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Is there a unique normal form for
each formula of predicate calculus?
I am aware of prenex normal forms of predicate calculus and of disjunctive and conjunctive normal forms of propositional calculus. But what is the name of the combination of the two, i.e. the formula is in prenex normal form and the (quantifier-f... |
How do i expand and simplify the vector expression
$(\vec{a}-\vec{b})^T.(\vec{a}-\vec{b})$ = ?
And if there are matrices A and B instead of vectors a and b, how do i multiply and simplify this expression?
$(A-B)^T.(A-B)$ = ?
Also in the normal equations for solving least squares problems, we have expressions of the typ... |
I have found some pretty complete lists (I think) of mathematical symbols here and here, but I don't see a symbol for the word "and" on either list. A person could easily just write the word "and" or use an ampersand, but I was wondering if there was an actual mathematical symbol for the word "and". Also, if anyone k... |
If there are two different proofs for one theorem, at some level are the two proofs the same, or can they be fundamentally different?
In other words, if you have two proofs of a theorem, can one show that the two proofs are expressing the same thing in different ways, and then remove the redundancies and generate a "sh... |
So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so here I'm asking for suggestions for visualization methods.
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Suppose I found a preprint on arXiv.org (like this one: http://arxiv.org/abs/math/0309146 ), and I need to check if what's in there is actually true. Any tips? I don't want to study an article only to later find that it contained an error <_<
More generally, should I spend my time on arXiv.org preprints? How serious is... |
Prove that the set of functions $f: \mathbb{Q} \rightarrow \{1,2,3\}$ is uncountably infinite.
I'm totally stuck on this one. We have just been shown Georg Cantor's diagonalization argument in class for why $\mathbb{R}$ is uncountably infinite (and I for the most part understand it---assemble an enumeration for the rea... |
The following question was part of a problem sheet in numerical maths:
Given the scalar equation $\ddot{y} = - \lambda y$, $\lambda > 0$
show that for $h > h_{critical}$, the approximate solution goes to infinity as $t$ goes to infinity. This behaviour is called blow-up.
The Stormer-Verlet-method is $y_{k+1} = -y_{k-1}... |
Show that if a manifold $M$ is contractible, then every vector bundle over $M$ is equivalent to the trivial bundle.
Got this as homework but I'm kind of lost in the hole vector bundle subject, so if anyone could give some hints to this one as well as to understanding the subject, it would be highly appreciated.
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Wikipedia says that if $a\le b$ and $b\le a$ in Rudin–Keisler order for ultrafilters $a$ and $b$, then $a$ and $b$ are Rudin–Keisler equivalent. How to prove this?
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In 2D or 3D I have a fix point y and Gaussian distribution of a random point x. I am now interested in the mean euclidean distance between x and y:
$E_x [ d(x,y) ] = \int _{-\infty }^{\infty } d(x,y) N(x; \mu, \Lambda )dx$
Many thanks in advance
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Given a sequence of sets, is there some well-defined notion of a limit of a set?
In other words, given some universe set $U$, I am wondering if there is a topology on $2^U$ (the powerset of $U$) such that the usual intersection and the union limits converge in that topology.
As an explicit example, let $U=\mathbb{N}$,... |
I'm trying to decipher a differential geometric comment on page 23-24 of Berline, Getzler, and Vergne's "Heat Kernels and Dirac Operators".
Take a trivial vector bundle $E \times M$ in a manifold $M$ with connection $\nabla = d + \omega$ where $\omega$ is an $End(E)$-valued 1 form. Let $g: GL(E) \to End(E)$ be the tau... |
Background
I am trying to extract data from scientific publications. Sometimes an experiment can test two factors, e.g. a standard two-way factorial ANOVA.
Call these factors $A$ and $B$. If factor $A$ has two levels and $B$ has three, there are six total treatments.
If the effects of $A$ and $B$ are significant, but ... |
for $f \in L^1_{loc}$, let the sharp-function be defined as
$f^\sharp(x) := \sup_{B \in x} |B|^{-1} \int_B |f(y) - |B|^{-1} \int_B f(z) dz | dy$
We define the space $BMO \subset L^1_{loc}$ via
$BMO := \{ f \in L^1_{loc} \;\;\;s.t.\;\;\; f^\sharp \in L^{\infty} \}$
and the BMO-(semi)norm on BMO as
$\Vert f \Vert_{BMO} ... |
I have a determinant to expand which is
$$\triangle =
\begin{bmatrix}
p& 1 & \frac{-q}{2}{}\\
1& 2 &-q \\
2& 2 & 3
\end{bmatrix} = 0 $$
But when I am expanding the determinant along the first row such as $ p(6+2q) - (3+2q) = 0 $ but when I am trying to expand along first column I am getting $p(6+2q) - (3+q) = ... |
It is clear that cyclic groups have the property that they cannot be written in a non-trivial way as an amalgamated free product or as an HNN-extension.
Can someone please provide us examples of torsion-free 2-generated groups having this property? Any comments related to this question are welcome!
[I'm aware of Serre'... |
While looking through our Analysis Script, I got suddenly aware that we had already learned quite an amount of rules and laws for calculating limits.
We got:
$\displaystyle \lim_{x \to a}\;c = c$
$\displaystyle \lim_{x \to a}\;x = a$
If the limits exist, one gets with corresponding limits $u,v$
$\displaystyle \lim_{... |
I want to find out, how many ways I can produce an average 4.6 of 13 numbers, when I have only numbers {1,2,3,4,5}.
I thought, that it could be done by generating functions, but since my knowledge on them isn't very deep yet, I'm a little lost. Also I have a hard time dealing with the average not being integer.
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I would appreciate it if someone could explain the difference(s) between a
$C^\infty$ and a $C^\omega$ surface embedded in $\mathbb{R}^3$.
I ran across these terms in M. Berger's Geometry Revealed book (p.387).
The context is: There are examples of two different $C^\infty$ compact surfaces
that are isometric, but no kn... |
I have a natural number $m$, and a polynomial $p: \mathbb{Z}^n \rightarrow \mathbb{Z}$, and I need to find a set $(x_1, x_2,...,x_n)$ so that $p(x_1, x_2,...,x_n) \bmod m = 0$.
First: Does every non-constant polynomial have such a set? And more importantly: Does there exist an (efficient) procedure to determine it, in ... |
In my algebraic topology class we showed how to calculate the homology groups of $S^n$, using the tools of singular homology, however we did not discuss other ways of doing it; my question - is there any relatively simple way of doing this, using simplicial homology? I tried thinking about this for a bit, but couldn't ... |
The hypothesis of theorem 5.5 in page 152 of Hungerford's Algebra requires that the homomorphism maps the identity of R to the identity of S, but the proof "never" use this hypothesis. Is the theorem still true if the stronger hypothesis is removed. I am so sorry that I currently do not have link to the book.
Theorem 5... |
Let $v_n$ be the volume of the unit ball in $\mathbb{R}^n$. Show by using Fubini's theorem that $$v_n = 2\,v_{n-1}\,\int_0^1{(1-t^2)^{(n-1)/2}\,dt}\,.$$
I've been trying to do this inductively, and I've been thinking about the integral as kind of a generalization of the integral formula for the area of (part of) the u... |
Can you give examples of combinatorial construction riddles, approachable by gifted high school students?
Examples:
Find a finite set $A$ and $B \subset 2^A$ such that any element of $A$ is covered by at least 100 sets in $B$, but $B$ cannot be partitioned into 2 disjoint sets, each covering $A$.
Find the shortest seq... |
It's late at night and I'm tired, but I just stumbled across this while doing my homework. Any chance this is new? Or, maybe, did I just somehow transform it and it is still basically the same formula? In that case, forgive me, please.
Anyway, here it is. It is a recursive function (thus, of limited use!?) to calculate... |
Greetings All
I have some test matlab code which can scale and do rotation but the
translation (tl value) doesn't seem to be working.
I expected the entire object to be moved over x=2 and y=0 but it
doesn't move any ideas?
clf; clear all
rp=[0 4 2 -1 0; 0 1 -4 -3 0;0 0 0 0 0] %x-y values
scalenum=2 %scale factor... |
So I have a test on abstract algebra tomorrow. It is basically on rings and polynomials in a ring/field.
I just have a couple of questions about terms. The definition of an isomorphic ring states that
there is a function $f: R \rightarrow S$ such that:
i) $f$ is injective;
ii) $f$ is surjective;
iii) $f(a+b) = f(a)+f(... |
I'm trying to prove that if $E \subset \mathbb{R}$ has finite measure and $\varepsilon \in \mathbb{R}$ such that $\varepsilon>0$, then $E$ is the disjoint union of a finite number of measurable sets, each of which has measure at most $\varepsilon$.
Here is what I've done:
Let $\varepsilon \in \mathbb{R}$ such that $\v... |
Given a Noetherian ring $R$ and a proper ideal $I$ of it. Is it true that $$\bigcap_{n\ge 1} I^n=0$$ as $n$ varies over all natural numbers?
If not, is it true if $I$ is a maximal ideal? If not, is it true if $I$ is the maximal ideal of a local ring $R$? If not, is it true under additional assumptions on $R$ (like $R... |
Let $f$ be analytic on the open set $G\subset \mathbb{C}$. What's the best way of showing that the function $\phi:G \times G \to \mathbb{C}$ defined by $\phi(z,w)=[f(z)-f(w)]/(z-w)$ for $z \neq w$ and $\phi(z,z)=f'(z)$ is continuous?
This is obvious for $z\neq w$, but I'm having some trouble on the points $(z,z)$. I di... |
I have discovered that Groebner basis computations may help in a problem I am working on. However, I am having some very specific problems. First, the literature I have discovered on Groebner basis calculations are either extremely advanced or sacrifice completeness for simplicity. Secondly, to come to grips with these... |
This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list.
$1.$ Story of maths Part1 Part2 Part3 Part4
$2.$ Dangerous Knowledge Part1 Part2
$3.$ Fermat's Last Theorem
$4.$ The Importance of Mat... |
I am using the Mathieu equation
$$x''(t) + a x(t) + 2 q \cos(2 t) x(t) = 0,$$
as a model of a physical system. The conditions on the parameters $a$ and $q$ for stable solutions to exist are well known and are even implemented as built-in functions in systems such as Mathematica.
Now, I would like to extend my model ... |
I'm not much good at even basic number theory, so this is really mysterious to me. I think the actual question might want to restrict to finite Laurent series, but I don't think it matters: I'm pretty sure what I want to do is figure out what each homogeneous component is allowed to look like. Moreover, I'm pretty su... |
I have never seen a problem like this before, so I was wondering if anyone could give me help getting started. I'm studying for a quiz on Wednesday.
Find an equation of the sphere that passes through the origin and whose center is (5, 10, -9).
___ = 0
Note that you must put everything on the left hand side of the equa... |
For example:
Suppose vector $u = (-2,3)$ and
vector $v = (-5,3)$ then:
$(u + v = ?)$ and
$(u - v = ?)$ and
$(v - u = ?)$ and
$(6u = ?)$ and
$(-1/8v = ?)$ and
$(3u - 4v = ?)$
|
Suppose one is trying to devise a method to generate random matrices with a certain distribution. How does one verify that the generated matrices follow the desired distribution? In particular, I am interested in uniformly distributed random orthogonal matrices, but I am curious how people test their codes for correctn... |
I didn't think it was possible to have a finite area circumscribing an infinite volume but on page 89 of Nonplussed! by Havil (accessible for me at Google Books) it is claimed that such is the goblet-shaped solid generated by revolving the cissoid y$^2$ = x$^3$/(1-x) about the positive y-axis between this axis and the ... |
I have found formulas online for the the roots of a cubic polynomial with real coefficients, but they all said this they would not work for polynomials with complex coefficients. I need to solve the following:
$x^3+cx^2+\alpha$ where $c>0$ and $\alpha\in\mathbb{C}$. Anyone have the formula
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Given is an almost-square matrix $A$ with $n$ columns and $n-1$ rows with maximum rank. The solutions of the homogeneous system $Ax = 0$ form a 1-dimensional subspace of $\mathbb{R}^n$.
I've discovered the following which I believe to be true but I can't prove: the components of the vector $x$ that spans the (1D) solut... |
A first-order sentence is (logically) valid iff it's true in every interpretation. And it's valid iff it can be deduced from the FO axioms alone.
One normal case of showing that a FO sentence is true is deducing it (syntactically).
I guess that indirect proofs have to be interpreted more "semantically":
"Assume that ~... |
I am reading an article and it is written there:
If $A$ is a $k$-linear category (possibly without direct sums) we can embed it in the
additive category $A \times \mathbb{N}$, where a morphism $(x,m) \mapsto (y, n)$ is an $n \times m$ matrix
with entries in $A(x, y) = Hom_A(x, y)$. Of course if $A$ is additive then $A... |
Can anyone explain me as to why Tensor Products are important, and what makes Mathematician's to define them in such a manner. We already have Direct Product, Semi-direct products, so after all why do we need Tensor Product?
The Definition of Tensor Product at Planet Math is confusing.
Definition: Let $R$ be a commuta... |
Summations can really get complicated - esp. when you have convoluted n-fold summations with all kinds of different indices.
My question: Is there some software (or add-on) with which you can find and prove (complicated) summation identities?
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Simplifying a business example, I have the following situation:
Some objects should be distributed in a graph in most "linear" way possible for a given "thermometer".
Say, a voyager visits some cities. Several cities are visited multiple times.
So, we have list of cities in ordinate axis (that may be duplicated), and T... |
$$f_T(t;B,C) = \frac{\exp(-t/C)-\exp(-t/B)}{C-B}$$
where our mean is $C+B$ and $t>0$.
so far i have found my log likelihood functions and differentiated them as follows:
$$dl/dB = \sum[t\exp(t/C) / (B^2(\exp(t/c)-\exp(t/B)))] +n/(C-B) = 0$$
i have also found a similar $dl/dC$.
I have now been asked to comment what you... |
Are there any ways to visualize(plot/draw) points on a high dimensional (ex: dimension = 5) manifold?
|
In Poisson distribution, the probability of inter arrival time to be t or less is:
$$ P(X\leq t)= 1 - P(X>t) = 1 - P(0 \mbox{ arrivals in } t) = 1 - e^{-\lambda t} $$
and probability of one arrival in t is:
$$ P(k=1)= \lambda t e^{- \lambda t} $$
I wonder how the exponential distribution can be derived from Poisson to... |
I know that the answer is $C(8,2)$, but I don't get, why.
Can anyone, please, explain it?
|
Nakayama's lemma states that given a finitely generated $A$-module $M$, and $J(A)$ the Jacobson radical of $A$, with $I\subseteq J(A)$ some ideal, then if $IM=M$, we have $M=0$.
I've read the proof, and while being relatively simple, it doesn't give much insight on why this lemma should be true, for example - is there ... |
I have a quadrilateral whose four sides are given
$2341276$, $34374833$, $18278172$, $17267343$ units.
How can I find out its area? What would be the area?
|
I am trying to find an accurate way of calculating the capacity of an underground tank at a given depth. The tank manufacturer has provided a strapping table for the tank which tells me the capacity at various depths.
Gauge Depth (cm) / Capacity (Liters)
2cm = 29.8 liters
12cm = 240.4 liters
...
66cm = 2372.8 liters... |
Why is the following tuple of vectors not a basis of $\mathbb{R}^{2}$? $\left ( \left ( \begin{array}{c} 1\\ 0\\0 \end{array} \right ), \left ( \begin{array}{c} 0\\ 1\\0 \end{array} \right ) \right )$
I would've thought that just like $\left ( \left ( \begin{array}{c} 1\\ 0\end{array} \right ), \left ( \begin{array}... |
I need a little bit of help, I found that theorem, but the book doesn't prove it and gives a reference to another book that I don't have; does anyone have an idea?
Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. Show that $M$ is free if the localizations $M_m$ have the same rank for all maximal i... |
Suppose $f(x_1,x_2,\dots,x_n)$ is a multivariable function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that for all partial derivatives, $1\le i \le n$,
$$\frac{\partial f}{\partial x_i}(q_1,q_2,\ldots,q_n) \ge 0$$
for all $q_i \ge r_i$.
Is f a non-decreasing function in the set of all points $q_i \ge r_i$ for all... |
(Application of the rank-nullity theorem) Suppose $S,T: V\to V$ are linear transformations of a finite dimensional vector space $V$, and that the composition $ST\colon V\to V$ is invertible. Show that then $T$ is one-to-one, and deduce that it is invertible. Show also that $S$ is invertible. Deduce that $ST$ is invert... |
In contrast to the possibility of taking an arbitrary sequence of elements of submodules in the definition of direct product, the definition for the direct sum of submodules of a module requires the indexed elements to vanish cofinitely(i.e. except finitely many times).
More precisely, Let $R$ be a ring, and $\{M_i : ... |
I was reading this book about medicine and I found this paragraph in it
A person's daily risk of getting lung illness ... is $0.5$ percent. The daily risk of getting a bad lung illness ... plus treatment is $0.05$ percent. So, you are looking at the difference between a $99.5$ percent chance of staying well, and a $9... |
If $N$ points on the circumference of a circle are chosen at random, what is the probability $F(\theta)$ that the maximum gap between neighboring points is at least $\theta$? Because the gaps sum to $2\pi$, the maximum must be at least $2\pi/N$, so $$F(\theta)=1 \text{ for } \theta\le\frac{2\pi}{N}.$$ At the other e... |
I have an integral:
$$\int_0 ^a\int_0 ^b\int_0 ^a\int_0 ^b \sin(x)\sin(\bar{x})\sin(y)\sin(\bar{y})f(x,\bar{x},y,\bar{y}) \, dx \, dy \, d\bar{x} \, d\bar{y}$$
Where:
$$f= \dfrac{\sin(\sqrt{(x-\bar{x})^2+(y-\bar{y})^2})}{(x-\bar{x})^2+(y-\bar{y})^2}$$
Under the transformation $\bar{x}=x+u$ and $\bar{y}=y+v$ $\Rightarr... |
So, here's the problem (1.5.12, Probability and Statistics, Degroot and Schervish 3e) :
Let $ A_1, A_2, ... $ be an arbitrary infinite sequence of events, and let $ B_1, B_2, ... $ be another infinite sequence of events defined as follows: $B_1 = A_1, B_2 = A^c_1\cap A_2, B_3 = A^c_1 \cap A^c_2 \cap A_3 ... $
Prove th... |
I want to compute the integral
$\int_{[0,1]^m} \sqrt{\sum_{j=1}^m u_j^2} \ du_1 \dots du_m$.
Wolfram Alpha gives an exact result for $m = 2$, but fails to do so for higher values. I am particularly interested in the asymptotics.
I'm sure the value is a standard result, but don't know where to find it (no, I don't have... |
I know how to compute the probability that $0$ runs (or one or more) of $M$
consecutive successes will occur in a sequence of $N$ trials, each with probability $P$ of success.
Is there an explicit formula (mass function) or a method of calculation (for example in Maple) to find the probability that exactly $R$ runs of... |
I am reading the wikipedia article on the Laplacian matrix:
https://en.wikipedia.org/wiki/Laplacian_matrix
I don't understand what is the particular use of it; having the diagonals as the degree and why the negative adjacency elements off the diagonal? What use would this have?
Then on reading about its norm, first of ... |
If $z = x + D(x + y)$ and I let $g(x,y) = x + y$
Would I be right in saying that :
$z = x + f(g(x))$
and
$\frac{dz}{dx} = 1 + \frac{df}{dg}\cdot\frac{dg}{dx}$?
|
Take 13 integers. Prove that if any 12 of them can be partitioned into two sets of six each with equal sums, then all the integers are the same.
Does anyone know if the general case with 2n+1 integers and sets of n integers results in the same conclusion?
|
To compute the condition
$$ \kappa_{\mathrm{abs}}^\infty := \max\{\| W(t,t_0, y_0) \|: t_0 \leq t \leq T \}$$
of an initial value problem on $[t_0, T]$, I need to compute the Wronski matrix $W$.
The approximation of the solution is denoted by the discrete evolution operator $\Psi^{t_0, t}$ and the exact solution is de... |
The class number of $\mathbb{Q}(\sqrt{-23})$ is $3$, and the form
$$2x^2 + xy + 3y^2 = z^3$$
is one of the two reduced non-principal forms with discriminant $-23$.
There are the obvious non-primitive solutions $(a(2a^2+ab+3b^2),b(2a^2+ab+3b^2), (2a^2+ab+3b^2))$.
I'm pretty sure there aren't any primitive solutions, but... |
What about $GL(n,\mathbb C)$? Is it open, dense in $M(n,\mathbb C)$?
|
I would be glad if someone can help me understand the argument in the first paragraph of page 4 of this paper.
Especially I don't understand their first sentence,
"Using N bosons (fermions) distributed over m states, one can construct completely symmetric (antisymmetric) irreducible representations of the group U(m) a... |
Let us start by clarifying this a bit. I am aware of some proofs that irrationals/reals are uncountable. My issue comes by way of some properties of the reals. These issues can be summed up by the combination of the following questions:
Is it true that between any two rationals one may find at least one irrational?
... |
I am looking for an equation to calculate the volume of liquid in an underground tank based on a depth reading.
The tank is an ellipsoid shape with the following dimensions:
X-Axis Radius: 30.425 cm
Y-Axis Radius: 30.425 cm
Z-Axis Radius: 60.95 cm
I am trying to find an equation that would provide the volume... |
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$?
I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward.
Thanks.
|
Hoping to run into some mathcadders:
I have a worksheet which includes the following except (and error)
Any help?
|
The basic problem is that I want to be extremely clear about the sets
that mathematical manipulations and operations are taking place in, I am
hoping for someone who really understands this to read what I've written
closely and point out what is getting me all mixed up, though of course
reading &/or responding isn't ma... |
After reading the wiki page on Enigma's, I am still confused as to how it works. Are there some sources that run through how the enigma machine works that is easy to follow. Which also gives information about the how codes are encrypted and decrypted?
|
On the sound of sounding ridiculous, but in the line of "There are no stupid quetsions": Is there a way to express $\omega_1$ (and in general $\omega_k$ with $k >= 1$ as a Conway game (that is $<L|R>$, with L and R the left and right options). And is there a way to do it such that addition, multiplication, etc. makes s... |
Is this possible? I have a 4-connected 17-graph and am wondering whether there are another 3 like it that can be put together to make its complement. My current 4-connected example is here
|
In my stochastics class we were given the following problem ($\mathcal{B}(\mathbb{R})$ stands for the Borel $\sigma$-algebra on the real line and $\mathbb{R}$ stands for the real numbers):
Let $f : \Omega → \mathbb{R}$ be a function. Let $F = \{ A \subset \Omega : \text{ there exists } B \in \mathcal{B}(\mathbb{R}) \te... |
arctan(n+1) - arctan(n+2) where n varies from 0 to infinity
I'm having trouble figuring out this problem. I've calculated the first 5 terms of the series and ended up with the following:
-.0.32175 - 0.14189 - 0.07677 - 0.07677 - 0.04758 ......
To me, it is apparent that the series is converging on some number (perhaps ... |
For example, how many times ($N$) do I need to flip a coin ($p=0.5$) to have a $P=90\%$ probability of having observed $20$ heads. I empirically found that I need $N=20+28=48$.
Is it correct?
Is there an explicit formula for the Negative binomial inverse cumulative function?
|
I have this question:
Prove/disprove: The set of natural numbers (including zero) with usual topology is the continuous image of the Sorgenfrey line.
Can't we take the map $g: \mathbb{R}_{l} \rightarrow \mathbb{N}$ given by $g(x) = |[x]|$ ? (i.e the absolute value of the floor function).
|
If $R=k[t,t^{-1}]$ is a graded ring where $R_0=k$ is a field and $t\in R$ is a homogeneous element of positive degree which is transcendental over $k$, how can I prove that every graded $R$-module is free?
|
This is a problem that I'm attempting to solve (NOT homework!):
There are $10$ trees. Each one is planted in the center of the circle of space that has a diameter of $16$. The $10$ circles (each with diameter of $16$) are placed in the shape of an equilateral triangle ($4$ circles, then $3$ circles on top, $2$ circles ... |
This is a little exercise found in Robin Hartshorne's Euclid: Geometry and Beyond:
I believe I have found a solution, which only exists when $AB$ has length at least that of the chord bisected by $P$, and length at most the diameter of the circle. So suppose that is the case. I do the following construction. $\odot cX... |
How does one solve the equation $2e^x (x-1)=0$ ?
|
I am looking for information for some extended family members.
A medical procedure has a $4$% chance of stroke per year over $5$ years.
One family member has said that that is a $20$% chance of stroke over all ($5\times 4$%).
Others argue that it is less because $4$% each year is smaller.
I am not sure how to frame... |
Let $X = [0,1]^{\omega_1}$ (that is, all functions from the set of countable ordinals to the unit interval) have the product topology. Let $Y$ be all functions in $X$ with countable support (that is, all points $(x_\alpha) \in X$ with $x_\alpha = 0$ for all but countably many $\alpha \in \omega_1$). I am trying to find... |
Let $C^1[0,1]$ be space of all real valued continuous function which are continuously differentiable on $(0,1)$ and whose derivative can be continuously extended to $[0,1]$. For $f$ in that set define the norm of $f = \max{\{\|f(t)\|, \|f'(t)\|\}}$ where $t$ is in $[0,1]$. can you show this space is complete under thi... |
This should be pretty simple. Here is what I want to put into a formula/algorithm:
Divide the size of every object of an array by the number of times that object has been accessed in the last n seconds and discarding the highest quotient. (You can probably ignore the bold part)
I can program this just fine, but I am... |
I'm working through Stanford's computer vision course to refresh some of my math skills (http://vision.stanford.edu/teaching/cs223b/syllabus.html), and I've run into a problem while working with a simple 2d Support Vector Machine.
Basically, the SVM library I am using only returns the vector 'w', which is the normal to... |
How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$?
Can it be proved easily?
|
Is there a real function over the real numbers with this property $\ \sqrt{|x-y|} \leq |f(x)-f(y)|$ ?
My guess is no but can anyone tell me why?
This came up as a question of one of my collegues and i cant give an answer.
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$$\sum_{m=2}^{10^6}\ \sum_{n=1}^{\infty}\dfrac{m^{\{n\}}+m^{-{\{n\}}}}{m^n}$$
Where $\{n\}=\text{ integer nearest to the } \sqrt{n} $.
Looks like a non-trivial summation. How to properly approach this problem?
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Assume $M=N_1+N_2$ is a module, where $N_1,N_2$ are noetherian modules. How can I show that $M$ is also noetherian?
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It can be proved that arbitrary union of open sets is open. Suppose $v$ is a family of open sets. Then $\bigcup_{G \in v}G = A$ is an open set.
Based on the above, I want to prove that an arbitrary intersection of closed sets is closed.
Attempted proof: by De Morgan's theorem:
$(\bigcup_{G \in v}G)^{c} = \bigcap_{G \... |
There are ten questions on a piece of paper. Your task is to fill in each blank with a positive integer less than 10 such that there is no contradiction. You can reuse any digit.
The question is as follows:
Digit 0 appears __ times on this paper.
Digit 1 appears __ times on this paper.
Digit 2 appears __ times on this... |
Let me prove subset $A$ of $\mathbb{R}$ is open if and only if it is a countable union of open intervals.
For all $x \in A$ there is an $\epsilon > 0$ such that $(x-\epsilon,x+\epsilon)$ is an open interval contained in $A$. Now find rationals such that $r_{x} \in (x-\epsilon,x)$ and $s_{x} \in (x,x+\epsilon)$ and $A =... |
Let $M(n;\mathbb{R})$ denote the set of all $n \times n$ matrices with real entries (identified with $\mathbb{R}^{n^{2}}$ and endowed with its usual topology) and let $GL(n;\mathbb{R})$ denote the group of invertible matrices. Let $G$ be a subgroup of $GL(n;\mathbb{R})$. Define $$H = \biggl\{ A \in G \ \biggl| \ \exis... |
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