# Questions tagged [induction]

The induction tag has no usage guidance.

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### Terminology associated with mathematical induction

In "Number: The Language of Science" (1930), Tobias Dantzig refers to what we call the base case of mathematical induction as "the induction step" (and refers to what we call the ...

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### Zorn's lemma: old friend or historical relic?

It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...

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### Largest eigenvalue of a Laplacian matrix to lower bound the prime counting function?

Let $L_n$ be the Laplacian matrix of the undirected graph $G_n = (V_n, E_n)$ (which is defined here: Why is this bipartite graph a partial cube, if it is? ) with sorted spectrum:
$$\lambda_1 (G_n) \ge ...

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### What does "sup" mean in the context of a w type? [closed]

Like the constructor for a W type is called "sup" but I don't know what that expands to. Is it super? maybe supremum? Or is it just an arbitrary name, like dynamic programming?

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### Induction for quantum group

I am confused about a claim in the article Representation of quantum algebras by Henning Haahr Andersen, Patrick Polo and Wen Kexin. I probably misunderstood a definition, but I found two claims about ...

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### Mathematical induction and the counting function on $\mathbb{Z}_p^2$

Let $\mathbb{Z}_p$ be a finite field of order $p$ and $\mathbb{Z}_p^2$ be a $2$-dimensional vector space over $\mathbb{Z}_p$. We consider the distance $\lVert \cdot \rVert:\mathbb{Z}_p^2\to \mathbb{Z}...

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### Nicely motivated papers or book chapters on the formula for the sum of the $k$-th powers of the first natural numbers [closed]

Do you know of a text where I can find a nicely motivated proof of the formula for $1^{k}+2^{k}+\cdots+n^{k}$?
At the very beginning of page 68 of Professor H. S. Wilf's generatingfunctionology, one ...

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### How to structure a proof by induction in a maths research paper?

I am 16 years old at the time of writing (so I have no supervisors to seek advice from) and I have written a mathematics research paper, which I plan on submitting to a journal for publication. I ...

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### Seek help to formalize an argument to positiveness of function defined inductively by integral [closed]

I have on domain $[0,\infty)$ a known and positive function $f(x)$ and two unknown functions $g(x), h(x)$ that start positive when $x=0$.
I also know that if $h(x)$ is positive, then $g(x)$ is also ...

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### Can you define inductive data types over categories other than Set?

Can you define inductive data types over categories other than $\mathbb{Set}$?
What does it look like? How about for a specific example like the category of monoids? If you were clever could you write ...

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### Is the Frog game solvable in the root of a full binary tree?

This is a cross-post from math.stackexchange.com$^{[1]}$, since the bounty there didn't lead to any new insights.
For reference,
The Frog game is the generalization of the Frog Jumping (see it on ...

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### Generalizations of classical tiling problem

A classic problem using an inductive construction is to show that the $2^n \times 2^n$-square, with a missing corner, can be tiled with L-triominoes.
The proof goes like this:
It is true for $n=1$, ...

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### Limit of alternated row and column normalizations

Let $E_0$ be a matrix with non-negative entries.
Given $E_n$, we apply the following two operations in sequence to produce $E_{n+1}$.
A. Divide every entry by the sum of all entries in its column (...

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### Models of arithmetical theory R + induction in which successor is not injective

Consider the arithmetical theory sometimes denoted by $\mathsf{R}$. The non-logical vocabulary of $\mathsf{R}$ consists of '$0$', '$S$', '$+$' and '$\times$'. The axioms of this theory are all true ...

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### An iterative argument involving $f(n + 1) - f(n) $

I am working with an argument involving an inequality of the form:
$$ f(n + 1) \leq f(n) + C (f(n))^{1 - \frac{1}{\gamma}} \qquad(\ast)$$
where $f$ is a positive function, $\gamma > 0$ and $C >...

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### (Types of) induction on infinite chains

This question may be trivial, or overly optimistic. I do not know (but I guess the latter...). I am a group theorist by trade, and the set-up I describe cropped up in something I want to prove. So ...

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### Why restrict to $\Sigma_1^0$ formulas in $RCA_0$ induction?

I recently asked this question over on math.se, warmly welcomed by crickets. I hope it's appropriate here.
I'm reading Stillwell's Reverse Mathematics, and the induction axiom was just introduced.
...

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### Is There an Induction-Free Proof of the 'Be The Leader' Lemma?

This lemma is used in the context of online convex optimisation. It is sometimes called the 'Be the Leader Lemma'.
Lemma:
Suppose $f_1,f_2,\ldots,f_N$ are real valued functions with the same ...

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### Cases where multiple induction steps are provably required

I am looking for references for theorems of the form:
1) Any proof of theorem $X$ requires $n$ applications of induction axioms
and especially
2) Any proof of theorem $X$ requires $n$ nested ...

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### Are Regularity schema and $\in$-induction schema equivalent in intuitionistic logic?

In posting "Does Regularity schema imply $\in$-induction when added to first order Zermelo set theor?"
the answer was that they are equivalent in classical first order logic with membership "$\in$".
...

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### Does Regularity schema imply $\in$-induction when added to first order Zermelo set theory?

That $\in$-induction fails to be a theorem schema of first order Zermelo + Foundation (see here), then it appears that it is more eligible to replace axiom of Regularity (Foundation) by a Regularity ...

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### Is $\in$-induction provable in first order Zermelo set theory?

Are there models of first order Zermelo set theory (axiomatized by: Extensionaity, Foundation, empty set, pairing, set union, power, Separation, infinity) in which $\in$-induction fail?
I asked this ...

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### How to estimate a recursive inequality with an upper bound

The below is a simplification of part of a proof I'm working on, in numerical analysis. It is similar to a paper that I studied some months ago, for which I got some advice here on MathOverflow.
I ...

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### Independent/Easy fraction of sentences over PA

Let $S(n)$ be the set of all sentences over PA of length at most $n$ (counting the quantifier symbols, boolean connectives, arithmetic operations and constants, and counting each variable as length $1$...

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### Towers of induced functions

Suppose $\mathbb{X}$ and $\mathbb{Y}$ are classes, and $f:\mathbb{X}\rightarrow\mathbb{Y}$. It seems like pretty standard course to consider an 'induced function' $f:\mathcal{P}\mathbb{X}\rightarrow\...

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### Proving the existence of a sequence with recursive growth constraints

Fix $0 < c < 1/3$. Show that there exists a strictly increasing sequence $a_k > 0$, a sequence $b_k \geq 0$, and an infinite set $K \subseteq \mathbb{Z}_{>0}$ such that
\begin{align}\...

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### Mathematical induction vis-a-vis primes

One of the most used proof-techniques is mathematical induction, and one of the oldest subjects is the study of prime numbers. Thanks to Euclid, we can consider the primes as a infinite monotone ...

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### Inductive Definitions in Category Theory

I'm trying to pin down a notion of inductive definability in category-theoretic terms.
The sorts of inductively defined sets (and classes) I'm most interested in are those that admit of induction and ...

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### When must one strengthen one's induction hypothesis?

My questions are about the phenomenon that in order to prove a fact $\forall x \phi(x)$ by induction, sometimes straightforward induction "does not work" and instead one "must" use a "stronger" ...

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### Is Cauchy induction used for proofs other than for AM–GM?

The proof by Cauchy induction of the arithmetic/geometric-mean inequality is well known. I am looking for a further theorem whose proof is much neater by this method than otherwise.

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### Affine analog of the theory of sheets

In the study of adjoint orbits in a complex semi-simple lie algebra, there is a well known object known as a "sheet". These are the irreducible components of the union of orbits of the same dimension. ...

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### Problem with a proof in Wellfounded trees in categories

I'm reading the paper Wellfounded trees in categories by Moerdijk and Palmgren. I'm having trouble understanding the proof of Theorem 7.2 (page 216), i.e. that in $\mathbf{ML}_{< \omega} \mathbf{W}$...

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### 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}) \stackrel{n}{\...

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### 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 ...

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### 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 ...

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### 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 problems/questions ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 z\bigr((Sx=...

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### 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 ...

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### 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 ...

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### 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 $\eta: C\to k$ ...

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### 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 $M(P)$ ...

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### 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, ...

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### 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 (...

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### 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 ...

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### 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 \\ ...

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### 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 ...

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### 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 ...