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How many integers in {1,2 ..., 100} are divisible by 2, 3, 8 or 10 using inclusion exclusion principle

When I use brute force approach to answer this question it gives me exactly 67 ...
Koussay Dhifi's user avatar
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5 views

Convergence of series regarding symmetric difference

I am trying to understand the following exercise ($\Delta$ means the symmetric difference i.e. $A \Delta B$ contains all elements which are either in $A$ or in $B$): Let $X$ be a non-empty-set and $...
TeX_User's user avatar
1 vote
0 answers
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Examples of closed categories which are not monoidal closed?

There is a solid definition of "closed category" axiomatizing the idea that we can assign something resembling a hom-object to each pair of objects of a category. However, I am struggling to ...
Morgan Rogers's user avatar
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0 answers
7 views

Law of Total Probability and Conditional Probability

There is something that's a bit confusing for me in the topic of Conditional Probabilities. Assume we have the Universal Set as follows, Which is partitioned into $F,F^c$. Now if we want to calculate ...
Vacation Due 20000's user avatar
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0 answers
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Show multivariate version of Slutsky's theorem: $Z_nX_n\overset{d}{\to}AX$

Let $Z_n$ be a $(d \times d)$-probability matrix and $A$ be a $(d \times d)$-constant matrix. If $X_n\overset{d}{\to}X$ and $Z_n\overset{p}{\to}A$, then $Z_nX_n\overset{d}{\to}AX.$ my attempt If $A$ ...
ytnb's user avatar
  • 590
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0 answers
12 views

Categorical condition for the cokernel of a "diagonal" map

Let $C$ be a category (additive or abelian, say) and $X$ an object. Given a monomorphism of $X$ into $X \oplus X$ (like a diagonal morphism), what kind of conditions are necessary on $C$ to guarantee ...
user39269's user avatar
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7 views

Dense inclusions for $L^{\infty}$-Bochner spaces

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. It is known that $ L^{\infty}(\Omega)$ is dense in $L^{1}(\Omega)$ in the case that $\Omega$ is bounded, since $C_c^{\infty}(\Omega)$ is dense in ...
kumquat's user avatar
  • 169
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Does the fourier transform of the product of test function and bounded function belong to $L^p$?

Let $\mathcal D(\mathbb R^n)$ be the set of test functions, i.e., functions in $C^\infty(\mathbb R^n)$ with compact support. Let $f\in \mathcal D(\mathbb R^n)$, and $m:\mathbb R^n\to \mathbb C$ be a ...
daㅤ's user avatar
  • 3,274
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0 answers
6 views

Compensating acceleration vectors from IMU orientation

I have an IMU sensor giving me a quaternion of its orientation. I want to be able to mount the IMU as I want on the airplane frame, and still be able to get a quaternion that would be the same as if ...
Anselme's user avatar
2 votes
0 answers
11 views

Is there a Log-Sobolev inequality for Lebesgue measure on $[0,1]$ on compact subsets of $\mathbb R^n$?

After searching on the internet for long enough, I would like to pose the question here. I hope there is no duplicate (if there is please let me know) Is it true that, there is a universal constant $C&...
mathnoob's user avatar
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-1 votes
0 answers
7 views

Martingale proof explanation

Given the process $f(t,W_t)=\lambda sinh(\lambda W_t)exp(-\lambda^2t/2)$, where $W_t$ is a brownian motion and $\lambda$ is a constant, can you explain how the following steps where derived? $$\mathbb{...
DivertingPie's user avatar
1 vote
2 answers
19 views

In a graded algebra $B$, is $B_+$ always prime?

I'm sure that this question is very easy but I couldn't really find an answer on the internet. Let $B$ be a graded $A$-algebra. We have $B=\bigoplus_{n\in\mathbb N}B_n$, and we have the ideal $B_+:=\...
Ezio Greggio's user avatar
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0 answers
13 views

How to show that the increment operation for natural numbers is well defined?

Let $a,b$ be natural numbers and $++$ be an increment operation. Based on Peano axioms alone, how to show that if $a=b$, then $a++ = b++$? Do note that the book I am reading (Real Analysis by Terrence ...
Stokolos Ilya's user avatar
-2 votes
0 answers
27 views

What are the positive integers such that n³-31/n²-7 equals to an integer [closed]

I've tried rewriting it but it did not help,any help is appreciated.
complexity's user avatar
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0 answers
15 views

T/F: If $\alpha\in\mathbb{R}_{>1}$ is not Pisot, then $\{\lfloor\alpha^n\rfloor:n\in\mathbb{N}\}$ contains infinitely many odd and even integers

True or false: If $\alpha\in\mathbb{R}_{>1}\setminus\mathbb{N},$ then the set $\{ \lfloor \alpha^n \rfloor: n\in\mathbb{N} \}$ contains infinitely many even integers and infinitely many odd ...
Adam Rubinson's user avatar
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6 views

monotonicity of Dirichlet energy under different Dirichlet boundary conditions.

Let $\Omega$ be a domain with two smooth boundaries in $\mathbb{R}^n$. Suppose $u_1$ is a harmonic function in $\Omega$ satisfying that $u=0$ in $\partial_1\Omega$ and $u=1$ in $\partial_2\Omega$. ...
STUDENT's user avatar
  • 816
-1 votes
1 answer
21 views

Division , is it the quantity contained by one group?

Disclaimer: This might sound stupid Division means dividing something into groups, that is how most people define division. So by definition it is how much of the number belongs to a single group when ...
nandu's user avatar
  • 11
0 votes
1 answer
37 views

Maximum value of $a_{1}a_{2}+a_{2}a_{3}+\cdots +a_{7}{a_{8}}$

If $a_{1},a_{2},\cdots \cdots ,a_{8}$ all are non negative numbers and $a_{1}+a_{2}+\cdots +a_{8}=16$. Then maximum of $\displaystyle a_{1}a_{2}+a_{2}a_{3}+\cdots +a_{7}a_{8}$ What I try : $\...
jacky's user avatar
  • 5,202
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0 answers
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Minimum Spanning Tree (MST): Cut property direct proof

I'm trying to fully understand the cut property in the concept of minimum spanning trees (MST) in graph theory and graph algorithms. It seems that all the literature out there proves this theorem via ...
Michel H's user avatar
  • 342
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0 answers
28 views

Writing/Typing Math in 2024 [closed]

When I was in math grad school three decades ago, everyone took notes, solved problems, and did research with pen and paper. For publication, 99% of work was prepared in LaTeX. (I'm more than a little ...
Elliotte Rusty Harold's user avatar
0 votes
1 answer
13 views

The Relationship Between Field Extension Tower and Tensor Product

Let $F$ be a field. Consider field extension $ F \subset L \subset K $. Let $ \alpha_1, \dots , \alpha_m $ be an $F$-basis of $ L $, $\beta_1, \dots, \beta_n$ an $L$-basis of $ K $. By the theory of ...
Long-Ping Li's user avatar
0 votes
1 answer
26 views

Is there a geometric interpretation to this complex equation?

$$ |z - i| + 2 \overline{z} - 1 = |\overline{z} - i| + z + 3i $$ I tried solving this by plugging in z = x + yi, but I feel like that gets unnecessarily complex. Is there a geometric interpretation ...
Mixoftwo's user avatar
  • 133
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0 answers
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Factorization over cyclotomic extensions of finite fields

I know that $\Phi _{8}(x)=x^4+1=(x^2+18x+1)(x^2+5x+1)$ in $(\mathbb{Z}/\langle 23\rangle)[x]$ and also that $x^4+1$ splits in $(\mathbb{Z}[e^{i\pi /4}]/\langle 23\rangle)[x]$ as $x^4+1=(x-e^{\pi i/4})(...
1278dome's user avatar
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0 answers
7 views

Small examples of non-transitive Automorphism groups of Steiner Systems

I'm currently doing research for a bachelor's seminar talk. I have found a result from E. Mendelsohn, "On the groups of automorphisms of Steiner triple and quadruple systems" stating that ...
dilemmma's user avatar
1 vote
1 answer
27 views

Compact + absolute convergence of eigenvalues $\Rightarrow$ trace class?

Let $H$ be a complex Hilbert space. Lidskii’s theorem says that if an operator $T \in B(H)$ is trace class, then $\operatorname{tr}(T) = \sum_i \lambda_i$, where the sum includes all nonzero ...
WillG's user avatar
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Smith normal form of 3x3 matrices

Let $A=\begin{pmatrix}a & d & e\\ 0 & b & f\\ 0 & 0 & c \end{pmatrix}$ be an integer matrix such that $abc=n$, $0 \leq d < b$ and $0 \leq e,f < c$. I'm trying to find an ...
user23571119's user avatar
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Can any one teach me about proof of Theorem 7 in page 282 of the Evans's PDE book more friendly?

I think that I am begginer of PDE. I am reading the Evan's PDE, p.282, Theorem 7 and some question arises : Q.1. First underlined statement : Where did $C$ come from? In previous sentence Evans wrote ...
Plantation's user avatar
  • 2,656
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0 answers
24 views

Why are split monomorphisms equalizers?

The nlab states about a split monomorphism $m: A \rightarrow B$ with a retraction $r$: Any split monomorphism is a monomorphism, in fact a regular monomorphism (it is the equalizer of $𝑚∘𝑟$ and $...
SaschaH's user avatar
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0 answers
10 views

Steps on solving point to plane exercises

I just want confirmation that the steps I've took to solve these two exercises are correct Exercise 1 Consider in R3 the line l defined by: \begin{cases} x = 2 + 3t \\ y = 2 - 2t \\ z = 1 + t \\ t \...
zaxunobi's user avatar
  • 131
0 votes
2 answers
27 views

Jacobson radical and invertible element

Let $I_1,I_2$ be ideals of a ring $R$ such that $I_1+I_2=R$ and their intersection is contained in $J(R)$ (the Jacobson radical of $R$). Show that if $x_2$ is an element of $I_2$ s.t. $x_2+I_1$ is ...
user686685's user avatar
-1 votes
1 answer
26 views

Proving opposite statements in inequalities

I was solving the following inequality: "Considering $a,b,c>0$ satisfying $abc=1$, prove that: $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge a+b+c$$ I tried to use Hölder: $$\left(\frac{a}{b}+\frac{...
Francisco Sierra's user avatar
1 vote
0 answers
10 views

Holomorphic interpolation of a square summable sequence

My question is as follows. For some $0 < R < 1$, consider the set $X = \{Re(z) > 0 \} \setminus D(0,R)$. Given at each integer $n \geq 1$ a value $a_n \in \mathbb{R}$, such that $(a_n)_n \in ...
user1274777's user avatar
-2 votes
0 answers
12 views

Concept of the likelihood function

I have a few doubts regarding how to interpret the likelihood function. 1)If the likelihood of a single data point is the value of the y-axis at that point,then what is the likelihood of multiple data ...
stat1809's user avatar
6 votes
1 answer
64 views

Does this inequality holds $ \int_{[0,1]} (f')^2 e^f dx \geq C \int_{[0,1]} f dx$ for some constant $C>0$?

Prove or disprove the following: There exists a universal constant $C>0$ such that $$ \int_0^1 (f'(x))^2 e^{f(x)}dx \geq C \int_0^1f(x)dx $$ for all $f$ such that $ \int_0^1 e^{-f(x)}dx = 1$. This ...
mathnoob's user avatar
  • 129
-1 votes
1 answer
11 views

Number of non-isomorphic sub-trees of a tree with n vertices [closed]

Does any of you know any upper bound for it better than $O(2^n)$? I try to create a dynamic programming algorithm which goes through all subtrees of a given tree. I wonder whether it is possible to go ...
Mich Szyfel's user avatar
1 vote
0 answers
22 views

How axioms of inner product ensure that an instantiation/realization capture notion of angle correctly?

Axiomatic definition of inner product can lead to various instantiations like euclidean inner product or complex inner product or weighted inner product etc. Whatever the special case, we can be sure, ...
irman 's user avatar
  • 11
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0 answers
17 views

Fully spanning subset

Consider a finite multiset of integers denoted by M. I define a subset of that multiset denoted by A. I say that A is a Fully Spanning Sub-set of M if, for every item x in M, if ...
JoeHills's user avatar
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0 answers
22 views

Prove that for $n$ sufficiently large and $k \leq n$, there exists $m$ having exactly $k$ divisors $\le n$

Prove that for all sufficiently large positive integers $n$ and a positive integer $k \leq n$, there exists a positive integer $m$ having exactly $k$ divisors in the set $\{1,2, \ldots, n\}$. here is ...
Saucitom's user avatar
-2 votes
0 answers
30 views

Prove, without using derivatives, that xsin(1/x) is not Lipschitz in (0,1] [closed]

I know that it is not Lipschitz by using derivatives, but I can't prove it without.
Salvo Profita's user avatar
1 vote
1 answer
39 views

Oven temperature

This question is a crosspost from this one on another StackExchange subsite due to the recommendation given there. The question arose in the kitchen, not while doing homework. The motivation Consider ...
user7427029's user avatar
0 votes
0 answers
14 views

Equality of two completions

I have the following question. Suppose $R$ is Noetherian ring, $I$ is ideal in $R$ and $S$ is multiplicatively closed set. Let $(I^n\colon\langle S\rangle) = \varphi^{-1}(I^nS^{-1}R),$ where $\varphi\...
abcd1234's user avatar
1 vote
0 answers
44 views

Why do Fibonacci sequences result from this process?

If we have two columns of numbers made by the following rule, we get two Fibonacci sequences. Is there a straightforward way that enables us to just 'see' why this would happen. If anyone can find ...
John Hunter's user avatar
0 votes
0 answers
14 views

Inaccurate Eigenvectors from LAPACK's ZHEEVR routine

I was trying to replace Scipy's eigh routine in my cython code with C-level LAPACK. Essentially I want to compute eigenvectors corresponding to the largest and smallest eigenvalues of a hermitian ...
mgns's user avatar
  • 194
0 votes
0 answers
8 views

Under what conditions is the matrix norm of $||(R + A^T P A)^{-1}B^T P||$ finite for all positive semi-definite matrices $P$?

Assume $R$ is a positive definite matrix, $A$ and $B$ are square matrices, and $P$ is a positive semi-definite matrix. I would like to know the conditions under which $\sup_{P \geq 0} \|(R + A^T P A)^...
Weiyan's user avatar
  • 1
0 votes
0 answers
25 views

Finding area of triangle in Argand Plane

Given $|z|^2=4$, find the area of the triangle formed by the complex numbers $z$, $\omega z$, $z+\omega z$ (here, $\omega$ is the complex cube root of unity). I understand the solution which claims ...
Satyam sharma's user avatar
0 votes
1 answer
27 views

Excess kurtosis of binomial distribution

So I been at this for hours. I don't know where my simplification is going wrong but here it is: $$\frac{n[(n-1)(n-2)(n-3)p^4 + 6(n-1)(n-2)p^3 +7(n-1)p^2 + p] - 4(n[(n-1)(n-2)p^3 +3(n-1)p^2 + p + 6(np)...
maria guallpa's user avatar
0 votes
1 answer
39 views

Double-checking my formula derivation

The formula for the surface area of a right antiprism with regular n-gonal bases B isn't on the wikipedia page for antiprisms so I derived it myself. Based on the diagram below I got that Surface ...
Nate's user avatar
  • 97
2 votes
3 answers
79 views

Use $\,\varepsilon-\delta\,$ definition of limits to show that $\lim\limits_{(x,y)\to(-1,2)}\frac{x^3+y^3}{x^2+y^2}=\frac{7}{5}$

We need to show that for every $\varepsilon >0$ there exists a $\delta >0$ such that $$\left\lvert\frac{x^3+y^3}{x^2+y^2}-\frac{7}{5}\right\rvert<\varepsilon $$ whenever $$0<\sqrt{(x+1)^2+(...
Afzal Ansari's user avatar
2 votes
1 answer
37 views

Any two disjoint subsets in a family of subsets intersect, prove that any maximal such family of subsets must contain $2^{n-1}$ subsets

Here is the problem: Let $\mathcal{F}$ be a family of subsets of an $n$-element set $X$ with the property that any two members of $\mathcal{F}$ meet, i.e., $A \cap B \neq \emptyset$ for all $A, B \in \...
szpolska's user avatar
1 vote
0 answers
35 views

Let $f,g\in C[0,1]$ and $U= \{h\in C[0,1]:f(t)<h(t)<g (t),\forall t\in [0, 1]\}$ in $X= (C[0,1], \| .\|_{\infty} ).$ Is $U$ a ball in $X?$

Let $f,g: [0,1]\to\Bbb R$ be continuous and $f(t) < g (t)$ for all $t\in [0,1].$ Consider the set $$U= \{h\in C[0,1] : f (t) < h(t) < g (t) ,\text{ for } t\in [0, 1]\}$$ in the space $X= (C[0,...
Thomas Finley's user avatar

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