Proof Strategy Research Articles

Well-posedness of Hamilton-Jacobi equations in the Wasserstein space: non-convex Hamiltonians and common noise

. We establish the well-posedness of viscosity solutions for a class of semi-linear Hamilton-Jacobi equations set on the space of probability measures on the torus. In particular, we focus on equations with both common and (non-degenerate) idiosyncratic noise, and with Hamiltonians which are not necessarily convex in the momentum variable. Our main results show (i) existence, (ii) the comparison principle (and hence uniqueness), and (iii) the convergence of finite-dimensional approximations for such equations. Our proof strategy for the comparison principle is to first use a mix of existing techniques to prove a “partial comparison” result, i.e., a comparison principle which holds when either the subsolution or the supersolution is Lipschitz with respect to a certain very weak metric. Our main innovation is then to develop a strategy for removing this regularity assumption. In particular, we use some delicate estimates for a sequence of finite-dimensional PDEs to show that under certain conditions there exists a viscosity solution which is Lipschitz with respect to the relevant weak metric. We then use this existence result together with a mollification procedure to establish a full comparison principle, i.e., a comparison principle which holds when the subsolution and supersolution under consideration are only semi-continuous. Finally, we apply these results to mean-field control problems with common noise and zero-sum games over the Wasserstein space.

On Extending Incorrectness Logic with Backwards Reasoning

This paper studies an extension of O'Hearn's incorrectness logic (IL) that allows backwards reasoning. IL in its current form does not generically permit backwards reasoning. We show that this can be mitigated by extending IL with underspecification. The resulting logic combines underspecification (the result, or postcondition, only needs to formulate constraints over relevant variables) with underapproximation (it allows to focus on fewer than all the paths). We prove soundness of the proof system, as well as completeness for a defined subset of presumptions. We discuss proof strategies that allow one to derive a presumption from a given result. Notably, we show that the existing concept of loop summaries -- closed-form symbolic representations that summarize the effects of executing an entire loop at once -- is highly useful. The logic, the proof system and all theorems have been formalized in the Isabelle/HOL theorem prover.

Relative energy method for weak–strong uniqueness of the inhomogeneous Navier–Stokes equations far from vacuum

We present a weak–strong uniqueness result for the inhomogeneous Navier–Stokes equations in Rd (d=2,3) for bounded initial densities that are far from vacuum. Given a strong solution, i.e. a solution satisfying the equation as an identity in L2, and a Leray–Hopf weak solution, we establish that they coincide if the initial data agree. Our proof strategy is based on the relative energy method and new W-1,p-type stability estimates for the density. A key point lies in proving that every Leray–Hopf weak solution originating from initial densities far from vacuum remains distant from vacuum at all times.

Contractibility of the Rips complexes of Integer lattices via local domination

We prove that for each positive integer n n , the Rips complexes of the n n -dimensional integer lattice in the d 1 d_1 metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n 2 ( 2 n − 1 ) n^2(2n-1) , with the bounds arising from the Jung constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension n n and a fixed scale r r , this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for n = 1 , 2 , 3 n=1,2,3 , the Rips complex of the n n -dimensional integer lattice at scale greater or equal to n n is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions n n .

Dual process in the two-parameter Poisson–Dirichlet diffusion

The two-parameter Poisson–Dirichlet diffusion takes values in the infinite ordered simplex and extends the celebrated infinitely-many-neutral-alleles model, having a two-parameter Poisson–Dirichlet stationary distribution. Here we identify a dual process for this diffusion and obtain its transition probabilities. The dual is shown to be given by Kingman’s coalescent with mutation, conditional on a given configuration of leaves. Interestingly, the dual depends on the additional parameter of the stationary distribution only through the test functions and not through the transition rates. After discussing the sampling probabilities of a two-parameter Poisson–Dirichlet partition drawn conditionally on another partition, we use these notions together with the dual process to derive the transition density of the diffusion. Our derivation provides a new probabilistic proof of this result, leveraging on an extension of Pitman’s Pólya urn scheme, whereby the urn is split after a finite number of steps and two urns are run independently onwards. The proof strategy exemplifies the power of duality and could be exported to other models where a dual is available.

Provable Security for the Onion Routing and Mix Network Packet Format Sphinx

Onion routing and mix networks are fundamental concepts to provide users with anonymous access to the Internet. Various corresponding solutions rely on the Sphinx packet format. However, flaws in Sphinx's underlying proof strategy were found recently. It is thus currently unclear which guarantees Sphinx actually provides, and, even worse, there is no suitable proof strategy available. In this paper, we restore the security foundation for all these works by building an analytical framework for Sphinx. We discover that the previously-used Decisional Diffie-Hellman (DDH) assumption is insufficient for a security proof and show that the Gap Diffie-Hellman (GDH) assumption is required instead. We apply it to prove that a slightly adapted version of the Sphinx packet format is secure under the GDH assumption. We are thus, to the best of our knowledge, the first to provide a detailed, in-depth security proof for Sphinx that holds. Our adaptations to Sphinx are necessary, as we demonstrate with an attack on sender privacy that would otherwise be possible in Sphinx's adversary model.

Sharp Gaussian upper bounds for Schrödinger semigroups on the half-line

In 1998, V. Liskevich and Y. Semenov proved sharp Gaussian upper bounds for Schrödinger semigroups on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} with potentials satisfying a global Kato class condition. Using similar basic ideas we show sharp Gaussian upper bounds for Schrödinger semigroups on the half-line, also assuming a suitable global Kato class condition. Our proof strategy includes a new technique of weighted ultracontractivity estimates.

The 'Summa Halensis' on the Composition of the Human Body

The author of the Summa Halensis claims that the human body is maximally composite and argues for this using a proof strategy that intends to deduce the body’s composition from the human soul’s immateriality. This study examines that claim and argument, which is given both in a shorter and a longer form. The core of the article consists in a careful reconstruction of both forms, along with an enquiry into its Jewish Neoplatonic sources (first and foremost the Fons vitae) and its appearance in zoological commentaries contemporary to the Summa written by Peter of Spain and Albert the Great. It emerges that the argument brings into play various features of the Summist’s hylomorphic theory, especially a pluralism about substantial forms.

Mathematical Approach towards Adaptive Machine Learning Models for Dynamic Security Threats in Industrial IoT

The always changing nature of security dangers within the Industrialization Internet of Things (IIoT) makes it exceptionally difficult for standard security strategies to work. Since of this, versatile machine learning (ML) models got to be made that can respond to unused dangers in genuine time. This study proposes the utilize of versatile machine learning procedures as a better approach to form IIoT situations more secure. To begin with, the paper addresses the most security issues that IIoT frameworks have, cantering on how the frameworks must be able to alter to address unused dangers. Cutting edge online dangers are exceedingly cleverly and alter quickly, frequently resisting conventional security measures. This appears how vital it is to discover arrangements that can alter rapidly. The proposed customizable machine learning show employments real-time information streams sent by IIoT gadgets to keep risk recognizable proof and defence strategies up to date. These models utilize strategies such as design acknowledgment, peculiarity discovery, and prescient analytics to distinguish bizarre behavior which will indicate a security breach. By learning from past information and adjusting to modern designs and patterns, the models can distinguish known and obscure dangers more precisely and rapidly. A key angle that creates this customizable machine learning models work so well is the truth that they can operate autonomously in IIoT situations. They complement standard rule-based strategies with energetic data-driven bits of knowledge and work consistently with current security frameworks. This combination permits you to halt dangers some time recently they happen and react rapidly, decreasing the probability of a security occurrence and the harm it may cause. The study also covers the technical pieces and strategies required to convey adaptable machine learning in an IIoT environment. She talks approximately the challenges that emerge in collecting and planning information, preparing models, and drawing conclusions in genuine time. She emphasizes the significance of having a high-performance, versatile framework that can handle huge sums of information of different sorts. The study talks about important issues like privacy, data security, and following the rules when using flexible machine learning models in IIoT ecosystems that are sensitive. It supports open and responsible methods to make sure that release is done in an ethical way and that operations are safe from threats from enemies.

Construction of multi‐bubble blow‐up solutions to the L2$L^2$‐critical half‐wave equation

AbstractThis paper concerns the bubbling phenomena for the ‐critical half‐wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow‐up solutions concentrating exactly at these singularities. This provides the first examples of multi‐bubble solutions for the half‐wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single‐bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non‐linear Schrödinger equations (NLS). However, unlike the single‐bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non‐local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half‐wave operator, and find that there exists a narrow room between the orders and for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi‐bubble non‐local structure.

Enhanced water decontamination via photogenerated electron delocalization of π → π* and D-π-A synergistically

Mixed-dimensional van der Waals heterojunctions (MD-vdWhs), known for exceptional electron transfer and charge separation capabilities, remain underexplored in photocatalysis. In this study, we leveraged the synergistic effect of intermolecular π → π* and D-π-A dual channels to fabricate novel MD-vdWhs. Owing to the synergistic effect, it exhibits superior electron transfer and delocalization ability, thereby enhancing its photocatalytic performance. The Optimal photocatalyst can degrade 98.78 % of 20 mg/L tetracycline (TC) within 15 min. Additionally, we introduced a novel proof strategy for investigating the photoelectron transfer path, creatively demonstrating the synergistic dual channels effect, which can be attributed to the carbonyl density and light-excitation degree. Notably, even under low-power light sources, it achieved complete inactivation of Escherichia coli within just 7 mins, far surpassing current cutting-edge research. This theoretical framework holds promise for broader applications within related studies.

Axiomatization of modal logic with counting

Abstract Modal logic with counting is obtained from basic modal logic by adding cardinality comparison formulas of the form $ \#\varphi \succsim \#\psi $, stating that the cardinality of successors satisfying $ \varphi $ is larger than or equal to the cardinality of successors satisfying $ \psi $. It is different from graded modal logic where basic modal logic is extended with formulas of the form $ \Diamond _{k}\varphi $ stating that there are at least $ k$-many different successors satisfying $ \varphi $. In this paper, we investigate the axiomatization of ML(#) with respect to different frame classes, such as image-finite frames and arbitrary frames. Drawing inspiration from existing works, we employ a similar proof strategy that uses the characterization of binary relations on finite Boolean algebras capable of representing generalized probability measures or finite (respectively arbitrary) cardinality measures. Our main result shows that any formula not provable in the Hilbert system can be refuted within a finite (respectively arbitrary) cardinality measure Kripke frame with a finite domain. We then transform this finite (respectively arbitrary) cardinality measure Kripke frame into a Kripke frame in the corresponding class, refuting the unprovable formula.

Code to Qed, the Project Manager's Guide to Proof Engineering

Despite growing efforts and encouraging successes in the last decades, fully formally-verified projects are still rare in the industrial landscape. The industry often lacks the tools and methodologies to efficiently scale the proof development process. In this work, we give a comprehensible overview of the proof development process for proof developers and project managers. The goal is to support proof developers by rationalizing the proof development process, which currently relies heavily on their intuition and expertise, and by facilitating communication with the management line. To this end, we concentrate on the aspect of proof manufacturing and highlight the most significant sources of proof effort. We propose means to mitigate the latter through proof practices (proof structuring, proof strategies, and proof planning), proof metrics, and tools. Our approach is project-agnostic, independent of specific proof expertise, and computed estimations do not assume prior similar developments. We evaluate our guidelines using a separation kernel undergoing formal verification, driving the proof process in an optimised way. Feedback from a project manager unfamiliar with proof development confirms the benefits of detailed planning of the proof development steps, clear progress communication to the hierarchy line, and alignment with established practices in the software industry.

A Variational Perspective on Auxetic Metamaterials of Checkerboard-Type

The main result of this work is a homogenization theorem via variational convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumptions of physical growth and non-self-interpenetration. While the obtained energy estimates are rather standard, determining the effective deformation behavior, or in other words, characterizing the weak Sobolev limits of deformation maps whose gradients are locally close to rotations on the stiff components, is the challenging part. To this end, we establish an asymptotic rigidity result, showing that, under suitable scaling assumptions, the attainable macroscopic deformations are affine conformal contractions. This identifies the composite as a mechanical metamaterial with a negative Poisson’s ratio. Our proof strategy is to tackle first an idealized model with full rigidity on the stiff tiles to acquire insight into the mechanics of the model and then transfer the findings and methodology to the model with diverging elastic constants. The latter requires, in particular, a new quantitative geometric rigidity estimate for non-connected squares touching each other at their vertices and a tailored Poincaré type inequality for checkerboard structures.

Global existence for an isotropic modification of the Boltzmann equation

Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as isotropic Boltzmann by analogy with the isotropic Landau equation introduced by Krieger and Strain (2012) [35]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau.Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the “very soft potentials” regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of Gualdani and Guillen (2022) [22] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities.

Identity-Based Encryption with (Almost) Tight Security in the Multi-instance, Multi-ciphertext Setting

We construct an identity-based encryption (IBE) scheme that is tightly secure in a very strong sense. Specifically, we consider a setting with many instances of the scheme and many encryptions per instance. In this setting, we reduce the security of our scheme to a variant of a simple assumption used for a similar purpose by Chen and Wee (CRYPTO 2013, Springer, 2013). The security loss of our reduction is O(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{O} (k)$$\\end{document} (where k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k $$\\end{document} is the security parameter). Our scheme is the first IBE scheme to achieve this strong flavor of tightness under a simple assumption. Technically, our scheme is a variation of the IBE scheme by Chen and Wee. However, in order to “lift” their results to the multi-instance, multi-ciphertext case, we need to develop new ideas. In particular, while we build on (and extend) their high-level proof strategy, we deviate significantly in the low-level proof steps.

A Conversion Formula for One Kind Fourth Power Mean of Kloosterman Sums and the Character Sums Module p

The primary contribution of this paper is to research one kind special Kloosterman sums using analytic method. We translate one kind fourth power mean of the Kloosterman sums into the character sums of one kind polynomials. It is possible to construct an exact formula for the fourth power mean of one kind Kloosterman sums by computing the character sums of the polynomials. The paper is novel because of its proof and conversion strategies.

Improved covering results for conjugacy classes of symmetric groups via hypercontractivity

Abstract We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as normal sets. We show that for any $\epsilon>0$ , there exists $n_0$ such that if $n>n_0$ and A is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon }}$ , then $A^2 \supseteq A_n$ . This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$ . Our proof strategy combines two types of techniques. The first is ‘traditional’ techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck–Shalev, Larsen–Shalev, and more recently, Larsen–Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

Extensible Proof Systems for Infinite-State Systems

This article revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.

Bounds and Constructions of Parent Identifying Schemes via the Algorithmic Version of the Lovász Local Lemma

The usefulness of Identifiable Parent Property (IPP) schemes in diverse scenarios has led to several distinct but related concepts. This work focuses on three of these concepts: “classical” IPP codes, Multimedia IPP codes, and IPP set systems. Although several existence bounds for all of the above schemes are known, constructions are scarce. In this paper, we present explicit constructions of all mentioned IPP notions, in the form of combinatorial objects. Our discussion follows a systematic procedure. First, we use the Lovász Local Lemma (LLL) to obtain existence bounds for the object to be constructed. The bounds derived essentially match the previously best-known ones. Additionally, our proof strategy enables for further development. It allows us to use the Moser-Tardos algorithmic version of the LLL in order to construct, with polynomial complexity, the actual objects. Moreover, we extend the results of Giotis et al. to precisely establish the computational complexity of the proposed algorithms.