Comment by Vinay Karthik on Proof required for a sufficient condition to find...
I am sorry for all the confusions. Here a is an arbitrary integer. And yes, as I am learning maths, I certainly didn't know about Carmichael numbers. Though I understood your explanation @lulu. Thank...
View ArticleComment by Vinay Karthik on Calculating $e^{j2\pi t}$ in two ways gives...
Hope this solves your problem. math.stackexchange.com/a/912686/1277478
View ArticleComment by Vinay Karthik on Prove $(0\le a\lt b)\Rightarrow...
Hope my answer here helps to solve your problem. math.stackexchange.com/a/4863352/1277478
View ArticleComment by Vinay Karthik on amount of numbers in [10000] of which the sum of...
Hope this helps you. math.stackexchange.com/q/2028849/1277478
View ArticleComment by Vinay Karthik on Doubt regarding unboundedness of complex valued...
Yes. $z \to \infty \implies \lvert z \rvert \to \infty \implies \lvert \frac{1}{z} \rvert \to 0 \implies \frac{1}{z} \to 0$. How should I proceed from there ?
View ArticleComment by Vinay Karthik on How to prove that any even number, divided by 2 a...
Hope this helps math.stackexchange.com/a/25919/1277478
View ArticleComment by Vinay Karthik on Weird Functional Equation problem on the irrationals
. is the simple product of real numbers. I shall edit it once
View ArticleComment by Vinay Karthik on Weird Functional Equation problem on the irrationals
Excellent @RobertIsrael ! I didn't think of a substitution. I simply equated the arguments of f in the first equation of your solution to get $2t^2 = 1$. From there my solution follows.
View ArticleComment by Vinay Karthik on An unparalleled problem on matrices ....
@Volk, But $d \ne 1$, as $1$ is not a proper divisor of $n$, so how can we conclude that every matrix $M_i \in \mathcal{M}$ has the same determinant ?
View ArticleAnswer by Vinay Karthik for How to determine whether a curve can be smoothly...
The triangle is essentially a disjoint union of$3$ consecutive smooth curves ($i.e.$ the sides of the triangle) . You can read the first 2 - 3 paragraphs in the writeup attached below for more...
View ArticleAnswer by Vinay Karthik for In writing a proof how are we certain whether...
Actually, the proof $a < b => √a < √b$ is sufficient to prove your question.Just take $a = 0$ and $b = a$ for proving $0 < a => 0 < √a$ in the above statement. For the equality case...
View ArticleAnswer by Vinay Karthik for Timing of the Taylor expansion
The result remains the same for the case where you $precisely$ replace a convergent power series by its sum function, or vice versa , $i.e.$ you don't neglect any of the higher order terms in the...
View ArticleAnswer by Vinay Karthik for Suppose that Ax = b and Cx = b have the same...
Let A $\epsilon \space \mathbb{R}^{m * n}$Let $e_i$ denote the basic column vectors in $\mathbb{R}^{n * 1}$, for each $1 \le i \le n$.Let $b_i := Ae_i$, for each $1 \le i \le n$.By our hypothesis,...
View ArticleAnswer by Vinay Karthik for Set theory definition doubt
The phrase “well - defined” essentially implies that any random thing in the world has 2 options, either it lies in the set or it doesn’t , and this classification is true, regardless of who observes...
View ArticleAnswer by Vinay Karthik for Inequality with logarithms and radicals of order 4
In case of $x \gt y$, we have :$y \lt x \implies \log_x y \lt 1 \implies \frac{1}{\log_y x} \gt 1 \implies y^{\frac{1}{\log_y x}} \gt y$.Note here the inequality gets reversed in the second implication...
View ArticleAnswer by Vinay Karthik for Find the number of rectangles not containing the...
Label the vertical lines in the grid from left to right, starting from $1$ upto $9$. Similarly label the horizontal lines from top to bottom, from $1$ to $6$.Now, the black square is formed by...
View ArticleAnswer by Vinay Karthik for $\mathcal M = \{z \in \mathbb C | |z|=r \}$ ,...
The key idea is to use the position vectors of the points on the circle.2 vectors with magnitude $r$ will add up to a vector with magnitude $r$ iff the angle between the vectors is 120°.So the angle...
View ArticleAnswer by Vinay Karthik for if $\tan(\cot(x))=\cot(\tan(x))$ find $\sin(2x)$
Here is the mistake.$\cot A = \cot B \nRightarrow A = B$. We have :$\cot A = \cot B \implies \tan A = \tan B \implies \sin ({A - B}) = 0 \implies A - B = n\pi$.The second implication in the above line...
View ArticleAnswer by Vinay Karthik for Weird Functional Equation problem on the irrationals
Here is my solution, after taking suggestions from @RobertIsrael in the comments.Let $x = y = \frac{1}{\sqrt{2}}$, in the original equation. After simple algebra, we get $f(\sqrt{2}) = 0$. Similarly we...
View ArticleAnswer by Vinay Karthik for Show that if $x>1$,...
Here is the solution.$\log_e \sqrt{x^2 - 1} - \log_e x = \frac{1}{2}\log_e \left({1 - \frac{1}{x^2}}\right)$.Now use the Taylor series expansion of $\log_e {(1 + x)}$ and replace $x$ by $\frac{-1}{x^2}$.
View ArticleA stronger version of the Rosen’s subsequence theorem.
The following question was asked in my combinatorics exam -“Let $n$ be a positive integer. Exhibit an arrangement of integers between $1$ to $n^2$ which has no increasing or decreasing subsequence of...
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