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H: proving that there is no solution for $n$ in positive integers except $n = 1$ for $2^n = nk + 1$
If $k$ is a positive integer, how to prove that there is no solution for $n$ in positive integers except $n = 1$ for $2^n = nk + 1$
AI: The reason is that the only solution to the congruence $2^n\equiv 1\mod n$ is $n=1$... |
H: Getting the recurrence formula with a condition
Get the recurrence formula of
$$U_n=2(-3)^n-5n(-3)^n$$
For $$n \geq 1$$
What am I supposed to do with this condition $n\geq 1$?
AI: $U_{n}=(-3)^{n}(2-5(n))$
$U_{n+1}=(-3)^{n+1}(2-5(n+1)) = 3(-3)^n(2-5n-5) $
$U_{n+1}= -3U_{n}+5(-3)^{n+1}$ |
H: Poincare inequality question
Help me please with this question:
Is it true that for all $N>0$ exists $u \in C^{1}[0,1]$ such that $u(0)=0$ and $\frac{\int_{0}^{1}u^2(x)dx}{\int_{0}^{1}\left [ u'(x) \right ]^2dx}>N$;
what happens without assuming that $u(0)=0$?
Thanks a lot!
AI: If $u\in C^1[0,1]$ satisfies $u(0)=0$... |
H: The relation between formal logic and proof writing
I was reading vellmans how to prove it and he forms a link between formal logic and proof writing. For instance, he decomposes if p then q to not(p and not q) and similarly for other such proof writing statements. However what I don't follow is What's the motivati... |
H: Does $k$-th power of $p$ divide ${}_n\!C_r$ if the previous divides $n$?
Does $p^k$ divide ${}_n\!C_r$ for all integer r if $p^k|n$ where $0\leq r \leq n$ and $p$ is prime?
AI: The answer is no. Take $p=3$, $n=6$, $k=1$, $r=3$. Then $p^k\mid n$ , but $p^k\nmid{}_n C_r$ |
H: What is Kosambi-Cartan-Chern (KCC) theory?
I'm reading a paper that's basically about stability analysis of the Lane-Emden differential equation. The authors make use of "Kosambi-Cartan-Chern (KCC) theory". I've been trying to find out what this "theory" is about and haven't really gotten anywhere except for a an... |
H: In set theory, what does the symbol $\mathfrak d$ mean?
What's meaning of this symbol in set theory as following, which seems like $b$?
I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma?
Thanks for any help:)
AI: The symbol $\mathfrak d$ is used to denote the... |
H: Immersed curve in $\mathbb{R}^2$ and regular curve
I read that a curve $\gamma:S^1 \to \mathbb{R}^2$ is an immersion iff it's regular. So it's an immersion iff $\gamma'(t) \neq 0$ for all $t \in S^1$.
An immersed curve is one whose derivative is an injective linear map. When the domain is single-variabled, this is... |
H: Top cohomology detecting compactness
Could someone point me to a standard reference for the fact that the top cohomology $H^n(M,A)$ of an $n$-dimensional manifold $M$ is non-trivial for local coefficients $A$ if and only if the manifold is compact?
EDIT: It seems that there are some issues when $M$ is non-orientabl... |
H: Formula to this pattern? $1$, $11$, $21$, $1211$, $111221$, $\ldots$
I have this pattern:
1
11
21
1211
111221
I'm guessing it's a fibo pattern, been at it for hours now. Anyone know?
AI: It's known as the look-and-say sequence. |
H: Stronger condition lead to weaker result?
Suppose there are two theorems $A \Rightarrow B$ and $C \Rightarrow A$. Then we have $C \Rightarrow B$.
Now comparing $A \Rightarrow B$ and $C \Rightarrow B$, we know that $C \Rightarrow A$ means C is a stronger condition than A. Is it to say $A \Rightarrow B$ is stronger o... |
H: To show $X$ is a complete vector field on $M$
Well, I have solved myself the problem : every smooth vector field on a compact manifold is complete.
Now I have got this problem which I am not able to progress:
let $X$ is a vector field on $M$, suppose $\exists \epsilon >0\ni (-\epsilon,\epsilon) \subsetneq (a(m),b(m... |
H: Functional Inequality question where $\int^x_0\frac1{f'(t)}dt = \int^x_02f(t)dt $ ,$ 0 \leq x \leq 1$ and $f(0) = 0$
$$\int^x_0\frac1{f'(t)}dt = \int^x_02f(t)dt \tag{1}$$ where $0 \leq x \leq 1$ and $f(0) = 0$
I need to prove that $$f(\frac1{\sqrt{2}})> \frac1{\sqrt{2}}$$
$$f(\tan (x))> \tan(x) > x , x \in (0,\fra... |
H: Compute $\sum_0^{n-1}2^i11^{n-i-1}\bmod10^9$ when $n=13^{17}$
Given the following function $f$
$f(1)=1$
$f(n)=11\cdot f(n-1)+2^{n-1}$
I would like to compute $f(13^{17})\mod 10^9$ and ended up using the following :
$f(n)=\sum_{i=0}^{n-1}({11^{n-(i+1)}\cdot 2^i})$
though I am able to quickly compute a single term ... |
H: for $n=x^2+3y^2$ , $n=\prod p^{a(p)}$ , $a(p)$ is even for all $p \equiv 2 \pmod 3$ where $p$ is prime
I'd really like your help with the following Number Theory question:
I need to show that if I can write an integer $n=x^2+3y^2$ so in the factorization of $n$ to primes, every $p \equiv 2\pmod 3$ would be with a e... |
H: In what base does "100" equal "4 in base 10"
I was reading this answer to an amusing comic related question: https://math.stackexchange.com/a/166891/35132 and I understand that in the linked answer, the examples of how four may be expressed used base (expressed in decimal!!) is 10, 4, 3 for 4, 10, 11.
What I can't ... |
H: Isomorphism $KG \cong K[x]/(x^p-1)$
Why is $KG \cong K[x]/(x^p-1)$, for $K=\mathbb{Z}/p \mathbb{Z}$ and $G = \langle g | g^p=1 \rangle$ a ring isomorphism?
If I take $g \mapsto [x]$, I'd have an group homomorphism. And than an algebra(?) homomorphism f: $KG \rightarrow K[x]$, $a_g g \mapsto a_g[x]$, with $a_g \in K... |
H: Change of order of summation.
I feel like an idiot for asking this, so bear my stupidity.
I have the sum $\sum_{n\leq N} \sum_{p | n ; \ p \ prime} 1$, and I want to change the order of summation of these two sums I think it should be:
$$\sum_{n | N} \sum_{p\leq n ; \ p \ prime} 1 = \sum_{n | N} \pi(n)$$
Is this ri... |
H: What is the algorithm for solving an equation like this one?
The solutions of the equation : $\sqrt{x+2\sqrt{x-1}} + \sqrt{x-2 \sqrt{x-1}} = 2$ are:
A) $x=1$;
B) $x=2$;
C) $x\in [1,2]$;
D) $x\in \begin{bmatrix}
\frac{3}{2},2
\end{bmatrix}$;
E) $x=\frac{3}{2}$;
AI: As you have some answer possibilities yo... |
H: Written as disjuctions, conjunctions and negations?
With a domain from -2 to 2 I'm trying to write the following using disjunctions conjunctions and nagations. I'm not sure how correct I am and wanted to know if I did them correct? Could someone help with the last one I just cant figure out how to start that expres... |
H: interchange sum and integral
suppose I have a family of i.i.d standard normal random variables $Y_{n,k}$ and I define $X^N_t:=\sum_{n=0}^N\sum_{k=1}^{2^n}Y_{n,k}\phi_{n,k}(t)$ for $t\in [0,1]$ where $\phi_{n,k}$ are the Schauder functions.Furthermore, I know that $(X_t^N)$ is a martingale bounded in $L^2$ and there... |
H: Work out the number of edges given conditions of a graph
Given a set of nodes $V$, and a parameter $c \lt |V|$,
How can we show that whether we can derive an un-directed graph $G=(V,E)$ where the degree of each node equals to $c$?
If a graph as above exists, how many un-directed edges are there in $G$?
AI: For 2.,... |
H: Tell if $(\mathbb Z_6, \odot)$ is a semigroup and if the identity element belongs to it
Let the operation $\odot$ be defined in $\mathbb Z_6$ as follows:
$$a \odot b = a +4b+2$$
check if $(\mathbb Z_6, \odot)$ is a semigroup and if the identity element belongs to it.
This is the way I have solved this exercise:
Let... |
H: Convergence Properties of the Taylor Series for $\frac{1+z}{1-z}$
I just ran into the following exercise:
Find and state the convergence properties of the Taylor series for the following:
$$\frac{1+z}{1-z}$$
around $z_0=i$.
First of all, let
$$f(z)=\frac{1+z}{1-z}.$$
Then we have that
$$\begin{align}
f^{(1)}(... |
H: Quadric surface graphing applet
Does anybody know of any online tool/applet that can be used to graph quadric surfaces? i.e. If I want an elliptic paraboloid, I can click on "elliptic paraboloid" and enter my own specified values for a, b, and c. I couldn't find one on google, any ideas?
AI: The POV-Ray 3D renderer... |
H: Density function of a function of random variable by expectation?
Suppose $Z$ is a continuous random variable on $\mathbb{R}^n$.
$f: \mathbb{R}^n \to \mathbb{R}^+ \cup \{0\}$ is a function, such that $\mathrm{E} Z = \int_{\mathbb{R}^n} z \times f(z)dz$. Then we know $f$ is not necessarily the density function of ... |
H: Is there an infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$ analogous to the Rogers-Ramanujan identity?
Given
$$
\left(\frac{\eta(5\tau)}{\eta(\tau)}\right)^{6}\;\; =\;\; \frac{r^5}{1-11r^5-r^{10}},\;\;\;\;\;\text{with}\;\;r\; =\; q^{1/5} \prod_{n=1}^\infty \frac{(1-q^{5n-1})(1-q^{5n-4})}{(1-... |
H: How to explain what it means to say a function is "defined" on an interval?
I am having difficulty in explaining the terminology "defined" to the students I am assisting. Here is the sentence: If a real-valued function $f$ is defined and continuous on the closed interval $[a,b]$ in the real line, then $f$ is bounde... |
H: The role of maps in algebraic structures?
In group theory we have encountered the concepts of homomorphisms as maps from one group to another, and the same thing with ring and field theory and we ave called them respectively homomorphisms of groups (of rings, and of fields). Also in linear algebra we have linear ma... |
H: entire function is constant
Let $ f $ a complex entire function such that:
$$ |f(z)| \leq \sqrt{2|z|} + \frac{1}{\sqrt{2|z|}} \quad \forall z \neq 0 $$
Prove that $ f$ is constant.
Thank's in advance!
AI: Let $g(z)=\frac{f(z)-f(0)}{z}$. Then $g(z)$ is entire and for all $z$ with $|z| \geq 1$ you have
$$ \left| g(z)... |
H: Jacobian physical meaning
Possible Duplicate:
What is Jacobian Matrix?
Is there any physical intuition for the Jacobian?
I understand that it is the matrix of partial derivatives and how to construct it. What I want to know is
what's the use of it? Application wise
is there a nice intuitive explanation for it?... |
H: Determine largest area of house using geometry
The question is as follows. Given a house as shown in the figure below
$AB = 3a\,,\,BC = 10\,,\,CD = a\,,\,DE= a\,,\,EF=2a,$ and $FA=10-a$
Where a is some arbitrary constant such that $a \in [0,10]$
Now one could show that the area of the figure is given by the functi... |
H: Interpretation of the Fourier coefficients?
Suppose I have a discrete function $f( x_i ) = y_i$.
I can use these pairs $(x_i, y_i)$ as complex number $z_i = x_i + j \, y_i$.
Now, having this set $z_i$, I can apply discrete Fourier transform, as show in Wikipedia.
Now, suppose the calculated Fourier coefficients ar... |
H: How many Fourier coefficients are enough ( in discrete fourier transform)
Probably there's a similar question, but I could't find it through all questions about FT
As the title says, how many Fourier coefficients are enough, to be able to "resume" the original function, using inverse discrete Fourier transform?
F... |
H: relating flatness, equidimensional, and complete intersection
I am a bit confused and am trying to clarify some notions. First consider the following well-known statement.
A dominant map $f:X\rightarrow Y$ between regular varieties is flat if and only if it is equidimensional.
Question 1. Doesn't a regular variet... |
H: Why does $L^2$ convergence not imply almost sure convergence
What's wrong with this argument?
Let $f_n$ be a sequence of functions such that $f_n \to f$ in $L^2(\Omega)$. This means $$\lVert f_n - f \rVert_{L^2(\Omega)} \to 0,$$ i.e.,
$$\int_\Omega(f_n - f)^2 \to 0.$$
Since the integrand is positive, this must mea... |
H: Solving Fibonaccis Term Using Golden Ratio ConvergEnce
While solving this problem, I discovered that there is a relationship between the Fibonacci sequence and the golden ratio. After I got the correct answer via brute force, I discovered this relationship. One of the posters said this:
The nth Fibonacci number ... |
H: Error estimation in $H^1(\sigma)$
I have a question. If $u$ and $u_h$ are the solutions of the continuous and discrete variational equations, respectively, then for $u\in H^1_0$, how does one prove that $\lim_{h\rightarrow 0} \|u - u_h\| = 0$ where the norm is taken in the $H^1(\sigma)$. Sigma is the domain.
Thank ... |
H: Finding arithmetic mean, standard deviation, mode and median
On the market quality of the fruit was measured and following results came out:
Quality of Fruit( in measuring units ) 65 70 75 80 85 90 95 100
Number 2 3 2 5 8 7 5 3
Define:
a) arithmetic mean and standard de... |
H: Visualization of Cantor set by Mathematica or Maple!
We all know what it is the Cantor set. The Cantor set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third $(\frac{1}{3}, \frac{2}{3})$ from the interval $[0, 1]$, leaving two line segmen... |
H: Conformal mappings:general tips
Has anyone any general tips for finding conformal tranformations from a domain to another?
For example from $D=\{ z=x+iy | x^2+y^2<1, x^2-x+y^2>0 \}$ to the unit disc. This set is a disc
minus a smaller disc, so what should i imagine to try? If someone could provide his reasoning
ste... |
H: Is it unlikely to get the same number of heads/tails?
A question in probability by a non-mathematician:
A fair coin is tossed $2N$ times. Is it unlikely that we get exactly $N$ heads and $N$ tails?
From one side, this must be the most likely result! But intuitively, if someone reports to me that they threw 2,000,00... |
H: Circular Rotation
I'm trying to emulate planetary rotation. I have a 'planet' rotating around the 'sun', but when trying to rotate the 'moon' around the 'planet', the motion is skewed.
If I stop the motion of the 'planet', the rotation is fine. I think this is just outside of my understanding.
I have demo at http:/... |
H: Showing that an entire function is a polynomial
Let $f(z)$ be an entire function, $R_n$ a sequence of positive real numbers tending to $\infty$ such that $f(z) \neq 0$ on $|z|=R_n$ and there exists $M>0$ such that
$$\int_{|z|=R_n} \left|\frac{f'(z)}{f(z)}\right| ~dz<M$$
for all $n$. Show that $f$ is a polynomial.
... |
H: Why is decomposing rational functions by assigning selected numerical values to x mathematically consistent?
Possible Duplicate:
How does partial fraction decomposition avoid division by zero?
Say you have the rational function:
$\frac{x^2 + 1}{(x-1)(x-2)(x-3)}$
This means that the function is undefined when x i... |
H: Is there more than one infinitessimal among the hyperreal numbers
Take $\mathbb{H}=\mathbb{R}^\mathbb{N}/\mathcal{U}$, where $\mathcal{U}$ is some ultrafilter. Questions:
Are there more than one independent infinitessimal in this field. This means $\epsilon_1 > 0$ and $\epsilon_2 > 0$ such that $0 < \epsilon_1 < \... |
H: Inequality between chromatic number and number of edges of a graph
I have not been able to find a proof to the statement that if a graph $G$ has $\chi(G)=k$, then it must have at least $\binom{k}{2}$ edges. Would you be able to show me a simple proof?
AI: Suppose we have a $k$-coloring of our graph, with $\chi(G)=k... |
H: invariance of cross product under coordinates rotation
Question goes as
If $\vec A$ and $\vec B$ are invariant under rotation, the prove that $ \vec A \times \vec B $ is also invariant.
However solution of on the other page is not given. Says that if you replace A and B with A' and B' and i,j,k with i', j', k'... |
H: Value of $n$ for which an improper integral is convergent.
A question from the Calculus book that I'm self-studying is asking me to determine the value of $n$ for which the improper integral below is convergent:
$$\int_1^{+\infty}\left( \frac{n}{x+1} - \frac{3x}{2x^2 + n} \right ) dx$$
My attempt is below:
Using th... |
H: r-colorable simple graph
What $r$ colorable simple graph on $n$ vertices has the most edges? Is there a unique such graph? I am told this has something to do with Turan's theorem...
(This is the progress I have made- deleting a vertex $u$ and replacing it with a copy of a vertex $v$ preserves the number of vertices... |
H: Multiple variables for a logical expression?
I wanted to know if what I did is even on the correct path for how this question is worded. How can you have two variables when it's dealing with a single unhappy person? I'm guessing the third way will just be using De Morgan’s Laws for quantifiers?
Translate each of t... |
H: On $T_2$, first countable, countably compact space
As we know,
For every $T_2$, first countable, compact space, its cardinality is not more than $2^\omega$. (See chapter 3 of Engelking's book.)
However, I want to know whether the result is same for the $T_2$, first countable, countably compact space, i.e., fo... |
H: Am I too young to learn more advanced math and get a teacher?
I am still 15 years old, but I am very interested in pure math. I have been teaching myself though books, from the internet and from others for the past year or so. I haven't mastered all the topics that are covered in university, just the ones that ha... |
H: When standard deviation is unknown?
I am reading a book about Statistics and I have encountered a text line:
...except in the case where standard deviation of the basic set is unknown...
I am not really sure what it means, could you please help me. For what types of data can't we define the standard deviation?
AI... |
H: Determining the minimum number of pixels on the boundary of a circle drawn in discrete space
I am trying to draw a circle in discrete space (actual image pixel space). I have the center (x,y) and radius r of a circle that I am supposed to draw. The manner in which I draw this circle is the following:
Starting from ... |
H: Unit distance graph in $\mathbb{R}^2$
Suppose $G$ is the simple graph with vertex set $\mathbb{R}^2$ created by connecting two points iff they are distance one from each other in the plane. How can we prove that $4\le \chi(G)\le7$? I think I have seen this problem somewhere before but can't pinpoint it. No hints pl... |
H: Difference between permutation and combination?
Permutation:
$$P(n,r) = \frac{n!}{(n-r)!}$$
Combination:
$$C(n,r) = \frac{n!}{(n-r)!r!}$$
Apparently, you use combination when the order doesn't matter. Great. I see how a combination will give you all the possible well, combinations. However, I don't see what exactly... |
H: Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$
Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals.
Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x... |
H: Counting men and women around a circular table such that no 2 men are seated next to each other
Possible Duplicate:
Circular Permutation
I was looking through old homework to work on my counting skills, and I had this problem that I didn't get right.
Question: In how many ways can 4 men and 5 women be seated aro... |
H: $f, g$ entire functions with $f^2 + g^2 \equiv 1 \implies \exists h $ entire with $f(z) = \cos(h(z))$ and $g(z) = \sin(h(z))$
I am studying for a qualifier exam in complex analysis and right now I'm solving questions from old exams. I am trying to prove the following:
Prove that if $f$ and $g$ are entire functions... |
H: Multiple variable quantifications in english?
Problem 1 is really simple but 2 is confusing me a little. I'm going on the assumption that $x$ and $y$ are both students and they are saying $z$ is the class? Is it really as simple as what I wrote or am I missing something?
Let $C(x, y)$ mean that student $x$ is enro... |
H: Implicit differentiation question
Differentiate given
$$\frac{y}{x-y}=x^2+1$$
Initially I wanted to use the quotient rule to solve this, but then I tried differentiating it as it is:
$$\frac {y_\frac{dy}{dx}}{1-y_\frac{dy}{dx}}=2x$$
$$\frac{dy}{dx}(y y^{-1})=2x$$
$$\frac{dy}{dx}=\frac{2x}{yy^{-1}}$$
$$\frac{dy}{dx}... |
H: Limit of a sequence of real numbers
If $(a_n), (b_n)$ are two sequences of real numbers so that $(a_n)\rightarrow a,\,\,(b_n)\rightarrow b$ with $a, b\in \mathbb{R}^+$. How to prove that $a_n^{b_n}\rightarrow a^b$ ?
AI: Since $a_n\to a$ and $a>0$ by assumption, we have $a_n>0$ for $n\geq N$ for some sufficiently l... |
H: proof for members of ideals
If $I$ is an ideal, could you show that if $ x\in I$ and $y\notin$ I, then $x+y \notin I$? It seems like an intuitively obvious statement and yet my rigor is failing me. So if you could show me all the steps of the proof that would be much appreciated.
AI: Assume for the sake of contradi... |
H: Rules of Inference argument has multiple steps?
I need to write the rules of inference and explain which rules of inference are used for each step. This looks like it's just Modus Ponens from what I wrote the equations as, am I missing some steps or is this really that simple?
“There is someone in this class who h... |
H: Ring map-integers proof
Prove that there is a ring map from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}/m\mathbb{Z}$ iff $m|n$.
I am not able to prove this with the appropriate rigour, could someone show the steps for a proof?
AI: $\def\Z{\mathbb Z}$
First suppose that there is a ring morphism $\phi\colon\Z/n\Z \to \Z... |
H: Exponentiation of reals
Let $1<b\in \mathbb{R}$ and $x\in \mathbb{R}$.
I want to prove that $\sup${$b^t\in \mathbb{R}$|$x≧t\in \mathbb{Q}$} = $\inf${$b^t\in \mathbb{R}$|$x≦t\in \mathbb{Q}$}.
I have proved that $\sup$≦$\inf$, but dont know how to show that $\inf$≦$\sup$..
AI: We know that $f(t) = b^t$ is continuous ... |
H: Rings, ideals and units-proof
Let $R$ be a ring and let $J$ be an ideal of $R$. Assume $J$ contains
a unit of $R$. Prove $J=R$, rigorously.
Again I feel here as if the statement is intuitive so I am not able to make it rigorous. If you could show all steps in the proof that would be nice.
AI: If $J$ contains a ... |
H: How to confirm if my explicit formula is right?
I have to determine an explicit formula for
$$a_n=5a_{n-1}+6a_{n-2}$$
Initial values are $$a_0=2\\a_1=-1\\n>=2$$
My answer is
$$a_n = \frac{1}{7}\cdot 6^n+\frac{13}{7}\cdot (-1)^n$$
Which I suspect is wrong. But, how to "test" it?
AI: You are right. All solutions of t... |
H: Rigorous definition of what it means to be the same group?
Consider the following groups: $(\mathbb{Z}_4,+)$, $(U_5,.)$, $(U_8,.)$ and the set of symmetries for a rhombus if I am not mistaken the first and last are equivalent. What other justifiable equivalencies and nonequivalencies are there and what does it mean... |
H: Ring map one-to one kernel-proof
How can we show that a ring map $f: R\rightarrow S$ is one-to-one iff $\ker(f)=\{0\}$? I have seen this for a while axiomatically so I am unsure of my rigor.
AI: $(\Rightarrow)$ If $f$ is injective, then $\text{ker}(f) = \{0\}$ since $f(0) = 0$ for all homomorphisms and injectivity.... |
H: Prove that (X,Y) is bivariate normal if X is normal and Y conditionally on X is normal
$X$ has a normal distribution. The conditional distribution of another random variable $Y$ given that $X=x$ is a normal distribution with mean $ax+b$ and variance $t^2$, where $a$, $b$, and $t^2$ are constants. How can I prove t... |
H: Evaluating $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\, \mathrm{d}x$
I have to evaluate:
$$\int_{0}^{\pi/2}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\, \mathrm{d}x. $$
I can't get the right answer! So please help me out!
AI: Let $I$ denote the integral and consider the substit... |
H: Find the maximum of a linear function, given other linear equations with infinite solutions
$a_{1,1}x_1 + a_{1,2}x_2 + \dots + a_{1,20}x_{20}\leq b _1$
$a_{2,1}x_1 + a_{2,2}x_2 + \dots + a_{2,20}x_{20} \leq b_2$
$x_1 \geq 0, x_2 \geq 0, \dots, x_{20} \geq 0$
$f(x) = a_{3,1}x_1 + a_{3,2}x_2+ \dots + a_{3,20}x_{20} ... |
H: Help deriving a vorticity equation
I am reading Majda & Bertozzi (Vorticity and Incompressible Flow). In page 12 the following equation appears:
$$\frac{D \Omega}{Dt} + \Omega \mathcal{D} + \mathcal{D} \Omega = \nu \Delta \Omega$$
where $\frac{D}{Dt}$ is the convective/lagrangian/material derivative. $\Omega$ and $... |
H: Construction of $\Bbb R$ from $\Bbb Q$
As it is true that we can construct all rational numbers $\Bbb Q$ from the set of integers $\Bbb Z$, is it possible to construct the set of real numbers $\Bbb R$ from $\Bbb Q$? If yes, how? Is there any procedure? And if no, is there any proof that we can't ? Thanks!
AI: Consi... |
H: Ramification of primes
Let $K \subset L$ be two fields with ring of integers $\mathcal O_K$ and $\mathcal O_L$.
If a prime $p$ is totally ramified in $\mathcal O_K$, is it true that $p$ is also ramified in $\mathcal O_L$?
AI: If $p$ is (not necessarily totally) ramified in $K$ then there is a prime ideal $\mathfrak... |
H: trivial Picard group
let $S=\operatorname{Spec}(A)$ be an affine scheme. For which ring $A$, not field is it known that $H^1(S,\mathcal{O}_S^{*})$ is trivial?
If $X\to S$ is a finite map and $H^1(S,\mathcal{O}_S^{*})$ is trivial, is it true that also $H^1(X,\mathcal{O}_X^{*})$ is trivial?
Thanks
AI: Sample answers ... |
H: $f$ is irreducible in $\Bbb F[x]$
Let $\Bbb F$ be a field of characteristic $p\gt 0$ and $f(x)=x^{p^n}-c \in\Bbb F[x]$ where $n$ is a positive integer. If $c \notin \{a^p:a\in \Bbb F \}$, show that $f$ is irreducible in $\Bbb F[x]$.
I recently started studying Field theory, so I don't know How to approach this pr... |
H: The Betti numbers of a triangulated ball relative to a disk in its boundary
Fix $n \geq 1$ and let $B$ denote a triangulated closed $n$-ball. Let $D$ be a subset of the boundary of $B^n$ that is homeomorphic to the closed $(n-1)$-ball and such that is properly triangulated by the same triangulation as well.
I would... |
H: Proof of $R/I$ integral over $S/(S \cap I)$
Can you tell me if my reasoning is correct?
I want to prove if $S \subset R$ are rings and $R$ is integral over $S$ and $I$ is an ideal of $R$ then $R/I$ is integral over $S/ (S\cap I)$.
Let $R$ be integral over $S$. $(S \cap I) \subset I$ is an ideal of $S$ and hence of ... |
H: How do I calculate a change of coordinates given two lines as new axis?
Suppose we have an old coordinate system using the variables $x$ and $y$. We are given two equations for lines to form the axis for new coordinates. e.g. The line $z=0$ is given by the equation $y = m_1 x + b_1$, and the line $w=0$ is given by... |
H: Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})[X]$
Is it possible to determine how many irreducible factors has $X^p-1$ in the polynomial ring $(\mathbb{Z}/q \mathbb{Z})[X]$ has and maybe even the degrees of the irreducible factors? (Here $p,q$ are primes with $\gcd(p,q)=1$.)
AI: It has one factor of ... |
H: Decreasing functions
Let $u:[a,b]\to \mathbb{R}$ be a continuous and a. e. differentiable function (with respect to the Lebesgue measure).
Is it true that $u' < 0$ a. e. in $[a,b]$ implies $u$ strictly decreasing everywhere in $[a,b]$?
(New question added on 12/21/2012)
I know the answer is negative (thanks to Jon... |
H: How can I disprove the following statements?
1) $ a \neq b , a^0 = b^0, a = b$
Usually we do $ a^n = b^n \implies a= b, $ can't we do it when $n$ is zero?
2) $ i = \sqrt{ -1 \over 1} = \sqrt{1 \over -1 } = {1 \over i} \implies -1 = 1$
What's wrong with the above statements?
AI: What you're doing is basically th... |
H: Is this polynomial positive?
Let $p\geq 2$, and $p$ is not a half odd integer. $t\in R$.
Is the following polynomial positive:
$$
T_k(t)=\left(\frac t2\right)^p\sum_{j=0}^k\frac{\left(-\frac{t^2}{4}\right)^j\Gamma(p+1)}{j!\Gamma(p+j+1)}.
$$
Thank you for your help
AI: Consider series
$$
T(t)=\left(\frac{t}{2}\right... |
H: Derivative of Multi Variable Equation for Diablo 3 Damage Equation
It has been years since I took calculus and I am trying to figure out if it is possible to calculate the derivative of a mutli variable equation and what the result would be. This equation is from the video game Diablo 3 that represents your damage ... |
H: Heat equation with bounded or compactly supported initial data
Let $u$ denote to the solution of the heat equation
$$\begin{cases} u_t(x,t)-\Delta u(x,t) & = & 0 & t>0 \\ u(x,0) & = & g(x) \end{cases}$$
where $x\in\mathbb{R}^n$.
I want to show that
if $||g||_\infty<\infty$ then $u$ tends to some constant as $t\to\... |
H: Probability that 2 sequences of k trials (success/failure) are identical.
A single trial has probability p of success and (1-p) of failure. For simplicity, we assume p = 1/2 = (1-p). An experiment is defined as a sequences of trials until 4 consecutive success or failures.
Suppose two such experiments were conducte... |
H: Finding the equation of a curve that has a perpendicular distance of $d$ from another curve
Let's say we have an equation of a curve as $y = f(x)$.
I want to find the curve $y = g(x)$ where $(x_1, f(x_1))$ has a perpendicular distance of $d$ from that curve.
Doing this with straight lines is pretty easy. For exampl... |
H: Maximum likelihood estimate for pdf
I am attempting a problem from Larsen and Marx, 4th edition that asks to find the maximum likelihood estimate for $\theta$ in the pdf: $$ f(y; \theta) = \dfrac{2y}{1-\theta^{2}}, \theta \leq y \leq 1$$
It also states that a random sample of size 6 yielded measurements 0.70, 0.63... |
H: What does "increases in proportion to" mean?
I came across a multiple choice problem where a function $f(x) = \frac{x^2 - 1}{x+1} - x$ is given. One has to choose the statement that is correct about the function. The different statements about the function included:
(1) the function increases in proportion to $x^2... |
H: Question about polynomials and congruences
Let be $m(X), n(X), a(X), b(X), m(X), n(X), g(X) \in F[X]$ a polynomials, where $F$ is a finite field with characteristic 2. Let be a relations $a \equiv b \mod g(X)$ and $m \equiv n \mod g(X)$. Are there any relation, property, etc, between $a(X), b(X), m(X), n(X)$?.
AI: ... |
H: Division and number scaling
I'm trying to implement an interactive (secure) protocol which operates only on integers. Here's what I have:
$$ f(x) = \sum_i{a_i K_i} + b $$
$$ K_i = \dfrac{1}{1 + \gamma \|x - s_i\|^2}$$
where $ 0 < i \le M $ and $$ \|x - s_i \|^2 = \sum_{j=1}^{N}{(x_j - s_{ij})^2}.$$
Because the serv... |
H: Chronicles of Discoveries
First I'd like to bring an example to make myself more clear.
I know what the Jacobian matrix is and where, how and why it is used (some examples, at least) . But still I can't get it's geometrical interpretation. And I don't understand how and why it works.
I wonder how Jacobian matrix ... |
H: Could someone remind me why is incorrect to switch an infinite sum and an integral?
Could someone jog my memory on this?
The order of operation between an $\int$ and $\sum_{n\in \mathbb{N}}$ is not always interchangable? Note that the sum is an INFINITE sum
Why is it that $\int \sum_{n \in \mathbb{N}} \neq \sum_{n ... |
H: Limit of the sequence of rational numbers above a given real
I need to prove that for any $a \in\mathbb{R}^+$ the sequence $S[a]_n=a+b_n$, where $b_n = \min\{|{x \over n}-a|\;\colon\;x \in \mathbb{N}\}$, converges to $a$. The only way I know how to prove convergence is with an epsilon-delta argument but I don't thi... |
H: Find $\lim \ a_n$ if $a_n = \frac{1}{\sqrt[3]{n^3+1}} + \frac{1}{\sqrt[3]{n^3+2}}+\cdots+\frac{1}{\sqrt[3]{n^3+n}}.$
I would greatly appreciate some help in finding $\lim \ a_n$ if
$$a_n = \frac{1}{\sqrt[3]{n^3+1}} + \frac{1}{\sqrt[3]{n^3+2}}+\cdots+\frac{1}{\sqrt[3]{n^3+n}}.$$
AI: HINT:
Note that $$\dfrac1{\sqrt[3... |
H: Number of $1$s in a binary grid
Consider a binary grid of size $4\times 4$, each of cell can either have $0$ or $1$. Among all possible $2^{16}$ arrangement how many arrangement of such grid exist in which each row and column contains even number of $1$s.
Solution which I thought
There will be $2$ possibilitie... |
H: Explain Carmichael's Function To A Novice
I understand that the Carmichael Function (I'm going to call λ) is essentially the smallest positive integer m, where $a^m$ is congruent $1 \pmod n$ for all $a$ co-prime to $n$ and less than $n$.
6 makes sense to me. The only co-prime is 5 and $5^2 = 25\equiv 1 \pmod 6$. So... |
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