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It is well known that the Fundamental Theorem of Algebra doesn’t hold true in the case of complex harmonic polynomials. The motive of the manuscript is to explore the zeros of a complex harmonic polynomial F(z) of degree (n + m) . In particular, we prove that the sum of the orders of the zeros of F(z) is - (n + m) . We also show that F(z) has a total (n + m + 2) or (n + m + 2 k + 2) number of zeros under different conditions on the coefficients. In addition to the count of zeros, we locate the zeros of F(z) and obtain that all non-trivial zeros of F(z) lie in the given annular region. [ABSTRACT FROM AUTHOR]

We present an elementary proof of the fundamental theorem of algebra, in which only the most basic notions of calculus are used. Considering discrete quantities instead of continuous functions allows us to obtain a proof that is fairly approachable but maintains the mathematical rigor. [ABSTRACT FROM AUTHOR]

We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting, thus obtaining a) quantitative central limit theorems for approximation of arbitrary non-degenerate Gaussian random elements taking values in a separable Hilbert space and b) fourth moment bounds for approximating sequences with finite chaos expansion. Our results rely on an infinite-dimensional version of Stein's method of exchangeable pairs combined with the so-called Gamma calculus. Two applications are included: Brownian approximation of Poisson processes in Besov-Liouville spaces and a functional limit theorem for an edge-counting statistic of a random geometric graph. [ABSTRACT FROM AUTHOR]

This article explores the accessibility of symbolic computations, such as using the Wolfram Mathematica environment, in promoting the shift from informal experimentation to formal mathematical justifications. We investigate the accuracy of computational results from mathematical software in the context of a certain summation in trigonometry. In particular, the key issue addressed here is the calculated sum ∑ n = 0 44 tan ⁡ 1 + 4 n ° . This paper utilizes Wolfram Mathematica to handle the irrational numbers in the sum more accurately, which it achieves by representing them symbolically rather than using numerical approximations. Can we rely on the calculated result from Wolfram, especially if almost all the addends are irrational, or must the students eventually prove it mathematically? It is clear that the problem can be solved using software; however, the nature of the result raises questions about its correctness, and this inherent informality can encourage a few students to seek viable mathematical proofs. In this way, a balance is reached between formal and informal mathematics. [ABSTRACT FROM AUTHOR]

Since Geronimo Cardano, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We discovered two algorithms that lead to the same resolvente, each with two solutions; therefore, six formal solutions appear to solve an algebraic equation of degree four. Given that a square was utilized to derive the solution in both instances, it is critical to verify each solution. This check reveals that the four Cardanic solutions are the only four solutions to an algebraic equation of degree four. This demonstrates that Carl Friedrich Gauss' (1799) fundamental theorem of algebra is not simple, despite the fact that it is a fundamental theorem. This seems to be a novel insight.

In the paper, we discussed the distribution of unique range sets and its elements over the extended complex plane from a different point of view and obtained some new results regarding the structure and position of unique range sets. These new results have immense applications like classifying different subsets of C to be or not to be a unique range set, exploring the fact that every bi-linear transformation preserves unique range sets for meromorphic functions, providing simpler and shorter proofs of existence of some unique range sets, unfolding the fact that zeros or poles of any meromorphic function lie in a unique range set, in particular, identifying the Fundamental Theorem of Algebra to a more specific region and many more applications. We have also posed some open questions to unveil the mysterious arrangement of the elements of unique range sets. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Polytechnic is the property of Journal of Polytechnic and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article explores the accessibility of symbolic computations, such as using the Wolfram Mathematica environment, in promoting the shift from informal experimentation to formal mathematical justifications. We investigate the accuracy of computational results from mathematical software in the context of a certain summation in trigonometry. In particular, the key issue addressed here is the calculated sum ∑n=044tan⁡1+4n°. This paper utilizes Wolfram Mathematica to handle the irrational numbers in the sum more accurately, which it achieves by representing them symbolically rather than using numerical approximations. Can we rely on the calculated result from Wolfram, especially if almost all the addends are irrational, or must the students eventually prove it mathematically? It is clear that the problem can be solved using software; however, the nature of the result raises questions about its correctness, and this inherent informality can encourage a few students to seek viable mathematical proofs. In this way, a balance is reached between formal and informal mathematics.

We give a simple proof of the existence of a complex eigenvalue of real or complex square matrices with Linear Algebra accessible to students beginning a first course on the subject, namely using just the Cauchy–Schwarz inequality in C n , linear independence and the matrix product, together with the Weierstrass extreme value theorem. It yields an equally simple proof of the Fundamental Theorem of Algebra with Linear Algebra. [ABSTRACT FROM AUTHOR]

In this article, the authors refer to the interesting nature and properties of the number 24 and its occurrence and relevance in different contexts and conjecture that it is the raison d’ˆetre for its occurrence in the definition of the τ -function of Ramanujan. [ABSTRACT FROM AUTHOR]

The purpose of the article is to familiarize teachers and students with the fundamental theorem of algebra for real polynomials and its proof. Polynomials of small degrees are considered separately. It is showed how equations of the third and fourth degree can be solved. An algorithm for solving cubic equations is proposed.

We prove a version of the fundamental theorems of Morse theory in the setting of finite partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell complexes and derive the Morse-Pitcher inequalities in that context. [ABSTRACT FROM AUTHOR]

Copyright of Lietuvos Matematikos Rinkinys is the property of Vilnius University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss's first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra. [ABSTRACT FROM AUTHOR]

Combinatorial properties of zeons have been applied to graph enumeration problems, graph colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series of elementary zeon functions are naturally reduced to finite sums by virtue of the nilpotent properties of zeons. Further, the zeon extension of any analytic complex function has zeon polynomial representations on associated equivalence classes of zeons. In this paper, zeros of polynomials over complex zeons are considered. Existing results for real zeon polynomials are extended to the complex case and new results are established. In particular, a fundamental theorem of zeon algebra is established for spectrally simple zeros of complex zeon polynomials, and an algorithm is presented that allows one to find spectrally simple zeros when they exist. As an application, inverses of zeon extensions of analytic functions are computed using polynomial methods. [ABSTRACT FROM AUTHOR]

The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or Jordan normal forms. Perturbation theory and various regularization techniques help address some numerical difficulties of computation eigenvectors, but often rely on per se uncomputable quantities, such as a minimal gap between eigenvalues. In this note, the eigenvalue problem is revisited within constructive analysis allowing to explicitly consider numerical uncertainty. Exact eigenvectors are substituted by approximate ones in a suitable format. Examples showing influence of computation precision are provided. [ABSTRACT FROM AUTHOR]

Copyright of ArcHistoR is the property of Universita degli Studi Mediterranea di Reggio Calabria and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract Complex rational curves have been used to represent circular splines as well as many classical curves including epicycloids, cardioids, Joukowski profiles, and the lemniscate of Bernoulli. Complex rational curves are known to have low degree (typically half the degree of the corresponding rational planar curve), circular precision, invariance with respect to Möbius transformations, special implicit forms, an easy detection procedure, and a fast algorithm for computing their μ -bases. But only certain very special rational planar curves are also complex rational curves. To construct a wider collection of curves with similar appealing properties, we generalize complex rational curves to hyperbolic and parabolic rational curves by invoking the hyperbolic and parabolic numbers. We show that the special properties of complex rational curves extend to these hyperbolic and parabolic rational curves. We also provide examples to flesh out the theory. [ABSTRACT FROM AUTHOR]

In this paper, the single-variable fuzzy polynomial equations are studied. We firstly define two solution types for the equations, called solution and r-cut solution. Then, sufficient and necessary conditions are proposed for existence of the solution and r-cut solution of the equations, respectively. Also, a new algorithm is designed to find all the solutions and r-cut solutions of the equations using algebraic computations. Based on Descartes' rule of signs, we express and prove a fuzzy version of fundamental theorem of algebra to obtain the number of real roots of a single-variable fuzzy polynomial. Moreover, we present an upper bound on the number of solutions of the equations and show that each single-variable fuzzy polynomial equation has at most two distinct solutions. [ABSTRACT FROM AUTHOR]

Assume Mn is the n‐dimensional permutation module for the symmetric group Sn, and let Mn⊗k be its k‐fold tensor power. The partition algebra Pk(n) maps surjectively onto the centralizer algebra EndSn(Mn⊗k) for all k,n∈Z⩾1 and isomorphically when n⩾2k. We describe the image of the surjection Φk,n:Pk(n)→EndSn(Mn⊗k) explicitly in terms of the orbit basis of Pk(n) and show that when 2k>n the kernel of Φk,n is generated by a single essential idempotent ek,n, which is an orbit basis element. We obtain a presentation for EndSn(Mn⊗k) by imposing one additional relation, ek,n=0, to a presentation of the partition algebra Pk(n) when 2k>n. As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group Sn. We show under the natural embedding of the partition algebra Pn(n) into Pk(n) for k⩾n that the essential idempotent en,n generates the kernel of Φk,n. Therefore, the relation en,n=0 can replace ek,n=0 when k⩾n. [ABSTRACT FROM AUTHOR]

There are several proofs of the fundamental theorem of algebra, mainly using algebra, analysis and topology. The aim of this article is to discuss the fundamental theorem of algebra as an application of Nevanlinna's second fundamental theorem. [ABSTRACT FROM AUTHOR]

In this article we give a new proof of the Fundamental Theorem of Algebra. Our proof is algebraic. We simplify the known proof of the Fundamental Theorem considering special case of polynomials of odd degree with real coeffcients. This case allows us to apply the method of mathematical induction to get the proof in general case without using infiniteness of the field. [ABSTRACT FROM AUTHOR]

The sharp asymptotic constants in L²-Markov inequalities are given for generalized Jacobi weights. They are expressed in terms of the smallest positive zeros of Bessel functions of the first kind and via the equilibrium measure of the ground set. [ABSTRACT FROM AUTHOR]

Abstract In this paper we obtain Liouville type theorems for positive supersolutions of the elliptic system { − Δ u + | ∇ u | q = λ f (v) − Δ v + | ∇ v | q = μ g (u) in exterior domains of R N , where q > 1 and the functions f and g behave like a power near zero or infinity. We show that positive supersolutions do not exist in some ranges of the parameters involved, which are optimal in the special case where f (v) = v p and g (u) = u s , with p , s > 0. [ABSTRACT FROM AUTHOR]

We study λ-hypersurfaces that are critical points of a Gaussian weighted area functional ... dA for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete λ-hypersurfaces in terms of the norm of the second fundamental form |A|. Second, we show that in one dimension, the only smooth complete and embedded λ-hypersurfaces in ℝ² with λ ≥ 0 are lines and round circles. Moreover, we establish a Bernstein-type theorem for λ-hypersurfaces which states that smooth λ-hypersurfaces that are entire graphs with polynomial volume growth are hyperplanes. All the results can be viewed as generalizations of results for self-shrinkers. [ABSTRACT FROM AUTHOR]

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman. [ABSTRACT FROM AUTHOR]

Let OSp ( V ) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension ( m | 2 n ) . Lehrer and Zhang showed that there is a surjective algebra homomorphism F r r : B r ( m − 2 n ) → End OSp ( V ) ( V ⊗ r ) , where B r ( m − 2 n ) is the Brauer algebra of degree r with parameter m − 2 n . The second fundamental theorem of invariant theory in this setting seeks to describe the kernel Ker F r r of F r r as a 2-sided ideal of B r ( m − 2 n ) . In this paper, we show that Ker F r r ≠ 0 if and only if r ≥ r c : = ( m + 1 ) ( n + 1 ) , and give a basis and a dimension formula for Ker F r r . We show that Ker F r r as a 2-sided ideal of B r ( m − 2 n ) is generated by Ker F r c r c for any r ≥ r c , and we provide an explicit set of generators for Ker F r c r c . These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra End osp ( V ) ( V ⊗ r ) over the orthosymplectic Lie superalgebra osp ( V ) to be isomorphic to B r ( m − 2 n ) . [ABSTRACT FROM AUTHOR]

Let p > 2 . We show how the fundamental theorem of surface theory for surfaces of class W loc 2 , p ( ω ) over a simply-connected open subset of R 2 established in 2005 by S. Mardare can be extended to surfaces of class W 2 , p ( ω ) when ω is in addition bounded and has a Lipschitz-continuous boundary. Then we establish a nonlinear Korn inequality for surfaces of class W 2 , p ( ω ) . Finally, we show that the mapping that defines in this fashion a surface of class W 2 , p ( ω ) , unique up to proper isometries of E 3 , in terms of its two fundamental forms is locally Lipschitz-continuous. [ABSTRACT FROM AUTHOR]

The theories of (Hopf) bialgebras and weak (Hopf) bialgebras have been introduced for vector space categories over fields and make heavily use of the tensor product. As first generalisations, these notions were formulated for monoidal categories, with braidings if needed. The present authors developed a theory of bimonads and Hopf monads H on arbitrary categories A , employing distributive laws, allowing for a general form of the Fundamental Theorem for Hopf algebras. For τ -bimonads H , properties of braided (Hopf) bialgebras were captured by requiring a Yang–Baxter operator τ : H H → H H . The purpose of this paper is to extend the features of weak (Hopf) bialgebras to this general setting including an appropriate form of the Fundamental Theorem. This subsumes the theory of braided Hopf algebras (based on weak Yang–Baxter operators) as considered by Alonso Álvarez and others. [ABSTRACT FROM AUTHOR]

Let 0 < α , β < 2 be any real number. In this paper, we investigate a class of fractional elliptic systems of the form { ( − △ ) α / 2 u ( x ) = f ( v ( x ) ) , ( − △ ) β / 2 v ( x ) = g ( u ( x ) ) , x ∈ R + n , u , v ≡ 0 , x ∉ R + n . Applying the iteration method and the direct method of moving planes for the fractional Laplacian, without any decay assumption on the solutions at infinity, we prove the Liouville theorem of nonnegative solutions under some natural conditions on f and g . [ABSTRACT FROM AUTHOR]