Proof Of Existence Research Articles

Ergodic Archimedean dimers

We study perfect matchings, or close-packed dimer coverings, of finite sections of the eleven Archimedean lattices and give a constructive proof showing that any two perfect matchings can be transformed into each other using small sets of local ring-exchange moves. This result has direct consequences for formulating quantum dimer models with a resonating valence bond ground state, i.e., a superposition of all dimer coverings compatible with the boundary conditions. On five of the composite Archimedean lattices we supplement the sufficiency proof with translationally invariant reference configurations that prove the strict necessity of the sufficient terms with respect to ergodicity. We provide examples of and discuss frustration-free deformations of the quantum dimer models on two tripartite lattices.

Abdüllatif Harpûtî’nin İsbât-ı Vâcib Delilleri

Abdullatif Harpūtī (d.1916) was a late Ottoman scholar who worked as a professor of "Kalām" at the Istanbul Darulfunun. Recognizing the deficiencies in the science of kalām and arguing that some changes should be made in terms of method and content, Harpūtī states that the time has come for the third compilation in the science of kalām and that this compilation is obligatory upon the scholars of the century. He himself tried to serve as a model for this period with his book Tanqīḥ al-kalām. His study, which consists of muqaddimah (preface), divinity, prophecy, sam'iyyāt, and imāmate chapters, deals with the currents of thought that emerged in the Western world by bringing evidence against them. Harpūtī states that the scientific data on matters of faith, which are put forward by rational methods, change over time. For this reason, while writing chapters such as divinity and prophecy, he used different information in light of the scientific data of his time. He also included and refuted the views of movements such as materialism, positivism, and atheism, which were newly widespread in his time. In this paper, we have analyzed the proofs used by al Harpūtī on the subject of ithbāt al-wājib. Harpūtī analyzed the proofs of ithbāt al-wājib under the subheadings of "the proofs of philosophers and muhaqqiq theologians", "the path followed by most theologians", "the path followed by the prophets and asfiyān", and "the recent theologian philosophers and ithbāt al-wājib”. These subheadings do not clearly indicate which of the classical proofs he means. By "the proof of the philosophers and the muhaqqiq theologians" Harpūtī meant the evidence of possibility (imkān); by "the path followed by most theologians" he meant the proof of existence (huduth); and by “the path followed by the Prophets and asfiyān" he meant the evidence of purpose and order. Under the title “The recent theologian philosophers and ithbāt al-wājib", he collected the proof of possibility, the proof of motion, and the proofs of purpose and order. The proof of the disposition (fıtrah) that is included in our study is not to be found in the titles. However, from Harpūtī 's explanations, we have determined that he also used this argument. This study aims to reveal the proofs of ithbāt al-wājib used by al Harpūtī . This research is significant for us to comprehend what evidence Harpūtī , who opened the door to the new science of Kalām and is considered one of the important figures of the new science of Kalām, used for proof. The documentation method was used in this research, and the data obtained were analyzed and reflected in the text. As a result of the research, it was determined that al Harpūtī used the proofs of possibility, existence, purpose and order, motion, and disposition.

All maximal gauged supergravities with uplift

Abstract Generalized parallelizable spaces permit the uplift of many maximal gauged supergravities to ten or eleven dimensions. Although some of the former are explicitly known, the literature is still lacking their systematic construction. We resolve this issue by going beyond the known existence proofs and present the explicit construction of all generalized parallelizable spaces for maximal gauged supergravities in four or more dimensions. In the process, all embedding tensors that can be realized without breaking the section condition of exceptional field theory are identified. This addresses the old question: “Which maximal gauged supergravities can be embedded in the low-energy limit of superstring/M-theory” and provides the basis to explore solution-generating techniques based on generalized dualities.

On Unsteady Internal Flows of Incompressible Fluids Characterized by Implicit Constitutive Equations in the Bulk and on the Boundary

Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of incompressible fluids is nowadays available not only for Navier–Stokes fluids but also for various fluid models where the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient is nonlinear. The majority of such studies however concerns models where such a dependence is explicit (the stress is a function of the velocity gradient), which makes the class of studied models unduly restrictive. The same concerns boundary conditions, or more precisely the slipping mechanisms on the boundary, where the no-slip is still the most preferred condition considered in the literature. Our main objective is to develop a robust mathematical theory for unsteady internal flows of implicitly constituted incompressible fluids with implicit relations between the tangential projections of the velocity and the normal traction on the boundary. The theory covers numerous rheological models used in chemistry, biorheology, polymer and food industry as well as in geomechanics. It also includes, as special cases, nonlinear slip as well as stick–slip boundary conditions. Unlike earlier studies, the conditions characterizing admissible classes of constitutive equations are expressed by means of tools of elementary calculus. In addition, a fully constructive proof (approximation scheme) is incorporated. Finally, we focus on the question of uniqueness of such weak solutions.

Psychological AI: Designing Algorithms Informed by Human Psychology

Psychological artificial intelligence (AI) applies insights from psychology to design computer algorithms. Its core domain is decision-making under uncertainty, that is, ill-defined situations that can change in unexpected ways rather than well-defined, stable problems, such as chess and Go. Psychological theories about heuristic processes under uncertainty can provide possible insights. I provide two illustrations. The first shows how recency—the human tendency to rely on the most recent information and ignore base rates—can be built into a simple algorithm that predicts the flu substantially better than did Google Flu Trends’s big-data algorithms. The second uses a result from memory research—the paradoxical effect that making numbers less precise increases recall—in the design of algorithms that predict recidivism. These case studies provide an existence proof that psychological AI can help design efficient and transparent algorithms.

Admissible Ordering on Monomials is Well-Founded: A Constructive Proof

Admissible Ordering on Monomials is Well-Founded: A Constructive Proof

Periodic traveling wave solutions of discrete nonlinear Klein‐Gordon lattices

We prove the existence of periodic traveling wave solutions for general discrete nonlinear Klein‐Gordon systems, considering both cases of hard and soft on‐site potentials. In the case of hard on‐site potentials, we implement a fixed‐point theory approach, combining Schauder's fixed‐point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for nontrivial solutions to exist, energy (norm) thresholds for their existence, and upper bounds on their velocity. In the case of soft on‐site potentials, the proof of existence of periodic traveling wave solutions is facilitated by a variational approach based on the mountain pass theorem. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.

Existence of EFX for two additive valuations

Fair division of indivisible items is a well-studied topic in Economics and Computer Science. The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Envy-freeness is one of the most widely studied notions of fairness. Since complete envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent’s bundle. However, despite significant efforts by many researchers for several years, it is known that a complete EFX allocation always exists only in limited cases. In this paper, we show that a complete EFX allocation always exists when each agent is of one of two given types, where agents of the same type have identical additive valuations. This is the first such existence result for non-identical valuations when there are any number of agents and items and no limit on the number of distinct values an agent can have for individual items. We give a constructive proof, in which we iteratively obtain a Pareto dominating (partial) EFX allocation from an existing partial EFX allocation.

Extravesical ureteral ectopia from an iliac dystopic hypoplastic left kidney in a girl

Extravesical ureteral ectopia (EUE) is a rare cause of urinary incontinence (UI) in girls. In 80% of cases, ectopia is associated with ureteral duplication. We are reporting a case of a 17-year-old girl with UI by EUE from an iliac dystopic hypoplastic left kidney. The girl received outpatient treatment for UI and left kidney agenesia since the age of 5, when her parents for the first time noticed the UI with background normal urination. The child underwent a complete range of medical tests: CT, cystoscopy, renal nuclear scan, uroflowmetry, ureteral cystometry and profilometry. Although the examination revealed no proof of renal tissue existence on the left side, a pad test with a dye was performed to exclude an EUE. The test gave a positive result, which determined further diagnostic search. Vascular CT was performed and made it possible to visualize a severely hypoplastic left kidney dystopic towards the iliac vessels. Laparoscopic nephrectomy was performed and UI manifestations were corrected. Availability of a wide range of visualisation methods in the urologist's toolkit does not always allow to verify hypoplastic renal dystopia when the kidney is drained by a ureter with an extravesical ectopic orifice. When a reasonable suspicion of an EUE is combined with lack of information about its location and dimensions after visualizing tests, a pad test with a dye appears appropriate prior to a diagnostic laparoscopy in our opinion. Once the UU is confirmed by the dye test results, further diagnostic search using laparoscopic access is justified and can be transformed into a treatment procedure if the kidney is revealed.

Development of ACERA Learning Model Based on Proof Construction Analysis

Development of ACERA Learning Model Based on Proof Construction Analysis

Technology optimization for patient safety: a blockchain-based anesthesia record system architecture

Patient safety is acknowledged as a primary aim of anesthesiology. Anesthesia records constitute the main document of the intraoperative course of anesthesia administration. In this paper, we postulate that anesthesia record systems should be based on an integral tamper-proof design and provide specific technology characteristics to ensure data immutability, accessibility and transparency. Issues and limitations regarding current anesthesia record technologies are reviewed. We introduce a novel anesthesia record system designed for patient safety optimization which integrates dedicated hardware, blockchain technology and decentralized storage solutions. We propose an oracle network in which anesthesiologists run independent Sybil-resistant nodes which broadcast biosensor time series to decentralized storage systems and generate proofs of existence on public blockchains. Records are biometrically signed and incorporate information on the temporo-spatial relation between the anesthetized patient and the professional in charge through a unique personal-transponder wearable device. Compatibility for data science and machine learning implementation are discussed. Finally, we evaluate future impact and technological potential.

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.