All Questions
0
votes
0answers
6 views
Recursive definitions of sequence $a_n = n(n+1)$ and $a_n = n^2$
Question:
Recursive definitions of sequence $a_n = n(n+1)$ and $a_n = n^2$.
My Attempt:
For the first one,$a_n = n(n+1)$, I first manually generate a sequence using $n \geq 1$,
$$2, 6, 12, 20, 30, ...
0
votes
0answers
6 views
Nomenclature Question: Algebraic Studies
What are magmas, monoids, groups, rings, modules, fields, algebras, etc. called? I have been calling them "algebraic structures", but I want to know their real/recognized and general name. I would ...
0
votes
0answers
4 views
Local Rank in Stable groups
Let $G$ be a stable group (better still - a definable group in a stable structure), $a,b$ be independent over $A$, and $\Delta$ an invariant finite set of formulas.
Why is $R_\Delta(tp(a\cdot ...
1
vote
2answers
18 views
Integration by substitution, why do we change the limits?
I've highlighted the part I don't understand in red. Why do we change the limits of integration here? What difference does it make?
Source of Quotation: Calculus: Early Transcendentals, 7th Edition, ...
0
votes
0answers
9 views
Criteria for being a submanifold
Let $N$ be a subset of a smooth manifold $M$. I want to proof the following criteria:
Every connected component of $N$ is a locally closed submanifold of $M$ if (and only if?) for every point $n ...
0
votes
0answers
5 views
Iterative maximization of a multivalued function - Is it possible?
I wonder if such a maximization technique possible for a multivariable, well-behaving function like $f(x_1,x_2,...,x_N)$ starting at an arbitrary point $(p_1,p_2,...,p_N)$: We first the take the ...
-1
votes
0answers
7 views
Orthogonal Trajectories in Differential Equations
A curve rises from the origin in the $xy$ plane into the first quadrant. The area under the curve $(0,0)$ to $(x,y)$ is $\frac13$ the area of rectangle with these point as opposite vertices. Find the ...
0
votes
2answers
18 views
Sets as extremely trivial groups
A group is a structure defined upon an underlying set which is endowed with a single binary operator that has some rules attached to it. I was wondering whether one could describe a set itself as ...
0
votes
1answer
8 views
an ideal of matrix ring which is projective
Let $K$ be a field and
$$
A=\left\{
\begin{pmatrix}
a&b&c\\
d&e&f\\
0&0&g
\end{pmatrix}
:a,\dots,g\in K
\right\},
$$
then
$$
J=\left\{
\begin{pmatrix}
0&0&c\\
...
1
vote
0answers
19 views
Understanding the proof of Taylor's theorem
I'm trying to understand the proof of Taylor's theorem from here:
I already made a question about the remainder part of the theorem and got an answer for it here: Remainder term in Taylor's ...
0
votes
1answer
15 views
Expansion and factoring with trig identities.
I have a question about this expansion..
Lets say I have $(\cos u + i\sin u）^2$ and after expanding, we get $\cos^2 u +2i\sin u\cos u + i^2\sin^2u$. However what if we have $\cos u 2i\sin u$ ...
0
votes
0answers
15 views
Area of a triangle whose each side is less than 2 and greater than1.
What is the area of a triangle if each of its sides is greater than 1 and less than 2?
My Try:Let a,b,c be the sides of triangle,then
...
0
votes
0answers
14 views
If a series converges then the power series converges for all z
How can I prove that if $\sum \limits_{k=1}^{\infty} c_n$ , $c_n\in \mathbb{C}$, converges then $\sum \limits_{k=1}^{\infty} c_n \frac{z^n}{1-z^n}$ converges for all z in $\mathbb{C}$ with ...
0
votes
1answer
23 views
Is it always a tautology?
If any two compound propositions $P$ and $Q$ are equivalent, then is the proposition formed from their biconditional $P \leftrightarrow Q$ always a tautology?
0
votes
0answers
10 views
Implicit Function Theorem (Two Variables)
While reading through a paper I came a cross a result (due to the Implicit Function Theorem) that I cannot derive.
Let $u(x,y)$ be the solution of a PDE ($x$ and $y$ are independent variables). We ...
0
votes
0answers
6 views
Top and bottom power spectral density of a height profile
Imagine I have a simple 1D height profile which is NOT symmetric. Now,
what is truly important for me is to know what are the frequency content
of the top profile (i.e. a cut profile above the ...
1
vote
2answers
38 views
How to find sums like these
How do i find sums like these?--
$$S=\displaystyle\sum_{k=0}^{39} \dbinom{200}{5k}$$
that is, when there is a summation of binomial coefficients, but with jumps of some terms..?
1
vote
1answer
14 views
Example of an inverse semigroup
An inverse semigroup $S$ is a semigroup in which for each $x\in S$ there exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. I'm trying to find an explicit example(which is not a group) of such ...
1
vote
1answer
49 views
Am confused by integral from Apostol
I have been attempting to solve the following Integral from Apostol "Calculus" Volume 1, Section 6.22, Question 46. The integral is
$$
\int \sqrt{(x-a)(b-x)}\, dx,\,\,\, a\neq b
$$
Points to Note:
...
5
votes
1answer
43 views
Find all positive integers $n$ such that sum of digits of $2^n$ is equal to $n$.
For example, $2^5=32$ and $3+2=5$. Similarly, it can be shown that it works for $2^{70}$.
Using basic results in modular arithmetic, one can show that $n$ has to be either of the form $18k+5$ or ...
0
votes
2answers
33 views
Continuous function which has only rational values.
If $f:[a,b]\to\mathbb{R}$ is a continuous function and $f(x)\in\mathbb{Q}$ for all $x\in[a,b]$ then what can say about $f$?
My try: I think f should be constant, if it is not constant then it ...
0
votes
1answer
27 views
What is the smallest non-trivial Hilbert space?
I came to know without proof or explanation that smallest non-trivial Hilbart space is generated by two basis vectors. What is its proof?
One example I know. Denote $a = (0 , 1)$ and $b = (1 , 0)$. ...
0
votes
0answers
3 views
How can I find the projection of a sphere onto a direction vector subject to constraints?
Given a direction defined by a unit vector, I can find the projection of a point on that direction by using the dot product. The projection interval of a set of points is the max and min values of the ...
0
votes
1answer
9 views
Looking for a approximation/solution to my mortgage calculator function
I'm working on a little function, $t(A,y,r)$ that calculates the monthly payment of a fixed-rate mortgage, where $A$ is the amount borrowed, $y$ is the number of years over which the loan will be ...
0
votes
0answers
12 views
Line not intersecting circle, maximum value of expression involving radius
If line $y+x=2$ do not intersect any member of circles $x^2 + y^2 -ax = 0$ at two distinct points where a is parameter, then maximum value of $|a + 4|$.
My try:
Since the line does not intersect ...
0
votes
1answer
29 views
Questionable Intervals
Find the numbers $X_1 , X_2 , \ldots , X_{10} $ such:
$X_1$ is in the interval $[0,1]$.
If we divide the interval $[0,1]$ in halves,each half consists of only one of $X_1$ or $X_2$.
If we divide ...
1
vote
0answers
12 views
Why does a p-subgroup of a group have to lie inside the normalizer of a p-sylow subgroup?
Prove that a p-subgroup is always contained within a p-sylow subgroup of a group $G$.
Lang's Algebra mentions this fact on pg 35. However, he starts the proof by assuming that every p-subgroup at ...
1
vote
1answer
24 views
How find the minimum of the $|w^3+z^3|$,if $|z+w|=1,|z^2+w^2|=14$
let complex $z,w$ such
$$|z+w|=1,|z^2+w^2|=14$$
find the minimum of the value
$$|w^3+z^3|$$
My idea: let
$$z=a+bi,w=c+di\Longrightarrow z+w=(a+c)+(b+d)i,z^2+w^2=(a^2+b^2+c^2+d^2)+2(ab+cd)i$$
then we ...
0
votes
0answers
26 views
Proof of the sum of an alternating series similar to the alternating harmonic series [on hold]
Hi
I know I saw the answer to the following question somewhere
online, I think it was stackexchange but I'm not absolutely
certain..repeated searching on this website and on google have been
...
5
votes
0answers
115 views
Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)
Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer.
I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
2
votes
3answers
56 views
How addition works
Lets say i am doing 12 + 13 by using the addition method that we know. i mean first we write 13 below 12, then we do 2+3 and then 1+1. The result can be validated as 25 (or true) by doing the counting ...
1
vote
1answer
15 views
Inequality containing finite sum.
For what value of k the following inequality holds?
$\sum_{i=1}^{n}a_{i}^3<k|\sqrt{\sum_{i=1}^{n}a_{i}}|$
I don't have any idea to solve this.
1
vote
3answers
67 views
Does $f(dx)$ have any meaning?
Simple question, does a differential $dx$ have any meaning composed in a function $f$, such as $\sqrt{dx}$, where $f(x)\neq x$?
1
vote
0answers
13 views
complex-valued affine function and complex-valued linear functional
If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...
2
votes
1answer
62 views
Integration by Euler's formula
How do you integrate the following by using Euler's formula, without using integration by parts? $$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}$$
I did integrate it by parts, by ...
0
votes
2answers
37 views
Number between less than 15 or equal to 15 or between the two options? [duplicate]
Is $14.9999999$ ad infinitum $\lt 15$, $= 15$, or in between?
5
votes
1answer
95 views
How find the value of the $x+y$
Question:
let $x,y\in \Bbb R $, and such
$$\begin{cases}
3x^3+4y^3=7\\
4x^4+3y^4=16
\end{cases}$$
Find the $x+y$
This problem is from china some BBS
My idea: since
...
1
vote
0answers
34 views
Manifolds and Varieties
In my language, spanish, the word for (topological) manifold and (algebraic) variety is the same: "Variedad". This happens also in some other languages, like french (variété) or portuguese ...
0
votes
1answer
44 views
Proving that the limit of 1/x as x approaches negative infinity equals 0
I get stuck in trying to find a proper epsilon. I know that it is supposed to be $-1/\epsilon$ but I don't understand how to manipulate the inequality $\epsilon > -1/x$. Please help.
0
votes
1answer
21 views
Is there a closed form solution for slope lines of bilinear function?
Given a bilinear function $f(x,y) = a + bx + cy + dxy$, is there a closed form solution for a slope line passing through point $(x_0, y_0, f(x_0, y_0))$? It can exclude degenerate cases, e.g. $b = c = ...
1
vote
3answers
36 views
What kind of function space does the a set of linearly independent exponential form?
I'm confused because you can define a basis $\{1, x, x^2,\ldots\}$ as the basis for the space of polynomials, $\{1, \sin(x), \cos(x),\ldots\}$ as the basis for fourier series, but little words are ...
2
votes
3answers
266 views
In a class of 65, there are twice as many maths students as biology students.
I have a task:
In a class of 65, there are twice as many maths students as biology students. If 12 biology students do not take maths and 15 students take neither of these subjects, how many students ...
1
vote
2answers
32 views
The $i^{th}$ prime in a given ring R
When I say that $p_1=2$, I mean that the first prime in the standard ring of integers $(\mathbb{Z},*,+)$ is $2$. I was wondering whether the notion of ordering the primes like this can be generalized ...
1
vote
0answers
19 views
Kahler forms of a smooth affine algebra vanish eventually?
If $k$ is a Noetherian ring, then do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$?
I mean is: $\Omega^{d+1}_{A|k}\cong 0$?
3
votes
0answers
24 views
$z\in\mathfrak R$ iff for every $a\in A$ there is $w$ for which $z+w=zaw=waz$.
In his BAII, Jacobson gives the following exercise, which he attributes to McCrimmon.
Show that $z\in\mathfrak R(A)$ iff for each $a\in A$ there exist $w\in A$ such that $z+w=zaw=waz$.
I have ...
1
vote
0answers
25 views
Chow Groups of $\mathbb{P}^n$
I'm trying to see that $A_k(\mathbb{P}^n)=\mathbb{Z}$ for all $k$. I am trying to do this with induction on $n$, by applying the excision sequence $$A_k(Y) \xrightarrow{i_*} A_k(X) \xrightarrow{j^*} ...
0
votes
0answers
18 views
Sum of Nonnegative Matrix and Diagonal Matrix
Setup: Let $D = D^T > 0$ be a positive definite and diagonal $n\times n$ matrix, and let $A = A^T \in \mathbb{R}^{n\times n}$ be nonnegative with zero diagonals. That is, $a_{ij} \geq 0$ for $i\neq ...
0
votes
0answers
24 views
linear algebra question related to basis, kernel and linear transformation [duplicate]
Let V be a 2-dimensional vector space, and let α=e1,e2 be a basis for V. Define a linear transformation T:V→V by declaring that:
T(e1+e2)=2e1−e2
T(e2)=4e1−2e2.
a. Find [T]α,α. (one alpha is upper ...
1
vote
1answer
34 views
Absorbing at Each point means Open?
In a topological vector space, it seems that open sets have the property that they are absorbing at each of their points, since $(\alpha,x)\to \alpha x$ is continuous.
I am wondering if the converse ...
0
votes
0answers
7 views
Covariance matrix of distance matrix from an estimated squared distance matrix
I would like to perform fuzzy clustering on a data set $\boldsymbol{X}$ that contains missing elements. The fuzzy clustering algorithm requires the computation of covariance matrix $\boldsymbol{C}$ of ...
