Proof Technique Research Articles

A sequent calculus for first-order logic formalized in Isabelle/HOL

Abstract We formalize in Isabelle/HOL soundness and completeness of a one-sided sequent calculus for first-order logic. The completeness is shown via a translation from a semantic tableau calculus, whose completeness proof we base on the theory entry ‘First-Order Logic According to Fitting’ by Berghofer in the Archive of Formal Proofs. The calculi and proof techniques are taken from Ben-Ari’s textbook Mathematical Logic for Computer Science (Springer, 2012). We thereby demonstrate that Berghofer’s approach works not only for natural deduction but also constitutes a framework for mechanically checked completeness proofs for a range of proof systems.

A physics‐informed order‐of‐magnitude approach to handling dynamic iterations applied to models of physical systems: Theoretical framework

In this paper, we consider ‐dimensional systems of differential equations applied to model a range of physical phenomena, where represents any positive integer. The defining characteristic that we focus on are the magnitudes of the inherent physical parameters, which are frequently of different orders of magnitude for various physical systems. This property, inherent to the given physical problems, can be exploited to reduce the problems into a set of much simpler subproblems, expanding our ability to gain physical insight in terms of the underlying simpler subproblems. These orders of magnitude motivate the choice of a specific class of transformations that can be chosen to reduce the number of required dynamic iterations in a sequence‐based approach to examining such systems. This is also useful for the development of proof techniques for proving properties of mathematical models, such as their well‐posedness.

Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise

In the past decade, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that has discontinuities in space has begun. In the majority of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and that the diffusion coefficient is globally Lipschitz continuous and nondegenerate at the discontinuities of the drift coefficient. Under this type of assumptions the best Lp-error rate obtained so far for approximation of scalar SDEs at the final time is 3/4 in terms of the number of evaluations of the driving Brownian motion. In the present article, we prove for the first time in the literature sharp lower error bounds for such SDEs. We show that for a huge class of additive noise driven SDEs of this type the Lp-error rate 3/4 can not be improved. For the proof of this result we employ a novel technique by studying equations with coupled noise: we reduce the analysis of the Lp-error of an arbitrary approximation based on evaluation of the driving Brownian motion at finitely many times to the analysis of the Lp-distance of two solutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their values at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal random variables (Y,Z) by providing explicit lower bounds for the covariance Cov(f(Y),g(Z)) in case of piecewise Lipschitz continuous functions f and g. In addition it turns out that our proof technique also leads to lower error bounds for estimating occupation time functionals ∫01f(Wt)dt of a Brownian motion W, which substantially extends known results for the case of f being an indicator function.

313 Passive Flow-Reversal With Balloon-Guide Catheter for Transfemoral Carotid Artery Stenting: Proof of Concept, Technical Description and Experience

INTRODUCTION: The use of modern transfemoral balloon guide catheters (BGC) for flow reversal during carotid artery stenting (CAS) is scarcely described in literature but represents a promising and efficient technique for embolic protection. METHODS: We performed a retrospective analysis of patients ≥ 18-year-old who underwent elective CAS with the use of flow-reversal through the Walrus BGC. The BGC is positioned in the common carotid artery and inflated, followed by opening of the stop-cock for passive reversal of flow. We utilized a robotic transcranial Doppler monitoring during this technique for proof of concept. Patients’ characteristics, procedural details and clinical follow-up were evaluated. RESULTS: One-hundred and ten patients were included. Mean age was 69.1 years ± SD 9.6 and 30 (27.3%) were female. Most common comorbidities were hyperlipidemia (81.8%) and hypertension (73.6%). 54 (49.1%) patients were symptomatic. 102 (92.7%) patients had stenosis ≥70%. Contralateral stenosis ≥50% was present in 54 (49.1%) patients. Distal filters were used in 91 (82.7%) cases. Angioplasty was performed in 86 (78.2%) patients. Stenting was successful in 100% of cases. There were no periprocedural TIA or stroke. Stroke or TIA and mortality were both observed in only two (1.8%) patients during 30-days follow-up, of which one was due to non-compliance with antiplatelet regimen. CONCLUSIONS: In our institution’s experience, this technique was safe, feasible and efficient, with 100% technical success and no periprocedural thromboembolic complications. More extensive studies are needed to establish the role of proximal protection with flow-reversal using modern BGCs.

On the stability of unevenly spaced samples for interpolation and quadrature

Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper, the question of how the uneven distribution of the samples impacts the quality of interpolation and quadrature is analyzed. Starting with equally spaced nodes on $$[-\pi ,\pi )$$ with grid spacing h, suppose the unevenly spaced nodes are obtained by perturbing each uniform node by an arbitrary amount $$\le \alpha h$$ , where $$0 \le \alpha < 1/2$$ is a fixed constant. A discrete version of the Kadec-1/4 theorem is proved, which states that the nonuniform discrete Fourier transform associated with perturbed nodes has a bounded condition number independent of h, for any $$\alpha <1/4$$ . Then, it is shown that unevenly spaced quadrature rules converge for all continuous functions and interpolants converge uniformly for all differentiable functions whose derivative has bounded variation when $$0\le \alpha <1/4$$ . Though, quadrature rules at perturbed nodes can have negative weights for any $$\alpha >0$$ , a bound on the absolute sum of the quadrature weights is provided, which shows that perturbed equally spaced grids with small $$\alpha $$ can be used without numerical woes. While the proof techniques work primarily when $$0 \le \alpha < 1/4$$ , it is shown that a small amount of oversampling extends our results to the case when $$1/4\le \alpha <1/2$$ .

Critical Thinking Skills Development: An Analysis of Mathematics Curriculum 2006 (Grade-wise)

The current qualitative study was conducted to analyze the mathematics curriculum across grade levels for students’ critical thinking skills development. Qualitative content analysis was used to analyze the document with the facilitation of NVivo software. As per the analysis, Grades I-III cover core math concepts at an introductory level using real-life examples presented concretely. Then grades IV-VIII revisit these same ideas with more depth and add logical reasoning. Finally (IX-X) and higher secondary (XI-XII) levels aim to build a rigorous conceptual understanding of complex mathematical topics like calculus, vectors, and complex numbers as well as non-routine problem-solving capacity utilizing proof techniques and technology tools like MAPLE software. The research results show that the curriculum is indeed effective for 21st-century skills development such as critical thinking, problem-solving, and technological literacy. Students who complete this systematic program will be prepared for advanced studies or careers that require mathematical proficiency.

Privacy-Preserving Public Route Planning Based on Passenger Capacity

Precise route planning needs huge amounts of trajectory data recorded in multimedia devices. The data, including each user’s location privacy, are stored as cipher text. The ability to plan routes on an encrypted trajectory database is an urgent necessity. In this paper, in order to plan a public route while protecting privacy, we design a hybrid encrypted random bloom filter (RBF) tree on encrypted databases, named the encrypted random bloom filter (eRBF) tree, which supports pruning and a secure, fast k nearest neighbor search. Based on the encrypted random bloom filter tree and secure computation of distance, we first propose a reverse k nearest neighbor trajectory search on encrypted databases (RkNNToE). It returns all transitions, in which each takes the query trajectory as one of its k nearest neighbor trajectories on the encrypted database. The results can be the indicator of a new route’s capacity in route planning. The security of the trajectory and query is proven via the simulation proof technique. When the number of points in the trajectory database and transition database are 1174 and 18,670, respectively, the time cost of an R2NNToE is about 1200 s.

Blockchain-Assisted Personalized Car Insurance With Privacy Preservation and Fraud Resistance

It is well known that auto insurance companies (ICs) use personalized car insurance (PCI) to continuously track drivers' behavior to determine their auto premiums. However, drivers inevitably have concerns about the transparency of data collection/processing and the potential privacy leakage. In this paper, we propose a new PCI scheme to achieve privacy preservation and transparency with the assistance of a consortium blockchain. Specifically, a blockchain is first established by a group of consortium members, and each IC can deploy insurance contracts on the blockchain to support public verification of data collection/processing and thus fulfill the transparency requirement. Then a verifiable and privacy-preserving driving behavior evaluation protocol is designed by tailoring partially homomorphic encryption and zero-knowledge proof techniques. Drivers can use the protocol to interact with ICs through the contracts, and ICs can learn drivers' behavior and set corresponding auto premiums by analyzing encrypted driving data. Furthermore, a third-party auditor (TPA) is authorized by drivers and ICs to audit encrypted driving data on the contracts and resist fraud attacks. We model the contract-based auditing as a recursive inspection game where TPA can minimize the number of audits to detect data fraud and penalize malicious drivers according to Nash equilibrium. Therefore, the proposed PCI scheme can guarantee that most of the collected driving data are not biased. Formal simulation-based security analysis is given to prove the security of the proposed scheme, and a proof-of-concept prototype is also developed on an open-source blockchain to demonstrate the feasibility.

Stability tests and solution estimates for non-linear differential equations

This article deals with certain systems of delay differential equations (DDEs) and a system of ordinary differential equations (ODEs). Here, five new theorems are proved on the fundamental properties of solutions of these systems. The results on the properties of solutions consist of sufficient conditions and they dealt with uniformly asymptotically stability (UAS), instability and integrability of solutions of unperturbed systems of DDEs, boundedness of solutions of a perturbed system of DDEs at infinity and exponentially stability (ES) of solutions of a system of nonlinear ODEs. Here, the techniques of proofs depend upon the Lyapunov- Krasovski? functional (LKF) method and Lyapunov function (LF) method. For illustrations, in particular cases, four examples are constructed as applications. Some results of this paper are given at first time in the literature, and the other results generalize and improve some related ones in the literature.

Defective Ramsey Numbers and Defective Cocolorings in Some Subclasses of Perfect Graphs

In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to k-dense and k-sparse sets, both commonly called k-defective sets. We focus on the computation of defective Ramsey numbers restricted to some subclasses of perfect graphs. Since direct proof techniques are often insufficient for obtaining new values of defective Ramsey numbers, we provide a generic algorithm to compute defective Ramsey numbers in a given target graph class. We combine direct proof techniques with our efficient graph generation algorithm to compute several new defective Ramsey numbers in perfect graphs, bipartite graphs and chordal graphs. We also initiate the study of a related parameter, denoted by \(c^{{\mathcal {G}}}_k(m)\), which is the maximum order n such that the vertex set of any graph of order n in a class \({\mathcal {G}}\) can be partitioned into at most m subsets each of which is k-defective. We obtain several values for \(c^{{\mathcal {G}}}_k(m)\) in perfect graphs and cographs.

On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator

Abstract In this article, we considered the pseudo-parabolic equation with Caputo-Fabrizio fractional derivative. This equation has many applications in different fields, such as science, technology, and so on. In this article, we gave the formula of mild solution, which is represented in the form of Fourier series by some operators . In the linear case, we investigated the continuity of the mild solution with respect to the fractional order. For the nonlinear case, we investigated the existence and uniqueness of a global solution. The main proof technique is based on the Banach fixed point theorem combined with some Sobolev embeddings. For more detailed, we obtained two other interesting results: the continuity of mild solution with respect to the derivative order and the convergence of solution as the coefficient k approaches to zero.

MaxSAT Resolution and Subcube Sums

We study the MaxSAT Resolution (MaxRes) rule in the context of certifying unsatisfiability. We show that it can be exponentially more powerful than tree-like resolution, and when augmented with weakening (the system MaxResW), p -simulates tree-like resolution. In devising a lower bound technique specific to MaxRes (and not merely inheriting lower bounds from Res), we define a new proof system called the SubCubeSums proof system. This system, which p -simulates MaxResW, can be viewed as a special case of the semi-algebraic Sherali–Adams proof system. In expressivity, it is the integral restriction of conical juntas studied in the contexts of communication complexity and extension complexity. We show that it is not simulated by Res. Using a proof technique qualitatively different from the lower bounds that MaxResW inherits from Res, we show that Tseitin contradictions on expander graphs are hard to refute in SubCubeSums. We also establish a lower bound technique via lifting: for formulas requiring large degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums.

Eigenvalues and cycles of consecutive lengths

AbstractAs the counterpart of classical theorems on cycles of consecutive lengths due to Bondy and Bollobás in spectral graph theory, Nikiforov proposed the following open problem in 2008: What is the maximum such that for all positive and sufficiently large , every graph of order with spectral radius contains a cycle of length for each integer . We prove that , improving the existing bounds. Besides several novel ideas, our proof technique is partly inspired by the recent research on Ramsey numbers of star versus large even cycles due to Allen, Łuczak, Polcyn, and Zhang, and with aid of a powerful spectral inequality. We also derive an Erdős–Gallai‐type edge number condition for even cycles, which may be of independent interest.

Application of Jeevamrit Improves Soil Properties in Zero Budget Natural Farming Fields

Zero Budget Natural Farming (ZBNF), utilizing natural resources, multiple cropping systems, and cow-dung- and urine-based products to improve soil biology, has been practiced by thousands of farmers in India. However, without any scientific proof, this traditional and ancient technique is mocked as a bugged theory in the scientific community. In the current study, we have investigated the effect of Jeevamrit—cow-dung- and urine-based formulation—on soil chemical and microbial properties of the ZBNF field coupled with metagenomic analysis and the economics of ZBNF. The percentage increase in soil properties, such as organic carbon, available phosphorus, and available potassium, was recorded up to 46%, 439%, and 142%, respectively, while micronutrients, such as Zn, Fe, Cu, and Mn, also increased up to 98%, 23%, 62%, and 55%, respectively, from 2017 to 2019. Whole genome metagenomic analysis revealed that Proteobacteria were dominantly present, and bacterial phyla including Bacillus, Pseudomonas, Rhizobium, and Panibacillus. On the other hand, Ascomycota was the dominating fungal phyla present in the soil sample. Further, functional analysis showed a high representation of genes/enzymes involved in amino acids and carbohydrate metabolism contributing to soil fertility, plant growth, defense, and development. Additionally, the cost–benefit ratio of ZBNF was double the farmer’s practice when tested with the rice and wheat cropping system. The results from this study provide a new proof of concept and understanding of the potential of the ZBNF component, i.e., Jeevamrit, in improving soil properties.

The Path to Durable Linearizability

There is an increasing body of literature proposing new and efficient persistent versions of concurrent data structures ensuring that a consistent state can be recovered after a power failure or a crash. Their correctness is typically stated in terms of durable linearizability (DL), which requires that individual library operations appear to be executed atomically in a sequence consistent with the real-time order and, moreover, that recovering from a crash return a state corresponding to a prefix of that sequence. Sadly, however, there are hardly any formal DL proofs, and those that do exist cover the correctness of rather simple persistent algorithms on specific (simplified) persistency models. In response, we propose a general, powerful, modular, and incremental proof technique that can be used to guide the development and establish DL. Our technique is (1) general , in that it is not tied to a specific persistency and/or consistency model, (2) powerful , in that it can handle the most advanced persistent algorithms in the literature, (3) modular , in that it allows the reuse of an existing linearizability argument, and (4) incremental , in that the additional requirements for establishing DL depend on the complexity of the algorithm to be verified. We illustrate this technique on various versions of a persistent set, leading to the link-free set of Zuriel et al.

A unified proof technique for verifying program correctness with big-step semantics

The deductive verification of computer programs helps ascertain the absence of errors for all possible inputs and executions. Computer programs can be written in a plethora of languages. The identification of mechanisms for reasoning about the common features of different languages gives rise to language-independent program verification techniques. Existing language-independent verification techniques either work with small-step operational semantics or require the definition of a special-purpose logic. In addition, these techniques have only been demonstrated in verification tasks where the code of the target program is completely known. We present a language-independent verification technique that works with big-step operational semantics and a general-purpose logic. We illustrate the technique by verifying programs in languages of different paradigms. We use the technique to develop a method for verifying blockchain smart contracts that could call into unknown code and be reentered from the callee. All the theoretical results and verification examples are formalized in the Coq proof assistant.

A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Discrete‐time case

Fixed costs of ordering items or setting up a production process arise in many real‐life scenarios. In their presence, the most widely used ordering policy in the stochastic inventory literature is the [Formula: see text] policy. Optimality of [Formula: see text] policies and [Formula: see text]‐type policies have been examined for various inventory models, including those with the inventory level being reviewed in every period or continuously, finite and infinite horizons, discounted‐cost and average‐cost criteria, backlogging and lost‐sales practices, standard and generalized demands and/or costs, deterministic and stochastic lead times, single‐product and multi‐product settings, and coordinated pricing‐inventory decisions. We comprehensively survey the vast literature accumulated over seven decades in two papers. This paper is devoted to discrete‐time models, and the companion paper, also published in this journal issue, reviews continuous‐time models. We go over model specifications, proof techniques, specific results, and limitations of the articles published in the literature. We conclude each paper by providing corresponding suggestions for extensions and directions for future research.

Relative cluster tilting subcategories in an extriangulated category

&lt;abstract&gt;&lt;p&gt;Let $ \mathscr{B} $ be an extriangulated category which admits a cluster tilting subcategory $ \mathcal{T} $. We firstly introduce notions of $ \mathcal{T} $-cluster tilting subcategories and related subcategories. Then we prove there is a correspondence between $ \mathcal{T} $-cluster tilting subcategories of $ \mathscr{B} $ and support $ \tau $-tilting pairs of $ mod \underline{\Omega(\mathcal{T}}) $, which recovers several main results from the literature. Note that the generalization is nontrivial and we give a new proof technique.&lt;/p&gt;&lt;/abstract&gt;

A generalization of Kruskal’s theorem on tensor decomposition

Abstract Kruskal’s theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a ‘splitting theorem’ for sets of product tensors, in which the k-rank condition of Kruskal’s theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e., is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal’s theorem. While several extensions of Kruskal’s theorem are already present in the literature, all of these use Kruskal’s original permutation lemma and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions and can certify uniqueness below this threshold. We obtain several other useful results on tensor decompositions as consequences of our splitting theorem. We prove sharp lower bounds on tensor rank and Waring rank, which extend Sylvester’s matrix rank inequality to tensors. We also prove novel uniqueness results for nonrank tensor decompositions.

A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Continuous‐time case

Fixed costs of ordering items or setting up a production process arise in many real‐life scenarios. In their presence, the most widely used ordering policy in the stochastic inventory literature is the [Formula: see text] policy. Optimality of [Formula: see text] policies and [Formula: see text]‐type policies have been examined for various inventory models, including those with the inventory level being reviewed in every period or continuously, finite and infinite horizons, discounted‐cost and average‐cost criteria, backlogging and lost‐sales practices, standard and generalized demands and/or costs, deterministic and stochastic lead times, single‐product and multi‐product settings, and coordinated pricing‐inventory decisions. We comprehensively survey the vast literature accumulated over seven decades in two papers. This paper is devoted to continuous‐time models, and the companion paper, also published in this journal issue, reviews discrete‐time models. We go over model specifications, proof techniques, specific results, and limitations of the articles published in the literature. We conclude each paper by providing corresponding suggestions for extensions and directions for future research.