Classical Theorem Research Articles

Spanning trees for many different numbers of leaves

Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.

Harmonic curves and the beauty of projective geometry

AbstractThe purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of some classic theorems and a slight reformulation of its axiomatics.

Korn and Poincaré-Korn inequalities: A different perspective

We present a concise point of view on the first and the second Korn’s inequality for general exponent p p and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for p = 2 p = 2 , uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent 1 > p > ∞ 1>p>\infty remains the same, the only change being invoking now the q q -Riesz representation theorem (with q q the harmonic conjugate of p p ). We also complement the analysis with elementary derivations of Poincaré-Korn inequalities in bounded and unbounded domains, which are essential tools in showing the coercivity of variational problems of elasticity but also propedeutic to the proof of the first Korn inequality.

Infinite combinatorics revisited in the absence of Axiom of choice

AbstractWe investigate whether classical combinatorial theorems are provable in ZF. Some statements are not provable in ZF, but they are equivalent within ZF. For example, the following statements (i)–(iii) are equivalent: $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1 , $${\omega }_1\rightarrow ({\omega }_1,{\omega }+1)^2$$ ω 1 → ( ω 1 , ω + 1 ) 2 , any family $$\mathcal {A}\subset [{On}]^{<{\omega }}$$ A ⊂ [ On ] < ω of size $${\omega }_1$$ ω 1 contains a $$\Delta $$ Δ -system of size $${\omega }_1$$ ω 1 . Some classical results cannot be proven in ZF alone; however, we can establish weaker versions of these statements within the framework of ZF, such as $${{\omega }_2}\rightarrow ({\omega }_1,{\omega }+1)$$ ω 2 → ( ω 1 , ω + 1 ) , any family $$\mathcal {A}\subset [{On}]^{<{\omega }}$$ A ⊂ [ On ] < ω of size $${\omega }_2$$ ω 2 contains a $$\Delta $$ Δ -system of size $${\omega }_1$$ ω 1 . Some statements can be proven in ZF using purely combinatorial arguments, such as: given a set mapping $$F:{\omega }_1\rightarrow {[{\omega }_1]}^{<{\omega }}$$ F : ω 1 → [ ω 1 ] < ω , the set $${\omega }_1$$ ω 1 has a partition into $${\omega }$$ ω -many F-free sets. Other statements can be proven in ZF by employing certain methods of absoluteness, for example: given a set mapping $$F:{\omega }_1\rightarrow {[{\omega }_1]}^{<{\omega }}$$ F : ω 1 → [ ω 1 ] < ω , there is an F-free set of size $${\omega }_1$$ ω 1 , for each $$n\in {\omega }$$ n ∈ ω , every family $$\mathcal {A}\subset {[{\omega }_1]}^{{\omega }}$$ A ⊂ [ ω 1 ] ω with $$|A\cap B|\le n$$ | A ∩ B | ≤ n for $$\{A,B\}\in {[\mathcal {A}]}^{2}$$ { A , B } ∈ [ A ] 2 has property B. In contrast to statement (5), we show that the following ZFC theorem of Komjáth is not provable from ZF + $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1 : (6$$ ^*$$ ∗ ) every family $$\mathcal {A}\subset {[{\omega }_1]}^{{\omega }}$$ A ⊂ [ ω 1 ] ω with $$|A\cap B|\le 1$$ | A ∩ B | ≤ 1 for $$\{A,B\}\in {[\mathcal {A}]}^{2}$$ { A , B } ∈ [ A ] 2 is essentially disjoint. A function f is a uniform denumeration on$${\omega }_1$$ ω 1 iff $${\text {dom}}(f)={\omega }_1$$ dom ( f ) = ω 1 , and for every $$1\le {\alpha }<{\omega }_1$$ 1 ≤ α < ω 1 , $$f({\alpha })$$ f ( α ) is a function from $${\omega }$$ ω onto $${\alpha }$$ α . It is easy to see that the existence of a uniform denumeration of $${\omega }_1$$ ω 1 implies $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1 . We prove that the failure of the reverse implication is equiconsistent with the existence of an inaccessible cardinal.

Spectrum of inner product type integral transformers on Hilbert spaces

In this paper, we have determined the spectrum of inner product type integral transformers on the Hilbert spaces. This has been achieved using the abelian properties of sets and the concept of commutators as in the von Neumann algebras. We have also shown that the spectrum of the inner product type integral transformers is contained in the closure of its numerical range using the classical spectral inclusion theorem.

Frobenius Local Rings of Order p4m

Suppose R is a finite commutative local ring, then it is known that R has four positive integers p,n,m,k called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field Fpm and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order p4m. Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants p,n,m,k. Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings.

The Lomonosov type theorems and the invariant subspace problem for non-archimedean Banach spaces

In this paper, we study the existence of invariant (and even hyperinvariant) subspaces of bounded operators on a non-archimedean Banach space E=(E,‖.‖) over a valued field K equipped with a non-trivial non-archimedean valuation |.|. Specifically, we consider compact operators and operators that commute with a compact operator. First we show that if E has a base, then any compact operator T such that limn⁡‖Tn‖1n>0 has a finite-dimensional hyperinvariant subspace. Next we show that if K is locally compact, then every compact operator T on E has a hyperinvariant subspace. Afterward, assuming that K is spherically complete or E is of countable type, we provide a necessary condition for a bounded operator on E to have a hyperinvariant subspace. We demonstrate that the classical Lomonosov Invariant Subspace theorem does not hold in the case where K is non-spherically complete. Finally, we prove Lomonosov type theorem for spectral quasinilpotent operators, when K is locally compact.

Instance-optimal information-based voting

The classical Condorcet Jury Theorem considers a voting scenario in which there exists a candidate whose election would be ideal for each voter; each voter, though, has only a limited understanding of the world and is thus unable to determine exactly who this candidate is. The main question in this scenario is whether the voters, acting individually, can cast their ballots so that the unknown optimal candidate wins the election, and the welfare of the group of voters is maximized.In this setting, each candidate is represented by a known probability distribution over signals about the world that the voters can perceive, that is, over bits. One of these candidates is chosen (secretively, by an adversary) to be the ideal candidate. Afterwards, each voter samples this unknown candidate's distribution once and casts a ballot with the hope that the unknown ideal candidate wins the election.In this paper, we consider the famous Condorcet voting system, as well as some of its variants. First, we give a positive answer to an open question of Chierichetti and Kleinberg [8], and show that, with Condorcet voting, there exists a uniform voting strategy that makes the group of voters succeed with probability 1−δ provided that Θ(ϵtv−2⋅ln⁡δ−1) voters take part in the election — here, ϵtv is the minimum total variation distance between the distributions of two candidates.We also give a uniform voting strategy for the Copeland voting system (a variant of Condorcet) that makes the group succeed with probability 1−δ with Θ(ϵH−2⋅ln⁡δ−1) voters, where ϵH is the minimum Hellinger distance between the distributions. Our uniform Copeland strategy, then, is an instance-optimal hypothesis testing algorithm: constants aside, the strategy is as efficient as the optimal omniscient algorithm which determines the unknown candidate after having directly observed each of the signals perceived by the voters. Then, we “derandomize” our uniform Copeland strategy, and obtain a Condorcet strategy that achieves instance-optimality at the cost of losing uniformity; finally, we prove that this loss of uniformity is necessary: no uniform Condorcet strategy can achieve instance-optimality, in general.Thus, the right voting strategies let these classical combinatorial voting systems attain the same efficiency of centralized, optimal, hypothesis testers.

Modeling the distribution of jet fuel price returns based on fat-tail stable Paretian distribution.

Jet fuel plays a crucial role as an essential energy source in aerospace and aviation operations. The recent increase in fuel prices has presented airlines with the new challenge of managing jet fuel costs to ensure consistent cash flow and minimize operational uncertainties. The conventional risk prediction models used by airlines often assume that risks are normally distributed according to the classical Central Limit Theorem, which can lead to under-hedging. This paper proposes an innovative approach using the stable Paretian model to analyze the price return of jet fuel in large samples. It comprehensively compares the fitting effect of the stable Paretian distribution with that of the normal distribution based on specific criteria and non-parametric significance tests. Furthermore, it investigates the accuracy of risk measures such as Value at Risk (VaR) and Conditional Value at Risk (CVaR) predicted by both models. In addition to comparing differences in VaR between predicted values and actual values, this paper provides a more comprehensive comparison of risk measures under rolling window forecast situation. Results suggest that despite indistinguishable results in VaR backtest, the stable Paretian distribution has a overall better fitting effect as well as a less biased predicted CVaR based on the AIC of -14099.46, BIC of -14110.98, p = 0.58 in Kolmogorov-Smirnov test and p = 0.46(0.92) in the 0.01(0.05) significance level of Expected Shortfall Regression Test. This might be explained by its ability to capture asset return dynamics while maintaining shape stability with few parameters. This research can provide valuable insights for guiding airlines' risk management decisions. its ability to capture asset return dynamics while maintaining shape stability with few parameters. This research can provide valuable insights for guiding airlines' risk management decisions.

Mikhlin-Type Hp Multiplier Theorem on the Heisenberg Group

The classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type Hp multiplier theorem on the Heisenberg group. If an operator-valued function M(λ) satisfies certain conditions, the right-multiplier operator TM is bounded on the Hardy space Hp(Hn), which is defined in terms of maximal functions, and elements can be decomposed into atoms or molecules. The paper also discusses the relationship with other results and open problems.

Submanifolds with constant scalar curvature in space forms

Let [Formula: see text] be an [Formula: see text]-dimensional oriented compact submanifold with constant normalized scalar curvature [Formula: see text] in the space form [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the mean curvature and the Ricci curvature of [Formula: see text] respectively. By applying Cheng-Yau’s self-adjoint operator, we first prove that if [Formula: see text] is a hypersurface in a unit sphere, and [Formula: see text], then [Formula: see text] is totally umbilical. Furthermore, we investigate the submanifolds in [Formula: see text] with flat normal bundle satisfying [Formula: see text], and obtain a complete classification theorem.

Relative monadicity

We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense ▪-functor j:A→E, a ▪-functor r:D→E is j-monadic if and only if r admits a left j-relative adjoint and creates j-absolute colimits. This provides a refinement of the classical monadicity theorem – characterising those categories whose objects are given by those of E equipped with algebraic structure – in which the arities of the algebraic operations are valued in A. In particular, when j=1, we recover a formal monadicity theorem. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the (j-relative) monadicity of the composite of a ▪-functor with a (j-relatively) monadic ▪-functor.

Diffraction tomography for incident Herglotz waves

Abstract Diffraction tomography is an inverse scattering technique used to reconstruct the spatial distribution of the material properties of a weakly scattering object. The object is exposed to radiation, typically light or ultrasound, and the scattered waves induced from different incident field angles are recorded. In conventional diffraction tomography, the incident wave is assumed to be a monochromatic plane wave, an unrealistic simplification in practical imaging scenarios. In this article, we extend conventional diffraction tomography by introducing the concept of customized illumination scenarios, with a pronounced emphasis on imaging with focused beams. More specifically, we consider incident Herglotz waves and extend the classical Fourier diffraction theorem to this setting. This yields a new two-step reconstruction process which we comprehensively evaluate through numerical experiments.

Nonlinear twofold saddle point-based mixed finite element methods for a regularized μ(I)-rheology model of granular materials

We propose and analyze new mixed finite element methods for a regularized μ(I)-rheology model of granular flows with an equivalent viscosity depending nonlinearly on the pressure and the Euclidean norm of the symmetric part of the velocity gradient. To this end, and besides the velocity, the pressure and the aforementioned strain rate, we introduce a modified stress tensor that includes the convective term, and the skew-symmetric vorticity, as auxiliary tensor unknowns, thus yielding a mixed variational formulation within a Banach spaces framework. Then, the pressure is obtained through an iterative postprocess suggested by the incompressibility condition of the fluid, which allows us to express this unknown in terms of the aforementioned stress and the velocity. A fixed-point strategy combined with a solvability result for a class of nonlinear twofold saddle point operator equations in Banach spaces, is employed to show, along with the classical Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. In particular, PEERS and AFW elements of order ℓ⩾0 for the stress, the velocity, and the skew-symmetric vorticity, and piecewise polynomials of degree ⩽ℓ+n (resp. ⩽ℓ+1) for the strain rate with PEERS (resp. with AFW), yield stable Galerkin schemes. Optimal a priori error estimates are derived and associated rates of convergence are established. Finally, numerical results confirming the latter and illustrating the good performance of the methods, are reported.

On countably saturated and pre-dyadic spaces

In this paper, we introduce a new class of spaces which extends the class of dyadic compacta. This permits to extend a classical theorem of Engelking and Pełczyński. Technically, our arguments are based on the concept of a countably saturated space. This is a space in which the closure of every countable subset is a metrizable compactum. The existence of a dense countably saturated subspace in compact Hausdorff spaces is shown below to play a crucial role.

On algebraic dependencies between Poincaré functions

Abstract Let A be a rational function of one complex variable of degree at least two, and $z_0$ its repelling fixed point with the multiplier $\unicode{x3bb} .$ A Poincaré function associated with $z_0$ is a function meromorphic on ${\mathbb C}$ such that , and In this paper, we study the following problem: given Poincaré functions and , find out if there is an algebraic relation between them and, if such a relation exists, describe the corresponding algebraic curve $f(x,y)=0.$ We provide a solution, which can be viewed as a refinement of the classical theorem of Ritt about commuting rational functions. We also reprove and extend previous results concerning algebraic dependencies between Böttcher functions.

Applications of Extended Kummer’s Summation Theorem

In the theory of hypergeometric series and generalized hypergeometric series, classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series F12; those of Watson, Dixon, Whipple, and Saalschutz for the series F23; and others, play a key role. Applications of these classical summation theorems are well known. Berndt pointed out that a large number of interesting summations (including Ramanujan’s summations and the Gregory–Leibniz pi summation) can be obtained very quickly by employing the above-mentioned classical summation theorems. Also, several interesting results involving products of generalized hypergeometric series have been obtained by Bailey by employing the above-mentioned classical summation theorems. Recently, the above-mentioned classical summation theorems have been generalized and extended. In our present investigations, our aim is to demonstrate the applications of the extended Kummer’s summation theorem in establishing (i) extensions of Gauss’s second summation theorem and Bailey’s summation theorem; (ii) extensions of several summations (including Ramanujan’s summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) an extension of classical Dixon’s summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Leibniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey.

K-free and k-full numbers with prime factors in an arithmetic progression

We apply a classical Wirsing's theorem and study k-free numbers, k-full numbers and perfect powers with prime factors in an arithmetic progression.

Double Satake Diagrams and Canonical Forms in Compact Symmetric Triads

In this paper, we first introduce the notion of double Satake diagrams for compact symmetric triads. In terms of this notion, we give an alternative proof for the classification theorem for compact symmetric triads, which was originally given by Toshihiko Matsuki. Secondly, we introduce the notion of canonical forms for compact symmetric triads, and prove the existence of canonical forms for compact simple symmetric triads. We also give some properties for canonical forms.

Cartesian closed varieties I: the classification theorem

In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all unary theories—whose models are sets equipped with an action by a monoid M—and all hyperaffine theories—whose models are sets with an action by a Boolean algebra B. We improve on Johnstone’s result by showing that an equational theory is cartesian closed just when its operations have a unique hyperaffine–unary decomposition. It follows that any non-degenerate cartesian closed variety is a variety of sets equipped with compatible actions by a monoid M and a Boolean algebra B; this is the classification theorem of the title.