0
votes
0answers
3 views

$n$-vertex $3$-edge-colored graphs with exactly $6$ automorphisms which preserve edge color classes, but permute the edge colors distinctly?

In each of these $3$-edge-colored graphs, there are exactly $6$ automorphisms which preserve the set of edge color classes: (These automorphisms don't necessarily map e.g. green edges to green ...
0
votes
0answers
3 views

looking for some exercise to test understanding of convector

I am trying to understand the concept of "convector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
1
vote
0answers
17 views

Sets of Egyptian fractions which sum to 1

Let an 'Egyptian unity sum set' be a set of positive integers {a, b, c ...} such that their Egyptian fractions sum to 1; and none of the elements are equal. That is: 1/a + 1/b + 1/c ... = 1 Let the ...
1
vote
0answers
13 views

The relationship between subfields and subgroups of a finite field.

I am trying to get my head around the structure $GF(p^n)$ when viewed as a vector space of dimension $n$ over $GF(p)$ (mainly the relationship between the additive and multiplicative structures). I'm ...
1
vote
1answer
22 views

If $x^a \equiv x^b \bmod p$, what can we say about $a$ and $b$?

If $x^a \equiv x^b \bmod p$, what can we say about $a$ and $b$, for $p$ prime? Is there any way to show the relationship between $a$ and $b$ specifically? It doesn't seem to be the case that $ a ...
0
votes
0answers
12 views

Approximation, $C^{1}$ function

I have a question about approximation by $C^{1}$ fuctions. Let $f:[0,1] \to \mathbb{R}$ be a Lipschitz continuous function. Question Let $\epsilon>0$. Can we find a $C^{1}$ fucntion $G$ ...
1
vote
0answers
13 views

how many strings are length 10 with 5 1's and 5 0's are there?

I used generating functions and got $A(x,y)=(x+y)^{10}$. Then I found the coefficient of the $x^5y^5$ and got $252$. Is that the correct answer?
0
votes
0answers
12 views

Proof of the ultraparallel theorem in the Beltrami Klein model

I was reading (and editing) the proof mentioned at https://en.wikipedia.org/wiki/Ultraparallel_theorem#Proof_in_the_Beltrami-Klein_model and noticed it is not correct. (the ultra parallel theorem is ...
1
vote
1answer
18 views

What's the time complexity of $T(n) =\sqrt{99nT(\sqrt {n})+100n}$?

What's the time complexity of $T(n) =\sqrt{99nT(\sqrt {n})+100n}$ , I don't have an idea for solve the question. My attempt : $\frac{T(n)}{\sqrt {n}}^2 =99T(\sqrt {n})+100 $ and $\ s(k)= ...
0
votes
0answers
13 views

Dual group of $\mathbb Z$

We know $\hat{\mathbb Z}=\mathbb T$ and the map $\alpha\longmapsto\chi_{\alpha}$ is an isomorphism of $\mathbb T$ on to the character group of $\mathbb Z$, but i cant prove this map is continuous? I ...
2
votes
0answers
7 views

Model-theoretic characterization of local modal correspondence

I've been reading van Benthem's dissertation (available on ILLC's website) on modal correspondence theory. In Section I.3, he develops a model-theoretic characterization of modal formulas having ...
1
vote
1answer
14 views

Vector in function format

Not sure how to interpret the follow: Find the intersection point(s) of the line $r(t)=(0, -2, -1)+t(1, 1, 1)$ and the plane $x+2y-4z=-3$ Does $r(t)=(0, -2, -1)+t(1, 1, 1)$ mean $r=(0, -2t, -t)$?
2
votes
0answers
15 views

Factoring bivariate quadratics with real coefficients (for high school students).

I was tutoring a Year 10 student last night (he's learning about quadratics). Unfortunately, we ran into a class of problems that I couldn't explain how to solve (beyond simply guessing and checking), ...
-5
votes
0answers
32 views

Julia and Mandelbrot Sets [on hold]

I need to know how escape, prisoner, Julia and Mandelbrot sets work. Are they all in one sequence or are they separate.
-1
votes
0answers
8 views

Bayesian statistics proof for a continuous uniform distribution

I know that if the prior distribution is chosen to be a continuous uniform distribution, then the exact posterior distribution will simply the normalized version of the likelihood function. I was ...
1
vote
1answer
24 views

Need help with Logic Question

ATTEMPT Given F $\subseteq$ G Writing out logical form of Goal we have $\exists$A$\in$F(x$\in$A) $\rightarrow$ $\exists$A$\in$G(x$\in$A) Now assuming(putting to list of givens) ...
-1
votes
1answer
37 views

Modulus Problem [on hold]

I do not understand how to solve such a question: $$|x+1| - |x| + 3|x-1| -2|x-2| = x+2.$$ How would you go about all the possibilities with which sign the modulus could take? Appreciate any help!
0
votes
1answer
9 views

Finding the Control Point in a bezier curve

This is a basic (and probably a stupid) question, math is not my forte and I don't know much about math, in this site: http://www.ams.org/samplings/feature-column/fcarc-bezier in the bezier curves ...
0
votes
2answers
22 views

order and cycles of perfect shuffle of 52 cards

This is the shuffle: $$1,2,\cdots,52$$ is turned into $$1,27,2,28,\cdots,26,52$$ when I try to write the cycles of this shuffle, I get a LOT of cycles, for example: $$(2\ 3)(27 \ 2)(26\ ...
0
votes
0answers
9 views

When can we say an algorithm is parameter free?

I was reading the paper by Dr. Francesco at http://arxiv.org/abs/1406.3816. The author presented a kernel version of parameter-free algorithm. But, any kernel, AFAIK, takes some parameter(RBF ...
4
votes
3answers
55 views

People sitting in a circle chewing gum

Ten people are sitting in a circle of ten chairs, chewing gum. Each person spits out his or her gum and places it either under his or her own chair or under an immediately adjacent chair. How many ...
0
votes
1answer
21 views

Critical point but not an extremum or saddle point

Let $f: R^2\to R$. Now, a critical point does not mean $f$ has a local (or global) extrema. Of course it could be a saddle point. Does anyone have an example of a function $f: R^2\to R$ that has a ...
3
votes
4answers
34 views

Basic of Partial Differential Equation

I pretty new to calculus and I am trying to understand the following transformations: $2uu_{t} = \frac{\partial }{\partial t}u^{2} $ $2u_{t}u_{tt} = \frac{\partial }{\partial t}u_{t}^{2} $ $2uu_{xx} ...
0
votes
1answer
17 views

number of combinations/permutations

if I have $n$ drawers and in each drawer I can only have 1 pen or 1 pencil for example if i have $3$ drawers the possible ...
0
votes
1answer
13 views

Question involving the PDF of a function of a random variable.

I'm trying to understand the setup for problem 3.1, from M.G. Bulmer's Principles of Statistics (Dover, 1967). Suppose that $X$ is a continuous random variable, and that $Y$ is a linear function ...
1
vote
4answers
33 views

How to show that the cycle $(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2)$

I generally do not have any problem multiplying cycles, but I've seen on Wikipedia that $$(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2). $$ I started following the path of $2$ on the right: ...
0
votes
0answers
22 views

Is the following a legitimate proxy for the Axiom of Replacement?

I'm working on an interface between set theory and plural logic. Here's my question: If one were to endow set theory with the expressive resources of plural quantification, could the following count ...
1
vote
2answers
21 views

maximum likelihood estimator for theta [on hold]

I was wondering if someone could please just get me started on this question i'm just a bit stuck: $$ f(y_1,y_2,\ldots,y_n\mid \theta)\propto \exp\left[\frac{−1}8 \sum_i (y_i−\theta)^2\right] $$ Any ...
0
votes
0answers
5 views

Time-series: High autocorrelation parameters?

I plotted the autocorrelation and partial autocorrelation for two of my time series data in R. But it seems that one of the autocorrelation plots of the two has much higher autocorrelation parameters ...
1
vote
0answers
38 views

Prove Pythagoras' Theorem by using the area of a kite [on hold]

Given a kite which can be dissected into two right triangles, use the area of the kite to construct a proof of Pythagoras' Theorem, preferably without using any figure not contained in the kite. If ...
-1
votes
2answers
30 views

Exponential distribution and expectation [on hold]

Given that $X ∼ Exp(λ)$, compute $\mathbb{E}[e^{−(X−\lambda/2)^2} ]$. Your answer should not be left as an integral. so you would get $\mathbb{E}[e^{-x^2+x\lambda-\lambda^2/4} ]$? Can this question ...
1
vote
1answer
26 views

A more elegant approach to proving independence between $X_{(3)}$ and $X_{(2)}-X_{(3)}$

For $X_1,X_2,X_3 \sim$i.i.d exponential ($\lambda$), I am trying to show independence between $X_{(3)}$ and $X_{(2)}-X_{(3)}$ where $X_{(3)}$ is the third largest observation, i.e. the minimum in this ...
2
votes
3answers
104 views

What is the accepted syntax for a negative number with an exponent?

A friend is taking a college algebra class and they are teaching him that $-3^2 = -9$ Their explanation is: $-3^2 = -(3^2) = -9$ It has been a long time for me but I thought that in the absence ...
4
votes
1answer
73 views

Prove that every integer $n\geq 7$ can be expressed as a sum of distinct primes. [on hold]

My teacher said to use Bertrand's postulate and I have tried this for so long and I seem to go nowhere. Help would be appreciated.
0
votes
0answers
17 views

Looking for an alternative proof of the angle difference expansion

I have thought about this for a while and have no progress. Does there exist a purely Euclidean Geometric proof of the Angle Difference expansion for Sine and Cosine, for Obtuse angles?
1
vote
0answers
10 views

why should add one column using Moore-Penrose pseudoinverse

I have a code from someone that I dont understand: This code is written in matlab and the function is to estimate linear geometric transformation [1] of a matrix using pinv. The size of first matrix ...
2
votes
2answers
31 views

Existence of a solution for a nonlinear ODE on $[0,\infty)$

I'd like to prove that the solution to the following IVP exists on $[0,\infty)$. The IVP is given by $$ \begin{cases} y'(t) = y^2 \cos(t)-ye^t \\ y(0)= y_0 \end{cases} $$ where $y_0 ...
0
votes
0answers
18 views

Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
0
votes
1answer
10 views

Free product of two algebras and actions of algebras.

Let $A, B$ be two algebras. Suppose that $A$, $B$ acts on $V$. Then we have two maps $$ \delta_1: A \otimes V \to V, \\ \delta_2: B \otimes V \to V, $$ which satisfy the axioms of actions. Do we ...
0
votes
3answers
34 views

Probability on selecting balls

If I have B black balls and W white balls in a bag, what is the probability that the last one I select is white? How shall I solve this problem? I am not sure how to make a start, is it correct ...
3
votes
1answer
36 views

Prove or disprove that a series is convergent

I was given the following task which I struggle with. Prove the following statement, or disprove it by giving a counter example: if $\sum_{n=1}^\infty a_n$ is convergent then $\sum_{n=1}^\infty ...
3
votes
3answers
137 views

What is the symbol for primes?

Although there isn't much difference between $\mathbb{Z},\mathbb{N},\mathbb{I}$, they are well known, each one gets it own distinguished symbol. Are there any reason that primes don't get their own ...
0
votes
1answer
13 views

A sheet of cardboard measures 15cm by 7cm. Four equal squares are cut out of the corners and the sides are turned up to form an open box. [on hold]

a) If the edge of the cut out square is x cm, express the dimensions of the box in terms of x. b) What are the restrictions placed on the values of x? (i.e. the implied maximal domain).
0
votes
0answers
15 views

isotopy equivalence of maps

In the book Encyclopaedia of Mathematics, Vol. 6, Question: I do not understand the part Does this mean $F_1\circ f_0=f_1: X\to Y$ or as subsets of $Y$, $$ \{y\in F_1(f_0(X))\}=\{y\in f_1(X)\}? ...
0
votes
0answers
9 views

$U^TA_1V$ is a rank-one matrix?

To give a little bit of context, the question I am asking is related to SVD decomposition. More specifically, we are trying to prove that the best rank one approximation for $A_1$ is $\sigma_1 u_{1} ...
-1
votes
1answer
27 views

How can I find an algebraic formula to test whether two line segments intersect or not?

Suppose, $AB$ and $CD$ are two line segments. And, they have slopes $m_1$ and $m_2$ respectively. They will intersect with each other if, $m_1 \ne m_2 .$ Suppose, $A(x1, y1); B(x2, y2); C(x3, y3); ...
2
votes
2answers
25 views

Continued Fraction Counting Problem

The house of my friend is in a long street, numbered on this side one, two, three, and so on. All the numbers on one side of him added up exactly the same as all the numbers on the other side of him. ...
3
votes
1answer
33 views

at lest one of 100 consecutive integers is relatively prime to all natural numbers less or equal 100

for an arbitrary integer $n$ define $A_n=\{i|n \leq i \leq n+99 \text{ where i is an integer}\}$ (i.e. $A_n$ is 100 consecutive integers) is it true that for any integer n there is an element in ...
0
votes
0answers
8 views

Statistical significance of deviation from a complex-valued model

I have complex-valued data. At each one of about 100 linearly spaced $x$ values, I have a corresponding measurement of a complex quantity with well-defined Gaussian uncertainties on both the real and ...
1
vote
0answers
18 views

Question About Filled Julia and Julia Sets

Question: Let $Q_{c}(z) = z^{2} +c $ which $ c \in \mathbb{C}$ and suppose that $z_{0} \in K _{c}$ for the filled Julia Set, $K_{c}$ of $Q_{c}$. Suppose further that $z_{1} = Q_{c}(z_{0})$ and it ...

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