Classical Theorem Research Articles

Multilateral Supervaluationism and Classicality

Abstract Incurvati and Schlöder (Journal of Philosophical Logic, 51(6), 1549–1582, 2022) have recently proposed to define supervaluationist logic in a multilateral framework, and claimed that this defuses well-known objections concerning supervaluationism’s apparent departures from classical logic. However, we note that the unconventional multilateral syntax prevents a straightforward comparison of inference rules of different levels, across multi- and unilateral languages. This leaves it unclear how the supervaluationist multilateral logics actually relate to classical logic, and raises questions about Incurvati and Schlöder’s response to the objections. We overcome the obstacle, by developing a general method for comparisons of strength between multi- and unilateral logics. We apply it to establish precisely on which inferential levels the supervaluationist multilateral logics defined by Incurvati and Schlöder are classical. Furthermore, we prove general limits on how classical a multilateral logic can be while remaining supervaluationistically acceptable. Multilateral supervaluationism leads to sentential logic being classical on the levels of theorems and regular inferences, but necessarily strictly weaker on meta- and higher-levels, while in a first-order language with identity, even some classical theorems and inferences must be forfeited. Moreover, the results allow us to fill in the gaps of Incurvati and Schlöder’s strategy for defusing the relevant objections.

Maintaining CMSO 2 properties on dynamic structures with bounded feedback vertex number

Let φ be a sentence of \(\mathsf {CMSO}_2 \) (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph G that is updated by edge insertions and edge deletions, maintains whether φ is satisfied in G . The data structure is required to correctly report the outcome only when the feedback vertex number of G does not exceed a fixed constant k , otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time \(\mathcal {O}_{\varphi,k}{(\log n)} \) . If we additionally assume that the feedback vertex number of G never exceeds k , this update time guarantee is worst-case. By combining this result with a classic theorem of Erdős and Pósa, we give a fully dynamic data structure that maintains whether a graph contains a packing of k vertex-disjoint cycles with amortized update time \(\mathcal {O}_{k}{(\log n)} \) . Our data structure also works in a larger generality of relational structures over binary signatures.

$L_{\kappa\omega}$-equivalence of abelian groups with partial decomposition bases

We consider the class of abelian groups possessing partial decomposition bases in the language L_{\kappa \omega} for uncountable cardinals \kappa . Jacoby, Leistner, Loth and Strüngmann developed a numerical invariant deduced from the classical global Warfield invariant and proved that if two groups have identical modified Ulm invariants and Warfield invariants up to \omega\delta for some ordinal \delta , then they are equivalent in L_{\infty\omega}^{\delta} . Subsequently, Jacoby and Loth showed that the converse is true for appropriate \delta . In this paper we prove that the modified Warfield invariant up to \kappa is expressible in L_{\kappa\omega} , thus a complete classification theorem in L_{\kappa \omega} is obtained. This generalizes a result of Barwise and Eklof.

Titchmarsh theorems on Damek–Ricci spaces via moduli of continuity of higher order

A classical theorem of Titchmarsh relates the [Formula: see text]-Lipschitz functions and decay of the Fourier transform of the functions. In this note, we prove the Titchmarsh theorem for Damek–Ricci space (also known as harmonic [Formula: see text] groups) via moduli of continuity of higher orders. We also prove an analogue of another Titchmarsh theorem which provides integrability properties of the Fourier transform for functions in the Hölder Lipschitz spaces.

On-shell approach to black hole mergers

We develop an on-shell approach to study black hole mergers. Since, asymptotically, the initial and final states can be described by point-like spinning particles, we propose a massive three-point amplitude for the merger of two Schwarzschild black holes into a Kerr black hole. This three-point amplitude and the spectral function of the final state are fully determined by kinematics and the model-independent input about the black hole merger which is described by a complete absorption process. Using the Kosower-Maybee-O’Connell (KMOC) formalism, we then reproduce the classical conservation laws for momentum and angular momentum after the merger. As an application, we use the proposed three-point to compute the graviton emission amplitude, from which we extract the merger waveform to all orders in spin but leading in gravitational coupling. Up to sub-subleading order in spin, this matches the classical soft graviton theorem. We conclude with a comparison to black hole perturbation theory, which gives complementary amplitudes which are non-perturbative in the gravitational coupling but to leading order in the extreme mass ratio limit. This also highlights how boundary conditions on a Schwarzschild background can be used to rederive the proposed on-shell amplitudes for merger processes.

Compactifications of pseudofinite and pseudoamenable groups

We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing’s work, the Jordan–Schur theorem, and a (relatively) more recent result of Kazhdan (1982) on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudoamenable groups to compact Lie groups. Together with the stabilizer theorems of Hrushovski (2012) and Montenegro et al. (2020), we obtain a uniform (but non-quantitative) analogue of Bogolyubov’s lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.

Foundations of Matroids, Part 1: Matroids without Large Uniform Minors

The foundation of a matroid is a canonical algebraic invariant which classifies, in a certain precise sense, all representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and hyperfields which are special cases of both tracts (as defined by the first author and Bowler) and ordered blue fields (as defined by the second author). Using deep results due to Tutte, Dress–Wenzel, and Gelfand–Rybnikov–Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain “cross-ratios” generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank configurations arising in projective geometry. Although the presentation of the foundation is valid for all matroids, it is simplest to apply in the case of matroids without large uniform minors. i.e., matroids having no minor corresponding to five points on a line or its dual configuration. For such matroids, we obtain a complete classification of all possible foundations. We then give a number of applications of this classification theorem, for example: We prove the following strengthening of a 1997 theorem of Lee and Scobee: every orientation of a matroid without large uniform minors comes from a dyadic representation, which is unique up to rescaling. For a matroid M M without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila–Rincón–Williams: if M M is positively oriented then M M is representable over every field with at least 3 elements. Two matroids are said to belong to the same representation class if they are representable over precisely the same pastures. We prove that there are precisely 12 possibilities for the representation class of a matroid without large uniform minors, exactly three of which are not representable over any field.

Irreducible representations of the crystallization of the quantized function algebras $C(\operatorname{SU}_{q}(n+1))$

The crystallization of the C^{*} -algebras C(\operatorname{SU}_{q}(n+1)) was introduced by Giri and Pal in [Proc. Indian Acad. Sci. Math. Sci. 134 (2024), article no. 30] as a C^{*} -algebra C(\operatorname{SU}_{0}(n+1)) given by a finite set of generators and relations. Here we study the representations of the C^{*} -algebra C(\operatorname{SU}_{0}(n+1)) and prove a factorization theorem for its irreducible representations. This leads to a complete classification of all irreducible representations of this C^{*} -algebra. As an important consequence, we prove that all the irreducible representations of C(\operatorname{SU}_{0}(n+1)) arise exactly as q\to 0+ limits of the irreducible representations of C(\operatorname{SU}_{q}(n+1)) . We also present a few other important corollaries of the classification theorem.

The Random Arnold Conjecture: A New Probabilistic Conley-Zehnder Theory for Symplectic Maps

Inspired by the classical Conley-Zehnder Theorem and the Arnold Conjecture in symplectic topology, we prove a number of probabilistic theorems about the existence and density of fixed points of symplectic strand diffeomorphisms in dimensions greater than 2. These are symplectic diffeomorphisms Φ=(Q,P):Rd×Rd→Rd×Rd on the variables (q, p) such that for every p∈Rd the induced map q↦Q(q,p) is a diffeomorphism of Rd. In particular we verify that quasiperiodic symplectic strand diffeomorphisms have infinitely many fixed points almost surely, provided certain natural conditions hold (inspired by the conditions in the Conley-Zehnder Theorem). The paper contains also a number of theorems which go well beyond the quasiperiodic case. Overall the paper falls within the area of stochastic dynamics but with a very strong symplectic geometric motivation, and as such its main inspiration can be traced back to Poincaré’s fundamental work on celestial mechanics and the restricted 3-body problem.

Localization and Flatness in Quantale Theory

The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a notion of “flat quantale morphism” as an abstraction of flat ring morphisms. For this, we start from a characterization of the flat ring morphism in terms of the ideal residuation theory. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat ring morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional coherent quantales are obtained, formulated in terms of Going-down, Going-up, and localization. We also prove two characterization theorems for the coherent quantales of dimension at most one. The results of the paper can be applied both in the theory of commutative rings and to other algebraic structures: F-rings, semirings, bounded distributive lattices, commutative monoids, etc.

The 2-Sheeted, 3-Sheeted, and Universal Coverings of Corresponding 2-Oriented Graph of Rank-2 Free Group

This paper reviews results from classical covering theory and applies the classification theorem to provide a comprehensive classification of the connected 2-sheeted and 3-sheeted covering spaces of the bouquet of 2 circles, up to isomorphism without basepoints. These coverings are represented as 4-valent graphs, labeled by the two generators of the rank-2 free group, which serves as the fundamental group of the bouquet of 2 circles. The result directly addresses a question posed by Hatcher, a problem often answered incompletely in previous attempts. This paper provides all possible cases and includes detailed illustrations to accompany the classifications. Additionally, the paper explores the X-digraphs associated with the corresponding index-2 and index-3 subgroups of the rank-2 free group. These subgroup graphs introduce powerful combinatorial tools to examine the subgroups of free groups. In the broader context of geometric group theory, the paper concludes by discussing the Cayley graph of the rank-2 free group as a universal covering space.

On Lurie’s theorem and applications

Lurie’s theorem states that there exists a sheaf of ring spectra on the site of formally étale Deligne–Mumford stacks over the moduli stack of p-divisible groups of height n, which agrees with the classical Landweber exact functor theorem (LEFT) on affines. In other words, this theorem is a global, higher categorical refinement of the LEFT. In recent work, Lurie has introduced many of the ingredients one needs to prove this theorem, and in this article, we gather these ingredients together and prove Lurie’s theorem. Applications of this theorem to Lubin–Tate theories, topological modular and automorphism forms, and Adams operations are also discussed.

A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set

By analyzing proofs of the classical Riesz-Kantorovich theorem, the Mazón-Segura de León theorem on abstract Uryson operators and the Pliev-Ramdane theorem on C-bounded orthogonally additive operators on Riesz spaces, we find the most general (to our point of view) algebraic structure, which we call a complementary space, for which the theorem can be generalized with a similar proof. By a complementary space we mean a PO-set $G$ with a least element $0$ such that every order interval $[0,e]$ of $G$ with $e \neq 0$ is a Boolean algebra with respect to the induced order. There are natural examples of complementary spaces: Boolean rings, Riesz spaces with the lateral order. Moreover, the disjoint union of complementary spaces is a complementary space. Our main result asserts that, the set of all additive (in certain sense) functions from a complementary space to a Dedekind complete Riesz space admits a natural Dedekind complete Riesz space structure, described by formulas which are close to the classical Riesz-Kantorovich ones. This theorem generalizes the above mentioned Mazón and Segura de León and Pliev-Ramdane theorems. In the final section, we construct a model of market with an arbitrary commodity set, connected to a complementary space.

Symmetries of the gravitational scattering in the absence of peeling

The symmetries of the gravitational scattering are intimately tied to the symmetries which preserve asymptotic flatness at null infinity. In Penrose’s definition of asymptotic flatness, a central role is played by the notion of asymptotic simplicity and the ensuing peeling behavior which dictates the decay rate of the Weyl tensor. However, there is now accumulating evidence that in a generic gravitational scattering the peeling property is broken, so that the spacetime is not asymptotically-flat in the usual sense. These obstructions to peeling can be traced back to the existence of universal radiative low frequency observables called “tails to the displacement memory”. As shown by Saha, Sahoo and Sen, these observables are uniquely fixed by the initial and final momenta of the scattering objects, and are independent of the details of the scattering. The universality of these tail modes is the statement of the classical logarithmic soft graviton theorem. Four-dimensional gravitation scattering therefore exhibits a rich infrared interplay between tail to the memory, loss of peeling, and universal logarithmic soft theorems.In this paper we study the solution space and the asymptotic symmetries for logarithmically-asymptotically-flat spacetimes. These are defined by a polyhomogeneous expansion of the Bondi metric which gives rise to a loss of peeling, and represent the classical arena which can accommodate a generic gravitational scattering containing tails to the memory. We show that while the codimension-two generalized BMS charges are sensitive to the loss of peeling at I+, the flux is insensitive to the fate of peeling. Due to the tail to the memory, the soft superrotation flux contains a logarithmic divergence whose coefficient is the quantity which is conserved in the scattering by virtue of the logarithmic soft theorem. In our analysis we also exhibit new logarithmic evolution equations and flux-balance laws, whose presence suggests the existence of an infinite tower of subleading logarithmic soft graviton theorems.

Generalizing the Classical Remainder Theorem: A Reflection-Based Methodological Strategy

The framework of this paper is the presentation of a case study in which university students are required to extend a particular problem of division of polynomials in one variable over the field of real numbers (as generalizing action) clearly influenced by prior strategies (as reflection generalization). Specifically, the objective of this paper is to present a methodology for generalizing the classical Remainder Theorem to the case in which the divisor is a product of binomials (x−a1)n1(x−a2)n2⋯(x−ak)nk, where a1,a2,⋯,ak∈R and n1,n2,⋯,nk∈N. A first approach to this issue is the Taylor expansion of the dividend P(x) at a point a, which clearly shows the quotient and the remainder of the division of P(x) by (x−a)k, where the degree of P(x), say n, must be greater than or equal to k. The methodology used in this paper is the proof by induction which allows to obtain recurrence relations different from those obtained by other scholars dealing with the generalization of the classical Remainder Theorem.

Smooth zero entropy flows satisfying the classical central limit theorem

Smooth zero entropy flows satisfying the classical central limit theorem

Quillen stratification in equivariant homotopy theory

We prove a version of Quillen’s stratification theorem in equivariant homotopy theory for a finite group G, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules. Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points. We then apply our methods to the case of Borel-equivariant Lubin–Tate E-theory En_, for any finite height n and any finite group G, where we obtain a sharper theorem in the form of cohomological stratification. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing ⊗-ideals of the category of equivariant modules over En_, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.

New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties

In this paper we derive new existence results for conjoined bases of singular linear Hamiltonian differential systems with given qualitative (Sturmian) properties. In particular, we examine the existence of conjoined bases with invertible upper block and with prescribed number of focal points at the endpoints of the considered unbounded interval. Such results are vital for the theory of Riccati differential equations and its applications in optimal control problems. As the main tools we use a new general characterization of conjoined bases belonging to a given equivalence class (genus) and the theory of comparative index of two Lagrangian planes. We also utilize extensively the methods of matrix analysis. The results are new even for identically normal linear Hamiltonian systems. The results are also new for linear Hamiltonian systems on a compact interval, where they provide additional equivalent conditions to the classical Reid roundabout theorem about disconjugacy.

Hawking-Type Singularity Theorems for Worldvolume Energy Inequalities

AbstractThe classical singularity theorems of R. Penrose and S. Hawking from the 1960s show that, given a pointwise energy condition (and some causality as well as initial assumptions), spacetimes cannot be geodesically complete. Despite their great success, the theorems leave room for physically relevant improvements, especially regarding the classical energy conditions as essentially any quantum field theory necessarily violates them. While singularity theorems with weakened energy conditions exist for worldline integral bounds, so-called worldvolume bounds are in some cases more applicable than the worldline ones, such as the case of some massive free fields. In this paper, we study integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities. Under the additional assumption of a—potentially very negative—global timelike Ricci curvature bound, a Hawking-type singularity theorem is proved. Finally, we apply the theorem to a cosmological scenario proving past geodesic incompleteness in cases where the worldline theorem was inconclusive.

Logarithmic soft theorems and soft spectra

Using universal predictions provided by classical soft theorems, we revisit the energy emission spectrum for gravitational scatterings of compact objects in the low-frequency expansion. We calculate this observable beyond the zero-frequency limit, retaining an exact dependence on the kinematics of the massive objects. This allows us to study independently the ultrarelativistic or massless limit, where we find agreement with the literature, and the small-deflection or post-Minkowskian (PM) limit, where we provide explicit results up to O(G5). These confirm that the high-velocity limit of a given PM order is smoothly connected to the corresponding massless result whenever the latter is analytic in the Newton constant G. We also provide explicit expressions for the waveforms to order ω−1, log ω, ω(log ω)2 in the soft limit, ω → 0, expanded up to sub-subleading PM order, as well as a conjecture for the logarithmic soft terms of the type ωn−1(log ω)n with n ≥ 3.