Independent Polynomial Research Articles

SEMI-ANALYTIC FIBONACCI POLYNOMIAL SOLUTION FOR VOLTERRA–FREDHOLM INTEGRAL EQUATION WITH ERROR ANALYSIS

Herein, a spectral scheme is implemented and analyzed for numerically handling general Volterra–Fredholm integral equations (VFIEs), for this purpose, the linearly independent Fibonacci polynomials are utilized as basis functions for the solution, then the spectral collocation process is used to transform the integral equation into a system of algebraic equations with undetermined coefficients. The error, convergence and stability analyses of the scheme are discussed in-depth, some numerical examples are exhibited to ensure the applicability, efficiency and accuracy of the solver.

A new basis for the U-invariants of binary forms

Let K be a field of characteristic zero. The K-vector space Vn of all binary forms of degree n with coefficients in K carries a natural action of the group via substitutions of variables x and y. The invariant polynomials under the action of certain subgroups of were studied intensively in the 19th century. If U is the subgroup of the upper triangular unipotent matrices, with the aid of heavy computer calculations, we know the algebra of U-invariant polynomials in only for It appears to be a hopeless task to get much further along these lines. However, it is known that after inverting the U-invariant a 0, the algebra has a very special form, namely it is generated by algebraically independent polynomials together with In this note, we give an explicit set of such polynomials which are of degrees 2 and 3. We also extend these results to U-invariants of several binary forms.

Independent domination polynomial of zero-divisor graphs of commutative rings

Independent domination polynomial of zero-divisor graphs of commutative rings

Independent component analysis-based arbitrary polynomial chaos method for stochastic analysis of structures under limited observations

Reasonable modelling of non-Gaussian structural inputs from limited observations as well as efficient propagation of system response are of paramount importance in the uncertain analysis of real engineering problems. In this paper, we develop an independent component analysis-based (ICA-based) arbitrary polynomial chaos (aPC) method for the construction of non-Gaussian random model and associated propagation of response under limited observations. Our method firstly introduces the ICA in the context of Karhunen-Loeve (KL) representation of observations of uncertain parameters. By further implementing the aPC expansion on ICA variables, the associated aPC-based response propagation is developed. Since the ICA variables exhibit minimized dependence in all of the orthogonal transformation of KL coefficients, the curse of dimensionality and the inaccurate modelling encountered in the existing PC-based methods can be overcome in the developed ICA-based input model. Also, since the aPC expansion enables to directly construct PC representation with respect to arbitrary probability measure, the convergence of conventional PC-based response propagation is significantly accelerated, resulting an efficient aPC-based response analysis. In this way, the current work provides an effective framework for the reasonable stochastic modelling and efficient response propagation of real-life engineering systems with limited observations. Two numerical examples are presented, including the stochastic analysis of structures subjected to random seismic ground motion, highlighting the effectiveness of the proposed method.

Bivariate Lagrange interpolation at the checkerboard nodes

In this paper, we derive an explicit formula for the bivariate Lagrange basis polynomials of a general set of checkerboard nodes. This formula generalizes existing results of bivariate Lagrange basis polynomials at the Padua nodes, Chebyshev nodes, Morrow-Patterson nodes, and Geronimus nodes. We also construct a subspace spanned by linearly independent bivariate vanishing polynomials that vanish at the checkerboard nodes and prove the uniqueness of the set of bivariate Lagrange basis polynomials in the quotient space defined as the space of bivariate polynomials with a certain degree over the subspace of bivariate vanishing polynomials.

On the Independent Neighborhood Polynomial of the Rooted Product of Two Trees

Let G be a connected graph. We say that a given graph is a tree if every pair ofvertices is connected by a unique path. The rooted product of two trees is relevant to tree, as the obtained product is another tree. In this paper, we establish the independent neighborhood sets of a tree and obtain its corresponding independent neighborhood polynomial. Furthermore, the independent neighborhood polynomial of the rooted product of two trees were determine using their independent neighborhood sets.

On several notions of complexity of polynomial progressions

AbstractFor a polynomial progression $$ \begin{align*}(x,\; x+P_1(y),\ldots,\; x+P_{t}(y)),\end{align*} $$ we define four notions of complexity: Host–Kra complexity, Weyl complexity, true complexity and algebraic complexity. The first two describe the smallest characteristic factor of the progression, the third refers to the smallest-degree Gowers norm controlling the progression, and the fourth concerns algebraic relations between terms of the progressions. We conjecture that these four notions are equivalent, which would give a purely algebraic criterion for determining the smallest Host–Kra factor or the smallest Gowers norm controlling a given progression. We prove this conjecture for all progressions whose terms only satisfy homogeneous algebraic relations and linear combinations thereof. This family of polynomial progressions includes, but is not limited to, arithmetic progressions, progressions with linearly independent polynomials $P_1, \ldots ,\!P_t$ and progressions whose terms satisfy no quadratic relations. For progressions that satisfy only linear relations, such as $$ \begin{align*}(x,\; x+y^2,\; x+2y^2,\; x+y^3,\; x+2y^3),\end{align*} $$ we derive several combinatorial and dynamical corollaries: first, an estimate for the count of such progressions in subsets of $\mathbb {Z}/N\mathbb {Z}$ or totally ergodic dynamical systems; second, a lower bound for multiple recurrence; and third, a popular common difference result in $\mathbb {Z}/N\mathbb {Z}$ . Lastly, we show that Weyl complexity and algebraic complexity always agree, which gives a straightforward algebraic description of Weyl complexity.

A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in $m+1$ variables with coefficients in the finite field ${\mathbb{F}}q$ with $q$ elements, when $d<q$. It is shown that this formula holds in the affirmative for several values of $r$. In the general case, we give explicit lower and upper bounds for $e_r(d,m)$ and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal combinatorics such as the Clements-Lindstrom Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed-Muller codes are also included.

Canonical representation of three-qubit states with real amplitudes

Let us say that a three-qubit state u 000|000⟩ + u 001|001⟩ + ⋯ + u 111|111⟩ is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state |ϕ⟩, there exist three angles θ 0, θ 1 and θ 2 such that R y (θ 2) ⊗ R y (θ 1) ⊗ R y (θ 0)|ϕ⟩ is a three-qubit of the form λ 1|000⟩ + λ 2|011⟩ + λ 3|101⟩ + λ 4|110⟩ + λ 5|111⟩ with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (Acín et al 2000 Phys. Rev. Lett. 85 1560; Acín et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643).

Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces

The theory of analytic functions is an important section of nonlinear functional analysis.In many modern investigations topological algebras of analytic functions and spectra of suchalgebras are studied. In this work we investigate the properties of the topological algebras of entire functions,generated by countable sets of homogeneous polynomials on complex Banach spaces.&#x0D; Let $X$ and $Y$ be complex Banach spaces. Let $\mathbb{A}= \{A_1, A_2, \ldots, A_n, \ldots\}$ and $\mathbb{P}=\{P_1, P_2,$ \ldots, $P_n, \ldots \}$ be sequences of continuous algebraically independent homogeneous polynomials on spaces $X$ and $Y$, respectively, such that $\|A_n\|_1=\|P_n\|_1=1$ and $\deg A_n=\deg P_n=n,$ $n\in \mathbb{N}.$ We consider the subalgebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ of the Fr\'{e}chet algebras $H_b(X)$ and $H_b(Y)$ of entire functions of bounded type, generated by the sets $\mathbb{A}$ and $\mathbb{P}$, respectively. It is easy to see that $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ are the Fr\'{e}chet algebras as well.&#x0D; In this paper we investigate conditions of isomorphism of the topological algebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y).$ We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra $H_{bs}(L_{\infty})$ of entire functions of bounded type on $L_{\infty}[0,1]$ which are symmetric, i.e. invariant with respect to measurable bijections of $[0,1]$ that preserve the measure. We prove that$H_{bs}(L_{\infty})$ is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space $\ell_{\infty}.$

DESIGNING AND OPTIMIZATION OF NAPROXEN SODIUM DEFORMABLE VESICULAR SYSTEMS THROUGH FACTORIAL DESIGN: BOX BEHENKEN MODEL

Objective: The objective of this investigation was to develop and statistically optimize deformable vesicles such as transfersomes and transethosomes of Naproxen sodium by employing 33factorial designs through software Design expert version 12 (Box–Behnken design) for dermal delivery.&#x0D; Methods: The levels of the drug, phosphatidylcholine, and span 80 (independent variables) were varied to study the influence on vesicle size and % entrapment efficiency (dependent variables) of transfersomes and for transethosomes, the levels of phosphatidylcholine, ethanol, and span 80 were selected as independent variables Second-order quadratic polynomial equation, 2D and 3D contour plots represented the relationship between variables and desired response. The optimization process was carried out using desirability plots and point prediction techniques.&#x0D; Results: Results of the present study demonstrated that optimized transfersomes and transethosomes showed vesicle sizes of 114.91 nm and 102.91 nm respectively, while entrapment efficiency of 80.11 % and 86.97%, respectively. Both formulations showed high zeta potential values indicating the stability of the optimized formulation. ANOVA statistical results showed a significant difference (P&lt;0.05).&#x0D; Conclusion: The results indicated that the independent variable plays a crucial role in optimizing a formulation that can be used for further research studies. Present preliminary study data provided strong evidence that the optimized deformable vesicular formulations through box Behnken factorial design can be a potentially useful drug carrier for naproxen sodium dermal delivery with minimum vesicle size and efficient entrapment efficiency.

On the Independent Neighborhood Polynomial of the Cartesian Product of Some Special Graphs

Two vertices x, y of a graph G are adjacent, or neighbors, if xy is an edge of G. A set S of vertices in a graph G is a neighborhood set if G =[v∈ShN[v]i where hN[v]i is the subgraph induced by v and all the vertices adjacent to v. If no two of the elements of S are adjacent, then S is called an independent neighborhood set. The independent neighborhood polynomial of G of order m is Ni(G, x) = Xm j=ηi(G) ni(G, j)xj where ni(G, j) is the number of independent neighborhood set of G of size j and ηi(G) is the minimum cardinality of an independent neighborhood set of G. This paper investigates the independent neighborhood polynomial of the Cartesian product of some special graphs.

The Independent Set Polynomials of a class of Graphs

In this paper, a special graph is defined. According to the structural characteristics of the special graph , the independent set polynomial of a special graph is derived by using the properties and lemma of independent set polynomial of simple graph. The Merrifield-Simmons index of this kind of special graph can be obtained.

Mollified finite element approximants of arbitrary order and smoothness

The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily shaped polytopes. We propose the mollified basis functions of arbitrary order and smoothness for partitions consisting of convex polytopes. On each polytope an independent local polynomial approximant of arbitrary order is assumed. The basis functions are defined as the convolutions of the local approximants with a mollifier. The mollifier is chosen to be smooth, to have a compact support and a unit volume. The approximation properties of the obtained basis functions are governed by the local polynomial approximation order and mollifier smoothness. The convolution integrals are evaluated numerically first by computing the boolean intersection between the mollifier and the polytope and then applying the divergence theorem to reduce the dimension of the integrals. The support of a basis function is given as the Minkowski sum of the respective polytope and the mollifier. The breakpoints of the basis functions, i.e. locations with non-infinite smoothness, are not necessarily aligned with polytope boundaries. Furthermore, the basis functions are not boundary interpolating so that we apply boundary conditions with the non-symmetric Nitsche method as in immersed/embedded finite elements. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson and elasticity problems.

On the independent domination polynomial of a graph

An independent dominating set of the simple graph G=(V,E) is a vertex subset that is both dominating and independent in G. The independent domination polynomial of a graph G is the polynomial Di(G,x)=∑Ax|A|, summed over all independent dominating subsets A⊆V. A root of Di(G,x) is called an independence domination root. We investigate the independent domination polynomials of some generalized compound graphs. As consequence, we construct graphs whose independence domination roots are real. Also, we consider some graphs related to paths and study the number of their independent dominating sets.

Polynomial configurations in sets of positive upper density over local fields

Let F(x) = (f1(x), …, fm(x)) be such that 1, f1, …, fm are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, de Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of ℝm with respect to the portion of the graph of F defined by a ≤ log åså ≤ T is at most O(1/(T — a)). We conclude that if I ⊆ ℝm has positive upper density, then the difference set I — I contains vectors of the form F(s)for an unbounded set of values s ∈ ℝ. It follows that the Borel chromatic number of the Cayley graph of ℝm with respect to the set {±F(s): s ∈ ℝ} is infinite. Analogous results are also proven when ℝ is replaced by the field of p-adic numbers ℚp. At the end, we will also show the existence of real analytic functions f1, …, fm, for which the analogous statements no longer hold.

Sums of random polynomials with independent roots

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let p and q be two independent random polynomials of degree n, whose roots are chosen independently from the probability measures μ and ν in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum p+q as n tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of μ and ν. More generally, we consider the sum of m independent degree n random polynomials when m is fixed and n tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials.

Data-driven sensitivity indices for models with dependent inputs using polynomial chaos expansions

Uncertainties exist in both physics-based and data-driven models. Variance-based sensitivity analysis characterizes how the variance of a model output is propagated from the model inputs. The Sobol index is one of the most widely used sensitivity indices for models with independent inputs. For models with dependent inputs, different approaches have been explored to obtain sensitivity indices in the literature. Typical approaches are based on procedures of transforming the dependent inputs into independent inputs. However, such transformation requires additional information about the inputs, such as the dependency structure or the conditional probability density functions. In this paper, data-driven sensitivity indices are proposed for models with dependent inputs. We first construct ordered partitions of linearly independent polynomials of the inputs. The modified Gram-Schmidt algorithm is then applied to the ordered partitions to generate orthogonal polynomials with respect to the empirical measure based on observed data of model inputs and outputs. Using the polynomial chaos expansion with the orthogonal polynomials, we obtain the proposed data-driven sensitivity indices. The sensitivity indices provide intuitive interpretations of how the dependent inputs affect the variance of the output without a priori knowledge on the dependence structure of the inputs. Four numerical examples are used to validate the proposed approach.

ON CONSTANT COEFFICIENT PDE SYSTEMS AND INTERSECTION MULTIPLICITIES

In this paper we consider the concept of the multiplicity of intersection points of plane algebraic curves $p,q=0,$ based on partial differential operators. We evaluate the exact number of maximal linearly independent differential conditions of degree $k$ for all $k\ge 0.$ On the other hand, this gives the exact number of maximal linearly independent polynomial and polynomial-exponential solutions, of a given degree $k,$ for homogeneous PDE system $p(D)f=0,$ $q(D)f=0.$

An Erdős-Ko-Rado Type Theorem via the Polynomial Method

A family F is an intersecting family if any two members have a nonempty intersection. Erdős, Ko, and Rado showed that | F | ≤ n − 1 k − 1 holds for a k-uniform intersecting family F of subsets of [ n ] . The Erdős-Ko-Rado theorem for non-uniform intersecting families of subsets of [ n ] of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that | F | ≤ n − 1 k − 1 + n − 1 k − 2 + ⋯ + n − 1 0 holds for non-uniform intersecting families of subsets of [ n ] of size at most k. In this paper, we prove that the same upper bound of the Erdős-Ko-Rado Theorem for k-uniform intersecting families of subsets of [ n ] holds also in the non-uniform family of subsets of [ n ] of size at least k and at most n − k with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials.