Simple Inductive Proof Research Articles

Lagrange Inversion Formula by Induction

We present a simple inductive proof of the Lagrange Inversion Formula.

A Full and Detailed Proof for the Riemann Hypothesis & the Simple Inductive proof of Goldbach’s Conjecture

As in my previous two papers [2] & [3] about the boundary of the prime gap still cause some misunderstanding, I here in this paper tries to clarify those detailed steps in proving such boundary of the prime gap for a contradiction. Indeed, the general idea of my designed proof is to make all of the feasible case of the Riemann Zeta function with exponents ranged from 1 to s = u + v*I becomes nonsense (where u, v are real numbers with I is imaginary equals to (-1)1/2 except that u = 0.5 with some real numbers v as the expected zeta roots. Once if we can exclude all other possibilies unless u = 0.5 with some real numbers v in the Riemann Zeta function’s exponent “s”, then the Riemann Hypothesis will be proved immediately. The truth of the hypothesis further implies that there is a need for the shift from the line x = 0 to the line x = 0.5 as all of the zeta roots lie on it. However, NOT all of the points on x = 0.5 are zeros as we may find from the model equation that has been well established in [2]. One of my application is in the quantum filtering for an elimination of noise in a quantum system but NOT used to filter human beings like the political counter-parts.In general, this author suggests that for all of the proof or disproof to any cases of hypothesis, one may need to point out those logical contradictions [14] among them. Actually, my proposition works very well for the cases in my disproof of Continuum Hypothesis [15] together with the proof in Riemann Hypothesis

Bijections Between Walks Inside a Triangular Domain and Motzkin Paths of Bounded Amplitude

This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0°, 60°, 120°, 180°, 240°, 300°) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height.
 We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a 2n-to-one function from generic triangular walks to triangular walks only using directions 0°, 120°, 240°. The second is based on an extension of Mortimer and Prellberg's results to triangular walks starting not only at a corner of the triangle, but at any point inside it. It has a linear-time complexity and is in fact adjustable: by changing some set of parameters called a scaffolding, we obtain a wide range of different bijections.
 Finally, we extend our results to higher dimensions. In particular, by adapting the previous proofs, we discover an unexpected bijection between three-dimensional walks in a pyramid and two-dimensional simple walks confined in a bounded domain shaped like a waffle.

Improved Bounds for Guarding Plane Graphs with Edges

An edge guard set of a plane graph G is a subset $$\varGamma $$ of edges of G such that each face of G is incident to an endpoint of an edge in $$\varGamma $$ . Such a set is said to guardG. We improve the known upper bounds on the number of edges required to guard any n-vertex embedded planar graph G: (1) We present a simple inductive proof for a theorem of Everett and Rivera-Campo (Comput Geom Theory Appl 7:201–203, 1997) that G can be guarded with at most $$\frac{2n}{5}$$ edges, then extend this approach with a deeper analysis to yield an improved bound of $$\frac{3n}{8}$$ edges for any plane graph. (2) We prove that there exists an edge guard set of G with at most $$\frac{n}{3} + \frac{\alpha }{9}$$ edges, where $$\alpha $$ is the number of quadrilateral faces in G. This improves the previous bound of $$\frac{n}{3} + \alpha $$ by Bose et al. (Comput Geom Theory Appl 26(3):209–219, 2003). Moreover, if there is no short path between any two quadrilateral faces in G, we show that $$\frac{n}{3}$$ edges suffice, removing the dependence on $$\alpha $$ .

The Art Gallery Theorem, Revisited

The art gallery theorem asserts that any polygon with n vertices can be protected by at most ⌊n/3⌋ stationary guards. The original proof by Chvátal uses a nonroutine and nonintuitive induction. We give a simple inductive proof of a new, more general result, the constrained art gallery theorem: If V* and E* are specified sets of vertices and edges that must contain guards, then the polygon can be protected by at most ⌊(n + 2|V*| + |E*|)/3⌋ guards. Our result reduces to Chvátal's art gallery theorem when V* and E* are empty. We give a second short proof of this generalization in the spirit of Fisk's proof of the art gallery theorem using graph colorings.

Exponential laws for ultrametric partially differentiable functions and applications

We establish exponential laws for certain spaces of differentiable functions over a valued field \(\mathbb{K}\). For example, we show that $$C^{(\alpha ,\beta )} \left( {U \times V,E} \right) \cong C^\alpha \left( {U,C^\beta \left( {V,E} \right)} \right)$$ if α ∈ (ℕ0 ∪ {∞})n, β ∈ (ℕ0 ∪ {∞})m, \(U \subseteq \mathbb{K}^n\) and \(V \subseteq \mathbb{K}^m\) are open (or suitable more general) subsets, and E is a topological vector space. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by Enno Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C r-functions on (ℤ p )n in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.

Broken circuits in matroids-Dohmen's inductive proof

Dohmen [4] gives a simple inductive proof of Whitney’s famous broken circuits theorem. We generalise his inductive proof to the case of matroids.

QUADRATIC RECIPROCITY VIA RESULTANTS

We give a simple inductive proof of quadratic reciprocity using resultants.

Short proofs of theorems of Mirsky and Horn on diagonals and eigenvalues of matrices

A theorem of Mirsky provides necessary and sufficient conditions for the existence of an N-square complex matrix with prescribed diagonal entries and prescribed eigenvalues. A simple inductive proof of this theorem is given.

A simple (inductive) proof for the non-existence of 2-cycles of the [formula omitted] problem

A 2-cycle of the 3 x + 1 problem has two local odd minima x 0 and x 1 with x i = a i 2 k i − 1 . Such a cycle exists if and only if an integer solution exists of a diophantine system of equations in the coefficients a i . We derive a numerical lower bound for a 0 ⋅ a 1 , based on Steiner's proof for the non-existence of 1-cycles. We derive an analytical expression for an upper bound for a 0 ⋅ a 1 as a function of K and L (the number of odd and even numbers in the cycle). We apply a result of de Weger on linear logarithmic forms to show that these lower and upper bounds are contrary. The proof does not use exterior lower bounds for numbers in a cycle and for the cycle length.

Variations on the Roy-Gallai theorem

A generalization of the Roy-Gallai Theorem on the chromatic number of a graph is derived which is also an extension of several other results of Berge and of Li. A simple inductive proof is given which provides a direct way of deriving the Theorem of Li. We also show that some classical results valid for optimal colorings cannot be transposed to suboptimal colorings. We finally investigate some elementary properties which are also valid in suboptimal colorings.

An improved linear edge bound for graph linkages

A graph is said to be k- linked if it has at least 2 k vertices and for every sequence s 1,…, s k , t 1,…, t k of distinct vertices there exist disjoint paths P 1,…, P k such that the ends of P i are s i and t i . Bollobás and Thomason showed that if a simple graph G on n vertices is 2 k-connected and G has at least 11 kn edges, then G is k-linked. We give a relatively simple inductive proof of the stronger statement that 8 kn edges and 2 k-connectivity suffice, and then with more effort improve the edge bound to 5 kn.

Simple Inductive Proofs of the Fishburn and Mirkin Theorem and the Scott-Suppes Theorem.

This paper presents new proofs of two classic characterization theorems for families of ordered sets. The first is that any finite poset with no restriction isomorphic to \(\underline 2 + \underline 2 \) has an interval representation. The second is that any finite poset with no restriction isomorphic to \(\underline 2 + \underline 2 \) or to \(\underline 3 + \underline 1 \) has a unit interval representation. Both proofs are straightforward and inductive.

Syntactic Type Soundness Results for the Region Calculus

The region calculus of Tofte and Talpin is a polymorphically typed lambda calculus with annotations that make memory allocation and deallocation explicit. It is intended as an intermediate language for implementing Hindley-Milner typed functional languages such as ML without traditional trace-based garbage collection. Static region and effect inference can be used to annotate a statically typed ML program with memory management primitives. Soundness of the calculus with respect to the region and effect system is crucial to guarantee safe deallocation of regions, i.e., deallocation should only take place for objects which are provably dead. The original soundness proof by Tofte and Talpin requires a complex co-inductive safety relation. In this paper, we present two small-step operational semantics for the region calculus and prove their type soundness with respect to the region and effect system. Following the standard syntactic approach of Wright, Felleisen, and Harper, we obtain simple inductive proofs. The first semantics is store-less. It is simple and elegant and gives rise to perspicuous proofs. The second semantics provides a store-based model for the region calculus. Albeit slightly more complicated, its additional expressiveness allows us to model operations on references with destructive update. A pure fragment of both small-step semantics is then proven equivalent to the original big-step operational approach of Tofte and Talpin. This leads to an alternative soundness proof for their evaluation-style formulation.

Syntactic Type Soundness for the Region Calculus

The region calculus of Tofte and Talpin is an annotated polymorphically typed lambda calculus which makes memory allocation and deallocation explicit. It is intended as an intermediate language in a compiler for ML-like languages. The region annotations are obtained by static region and effect inference, which makes it an attractive alternative for garbage collection. Soundness of the region and effect system is crucial to guarantee safe deallocation of regions, i.e. deallocation should only take place for objects which are provable dead.Tofte and Talpin have proved type soundness of the region calculus using rule-based co-induction. This proof is quite complicated and not very intuitive. Much of the problem lies in the low-level big-step operational semantics which involves manipulations of an explicit store and which has a co-inductive definition. In this paper, we present a small-step operational semantics for the region calculus, based on syntactic rewriting. We prove type soundness following the approach of Wright and Felleisen, leading to very simple inductive proofs.

Large Momentum Bounds from Flow Equations

We analyse the large momentum behaviour of 4-dimensional massive euclidean Phi-4-theory using the flow equations of Wilson's renormalization group. The flow equations give access to a simple inductive proof of perturbative renormalizability. By sharpening the induction hypothesis we prove new and, as it seems, close to optimal bounds on the large momentum behaviour of the correlation functions. The bounds are related to what is generally called Weinberg's theorem.

Bounds to the Chromatic Polynomial of a Graph

In this paper, we present a simple inductive proof of some recently published bounds to the chromatic polynomial of a graph.

Gauge independence of theS-matrix in the causal approach

The gauge dependence of the time-ordered products for Yang-Mills theories is analysed in perturbation theory by means of the causal method of Epstein and Glaser together with perturbative gauge invariance. This approach allows a simple inductive proof of the gauge independence of the physical S-matrix.

Hook lengths in a skew Young diagram

Regev and Vershik (Electronic J. Combinatorics 4 (1997), #R22) have obtained some properties of the set of hook lengths for certain skew Young diagrams, using asymptotic calculations of character degrees. They also conjectured a stronger form of one of their results. We give a simple inductive proof of this conjecture. Very recently, Regev and Zeilberger (Annals of Combinatorics, to appear) have independently proved this conjecture.

Formal systems of dialogue rules

Section 1 contains a survey of options in constructing a formal system of dialogue rules. The distinction between material and formal systems is discussed (section 1.1). It is stressed that the material systems are, in several senses, formal as well. In section 1.2 variants as to language form (choices of logical constants and logical rules) are pointed out. Section 1.3 is concerned with options as to initial positions and the permissibility of attacks on elementary statements. The problem of ending a dialogue, and of infinite dialogues, is treated in section 1.4. Other options, e.g., as to the number of attacks allowed with respect to each statement, are listed in section 1.5. Section 1.6 explains the concept of a ‘chain of arguments’. From section 2 onward four types of dialectic systems are picked out for closer study: D, E, Di and Ei. After a preliminary section on dialogue sequents and winning strategies, the equivalence of derivability in intuitionistic logic and the existence of a winning strategy (for the Proponent) on the strength of Ei is shown by simple inductive proofs. Section 3 contains a — relatively quick — proof of the equivalence of the four systems. It follows that each of them yields intuitionistic logic.