User clay

6 votes

1 answer

2k views

Prove that finite dimensional $V$ is the direct sum of its generalized eigenspaces $V_\lambda$

asked Apr 5, 2015 at 17:30

6 votes

1 answer

3k views

Find kernel generators for ring maps

asked May 8, 2015 at 0:54

6 votes

2 answers

135 views

If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$

asked Aug 10, 2017 at 21:05

6 votes

1 answer

245 views

Explain Newton Method Variation for Finding Poles of a Function

asked Oct 2, 2017 at 18:48

5 votes

2 answers

2k views

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$. [duplicate]

asked Jul 10, 2014 at 14:24

4 votes

2 answers

1k views

What is an irreducible polynomial in $\mathbb{Z}$ that has root $\sqrt{2}+\sqrt{3}$?

asked Apr 27, 2015 at 20:43

4 votes

0 answers

69 views

Solve linear system of equations in $\mathbb{Z}/m\mathbb{Z}$

asked Nov 14, 2016 at 23:17

4 votes

2 answers

235 views

Textbook Proposition on Product of Real Analytic Functions

asked Dec 5, 2018 at 15:38

4 votes

1 answer

379 views

Show that $F(x) = \sum\limits_{n=1}^\infty 2^{-n} f(x - r_n)$ is integrable

asked Mar 25, 2019 at 16:59

4 votes

0 answers

160 views

Find Adjoint of $L = p(x) \frac{d^2}{dx^2} + r(x) \frac{d}{dx} + q(x)$

asked Oct 31, 2019 at 16:05

4 votes

1 answer

344 views

Show that the collection of sets for which the inner measure equals the outer measure $\mu_(A) = \mu^(A)$ is a $\sigma$-algebra.

asked Sep 6, 2021 at 17:53

4 votes

1 answer

305 views

Karatzas and Shreve Exercise 1.8: If paths of process $X$ are RCLL almost surely, show that $A$ can fail to be in $\mathscr{F}_{t_0}^X$

asked Mar 12, 2022 at 0:35

4 votes

1 answer

197 views

Karatzas & Shreve 2.6: Show that the hitting time $H_{\Gamma}$ is an optional time.

asked Mar 15, 2022 at 23:52

4 votes

1 answer

207 views

Karatzas + Shreve 2.19: Show that $Y_t = \int_0^t f(s, X_s) \, ds; t \ge 0$ is progressively measurable

asked Mar 21, 2022 at 14:09

3 votes

2 answers

931 views

Let $Y_n$ be i.i.d with $EY_n = 1, P(Y_n = 1) < 1, X_n = \prod_{k=1}^n Y_k$. Show martingale convergence of $X_n \to 0$ a.s.

asked Feb 18, 2022 at 21:13

3 votes

1 answer

778 views

Show that a local martingale is a true martingale if and only if it is a process of class DL

asked Apr 15, 2022 at 15:11

3 votes

1 answer

201 views

Find eigenvalues of $x^2 \frac{d^2\phi}{dx^2} + x \frac{d\phi}{dx} + \lambda \phi = 0$ with boundary conditions $\phi(1) = \phi(b) = 0$

asked Oct 30, 2019 at 6:17

3 votes

1 answer

192 views

For the given topologies, which sequences converge to which limits?

asked Feb 13, 2020 at 6:53

3 votes

2 answers

124 views

For $0 < \alpha < 1$, construct an open set $E \subset [0,1]$ which is dense in $[0,1]$ and $m(E) = \alpha$.

asked Mar 7, 2019 at 6:43

3 votes

2 answers

231 views

Show that Ratio Test gives no information for $3^{-1} + 5^{-1} + 3^{-2} + 5^{-2} + 3^{-3} + 5^{-3} + \cdots$

asked Oct 9, 2018 at 2:20

3 votes

1 answer

114 views

Prove $e^{-ax} \le (1-x)^a + \frac{1}{2}ax^2$ for $a > 1$ and $0 \le x \le 1$:

asked Nov 15, 2016 at 22:25

3 votes

1 answer

1k views

Homework Problem on Absolute + Relative Error [closed]

asked Sep 18, 2017 at 22:04

3 votes

2 answers

196 views

Prove that if $p$ is a prime and $n \in \mathbb{Z}_{>0}$, then $p \mid n^p - n$

asked Aug 10, 2017 at 17:19

3 votes

3 answers

3k views

Prove that $3^n > 2n^2 + 3n$ for $n \in [4,\infty) \cap\mathbb{N}$

asked Jun 14, 2014 at 17:12

3 votes

3 answers

977 views

Prove $A = (A \setminus B) \cup (A \cap B)$

asked Jun 22, 2014 at 0:38

3 votes

1 answer

1k views

Determine the matrix of the operator $T: M \mapsto AMB$ on the space $F^{2\times 2}$ of $2 \times 2$ matrices

asked Mar 24, 2015 at 1:30

2 votes

1 answer

186 views

Let $a_1=1$, $a_2=3$ , and for $n \ge 2$ let $a_n=a_{n-1}+a_{n-2}$. Show that $a_n < \left(\frac{7}{4}\right)^n$ for all natural numbers.

asked Jun 15, 2014 at 21:17

2 votes

2 answers

7k views

Variance measure for categorical data?

asked Feb 5, 2015 at 21:29

2 votes

1 answer

301 views

Show that $(\mathbb{Z}/n)^\times$ cannot be cyclic unless $n$ is either a prime power or twice a prime power.

asked May 11, 2015 at 1:02

2 votes

2 answers

648 views

Generalization of Chinese Remainder Theorem

asked May 11, 2015 at 20:06

Stack Exchange Network

186 Questions

Prove that finite dimensional $V$ is the direct sum of its generalized eigenspaces $V_\lambda$

Find kernel generators for ring maps

If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$

Explain Newton Method Variation for Finding Poles of a Function

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$. [duplicate]

What is an irreducible polynomial in $\mathbb{Z}$ that has root $\sqrt{2}+\sqrt{3}$?

Solve linear system of equations in $\mathbb{Z}/m\mathbb{Z}$

Textbook Proposition on Product of Real Analytic Functions

Show that $F(x) = \sum\limits_{n=1}^\infty 2^{-n} f(x - r_n)$ is integrable

Find Adjoint of $L = p(x) \frac{d^2}{dx^2} + r(x) \frac{d}{dx} + q(x)$

Show that the collection of sets for which the inner measure equals the outer measure $\mu_(A) = \mu^(A)$ is a $\sigma$-algebra.

Karatzas and Shreve Exercise 1.8: If paths of process $X$ are RCLL almost surely, show that $A$ can fail to be in $\mathscr{F}_{t_0}^X$

Karatzas & Shreve 2.6: Show that the hitting time $H_{\Gamma}$ is an optional time.

Karatzas + Shreve 2.19: Show that $Y_t = \int_0^t f(s, X_s) \, ds; t \ge 0$ is progressively measurable

Let $Y_n$ be i.i.d with $EY_n = 1, P(Y_n = 1) < 1, X_n = \prod_{k=1}^n Y_k$. Show martingale convergence of $X_n \to 0$ a.s.

Show that a local martingale is a true martingale if and only if it is a process of class DL

Find eigenvalues of $x^2 \frac{d^2\phi}{dx^2} + x \frac{d\phi}{dx} + \lambda \phi = 0$ with boundary conditions $\phi(1) = \phi(b) = 0$

For the given topologies, which sequences converge to which limits?

For $0 < \alpha < 1$, construct an open set $E \subset [0,1]$ which is dense in $[0,1]$ and $m(E) = \alpha$.

Show that Ratio Test gives no information for $3^{-1} + 5^{-1} + 3^{-2} + 5^{-2} + 3^{-3} + 5^{-3} + \cdots$

Prove $e^{-ax} \le (1-x)^a + \frac{1}{2}ax^2$ for $a > 1$ and $0 \le x \le 1$:

Homework Problem on Absolute + Relative Error [closed]

Prove that if $p$ is a prime and $n \in \mathbb{Z}_{>0}$, then $p \mid n^p - n$

Prove that $3^n > 2n^2 + 3n$ for $n \in [4,\infty) \cap\mathbb{N}$

Prove $A = (A \setminus B) \cup (A \cap B)$

Determine the matrix of the operator $T: M \mapsto AMB$ on the space $F^{2\times 2}$ of $2 \times 2$ matrices

Let $a_1=1$, $a_2=3$ , and for $n \ge 2$ let $a_n=a_{n-1}+a_{n-2}$. Show that $a_n < \left(\frac{7}{4}\right)^n$ for all natural numbers.

Variance measure for categorical data?

Show that $(\mathbb{Z}/n)^\times$ cannot be cyclic unless $n$ is either a prime power or twice a prime power.

Generalization of Chinese Remainder Theorem