathbb{C}) \to L^2(\mathbb{T},\mathbb{C})$ as
\begin{...
- 6,808
-1
votes
0
answers
8
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Finance PV Annuity
Jenna wants to determine the present value of an annuity she is considering to purchase. The annuity promises to pay her $5,000 at the end of each year for the next 10 years. If the discount rate (or ...
0
votes
0
answers
26
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Cauchy Riemann using polar form of $f(z)$? (NOT polar form of $z$)
Every time I search for a polar form of the Cauchy-Riemann equations, I find answers relating the derivatives of the real and complex parts of $f(z)$ to each other. That is: $\dfrac{\partial u}{\...
- 560
1
vote
1
answer
36
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When can we say that $a \ge b \ge c$ without loss of generality?
I have seen a lot of problems solved by the method, we can say that $a \ge b \ge c$. without loss of generality". My question is when can we say this about a,b,c, because I realize through direct ...
- 453