Reputation
13,885
Top tag
Next privilege 15,000 Rep.
Protect questions
Badges
2 28 92
Impact
~334k people reached

2d
comment inverse of sum of diagonal matrix and eigendecomposition
I don't think that you can compute the inverse of a sum, in terms of the inverses of the two matrices.
Apr
12
comment Set of points at which a function coincides with its convexification is compact?
Yes, I guess lower semicontinuous could work.
Apr
12
comment $x\over(1-x)$ $y\over(1-y)$ $z\over(1-z)$ >= 8 when $x ,y ,z $ are positive proper fractions and $x+y+z = 2$
What does proper fractions mean? Are they rational numbers?
Apr
11
comment Proving properties about matrix $A$ s.t. $A^2 = -I$
The equation you get for $\lambda$ is $\lambda^2 = -1$. This has roots $\pm i$. The sum of the eigenvalues is equal to the trace (the sum of the elements on the diagonal), and since $A$ has real entries, the trace is real.
Apr
11
comment For $n \ge 3$, every subgroup of $A_n$ with index $n$, is isomorphic to $A_{n-1}$
I guess you could use the fact that for $n \geq 5$, normal subgroups of $S_n$ are $\{e\},A_n,S_n$. Prove that a subgroup of $A_n$ with index $n$ is a normal subgroup of a group isomorphic to $S_{n-1}$. Then its cardinality implies that it can only be $A_{n-1}$.
Apr
10
comment Learning modern differential geometry before curves and surfaces
I don't see why what you are doing could not be "safe". But it's just like talking about derivatives, without actually finding a derivative of a concrete function like $\exp,\sin,\ln$. Differential geometry was not a success for me, and as I look back, is this lack of concrete examples and connection with the curves and surfaces which kept me from learning more. I liked this course on curves and surfaces: alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf If you like differential geometry, you'll like curves and surfaces.
Apr
10
comment Learning modern differential geometry before curves and surfaces
If you're comfortable with geometry on manifolds, it costs nothing to take a look into curves and surfaces. How can you speak of the tangent space to a manifold without being able to work with tangents to curves, or tangent planes to surfaces? How do you compute the curvature of a curve, or the mean curvature of a surface at a point? These are basic questions you need to know from curves and surfaces...
Apr
9
comment Please help solve for the variables A and B
Replace B in the first equation. Then you'll have just an equation in A
Apr
9
comment Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square
@Elaqqad: The article you cite proves this only for polynomials, as far as I see.
Apr
9
comment Unique solution of nolinear equation set
Yes, that's true.
Apr
9
comment Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square
Does $\Bbb{N}^+$ contain zero?
Apr
9
comment Symmetric matrix with zero elements below the anti-diagonal
Note that doing two line swaps (i.e. multiplication with a permutation matrix) this turns into a lower triangular matrix.
Apr
8
comment Convergence of Cesàro means for a monotonic sequence
@thomas Why not? Since $a_n$ is monotone, if it diverges, it must have a limit equal to $\pm \infty$. Then the chain of the inequalities says that the middle partial limits must both be equal to the same infinite limit. This contradicts the hypothesis on the convergence of the mean.
Apr
8
comment Proving summations involving the Legendre symbol
Are those quadratic residues?
Apr
8
comment A sequence for which there is $k$ such that for each $\epsilon >0$ we have that $|x_n -l | < \epsilon$ for all $n \geq k$
Yes, that's the conclusion.
Apr
8
comment Let $f: [0,\infty] \to \mathbb{R}$ be continuous such that its limit tends to $0$ as $x \to \infty$. Prove that $f$ is uniformly continuous.
Another way to do it: Since $f$ has a limit at $+\infty$, you can say that $f(x) = f(\tan y)$ with $y \in [0,\pi/2]$. If you denote $g: [0,\pi/2],\ g(y) = f(\tan y)$, then $g$ is continuous. Furthermore, $f(x) = g(\arctan x)$. Now use the fact that $g$ is uniformly continuous and that $|\arctan x-\arctan y| \leq |x-y|$.
Apr
8
comment Let $f: [0,\infty] \to \mathbb{R}$ be continuous such that its limit tends to $0$ as $x \to \infty$. Prove that $f$ is uniformly continuous.
I would argue differently. For $\varepsilon>0$ choose $N$ such that $|f(x)-f(y)|<\varepsilon/2$ for $x,y >N$. Then for $[0,N]$ choose the $\delta$ corresponding to $\varepsilon/2$ from the uniform continuity property. This $\delta$, then works for all $x,y$ since if $x,y>N$ or $x,y<N$ the property is obviously satisfied. If $x<N<y$, then apply the uniform continuity bound on $|f(x)-f(N)| <\varepsilon/2$ and $|f(N)-f(y)|<\varepsilon/2$.
Mar
31
comment Why is $E[|Y|^q]=E[\liminf_{n \to \infty} |Y_n|^q]$
Look at Fatou's Lemma: en.wikipedia.org/wiki/Fatou%27s_lemma at the paragraph Convergence in measure.
Mar
23
comment A quick way to prove the inequality $\frac{\sqrt x+\sqrt y}{2}\le \sqrt{\frac{x+y}{2}}$
@Stef: To such a question (no background, no reason, no details) I did not want to give an explicit answer. Anyone who stared more than two minutes at the inequality would square both sides, and start reducing what can be reduced.
Mar
23
comment A quick way to prove the inequality $\frac{\sqrt x+\sqrt y}{2}\le \sqrt{\frac{x+y}{2}}$
Just square the inequality $\frac{a+b}{2} \leq \sqrt{\frac{a^2+b^2}{2}}$. It is one of the most basic inequalities. All mean-type inequalities reduce to $(a-b)^2 \geq 0$.