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H: Errata for Mendelson's Introduction to Mathematical Logic (5th ed)?
I'm hoping that there exists a prepared errata list for Elliott Mendelson's Introduction to Mathematical Logic (5th ed). If so, how does one locate it? (It does not seem to be available from the publisher (CRC Press). If no such errata resource ... |
H: Prove that $(\mathbb{Z},d)$ is a metric space
I got this from Mendelson:
Let $\mathbb {Z}$ be the set of integers.Let $p$ be a positive prime integer. Given distinct integers $m$, $n$ there´s a unique integer $t=t(m,n)$ such that:
$$ m-n=p^tk $$
where $k$ is an integer not divisible by $p$. Define a function $... |
H: Least quadratic non residue algorithm
I am trying to implement Tonelli-Shanks algorithm and at one of the steps I have to find the least quadratic non residue. I've searched the web for a while for some kind of algorithm but so far I've seen only paper with lot's of math behind them.
Are there any algorithms for fi... |
H: Compactness of $\mathcal K$ in the Hausdorff distance
Possible Duplicate:
The Class of Non-empty Compact Subsets of a Compact Metric Space is Compact
Let $(M,d)$ be a metric space and let $\mathcal K(M)$ denote the set of all non-empty compact subsets of $M$. This collection is a metric space when equipped with ... |
H: Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind
Consider the integral
$$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x$$
For $n=3$ we have
$$(-\gamma^2-2\zeta(3)-3\zeta(2)\gamma)\genfrac{[}{]}{0pt}{}{k}{1}+3(\gamma^2+\zeta(2))\genfrac{[}{]}{0pt}{}{k}{2}-6\gamma\genfra... |
H: Uniform convergence of infinite series
Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets of $\mathbb{D}$.
I met this problem on today's qual. Here is what I have so f... |
H: There exist infinitely many values $n\in \mathbb{N}$ such that $8n^{2}+5$ is divisible by $7$ and $11$, without using diophantine equations
Show that there exist infinitely many values $n\in \mathbb{N}$ such that $8n^2+5$ is divisible by $7$ and $11$, without using diophantine equations.
I have $8n^2+5=8(n^2-9)+77$... |
H: Let $M$ be a maximal ideal in $R$ such that for all $x\in M$, $x+1$ is a unit. Show that $R$ is a local ring with maximal ideal $M$
So I'm trying to prove that if $M$ is a maximal ideal in $R$ such that for all $x\in M$, $x+1$ is a unit, then $R$ is a local ring with maximal ideal $M$, that is to say $R$ has a uniq... |
H: Quadrilateral Top Point
Suppose you are given a quadrilateral - orient so that one point is the "bottom" (ie. like a diamond). Given three points: the bottom point, the left point and right point, I want to solve for the top point. The only other things I know are the lengths of the top left and right sides. I know... |
H: Bounded power series without convergence on the boundary
Does there exist a holomorphic function $f\in H^\infty$ (i.e. holomorphic and bounded on the open unit disc) such that its power series expansion at $0$ doesn't converge anywhere in $\{|z|=1\}$?
AI: To every $f \in H^\infty$ there corresponds $f^\star \in L^\... |
H: Showing $f(x)=\sum_{n=1}^{\infty}{\sin\left(\frac{x}{n^2}\right)}$ is continuous.
Let
$$f(x)=\sum_{n=1}^{\infty}{\sin\left(\frac{x}{n^2}\right)}.$$
a) Show that the series converges for $x\in [0,\pi/2]$.
b) Show that $f$ is monotone and continuous on this interval.
This is what I have for (a). Is it right? I am st... |
H: Given the pairwise distances between $n$ points, how can I find plausible coordinates for the points?
If I have three points $A, B, C$, and I know the distances between $A$ and $B$, $B$ and $C$, and $A$ and $C$,
(1) How can I find (one possible value for) the coordinates of $A$, $B$, and $C$?
(2) If the distance fu... |
H: The continuity of multivariable function
$F$ is a function on $\mathbb R^n$ such that for every smooth curve $\gamma:[0,1] \rightarrow \mathbb R^n, \gamma(0)=0 $, we have $\mathop {\lim }\limits_{t \to 0} F(\gamma (t)) = 0$, is it necessary that $\mathop {\lim }\limits_{\left| x \right| \to 0} F(x) = 0$ ?
AI: Good ... |
H: Halting problem on finite set of programs
As I understand the halting problem, it imply the fact that there doesn't exist one program which can answer the halting problem for every computable program and it rely on Cantor diagonalization to build the proof.
However, Cantor diagonalization would not seem to be pract... |
H: How to write a good mathematical paper?
I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar.
A good time always goes too fast! Two years are fled. In the third year of PHD, my major is general topology and I'm facing with graduation from PHD. I... |
H: Describing the effect on $ax^2$ by manipulating $a$
Please take, for example, $y = x^2$ and $y = 2x^2$.
Graphs: Wolfram Alpha
What is the most appropriate way to describe the effect of $a$? "$a$ causes the parabola to open at $1/a$ the rate of $y = x^2$"?
AI: First of all, the sign of $a$ impacts if it opens up or ... |
H: Computing sign table for $2^k$ factorial experiment design
all!
I need to compute the sign table for a generic $2^k$ factorial design. For $k$ factors we compute $2^k$ experiments and need to compute a $2^k \times 2^k$ matrix, as the following example for $k=3$:
\begin{matrix}
& I & A & B & C & AB & AC & BC & ABC\... |
H: Contragredient representation of a finite group
I am reading Serre's Linear Representations of Finite Groups and in an exercise in there he asks to show if $\rho$ is a representation of a finite group on $\textrm{GL}(V)$ with $V$ a finite dimensional complex vector space that there exists a unique linear representa... |
H: Derivatives of vectors involving the expectation operator - Part I
So, I am trying to take the derivative of the following equation, because it is needed in an optimization problem. I want to make sure I am on the right track. The equation is:
$$
-3 \mathbb E[(w^Tz)^2]^2
$$
So my question is, what is:
$$
\frac{\del... |
H: Why isn't this a regular language?
I'm stuck as to figuring out why $L_1$={$n^p$ | $p$ = a prime number} is not a regular language but $L_2$={$n^p$ | $p$ = a prime number bounded by some fixed number f} is. I can see that $L_2$ is a finite language so it's a regular language. But I can't seem to find a way to bui... |
H: Subgame Perfect Nash Equilibrium
My homework question is summarized below:
There are 7 players (say P1,P2,...,P7) trying to split 100 dollars. The game starts with P1 proposing an allocation of the 100 dollars to each player, that is, P1 assigns a dollar amount [0, 100] to each player, including himself. Then the... |
H: Is this operator monotone?
Consider a convex optimization problem.
$$\min_{u\in\Re^k} f(u)$$
s.t. $g_i(u)\leq0,\ i=1,\ldots,m$
Let $F(x)=F(u,\lambda)=(f'(u)+\sum_{i=1}^m\lambda_ig_i'(u),-g_1(u),\ldots,-g_m(u)):\Re^n\rightarrow\Re^n$, ($n=k+m$)
$G=\{x=(u,\lambda)\in\Re^n:\lambda\geq0\}$.
$f$ and $g_i$ are all conve... |
H: 'Quantum' approach to classical probability
Quantum mechanics defines a state of a system as a superposition of 'classical' states with complex coefficients, thus reducing many problems to linear algebra. Can classical statistics be approached to that way? Is this approach useful?
AI: Yes. You can define a(n algebr... |
H: Solving $|a| < |b|$
I apologize if this question in general, but I've been having trouble finding solutions as Google discards absolute value signs and inequality symbols.
I am looking for a way to eliminate absolute value functions in $|a| < |b|$.
I can solve $|a| < b$ and $|a| > b$, but I am unsure what method / ... |
H: Generating random string
I'm working on a small web service that will require users to upload files. The file type is P12. When the file is uploaded and I need to save it on my server, I need to generate a random name for it. I can have the server generate a long random string of letters and numbers called a UUID. ... |
H: What is the structure of invariant matrix polynomials?
Let the field $\mathbb F$ be either $\mathbb R$ or $\mathbb C$ and $M_n(\mathbb F)$ all $n \times n$ matrixes.
We denote by $I_n(\mathbb F)$ the space of all functions:
$$P : M_n(\mathbb F) \rightarrow \mathbb F$$
which are polynomial (in the sense that $P(A)$ ... |
H: A basic question related with the positive definite matrix
I have a one doubt related with positive definite matrices.
Suppose that we have an arbitrary non zero matrix $A$ . Can we find such matrix $B$ which may depend on $A $such that product $AB$ is always a positive definite matrix irrespective of the nature o... |
H: Determing if two k subsets are disjoint given the product of their elements
Consider the following problem (phrased with the use of a black box).
You choose $n$ numbers $X = \{x_1,\ldots,x_n\}$ and pass it to a black box that returns a list $Y = \{y_1,\ldots,y_m\}$ where each of the elements is a product of the el... |
H: Approximate rational dependence
After seeing question Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer? I wonder if there is an algorithm that can find approximate rational dependence?! I pick any irrational numbers $\alpha$, $\beta$, $\gamma$, $ \delta$ and the task is to find successively good approxi... |
H: Any software package with Mordell equations implemented?
I admit I have no idea how to tag this post, but I'm looking for a CAS/number theory software package that would implement a decent algorithm for computing the integral solutions to
$x^2 = y^3 - k$,
$k \in \mathbb{Z}_+$ a parameter taking values up to order o... |
H: Order of a set $X$ acted upon transitively by the Symmetric Group
Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$.
Show that either $|X| \le 2$ or $|X| \ge n$.
Small steps towards the solution:
As $S_n$ acts transitively on a set... |
H: Graph isomorphism as permutation matrix.
The automorphism group of a graph (lets us consider undirected) is the set of all permutation on vertices that
preserve the adjacency. It is claimed: automorphism group of graph may be equivalently
defined as the set of permutation matrices $\pi$ which commute with the adjac... |
H: What are the solutions of $x^n+a^n=0$?
For what values $a$, the equation $x^n+a^n=0$ has $n$ different solutions? what are the solutions?
(the question refers to complex solutions).
AI: By the Fundamental theorem of algebra, since for every value of $n>0$ this is a polynomial of degree $n>0$ there are exactly $n$ r... |
H: Are there diagonalisable endomorphisms which are not unitarily diagonalisable?
I know that normal endomorphisms are unitarily diagonalisable. Now I'm wondering, are there any diagonalisable endomorphisms which are not unitarily diagonalisable?
If so, could you provide an example?
AI: $$\begin{pmatrix} 1 & 2 \\ 1 &... |
H: Are compacta in a complete infinite dimensional normed space nowhere dense?
Let $X$ be an infinite dimensional Banach space. I want to show that any compact subset $\varnothing\neq A\subset X$ is nowhere dense.
I've been able to prove the statement for $X=(C[0,1],\|\cdot\|_\infty)$ by using an Arzelà-Ascoli argumen... |
H: Does a polynomial exist such that $|P(x)| < a$ for some real $a$ and all real $x$?
I'm pretty sure the answer is in the negative.
Can someone show me the proof?
AI: The answer is no.
Write
$$
P(x)=ax^n+\text{terms of lower degree}
$$
with $a\neq0$ and $n\geq1$. Then
$$
\lim_{x\to\infty}P(x)=\pm\infty
$$
according ... |
H: Rational Functions are Determined By Locations and Multiplicities of Zeros and Poles (why?)
There is a theorem that says rational functions in the extended complex plane are exactly the meromorphic functions.
After this, my textbook draws the corollary:
"...as a consequence, a rational function is determined up to ... |
H: A question about the regular languages being closed under Boolean operation (how to generalize)
I know that if $L_{1},L_{2}$ are regular languages then so is $L_{1}\cap L_{2},L_{1}\cup L_{2}$
are regular languages, I also know that $L$ is regular $\implies L^{c}$
is regular
.
It is easy to generalize in this manner... |
H: Solving $\phi (n) < (n-1) \cdot \frac{15499}{94744} $
I am working on challenge 243 from Project Euler (PE 243). The question is:
$$\text{Solve } \phi (n) < (n-1)\cdot \frac{15499}{94744}$$
I can calculate $\phi(n)$ for any $n$, but I think the $n$ which solves the problem is larger than the range I can brute for... |
H: How do I transform the equation based on the condition?
If $q$ and $w$ are the roots of the equation $$2x^2-px+7=0$$
Then $q/w$ is a root of ?
P.s:- It is an another question of How do I transform the equation based on this condition?
AI: The following result (and its generalizations to higher degree polynomials)... |
H: Nonzero elements of splitting field
Let $F$ be a splitting field of $x^{p^{n}} - x \in \mathbb{Z}_p[x]$.
How is it that the nonzero elements multiply to $-1$ and sum to $0$? I don't get how we get that result.
AI: We appeal to the structure theorem for finite fields: the field $\mathbb{F}_{p^n}$ is precisely the sp... |
H: Pseudocompactness does not imply compactness
It is well known that compactness implies pseudocompactness; this follows from the Heine–Borel theorem. I know that the converse does not hold, but what is a counterexample?
(A pseudocompact space is a topological space $S = \langle X,{\mathfrak I}\rangle$ such that ever... |
H: Constrained optimization: equality constraint
I have this very general problem (for $n>2$):
$$
\begin{align}
& \max Z = f(x_1,\ldots ,x_n) \\[10pt]
\text{s.t. } & \sum_{i=1}^{n} x_i = B \\[10pt]
& x_i \geq 0
\end{align}
$$
Assume that the function $f(x_1,\ldots,x_n)$ is not a "nice" function, non-linear, non-convex... |
H: Cantor Bendixson Theorem
Cantor Bendixson Theorem: Every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.
This definition differs a bit from that in wikipedia.
I have proved that 'If $X$ is a separable metric space and $E$ is a uncountable s... |
H: General solution of constant-coefficient second-order linear ODE
I am trying to look a bit deeper into the mathematics the equation of motion used in physics and engineering. I have some specific questions at the end, but please correct me if I make a mistake in
my statement.
I have an ordinary, second-order, line... |
H: Continuity of $f:X\to [0,1]$ where $X$ is homeomorphic to the Cantor set.
This is an exercise from Mendelson's Introduction to Topology, page 101.
THEOREM
For each $n\in \Bbb N$, let $X_n$ be the discrete two-point topological space $\{0,2\}$. Define the product space $X=\prod\limits_{n\in \mathbb N}X_n\;$. Let $f:... |
H: What does asymptotically optimal mean for an algorithm?
What does it mean to say that
heap sort and merge sort are asymptotically optimal comparison sorts .
I know What the Big O , Big Omega($\omega)$ and Theta($\theta$) notations are and I also know why these two sorts are called comparison sorts ? But I am not ... |
H: Congruence for embeddings of a poset into semilattices
Let $\mathfrak{A}$ be a poset, $\mathfrak{B}$ and $\mathfrak{C}$ be meet-semilattices with least elements. Let $f:\mathfrak{A}\rightarrow\mathfrak{B}$ and $g:\mathfrak{A}\rightarrow\mathfrak{C}$ are order embeddings.
Can we warrant that $f(x)\cap^{\mathfrak{B}}... |
H: Questions regarding the quadratic form $q: \bigwedge^2 \mathbb{R}^4 \to \bigwedge^4 \mathbb{R}^4, x \mapsto x \wedge x$
An exercise of my last year's linear algebra class asks as follows:
Determine the type of the quadratic form $q: \bigwedge^2 \mathbb{R}^4 \to \bigwedge^4 \mathbb{R}^4, x \mapsto x \wedge x$.
Que... |
H: Second degree Diophantine equations
I found a question whether there are general methods to solve second degree Diophantine equations. I was unable to find an answer so is this known? In particular, the original writer wants to know whether one can find all integers satisfying $x^2 + x = y^2 + y + z^2 + z$.
AI: Abo... |
H: Quantum Information: Tensor with Outer Product
If $\mid\phi\rangle=a_0\mid 00\rangle + a_1\mid 01\rangle +a_2\mid 10\rangle +a_3\mid 11\rangle$ and $P_0=\mid 0\rangle\langle 0\mid \otimes I$, how do we show that $\langle\phi \mid P_0\mid \phi\rangle =\langle\phi \mid a_0\mid 00\rangle+\langle \phi\mid a_1\mid 01\r... |
H: Geometric meaning of a nondegenerate critical point
Let $f\!:M\!\rightarrow\!\mathbb{R}$ be a smooth function on a manifold and $p\!\in\!M$. Is there any way to geometrically/visually characterize the conditions
$p$ is a critical point (i.e. $D(f)_p\!=\!0$) and
$p$ is a nondegenerate critical point (i.e. $\det D... |
H: Figuring out the percentage of A in relation to B
I have two numbers
13.49 and 7.8
I want to know how to figure out how much 7.8 is of 13.49 in percentage
e.g. 7.8 is 80% of 13.49
What is the best way to do this?
AI: The percentage is $\frac {7.8}{13.49}\cdot 100 \%\approx 57.82 \%$ Note that multiplying by $1... |
H: Class of manifolds is a set?
Is the class of all 2-countable manifolds a set?
I think so: each such space is a countable union of sets of cardinality $|\mathbb{R}^n|\!=\!|\mathbb{R}|$, i.e. a manifold has cardinality continuum, and there are less than $\mathbb{R}^\mathbb{R}$ ways of gluing it together.
Why then is... |
H: Projecting a vector component out of vector $a$
Suppose a vector $a$ is given. What is precisely meant by projecting the component of vector $b$ out of $a$? Does that mean that the resulting vector $a_1$ (obtained by "projecting out the component of $b$ from $a$") is orthogonal to $b$? How could one achieve this?
I... |
H: Subgroup generated by additive integers: Expressing d as an integer combination of ra + sb.
$$a = 123,\ b = 321.$$
The first part of the question I'm trying to solve wants me to find $d = \gcd(a,b)$. I can quickly solve this by using the Euclidean algorithm, and $d$ is $3$.
However, the second part requires me to e... |
H: Given a random sequence of input points does it always produce a random output, excluding $f(x)$ = constant?
Assume I have $y = f(x) \ne \mathrm{constant} $ and $(x_1 , \ldots ,x_N)$ a sequence N random input points, is there a set of criteria or a theorem that tells me that the output sequence $(y_1, \ldots, y_N)... |
H: Existence/Finiteness in $\mathbb{R}^2_+$ using Fubini or Tonelli
I am having some trouble understanding how to apply Fubini and/or Tonelli Theorems to determine whether a Lebesgue integral over $\mathbb{R}^2_+$ exists and if it is finite.
If someone could help me by showing the explicit steps for the examples below... |
H: Finding an interpolating polynomial and natural cubic spline for a given accuracy
I'm trying to make an exercise but I don't know how to start. Is there somebody that can give me a hint so that I can start with the exercise. The exercise is:
Consider the function $f(x) = \sin(x)$ on the interval $[0,\pi]$. Evaluate... |
H: Global irreducible decomposition of an analytic set
Let $M$ be a complex manifold (or a complex analityc space) and $Z$ be an analytic subset of $M$. By Noetherianity of the rings of germs of analytic functions at a point we know that $Z$ has finitely many irreducible germs at any $p \in Z$.
What about the "global"... |
H: Evaluate $\int_0^\infty (u\log(a^2+u^2))/(e^u-1)\, du$
Does the following definite integral have a known "closed form" value?
$$\int_0^\infty\frac{u\log(a^2+u^2)}{e^u-1}~du,$$
or can anyone see a way to integrate it?
AI: I was only able to find the following formula: for $a = 2\pi\alpha$,
$$\begin{align*}
\int_{0}^... |
H: Solving inequality $x^3-4x>0$
I was sort of finding the roots by doing $x^3-4x>0=x(x^2-4)$
$x = 0, x = -2,x = 2$ for $$x(x-2)(x+2)>0$$
Then I stopped and thought; maybe I shouldn't be doing that? I am doubting myself! Can anyone confirm if I am doing the right thing to solve the equality?
AI: The polynomial $x^3-4... |
H: Let X and Y be random variables with joint pdf $f(x,y)=x+y$ for $0
I am using the fact that $X$ and $Y$ are independent if and only if $f_X(x)f_Y(y)=f(x,y)$. So I have
$$f_{X}
(x)=\int_{0}^{1}f(x,y)dy\\=\int_{0}^{1}x+ydy\\=[xy+\frac{y^{2}}{2}]_{y=0}^{y=1}\\=x+\frac{1}{2}$$
and by basically exactly the same math, $f... |
H: References for elliptic curves
I just finished reading Silverman and Tate's Rational Points on Elliptic Curves and thought it was very interesting.
Could any of you point me to some more references (e.g. books, articles) on elliptic curves? Someone suggested Silverman's The Arithmetic of Elliptic Curves, but I don'... |
H: If the sum of positive integers $a$ and $b$ is a prime, their gcd is $1$. Proof?
I feel this is an intuitive result.
If, for example, I was working with the prime number $11$, I could split it in the following ways: $\{1, 10\}$, $\{2, 9\}$, $\{3, 8\}$, $\{4, 7\}$, $\{5, 6\}$.
Then clearly there is no way that the $... |
H: If $V_0$ is the subspace of matrices of the form $C=AB-BA$ for some $A,B$ in a vector space $V$ then $V_0=\{A\in V|\operatorname{Trace} (A)=0\}$
If $V_0$ is the subspace consisting of matrices of the form $C=AB-BA$ for some $A,B$ in a vector space $V$ then $V_0=\{A\in V|\operatorname{Trace}(A)=0\}$.
The problem abo... |
H: Irreducible minimal polynomial implies every invariant subspace has an invariant complement
Full version of the problem is following:
Let T be a linear transformation on a finite dimensional vector space $V$ over a field $\mathbb{F}$. If the minimal polynomial $p_t$ of T is irreducible, then every T invariant subs... |
H: General topology multiple choice question
Let $X$ be a normal Hausdorff space. Let $A_1$, $A_2$, and $A_3$ be closed subsets of $X$ which are pairwise disjoint. Then there always exists a continuous real valued function $f$ on $X$ such that $f(x) = a_i$ if $x$ belongs to $A_i$, $i=1,2,3$
iff each $a_i$ is either 0... |
H: A bit of direct sum confusion
So I'm reading through Serre's "Linear Representations of Finite Groups," and I'm a bit confused by what's probably a fairly minor point. However, subsequent proofs are hinging on it, so I figure I'll turn to you guys for clarification.
So he defines the direct sum (I guess in this ca... |
H: Equivalent definition for a collection of simplices to be a simplicial complex
I am reading the following lemma from Munkres' Elements of Algebraic Topology:
Lemma 2.1 A collection $K$ of simplices is a simplicial complex if and only if the follow hold:
Every face of a simplex of $K$ is in $K$.
Every pair of dis... |
H: Error in manual derivative calculation
I need to compute the derivate of
$g(x) = \ln\left(\frac{1}{x \sqrt{x+1}}\right)$
Simplifying this I get
$g(x) = \ln(1) - (\ln x + \frac{1}{2}\ln(x+1))$
Evaluating the derivate I get
$g'(x) = 0-(\frac{1}{x} + \frac{1}{2x+2})$
I have compared my result to WolframAlphas answer a... |
H: Does this graph have two DISTINCT Euler circuits but no Hamiltonian circuit?
I know it doesn't have a Hamiltonian circuit because vertices c and f will be traversed twice in order to return to a. Just confirming this.
I mainly want to know whether I have the definition of distinct Euler circuits in a graph right, a... |
H: How to prove $x^{n}$ is not uniformly continuous
How to prove $x^{n}$ is not uniformly continuous on the interval $[0, +\infty)$ for $n \in \mathbb{N}$ and $n>1$?
AI: You want to show that there exists some $\epsilon>0$ such that there is no $\delta>0$ for which
$$|x-y|<\delta\implies |x^n-y^n|<\epsilon.$$
Try $\ep... |
H: Regular expression arithmetics
What are the rules of regular expression arithmetics ?
For example: Let $\Sigma=\{0,1\}$
$1. 1+01=(\epsilon+0)1$.
$2. (\epsilon+00)^*=(00)^*$
AI: Off the top of my head you have at least the following:
$\epsilon\alpha=\alpha=\alpha\epsilon$.
$\alpha+\beta=\beta+\alpha$.
$\alpha+... |
H: Using implicit differentiation on trigonometric fraction and logarithm
I am supposed to solve the following using implicit differentiation.
$$\sin\left(\frac{x}{y}\right)+\ln(y)=xy$$
This is what I have so far:
$$\cos\left(\frac{x}{y}\right)\frac{d}{dx}\left(\frac{x}{y}\right)+\frac{d}{dx}\big(\ln(y)\big)=y+x\frac{... |
H: Product integral of $x$ between $1/a$ and $a$ "should" equal 1 but doesn't
I was looking at the Product Integral. I'm defining it here with the Gamma symbol like this:
$\Gamma _a^bf(x)dx=e^{\int_a^b \log (f(x)) \, dx}$
It occured to me that the product integral of $f(x) = x$ in the range $[\frac {1} {a}, a]$ "shoul... |
H: Generator of multiplicative group of $\mathbb{Z}/p\mathbb{Z}$
Possible Duplicate:
Finding a primitive root of a prime number
Given a field $F_p = \mathbb{Z}/p\mathbb{Z}$ I want to find all $x$ such that $\{x^n|n\in \mathbb{N}\} = F_p \setminus \{0\}$. Note that $x=2$ seems to work for a lot of primes, but only $... |
H: Evaluating $\int_0^1 \! C(x) \, \mathrm dx$ through integration by parts
$$
\int_0^1 \! C(x) \, \mathrm{d} x.
$$
where $C(x) = \int_0^x \cos(t^2) \, \mathrm{d} t$.
I am really not quite sure how to go about this one, especially given that it needs to be calculated using integration by parts.
My lecturer has an e... |
H: Area enclosed by lines and curve .
I am given $y=x$ and $y=\frac{x}{8}$ for the lines. The curve is $y=\frac{1}{x^2}$ . This is more of a question about the question though. I have no idea what it is asking.
My instinct is to integrate over some limits but I am not sure where this circle features and whether I am ... |
H: $m \times n$ persons stand in $m$ rows and $n$ columns
$m \times n$ persons stand as a rectangle of $m$ rows and $n$ columns.
Each one is taller than the one next to his left.
Now, if commander orders each column to sort by height, prove that after such sorting, each one is still taller than the one next to his lef... |
H: Geometry/Triangles Problem
Let $A,B,C$ be a triangle's vertices (FIG.1), and $D$ is a point on $\overline {BC}$. It is given that $\angle B= 2 \angle C $, $\overline{AB}=\overline{DC}$ and also that $\overline{AD}$ is the bisector of angle $A$.. We have to prove that angle $\angle BAC=72°$.
$\hspace{0.54 cm}$
$$\... |
H: Choose 100 numbers from 1~200 (one less than 16) - prove one is divisible by another!
Prove that if 100 numbers are chosen from the first 200 natural
numbers and include a number less than 16, then one of them is
divisible by another.
How to prove this? many thanks....
AI: Assign pigeonholes by considering th... |
H: Boolean Algebra simplification - odd number terms
I'm new to boolean algebra and having problems simplifying expressions with odd number terms,
Expressions such as:
1.
A'B'C'D + A'B'CD + AB'C'D + AB'CD + ABC'D
2.
A'BC + AB'C' + A'B'C' + AB'C + ABC
Here is my logic for both expressions:
A'B'C'D + A'B'CD + AB'C... |
H: Inequality. $a^2+b^2+c^2 \geq a+b+c$
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that
$a^2+b^2+c^2 \geq a+b+c$.
Thanks
AI: Using Cauchy-Schwarz inequality we get
$$
a+b+c=a\cdot 1+b\cdot 1+c\cdot 1+\leq\sqrt{a^2+b^2+c^2}\sqrt{1^2+1^2+1^2}\tag{1}
$$
From AM-GM we obtain $a^2+b^2+c^2\geq 3\sqrt[3]... |
H: An eigen value problem
Let $A$ be a $3×3$ matrix with $\operatorname{trace} (A) = 3$ and $\det (A) = 2$. If $1$ is an eigenvalue of $A$, then what are the eigenvalues of the matrix $A^2 - 2I$?
AI: Hint: Can you find the other eigenvalues of $A$ ? what do you know about the sum of all eigenvalues of $A$ ? what... |
H: Probability density function. Find $k$
Let $f_{X,Y}(x,y) = kxy$, for $0 ≤ x, y ≤ 1,$ otherwise $f_{X,Y}(x,y) = 0$.
(a) Determine $k$ such that $f_{X,Y}(x,y)$ is a PDF.
AI: Selected from PDF at WikiPedia.
The probability density function is nonnegative everywhere, and its integral over the entire space is equal to ... |
H: Coin toss sequences
Suppose $20$ tosses of a coin yield $8$ heads, $12$ tails.
Let $X$ be the # of sequences of a head followed by exactly $2$ tails.
Let $Y$ be the # of sequences of a head followed by at least $2$ tails.
What are the values of $E[X]$ and $E[Y]$ ?
My efforts are meeting dead ends.
AI: The probabili... |
H: Question about definition of Sobolev spaces
I'm trying to understand the following definition:
which can also be found here on page 136.
Question 1: Closure with respect to what norm? It's not given in the definition.
Question 2: Do I have this right: I can view $C^\infty$ as a dense subspace of $H^k$ via the map ... |
H: What is the formula for this exponentially growing "stairs"?
I'm looking for a formula that, given a linear $x$ input, would yield values of $y$ in a "stairs" shape so to speak, in such a way that as the value of $x$ grows higher, the difference between each step is bigger, and it also takes longer to reach the nex... |
H: Extension of an embedding to a field extension by a transcendental element
Let $K$ be a subfield of a field $K'$ and suppose we have an embedding $\phi:K \rightarrow L$ of $K$ into an algebraically close field $L$. Let $x \in K'$. If $x$ is algebraic over $K$, then we can extend $\phi$ to $K(x)$. What if $x$ is tra... |
H: Find the number of different sets C such that $B\subset C\subset A$
Assume that A consists of n elements and $B\subset A$ consists of k elements. Find the number of different sets C such that $B\subset C\subset A$. I am reading the A. Shen and N. K. Vereshchagin book, Basic Set Theory.
I try to use Combinatorics fo... |
H: How to see $\sin x + \cos x$
$$\sin x + \cos x = \sqrt{2} \sin(x + \pi/4)$$
Is there an easy way to visualize this identity or to convert the left-hand side to the right-hand side?
In general, can $p \sin x + q \cos x$ for some integers $p$ and $q$ be easily expressed as $\sin($something$)$ or $\cos($something$)$?... |
H: Evaluating $\int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx$
Let $\alpha \in \mathbb R$ and let's call $I:=[0,1]$. Evaluate
$$
\int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx.
$$
Well, the case $n=1$ is easy and the integral equals $\frac{1}{\alpha+1}$, for every $\mathbb R \... |
H: Surface integral
Without getting into the whole question, I was asked to evaluate a surface integral
$\iint\limits_S f(x,y,z) da$
where S is the cylinder $x^2 + y^2 = x$
between $z=a$ and $z=b$
Now normally I would parametize this as a cylinder and it would be easy peasy but I'm worried about the equation of the... |
H: Newton algorithm for a function in $\mathbf{R}^n\rightarrow \mathbf{R}$
I am curious on how the Newton algorithm would work to solve an equation of the type: $f(x_1,\dots,x_n)=0$.
As far as I understand, in dimension $1$, one solves $f(x)=0$ by starting with some $x_0$, and building the sequence $(x_n)$ defined by ... |
H: Roots of rational equation with multiple variables?
Let's say we have a rational polynomial in $k$ variables. We are only interested in rational solutions. If $k = 1$, name the variables ${x}$, if $k = 2$, name them ${x,y}$.
For $k = 1$, it can be done very fast. The Rational Root Theorem gives a set of candidates.... |
H: Showing $\lim_{n \to +\infty} \log(n!)/(n\log n) = 1$ without using Stirling approximation
As a passage of a bigger limit I have to show that $$ \lim_{ n \to \infty } \frac{\log(n!)}{n\log(n)} = 1. $$
I think it could be done using Stirling approximation, but I'm wondering if there's a way without that formula.
AI:... |
H: Proof by induction of Bernoulli's inequality $ (1+x)^n \ge 1+nx$
I am working on getting the hang of proofs by induction, and I was hoping the community could give me feedback on how to format a proof of this nature:
Let $x > -1$ and $n$ be a positive integer. Prove Bernoulli's inequality:
$$ (1+x)^n \ge 1+nx$$
Pro... |
H: Some questions about distribution theorem
Given an equation $P(D)u=0$, where $P$ is a polynomial (not equal to a constant). Here are some basic information about the distributional solution $u$:
If $P$ has at least one real root, then there exist a (non zero) solution $u \in S'$(temperate distribution);
It never h... |
H: Express Expectation and Variance in other terms.
Let $X \sim N(\mu,\sigma^2)$ and
$$f_X(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$
where $-\infty < x < \infty$.
Express $\operatorname{E}(aX + b)$ and $\operatorname{Var}(aX +b)$ in terms of $\mu$, $\sigma$, $a$ and $b$, where $a$ and $b$ ar... |
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