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Formulating a Schubert problem as solutions to a system of equations in either Plücker space or local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale's -theory., Author(s): Jonathan D. Hauenstein[sup.1] , Nickolas Hein[sup.2] , Frank Sottile[sup.3] Author Affiliations: (1) Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, 46556, Notre Dame, IN, USA [...]

The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifications in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and generalizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.

The authors present two calculi for intuitionistic logic that eliminates the need for global metarules.

The article solves the ladder problem-using envelope, which is a direct approach wherein maximum line that will fit around the corner is calculated. The traditional approach was to find minimal line that cannot go around the corner.

Concepts of calculus are used to calculate the floor space required to accommodate the opening and closing of a bifold door mounted on a closet. A parameterization of the floor space reveals that the same astroidal and elliptical curves occur, irrespective of the length of the door panels.

A completeness proof for the implicative fragment of intuitionistic calculus with respect to semantics of computability logic, which understands intuitionist implication as interactive algorithmic reduction, is provided.

A direct proof of completeness of multiplicative proof-nets, MLL starting from a set of simple generators is provided by examining 'weak-distributivity' as a rewriting rule.

Intensional models for the classical theory of types are defined and an intensional type logic, ITL is resulted. A cut-free sequent calculus for type theory is presented and the completeness of this calculus with respect to the class of intensional models through a model existence theorem is shown.

A planimeter is a device used to measure the area enclosed by a curve and the most familiar is the polar planimeter. Proofs using Green's theorem for the rolling and polar planimeters are illustrated, which can be used in a vector calculus course. Both these planimeters are available in mechanical and electronic versions for commercial use.

Robust Lipschitzian properties of set-valued mappings and marginal functions play a crucial role in many aspects of variational analysis and its applications, especially for issues related to variational stability and [...]

Byline: Troy Story (1) Keywords: applied differential geometry; exterior calculus Abstract: Mathematical models of dynamics employing exterior calculus are shown to be mathematical representations of the same unifying principle namely, the description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus, to a set of characteristic differential equations and a characteristic tangent vector which define transformations of the system. This principle, whose origin is Arnold's use of exterior calculus to describe Hamiltonian mechanics and geometric optics, is applied to irreversible thermodynamics and the dynamics of black holes, electromagnetism and strings. It is shown that 'exterior calculus' models apply to systems for which the direction of change is given by a characteristic tangent vector and 'conventional calculus' models apply to systems whose direction of change is arbitrary. The relationship between the two types of models is shown to imply a technical definition of equilibrium for a dynamic system. Author Affiliation: (1) Department of Chemistry, Morehouse College, Atlanta, GA, 30314, USA Article History: Registration Date: 11/01/2005

The authors provide direct proof for admissibility of cut and other rules for a sequent calculus in intuitionistic logic. They name this calculus G4i and show how to extend the proof with first-order syntax and multiple succedents.

Bernhard Bolzano and Augustin Louis Cauchy defined the concepts of continuity and limit to expand the uses of calculus in the early 19th century. They initiated the use of delta and epsilon to prove statements that involved limits and continuity in the context of infinite processes.

Trochoids are a family of curves created by a tack in a wheel that is traveling around a circle. In this article, an alternate model is presented that uses a tack in a bungee cord that is fixed to a circular track to create series of trochoids.

The authors consider the modal-calculus in terms of interpolation, the Lyndon Theorem, and the Los-Tarski Theorem. All 3 classical theorems are proven for the mu-calculus.

This article introduces students to the study of fractional calculus and derivatives. Problems involving fractional derivatives of exponential functions, trigonometric functions, and iterated integrals are presented.

ABSTRACT : A number is said to be a Smarandache Lucky Number if an incorrect calculation leads to a correct result. For example, the fraction 42/21 = (4-2)/(2-1) = 2/1 [...]

The author argues that there is a topology that can examine partial derivatives of a function. Proof of this concept and implications for the study of multivariable calculus are discussed.

The partition calculus of ordinals is discussed, with a focus on complementary nonarrow consistency proofs. The negative partition relation proved by Galvin is shown to be consistent; ccc forcing is also examined.

Infinitesimals have long baffled mathematicians since these are easy to manipulate algebraically yet difficult to understand. A method for the non-nonstandard analysis of infinitesimals is discussed. The method is based on existing calculus texts and allows the natural construction of infinitesimals. The manipulation of sequences of real numbers without using the Axion of Choice and Transfer Principle is described.

A basic theorem of calculus for gauge integrals was proven by describing gauge integrals and incorporating the concept of the parametric derivative. One of the conditions for which F can be considered as a parametric derivative of f on {a,b} is that a differentiable function phi should exist and map an arbitrary interval {alpha, beta} onto {a,b}. The theorem stipulates that a function f(x) is gauge integrable over an interval {a,b} and that the integral from a to b of f(x)dx is equal to F(b) F(a), provided that f(x) is a parametric derivative of F(x) on the same interval.

The calculus formula F'(j) = {F(j+h)-F(j-h)}/2h was used to approximate the value of slowly convergent series. Generalizations were then made regarding more precise calculations of derivatives and the alternating series summation from one to infinity of (-1)(super j+1)a(sub j). The method was then compared with Euler-Maclaurin summation formula. The method generates certain error bounds of the form f(super m-2)(k), which are similar to error bounds that result in the summation estimates using the Euler-Maclaurin summation formula.

Calculus is characterized by concepts of geometric series. Concerns associated with finite geometric series can be generalized by direct multiplication and a standard definition of derivatives. Integrals directly correlate with Riemann sums during the computation of simple integrals with uniform partitions. Moreover, geometric dissections of intervals can be solved by utilizing the Riemann sum definitions characterized by positive integral powers.

An alternative to the Mean Value Theorem called the Increasing Function Theorem (IFT) is proposed. IFT states that if f' is greater than or equal to 0 on an interval, then f is increasing on that interval. IFT is claimed to be easier to state, more intuitive and more effective. The IFT shows how the theoretical structure of calculus can be made more thoughtful and sensible.

The Increasing Function Theorem (IFT) has been proposed as an easier, more intuitive and more effective replacement to the Mean Value Theorem (MVT). Although the IFT does have its strengths, it cannot be used to demean the value of MVT. MVT provides an effective test of the effectiveness of Weierstrass' definition of limit and continuity and can be used more often than IFT in establishing various kinds of results.

Infinitesimals, the infinitesimally small numbers, cannot be seen or observed through practical means, but can be visualized using mathematical tools. All the elementary properties of addition and multiplication of real numbers are also valid for infinitesimals. These elementary properties of infinitesimals help in proving the continuity of quadratic functions. Calculus can be better understood by taking into account the concept of infinitesimals. A brief description of the real number system in conjunction with Tennenbaum's theorem for polynomials leads to better understanding of infinitesimals.

Gerardus Mercator's map-projection technique of the 16th century can be used to solve the calculus problem of multivariable arc length. Mercator's map enabled rhumb lines, which are constant bearing paths connecting two points. These rhumb lines can be parametrized using rectangular coordinates and trigonometric identities. An improper integral then gives arc length in real three-dimensional space.

The importance of tangents in calculus is emphasized by the relation between tangents and derivatives. The graph of any function F possesses a tangent at the location which has the coordinates (c, F(c)), if the derivative F(c) is present. The reverse of this does not exist. Tangents are also related to the derivatives through tangent vectors. The tangent of a simple arc has a line at P, when the simple arc possesses a non-zero tangent vector at the location P. The reverse generally does not exist, but under certain conditions, when P is a cusp, it may exist.

Concavity and convexity of functions in calculus can be defined without resort to derivatives. This approach uses the concept of lower bounding line. Convexity is first defined and then the definition is expanded to include concavity and differentiability. Five conditions, one lemma and one theorem are used.

Operations of addition and restriction are demonstrated to be continuous, given the usual constrain qualifications, and these results are applicable for convex programming. The results are valid for normed linear spaces without completedness being necessary. The approximation results can also be applied in a situation where X is a Banach space, giving a generic theorem which can be used for problems of the constrained convex type.

Summation by parts can be applied to calculus and number theory. This technique is the discrete analog of integration by parts and can be applied to geometric figures composed of areas of rectangles. The technique is applicable to many topics in calculus including the definite integral, geometric series, arithmetic sequences, derivatives of powers and reduction formulae. From number theory, sequences of Fibonacci numbers are examined.

A survey has been done to prove that the principle of universality is a generic phenomenon in analysis. It has been believed that any irregularly behaving process in analysis is capable of producing a universal element. Universality was first observed in 1914 by Fekete. His studies showed that there exists a real power series that only diverges at every point. Thus, ot every continuous function g on [-1,1] with g(0)=0 there exists an increasing sequence of integers.

An inductive method simplifies the application of L'Hopital's rule to problems involving a tower of exponents, giving a limit of x to the xth power ... to the xth power as x tends to zero from the right. The expression tends to zero for odd number of x and tends to one for even number of x. The n-th hyperpower of x is evaluated using parentheses nesting and standard calculus principles. A theorem for determining the limits of x raised to a tower of exponents is presented and an inductive proof is considered.

The author develops a witnessing theorem and employs it to explain the different fragments of the quantified propositional calculus proof system. The formulas of the quantified propositional proof system are shown to be closely related to the different theorems of Buss's theory.

The article discusses a new approach, which employs Klop's l-translation and perpetual reductions for proving the strong cut-elimination in the intuitionistic sequent calculus. The method is shown to be extremely effective and accurate and has been used for a large number of systems.

Derivations of two infinite products of 'pi', Wallis's and Vieta's, are presented. A method to accelerate the convergence of Vieta using calculus is presented, and it is shown that the accelerated Vieta's formula can be used to approximate 'pi' to more than 300,000 digits.

A concise note on Bernoulli numbers and Bernoulli polynomials is presented. Some of the power-sum formulas are presented, and their proofs are given using elementary calculus concepts.

A detailed and expanded presentation of those parts of the Geometriae pars universalis that refer to the basic ideas of the calculus while keeping Gregory's own notation is provided. The basic features of the calculus such as tangents, areas and their relationships are discussed.

A new method to establish the monotonicity of a quotient of two functions is discussed. A new rule that applies to a wide class of quotient of functions is developed which is validated by application of the Cauchy mean value theorem, also known as the generalized mean value theorem.

Existence of divergent samples of 'harmonic-like' series is proved. An uncountably infinite family of examples of the divergence of balanced harmonic like series is constructed.

A theorem about volume found in Courant's calculus text is explained. It generalizes both Cavalieri's Principle and the Theorem of Pappus as a means for computing the volume swept out by a moving planar region.

Both S4 and the Intuitionistic propositional calculus Int are P-SPACE complete, enabling a polynomial translation between the systems. A solution for a sound and faithful polynomial translation from S4 into Int is presented, which is based on Kripke semantics and which describes model checking for S4 using formulas of Int.

The overall philosophy as well as some specific details of improvements that were undertaken to improve the learning of calculus sequence at a mid-sized state university starting in 1997 are described. The project and its underlying philosophy could serve as a blueprint for endeavors like it at similar institutions.