Normal Form Research Articles

A diffusive plant-sulphide model: spatio-temporal dynamics contrast between discrete and distributed delay

Abstract This paper studies the spatio-temporal dynamics of a diffusive plant-sulphide model with toxicity delay. More specifically, the effects of discrete delay and distributed delay on the dynamics are explored, respectively. The deep analysis of eigenvalues indicates that both diffusion and delay can induce Hopf bifurcations. The normal form theory is used to set up an exact formula that determines the properties of Hopf bifurcation in a diffusive plant-sulphide model. A sufficiently small discrete delay does not affect the stability and a sufficiently large discrete delay destabilizes the system. Nonetheless, a sufficiently small or large distributed delay does not affect the stability. Both delays cause instability by inducing Hopf bifurcation rather than Turing bifurcation.

Mathematical Model of a Nonlinear Electromagnetic Circuit Based on the Modified Hamilton–Ostrogradsky Principle

This paper presents a mathematical model of a typical lumped-parameter electromagnetic assembly, which consists of two subassemblies: one includes a magnetic circuit and the other with selected elements of electric circuits. An interdisciplinary research approach is used, which assumes the use of a modified integral method based on the variational Hamilton–Ostrogradsky principle. The modification of the method is the extension of the Lagrange function by two components. The first one reflects the dissipation of electromagnetic energy in the system, while the second one reflects the effect of external non-potential forces acting on the electromagnetic system. This approach allows for the avoidance of the inconvenience of the classical theory, which assumes the decomposition of the entire integrated system into individual electrical subsystems. The state equations of the electromagnetic subassembly are presented solely on the basis of the energy approach, which in turn allows taking into account various latent motions in the system, because the equations are derived based on non-stationary constraints between subsystems. The adopted theory allows for the formulation of the model of the system in a vector form, which gives much more possibilities for the analysis of higher-order electromagnetic circuits. Another important advantage is that the state equations of the considered electrical object are given in Cauchy normal form. In this way, the equations can be integrated both explicitly and implicitly. The results of computer simulations are presented in graphical form, analysed, and discussed.

SAT Solving Using XOR-OR-AND Normal Forms

This paper introduces the XOR-OR-AND normal form (XNF) for logical formulas. It is a generalization of the well-known Conjunctive Normal Form (CNF) where literals are replaced by XORs of literals. As a first theoretic result, we show that every CNF formula is equisatisfiable to a formula in 2-XNF, i.e., a formula in XNF where each clause involves at most two XORs of literals. Subsequently, we present an algorithm which converts Boolean polynomials efficiently from their Algebraic Normal Form (ANF) to formulas in 2-XNF. Experiments with the cipher ASCON-128 show that cryptographic problems, which by design are based strongly on XOR-operations, can be represented using far fewer variables and clauses in 2-XNF than in CNF. In order to take advantage of this compact representation, new SAT solvers based on input formulas in 2-XNF need to be designed. By taking inspiration from graph-based 2-CNF SAT solving, we devise a new DPLL-based SAT solver for formulas in 2-XNF. Among others, we present advanced pre- and in-processing techniques. Finally, we give timings for random 2-XNF instances and instances related to key recovery attacks on round reduced ASCON-128, where our solver outperforms state-of-the-art alternative solving approaches.

On the bifurcations in a quadrotor unmanned aerial vehicle dynamical system using normal form theory

On the bifurcations in a quadrotor unmanned aerial vehicle dynamical system using normal form theory

Verifying chip designs at RTL level

As chip designs become increasingly complex, the potential for errors and defects in circuits inevitably rises, posing significant challenges to chip security and reliability. This study investigates the use of the SAT-based bounded model checking (BMC) for Propositional Projection Temporal Logic (PPTL) to verify Verilog chip designs at the register transfer level (RTL). To this end, we propose an algorithm to implement automated extraction of state transfer relations from AIGER netlist and construction of Kripke structure. Additionally, we employ PPTL with the full regular expressiveness to describe the circuit properties to be verified, especially the periodic repetitive properties. This is not possible with Linear Temporal Logic (LTL) and Computational Tree Logic (CTL). By combining the PPTL properties with finite system paths and transforming them into conjunctive normal forms (CNFs), we utilize an SAT solver for verification. Experimental results demonstrate that our verification tool, SAT-BMC4PPTL, achieves higher verification efficiency and comprehensiveness.

Asymptotic behavior of solutions of nonlinear discrete systems in critical cases

AbstractThe paper is dedicated to the analysis of the asymptotic behavior of essentially nonlinear discrete systems whose linearization possesses eigenvalues on the unit circle. For these systems, the paper establishes sufficient conditions for the asymptotic stability regardless of terms higher than the third order. Moreover, it derives an asymptotic stability criterium for a reduced critical subsystem. The proof incorporates a constructive approach for deriving Lyapunov functions using the center manifold reduction and normal form method. The key result of the paper asserts that, given the obtained stability conditions, the solutions of the system converge to the origin with a polynomial decay rate. The paper also introduces a method for evaluating the coefficients of the decay rate estimate. To illustrate these findings, we apply the obtained result to analyzing nonlinear business cycle and predator-pray models.

Small-amplitude synchronization in driven Potts models

We study driven q-state Potts models with thermodynamically consistent dynamics and global coupling. For a wide range of parameters, these models exhibit a dynamical phase transition from decoherent oscillations into a synchronized phase. Starting from a general microscopic dynamics for individual oscillators, we derive the normal form of the high-dimensional Hopf bifurcation that underlies the phase transition. The normal-form equations are exact in the thermodynamic limit and close to the bifurcation. Exploiting the symmetry of the model, we solve these equations and thus uncover the intricate stable synchronization patterns of driven Potts models, characterized by a rich phase diagram. Making use of thermodynamic consistency, we show that synchronization reduces dissipation in such a way that the most stable synchronized states dissipate the least entropy. Close to the phase transition, our findings condense into a linear dissipation-stability relation that connects entropy production with phase-space contraction, a stability measure. At finite system size, our findings suggest a minimum-dissipation principle for driven Potts models that holds arbitrarily far from equilibrium. Published by the American Physical Society 2024

Bridging Data Models in Health Care With a Novel Intermediate Query Format for Feasibility Queries: Mixed Methods Study.

To advance research with clinical data, it is essential to make access to the available data as fast and easy as possible for researchers, which is especially challenging for data from different source systems within and across institutions. Over the years, many research repositories and data standards have been created. One of these is the Fast Healthcare Interoperability Resources (FHIR) standard, used by the German Medical Informatics Initiative (MII) to harmonize and standardize data across university hospitals in Germany. One of the first steps to make these data available is to allow researchers to create feasibility queries to determine the data availability for a specific research question. Given the heterogeneity of different query languages to access different data across and even within standards such as FHIR (eg, CQL and FHIR Search), creating an intermediate query syntax for feasibility queries reduces the complexity of query translation and improves interoperability across different research repositories and query languages. This study describes the creation and implementation of an intermediate query syntax for feasibility queries and how it integrates into the federated German health research portal (Forschungsdatenportal Gesundheit) and the MII. We analyzed the requirements for feasibility queries and the feasibility tools that are currently available in research repositories. Based on this analysis, we developed an intermediate query syntax that can be easily translated into different research repository-specific query languages. The resulting Clinical Cohort Definition Language (CCDL) for feasibility queries combines inclusion criteria in a conjunctive normal form and exclusion criteria in a disjunctive normal form, allowing for additional filters like time or numerical restrictions. The inclusion and exclusion results are combined via an expression to specify feasibility queries. We defined a JSON schema for the CCDL, generated an ontology, and demonstrated the use and translatability of the CCDL across multiple studies and real-world use cases. We developed and evaluated a structured query syntax for feasibility queries and demonstrated its use in a real-world example as part of a research platform across 39 German university hospitals.

Normal form of Turing–Turing bifurcation for the diffusive Bazykin system with prey-taxis

In this paper, we introduce prey-taxis to the diffusive Bazykin system and study the codimension-two Turing–Turing bifurcation of this modified system. For the local system, i.e. without diffusion terms and the prey-taxis term, we investigate the stability of the unique positive equilibrium. Then, for the diffusive Bazykin system with prey-taxis, we examine the stability of the unique positive constant steady state, Turing instability, and Turing bifurcation. Moreover, by choosing the random diffusion coefficients as bifurcation parameters, we investigate the Turing–Turing bifurcation of the diffusive Bazykin system with prey-taxis. Furthermore, to demonstrate and classify the spatiotemporal dynamics near the Turing–Turing bifurcation point, we derive an algorithm for calculating the third-order truncated normal form of this bifurcation using the center manifold theorem and normal form theory. Finally, to verify the theoretical analysis results and demonstrate the effectiveness of the derived algorithm, we perform several numerical simulations using matlab software. We also investigate the influence of the prey-taxis coefficient on the Turing instability of the diffusive Bazykin system with prey-taxis.

Normalization of Singular Contact Forms and Primitive 1-forms

AbstractA differential 1-form $$\alpha $$ α on a manifold of odd dimension $$2n+1$$ 2 n + 1 , which satisfies the contact condition $$\alpha \wedge (d\alpha )^n \ne 0$$ α ∧ ( d α ) n ≠ 0 almost everywhere, but which vanishes at a point O, i.e., $$\alpha (O) = 0$$ α ( O ) = 0 , is called a singular contact form at O. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form $$\omega $$ ω in dimension 2n, i.e., differential 1-forms $$\gamma $$ γ which vanish at a point and such that $$d\gamma = \omega $$ d γ = ω , and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin (1975), Webster (Amer. J. Math. 109, 807–832 (1987)) and Zhitomirskii (1986, 1992). Besides the classical normalization techniques, such as the step-by step normalization methods based on the cohomological equations and the Moser path method, we also use the toric approach to the normalization problem for dynamical systems (Jiang et al. 2019; Zung, Ann. Math. 161, 141–156 2005; Zung 2016; Zung, Arch. Rational Mech. Anal. 229, 789–833 (2018)).

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

A Low-Level Look at A-Normal Form

A-normal form (ANF) is a widely studied intermediate form in which local control and data flow is made explicit in syntax, and a normal form in which many programs with equivalent control-flow graphs have a single normal syntactic representation. However, ANF is difficult to implement effectively and, as we formalize, difficult to extend with new lexically scoped constructs such as scoped region-based allocation. The problem, as has often been observed, is that normalization of commuting conversions is hard. This traditional view of ANF that normalizing commuting conversions is hard, found in formal models and informed by high-level calculi, is wrong. By studying the low-level intensional aspects of ANF, we can derive a normal form in which normalizing commuting conversion is easy, does not require join points, or code duplication, or renormalization after inlining, and is easily extended with new lexically scoped effects. We formalize the connection between ANF and monadic form and their intensional properties, derive an imperative ANF, and design a compiler pipeline from an untyped -calculus with scoped regions, to monadic form, to a low-level imperative monadic form in which A-normalization is trivial and safe for regions. We prove that any such compiler preserves, or optimizes, stack and memory behaviour compared to ANF. Our formalization reconstructs and systematizes pragmatic choices found in practice, including current production-ready compilers. The main take-away from this work is that, in general, monadic form should be preferred over ANF, and A-normalization should only be done in a low-level imperative intermediate form. This maximizes the advantages of each form, and avoids all the standard problems with ANF.

Mitigating Normal Form Violations Using Resources-Events-Agents, Resources-Events-Agents-Locations, and Hierarchy: A Knowledge-Based Approach

ABSTRACT Resources-Events-Agents (REA) accounting database systems knowledge can facilitate identification of normal form in databases that are not REA-compliant. This paper examines several database snippets, drawn from Oracle database materials, that have normal form violations and determines that knowledge of REA/REAL entity models (REA-Compliance) provides information that can be used to find those normal form violations in relational tables. We find that REA/REAL knowledge provides a capability that has not been fully leveraged, providing a link between the real-world and academic accounting database systems. Data Availability: Data are published in the paper. JEL Classifications: M4.

Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

Generation of affinity maps for thiazolidinediones with human serum albumin using affinity microcolumns. II. Effects of advanced glycation end products on multisite drug binding

Multisite protein interactions by the thiazolidinedione-class drugs pioglitazone and rosiglitazone were examined by using high-performance affinity microcolumns that contained normal human serum albumin (HSA) vs HSA that had been modified to form advanced glycation end products by glyoxal (Go) or methylglyoxal (MGo). The results were used to generate an affinity map for these drugs at several key regions on HSA. Strong binding (∼105 M−1) by these drugs was seen at both Sudlow sites I and II. About a 50 % decrease in the affinities at Sudlow site II was observed for pioglitazone for Go-modified HSA, while either a 47 % decrease or 1.6-fold increase in affinity was seen for MGo-modified HSA, depending on the extent of modification. The binding affinity for rosiglitazone at Sudlow site II had a 40–83 % decrease for Go-modified HSA and either a non-significant change or 1.4-fold increase for MGo-modified HSA. At Sudlow site I, pioglitazone gave a 41 % decrease in affinity for either Go or MGo-modified HSA, and for rosiglitazone up to a 55 % decrease or 1.3-fold increase in affinity was noted. Positive allosteric effects were seen by these drugs with the tamoxifen site of HSA, and neither drug had any notable binding at the digitoxin site for the normal or modified forms of HSA. Rosiglitazone also had weak interactions at a site in subdomain IB, which increased in affinity by up to 5.0-fold with the Go- or MGo-modified HSA. This study illustrated how affinity microcolumns can be used to provide a detailed analysis of solute-protein systems that involve complex interactions. The data obtained should also be valuable in providing a better understanding of how drug interactions with HSA and other proteins can be altered by modifications of these binding agents in diseases such as diabetes.

Complex dynamics of an SIHR epidemic model with variable hospitalization rate depending on unoccupied hospital beds

In this paper, we propose an susceptible-infectious-hospitalized-recovered (SIHR) epidemic model with a nonlinear hospitalization rate depending on the number of unoccupied hospital beds. Note that the number of all hospital beds is used as a measure of all available medical resources. The basic reproduction number is calculated using the next-generation matrix method. We analyze the existence of endemic equilibria and discuss the global stability of the disease-free equilibrium. Existence and stability of endemic equilibria indicate possible occurrences of bifurcations. We confirm the appearance of backward bifurcation, saddle–node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation using normal form theory and central manifold theory. Numerical simulations show that the dynamic behavior of the model undergoes a transition from forward bifurcation to backward bifurcation and saddle–node bifurcation when the number of total hospital beds is reduced. Our findings suggest that when the number of total hospital beds falls below a threshold, backward bifurcation will occur, meaning that the disease cannot be eliminated even if the basic reproduction number is below unity. Therefore, the number of hospital beds should be increased beyond the bed threshold during an outbreak of a disease, which has important implications for disease control.

Multiteam Semantics for Interventionist Counterfactuals: Probabilities and Causation

AbstractIn (Barbero and Sandu 2020 Journal of Philosophical Logic, 50, 471-521), we showed that languages encompassing interventionist counterfactuals and causal notions based on them (as e.g. in Pearl’s and Woodward’s manipulationist approaches to causation) as well as information-theoretic notions (such as learning and dependence) can be interpreted in a semantic framework which combines the traditions of structural equation modeling and of team semantics. We now present a further extension of this framework (causal multiteams) which allows us to talk about probabilistic causal statements. We analyze the expressivity resources of two causal-probabilistic languages, one finitary and one infinitary. We show that many causal-probabilistic notions from the literature on causal inference can already be expressed in the finitary language, and we prove a normal form theorem that throws a new light on Pearl’s “ladder of causation”. In addition, we provide an exact semantic characterization of the infinitary language, which shows that this language captures precisely those causal-probabilistic statements that do not commit us to any specific interpretation of probability; and we prove that no usual, countable language is apt for this task.

Learning the latent dynamics of fluid flows from high-fidelity numerical simulations using parsimonious diffusion maps

We use parsimonious diffusion maps (PDMs) to discover the latent dynamics of high-fidelity Navier–Stokes simulations with a focus on the two-dimensional (2D) fluidic pinball problem. By varying the Reynolds number Re, different flow regimes emerge, ranging from steady symmetric flows to quasi-periodic asymmetric and chaos. The proposed non-linear manifold learning scheme identifies in a crisp manner the expected intrinsic dimension of the underlying emerging dynamics over the parameter space. In particular, PDMs estimate that the emergent dynamics in the oscillatory regime can be captured by just two variables, while in the chaotic regime, the dominant modes are three as anticipated by the normal form theory. On the other hand, proper orthogonal decomposition/principal component analysis (POD/PCA), most commonly used for dimensionality reduction in fluid mechanics, does not provide such a crisp separation between the dominant modes. To validate the performance of PDMs, we also compute the reconstruction error, by constructing a decoder using geometric harmonics (GHs). We show that the proposed scheme outperforms the POD/PCA over the whole Re number range. Thus, we believe that the proposed scheme will allow for the development of more accurate reduced order models for high-fidelity fluid dynamics simulators, relaxing the curse of dimensionality in numerical analysis tasks such as bifurcation analysis, optimization, and control.

Binomial Transforms of k-Narayana Sequences and Some Properties

The aim of the study is to obtain new binomial transforms for the $k-$ Narayana sequence. The first of these is the binomial transform, which is its normal form, and in the first step, after finding the recurrence relation of this new binomial transform, the generating function and Binet formula were obtained. Finally, Pascal's triangle was calculated. In the rest of the article, $k-$binomial transform was performed for the $k-$ Narayana sequence and the recurrence relation, generating function, Binet formula and Pascal's triangle were examined for the new sequence obtained. Then, by performing the falling binomial transform and the rising binomial transform, the features listed above were found again for these sequences.

An [formula omitted]-adic algorithm for bivariate Gröbner bases

Let A be a domain, with m⊆A a maximal ideal, and let F⊆A[x,y] be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field K of A). We present a randomized m-adic algorithm to recover the lexicographic Gröbner basis G of 〈F〉⊆K[x,y], or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set F of cardinality greater than two. We use this result to bound the size of the coefficients of G, and to control the probability of choosing a good maximal ideal m⊆A. We give a complete cost analysis over number fields (K=Q(α)) and function fields (▪), and we obtain a complexity that is less than cubic in terms of the dimension of K/〈G〉 and softly linear in the size of its coefficients.