Classical Theorem Research Articles

On Lurie’s theorem and applications

AbstractLurie’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.

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.

Mixed multiplicity and converse of Rees' theorem for modules

In this paper, we prove the converse of Rees' mixed multiplicity theorem for modules, which extends the converse of the classical Rees' mixed multiplicity theorem for ideals given by Swanson-Theorem 1.3. Specifically, we demonstrate the following result:Let (R,m) be a d-dimensional formally equidimensional Noetherian local ring and E1,…,Ek be finitely generated R-submodules of a free R-module F of positive rank p, with xi∈Ei for i=1,…,k. Consider S, the symmetric algebra of F, and IEi, the ideal generated by the homogeneous component of degree 1 in the Rees algebra [R(Ei)]1. Assuming that (x1,…,xk)S and IEi have the same height k and the same radical, if the Buchsbaum-Rim multiplicity of (x1,…,xk) and the mixed Buchsbaum-Rim multiplicity of the family E1,…,Ek are equal, i.e., eBR((x1,…,xk)p;Rp)=eBR(E1p,…,Ekp,Rp) for all prime ideals p minimal over ((x1,…,xk):RF), then (x1,…,xk) is a joint reduction of (E1,…,Ek).In addition to proving this theorem, we establish some properties that relate joint reduction and mixed Buchsbaum-Rim multiplicities.

On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras

In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions C[a,b] equipped with norms stronger than the supremum norm. We are particularly interested in Hölder spaces since they are natural ranges of integral operators of fractional order. Our goal is to preserve the expected regularity of the fixed points (solutions of the equations) when investigating their uniqueness, without assuming a contraction condition on the space under study. We claim some symmetry between the case of the obtained results and the case of the classical Banach fixed-point theorem in such spaces, even for operators which are not necessarily contractions in the sense of the norm of these subspaces. This result is of particular interest for the study of quadratic integral equations, and as an application example we prove the uniqueness theorem for such a kind equations with general fractional order integral operators, which are not necessarily contractions, in a suitably constructed generalized Hölder space.

Harmonic curves and the beauty of projective geometry

The 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.

A Frobenius integrability theorem for plane fields generated by quasiconformal deformations

We generalize the classical Frobenius integrability theorem to plane fields of class CQ, a regularity class introduced by Reimann [9] for vector fields in Euclidean spaces. Reimann showed that a CQ vector field is uniquely integrable and its flow is a quasiconformal deformation. We prove that an a.e. involutive CQ plane field (defined in a suitable way) in Rn is integrable, with integral manifolds of class C1,Q.

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.

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.

Wigner’s theorem for stabilizer states and quantum designs

We describe the symmetry group of the stabilizer polytope for any number n of systems and any prime local dimension d. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. In the case of qudits, the structure is somewhat richer: For n = 1, it is a wreath product of permutations of bases and permutations of the elements within each basis. For n > 1, the symmetries are given by affine symplectic similitudes. These are the affine maps that preserve the symplectic form of the underlying discrete phase space up to a non-zero multiplier. We phrase these results with respect to a number of a priori different notions of “symmetry,” including Kadison symmetries (bijections that are compatible with convex combinations), Wigner symmetries (bijections that preserve inner products), and symmetries realized by an action on Hilbert space. Going beyond stabilizer states, we extend an observation of Heinrich and Gross [Quantum 3, 132 (2019)] and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments. In particular: the symmetries of a set that behaves like a 3-design preserve Jordan products and are therefore realized by conjugation with unitaries or anti-unitaries. (The structure constants of the Jordan algebra are encoded in an order-three tensor, which we connect to the third moments of a design). This generalizes Kadison’s formulation of the classic Wigner theorem on quantum mechanical symmetries.