Simple Proof Research Articles

Strong Greedoid Structure of $r$-Removed $P$-Orderings

Inspired by the notion of $r$-removed $P$-orderings introduced in the setting of Dedekind domains by Bhargava, we generalize it to the framework of arbitrary ultrametric spaces. We show that sets of maximal "$r$-removed perimeter" can be constructed by a greedy algorithm and form a strong greedoid. This gives a simplified proof of several theorems previously obtained by Bhargava, as well as generalises some results of Grinberg and Petrov who considered the case $r=0$ corresponding, in turn, to simple $P$-orderings.

Local complete intersections and Weierstrass points

This work presents a simple proof that the moduli space of complete integral Gorenstein curves with a prescribed symmetric Weierstrass semigroup becomes a weighted projective space, even for fields of positive characteristic, when the associated monomial curve is a local complete intersection.

From Condorcet’s paradox to Arrow: yet another simple proof of the impossibility theorem

From Condorcet’s paradox to Arrow: yet another simple proof of the impossibility theorem

A simple proof of the rationality of Takagi-like functions

A simple proof of the rationality of Takagi-like functions

Generalized quadratic Gauss sums and their 2mth power mean

Abstract The main purpose of this article is to study the problem of calculating the 2mth power mean of the generalized quadratic Gauss sums, and using the analytic method and an interesting combinatorial identity to give a sharp asymptotic formula for the 2mth power mean. Thus, a new simple proof of this existing result [N. Bag, A. Rojas-León, and W. P. Zhang, On some conjectures on generalized quadratic Gauss sums and related problems, Finite Fields Appl. 86 (2023), 10213] is provided.

The Fourier transform on valuations is the Fourier transform

Alesker has proved the existence of a remarkable isomorphism of the space of translation-invariant smooth valuations that has the same functorial properties as the classical Fourier transform. In this paper, we show how to directly describe this isomorphism in terms of the Fourier transform on functions. As a consequence, we obtain simple proofs of the main properties of the Alesker–Fourier transform. One of these properties was previously only conjectured by Alesker and is proved here for the first time.

A short note on the Love number of extremal Reissner-Nordstrøm and Kerr-Newman black holes

We provide a simple proof of why the Love number vanishes for extremal Reissner-Nordstrøm and Kerr-Newman black holes. The argument is based on a conformal inversion isometry of the spacetime connecting the horizon with large distances.

Face‐simple minimal quadrangulations of surfaces

AbstractFor each surface besides the sphere, projective plane, and Klein bottle, we construct a face‐simple minimal quadrangulation, that is, a simple quadrangulation on the fewest number of vertices possible, whose dual is also a simple graph. Our result answers a question of Liu, Ellingham, and Ye while providing a simpler proof of their main result. The inductive construction is based on an earlier idea for finding near‐quadrangular embeddings of the complete graphs using the diamond sum operation.

Lagrange expansion formula and its application in combinatorial mathematics

There is a uniqueness theorem for an implicit function. When a function satisfies certain conditions, the equation can determine a unique implicit function. However, we cannot determine the display expression of the implicit function according to this theorem. The Lagrange expansion formula just compensates for this disadvantage, which explicitly gives the display expression of the implicit function. This paper will give a simpler proof of the Lagrange expansion formula and provide examples to illustrate its application in combinatorial mathematics.

On Ramanujan’s modular equations of degree three

The object of this article is to give new and simple proof of all but one Ramanujan’s modular equations of degree 3.

On some Schläfli type modular equations

On page 90 of his first notebook Ramanujan has recorded many Schl¨afli type modular equations. This article aims to give new and simple proof of Sch¨alflitype modular equation of degree 11 which was recorded by Ramanujan and a degree 23 modular equation. We use parametrization method to prove these modular equations.

Corrigendum to “A simpler proof of toroidalization of morphisms from 3-folds to surfaces”

Corrigendum to “A simpler proof of toroidalization of morphisms from 3-folds to surfaces”

Compositionality and Observational Refinement for Linearizability with Crashes

Crash-safety is an important property of real systems, as the main functionality of some systems is resilience to crashes. Toward a compositional verification approach for crash-safety under full-system crashes, one observes that crashes propagate instantaneously to all components across all levels of abstraction, even to unspecified components, hindering compositionality. Furthermore, in the presence of concurrency, a correctness criterion that addresses both crashes and concurrency proves necessary. For this, several adaptations of linearizability have been suggested, each featuring different trade-offs between complexity and expressiveness. The recently proposed compositional linearizability framework shows that to achieve compositionality with linearizability, both a locality and observational refinement property are necessary. Despite that, no linearizability criterion with crashes has been proven to support an observational refinement property. In this paper, we define a compositional model of concurrent computation with full-system crashes. We use this model to develop a compositional theory of linearizability with crashes, which reveals a criterion, crash-aware linearizability , as its inherent notion of linearizability and supports both locality and observational refinement. We then show that strict linearizability and durable linearizability factor through crash-aware linearizability as two different ways of translating between concurrent computation with and without crashes, enabling simple proofs of locality and observational refinement for a generalization of these two criteria. Then, we show how the theory can be connected with a program logic for durable and crash-aware linearizability, which gives the first program logic that verifies a form of linearizability with crashes. We showcase the advantages of compositionality by verifying a library facilitating programming persistent data structures and a fragment of a transactional interface for a file system.

Global Existence in the Critical Space for the Space-Time Monopole Equations

We prove the global existence of solutions in the critical space \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1({\\mathbb {R}})$$\\end{document} for space-time monopole equations under the Lorenz gauge condition. The existence of a global solution was showed in our previous paper. Here we adapt a different function space to allow for a simplified proof. We first prove the local existence in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1({\\mathbb {R}})$$\\end{document} and extend the local solution to the global one.

A generalisation of b a < a b when e ≤ a < b

In this short note we give a simple proof of a generalization of the inequality obtained in Chakraborty [(2022). A visual proof that International Journal of Mathematical Education in Science and Technology, 55(5), 1291–1292], as well as of similar inequalities.

Invariant idempotent ⁎-measures generated by iterated function systems

It is known that every continuous t-norm ⁎ generates a functor of the so-called ⁎-measures in the category of compact Hausdorff spaces. Similarly to the case of the hyperspace functor and the probability measure functors one can define the notion of invariant ⁎-measure for iterated function systems of contractions on compact metric spaces.We provide a simple proof of existence and uniqueness of invariant ⁎-measures. Some examples of invariant ⁎-measures, for different t-norms ⁎, are presented.

Perspective on complexity measures targeting read-once branching programs

A model of computation for which reasonable yet still incomplete lower bounds are known is the read-once branching program. Here variants of complexity measures successful in the study of read-once branching programs are defined and studied. Some new or simpler proofs of known bounds are uncovered. Branching program resources and the new measures are compared extensively. The new variants are developed in part in the hope of tackling read-k branching programs for the tree evaluation problem [8]. Other computation problems are studied as well. In particular, a common view of a function studied by Gál [11] and a function studied by Bollig and Wegener [3] leads to the general combinatorics of blocking sets. Technical combinatorial results of independent interest are obtained. New leads towards further progress are discussed. An exponential lower bound for non-deterministic read-k branching programs for the GEN function [17] is also derived, independently from the new measures.

The International Arbitration Award as a Simple Proof Requirement in Bankruptcy

The International Arbitration Award as a Simple Proof Requirement in Bankruptcy

Statistically optimal firstorder algorithms: a proof via orthogonalization

Abstract We consider a class of statistical estimation problems in which we are given a random data matrix ${\boldsymbol{X}}\in{\mathbb{R}}^{n\times d}$ (and possibly some labels ${\boldsymbol{y}}\in{\mathbb{R}}^{n}$) and would like to estimate a coefficient vector ${\boldsymbol{\theta }}\in{\mathbb{R}}^{d}$ (or possibly a constant number of such vectors). Special cases include low-rank matrix estimation and regularized estimation in generalized linear models (e.g. sparse regression). Firstorder methods proceed by iteratively multiplying current estimates by ${\boldsymbol{X}}$ or its transpose. Examples include gradient descent or its accelerated variants. Under the assumption that the data matrix ${\boldsymbol{X}}$ is standard Gaussian, Celentano, Montanari, Wu (2020, Conference on Learning Theory, pp. 1078–1141. PMLR) proved that for any constant number of iterations (matrix vector multiplications), the optimal firstorder algorithm is a specific approximate message passing algorithm (known as ‘Bayes AMP’). The error of this estimator can be characterized in the high-dimensional asymptotics $n,d\to \infty $, $n/d\to \delta $, and provides a lower bound to the estimation error of any firstorder algorithm. Here we present a simpler proof of the same result, and generalize it to broader classes of data distributions and of firstorder algorithms, including algorithms with non-separable nonlinearities. Most importantly, the new proof is simpler, does not require to construct an equivalent tree-structured estimation problem, and is therefore susceptible of a broader range of applications.

Geodesic vector fields, induced contact structures and tightness in dimension three

In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the ‘mixed’ sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact 3-manifolds with isometric Reeb flows (also known as R-contact manifolds) up to diffeomorphism.