Questions tagged [induction]

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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 ...
45 votes
7 answers
6k views

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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2 votes
0 answers
158 views

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 ...
1 vote
0 answers
110 views

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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5 votes
0 answers
209 views

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 ...
1 vote
1 answer
122 views

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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0 votes
1 answer
116 views

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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14 votes
2 answers
2k views

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 ...
-1 votes
1 answer
54 views

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 ...
2 votes
0 answers
91 views

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 ...
17 votes
0 answers
377 views

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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1 vote
0 answers
292 views

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 $n=1$, ...
6 votes
2 answers
153 views

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 (...
3 votes
1 answer
147 views

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 ...
1 vote
1 answer
177 views

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 >...
2 votes
2 answers
379 views

(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 ...
5 votes
1 answer
284 views

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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9 votes
1 answer
2k views

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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6 votes
2 answers
632 views

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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0 votes
2 answers
195 views

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 "$\in$". ...
1 vote
2 answers
270 views

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 ...
6 votes
1 answer
417 views

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 ...
3 votes
2 answers
445 views

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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7 votes
1 answer
224 views

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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-2 votes
1 answer
336 views

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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1 vote
0 answers
141 views

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}\...
5 votes
4 answers
643 views

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 ...
6 votes
1 answer
511 views

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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3 votes
0 answers
492 views

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" ...
23 votes
5 answers
3k views

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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3 votes
1 answer
331 views

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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4 votes
0 answers
121 views

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 $\mathbf{ML}_{< \omega} \mathbf{W}$...
1 vote
0 answers
67 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}) \stackrel{n}{\...
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3 votes
0 answers
507 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 ...
18 votes
7 answers
2k 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 ...
3 votes
2 answers
681 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 problems/questions ...
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3 votes
2 answers
930 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 ...
36 votes
3 answers
4k 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 ...
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8 votes
1 answer
829 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 ...
1 vote
3 answers
3k 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 z\bigr((Sx=...
7 votes
4 answers
1k 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 ...
8 votes
2 answers
2k 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
0 answers
148 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
1 answer
836 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
2 answers
979 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, ...
29 votes
11 answers
9k 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 (...
11 votes
3 answers
2k 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
1 answer
463 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 \\ ...
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3 votes
2 answers
2k 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 ...
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6 votes
1 answer
822 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 ...