0
votes
0answers
6 views

Inverse in a functional space

I would like to understand why the inverse of a bounded operator must to be bounded too? In other context, all bijective function have an inverse but when we deal with a bounded operator it have to be ...
0
votes
0answers
5 views

Matrix operation repeat matrix members

I am going to use C++ Armadillo library which handles matrices to generate matrix $B$ and $C$ from matrix $A$. $$ A=[M_0,M_1,\ldots,M_{n-1}]^T $$ $$ ...
1
vote
0answers
19 views

Using a decimal addition table for subtracting

I'm reviewing the math I missed in school and I've come across a decimal addition table in a book along with a description of how to use the same table for subtracting. I'm having trouble parsing the ...
1
vote
0answers
22 views

When should I resort to Eulers identity?

I'm working on the following excercise: Calculate: $$\int_0^{+\infty} \frac{x^{\frac{1}{3}}\sin (x+\frac{\pi}{3})}{x^2+1}\operatorname dx$$ Using the contour-integral $\int_{\Gamma} ...
3
votes
0answers
25 views

Topology and Measures

I apologize if this question is a bit vague; I'm just wondering if there is a concept like what I'm talking about, or if I'm just lost. I'll start with just some thoughts. I looked a bit, and I don't ...
0
votes
1answer
19 views

Probabilistic arguments in calculus

As the book Probabilistic Techniques in Analysis by R. Bass shows, there is a huge interplay between analysis and probability. However, I would like to see some examples of "more basic" relationships ...
0
votes
5answers
64 views

Prove $a^{2}+2ab+3b^{2}+2a+6b+3 \geq 0$

If $a,b\in\mathbb{R}$ prove that the following inequality holds: $a^{2}+2ab+3b^{2}+2a+6b+3 \geq 0$
0
votes
2answers
46 views

simplify expression to find a limit

find the limit of $\large\frac{n}{\sqrt[n]{n!}}$ using the ratio test $$\large \frac{(\frac{(n+1)^{n+1}}{(n+1)!})^\frac{1}{n+1}}{(\frac{n^n}{n!})^\frac{1}{n}}$$ I have added $\frac{n}{n}$ and ...
1
vote
0answers
9 views

Existence of a maximum matching containing v vertice in a graph

Let be $v$ vertice of a graph $G$, which is not isolated ($v$ is not unconnected). Prove the existence of a maximum matching in which $v$ is saturated (matched). Thanks in advance!
1
vote
0answers
16 views

Is Dodecahedron tesselation somehow possible?

In this video (at 3:25) there is an animation of planets inside a dodecahedron matrix (or any data-structure that best fit this 3d mosaic). I tried reproducing it with 12 sided dices, or in Blender, ...
7
votes
3answers
46 views

$\sin 4x +\sqrt{3} \sin 3 x + \sin 2 x=0$

This question is from a 2012 VMK entrance exam I was trying to solve it first by expanding $\sin 4 x = 2 \sin 2 x \cos 2x$, then by noticing that if divided by 2, one can get, e.g. $ ...
2
votes
4answers
40 views

Prove that a function is continuous for every $x \in R$

Prove that the function: $$ f(x)=\frac{\sqrt{x^2-x+1}}{|\sin(x)-4|-2} $$ is defined for every $x \in R$ and continuous in every $x \in R$, So I said that in order for this function to be defined we ...
2
votes
0answers
15 views

Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$ \int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{$\star$} $$ Can you please teach me a rigorous ...
1
vote
1answer
19 views

How to solve ODE's $\dot{x}=ax+by$ and $\dot{y}=bx+cy$?

I need help in solving a system of ODE's $$x'(t)=ax(t)+by(t) \mbox{ and } y'(t)=bx(t)+cy(t)$$ where $a,b \in \mathbb{R}$ and $x,y$ denote standard co-ordinates in $\mathbb{R}^2$. I checked on ...
0
votes
2answers
25 views

Circle with center point and tangential to lines

I have defined Points all points (3 blue, and one green). All points have the same distance to A point. Yellow lines are bisectors. I have equations of AB and ...
0
votes
0answers
8 views

equation for entropy of angles around a circle?

Hope you can help.. I'm using an random (even) distribution of angles to represent agents converging on a target value. (e.g zero degrees, where agents are at 4, 58, 99 & 202 degrees) I need to ...
0
votes
0answers
23 views

How to fix error Msg 1801, Level 16, State 3, Line 3 [on hold]

I want add script file in sqlserver2012 but when execute i have error: Msg 1801, Level 16, State 3, Line 3 Please help me Thanks
2
votes
0answers
25 views

An infinite series, in which each term is a definite integral of a complicated transcendental function

I am reading a paper (sorry, no e-copy) with a number of infinite series, in which each term of the infinite series is an integral whose integrand is a complicated transcendental function involving ...
5
votes
0answers
50 views

Find the values of the positive integers $n$ such that: $\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$ is positive integer

My question is as follow: Find the values of the positive integers $n$ such that: $$\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$$ is positive integer. I can see that for $n=1$ (among some ...
0
votes
3answers
41 views

What happens to the angles of an isosceles triangle if one vertex is at infinity?

My son and I were trying to decide whether an isosceles triangle can ever have 90 degree base angles. I would argue that if the two equal length sides are both infinitely long, they must have 90 ...
1
vote
1answer
10 views

an example for an arbitrary graph $G$ with even vertices which $\forall S \in V(G) , |N(S)|\geq |S| $ but there is no complete matching .

I want to say an example for an arbitrary graph $G$ with even vertices which $\forall S \in V(G) , |N(S)|\geq |S| $ but there is no complete matching . I have tried so many shapes but I couldn't ...
-2
votes
0answers
42 views

How do you find the square root of a binomial? [on hold]

Mu Alpha Theta Nunn FL #25 Find the coefficient of the 4th term in the binomial expansion of $(9x-2y)^{\frac12}$. How I started: I first thought that there should be a $3\cdot\sqrt{x}$ term at the ...
0
votes
2answers
26 views

Can an arbitrary constant in the solution of a differential equation really take on any value?

Consider the first order differential equation $y' = -2y^{\frac{3}{2}}$. It has $y = \dfrac{1}{(x+c)^2}$ as the solution. Now, if I divide both the numerator and denominator by $c^2$ (assuming $c ...
-5
votes
0answers
33 views

Michael Artin Homework. [on hold]

Prove that a rigid motion is bijective. | m(X)-m(Y) | = |X-Y| how can i prove this ? I have no idea how to start . Its from Michael Artin book .
0
votes
2answers
25 views

Properties of partial derivatives.

Is the following statement $$\frac{\partial^{2}f}{\partial x\partial y}=\frac{\partial^{2}f}{\partial y\partial x}$$ always true? If not what are the conditions for this to be true?
0
votes
3answers
32 views

Probability : How to approach this question : [on hold]

There are 100 boxes in front of you. You have 100 balls in your pocket which you throw one by one towards the boxes in front of you. Each ball will definitely end up in a box and has equal probability ...
0
votes
1answer
14 views

Permutation/Combination question on dice

Question: Three dice (six faces: each face -> number 1 to 6) are rolled. What is the number of possible outcomes such that at least one dice shows number 2? My attempt: One dice has to show ...
1
vote
2answers
26 views

Free finitely generated

Let $A$ be a ring and consider the free modules $A^{\oplus n}$, $A^{\oplus k}$, with $n,k\in \mathbb{N}$. Can $A^{\oplus n}$ be isomorphic to $A^{\oplus k}$ if $k\neq n$? Thanks in advance for the ...
2
votes
0answers
16 views

What is the combination of Complex, Split-Complex and Dual Numbers

If $a+bi:i^2=-1$ is a complex number, $a+cj:j^2=+1$ is a split-complex number, and $a+d\epsilon:\epsilon^2=0$ is a dual number; what is the term for the combination ...
0
votes
2answers
18 views

Solving for unknowns in parametric equation

I have the parametric equation of a circle: $$f(u) = \langle a \cos(u) + b, a \sin(u) + c\rangle,$$ and because the equation has $3$ unknowns $a,b$ and $c$, I have been given $3$ points $p_0, p_1$, ...
1
vote
0answers
14 views

Solve the integral equation then solve the differential equation

$y(x) = 3 + 2\int\limits_1^x t\times y(t) \times dt $ First I solved for the integral equation then the I'm told to differentiate and I get ${dy \over dx} = 2\times x \times y(x) $ Then I see ...
-6
votes
3answers
41 views

How to solve this inequality? $2\cos(x+1)>0$ [on hold]

Please help me answer this question. How can I solve the following inequality? Solve the following inequality: $2\cos(x+1)>0$. Thank you.
2
votes
1answer
12 views

What is partial derivative of distance to line equation?

The distance from a point to a line is given by the equation: $$ \mbox{distance}\ = \frac{|ax + by +c|}{\sqrt{a^2 + b^2}}$$ What are the partial derivatives of this equation with respect to $a$, ...
1
vote
0answers
16 views

Traveling salesman neighborhood

I am solving some TSP problems and i got this one and i am not pretty sure about my answer. By seeing TSP as a formal combinatorial problem, i have that the Feasible solutions $F$ is the set defined ...
1
vote
0answers
16 views

Sum of Coefficients and Number of Terms in Trinomials and Quadrinomials

I already know how to find the sum of coefficients in a binomial, but how do you do it for a trinomial/quadrinomial (after like terms are added)? Example Problem: $(wa+xb+yc+zd)^n$ (all variables are ...
2
votes
1answer
18 views

Number of edges in a graph with n vertices and k connected components

Let $m$ be te number of edges, $n$ the number of vertices and $k$ the number of connected components of a graph G. Prove that: $m$ $\leq$ $\frac{(n-k+1)*(n-k)}{2}$ Thanks!
3
votes
3answers
43 views

Find $ \int \frac {1-x^2}{1+3x^2+x^4} \, \mathrm{d}x $

Today, the CalcBee sample problems got released. The following problem was my creation and I wanted to see how many solutions people can come up with. The result is very beautiful and I thought it ...
3
votes
1answer
27 views

Question on how to work with “differential”

I have the following equation which is from a famous paper in economics: $$ \sum_{i=1}^{n} x_idx_i=\frac{1}{2} d\Big[\sum_{i=1}^n x_{i}^2\Big]=XHdX+\frac{1}{2}d\big[X^2H\big] $$ Can you tell me how ...
1
vote
0answers
25 views

Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...
1
vote
1answer
30 views

Why can't the definition of convergence be alterted to this one?

I am trying to find out of a seqence with the following property is convergent: Let $(r_n)$ be a sequence of real numbers. Suppose there is a number $r\in\mathbb{R}$ such that for any ...
3
votes
0answers
14 views

Game to maintain distinct number of balls in glasses

There are $n$ glasses, containing $n+1,n+2,\ldots,2n$ balls, respectively. Two players $A$ and $B$ play a game, alternately taking turns with $A$ going first. In each move, the player must choose some ...
5
votes
4answers
59 views

Evaluating $\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$

Evaluating $$\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$$ using $ux=\sqrt{x^2-1}$ I try to $u^2x^2=x^2-1$ $x^2=\frac{-1}{u^2-1}$ However I cant get rid of $x$ because derivative has $x\;dx$. How can I ...
3
votes
0answers
27 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
0
votes
1answer
36 views

How to compute $\int_0^1\int_0^1 |x-y|dxdy$? [on hold]

Can anyone help me to solve $\int_0^1\int_0^1 |x-y|dxdy$ ? Thanks.
1
vote
2answers
24 views

Size of the orbits of a normal subgroup

So this is the question: Let $H$ be a finite subgroup of $G$, and let $(h,h')(x)=hxh^{-1}$ define an achtion of $H\times H$ on $G$, prove that $H$ is a normal subgroup of $G$ if and only if every ...
2
votes
0answers
8 views

How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere of radius $R$?

Question How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere $\mathbb S $ of radius $R \quad (R \gg a)$? What I have thought so far Since the area of the ...
4
votes
1answer
18 views

Trigonometic Substitution VS Hyperbolic substitution

The following tables were taken from University of Pennsylvania's page about Calculus: Trigonometric Substitution Hyperbolic Substitution As you can see, the forms $1+x^2$ and $x^2-1$ are repeated ...
6
votes
1answer
18 views

Is every measurable set a measure-independent limit of open sets

My main question is Q1. Let $B$ be a Borel-measurable subset of $\mathbb R$. Is there a sequence of open sets $U_n$ independent of any measure such that for all Borel probability measures ...
1
vote
0answers
24 views

Finding a surjective homomorphism

I have to show that for all groups with $2007(=3^2\times223$) elements that there exists a surjective homomorphism to a group of 9 elements. Obviously a group with 2007 elements has a subgroup of ...
1
vote
4answers
31 views

Calculate a limit of exponential function

Calculate this limit: $$ \lim_{x \to \infty } = \left(\frac{1}{5} + \frac{1}{5x}\right)^{\frac{x}{5}} $$ I did this: $$ ...

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