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### Approximation of a martensitic laminate with varying volume fractions

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

### Several Differentiation Formulas of Special Functions. Part IV

Formalized Mathematics

In this article, we give several differentiation formulas of special and composite functions including trigonometric function, polynomial function and logarithmic function.

### Several Differentiation Formulas of Special Functions. Part VI

Formalized Mathematics

In this article, we prove a series of differentiation identities [3] involving the secant and cosecant functions and specific combinations of special functions including trigonometric, exponential and logarithmic functions.

### Several Differentiation Formulas of Special Functions. Part V

Formalized Mathematics

In this article, we give several differentiation formulas of special and composite functions including trigonometric, polynomial and logarithmic functions.

### Integrability Formulas. Part I

Formalized Mathematics

In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, and the polynomial function.

### Basic Properties of Even and Odd Functions

Formalized Mathematics

In this article we present definitions, basic properties and some examples of even and odd functions [6].

### Integrability Formulas. Part III

Formalized Mathematics

In this article, we give several differentiation and integrability formulas of composite trigonometric function.

### Approximation of a Martensitic Laminate with Varying Volume Fractions

ESAIM: Mathematical Modelling and Numerical Analysis

We give results for the approximation of a laminate with varying volume fractions for multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfy the corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energy for the approximation of the limiting macroscopic deformation and...

### Several Differentiation Formulas of Special Functions. Part III

Formalized Mathematics

In this article, we give several differentiation formulas of special and composite functions including trigonometric function, inverse trigonometric function, polynomial function and logarithmic function.

### Difference and Difference Quotient

Formalized Mathematics

In this article, we give the definitions of forward difference, backward difference, central difference and difference quotient, and some of their important properties.

### Simple Continued Fractions and Their Convergents

Formalized Mathematics

The article introduces simple continued fractions. They are defined as an infinite sequence of integers. The characterization of rational numbers in terms of simple continued fractions is shown. We also give definitions of convergents of continued fractions, and several important properties of simple continued fractions and their convergents.

### Integrability Formulas. Part II

Formalized Mathematics

In this article, we give several differentiation and integrability formulas of special and composite functions including trigonometric function, and polynomial function.

### Difference and Difference Quotient. Part II

Formalized Mathematics

In this article, we give some important properties of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions [11].MML identifier: DIFF 2, version: 7.8.09 4.97.1001

### Generators for algebras dense in ${L}^{p}$-spaces

Studia Mathematica

For various ${L}^{p}$-spaces (1 ≤ p < ∞) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in ${L}^{p}\left(\mu \right)$. For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates...

### Some Operations on Quaternion Numbers

Formalized Mathematics

In this article, we give some equality and basic theorems about quaternion numbers, and some special operations.

### Several Integrability Formulas of Special Functions. Part II

Formalized Mathematics

In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, the hyperbolic function and the polynomial function [3].MML identifier: INTEGR11, version: 7.11.01 4.117.1046

### Basic Properties of Periodic Functions

Formalized Mathematics

In this article we present definitions, basic properties and some examples of periodic functions according to [5].

### Several Integrability Formulas of Some Functions, Orthogonal Polynomials and Norm Functions

Formalized Mathematics

In this article, we give several integrability formulas of some functions including the trigonometric function and the index function [3]. We also give the definitions of the orthogonal polynomial and norm function, and some of their important properties [19].MML identifier: INTEGRA9, version: 7.11.01 4.117.1046

### A Theorem on Directed Quasi-Ordered Sets and Some Remarks

Publications du Département de mathématiques (Lyon)

### Global dynamics of an epidemic model with a ratio-dependent nonlinear incidence rate.

Discrete Dynamics in Nature and Society

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