0
votes
0answers
21 views

How do I know if a given polynomial is a quasi polynomial?

How do I know if a given polynomial is a quasi polynomial? For example, if I'm given the polynomial: $e^x\tan(x)$ or the polynomial $e^{(i-t)}t^3$, my gut feeling is that they're both not quasi ...
1
vote
2answers
22 views

Finding median of union of two sorted (ordered) lists

We are given two sorted list of numbers $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$. Question is, how to find a median for list $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$. Algorithm should ...
1
vote
3answers
50 views

Topology; Definition of the open ball and open sets confuses me

I just picked up T.W Gamelin’s book on topology. I started reading and got confused when I came to the definition of an open ball on the second page. It says $B(x;r) =$ All $y$ in the set $X$ such ...
2
votes
0answers
38 views

Is this proof clear, complete and readable?

I am trying to prove this statement: $C^2/U(1) $ can be identified with $R^3$ so that the image of the $U(1)$ fixed point is $(0,0,0)$. And I was wondering if someone could tell me if the following ...
0
votes
1answer
21 views

How to this Simplify Boolean Expression

Im very weak in math and logic, and currently try doing K-map, and got this as result: $(C'B')+(CB)$ my question is, can this be more simple? I tried it myself, but I got $0$. $(C'C)+(B'B)$ I just ...
3
votes
1answer
56 views

formula for summation notation involving variable powers

I need help finding the formula for this summation notation: $$\sum_{k=1}^n{k^{2k} }$$ or $$1^2 + 2^4 +3^6 +.....+n^{2n} $$ And preferably not involving calculus.
0
votes
0answers
26 views

Trying to prove concurrence of altitudes of a triangle.

I know that this question had been asked before, but I am not exactly following what the answers say. Doing my own way here: I am puuzzled how to continue? I named the points A,B,C, and the foot of ...
2
votes
1answer
53 views

Irreducible and prime elements

In my commutative algebra lecture notes it says: A non-zero element $p$ of a ring $R$ which is not a unit of $R$ is called a prime element if $p=ab$ implies $a$ is a unit or $b$ is a unit. Is this ...
1
vote
2answers
88 views

Permutation of positive real numbers

Consider a set of positive real numbers $\{P_1,P_2,\dots,P_n\}$ and a permutation of this set $\{Q_1,Q_2,\dots,Q_n\}$. Is it possible to find a permutation such that ...
0
votes
0answers
16 views

Evaluate the sum $n$ of geometric random variables

Let $X_i\sim G\left (1-\frac{1-i}{n}\right)$. Evaluate $ \sum_{n=1}^n X_i$ My Try: $$ \sum_{i=1}^n X_i = \sum_{i=1}^n \sum_{k=1}^\infty \left(\frac {i-1}{n}\right)^{k-1}\left( 1 - ...
1
vote
0answers
10 views

Entropy proerty

Let $a,b,c>0$ be distinct postive reals. Define four different probability distributions: $$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ ...
2
votes
2answers
36 views

If given $\sum_{r=1}^{m-1}\binom r3$, how does the summation evaluate when $n<r$ in $\binom nr$?

Correct me if I'm running the summation correctly - $$\sum_{r=1}^{m-1}\binom r3=\binom 13+\sum_{r=2}^{m-1}\binom r3$$ $$\sum_{r=1}^{m-1}\binom r3=\binom 13+\binom 23+\sum_{r=3}^{m-1}\binom r3$$ ...
1
vote
2answers
27 views

Transforming linear combination of the cosine and sine function

In the proof of Transforming $a\cos(x)+b\sin(x)$ to $r\cos(\phi-x)$ ...
4
votes
2answers
74 views

No. of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$

Number of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$ $\bf{My\; Try::}$Using Hit and trial $x=0$ and $x=1$ are solution of above exponential equation. Now we will ...
2
votes
2answers
14 views

Find the area of the cyclic quadrilateral given the two diagonals

One diagonal of a cyclic quadrilateral coincides with a diameter of a circle whose area is 36$\pi$ $cm^2$. If the other diagonal which measures 8 $cm$ meets the first diagonal at right angles, find ...
-2
votes
1answer
37 views

Evaluating a complex integral using the Cauchy integral formula [on hold]

I need to evaluate the following integral counterclockwise: $$\oint_{\left | z \right |=\frac{1}{2}} \frac{dz}{(z-1)\sin z} $$ using the Cauchy integral formula
4
votes
2answers
248 views

Proving an intuitive fact about sets in the plane

The entire 2-dimensional plane is covered by 3 sets: Blue, Green and Red. It is given that: All sets are closed. All sets are interior-disjoint (but may meet at their boundaries). Blue is bounded. ...
1
vote
2answers
41 views

$\Bbb{Z}[X]/\langle f\rangle$ is a finitely generated $\Bbb{Z}$-module

Let $f$ be a monic polynomial in $\Bbb{Z}[X]$. Show that $\Bbb{Z}[X]/〈f〉$ is a finitely generated $\Bbb{Z}$-module. I don't even know how to start. If $g\in\Bbb{Z}[X]/〈f〉$, we are trying to find ...
1
vote
2answers
30 views

Changing order of summation - proof

How was the right side of equation obtained from its left side? I could obviously guess immediately that this is true, but mathematics is not about guessing. Are there any intermediate steps between ...
0
votes
1answer
17 views

suppose a graph $G$ is 3-regular with bridge (cut edge),do we have $\chi^{'}(G)=4$?

suppose a graph $G$ is 3-regular with bridge (cut edge),do we have $\chi^{'}(G)=4$? I think that it is right but I couldn't prove it,can you give me some hint or guidance about it,thanks a lot.
3
votes
2answers
29 views

Cardinal Arithmetic proof issues.

Let $X$ be a finite set and let $x$ be an object which is not an element of $X$. Then $X \cup \{x\}$ is finite and $|X \cup \{x\}| = |X| + 1$. Proof. Let X be a finite set with cardinality n, ...
1
vote
0answers
26 views

Eisenbud/Haris, Exercice I-53: morphism between global spectra

Let $X=\operatorname{Spec}(\mathscr{F})$ and $Y=\operatorname{Spec}(\mathscr{G})$ two global spectra over a scheme $S$ and let $f:X\to Y$ be a morphism. I want to show that for all prime ideal sheaf ...
1
vote
1answer
23 views

A function differentiable only at $0$ and for $|z|=1$

I need to find a polynomial function that is differentiable at the origin where $f'(0)=1$ and at every point $|z|=1$ but at no other point in the complex plane. I just have no clue how to solve ...
1
vote
1answer
34 views

Finite coloring of an interval

Two real functions, $f$ and $g$, are defined on the interval $[-1,1]$. Each point $x$ in the interval is colored in one of 3 colors: Red - if $f(x)>g(x)$ Blue - if $f(x)=g(x)$ Green - if ...
1
vote
2answers
36 views

Does $\displaystyle\liminf_{n\to \infty} -a_{n}= -\displaystyle\limsup_{n\to \infty}a_{n}$?

Let $(a_{n})$ be a bounded sequence. How to prove $$\displaystyle\liminf_{n\to \infty} -a_{n}= -\displaystyle\limsup_{n\to \infty}a_{n}$$ I don't how formally prove this..can someone guide me? tnx!
1
vote
2answers
39 views

Extending Euler's Theorem gives minus 1 - why?

Euler's Theorem states that for some coprimes $n$ and $a$: $a^{\phi(n)} \equiv 1 \mod n$ Example: $ a = 10, p=7, q=11, n=p*q=77, \phi(n) =(p-1)*(q-1)= 60$ $10^{60} \equiv 1 \mod 77$ When I take ...
1
vote
1answer
27 views

Existence of finite sets of infinite set without using AC

Is it possible to prove that every infinite set $B$ has a subset of cardinality $n$, for every natural $n$, without using AC? I know how to prove this claim by induction. In the induction step I chose ...
0
votes
0answers
18 views

Find an orthonormal basis for the subspace w

Find an orthonormal basis for the subspace $ W = \text{span} \{(3, 0, 4, 0),(0, −2, 1, 0),(0, −3, 0, 1)\}$ of $\mathbb{R}^4$ Without using Gram-Schmidt process.
2
votes
1answer
32 views

Solving cauchy hyperbolic second order pde

I'm currently taking a course in partial differential equations. I'm trying to solve the following problem (which is, as far as I can tell, a bit above the level of the course): $$\begin{align} ...
2
votes
1answer
36 views

Deriving angle from sin or cos

How can I derive the value in degrees of an angle starting from either the cos or sin value? $$ \cos(t) = c_{1} \quad \text{or} \quad \sin(t) = c_{2} $$
4
votes
1answer
42 views

Proving that if the semigroup (A, *) is a group, then the relation is an equivalence relation.

I'm aware that posting exam questions is probably frowned upon, but this isn't homework, I think I'm genuinely misunderstanding some part of the algebra. The question is this: Throughout this ...
0
votes
2answers
27 views

Intersections of two parabolas given focii

As part of Voronoi's algorithm, I need to calculate the intersection of two parabolas to compute a breakpoint at run time. I've spent literally 8 hours on this, and I've only gotten my equations to ...
1
vote
2answers
23 views

Show that $\left| \mathbb{P}(X=m)-\mathbb{P}(Y=m) \right| \le \mathbb{P}(Y\ne X)$

Let $X,Y$ two random variables of the same probability space. Show that $$\left| \mathbb{P}(X=m)-\mathbb{P}(Y=m) \right| \le \mathbb{P}(Y\ne X)$$ I think I need to start from LHS and split it ...
4
votes
2answers
27 views

Number of teams and matches

This question has two parts. Given n players, how many different teams can be created with at least one and at most n-1 players? For example, given the four players A, B, C, and D, the following ...
0
votes
0answers
11 views

Minimum weight vertex cover test set

I want to test the efficiency of an approximate algorithm for finding the minimum weight vertex cover. Is there an online test suite that contains weighted graphs with known vertex covers? If not, ...
2
votes
3answers
28 views

Explanation solution partial-fraction of $\frac{x^2 + 2}{x^2 - 1}$

The partial fraction of $\dfrac{x^2+2}{x^2-1}$ is $1 + \dfrac{3}{2}\cdot(\dfrac{1}{x-1}-\dfrac{1}{x+1})$. I understand how you get $\dfrac{3}{2}\cdot(\dfrac{1}{x-1}-\dfrac{1}{x+1})$ but from where ...
3
votes
3answers
85 views

Convergence of summable sequences

If $(a_n)$ is a sequence such that $$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$ How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
4
votes
0answers
49 views

Is there a logic to formalize the concept of “understanding”

The question may seem little bit weird given that philosophers have been struggling to have a full grasp on the concept of "understanding". But I'm wondering if there are any logics (modal-based or ...
4
votes
1answer
68 views

Find the inverse function 3

Find the inverse function for the following function: $$f(x) = \log_{\sqrt{4-x^2} } \left(x^3+5x^2-x \right)$$ Thanks.
4
votes
1answer
52 views

Homomorphism between finite groups

I have to prove or disprove the following statement: If $\phi:G \rightarrow H$ is a homomorphism between finite groups, with non-trivial image (i.e. $\phi(G)\neq\{e_H\}$), then $\#G$ and $\#H$ ...
2
votes
1answer
27 views

Proof, that every nonempty set of integers, not all zero, has a greatest common divisor

I'm searching for a proof or (better) a way to understand the proof from the book "Elementary methods in number theory", that every nonempty set of integers, not all zero, has a greatest common ...
1
vote
3answers
24 views

Find sum of quartic function coefficients by its plot

the plot of quartic function $y=ax^4-x^2+bx+c$ is given: I need to find a sum of $$\frac{a}{|a|} + 2\frac{b}{|b|} + 4\frac{c}{|c|} $$ How to do it?
3
votes
2answers
26 views

Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the laplace transform of the function $f(t)=te^{-t}\sin(2t)$ using only the properties of laplace transform, meaning, use clever tricks and the table shown at ...
0
votes
1answer
10 views

Finding ellips equation by focuses and tangent line

The Ellips which has focuses in $(±3,0)$ and a tangent line $x+y-5=0$. I need to find ellips equation. I've founded these equations $\frac{x_{0}}{a^2} = \frac{1}{5}, \frac{y_{0}}{b^2} = \frac{1}{5}$ ...
0
votes
1answer
25 views

How do I solve $(1+x^2)y'=\frac{1}{y}, z\geq 0, y(0)=0$?

I have this assignment: $$(1+x^2)y'=\frac{1}{y}, z\geq 0, y(0)=0$$ There was a long time ago that I solved one of those, but if I remember it right, I would want to rewrite the equation on the ...
1
vote
0answers
29 views

Sylow subgroup of $S_{11}$

I want to construct some Sylow $3$-subgroup of $S_{11}$.This subgroup has $3^4$ elements. I know any Sylow $3$-subgroup is isomorphic to $(\mathbb{Z}/3\mathbb{Z})^3\rtimes P$ where $P$ is a Sylow ...
1
vote
1answer
26 views

Injective homomorphism between a finite group $G$ and $GL_n(\mathbb{F}_p)$ where $p$ is prime

I'm looking for a solution to the following problem: Given a natural number $n$, a prime number $p$ and a finite group $G$, I need to find an injective homomorphism between $G$ and the group ...
0
votes
1answer
11 views

Equation of heat conduction for spherical solid

What will be the correct Equation of heat conduction for a homogeneous spherical solid with constant thermal diffusivity K and no heat source?
1
vote
0answers
19 views

What does it mean for a probability to “increase exponentially”?

In Wikipedia's description of the Metropolis algorithm, I see the phrase: The probability of rejection increases exponentially as a function of the number of dimensions. This obviously can't ...
2
votes
5answers
68 views

Every sequence in $\mathbb{R}$ has a monotonic subsequence

I have trouble with this kind of infinite construction in topology. Can someone check my proof is sound? Let $s$ be a sequence in $\mathbb{R}$. Then $s$ has a monotonic subsequence. There are two ...

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