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Green's Theorem has the form:
$$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy $$
The Cauchy-Riemann equations have the following form:(Assuming $z = P(x,y) + iQ(x,y)$)
$$\frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y}, \frac{\partia... |
In propositional logic, for example:
$$\neg p \vee q.$$
If $p$ is true at the outset, does that mean it must be considered false when comparing with q in the disjunction?
P.S. I am unsure about tags for this question.
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Why is the following true?
If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
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It is my question when I was in Senior High School. Up to now, I have no idea about the correct explanation. I just accept it by faith :D
Here is the question:
If the symbol $\Delta x \to 0$ does not mean $\Delta x =0$, how can the points $A$ and $B$ get overlapped which in turn, how can the line joining $A$ and $B$ ... |
Given a Noetherian, local ring $(R,m)$, can we always find a system of parameters whose images in the cotangent space $m/m^2$ are linearly independent?
We can do this in the regular case, by just choosing a basis for the cotangent space and looking at their preimages in $m$ under the canonical map. By Nakayama's lem... |
I encountered a problem which is asks for the first negative term in the expansion of $(1+x)^{\frac{27}{5}}$, where $x$ is always positive,the solution suggested in my module is like this:
$\displaystyle \text{ For the first negative term: } n-r+1 \lt 0 \text{ or, } r \gt 6\frac{2}{5}$, since $n = \frac{27}{5}$
Hence,... |
Let H be a subgroup of G and V the trivial representation of C[H]. Prove that we have an isomorphism of C[G]-modules between $\mathrm{Ind}_{H}^{G}(V)$ and C[G/H].
Thanks for your help.
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my Calculus II class is nearing the end of the quarter and we've just started differential equations to get ready for Calculus III. In my homework, I came upon these problems.
One of the problems was:
Find the general solution to the differential equation
$$\frac{dy}{dt} = t^3 + 2t^2 - 8t.$$
The teacher just said to ... |
Find the value of:
$$ 1+ \biggl(\frac{1}{10}\biggr)^2 + \frac{1 \cdot 3}{1 \cdot 2} \biggl(\frac{1}{10}\biggr)^4 + \frac{1\cdot 3 \cdot 5}{1 \cdot 2 \cdot 3} \biggl(\frac{1}{10}\biggr)^6 + \cdots $$
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I know from an online calculator http://www.numberempire.com/derivatives.php that
$\lim_{x \to \infty} \left ( \frac{x+2}{x} \right )^{x}=e^{2}$. How do you calculate this step by step?
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Three planes A, B and C intersect at point P. The dihedral angle between A and B is $\theta$ and the dihedral angle between B and C is $\psi$.
Solve for the dihedral angle between by A and C.
Planes A and B intersect plane C in two lines that form angle QPR. That is, point Q lies in both A and C, and point R lies in... |
Could someone give me an example of a (smooth) $n$-manifold $(n=2, 3)$ which cannot be embedded (or immersed) in $\mathbb R^4$?
Thanks in advance!
S. L.
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In Linear Algebra, when computing an inner product $<x,y> = y^*Wx$, what is the name of the matrix W?
If it doesn't have a name, where can I find a practical explanation of how to construct it for a particular problem or space?
Is there a text on the subject that explains this simply? A note on my background, I am ... |
I am looking for a correct proof of this statement: If $G$ is a group such that $G/Z(G)$ is cyclic, then $G$ is commutative.
Proof: $G/Z(G)$ is isomorphic to $\operatorname{Inn}(G)$ and is cyclic, and then for every $a$ and $b$ in $G$ the inner isomorphisms $\gamma_a$ and $\gamma_b$ satisfy $\gamma_a \gamma_b = \gamma_... |
How to prove DeMorgan's Law?
$$A - (B \cup C) = (A - B) \cap (A - C)$$
$$A - (B \cap C) = (A - B) \cup (A - C)$$
EDIT:
Here is what I have tried so far:
Considering the first equation, assuming $x \in A - (B \cup C)$ then $x \in A$ and $x \not\in B$ and $x \not\in C$, while the right hand means ($x \in A$ and $x \n... |
A lot of textbooks said it was hard for human to accept zero when it was first introduced.
How could it be? It seems to me as natural as positive integer which represent there is no elements at all.
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I stumbled across this paragraph in a paper:
Hence, user b cannot decrypt C
directly. But using e and d , user b
can quickly factor N.
How is it possible to speedup the prime factorization when knowing e (public key) and d (private key)?
For clarification:
RSA provides us with these equations:
$n = pq$
$\phi = ... |
How possible methods for finding groups of given order we have?
Sylow theorems - and next? We know something about center of group, about possible orders of these elements and subgroups. But what is most effective way, to find all non-isomorphic groups of given order? Is there any other strong "weapon" for this (like S... |
Here is what my textbook told me:
Assuming the formula for surface $\Sigma$ is $$F(x,y,z) = 0$$
Suppose $X_0 = (x_0, y_0, z_0)$ is a point on the surface $\Sigma$ and we assuming F(x,y,z) is differentiable and $$\mathbf{J}F(X_0) = (\frac{\partial F(X_0)}{\partial x}, \frac{\partial F(X_0)}{\partial y}, \frac{\partial ... |
For $n > 2$, is it possible to embed $\underbrace{S^1 \times \cdots \times S^1}_{n\text{ times}}$ into $\mathbb R^{n+1}$?
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I have a uniform grid of square cells and a point on that grid. I want to calculate the intersection of cells on the grid, with a circular area centered on the point, with a radius R.
Does anyone know of an efficient way of doing this?
Thanks!
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Follow-up to Field reductions. part two
For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting equivalence classes on $K(\setminus a)$ or binary operations, or distance functions, or ...? Real numbers are represente... |
What is the limit of the series $1 \over (2n)!$ for n in $[0, \infty)$ ?
$$ \sum_{n = 0}^{\infty}{1 \over (2n)!}$$
I've ground out the sum of the 1st 1000 terms to 1000 digits using Python,
(see here ), but how would a mathematician calculate the limit? And what is it?
No, this isn't homework. I'm 73. Just curious. ... |
folks,
i am new to this forum and not a math expert. so please bear with me if am asking silly questions.
The question is "probability of getting 50 heads from tossing a coin 100 times".
So the answer for this is, I guess, ${100 \choose 50} (2 ^{-100})$.
So all am trying to get is easier way to calculate ${100 \choose... |
How to calculate the sides and hypotenuse length of the right triangle if I know
$ \text{bigger side} = 60$, $\text{one angle} = 60^o$ & $\text{second angle} = 30^o$ ($\text{third angle} = 90^o$)
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What are the necessary and sufficient conditions on a vector field $F$ for the divergence $\nabla\cdot F$ to exist at a given point.
EDIT
In Divergence in the second line under the heading "Application in Cartesian coordinates", why is it assumed that $\vec{F}$ to be a continuously differentiable vector field ?
EDIT 2
... |
Given $x(t) = f(t) \cdot g(t)$, what is the Fourier transform of $x(t)$? If possible, please explain your answer.
The motivation behind the question is homework, but this is a basic principle in the class that I never quite grasped properly. My current homework builds upon the principle of the question. So answering ... |
I have a couple questions that I'd like answered:
(1) How do I evaluate this integral?
(2) Is it possible to do a substitution? What about $u=e^{i \cdot t}$?
(3) What if 5 and 3 are replaced by other constant integers?
(4) What should be the most effective method to handle question (3), using the fewest calculat... |
I have a system that is modeled by the following differential equation:
$$db/dt = h_aj(t) + k(T_a-b(t))$$
where $db/dt$ is the rate of temperature change, $j(t)$ is an input, $h_a, k, T_a$ are all constants, and $b(t)$ is an output. Note that this is newtonian cooling with a heating input, $h_aj(t)$.
I want to find th... |
I have data for points of a 3D grid. The points of the grid are generated from three nonorthogonal vectors: each grid point has coordinates $\mathbf{q}_{ijk} = i \mathbf{a} + j \mathbf{b} + k \mathbf{c}$, where $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are nonorthogonal, noncoplanar vectors. (In cristallography, it's... |
Suppose I is an index set (not necessarily countable). Say we have a family of subsets $\{C_{i}: i \in I\}$ and suppose that this family is a cover of a space $X$, that is, for each $x$ in $X$ we have $x \in C_{j}$ for some $j \in I$.
Can we say the following? for each $x \in X$ there exists some natural number $n$ suc... |
Let's say that $f(x)=f^{1}(x)$ and that $f(f(x))=f^{2}(x)$. Moreover, $f^{n}(x)$ is the n-th iterate of $f(x)$, for $n \in \mathbb{N}$. I'm curious about extending iteration to larger number sets. For $n \in \mathbb{R}$, there's the concept flow (I think?). I don't understand the Wikipedia-article on this subject very ... |
Given a topological space (with nice enough conditions, maybe Hausdorff, compactly generated, or CW complex, I'm not sure) $X$ and a subspace $A\subset X$, is it true that $H^n(X,A)\cong H^n(X\backslash A)$ whenever there is an open set containing $A$ which can be retracted to $A$? Can this be shown easily using the M... |
Question: (a) Let $f: V \rightarrow W$ with $ V,W \simeq \mathbb{R}^{3}$ given by: $$f(x_1, x_2, x_3) = (x_1 - x_3, 2x_1 -5x_2 -x_3, x_2 + x_3).$$
Determine the matrix of $f$ relative to the basis $\{(0,2,1),(-1,1,1),(2,-1,1)\}$ of $V$ and $\{(-1,-1,0),(1,-1,2),(0,2,0)\}$ of $W$.
(b) Let $n \in \mathbb{N}$ and $U_n$ t... |
I'm looking for a graphic that shows the isomorphism theorems illustrated using a finite vector space. (that is a finite dimensional vector space over a finite field.) Something that shows nicely how the cosets "collapse" in to single elements or how cosets at different "levels" are matched up to form the isomorphisms.... |
I have a radius, R, for an aircraft traveling at velocity, V. If we start at point, (X,Y), what is the position of the point at time, t.
For example:
The aircraft is at point (0,0) and traveling at 250 knots and initiates a turn with a bank angle, phi, of 5 degrees. Assume that the aircraft can instantaneously rotate t... |
For a given radius, theta, and position, can you calculate a Bezier curve?
|
Let $F_n$ be an $n$-generator free group with a free basis $x_1,\ldots,x_n.$ Is it true that the stabilizer of $x_1$ in $\mathrm{Aut}(F_n)$ is generated by all left and right Nielsen moves $\lambda_{ij}$ and $\rho_{ij}$ such that $i \ne 1$ and by the element of order two $\epsilon_n$ such that $\epsilon_n(x_n)=x_n^{-1}... |
Problem: The series $\sum_{n=1}^{\infty} a_n$ diverge and positive. What can be said on the series $\sum_{n=1}^{\infty} \frac{a_n}{1+n^{2}a_n}$ ?
Approach:
First, I tried separating it into two cases: $a_n \to \infty$, $a_n \to A$ (where $A$ is some const value).
Tried the ratio test and got nowhere with that.
I thi... |
A regular sequence is an $n$-fold collection $\{r_1, \cdots, r_n\} \subset R$ of elements of a ring $R$ such that for any $2 \leq i \leq n$, $r_i$ is not a zero divisor of the quotient ring
$$ \frac R {\langle r_1, r_2, \cdots, r_{i-1} \rangle}.$$
Does the order of the $r_i$'s matter? That is, is any permutation of a... |
Consider a particle starting at the the origin and moving along the positive real line at a constant speed of 1. Suppose there is a counter which clicks at random time intervals following the exponential distribution with parameter $\lambda$ and whenever the counter clicks, the position $x > 0$ of the particle at that... |
Currently taking a logic class and trying to understand this.
You have two set $A$ and $B$.
Both sets are empty sets.
Is set $A$ a subset of the complement of set $B$?
Assume the context is the universal set.
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This is a follow up to a question I had asked earlier about a linear recurrence relationship satsified by $\lfloor(1+\sqrt{5})^n\rfloor$. I messed up there, and I actually meant to ask about $L(n)=\lfloor(1+\sqrt{3})^n\rfloor$.
Following Douglas' suggestion I have determined that the values (at least the first 1000) sa... |
$$\sum_{n=1}^{\infty}\frac{1}{2^n3^{(n-1)}}$$
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I am in the process of proving that if a space curve (in $R^3$) has infinite length and the curvature tends towards $0$ as the natural parameter $s$ tends to infinity, the curve must be unbounded - i.e. not contained in any sphere of finite radius. This seems correct intuitively, but I have no guarantee it is correct, ... |
Let $\{A_n\}$ be a sequence of events which may NOT be independent. We are asked to prove the following:
$$
P(A_n\ \text{i.o.}) = 1 \iff \text{for all $A$ with $P(A)>0$,}\ \sum_{n=1}^\infty P(A\cap A_n) = \infty.
$$
Here is what I have so far. I think if we assume that ∑ P(A ∩ An) = ∞, That means
P[(A ∩ An) i.o.]=1, ... |
I am an ESL learner and most of the time I am able to understand English but In my courses, we don't study mathematics, and I think that it is a mistake or wrong.
1-What are really the best sources (book or books) to study and review completely high school mathematics?
-titles
-authors
-edition
-publishers
2-What are r... |
Prove that any isomorphism between two cyclic groups always maps every generator to a generator.
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I have deleted the content of the original post.
The following exercise is inspired by answers to the questions this and this, but no knowledge of probability theory is required. The solution is straightforward assuming familiarity with the relevant theorems/properties. The proof is supposed to be informal.
Let $\hat \... |
Can someone please guide me to a way by which I can solve the following problem.
There is a die and 2 players. Rolling stops as soon as some exceeds 100(not including 100 itself). Hence you have the following choices: 101, 102, 103, 104, 105, 106.
Which should I choose given first choice.
I'm thinking Markov chains, ... |
If $(x_n)$ and $(y_n)$ are positive real sequences such that $(y_n)$ is bounded and $\lim\frac{(x_n)}{(y_n)} = 0$, prove that $\lim(x_n) = 0$
So since $(y_n)$ is bounded, there exists a real number $M$ such that for all n natural numbers, $(y_n) < |M|$
So by the limit definition :
$\lim|\frac{(x_n)}{(y_n)}-0| = \li... |
Let $H=\left\{ z\in\mathbb{C}\mid\Im\left(z\right)>0\right\}$ be the upper-half Poincare plane. Let $GL\left(2,\mathbb{R}\right)$ be the general linear group, $Z\left(GL\left(2,\mathbb{R}\right)\right)$ be the center of the general linear group and $O\left(2,\mathbb{R}\right)$ be the orthogonal subgroup of $GL\left(2,... |
I am trying to get a grasp on what a representation is, and a professor gave me a simple example of representing the group $Z_{12}$ as the twelve roots of unity, or corresponding $2\times 2$ matrices. Now I am wondering how $\operatorname{GL}(1,\mathbb{C})$ and $\operatorname{GL}(2,\mathbb{R})$ are related, since the e... |
I started reading about uniform spaces in Bourbaki, and the closure operation has been defined as such:
$$\bar{A}=\bigcap_{V\in\mathcal{U}}V(A)$$
Where $\mathcal{U}$ is the uniform structure of the space, and $V(A)$ is the set of all left-relatives of $A$. One thing I don't understand is why $\bar{A}=\bar{\bar{A}}$, a... |
Can someone please share references (websites or books) where I can find problems related with Fubini's theorem and applications of Radon-Nikodym theorem? I have googled yes and don't find many problems. What are the "typical" problems (if there are any) related with these topics? [Yes, exam is coming soon so I don't k... |
I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it involved de Rham cohomology, but I don't really know anything about the cohomology of Lie groups so this doesn't help me much. ... |
On a 2D plane, how to construct a graph with 7 vertexes and 21 edges? I tried various combination but couldn't seem to draw that kind of contrived graph on a paper.
But my understanding is that it is possible. So anyone can help me with it?
|
I have a network, with $V$ vertex, $E$ edges and $C$ cycles, which has “current” flowing in and out from the network, as shown below:
Each edge has a current value associated to it.
The equations governs the current flow in the network are the so-called “current conservation theorem”, which means that all the current ... |
Is Knopp's Theory and Application of Infinite Series out of date? It's looks terrific to me, but the Dover edition I bought new maybe a year ago: http://preview.tinyurl.com/2eprqps seems to be the same as an edition published in 1951 and may go back as far as 1921. 60 or 90 years is a lot of math years. How about it? D... |
I've never actually done a delta-epsilon proof, so I thought I'd try my hand at one. I decided to try it out for $f(x)=1/x$. If I understand correctly from the wikipedia article, I want to show for any $\varepsilon>0$, there exists a $\delta>0$ such that if $|x-c|<\delta$, then $|f(x)-f(c)|<\varepsilon$.
Anyway, I noti... |
Let $Z$ be a random variable with finite mean and consider the function $F: R \rightarrow R$ defined by $$ F(x) = E[\max(x+Z,0)]$$ How to prove that $F(x)$ is a continuous function of $x$?
This is not homework. A textbook I am reading states that $F(x)$ is continuous without providing any explanation, and I can't seem ... |
This is a slightly different question than the one asked here.
I understand that for non-planar graph there are edges in the graph which has more than 2 cycles sharing on it. But for planar graph, can we prove that it is always possible to find a complete set of cycle basis that each edge is shared by at most 2 cycles... |
Can this expression be simplified?
$ \sum_{i=0}^n (k^i)/i! $
EDIT:
This expression I have got as the number of possible ways to select n objects or less from k different infinite objects (you can select as many as you can from any type)...
I believe it must be equal to the number of possible ways to select n objects ... |
This is just a completely random question of the top of my head.
Let $\Omega$ be the set of bijections $\phi:\mathbb{N} \cup \{0\} \rightarrow \mathbb{Q}$.
We can imagine any $\phi \in \Omega$ to be equivalent to a wacky "staircase", where for $x \in \mathbb{R}^+$ we have $x \mapsto \phi(\lfloor x \rfloor)$.
If we walk... |
Many introductory books on vector spaces mention that the scalars need not be reals, and might even have sections discussing complex vector spaces or vector spaces over the integers mod 2. I have never seen any such book mention that all of the theory goes through as well if one restricts the scalars to be just ration... |
I've run across the recurrence $a_{n+1} = (a_n)^2 + 1$ in the past. Unfortunately, the referrence escapes me. However, my impression was that recurrences involving the product of previous terms (such as $a_{n+1} = (a_n)(a_{n-1})$) are difficult to solve. I'm wondering what is known for this very general problem.
(... |
A general problem for the Calculus of Variations asks us to minimize the value of a functional $A[f]$, where $f$ is usually a differentiable function defined on $\mathbb{R}^n$.
What if, however, the domain of $A$ is not actually all differentiable functions. Suppose there is a constraint equation on $f$, such as (for e... |
There are presumably three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:
axiomatically (with axioms concerning dimension)
by the abstract Euclidean group $E(n)$ (as its symmetry group, determining $E^n$ uniquely)
by presupposing a metric and requiring that the space is a... |
In a planar graph $G$, one can easily find all the cycle basis by first finding the spanning tree ( any spanning tree would do), and then use the remaining edge to complete cycles. Given Vertex $V$, edge $E$, there are $C=E-V+1$ number of cycles, and there are $C$ number of edges that are inside the graph, but not insi... |
(Pun definitely intended.)
Dear MSE-Community,
If I were to choose one artist that has made interesting works of art not only because of their beauty but also because of their connections to Mathematics, I would choose Escher. His works have always intrigued me. Some of his paintings are mind-boggling when looked at fo... |
For a constant, N, what value of x will maximize the cosine (or any trig) function?
\begin{equation}
1 = \cos{(Nx)}
\end{equation}
I am looking for the exact form, not the approximation because,
\begin{equation}
\frac{\arccos{(1)}}{N} = x = 0
\end{equation}
For example, WolframAlpha.com states that if N = 19.013, then,... |
I've been wondering about the following question:
Suppose that $P$ is a polynomial of degree $n$ with complex coefficients. Assume that $a_0, a_1, \dots, a_n \in \mathbb{C}$ are distinct. Are the polynomials $$P(x + a_0), P(x + a_1), \dots, P(x + a_n)$$ linearly independent?
|
Is it even worth the theorem below?
For every positive integer $n$, there is a real number $r$,
and $\frac{1}{12n+1} \lt r \lt\frac{1}{12n}$, such that:
$$ n! = \sqrt{2n\pi}\left(\frac{n}{e}\right)^n e^r.$$
I saw this statement on some sites, but got no further details.
I think the statement refers to an exact value ... |
Can we call the greatest integer function as a periodic function with no fundamental period or is it just non-periodic.
Please explain your answer.
To my understanding, if we consider $f(x) = [x]$ now, $f(3) = [3] = 3$ and $f(3+0.5) = [3.5] = 3$ So, can't we say that it is periodic (A constant function is periodic wit... |
What is the value of n $\in \mathbb{Z} $ for which the function $\displaystyle f(x) = \frac{\sin nx} { \sin \biggl( \frac{x}{n} \biggr) } \text { has } 4\pi $ as period?
Also could it be possible to solve this if we need $x\pi$ as period ?I am interested in learning the general approach for this particular type of the ... |
Let $B$ be the unit ball of $\ell^2(\mathbb{N})$, i.e. $B=\lbrace x\in \ell^2(\mathbb{N}): \|x\|\le 1\rbrace.$ For each $x=(x_1,x_2,\cdots)\in B$, let
$$f(x)=(1-\|x\|,x_1,x_2,\cdots).$$ Define $T:B\to 2^B$ by $$T(x)=B(f(x),r(x))\cap B, \mbox{ where }r(x)=\frac{1}{2}(\|x-f(x)\|).$$
Is it true that $$D(Tx,Ty)\le \|x-y\... |
Question: Let $V \subset M(n,n,\mathbb{R})$ be the set of all symmetric, real $(n \times n)$ matrices, that is $a_{ij} = a_{ji}$ for all $i,j$. Show that $V$ is a subspace of $M(n,n,\mathbb{R})$ and calculate dim$(V)$.
My attempt so far:
First part: To show that $V$ is a subspace I need to show:
(a) $ 0 \in V$ and
(... |
Given an angle $\theta$ (in radians), write $\theta = 2\pi t + \phi$, where $\phi$ is between $0$ and $2\pi$, $0\leq \phi\lt 2\pi$, and $t$ is an integer. Call $k=2\pi t$ the number of "full circles" or "full turns" corresponding to the angle.
If $\theta = \frac{13\pi}{4}$,
am I right that k=$2\pi$?
|
I heard that they are difficult. Is this true? Are they like the qualifying exams in grad school? For example, is the probability exam and the financial math exam comparable to qualifying exams (e.g. needs months and months of study)?
|
Let C be a category of sets, which has objects all sets and arrows all functions, with usual identity functions and the usual composition of functions. For any set S, the assignment s-s for all s in S describes the identity function. If S is a subset of Y, the assignment also describes the inclusion function from S to ... |
Can $n!$ be a perfect square when $n$ is an integer greater than $1$?
Clearly, when $n$ is prime, $n!$ is not a perfect square because the exponent of $n$ in $n!$ is $1$. The same goes when $n-1$ is prime, by considering the exponent of $n-1$.
What is the answer for a general value of $n$? (And is it possible, to p... |
Please excuse the naive question. I have had two classes now in which this theorem was taught and proven, but I have only ever seen a single (indirect?) application involving the quantum harmonic oscillator. Even if this is not the strongest spectral theorem, it still seems useful enough that there should be many nic... |
I'm not quite sure where such a concept fits. Suppose $X$ is a topological space. I know then that the diagonal $\Delta=\{(x,x)\ | x\in X\}$, so $\Delta\subseteq X\times X$. What then would a neighborhood of the diagonal be comprised of?
By this I mean, suppose $E\in\mathscr{N}_\Delta$, where $\mathscr{N}_\Delta$ is t... |
If topology is called rubber-sheet geometry, would it be accurate to describe the "cut and shuffle" topic of "piecewise isometries" as broken glass geometry ?
Isometry sounds more geometrical than topological. What would be a good name for "piecewise topology" ? If you take a topological space and break it up and put ... |
Sorry if this is a really naive question, but in my reading of a lot of textbooks and articles, there is a lot of mention of how many decimals we know of a certain number today, such as $\pi$ or $e$. An excerpt from my textbook:
In 1748, Leonard Euler used the sum of the infinite series of $e$ (mentioned in the book i... |
Theorem: If a function f is analytis throughout a simply connected domain D, then for every closed contour C lying in D, the integral is o along the contour.
This is a theorem from the text I am using without complete proof. The problem is that when the closed contour has an infinite number of self-intersection points.... |
I want to solve equations like this (mod $2^n$):
$$\begin{array}{rcrcrcr} 3x&+&4y&+&13z&=&3&\pmod{16} \\ x&+&5y&+&3z&=&5&\pmod{16} \\ 4x&+&7y&+&11z&=&12&\pmod{16}\end{array}$$
Since we are working over a ring and not a field, Gaussian elimination doesn't work. So how can I still solve these types of equations?
|
How to prove this: $$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$$
From Apostol's number theory text i know that $$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{x}} + A + \mathcal{O}\Bigl(\frac{1}{\log{x}}\Bigr)$$ But how can i use this to prove my claim.
|
Suppose I take $n$ sets $\{S_i\}$ where each $S_i$ is some subset of integers. Let $P$ indicate a set of integers obtainable by drawing exactly one factor from each $\{S_i\}$ and multiplying them together. What conditions on these sets do I need to ensure that
$P$ forms a contiguous interval
Each element of $P$ has a ... |
Just curious. You pay electronically, and you get a string of numbers. My question, what is the likelihood for a person to type in the same serial number to say, top up their mobile credit? What is the range of probability electronic payment companies are using?Then, does electronic payment in China have longer serial ... |
If $K$ is a field complete with respect to a non-archimedean absolute value, then the $n$th term test (checking whether the $n$th term of a series goes to zero) is sufficient to check convergence of a series in $K$.
I was wondering if we have any results that guarantee, under additional hypothesis, whether a series in ... |
Every Fourier transform formula that I know of has a corresponding Fourier series analog, except the multiplication formula
$$\int_{-\infty}^\infty f(x)\hat{g}(x) dx=\int_{-\infty}^\infty \hat{f}(y)g(y) dy.$$
So my question is whether a Fourier series analog for this formula exists.
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Let $f$ be Lebesgue measurable in $[0,1]$ and assume $f$ takes finitely many values. Assuming $f(x) - f(y)$ is Lebesgue measurable in $[0,1] \times [0,1]$ show that $f$ is integrable over $[0,1]$.
Stuck for a while with this one. (Not homework, just practice)
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I wanted to ask these questions here, feeling that this place place is without doubt is better than anywhere else, feeling hesitant and shy to ask such question.
For several reasons, the environment which I live in doesn't encourage the learning of natural sciences, it permits it but doesn't put efforts in teaching it.... |
Let $\rho : S_n \to \mathrm{GL}(\mathbb{C}^n)$, where $\rho(\sigma)(x_1, \ldots, x_n) = (x_{\sigma^{-1}(1)}, \ldots, x_{\sigma^{-1}(n)})$. How can you prove that $W = \{ (x_1, \ldots, x_n) : x_1 + \cdots + x_n = 0 \} \subset \mathbb{C}^n$ is an irreducible representation?
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A doubly-stochastic matrix is an $n \times n$ matrix $P$ such that
$$ \sum_{i=1}^n p_{ij} = \sum_{j=1}^n p_{ij} = 1 $$
where $p_{ij}\ge 0$. Can someone please suggest an algorithm for generating these matrices uniform randomly?
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I have a homework problem in my textbook that has stumped me so far. There is a similar one to it that has not been assigned and has an answer in the back of the textbook. It reads:
How many ways are there for $8$ men and $5$ women to stand in a line so that no two women stand next to each other?
The answer is $609638... |
Suppose $\frac{dx}{dt}=ax+b$ and then assume that $a=c+g$ where $g$ is a Wiener process.
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Is there a "SETI at Home" style initiative for calculating constants with distributed computing?
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For what values of $x = (x_0 x_1 \ldots x_n)$ the following inequality holds for all positive values of the vector $a = (a_0 a_1 \ldots a_n)$: $a x^T \ge 0$?
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