Set Of Polynomials Research Articles

Equilibrium measures on Julia sets of random quadratic polynomials

Abstract We consider sequences of compositions of quadratic polynomials f c n ( z ) = z 2 + c n . For such sequences one can naturally generalise the definitions of the Julia set and basin of infinity from the autonomous case. In this setting the Julia set depends on a sequence ω = ( c 0 , c 1 , … ) . We study the equilibrium (harmonic) measure on such Julia sets. In particular, we calculate the Hausdorff dimension of the equilibrium measure and study its dependence on the ‘scale of randomnes’.

Implicit Factorization with Shared Any Bits

At PKC 2009, May and Ritzenhofen proposed the implicit factorization problem (IFP). They showed that it is undemanding to factor two h-bit RSA moduli N1=p1q1, N2=p2q2 where q1, q2 are both αh-bit, and p1, p2 share uh>2αh the least significant bits (LSBs). Subsequent works mainly focused on extending the IFP to the cases where p1, p2 share some of the most significant bits (MSBs) or the middle bits (MBs). In this paper, we propose a novel generalized IFP where p1 and p2 share an arbitrary number of bit blocks, with each block having a consistent displacement in its position between p1 and p2, and we solve it successfully based on Coppersmith’s method. Specifically, we generate a new set of shift polynomials to construct the lattice and optimize the structure of the lattice by introducing a new variable z=p1. We derive that we can factor the two moduli in polynomial time when u>2(n+1)α(1−α^1/(n+1)) with p1, p2 sharing n blocks. Further, no matter how many blocks are shared, we can theoretically factor the two moduli as long as u>2αln(1/α). In addition, we consider two other cases where the positions of the shared blocks are arbitrary or there are k>2 known moduli. Meanwhile, we provide the corresponding solutions for the two cases. Our work is verified by experiments.

Extensive Remedial in Factoring Polynomials on STEM Learners: An Action Research

There is a widely belief that high school students find it difficult to factor various kind of polynomials. Different ways of factoring polynomials are also highly considered in this action research. This paper chose grade 11 Science Technology Engineering and Mathematics students as respondents of the study. The study will use Extensive Remedial as the intervention for the problem. Results and Interpretation are obtained by using various appropriate statistical formulas like t-test and standard deviation. The researchers conducted this action research to verify if an extensive remedial will have a positive impact on improving the students’ ability to factor different set of polynomials

Automorphism groups of cubic fivefolds and fourfolds

AbstractIn this paper, we introduce notions of partitionability and characteristic sets of homogeneous polynomials and give a complete classification of groups faithfully acting on smooth cubic fivefolds. Specifically, we prove that there exist 20 maximal ones among all such groups. As an application, we classify all possible subgroups of the automorphism groups of smooth cubic fourfolds.

A note on the Archimedean Positivstellensatz in real algebraic geometry

A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact semi-algebraic sets by means of smaller sets of squares or polynomials. A large number of examples is developed in detail.

A novel approach for the numerical solution of nonlinear Fredholm integral equations using Hosoya polynomial method

Abstract In this paper, we study the graph theoretical polynomial known as the Hosoya polynomial obtained from one of the standard classes of graphs called path. Using this polynomial applied for the numerical solution of the nonlinear Fredholm integral equation, which reduces in the algebraic system of equation with collocation points, then solving this system using Newton’s iterative with the help of MATLAB, we obtain the required approximate solution. The desired results in terms of a set of continuous polynomials over a closed interval [0, 1]. Illustrative applications show the efficiency, accuracy and validity of the proposed technique.

Monomial asymptotic polynomials and applications to polynomial optimization problems

The problem of polynomial optimization plays an important role in many fields such as physics, chemistry, and economics. This problem has received research attention from many mathematicians recently. In this paper, we study the polynomial optimization problem over a non-compact semi-algebraic set, for which its constraint set of polynomials G is asymptotic with a finite family of monomials. By changing variables via a suitable monomial mapping, we transform the problem under consideration into the polynomial optimization problem over a compact semi-algebraic feasible set. We then apply the well-known result that the optimal value of a polynomial over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semi-definite programs and the convergence is finite generically, to obtain results in the general case when the cone C(G) is unimodular. In particular, in the case of polynomials in two variables, we solve the problem quite completely without requiring C(G) to be unimodular

Orthonormal aberration decomposition for circular aperture and rectangular field with extension to multiple-aperture systems

A new set of orthogonal polynomials and the traditional Zernike polynomials are combined to construct the wavefront of the sparse aperture (SA) optical system. The new set of orthogonal polynomials in the rectangular domain is derived based on the Zernike polynomials. The modulation transfer functions (MTFs) of the SA system under different fields of view are calculated. The effects of some wavefront terms on the field-related MTF are analyzed. Imaging simulation and restoration are conducted for the SA system. The results indicate that the wavefront of the SA system can be represented by the combination of the new set of orthogonal polynomials and the Zernike polynomials. The former represents the field term, and the latter represents the pupil term of the wavefront. The field-related Wiener filter is constructed to restore the imaging results of the SA system, and the results show that the image quality can be improved greatly through restoration.

Effective battery pack cooling by controlling flow patterns with vertical and spiral fins

Battery Thermal Management System (BTMS) is essential for dissipating heat and controlling temperature distribution within the battery pack of an electric vehicle to maintain its optimal operating temperature. This research focuses on the influence of cooling fins in achieving balanced temperature distribution inside the battery pack. Vertical and spiral fins are attached around 21,700 cylindrical cells for this investigation. ANSYS FLUENTsoftwareis employed to conduct the numerical simulations, where a finite volume approach is used to solve the mass conservation, Navier–Stokes, and energy transport equations. The study also uses a set of polynomial functions to calculate the heat generation inside the cells. It assesses the effects of the Reynolds number, fin loop number, and outlet section of BTMS at numerous discharge rates. The findings indicate that the spiral fins significantly impact the thermal performance, reducing Tmax by 21.31 % due to the enhanced flow circulations within the BTMS. A high Reynolds number also positively affects Tmax by reducing the temperature uniformity caused by the shorter time for air-cooling contact with the cell surfaces. More specifically, at a Reynolds number ≥9433, a Z-type BTMS with the spiral fins can maintain the temperature uniformity at a temperature difference of 5 °C. Among the six different BTMSs analysed, a U-type BTMS is reported to have the best performance, reducing Tmax and ΔTmax by 4.11 % and 38.1 %, respectively. The pressure drop in this case is also reduced by 8.46 %, thus lowering the overall power consumption.

On a recent extension of a family of biprojective APN functions

APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension, done by Calderini et al. (2023), of a biprojective APN family introduced by Göloğlu (2022) defined on F22m. We show that this generalization yields functions equivalent to Göloğlu's original family if 3∤m. If 3|m we show exactly how many inequivalent APN functions this new family contains. We also show that the family has the minimal image set size for an APN function and determine its Walsh spectrum, hereby settling some open problems. In our proofs, we leverage a group theoretic technique recently developed by Göloğlu and the author in conjunction with a group action on the set of projective polynomials.

Even grade generic skew-symmetric matrix polynomials with bounded rank

We show that the set of m×m complex skew-symmetric matrix polynomials of even grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of even grade d and rank at most 2r. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [24].

Modeling unsaturated hydraulic conductivity of compacted bentonite using a constrained CatBoost with bootstrap analysis

Accurately determining the hydraulic conductivity of unsaturated bentonite is important for modeling subsurface thermo-hydro-mechanical and chemical processes. This study introduced a new hybrid model that employs a constrained categorial boosting (CatBoost) algorithm, combined with a genetic algorithm for hyperparameter tuning, to estimate the hydraulic conductivity of unsaturated compacted bentonite The performance of the constrained CatBoost model was benchmarked against a diverse set of data-driven baseline regression models, including lasso, elastic net, polynomial, k-nearest neighbors, decision tree, bagging tree, random forest, and CatBoost. The results indicated that the constrained CatBoost model offers a superior balance between model robustness and predictive accuracy in estimating the hydraulic conductivity of compacted bentonite-based materials during the wetting phase. The model effectively captured the U-shape relationship between hydraulic conductivity and suction, a key characteristic of bentonite behavior. Additionally, bootstrapping analyses confirmed the model's reliability under data variability, further validating its applicability in environmental and geotechnical applications.

Automatic Correction of Arithmetic Circuits in the Presence of Multiple Bugs by Groebner Basis Modification

One promising approach to verify large arithmetic circuits is making use of Symbolic Computer Algebra (SCA), where the circuit and the specification are translated to a set of polynomials, and the verification is performed by the ideal membership testing. Here, the main problem is the monomial explosion for buggy arithmetic circuits, which makes obtaining the word-level remainder become unfeasible. So, automatic correction of such circuits remains a significant challenge. Our proposed correction method partitions the circuit based on primary output bits and modifies the related Groebner basis based on the given suspicious gates, which makes it independent of the word-level remainder. We have applied our method to various signed and unsigned multipliers, with various sizes and numbers of suspicious and buggy gates. The results show that the proposed method corrects the bugs without area overhead. Moreover, it is able to correct the buggy circuit on average 51.9× and 45.72× faster in comparison with the state-of-the-art correction techniques, having single and multiple bugs, respectively.

Polynomial calculus for optimization

MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem.Weighted Polynomial Calculus is a natural generalization of the systems MaxSAT-Resolution and weighted Resolution. Unlike such systems, weighted Polynomial Calculus manipulates polynomials with coefficients in a finite field and either weights in N or Z. We show the soundness and completeness of weighted Polynomial Calculus via an algorithmic procedure.Weighted Polynomial Calculus, with weights in N and coefficients in F2, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in Z, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable.

On the geometry of zero sets of central quaternionic polynomials

Let R be the ring H[x1,…,xn] of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if p∈R vanishes at all the common zeros of I in Hn with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in Hn. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation

To compute the greatest common divisor (GCD) of a set of multivariate polynomials, modular algorithms are typically employed to prevent any growth in the coefficient polynomials in the intermediate expressions. However, when dealing with multivariate polynomials with a priori errors on their coefficients, using such modular algorithms becomes challenging. This is because any resulting approximate GCD computed in one variable may have perturbations depending on the evaluation point and may not be an image of the same desired multivariate approximate GCD. This necessitates computing it as given multivariate polynomials, and operating with large matrices whose size is exponential in the number of variables. In this paper, we present a new modular algorithm, suitable for dense cases and effective for sparse ones, called “SLRA interpolation”. This algorithm uses the multidimensional fast Fourier transform (FFT) and the structured low-rank approximation (SLRA) of non-square block diagonal matrices. The SLRA interpolation technique may reduce the time-complexity for one iteration in the computation of approximate GCD of several multivariate polynomials, especially for the sparse case.

Spectrahedral shadows and completely positive maps on real closed fields

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application we prove that the cone of copositive matrices of size n\geq5 is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model-theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital \mathbb{R} -linear completely positive map R\to R on a real closed field extension R of \mathbb{R} .

Spherical bispectrum expansion and quadratic estimators

We describe a general expansion of spherical (full-sky) bispectra into a set of orthogonal modes. For squeezed shapes, the basis separates physically-distinct signals and is dominated by the lowest moments. In terms of reduced bispectra, we identify a set of discrete polynomials that are pairwise orthogonal with respect to the relevant Wigner 3j symbol, and reduce to Chebyshev polynomials in the flat-sky (high-momentum) limit for both parity-even and parity-odd cases. For squeezed shapes, the flat-sky limit is equivalent to previous moment expansions used for CMB bispectra and quadratic estimators, but in general reduces to a distinct expansion in the angular dependence of triangles at fixed total side length (momentum).We use the full-sky expansion to construct a tower of orthogonal CMB lensing quadratic estimators and construct estimators that are immune to foregrounds like point sources or noise inhomogeneities. In parity-even combinations (such as the lensing gradient mode from TT, or the lensing curl mode from EB) the leading two modes can be identified with information from the magnification and shear respectively, whereas the parity-odd combinations are shear-only. Although not directly separable, we show that these estimators can nonetheless be evaluated numerically sufficiently easily.

Finite algebras in the design of multivariate cryptography algorithms

A new approach to the design of multivariate public-key cryptalgorithms is introduced. It envisages using non-linear mappings defined as squaring and cubic operations in finite fields represented as finite algebras. The developed approach allows significant reduction of the size of public key and thereby make post-quantum algorithms of multivariate cryptography much more practical. In the developed algorithms, the secret key includes a set of values of structural constants that determine the modifications of the finite fields used and the coefficients in the set of sixth degree polynomials that make up the public key

R I biorthogonal polynomials of Hahn type

ABSTRACT A finite family of R I polynomials is introduced and studied. It consists in a set of polynomials of 3 F 2 form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of R I type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions. The algebra associated to these operators is provided.