Codifferential method for minimizing nonsmooth DC functions

In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (DC functions) is proposed. The algorithm is based on the concept of codifferential. It is assumed that DC decomposition of the objective function is known a priori. We develop an algorithm to compute descent directions using a few elements from codifferential. The convergence of the minimization algorithm is studied and its comparison with different versions of the bundle methods using results of numerical experiments is given.     

Hyperbolic Calculus

The complex numbers are naturally related to rotations and dilatations in the plane. In this paper we present the function theory associate to the (universal) Clifford algebra forIR ^1,0 [1], the so called hyperbolic numbers [2,3,4], which can be related to Lorentz transformations and dilatations in the two dimensional Minkowski space-time. After some brief algebraic interpretations (part 1), we present a “Hyperbolic Calculus” analogous to the “Calculus of one Complex Variable”. The hyperbolic Cauchy-Riemann conditions, hyperbolic derivatives and hyperbolic integrals are introduced on parts 2 and 3. Then special emphasis is given in parts 4 and 5 to conformal hyperbolic transformations which preserve the wave equation, and hyperbolic Riemann surfaces which are naturally associated to classical string motions.     

算术与微积分

本文对算术中的度量进行了讨论,定义了度规、微分与度规积分。就像乘除法运算是加减法运算的推广一样,微积分运算是乘除法运算在度规是变量时的推广。     

Spectral Exterior Calculus

A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space dimension. To achieve this, the exterior calculus is reformulated entirely in terms of the eigenvalues and eigenfunctions of the Laplacian operator on functions. The exterior derivatives of these eigenfunctions (and their wedge products) are shown to form a frame (a type of spanning set) for appropriate L² spaces of k -forms, as well as higher-order Sobolev spaces. Formulas are derived to express the Laplace-de Rham operators on forms in terms of the eigenfunctions and eigenvalues of the Laplacian on functions. By representing the Laplace-de Rham operators in this frame, spectral convergence results are obtained via Galerkin approximation techniques. Numerical examples demonstrate accurate recovery of eigenvalues and eigenforms of the Laplace-de Rham operator on 1-forms. The correct Betti numbers are obtained from the kernel of this operator approximated from data sampled on several orientable and non-orientable manifolds, and the eigenforms are visualized via their corresponding vector fields. These vector fields form a natural orthonormal basis for the space of square-integrable vector fields, and are ordered by a Dirichlet energy functional which measures oscillatory behavior. The spectral framework also shows promising results on a non-smooth example (the Lorenz 63 attractor), suggesting that a spectral formulation of exterior calculus may be feasible in spaces with no differentiable structure. © 2020 Wiley Periodicals, Inc.     

Introducing Differential Calculus

In this paper we discuss three ways of introducing calculus all based on concepts which students would either already know or which can be introduced without much difficulty at this stage of their mathematical training. The aim of this paper is to persuade teachers of mathematics that topics in ‘higher mathematics’, specifically calculus, can and should be presented to students in terms of concepts with which they are familiar.     

Contact Tensor Calculus

Summary It is shown that a Tensor Calculus in which the group of transformations is the group of contact transformations can be developed by making a systematic use of existing theories of non-holonomic spaces. The generalizations of Riemannian Geometry are then considered in relation to the new Calculus.     

F—fuzzy演绎系统

本文建立了一种演绎系统FFCS,在该演义系统中可以处理具有模糊性的推理过程,区别与其他对模糊推理进行形式化的逻辑系统,FFCS对模糊假言推理FMP做了完全形式化的处理。     

解读“微积分算术”

2011年在科学网博客上出现的微积分算术,把抽象而高深的微积分看作函数的算术,只用几步高中代数,就能避开极限而又不失严格讲解微积分.首先,用等式讲解多项式的微积分;然后用不等式讲解显式初等函数的微积分.但是,某些读者可能会存在疑问:真的能让微积分的门槛降低,而又不失严格?它到底具备什么样的特点?对学生群体的定位如何?需要进一步的解读,这就是本文的目的.     

算术观点下的林氏微积分与传统教科书中微积分比较分析

微积分是大学里普及程度非常高的一门学科,数学系学生、理工科学生、文科学生都需要学习,传统教科书中的微积分复杂度较高,使很多学生望而生畏.而算术观点下的林氏微积分复杂度保持在乘法表的水平,大大降低了微积分的门槛,且直击微积分的核心:牛顿—莱布尼茨公式.以《数学分析》中的微积分部分为例,与林群的微积分做对比,以期为大学微积分的教学改革提供思考.     

A Bilinear Pseudodifferential Calculus

In this paper we construct a new class of bilinear pseudodifferential operators which contains both the Coifman-Meyer class as well as the non-translation invariant class closely related both to the bilinear Hilbert transform and previously studied in Bényi et al. (J. Geom. Anal. 16(3):431–453, 2006), Bényi et al. (J. Anal. Math., 2009), Bernicot (Anal. PDE 1:1–27, 2008) as well as the bilinear Marcinkiewicz class studied in Grafakos and Kalton (Stud. Math. 146(2):115–156, 2001). We prove boundedness on Sobolev spaces for these operators as well as establish a symbolic calculus that exhibits the nice behavior of our new class under transposition and composition with linear operators.     

Calculus demonstrations using MATLAB

The note discusses ways in which technology can be used in the calculus learning process. In particular, five MATLAB programs are detailed for use by instructors or students that demonstrate important concepts in introductory calculus: Newton's method, differentiation and integration. Two of the programs are animated. The programs and the graphical user interface have been specifically designed to help the student understand the processes behind these important introductory concepts. Each program has a series of demonstrations that show unusual, difficult or important cases.     

杰出人物论微积分

摘编14位杰出人物对微积分的精辟论述。     

Calculus of Functorial Morphisms

We prove that, if F, G: 𝒞 → 𝒟 are two right exact functors between two Grothendieck categories such that they commute with coproducts and U is a generator of 𝒞, then there is a bijection between Nat(F, G) and the centralizer of Hom_𝒟(F(U), G(U)) considered as an Hom_𝒞(U, U)-Hom_𝒞(U, U)-bimodule. We also prove a dual of this result and give applications to Frobenius functors between Grothendieck categories.     

Calculus on Poisson Manifolds

We prove the existence of a Lie bracket on the space of 1-formson a Poisson manifold. This gives rise to a calculus of skewsymmetriccontravariant tensors dual to Cartan's calculus of forms.     

Fundamental Theorem of Calculus

A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been laid out. Various classical examples of this theorem, such as the Green’s and Stokes’ theorem are discussed, as well as the theory of monogenic functions which generalizes analytic functions of a complex variable to higher dimensions.     

Calculus In Enveloping Algebras

Motivated by, but independent of, some recent work in quantumstochastic calculus, a theory of differential and integral calculusis developed which is intrinsic to the universal envelopingalgebra of a Lie algebra whose Lie bracket is obtained by takingcommutators in an associative algebra. The differential mapsatisfies a generalisation of Leibniz' formula called the Leibniz–Itôformula, which involves the associative multiplication. Thereis an analogue of the Taylor–Maclaurin expansion. Throughpassing to formal power series, a theory of product integralsis developed; such integrals are characterised by a group-likeproperty with respect to the coproduct.     

Basic Propositional Calculus I

We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E₁, a theory axiomatized by T → ⊥. The intersection CPC ∩ E₁ is axiomatizable by the Principle of the Excluded Middle A V ∨ ?A. If B is a formula such that (T → B) → B is not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula (T → B) → B is derivable exactly when B is provably equivalent to a formula of the form ((T → A) → A) → (T → A).