Short Proof Research Articles

Reeb–Anosov flows are skewed $$\mathbb {R}$$-covered, shorter proof

Reeb–Anosov flows are skewed $$\mathbb {R}$$-covered, shorter proof

Characterizing Nonuniform Hyperbolicity by Mather-Type Admissibility

AbstractWe consider linear cocycles acting on Banach spaces which satisfy the assumptions of the multiplicative ergodic theorem. A cocycle is nonuniformly hyperbolic if all Lyapunov exponents are non-zero, which is equivalent to the existence of a tempered exponential dichotomy. We provide an equivalent characterization of nonuniform hyperbolicity in terms of a Mather-type admissibility of a pair of weighted function spaces. As an application we give a short proof of the robustness of tempered exponential dichotomies under small linear perturbation.

On quaternionic bisectional curvature

AbstractIn this article we study the concept of quaternionic bisectional curvature introduced by Chow and Yang in (J Differ Geom 29(2):361–372, 1989) for quaternion-Kähler manifolds. We show that non-negative quaternionic bisectional curvature is only realized for the quaternionic projective space $$\mathbb {H}\textrm{P}^n$$ H P n . We also show that all symmetric quaternion-Kähler manifolds different from $$\mathbb {H}\textrm{P}^n$$ H P n admit quaternionic lines of negative quaternionic bisectional curvature. In particular this implies that non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature. Moreover we give a new and rather short proof of a classification result by A. Gray on compact Kähler manifolds of non-negative sectional curvature.

ON NONINNER AUTOMORPHISMS OF SOME FINITE P-GROUPS

Abstract We settle the noninner automorphism conjecture for finite p-groups ( $p> 2$ ) with certain conditions. Also, we give an elementary and short proof of the main result of Ghoraishi [‘On noninner automorphisms of finite nonabelian p-groups’, Bull. Aust. Math. Soc.89(2) (2014) 202–209].

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

A Telescopic Proof of Cayley’s Formula

We give a short proof of the fact that the number of labeled trees on n vertices is n n − 2 . Although many short proofs are known, we have not seen this one before.

A Simple Elementary Proof of Fermat's Last Theorem

In this article we attempt to describe a short proof of Fermat's Last Theorem. The Fermat's diophantine equation xn = yn + zn gives rise to an equation of (n-1)th degree, which can be proved to have no positive rational roots. This proves the case for any odd prime n. For n = 4 we use the principle of reductioad-absurdum along with a polynomial equation of degree 3. Further Beal's conjecture is examined and proved to be true.

Intrinsic universality in automata networks II: Glueing and gadgets

An automata network (AN) is a finite graph where each node holds a state from a finite alphabet and is equipped with a local map defining the evolution of the state of the node depending on its neighbors. This paper is the second of a series about intrinsic universality, i.e. the ability for a family of AN to simulate arbitrary AN. We develop a proof technique to establish intrinsic simulation and universality results which is suitable to deal with families of symmetric networks where connections are non-oriented. It is based on an operation of glueing of networks, which allows to produce complex orbits in large networks from compatible pseudo-orbits in small networks. As an illustration, we give a short proof that the family of networks where each node obeys the rule of the ‘game of life’ cellular automaton is strongly universal. In the third paper of the series, we heavily rely on this proof technique to show intrinsic universality results of various families with particular update schedules.

Unifying some classical results on Artinian rings and modules

In this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results.

Kempner-like Harmonic Series

Inspired by a question asked on the list mathfun, we revisit Kempner-like series, i.e., harmonic sums ∑ ′ 1 / n where the integers n in the summation have “restricted” digits. First we give a short proof that lim k → ∞ ( ∑ s 2 ( n ) = k 1 / n ) = 2 log 2 , where s 2 ( n ) is the sum of the binary digits of the integer n. Then we give a generalization that addresses the case where s 2 ( n ) is replaced with s b ( n ) , the sum of b-ary digits in base b: we prove that lim k → ∞ ∑ s b ( n ) = k 1 / n = ( 2 log b ) / ( b − 1 ) . Finally we indicate that other generalizations could be studied: the sum of digits in base 2 could be replaced with, e.g., the function a 11 ( n ) of—possibly overlapping—11 in the base-2 expansion of n, for which one can obtain lim k → ∞ ∑ a 11 ( n ) = k 1 / n = 4 log 2 .

A short proof for the parameter continuation theorem

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives a method for computing discriminants.

Gelation and Localization in Multicomponent Coagulation with Multiplicative Kernel Through Branching Processes

The multicomponent coagulation equation is a generalization of the Smoluchowski coagulation equation, where the size of a particle is described by a vector. Similar to the original Smoluchowski equation, the multicomponent coagulation equation exhibits gelation behavior when supplied with a multiplicative kernel. Additionally, a new type of behaviour called localization is observed due to the multivariate nature of the particle size distribution. Here we extend the branching process representation technique, which we introduced to study differential equations in our previous work, and apply it to find a concise probabilistic solution of the multicomponent coagulation equation supplied with monodisperse initial conditions. We also provide short proofs for the gelation time and characterisation the localization phenomenon.

A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization

Wasserstein distributionally robust optimization has emerged as a recent topic with broader applications in operations research and machine learning. Various proofs have been presented in the literature, each differing in assumptions and levels of generality. In “A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization,” Zhang, Yang, and Gao present a novel elementary proof that not only shortens existing frameworks but also offers surprising generalizations. Leveraging classical Legendre—Fenchel duality, they demonstrate that strong duality is contingent on a certain interchangeability principle. Moreover, they extend this duality result to encompass risk-averse optimization and globalized distributionally robust counterparts.

A uniform characterization of the Octonions and the Quaternions using commutators

Let $R$ be a ring which is not commutative. Assume that either $R$ is alternative, but not associative, or $R$ is associative and any commutator $v\in R$ satisfies: $v^2$ is in the center of $R.$ We show (using commutators) that if $R$ contains no divisors of zero and $\text{char}(R)\ne 2,$ then $R//C,$ the localization of $R$ at its center $C,$ is the octonions in the first case and the quaternions, in latter case. Our proof in both cases is essentially the same and it is elementary and rather self contained. We also give a short (uniform) proof that if a non-zero commutator in $R$ is not a zero divisor (with mild additional hypothesis when $R$ is alternative, but not associative (e.g.~that $(R,+)$ contains no $3$-torsion), then $R$ contains no divisors of zero.

Twin-width of planar graphs; a short proof

The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from the final resolution; there is a lower bound of 7 coming from a construction by Král’ and Lamaison (2022), and an upper bound of 8 by Hliněný and Jedelský (2022). The upper bound (currently best) of 8, however, is rather complicated and involved. In the paper we give a short and simple self-contained proof that the twin-width of planar graphs is at most 11. We believe that this short proof can also shed more light on the topic of upper bound(s) on the twin-width of planar and beyond-planar graphs in general.

A Short Proof of Bose–Einstein Condensation in the Gross–Pitaevskii Regime and Beyond

AbstractWe consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $$ N^{\kappa -1}$$ N κ - 1 , for some $$\kappa &gt;0$$ κ &gt; 0 . For the range $$ \kappa \in [0, \frac{1}{43})$$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $$ \kappa \in [0, \frac{1}{20})$$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

Short proofs of three results about intersecting systems

Short proofs of three results about intersecting systems

A short elementary proof of Beben and Theriault's theorem on homotopy fibers

Beben and Theriault proved a theorem on the homotopy fiber of an extension of a map with respect to a cone attachment, which has produced several applications. We give a short and elementary proof of this theorem.

Idempotents in nilpotent quotients and triangulated categories

We will prove that if I is a nilpotent ideal of an additive category A, then an idempotent e of the category A splits iff its image splits in A/I. Based on this fact, we give a short proof of a crucial proposition of Le and Chen.

Quantum Merkle Trees

Committing to information is a central task in cryptography, where a party (typically called a prover) stores a piece of information (e.g., a bit string) with the promise of not changing it. This information can be accessed by another party (typically called the verifier), who can later learn the information and verify that it was not meddled with. Merkle trees \cite{Merkle87} are a well-known construction for doing so in a succinct manner, in which the verifier can learn any part of the information by receiving a short proof from the honest prover. Despite its significance in classical cryptography, there was no quantum analog of the Merkle tree. A direct generalization using the Quantum Random Oracle Model (QROM) \cite{BonehDFLSZ11} does not seem to be secure. In this work, we propose the quantum Merkle tree. It is based on what we call the Quantum Haar Random Oracle Model (QHROM). In QHROM, both the prover and the verifier have access to a Haar random quantum oracle G and its inverse.Using the quantum Merkle tree, we propose a succinct quantum argument for the Gap-k-Local-Hamiltonian problem. Assuming the Quantum PCP conjecture is true, this succinct argument extends to all of QMA. This work raises a number of interesting open research problems.