Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing...
For a vector space $V$ over $\mathbb R$, I say a subset $S$ of $V$ is "anti-convex" if $\forall a,b\in S (a\ne b)$, $\exists t\in ]0,1[$, $b+t(a-b)\not\in S$. For example, all hollow circles...
View ArticleConvergence to Dirichlet function is not uniform
Let $r_{1},r_{2},...$ a sequence that includes all rational numbers in $[0,1]$. Define $$f_n(x)=\begin{cases}1&\text{if }x=r_{1},r_{2},...r_{n}\\0&\text{otherwise}\end{cases}$$this sequence...
View ArticleHow to calculate the following limits?
How is it possible calculate the following limits?$\displaystyle \lim_{x\to 0} \frac{\arctan x -x}{x^3}$$\displaystyle \lim_{x\to 0} \frac{\ln(1+x)-x}{x^2}$$\displaystyle \lim_{x\to 0} \frac{\sinh x...
View ArticleDoes the series have a closed form value?
Sum of y/(n^2+y^2)+1/y n from 1 to infinityCould 1/(2y)+\pi/2cot(\pi y) be the closed form? If yes, how?
View ArticleHow to get from the forward Fourier transform to the inverse Fourier transform
I want to find out the steps and intuition into how we come up with the Inverse Fourier transform. I can't just accept it as is, I want to find the complete derivation from the forward to the inverse....
View ArticlePointwise convergence if $f_n(x)=0$ whenever...
Given $(a_{n})\in\mathbb{R}$ let$f_{n}:[0,1]\to\mathbb{R}$s.t. $f_{n}=a_{n} $if $\frac{1}{n+1}<x<\frac{1}{n}$and $f_{n}=0$ otherwisefor every $n\in\mathbb{N}$.I need to find the pointwise...
View ArticleReversed form of Gronwall's inequality?
I am looking for a "reversed" form of Grönwall's inequality. Let's recall the usual requirements from Grönwall's inequality. First, denote by $I\subset\mathbb{R}$ an interval of the form $[a,b]$ with...
View ArticleHow to show a bijection between R^R and the power set of R
I want to solve Q15 and also Q14. I have to show that these sets have the same cardinality by showing a bijection map from one set to another. For Q15 I have thought that the bijection map betweeen R^R...
View ArticleHow to proove that $x \cos nx$ is equicontinuous?
How to prove that $x\cos(nx)$ is equicontinuous on $[0;1]$? I proved that it is equicontinuous on 0, however I cannot prove either it is equicontinuous on[0;1] or not. I also tried to do something with...
View ArticleConvergence of p_ji (k) in irreducible, aperiodic finite Markov chains
Image for the questionHello, I can't prove the blue step? Can someone see how? It is known: All terms are =>0, m_ii=Sum k=1 to inf kf_ii, f_ji=sum k=1 to inf f_ji (k)=<1 . Sum i=1 to N of p_ji(k)...
View ArticleInfinite Riemann integral implies infinite lower Riemann sum?
Suppose that we have a function $f$ defined on $[0,\infty)$ that is Riemann integrable on every bounded set $[0,a]$, but whose improper integral diverges: $$\int_{0}^{\infty} \!\!\!f(x)\,\text{d}x...
View ArticleProve that $\sum {{a_n}} $ converges iff the sequence of partial sums is...
Let $(a_n)$ be a sequence of nonnegative real numbers. Prove that $\sum {{a_n}} $ converges iff the sequence of partial sums is bounded. I don't know how to do this proof. Please help!
View ArticleClosed sets: definition(s) and applications
My textbook has not been very clear (at least to me) with respect to closed sets.I have the following understanding. These two definitions are equivalent with respect to closed sets:(1) A closed set is...
View Articlethree-point explicit schemes for the linear advection equation
Here is the full assignment:Consider three-point explicit schemes for the linear advection equation on the realline of the...
View ArticleLimit of a recursively-defined sequence
What is the limit of the sequence defined as$$(x)_{ n\geq 0},\qquad x_0 = 1,\quadx_{n+1} = x_{n} + \frac{2}{x_{n}}\ ?.$$Since it's an increasing sequence, I suppose it is also divergent with a limit of...
View ArticleInjectivity of Quadratic Matrix Function
Fix positive integers $D,N$ and $d$.Let $A$ and $W$ be $N\times N$ and $d\times d$ matrices respectively. Consider the map$$\begin{aligned}f:\mathbb{R}^{N\times d} & \to \mathbb{R}^{N\times...
View ArticleThe Promotion of Cauchy-Schwarz Inequality in Probability Theory
In probability theory, if we have $\xi$ and $\eta$ as random variables, the Cauchy Inequality can be designated as:$$E(\xi \eta) \leqslant \sqrt{E(\xi^2)E(\eta^2)}$$Then I have an idea to make an...
View ArticleAmbiguity in solving differential equations
Suppose we want to solve the differential equation $y'=x \sqrt{y}$. Easy right? Because you can transform the equation into a separable one. However, I think that there are more than meets the eye....
View ArticleA contradiction about an open set whose boundary is not of measure zero....
We construct a bounded open set $A$ in $\mathbf{R}$ such that $BdA$ does not have measure zero. The rational numbers in the open interval $(0,1)$ are countable; let us arrange them in sequence...
View ArticleODE satisfied by a special function
ContextI would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica. Then, I would like...
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