Proof Of Existence Research Articles

Non-self-similar collapse of three point vortices in a generalized two-dimensional fluid system

Abstract In this study, we investigate the non-self-similar collapse of three vortices in a point vortex model of a generalized two-dimensional (2D) fluid system. In the classical point vortex model, the existence of the self-similar collapse of three vortices has been well known from Gr\"obli (1877). Moreover, the non-existence of the non-self-similar collapse of three vortices in the classical point vortex model was proved by Hern\'andez-Gardu\~no and Lacomba (2007) and Hiraoka (2008). In this study, we apply the method similar to Hern\'andez-Gardu\~no and Lacomba (2007) to the point vortex model of the generalized 2D fluid system, which includes a real parameter $\alpha$, where $0 < \alpha \le 3$. We present proofs for the possible existence of the non-self-similar collapse of three vortices for $0 <\alpha < 2$ and for the non-existence of such collapse for $2 < \alpha \le 3$. Furthermore, as an application, we analyze in detail the self-similar and non-self-similar collapses of three vortices in the symmetric case of the surface quasi-geostrophic system ($\alpha=1$). A close relationship between the self-similar and non-self-similar collapses of the three vortices is identified.

A Rigorous Integrator and Global Existence for Higher-Dimensional Semilinear Parabolic PDEs via Semigroup Theory

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift–Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta–Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.

Progressful Interpreters for Efficient WebAssembly Mechanisation

Mechanisations of programming language specifications are now increasingly common, providing machine-checked modelling of the specification and verification of desired properties such as type safety. However it is challenging to maintain these mechanisations, particularly in the face of an evolving specification. Existing mechanisations of the W3C WebAssembly (Wasm) standard have so far been able to keep pace as the standard evolves, helped enormously by the W3C Wasm standard's choice to state the language's semantics in terms of a fully formal specification. However a substantial incoming extension to Wasm, the 2.0 feature set, motivates the investigation of strategies for more efficient production of the core verification artefacts currently associated with the WasmCert-Coq mechanisation of Wasm. In the classic formalisation of a typed operational semantics as followed by the W3C Wasm standard, both the type system and runtime operational semantics are defined as inductive relations, with associated type soundness properties (progress and preservation) and an independent sound interpreter. We investigate two more efficient strategies for producing these artefacts, which are currently all separately defined by WasmCert-Coq. First, the approach of Kokke, Siek, and Wadler for deriving a sound interpreter from a constructive progress proof --- we show that this approach scales to the W3C Wasm 1.0 standard, but results in an inefficient interpreter in our setting. Second, inspired by results from intrinsically-typed languages, we define a progressful interpreter which uses Coq's dependent types to certify not only its own soundness, but also the progress property. We show that this interpreter can implement several performance optimisations while maintaining these certifications, which are fully erasable when the interpreter is extracted from Coq. Using this approach, we extend the WasmCert-Coq mechanisation to the significantly larger Wasm 2.0 feature set, discovering and correcting several errors in the expanded specification's type system.

Maximal Simplification of Polyhedral Reductions

Reductions combine collections of input values with an associative and often commutative operator to produce collections of results. When the same input value contributes to multiple outputs, there is an opportunity to reuse partial results, enabling reduction simplification. Simplification often produces a program with lower asymptotic complexity. Typical compiler optimizations yield, at best, a constant fold speedup, but a complexity improvement from, say, cubic to quadratic complexity yields unbounded speedup for sufficiently large problems. It is well known that reductions in polyhedral programs may be simplified automatically, but previous methods cannot exploit all available reuse. This paper resolves this long-standing open problem, thereby attaining minimal asymptotic complexity in the simplified program. We propose extensions to prior work on simplification to support any independent commutative reduction. At the heart of our approach is piece-wise simplification, the notion that we can split an arbitrary reduction into pieces and then independently simplify each piece. However, the difficulty of using such piece-wise transformations is that they typically involve an infinite number of choices. We give constructive proofs to deal with this and select a finite number of pieces for simplification.

Parameter Identification and Prediction of the Rőssler System with Complete and Incomplete Information: Two Known and One Unknown State Variables

Parameters identification algorithms are formulated for the system of equations of Rőssler's type. The problem is solved for the cases of complete and incomplete information about functions of the system. If there is complete information about the state variables, it is possible to apply the algorithms discussed to any system of linear or nonlinear ordinary differential equations of arbitrary order in the Cauchy form that linearly depends on the unknown parameters (or groups of unknown parameters). The problems of parameter identification in the case of incomplete information about the state variables must be solved individually, depending on the possibility (or impossibility) of eliminating unknown steady states from the system of equations. Most real-world problems in fields such as chemical kinetics, mathematical ecology, predator-prey dynamics in game reserves, and the spread of infectious diseases belong to this class of problems, in which the algorithms discussed demonstrate their applicability. In the present paper the case of one unknown function is considered. Lemmas about possibilities on complete and incomplete parameter identification are formulated and proven. The algorithms of the parameter identification are formulated in the process of the constructive proofs of the lemmas. Numerical examples and graphs of solutions are considered which demonstrate efficiency and accuracy of the developed algorithms. In the proposed paper, the integration approach is used instead of the differential approach because it allows for the smoothing of discrete data, thereby reducing the estimation errors of the unknown parameters.

Students’ Proof Construction through Computer-Supported Collaborative Learning: The Perspectives of Habermas’ Theory of Rationality and Duval’s Theory of Registers of Semiotic Representation

Students’ Proof Construction through Computer-Supported Collaborative Learning: The Perspectives of Habermas’ Theory of Rationality and Duval’s Theory of Registers of Semiotic Representation

Marginally outer trapped tubes in de Sitter spacetime

We prove two results which are relevant for constructing marginally outer trapped tubes (MOTTs) in de Sitter spacetime. The first one (Theorem 1) holds more generally, namely for spacetimes satisfying the null convergence condition and containing a timelike conformal Killing vector with a “temporal function”. We show that all marginally outer trapped surfaces (MOTSs) in such a spacetime are unstable. This prevents application of standard results on the propagation of stable MOTSs to MOTTs. On the other hand, it was shown recently, Charlton et al. (minimal surfaces and alternating multiple zetas, arXiv:2407.07130), that for every sufficiently high genus, there exists a smooth, complete family of CMC surfaces embedded in the round 3-sphere S3. This family connects a Lawson minimal surface with a doubly covered geodesic 2-sphere. We show (Theorem 2) by a simple scaling argument that this result translates to an existence proof for complete MOTTs with CMC sections in de Sitter spacetime. Moreover, the area of these sections increases strictly monotonically. We compare this result with an area law obtained before for holographic screens.

RadarFormer: End-to-End Human Perception With Through-Wall Radar and Transformers.

For fine-grained human perception tasks such as pose estimation and activity recognition, radar-based sensors show advantages over optical cameras in low-visibility, privacy-aware, and wall-occlusive environments. Radar transmits radio frequency signals to irradiate the target of interest and store the target information in the echo signals. One common approach is to transform the echoes into radar images and extract the features with convolutional neural networks. This article introduces RadarFormer, the first method that introduces the self-attention (SA) mechanism to perform human perception tasks directly from radar echoes. It bypasses the imaging algorithm and realizes end-to-end signal processing. Specifically, we give constructive proof that processing radar echoes using the SA mechanism is at least as expressive as processing radar images using the convolutional layer. On this foundation, we design RadarFormer, which is a Transformer-like model to process radar signals. It benefits from the fast-/slow-time SA mechanism considering the physical characteristics of radar signals. RadarFormer extracts human representations from radar echoes and handles various downstream human perception tasks. The experimental results demonstrate that our method outperforms the state-of-the-art radar-based methods both in performance and computational cost and obtains accurate human perception results even in dark and occlusive environments.

Topological closure of formal powers series ideals and application to topological rewriting theory

We investigate formal power series ideals and their relationship to topological rewriting theory. Since commutative formal power series algebras are Zariski rings, their ideals are closed for the adic topology defined by the maximal ideal generated by the indeterminates. We provide a constructive proof of this result which, given a formal power series in the topological closure of an ideal, consists in computing a cofactor representation of the series with respect to a standard basis of the ideal. We apply this result in the context of topological rewriting theory, where two natural notions of confluence arise: topological confluence and infinitary confluence. We give explicit examples illustrating that in general, infinitary confluence is a strictly stronger notion than topological confluence. Using topological closure of ideals, we finally show that in the context of rewriting theory on commutative formal power series, infinitary and topological confluences are equivalent when the monomial order considered is compatible with the degree.

On Quantified Splitting Proof System for Propositional Calculi

In this paper, some new quantified propositional proof system is introduced and compared by proof complexities with other quantified and not quantified propositional proof systems. It is proved that the introduced system 1) is polynomially equivalent to its quantifier-free variant and 2) has exponential speed-up by sizes over some variants of the quantified resolution system. As the introduced system has a very simple proof construction strategy, it can be very useful not only in Logic, and therefore in Artificial Intelligence, but also in areas such as Computational Biology and Medical Diagnosis.

Comparing different types of instructional videos in a flipped proof-based classroom

BackgroundProofs are a key component in undergraduate mathematics, but understanding presented proofs and constructing proofs is a challenge for many students. Flipped undergraduate mathematics classrooms often employ instructional videos, yet little is known about their potential to help students understand and construct proofs.ObjectiveThis study investigates the potential of video-based proof presentations on student learning. We compared a video that presented the proof construction process (proof video); a video that heuristically presented the proof construction process, which modeled key decisions and named the phases of proof construction and activities (heuristic proof video); and a video that offered prompts during the proof construction process, where self-explanation prompts guided students through these phases and activities (prompted proof video).MethodsA between-subjects design was employed, involving 177 mathematics (teacher) students in a first-semester proof-based linear algebra course. Data were collected on students’ comprehension of the presented proof, their knowledge for proof construction, and their evaluative perceptions. Statistical analyses were performed using ANOVA (proof comprehension) and MANOVA (evaluative perceptions) to compare the groups. Qualitative content analysis was employed to identify different facets of knowledge for proof construction and the groups were contrasted using χ2-tests.ResultsWe found that independent of the video they watched, students achieved a rather local comprehension of the presented proof. The heuristic proof video showed potential for offering meta-knowledge of how to approach proof construction and knowledge on process-related activities that support individual phases of proof construction but required more time. Yet, while students perceived all videos positively, they liked the heuristic proof video best.ConclusionThe results provide insights into the design of instructional videos, suggesting that, in the early stages of learning about proofs, a heuristic proof video may help address the challenges students face.

Human dignity: a contract or an abstract?

Although human dignity has been the focus of many researchers, fundamental debates about its existence have often been ignored. Different views on human dignity and its existence can lead to divergent interpretations of human rights. In this study, we attempted to find an answer to the question of the nature of human dignity by examining and collecting the opinions of experts and analyzing and criticizing them. Our analysis showed that since dignity is linked to human existence and understanding, it has a subjective nature. Subjective existences have different types, including contractual and abstract. Contractual existence finds its way to objective entities through human thought, and it is changeable. An abstract existence, on the other hand, is created by perception of an objective entity in a constant way among human beings. Human dignity is consistent with the contractual type, because simply seeing a human does not bring to mind the existence of dignity and human rights. Once we accept the contractual nature of dignity, we must determine who bestowed this dignity on man. Through investigations, we came to the conclusion that God is the only one that can grant such privilege, and the existence of dignity for humans is a proof of God's existence.

State-dependent prox-regular sweeping process with a general nonconvex composed perturbation

We consider a sweeping process with a triple perturbation defined on a separable Hilbert space. The values of the moving set are time- and state-dependent prox-regular sets. The perturbation is given by the sum of three multivalued mappings having different semicontinuity properties with respect to the state variable. The first mapping with closed, possibly, nonconvex values is lower semicontinuous. The second one with closed convex values has weakly sequentially closed graph. The values of the third mapping can be both convex and nonconvex closed sets. This mapping has closed graph at the points where it is convex-valued. At a point therein its value is a nonconvex set, the mapping is lower semicontinuous on a neighborhood of this point. Usually, the latter mapping is called a mapping with mixed semicontinuity properties. We prove the existence of a solution to our sweeping process. To this aim, we propose a new method that is not related to the catching-up algorithm or its modifications often used in the existence proofs for sweeping processes. We use classical approaches based on a priori estimates and a fixed-point argument for multivalued mappings. Our existence result is completely new and it implies the existing results for the considered class of sweeping processes with state-dependent moving sets.

Adaptive human–machine theorem proving system

This paper presents a novel approach to constructing human–machine theorem proving systems. These systems integrate machine learning capabilities, human expert knowledge, and rigorous logical control for the effective construction and verification of proofs. The innovation of this approach lies in its openness: the user is given a tool for building such systems. Users can create theorem proving systems by selecting existing logical inference strategies or adding new ones in accordance with the provided interfaces or relationship agreements. The systems being built have a significant human--machine adaptive nature. During their operation, a meta-strategy is developed and trained based on the foundational elements of the existing strategies. The system is designed as a universal framework for managing various strategies, with the provision of a basic architecture and a library of strategies with the possibility of adding new ones. During the learning process, a system of structural characteristics is also accumulated and trained, on the basis of which a decision is made on the use of specific strategy elements at the next step. The presentation is conducted for the sequential calculus of minimal positive predicate logic as the most suitable for the deductive synthesis of programs and algorithms, for which it is proposed to use this approach.

Numerical analysis of the stress‐based formulation of linear elasticity

AbstractThe Beltrami–Michell equations of linear elasticity differ from the Navier–Cauchy equations, in that the primary field in former equations is the stress tensor rather than the displacement vector. Consequently, the equations can be used for circumstances where the displacement field is not of interest, for example in design, or when increased smoothness of the solution of the stress tensor is desired. In this work, we explore the stress‐based Beltrami–Michell equations for isotropic linear elastic materials. We introduce the equations in modern tensor notation and investigate their limitations. Further, we demonstrate how to symmetrise and stabilise their weak formulation, complemented by existence and uniqueness proofs. With latter at hand, we construct a conforming finite element discretisation of the equations, avoiding the need for intermediate stress functions. Finally, we present some numerical examples.

Language is a “quite useless” tool: A rejoinder to Fedorenko, Piantadosi, and Gibson’s “Language is primarily a tool for communication rather than thought”

Contrary to the prevailing assumption that language is “primarily a tool for communication rather than thought”, I argue that language is, to invoke Oscar Wilde, “quite useless”. Arguing from aesthetic philosophy and the minimalist program for linguistic theory, I conject that language, like art, is not “for” anything—it simply is, conforming to aesthetic rather than utilitarian principles. Of course, like art, language can be a powerful instrument of communication, but its function is not that of expressing thought; it creates thoughts, “primarily” for communicating with oneself, engaging in Popperian critical rationalism, making thoughts (e.g., sentences, constructive proofs) to match Platonic objects (e.g., propositions, classical proofs).

Global existence and time decay rate of classical solutions to a hybrid Vlasov-Fokker-Planck-MHD equations

In this paper, we prove the existence of global classical solutions to a kinetic-fluid system when initial data is a small perturbation of some given equilibrium state in R3. The system consists of the Vlasov-Fokker-Planck equation coupled with the compressible magnetohydrodynamics (MHD) equations via the nonlinear coupling terms of Lorenz force type. It describes the motion of energetic particles in a fluid with a magnetic field. The proof of global existence mainly relies on the energy method. Due to the complex nonlinear structure of Lorentz force, we need to establish a more refined uniform a prior estimates. Moreover, under additional conditions on initial data, the optimal time decay rate of solutions toward the equilibrium state can be obtained by using the Fourier analysis.

The Mean Value Theorem and It’s Application

The Mean Value Theorem (MVT) is the central theorem of differential Calculus. It’s the major tool to research on function, also bridging the gap between function and derivative. There are lot of researchers focus on it from ancient times until now. This article introduces details about two mean theorems include the Rolle’s Theorem and Lagrange Theorem. This paper uses the sample question and refutation to explain the conditions of the theorems, thus addressing the questions that many students will ask when learning definitions of these theorems. The article later examines the application of the MVT. The proof of the unique existence of roots and the inequality is included. The Mean Value Theorem reveals the link between the macroscopic, overall properties for a function on an interval and the microscopic, localized properties of the function at a point. The significance of the application for the mean value theorem is profound, not only promoting the development of mathematical analysis theory, but also playing an important role in multiple fields

Combinatorial Analysis of a Classic Card Trick

Summary We demonstrate how the analysis of Miraskill, a classic card trick by the legendary Canadian magician Stewart James (1908–1996), leads to a connection with the Legendre polynomials and provides a constructive proof of combinatorial identities that occur in quantum mechanics.

Extracting efficient exact real number computation from proofs in constructive type theory

Abstract Exact real computation is an alternative to floating-point arithmetic where operations on real numbers are performed exactly, without the introduction of rounding errors. When proving the correctness of an implementation, one can focus solely on the mathematical properties of the problem without thinking about the subtleties of representing real numbers. We propose a new axiomatization of the real numbers in a dependent type theory with the goal of extracting certified exact real computation programs from constructive proofs. Our formalization differs from similar approaches, in that we formalize the reals in a conceptually similar way as some mature implementations of exact real computation. Primitive operations on reals can be extracted directly to the corresponding operations in such an implementation, producing more efficient programs. We particularly focus on the formalization of partial and nondeterministic computation, which is essential in exact real computation. We prove the soundness of our formalization with regards to the standard realizability interpretation from computable analysis and show how to relate our theory to a classical formalization of the reals. We demonstrate the feasibility of our theory by implementing it in the Coq proof assistant and present several natural examples. From the examples we have automatically extracted Haskell programs that use the exact real computation framework AERN for efficiently performing exact operations on real numbers. In experiments, the extracted programs behave similarly to native implementations in AERN in terms of running time and memory usage.