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H: existence/uniqueness of solution and Ito's formula
Given the Ito SDE
$$
dX_t=a(X_t,t)dt + b(X_t,t) dB_t
$$
where $a(X_t,t)$ and $ b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions.
Given a function $f(X_t,t) ∈ C^2$ using Ito's formula I can derive the SDE
$$
df = \frac{\partia... |
H: Exterior Algebra of smooth differential forms
I'm a little bit confused about the exterior algebra of smooth differential forms $\Omega(M)$ on a manifold M. The definition of k-forms is clear to me, but I don't understand how to put them together, s.t. they form $\Omega(M)$ so to speak. Maybe you can help me to get... |
H: Help on this integral
I'd like to know why this holds
If one have $f(x_t,t)=x_t\mathrm{e}^{\theta t}$
$\int_0^tdf(x_t,t)=x_t\mathrm{e}^{\theta t}-x_0$
That shouldn't be only $x_t\mathrm{e}^{\theta t}$?
AI: this is definite integral, $\int_0^t df(x_s,s) = f(x_s,s)|_0^t = x_t e^{\theta t} - x_0 e^{\theta 0} = x_t e^{... |
H: Do closure operators on arbitrary posets give rise to complete lattices?
The notion of a closure operator is defined here for an arbitrary partially ordered set.
Now consider an arbitrary set $x$, and call its powerset $P.$ Furthermore, let us order $P$ by inclusion, thereby obtaining a complete lattice. Under thes... |
H: To prove that the following equation has no solution
The question is :
Prove that there are no real numbers $(x,y)$ that satisfy the equation
$$ 13+\ 12[\arctan(x)]=62[\ln(x)]+8[e^x]+4[\arccos(y)] $$
$ [\ ] \text{ denotes the greatest integer function} $
I tried writing the possible values of $\arctan(x), \arccos... |
H: limit of evaluated automorphisms in a Banach algebra
Let $\mathcal{A}=\operatorname{M}_k(\mathbb{R})$ be the Banach algebra of $k\times k$ real matrices and let $(U_n)_{n\in\mathbb{N}}\subset\operatorname{GL}_k(\mathbb{R})$ be a sequence of invertible elements such that $U_n\to 0$ as $n\to\infty$. Define $\sigma_n\... |
H: no. of solution of the equation $[x]^2+a[x]+b = 0$ is
If $a$ and $b$ are odd integer. Then the no. of solution of the equation $[x]^2+a[x]+b = 0$ is
where $[x] = $ greatest Integer function
My Try:: Let $[x] = y$. Then equation become $y^2+ay+b = 0$
Now If given equation has real Roots, Then $\displaystyle y = \f... |
H: Integrating secant squared times tangent
Integrate the function.
$$ \int \sec^2 x \tan x dx $$
I'm trying to get a proper substitution, but I couldn't get anything proper.
AI: This is a good one. Hope you are familiar with integration by substitution.
now case 1:
put $\\ \tan x=t$ so, $ dt=\sec^2xdx $
Substituting,... |
H: Trouble with complex numbers
Is my following calculation true?
$e^{a+ib}e^{\overline{a+ib}}=e^{a+ib}e^{a-ib}=e^{2a}$? for a,b real numbers
or in general, what is $\overline{{z}^{w}}$ if $z,w$ are complex numbers?
AI: yes $\color {green}{e^{a+ib}\bar{e^{a+ib}}=e^{a+ib}e^{a-ib}=e^{2a}} $ is true
Hint:$$z^w=e^{w \ l... |
H: log base 1 of 1
What is $\log(1)$ to the base of $1$?
My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$.
So I was wondering where I have gone wrong.
AI: The reason why it is not convenient to define $\log$ for the base of $1$ is simple:
$$\log_1... |
H: Is this notation for Stokes' theorem?
I'm trying to figure out what $\iint_R \nabla\times\vec{F}\cdot d\textbf{S}$ means. I have a feeling that it has something to do with the classical Stokes' theorem. The Stokes' theorem that I have says
$$
\int\limits_C W_{\vec{F}} = \iint\limits_S \Phi_{\nabla\times\vec{F}}
$$
... |
H: How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations
I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong?
$$(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$$
Thanks
AI: The eas... |
H: Proof of an equivalent definition of a continuous function
In Dudley, Real Analysis and Probability (2nd ed.), Theorem 2.1.2 states:
Given topological spaces $(X,\mathcal{T})$ and $(Y,\mathcal{U})$ and a function $f:X\to Y$, if for every convergent filter base $\mathcal{F}\to x$ in $X$, $f[[\mathcal{F}]]\to f(x)$ i... |
H: Logic Circuit Question
1) Write the boolean expression after every GATE
2) Write the boolean expression of GATE 3
3) Try to simplify the boolean expression of GATE3
I need to know if what I did its right + your advice if there is another way to answer those questions.
1) $GATE_1 = (A'B)' = A+B'$ |$ GATE_2 = 1 ⊕ ... |
H: Polygon Inequality
We know that to form a triangle the 3 sides should obey the triangle inequality . So is there any rule to be followed by the sides of $n$-sided convex polygon.
For Eg:-
$1,2,4$ cannot form a triangle so can we tell if we are given $n$ line segments can we make a $n$-sided convex polygon.
AI: The ... |
H: prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ when $n$ is odd
let $n$ be odd integer , prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $
it's an example which the text proves ! but i can't understand any thing from the argument !
but i tried to prove it by constructing the isomorphism... |
H: $L^2$-lower semicontinuity of an integral operator on $G(x,\nabla w(x))$
In the paper "A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations" by Nochetto, Savaré, Verdi
we find the following claim in Example 2.4:
Let $\mathcal H:=L^2(\Omega),\ p>1$
$$ \phi(w):=\int_... |
H: Solution of a Dirichlet problem on the unit disk
Find the solution of the Dirichlet problem:
$$\Delta u(r,\phi)=0, r<1, u(1,\phi)=f(\phi)$$
where $x=r\cos\phi$ and $y=r\sin\phi$ and
$$f(\phi)=\sin^3(\phi).$$
I start by doing the following:
Enter the polar coordinates $x=r\cos\phi$ and $y=r\sin\phi$. Deriving:
$$\be... |
H: normal distribution, two independent random variable
if $X$ and $Y$ are independent normal distribuited random variables and $T=2X-Y-1$ and $E[X]=E[Y]=1$ and $Var(X)=Var(Y)=4$, what is $Var(T)$?
I get $E[T]=E[2X-Y-1]=2-1-1=0$, but i don't know how to get $Var(T)$.
AI: This answer (which is more of an outline to an ... |
H: Product of two primitive polynomials
I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.
Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate ... |
H: Prove/disprove: In a graph with at least one component that does not contain a hamilton circuit, we can make it hamiltorian by adding a vertex
Prove/disprove: In a graph $G$ with at least one component that does not contain a Hamiltonian circuit, we can add a vertex $x$ and certain edges that connect it with certa... |
H: Find the probability $P(0.5 < X < 5)$
We select two balls without replacing from a box where there are 7 red balls and 3 green balls.
Be $X$ the random variable denoting the number of selected green balls.
Please compute $P(0.5 < X < 5)$ (if you can explain to me how to do $P(a<X<b)$ in general that will be great).... |
H: What does $\mathbb Z_+$ mean?
I am not so sure whether the meaning of $$\mathbb Z_+$$ is very clear. How many different definitions are there? Does the definition that is used depend on whether the writer is English or German?
In French maths, this notation doesn't exist.
AI: Many people would interpret this to mea... |
H: Constructing a number system
I have just started working through a book on higher algebra. I'm just at the beginning, where the authors introduce the notation and talk about the various number systems.
I found this particular paragraph confusing:- "The basic idea in the construction of new set of numbers is to tak... |
H: Period of derivative is the period of the original function
Let $f:I\to\mathbb R$ be a differentiable and periodic function with prime/minimum period $T$ (it is $T$-periodic) that is, $f(x+T) = f(x)$ for all $x\in I$. It is clear that
$$
f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\to 0} \frac{f(x+T+h) - f... |
H: CW-pairs are good pairs
Hatcher uses in a proof that every subcomplex of a CW-complex is a deformation retract of some neighborhood. In what way can I see this in the infinite dimensional case?
AI: Have you checked the appendix? I think there are more explanations there about CW complexes topology. |
H: Is it always true that "max $\ge$ average + sigma"?
Assume that $i$ from $1,\ldots,N$, $x_i \ge 0$ and:
$$\mathrm{avg} = \frac{\sum_i x_i}{N}$$
$$\sigma = \sqrt\frac{\sum_i{(x_i-\mathrm{avg})^2}}{N}$$
Is that true that:
$$\max_i x_i \ge \mathrm{avg} + \sigma\text{ ??}$$
REFERENCE
http://en.wikipedia.org/wiki/Avera... |
H: Let $G$ be the graph whose vertices are binary sequences of length 4, two vertices are adjacent if they have exactly 2 bits different. Is it planar?
Given $G$, a graph whose vertices are binary sequences of length 4. Two vertices are adjacent in $G$ if and only if they differ by exactly two bits. Is it planar?
He... |
H: Simple generalized integral
The integral to compute is $\displaystyle\int_0^\infty \frac{1}{3+x^2} \ \mathrm dx$.
I know how to compute the indefinite integral of this function - I obtained:
$$\frac{\sqrt{3}}{3} \arctan\left(\frac{x}{\sqrt{3}}\right).$$
But when I compute the definite integral it now gives me :
$$\... |
H: Why is it linearly dependent when the linear combination is zero only with none zero coefficients in 3D?
Title says it all. I'm asking the geometrical sense.
I know it is linearly independent if the linear combination of vectors is zero with all the coefficients are zero. And so do dependent.
Independent $\sum a_n... |
H: Using natural numbers $1,2,...n,$ in how many ways can the number $n$ be formed from the sum of **one** or more smaller natural numbers?
Using natural numbers $1,2,...n,$ in how many ways can the number $n$ be formed from the sum of one or more smaller natural numbers?
I thought it would be an easy problem but i ... |
H: Is There a Better Strategy for this Combination Scenario?
An elevator containing five people can stop at any of seven floors.
What's the probability that no two people get off at the same floor?
Assume that the occupants act independently
that all floors are equally likely for each occupant.
For the denominator... |
H: cardinality of all real sequences
I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, since the cardinality of all rational sequences is the same as the cardinality of... |
H: Open and Closed Set in Zariski Topology
I'm confused about the definition closed and open set in Zariski Topology, it is said that the set $$V(I)=\{P \in \operatorname{Spec}(R)\mid I \subseteq P\}$$ are the closed set in Zariski Topology. But it is said in James Munkres's Topology that a subset $U$ of $X$ is an ope... |
H: Is perspective transform affine? If it is, why it's impossible to perspective a square by an affine transform, given by matrix and shift vector?
I'm a bit confused. I want to program a perspective transformation and thought that it is an affine one, but seemingly it is not. As an example, I want to perspective a sq... |
H: Green fuctions and sturm liouville problem
I have the following problem. Find the Green function in the problem of Sturm-Liouville:
$$Ly=-y''.$$
$$y'(0)=y(0),\quad y'(1)+y(1)=0$$
I can not find the eigenvalues and the eigenfunctions but how do I find the Green's function?
The solution in the book is
$\frac{1}{3}... |
H: Concrete Mathematics Summation Question
I'm sorry if this question is too novice, but I am just beginning discrete math. I've been working through the book Concrete Mathematics (Graham,Knuth,Patashnik) and I reached a double summation that has me very confused. I'v been trying to work it out, and I think I have a s... |
H: What does the notation $[V]^2$ mean (in graph theory)?
In graph theory, a graph is a pair $G=(E,V)$ of sets satisfying $E\subseteq[V]^2$. But what is $[V]^2$?
I suppose that it is the same as $V\times V=V^2$, but I do not know where the square brackets come from.
Thanks in advance!
AI: In set theory (and graph theo... |
H: $f$ continuously differentiable implies even or odd?
Suppose that $f(x)$ is continuously differentiable for all $x\in\mathbb{R}$. Then $f$ is continuous and Riemann integrable such that $\int f(x)\,dx = F(x)+c$. My question is:
Can we conclude that $F$ must be either odd or even?
In other words, can we rule out ... |
H: How to isolate $v_m$?
Note: I am not asking anything pertaining to the physics of this question; only the mathematics. The physics is just given as a context to the problem for those interested, as opposed to simply saying "simplify this equation".
The double ball drop problem is as follows:
A ball of mass $m$ is p... |
H: The nested self-composition of $f(x) = \frac{\sqrt3}2x+\frac12\sqrt{1-x^2}$
The function $f(x)$ is defined, for $|x|\leqslant1$ by $$f(x)=\frac{\sqrt 3}{2}x+\frac{1}{2}\sqrt{1-x^2}.$$ Find an expression for $$f^n(x)=\underbrace{f \circ f \circ \cdots \circ f(x)}_{\text{n times}},$$where $n\in\mathbb{Z^+}$.
Now w... |
H: Find the jordan form of a given matrix
let $n,k\in\ N$ and $\lambda \in F$.
Find the rank and the Jordan form of matrix $$A={ J }_{ n }{ (\lambda ) }^{ k }$$
AI: $J_n(\lambda)^k$ is an upper-triangular matrix, and its diagonal elements will all be $\lambda^k$. Thus, its rank is $n$, and its Jordan form is $J_n(\la... |
H: Bound on Expectation of a convex function of a Random variable
My friend asked me the following question, which I at first thought was simple and straightforward:
If $X$ is an integrable random variable and $g$ is a convex function(all real valued), then is it true that $\forall a>0$
$$E[g(X)] < \infty \Rightarrow ... |
H: Find Gross from Net and Percentage
I would like know if a simple calculation exists that would allow me to determine how much money I need to gross to receive a certain net amount. For example, if my tax rate was 30%, and my goal was to take home 700, I would need to have a Gross salary of $1000.
AI: Let $x$ be the... |
H: Solve these equations simultaneously
Solve these equations simultaneously:
$$\eqalign{
& {8^y} = {4^{2x + 3}} \cr
& {\log _2}y = {\log _2}x + 4 \cr} $$
I simplified them first:
$\eqalign{
& {2^{3y}} = {2^{2\left( {2x + 3} \right)}} \cr
& {\log _2}y = {\log _2}x + {\log _2}{2^4} \cr} $
I then had:
$\... |
H: $f : \mathbb{R} \to \mathbb{R}$ be injective then $f^{ −1} (\mathbb{Q} \cap [0, 1])$ is
Let $f : \mathbb{R} \to \mathbb{R}$ be injective then $f^{ −1} (\mathbb{Q} \cap [0, 1])$ is
(a) measurable and its measure is 0.
(b) measurable and its measure is 1.
(c) measurable and its measure is ∞.
(d) need not be measurabl... |
H: Small doubt about Dirichlet's problem
Find the solution of dirichlet s problem:
$\Delta u(r,\phi)=0, r<1, u(1,\phi)=f(\phi)$
Where $x=r\cos\phi$ and $y=r\sin\phi$ and
$f(\phi)=\cos^2(\phi)$
I starting by doing following:
Enter the polar coordinates $x=r\cos\phi$ and $y=r\sin\phi$. Deriving:
$u_r=u_x\cos\phi+u_y\sin... |
H: Proof that binomial coefficients are integers - combinatorial interpretation
For any integers $k \le n$ here is an injective group homomorphism $$S_k \times S_{n-k} \rightarrow S_n$$ such that a tuple $(\sigma, \tau)$ permutes $\{1,...,n\}$ by letting $\sigma$ act on $\{1,...,k\}$ and $\tau$ act on $\{k+1,...,n\}$.... |
H: Let $f : [0, 1] \to [0, 1]$ be strictly increasing then
Let $f : [0, 1] \to [0, 1]$ be strictly increasing then
(a) f is continuous.
(b) If f is continuous then f is onto.
(c) If f is onto then f is continuous.
(d) None of the above
It should be injective I am sure, but right now no counter example is coming in my ... |
H: If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Is $f(x+iy) = \exp(x+iy)$?
$(1)$ If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Then, is $f(x+iy) = \exp(x+iy)$ for every $x$ and every $y$?
$(2)$ If $f$ is an entire function such that $f(iy) = iy... |
H: For $X_i$ iid $ Var( \sum_{i=1}^n X_i )= \sum_{i=1}^n Var (X_i)$?
My contention is that it's true. I thought of two ways of proving it, unsure which one is better (and/or correct):
Suppose $X_i \sim N(\mu, \sigma^2)$, then it's moment generating function (MGF) is
$$\Psi_{X_i}(t) = \exp\{\mu t + \frac12 \sigma^2 t^... |
H: Simple Differential Equation
We have
$$\frac{dx}{dt}=x-y-1$$
$$\frac{dy}{dt}=x-y+1$$
Express $y$ in terms of only $x$ (i.e. no $t$ term).
My professor gave me the hint "use $\frac{d}{dt}(x-y)$", but I don't know how this is supposed to help me.
AI: Use both $x-y$ and $x+y$. I get
$$\frac{d}{dt}(y-x) = 2 \imp... |
H: Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series
In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:
$$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$
But, where ... |
H: Why does the graph of $x^n$ only have an imaginary component if $n$ is not an integer?
I was playing around graphing equations and noticed that only non-integer exponents of x yield imaginary graphs. Try it on Wolfram Alpha. Why is that?
AI: An imaginary component is a component containing $i=\sqrt{-1}=(-1^{\frac{1... |
H: Probability of winning the game "1-2-3"
Ok, game is as follow, with spanish cards (you can do it with poker cards using the As as a 1)
You shuffle, put the deck face bottom, and start turning the cards one by one, saying a number each time you turn a card around ---> 1, 2, 3; 1, 2, 3; etc. If when you say 1 a 1 com... |
H: The fundamental group functor is not full. Counterexample? Subcategories with full restriction?
Anyone aware of a nice counterexample to "The fundamental group functor is full?" (Which is...false, right?) and are there a nontrivial subcategories on which its restriction is full?
I.e.
Can you think of an example of ... |
H: Proof that a normal subgroup $N$ of $G$ is the identity coset in the group of cosets of $N$
I am not sure how to prove the following:
Let $N$ be a group. Prove that for any $n\in N, nN=N$.
(Or maybe the following, but I'm not sure it's correct: $n\in N$ $\Leftrightarrow nN = N$)
I haven't been asked to prove this d... |
H: 2 questions regarding compactness and closed
Let $(X,d)$ be a metric space. Let $E$, $F$ be two disjoint non-empty subsets of $X$ with $E$ compact and $F$ closed.
Show that $\inf\{d(x,y): x\in E, y\in F\}>0$
Show that this does not longer true is $E$ is not compact: find two disjoint closed subsets $E$ and $F$ of ... |
H: Two problems about rings.
Somebody can to help me in such exercices:
(1) A ring R such that $a^2 = a$ for all $a\in R$ is called a Boolean ring. Prove that every
Boolean ring R is commutative and $a + a = 0$ for all $a \in R$.
(2) Let R be a ring with more than one element such that for each nonzero $a\in R$ t... |
H: How to do I calculate the conditional probability distribution?
The Chicago Cubs are playing a best-of-five-game series (the first team to win 3 games win the series and no other games are played) against the St. Louis Cardinals. Let X denotes the total number of games played in the series. Assume that the Cubs w... |
H: Primitive element (fields)
I'm re-reading the primitive element lemma and I can't reason the following concept. Let $f,g\in F[x]$ be in the polynomial ring of one variable over the field $F$. Let those two polynomials have a unique common simple root $\beta$. Then the largest unary common divisor of $f$ and $g$, wi... |
H: Ratio between trigonometric sums: $\sum_{n=1}^{44} \cos n^\circ/\sum_{n=1}^{44} \sin n^\circ$
What is the value of this trigonometric sum ratio: $$\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ} = \quad ?$$
The answer is given as $$\frac{\displaystyle\sum_{n=1}^{44} \co... |
H: Is $f'(x)-3f(x) = 0$ subspace of differentiable functions $f\colon (0,1)\to \mathbb{R}$
$V$ is space of differentiable functions $f(0,1) \to \mathbb{R}$ and $W$ is a subset of $f$ that meets $f'(x) - 3f(x) = 0$ for all $x\in (0,1).$
Is subset $W$ a subspace of $V$?
I know that I have to prove that it's closed under... |
H: calculate integral $\;2\int_{-2}^{0} \sqrt{8x+16}dx$
I want to calculate the integral $$2\int_{-2}^{0} \sqrt{8x+16}dx$$ The answer is $\;\dfrac {32}{6}\;$ but I don't know how to get it.
AI: $$I = F(x) = \int_{-2}^0 \sqrt{(8x + 16)}\,dx$$
We use substitution:
Let $u = 8x + 16,\;\;du = 8\,dx \implies dx = \dfrac ... |
H: Applying the contra positive of the finite intersection property
I'm reading a proof which has the following setting.
I have a family $D$ of compact sets with empty intersection. The next line takes a finite subset of $D$ with empty intersection.
This is clearly possible if families of compact sets enjoy the fini... |
H: The Fitting subgroup centralizes minimal normal subgroups in finite groups
Let $G$ be a finite group:
If $N$ is a minimal normal subgroup of $G$, then $F(G) \leq C_G(N)$.
Here $C_G(N)$ denotes the centralizer of $N$ in $G$, and
$F(G)$ denotes the Fitting subgroup of $G$.
AI: If $N$ is abelian, then $N \leq F$ and $... |
H: What are the last two digits of $3^{3^{100}}$?
What are the last two digits of $3^{3^{100}}$? I had this on an exam, just curious.
AI: From Fermat's little theorem, $a^{\phi(n)}\equiv 1\pmod n$ if $\gcd(a,n)=1$. With $a=3$ and $n=100$, we conclude $3^{40}\equiv1\pmod{100}$. Hence if $3^{100}=m\cdot 40+r$, we only n... |
H: Find the length of the parametric curve
Find the length of the parametric curve defined by:
$x=t+\dfrac{1}{t}$
and
$y=\ln{t^2}$
on the interval
$(1 \le t \le 4)$.
AI: Hint: The derivative of your first function is $1-\frac{1}{t^2}$. The derivative of the second is $\frac{2}{t}$. Note that
the sum of the squares of... |
H: Multiple Integral
I'd like to know when exactly we have the right to inverse the order of the variables in a multiple integral. Which are the cases which cause problems. (When $\int_a^b \int_c^d \int_e^f f(x,y,z) \, \mathrm dx\, \mathrm dy \, \mathrm dz \neq \int_c^d \int_a^b \int_e^f f(x,y,z) \, \mathrm dx\, \math... |
H: Compute dependent probability
I have
$X = \pmatrix{-2 & -1 & 0 & 1 & 2 \\
.05 & .2 & .3 & .4 & .05} $, $$F(X) =
\begin{cases}
0 & X \le -2 \\
.05 & -2 \le X < -1 \\
.25 & -1 \le X < 0 \\
.55 & 0 \le X < 1 \\
.95 & 1 \le X < 2 \\
1 & 2 \le X
\end{cases}
$$
I must compute:
$$
P(X > -2.1\ |\ X < 1.3) \\
$$
I kn... |
H: Sheafification of singular cochains
Let $S^k$ be the presheaf on a space $X$ that assigns to every open set $U$ the abelian group $S^k(U)$ of singular k- cochains on $U$. This is clearly not a sheaf. Consider the sheafification $F^k$ of each $S^k$. These sheaves form an exact resolution of the constant sheaf of int... |
H: Algebra simplification in mathematical induction .
I was proving some mathematical induction problems and came through an algebra expression that shows as follows:
$$\frac{k(k+1)(2k+1)}{6} + (k + 1)^2$$
The final answer is supposed to be:
$$\frac{(k+1)(k+2)(2k+3)}{6}$$
I walked through every possible expansion; I... |
H: About the sum of the digits of $k^{105}$.
I read here that
We cannot find an integer $k>2$ such that the sum of the digits of $k^{105}$ is $k$.
Does anyone know a proof of this?
AI: A dumb-but-working approach could look like this:
$k^{105}$ has $1+\lfloor\log_{10} k^{105}\rfloor=1+\lfloor 105\log_{10} k\rfloo... |
H: How many even numbers can you form which are greater than 100 subject to the following constraints?
How many 3 digit even numbers can be made from the numbers 0,1,2,3 which are greater than 100.
The book answer says 20 but I am getting 23.
AI: Your answer is correct.
As a first approximation there are $3$ choices f... |
H: A convergence test for improper integrals ($\mu$-test)
I came across a convergence test for improper integrals referred to as the $\mu$-test while I was looking through a textbook. I'm interested in understanding the idea behind the test since no explanation is given in the textbook.
Let $f(x)$ be unbounded at $a... |
H: Question on statistics - mean and standard deviation?
I'm not sure what to do here:
A class of 30 students were weighed. Their mean was found to be 58kg with a standard deviation of 5.5kg. What percentage of students will have a mass between 52.5 kg and 63.5 kg?
AI: For the answer, you are expected to assume that t... |
H: Integral inequation
In my statistics book Chebyshev's inequality is proven. In several steps this inequality is used:
$$ \int_a^{+\infty} \phi(x) f_X(x)dx \quad \geq \quad \phi(a) \int_a^{+\infty} f_X(x)dx $$
and also:
$$ \int_{-\infty}^{-a} \phi(x) f_X(x)dx \quad \geq \quad \phi(-a) \int_{-\infty}^{-a} f_X(x)dx $$... |
H: Coordinate geometry and translations: rotations and composites oh my!
Here's the problem:
Let $R_y$ be a reflection in the $y$-axis and $T : (x,y) \rightarrow (x-3,y+1)$. Which one of the following transformations is equivalent to $R_y \circ T$?
Here's my thought process:
okay, so this means that first we perfor... |
H: Minimal subgroups lie in the center so group is nilpotent
Let $G$ be a group of odd order. If every minimal subgroup lies in the center, prove that $G$ is nilpotent . Thanks!
AI: Proposition: Let $G$ be a finite group, $p$ an odd prime, such that every subgroup of order $p$ is contained in the center of $G$. Then ... |
H: Why is it impossible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that $|z_1- z_2|=|z_1-z_3|=|z_2-z_3|=|z_1-z_4|=|z_2-z_4|=|z_3-z_4|$?
A. It is possible to find distinct $z_1,z_2,z_3\in \mathbb C$ such that $|z_1-z_2|=|z_1-z_3|=|z_2-z_3|$.
Answer: True
B. It is possible to find distinct $z_1,z_2,z_3, z_4\... |
H: Find CDF of uniformly distributed variable
Suppose $X$ is uniformly distributed over $[-1,3]$ and $Y=X^2$. Find the CDF $F_{Y}(y)$
From definition, I know that X's PDF is
$\displaystyle f_{X}(x)=\begin{cases}
\frac{1}{4}, & -1\leq x\leq 3, \\
0, & \text{otherwise}.
\end{cases}$
Thus $\displaystyle F_{Y}(y)=P(X^2... |
H: How many Jordan normal forms are there for this characteristic polynomial?
Given the characteristic polynomial of a matrix $A \in \mathbb{C}^{6x6}$ with $p(A)=(\lambda-2)^2(\lambda-1)^4$, we were supposed to determine all Jordan normal forms that have this characteristic polynomial.
I determined 10 (is this correct... |
H: Can we see a ring $R$ as a subring of $S^{-1}R$?
I know that we can consider an integral domain $D$ as a subring of its quotient field, I'm wondering why we can't consider any commutative ring with identity as a subring of $S^{-1}R$ identifying $r\in R$ as $r/1_R\in S^{-1}R$.
Thanks in advance.
AI: This is because... |
H: What leads us to believe that 2+2 is equal to 4?
My professor of Epistemological Basis of Modern Science discipline was questioning about what we consider knowledge and what makes us believe or not in it's reliability.
To test us, he asked us to write down our justifications for why do we accept as true that 2 plus... |
H: Notation for set of all closed sets
Is there a common notation for the set of all closed sets of a topological space?
I have been using $(X,\tau)$ to denote a topological space with $\tau$ being the topology, set of all open sets. I was wondering if there is something like that this is used widely but for all the c... |
H: Irreducible characters form orthonormal basis of set of class functions
I am reading Serre's book (Linear Representations of Finite Groups). Theorem 6 in chapter 2 says that the irreducible characters $\chi_1,\dotsc,\chi_h$ of a finite group $G$ form an orthonormal basis of $H$, the set of class functions on $G$. I... |
H: Probability question/"puzzle"
Given I have some number X.
I draw a random number R from the uniform distribution on the unit interval and construct two new numbers Y and Z through the following procedure:
Y = R*X
Z = X - Y
What is the probability, that neither Y or Z (not exclusive OR) are below 1? As far as I see,... |
H: For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?
I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you tell $N\subseteq M_i$ for som... |
H: Probability puzzle - the 3 cannons
(Apologies if this is the wrong venue to ask such a question, but I don't understand how to arrive at a solution to this math puzzle).
Three cannons are fighting each other.
Cannon A hits 1/2 of the time. Cannon B hits 1/3 the time. Cannon C hits 1/6 of the time.
Each cannon fire... |
H: Confusion about the usage of points vs. vectors
As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification on the usage of vectors and the usage of points, because it seems l... |
H: A question about the validity of a notation
I am writing a paper and using such a notation. Do you think that it is mathematically a reasonable notation?
$$
\hat{{\cal{P}}}_{i}=\{\hat{Q}: \hat{Q}_i|G_i[q_1/q_0<t]\stackrel{i=1}{\underset{i=0}{\gtreqqless}}Q_i|G_i[q_1/q_0<t] \},
$$
Thank you very much.
EDIT: so I nee... |
H: Verify the real solution of a linear system of differential equation
I'm trying to solve $Y' = AY$ where $A= \left[
\begin{array}{ c c }
-2 & 6 \\
-3 & 4
\end{array} \right]$
I have found the eigenvalue $1 \pm 3i$ with eigenvector for $1+3i: $
$v = \left[
\begin{array}{ c c }
1-i \\
1
\e... |
H: A confusion on Axiom of infinity
I'm currently working "the elements of advanced mathematics" by steven g. krantz, currently on Chapter 5.
I came to "Axiom of Infinity" which roughly states:
$$\exists A \; s.t. \; \phi \in A \; and \; \forall a\in A, a\cup \left \{ a \right \} \in A$$
Now, doesn't this mean:
$$\exi... |
H: Question about inner product.
Let $V=C([-1, 1])$ and $$\langle f, g\rangle=\int_{-1}^1 f(x)g(x)dx$$
Let $W=\{f \in V \mid f\text{ is even}\}$. Find $W^\perp$.
Progress: I know that every odd function belongs to $W^\perp$ and I suspect $W^\perp=\{f \in V \mid f\text{ is odd}\}$.
AI: Let $g$ be an odd function. Then ... |
H: Why Gaussian Elimination only works over field?
When I was solving system of linear congruences (n variables, n equations), like this:
$AX \equiv b \pmod p$
I was told that ordinary Gaussian Elimination works if $p$ is prime. And I figured out that when $p$ is prime, integers $\pmod p$ form a field, otherwise it do... |
H: Let $f(x)= \frac {1}{\sqrt{|[|x|-1]|-5}}$ where $[ .]$ is greatest integer function, Find domain of $f(x)$ ??
Problem: Let
$$f(x)= \dfrac {1}{\sqrt{|\bigl[|x|-1\bigr]|-5}}$$
where $\bigl[.\bigr]$ is greatest integer function. Find domain of $f(x)$.
Solution: The function $f$ is defined for $|\bigl[|x|-1\bi... |
H: What is the kernel of the evaluation homomorphism?
I'm studying Sharp's Steps in Commutative Algebra, and I need a hint how to proceed with this exercise in the page 26:
First of all, I didn't understand even the notation, what did the author mean by $(X_1-\alpha_1,...,X_n-\alpha_n)$?
Thanks a lot.
AI: Basically t... |
H: Determining whether or not a group has an element of a specific order
If $|G| = 55$, must it have an element of order $5$ and/or $11$?
I'm not quite sure how to determine this. I know it could be possible by Lagrange's Theorem, but I'm stuck otherwise. Any help would be appreciated.
Edit: I haven't learned materia... |
H: Prove that if $a b c=1$, $a^2+b^2+c^2\ge 3$. (for $a,b,c\in\mathbb{R}$)
I can do this problem using calculus minimization techniques, using Lagrange multipliers to find the equations $a^2=b^2=c^2$, so with $abc=1$, $a$, $b$, and $c$, are either $-1$ or $1$ (making sure you don't end up with $abc$ negative). So if t... |
H: Definition of measurable functions defined w.r.t. topology
(Big) Rudin's "Real and Complex Analysis" defines (definition 1.3) a measurable function from a measurable space into a topological space as one that has the property that the inverse image of every open set in the range space is measurable in the domain sp... |
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