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H: Let $f:[0, 1] \rightarrow \mathbb{R}$ a continuous function. If $a>0$, show that:
Let $f:[0, 1] \rightarrow \mathbb{R}$ a continuous function. If $a>0$, show that:
$$\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)-f(0)}{x}dx=0$$
This question came from:How can one show that $ f(0)\ln(\frac{b}... |
H: Poincare Inequality implies Equivalent Norms
I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following in the book(page 266.) "In view of the Poincare Inequality, on $W_{0}^{1... |
H: A function which is continuous at everywhere in its domain, but diff only at one point
Suppose a real valued function $f:\mathbb R \to \mathbb R$ is continuous everywhere. Is it possible to construct $f$ that is differentiable at only one point? If possible give an example also.
AI: It is possible. Start with a fun... |
H: Showing that a set is open/closed
$\def\R{\mathbb R}$
Is the set
$$S=\{(x_1,x_2,x_3) \in \R^3 \mid e^{x_1} + x_2^2 <x_3 \} \subset \R^3$$ open or closed?
My attempt:
Let $f:\R^3 \to \R$, $f(x_1,x_2,x_3)$ =$e^x_1 + x_2^2-x_3$. Then $S=\{(x_1,x_2,x_3) \in \R^3 \mid f(x_1,x_2,x_3)<0 \}$. Also $f(x_1,x_2,x_3)$ is co... |
H: Are my answers correct? (finding intercepts, asymptotes, and extrema)
Are my answers correct?
a) (0, 4/3) and (2,0) and (-2, 0)
b) Horizontal asymptote: $y = 3$, Vertical asymptotes: $x = 3, x = -3$
c) Extremum is at $(0, 4/3)$, maximum.
AI: Yes, all of the answers are correct. In future, maybe try and show your w... |
H: Supremum over dense subset of banach space
Let $\{x_n\}$ be a countable dense subset of a Banach space $X$. How can I show that
$$\sup_{x \in X}f(x) = \sup_{n \in \mathbb{N}}f(x_n)$$
where $f$ is continuous and real-valued??
AI: For a continuous function, you always have $f(\overline{A}) \subset \overline{f(A)}$. S... |
H: When is round-robin scheduling possible and with in minimal time?
Suppose that you have six teams $x_0, x_1, x_2, x_3, x_4, x_5$. Can you schedule round-robin games between them so that if one game is played each day, the series of games can be completed in five weeks?
A naive attempt fails:
$$(x_0,x_1), (x_2, x_3)... |
H: Confirming proofs of properties of preimages
I do not think that I made terrible mistakes, nevertheless a conforming word would be good for me. Thank you!
Let $f\colon X\to Y$ be a map and $(B_i)_{i\in I}$ a family of subsets of $Y$ ($I$ is any index set). Overmore consider any $A,B\subset Y$. Show:
\begin{al... |
H: Prove that $x_{n+1}=\frac{2}{9}(x_n^3+3)$ converges
Let $x_1=1/2$ and $x_{n+1}=\frac{2}{9}(x_n^3+3)$ for $n\geq 1$. We want to prove that the sequence $(x_n)$ converges to real number $r\in (0,1)$ satisfying the equation $2r^3-9r+6=0$.
First part
For the sequence to converge, it must be bounded. A sequence $X:=(x_n... |
H: $\mathbb{Q}[X,Y]/(X,Y^{2}-1)$ is this a maximal ideal or a prime ideal?
$\mathbb{Q}[X,Y]/(X,Y^{2}-1)$ is this a maximal ideal or a prime ideal?
So far i got:
$\mathbb{Q}[X,Y]/(X,Y^{2}-1) = \left(\mathbb{Q}[Y]\right)X/(X,Y^{2}-1)$ is isomorphic to
$\mathbb{Q}[Y]/(Y^{2}-1)$ = $\mathbb{Q}[Y]/(Y+1)(Y-1)$. Because we n... |
H: Let $R$ be the set of all integers with alternative ring operations defined below. Show that $\Bbb Z$ is isomorphic to $R$.
For any integers $a,b$, define $a\oplus b=a + b - 1$ and $a\odot b=a + b - ab.$ Let $R$ be the ring of integers with these alternative operations. Show that $\Bbb Z$ is isomorphic to $R$.
Wh... |
H: The $n\times n$ matrix $(a_{ij})$ with $\sum_{j=1}^{n} a_{ij} = 1$ has an eigenvalue 1.
The Problem
Let $A = (a_{ij})$ be an $n\times n$ matrix such that
$$
\sum_{j=1}^n a_{ij} = 1
$$
for all $i = 1,2,\dots,n$. Show that $A$ has an eigenvalue $1$.
I tried to figure out from $A - \lambda I v = \mathbf{0}$ but now I... |
H: Lebesgue integral over discrete set
Let $X=\{x_1,x_2,\ldots\}$, and for any $A\subseteq X$ let $$\mu(A)=\sum_{x_n\in A}2^{-n}$$ Suppose $f$ is given by $f(x_n)=1/n!$. Find $$\int_X fd\mu$$
I tried to estimate from below by simple functions $s_k$ where $s_k(x_n)=1/n!$ for $n=1,2,\ldots,k$ and $s_k(x_n)=0$ otherwise.... |
H: Series $\sum_{i=1}^\infty2^{-i}/i!$
The series $\sum_{i=1}^\infty2^{-i}/i!$ is clearly convergent by the ratio test with $\sum_{i=1}^{\infty}2^{-i}$, but is it possible to calculate the exact sum?
AI: More generally, $$e^x=\sum_{i=0}^\infty \frac{1}{i!}x^i$$
Applying this with $x=2^{-1}$ gives the series you want, ... |
H: The existence of inequalities between the sum of a sequence and the sum of its members
Let $(a_n)_{n=m}^\infty$ be a sequence of positive real numbers. Let $I$ denote some finite subset of $M := \{m, m+1, \cdots \}$, i.e., $I$ is the index of some points of $(a_n)_{n=m}^\infty$.
Does there exist a real number $r$ s... |
H: Moment generating function of a function
I need to find MGF of $Z = \mu + \frac{1}{\lambda}X$ ($\mu, \lambda$ are constants). How do I do that?
UPDATE
If it helps:
$$f(x) = \frac{1}{\sqrt{2\pi}}\exp{ \{ -\frac{x^2}{2} \} }$$
So ...
$$M_X(t) = te^{t^2/2}$$
AI: The general rule is this:
$$ M_Z(t) = E[e^{tZ}] = E[e... |
H: Balancing objects of varying length in a collection of set length while maintaining order
I have an object each with a length associated with it. I can then have multiple of these objects and I want to put them into another collection/array with a certain set count. Order matters and I need every element in the top... |
H: Learning roadmap for Non-commutative Geometry
I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional Analysis is very little.Please advise which topics in Functional Analysis and ... |
H: Fatou's lemma on bounded domain and bounded range
Fatou's lemma states that if $f_1,f_2,\ldots$ are nonnegative measurable functions, and $f=\liminf f_i$, then $$\int_E fd\mu\leq \liminf\int_E f_id\mu$$ The two examples for strict inequality given in the link are where $f_n$ takes on a nonzero value on an arbitrari... |
H: Probability of drawing 3 balls
A box contains 8 red, 3 white, and 9 blue balls. If 3 balls are drawn at random determine the probability that
all 3 balls are red
all 3 balls are white
2 are red and 1 is white
at least 1 is white
What I tried for the first one is
3/20 * 2/19 * 1/18 = 6/6840
But they said it was... |
H: What is the $\lVert v\rVert$ sign mean in the context of vectors?
Suppose $V$ a inner product space, $u, v \in V$.
I need to prove this identity:
$$\lVert u+v\rVert^2 +\lVert u-v\rVert^2 = 2\left(\lVert u\rVert^2 +\lVert v\rVert^2\right) $$
what is the $\lVert v\rVert$ ("two lines") sign mean? thanks.
AI: It is the... |
H: $f(z)=f(iz)\forall z$, my question is : is there such $f$ exists?
$f$ be an entire function such that $|f(z)|\le c|z|^3\forall |z|\ge 3,f(z)=f(iz)\forall z$, my question is : is there such $f$ exists?
$f(z)=a_0+a_1z+a_2z^2+a_3z^3+a_4z^4+a_5z^5+\dots=a_0+a_1iz-a_2z^2-ia_3z^3+a_4z^4+ia_5z^5\dots$
$|g(z)|=|{f(z)\over ... |
H: Is $GL(n;R)$ closed as a subset of $M_n(R)$?
Let $M_n(R)$ denote the space of all $n×n$ matrices with real entries. The general linear group over real numbers,denoted $GL(n,R)$, is given by $GL(n,R)=${$A∈M_n(R)|det(A)\neq0$}. Is $GL(n,R)$ closed as a subset of $M_n(R)$?
Some thought of mine:Obviously,ther... |
H: Prove that a nonzero homomorphic image of a local ring is a local ring
The problem says that prove that a nonzero homomorphic image of a local ring is a local ring.
Could you give me a scratch of a proof for this or maybe a full answer if you don't mind?
AI: The reason is that for every ring $R$ and a proper ideal ... |
H: What are the properties of the roots of the incomplete/finite exponential series?
Playing around with the incomplete/finite exponential series
$$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$
for some values on alpha (e.g. solve sum_(k=0)^19 z^k/(k!) = 0 for z), I made a few obse... |
H: I have five eggs to color for Easter. I can color them red, yellow, or blue. How many ways are there to do this?
Not sure if my thinking is correct on this problem. I have five eggs to color for Easter. I can color them red, yellow, or blue. How many ways are there to do this?
I was thinking 5 * 5 choose 3 since y... |
H: Dot product over complex vectors: Conjugate first or second?
Does there exist a truly "standard" dot product over complex vectors?
Wikipedia and Wolfram's MathWorld indicate directly or indirectly that
the second argument is conjugated.
Matlab's dot product is the opposite. "When A and B are both column vectors,... |
H: O(p,q; C) isomorphic to the usual orthogonal group O(p + q; C) for complex field
I'm trying to make sense of this statement that appears on wiki:
"The group O(p,q) is defined for vector spaces over the reals. For complex spaces, all groups O(p,q; C) are isomorphic to the usual orthogonal group O(p + q; C), since th... |
H: If $Y \subseteq X$ is open, and $S \subseteq Y$ is nowhere dense in $Y$, then $S$ is nowhere dense in $X$
Let $X$ be a topological space, let $Y$ be an open subspace of $X$. Suppose that $S \subseteq Y$ is nowhere dense in $Y$, that is, $\operatorname{int}_Y(\operatorname{cl}_Y(S)) = \emptyset$. How to prove that $... |
H: What does it mean to "determine" an equivalence relation?
I don't understand the following problem:
What does it mean exactly that a number of pairs can "determine" an equivalence relation? Say if I have the following set: {1, 2, 3}, and a relation R that's true for (a, b) if a=b. Then would the pairs {1, 1}, {2, ... |
H: Prove that $[0,1]$ is not a compact subset of $\mathbb{R}$ with the lower limit topology, i.e. open sets are of the form $[a,b)$.
Prove that $[0,1]$ is not a compact subset of $\mathbb{R}$ with the lower limit topology, i.e. open sets are of the form $[a,b)$.
My question is will different topology affect compactnes... |
H: Balls and boxes probability problem
Here is another question from the book of V. Rohatgi and A. Saleh. I would like to ask help again. Here it goes:
Let A, B, and C be three boxes with three, four, and five cells, respectively. There are three yellow balls numbered 1 to 3, four green balls numbered 1 to 4, and five... |
H: How do I ascertain whether an equation defines a function?
Asides from graphing an equation are there any other ways of determining if an equation is a function?
AI: If you can re-write the equation as a dependent variable and a independent variable, so Y = an equation only involving X, then verify that each X dete... |
H: Explanation on a "different" proof that $C_c(\Omega)$ is dense in $L^p(\Omega)$.
Theorem: Let $\Omega\subset \mathbb{R}^n$ be an open set and $1\leq p < \infty$. The space $C_c(\Omega)$ is dense in $L^p(\Omega)$.
Haim Brezis has a French book called "Analyse fonctionnelle: theorie et applications" (my version, An... |
H: How to calculate this series/with factorial in the numerator?
I'm wondering if anyone could help me out with figuring out this series?
$$\sum_{k=1}^{\infty }\frac{2^{2k}}{(2k)!}$$
with the factorial in the bottom, what trick we need here to calculate it?
Thank you!!
AI: This is $\sum_{k=1}^{\infty }\frac{2^{2... |
H: Integrable function via Fubini
Suppose $f(x,y)\geq 0$ is a measurable function on $\mathbb{R}\times\mathbb{R}$, and $$\int_\mathbb{R}\left(\int_\mathbb{R}f(x,y)d\mu(x)\right)d\mu(y)<\infty$$
Do we have that $\dfrac{xyf(x,y)}{x^2+y^2}$ is integrable on $\mathbb{R}\times\mathbb{R}$.
By Fubini, we have that $$\int_{\... |
H: Greatest common divisor, no prime factor without power
If I try to get the GCD of 272 and 210 I end up with the following
$272 = 272/2=136/2=68/2=34/2=17$
$2^4 \times 17$
$210 = 210/2=105/5=21/3=7$
$2 \times 5 \times 3 \times 7$
Normally if there are 2 equal numbers with a different force, the one with the highes... |
H: The closed form of of the Dirac Delta Function
I know that $\delta(x)$ is the Dirac Delta function - satisfying -$$\int^{\infty}_{- \infty}\delta(x)dx=1$$Can anyone tell me a closed form of $\delta(x)$ . I guess it might be in a form of $e^{a(x)}$ for some $a(x)$
AI: The Dirac delta function is not a function. Eve... |
H: How to solve system of 4 equations with four unknowns?
I have got system of 4 equations as shown below and I am considering if there is any other method than brute force to solve them.
B + C + D = S1
A + C + D = S2
A + B + D = S3
A + B + C = S4
Values of S1-S4 are given:
S1 = 70
S2 = 75
S3 = 80
S4 = 75
I tried to... |
H: Alternating series problem
My mind is blanking completely on how to do this one. the base string of numbers i have to pull a series out of is
$$ \frac{2}{3} -\frac{2}{5} + \frac{2}{7} - \frac{2}{9} + \frac{2}{11} - ...$$
which i found to be
$$\sum_{n=1}^\infty(-1)^n( \frac{2}{2n+1} ) $$
I have to do an An to Bn... |
H: If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive, so $$\int_{0}^{1} \frac{f(x)}{f(x)+f(1-x)}dx=\frac{1}{2}$$???
all examples that I tested have worked.
AI: Since
$$
\int_{1/2}^1\frac{f(x)}{f(x)+f(1-x)}\,dx\stackrel{y=1-x}{=}\int_... |
H: Solving a system of equations with matrices
I have
\begin{align}
x_1 + 2x_2 & = 3 \\[0.5ex]
4x_1 + 5x_2 & = 6 \\[0.5ex]
9x_1 + 12x_2 & = 14
\end{align}
I get the reduced row echelon form of
$$\left[\begin{array}{cc|c}
1 & 0 & 0 \\[0.55ex]
0 & 1 & 0 \\[0.55ex]
0 & 0 & 1
\end{array... |
H: how to find center/radius of a sphere
Say you have an irregular tetrahedron, but you know the (x,y,z) coordinates of the four vertices; is there a simple formula for finding a sphere whose center exists within the tetrahedron formed by the four points and on whose surface the four points lie?
AI: Simple formula, ma... |
H: What is the CPLX button on a calculator?
On my cheap "dollar store" scientific calculator, it has a 2nd function button named "CPLX". When you press it, the calculator displays some text similar to the "DEG, RAD, GRAD" that says "CPLX". When in this mode, you can't add, subtract, multiply, or divide. The equals bu... |
H: Taking power of a simple function with disjoint domains
I have a question in mind:
Suppose I have a nonnegative simple function
$$\sum_{i=1}^{N}{a_i}{\chi_{E_i}}$$ where the $E_i$s are pairwise disjoint.
Then consider $$(\sum_{i=1}^{N}{a_i}{\chi_{E_i}})^p$$ where $1<p<\infty$.
Is it true that $$(\sum_{i=1}^{N}{a_i... |
H: Definition: finite type vs finitely generated
The mathematical term "finite type" appears more and more in the modern articles nowadays. But it is still hard to be found in the standard textbooks. I learned the definition of it from Stacks Project http://stacks.math.columbia.edu/tag/00F2 , it is defining on the rin... |
H: How can i solve this probability problem fast?
We have a box and we have on it 6 balls with numbers from 0 to 5. We push out 3 balls in the way that after pushing out a ball we turn it back again in the box. What is the probability that the sum of the numbers will be equal to 5. Thanks in advance.
AI: $x_1+x_2+x_3=... |
H: Prove that image and kernel of a matrix are invariant subspaces
A subspace $\mathcal{S}$ of $\mathbb{R}^n$ is called invariant for an $n \times n$ matrix $A$ if $Av$ lies in $\mathcal{S}$ whenever $v \in \mathcal{S}$.
The image and kernel of a matrix are invariant under the matrix.
If $A$ and $B$ are commuting mat... |
H: Prove that $\text{argmin}_x f(x) \leq \text{argmin}_x (f(x) - mx)$ for any $f(x)$ and $m\geq0$
How to prove that $\text{argmin}_x f(x) \leq \text{argmin}_x (f(x) - mx)$ for any function $f(x)$ and $m\geq0$?
AI: Hint: Let $y = \mathrm{argmin}_x f(x)$ and $z = \mathrm{argmin}_x (f(x) - mx)$. Write inequalities for $f... |
H: Expressing the negation of $[\neg(p\land\neg q)]\land\neg r$ without $\neg,\land,\lor$
Negate $[\neg(p\land\neg q)]\land\neg r$ and replace the resulting formula by an equivalent which does not involve $\neg, \land$ or $\lor$.
Can someone tell me how to get through this question? Help!
I start with $$\neg[[\neg(... |
H: Rate of change: area and perimeter
The side of rectangle $x = 20m$ increases at the rate of $5m/s$, the other side $y=30m$ decreases at $4m/sec$. What is the rate of change o the perimeter and area of the retangle?
If we put this rectangle in the cartesian plan (starting on the origin) , we can calculate it's perim... |
H: Morphism from a proper irreducible scheme into an affine scheme of finite type
Let $K$ be a field and $X$ be a proper irreducible $K$-scheme. Show that the image of any $K$-morphism $X \rightarrow Y$ into an affine $K$-scheme $Y$ of finite type consists of a single point.
I came across this exercise in my reading,... |
H: Find $\int_0^{2\pi}\frac{1-\frac{1}{4}\cos\theta}{1+\frac{1}{16}\cos^2\theta}d\theta$
How to evaluate the following integral?
$$\int_0^{2\pi}\frac{1-\frac{1}{4}\cos\theta}{1+\frac{1}{16}\cos^2\theta}d\theta$$
This is an exercise in complex analysis. It looks like a holomorphic function $f(z)$ that we integrate alon... |
H: Length of vector
How do you find the length of this vector?
$(5 e^{20}) \mathbf{ \hat i}-e^{-4}\mathbf{ \hat j}+\mathbf{\hat k}$
Every time I try to do this, it turns out very ugly. I need to find the length to solve for the unit tangent vector of $(e^5\mathrm t)\mathbf{ \hat i}+(e^{-1}\mathrm t)\mathbf{ \hat j}+t\... |
H: Does $d(x_{n+1},x_n)
Let $(X,d)$ be a complete metric space, $(x_n)_{n\in\mathbb{N}}\subset X$ such that $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ for all $n\in\mathbb{N}$. Since I cannot construct such sequence which is not convergent, I wonder if every such sequence is convergent? I first thought of defining a sequence $y_... |
H: Geometry: angle relative to x-axis for a tangent of a circle (see picture)
Excuse my poor description in the title, I think a picture is needed to explain my question:
Theta is the angle to the x-axis.
So my question is: given the radius of the circle, theta, and beta, what is theta'?
The problem arises when simul... |
H: Understanding Limit Points
I am doing some analysis and trying to understand the idea of a limit point. This is the way I am thinking of it. I would just like someone to confirm if I have the right idea, or to correct me if I am mistaken. Thanks.
The way I'm thinking of a limit point of $p$:
If there is a $q\neq p$... |
H: Ideals in a Noetherian ring
Let $R$ be a ring, let $\mathfrak{i}$ be an ideal of $R$, let $\{x_{\alpha}\}_{\alpha\in\mathcal{A}}$ be a set of generators for $\mathfrak{i}$, suppose $\mathcal{A}$ has infinitely many elements.
Now, assume $R$ Noetherian. Can I extract from $\{x_{\alpha}\}$ a finite set of generators ... |
H: Prove that $A\cap (B\setminus C)=(A \cap B)\setminus(A \cap C)$.
Problem: Prove that $A\cap (B\setminus C)=(A \cap B)\setminus(A \cap C)$.
I've tried it on my own:
\begin{align}
x&\in A\cap (B\setminus C) \\
&\Leftrightarrow (x\in A) \wedge (x\in B\setminus C) \\
&\Leftrightarrow (x\in A) \wedge (x\in B \wedge x\... |
H: What is the purpose of sets? Why do we use them?
All is in the title. Why sets? Why do we need them and where are they important?
AI: The purpose of sets is to house a collection of related objects. They are important everywhere in mathematics because every field of mathematics uses or refers to sets in some way. ... |
H: simplify the summation over i
It's a question on the 1000 Exercise in probability.
Let $X_1, X_2, X_3$ be independent random variables talking values on positive integers,
and having mass function given by $P(X_i=x)=(1-p_i)p_i^{x-1}$, for x =1,2,3,..., i=1,2,3
Show:
$$P(X_1<X_2<X_3)=\frac{(1-p_1)(1-p_2) p_2 p_3^2... |
H: What's the integral of a constant?
If the derivative of a constant is $0$ then what is the integral of a constant?
What is the integral of $0$?
AI: The integral of a constant $C$ with respect to $x$ is $Cx + A$, $A$ constant. Applying this rule to the constant function $y(x) = 0$, $\int {0}dx = 0+A = A$. |
H: Proving mutual orthogonality of vectors
Let three vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ in three-space be such that:
$$ a_ia_j + b_ib_j + c_ic_j = \delta_{ij} $$
where a vector name with subscript represents a component of the vector (the subscript can take the value 1,2 or 3) and $ \delta_{ij} $ is ... |
H: An interesting (unknown) property of prime numbers.
I don't know if this is the right place to ask this question. Please excuse my ignorance if it is not.
I like to play with integers. I have been doing this since my childhood. I spend a lot of time looking up new integer sequences on OEIS. Last week I stumbled upo... |
H: Prove $T_n(x)$ of Chebyshev Polynomial given the recurrence relation
Using the recursion formula for Chebyshev polynomials, show that $T_n(x)$ can be written as
$$T_n(x)=2^{n-1}(x-x_1)(x-x_2)...(x-x_n)$$
where $x_i$ are the $n$ roots of $T_n$
The recurrence relation: $T_0(x)=1$,$T_1(x)=x$, and $T_{n+1}(x)=2xT_n(x... |
H: Are these bases for a topology?
I have the following topology :
$$\tau= \Bigl\{U\subseteq \mathbb{R}^2: (\forall(a,b) \in U) (\exists \epsilon >0) \bigl([a,a+\epsilon] \times [b-\epsilon, b+\epsilon]\subseteq U\bigr)\Bigr\}$$
Are these a basis for the previous topology:
$\beta_1= \{[a,a+\epsilon] \times [b-\epsilo... |
H: Solving a system of equations with 3 unknows
X/Y = 5
X/Z = 2.5
Y/Z = 0.5
I don't want to check if this system is correct, but rather get actual values of X, Y, Z which could actually fit into this.
From my head I can think of examples like x= 5, y = 1, z = 2 and 50, 10, 20 would work as well here.
But what would... |
H: Laurent series of $z^{-3}$ at $z_0 = i$. Is there a way to do this by hand or is the question just evil?
I have to find the two Laurent series expansions of $\frac{1}{z^3}$ about $i$. The only approach I can think of is to do:
$$\frac{1}{z^3} = \frac{1}{(z-i)^3} \left( \frac{z-i}{z} \right) ^3 = \frac{1}{(z-i)^3} \... |
H: Why does the google calculator give $\tan 90^{\circ} = 1.6331779e^{+16}$?
I typed in $\tan 90^{\circ}$ in Google and it gave $1.6331779\mathrm{E}16$. How did it come to this answer? Limits? Some magic?
AI: The closest IEEE-754 double value to $\pi/2$ is $1.5707963267948965579989817342720925807952880859375$. The cos... |
H: Cross product of vector functions
I was trying to make sense of a problem when I stumbled upon this on yahoo answers. I was just wondering if it was correct. If it is, can you please maybe explain why?
${\bf r}'(t) = \langle -5 \cos t, -5 \cos t, -4 \sin t \rangle$
${\bf r}''(t) = \langle 5 \sin t, 5 \sin t, -4 \co... |
H: Probability of a pair of vertices being selected in a random subset of k vertices
Let $G$ be a graph of $n$ vertices. I select $k$ vertices uniformly at random. What is the probability that a specific pair of vertices are in the subset?
AI: There are $\binom{n}{k}$ equally likely ways to choose $k$ vertices.
Ther... |
H: Is this space a banach space?
Hi I want to find out whether $l^1$ with the norm $||x||:=sup_n |\sum_{i=1}^{n} x_i|$ is a Banach space. In case that you think that it is a Banach space, just say: It's a Banach space(and then I will first try to prove this), but in case that it is not, I would be grateful to you, if ... |
H: When does $xxyy = xyxy$ not imply $xy = yx$ in a ring?
This is a follow-up to a graded question on some homework. In attempting to prove that a ring R such that $x^2 = x$ for all $x \in R$ is commutative, I tried using the fact that $xy = x^2y^2 = (xy)^2$ for all $x, y\in R$, so $xxyy = xyxy$. I thought this was en... |
H: How should subscripts be used when evaluating midpoint, trapezoid, and simpsons rule?
I conceptually understand how these rules work, and I can visually see what I need to calculate, but the subscripts sometimes confuse me. Some start at $x_{0}$ and others start at $x_{1}$
If I'm looking at evaluating $\int f(x)dx... |
H: What is probability of independent events?
I have N similar hard drives. There is a probability f% that any one of them will crash. What is the probability that at least two of them will crash?
AI: With $p=\frac f{100}$, the probability of no crash at all is $(1-p)^N$ and the probability of exactly one crash is $Np... |
H: The roots of $z^4+z^3+1=0$ are in $\frac{3}{4}<|z|< \frac{3}{2}$.
How can we show that all the roots of the complex polynomial
$$p(z)=z^4+z^3+1$$
lie in $\frac{3}{4}<|z|< \frac{3}{2}$? This is an exercise in complex analysis.
AI: Broad hints: if $|z|\leq \frac34$, what can you say about $|z^4+z^3|$? (Try using the... |
H: Finding two different matrices that have the same product with another one? (AB=AC)
Say you have a matrix, $A$, and you want to find two matrices $B, C$ for which $AB=AC$, but being $B\neq C$. I know this is quite possible due to matrices' nature, but I can't quite find a method to find them except for just random ... |
H: A Better Way to Solve this Factorial Problem?
I had a problem that asked me to find which of the following is larger:
${2013 \choose 500}$ or ${2013 \choose 1500}$
Beneath is my proof. I think it is correct (though your verification and suggestions would nonetheless be appreciated). I haven't worked with factoria... |
H: A finite sum with cosines
I'm not able to compute the following sum :
$$\sum_{k=0}^n\frac{\cos(kx)}{(\cos(x))^k}$$
AI: You have $\cos(kx) = \Re(e^{ikx})$. Can you use a geometric series argument to do the calculation? |
H: $\mod 4$ properties of Fermat number
Let $k\in\Bbb N$.
Let $2^{2^k}+1$ be a composite Fermat number.
Let $p$ be a prime factor of $2^{2^k}+1$.
Then is $p\mod 4\equiv 1$?
AI: If $p$ is a prime divisor of $F_k = 2^{2^k}+1$, then we have
$$2^{2^k} \equiv -1 \pmod{p},$$
hence $2^{2^{k+1}} \equiv 1 \pmod{p}$ and the ord... |
H: What is the underlying structure that makes this analogy so good?
In "Linear Algebra Done Right", the author draws (in my opinion) a fantastic parallel between $\mathbb{C}$ and $\mathcal{L}(V)$ (where $V$ is an $\mathbb{F}$-inner product space). In this analogy, he establishes:
A complex number $z$ corresponds to... |
H: Do the algebraic properties of the exponential and log functions specify them uniquely in probability theory?
I come from a physics background and in classical mechanics, we construct a Hamiltonian function whose partial derivatives generates a vector field, two independent systems are assigned a total Hamiltonian ... |
H: If $\lfloor a \rfloor \le b$, what is $a$ less than?
I know that $\lfloor a \rfloor \le b$. Can I then conclude that $a \le b + 1$?
Is that the most precise way to describe a?
AI: $\lfloor a\rfloor\le b$ if and only if $a<b+1$. Note the strict inequality: if $a$ were equal to $b+1$, its floor would be $b+1$, not $b... |
H: What would be the algebraic representation to this verbal function?
You are exploring a hidden cave when suddenly, your flashlight goes out. Luckily, you have a 10-inch wax candle with you... Whew! You light the candle, but you must get out of the cave quickly, because every two minutes, one inch of the wax candle ... |
H: Open immersion from a proper scheme to a separated, irreducible scheme.
Fix a scheme $S$ and let $X$ and $Y$ be $S$-schemes. Assume that $X$ is proper over $S$ and $Y$ is separated over $S$. Let $f: X \rightarrow Y$ be an open immersion of $S$-schemes. If $Y$ is connected, show that $f$ is an isomorphism.
Here's an... |
H: Functions and Mapping question?
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying the
condition
$$2f(x) = f(x + y) + f(x + 2y)$$
for all real numbers $x$ and all non-negative real numbers $y$.
I just don't know where to start.
AI: HINT: If you set $x=0$, the condition requires that $$2f(0)=f(y)+... |
H: How to invert number of days since 1 JAN 2000
Given the day of a year $d_y$ and the year $y$, it is straightforward to calculate the number of days since 1 JAN 2000:
$D = d_y-1+(y-2000)*365+floor((y-2000)/4)-floor((y-2000)/100)+floor((y-2000)/400) $
But now, given $D$, I would like to invert and calculate $d_y$ and... |
H: Finding the equation of the tangent line at point P
Find an equation of the tangent line, at the point with coordinates (2,1), to the curve described by $2x^2y+xy^3-3x=4$
I'm not too sure how to solve for this question as I can't find anything in my textbook about it. I realize I need to find the derivative of the ... |
H: Probability that one out of four players gets all four aces
What is the probability that exactly one player gets all four aces if a randomly shuffled deck is dealt to four players?
My attempted working: ${4\choose1}\cdot\frac{1}{4!}\cdot\frac{1}{52\cdot51\cdot50\cdot49}\cdot{48\choose 9, 13, 13, 13}/{52\choose 13,... |
H: An equicontinous headache (problem)
There's this family of functions:
$$\mathcal H=\{ f_k\in\mathcal C^0([-1,1],\Bbb R) : f_k (x)=\begin{cases} -1 \;\text{if $x\in[-1,-\frac1k]$} \\ kx \;\text{if $x\in[-\frac1k,\frac1k]$} \\ 1 \;\text{if $x\in[\frac1k,1]$}\end{cases} \text{with $k\in\Bbb N$} \}$$
We have to show t... |
H: Prove: Dividing an odd number by 2 always produces a remainder of 1
How would I go about proving that for all n belonging to the natural numbers, if any given odd number n is divided by 2, then the remainder is at least 1?
I got a hint: Try to reduce the number of n, but I have no idea how that would help.
I was ... |
H: Do I need different constant names for different levels of integration?
I'm just getting my feet wet in integration, so pardon me if I misuse a term.
Let's take the anti-triple-derivative (I'm not sure if that's what it's actually called) of $8x$.
$y'''=8x$
$y''=4x^2+c$
$y'=\frac43x^3+cx+d$
$y=\frac13x^4+cx^2+dx+... |
H: How to show that a sequence is positive, monotonically decreasing and converges to 0
I have the following sequence
$$
y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots
$$
So first question is, how to show that it's always positive?
Second is, how to show that it's monotonically decreasing.
I think I hav... |
H: Base 10 notation
Why are the ten digits in base 10 noted $0, 1, 2 ,3 ,4 ,5 ,6 ,7 ,8 ,9$ ? I heard it has to do with angles, but since I can't see how $9$ has $9$ angles, this may not be the historical reason ?
AI: It has nothing to do with angles. The digits we have now are alterations of the Arabic versions, which... |
H: How are these definitions of Frechet derivatives related to each other?
First def. for f exists the fréchet-derivative if there is a continuous operator $T \in L(X,Y)$, such that $\lim_{h \rightarrow 0} \frac{f(x_0 + hv) - f(x_0)}{h} = Tv$ converges uniformly in $v$, vs.
Second def. $f(x_0+h)=f(x_0)+ Th + o(h)$.
Ho... |
H: Proof by Induction for a recursive sequence and a formula
So I have a homework assignment that has brought me great strain over the past 2 days. No video or online example have been able to help me with this issue either and I don't know where to turn.
I’m given
$a_0=0$
$a_n=2a_{n-1}+1$
After writing the first 6 t... |
H: Proving there are at least $N$ surjective functions from $A$ to $B$
Let $A = \{1,2,\ldots,m\}$; $B = \{1,2,\ldots,n\}$.
I have to prove that there are at least $\frac{m!}{(m-n+1)!}$ surjective functions from $A$ to $B$.
I've given it some thought, but I don't know how to work out the proof. I've looked at some simi... |
H: If $V$ is finite-dimensional and exist $\beta$ basis of $V$ such that $T(\beta)$ is a basis for $W$, then $T$ is a isomorphism?
Let $V$ and $W$ vector space over $F$ and $T : V \rightarrow W$ lineal.
The statement is false, but I can't find a counterexample.
AI: Consider $V=\Bbb{R}^2$ and $W=\Bbb{R}^1$, and let $\p... |
H: Sample size from population?
This is probably very rudimentary maths, but given a strict population size ($N = 20$ for example), is the sample size any number $<N$? For use in calculation of confidence intervals using a population size, all of the formulas use $n$ and not $N$, meaning I need a sample size rather th... |
H: why is one point set in a first countable $T_1$ space a $G_\delta$?
why is one point set in a first countable $T_1$ space a $G_\delta$ set?
I thought, in first countable space, there may be a point $x$ such that a local basis at x does not contain one point set ${x}$..
Thank you for attention in advance.
AI: Let $X... |
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