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H: Group structure from involutions, exercise devised by Richard Brauer.
The following is an idea originally communicated by Richard Brauer. I'm having difficulty following some of the combinatorial elements.
Let $G$ be a finite group containing exactly two conjugacy classes of involutions. Let $u_1$ and $u_2$ be non... |
H: Recurrence relation telescoping
Hi there I am trying to solve the following recurrence relation using telescoping. How would I go about doing it?
$$T(n) = \frac 2n \Big(T(0) + T(1) + \ldots+ T(n-1)\Big) + 5n$$
Assuming $n\ge 1$
AI: Rearrange $$T(n) = \frac 2n \Big(T(0) + T(1) + \ldots+ T(n-1)\Big) + 5n\tag{1}$$ to ... |
H: Cofinal subset of ordinals
Let $\alpha$ be an ordinal and suppose this ordinal has a cofinal subset.
Let the cofinality of $\alpha$ be $\beta$.
Thus $\beta$ is isomorphic with $B$, cofinal subset of $\alpha$.
Then $\beta$ is a cofinal subset of itself.
Now, let the cofinality of $\beta$ be $\gamma$.
Here, how to pr... |
H: Journals of math history?
In a related question to this one, in what journals do math historians publish their article in? Brian M. Scott provided a link to Judy Grabiner's, who is a math historian, home page and it seems that she publishes in general mathematical journals for the most part, such as the American Ma... |
H: combinations and probablity
A committee of 5 members is selected randomly from 10 parents, 16 students and 4 teachers. how can one find the probability that a teacher will be the chairman if the first person selected is to be made chairman.
AI: For a teacher to be Chair, we need to select a teacher on the first pi... |
H: Where is $f(x)={(x^2+2x-48)}/{x^2}$ increasing? decreasing?
The question is to find where the graph is increasing or decreasing.
The original function is $f(x)={(x^2+2x-48)}/{x^2}$
I know I need to find the prime of this function and I think it is this after using the quotient rule:
$2(-x^2-x+49)/x^3$
Finally, in ... |
H: Clarification on bounded convergence theorem.
For the proof of Bounded Convergence Theorem, I see how to get most all the information, but I don't see exactly why $f$ is measurable. I assume I am missing something completely obvious. Here is the theorem as I understand it so far.
Theorem: Let $\{ f_n \}$ be a seque... |
H: Angle of a triangle inscribed in a square
Say we have a square $ABCD$. Put points $E$ and $F$ on sides $AB$ and $BC$ respectively, so that $BE = BF$. Let $BN$ be the altitude in triangle $BCE$. What is $\angle DNF$?
I'm inclined to say that it's a right angle because that's what it looks like from what I've drawn,... |
H: probability of a horse winning a race.
Lets suppose ten horses are participating in a race and each horse has equal chance
of winning the race. I am required to find the following:
(a) the probability that horse A wins the race followed by horse B.
(b) the probability that horse C becomes either first or second ... |
H: Multiple roots of a polynomial over a field of characteristic $p$
I have to show for what value of the prime $p$ does the polynomial $ x ^4 + x + 6$
have a root of multiplicity $>1$ over the field of characteristic $p$.
$ p=2, 3, 5, 7 $
Please help.
For $F$ a field of characteristic $3$, $f(x)= x^4 + x = x(x^3+... |
H: a sequence question, 4 options
$x_n$, $y_n$ be two sequence satisfying $x_n\le y_n\le x_{n+2}$, then which are correct below?
$y_n$ is increasing
$x_n$ and $y_n$ converge together.
$x_n$ decreasing
$y_n$ bounded.
Well, I can see $x_1\le y_1\le x_3$, $x_2\le y_2\le x_4\dots$ by putting $n=1,2..$ but how to compare... |
H: are $f$ and $g$ bounded function?
$f$ and $g$ are entire function such that $|f^2+g^2|=1$ Then which of the following are correct?
$f$ and $g$ are constant.
$f$ and $g$ are bounded.
$f$ and $g$ have no zeroes on unit circle.
$ff'+gg'=0$
Well What I do is let $h(z)= f^2(z) + g^2(z)$ then clearly $h(z)$ is bounded ... |
H: Solve an absolute value equation simultaneously
My question is :
Solve simultaneously
$$\left\{\begin{align*}&|x-1|-|y-2|=1\\&y = 3-|x-1|\end{align*}\right.$$
What I did :
$y=3 - |x-1|$ is given.
Thus $y = 3-(x-1)$ or $y = 3-\left(-(x-1)\right),$ and so $$y = 4-x\qquad\mbox{ or } \qquad y = 2+x.$$
If $y = 2+x... |
H: What is the smallest integer which is $1\mod8$, $2\mod25$ and $7\mod11$?
Like the question says, what is the smallest integer which is $1\mod8$, $2\mod25$ and $7\mod11$?
I've worked out a number which is $1\mod8$ and $2\mod25$ by using that $$25 - 3 \times 8 = 1$$ so the number is $25 - 2\times 3 \times 8 = -23$ wh... |
H: Subring of polynomials
Let $k$ be a field and $A=k[X^3,X^5] \subseteq k[X]$.
Prove that:
a. $A$ is a Noetherian domain.
b. $A$ is not integrally closed.
c. $dim(A)=?$ (the Krull dimension).
I suppose that the first follows from $A$ being a subring of $k[X]$, but I don't know about the rest.
Thank you in advance.
AI... |
H: Laurent Series $\exp(1/z)/(1-z)$
I need some help finding the Laurent expansion and residue of
$$\dfrac{\exp \left(\frac1z \right)}{(1-z)}$$
So far I've done
$$\sum_{j=0}^\infty \frac{z^{-j}}{j!} \sum_{k=0}^\infty z^k = \sum_{j=0}^\infty \sum_{k=0}^\infty \frac{z^{k-j}}{j!}$$
but don't know where to go from here. ... |
H: Remembering Taylor series
Could anyone suggest a good way of memorizing Taylor series for common functions?
I have tried to remember them but never seem to be able to commit them to permanent memory.
AI: As have already been said there is no golden trick here. However it is sometimes useful to know some ways to man... |
H: How many elements of order $10$ are there in the symmetric group $S_7$?
I have a feeling you use Sylow's Theorems but I'm not sure where to start, any hints?
AI: If you write a permutation in disjoint cycle notation:
$(\alpha_1 \alpha_2 ... \alpha_{n_1})(\beta_1 ... \beta_{n_2})...$
then the order of the permutatio... |
H: Proof: $\alpha_1>\alpha_2$ in a triangle
I have to proof in a triangle, that $\alpha_1>\alpha_2$ holds. The inner point P (from where I draw the smaller triangle) is set randomly.
Does anyone have a suggestion where I have to start?
Greetings
AI: Given that the origin triangle is $\triangle ABC$, and the inner poi... |
H: Why can neither GeoGebra, nor MathWay solve this simple math problem?
$\frac{e^{0.75}}{-0.5^e+10000}$
This doesn't work either:
$\frac{2.72^{0.75}}{-0.5^{2.72}+10000}$
I can even solve it with a calculator.
AI: First of all, $(-0.5)^e$ is not the same as $-0.5^e$.
Secondly, $x^y$ is not well defined (at least not a... |
H: The limit of a sum with elements from 2 sequences
Let's consider the sequence of real numbers $(a_{n}),n\geq1,a_{1}>0$ that satisfies the following recurrence:
$$\frac{n(n+2)}{(n+1)}a_{n}-\frac{n^2-1}{n}a_{n+1}=n(n+1)a_{n}a_{n+1}$$
I'm supposed to calculate:
$$\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n} a_{i}b_{i... |
H: Relationship between smooth function and ordinary derivative.
The function $F(x)$ is said to be smooth at the point $x$, if
$$\lim_{h\to 0} \frac{F(x+h)+F(x-h)-2F(x)}{h}=0.$$
My question is that if function $F$ is differentiable at some point then how can we show that function $F$ is smooth?
AI: For once this is th... |
H: Running in the rain, good or bad idea?
I originally wanted to ask this on the Skeptiks site, but while having a look around I came across this site which seems to have some complicated maths, but not a reliable answer.
The conclusion it attempted to arrive at:
So here we have it - more mathematical advice to avoid... |
H: Easy questions about Linear equations
Prove that
Each field of characteristic zero contains a copy of the rational number field.
For an $n$ by $n$ matrix $A,$ if it is not invertible, then there exists an $n$ by $n$ matrix $B$ such that $AB=0$ but $B\ne0.$
For (1), I think I have to use the fact that each subfiel... |
H: Golden parallelepiped
Define a golden parallelepiped as a $d$-dimensional box with side lengths
$(1, \phi, \phi^2, \ldots, \phi^{d-1})$, where $\phi$ is the golden ratio:
The volume of the parallelepiped is $\phi^{d(d-1)/2}$, e.g., $\phi^3$ for $d=3$.
I wonder if there is a natural geometric explanation ... |
H: complex analysis multiple choice question
Suppose $\langle z_n \rangle_{n\ge0}$ is a sequence of complex numbers such that $\sum_{n=0}^\infty z_n$ converges,Given that $f$ is an entire function such that $f(z_n)=n$, then
$f\equiv 0$.
$f$ is unbounded.
No such $f$ exists.
$f$ has no zeroes.
AI: As Gerry pointed out... |
H: About a continuous function
I'm trying to solve this problem, but I don't have any idea. Can you help me?
Let X a compact metric space and $f:X\times\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function. Consider $m(t_0)=\max_x (f(x,t_0))$. Show that $m$ is continuous.
Thanks.
AI: Fix $t\in\Bbb R$. We just hav... |
H: Maximize volume of a box in first octant
So the question is
Find the volume of the largest rectangular box with 3 faces on the co-ordinate planes and one vertex in the plane $x+2y+3z=6$.
When I started typing this question, I didn't know what to do but as I got into it I tried some things and now actually have ... |
H: Perimeter of a triangle
A question states:
The length of each side of a certain triangle is an even number.If no two
sides have the same length what is the smallest perimeter the triangle
could have ?
According to the author the answer is 18 but i think the answer is 12. (Scalene triangle with 2+4+6 a... |
H: "Lock" a direction vector by preventing motion along a second vector
I have 2 unit vectors, o and v.
o is the orientation of a cylinder and v is a direction I wish to move inside this cylinder.
However, I want to allow v to only move perpendicular to the cylinder, so it cannot move along the axis defined as moving ... |
H: Show that $\left( \lnot a \implies a \right) \stackrel{?}{\implies} a$
Consider the following axiom:
$$\lnot a \implies a$$
Intuitively, this seems like a contradiction. But all implications hold if the LHS is false. Does this mean that:
$$a$$
is a valid conclusion? Or is there a contradiction in the axiom?
AI: Thi... |
H: Max size of the problem that can be solved in 2 hours if the algorithm takes n^2 microseconds
I have been going through my textbook trying to solve this problem but I can't seem to completely understand. It seems so simple yet I can't figure it out. How do I go about solving this?
AI: 2 hours is equal to $2\times 3... |
H: Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?
Can we even find examples of infinity in nature?
AI: If by nature mean physical universe, note that there is no proof that an infinite set exists. While some of us would like to believe that universe is infinite, this cannot be proved. You also have to rememb... |
H: Numerical solution of the Laplace equation on circular domain
I was solving Laplace equation in MATLAB numerically. However I have problems when the domain is not rectangular.
The equation is as follows:
$$
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0
$$
domain is circular
$$
x^2 + y^2 < ... |
H: Upper bound for complex polynomial
I have a polynomial $p$ of degree $n$ satisfying $\lvert p(z) \lvert \leq c\ \ \forall z\in\partial B_1(0)$. (Isn't this true for any polynomial?) Show $\lvert p(z)\lvert \leq c \lvert z\lvert^n \ \ \forall z\in \mathbb{C}\backslash B_1(0)$.
The obvious attempt would be $|p(z)|=|p... |
H: Strategies to find the set of functions $f:\mathbb R\to\mathbb R$ satisfing the functional equation: $f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$
My question is as follows: What methods can be used to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ satisfying a certain functional equation. An example of a case ... |
H: #(cardinality of irreps Lie algebras) > #(irreps of ssociative algebras). Proof?
I know that irreducible representations of associative $*$-algebras are fairly restricted: any $*$-algebra $A$ is isomorphic to a finite sum of simple algebras
$A\cong\oplus_{i=0}^{N}M_{n_i}(\mathbb{C})$
What's the cardinality of the i... |
H: Simple clarification - deduction using big-O notation
A set of lecture notes I'm reading on Halasz's theorem makes the following statement in a proof, which I can't quite follow - I was hoping someone might be able to clear up what I'm missing:
Therefore, $Re(\sum_p p^{-s} = \log \log x + O_M(1)$, where $O_M$ deno... |
H: Why would the author ask if I used the Associative Law to prove + is not equiv. to *?
I just started reading An Introduction to Mathematical Analysis by H.S. Bear and problem 1 goes as follows:
Problem 1: Show that + and * are necessarily different operations. That is, for any system (F, +, *) satisfying Axioms I,... |
H: Reflections generating isometry group
I was reading an article and it states that every isometry of the upper half plane model of the hyperbolic plane is a composition of reflections in hyperbolic lines, but does not seem to explain why this is true. Could anyone offer any insight? Thanks.
AI: An isometry $\phi:M\t... |
H: "Any one of" or "either"?
First there was this question:
Q. Find the number of numbers from 100 to 400 which are divisible by
either 2, 3, 5, and 7
And the obvious solution was to find
numbers divisible only by 2 i.e 100 102 104 ... 400
numbers divisible only by 3 Now we are left with only odd numbers i.e 101 1... |
H: Functional Analysis, Why this statement true?
Why this statement true?
If $f \in C^0([0,1], W^{2,2}(K)) $ then $ f \in C^0 ([0,1] \times K)$.
$ K \subset R^n $ , W : Sobolev Space.
AI: I don't think this is true. The Sobolev embedding theorem suggests this should only work for small $n$; in this case I think it sh... |
H: Prove: If $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.
Prove: If $X$ is a locally convex space, $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.
What I know:
If $L$ is a finite dimensional subspace, then $L$ is closed.
AI: If $M$ is not closed then this is not true in g... |
H: Combinatorics: Things always included together
Find the number of permutations of $n$ different things taken $r$ at a time so that two particular things are always included and are together?
Including two things initially, i have $(n-2)$ things from which I can choose $(r-2)$ things. Hence $\,^{(n-2)}C_{(r-2)}$ den... |
H: How to draw an ellipse using its center and two points on its perimeter
Suppose I have an ellipse centered at the origin, and two points on its perimeter which are not antipodal to one another (i.e. not negative to each other as vectors in $\mathbb R^2$). How can I draw the whole ellipse?
AI: The Wikipedia animatio... |
H: Counting ways of sitting in adjacent seats
In how many ways can $m$ people entering a theatre be seated in two rows, each containing $n$ seats with the condition that no two sit in adjacent seats in the first row?
AI: First calculate how many ways $j$ seats can be occupied from $n$: this is ${n \choose j}$ if $n\ge... |
H: Is the dual graph simple?
According to the book Topological Graph Theory by Gross and Tucker, given a cellular embedding of a graph on a surface (by 'surface' I mean here a sphere with $n\geq 0$ handles), one can define a dual multigraph by treating the faces of the original graph embedding as vertices and adding a... |
H: Eigenvalues of block matrices
Let $K$ be a field of characteristic 0, and consider the following block matrix
$$M=\left(\begin{array}{cc} A & B\\ -B&D\end{array}\right),$$
where each block is an $n\times n$ matrix with coefficients in $K$. I am looking for a relation between the eigenvalues of $M$ and those of $A$ ... |
H: Subspaces of all real-valued continuous functions on $\mathbb{R}^1$
I'll go ahead and give you the problem first, and then explain my trouble with it.
Which of the following subsets are subspaces of the vector space C(-$\infty$,$\infty$) defined as follows: Let V be the set of all real-valued continuous functions d... |
H: Moment of inertia - formula derivation: Missing $\frac{1}{2}$
I'm trying to deduce the formula of the moment of inertia of an object of rotation. The general formula for the moment of inertia is declared as:
$$J=m*r^2 =\sum{m_i * r_i^2}$$
If I replace $m_i$ of the $\sum{m_i * r_i^2}$ with $\int dm$ (where dm are t... |
H: Impossible to prove vs neither true nor false
First off I am not a logician, so I probably won't use the correct terms. Sorry !
I have heard, like most mathematicians, about questions like the continuum hypothesis, or the independance of the axiom of choice from ZF. These statements (continuum hypothesis or axiom o... |
H: Is $f(t)=t^\alpha$ for $\alpha\in(0,1)$ a sub-additive function?
Possible Duplicate:
Does $|x|^p$ with $0<p<1$ satisfy the triangular inequality on $\mathbb{R}$?
Is the function
$$f(t)= t^{\alpha},\quad \alpha\in (0,1)$$
a subadditive function?
My teacher said categorically that this is true. But I'm not ... |
H: Prove $\int 1-\prod_{i=1}^n (1- \mathbb{I}_{A_i}) d \mu= \mu ( \bigcup_{i=1}^n A_i )$
Let $(\Omega, \mathcal{A}, \mu)$ be a measurable space. $A_1, A_2,...,A_n \in \mathcal{A}$ are sets with finite measure.
I have to prove $\int 1-\prod_{i=1}^n (1- \mathbb{I}_{A_i}) d \mu= \mu ( \bigcup_{i=1}^n A_i )$. But I am on... |
H: An equality in a group
Let $G$ be a non-abelian group of automorphisms, where composition of automorphisms is the group operator. We have the following notation
$$g = g_1 \quad x_0 = g_1^{-1}g_2 \quad x_i = g^{-i}x_0g^i$$
where $g_1,g_2 \in G$. Now I want to show that
$$g_1^{-j}g_2^j=x_{j-1} x_{j-2}\cdots x_0$$
I h... |
H: calculating (nPr/q!) % m
for calculating the value of choosing r items from n items where q are of same kind, and we should take %m , i used the following relation
(nPr/q!) %m where m is prime
For calculating this
i calculated n!
n!%m
then, i calculated (n-r)! and multiplied it with q!, i.e temp =
(n-r)!*q!;
... |
H: How to show that $x \mapsto d(x,f(x))$ is continuous for a continuous $f$?
I'm busy with a topology course, but the following question has me somewhat stumped. The entire question is, let $(X,d)$ be a metric space and $f:X \rightarrow X$ be continuous. Show that $X \rightarrow R$, $x \mapsto d(x,f(x))$ is continuou... |
H: Polynomial irreducibility criterion
Given $f \in \mathbb{Z}[X]$ and $2\deg(f)+1$ distinct $a_i \in \mathbb{Z}$ such that $f(a_i)$ are prime numbers. Then $f$ is irreducible.
I'm trying to prove that and I am stuck. Any hints for a good starting point?
AI: Hint: Suppose $f=gh$ were a factorization where $g,h\in\m... |
H: Antiderivative $1/z$ on $\mathbb C$
Let $\Omega \subseteq \mathbb{C}$ be open and $\gamma:[\alpha,\beta]\rightarrow \Omega$ be a piecewise continuously differentiable and closed path.
Why does $z^{-1}$ have no antiderivative on $\mathbb{C}\setminus0$?
Why is $\int\limits_\gamma z^{-1} dz = 2\pi i \cdot \text{ind}_\... |
H: Need help with finding matrix inverse in $\mathbb{Z}/26\mathbb{Z}$
I am trying to learn the Hill Cipher and I am facing difficulties understanding how to find the inverse of a matrix in Modulo 26. What I've learnt so far is that I need to apply elementary row operations and apply modulo 26 after each operation unti... |
H: Are seminorms convex and a question on local base?
How can I prove that seminorms are convex?
Another question is that, when we talk about topological vector spaces, why do we emphasise neighborhoods about $0$? As far as I know, if we know the neighborhoods about $0$ then we can find all the open sets by translati... |
H: $3x^3 = 24$ quadratic equation
Completing the square I know by factoring
$$x^3 - 8 = 0\\
x-2 = 0$$
that one of the solutions is 2.
but the other solutions is $1 ± i \sqrt 3$.
Can someone explain to me how to get that?
AI: May be you know this already but it's not clear to me from your post that you have the progres... |
H: Euler's summation by parts formula
I'm beginning analytic number theory and I see this formula in Apostol's book : If $f$ has a continuous derivative $f'$ on the interval $[y,x]$, where $0 < y < x$, then
$$
\sum_{y < n \le x} f(n) = \int_y^x f(t) \, dt + \int_y^x (t-[t]) f'(t) \, dt + f(x) ([x] - x) - f(y) ([y]-y)... |
H: A function and its Fourier transform cannot both be compactly supported
I am stuck on the following problem from Stein and Shakarchi's third book.
I can't figure out how to use the hint productively. Once I know $f$ is a trigonometric polynomial, I see how to finish the problem, but I don't know how to conclude th... |
H: Finding the Derivative Of $f(x) = 7\ln(5xe^{-x})$
The original question is $f(x) = 7\ln(5xe^{-x})$
I'm not sure if I have to use the chain rule to figure out $\ln(5xe^{-x})$ because $5xe^{-x}$ is one term within ln.
My guess is that it's like this:
$$7(-(e^{-x-1})/e^{-x})$$
or just simply $-7$.
I'm specifically un... |
H: Calculating formula to store location of Lower Triangular Matrix
I am struggling with a problem from this textbook. The question is as follows:
Determine a formula h = f(i,j) to store location MATRIX[i][j] in h. Ensure to only store nonzero elements.
Then it asks how they can be stored in a single dimensional array... |
H: Why are there $3$ conjugacy classes of involutions in this centralizer?
I'm trying to work through a sketch proof attributed to Walter Feit on characterizing $S_5$.
Suppose $G$ is a finite group with exactly two conjugacy classes of involutions, with $u_1$ and $u_2$ being representatives. Suppose $C_1=C(u_1)\sime... |
H: Cauchy integral formula for convex sets
Again, I'm struggling with a proof.
Cauchy integral theorem for convex sets (Preliminary lemma)
Let $\Omega\subseteq \mathbb{C}$ open and convex, $ p\in\Omega,\ f$ : $\Omega\rightarrow \mathbb{C}$ continuous, $f\in H(\Omega\backslash \{p\})$ ($f$ analytic on $\Omega \backslas... |
H: For a reduced ring $A$, must $A[x]\setminus A$ be multiplicatively closed?
Suppose $A$ is a reduced commutative ring. Is $A[x]\setminus A$ is multiplicatively closed?
AI: Let $A$ be any reduced commutative ring with nonzero zero divisors, say $ab=0$ for some nonzero $a$ and nonzero $b$.
Since $(ax)(bx)=0$, $A[x]\se... |
H: Triangle related question
My question is:
In $\Delta ABC$, let $AE$ be the angle bisector of $\angle A$. If $\frac{1}{AE} = \frac{1}{AC} + \frac{1}{AB}$ then prove that $\angle A = 120^\circ$.
What I tried: I extended side $AB$ and took a point $M$ on it such that $AC$ is congruent to $AM$. Then I proved that $AE$ ... |
H: Power series expansion for analytic functions
Theorem
Let $\Omega\subseteq \mathbb{C}$ be open and $f\in H(\Omega)$ ($f$ analytic on $\Omega$). If $ C(z_{0},R)\subseteq\Omega$ (where $C(z_0,R)$ is the circle with origin $z_0$ and radius $R$), then we can represent $f$ on $C(z_{0},R)$ as a power series with converge... |
H: $\lim _{(x,y)\rightarrow (0,0)} \ln(\sin(x^2+y^2))=?$
I am working on this practice question for an upcoming exam.
I am not sure if I am oversimplifying this here (as the question is worth 3 marks):
In polar coordinates I have:
$\ln(\sin(r^2))$, so on any path, as $(x,y)\rightarrow(0,0)$, $r\rightarrow 0$.
For $r$... |
H: Parametric Plot for Ellipse
I have two vectors $a$ and $b$ and want to perform a parametric plot that generates an ellipse out of it. Normally you would do that for:
$$t \mapsto \cos(t) \cdot a + \sin(t) \cdot b$$
But when I look up parametric plot, I only find examples like:
$\sin(t), \cos(t)$
or
$t \cdot \sin(t)... |
H: Derivative of a function with variable range
Suppose for example that I have a function $g_y(x)$ such that
$g_y(x) = \begin{cases} y+x &\mbox{if } -x<y<1-x \\
1 & \mbox{if } y>1-x \\0 &\mbox{if } y<-x \end{cases}$
How, if possible, would I find $ \frac{dg}{dx} $?
Edit: Also, I need to have that $ x\in (0,1)$
AI: $$... |
H: Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
This question came up in a miscellaneous problem set I have been working on to refresh my memory on several topi... |
H: Simpson's rule failure on $\cos\pi x$
I think I know why but I can't represent this accurately with a graph.
I am supposed to show why Simpson's Rule is so far off for the integral $$\int_0^{20} \cos \pi x$$
I know that the answer should be zero because it repeats on that interval, 10 up and 10 down evenly.
I know... |
H: Evaluating integral similar to $\sin^{-1}(3x)$
I was just doing revision for an upcoming exam and I came across a question I do not know how to answer. It seems pretty simple, however I am having a blank moment. If anyone knows how to solve that I would really appreciate the help.
The question:
It is given that y =... |
H: The theorem on ordinary differential equations stated in the appendix of Kobayashi-Nomizu
The following theorem is stated in the appendix I of Foundation of differential geometry by Kobayashi-Nomizu.
They say the proof will be found in various text books on differential equations.
I checked several books, but could... |
H: The spectra of weighted shifts
Since weighted shifts are like the model-operators in operator theory and people have been studying them for so long, I think there should be quite a large literature on the spectra of such operators. However, after some search I hardly found anyone which gives a complete picture of w... |
H: Rudin's Principles of Analysis Theorem 1.11
I'm having a problem with this theorem. What if set $B$ is all $x$ such that $\sqrt{2} < x \le 2$, and $S$ is the set of all $y$ such that $\sqrt{2} < y \le 3$. $\sup L = \sqrt{2}$, which does not exist in $S$. Does this prove the theorem wrong by contradiction?
1.11 The... |
H: Lemma on bounding one metric by a multiple of another - clarification
My book (Principles of Topology by Fred Croom) states the following lemma: "Let $d_1$ and $d_2$ be two metrics for the set $X$ and suppose that there is a positive number $c$ such that $d_1(x,y) \le cd_2(x,y)$ for all $x,y\in X$. Then the identi... |
H: A (non-artificial) example of a ring without maximal ideals
As a brief overview of the below, I am asking for:
An example of a ring with no maximal ideals that is not a zero ring.
A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ring with no maximal ideals.
A homework question in my alg... |
H: Evaluate the partial derivatives of the following function:
The function $f: \mathbb{R} \longrightarrow \mathbb{R}$ is defined by the rule
$$f(x,y) = \begin{cases} \frac{x^5y}{x^4+y^2}, & (x,y) \neq (0,0), \\ 0, & (x,y)=(0,0). \end{cases}$$
Evaluate $f_x(0,y)$, $f_x(0,0)$, $f_y(x,0)$, and $f_y(0,0)$.
(The de... |
H: Dedekind Cut Proof
I am greatly confused with Dedekind cuts... I am trying to prove that this is a Dedekind cut:
If $D$ and $E$ are in $\mathbb{Q}$ and are Dedekind cuts, then prove that
$$D*E=(-\infty, 0] \cup \{r_1r_2\mid 0 < r_1 \in D, 0 < r_2 \in E\}$$
is a Dedekind cut as well.
My three propositions... |
H: Polar equation of a circle
A very long time ago in algebra/trig class we did polar equation of a circle where
$r = 2a\cos\theta + 2b\sin\theta$
Now I forgot how to derive this. So I tried using the standard form of a circle.
$$(x-a)^2 + (y - b)^2 = a^2 + b^2$$
$$(a\cos\theta - a)^2 + (b\sin\theta - b)^2 = a^2 + b^2... |
H: Question on some arithmetic calculations
When $6272$ is multiplied by $0.94$ the answer is $5895.68$. When it is divided by $1.06$ the answer is $\approx 5916.9811$. Why is it so?
Just as a little background, I am using the default Microsoft calculator for this calculation. I haven't pulled out Mathematica yet. Cou... |
H: How can we solve: $\sqrt{x} + \ln(x) -1 = 0$?
How could we solve $$\sqrt{x} + \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions?
AI: Both $\sqrt x$ and $\ln x$ are increasing functions of $x$, so $\sqrt x+\ln x=1$ can have at most one solution. As you... |
H: How can we solve: $\sqrt{x} - \ln(x) -1 = 0$?
How could we solve $$\sqrt{x} - \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions?
This question is inspired by my first question
How can we solve: $\sqrt{x} + \ln(x) -1 = 0$?. Here the situation is not cl... |
H: If a line bundle and its dual both have a section (on a projective variety) does this imply that the bundle is trivial?
Is there any reason that, on a projective variety X, if a line bundle L has a (non-zero) section and also its dual has a section then this implies that L is the trivial line bundle?
AI: Yes, there... |
H: lower bound for probability of no 2 balls per bin.
There are $n$ balls and $m$ bins and every ball is placed independently and uniformly at random into a bin. I'm trying to show that there exists a constant $c$ such that, if $m=c\sqrt{n}$ then with probability $\geqslant 1/2$, no two balls will collide into the sam... |
H: Question about distribution
Let $(f_k)_{1\le k\le \infty}\in L_{1}^\mathrm{loc}(\mathbb{R}^n)$ be a sequence of real valued functions such that $\operatorname{supp} f_k \subset \{|x|\le k^{-1}\}$, $$\int f_k (x)\,dx=1,k\in \{1,2,\ldots,\infty\}$$ Show that the sequence $(f_k^2)_{1\le k\le\infty}$ does not converge ... |
H: Citable Reference for Picard's Theorem in Banach Space
I was wondering if anyone knew of a legitimate citable reference where Picard's Theorem on the existence of solutions to ODEs in Banach space is proven? For some reason I can only find proofs for the case of $\mathbb{R}^n$ in any of the books I have looked at. ... |
H: Fields as a reflective subcategory of integral domains?
A subcategory $\mathbf A$ is reflective subcategory of $\mathbf B$ if for every $B\in\mathbf B$ there exists an $A_B\in\mathbf A$ and a $\mathbf B$-morphism $r_B \colon B \to A_B$ such that: for any $A\in\mathbf A$ and any $\mathbf B$-morphism $f \colon B \t... |
H: Derivative of $\sec^{-1}(\frac{x}{3})$
Derivative of $\sec^{-1}(\frac{x}{3})$
I have tried these types of problems with two different approaches and keep getting the same answer which seems to be wrong. I suspect I am doing something obvious incorrectly; however, I can't seem to figure it out.
First method:
$y = ... |
H: A problem of compactness and connectedness
Consider the subset $A$ and $B$ of $\mathbb{R}^2$ defined by
$A =\{(x, x\sin\frac{1}{x}) :x\in(0,1]\}$
$B = A\cup \{(0,0)\}$
I have to check for compactness and connectedness of $A$ and $B$.
Here is my attempt.
$A$ is bounded but not closed as 0 is the limit point of set ... |
H: Equating coefficients in a Fourier series
Suppose, for example, using Fourier series techniques to solve a differential equation leads to the following:
$a_0 + \sum_{n=1}^{\infty}a_n\sin(nx)+b_n\cos(nx)=4\sin x$
At this point, why can you equate the coefficients of $\cos(nx)$ and $\sin(nx)$ on the two sides the equ... |
H: Evaluating $f_x(0,0)$ and $f_y(0,0)$ using the limit definition of the partial derivative
Possible Duplicate:
Evaluate the partial derivatives of the following function:
The function $f:\mathbb R^2 \to \mathbb R$, defined as:
$$\left\{\begin{align*}&\frac{x^5y}{x^4+y^2}&&(x,y) \neq 0\\&0&&(x,y)=(0,0)\end{align*}... |
H: Summing the series $ \sum_{n=0}^{\infty} (-1)^n\int_0^1 \cos(nt^2)\mathrm dt$
I would like to find the sum of the series
$$ \sum u_{n}$$
where $$ u_{n}=(-1)^n\int_0^1 \cos(nt^2)\mathrm dt$$
Using the change of variable $t\rightarrow \sqrt{n}t$:
$$ u_{n}=\frac{(-1)^n}{\sqrt{n}} \int_0^{\sqrt{n}} \cos(t^2)\mathrm dt\... |
H: Integrating Reciprocals of Polynomials
I have seen integrals of the form $$\int \frac{1}{ax+b}dx$$ and $$\int \frac{1}{ax^{2}+bx+c} dx$$
But I cannot see how to integrate reciprocals of higher degree - does there exist a general solution to the integrals of reciprocals of cubics, quartics, and higher?
AI: Reciproca... |
H: A basic question related with compactness
I have to check for compactness of given subets of $\mathbb{R}^2$.
$A =\{(x, y) :xy = 1\}$
$B =\{(x, y) :x^2y^2 = 1\}$
$C =\{(x, y) :e^x = \cos y\}$
$D =\{(x, y) :\mid x\mid +\mid y \mid \leq 10^{100}\}$
The purpose of asking above question is not just to get answers. I nee... |
H: Working with Linear Operators on Vector Space
Let $\phi: V \rightarrow V$ be a linear operator on a vector space V over a field F
Prove that $V = \phi(V)\bigoplus NS (\phi )$
if and only if $\phi(V ) = \phi^2(V)$
AI: I'll use $\,\ker\phi\,$ instead of $NS$:
Suppose $\,V=\phi(V)\oplus\ker\phi\,$ and let$\,x\in\ph... |
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