First-order Formulas Research Articles

On the number of tuples of group elements satisfying a first-order formula

Abstract Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation x n = 1 x^{n}=1 in a finite group and the Solomon theorem (1969) on the number of solutions in a group to systems of equations with fewer equations than unknowns. Instead of systems of equations, we consider arbitrary first-order formulae in the group language with constants. Our result substantially generalizes the Klyachko–Mkrtchyan theorem (2014) on this topic.

A refined first-order expansion formula in [formula omitted]: Application to interpolation and finite element error estimates

The aim of this paper is to derive a refined first-order expansion formula in Rn, the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor’s formula. For a given function, the formula we derived is obtained by introducing a linear combination of the first derivatives, computed at n+1 equally spaced points. We show how this formula can be applied to two important applications: the interpolation error and the finite elements error estimates. In both cases, we illustrate under which conditions a significant improvement of the errors can be obtained, namely how the use of the refined expansion can reduce the upper bound of error estimates.

Robust secure resource optimization for active STAR-RIS systems

In this paper, we investigate the robust secure resource optimization for the active simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) system under the imperfect eavesdroppers channel state information. Considering the fairness, a max–min secure rate optimization problem is formulated based on several practical constraints. To deal with the original non-convex problem, an alternative iteration algorithm is proposed. First, the original problem is decomposed into two non-convex sub-problems. Next, the continuous convex approximation and S-procedure techniques are applied to deal with the non-convex and uncertainty constraints, respectively. Then, the first-order Taylor expansion formula is utilized to approximate the convex difference form of the objective function. Finally, two sub-problems are transformed into convex ones and alternately solved until convergence. Simulation results show that the secure rate of the proposed scheme is higher than the conventional schemes.

Enhancing interpolation and approximation error estimates using a novel Taylor-like formula

In this paper, we present an approach to enhance interpolation and approximation error estimates. Based on a previously derived first-order Taylor-like formula, we demonstrate its applicability in improving the P1-interpolation error estimate. Following the same principles, we have also developed a novel numerical scheme for the heat equation that provides a better error estimate compared to the classical implicit finite differences scheme. Numerical illustrations confirm this behavior.

Generalized kernel distance covariance in high dimensions: non-null CLTs and power universality

Abstract Distance covariance is a popular dependence measure for two random vectors $X$ and $Y$ of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that $X$ and $Y$ are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert–Schmidt) kernel distance covariance in high dimensions, in the distributional class of $(X,Y)$ with a separable covariance structure. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between $X$ and $Y$. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by $n\cdot \operatorname{dCor}^{2}(X,Y)/\sqrt{2}$ in the high-dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between $X$ and $Y$.

Possible Tetraquark Explanation for the J/ψK0S Structure Observed in Proton-Proton Collision Data at Center-of-Mass Energies of 7, 8, and 13 TeV

Bevelacqua Resources, 7531 Flint Crossing Circle SE, Owens Cross Roads, AL 35763 USA bevelresou@aol.com The LHCb Collaboration provided data that supports a J/ψK0S structure, denoted as Tθψs1(4000)0, using an amplitude analysis of B0 → J/ψϕK0S decay. The structure of the J/ψK0S is investigated using a first-order tetraquark mass formula. This mass relationship is based on weakly bound J/ψ plus K0S meson clusters. The first-order tetraquark mass formula provides a predicted mass of 3694 MeV/c2 that is within about 8% of the experimental value of 3991 MeV/c2 with a Jπ = 1+ prediction. The data does not distinguish between 1+ and 1¯ states. KEYWORDS: Tetraquark Model for the Tθψs1(4000)0; First-Order Mass Formula; Quark Model; Cluster Model.

Possible Description of the Excited States of the Neutral Charmed Omega in Terms of a First-Order Pentaquark Mass Formula

Bevelacqua Resources, 7531 Flint Crossing Circle SE, Owens Cross Roads, AL 35763 USA bevelresou@aol.com Recently proposed sscqq-bar pentaquark structures for the excited states of the neutral charmed omega are investigated using a first-order mass formula incorporating weakly interacting qq-bar meson and Ω0c baryon clusters. The results are compared to the predictions of the LHCb Collaboration. The first-order pentaquark mass formula has typically yielded results within about 10% of the experimental mass values. Using this criterion, the Ω0c(2770), Ω0c(3000), Ω0c(3050), Ω0c(3065), Ω0c(3090), Ω0c(3119), and Ω0c(3185) states could be described by the ssc + uu-bar and ssc + dd-bar pentaquarks with the meson clusters uu-bar and dd-bar having a pseudoscalar configuration. The Ω0c(3185) is also described by the ssc + ss-bar with the ss-bar in a pseudoscalar state. In a similar manner, the Ω0c(3327) state could be described by the ssc + uu-bar and ssc + dd-bar pentaquarks with the meson clusters uu-bar and dd-bar having a vector configuration, and a ssc + ss-bar pentaquark with the meson cluster ss-bar having a pseudoscalar configuration. The model is not sufficiently refined to distinguish between these states. The only state with a specified Jπ assignment is the Ω0c(2770). The Particle Data Group specifies a 3/2+ assignment. The first-order pentaquark allows both 1/2+ and 3/2+ values for all Ω0c excited states. KEYWORDS: Pentaquark, first-order mass formula, quark model, excited states of the neutral charmed omega

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid’s fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme.

Metric Temporal Equilibrium Logic over Timed Traces

AbstractIn temporal extensions of answer set programming (ASP) based on linear time, the behavior of dynamic systems is captured by sequences of states. While this representation reflects their relative order, it abstracts away the specific times associated with each state. However, timing constraints are important in many applications like, for instance, when planning and scheduling go hand in hand. We address this by developing a metric extension of linear-time temporal equilibrium logic, in which temporal operators are constrained by intervals over natural numbers. The resulting Metric Equilibrium Logic (MEL) provides the foundation of an ASP-based approach for specifying qualitative and quantitative dynamic constraints. To this end, we define a translation of metric formulas into monadic first-order formulas and give a correspondence between their models in MEL and Monadic Quantified Equilibrium Logic, respectively. Interestingly, our translation provides a blue print for implementation in terms of ASP modulo difference constraints.

Possible Hexaquark Explanation for the 4406.9 MeV/c2 Resonance in Electron-Positron Collisions at Center-of-Mass Energies from 4.23 to 4.70 GeV

Bevelacqua Resources, 7531 Flint Crossing Circle SE, Owens Cross Roads, AL 35763 USA The BESIII Collaboration observed two resonance structures in the e+ e¯ → π+ π− ψ(3823) reactions at center-of-mass energies from 4.23 to 4.70 GeV. The first resonance (R1) has a mass of 4406.9 MeV/c2 and is based on a π+ π¯ Ψ(3823) structure. A second resonance (R2) is based on a π+ π¯ γ ηc1 structure with a mass of 4647.9 MeV/c2. The spin and parity of these states was not provided in Ref. 1. A first-order hexaquark mass formula is applied to evaluate the possible mass and Jπ value of the candidate R1 hexaquark. The R2 state, with a photon included in the final state, is not consistent with the first-order hexaquark model, and is not evaluated. The first-order hexaquark model yields a R1 resonance mass within 1% of the experimental value. The model also suggests that the R1 has a 2¯ spin assignment. KEYWORDS: First Order Mass Formula, Hexaquark, Weak Coupling Model, π+ π¯ Ψ(3823) structure, and R1 Resonance at 4406.9 MeV/c2.

Numerical Study on Heat Transfer and Release Characteristics of Key Components in Electrically Heated Tobacco Products

Electrically heated tobacco products (EHTPs) could release effective aerosol components from tobacco materials at relatively low temperatures without a burning phenomenon. It is essential to grasp the temperature distribution and release mechanism of key components in heated tobacco materials. The existing experimental studies have provided initial insights into the thermodynamic behavior of tobacco materials under various conditions. However, current numerical models are still in their early stages of development, with the majority failing to correlate heat transfer with component release. Based on this, a coupled numerical model of gas flow, heat transfer, and the release of key components in the electrically heated tobacco product is established in this study, which exhibits improvements in revealing the internal heat and mass transfer characteristics in the porous media of tobacco and is capable of evaluating the influence of component contents and product design parameters. The release rates of water, glycerol, and nicotine components are quantitatively described by the first-order Arrhenius formula, and the transport of heat and gas flow is simulated using the Navier-Stokes equation. The accuracy of the model is validated through experiments, including temperature monitoring at multiple measurement points and determination of residual contents in the tobacco substrate after each puff. The simulation results suggest that an appropriate component ratio and tobacco filler mass can enhance both the release amount and release efficiency of key components, and reducing either the diameter or length of the tobacco section can help to improve the heat transfer performance. A slower heating rate matched with longer preheating times enables the complementary release of water and glycerol components, which helps to regulate the uniformity of component content in the aerosol to some extent. This study helps to provide suggestions for the design and optimization of electrically heated tobacco products.

A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2−1σ formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(τF2+τC4+hx2+hy2), where τF and τC are the time step sizes of fine grid and coarse grid respectively, while hx and hy are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.

Lifted algorithms for symmetric weighted first-order model sampling

Weighted model counting (WMC) is the task of computing the weighted sum of all satisfying assignments (i.e., models) of a propositional formula. Similarly, weighted model sampling (WMS) aims to randomly generate models with probability proportional to their respective weights. Both WMC and WMS are hard to solve exactly, falling under the #P-hard complexity class. However, it is known that the counting problem may sometimes be tractable, if the propositional formula can be compactly represented and expressed in first-order logic. In such cases, model counting problems can be solved in time polynomial in the domain size, and are known as domain-liftable. The following question then arises: Is it also the case for WMS? This paper addresses this question and answers it affirmatively. Specifically, we prove the domain-liftability under sampling for the two-variables fragment of first-order logic with counting quantifiers in this paper, by devising an efficient sampling algorithm for this fragment that runs in time polynomial in the domain size. We then further show that this result continues to hold even in the presence of cardinality constraints. To empirically validate our approach, we conduct experiments over various first-order formulas designed for the uniform generation of combinatorial structures and sampling in statistical-relational models. The results demonstrate that our algorithm outperforms a state-of-the-art WMS sampler by a substantial margin, confirming the theoretical results.

Possible Tetraquark Explanation for the ψ(4230), ψ(4360), and ψ(4415)

Bevelacqua Resources, 7531 Flint Crossing Circle SE, Owens Cross Roads, AL 35763 USA Ji et al. utilize heavy quark spin symmetry, to predict of the hadronic molecular characteristics of the ψ(4230), ψ(4360), and ψ(4415). Using symmetry arguments these states were identified by Ji et al. as DD1-bar, D*D1-bar, and D*D2*-bar bound states. The structure of the ψ(4230), ψ(4360), and ψ(4415) states are investigated using a first-order tetraquark mass formula. This mass relationship is based on weakly bound D plus D1-bar, D* plus D1-bar, and D* plus D2*-bar meson clusters. The first-order tetraquark mass formula provides predicted masses within about 3% of the experimental values, and the Jπ predictions are in agreement with values suggested by the data. However, the primitive angular momentum structure of the first-order mass model admit additional Jπ values. KEYWORDS: Tetraquark Model for the ψ(4230), ψ(4360), and ψ(4415); Mass Formula; Quark Model; Cluster Model.

Parameterized complexity of weighted team definability

AbstractIn this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analog of one of the most studied problems in parameterized complexity, the notion of weighted Fagin-definability, which is formulated in terms of satisfaction of first-order formulas with free relation variables. We focus on the parameterized complexity of weighted team definability for a fixed formula $\varphi$ of central team-based logics. Given a first-order structure $\mathcal{A}$ and the parameter value $k\in \mathbb N$ as input, the question is to determine whether $\mathcal{A},T\models \varphi$ for some team T of size k. We show several results on the complexity of this problem for dependence, independence, and inclusion logic formulas. Moreover, we also relate the complexity of weighted team definability to the complexity classes in the well-known W-hierarchy as well as paraNP.

A unified higher-order unsplit CFS-PML technique for solving second-order seismic equations using discontinuous Galerkin method

The absorbing boundary conditions play a key role in numerical simulations. The complex frequency-shifted perfectly matched layer (CFS-PML), one of the absorbing boundary conditions, has attracted widespread attention because it can efficiently attenuate near-grazing incident waves and evanescent waves. In practical application, the distinct boundary reflections still exist using the CFS-PML in a thin PML region. To further enhance the absorbing performance, we have developed a novel unified high-order unsplit CFS-PML technique. The high-order CFS-PML combined with the discontinuous Galerkin (DG) method is applied to solve the second-order seismic wave equation on an unstructured mesh. In this work, we establish a unified framework for high-order unsplit CFS-PML formulations. The classical CFS-PML is the first-order form of our proposed unsplit high-order approach. By using auxiliary variables and auxiliary ordinary differential equations (AODEs), a higher-order CFS-PML formulation can be reduced to a first-order CFS-PML formula. For an Nth-order AODE CFS-PML, it will lead to (2N-1) additional variables and (2N-1) auxiliary differential equations. Taking second- and third-order as examples, we have derived the high-order unsplit AODE CFS-PML expressions in detail. The DG method can effectively deal with complex geological structures, such as caves, cracks, faults, etc., which yields more accurate numerical solutions. We incorporate the high-order unsplit AODE CFS-PML technique into the DG method to approximate seismic wave equations. Their strengths are combined to calculate the propagating wavefield in more complex finite models and obtain better results. In addition, the stability conditions of the second- and third-order CFS-PMLs are provided. We perform the high-order unsplit CFS-PML implementation for seismic wave modeling in several models. The isotropic and anisotropic examples prove that the high-order unsplit CFS-PML enjoys better absorbing performance than the classical CFS-PML. The numerical result for the graben model certifies the feasibility and flexibility of our proposed high-order CFS-PML combined with the DG method.

Investigation of Uncertainty Propagation in the Resolved Resonance Range

This paper presents a comparative study using the first-order formula (also called “sandwich”) and the multivariate sampling methodologies for propagating uncertainty in the resolved resonance region. A distinctive aspect of this work is the generation of random cross sections by sampling Resonance Parameters (RPs) taking in consideration their correlations provided in nuclear data libraries via the Resonance Parameter Covariance Matrix (RPCM). SCOOBY (Sampling COvariance OBservatorY), a newly developed sampling tool, is presented in this paper. This tool relies on the GAIA-2 nuclear data processing code to read and correct the RPCM and generate random cross sections. The study compares the sandwich method that relies on sensitivity coefficients and covariance matrices, with the sampling method, which involves numerous Monte Carlo simulations with random cross sections. The comparison is tested on an ICSBEP benchmark, PU-MET-MIXED-002, chosen for its sensitivity to 239Pu cross sections. The results indicate that both methods quantify similar uncertainties, confirming the reliability of both the SCOOBY module and the GAIA-2 code. By using the PU-MET-MIXED-002 benchmark and sampling 239Pu resonance parameters, this work demonstrates the effectiveness of these methodologies in estimating uncertainty in criticality safety calculations. The findings highlight the robustness of these approaches in uncertainty propagation, suggesting the need for further research on additional benchmarks and expanded uncertainty propagation studies.

Bounded quantifier depth spectrum for random uniform hypergraphs

The notion of spectrum for first-order properties introduced by J. Spencer for Erdős–Rényi random graph is considered in relation to random uniform hypergraphs. In this work we study the set of limit points of the spectrum for first-order formulae with bounded quantifier depth and obtain bounds for its maximum value. Moreover, we prove zero–one k-laws for the random uniform hypergraph and improve the bounds for the maximum value of the spectrum for first-order formulae with bounded quantifier depth. We obtain that the maximum value of the spectrum belongs to some two-element set.

Proving Query Equivalence Using Linear Integer Arithmetic

Proving the equivalence between SQL queries is a fundamental problem in database research. Existing solvers model queries using algebraic representations and convert such representations into first-order logic formulas so that query equivalence can be verified by solving a satisfiability problem. The main challenge lies in "unbounded summations", which appear commonly in a query's algebraic representation in order to model common SQL features, such as projection and aggregate functions. Unfortunately, existing solvers handle unbounded summations in an ad-hoc manner based on heuristics or syntax comparison, which severely limits the set of queries that can be supported. This paper develops a new SQL equivalence prover called SQLSolver, which can handle unbounded summations in a principled way. Our key insight is to use the theory of LIA^*, which extends linear integer arithmetic formulas with unbounded sums and provides algorithms to translate a LIA^* formula to a LIA formula that can be decided using existing SMT solvers. We augment the basic LIA^* theory to handle several complex scenarios (such as nested unbounded summations) that arise from modeling real-world queries. We evaluate SQLSolver with 359 equivalent query pairs derived from the SQL rewrite rules in Calcite and Spark SQL. SQLSolver successfully proves 346 pairs of them, which significantly outperforms existing provers.

Levelwise construction of a single cylindrical algebraic cell

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.