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Let G be a graph and R + denote the set of non-negative real numbers. For a vertex v of G , E G ( v ) denotes the set of edges incident with v . Let φ : E ( G ) → R + and f : V ( G ) → R + . Then a generalized fractional f -factor of G is a real-valued function ω : E ( G ) → R + that satisfies 0 ≤ ω ( e ) ≤ φ ( e ) for every e ∈ E ( G ) and f ( v ) = ∑ e ∈ E G ( v ) ω ( e ) for every v ∈ V ( G ) . For two functions g , f : V ( G ) → R + with g ≤ f , we say that G has all generalized fractional ( g , f ) -factors if G has a generalized fractional h -factor for every h : V ( G ) → R + satisfying g ( x ) ≤ h ( x ) ≤ f ( x ) for all x ∈ V ( G ) . In this paper, we present a necessary and sufficient condition for a graph G to have all generalized fractional ( g , f ) -factors, and moreover, our proof is self-contained and does not use the ( g , f ) -factor theorem or the fractional ( g , f ) -factor theorem.

In this paper, we have a new matrix generalization with absolute matrix summability factor of an infinite series by using quasi-β-power increasing sequences. That theorem also includes some new and known results dealing with some basic summability methods

The aim of this paper is to consider an absolute summability method and generalize a theorem concerning $\left|\bar{N},p_{n}\right|_{k}$ summability of infinite series to ${\varphi-\mid{\bar{N},p_n;\delta}\mid}_k$ summability of infinite series by using almost increasing sequence. Furthermore, it is explained that a well known result dealing with $\left|\bar{N},p_{n}\right|_{k}$ summability is obtained when this generalization is restricted under special conditions.

We propose some generalizations of the classical Division Algorithm for polynomials over coefficient rings (possibly non-commutative). These results provide a generalization of the Remainder Theore...

In this chapter, we prove fundamental results on manifolds modeled on Hilbert space (more generally an infinite-dimensional normed linear space E such that E ≈ EN or \(E\approx E_f^N\)) or the Hilbert cube. We also prove the Toru’nczyk Factor Theorem, that is, for each complete metrizable ANR X with weight \(\leqslant \tau \) (\(\leqslant \aleph _0\)), the product of X with the Hilbert space of weight τ is a Hilbert manifold.

In this chapter, we focus on proving the Triangulation Theorem for Hilbert cube manifolds. The compact case is bound up with the Borsuk conjecture asserting that every compact ANR has the homotopy type of a finite simplicial complex. The proof is based on Wall’s work on the homotopy type of a finite simplicial complex (Theorem 4.1.6) and a result on simple homotopy equivalences (Theorem 4.1.5). We follow the course of Chapman’s Lecture Notes (Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. vol. 28, Amer. Math. Soc., Providence, 1975) and provide details for proofs. Combining the Triangulation Theorem with the Edwards Factor Theorem 3.8.1, we know that the Borsuk conjecture is true. We also prove the topological invariance of Whitehead torsion, which had been a longstanding problem.

We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division metho...

We use that the n-sphere for \(n\ge 2\) is simply-connected to prove the Poincare-Birkhoff-Witt Theorem.

In 1995, Palfy and Szabo stated the theorem that a projective space satisfies the six-cross theorem if and only if it is Desarguesian. A mistake in their proof is corrected. Moreover, for projective planes, the three-cross theorem of Palfy and Szabo is identified as the Reidemeister condition.

In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport–Hasse's theorem for the polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes.

Some results for the polynomial inequalities with restricted zeros are established with the help of a lemma proved by Dubinin. The estimates obtained, in particular, depend on the constant term and the leading coefficient of the polynomial P(z) and strengthen as well as generalize some known results earlier proved by Lax, Turan, Aziz and others.

We use suitably Page’s theorem to get effective results for interesting problems, by avoiding the ineffective Siegel’s theorem.

We propose an alternative proof of Pellet’s theorem for matrix polynomials that, unlike existing proofs, does not rely on Rouche’s theorem. A similar proof is provided for the generalization to matrix polynomials of a result by Cauchy that can be considered as a limit case of Pellet’s theorem.

In this paper we are concerned with a Gordan-type theorem involving an arbitrary number of inequality functions. We not only state its validity under a weak convexity assumption on the functions, but also show it is an optimal result. We discuss generalizations of several recent results on nonlinear quadratic optimization, as well as a formula for the Fenchel conjugate of the supremum of a family of functions, in order to illustrate the applicability of that theorem of the alternative.

This paper provides extension to the types of factoring of polynomials to illustrate the fact that multiplication with polynomials has no restrictions when it comes to the number of terms so does factoring. It aims to develop students’ perception about polynomials and to motivate them to continue discovering factorable polynomials.that are new to us.The idea originated from the expression x4 + x2 y2 + y4 which is a product of x2 + xy + y2 and x2 – xy + y2. Factor theorem, rational zeros of a polynomial, and symmetric and transitive properties of real numbers were used in this study in proving other factorable polynomials. The contents include the difference of perfect nth powers (ax)n – (by)n, the trinomial of the form xn + (xy)n/2 + yn, and the general form of a factorable polynomial p(x) = xn + xn – d yd + xn – 2d y2d + … + x2dyn – 2d + xdyn – d + yn for some positive integers n and d with a and b as real numbers.

Rouche’s Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The winding number provides a geometric interpretation relating to the conclusion of Rouche’s Theorem, but most undergraduate texts give no geometric insights that lead to an understanding of why Rouche’s Theorem holds. In addition, most texts do not inform students that a stronger version of the theorem exists. In this paper we present a simplified proof of the stronger version, which is a suitable topic for students to pursue as a short project, and provide a geometric argument for the weaker version. Finally, as a project for advanced students, we unpack a standard application of this theorem as used in control systems: the Nyquist stability criterion.

The solution of the motion equation of the rigid body systems with higher degrees of freedom 2 ≤ p° ≤ 10 is difficult. The presented method allows solving the motion equations of such systems by its transformation to higher degrees’ algebraic characteristic equations. The vibration of the system is then described by frequencies obtained from solution of characteristic equations. The proposed method follows Bezout's factor theorem, Bairstow-Hitchcock's method, method of synthetic division and other presuppositions given in the article. The solution is based on the determination of the complex zeros of polynomials.

An 84-year-old classical result of Ingham states that a rather general zero-free region of the Riemann zeta function implies an upper bound for the absolute value of the remainder term of the prime number theorem. In 1950 Tur´an proved a partial conversion of the mentioned theorem of Ingham. Later the author proved sharper forms of both Ingham’s theorem and its conversion by Tur´an. The present work shows a very general theorem which describes the average and the maximal order of the error terms by a relatively simple function of the distribution of the zeta zeros. It is proved that the maximal term in the explicit formula of the remainder term coincides with high accuracy with the average and maximal order of the error term.

In this paper, we impose restrictions on the complex coefficients of a polynomial in order to give bounds concerning the number of zeros in a specific region of the complex plane. Our results generalize and refine a good number of results in this area of research.

Summary This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].

In this paper, we have extended a known theorem dealing with vertical bar(N) over bar ,p(n)vertical bar(k) summability factors to the vertical bar A, p(n)vertical bar(k) summability under weaker conditions by using an almost increasing sequence

In this paper, we have generalized a new summability factor theorem for infinite series involving quasi power increasing sequences. Some new results are also deduced.

Let G be a graph with vertex set V and let g , f : V → Z + be two functions such that g ≤ f . We say that G has all ( g , f ) -factors if G has an h -factor for every h : V → Z + such that g ( v ) ≤ h ( v ) ≤ f ( v ) for every v ∈ V and ∑ v ∈ V h ( v ) ≡ 0 ( mod 2 ) . Two decades ago, Niessen derived from Tutte’s f -factor theorem a similar characterization for the property of graphs having all ( g , f ) -factors and asked whether there is a polynomial time algorithm for testing whether a graph G has all ( g , f ) -factors. (A characterization of graphs having all ( g , f ) -Factors, (Niessen, 1998).) In this paper, we show that it is NP-hard to determine whether a graph G has all ( g , f ) -factors, which gives a negative answer to the question of Niessen.

In this paper, we first present a family of iterative algorithms for simultaneous determination of all zeros of a polynomial. This family contains two well-known algorithms: Dochev-Byrnev’s method and Ehrlich’s method. Second, using Proinov’s approach to studying convergence of iterative methods for polynomial zeros, we provide a semilocal convergence theorem that unifies the results of Proinov (Appl. Math. Comput. 284: 102–114, 2016) for Dochev-Byrnev’s and Ehrlich’s methods.

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincare theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.