# All Questions

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### Classify $\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}/\left<(1,1,2)\right>$

I am following a solution to classify $G/H$, $G=\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z},H=\left<(1,1,2)\right>$, according to the theorem of finitely generated abelian groups. It claims that ...
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### How to prove whether the limit as $(x,y) \rightarrow (0,0)$ of $f(x,y) = \frac{x^2 y}{\sqrt{x^4 + y^2}}$ exists or doesn't, using delta epsilon

I don't get how I can cancel out the x^4 term in the denominator
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### What does $a\ne b \ne c$ mean?

Does $a\ne b \ne c$ mean $a \neq b \land b \ne c$ or $a \neq b \land b \ne c \land a\ne c$ ? These are two distinct statements when 2 $\;\not\!\!\!\implies(a \ne c)$, yet I am unaware if one ...
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1 vote
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### Explicit example of a Fourier transform of $f \in L^2(\mathbb R) \setminus L^1(\mathbb R)$

The Fourier transform shall be defined by $$\hat f(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i \cdot \xi x} dx$$ The Fourier transform is well-defined for $f \in L^1(\mathbb R)$, that is, $f$ ...
31 views

### Showing the function $f'(a)=f(a)f'(0)$

Let $f(x+y)=f(x)+f(y)$ for all $x$ and $y$. Show that if $f'(0)$ exists then $f'(a)$ exist and $f'(a)=f(a)f'(0)$. I can't really seem to make much progess, I let $x=y=0$ and got $f(0)=0$ and from ...
28 views

### Can the multiplicative product of bijective functions $\mathbb{R} \to \mathbb{R}$ be bijective?

Given two functions $f$ and $g$, which are bijective $\mathbb{R} \rightarrow \mathbb{R}$, can $h(x) = f(x)g(x)$ also be bijective on $\mathbb{R} \rightarrow \mathbb{R}$? I can prove no such $h$ exists ...
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### Kronecker sum of a matrix with its transpose

I know that if $A + A^T = B$, then $B$ is symmetric. How about the Kronecker sum? If $A \oplus A^T = B$, $B$ or $A$ have some particular properties?
9 views

### The number of integral values of a for which the equation x⁴ - (a + 2)x³ + 2ax²+ 4(a - 2)x -16= 0 has at least two positive roots; a ∈[-10, 10] is-

The number of integral values of a for which the equation x⁴-(a + 2)x³+2ax²+ 4(a - 2)x -16= 0 has at least two positive roots; a ∈[-10, 10] is/are
16 views

### Are there some famous conjectures related to perfect graphs?

I am reading the book "Graph Classes: A Survey", and I know that Perfect Graph Conjecture (PGC) and Strong Perfect Graph Conjecture (SPGC) once were famous conjectures about perfect graphs. ...
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1 vote
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### How to define the sum of cardinals in the definition of an inaccessible cardinal?

I'm reading the Wikipedia definition of an inaccessible cardinal and I'm trying to understand it. On Wikipedia, a (strongly) inaccessible cardinal $\kappa$ is defined in the following way: $\kappa$ ...
21 views

### Why in the Alternating Series Test we need to consider both partial sums with even and odd number of terms?

I'm sorry, I think this is a pretty stupid question, but can not we just prove it for 1 of them? I mean, by Definition of Series Convergence : if partial sum is convergent then the series is ...
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### Second place in the Turing Machine race

Given a deterministic Turing machine T which begins on an infinite blank strip, let its growth rate $G_T(t)$ represent the number of non-blank squares after the machine is run for $t$ time steps. It ...
30 views

### Check that $T-\lambda I$ is bounded

I have a question for Spectrum for a bounded linear operator and its adjoint on a Banach space are same. I have to show that spectrum for a bounded linear operator and its adjoint on a Banach space ...
12 views

### Additive white Gaussian noise (AWGN)

In my research work regarding wireless communication, I came across many research papers wherein AWGN is assumed to be modelled as "complex Gaussian with zero mean and unit variance". I ...
1 vote
28 views

### Does infinite system of measurable subsets of a measurable set have a measurable intersection

Let $U$ be a finitely-measurable set, $G_1, G_2, \dots$ are its measurable subsets such that for every $i$ $\mu(G_i) \ge \alpha > 0$. Does there necessarily exist a measurable subset $\Gamma$ such ...
44 views

### Understanding an inequality in the proof

I believe my question is on some very elementary computations (i.e., this is probably answerable even if you don't know what is the subgaussian norm $\| \cdot \|_{\psi_2}$ etc. However, do let me know ...
11 views

### Multiplicative bijective continuous map from complex number to complex number.

Is there any special form of this kind of map. Obviously it will take 1 to 1 and roots of unity to roots of unity. But how we can go further, how it can acts on real numbers. Any hint will be very ...
40 views

### How could you solve the equation: 2^x + .5 = 4 - x^2 without graphing it

How could you solve for $x$ without graphing it, assuming you have a non-graphing calculator $$2^x+0.5=4-x^2$$
31 views

### $(a^m+b^m)\mid(a^n+b^n) \iff m\mid n$ [duplicate]

Prove that $(a^m+b^m)\mid(a^n+b^n) \iff m\mid n$. Here $a, b, m, n\in\mathbb{Z}^+$, $m\leq n$ and $(a, b)=1$. This is a questions from a number theory book that I am recently studying. I have read ...
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### Fundamental axioms of logic used to interpret most of modern mathematics

so i'm going through Terrence Tao's analysis 1 and he has a clear emphasis on rigor, however its kind of a contradiction when he didn't even go over the basics of mathematical logic. A mathematical ...
30 views

### Advice: best not to use the next quantum "detectors" on Earth [closed]

It will be best not to use the next gen quantum "detectors" on Earth. <br> A good example of an early quantum sensor is an avalanche photodiode (APD). APDs have been used to detect ...
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### Let $X=(X_1,\dots,X_n)$ be a random variable on $(R^{n},B^{n},P)$, calculate the conditional disribution of X given $\sigma(f), f(x)=x^TD^{-1}x$

Let $X=(X_1,\dots,X_n)$ be a random variable on $(R^{n},B^{n},P)$, and follow the n-dimensional multivariate norm distribution $N(0,D)$. Let$$f(x)=x^TD^{-1}x,x\in R^{n}.$$ So how to get the ...
16 views

### Covariance of the product of two random variables with another random variable

Let X, Y and V be three binomial random variables, that are NOT independent from each other. My objective is to find an expression for the covariance \begin{align}\operatorname{Cov}(XY,V)&\end{...