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-5
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0answers
37 views

Mathematic Theory of Computation [on hold]

Let Sigma be an alphabet. Use the principle of induction on Sigma^*, to prove that |w v| = |w| + |v| for all w, v in Sigma^.
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0answers
34 views

How to show Well Founded Induction false? [closed]

The abstract reduction system ({a,b,c,d},→) where the → is defined as: http://i.stack.imgur.com/TS0Ud.png Let Q be a monadic predicate on {a,b,c,d} such that Q(a) = Q(b) = false and Q(c) = Q(d) = ...
1
vote
0answers
39 views

A curious example envolving moment's convergence

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...
3
votes
0answers
413 views

Two (strictly related) proofs by induction of inequalities

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
10
votes
6answers
513 views

Unconventional types of induction

Induction is one of the most common tools is mathematics, and everybody knows the ordinary induction and the strong induction. However, in some proofs induction is applied in an unexpected and elegant ...
0
votes
1answer
152 views

notable inductive proofs relating to fractals

what are notable/ prominent inductive proofs relating to fractals? the motivation for this question is: fractals are very difficult mathematical objects to work with, and many ...
3
votes
2answers
691 views

Neither Even Nor Odd Natural Numbers

Modular arithmetic (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. In Even XOR Odd Infinities? I asked if this statement ...
34
votes
3answers
3k views

the following inequality is true,but I can't prove it

The inequality is \begin{equation*} \sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right) \end{equation*} for all integer $d\geq 1$. I use computer to verify ...
4
votes
1answer
261 views

ERA, PRA, PA, transfinite induction and equivalences

I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exist. For instance I'm ...
2
votes
3answers
2k views

Why Does Induction Prove Multiplication is Commutative?

Andrew Boucher's General Arithmetic (GA2) is a weak sub-theory of second order Peano Axioms (PA2). GA has second order induction and a single successor axiom: $$\forall x \forall y \forall ...
7
votes
4answers
845 views

Examples of “exotic” induction

Next week I am going to teach two lessons on induction to very motivated students from high schools. At some point I would like to talk about ordered sets, well-ordered sets, and mention the fact that ...
5
votes
2answers
490 views

Homotopy Transfer Theorem for Differential Graded Associative Algebras

As in Algebra+Homotopy=Operad by Bruno Vallette, let $A$ with multiplication $\nu$ be a differential graded associative algebra equipped with degree +1 map $h$ and let $H$ be a chain complex such that ...
1
vote
0answers
76 views

Coinduction and corestriction are quasi-inverse equivalences for comodules?

I'm reading http://arxiv.org/abs/math/0310337. There the following statement is given without proof: Let $k$ be a field. Let $C$ be a counitary coaugmented coalgebra, i.e. there is $\eta: C\to k$ ...
0
votes
1answer
476 views

Maximum sum of 3 consecutive numbers in a permutation [closed]

Given that $X = \{0, 1, 2, ..., 7, 8, 9\}$, and $P$ is a permutation on $X$. Let $M(P)$ be the maximum sum of 3 consecutive elements. For example, if $P = (0, 2, 4, 1, 5, 7, 9, 3, 8, 6)$, then $M(P)$ ...
8
votes
2answers
541 views

Order statistics (e.g., minimum) of infinite collection of chi-square variates?

Hi everyone, This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, ...
9
votes
10answers
4k views

Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please). 2) There are proofs ...
9
votes
3answers
967 views

Easier induction proofs by changing the parameter

When performing induction on say a graph $G=(V,E)$, one has many choices for the induction parameter (e.g. $|V|, |E|$, or $|V|+|E|$). Often, it does not matter what choice one makes because the proof ...
4
votes
1answer
261 views

For which classes of functions this inverse function formula gives a closed form expression?

Lets consider this method of finding inverse function: $$f^{-1}(x) = \sum_{k=0}^\infty A_k(x) \frac{(x-f(x))^k}{k!}$$ where coefficients $A_k(x)$ recursively defined as $$\begin{cases} A_0(x)=x \\ ...
3
votes
2answers
665 views

How to restore the original formula from a binomial-like expansion?

I encountered with a recursive formula of the following kind: $$A(0,x)=1$$ $$A(n,x)= \sum _{j=0}^{n-1} \binom{n-1}{j} A(n-j-1,x) \sum _{k=0}^{x-1} A(j,k)$$ The sum terms can be re-arranged so to ...
6
votes
1answer
692 views

Symmetric Proof that Product is Well-Founded

This is a fairly minor, technical question, but I'll toss it out in case someone has a good idea on it. Suppose $(X,<_X)$ and $(Y,<_Y)$ are well-founded orderings (not necessarily linearly ...
27
votes
4answers
2k views

A principle of mathematical induction for partially ordered sets with infima?

Recently I learned that there is a useful analogue of mathematical induction over $\mathbb{R}$ (more precisely, over intervals of the form $[a,\infty)$ or $[a,b]$). It turns out that this is an old ...