instruction stringlengths 22 15.8k |
|---|
The Collatz function $T$ is defined on the set $\mathbb{Z}^+$ of positive integers
as: $T(n)=n/2$ if $n$ is even, and $T(n)=3n+1$ if $n$ is odd. Let $T^k$ be the $k$th iteration
of $T$. We say $n$ terminates if $T^k(n)=1$ for some $k$.
Let $n$ be an integer of the form $$3^{2^k(j-1)}+3^{2^k(j-2)}+\cdots +3^{2^k}+1$$
wh... |
My understanding right now is that an example of conditional independence would be:
If two people live in the same city, the probability that person A gets home in time for dinner, and the probability that person B gets home in time for dinner are independent; that is, we wouldn't expect one to have an effect on the ot... |
I am a student taking a "discrete maths" course. Teacher seems to jump from one subject to another rapidly and this time he covered ring theory, Z/nZ, and polynomial rings.
It is hard for me to understand anything in his class, and so the reports he gives become very hard.
I did my best to find answers using google, bu... |
I'm exploring the use of FFTs for multiplication, but even with simple examples it seems to go wrong. For example, here I'm trying to multiply $1$ by $2x$ (code is in matlab, but I think you can probably understand it even if you don't know the language):
# x and y are two polynomials, x(n) = 1; y(n) = 2n
x = [0 0 0 1]... |
Could you help me to show: $g(x,\epsilon)+f(x,\epsilon)=O(|\phi(x,\epsilon)|+|\psi(x,\epsilon)|)$
but $g(x,\epsilon)+f(x,\epsilon)\neq O(\phi(x,\epsilon)+\psi(x,\epsilon))$ (both when $\epsilon\to0$), where $O$ stands for the big Oh notation?
|
There is an ancient problem, I remember reading once in a book while I was a kid, that says: there was a father who had $N$ sons and $T$ cows. He divided the cows among the sons in the following order: for the first son, $1 + 1/7$ of the remaining; for the second son $2+ 1/7$ of the remaining; ..., for the $i$'th son $... |
Suppose $s_{1}$ are the numbers with 10-digits that have $1$ prime factor.
Suppose $s_{2}$ are the numbers with 10-digits that have $2$ prime factors.
Suppose $s_{n}$ are the numbers with 10-digits that have $n$ prime factors.
Leading $0$s are allowed. Which of $s$ would have the smallest expected value of tries to g... |
I've encountered this whilst writing an optimisation benchmark for some heuristic search algorithms. Feels like there should be a basic solution out there!
A square-grid graph is constructed from $n \times m$ vertices embedded into the cartesian plane, such that each vertex corresponds to a pair of integers. Say that a... |
I am familiar with using cross products in physics to answer questions about force and torque, but are there applications to other scientific fields? The examples that I've seen in most calculus textbooks deal only with physics or engineering questions.
|
You need to arrange a list of participants - four administrators and four students - who will sit behind a table in the order listed. In how many ways can you list them if you must alternate students and administrators?
This question is from Discrete Mathematics for Computer Scientists by Stien, Drysdale and Bogart.
H... |
Is it a mistake ab initio to think of categories as of models of category theory, just as we think of (inner) models-of-set-theory as of models of set theory, graphs as of models of graph theory, groups as of models of group theory, topologies as of models of topology, and so on?
There is one big difference, and maybe ... |
Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$
I understand that in order to find the solution, I need to use partial derivatives. However, the method in my textbook works for simpler proble... |
For $\alpha\in\mathbb{R}$, it seems that we have $\displaystyle \sum_{k \ge 0}\frac{1}{k^2+k-\alpha} = \frac{\pi}{\sqrt{4\alpha+1}}\tan\left(\frac{1}{2}\pi\sqrt{4\alpha+1}\right).$
I've tried many techniques and ways -- partial fractions, turning it into an integral, finding the partial sum and so on -- to show it, bu... |
I'd like to know how to derive these functions (I know the answers, I want to know how to get there)
\begin{align*}
f(x) &= \arcsin\left(\frac{x}{3}\right)\\
f(x) &= \arccos(2x+1)\\
f(x) &= \arctan(x^2)\\
f(x) &= \mathrm{arcsec}(x^7)\\
\end{align*}
etc.
|
In Euclidean geometry, I frequently use concepts related to invariance under scaling. For example, I know that if two squares have different side lengths, the ratio of their side lengths is the square root of the ratio of their areas.
I became curious about the justification of this principle recently when I was asked... |
Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are
distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$,
form a linearly independent set of functions.
Any tips on how to go about this proof. I tried a working from the
definition of an exponential and combin... |
My solution concerning a problem about Ergodic Recurrence requires me to prove that $\|P_T 1_B\| > 0$.
Where $P_T$ is the projection onto the space $I := \{f \in L^2 : f \circ T = f\}$, $T$ is a measure preserving mapping (so for all $B$ measurable $\mu(T^{-1} B) = \mu(B)$) and $B$ is of positive measure.
Can someone h... |
I've found in my solution book multiple references to this "dimension theorem" but have been unlucky looking for a more extensive description of this. When does this hold? Are there any other hypothesis needed?
Thanks.
|
If $z=f(x,y)$, then total derivative is $\mathrm{d}z=\frac{\partial f}{\partial x}\mathrm{d}x+\frac{\partial f}{\partial y}\mathrm{d}y$. If $\mathrm{d} z=0$, how do you show that $z$ is a constant?
|
Here I have a question about time scales in dynamic systems - for reference you can look at a previous question that spurs this one:
Minimizing the cost of a path in a dynamic system
That question was about finding the minimum cost path from $0$ to $c$ on the real line, where cost is quadratic in the size of the step a... |
I have this eq.
$$
\frac{x}{x-1} + e^{(x-1)/\epsilon} - \epsilon = 0.
$$
I need to derive an asymptotic expansion for any bounded roots $x(\epsilon)$ this equation might have, to find the coefficient of the $n$th (‘general’) term and show that asymptotic expansion converges for all $\epsilon$ in a neighborhood of zero.... |
This is a question on how to make sense of an equation (specifically, a description of a distribution) that, no matter how I slice it and dice it, always has some left over pieces that don't fit.
The equation in question describes a random variable $G$ drawn from a Dirichlet process $\textrm{DP}(\alpha_0,G_0)$:
\begin{... |
Q: Find the volume of the solid of revolution obtained by revolving the region bounded by the line $y = 2$ and the curve $y = \sec^2x$, $-90 \lt x \lt 90$, around the x-axis.
Attempted solution: $(1/3) \tan x ( \sec^2x + 2)$
I'm sure I did the integral correctly, but now when I need to sub in for $90, \tan(90)$ does no... |
Let $\mathbb{A}_{2}$ denote the 2-affice space over $\mathbb{C}$. Now let $Y=V(y-x^{2})$, here $V$ denotes the zero-locus. Now by definition the coordinate ring is the quotient ring $k[x,y]/I(Y)$ where $I(Y)$ is the ideal generated by all polynomials vanishing on $Y$.
Now the author computes $k[x,y]/I(Y)$, then it com... |
The BNF is defined as follows:
<S> -> <S>a<S>a<S> | b
This is my review question for a quiz, and according to my teacher, this grammar is ambiguous. However, I realized this grammar is not.
For example, given a sentence:
babababab
There are three ways to generate this sentence depending on which <S> we use, but its... |
If $f:X\rightarrow Y$ is a surjective morphism of schemes and $g:X'\rightarrow Y$ is another morphism of schemes, one can show that $p_{2}:X\times_{Y}X'\rightarrow X'$ is also surjective. The proof that I have seen (in Mumford's Red Book, for instance) goes as follows: Given $P\in X'$, we can find points $R\in Y$ and... |
How to find the real valued sufficient statistic of $\theta$ for a random sample from a distribution with the probability density function written below?
$$f(x;\theta) = \theta ax^{a−1} \exp(−\theta x^a), x > 0, \theta > 0, a > 0$$
|
Prove that if $p^{a}$ is a factor of the canonical factorization of ${{2n}\choose{n}}$ then $p^{a} < 2n$?
My attempt:
$${{2n}\choose{n}} = \frac{(2n)!}{n!n!}$$
Let $a_1$ be the highest of power of $(2n)!$
Let $a_2$ be the highest of power of $n!$
So the highest power of $\frac{(2n)!}{n!n!}$ = $a_1 - 2a_2$
where $a_1 <=... |
I want to find the irreducible components of the algebraic set $Y$, in $\mathbb{A}^{3}$ given by the zero-locus of the equations $x^{2}-yz$ and $xz-z$. I also want to compute the dimension of $Y$.
Well if $xz-z=0$ then $x=1$ or $z=0$. If $z=0$ then $x=0$ so we obtain the $y-axis$.
Now if $x=1$ then $yz-1=0$ hence:
$Y ... |
I'm working on an integration by parts problem, and I'm trying to substitute to simplify the equation:
$$\int_\sqrt{\frac{\pi}{2}}^\sqrt{\pi} \theta^3 \cos(\theta^2) d\theta$$
Using the substitution rule for definite integrals, I substitute $\theta^2 = t$ and apply the same to the limits of integration:
$$\int_\frac{\p... |
I'm having trouble recasting the following question in a form amenable to the calculus of variations.
Question: Given a set of $n$ points $P=\{(x_1,y_1),..(x_n,y_n)\}$ what is the curve passing through these points such that:
(C1) its length is minimal
(C2) the maximum curvature it takes on, as it interpolates the abo... |
I'm having difficulty understanding how to find Hydrostatic Force on various shapes. One problem is a right triangle with a height of 3ft and a width of 7ft. Its submerged 1ft below the surface with the 7ft side facing the surface. What do I do?
I understand the idea. My problem is more about setting up the integral.
... |
I'm reading this book and cannot make sense of the proof presented here: http://bit.ly/h8qAOJ
I have some understanding of abstract algebra but I'm not even sure if that's what we're dealing with here.
Some questions:
What is a homogeneous morphism? Is it the same as a homomorphism?
What does the notation "f [element... |
I am trying to understand how $\epsilon$ transitions work.
From what I've read, when you "go" to a state S that has arrows pointing outwards with $\epsilon$'s in it, you automatically go to those states too. That is, in the following diagram, when you are about to give the first digit as input to this FSA, you must con... |
following @Sivaram Ambikasaran's answer
for SVD, I get the computing (using MATLAB)of:
A = 2 1 3
1 2 5
3 5 4
[U,S,V] = svd(A)
U = -0.3793 -0.2964 -0.8765
-0.5571 -0.6832 0.4721
-0.7387 0.6674 0.0941
S = 9.3111 0 0
0 2... |
Does there exist $n$, and $r<2n-2$, such that the $n\times n$ square grid can be connected with an unbroken path of $r$ straight lines?
Note: This has essentially already been asked - see this MSE thread. I am posting this question because I wanted to explicitly focus on one of the unanswered parts.
Notice that $2n-... |
The question I'm trying to figure out states that I have 3 points P1, P2 and P3 in space. In one frame (Frame A I called it) those points are: Pa1, Pa2 and Pa3, same story for Frame B (namely: Pb1, Pb2 and Pb3).
Whats the rotation matrix from one to the other? That's literally all the information I have. What was s... |
How can I prove that $\displaystyle{\lim_{(x, y) \to (0, 0)} \frac{x^3 - y^3}{x^2 + y^2}}=0$? The method I've been taught is the pinching one, where you compare the absolute value of the function to greater limits that are known to equal zero, but I haven't managed to find a comparison that works. Could someone point... |
I recently read this pdf: Warning Signs of a Possible Collapse of Contemporary Mathematics, and I'm having some trouble understanding the issues it raises.
The author says that the consistency of Peano Arithmetic cannot be proved within the system, which is a direct application of Gödel's second incompleteness theorem.... |
It is known the following equivalence: Let $Y \subset \mathbb{A}^{n}$ be an algebraic set. Then $Y$ is irreducible if and only if $I(Y)$ is prime, where:
$I(Y) = \{f \in K[x_{1},x_{2},..,x_{n}]: \textrm{for all p in Y}, \ f(p)=0 \}$.
So as an example, the author considers $J = \langle (xz-y^2,x^3-yz) \rangle \subset k[... |
If two topological spaces have the same homotopy type, how do I prove that their complements in an ambient space are homeomorphic?
|
Given is a second-order tensor $T$, and three arbitrary vectors, $u$, $v$ and $w$, defined in Euclidean point space $\mathcal{E}$.
Prove that the determinant of the tensor $T$
$\det T=\frac{Tu.(Tv \times Tw)}{u.(v \times w)}$
is independent of $u$, $v$ and $w$, as long as they form a base in the 3-dimensional vector s... |
It goes like this:
Let $G$ be a group, $H < G$, and $N \lhd G$ be the smallest normal subgroup containing $H$. Then any $f \in \operatorname{Hom}(G,G')$ such that $H$ is in its kernel uniquely factors through $G/N$.
It's a simple proposition, but it's so weirdly formulated that I spent the whole day picking it apart.
B... |
Let $G = (V, E)$ and $G' = (V', E')$ be two graphs, and let $f: V \rightarrow V'$ be a surjection. Then $f$ is a covering map from $G$ to $G'$ if for each $v \in G$, the restriction of $f$ to the neighbourhood of $v$ is a bijection onto the neighbourhood of $f(v) \in V'$ in $G'$.
My question (homework) is how to easil... |
I could not understand the concept while googling. can anybody provide help?
what will be the determinant of the following matrix?
$$
\left[\begin{array}{cccc}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12 \\
13 & 14 & 15 & 16
\end{array} \right]$$
Thanks.... :)
|
How to integrate $x^{1/2}e^{-x}$ using integration by parts?
Answer should be $\left(-\sqrt{x} e^{-x}+(1/2)\sqrt{\pi} \mbox{erf}(\sqrt{x})\right)+c$
|
This is a continuation of my previous question and inspired by Arturo Margidin's reply.
Suppose there are a collection of measurable spaces $(X_i, \mathbb{S}_i), i \in I$. Let $\mathbf{X}=\prod_{i\in I}X_i$, and let $\mathbb{S}=\prod_{i\in I} \mathbb{S}_i$ be the product $\sigma$-algebra.
Suppose $A_{i_0}\subseteq X_{... |
In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) or a 9 followed by nine factorial symbols. Both are undoubtedly huge, but I haven't been able to produce an argument that... |
This is inspired by Carl Offner's reply to one of my previous questions and my previous question about marginal and joint measures.
Given a measure $\mu$ on product
$\sigma$-algebra $\prod_{i \in I}
\mathbb{S}_i$ of a collection of
measurable spaces $(X_i,
\mathbb{S}_i), i \in I$, does there
exist a measure $\mu_i$ on... |
I'm supposed to construct a DFA which accepts
{ w | w is a word except 'aa' and 'aaa' }
Is this the correct solution? The thick line state is supposed to be the end state.
EDIT
Sry, somehow mixed two different exercises. Corrected.
EDIT 2
Here's the corrected solution(?)!
|
Let $X$ be a complex manifold equipped with a smooth hermitian metric $h$. We can define a sub-fibration $B \to X$ of the tangent bundle $T_X$ by requiring that the fiber over a point be the unit ball in $T_X$ at that point, i.e.
$$ B_x = \{ \xi \in T_{X,x} \, | \, \| \xi \|_{h(x)} = 1 \}. $$
I want to see that if $X$ ... |
Let $p$ be a prime and let $F=\mathbb{Z}/p\mathbb{Z}$. Can a nonzero multivariate polynomial $f\in F[x_1,....,x_n]$ such that $\mathrm{deg}_if< p$ for all $i=1,\ldots,n$ be identically zero as a polynomial function $F^n\to F$, where $\mathrm{deg}_if$ stands for the degree of $f$ in $x_i$?
|
Variable $z$ varies from $-50.0$ to $18.0$. Find the equation that gives $d$ such that when $z=-50$, $d = 0.1$; and when $z = 18.0$, $d = 1.0$. The values of $d$ vary from 0.1 to 1 uniformly depending on the value of $z$ with the boundary conditions given above.
|
Background
I am conducting a quantitative synthesis (meta-analysis) that includes previously published data. For the model that I am using, an exact or inflated estimate of SE is acceptable.
Question
Given a 95% Confidence Interval (measured from mean to upper or lower confidence limit), $\alpha$, and $n$, is it correc... |
Is it true that every inner product in $\ l_2 $ is of the form $\langle x,y\rangle_a =\sum_{n=1} ^ {\infty} {a_n x_n y_n}$ ? (Of course $\ x=(x_n) , y=(y_n) $ are in $\ l_2 $ .)
|
Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible
$$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i \rangle$$
Now let $L(H)$ be the set of bounded operators, $N(H)$ the set of nuclear operators. We want to show
$$i: L(H) \... |
Or, in general, what does the magnitude of the cross product mean? How would you prove or disprove this?
|
Similar questions:
Chance of 7 of a kind with 10 dice
Probability of getting exactly $k$ of a kind in $n$ rolls of $m$-sided dice, where $k\leq n/2$
Probability was never my thing, so please bear with me.
I've reviewed the threads above to the best of my ability, but I still wonder how to go about finding a match of 3 ... |
Let's say that I have a variable $j$ defined by the following formula:
$$j=\frac{n(n+2) + m}{2}$$
where $n$ and $m$ are two parameters, both integers, that satisfy the following conditions:
$n\in \left[0,1,2,...\right]$
$m\in\left[-n,n\right]$
$\left(n-m\right)$ even
We know that, subject to the above conditions, the... |
I'm having some trouble while trying to understand one task..
The task is as follows:
$$\ddot{x}(t) + \dot{x}(t) + 2x(t) = \sin(\omega t)$$
where $x(0) = 7, t\geq 0$
The solution is in the following form:
$$x(t) = f(t) + A\sin(\omega t + \varphi)$$
And the task is: find $\omega$ so that $A$ is max.
My understanding of ... |
This is a question about the proof of Proposition 1.4 in Farb and Margalit's "Primer on Mapping Class Groups" (in v. 5.0, it is on page 37 in the PDF, which you can download here). The proposition states
Let $\alpha$ be a non-nullhomotopic simple closed curve on the (hyperbolic) surface $S$; then $[\alpha]\in\pi_1(S)... |
Since $\varphi(p)=p-1$ is even the p'th cyclotomic field contains some quadratic field. Hecke says that in fact every quadratic field is contained by some cyclotomic field.
What is this theorem called and how is it proved?
|
I'm completely new with the sum notation and Levi-Civita-Symbols. Can somebody tell me if what I did was correct?
Let $a,b \in \mathbb{R}^3$.
I want to prove $$ (b \wedge a) = -(a \wedge b), $$ so I did $$(b \wedge a)_i = \epsilon_{ijk}b_ja_k = - \epsilon_{ikj}b_ja_k = -\epsilon_{ikj}a_kb_j = - (a \wedge b)_i.$$
|
Player A, picks $n_1$ integers $a_1$,...,$a_{n_1}$ uniformly at random from $1$..$N$,
and player B picks $n_2$ integers $b_1$,...,$b_{n_2}$ the same way.
Given $N$, $n_1$, $n_2$, and $d$, what is the expected number of pairs ($a_i$,$b_j$) where |$a_i$-$b_j$| $=<$ $d$,
A. if A and B are allowed to select duplicates in... |
Let $\tau_a=\inf\{t: B_t=a\}$, the hitting time of the standard Brownian motion to reach the boundary $a$.
This is easily derived
$$E(e^{-\lambda \tau_a})=e^{-|a|\sqrt{2\lambda}}$$
But I am having a problem of using this formula to get the moments of $\tau_a$ by matching the coefficients of terms for each power of $\la... |
Let $\gamma: I \rightarrow \mathbb{R}^n$ be a smooth curve. We define a Frenet frame to be an orthonormal frame $X_1, \ldots X_n$ such that for all $1 \leq k \leq n$, $\gamma^{(k)}(t)$ is contained in the linear span of $X_1(t), \ldots, X_k(t)$, $t \in I$.
What is an example of a smooth curve $\gamma$ with no Frenet fr... |
I've a confession to make. I've been using PDF's and PMF's without actually knowing what they are. My understanding is that density equals area under the curve, but if I look at it that way, then it doesn't make sense to refer to the "mass" of a random variable in discrete distributions. How can I interpret this? Why d... |
Polynomials of degree 2, if the first term is positive, starts descreasing.
Polynomials of degree 3, if the first term is positive, starts increasing.
is this constant?
like..
Polynomials of degree 4, if the first term is positive, starts descreasing.
Polynomials of degree 5, if the first term is positive, starts in... |
Suppose I have a sequence of real numbers such that AGM $(a_{n},a_{n+1}) = 1/2^n$. Would that sequence be unique and how would I go about finding the individual $a_{n}$? (AGM is the arithmetic geometric mean)
|
Let $\pi: X\to C$ be a fibration in curves where $C$ is a non-singular curve and $X$ a regular, integral surface and the generic fiber $X_\eta$ is a non-singular curve over $k(C)$ (these hypotheses might be stronger than necessary, but I just threw a bunch on to make it as nice as possible).
Now a point on $X_\eta$, sa... |
Everyone is familiar with distributivity of multiplication over addition of real numbers. The distributivity of two binary operations sometimes goes both ways (e.g. max and min, or for lattices in general.)
Out of curiosity, I looked at the set of real numbers for which addition distributes over multiplication. A simpl... |
Can someone point me to the full derivation of the Fourier Series? I'm having problems understanding how the a's and b's coeffients are worked out.
|
I am given a 2D convex polygon ( given in terms of coordinate of the vertices), and I want to lay it on a grid with an undetermined uniform square grid size, $x$.
Now, my task is to choose $x$ such that, regardless of the size and the shape of the polygon, the number of the grid, $G_n$, inside, or intersecting the poly... |
What is the importance of eigenvalues/eigenvectors?
|
I was wondering what would you classify the damping state (under damped, over-damped, critically damped) a second-order ode system with a negative damping ratio? To me it doesn't make much sense since a negative damping ratio results in an unstable system.
The system I have is:
$G(s) = \frac{5}{s^2-6s+16}$
So solving f... |
I know that $\gcd(a,b)$ divides $a$ and $b$, and must also then divide $(a)(t)$ ($t$ being some integer). This makes sense to me, but how do I prove it? It seems that the addition of $(a)(t)$ is a continuation of the linear combination of $\gcd(a,b) = av + bu$ for some $v$, $u$ being elements of $\mathbb{Z}$.
Any h... |
I'm having a terrible time trying to understand Fourier transforms. I'm very visual so leaving the $X,Y,Z,t$ domain is not working form me :)
I'm trying to figure out the basics at the moment. Like, taking a Sine wave (they're odd right?) and converting it into its real and imaginary numbers. I'm pretty sure I got t... |
Is it possible, given a point in the interior of an angle to construct a circle through that point tangent to the rays forming the angle?
Suppose you have a point $A$ in the interior of an angle with vertex $O$. You can bisect the angle, take a point $I$ on the angle bisector, and drop perpendiculars $IP$ and $IQ$ to e... |
I ran into this problem with my abstract algebra homework the other night, and I think I am really close to solving it. By the definition of modulo n, I know that
c=a + rm, m being some integer
c=b + sn, n being some integer
I want to prove a=b + (rm + sn)
I have tried algebraically manipulating these equalities, but ... |
The quaternion group of order 8 has an irreducible two dimensional representation over $\mathbb{C}$ but how does one show that this representation cannot be defined over $\mathbb{R}$?
|
I have been doing some casual internet research on biclusters. (I have read the Wiki article several times.) So far, it seems as if there are few definitions or standard terminology.
I was wondering if there were any standard papers or books that anybody who is interested in algorithms for finding biclusters should ... |
Suppose $z = x + y$ and in addition, there is a function $f(z)=F(x,y)$ for all values. Is there a way to relate,
$$\frac{df}{dz}, \frac{\partial F}{\partial x}\text{ and }\frac{\partial F}{\partial y}\textrm{?}$$
In general, suppose $z = g(x,y)$, what is the relationship (if any)?
|
Let $f_C$ be the convex envelope of $f$ on a non-empty convex set $C \subset \mathbb{R}^n$
I need to show that $\{ x^∗ \in C : f(x^∗) \leq f(x), \forall x \in C \} \subset \{x^∗ \in C : f_C(x^∗) \leq f_C(x), \forall x \in C \}.$
Right now, I can show that: $\{ x^∗ \in C : f(x^∗) \leq f(x), \forall x \in C \} \subseteq ... |
This is a problem from Borevich and Shafarevich (p.180).
If $\mathfrak{O} = k[x]$, the problem is to find the set of valuations $A$ of the field $K = k(x)$ which satisfies the conditions (1),(2),(3) below. I think I want to find the largest such set. I also want to do the same when $\mathfrak{O} = k[1/x]$.
Can some... |
As a step in proving that the union of intervals $B_E \subset [0,1]$ (where $E$ is the set of infinite Bernoulli sequences, e.g. 0.0110...., such that some finite pattern repeats infinitely often) is a Borel set, the authors in this book build the set as follows. First, they define $E_n$ as the set of sequences where t... |
I was reading this paragraph and it got me thinking:
The closed ends of the honeycomb cells
are also an example of geometric
efficiency, albeit three-dimensional
and little-noticed. The ends are
trihedral (i.e., composed of three
planes) sections of rhombic
dodecahedra, with the dihedral angles
of all ad... |
A very basic ring theory question, which I am not able to solve. How does one show that
$\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$.
Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$, if $a,b$ are relatively prime.
My attempt was to define a map, $\varphi:\mathbb{Z}[i] \to... |
Let n be a natural number with unique prime factorization $p^m$... $q^k$ . Show that n can be written as a square if and only if all (m, ...k) are even
|
Consider a simple linear equation of the form:
$n=\frac{2x+2}{3}$
Let $n$ and $x$ represent something that comes in whole positive quantities (for example physical objects).
How can I
Define the equation only for $n$ and $x$ that are a part of natural numbers (whole numbers $>0$)
Solve the equation satisfying the abov... |
Lusin's theorem states that for every $\varepsilon$, for every borel measure $\mu$, for every function $f:\mathbb{R}^n\to\mathbb{R}^m$, for every open set $A$ of finite measure, there exists a compact set $K$ such that $\mu(A-K)<\varepsilon$ and $f$ restricted to $K$ is continue.
Now, I'm wondering about what's the for... |
I know a similar question was asked before, but I wanted to know if this can be extended to any number system by a generic formula.
For example, given a number X in number system A, how many digits would it have in number system B?
I'm looking for a sort of universal formula in terms of X, A and B.
Thanks
|
I need your help to solve this question
A subset $X \subset k^n$ ($k$ a field) is called algebraic if there exist polynomials $f_1, \dots, f_m \in k[t_1,\dots,t_n]$ such that
$$X = \{x \in K^n | f_1(x) = ...=f_m(x)=0\}.$$
The coordinate ring $k[X]$ of $X$ is the ring of all functions $f: X\to k$ that can be represente... |
This is a problem from Borevich and Shafarevich (p.180).
First, a definition:
A theory of divisors on a domain $\mathfrak{O}$ consists of a commutative semigroup $D$ (semigroup = associative multiplication with identity) having the property of unique factorization into irreducibles along with a homomorphism $\alpha \... |
It's easy to show $C[0,1]$ is not dense in $L^{\infty}[0,1]$ in the norm topology, but $C[0,1]$ is dense in $L^{\infty}[0,1]$ in the weak*-topology when take $L^{\infty}$ as the dual of $L^{1}$. how to prove it?
|
what is the solution for max y$^T$ $\Sigma$ y in terms of maximum eigen value and the associated eigen vectors ?
|
It is well known that the symmetric property is $d(x,y)=d(y,x)$ is not necessary in the definition of distance if the triangle inequality is carefully stated. On the other hand there are examples of functions satifying
(1) $d(x,y)\geq 0$ and $d(x,y)=0$ if and only if $x=y$
(2) $d(x,y)\leq d(x,z)+d(z,y)$
which are not s... |
Find $g$
a)
If $g∘(2f)=f+h$ where $f(x)=2x + 5$ and $h(x)=x^3 -2x$
b)
If $(2f)∘g=f+h$ where $f(x)=\ln(x+2)$ and $h(x)=\sin(x^2)$
Thanks
|
According to Batyrev, there are exactly 18 types of smooth toric Fano threefolds. As projective toric varieties, these are defined by the normal fan of certain 3-dimensional polytopes. Does anyone know of a reference where these polytopes are listed, e.g., as convex hulls of points?
|
The recurrence $a_{n+1}=a_n(n-1/2)$ is related to $\Gamma(n+1/2)$ ( not difficult to prove) and it could be represented in a way like $\frac {(2n-1)!!} {2^n}$
Also I know that
$(2n-1)!!$ is the number of permutations of 2n whose cycle type consists of n parts equal to 2; these are the involutions without fixed point... |
Given a social choice function $F$, a subset $B\subset A$ of the candidates and a coalition $S\subset N$ of the voters, $\beta$-effectiveness of $S$ for $B$ is equivalent to $N\setminus S$ not being $\beta$-effective for $A\setminus B$.
However, for a social choice correspondence it is not true - the fact that $N\setmi... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.