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H: What's the recurrence relation to this problem?
A machine can perform $3$ types of operation $A$, $B$ and $C$. The memory is initially $0$. A Program $P$ is a series of these operations. If the machine does $A$, it will add $1$ to the memory's current value. If it does $B$, it will subtract $1$ from the memory. If ... |
H: Calculating the following definite integral
How may I approach and compute the following integral? Could polylogarithm functions and complex numbers be avoided?
$$ \int_0^{\frac{\pi}{2}} \frac{x}{\tan x} dx$$
AI: Integrate by parts with $u=x$ and $\displaystyle dv=\frac{dx}{\tan x}$ so $$ \int^{\pi/2}_0 \frac{x}{\t... |
H: Solving an equation for B
I am having an issue solving this. I wrote this equation to find the with of a bar for a bar chart that is comparative.
\begin{equation}
D-P*(N+1) = ((N-1)*M)*B*0.75+N*B
\end{equation}
AI: No guarantee that the result will be an integer, but just distribute the $B$ out of the terms on the ... |
H: Parallelogram With non-90 angles
I realized that in-case of a parallelogram with a non-90 degree angles the two diagonals would never be equal. Is my assumption correct ?
AI: The diagonals of a parallelogram bisect each other, so if the diagonals have equal length, then they cut the parallelogram into four isosce... |
H: $10+10\times 0$ equals $0$ or $10$
I thought $10+10\times 0$ equals $0$ because:
$$10+10 = 20$$
And
$$20\times 0 = 0$$
I know about BEDMAS and came up with conclusion it should be $$0$$ not $$10$$
But as per this, answer is $10$, are they right?
AI: There is a "precedence of operations": some operations should be d... |
H: Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant
If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$?
I remember seeing such a formula, but can not remember where.... |
H: If each matrix representing $T$ contains a $0$, then $T$ is a scalar
Let $V$ be a finite-dimensional vector space over $\mathbb F$ ($\text{char}\ \mathbb F =0$) and $T$ is a linear transformation such that for any basis the matrix of linear transformation has at least one zero.
I want to show that there exists $c\i... |
H: General method to find inf, sup, maxs and mins of a function
Could someone explain how to find inf, sup, max and min values of a function (real-valued functions of real variable, generally continuous/differentiable, with some possible points of discontinuity)?
Some examples:
$$y = \frac{x}{2}\sqrt{\frac{\log{x} + 1... |
H: Why should coordinate transformations be reversible?
Intuitively I understand why coordinate transformation should be reversible. New coordinates should cover the same area covered by the initial coordinates, i.e. there should be one-to-one mapping.
But still, are reversible transformations used only because it is ... |
H: Cauchy's Residue Theorem on a Singularity Outside a Contour
I recently ran into the following exercise:
Evaluate
$$\oint_\Gamma\frac{\cos z}{(z-\pi)^2}dz,$$where $\Gamma$ is a complete circuit of the circle $|z|=1$.
Clearly, the singularity lies outside the contour:
However,... |
H: Number of elements in $\mathbb{Z}_p[x]/ \langle f \rangle$
I want to determine the number of elements in $\mathbb{Z}_p[x]/ \langle f \rangle$ where $f \in \mathbb{Z}_p[x]$ is an irreducible polynomial with $k$ degree bigger than 2. Is the number of elements always $p^k$? And are the equivalence classes $[f_1],[f_2]... |
H: Recurrence Relation for the nth Cantor Set
Wikipedia gives a recurrence relation for the $nth$ Cantor Set: $C_n = \frac{C_{n-1}}{3} \cup(\frac{2}{3}+\frac{C_{n-1}}{3})$.
http://en.wikipedia.org/wiki/Cantor_Set#Construction_and_Formula_of_the_ternary_set
How does one verify such a relation? I considered using induc... |
H: What do you call a graph of value vs rank of the value?
What do you call a graph of value vs rank of the value?
For example, say I make a census of the 2010 net income for a bunch of people.
The graph of net income vs age gives me a: scatterplot
The graph of net income vs location (latitude, longitude) gives me a:... |
H: Optimization, solving for the 'error' coefficient
Given a modified regression equation:
$\hat Y = \exp(\beta_0 + \sum\beta_ix_i + \varepsilon)*F$
where:
$\hat Y = 11353$
$\beta_0 = 8.693021$
$\sum\beta_ix_i = 5.95487177696$
$F = 0.21829$
what is:
$\varepsilon =$
AI: $$ \hat Y = \exp(\beta_0 + \sum\beta_ix_i + \vare... |
H: Sequence of Uniformly Bounded functions
Consider a sequence $\{ f_k \}_{k=1}^{\infty}$ of locally-bounded functions $f_k: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$.
Assume the following.
For any sequence $\{X_k\}_{k=1}^{\infty}$ of compact sets $X_k \subset \mathbb{R}^n$ such that $X_k \subseteq X_{k+1}$ and $X... |
H: Lines in the plane and recurrence relation
I am trying to solve the following problem from Cohen's Basic Techniques of Combinatorial Theory:
A collection of $n$ lines in the plane are are said to be in general position if no two are parallel and no three are concurrent. Let $a_n$ be the number of regions into whic... |
H: Computing the value of $(\frac{p-1}{2})!$ modulo $p$.
I want to prove that for $p \geq 3$, and for $a=(\frac{p-1}{2})!$, if $p \equiv1\pmod 4$, then $a^2\equiv -1 \pmod p$, and if $p \equiv 3\pmod4$, then $a \equiv \pm 1 \pmod p$.
For the first part, I used Wilson's theorem which says that for prime $p$, $(p-1)!=-1... |
H: Proving $\mathrm e <3$
Well I am just asking myself if there's a more elegant way of proving $$2<\exp(1)=\mathrm e<3$$ than doing it by induction and using the fact of $\lim\limits_{n\rightarrow\infty}\left(1+\frac1n\right)^n=\mathrm e$, is there one (or some) alternative way(s)?
AI: What answer you find most eleg... |
H: Does this intuition for "calculus-ish" continuity generalize to topological continuity?
In the past, I've always motivated continuity of a function from (some subset of) $\mathbb R$ to $\mathbb R$ based on the (incomplete) definition $\lim_{x \to c} f(x) = f(c)$; continuity at isolated points was never really too h... |
H: Can ≈ be used to express an arbitrarily rounded rational number?
Is it a formally acceptable use of ≈ to express a rational number rounded to an arbitrary number of significant digits?
For example, $\frac{4}{7}\approx0.57.$
If formally acceptable, is it expected?
AI: I see $\approx$ to mean close enough for the pur... |
H: HCF/LCM problem
Find the greatest number that will divide $x$, $y$ and $z$ leaving the same remainder in each case.
Now the solution for this is obtained by finding the HCF of $(x – y)$, $(y – z)$ and $(z – x)$.
Can you tell me why is that?
AI: HINT: $x,y$ have the same reminder when divided by $d$ if and only if $... |
H: Example of a function $f: [a, b] \to \mathbb{R}$ that is unbounded.
Can anyone give me an example of a function $f\colon[a,b]\to\mathbb{R}$ that is unbounded?
AI: There can be no continuous function that has that property (since a continuous function defined on a finite closed interval must achieve a maximum and a ... |
H: Normal subgroups of infinite symmetric group
I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the set $\mathbb{N}$) has exactly 4 normal subgroups." Does anyone have an... |
H: Not every metric is induced from a norm
I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function
$d(u,v) = \lVert u - v \rVert$, $u,v \in V$.
My question is whether every metric on a linear space can be induced by norm? I know answer is no but I need pro... |
H: F distribution: $P\{S_1\geq 3S_2\}$ with $n_1 = 7, n_2 = 13$
Using the $F$ distribution table:
I think it should be $ P\{\frac{S_1}{S_2} \geq3\} = 1 - P\{\frac{S_1}{S_2} \leq 3\} = 1 - 0.05 = 0.95$. But my notes say it should be $0.05$?
AI: Let $X = \frac{S_1}{S_2}$ be F-distributed random variable. Then
$$
f_... |
H: How to show that $0 \times 2 = 0$?
Here is the suggested proof:
$0 \cdot 2=2 \cdot 0 =(1+1) \cdot 0 = 1\cdot 0 + 1\cdot 0 = 0 + 0$.
But my question lies in this step.
Here is the definition of Zero: $0+a = a$ (for any number $a$)
therefore: $0+5=5$ or $0+1639=1639$,
but can we say $0+0 =0$ ? I mean, that "any"... |
H: Homeomorphism between a normed space and its open unit ball
I have studied that every normed space $X$ is homeomorphic to its open unit ball $B$. I want to know what conclusion can we draw from this statement? Does it mean that every normed space is open ball? I am confused with this statement. Could anyone clear m... |
H: Sard's Theorem For Constant functions
It states:
Let $g:A \to R^n$ be continuously differentiable, where $A \subset R^n$ is open, and let $B=${${x \in A: \det g'(x)=0}$}. Thne $g(B)$ has measure $0$.
Okay.... obviously this theorem is right... but why don't constant functions violate this? After all, the deriva... |
H: trying to understand how to solve this kind of limits.
Ok Right now Im doing some excersises... and now I get stuck on this one..
$$\lim_{x\to2} \frac{2^{x+1}-8}{4-2x}$$
tried to do this
$$\lim_{x\to2} \frac{2^{x}\cdot 2-8}{4-2x}$$
but it's the same thing...
AI: After factoring $-2$ from the denominator, and $2$... |
H: A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$
Prove that a finite-dimensional extension field $K$ of $F$ is normal if and only if it has this property: Whenever $L$ is an extension field of $K$ and $\sigma :K\rightarrow L$ an injective homomorphism such that $\sigma(c) = c$... |
H: Curry-Howard correspondence
I read that the Curry-Howard correspondence introduces an isomorphism between typed functions and logical statements. For example, supposedly the function
$$\begin{array}{l}
I : \forall a. a \to a\\
I = \lambda x. x
\end{array}$$
can be interpreted as:
$$p \implies p \text{ for any propo... |
H: How to solve infinite repeating exponents
How do you approach a problem like (solve for $x$):
$$x^{x^{x^{x^{...}}}}=2$$
Also, I have no idea what to tag this as.
Thanks for any help.
AI: I'm just going to give you a HUGE hint. and you'll get it right way. Let $f(x)$ be the left hand expression. Clearly, we have th... |
H: Maximum numbers of underdetermined solution of $Ax=b$
Given $m \times n$ real matrix $A$, where $m<n$, we know that the nullity of $A$ is the dimension of the kernel $W=\{w| Aw=0\}$. Also all solutions of linear equation $Ax=b$ for $b\neq 0$ can be described as
$$\mathcal{A}(v):=\{v+w | Aw=0\}=v+W$$
where $v$ sati... |
H: Multiplication of Matrix Properties - bi-symmetric, symmetric and anti-symmetric
Once again, can the great maths minds here please give a hand to explain or solve this problem. Much appreciated!
How can you prove that a bi-symmetric matrix multiplied by symmetrical vector will give me a symmetrical vector?
Vice ve... |
H: Probability of someone being born the same day as you in a group of N people.
I got into a discussion with a friend about this simple question:
Provided that I went to a class with $N$ people, where everyone was born the same year in the same city where there are $K$ hospitals, what's the probability that someone w... |
H: Next generation numbers
1: Discovering of negative numbers.
Assume a and b are positive integers
$x+a=b$ ----> if $b>a$ then $x$ is positive integer
$x+a=b$ ----> if $b=a$ then $x=0$
$x+a=b$ ----> if $b<a$ then $x=b-a$ is negative integer
2: Discovering of rational numbers.
$x+x+....+x=a.x=b$ ----> $b ... |
H: Duality of a finitely generated projective modules
Let $M$ and $N$ be a finitely generated projective module over a ring $R$. Suppose we have a non degenerate bilinear pairing $\langle \ \cdot \ ,\ \cdot\ \rangle: M \times N \to R$.
I want to show $M$ is isomorphic to the dual $N^*$ of $N$.
The injectivity of $M$ i... |
H: How to invert a polynomial function such as $f(x)=x^{\beta}\left((x-1)^6+1\right)$?
I'm trying to replicate a simulation study in a paper. For that I would need the inverse of this function:
$f(x)=x^{\beta}\left((x-1)^6+1\right), x\in[0,1]$
Plugging this unto Maxima returns:
solve(y=x^(beta)*((x-1)^6+1),x);
beta... |
H: How to handle group schemes by points?
I find it is very inconvenient to handle group schemes by its defination(i.e. everything is defined by morphism). And I have noticed that for group varieties, one can treat them as actual groups(i.e. talking about elements). Moreover, in the book Geometric Invariant Theory by ... |
H: 3 consecutive numbers
I was playing with some numbers and just realized that:
For any 3 consecutive numbers X, Y and Z: $Y^2$ = (X*Z) + 1
For eg: Consider numbers 171, 172 and 173
$172^2$ = 29584
and
171*173 = 29583
Can anyone tell me if there is any proof for this and what it is known as?
AI: Let the middle numb... |
H: How to use the Malgrange-Ehrenpreis-Theorem
In the Wikipedia article of this theorem
http://en.wikipedia.org/wiki/Malgrange%E2%80%93Ehrenpreis_theorem
it is said that i could be used to prove that
$P(\partial/\partial x_i)u(x)=f(x)$
has a solution for any distribution $f$, how this prove works?
AI: It is not somet... |
H: $K\subset Y\subset X$
Here is a theorem in Rudin's Principles of Mathematical Analysis:
$K\subset Y\subset X$.Then $K$ is compact relative to $X$ if and only
if $K$ is relative to $Y$.
I read the proof in the book but I tried to construct a different proof:Here it is:
"Proof": If $K$ is compact relative to $Y$... |
H: Wrong proof of convergence almost everywhere
Can you tell me where the mistake is?
If $(f_n) \in L^1$ is a sequence of functions such that $\sum_n \|f_n\|_1 < \infty$ I can prove that $f(x) = \sum_{n=1}^\infty f_n(x) < \infty$ for all $x \in X$.
To this end, let $n \in \mathbb N$, $x \in X$. Then
$$ \sum_{k=1}^n f... |
H: Drying blood - an algorithm for calculating the geometry of blood stains
Motivation
A bucket full of blood gets spilled over the floor. Question: What shape will the dried blood stains have?
Abstraction
The blood is modeled by a set of interacting particles (e.g. SPH). As time goes to infinity, the blood particles... |
H: Calculate $\int f(x) f''(x)dx$
I have a curiosity. If
$\int f(x) f'(x)dx=\int f(x) df(x)=\frac{\left(f(x)\right)^{2}}{2}+C$
what is the result of:
$\int f(x) f''(x)dx$
AI: $$
f(x)f'(x)-\int f'(x)^2 \, dx \ ?
$$ |
H: Matrix equation involving Kronecker product
I would like to know if exists an algorithm to find the solutions (if any) of the following matrix equation:
$$A\otimes B=B\otimes A$$
in which the $N \times N$ matrix $B$ is given and the $A$ ($A\neq0$) is the unknown $N\times N$ matrix. The conditions on B are:
$$B\neq ... |
H: Some approximations for $\arccos(1/(1+x))$
I was trying to calculate the maximum ground distance you can see on mountains, with your elvation given.
After some simple geometry, I was able to come up with the following formula:
Let $h$
be your elevation, $d(h)$
be the maximal distance you can see, then
$$d(h)=2... |
H: Solving non linear differential equation.
$\frac{d}{dt}\left(\frac{x'(t)}{x(t)}\right)=x(t)-x^2(t)$ where $x'(t)=\frac{d}{dt}x(t)$
What reasoning (if it exists) I can apply to solve this differential equation?
Thanks.
AI: Let $x=e^u$; then $u$ satisfies de differential equation
$$
u''=e^u-e^{2u}.
$$
Multiply by $u´... |
H: Does "regular" implies collectionwise hausdorff?
Does "regular" implies collectionwise hausdorff?
A topological space is said to be collectionwise Hausdorff if given any closed discrete collection of points in the topological space, there are pairwise disjoint open sets containing the points.
I believe it and t... |
H: Equivalent definitions on SVD property
Please have a look at the property of SVD - section "Singular values, singular vectors, and their relation to the SVD" from http://en.wikipedia.org/wiki/Singular_value_decomposition.
Could it be state that "and" between
$$Mv = \sigma\cdot u \quad\text{ and }\quad M^Tu = \sigma... |
H: find inverse function in given domain
Suppose that we have function $y=\sin(x)$ we need to find its inverse function, assuming that $D(f)=[-\pi/4.\pi/4]$
I know that inverse of $\sin(x)$ is $\arcsin(x)$, it would be answer of a given function too, but why do I need $D(f)=[-\pi/4.\pi/4]$? I don't know, should I int... |
H: About the number of functions passing for N points
Given one point $P\equiv (x_0,y_0)$ on a two-dimensional space $\Re^2$, there are infinite functions passing for that point. The question is: given $2$ points $P_0\equiv(x_0,y_0)$ and $P_1\equiv(x_1,y_1)$ is it still correct to say they are infinite? More in genera... |
H: How do I check the continuity of this function?
I am doing basic calculus, can someone tell me how to find the continuity here at $x=1$
$f(x)=\begin{cases}5x-4 & 0<x \leq 1\\4x^2 -3x & 1 < x < 2.\end{cases}$
AI: Suppose $a,b\in\Bbb R$ with $a<b$, let $I=(a,b)$, $c\in I$, $f:I\to\Bbb R$. Then we say that $f$ is cont... |
H: When does non-negativity of the integral of a function imply that the function itself is non-negative?
Let $(\Omega,\Sigma)$ be a measurable space and $(\omega_k)_{k\in\mathbb{N}}$ a sequence of elements of $\Omega$. Let
$$
\mathcal{M}:=\left\{\sum_{k=1}^\infty a_k\cdot\delta_{\omega_k}: \quad(a_k)_{k\in\mathbb{N}... |
H: Which do you recommend for learning how to write proofs — How to Prove it by Velleman, or How to Solve it by Polya?
Which of these two books is suited for a student looking to learn how to write proofs?
I have a working knowledge of calculus and linear algebra, but I'm not good at writing proofs. My intention is to... |
H: Radius of Convergence of power series, with n! and $n^n$
During revision, I came across this problem:
The set of real numbers $x$ for which the series $$\sum_{n=1}^{\infty}{\frac{n!x^{2n}}{n^n(1+x^{2n})}}$$ converges is __.
I tried using the ratio test, but got stuck in the process of simplification.
(The answer is... |
H: Some equivalences for convex sets
For a subset $A\subset V$ of a vector space over $\mathbb{R}$, let
$\mbox{conv}(A) := \left\{ \sum_{i=1}^n a_i x_i\, \middle| \,x_i\in A, a_i \ge 0\text{ with } \sum_{i=1}^n a_i = 1 \right\}$.
I want to show that if for all $x,y\in A$ and $\alpha\in[0,1]$ the vector $(1-\alpha)x +... |
H: How to find the equation of a line tangent to a function
How it's possible to find the equation to a line tangent to a function in a point where the derivative of the function is an indeterminate form?
I'm analyzing this function:
$$y = \frac{x^2}{1+\log|x|}$$
And the first derivative is:
$$y\,' = \frac{x(1+2\log|x... |
H: Which function can be a particular solution for $y''+a(t)y'+b(t)y=0$?
It's seems like a really simple question, but I can't understand how to solve it.
I am requested to decide which function $y=t^2$, or $y=t^2+1$ can be used for a particular solution of an order two equation: $y''+a(t)y'+b(t)y=0$ with the continue... |
H: Are there problems that are optimally solved by guess and check?
For example, let's say the problem is: What is the square root of 3 (to x bits of precision)?
One way to solve this is to choose a random real number less than 3 and square it.
1.40245^2 = 1.9668660025
2.69362^2 = 7.2555887044
...
Of course, this is ... |
H: Roots in a finite field
Given a finite field $|K|=q$ and an irreducible $f \in K[x]$ with $\deg(f)=n$ with $\alpha$ as a root. My candidates for the roots are $\alpha, \dots , \alpha^{q^{n-1}}$.
Assuming $\alpha^{q^i} = \alpha^{q^j}$ I want to conclude that $\alpha = \alpha^{q^{j-i}}$ or more precisly I want to con... |
H: What is the quantified definition for proper subset
We're thinking its:
$ A \subset B \leftrightarrow \forall x [x \in A \rightarrow x \in B] \land \exists x [x \notin A \land x \in B] $
is this OK?
Thanks,
z.
AI: As Brian says: Yes, that works. ${}{}{}{}{}{}{}{}{}{}{}{}{}{}$ |
H: Difference between supremum and upper bounds and between infimum and lower bounds
I'm having some difficulties catching the difference between upper bound and supremum and, similarly, between lower bound and infimum.
Let's take a look at this set:
$A=\{x\in \mathbb Q | 0<x\leq\sqrt2\}$
So the supremum is $\sup(A)=\... |
H: Number of natural and real numbers
Possible Duplicate:
The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$
I was reading Rubin and came across the fact that $2^{\aleph_0}$ is the cardinality of reals.
I follow the proof but I cant seem to understand physically what this attempts to sa... |
H: Which manifolds have a circle as their boundary?
The boundary of a disk or of a Möbius band is a circle.
Which other manifolds share that property?
AI: For the compact case, I believe the answer is, as you said in the comments above, any closed surface with a single puncture, i.e. a disk removed. I claim that this ... |
H: Why is this prime factor not counted
I came across a question:
If j is divisible by 12 and 10, is it divisible by 24 ?
The example draws the following factor tree which I agree with
But then it states that
There are only two 2's that are definitely in the prime factorization of j. because the 2 in the prime fa... |
H: The image of an ideal under a homomorphism may not be an ideal
This is an elementary question about ideals. Consider a ring homomorphism
$$
f: \mathbb{Z} \rightarrow \mathbb{Z}[x],
$$
and consider the ideal $\left< 2\right>$ in $\mathbb{Z}$. When why is it that $f(\left< 2\right>)$ is not an ideal?
Some websites... |
H: Proving $\inf\limits_{f\in\Gamma} \{ F(y(t))(f(t))\}= -\|F(y(t))\| $
I would like to proof the next claim:
Let $X$ a Banach space, $F\colon X\to X^*$ a linear continuous function,
$$
\Gamma:=\{f\in (\mathcal{C}([0,1],X)\,:\, f(0)=f(1)=0\mbox{ and }\|f\|\leq 1\}
$$ and fix $t\in [0,1]$, $y\in \Gamma$. Then $F(y(t))... |
H: Dimension of a cyclic submodule of a finite group representation
Let $\rho: G \to GL(V)$ be a finite (complex) group representation. What is the maximum dimension of $span\{\rho_gv | g \in G\}$ over all $v \in V$? This quantity is not necessarily the degree of $V$: for example, if $V$ is the direct sum of two isomo... |
H: Counter-example that $I\cup J$ in a ring $R$ may not be an ideal
I've been doing some reading about ideals and here is another question (to which I couldn't yet find or construct a counterexample).
Let $I, J$ be ideals in a ring $R$. Then $I\cup J$ is contained in $I+J$ but it may not be an ideal since it may not... |
H: divide triangle into ratio 1:2
I have a problem related to triangles. Please give me some hint to progress.
Suppose we have the following coordinates $A(-2,3)$,$B(1,-1)$,$C(-1,-1)$.
From point $A$, draw a line which divides the area of triangle $ABC$ in the ratio $1:2$.
I have calculated the length of the sides... |
H: Prove or disprove an inequality
Let's consider the following equation where $m,n$ are real numbers:
$$ x^3+mx+n=0 $$
I need to prove/disprove without calculus that for any real root of the above equation we have that:
$$ m^2-4 x_1 n \ge 0$$
AI: Suppose that $x_1$ is a real root of the cubic, and consider the quadr... |
H: How can we take a power series and multiply each term, i.e. $c_n x^n$ by $y^n$?
In other words, given a power series $f(x)$, is there an alternative to taking $\lim_{x\to{x y}}f(x)$? I ask this because I thought that there may be a way to replace the limit by integration, or some other operation that I know how to... |
H: How to get the characteristic equation?
In my book, this succession defined by recurrence is presented:
$$U_n=3U_{n-1}-U_{n-3}$$
And it says that the characteristic equation of such is:
$$x^3=3x^2-1$$
Honestly, I don't understand how. How do I get the characteristic equation given a succession?
AI: Here’s a rote ru... |
H: Proving that an integer is even if and only if it is not odd
There is this question, but the definition of "even" and "odd" that I am using uses integers instead of just natural numbers; i.e.,
An integer $n$ is even iff there is some integer $k$ such that $n=2k$.
An integer $n$ is odd iff there is some integer $k$... |
H: Bidding Item Problem
You are bidding on an item that has an unknown value uniformly
distributed between 0 and 1. You do not know the true value of the
item, but you know that if you end up winning the bid for the item,
the item will increase its value to 2x its original value. Your bid
can only go through ... |
H: Number of elements in a finite field extension for finite fields
Given an arbitrary finite field $K$ (not necessarily $\mathbb{F}_p$ with $p \in \mathbb{P}$) with $|K| = q$ and an irreducible polynomial $f$ with $\alpha$ as root and degree of $n$. Is $|K(\alpha)| = q^n$ and why? Its clear to me for $K$ isomorphic t... |
H: Variable Substitution
I need to show that next equation stands:
$$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = \frac{1}{r}\frac{\partial}{\partial r} (r\frac{\partial u}{\partial r} ) +\frac{1}{r^2}(\frac{\partial^2 u}{\partial \theta^2} )$$
where
$u=f(x,y)$ , $x = r\cos\theta$ , $y = r\sin... |
H: Series convergence, finding values that cause convergence.
I am trying to find the $x$ values that make this series converge:
$$\sum_{n = 1}^\infty (x+2)^n.$$
To me it seems like $x = -2$ would make the series converge but that is a wrong answer, I am not sure why either.
AI: Recall the geometric progression $$\sum... |
H: set in $\mathbb{R}$ which is not a Borel-set
Possible Duplicate:
Lebesgue measurable but not Borel measurable
Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly
if i start from the topology of $\mathbb{R}$, i.e. all open sets, and then build the closure under countable union and complement i get t... |
H: Proof that the Irrationals are Countable
Proof: Between any two irrationals lies a rational, by the Density of the rationals in the real number system. There are only countably many rationals; therefore, there are only countably many pairs of irrationals. Therefore the number of irrationals is countable since the c... |
H: Primality and repeated digits
I recently worked on problem 51 through project euler, I solved it essentially through brute-force but afterwards I viewed the forum and there were some more clever solutions.
For those unfamiliar with project euler, the problem reads:
By replacing the 1st digit of *3, it turns out th... |
H: Generators of von Neumann algebras
Let us suppose that M is a von Neumann algebra on some hilbert space $H$ such that $M = A''$ for some $C^*$-algebra $A\subseteq B(H)$.
I am wondering when $A$ will be dense in $M$ and by which topology. I also want to know that if I have two functional $S,T \in M^*$ such that $S|... |
H: Help with binomial theorem related proof
I'm currently working through Spivak on my own. I'm stuck on this proof, and the answer key is extremely vague on this problem. I think I'm missing a manipulation involving sums.
Prove that $\displaystyle\sum_{k=0}^{l}\dbinom{n}{k}\dbinom{m}{l-k}= \dbinom{n+m}{l}$.
As a hint... |
H: Complex series: $\frac{z}{(z-1)(z-3)} = -3 \sum\limits_{n=0}^\infty \frac {(z-1)^n}{2^{n+2}} - \frac{1}{2(z-1)}$ for $0 < |z-1| < 2$
Show that when $0 < |z-1| < 2$,
$$\frac{z}{(z-1)(z-3)} = -3 \sum_{n=0}^\infty \frac {(z-1)^n}{2^{n+2}} - \frac{1}{2(z-1)}$$
I thought to attack this using a partial fraction deco... |
H: How smooth is the distribution function of a convex polynomial?
Here is a prototype of the problem I have in mind: Let $P:\mathbb{R}^2\rightarrow\mathbb{R}$ be a strictly convex, nonnegative polynomial such that $P(0,0)=0$. Let $\alpha\geq 0$, and consider the following version of its distribution function (in the ... |
H: Simplifying to a certain expression structure
I have this expression:
$$4n^2-n+(8(n+1)-5)$$
And I know it is equivalent to this:
$$4(n+1)^2-(n+1)$$
I need to simplify my expression to get the same structure as that one. However, no matter what I try, I don't end up with that structure.
My question is, well, how to ... |
H: Differentiation under the integral sign.
I'm working through an integral suggested for practice at the end of the Wikipedia article on differentiation under the integral sign, and I'm stuck.
I am attempting to evaluate this integral:
$$\int_0^{\pi/2} \frac{x}{\tan x} \ dx.$$
The article suggests the following param... |
H: Prove/Show that a number is square if and only if its prime decomposition contains only even exponents.
Prove/Show that a number is square if and only if its prime decomposition contains only even exponents.
How would you write a formal proof for this.
AI: Suppose $a = p_1^{r_1}...p_n^{r_n}$. If $r_i$ are all eve... |
H: Calculating Variance from Math Problem
I was reading this post here when one particular problem caught my attention for a while -
The banker shuffles a standard deck of 52 cards and slowly deals them
face up. The dealt cards are left in full view where they can be
inspected at any time by the player. Whenever ... |
H: Intersection between a cylinder and an axis-aligned bounding box
Given a 3D axis-aligned bounding box (represented as its minimum point and maximum point) and a 3D cylinder of infinite length, what's the best way to test for intersection?
AI: Suppose you are given a box defined by
$$x_{min}\leq x\leq x_{max},y_{min... |
H: Find a single-valued analytic branch of $\sqrt{z^2-1}$ in $\mathbb{C} \backslash [-1,1]$.
I have the following question:
Show there is a single-valued analytic branch $f(z)$ for $\sqrt{z^2-1}$ in $\mathbb{C}\backslash [-1,1]$ such that $f(x) < 0$ for $x>1$. Here $[-1,1]$ denotes a closed interval in $\mathbb{R}$.... |
H: Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them
Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them.
Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ because any other number will be co... |
H: Is '$10$' a magical number or I am missing something?
It's a hilarious witty joke that points out how every base is '$10$' in its base. Like,
\begin{align}
2 &= 10\ \text{(base 2)} \\
8 &= 10\ \text{(base 8)}
\end{align}
My question is if whoever invented the decimal system had chosen $9$ numbers or $11$, ... |
H: Centroid for a rectangular section inclined at an angle theta
How to find Centroid for a rectangular section inclined at an angle theta? Is there any general formula available?
AI: The location of the centroid of a body doesn't change with rotation . If you are working in a coordinate system, then only the coordina... |
H: Is there a standard operation to "rotate rings on matrices"?
Is there some standard operation to "rotate rings on matrices"? Look at the image below:
The numbers around the four empty squares are what I'm calling ring, In the second matrix, this ring has been rotated counterclockwise. I'm aware that ring may not b... |
H: Unstable linear inverse problem: which "dampening" Tikhonov matrix should I use?
A linear inverse problem is given by:
$\ \mathbf{d}=\mathbf{A}\mathbf{m}+\mathbf{e}$
where d: observed data, A: theory operator, m: unknown model and e: error.
The Least Square Error (LSE) model estimate is given by:
$\ \mathbf{\tilde{... |
H: solving the congruence: $5^n\equiv3^n+2 \pmod{11}.$
I'd really like your help with solving the following congruence:
$$5^n\equiv3^n+2 \pmod{11}.$$
I don't know with what to start.
Any help?
Thanks
AI: Just note that 11 is prime so you only need to consider n modulo 10 here. Just list $5^n$ and $3^n$ modulo 11 for ... |
H: How do I have to interpret $\mathrm{nm}^2$?
My short question: How many $\mathrm{m}^2$ are $\mathrm{nm}^2$? Do I have to interpret it as $\mathrm{nm}^2=(\mathrm{nm})^2=(10^{-9}\mathrm{m})^2=10^{-18}\mathrm{m}^2$ or shall it be $\mathrm{nm}^2=\mathrm{n}(\mathrm{m}^2) = 10^{-9} \mathrm{m}^2$? What is the right conven... |
H: Proving that an expression divides a number
How do you prove that
$$n(n+1)(n+2)$$
is divisible by 6 by using the method of mathematical induction?
According to my book
$$\begin{aligned}
(n+1)(n+2)(n+3) &= n(n+1)(n+2)+3(n+1)(n+2)\\
&= 6k + 3*2k'\\
&= 6(k+k')\\
&=6k''
\end{aligned}$$
But I wonder, where does that k c... |
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