Form Of Theorem Research Articles

Denotational recurrence extraction for amortized analysis

A typical way of analyzing the time complexity of functional programs is to extract a recurrence expressing the running time of the program in terms of the size of its input, and then to solve the recurrence to obtain a big-O bound. For recurrence extraction to be compositional, it is also necessary to extract recurrences for the size of outputs of helper functions. Previous work has developed techniques for using logical relations to state a formal correctness theorem for a general recurrence extraction translation: a program is bounded by a recurrence when the operational cost is bounded by the extracted cost, and the output value is bounded, according to a value bounding relation defined by induction on types, by the extracted size. This previous work supports higher-order functions by viewing recurrences as programs in a lambda-calculus, or as mathematical entities in a denotational semantics thereof. In this paper, we extend these techniques to support amortized analysis, where costs are rearranged from one portion of a program to another to achieve more precise bounds. We give an intermediate language in which programs can be annotated according to the banker's method of amortized analysis; this language has an affine type system to ensure credits are not spent more than once. We give a recurrence extraction translation of this language into a recurrence language, a simply-typed lambda-calculus with a cost type, and state and prove a bounding logical relation expressing the correctness of this translation. The recurrence language has a denotational semantics in preorders, and we use this semantics to solve recurrences, e.g analyzing binary counters and splay trees.

Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter. Without imposing on any non-degeneracy assumption, we prove a formal KAM theorem for the mappings, which implies many previous KAM-type results under some non-degeneracy conditions. Moreover, by this formal KAM theorem, we can also obtain some new interesting results under some weaker non-degeneracy conditions. Thus, the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.

The intrinsic formality of $E_n$-operads

We establish that E_n -operads satisfy a rational intrinsic formality theorem for n\geq 3 . We gain our results in the category of Hopf cooperads in cochain graded dg-modules which defines a model for the rational homotopy of operads in spaces. We consider, in this context, the dual cooperad of the n -Poisson operad \mathsf{Pois}_n^c , which represents the cohomology of the operad of little n -discs \mathsf{D}_n . We assume n\geq 3 . We explicitly prove that a Hopf cooperad in cochain graded dg-modules \mathsf{K} is weakly-equivalent (quasi-isomorphic) to \mathsf{Pois}_n^c as a Hopf cooperad as soon as we have an isomorphism at the cohomology level H^*(\mathsf{K})\simeq\mathsf{Pois}_n^c when 4\nmid n . We just need the extra assumption that \mathsf{K} is equipped with an involutive isomorphism mimicking the action of a hyperplane reflection on the little n -discs operad in order to extend this formality statement in the case 4\mid n . We deduce from these results that any operad in simplicial sets \mathsf{P} which satisfies the relation H^*(\mathsf{P},\mathbb{Q})\simeq\mathsf{Pois}_n^c in rational cohomology (and an analogue of our extra involution requirement in the case 4\mid n ) is rationally weakly equivalent to an operad in simplicial sets \mathtt L \mathtt G_{\bullet}(\mathsf{Pois}_n^c) which we determine from the n -Poisson cooperad \mathsf{Pois}_n^c . We also prove that the morphisms \iota: \mathsf{D}_m\rightarrow\mathsf{D}_n , which link the little discs operads together, are rationally formal as soon as n-m\geq 2 . These results enable us to retrieve the (real) formality theorems of Kontsevich by a new approach, and to sort out the question of the existence of formality quasi-isomorphisms defined over the rationals (and not only over the reals) in the case of the little discs operads of dimension n\geq 3 .

Formalization of Euler–Lagrange Equation Set Based on Variational Calculus in HOL Light

As the theoretical foundation of Lagrangian mechanics, Euler–Lagrange equation sets are widely applied in building mathematical models of physical systems, especially in solving dynamics problems. However, their preconditions are often not fully satisfied in practice. Therefore, it is necessary to verify their applications. The purpose of the present work is to conduct such verification by establishing a formal theorem library of Lagrangian mechanics in HOL Light. For this purpose, some basic concepts such as functional variation and the necessary conditions for functional extreme are formalized. Then, the fundamental lemma of variational calculus is formally verified and some new constuctors and destructors are proposed. Finally, the Euler–Lagrange equation set is formalized. To validate the formalization, the formalization results are applied to verify the least resistance problem of gas flow. The present work not only lays a necessary and solid foundation for application involving Lagrangian mechanics but also extends the HOL Light theorem library.

The Existence of Singular Traces on Simply Generated Operator Ideals

We present direct and self-contained proofs of the existence of singular traces on sufficiently large classes of operator ideals that live on the separable infinite-dimensional real Hilbert space. Our only tool is Banach’s version of the extension theorem of linear forms. Thanks to the use of dyadic representations of operators, all proofs have become straightforward.

The Hochschild cohomology ring of a global quotient orbifold

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields TXpoly on X, and is generated as a TXpoly-algebra by the sum of the determinants det⁡(NXg) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne–Mumford stacks in general. We discuss, in the case of a symplectic group action on a symplectic variety X, relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, we employ compatible trivializations of the determinants det⁡(NXg), which requires (for the cup product) a nontrivial normalization missing in previous literature.

Super formal Darboux-Weinstein theorem and finite W-superalgebras

Let v=v0¯+v1¯ be a Z2-graded (super) vector space with an even C×-action and χ∈v0¯⁎ be a fixed point of the induced action. In this paper we prove an equivariant Darboux-Weinstein theorem for the formal polynomial algebras Aˆ=S[v0¯]∧χ⊗⋀(v1¯). We also give a quantum version of it. Let g=g0¯+g1¯ be a basic Lie superalgebra and e∈g0¯ be a nilpotent element. We use the equivariant quantum Darboux-Weinstein theorem to give a Poisson geometric realization of the finite W-superalgebra U(g,e) in the sense of Losev. An indirect relation between finite W-(super)algebras U(g,e) and U(g0¯,e) is presented. Finally we use such a realization to study the finite-dimensional irreducible modules over U(g,e).

Formalization of continuous Fourier transform in verifying applications for dependable cyber-physical systems

Continuous Fourier transform (CFT) is widely used and is often directly applied in cyber-physical systems (CPS) without checking its preconditions. This inevitably leads to unexpected defects and even errors. Thus, verification is necessary for the CFT-based engineering design to ensure a dependable cyber physical system. HOL4 (Higher Order Logic 4) is a formal theorem prover that prevails in software and hardware verification. However, there is no CFT theorem library in current HOL4. In this paper, the definition and some frequently used properties of CFT are formalized and verified in HOL4. Based on this, we formally model a basic theorem library of CFT. As a case study, the CFT library is employed to verify the frequency response of an RLC circuit, which is a critical application for dependable CPS. The formalization of the CFT and its properties and the construction of the formal CFT theorem library can effectively extend the function of the HOL4 system. The obtained formal results can be applied in various CFT-based cyber-physical systems.

On a Characterization Theorem on a -adic Solenoids

According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in independent random variables given another. We prove an analogue of this theorem for linear forms in two independent random variables with values in an a-adic solenoind without elements of order 2, assuming that the characteristic functions of the random variables do not vanish, and coefficients of the linear forms are topological automorphisms of the a-adic solenoid.

Formality and Kontsevich–Duflo type theorems for Lie pairs

Kontsevich's formality theorem states that there exists an L∞ quasi-isomorphism from the dgla Tpoly•(M) of polyvector fields on a smooth manifold M to the dgla Dpoly•(M) of polydifferential operators on M, which extends the classical Hochschild–Kostant–Rosenberg map. In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and g-manifolds (that is, manifolds endowed with an action of a Lie algebra g). The spaces tot(Γ(Λ•A∨)⊗RTpoly•) and tot(Γ(Λ•A∨)⊗RDpoly•) associated with a Lie pair (L,A) each carry an L∞ algebra structure canonical up to L∞ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups HCE•(A,Tpoly•) and HCE•(A,Dpoly•) admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an L∞ quasi-isomorphism from tot(Γ(Λ•A∨)⊗RTpoly•) to tot(Γ(Λ•A∨)⊗RDpoly•) whose first Taylor coefficient is equal to hkr∘(tdL/A∇)12. Here the cocycle (tdL/A∇)12 acts on tot(Γ(Λ•A∨)⊗RTpoly•) by contraction. Furthermore, we prove a Kontsevich–Duflo type theorem for Lie pairs: the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the Lie pair (L,A) is an isomorphism of Gerstenhaber algebras from HCE•(A,Tpoly•) to HCE•(A,Dpoly•). As applications, we establish formality and Kontsevich–Duflo type theorems for complex manifolds, foliations, and g-manifolds. In the case of complex manifolds, we recover the Kontsevich–Duflo theorem of complex geometry.

A PARAMETRIC, RESOURCE-BOUNDED GENERALIZATION OF LÖB’S THEOREM, AND A ROBUST COOPERATION CRITERION FOR OPEN-SOURCE GAME THEORY

AbstractThis article presents two theorems: (1) a generalization of Löb’s Theorem that applies to formal proof systems operating with bounded computational resources, such as formal verification software or theorem provers, and (2) a theorem on the robust cooperation of agents that employ proofs about one another’s source code as unexploitable criteria for cooperation. The latter illustrates a capacity for outperforming classical Nash equilibria and correlated equilibria, attaining mutually cooperative program equilibrium in the Prisoner’s Dilemma while remaining unexploitable, i.e., sometimes achieving the outcome (Cooperate, Cooperate), and never receiving the outcome (Cooperate, Defect) as player 1.

Equivariant formality of Hamiltonian transversely symplectic foliations

Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation which satisfies the transverse hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic $d\delta$-lemma in this setting. As an application, we show that if the foliation is also Riemannian, then there exists a natural formal Frobenius manifold structure on the equivariant basic cohomology of the foliation.

A theorem of the Square Root of the Brownian Motion

We present a formal theorem of the square root of the Brownian motion. In doing so, we show that this process can be presented as a typical complex random variable. In addition, we introduce the basic properties of this process.

Dynamics in a delayed diffusive cell cycle model

In this paper, we construct a delayed diffusive model to explore the spatial dynamics of cell cycle in G2/M transition. We first obtain the local stability of the unique positive equilibrium for this model, which is irrelevant to the diffusion. Then, through investigating the delay-induced Hopf bifurcation in this model, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. Applying the normal form and center manifold theorem of functional partial differential equations, we also determine the stability and direction of these bifurcating periodic solutions. Finally, numerical simulations are presented to validate our theoretical results.

Relationships among charges, fields, and potential on spherical surfaces boundary value problems

Abstract This paper suggests a new point of view for the Poisson equation and its solution for the potential and field on the d dimensional sphere, S d , on which point charges are distributed. The available solutions for the potential on multidimensional spheres in the literature are purely mathematical, while the solution suggested here is motivated by physical intuition and requires minimal background; namely, basic laws of electrostatics and dimensional analysis. In this study, the modified Coulomb’s law is presented by means of “dimensional reduction” and the use of equivalence between a point charge on S d and a charged ray in R d + 1 . Besides formal detailed solutions and theorems, this paper presents concrete physical examples (unstudied or studied partly), such as distribution of charges/sources on a two-sphere; Dirichlet problem for currents on a truncated sphere; and fields and potentials created by “infinite” cones. Well-known statements about special charge distributions in Euclidean space must be reformulated and amended when dealing with the case of charges embedded in a spherical manifold.

Khovanov homology from Floer cohomology

This paper realises the Khovanov homology of a link in S 3 S^3 as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k \mathbf {k} of characteristic zero. Here we prove the symplectic cup and cap bimodules, which relate different symplectic arc algebras, are themselves formal over k \mathbf {k} , and we construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over Z \mathbb {Z} in a manner compatible with the cup bimodules. It follows that Khovanov cohomology and symplectic Khovanov cohomology co-incide in characteristic zero.

Gravity formality

We show that Willwacher's cyclic formality theorem can be extended to preserve natural Gravity operations on cyclic multivector fields and cyclic multidifferential operators. We express this in terms of a homotopy Gravity quasi-isomorphism with explicit local formulas. For this, we develop operadic tools related to mixed complexes and cyclic homology and prove that the operad M⥀ of natural operations on cyclic operators is formal and hence quasi-isomorphic to the Gravity operad.

Summability experiments with a class of divergent inverse Kontorovich–Lebedev Transforms

The computation of inverse Kontorovich–Lebedev transforms is examined with the help of the extrapolative mW-transformation first presented by Sidi in user-friendly form in Sidi (1988). Recently published improved values for the Macdonald function of purely imaginary order have enabled this strategy to give 10 digit accuracy in the inversion, including an example with competing oscillations having different phase functions. Moreover the method is extended, with similar accuracy, to cases where the integral diverges but is summable in the sense of Abel. Further probing with divergent integrals that are not Abel summable but instead summable in the sense defined by Jones (1980) is also yielding encouraging results and some analysis, culminating in a formal theorem of convergence in that sense, is presented to support the interpretation and computation of these.

HybridFTW: Hybrid Computation of Dynamic Time Warping Distances

In this paper, we propose an efficient approach that computes the dynamic time warping (DTW) distance in time-series similarity search. The DTW distance is known to offer the high accuracy in similarity search, but it has difficulty in supporting the large database due to its high computational complexity. Recently, FastDTW and FTW have been proposed for efficient computation of DTW distances, but they have still performance limitations. In this paper, we propose a hybrid approach, called HybridFTW , which combines the advantages of both FastDTW and FTW. First, HybridFTW takes the advantage of FastDTW that provides fast computation through the limitation of allowable ranges. We call these allowable ranges dynamic (warping) bands , which reduce the computation spaces on the fly, and we reanalyze previous FastDTW and FTW in the viewpoint of static and dynamic bands. Second, HybridFTW also takes the advantage of FTW that exploits the early abandon effect by using the segment-based tight lower bound. To maximize the synergy of combining two methods, we obtain the dynamic band of FastDTW during the process of computing the lower bound in FTW. Using HybridFTW, we next propose range search and $k$ -NN search algorithms and prove their correctness through formal theorems. Experimental results on real and synthetic data sets show that HybridFTW improves the search performance by up to 38 times over FastDTW and by up to 12 times over FTW.

Orbital Linearization of Smooth Completely Integrable Vector Fields

The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.