Inequality

Consider the function $f$ defined on $\mathbb{R}^2$ by
$f(x, y) = x^4 + y^4 - 2(x - y)^2.$
Show that there exist $(\alpha, \beta) \in \mathbb{R}^2$ (and determine them) such that
$\forall (x, y) \in \mathbb{R}^2, f(x, y) \geq \alpha \| (x, y) \|^2 + \beta,$ where $\| \cdot \|$ denotes the Euclidean norm.

0 replies

Daytuz

Today at 4:02 AM

0 replies

AMM 12481 (Neat Generalization of Maximum Modulus Principle)

kgator 0

Today at 3:49 AM

Source: American Mathematical Monthly Volume 131 (2024), Issue 7: https://doi.org/10.1080/00029890.2024.2351727

12481. Proposed by Bernhard Elsner, Université de Versailles Saint-Quentin-en-Yvelines, Versailles, France, and Eric Müller, Villingen-Schwenningen, Germany. Let $f_1, \ldots, f_n$ be holomorphic functions on $U$ , where $U$ is an open, connected subset of $\mathbb{C}$ . Suppose that the function $g : U \rightarrow \mathbb{R}$ given by $g(z) = |f_1(z)| + \cdots + |f_n(z)|$ takes a maximum value in $U$ . Must each function $f_k$ be constant on $U$ ?

0 replies

kgator

Today at 3:49 AM

0 replies

Integrals problems and inequality

tkd23112006 16

N Today at 2:46 AM by Alphaamss

Let f be a continuous function on [0,1] such that f(x) ≥ 0 for all x ∈[0,1] and
$\int_x^1 f(t) dt \geq \frac{1-x^2}{2}$ , ∀x∈[0,1].
Prove that:
$\int_0^1 (f(x))^{2021} dx \geq \int_0^1 x^{2020} f(x) dx$

16 replies

tkd23112006

Feb 16, 2025

Alphaamss

Today at 2:46 AM

Galois group

ILOVEMYFAMILY 5

N Today at 1:49 AM by ILOVEMYFAMILY

Let $K$ be a field. Find the Galois groups

$a) \text{Gal}(K(x), K)$

$b) \text{Gal}(K(x,y), K)$

5 replies

ILOVEMYFAMILY

Mar 11, 2025

ILOVEMYFAMILY

Today at 1:49 AM

Constant term of minimal polynomial algebraic element

M4tchash3l 1

N Today at 12:00 AM by alexheinis

Suppose $a \in \mathbb{R}$ and $a \neq 0$ and there exists a positive integer $n$ such that $a^n \in \mathbb{Q}$ . Let $p(x)$ be minimal polynomial $a$ over $\mathbb{Q}$ . Prove that $p(0) = \pm a^{\deg(p)}$

1 reply

M4tchash3l

Yesterday at 9:31 PM

alexheinis

Today at 12:00 AM

Miklos Schweitzer 1982_10

ehsan2004 1

N Yesterday at 8:13 PM by bloodborne

Let $p_0,p_1,\ldots$ be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let $A_i$ denote the event that the number $i$ has been selected and that it is in the same place in both lines. Prove that the events $A_i \;(i=1,2,\ldots)$ are mutually independent, and $P(A_i)=p_i$ .

T. F. Mori

1 reply

ehsan2004

Jan 31, 2009

bloodborne

Yesterday at 8:13 PM

Do these have a closed form?

Entrepreneur 0

Yesterday at 7:56 PM

Source: Own

$\int_0^\infty\frac{t^{n-1}}{(t+\alpha)^2+m^2}dt.$ $\int_0^\infty\frac{e^{nt}}{(t+\alpha)^2+m^2}dt.$ $\int_0^\infty\frac{dx}{(1+x^a)^m(1+x^b)^n}.$

0 replies

Entrepreneur

Yesterday at 7:56 PM

0 replies

Integrate the reciprocal of a geometric series

IHaveNoIdea010 2

N Yesterday at 4:47 PM by GreenKeeper

Determine the exact value of $\int_{0}^{\infty} \frac{1}{\sum_{n=0}^{10} x^n} \,dx$

2 replies

IHaveNoIdea010

Friday at 2:31 PM

GreenKeeper

Yesterday at 4:47 PM

Derivative of function R^2 to R^2

Sifan.C.Maths 1

N Yesterday at 3:38 PM by alexheinis

Source: Internet

Give a function $f:\mathbb{R}^2 \to \mathbb{R}^2: f(x,y)=(x^2+xy,y^2+x)$ . Calculate the first and second derivative of the function at the point $(1,-1)$ .

1 reply

Sifan.C.Maths

Yesterday at 7:09 AM

alexheinis

Yesterday at 3:38 PM

Initial Value Problem

TheFlamingoHacker 2

N Yesterday at 3:30 PM by Mathzeus1024

Set up the IVP that will give the velocity of a $60$ kg sky diver that jumps out of a plane with no initial velocity and an air resistance of $0.8|v|$ . For this example assume that the positive direction is downward.