6
votes
1answer
122 views

For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$?

Inspired by this question I ask this. For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$? The original question concerned $a=e$, the usual ...
0
votes
1answer
68 views

compactness and subspaces

Let $T$ be a topological space. Let $K \subset Y \subset T$. Let $Y$ have the subspace topology. Let $K$ be compact in $Y$. Is $K$ compact in $T$ ? Well, for one hand, i think that if we think about ...
3
votes
2answers
112 views

Derive the solution to the Lagrangian $ \mathcal L= y(x)\sqrt{1+y'(x)^2}$

I am supposed to derive the solution to the Lagrangian $$ \mathcal L= y(x)\sqrt{1+y'(x)^2}$$ Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
4
votes
1answer
311 views

Is the Fractal Dimension of a Space-Filling Curve in a Plane Always 2?

I have been playing around with space-filling curves that completely fill the unit square. All of them that I have seen have a fractal dimensional of 2. Makes sense that it would be 2, but a Google ...
2
votes
1answer
57 views

$15a ≡ ca \pmod{25}$, then $15 ≡ c \pmod{25}$

For which numbers $a$ is it true that if $15a ≡ ca \pmod{25}$, then $15 ≡ c \pmod{25}$? I know that this means that $a\frac{15-c}{25}=k_1\in \mathbb{Z}$ and $\frac{15-c}{25}=k_2\in \mathbb{Z}$, but ...
0
votes
1answer
67 views

Chi square independence test

How to work out chi square independence in the following table? Below is the observed and expected data concerning 7 themes displayed in a newspaper over a period of 3 months. I understand how to ...
1
vote
1answer
88 views

Which line is better fit with least squares matrices

Given two best fit curves determined by the least squares method, how can I determine which line is a better fit. Specifics $X\boldsymbol\beta=\bf{y}$ is the representation of the "perfect fit" ...
2
votes
2answers
496 views

Floor function inequality of multiplication

In a final step of a homework, I want to deduce that $$n\lfloor(n-1)!e\rfloor+2\le \lfloor n!e\rfloor+1$$ I'm unable to see whether this is true in general that $$n\lfloor a\rfloor+1\le \lfloor ...
1
vote
1answer
384 views

Visual difference between strictly concave and not strictly concave

Ok, this is an elementary question that has been bothering me for a while. I have a function like $z(x,y)=x^{0.6}y^{0.4}$. Defined for $x,y\in R^{++}$ Plainly spoken, this function is concave but ...
0
votes
3answers
2k views

Finding Best-fit Curve from Points

I am given the set of data points: $(1,2), (0,1), (-1,0), (-2,3)$ I am trying to find the best fit curve in the space: $f(x) = ax^2 + bx + c \;\;\;a,b,c \in R$ How do I go about doing this? I was ...
3
votes
2answers
126 views

Elliptic regularity - nonlinear case

Let's say I have a weak solution $u \in H^1(\Omega)$, $\Omega \subset \mathbb{R}^n$ open, to the equation $$ \Delta u = e^u, $$ let's also assume that $e^u \in L^\infty(\Omega)$. Does it follow that ...
3
votes
1answer
588 views

Characterization of Harmonic Functions on the Punctured Disk

The following is an old qual problem I came across. If $h$ is harmonic on $D-\{0\}$, where $D$ is the unit disk, show that $h(z) = \Re(f(z)) + c \log|z|$ for where $f$ is analytic on $D- \{0\}$. ...
1
vote
0answers
29 views

I need a resource for basic convex optimization algorithms.

I'm trying to decide whether or not a certain CS problem can be solved in polynomial time. I've got it reduced down to a basic convex optimization problem, but I can't for the life of me find a good ...
1
vote
2answers
46 views

Numerical integration: example [closed]

Consider $I = \int_a^b f(x) dx = 267.25$ and the approximations $\overline{I}_8 = 295.27$, $\overline{I}_{16} = 274.15$, $\overline{I}_{32} = 268.97$ and $\overline{I}_{64} = 267.68$ of $I$ given by ...
6
votes
3answers
234 views

Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies

Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime. Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
0
votes
1answer
159 views

Stochastic processes-Brownian Motion

I hope someone can help me with the following exercise... Show that $ \int_0^t s \, dB_s =tB_t-\int_0^t B_s \, ds $, for each $t>0$, where $B$ is a Brownian motion. Thanks in advance!!!
3
votes
1answer
148 views

A sequence of analytic functions with converging path integrals converges how?

Suppose you have a sequence $\{f_n\}_{n \in \mathbb Z^+}$ of analytic functions on a compact set $K$ such that for every smooth curve $\gamma$ in $K$ the sequence $\{\int_\gamma f_n\}_{n \in \mathbb ...
3
votes
3answers
976 views

Path independence of an integral?

I'm studying for a test (that's why I've been asking so much today,) and one of the questions is about saying if an integral is path independent and then solving for it. I was reading online about ...
0
votes
2answers
588 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
4
votes
1answer
166 views

Smallest nonhamiltonian 2-connected bicubic graph with chromatic index 3

I found this rather trivial example for a bicubic nonhamiltonian 2-connected graph with chromatic index 3: $\hskip0.5in$ Is this the smallest one? If not can you construct a smaller one?
1
vote
1answer
62 views

Uniformly distributed variable

I've stumbled upon this correlation between two different variables thread and I can't understand this equation: $$\mathbb{E}\cos(2\Phi+\lambda \cdot (s+t)) = \frac{1}{2\pi} \int_0^{2\pi} \cos(2x+ ...
5
votes
4answers
416 views

Differentiating $\;y = x a^x$

My attempt: $$\eqalign{ y &= x{a^x} \cr \ln y &= \ln x + \ln {a^x} \cr \ln y &= \ln x + x\ln a \cr {1 \over y}{{dy} \over {dx}} &= {1 \over x} + \left(x \times {1 \over ...
0
votes
1answer
1k views

Improved Euler method and local error

Given the differential equation $t'=g(x,t)$ Use the improved Euler method by analysing the local error and its stability for the equation $t'=\delta t$, where $\delta$ is a complex ...
6
votes
1answer
152 views

Invariance of strategy-proof social choice function when peaks are made close from solution

A question emerging from reading Schummer, J., & Vohra, R. V. (2002). Strategy-proof Location on a Network. Journal of Economic Theory, 104(2), 405–428. The setting is as follows: A finite set ...
3
votes
3answers
255 views

Approximating an infinite sum of only odd terms by a definite integral

Consider the infinite Sum $S=\sum\limits_{n\ \text{odd}}^{\infty}\left(\dfrac{1}{nt}\right)^2\left[1-i\left(nt\right)^2\right]^{-1}$ Is there a way to approximate this sum as a contour integral? In ...
1
vote
2answers
134 views

If $[G:N]$ and $|H|$ are relatively primes, then $H\lt N$.

I'm trying to show that if $N$ is a normal subgroup of $G$ and $H$ is a subgroup of $G$ such that $[G:N]$ and $|H|$ are relatively primes, then $H\lt N$. I've already found $|H|||N|$ but I couldn't go ...
1
vote
1answer
66 views

The integral $\int_0^1\dfrac{(-x)^n}{1+x} dx $

How can I prove that: $\forall x \in \mathbb{N}\setminus {0} \quad \dfrac{-1}{n+1}\le \int_0^1\dfrac{(-x)^n}{1+x} dx \le \dfrac{1}{1+n}$ $\lim_{n\to+\infty}\Sigma_{i=1}^{n}\dfrac{(-1)^{i-1}}{i}$. ...
1
vote
2answers
994 views

Integral Question $\int\frac{\sin^4(x)}{\cos^2(x)}\,dx$

What you are suggesting to do? Convert $\sin^4(x)\Rightarrow (1-\cos^2(x))^2\,dx?$ $$ ∫\frac{\sin^4(x)}{\cos^2(x)}\,dx$$ Thanks!
2
votes
1answer
288 views

p-adic isomorphism $\mathbb{Q}_p\not\cong \mathbb{Q}_q\iff p\ne q$ [duplicate]

In class I learn that $\mathbb{Q}_3\not\cong\mathbb{Q}_5$ because one of them has $\sqrt{2}$ the other doesn't. Also professor asks us to find reference that $\mathbb{Q}_p\not\cong \mathbb{Q}_q\iff ...
1
vote
1answer
230 views

Tensor product of two cyclic modules

Given a commutative ring $A$ (with identity) and two cyclic $A$-modules, $M$ and $N$ with generators $x$ and $y$, respectively. How do you show that $\mathrm{Ann}(x\otimes_A y) = \mathrm{Ann}(x) + ...
2
votes
1answer
77 views

RSA cryptography question

RSA user Alice has a public key ($n_A=pq$,$e_A$), where $p$ and $q$ are different primes such that the least common multiple $l$ of $p-1$ and $q-1$ is relatively small (i.e. $l$ is close to ...
2
votes
1answer
62 views

Prove $X =\left \{(x, y) \in \mathbb{R}^3 \times \mathbb{R}^3 \ | \ |x| = 1, |y| = 1, x\cdot y = \frac{1}{2}\right\}$ is a manifold

I am having trouble with the following qualifying exam problem and I would appreciate any help. Thank you. Let $X$ be the set of pairs of unit vectors $(x, y)$ in $\mathbb{R}^3$ such that $x \cdot y ...
4
votes
4answers
291 views

Proof that $P(A \mid C) = 1$ implies $P(A \mid B∩ C) = 1$

This isn't homework. I'm reading Probabilistic Graphical Models by Koller et al.$\space$, and an easy problem in chapter $3$ made me think of a more general problem (which I'm now stuck on). I have ...
1
vote
1answer
87 views

Divisibility problem.

In line written squares of natural numbers from 1 to 2012. How many of these numbers have a remainder when divided by 17, which is divisible by 3?
4
votes
4answers
134 views

Limit of an integral.

$$\lim_{n\to\infty} \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^n}\,dx $$ Mathematica tells me the answer is 0, but how can I go about actually proving it mathematically?
1
vote
1answer
65 views

finding the P matrix (diagonalization of a matrix)

I'm trying to find the diagonalization of a matrix : this is my matrix : $$ A =\begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 1 & 1 & 1 \\ ...
4
votes
1answer
107 views

Compute the splitting field

I have to compute the splitting field of $x^6-1 \in Q[x]$. I know that $x^6-1=(x+1)(x-1)(x^2-x+1)(x^2+x+1)$ but I don't know what to do after that. Please help me.
4
votes
1answer
445 views

A question on torsion sheaves

Im not sure if Ive got this right: Let X be an integral scheme and $\mathcal{F}$ a coherent sheaf. Then $\mathcal{F}$ is torsion if and only if it is not supported at the generic point. It is is easy ...
1
vote
1answer
88 views

How to find the order of these groups?

I don't know why but I just cannot see how to find the orders of these groups: $YXY^{-1}=X^2$ $YXY^{-1}=X^4$ $YXY^{-1}=X^3$ With the property that $X^5 = 1$ and $Y^4 =1$ How would I go about ...
8
votes
1answer
1k views

Vague definitions of ramified, split and inert in a quadratic field

Our lecturer defined the following: Let $K=\mathbb Q(\sqrt d)$ be a quadratic field and $p$ a prime number, then (1) $\ p$ is ramified in $K$ if $\mathcal O_K⁄(p)\cong \mathbb F_p [x]⁄(x^2)$ ...
3
votes
3answers
2k views

Extreme Value Theorem proof help

Extreme Value Theorem: If $f$ is a continuous function on an interval [a,b], then $f$ attains its maximum and minimum values on [a,b]. Proof from my book: Since $f$ is continuous, then $f$ has the ...
2
votes
2answers
445 views

Complex analytic function and Cauchy-Riemann conditions question

here again asking for some guidance Let me state the problem and working on and then my question, So I have two functions: $$u(x, y) = \frac{x(1+x)+y^2}{(1+x)^2 + y^2}$$ and $$v(x, y) = ...
1
vote
1answer
178 views

Does the following Integral have a Closed Form Solution?

Is there a solution to the following Integral that can get rid of the integral, but can leave the answer in terms of $\Phi()$ (the standard normal CDF) and $\phi $ (the standard normal PDF).? $ ...
2
votes
1answer
38 views

Proportionality In two Values Equal to $0$

If two values $m$ and $n$ are in direct variation, then $m \propto n$ If the constant of proportionality is $q$ between them, then $m = qn$ If $m$ and $n$ both are equal to zero or $m = ...
3
votes
0answers
97 views

Finite locally groups

Let be $V\leq \operatorname{Aut}\left( G\right),\ N\vartriangleleft G$ and $V$-invariant. Consider the semidirect product of $V$ with $G$. Let $C_{G}(V)=\{g\in G:g^{v}=g\}$ and $[G,V]=\langle ...
2
votes
2answers
55 views

Prove the following: Product of Roots

$1^{(1/1)} \cdot 2^{(1/2)} \cdot 3^{(1/3)} \cdot 4^{(1/4)} \cdot 5^{(1/5)} $.... diverges well I don't really know if it does but my gut tells me it does: I can take the log of this product to ...
1
vote
1answer
59 views

Simplify Expression Question

Anyone can tell me if I can simplify this expression more? I Simplified this function => $minterm(1,3,4,6,7,9,10,11,12,15)$ to this expression: $W'X'Z+W'Z'X+WYZ+W'XYZ+WX'Y'Z+WX'YZ'+WXY'Z'$ Thanks!
0
votes
2answers
62 views

How to simplify or factor this equation

$$1 = x + 2\cdot x$$ How can I simplify this formula for $x$.
1
vote
1answer
35 views

Prove convergance of one series by using another

Question: $\sum_{n=1}^\infty a_n , \sum_{n=1}^\infty b_n$ are 2 positive series that satisfy: $\frac {a_{n+1}}{a_n} \le \frac {b_{n+1}}{b_n}$ Show that if $ \sum_{n=1}^\infty b_n$ converges then $ ...
-1
votes
1answer
96 views

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$?

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$? Justify your answer! Thanks.

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