All questions with tag [math: alternative-proof]


Equivalences to "D-finite = finite"

By a D - limited set, we suggest a set confessing no shot from the all-natural numbers (or equivalently, a set not in bijection with any kind of correct part). I have actually run into an evidence that the adhering to are equal in ZF: A. Every D - limited set is limited. B. D - limited unions of D - limited collections are D - limited. C. ...
2022-07-25 15:57:22

Tensor-free proof that for finite modules over reduced Noetherian rings, locally free = projective

Is there an elegant tensor-free proof of the fact that over a reduced Noetherian ring $A$, every finitely-generated $A$-module which is locally free, is projective? EDIT: I would be content with the case where $A$ is a local ring.
2022-07-24 09:19:19

Direct approach to the Closed Graph Theorem

In the context of Banach spaces, the Closed Graph Theorem and the Open Mapping Theorem are equivalent. It seems that usually one proves the Open Mapping Theorem using the Baire Category Theorem, and then, from this theorem, proves the Closed Graph Theorem. I was wondering about a more direct approach to the Closed Graph Theorem. What I want to s...
2022-07-22 17:57:51

Poincaré-Hopf theorem using Stokes

The wiki entry on the Poincaré-Hopf theorem asserts that it "relies greatly on indispensable, and also, specifically, Stokes' theorem". Nonetheless, in the illustration of evidence offered there which is essentially the one in Milnor is Topology from the Differentiable Viewpoint there is no integration. Does the evidence come to be l...
2022-07-22 17:53:58

How is Leibniz's rule for the derivative of a product related to the binomial formula?

Possible Duplicate: ¢ “Binomial theorem”-like identities The binomial formula defines the development of the $n$th power of the amount $(a+b)$: $$(a+b)^n = \sum_{k = 0}^n {n\choose k}a^kb^{n-k}$$ In calculus, there is a generalization of the item regulation called Leibniz's rule, which defines the development of the $n$th by-product of the...
2022-07-22 15:22:33

Is there any way to give sense to a geometric/visual proof?

Suppose one is given the following visual proof that $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} = 1$$ which is the following construction over $[0,1]\times[0,1]$ What this is suggesting is that the sum of the areas of the triangles, which is $1/2,1/4,1/8,\dots$ indeed covers the unitary square with area 1. What I want to know if i...
2022-07-20 17:27:54

Disjointness of stars in a simplicial complex in $\ell_2$

Definitions Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most $n+1$ points $x$ such that $f(x)\neq 0$. Functions $f$ such that $f(x) = 1$ for some $x$ are called vertices. A simplex is a convex hull of some vertic...
2022-07-17 17:20:01

Complex Analysis: Liouville's theorem Proof

I'm being asked to find an alternate proof for the one commonly given for Liouville's Theorem in complex analysis by evaluating the following given an entire function $f$, and two distinct, arbitrary complex numbers $a$ and $b$: $$\lim_{R\to\infty}\oint_{|z|=R} {f(z)\over(z-a)(z-b)} dz $$ What I've done so far is I've tried to apply the cauchy i...
2022-07-16 17:27:17

Is this reading of the proposition I.4 of the Elements valid?

I have actually researched what Heath, Russell, and also what others have actually claimed concerning the suggestion 4 of guide I of the Elements. Until now, I recognize the "problems" they see with making use of superposition, yet I still can not cover my head around the issue of I can make use of synchronizing numbers to make the evi...
2022-07-15 03:56:20

Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction.

Possible Duplicate: ¢ Summation of a factorial This formula is offered: ¢ $$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$ I've addressed it making use of mathematical induction yet I'm interested what can be the various other feasible means to confirm it.
2022-07-14 01:41:53

How can one prove that integrating $\cos{(ix)}\cos{(jx)}$ cancels in Fourier Analysis?

This portion of the Wikipedia entry on Fourier Analysis information a formula, and also later on claims that the terms for $j \ne k$ disappear. Could a person please give an evidence of this? I in fact would love to see greater than one evidence, yet the important things I'm worried about is obtaining a much deeper instinct of the internal oper...
2022-07-11 23:27:09

How to prove $\sum_{i=1}^{n} i = \frac{n}{2}(n+1)$ without words?

Possible Duplicate: ¢ Proof for formula for sum of sequence $1+2+3+\ldots+n$? Is there an image evidence for $\sum_{i=1}^{n} i = \frac{n}{2}(n+1)$?
2022-07-11 23:24:24

Proof of the duality of the dominance order on partitions

Could any person give me with a wonderful evidence that the dominance order $\leq$ on dividings of an integer $n$ pleases the following: if $\lambda, \tau$ are 2 dividings of $n$, after that $\lambda \leq \tau \Longleftrightarrow \tau ' \leq \lambda '$, where $\lambda'$ is the conjugate dividing of $\lambda$ (i.e. the tra...
2022-07-08 06:39:19

What are various proofs good for?

There are a lot of questions around below, which are confirmed to be appropriate or incorrect in numerous means. I ask yourself, what one can pick up from these varying means of just how to confirm something, although that: The even more evidence, the far better. Allow is claim a declaration is something like a means $A\to Z$. One evidence migh...
2022-07-07 02:52:18

Hahn-Banach theorem: 2 versions

I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as: Let $E$ be a vector space, $p: E \rightarrow \mathbb{R}$ be a sublinear function and $F$ be a subspace of E. Let $f: F\rightarrow \mathbb{R}$ be a linear function dominated by $p$ (by which I mean $\forall x \in F: f(x) \leq p(x)$). Then $f$ has a line...
2022-07-06 00:33:02

proof of l'Hôpital's rule that minimizes special-casing

A simple form of l'Hôpital's rule looks like this: If $u$ and $v$ are functions with $u(0)=0$ and $v(0)=0$, the derivatives $\dot{v}(0)$ and $\dot{v}(0)$ are defined, and the derivative $\dot{v}(0)\ne 0$, then \begin{align*} \lim_{x\rightarrow 0} \frac{u}{v} &= \frac{\dot{u}(0)}{\dot{v}(0)} \qquad . \end{align*} To me, the clearest way...
2022-07-04 17:40:26

Justifying exchange of limits in a double sum - a dubious proof

It is a standard theorem (given in Rudin's Principles of Mathematical Analysis, page 175, and many other places), that if $\{a_{ij}\}$ is a doubly indexed sequence, and $$\sum_{j=1}^\infty |a_{ij}| = b_i$$and $\sum b_i$ converges, then $$\sum_{j=1}^\infty\sum_{i=1}^\infty a_{ij} = \sum_{i=1}^\infty\sum_{j=1}^\infty a_{ij}$$ Rudin gives a proof ...
2022-07-04 17:39:11

Found a simpler proof, now how do I know if it's original?

I've located a less complex evidence for some identity/theorem, hypothetically talking, certainly ;) How do I recognize if it hasn't been done prior to? For vital outcomes it is rather very easy to locate. Incidentally, I assume I'm the first one to make use of the "alternative - proof" tag. I really feel initial today. lol.
2022-07-01 17:27:01

Sum of irrational numbers

Well, in this inquiry it is claimed that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$ , and also the proprietor requests for "alternative proofs" which do not make use of sensible origin theory. I created a solution, yet I simply confirmed $\sqrt[100]{\sqrt3 + \sqrt2} \notin \mathbb{Q}$ and also $\sqrt[100]{\sqrt3 - \sq...
2022-07-01 17:18:45

How to deduce open mapping theorem from closed graph theorem?

These 2 theories are equal yet I can not identify just how to reason the open mapping from the shut chart. Can any person offer a tip or some reference?
2022-07-01 16:02:12