Terms Of Polynomials Research Articles

Emergent potentials and non-perturbative open topological strings

We show that integrating out M2 branes ending on M5 branes inside Calabi-Yau manifolds captures non-perturbative open topological string physics. The integrating out is performed using a contour integral in complexified Schwinger proper time. For the resolved conifold, this contour can be extended to include the zero pole, which we argue captures the ultraviolet completion of the integrating out and yields the tree-level polynomial terms in the free energy. This is a manifestation of the Emergence Proposal, and provides further evidence for it. Unlike the case of closed strings, where the emergent terms are kinetic terms in the action, for these open strings it is tree-level potential terms which are emergent. This provides a first quantitative example of the proposal that classical tree-level potentials in string theory emerge from integrating out co-dimension one states.

Well-Posedness of the Schrödinger–Korteweg–de Vries System with Robin Boundary Conditions on the Half-Line

The Schrödinger–Korteweg–de Vries (SKdV) system can describe the nonlinear dynamics of phenomena such as Langmuir and ion acoustic waves, which are highly valuable for studying wave behavior and interactions. The SKdV system has wide-ranging applications in physics and applied mathematics. In this article, we investigate the local well-posedness of the SKdV system with Robin boundary conditions and polynomial terms in the Sobolev space. We want to enhance the applicability of this type of SKdV system. Our verification process is as follows: We estimate Fokas solutions for the Robin problem with external forces. Next, we define an iteration map in suitable solution space and prove the iteration map is a contraction mapping and onto some closed ball B(0,r). Finally, by the contraction mapping theorem, we obtain the uniqueness solution. Moreover, we show that the data-to-solution map is locally Lipschitz continuous and conclude with the well-posedness of the SKdV system.

Efficient Study Design and Analysis of Longitudinal Dose-Response Data Using Fractional Polynomials.

Correctly characterising the dose-response relationship and taking the correct dose forward for further study is a critical part of the drug development process. We use optimal design theory to compare different designs and show that using longitudinal data from all available timepoints in a continuous-time dose-response model can substantially increase the efficiency of estimation of the dose-response compared to a single timepoint model. We give theoretical results to calculate the efficiency gains for a large class of these models. For example, a linearly growing Emax dose-response in a population with a between/within-patient variance ratio ranging from 0.1 to 1 measured at six visits can be estimated with between 1.43 and 2.22 times relative efficiency gain, or equivalently, with 30% to a 55% reduced sample size, compared to a single model of the final timepoint. Fractional polynomials are a flexible way to incorporate data from repeated measurements, increasing precision without imposing strong constraints. Longitudinal dose-response models using two fractional polynomial terms are robust to mis-specification of the true longitudinal process while maintaining, often large, efficiency gains. These models have applications for characterising the dose-response at interim or final analyses.

On extending Powell, Stock, and Stoker (1989) to indexes with functionally dependent covariates

In this paper, we extend Powell et al.’s (1989) results to a situation where the index contains functionally dependent covariates. This allows one to obtain semiparametric, root-N consistent, estimates of index parameters when the index includes polynomial or interaction terms.

Optimizing estimated ultimate recovery in horizontal wells through data-driven analytics

The oil and gas industry has undergone significant changes in completion operation design and execution, largely due to the advancements in machine learning techniques. In our study, we analyzed a dataset of approximately 200 horizontal wells from the Wolfcamp formations, categorized into two groups based on productivity and classification as parent versus child (infill) wells. Using response surface methodology (RSM), a data analytics technique that models the relationship between explanatory and response variables with up to second-order polynomial terms, we illustrate how to optimize horizontal well performance, specifically focusing on Estimated Ultimate Recovery (EUR). We selected a pool of 5 input variables for analysis, including initial oil production of 180 days, injected fluid volume in barrels, number of perforations in each cluster, volume of Hydrochloric Acid injected during frack job in gallons, and volume of injected Linear Gel in gallons.The selections were made in collaboration with industry professionals and validated using data analytics-based methods that rank variable importance. By pinpointing potential strategies to achieve a globally optimal EUR through the adjustment of completion variables (perforations and hydrochloric acid should be set at their data-recorded maximum values of 7.69 and 10.82 on a logarithmic scale), the study aims to simplify the challenging task of maximizing EUR from producing wells, offering an additional avenue for estimating reserves. It emphasizes the adaptability of this approach in allowing different completion designs to be dynamically tailored for optimizing EUR, each with its unique set of inputs. Ultimately, these diverse designs can be synthesized through meta-analysis. In summary, the study provides a practical framework for EUR estimation, readily applicable by operators in unconventional exploration, representing a transformative milestone in the industry. The subsequent phase involves collecting additional data from this specific field area and employing a range of models to improve precision and specificity.

Analytical computation of stress intensity factor for multi-physics problems

This work presents a methodology for the analytical calculation of the stress intensity factor when the stress distribution on the crack surfaces is non-homogeneous. At first, a polynomial function is used to express the non-homogenous stress distribution. Subsequently, the principle of superposition of effects is applied, and the stress intensity factor is computed by multiplying each polynomial term by its respective geometric factor. Finite element fracture model is used to compute the geometric factor of the single polynomial grade. To explain the method, a spherical body is considered, with central and superficial cracks. Each geometric factor depends on a normalized geometrical parameter (the ratio between the crack length and sphere radius). The proposed methodology is applied to determine the stress intensity factor in the case of a crack driving force caused by diffusive fields, such as the concentration gradient in particles of electrodes active material in lithium-ion batteries. The methodology allows to speed up the fracture computation, then it is used to give electrode design guidelines to limit the fracture likeliness and mechanical degradation in lithium-ion batteries, as well as it is the basis for the development of algorithms assessing the capacity loss and the remaining useful life of lithium-ion batteries in real-time.

Computing nodes for plane data points by constructing cubic polynomial with constraints

To construct a parametric polynomial curve for interpolating a set of data points, the interpolation accuracy and shape of the constructed curve are influenced by two principal factors: parameterization of the data points (computing a node for each data point) and interpolation method. A new method of computing nodes for a set of data points was proposed. In this paper, the functional relationship between data points and corresponding nodes in cubic polynomials was established. Using this functional relationship, a functional cubic polynomial with one degree of freedom can pass through four adjacent data points. The degree of the freedom can be represented by two adjacent node intervals can be obtained by minimizing the cubic terms of the cubic polynomial. Since each node is computed in different node spaces, a method for constructing a quadratic curve is presented, which transforms all the quadratic curves into a unified form to compute nodes. Nodes computed using the new method exhibit quadratic polynomial precision, i.e., if the set of data point is taken from a quadratic polynomial F(t), the nodes by the new method are used to construct a interpolation curve, an interpolation method reproducing quadratic polynomial gives quadratic polynomial F(t). The primary advantage of the proposed method is that the constructed curve has a shape described by data points. Another advantage of the new method is that the nodes computed by it have affine invariance. The experimental results indicate that the curve constructed by the nodes using the new method has a better interpolation accuracy and shape compared to that constructed using other methods.

Circuit complexity of quantum access models for encoding classical data

How to efficiently encode classical data is a fundamental task in quantum computing. While many existing works treat classical data encoding as a black box in oracle-based quantum algorithms, their explicit constructions are crucial for the efficiency of practical algorithm implementations. Here, we unveil the mystery of the classical data encoding black box and study the Clifford + T complexity in constructing several typical quantum access models. For general matrices (even including sparse ones), we prove that sparse-access input models and block-encoding both require nearly linear circuit complexities relative to the matrix dimension. We also give construction protocols achieving near-optimal gate complexities. On the other hand, the construction becomes efficient with respect to the data qubit when the matrix is a linear combination of polynomial terms of efficiently implementable unitaries. As a typical example, we propose improved block-encoding when these unitaries are Pauli strings. Our protocols are built upon improved quantum state preparation and a select oracle for Pauli strings, which hold independent values. Our access model constructions provide considerable flexibility, allowing for tunable ancillary qubit numbers and offering corresponding space-time trade-offs.

Distribution of constant terms of irreducible polynomials in ℤ_p[x] whose degree is a product of two distinct odd primes

We obtain explicit formulas for the number of monic irreducible polynomials with prescribed constant term and degree $q_1q_2$ over a finite field, where $q_1$ and $q_2$ are distinct odd~primes. These formulas are derived from work done by Yucas. We show that the number of polynomials of a given constant term depends only on whether the constant term is a $q_1$-residue and/or a $q_2$-residue in the underlying field. We further show that as $k$ becomes large, the proportion of irreducible polynomials having each constant term is asymptotically equal. This paper continues work done in [1].

Identification for Tree-Shaped Structural Causal Models in Polynomial Time

Linear structural causal models (SCMs) are used to express and analyze the relationships between random variables. Direct causal effects are represented as directed edges and confounding factors as bidirected edges. Identifying the causal parameters from correlations between the nodes is an open problem in artificial intelligence. In this paper, we study SCMs whose directed component forms a tree. Van der Zander et al. give a PSPACE-algorithm for the identification problem in this case, which is a significant improvement over the general Gröbner basis approach, which has doubly-exponential time complexity in the number of structural parameters. However, they do not show that their algorithm is complete. In this work, we present a randomized polynomial-time algorithm, which solves the identification problem for tree-shaped SCMs. For every structural parameter, our algorithms decides whether it is generically identifiable, generically 2-identifiable, or generically unidentifiable. (No other cases can occur.) In the first two cases, it provides one or two fractional affine square root terms of polynomials (FASTPs) for the corresponding parameter, respectively. In particular, our algorithm is not only polynomial time, but also complete for for tree-shaped SCMs.

GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equations

Traditional machine learning is generally treated as a black-box optimization problem and does not typically produce interpretable functions that connect inputs and outputs. However, the ability to discover such interpretable functions is desirable. In this work, we propose GINN-LP, an interpretable neural network to discover the form and coefficients of the underlying equation of a dataset, when the equation is assumed to take the form of a multivariate Laurent Polynomial. This is facilitated by a new type of interpretable neural network block, named the “power-term approximator block”, consisting of logarithmic and exponential activation functions. GINN-LP is end-to-end differentiable, making it possible to use backpropagation for training. We propose a neural network growth strategy that will enable finding the suitable number of terms in the Laurent polynomial that represents the data, along with sparsity regularization to promote the discovery of concise equations. To the best of our knowledge, this is the first model that can discover arbitrary multivariate Laurent polynomial terms without any prior information on the order. Our approach is first evaluated on a subset of data used in SRBench, a benchmark for symbolic regression. We first show that GINN-LP outperforms the state-of-the-art symbolic regression methods on datasets generated using 48 real-world equations in the form of multivariate Laurent polynomials. Next, we propose an ensemble method that combines our method with a high-performing symbolic regression method, enabling us to discover non-Laurent polynomial equations. We achieve state-of-the-art results in equation discovery, showing an absolute improvement of 7.1% over the best contender, by applying this ensemble method to 113 datasets within SRBench with known ground-truth equations.

On regularized polynomial functional regression

This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity conditions, and regularization techniques. In doing so, it extends and generalizes several findings from the context of linear functional regression as well. We also provide numerical evidence that using higher order polynomial terms can lead to an improved performance.

Compact freeform-surface-based Offner imaging spectrometer with both a long-slit and broadband.

Current imaging spectrometers with conventional optical elements face major challenges in achieving a large field of view (FOV), broadband and compact structure simultaneously. In this paper, a compact freeform-surface-based Offner imaging spectrometer with both a long-slit and a broadband (CISLS) is proposed. To keep a long slit and an anastigmatic imaging, the slit off-axis amount of the initial system is within a specific range theoretically. While to achieve a compact structure, the slit off-axis amount should be away from the specific range and as small as possible. Based on the vector aberration theory and the analytical study, Zernike polynomial terms Z5 and Z6 introduce the astigmatism independent of FOV. They are utilized to well balance the astigmatism when the slit off-axis amount is away from the specific range, helping a miniaturization of the system. Other Zernike polynomial terms below the eighth order introduce the astigmatism related to FOV. They contribute to balancing the astigmatism that produced with the increasing of the FOV, thus achieving a wide FOV. The design results show that the proposed CISLS with a high spectral resolution of 2.7 nm achieves a long slit of 30 mm in length but a small size of only 60 mm × 64 mm × 90 mm in volume under a broadband from 400 nm to 1000 nm.

Efficient multipole representation for matter-wave optics

Technical optics with matter waves requires a universal description of three-dimensional traps, lenses, and complex matter-wave fields. In analogy to the two-dimensional Zernike expansion in beam optics, we present a three-dimensional multipole expansion for Bose-condensed matter waves and optical devices. We characterize real magnetic chip traps, optical dipole traps, and the complex matter-wave field in terms of spherical harmonics and radial Stringari polynomials. We illustrate this procedure for typical harmonic model potentials as well as real magnetic and optical dipole traps. Eventually, we use the multipole expansion to characterize the aberrations of a ballistically interacting expanding Bose–Einstein condensate in (3 + 1) dimensions. In particular, we find deviations from the quadratic phase ansatz in the popular scaling approximation. The scheme is data efficient by representing millions of complex amplitudes of a field on a Cartesian grid in terms of a low order multipole expansion without precision loss. This universal multipole description of aberrations can be used to optimize matter-wave optics setups, for example, in matter-wave interferometers.

Shear buckling of ship structures rectangular elements

In the work to determine the spectrum of critical loads and the corresponding forms of a rectangular clamped panel (plate) buckling under the action of balanced tangential forces on its contour, the Bubnov-Galerkin method using polynomials in two coordinates is proposed. This problem of the ship skin element pure shift does not have an exact closed solution, and the known approximate solutions require an analysis of their accuracy and reliability. The aim of the work is to obtain and analyze analytical solutions using polynomials of various degrees. Approximating deflection functions satisfying all the boundary conditions of the problem are represented sequentially by polynomials of 10th, 12th, 14th, 16th and 18th degrees in two coordinates with undefined coefficients. The solution of the main differential equation of the problem is found approximately in the integral sense, as a result of which homogeneous systems of linear algebraic equations with respect to unknown coefficients of polynomials are obtained. These systems contain a shear load as a parameter. To obtain eigenvalues (critical loads), the determinants of the systems are equated to zero. Numerical results are obtained in the Maple analytical computing system. For each approximation (polynomial), a power equation with respect to the critical load, the solution of which is paired values differing in signs is obtained. The forms of buckling are oblique waves. For a ship skin square panel, the first form of buckling is a single bulge along the diagonal of the panel. The second form is obtained in the form of two bulges directed in opposite directions (symmetrically-antisymmetrically with respect to the diagonals), etc. The numerical results are compared with the results of other authors. It is established that with an increase in the number of the polynomial terms, the initial critical loads and forms of buckling are specified, first of all.

Smirnov and Bernstein-type inequalities, taking into account higher-order coefficients and free terms of polynomials

Smirnov and Bernstein-type inequalities, taking into account higher-order coefficients and free terms of polynomials

Application of the Polynomial Chaos Expansion to the Uncertainty Propagation in Fault Transients in Nuclear Fusion Reactors: DTT TF Fast Current Discharge

Nuclear fusion reactors are composed of several complex components whose behavior may be not certain a priori. This uncertainty may have a significant impact on the evolution of fault transients in the machine, causing unexpected damage to its components. For this reason, a suitable method for the uncertainty propagation during those transients is required. The Monte Carlo method would be the reference option, but it is, in most of the cases, not applicable due to the large number of required, repeated simulations. In this context, the Polynomial Chaos Expansion has been considered as a valuable alternative. It allows us to create a surrogate model of the original one in terms of orthogonal polynomials. Then, the uncertainty quantification is performed repeatedly, relying on this much simpler and faster model. Using the fast current discharge in the Divertor Tokamak Test Toroidal Field (DTT TF) coils as a reference scenario, the following method has been applied: the uncertainty on the parameters of the Fast Discharge Unit (FDU) varistor disks is propagated to the simulated electrical and electromagnetic relevant effects. Eventually, two worst-case scenarios are analyzed from a thermal–hydraulic point of view with the 4C code, simulating a fast current discharge as a consequence of a coil quench. It has been demonstrated that the uncertainty on the inputs (varistor parameters) strongly propagates, leading to a wide range of possible scenarios in the case of accidental transients. This result underlines the necessity of taking into account and propagating all possible uncertainties in the design of a fusion reactor according to the Best Estimate Plus Uncertainty approach. The uncertainty propagation from input data to electrical, electromagnetic, and thermal hydraulic results, using surrogate models, is the first of its kind in the field of the modeling of superconducting magnets for nuclear fusion applications.

Model reduction for stochastic systems with nonlinear drift

In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subspace is identified leading to a balancing related highly accurate reduced order SDE. We provide an algebraic error criterion and an error analysis of the propose model reduction schemes. The paper is concluded by applying our method to spatially discretized reaction diffusion equations.

Comment on “Effects of (1+2)-dimensional circularly symmetric and static traversable wormhole with cosmic string on oscillator field”

We analyze the mathematical results and physical conclusions put forward in a recent paper about “the relativistic quantum dynamics of an oscillator field described by the Klein–Gordon equation in a circularly symmetric and static (1+2)-dimensional traversable wormhole spacetime with cosmic strings”. We show that the existence of allowed oscillator frequencies conjectured by the author is a mere artifact of the truncation of a power series with the purpose of obtaining bound-state solutions in terms of polynomials. Our theoretical analysis suggest that the eigenvalues are continuous, increasing functions of the oscillator frequency, conclusion that is verified by a straightforward numerical calculation.

Energy spectrum and applications of Eckart plus Hellmann potential in hyperspherical coordinates

We investigate the approximate bound-state solutions of the N-dimensional Schrödinger equation with a combination of Eckart and Hellmann potentials via the asymptotic iteration approach. By the use of the Greene-Aldrich approximation, which is valid for small values of the screening parameter, we establish the N-dimensional energy spectrum and the N-dimensional radial wave function in approximate analytic form. In hyperspherical coordinates, the normalized radial wave function is expressed in terms of hypergeometric and Jacobi polynomials. To double-check the energy spectrum in hyperspherical coordinates, we utilize polynomial solutions in view of the asymptotic iteration approach. The expressions for the expectation values of inverse position, square of inverse position, kinetic energy, and square of momentum are derived in hyperspherical coordinates by using the Hellmann-Feynmann theorem. We also present the analytically calculated energy eigenvalues for Eckart plus Hellmann potential. We deduce special forms of the relevant potential, such as Eckart, Hellmann, and Hulthén potentials.