0
votes
0answers
3 views

recursive automata, or recursion mod 2.

Consider the list of length $m$ $(1,0,\dots 0)$ we call this list $l_1$, we now define a sequence of lists recursively, where $l_1$ is the previous list, and if $l_n$ is the list $(a_1,a_2\dots a_n)$ ...
0
votes
0answers
12 views

What algebra book to read after Artin's Algebra?

Could I directly go to Lang's Algebra? Or should I supplement some gaps by Dummit and Foote?
0
votes
0answers
8 views

If $p(z)$ is an injective polynomial $\Longrightarrow$ $p(z)=az+b$

If $p(z)$ is an injective polynomial, how to prove that $p(z)=az+b$ with $a\neq 0$. $p(z)\in\mathbb{C}[z]$. Any hint would be appreciated.
0
votes
0answers
4 views

A problem about martingale with filtration.

Is there something wrong with this statement? Why $\sigma(M_n)$ is a filtration instead of $\{\sigma(M_1,\cdots ,M_n)\}$?
0
votes
0answers
2 views

$L^{1}$ norm of a horizontally shifted measurable function

Suppose we are in $(\mathbb{R}, \mathcal{B}(\mathbb{R}), m)$, where $m$ is Lebesgue measure and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Also, suppose $g: \mathbb{R} ...
0
votes
0answers
6 views

Condition under which a locally convex topological vector space becomes a normed linear space

Is this true that a locally convex topological (Hausdroff) vector space becomes a normed space when its local base has only one element, so only one minkowski functional and so only one seminorm and ...
-1
votes
0answers
11 views

calculus-differentiation [on hold]

a) A body moves along the $x$-axis with displacement $x$ cm where $x(t)=10[t-\ln(t+1)]$, $t>0$ and $t$ is the time in seconds. Find the object's velocity and acceleration after being in motion for ...
1
vote
1answer
15 views

Looking for a matrix A(t)

I need your help, I'm looking for a contraexample, I need to give a matrix A(t), such that $$e^{\int_0^tA(s)ds}$$ is not a matrix solution for $x'=A(t)x$. I really don't have any clue what can it be. ...
-1
votes
0answers
13 views

Solving two differential equations

del^2 B = - mu_0^2 * epsilon_0^2 * partial derivative of B with respect to t del^2 E = mu_0^2 * epsilon_0^2 * partial derivative of B with respect to t
0
votes
2answers
28 views

Find the smallest value of $f(x, y, z)$

Find the smallest value of $f(x, y, z) = \sqrt{x^2 + 1} + \sqrt{(y - x)^2 + 4} +\sqrt{(z - y)^2 + 1} + \sqrt{(10 - z)^2 + 9}$ I found this question while looking from some exam papers and have no ...
2
votes
2answers
27 views

Prove second derivative of $g$ is proportional to $g^2$

From Apostol's Calculus Vol. 1, chapter 6.26, exercise 30: Let $f(x) = \int_0^x (1+t^3)^{-1/2} dt$. $a)$ Prove $f$ is strictly monotonic. $b)$ Let $g$ be the inverse of $f$. Show that the ...
0
votes
0answers
3 views

Solving implicit theta-method function numerically faster than using fixed point iteration

When we are using the theta-method to solve an IVP. We have the equation: $$y(x_{n+1}) \approx y(x_n) + h[(1-\theta) f(x_n, y(x_n)) + \theta f(x_{n+1}, y(x_{n+1})]$$ where $f(x,y)=y'(x)$ The only ...
0
votes
0answers
31 views

How to solve the following integral?

I think the answer is D. but do is missing the + C? Can I multiply by (-1) to get the to answer number C? or I'm just completely confuse.
0
votes
2answers
40 views

Show that there is no surjective ring homomorphism from $\mathbb Z_2[x]$ to $\mathbb Z_2 \times \mathbb Z_2\times \mathbb Z_2$

I saw this question as a bonus from a past exam, and here's my solution for verification. I argued like so. I said suppose there is such a surjective homomorphism $f$, then $f(0)=(0,0,0)$, $f(1)= ...
9
votes
0answers
22 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
1
vote
1answer
14 views

Alternative way of proving the subgroup of rotations is normal in $\mathbb D_4$

I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used ...
0
votes
1answer
16 views

Proving that an equilateral triangle in the plane cannot have vertices on integer lattice points

I am hoping a few of you mathematicians more experienced with writing proofs might give me some guidance here and possibly give me some ideas about how to restructure the following into a more ...
2
votes
2answers
17 views

Is it meaningless to say $M\prec N$ for two proper class models?

Kunen in page 88 of his "Set Theory" book says: ... For a specific given $\varphi$, the notion $M\prec_{\varphi}N$ (i.e. $\forall \overline{a}\in M~~~M\models \varphi ...
0
votes
1answer
20 views

Normal Distribution Worded Problem

Standard deviation = 2.5 mL 98% of bottles must be between 998 mL and 1000mL Pr( 998 < x < 1000) = 0.98 This is a technology exam question, therefore to find the mean I used the method: ...
-1
votes
4answers
47 views

Please help with derivative question

Which is the derivative of $\ln(e^{\ln x})$? a) $\ln x$ b) $x$ c) $e^x$ d) $\ln(\ln x)$ e) Other I'm pretty sure the answer is $1/x$ but I'm not confident enough to mark none of the above...
-3
votes
0answers
16 views

There are mn+1 different integers randomly arranged [duplicate]

,then,there either exist m integers arranged in the order that one behind is bigger than one infront,or....I have tried look the order in the view of set theory.
7
votes
2answers
55 views

When does $(e^a)^b = e^{ab}$ hold?

For a complex number $A$ and a real number $B$, when does the well-known formula $(e^A)^B = e^{AB}$ fail? Or does it hold at all for complex A? Since $e^{2\pi i} = 1$, if this formula holds for ...
0
votes
0answers
12 views

What does a presentation on block design and Latin squares consist of?

I read the wikipedia pages of both and I just cannot understand these two concepts. I have a presentation on both of these topics next week and I need some headway on both of these topics.
1
vote
2answers
19 views

Prove that function is inner product

$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$. I need to show this function is an inner product: $$\langle p,q\rangle=\sum_{j=0}^n ...
-1
votes
2answers
19 views

Triangle Inequality on complex numbers

Problem Let $z= x + iy$, then prove that: $$|x| + |y| \le 2 ^{1/2} |z|$$ Progress I've tried to write $|z|$ as $(x^2 + y^2)^{1/2}$, and to make some algebra after this, but I'm really new at ...
0
votes
0answers
12 views

Closed Connected Subgroup of $SO(5)$

I was reading a paper in which a part of it they want to classify the closed connect subgroups of $SO(5)$. What they write is this: Let $G^0$ be a closed connected subgroup of $SO(5)$. Let $T$ be a ...
0
votes
1answer
20 views

Distribution of sample median for a discrete random variable

Say I have a set $S = \{x_1, \dots, x_m\}$, where $1 \le x_i \le n$, all distinct, with median $M$. I take a sample $T$ of size $t$ from $S$, with replacement. I call $Y$ the median of $T$. What is ...
0
votes
1answer
12 views

How to Find The Roots of this Quadratic Given Sum & Product

My question is: The sum of the roots of a quadratic is $55/72$, and the products of the roots is $-25/12$. Find the roots. How I'm trying to do it so far: (Also, please correct my thought process if ...
2
votes
0answers
24 views

Combination question why doesn't my method work?

A Mathematics department consists of 5 female and 5 male teachers. How many committees of 3 teachers can be chosen which contain at least one female and at least one male? a) 100 b) 120 c) 200 d) ...
10
votes
2answers
103 views

When does L' Hospital's rule fail?

This thought jumped out of me during my calculus teaching seminar. It is well known that the classical L'Hospital rule claims that for the $\frac{0}{0}$ indeterminate case, we have: $$ ...
0
votes
1answer
18 views

Analysis inequality of norms problem

This seems to be a bit of an odd one. I have worked out a possible answer, but I have a feeling I am going about this the wrong way. Help would be appreciated. Find $m,M\in \mathbb{R}$ so that for ...
1
vote
1answer
37 views

What are some examples of coolrings that cannot be expressed in the form $R[X]$?

Let $K$ denote a field. Then the polynomial ring $K[x]$ has the property that the sum of two units is either a unit, or zero. I'll bet there's heaps of other examples, though. So let a coolring be a ...
0
votes
2answers
37 views

$G=\{f_n(x):n\in \mathbb{Z}\}$ is cyclic

Define $f_n(x)=x+n \;\;\forall n \in \mathbb{Z}$. Let $G=\{f_n:n\in \mathbb{Z}\}$. I proved that $G$ is a subgroup of $S_\mathbb{R}$ ($f_n$ is a permutation of $\mathbb{R}$), and now I am trying to ...
1
vote
1answer
27 views

Find slope of tangent line using m$_{tan}=\lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$ and the point $P = (5,\frac{2}{5})$

Perhaps I'm missing something simple here, but every time I attempt this problem I get the same answer that does not make sense. The question says, use the definition m$_{tan}=\lim\limits_{x\to ...
0
votes
1answer
26 views

Showing that a Unit Speed Curve is a Circle.

In my recent differential geometry tutorial, we were given the question: Given the unit speed curve, $$\boldsymbol{r}(s)=\left(\frac{4}{5}\cos(s),1-\sin(s),-\frac{3}{5}\cos(s)\right)$$ show that ...
2
votes
1answer
74 views

Proof that odd + odd = even

In math class today we started talking about proofs that odd + odd is even. We went over the basic proof (using 2k+1 and equations etc) and I realized that the only reason that this property exists is ...
0
votes
0answers
16 views

Upper bound on the covariance of two gamma processes?

Given two binary gamma processes, $X = \Gamma(t; \gamma_1, \lambda_1)$ and $Y = \Gamma(t; \gamma_2, \lambda_2)$, what is their maximum covariance? Applying this answer, it would seem that it is the ...
0
votes
0answers
11 views

Prove that mean square error equals expected conditional variance

Hey there stack exchange, I'm a first year grad student in Statistics. The book I'm using mentioned conditional variance, and I wanted to read up more about it. I dove down the google rabbit hole and ...
1
vote
2answers
36 views

Proper way to express 0 in this case?

If 0=(x-a)(x-b)(x-c)...(x-x)..=0. So it's a product sum that we write with pi instead of sigma but how? There should be indexes but I'm not convinced that I understand what notation to use. $$\prod_{ ...
2
votes
1answer
12 views

Count numbers in a range “A to B” which the number of its divisors equal to N

I'am looking for efficient algorithm to find the number of divisors for Numbers in a Hugh Rang up to 10^9. Such task is presented in those two problems: NDIV, and spoj NFACTOR I used prime ...
0
votes
1answer
13 views

Probability of Birth Process

Suppose a simple birth process with birth rate $\beta$ starts with two individuals. What is the probabilities that at time $t$ the population contains two individuals?
0
votes
3answers
30 views

How to find the point of intersection with three equations?

Given the following equations with three variables $a, b, c$ $a-5b+4c=-3$ $2a-7b+3c=-2$ $-2a+b+7c=-1$ How can I determine the point (if it exists) at which all three lines intersect?
5
votes
1answer
50 views

Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$?

The exercise asks me this: Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$? ps: $f: \mathbb{R}\to \mathbb{R}$ I really don't know how to start :c, I appreciate hints.
0
votes
1answer
13 views

algorithm to see if some sign(s) of terms will sum to zero

I was trying to develop an algorithm so see if a given set of terms $a,b,c,d$ can sum to zero given any possible combination of signs $+/-$ on those terms. How does one compute how many distinct ...
0
votes
0answers
24 views

Modular forms are arithmetic objects

What does arithmetic object exactly means? In an article, I found the following statement: modular forms are arithmetic objects. What this should means? Bests.
2
votes
1answer
42 views

Does this argument rely on countable choice?

Consider the following Theorem: Any algebraic field extension $K|F$ of infinite degree contains finite subextensions of arbitrarily high degree. Proof: We'll prove that, for any n, there's a ...
0
votes
1answer
23 views

Proof of one instance of the Axiom of Choice from another.

I'm trying to show $(i)\implies(ii)$: $(i)$ For any relation $R$, there exists a function $H\subseteq R$, with $domH = domR$. $(ii)$ For any set $I$ and any function $H$ with domain $I$, if ...
1
vote
2answers
32 views

Geometric interpretations of an equality.

I need to prove that for complex numbers $w_1, w_2$ and $w_3$ if: $$\frac{w_2-w_1}{w_3-w_1}=\frac{w_1-w_3}{w_2-w_3}$$ then: $$|w_2-w_1|=|w_3-w_1|=|w_2-w_3|$$ by geometric interpretation of the given ...
2
votes
1answer
30 views

For any series that diverges, does there exist a sequence that converges to 0 yet the product diverges

Suppose $\sum_{n=1}^{\infty} a_n = \infty$. Does there exists a sequence $\{b_n\}$ such that $ \lim_{n \rightarrow \infty} b_n = 0$ where $\sum_{n=1}^{\infty} a_n b_n = \infty$? I am able to prove ...
0
votes
1answer
38 views

What is difference between ≈ and ~?

I'm reading a quantum mechanics book, and it has the following equation: $$ \Delta x \approx \frac{\lambda}{\sin\alpha} \sim \frac{h}{mc\sin\alpha} $$ What is the difference between $\approx$ and ...

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