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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 [...]

We offer a formulation of first-order logic that is diagrammatic in nature, retains a basic part of C. S. Peirce's own graphics, renders quantification both graphical and easily readable, and seems a natural approach to what lies at or near the foundations of his science of Semeiotic. The approach is in agreement with Peirce's propositional graphics (Alpha EG), but different in its treatment of quantifiers. Our syntax is formulated with the 'atoms' of the usual predicate calculus given a fine structure in terms of a bonding operation on monadic, dyadic, and triadic building blocks. A suitable semantics is introduced, based on relative product and culminating with a proof of a precise formulation of the famous Peircean Non-Reduction Thesis (NRT). (In particular, in Theorem B', we document precisely the quantificational aspect of NRT: quantification is a multivalent operation, with full expressivity occurring only for valencies ≥ 3.) The principles NRT and RCT (Relational Completeness Thesis) are implicit in predicate logic, whatever form it might take; but their explicit presentation has generally been either neglected (classical predicate calculus) or attended by technical complexities (Peirce's Beta EG, Burch's PAL). Here, the proofs are both explicit and relatively simple. Other contexts in which our 'molecular' treatment of relations may be advantageous will be indicated. Keywords: Charles Peirce, diagrammatic notation, relational bonding, quantifier valency, orbits, non-reduction theorem, Semeiotic., Preface There are at least a few plausible grounds for our use of the term Beta in our title, notwithstanding that there is a key departure, in our framework, from [...]

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.

In this note, the authors propose a generalization of the well known Wiener-Hopf design method of optimal controllers and filters, applicable to a certain class of systems described by fractional order differential equations, the so called rational order systems that, in the Laplace domain, are described by transfer functions which are quotients of polynomials in [s.sup.[alpha]], [alpha] = (1/q), q being a positive integer. As can be verified in the literature, such transfer functions arise in the characterization of some industrial processes and physical systems which can be adequately modeled using fractional calculus, or when modeling some distributed parameter systems by finite dimensional models. A brief exposition of the standard Wiener-Hopf method, and some fundamental considerations about rational order systems are given before presenting the proposed procedure. Illustrative examples are discussed. Index Terms--Fractional systems, optimal controllers, rational order, spectral factorization, Wiener-Hopf.

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.

Abstract -- Simple interval calculation is a method for linear modeling and for constructing interval estimates of predictions in a multivariate calibration. Simple interval calculation gives results in a convenient [...]

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 [...]

The ML module system provides powerful parameterization facilities, but lacks the ability to split mutually recursive definitions across modules and provides insufficient support for incremental programming. A promising approach to solve these issues is Ancona and Zucca's mixin module calculus CMS. However, the straightforward way to adapt it to ML fails, because it allows arbitrary recursive definitions to appear at any time, which ML does not otherwise support. In this article, we enrich CMS with a refined type system that controls recursive definitions through the use of dependency graphs. We then develop and prove sound a separate compilation scheme, directed by dependency graphs, that translates mixin modules down to a call-by-value [lambda]-calculus extended with a nonstandard 1et rec construct. Categories and Subject Descriptors: D.3.1 [Programming Languages]: Formal Definitions and Theory--Semantics; D.3.2 [Programming Languages]: Language Classifications--Applicative (functional) languages; object-oriented languages; D.3.3 [Programming Languages]: Language Constructs and Features Inheritance;modules;packages;recursion;F.3.2[LogicsandMeanings of Programs]: Semantics of Programming Languages--Operational semantics; F.3.3 [Logics and Meanings of Programs]: Studies of Program Constructs--Type structure General Terms: Languages, Theory Additional Key Words and Phrases: Mixins, modules, recursion, type systems

In programming language and software engineering, the main mathematical tool is de facto some form of predicate logic. Yet, as elsewhere in applied mathematics, it is used mostly far below its potential, due to its traditional formulation as just a topic in logic instead of a calculus for everyday practical use. The proposed alternative combines a language of utmost simplicity (four constructs only) that is devoid of the defects of common mathematical conventions, with a set of convenient calculation rules that is sufficiently comprehensive to make it practical for everyday use in most (if not all) domains of interest. Its main elements are a functional predicate calculus and concrete generic functionals. The first supports formal calculation with quantifiers with the same fluency as with derivatives and integrals in classical applied mathematics and engineering. The second achieves the same for calculating with functionals, including smooth transition between pointwise and point-free expression. The extensive collection of examples pertains mainly to software specification, language semantics and its mathematical basis, program calculation etc., but occasionally shows wider applicability throughout applied mathematics and engineering. Often it illustrates how formal reasoning guided by the shape of the expressions is an instrument for discovery and expanding intuition, or highlights design opportunities in declarative and (functional) programming languages. Categories and Subject Descriptors: D.2.1 [Software Engineering]: Requirements/Specifications; D.3.1 [Programming Languages]: Formal Definition and Theory; E.1 [Data Structures]; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic; F.4.3 [Mathematical Logic and Formal Languages]: Formal Languages; I.l.1 [Symbolic and Algebraic Manipulation]: Expressions and Their Representation; I. 1.3 [Symbolic and Algebraic Manipulation]: Languages and Systems; K.3.2 [Computers and Eduction]: Computer and Information Science Education General Terms: Design, Documentation, Languages, Theory, Verification Additional Key Words and Phrases: Analysis, binary algebra, calculational reasoning, databases, declarative languages, elastic operators, function equality, functional predicate calculus, generic functionals, Leibniz's principle, limits, programming languages, program semantics, quantifiers, recursion, software engineering, summation

When an algorithm in dimension one is extended to dimension d, in nearly every case its computational cost is taken to the power d. This fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality. For numerical analysis in dimension d, we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy. Basic linear algebra operations can be performed in this representation using one-dimensional operations, thus bypassing the exponential scaling with respect to the dimension. Although not all operators and algorithms may be compatible with this representation, we believe that many of the most important ones are. We prove that the multiparticle Schrodinger operator, as well as the inverse Laplacian, can be represented very efficiently in this form. We give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground-state eigenfunction for a simplified Schrodinger operator with 30 particles. We conjecture and provide numerical evidence that functions of operators inherit this property, in which case numerical operator calculus in higher dimensions becomes feasible.

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.

The calculus of stationary points or indirect optimization is noted for rapid speed. It's speed is realized in an analytical representation of the Hessian and a polynomial fit of both the objective function and the constraints. Constraints are treated as active using Lagrangian multipliers. Because the algorithm's speed is so rapid, the problem can be repeated multiple times with random starting guesses to circumvent the possibility of encountering local wells. Both an analytic and an actual field problem are considered by way of providing examples of the technique. Index terms - optimization, functional approximations, gradients

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 application of the Jones calculus to optical circuits in which counterpropagating light beams are present is discussed, with particular attention to the conventions associated with the Jones calculus for the description of the polarization state of an optical beam and to the reference system adopted. The reference system adopted here differs from that used by Jones, but it exploits a simpler formalism when used to describe complex optical systems where multiple reflections take place. The differences between these two reference systems are pointed out with particular attention to the behavior of nonreciprocal optical devices and reflectors. An application of the results to a simple optical circuit is presented.

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.