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In this paper, we use differential operator to present new identities and provide alternative proofs for certain established identities related to geometric polynomials. Additionally, by using the differential operator associated with geometric polynomials, Rolle's theorem and a theorem of Wang and Yeh, we present some results on the real rootedness of polynomials. [ABSTRACT FROM AUTHOR]

The classical Rolle's theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line. [ABSTRACT FROM AUTHOR]

Suppose A = { a 1 , ... , a n + 2 } ⊂ Z n has cardinality n + 2 , with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = (f 1 , ... , f n) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixedn, counts exactly the number of real roots of F in time polynomial in log (d H) . We also discuss a number-theoretic hypothesis that would imply a further speed-up to time polynomial in n as well. [ABSTRACT FROM AUTHOR]

For any real polynomial $p(x)$ of even degree k , Shapiro ['Problems around polynomials: the good, the bad and the ugly $\ldots $ ', Arnold Math. J. 1 (1) (2015), 91–99] proposed the conjecture that the sum of the number of real zeros of the two polynomials $(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)$ and $p(x)$ is larger than 0. We prove that the conjecture is true except in one case: when the polynomial $p(x)$ has no real zeros, the derivative polynomial $p{'}(x)$ has one real simple zero, that is, $p{'}(x)=C(x)(x-w)$ , where $C(x)$ is a polynomial with $C(w)\ne 0$ , and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros. [ABSTRACT FROM AUTHOR]

Generalized complex number systems generalize all unital real algebras of dimension two, including complex numbers, dual numbers, and double numbers. In this article, we have shown that the real part and the imaginary part of a holomorphic function of a generalized complex variable satisfy several forms of mean value theorem individually. In particular, as a corollary, these results hold true for complex holomorphic functions. [ABSTRACT FROM AUTHOR]

This paper proposes a generalization to the ordinary derivative, the deformable derivative. For this, we employ a limit approach like the ordinary derivative but use a parameter varying over the unit interval. The definition makes the deformable derivative equivalent to the ordinary derivative because one’s existence implies another. Its intrinsic property of continuously deforming function to its derivative, together with the graphical illustration of linear expression of the function and its derivative, renders sufficient substances to name it deformable derivative. We derive Rolle’s, Mean-value and Taylor’s theorems for the deformable derivative by establishing some of its basic properties. We also define the deformable integral using the fundamental theorem of calculus and discuss associated inverse, linearity, and commutativity property. In addition, we establish a connection between deformable integral and Riemann-Liouville fractional integral. As theoretical applications, we solve some fractional differential equations. [ABSTRACT FROM AUTHOR]

Consider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes' rule of signs gives compatibility conditions between s and the pair (r+,r−), where r+ is the number of positive roots and r− the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r+,r−) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for (s ; r 0 + , r 0 − ; r 1 + , r 1 − ; ... ; r d − 1 + , r d − 1 −) , where r i + (resp. r i − ) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle's theorem. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we give a complete analysis of the equilibrium points of background neural networks with uniform firing rates. By using continuity, monotonicity of some functions and Rolle’s theorem, the number of equilibrium points and their locations are obtained. Moreover, some novel sufficient conditions are given to guarantee the stability of the equilibrium points for the network model by utilizing Taylor’s theorem. A simulation example is conducted to illustrate the theories developed in this paper.

This paper presents a simple extension of Rolle’s Theorem. This extension allows determining the amount of numbers ξi in which f'(ξi) = 0 in a given interval, using the characteristics of the function f in that interval. The extension has been proved, and the geometric interpretation has been presented. Illustrative examples have also been developed for each case that can be obtained by applying the extension. Finally, the study examines the relation of this theorem with the problem of multiple internal rates of return (IRR).

The main interest of this paper is to describe new computational techniques for solving nonlinear problems using g -fractional differential operators. First, we introduce the concept of the g -conformable fractional differential operator on g -semiring. Then, the mean value theorem and Rolle’s theorem for g -conformable fractional differential operator are investigated. Moreover, we consider the exact solution of g -fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete analysis of the equilibrium points of background neural networks with uniform firing rates. By using continuity, monotonicity of some functions and Rolle's theorem, the number of equilibrium points and their locations are obtained. Moreover, some novel sufficient conditions are given to guarantee the stability of the equilibrium points for the network model by utilizing Taylor's theorem. A simulation example is conducted to illustrate the theories developed in this paper. [ABSTRACT FROM AUTHOR]

This article presents a generalization of Myers' theorem and when the boundary assumption f'(a)=f'(b) is removed, and to prove this result for holomorphic functions of one complex variable. After that, the equivalence of Rolle's and mean value theorems in the complex plane are proved.

We discuss the history of the famous Rolle's theorem "If a function is continuous at [a, b], differentiable in (a, b), and f (a) = f (b), then there exists a point c in (a, b) such that f'(c) = 0", and that of the related theorem on the root interval, "If a function is continuous on [a, b] and has different signs at the ends of the interval, then there exists a point c in (a, b) such that f (c) = 0". [ABSTRACT FROM AUTHOR]

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ( f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L. On the basis of this, for differentiable functions in the classes L, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω. For classes of functions defined by their smoothness characteristics Λ( f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths. [ABSTRACT FROM AUTHOR]

In the space L, we study a number of smoothness characteristics of functions; our study is based on the use of the generalized shift operator τ. For the case inwhich τ is the Steklov operator S, we obtain exact constants in Jackson-type inequalities for some classes of 2 π-periodic functions. We also calculate the exact values of the n-widths of function classes defined by the smoothness characteristics under consideration. [ABSTRACT FROM AUTHOR]

In this paper, we give some new mean value theorems, which are generalizations of Flett, Myers and Tong's theorems. [ABSTRACT FROM AUTHOR]

The maximum power transfer theorem (MPTT) is proved in a number of ways. The paper asserts pedagogical value and is aimed at evoking critical thinking among students. [ABSTRACT FROM AUTHOR]

This article presents a generalization of Myers' theorem and when the boundary assumption f'(a) = f'(b) is removed, and to prove this result for holomorphic functions of one complex variable. After that, the equivalence of Rolle's and mean value theorems in the complex plane are proved. [ABSTRACT FROM AUTHOR]

The interplay of physical intuition, computational evidence, and mathematical rigor in a simple trajectory model is explored. A thought experiment based on the model is used to elicit student conjectures on the influence of a physical parameter; a mathematical model suggests a computational investigation of the conjectures, and rigorous analysis at the level of elementary calculus is used to settle the issue. [ABSTRACT FROM AUTHOR]

In this note, by using elementary calculus on time scales, we recall the direct proof of the equivalence between Rolle's and generalized (Cauchy's) mean value theorems on time scales. [ABSTRACT FROM AUTHOR]

In this article, by using Rolle's theorem, we establish some results related to the mean value theorem for integrals. Our results are different from the set of integral mean value theorems which are given by Wayment [An integral mean value theorem, Math. Gazette 54 (1970), pp. 300-301] and Sahoo [Some results related to the integral mean value theorem, Int. J. Math. Ed. Sci. Tech. 38(6) (2007), pp. 818-822]. The importance of our results are illustrated by interesting examples. [ABSTRACT FROM AUTHOR]

The article discusses the complex zeroes in the Rolle's theorem. It includes some geometric and other elements of the distribution of the zeroes of polynomials. It cites the familiar theorem proven by mathematician Michel Rolle in his book of polynomials which assumes its maximum and minimum values on closed and bounded intervals.

Lesson Study is a process by which a team of instructors works collaboratively to plan, teach, observe and revise one particular lesson. A successful lesson study will not only result in a student-focused lesson plan that maximizes student understanding, but will also encourage teaching-centered collaboration among colleagues and increase the participants' understanding of student learning. A group of four instructors from the University of Wisconsin-Stout participated in a lesson study for Rolle's Theorem and the Mean Value Theorem beginning in the spring of 2007. This article summarizes the process of lesson study, the lesson itself, observations from the first round of teaching and possible revisions to the lesson. Included as an appendix are all of the materials of the preliminary lesson plan, as well as materials for the lesson study itself. [ABSTRACT FROM AUTHOR]

Abstract: This work concerns the extension of a weak form of the Rolle''s theorem to locally convex spaces that satisfy an axiom of separation. The result provides a condition for asserting the uniqueness of a solution to nonlinear functional equations, including nonlinear integro-differential equations. We use the extended Rolle''s theorem to prove the uniqueness of a solution to a nonlinear, fractional differential equation. [Copyright &y& Elsevier]

We establish approximate Rolle''s theorems for the proximal subgradient and for the generalized gradient. We also show that an exact Rolle''s theorem for the generalized gradient is completely false in all infinite-dimensional Banach spaces (even when they do not possess smooth bump functions). [Copyright &y& Elsevier]

In this paper, some basic rules of calculus on R are presented for arbitrary time scales. This allows us to prove Rolle's and mean value theorems on a time scale. [ABSTRACT FROM AUTHOR]

In an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle''s Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle''s Theorem. [Copyright &y& Elsevier]

This paper deals with global injectivity of vector fields defined on euclidean spaces. Our main result establishes a version of Rolle's Theorem under generalized Palais-Smale conditions. As a consequence of this, we prove global injectivity for a class of vector fields defined on n-dimensional spaces. [ABSTRACT FROM AUTHOR]