Proof Of Existence Research Articles

Sparse Gaussian processes for solving nonlinear PDEs

This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by formulating the problem of finding a reproducing kernel Hilbert space (RKHS) to approximate a PDE solution. The approximated solution lies in the span of base functions generated by evaluating derivatives of different orders of kernels at sample points. However, the RKHS specified by GPs can result in an expensive computational burden due to the cubic computation order of the matrix inverse. We conjecture that a solution exists on a “condensed” subspace that can achieve similar approximation performance, and we propose a SGP-based method to reformulate the optimization problem in the “condensed” subspace. This significantly reduces the computation burden while retaining desirable accuracy. The paper rigorously formulates this problem and provides error analysis and numerical experiments to demonstrate the effectiveness of this method. The numerical experiments show that the SGP method uses fewer than half the uniform samples as inducing points and achieves comparable accuracy to the GP method using the same number of uniform samples, resulting in a significant reduction in computational cost.Our contributions include formulating the nonlinear PDE problem as an optimization problem on a “condensed” subspace of RKHS using SGP, as well as providing an existence proof and rigorous error analysis. Furthermore, our method can be viewed as an extension of the GP method to account for general positive semi-definite kernels.

A constructive proof of the Fulkerson–Ryser characterization of digraphic sequences

Analogous to the Erdos–Gallai inequalities that give necessary and sufficient conditions for the existence of a graph on [Formula: see text] vertices with prescribed vertex degrees, the Ryser–Fulkerson inequalities give necessary and sufficient conditions for the existence of a digraph on [Formula: see text] vertices with prescribed indegrees and outdegrees of the vertices. In this note, we give a short constructive proof of the sufficiency of the Ryser–Fulkerson inequalities.

Optimal measurement structures for contextuality applications

The Kochen-Specker (KS) theorem is a cornerstone result in the foundations of quantum mechanics describing the fundamental difference between quantum theory and classical non-contextual theories. Recently specific substructures termed 01-gadgets were shown to exist within KS proofs that capture the essential contradiction of the theorem. Here, we show these gadgets and their generalizations provide an optimal toolbox for contextuality applications including (i) constructing classical channels exhibiting entanglement-assisted advantage in zero-error communication, (ii) identifying large separations between quantum theory and binary generalized probabilistic theories, and (iii) finding optimal tests for contextuality-based semi-device-independent randomness generation. Furthermore, we introduce and study a generalization to definite prediction sets for more general logical propositions, that we term higher-order gadgets. We pinpoint the role these higher-order gadgets play in KS proofs by identifying these as induced subgraphs within KS graphs and showing how to construct proofs of state-independent contextuality using higher-order gadgets as building blocks. The constructions developed here may help in solving some of the remaining open problems regarding minimal proofs of the Kochen-Specker theorem.

Social networks, empowerment, and wellbeing among Syrian refugee and Jordanian women: Implications for development and social inclusion

In response to large-scale refugee crises, frameworks for development assistance have promoted women’s empowerment, wellbeing, and social inclusion. A productive research agenda lies in analyzing social networks: it is unknown how women structure their social ties within refugee and host communities, and whether social networks matter for their sense of empowerment and wellbeing. In 2022, we surveyed Syrian refugee (n = 106) and Jordanian (n = 109) women from poor households across five neighborhoods in Amman. We implemented a standard network survey instrument (PERSNET) to assess network structure and composition. We tested associations with six measures (PE, MRS, MTL, Cantril, PWB, MSPSS) of psychological empowerment and wellbeing. We then conducted participatory network mapping (Net-Map) to assess local meanings of empowerment and visually map the pathways between social actors, community-based work, and psychological outcomes. Survey data show that networks were highly homogeneous, smaller for Syrians than Jordanians (p = 0.0001), and smaller for women in very poor households (p < 0.0001). As network size increased, so did levels of psychological empowerment (p = 0.02), motivation to lead (p = 0.007) and perceived social support (p = 0.001). Notably, as networks became increasingly kin-based, empowerment levels decreased (p = 0.003). Networks were more diverse for community volunteers, who named fewer female, married, and kin-based peers (p ≤ 0.05), and reported higher levels of resourcefulness (p = 0.01) and psychological wellbeing (p = 0.002). Qualitative data show that women, who described empowerment as “ability” and “proof of existence,” drew upon volunteering work to diversify their networks outside the home. Such evidence matters for development initiatives that build programs for women to work, learn, and socially interact. We conclude that expanding opportunities for volunteer work is one way of diversifying social networks and empowering urban poor women. Our research helps better understand how women can be supported to diversify their social ties, take community leadership roles, and respond to social change.

Markov Decision Processes with Time-Varying Geometric Discounting

Canonical models of Markov decision processes (MDPs) usually consider geometric discounting based on a constant discount factor. While this standard modeling approach has led to many elegant results, some recent studies indicate the necessity of modeling time-varying discounting in certain applications. This paper studies a model of infinite-horizon MDPs with time-varying discount factors. We take a game-theoretic perspective – whereby each time step is treated as an independent decision maker with their own (fixed) discount factor – and we study the subgame perfect equilibrium (SPE) of the resulting game as well as the related algorithmic problems. We present a constructive proof of the existence of an SPE and demonstrate the EXPTIME-hardness of computing an SPE. We also turn to the approximate notion of epsilon-SPE and show that an epsilon-SPE exists under milder assumptions. An algorithm is presented to compute an epsilon-SPE, of which an upper bound of the time complexity, as a function of the convergence property of the time-varying discount factor, is provided.

Constructing knots with specified geometric limits

It is known that any tame hyperbolic 3-manifold with infinite volume and a single end is the geometric limit of a sequence of finite volume hyperbolic knot complements. Purcell and Souto showed that if the original manifold embeds in the 3-sphere, then such knots can be taken to lie in the 3-sphere. However, their proof was nonconstructive; no examples were produced. In this paper, we give a constructive proof in the geometrically finite case. That is, given a geometrically finite, tame hyperbolic 3-manifold with one end, we build an explicit family of knots whose complements converge to it geometrically. Our knots lie in the (topological) double of the original manifold. The construction generalises the class of fully augmented links to a Kleinian groups setting.

Well-posedness of fractional Moreau’s sweeping processes of Caputo type

In this paper, we study the well-posedness (in the sense of existence and uniqueness of a solution) of a sweeping process involving fractional order derivative of Caputo type in Hilbert spaces. The convex constraint set C(t) is assumed to satisfy a Hölder continuous variation. The notion of a solution of this fractional sweeping process is proposed, the existence and uniqueness of such solution are proved and a stability result is provided, as well. More precisely, we present new arguments for the proof of existence of the fractional sweeping process. It is still based on the ideas of the catching-up algorithm of J.J. Moreau (see Kunze and Monteiro Marques (2000) and Moreau (1977)). We establish a catching-up algorithm for fractional sweeping process. This new algorithm gives us a sequence of approximate solutions. The technical problems are to prove that we can extract a weakly convergent subsequence, and to check that this limit function is a solution of the fractional sweeping process. Here we propose a new approach to overcome these problems. The existence of solutions follows, to some extent, from an important differential inequality on the Caputo fractional derivative of the squared norm CD0α‖u(t)‖2≤2u(t),CD0αu(t), which was studied in Aguila-Camacho et al. (2014), Gomoyunov (2018) and Kamenskii et al. (2021), that applied to two approximate solutions, by the monotonicity of normal cone NC(t)(⋅) and the minus sign on the right hand side, yields that their distance is nonincreasing. This reasoning allows us to prove that the sequence of approximate solutions is of the fractional sweeping process, converging to a solution. Our approach is constructive and the algorithm is implemented numerically. The results of the paper are applied in the study of an application to non-regular electrical circuit containing nonsmooth electronic device like diode and fractional inductor, which represent an additional novelty of our paper. Finally, two numerical simulations to valid the theoretical results are given.

RENOVATING ENGINEERING DEPARTEMENTS’ CREATION HERITAGE TO MEET CONTEMPORARY CHALLENGES: FRUGAL VALIDATION PATTERNS AND CONSTRUCTIVE PROOF LOGICS FOR NEW ENGINEERING RULES

AbstractEngineering departments design infrastructure by applying rule systems. The latter are an old creation heritage, based on decades of engineering, that makes it possible to design and govern the operation of the physical heritage which is the infrastructure. Replacing the infrastructure is not sustainable in the meaning of grand challenges; renovating it by applying engineering rules is but could appear too expensive. The literature highlights situations where renovation by respecting the state of the art is too costly and so is the validation of new renovation rules. Are there forms of frugal validation that allow for sustainable renovation of existing systems? This paper tries to explore a third way, a renovation of the physical heritage from a renovation of the system of rules, conceiving in the system of rules, new propositions, and their validation. Using the C-K theory, a case analysis was performed within the French national rail network manager (SNCF Network), a company that has a historical engineering heritage and is at the same time implementing a renovation of it. The paper shows that the renovation of engineering departments’ creation heritage can go through frugal validation patterns and constructive proof logics.

A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators

The purpose of the research is the study of a nonconstant gradient constrained problem for nonlinear monotone operators. In particular, we study a stationary variational inequality, defined by a strongly monotone operator, in a convex set of gradient-type constraints. We investigate the relationship between the nonconstant gradient constrained problem and a suitable double obstacle problem, where the obstacles are the viscosity solutions to a Hamilton–Jacobi equation, and we show the equivalence between the two variational problems. To obtain the equivalence, we prove that a suitable constraint qualification condition, Assumption S, is fulfilled at the solution of the double obstacle problem. It allows us to apply a strong duality theory, holding under Assumption S. Then, we also provide the proof of existence of Lagrange multipliers. The elements in question can be not only functions in L2, but also measures.

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Revisiting the Lorenz ’63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}, we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average ‘transport,’ which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the nonzero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the nonzero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

We prove exponential expression rates of deep operator networks (deep ONets) between infinite-dimensional spaces that emulate the coefficient-to-solution map of linear, elliptic, second-order, divergence-form partial differential equations (PDEs). In particular, we consider problems set in -dimensional periodic domains, , with analytic right-hand sides and coefficients. Our analysis covers diffusion-reaction problems, parametric diffusion equations, and certain elliptic systems such as linear isotropic elastostatics in heterogeneous materials. In the constructive proofs, we leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the branch and trunk ONet architecture of Chen and Chen [IEEE. Trans. Nural Netw., 4 (1993), pp. 910–918] and of Lu et al. [Nat. Mach. Intell., 3 (2021), pp. 218–229], we construct deep ONets which emulate the coefficient-to-solution map to a desired accuracy in the norm, uniformly over the PDE coefficient set. The ONets have size , where is the approximation accuracy, for some depending on the physical space dimension.

Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward

Long-range entanglement--the backbone of topologically ordered states--cannot be created in finite time using local unitary circuits, or equivalently, adiabatic state preparation. Recently it has come to light that single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases. Here we show how this observation imposes a complexity hierarchy on long-range entangled states based on the minimal number of measurement layers required to create the state, which we call "shots". First, similar to Abelian stabilizer states, we construct single-shot protocols for creating any non-Abelian quantum double of a group with nilpotency class two (such as $D_4$ or $Q_8$). We show that after the measurement, the wavefunction always collapses into the desired non-Abelian topological order, conditional on recording the measurement outcome. Moreover, the clean quantum double ground state can be deterministically prepared via feedforward--gates which depend on the measurement outcomes. Second, we provide the first constructive proof that a finite number of shots can implement the Kramers-Wannier duality transformation (i.e., the gauging map) for any solvable symmetry group. As a special case, this gives an explicit protocol to prepare twisted quantum double for all solvable groups. Third, we argue that certain topological orders, such as non-solvable quantum doubles or Fibonacci anyons, define non-trivial phases of matter under the equivalence class of finite-depth unitaries and measurement, which cannot be prepared by any finite number of shots. Moreover, we explore the consequences of allowing gates to have exponentially small tails, which enables, for example, the preparation of any Abelian anyon theory, including chiral ones. This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.

Causes of proof construction failure in proof by contradiction

Failure to deduce false suppositions in proof by contradiction is still considered “more difficult” than proving the conditional to in proof by contraposition. This study aims to identify the types of proof construction failures based on the action steps of proof by contradiction, then offer a framework of construction failure hypothesis specifically used in proof by contradiction. The research data were collected and analyzed from the work of students who have agreed to be research participants, a total of 83 students. The results of the analysis of student work successfully identified four types of failures, namely formulating suppositions, constructing and manipulating suppositions, identifying contradictions, and disproving suppositions. These four types of failures then became the material for the development of the hypothesis framework of a failure to construct proof by contradiction, which consists of 17 hypothesis nodes divided into three main hypotheses, namely: operational (action), affective (emotional), and foundational (logical reasoning). The failure hypothesis framework justifies that the sources of the failure of proof construction in proof by contradiction are understanding of the act of producing a proof by contradiction, emotionality towards the coherence of the construction steps, disproving suppositions, beliefs, use of appropriate definitions-theorems and axioms, and cognitive tension in proof by contradiction; and formal logic of the act of producing a proof by contradiction, as well as differences in the underlying logic with other acts.

Automated Detection of Under-Constrained Circuits in Zero-Knowledge Proofs

As zero-knowledge proofs gain increasing adoption, the cryptography community has designed domain-specific languages (DSLs) that facilitate the construction of zero-knowledge proofs (ZKPs). Many of these DSLs, such as Circom, facilitate the construction of arithmetic circuits, which are essentially polynomial equations over a finite field. In particular, given a program in a zero-knowledge proof DSL, the compiler automatically produces the corresponding arithmetic circuit. However, a common and serious problem is that the generated circuit may be underconstrained, either due to a bug in the program or a bug in the compiler itself. Underconstrained circuits admit multiple witnesses for a given input, so a malicious party can generate bogus witnesses, thereby causing the verifier to accept a proof that it should not. Because of the increasing prevalence of such arithmetic circuits in blockchain applications, several million dollars worth of cryptocurrency have been stolen due to underconstrained arithmetic circuits. Motivated by this problem, we propose a new technique for finding ZKP bugs caused by underconstrained polynomial equations over finite fields. Our method performs semantic reasoning over the finite field equations generated by the compiler to prove whether or not each signal is uniquely determined by the input. Our proposed approach combines SMT solving with lightweight uniqueness inference to effectively reason about underconstrained circuits. We have implemented our proposed approach in a tool called QED 2 and evaluate it on 163 Circom circuits. Our evaluation shows that QED 2 can successfully solve 70% of these benchmarks, meaning that it either verifies the uniqueness of the output signals or finds a pair of witnesses that demonstrate non-uniqueness of the circuit. Furthermore, QED 2 has found 8 previously unknown vulnerabilities in widely-used circuits.

How do explicit and implicit evaluations shift? A preregistered meta-analysis of the effects of co-occurrence and relational information.

Based on 660 effect sizes obtained from 23,255 adult participants across 51 reports of experimental studies, this meta-analysis investigates whether and when explicit (self-reported) and implicit (indirectly revealed) evaluations reflect relational information (how stimuli are related to each other) over and above co-occurrence information (the fact that stimuli have been paired with each other). Using a mixed-effects metaregression, relational information was found to dominate over contradictory co-occurrence information in shifting both explicit (mean Hedges' g = 0.97, 95% CI [0.89, 1.05], 95% PI [0.24, 1.70]) and implicit evaluations (g = 0.27, 95% CI [0.19, 0.35], 95% PI [-0.46, 1.00]). However, considerable heterogeneity in relational effects on implicit evaluation made moderator analyses necessary. Implicit evaluations were particularly sensitive to relational information (a) in between-participant (rather than within-participant) designs; when (b) co-occurrence information was held constant (rather than manipulated); (c) targets were novel (rather than known); implicit evaluations were measured (d) first (rather than last) and (e) using an affect misattribution procedure (rather than an Implicit Association Test or evaluative priming task); and (f) relational and co-occurrence information were presented in temporal proximity (rather than far apart in time). Overall, the present findings suggest that both implicit and explicit evaluations emerge from a combination of co-occurrence information and relational information, with relational information usually playing the dominant role. Critically, variability in these effects highlights a need to refocus attention from existence proof demonstrations toward theoretical and empirical work on the determinants and boundary conditions of the influences of co-occurrence and relational information on explicit and implicit evaluations. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

PEMANFAATAN MEDIA SOSIAL SEBAGAI MEDIA PROMOSI DAN SUMBER INFORMASI TEREKAM PADA UKM BATIK BAMBU KENANGA

The role of social media is currently developing not only as a means of showing oneself or proof of existence. However, business people are starting to use social media as a promotion, because this media offers low costs with a wide market reach. Not only as promotional media, social media can also be used as document storage in an orderly and chronological manner, according to the time at which the documents were made. This is the aim of this activity, which is to maximize the role of social media for batik SMEs. Relatively low cost, easy operation (can be done via HP), and does not require skills that must be experts in the IT field. This activity was carried out at Batik Bambu Kenanga, located in Turen District, Malang Regency, East Java. The results show that the guidance and training provided was very useful and taught Batik Bambu Kenanga to compile promotional content as well as archive important documents, especially the batik products they produce.

A Comprehensive Formalization of Propositional Logic in Coq: Deduction Systems, Meta-Theorems, and Automation Tactics

The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ the Coq interactive theorem prover to methodically formalize the language, semantics, and syntax of propositional logic, a fundamental aspect of mathematical reasoning and proof construction. We construct four Hilbert-style axiom systems and a natural deduction system for propositional logic, and establish their equivalences through meticulous proofs. Moreover, we provide formal proofs for essential meta-theorems in propositional logic, including the Deduction Theorem, Soundness Theorem, Completeness Theorem, and Compactness Theorem. Importantly, we present an exhaustive formal proof of the Completeness Theorem in this paper. To bolster the proof of the Completeness Theorem, we also formalize concepts related to mappings and countability, and deliver a formal proof of the Cantor–Bernstein–Schröder theorem. Additionally, we devise automated Coq tactics explicitly designed for the propositional logic inference system delineated in this study, enabling the automatic verification of all tautologies, all internal theorems, and the majority of syntactic and semantic inferences within the system. This research contributes a versatile and reusable Coq library for propositional logic, presenting a solid foundation for numerous applications in mathematics, such as the accurate expression and verification of properties in software programs and digital circuits. This work holds particular importance in the domains of mathematical formalization, verification of software and hardware security, and in enhancing comprehension of the principles of logical reasoning.

Observation of the radiative decay of the 229Th nuclear clock isomer.

The radionuclide thorium-229 features an isomer with an exceptionally low excitation energy that enables direct laser manipulation of nuclear states. It constitutes one of the leading candidates for use in next-generation optical clocks1-3. This nuclear clock will be a unique tool for precise tests of fundamental physics4-9. Whereas indirect experimental evidence for the existence of such an extraordinary nuclear state is substantially older10, the proof of existence has been delivered only recently by observing the isomer's electron conversion decay11. The isomer's excitation energy, nuclear spin and electromagnetic moments, the electron conversion lifetime and a refined energy of the isomer have been measured12-16. In spite of recent progress, the isomer's radiative decay, a key ingredient for the development of a nuclear clock, remained unobserved. Here, we report the detection of the radiative decay of this low-energy isomer in thorium-229 (229mTh). By performing vacuum-ultraviolet spectroscopy of 229mTh incorporated into large-bandgap CaF2 and MgF2 crystals at the ISOLDE facility at CERN, photons of 8.338(24) eV are measured, in agreement with recent measurements14-16 and the uncertainty is decreased by a factor of seven. The half-life of 229mTh embedded in MgF2 is determined to be 670(102) s. The observation of the radiative decay in a large-bandgap crystal has important consequences for the design of a future nuclear clock and the improved uncertainty of the energy eases the search for direct laser excitation of the atomic nucleus.

Newtonian Fractional-Dimension Gravity and Galaxies without Dark Matter

We apply Newtonian fractional-dimension gravity (NFDG), an alternative gravitational model, to some notable cases of galaxies with little or no dark matter. In the case of the ultra-diffuse galaxy AGC 114905, we show that NFDG methods can effectively reproduce the observed rotation curve using a variable fractional dimension DR, as was performed for other galaxies in previous studies. For AGC 114905, we obtain a variable dimension in the range D≈ 2.2–3.2, but our fixed D = 3 curve can still fit all the experimental data within their error bars. This confirms other studies indicating that the dynamics of this galaxy can be described almost entirely by the baryonic mass distribution alone. In the case of NGC 1052-DF2, we use an argument based on the NFDG extension of the virial theorem applied to the velocity dispersion of globular clusters showing that, in general, discrepancies between observed and predicted velocity dispersions can be attributed to an overall fractal dimension D&lt;3 of the astrophysical structure considered, and not to the presence of dark matter. For NGC 1052-DF2, we estimate D≈2.9, thus confirming that this galaxy almost follows standard Newtonian behavior. We also consider the case of the Bullet Cluster merger (1E0657-56), assumed to be one of the strongest proofs of dark matter existence. A simplified but effective NFDG model of the collision shows that the observed infall velocity of this merger can be explained by a fractional dimension of the system in the range D≃ 2.4–2.5, again, without using any dark matter.

Equilibrium arrivals to a last-come first-served preemptive-resume queue

We consider a queueing system which opens at a given point in time and serves a finite number of users according to the last-come first-served discipline with preemptive-resume (LCFS-PR). Each user must decide individually when to join the queue. We allow for general classes of user preferences and service time distributions and show existence and uniqueness of a symmetric Nash equilibrium. Furthermore, we show that no continuous asymmetric equilibrium exists, if the population consists of only two users, or if arrival strategies satisfy a mild regularity condition. For an illustrative example, we implement a numerical procedure for computing the symmetric equilibrium strategy based on our constructive existence proof for the symmetric equilibrium. We then compare its social efficiency to that obtained if users are instead served on a first-come first-served (FCFS) basis.