Theorem States Research Articles

Generalizations of the Eneström–Kakeya Theorem Involving Weakened Hypotheses

The well-known Eneström–Kakeya Theorem states that, for P(z)=∑ℓ=0naℓzℓ, a polynomial of degree n with real coefficients satisfying 0≤a0≤a1≤⋯≤an, then all the zeros of P lie in |z|≤1 in the complex plane. Motivated by recent results concerning an Eneström–Kakeya “type” condition on real coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients and concerning the moduli of the coefficients. In this way, our results generalize the other recent results.

Further extensions of Haglund-Remmel-Wilson identity

MacMahon's equidistribution theorem states that the permutation statistics inversion number and major index are equidistributed. In 2015, Remmel and Wilson proved a conjectured identity of Haglund which is an extension of MacMahon's equidistribution theorem to ordered set partitions. Recently, Liu extended this identity to k-Stirling permutations and posed a conjecture concerning an ascent analogue of his extension. In this paper, we shall present a combinatorial proof of this conjecture. Furthermore, we derive an analogous result for another maj-like statistic introduced by Liu.

Multiply minimal points for the product of iterates

The multiple Birkhoff recurrence theorem states that for any $d\in\mathbb N$, every system $(X,T)$ has a multiply recurrent point $x$, i.e. $(x,x,\ldots, x)$ is recurrent under $\tau_d=:T\times T^2\times \ldots \times T^d$. It is natural to ask if there always is a multiply minimal point, i.e. a point $x$ such that $(x,x,\ldots,x)$ is $\tau_d$-minimal. A negative answer is presented in this paper via studying the horocycle flows. However, it is shown that for any minimal system $(X,T)$ and any non-empty open set $U$, there is $x\in U$ such that $\{n\in{\mathbb Z}: T^nx\in U, \ldots, T^{dn}x\in U\}$ is piecewise syndetic; and that for a PI minimal system, any $M$-subsystem of $(X^d, \tau_d)$ is minimal.

The Green-Tao Theorem and the Infinitude of Primes in Domains

We first prove an elementary analogue of the Green-Tao Theorem. The celebrated Green-Tao Theorem states that there are arbitrarily long arithmetic progressions in the set of prime numbers. In fact, we show the Green-Tao Theorem for polynomial rings over integral domains with several variables. Using the Generalized Polynomial van der Waerden Theorem, we also prove that in an infinite unique factorization domain, if the cardinality of the set of units is strictly less than that of the domain, then there are infinitely many prime elements. Moreover, we deduce the infinitude of prime numbers in the positive integers using polynomial progressions of length three. In addition, using unit equations, we provide two more proofs of the infinitude of prime numbers. Finally, we give a new proof of the divergence of the sum of reciprocals of all prime numbers.

The arithmetical combinatorics of k,l-regular partitions

For all positive integers k,l,n, the Little Glaisher theorem states that the number of partitions of n into parts not divisible by k and occurring less than l times is equal to the number of partitions of n into parts not divisible by l and occurring less than k times. While this refinement of Glaisher theorem is easy to establish by computation of the generating function, there is still no one-to-one canonical correspondence explaining it. Our paper brings an answer to this open problem through an arithmetical approach. Furthermore, in the case l=2, we discuss the possibility of constructing a Schur-type companion of the Little Glaisher theorem via the weighted words.

A Napoleon-like Theorem for Quadrilaterals

Napoleon’s theorem states that if we place equilateral triangles externally on the sides of an arbitrary triangle, the centers of these equilateral triangles will form an equilateral triangle. In this note, we introduce a Napoleon-like, or simply Napoleonic result for quadrilaterals.

On the Gleason-Kahane-Żelazko Theorem for Associative Algebras

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that Lambda (textbf{1})=1, is multiplicative, that is, Lambda (ab)=Lambda (a)Lambda (b) for all a,bin A. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra A over a field of at least 3 elements, and having an ideal of codimension 1, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If A is a commutative algebra, then the localization A_{P} is a GKŻ-algebra for every prime ideal P of A. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra Asubseteq {mathbb {F}}^{X} over a subfield {mathbb {F}} of {mathbb {C}}, contains all the bounded functions in {mathbb {F}}^{X}, then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in (0,infty ) satisfy the GKŻ property, while the algebra of compactly supported distributions does not.

Two Theorems and Important Insight on How the Preferred Mechanism of Free Radical Scavenging Cannot Be Settled. Comment on Pandithavidana, D.R.; Jayawardana, S.B. Comparative Study of Antioxidant Potential of Selected Dietary Vitamins; Computational Insights. Molecules 2019, 24, 1646.

Totally ignoring that the five enthalpies of reaction-bond dissociation enthalpy (BDE), adiabatic ionization potential (IP), proton dissociation enthalpy (PDE), proton affinity (PA), and electron transfer enthalpy (ETE)-characterizing the three free radical scavenging mechanisms-direct hydrogen atom transfer (HAT), sequential electron transfer proton transfer (SET-PT), and stepwise proton loss electron transfer (SPLET)-are not independent of each other, a recent publication on the antioxidant activity of dietary vitamins compared various vitamins and "found" different quantities, which should be strictly equal by virtue of energy conservation. Aiming to clarify this point, as well as to avoid such mistakes in future studies and to unravel errors in the previous literature, in the present paper we formulate two theorems that any sound results on antioxidation should obey. The first theorem states that the sums of the enthalpies characterizing the individual steps of SET-PT and SPLET are equal: IP+PDE = PA+ETE (=H2). This is a mathematical identity emerging from the fact that both the reactants and the final products of SET-PT and SPLET are chemically identical. The second theorem, which is also a mathematical identity, states that H2 - BDE = IPH > 0, where IPH is the ionization potential of the H-atom in the medium (e.g., gas or solvent) considered. Due to their general character, these theorems may/should serve as necessary sanity tests for any results on antioxidant activity, whatever the method employed in their derivation. From a more general perspective, they should represent a serious word of caution regarding attempts to assign the preferred free radical scavenging pathway based merely on thermochemical descriptors.

The Case of Equality for the Spacetime Positive Mass Theorem

The rigidity of the spacetime positive mass theorem states that an initial data set (M, g, k) satisfying the dominant energy condition with vanishing mass can be isometrically embedded into Minkowski space. This has been established by Beig–Chruściel and Huang–Lee under additional decay assumptions for the energy and momentum densities \(\mu \) and J. In this note, we give a new and elementary proof in dimension 3 which removes these additional decay assumptions. Our argument uses spacetime harmonic functions and Liouville’s theorem. We also provide an alternative proof based on the Killing development of (M, g, k).

Breaking through the Mermin-Wagner limit in 2D van der Waals magnets

The Mermin-Wagner theorem states that long-range magnetic order does not exist in one- (1D) or two-dimensional (2D) isotropic magnets with short-ranged interactions. Here we show that in finite-size 2D van der Waals magnets typically found in lab setups (within millimetres), short-range interactions can be large enough to allow the stabilisation of magnetic order at finite temperatures without any magnetic anisotropy. We demonstrate that magnetic ordering can be created in 2D flakes independent of the lattice symmetry due to the intrinsic nature of the spin exchange interactions and finite-size effects. Surprisingly we find that the crossover temperature, where the intrinsic magnetisation changes from superparamagnetic to a completely disordered paramagnetic regime, is weakly dependent on the system length, requiring giant sizes (e.g., of the order of the observable universe ~ 1026 m) to observe the vanishing of the magnetic order as expected from the Mermin-Wagner theorem. Our findings indicate exchange interactions as the main ingredient for 2D magnetism.

Revisiting variable-value population principles

AbstractWe examine a general class of variable-value population principles. Our particular focus is on the extent to which such principles can avoid the repugnant and sadistic conclusions. We show that if a mild limit property is imposed, avoidance of the repugnant conclusion implies the sadistic conclusion. This result generalizes earlier observations by showing that they apply to a substantially larger class of principles. Our second theorem states that, under the limit property, the axiom of mere addition also conflicts with avoidance of the repugnant conclusion. This result is a consequence of a similar observation that appears in the earlier literature.

Toward an asymptotic description of Prandtl–Batchelor flows with corners

The Prandtl–Batchelor theorem states that the vorticity in a steady laminar high Reynolds (Re) number flow containing closed streamlines should be constant; however, apart from the simple case of circular streamlines, very little is known about how to determine this constant (ω0). This paper revisits earlier work for flow driven by a surrounding smooth moving boundary, with a view to extending it to the case where the enclosing boundary has corners; for this purpose, a benchmark example from the literature for flow inside a semi-circle is considered. However, the subsequent asymptotic analysis for Re≫1 and numerical experimentation lead to an inconsistency: the asymptotic approach predicts boundary-layer separation, whereas a linearized asymptotic theory and computations of the full Navier–Stokes equations for Re≫1 do not. Nevertheless, by considering a slightly modified problem instead, which does not suffer from this inconsistency, it is found that, when extrapolating the results of such high-Re computations to infinite Re, the agreement for ω0 is around 5%, which is roughly in line with previous comparisons of this type. Possible future improvements of the asymptotic method are also discussed.

On Subsampling Procedures for Support Vector Machines

Herein, theoretical results are presented to provide insights into the effectiveness of subsampling methods in reducing the amount of instances required in the training stage when applying support vector machines (SVMs) for classification in big data scenarios. Our main theorem states that under some conditions, there exists, with high probability, a feasible solution to the SVM problem for a randomly chosen training subsample, with the corresponding classifier as close as desired (in terms of classification error) to the classifier obtained from training with the complete dataset. The main theorem also reflects the curse of dimensionalityin that the assumptions made for the results are much more restrictive in large dimensions; thus, subsampling methods will perform better in lower dimensions. Additionally, we propose an importance sampling and bagging subsampling method that expands the nearest-neighbors ideas presented in previous work. Using different benchmark examples, the method proposed herein presents a faster solution to the SVM problem (without significant loss in accuracy) compared with the available state-of-the-art techniques.

Variational Methods for Atoms and the Virial Theorem

In the case of the one-electron Dirac equation with a point nucleus, the virial theorem (VT) states that the ratio of the kinetic energy to potential energy is exactly −1, a ratio that can be an independent test of the accuracy of a computed solution. This paper studies the virial theorem for subshells of equivalent electrons and their interactions in many-electron atoms. This shows that the linear scaling of the dilation is achieved through the balancing of the contributions to the potential of an electron from inner and outer regions that some Slater integrals impose conditions on a single subshell, but others impose conditions between subshells. The latter slows the rate of convergence of the self-consistent field process in which radial functions are updated one at a time. Several cases are considered. Results are also extended to the nonrelativistic case.

Infinite-time incompleteness of noncompact Yamabe flow

We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.

A generalization of the Hopf degree theorem

The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth n n -manifold M M to the n n -sphere are classified by their degree. Such a map is equivalent to a section of the trivial n n -sphere bundle over M M . A generalization of the Hopf theorem is obtained for sections of nontrivial oriented n n -sphere bundles over M M .

A set-theoretic proof of the representation of MV-algebras by sheaves

ABSTRACT In this paper, we provide a set-theoretic proof of the general representation theorem for MV-algebras, which was developed by Dubuc and Poveda in 2010. The theorem states that every MV-algebra is isomorphic to the MV-algebra of all global sections of its prime spectrum. We avoid using topos theory and instead rely on basic concepts from MV-algebras, topology and set theory.

Examining the limits of the Condorcet Jury Theorem: Tradeoffs in hierarchical information aggregation systems

Condorcet’s Jury Theorem states that the correct outcome is reached in direct majority voting systems with sufficiently large electorates as long as each voter’s independent probability of voting for that outcome is greater than 1/2. Previous research has found that switching to a hierarchical system always leads to an inferior result. Yet, in many situations direct voting is infeasible (e.g., due to high implementation or infrastructure costs), and hierarchical voting may provide a reasonable alternative. This paper examines differences in accuracy rates of hierarchical and direct voting systems for varying group sizes, abstention rates, and voter competences. We derive three main results. First, we prove that indirect two-tier systems differ most from their direct counterparts when group size and number are equal (i.e., when each equals [Formula: see text], where Nd is the total number of voters in the direct system). In multitier systems, we prove that this difference is maximized when group size equals [Formula: see text], where n is the number of hierarchical levels. Second, we show that while direct majority rule always outperforms indirect voting for homogeneous electorates, hierarchical voting gains in accuracy when either the number of groups or the number of individuals within each group increases. Third, we prove that when voter abstention and competency are correlated within groups, hierarchical systems can outperform direct voting. The results have implications beyond voting, including information processing in the brain, collective cognition in animal groups, and information aggregation in machine learning.

Relationship between the wave function of a magnet and its static structure factor

We state and prove two theorems about the ground state of magnetic systems described by very general Heisenberg-type models and discuss their implications for magnetic neutron scattering. The first theorem states that two models cannot have the same correlator without sharing the corresponding ground states. According to the second theorem, an $N$-qubit wave function cannot reproduce the correlators of a given system unless it represents a true ground state of that system. We discuss the implications for neutron scattering inverse problems. We argue that the first theorem provides a framework to understand neutron-based Hamiltonian learning. Furthermore, we propose a variational approach to quantum magnets based on the second theorem where a representation of the wave function (held, for instance, in a neural network or in the qubit register of a quantum processor) is optimised to fit experimental neutron-scattering data directly, without the involvement of a model Hamiltonian.

Packing Directed Cycles Quarter- and Half-Integrally

The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size $$\cal{O}(k\log k)$$ . The analogous result for directed graphs has been proven by Reed, Robertson, Seymour, and Thomas, but their proof yields a nonelementary dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing. We show that we can obtain a polynomial bound if we relax the disjointness condition. More precisely, we show that if in a directed graph G there is no family of k cycles such that every vertex of G is in at most two (resp. four) of the cycles, then there exists a feedback vertex set in G of size $$\cal{O}(k^{6})$$ (resp. $$\cal{O}(k^{4})$$ ). We show also variants of the above statements for butterfly minor models of any strongly connected digraph that is a minor of a directed cylindrical grid and for quarter-integral packings of subgraphs of high directed treewidth.