Polynomials In Variables Research Articles

A refinement of Reznick's Positivstellensatz with applications to quantum information theory

In his solution of Hilbert's 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin's result can always be chosen as an N-th power of the squared norm of the variables and gave explicit bounds on N. By using concepts from quantum information theory (such as partial traces, optimal cloning maps, and an identity due to Chiribella) we give simpler proofs and minor improvements of both real and complex versions of this result. Moreover, we discuss constructions of Hilbert identities using Gaussian integrals and we review an elementary method to construct complex spherical designs. Finally, we apply our results to give improved bounds for exponential quantum de Finetti theorems in the real and in the complex setting.

Some identities related to degenerate r-Bell and degenerate Fubini polynomials

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler polynomials. This paper is focused on the study of some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind. Especially, we express the power series in terms of the degenerate r-Bell polynomials, of the degenerate r-Stirling numbers of the second kind and of the degenerate Fubini polynomials.

High-Speed Variable Polynomial Toeplitz Hash Algorithm Based on FPGA.

In the Quantum Key Distribution (QKD) network, authentication protocols play a critical role in safeguarding data interactions among users. To keep pace with the rapid advancement of QKD technology, authentication protocols must be capable of processing data at faster speeds. The Secure Hash Algorithm (SHA), which functions as a cryptographic hash function, is a key technology in digital authentication. Irreducible polynomials can serve as characteristic functions of the Linear Feedback Shift Register (LFSR) to rapidly generate pseudo-random sequences, which in turn form the foundation of the hash algorithm. Currently, the most prevalent approach to hardware implementation involves performing block computations and pipeline data processing of the Toeplitz matrix in the Field-Programmable Gate Array (FPGA) to reach a maximum computing rate of 1 Gbps. However, this approach employs a fixed irreducible polynomial as the characteristic polynomial of the LFSR, which results in computational inefficiency as the highest bit of the polynomial restricts the width of parallel processing. Moreover, an attacker could deduce the irreducible polynomials utilized by an algorithm based on the output results, creating a serious concealed security risk. This paper proposes a method to use FPGA to implement variational irreducible polynomials based on a hashing algorithm. Our method achieves an operational rate of 6.8 Gbps by computing equivalent polynomials and updating the Toeplitz matrix with pipeline operations in real-time, which accelerates the authentication protocol while also significantly enhancing its security. Moreover, the optimization of this algorithm can be extended to quantum randomness extraction, leading to a considerable increase in the generation rate of random numbers.

Computing the Connected Components of the Complement to the Amoeba of a Polynomial in Several Complex Variables

Computing the Connected Components of the Complement to the Amoeba of a Polynomial in Several Complex Variables

SL2 quantum trace in quantum Teichmüller theory via writhe

Quantization of the Teichm\"uller space of a punctured Riemann surface $S$ is an approach to $3$-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop $\gamma$ in $S$ gives rise to a natural trace-of-monodromy function $\mathbb{I}(\gamma)$ on the Teichm\"uller space. For any ideal triangulation $\Delta$ of $S$, this function $\mathbb{I}(\gamma)$ is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arcs of $\Delta$. An important problem was to construct a quantization of this function $\mathbb{I}(\gamma)$, namely to replace it by a noncommutative Laurent polynomial in the quantum variables. This problem, which is closely related to the framed protected spin characters in physics, has been solved by Allegretti and Kim using Bonahon and Wong's ${\rm SL}_2$ quantum trace for skein algebras, and by Gabella using Gaiotto, Moore and Neitzke's Seiberg-Witten curves, spectral networks, and writhe of links. We show that these two solutions to the quantization problem coincide. We enhance Gabella's solution and show that it is a twist of the Bonahon-Wong quantum trace.

Adaptive group-wise modeling of thermally induced errors of a turning center

Traditional multivariate regression analysis-based thermal error models use only one polynomial of several temperature variables to predict thermal errors, which will produce lower local prediction accuracy for a longer machining process with sudden changes of machining parameters, and hence the group-wise modeling method is proposed in this paper. Resorting to hard break points and soft break points, the grouping work is completed in two steps: hard grouping and soft grouping. The positions of hard break points are optimized using the genetic algorithm toolbox in Matlab software to realize adaptive grouping. The mechanism for updating the thermal error model coefficients vectors for different soft groups is developed. The modeling test is carried out on a turning center for which the positions of thermal key points are optimized. The prediction results for radial and axial thermal errors show that four hard break points can basically meet the requirements at the di value of 80%, so the group-wise modeling method is helpful to advance the prediction accuracy of thermal errors.

COMPUTING THE CONNECTED COMPONENTS OF THE COMPLEMENT TO THE AMOEBA OF A POLYNOMIAL IN SEVERAL COMPLEX VARIABLES

In this paper, we propose a method for computing and visualizing the amoeba of a Laurent polynomial in several complex variables, which is applicable in arbitrary dimension. The algorithms developed based on this method are implemented as a free web service (http://amoebas.ru), which enables interactive computation of amoebas for polynomials in two variables, as well as provides a set of computed amoebas and their cross-sections in higher dimensions. The correctness and running time of the proposed algorithms are tested using a set of optimal polynomials in two, three, and four variables, which are generated using Mathematica computer algebra system. The developed program code makes it possible, in particular, to generate optimal hypergeometric polynomials in an arbitrary number of variables supported in an arbitrary zonotope given by a set of generating vectors.

Meromorphic functions sharing small functions with their difference polynomial in several variables

The aim of this paper is to deal with the uniqueness problem on meromorphic functions in $ \mathbb{C}^{m} $ sharing small functions with their difference polynomial, and the results obtained can be seen as some extensions of previous results from one complex variable to several complex variables.

Lidstone interpolation III. Several variables

A polynomial in a single variable is uniquely determined by its derivatives of even order at 0 and 1. More precisely, such an univariate polynomial can be written and a finite sum of f ( 2 n ) ( 0 ) Λ n ( 1 − z ) f^{(2n)}(0) \Lambda _n(1-z) and f ( 2 n ) ( 1 ) Λ n ( z ) f^{(2n)}(1) \Lambda _n(z) , ( n ≥ 0 n\ge 0 ), where the Λ n ( z ) \Lambda _n(z) are the Lidstone polynomials defined by the conditions ( d d z ) 2 k Λ n ( 0 ) = 0 and ( d d z ) 2 k Λ n ( 1 ) = δ k , n , k ≥ 0 , n ≥ 0. \begin{equation*} \left (\frac {\mathrm {d}}{\mathrm {d}z}\right )^{2k} \Lambda _n(0)=0\text { and } \left (\frac {\mathrm {d}}{\mathrm {d}z}\right )^{2k} \Lambda _n(1)=\delta _{k,n},\quad k\ge 0, \; n\ge 0. \end{equation*} We generalize this theory to n n variables, replacing the two points 0 0 , 1 1 in C \mathbb {C} with n + 1 n+1 points e _ 0 , e _ 1 , … , e _ n {\underline {e}}_0,{\underline {e}}_1,\dots ,{\underline {e}}_n in C n \mathbb {C}^n , where e _ 0 {\underline {e}}_0 is the origin of C n \mathbb {C}^n and e _ 1 , … , e _ n {\underline {e}}_1,\dots ,{\underline {e}}_n the canonical basis of C n \mathbb {C}^n . By selecting a suitable subset of even order derivatives at these n + 1 n+1 points, we show that any polynomial in n n variables has a unique expansion. We obtain generating series for these sequences of polynomials and we deduce an expansion for entire functions in C n \mathbb {C}^n of exponential type > π >\pi . We extend to several variables results due to Lidstone [Proceedings Edinburgh Math. Soc., II. Ser. (2)2, 16-19 (1930)], Poritsky [Trans. Amer. Math. Soc. 34 (1932), pp. 274–331], Whittaker [Proc. London Math. Soc. (2) 36 (1934), pp. 451–469], Schoenberg [Bull. Amer. Math. Soc. 42 (1936), pp. 284–288], Buck [Proc. Amer. Math. Soc. 6 (1955), pp. 793–796]. We also show that our results are, to a certain extent, best possible.

Bi-fidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos expansion

Digital twin models allow us to continuously assess the possible risk of damage and failure of a complex system. Yet high-fidelity digital twin models can be computationally expensive, making quick-turnaround assessment challenging. Toward this goal, this article proposes a novel bi-fidelity method for estimating the conditional value-at-risk (CVaR) for nonlinear systems subject to dependent and high-dimensional inputs. For models that can be evaluated fast, a method that integrates the dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) approximation with a standard sampling-based CVaR estimation is proposed. For expensive-to-evaluate models, a new bi-fidelity method is proposed that couples the DD-GPCE with a Fourier-polynomial expansion of the mapping between the stochastic low-fidelity and high-fidelity output data to ensure computational efficiency. The method employs measure-consistent orthonormal polynomials in the random variable of the low-fidelity output to approximate the high-fidelity output. Numerical results for a structural mechanics truss with 36-dimensional (dependent random variable) inputs indicate that the DD-GPCE method provides very accurate CVaR estimates that require much lower computational effort than standard GPCE approximations. A second example considers the realistic problem of estimating the risk of damage to a fiber-reinforced composite laminate. The high-fidelity model is a finite element simulation that is prohibitively expensive for risk analysis, such as CVaR computation. Here, the novel bi-fidelity method can accurately estimate CVaR as it includes low-fidelity models in the estimation procedure and uses only a few high-fidelity model evaluations to significantly increase accuracy.

REKONSTRUKSI MODEL PROGRAM NONLINIER DENGAN FUNGSI POLINOMIAL MENJADI BENTUK PROGRAM KUADRATIK

The purpose of this research to revisits methods that are more effective in nonlinear Optimization with single variable polynomial functions at high degrees. Models with linear objective functions and constraint functions are Polynomials of third, fourth and fifth degree reconstructed into subproblems that are easier to solve, namely quadratic programs, using bilinear Auxiliary Functions and solved by MATLAB simulations. The method used is the development of Tawarmalani & Sahinidis' research regarding relaxation with Auxiliary Functions. Examples of nonlinear Optimization with polynomial functions are also given to illustrate the implementation of this algorithm. The results of the research show that the application of the development reconstruction method produces a global solution that is no better than the solution to the original problem so that it is not an effective alternative method to use.

An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for two‐dimensional compressible flows on adaptive quadrilateral meshes

AbstractIn this article, a discontinuous Galerkin (DG) scheme on the adaptive quadrilateral meshes is proposed to simulate two‐dimensional compressible flows in the direct arbitrary Lagrangian–Eulerian (ALE) framework. In our scheme, the Euler equations are discretized in the reference element with the help of a bilinear map. A kind of Taylor expansion basis functions in the reference element is used to construct the interpolation polynomials of variables. We describe the property that the material derivatives of the basis functions used in the DG discretization are equal to zero, with which the scheme is simplified. Furthermore, the mesh velocity in our ALE framework is obtained by implementing the approach of mesh movement based on the variational principle from [Adaptive mesh methods for one‐ and two‐dimensional hyperbolic conservation laws. SIAM J Numer Anal. 2003;41:487–515]. This approach of mesh movement automatically concentrates the mesh nodes near the regions with large gradient values of the variables and can greatly improve the resolution of the solution near these regions. In addition, a WENO (weighted essentially non‐oscillatory) reconstruction helps our scheme remove the numerical oscillations. Some numerical examples are presented to demonstrate the accuracy, high resolution, and robustness of our scheme.

Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture

Abstract An n-vertex graph is called C-Ramsey if it has no clique or independent set of size $C\log _2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.

Structural Interval Reliability Algorithm Based on Bernstein Polynomials and燛vidence Theory

Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors. Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid. The uncertain parameters mainly exist in the form of intervals. This method requires a lot of calculation and is often difficult to achieve efficiently. In order to solve this problem, this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty. Based on the non-probabilistic reliability index method, the extreme value of the limit state function is obtained using the properties of Bernstein polynomials, thus avoiding the need for a lot of sampling to solve the reliability analysis problem. The method is applied to numerical examples and engineering applications such as experiments, and the results show that the method has higher computational efficiency and accuracy than the traditional linear approximation method, especially for some reliability problems with higher nonlinearity. Moreover, this method can effectively improve the reliability of results and reduce the cost of calculation in practical engineering problems.

Mathematical modeling and analysis of the thermomechanical behavior of a bimetallic spherical element during technological processing with an electromagnetic impulse

The initial-boundary problem of thermomechanics for a bimetallic spherical element under homogenous non-stationary electromagnetic action is formulated. The azimuthal component of the magnetic field strength vector, temperature, and the radial component of the displacement vector were selected as key functions. To find the key functions, the approximation of their distributions in the constituent layers of the spherical element by quadratic polynomials in the radial variable is proposed. This approximation makes it possible to accurately satisfy the specified boundary conditions both on the surfaces of this element and on the surface of the connection of the component layers, where ideal electromagnetic, thermal and mechanical contacts take place. The coefficients of the polynomials approximating the key functions in the component layers of the spherical element are given by a linear combination of functions describing all given boundary conditions and integral characteristics of the key functions on both component layers. As a result, the original initial-boundary value problems for the defining functions are reduced to Cauchy problems for their integral (summed over the package of layers) characteristics. With the use of Laplace transformation in time, the general solutions of these problems under homogeneous non-stationary electromagnetic action are recorded. The change in time of radial and azimuthal stresses and stress intensity was numerically analyzed by technological processing with an electromagnetic impulse, as well as the performance and properties of the contact connection of the bimetallic spherical element were investigated.

Norms on complex matrices induced by random vectors

AbstractWe introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.

Distributionally Robust Optimization with Moment Ambiguity Sets

This paper studies distributionally robust optimization (DRO) when the ambiguity set is given by moments for the distributions. The objective and constraints are given by polynomials in decision variables. We reformulate the DRO with equivalent moment conic constraints. Under some general assumptions, we prove the DRO is equivalent to a linear optimization problem with moment and psd polynomial cones. A Moment-SOS relaxation method is proposed to solve it. Its asymptotic and finite convergence are shown under certain assumptions. Numerical examples are presented to show how to solve DRO problems.

The Jones polynomial of collections of open curves in 3-space

Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems, such as in polymers and biopolymers. In this manuscript, we give a novel definition of the Jones polynomial that generalizes the classic Jones polynomial to collections of open curves in 3-space. More precisely, first we provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined single variable polynomial that is a topological invariant, which, for link-type linkoids, coincides with that of the corresponding link. Using the framework introduced in (Panagiotou E, Kauffman L. 2020Proc. R. Soc. A476, 20200124. ((doi:10.1098/rspa.2020.0124)), this enables us to define the Jones polynomial of collections of open and closed curves in 3-space. For collections of open curves in 3-space, the Jones polynomial has real coefficients and it is a continuous function of the curves’ coordinates. As the endpoints of the curves tend to coincide, the Jones polynomial tends to that of the resultant link. We demonstrate with numerical examples that the novel Jones polynomial enables us to characterize the topological/geometrical complexity of collections of open curves in 3-space for the first time.

Sums of Separable and Quadratic Polynomials

We study separable plus quadratic (SPQ) polynomials, that is, polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sums of squares, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial and (ii) a sum of squares. We establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials but negative already for bivariate quartic SPQ polynomials. We use our decomposition result for convex SPQ polynomials to show that convex SPQ polynomial optimization problems can be solved by “small” semidefinite programs. For question (ii), we provide a complete characterization of the answer based on the degree and the number of variables of the SPQ polynomial. We also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. We end by presenting applications of SPQ polynomials to upper bounding sparsity of solutions to linear programs, polynomial regression problems in statistics, and a generalization of Newton’s method that incorporates separable higher order derivative information. Funding: This work was partially supported by an AFOSR MURI award, the Defense Advanced Research Projects Agency Young Faculty Award, the Princeton School of Engineering and Applied Sciences Innovation Award, the National Science Foundation Faculty Early Career Development Program Award, the Google Faculty Award, and the Sloan Fellowship.

Cosmology of a Polynomial Model for de Sitter Gauge Theory Sourced by a Fluid

In the de Sitter gauge theory (DGT), the fundamental variables are the de Sitter (dS) connection and the gravitational Higgs/Goldstone field ξA, where A is a 5 dimensional index. Previously, a model for DGT was analyzed, which generalizes the MacDowell–Mansouri gravity to have a variable cosmological constant, Λ=3/l2, where l is related to ξA by ξAξA=l2. It was shown that the model sourced by a perfect fluid does not support a radiation epoch and the accelerated expansion of the parity invariant universe. In this paper, I consider a similar model, namely, the Stelle–West gravity, and couple it to a modified perfect fluid, such that the total Lagrangian 4-form is polynomial in the gravitational variables. The Lagrangian of the modified fluid has a nontrivial variational derivative with respect to l, and as a result, the problems encountered in the previous study no longer appear. Moreover, to explore the elegance of the general theory, as well as to write down the basic framework, I perform the Lagrange–Noether analysis for DGT sourced by a matter field, yielding the field equations and the identities with respect to the symmetries of the system. The resulted formula are dS covariant and do not rely on the existence of the metric field.