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If $x$ does not equal $1$ then either $x$ is not a perfect square or $x+3$ is not a perfect square. I know how to prove a perfect square but no idea on this
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Prove or Disprove, If $ T: \mathbb R^n \rightarrow \ell_2 $ is linear, then It preserves total boundedness $ T(A) $ is totally bounded in $\ell_2$ when $A \subseteq\mathbb R^n$ is bounded.
I think this is not true. am i right?
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Suppose that $a$ and $b$ are some floating point numbers such that $a\lt b$. How can I show that $$a\le \text{fl}\left(\frac{a+b}{2}\right)\le b$$
specifically in IEEE standard floating point arithmetic? In this, for example $\text{fl}(a)$ refers to the floating point number closest so some value $a$.
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I understand geometrically why the Gaussian integers modulo $i+1$ is $\mathbb{Z}_{2}$, using lattices. Is there an algebraic isomorphism construction, however?
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Let $\mathbb{X}$ be a normed space that is complete and $\mathbb{Y}$ be another normed space which is not complete. Then can a bounded linear map $A:\mathbb{X} \to \mathbb{Y}$ be bijective or not?
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Let $a,b,c,d$ be positive integers. Suppose that $$\frac cd=\frac ab.$$ I want to prove that if $a$ and $b$ are relative primes, then $c/a=d/b$ is an integer.
That is, the only way a fraction can be represented in a way other than its simplest form is to multiply both the numerator and the denominator by the same integ... |
First of all, I apologize for my amateurness and inexperience. Although I always enjoyed math, only two years ago I started experimenting with continued fractions and gained a deep reverence for them.
I first "discovered" the trivial but beautiful identities linking the golden ratio to the n-th Lucas number through raw... |
Mrs. Grundy has two children. Given that Mrs. Grundy has at least one child born on a Monday, what is the probability that both her children were born on Mondays?
Assume that each child was equally likely to be born on any day of the week, and that the two birthdays are independent (Mrs. Grundy doesn't have twins!).
I ... |
I understand that when $G$ has exactly one maximal subgroup (inclusion-wise), then $G$ has to be cyclic.
But is it possible to determine all possible groups with exactly one maximal subgroup?
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Can we draw infinitely many hexagons, not necessarily convex, on the plane so that any three of them share a common interior point, but no four of them does? For four hexagons this is possible, using concavity to get the fourth hexagon to avoid the area when the other three overlap but hitting the area where any pair o... |
I have a proof that I'm working on and it goes like this:
Assume $a_k > 0$ and $a_k$ is monotonically decreasing. Show that:
$$\sum_{k=1}^\infty a_k < \infty \iff \sum_{k=0}^\infty b_k < \infty$$
where $b_k = 2^k a_{2^k}$
My problem is that I don't know where I can start versus what I have to show beforehand. Do I star... |
Suppose that $f$ is differentiable and that $\nabla f(x) = 0 \forall x \in \mathbb{R}^n$. Prove that there is a $v \in \mathbb{R}$ such that $f(x) = v \\ \forall x \in \mathbb{R^n}$
could anyone offer some guidance, I am completely stuck.
here's what I have thus far:
Fix $x,y \in \mathbb{R^n}$. Let $p:[0,1] \rightarrow... |
In the gambling game "craps," a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2, 3 or 12. If the sum is 4, 5, 6, 8, 9, or 10, that number i... |
The above applies $\forall x,y \in \mathbb{R}$
I've tried: $x + y \ge 0$
$$x + y \ge x$$
$$ (x + y)^2 \ge 2xy$$
$$\frac{(x + y)^2}{2} \ge xy$$
But the closest I get is $\dfrac{x+y}{\sqrt{2}} \ge \sqrt{xy}$
Any ideas?
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I just don't understand how this equation breaks down like this.
The second step... $[k^2 + k + 2k + 1]$ is perplexing, but breaking that down to $(k+1)(k+2)$ has completely baffled me. I would expect this to be equivalent to $[k^2 + 2k + 1k + 2]$.
Any help would be greatly appreciated.
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Consider the set $\mathrm{GL}(n,\mathbb{R}) = \{ \ A \in M_{n \times n}(\mathbb{R}) \ | \ \mathrm{det}(A) \neq 0 \ \}$.
I'm trying to show that this is smooth submanifold of $\mathbb{R}^{n^{2}} \cong M_{n \times n}(\mathbb{R})$.
I am confused about what atlas to use to prove this. My thoughts were to find a set $U \su... |
Is there an easy way how to characterize the groups with exactly two automorphisms?
I was able to find the following finite groups with exactly two automorphisms: $\mathbb{Z}_3$, $\mathbb{Z}_4$, $\mathbb{Z}_6$.
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The number of storms in the upcoming rainy season is assumed to be Poisson distributed, but with a parameter $\Lambda$ that is also random and uniformly distributed on $(0,5)$. That is, $\Lambda \sim$ unif$(0,5)$ and given that $\Lambda = \lambda$, the conditional distribution of the number of storms $N$ is Poisson wit... |
While thinking about geodesic lines I started exploring subsets of a metric space that have the following property.
$ \forall a,b,c \in L, d(a,c)> d(a,b) \land d(a,c) > d(b,c) \implies d(a,c) = d(a,b) + d(b,c) $
Where L is the subset I'm investigating. In $\mathbb{R}^n$ with a Pythagorean metric straight lines have thi... |
How many incongruent primitive roots does $13$ have:
So far I know that $\phi(\phi(13)) = \phi(12) = 4,$ so there would be $4$ primitive roots. And a RRS mod $13$ is $ \{1,2,3,...,12 \}.$ But I am not sure where to go from here to find the incongruent primitive roots. Any help would be appreciated!
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Given a square matrix $M$, we know the exponential of $M$ is
$$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$
and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(I-M)^k}{k}$$
The derivative of $\exp(M)$ should be itself. It is easy to prove if $\frac{dM}{M}=I$.
But how to calculate the derivative of $\log(M)$? By the... |
Suppose we have a collection of independent Boolean random variables $X_i$ and $Y_i$ (for $1 \le i \le N$), and are told $p_i = P(X_i = 1)$ for all $i$. We are now given a set of values $x_i$ that was produced either by $X_i$ or by $Y_i$ (but we don't know which).
We would like to determine how likely it is that the $x... |
Let S = {a,b,c}. Recall that a relation on S is a subset of S×S. Give
an example of a relation R on S that is reflexive and:
a. Symmetric but not anti-symmetric.
b. Anti-symmetric but not symmetric.
c. Neither symmetric nor anti-symmetric.
d. Both symmetric and anti-symmetric.
My Answer:
S = {a,b,c}
Relation: = ‘|x| ... |
Let $C \subseteq [0,1]$ be uncountable. Let $A$ be the set of all $a \in (0, 1) $ such that $ C \cap [a,1] $ is uncountable and set $\alpha = \sup A$. Is $C \cap [\alpha, 1]$ an uncountable set?
I know that $a$ must exist, but I'm not sure how to go about proving/disproving this statement.
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Given $f(x)=x^4+2x^3+2x^2-2x-3$, where $x-1$ is a factor of $f(x)$, how is it possible to solve $f(x)$ without the Rational Root Theorem?
Here's my progress:
$$f(x)=x^4+2x^3+2x^2-2x-3$$
$$f(x)=(x-1)(x^3+3x^2-5x+3)$$
And there's where I got stuck.
I cheated, though, and tried solving this equation with the Rational Root... |
let $x= 0.1234567891011121314\ldots$
Let $N(k,n)$ be the number of occurence of the digit $k$ ($0 \leq k \leq 9$) in the first $n$ decimals of $x$.
If $\lim_{n \to \infty} \frac {N(k, n)}{n}$ exists then that limit is called the frequency of $k$ (in base $10$).
If all digits have frequency $1/10$ then the number is sa... |
Say I have a function $f(x) = \sqrt{x}$ for $x$ greater than or equal to $0$. If $x$ approaches $0$ from the left, can we say the limit exists? values less than $0$ arent even in the domain of our function and while its clear to see the function exists at $x=0$. I dont think we can evaluate for values outside the domai... |
Create a tetrahedron with corners $A=(1,2,4)$, $B = (1,0,2)$, $C = (2,1,3)$ and $D = (4,1,1)$. Determine the angle $\alpha$ between the edge $AB$ and the side $BCD$.
The first step I make is displacing the tetrahedron to place $B$ in origin. Thus we get the new coordinates $A' = (0,2,2)$, $B=(0,0,0)$, $C'=(1,1,1)$, $D... |
If $|z_{1}|=2,|z_{2}|=3,|z_{3}|=4$,then find maximum value of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$.
My attempt:I considered 3 circles having centre origin and radii as $2,3,4$. Then I tried invoking the triangle inequality $|z_{1}-z_{2}|\leq|z_{1}|+|z_{2}|$ where equality occurs if origin,$z_{1},z_{2}$ are... |
$$ C=\frac{h}{m(t_1-t_2)} $$
Solve for $t_1$.
The correct answer I have been given is, $t_1=t_2-h/(mC)$. I just need help in the steps taken to reach this.
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Say we have a standard deck of $52$ cards. Probability of drawing the King of Hearts is $\frac{1}{52}$ obviously.
But lets say we were to make $30$ draws with replacement (so each time the card is drawn, it is put back in the deck and and the deck is shuffled). What are the odds that the King of Hearts card was drawn... |
If $g ∈ \mathrm{SO}(3)$ is the rotation about axis $p$ by angle $α$, and $h$ is a rotation mapping $p$ to another line $q$, then $g$ conjugated by $h$ is the rotation about $q$ by the same angle $α$. I've seen this fact presented as obvious, but I don't find 3D rotations trivial to visualize in general, and would like ... |
I have "solved" the following question using prove by contradiction. But it seems a bit off to me:
Let {$x_k$} be a sequence satisfying $x_{k+1}\le(1-\beta)x_k$ for $0\lt\beta\lt 1$, and $x_0\le C$. Prove that $x_k\le \varepsilon$ for all
$k\geq \beta^{-1}log(\frac{C}{\varepsilon})$
What I did is as follows:
Assume f... |
I am unsure how to prove that $P(A_1 \cup \cdots \cup A_n) \leq \sum_{k=1}^n P(A_k)$. The left hand side looks like the the general inclusion-exclusion formula, but the right side does not. However, it seems like a proof by induction would work well for this problem, but I'm not sure how to make the jump after the base... |
If $(a_n) \rightarrow 0 $ and $(b_n - a_n) \rightarrow 0$, then $(b_n) \rightarrow a$.
I can't think of any counterexamples to this conjecture but I'm not sure how to prove it. Any help would be appreciated.
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In the Mandelbrot Set, c=0 has a periodicity of 1 and is surrounded by a cartoid of non-periodic points that asymptotically approach periodicty 1. Extending left along the real axis are connected lobes of respective periodicity $2^1,2^2,2^3,2^4,2^5...$ Where exactly does that series of lobes end?
The center of the $... |
I already know about the Harmonic series: $$\sum_{n = 1}^{\infty} \frac 1n = 1 + \frac 12 + \frac 13 + \frac 14 + \frac 15 + \frac 16 + \cdot \cdot \cdot$$ But is there a name for this infinite product series: $$\prod_{n = 1}^{\infty}n = 1 \times 2 \times 3 \times 4 \times 5 \times \cdots$$ It is not that I need any he... |
I don't understand why Gauss's lemma is invoked in the proof in Dummit and Foote that $\Phi_n(x)$ (the $n$th cyclotomic polynomial) belongs to $\mathbb{Z}[x]$. I'm an analyst and I wanted to remind myself about cyclotomic polynomials. The following is the proof as I've written it, and because it's been a while since I'... |
Let $f:X \rightarrow Y$ be a continuous map of topological spaces and $B \subseteq Y$
Is it true that $f^{-1}(\overline{B})=\overline{f^{-1}(B)}$?
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I am trying to show that any solution of $y'' = -(1+e^x)y$ goes to $0$ as $x \to \infty$.
I was able to show that the solutions hit $0$ infinitely often.
It seems (empirically) that the solutions decay like $e^{-x/4}$.
More specifically, $e^{x/4}y(x)$ is bounded (away from both $0$ and $\pm \infty$) as $x \to \infty$.
... |
How do you prove
$$1^2 + 2^2 + 3^2 + \cdots + n^2 = \dfrac{n(n + 1)(2n + 1)}{6}$$
without induction?
I'm looking for a combinatorial proof of this.
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How to solve
$$\frac {ds}{dt}+s=|t|,~s(0)=1.$$
Because i have never seen before such a inhomogeneous ODE, where on right hand side there is modulus function. What i did to solve it, i broken up right hand side in to two point function by definition of modulus function, but still i did not get my apple. Any help will ... |
Prove that if $k$ is any odd integer and $m$ is any even integer, then, $k^2 + m^2$ is odd.
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So I have an answer.. but the basis step doesn't make any sense to me. It is possible that I do not understand the syntax used.
Let $P(n)$ be the predicate $3 | (n^3 - n)$ for every positive integer $n$.
For $n = 1, P(1)$ is $3 | (1 - 1)$.
This can be written as $(1 - 1) = 3\cdot0$ which is clearly true.
... |
all!
I'm working on some homework problems from my linear algebra class and we're solving just basic matrix equations, but the solutions have what look like transpose symbols attached to the solution set. I'm getting the correct solutions, I'm just not sure why the symbol is there. If you could enlighten me, I'd be m... |
Consider the vector space $F_n^2$ , and, as usual, let $e_i$ denote
the vector with 1 at the $i$th coordinate, and 0 at all others. Call a
vector of the type $e_i + e_j$ an edge vector, (think of it as
characterizing the edge between i and j in the graph on the vertex set
${1, 2, \dots, n}$. Let $W$ be the $n−... |
Let $a \in \mathbb{Z}[i]$ such that $N(a)$ is a prime or the square of a prime congruent to 3 modulo 4 in $\mathbb{Z}$. That is, $N(a)=p$ or $p^2$ where $p \equiv 3 \bmod 4$. Prove that $a$ is a prime element.
I know that if $N(a)$ is a prime then $a$ is prime as a Gaussian integer. How do I show that the ideal gener... |
I am trying to understand a step in the theory section of my differential equations textbook. The author writes,
For example, suppose we transform the first order differential
equation $\frac{dy}{dx} = f(x,y)$ by the substition $y = g(x,u)$,
where $u$ is regarded as a function of the variable $x$. If $g$
possess... |
Let $H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\}$ for all $g \in G \setminus H$. Then is it true that $$|\cup_{g \in G \setminus H}gHg^{-1}|>\dfrac 12 |G|+1$$
If not, then is it at least true that
$$|\cup_{g \in G}gHg^{-1}|>\dfrac 12 |G|+1$$
(I know the upper bound $|\cup_{g \in G}gHg^{... |
Suppose $U$ and $W$ are subspaces of a vector space $V$ such that $\dim(U) =\dim(W)$ and $U\ne W$, how to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$?
My approach is to use $\dim(U+W)=\dim(U)+\dim(W)-\dim(U\cap W)$.
Since $U\ne W$, $\dim(U\cap W)=0$. Now I need to prove that $\dim(U+W)=\dim(V)$.
So $\dim(U+W)=\dim... |
I have the complex number $z = 5 + 6i$
in polar form
$$z = \sqrt{61} (\cos \theta + i\sin \theta)$$
and
$$\theta = \tan^{-1}\left(\frac{6}{5}\right) = 0.87605805059 \text{ rad}$$
But I need that value in multiples of $\pi$ so it becomes easier to handle in a sum with another complex value.
|
Use the identities
$$\cos 2x=2\cos^2 x -1=1-2\sin^2 x$$
$$\sin x=\cos \left(\frac{\pi}{2}-x\right)$$
to help evaluate $$\int_{0}^{\pi/2} \sqrt{1-\sin x} \; dx$$
I've already done $\int_{0}^{\pi/2} \sqrt{1+\cos x} , dx$ but I'm stumped on this one, thanks.
|
I was faced with a pretty tricky math problem from my school. Here is the question.
If $f(x-4) = x^3 + 2x^2 + 7x + 1$, determine $f(x)$.
My first instinct is to graph the initial equation and shift it $4$ points to the right, I saw the $(x-4)$. Would this be the correct way to start?
All help is appreciated. Thanks in ... |
Let $n$ be an odd integer more than one. Prove that the sequence $$\binom{n}{1}, \binom{n}{2}, \ldots,\binom{n}{\frac{n-1}{2}}$$
contains an odd number of odd numbers.
I tried writing out the combination form as $$\frac{(2k+1)!}{(m!)((2k+1)-m)!}.$$ How do I use this to show that the sequence contains an odd number ... |
I am supposed to use equivalencies to find the prenex DNF for the wff:
$\exists xp(x) \land \exists xq(x) \rightarrow \exists x(p(x) \land q(x))$
It's been awhile since I've done something like this so I might need a refresher, but the answer I am getting is:
$\lnot \exists xp(x) \lor \lnot \exists xq(x) \lor \exists x... |
How many different varieties of pizza can be made if you have the following choice: small medium, large; thin, hand tossed, pan; and $12$ toppings (cheese is an automatic), from which you may select from $0-12$.
What I did for this was $3C1 \cdot 3C1 \cdot 13C1$. However, I know my answer is incorrect. Is it because... |
I started off with: one of $V_1$ or $V_2$ must be countably infinite. WLOG (without loss of generality) Let $V_1$ be countably infinite. I am not sure how to proceed after this.
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A graph is $k$-regular if every vertex has degree $k$. Show that if a bipartite graph $G = (V, E)$ with bipartition $V = A \cup B$ is k-regular, then $|A| = |B|$.
I dont understand it. Please explain and show the proof so I can understand.
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I have this ode as the following,
$$\lambda(t)^\prime+p(t)\lambda(t)=G(t)$$
I need to solve for $\lambda(t)$. Any suggestion or hint is much appreciated.
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The question asks me to find the coefficient of the term $x^6y^4$ in the expression $(xy^2+x^2+3y)^7$. This was pretty simple. This is how I did it:
$$(xy^2+x^2+3y)^7 = \sum_{a+b+c = n} (xy^2)^a + (x^2)^b + (3y)^c $$
$$ x^{(a+2b)}, y^{(2a+c)}, 3^c = x^6y^4 $$
From this we get, $$ a+2b = 6, 2a+c = 4, a+b+c = 7$$
Solving... |
Show that the two pairs of lines $12x^2+7xy-12y^2=0, 12x^2+7xy-12y^2-x+7y-1=0$ form a square.
I know that both the equations represent a pair of straight lines. Also the first equation represents a pair of straight lines through origin which are as follows
$(4x-3y)(3x+4y)=0$
But how should I calculate for the second eq... |
I would like to find a pair of relatively prime polynomials $p,q \in k[x]$ (where $k$ is a field) such that $$\frac{p(x)}{q(x)} = \frac{p \left( \frac{1}{1-x} \right)} {q\left( \frac{1}{1-x} \right)} .$$
I believe that $p$ and $q$ should have at most degree 3.
A similar situation is the following:
Take $f(x) = \frac{... |
Problem Statement
Show how the $L_1$-sparse reconstruction problem:
$$\min_{x}{\left\lVert x\right\rVert}_1 \quad \text{subject to} \; y=Ax$$
can be reduced to a linear programming problem of form similar to:
$$\min_{u}{b^Tu} \quad \text{subject to} \; Gu=h, Cu\le e.$$
We can assume that $y$ belongs to the range of $A$... |
Hello all I'm teaching myself cryptography, and I'm struggling with polynomial arithmetic over finite fields. I've some what been able to teach myself how to do the arithmetic over $GF(2)$, but when the finite field increase to higher numbers like $GF(101)$ I struggle with long division aspect as it relates to the 101.... |
How can I prove that the following polynomial expression is divisible by 3 for all integers $k$?
$$k^3 + 3k^2 + 2k$$
|
Let $X\equiv\mathbb N^{\mathbb N}$ denote the set of all sequences of positive integers. For a fixed $n\in\mathbb N$ and $(y_1,\ldots,y_n)\in\mathbb N^n$, define the “point cylinder” as follows:
$$C_{y_1,\ldots,y_n}\equiv\{ (x_k)_{k\in\mathbb N}\in X\,|\,x_1=y_1,\ldots, x_n=y_n\}.$$ That is, a point cylinder contains a... |
Let $U$ be uniformly distributed on the interval [$\frac{1}{3},1$]. Let $X$ be a random variable such that the conditional distribution of $X$ given $U = p$ is Geometric with parameter $p$.
Show that $P(X=2) = \frac{2}{3} - \frac{13}{27}$
So this is how I approached the problem:
$P(X=k) = \int_{a}^{b} p(1-p)^{k-1} dx$ ... |
What is the most general mobius transformation that maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$.
I want to find the most general form of such a linear transformation, I'll denote it $T$. My reasoning is as follows:
The origin must be fixed since the origin in the $w$-plane is the only point where the two conc... |
I have a doubt related to angles which I am a bit embarrassaed to ask since I know is something of basic geometry, but nevertheless my question is the following:
As I understand it, an angle between two rays with a common vertex is related to the "openness" that exists between these rays. So, one way to measure this "o... |
If it is valid to interpret multiplication by a complex number as a dilative rotation, does that mean that it can be viewed as a function $$f: R^2 \rightarrow R^2$$
making it a linear operator?
|
If $R$ be a Union of zero measure (lebesgue) sets , what can we say about the cardinal of index set? Does this question related to continuum hypothesis? Thanks.
|
We are attempting to show that, for $t_0,t_1\in\mathbb{R}$, and $t_\sigma=(1-\sigma)t_0+\sigma t_1$ for any $\sigma\in[0,1]$, then the following inequality holds for any $u\in H^{t_1}(\mathbb{R}^d)$:
$\|u\|_{H^{t_\sigma}(\mathbb{R}^d)}\leq\|u\|^{1-\sigma}_{H^{t_0}(\mathbb{R}^d)}\cdot \|u\|^\sigma_{H^{t_1}(\mathbb{R}^d)... |
Questions
We define $A(x)= \text{number of prime factors of x}$
For example $A(2 \times 3^2) = 3$
I noticed when
$s_k = \frac{N!}{\prod_j n_j}$ and $\sum_{j} n_j = N$
$$ s_1 < s_2 \implies A(s_1) < A(s_2)$$
Why is this so?
Also What is the asymptotic relation of $A(x)$ as $x \to \infty$?
My attempt for asymptotic rela... |
The number of accidents follows a Poisson distribution with mean $12$. Each accident generates $1,2,$ or $3$ claimants with probabilities $\frac{1}{2},\frac{1}{3},\frac{1}{6}$ respectively. Calculate the variance in the total number of claimants.
This is a SOA exam question so I have been provided the answer and they... |
Let $\{f_n\}$ be a sequence of Riemann integrable functions.
What are the lightest conditions on $f_n$ to guarantee the following?
$$
\lim_n \int_a^b f_n\,\mathrm dx=\int_a^b \lim_n f_n \, \mathrm dx
$$
I'm aware of dominated convergence theorem, but are there lighter sufficient conditions for the above to hold? Maybe ... |
I have observed, by calculation, that the sum of the reciprocals of all the n-digit prime numbers is approximately 1/n, and that this becomes increasingly accurate as n increases. Is there a simple way to prove this?
I have also observed that this appears to hold true whether one is working in base 10, or in base 2, or... |
For simplicity, I'll give the example as splitting 3 bins of balls into 4 bins.
Bin 1 contains $N_1$ blue balls, bin 2 $N_2$ red balls, and bin 3 $N_3$ green balls
How many combinations of ways are there to split these balls into 4 distinct output bins?
How do you generalize this problem for $N$ bins of balls into $M... |
Problem:
A rectangular window is to be cut into a door 6.5 ft tall so that the window is 6 inches less than twice the distance above the window. The window must be 18 inches wide and have an area of at least 324 square inches. What is the maximum distance between the top of the door and the top of the window?
My attemp... |
The question is:
Let $S = \left\{a,b,c\right\}$. Recall that a relation on $S$ is a subset of $S\times S$. Give
an example of a relation $R$ on $S$ that is reflexive and:
a. Symmetric but not anti-symmetric.
b. Anti-symmetric but not symmetric.
c. Neither symmetric nor anti-symmetric.
d. Both symmetric and anti-symme... |
I have some issues understanding factorization.
If I have the expression $x^{2}-x-7$ then (I was told like this) I can put this expression equal to zero and then find the solutions with the quadratic formula, so it gives me $x_{0,1}= 1 \pm 2\sqrt{2}$ then $$x^{2}-x-7 = (x-1-2\sqrt{2})(x-1+2\sqrt{2}).$$
That is correct ... |
I'm reading a proof where it requires to show that a topology is not induced by a metric. My question is: What does it mean for a topology to be induced / not induced by a metric?
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Let $f:[0,1] \rightarrow [0,1]$ be a continuous function. Show that there exists c in $[0,1]$ such that $f(c)=c^2$
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Why does $i$ only get involved in factoring quadratics like $x^2+9$, but not $x^2-9$? Why does the $+$ sign lead to complex numbers?
|
If $\gcd(A,B,C)=1$, can we find $h$ s.t. $\gcd(A,B+hC)=1$?
I have tried but I find I am not able to prove this. Maybe I do not know some important thing? Could someone help? Thanks!
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Show that there exist two disjoint closed sets $A, B \subset \mathbb{R}$ such that there does not exist a positive $ϵ$ with $d(x, y)\ge ϵ$ where $x∈A$ and $y∈B$.
I have already proved that if $A$ is compact and $B$ is closed, then there exist a positive $ϵ$ such that $d(x, y)\geϵ$. So for the question that we need to... |
Abbreviate: $D =$ a person is diseased, $+ =$ the test is positive.
Assume: $\Pr(D) = 0.001, \; \Pr(+|D)=0.99, \; \Pr(+|D^C) = 0.01 \qquad ($$\iff$ $ \Pr(-|D^C) = 0.99).$
1. Use Bayes's Theorem: $\Pr(D|+) = 0.0 \; 90 \; 1639... \quad $ (confirmed; scroll down 30% hereof).
2. Quantify the above probabilities for 1000 ... |
I am having trouble understanding a problem of applying the concepts of permutations and combinations in an example that I found while reading my textbook.
Basically, it wants the number of elements in the set, S, of all lists L of length n that satisfy a given property. Each element of the list is selected from a fin... |
Let $a_1\le a_2\leq\ldots$ be an infinite sequence of positive integers. A positive integer $n$ is called good if $i=na_i$ for some index $i$. For which $(m,n)$ is it true that if $m$ is good, then $n$ must be good too? If $m=n$ it's obviously true. We're looking at the integers in the sequence $1/a_1,2/a_2,3/a_3,\ldot... |
I want to show if a linear transformation $T$ has rank $k$ then there exist $\delta$ such that the open ball centered at $T$ with radius $\delta$ contains only linear transformations that has rank greater than or equal to k
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Let $z:\triangle^{L-1}\to \mathbb{R}^L$ be continuous. Define $f:\triangle^{L-1} \to \triangle^{L-1}$ be defined component wise as
$$
f_l(s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum_{l=1}^L \max\{0,z_l(s)\}}
$$
Then $z$ being continuous implies that $f$ is continuous.
Is this obvious (to someone who, unlike me, understan... |
Question: Point P ($x_1 , y_1$) is on the hyperbola $\frac {x^2}{a^2}$ - $\frac {y^2}{b^2}$ = 1
Prove that the equation of the tangent at P is
$$ \frac {xx_1}{a^2} - \frac {yy_1}{b^2} = 1 $$
What I have attempted:
Equation of a tangent is:
$$ (y-y_0) = m(x-x_0) $$
In this case we have $x_0 = x_1$ and $y_0 = y_1$ ... |
I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to.
Thank you.
$$\lim_{n \to \infty} \sum_{i = 1}^n \frac{n}{n^2 + i^2}$$
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I wonder if someone help me with this:
I have $\pi_1+\pi_2+ \pi_3 +\pi_4=A$ and $\pi_1\pi_2\pi_3\pi_4=B$ where $\pi_i \;\forall i=1,2,3,4$ are unknown but $A,B$ are known numbers. Can I find for the following in terms of $A$ and $B$:
$1/\pi_1+1/\pi_2+1/\pi_3+1/\pi_4=?$
Any help is appreciated.
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Okay, checking to see if i'm on the right track. I essentially did the same prove for Euclid's lemma but exchanged the $d$ for the $p$. Is that the right idea? Or am I missing something?
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I am currently studying real analysis from rudin and really like the chapter on special functions. But Rudin does not give much knowledge about those topics. Reading the references I found book by Emil artin on The Gamma function and really liked it. Are there more books like Emil artin's gamma function on special func... |
I am trying to show that $W^{1,2,0}(U)$ is dense in $H^1(U)$ for an open, bounded set $U\subset\mathbb{R}.$ Where $W^{1,2,0}(U)$ is defined to be the space of smooth functions $f:U\to\mathbb{C}$ such that $f'$ exists and are in $L^2(U)$ and the norm is given by $\|f\|_{H^1(U)}=\|f\|_2+\|f'\|_2$
I have something written... |
What is meant by a 'pure' wave?
I know it might sound like a basic question, but I've never been taught this.
I saw that a sine wave is a pure wave. I tried Googling what a pure wave is, but all I get is links regarding Pure Wave Inverters for sale...which is not what I'm looking for.
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$13$ cards are chosen at random with no replacement from a deck of $52$ cards.
find the probability there are $5$ spades chosen, $4$ hearts, $3$ diamonds and $1$ club.
I got $(13!/(5!*4!*3!))/(52!/(13!*39!)$
I'm not sure if I had this correctly or not but my answer seems to be way off.
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He talks about a function $f(a)$ for which all the derivatives vanish. He shows inside a circle within our domain $\Omega$, for any circle $C$ we take, there $f$ is identicaly zero.
Then he shows that $f$ is identically zero in $\Omega$. Let $E_1$ be the set where $f$ and all its derivatives vanish, and $E_2$ be the s... |
Suppose $Q$ is cyclic $(h,q)$ code over $F_u$ such that $\gcd(h,u) = 1$.
Prove that degree of the generator polynomial $m(x)$ of $Q$ is $h - q$.
Why do we need the condition $\gcd(h,u) = 1$ ?
Any ideas?
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