A Padé approximant is a type of rational function used to approximate a given function, typically a power series. It is defined as the ratio of two polynomials, \( P(x) \) and \( Q(x) \), where \( P(x) \) is of degree \( m \) and \( Q(x) \) is of degree \( n \).

Partial fractions is a mathematical technique used to decompose a rational function into a sum of simpler fractions, called partial fractions. This method is particularly useful in algebra, calculus, and differential equations, as it simplifies the process of integrating rational functions. A rational function is typically expressed as the ratio of two polynomials, say \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

The Hartogs–Rosenthal theorem is a result in the field of functional analysis, particularly dealing with Banach spaces. It describes a certain property of bounded linear operators between infinite-dimensional Banach spaces.

Legendre rational functions are a family of rational functions constructed from Legendre polynomials, which are orthogonal polynomials defined on the interval \([-1, 1]\). These functions are used in various areas of mathematics, including numerical analysis and approximation theory.

A linear fractional transformation (LFT), also known as a Möbius transformation, is a function that maps the complex plane to itself. It is defined by the formula: \[ f(z) = \frac{az + b}{cz + d} \] where \(a\), \(b\), \(c\), and \(d\) are complex numbers, and \(ad - bc \neq 0\) to ensure that the transformation is well-defined and non-degenerate.

The Rvachev function, also known as the Rvachev test function, is a mathematical function often used in optimization and benchmarking for algorithms, particularly in the fields of global optimization and numerical analysis. It is known for having multiple local minima, which makes it a challenging function for optimization techniques.

The term "approximate limit" can refer to different concepts depending on the context in which it's used. Here are a couple of interpretations: 1. **Mathematics (Calculus and Analysis)**: In the context of calculus, the limit of a function as it approaches a particular value can sometimes be computed or understood using approximate values or numerical methods.

A Baire function is a specific type of function that arises in the field of descriptive set theory, which is a branch of mathematical logic and analysis. Baire functions are defined on the real numbers (or other Polish spaces) and can be categorized based on their levels of complexity. ### Definition: Baire functions are defined using the idea of Baire classes.

In mathematical analysis, a **Baire-1 function** (or **Baire class 1 function**) is a special type of function that is defined in terms of its pointwise limits of continuous functions.

Particle size refers to the physical dimensions of solid particles, which can be expressed in terms of diameter or volume. It is a critical parameter in various fields, including materials science, pharmaceuticals, environmental science, and engineering, as it can influence the properties and behavior of materials and substances. Particle size can be measured in several ways: 1. **Diameter**: Often expressed in micrometers (µm), nanometers (nm), or millimeters (mm).

Georg Cantor's first significant work on set theory is often considered to be his 1874 article titled "Über eine Eigenschaft der reellen Zahlen" (translated as "On a Property of the Real Numbers"). In this paper, Cantor introduced the concept of sets and laid the groundwork for later developments in set theory, including his work on different types of infinities and cardinality.

Carleman's inequality is a mathematical result in the field of functional analysis and approximation theory. It provides a bound on the norms of a function based on the norms of its derivatives. Specifically, it is often used in the context of the spaces of functions with certain smoothness properties. One of the most common forms of Carleman's inequality is related to the Sobolev spaces and is used to show the equivalence of certain norms.

Cousin's theorem is a concept in complex analysis, specifically in the context of holomorphic functions and their properties. It is named after the French mathematician François Cousin. The theorem has two main formulations, often referred to as Cousin's first and second theorems.

The Dini derivative is a concept used in mathematical analysis, particularly in the study of functions and their behavior. It defines a way to quantify the rate of change of a function along a certain direction while taking into account a generalized notion of limit.

ETrice is a model-based software development framework that is primarily used for designing and implementing distributed systems and applications. It is built around the concepts of the Actor model, where components (or "actors") communicate with each other via message passing, making it particularly suitable for applications that require high levels of concurrency and scalability. ETrice provides a set of tools and methodologies to facilitate the specification, design, and implementation of systems.

Hadamard's lemma is a result in the field of differential calculus that relates to the expansion of a function in terms of its derivatives. Specifically, it provides a formula for expressing the value of a function at a point in terms of its Taylor series expansion around another point.

An interleave sequence refers to a technique of merging or combining elements from multiple sequences in such a way that the elements from each sequence are alternated in the final output. This concept is often used in computer science, particularly in data processing, algorithms, and digital communication, where it can help in improving data throughput and error correction.

The term "Invex function" refers to a specific class of functions used in optimization theory, particularly in the context of mathematical programming and convex analysis. Invex functions generalize convex functions and are often characterized by certain properties that make them useful in optimization problems.

The Least Upper Bound (LUB) property, also known as the supremum property, is a fundamental concept in real analysis and is one of the defining characteristics of the real numbers. The LUB property states that for any non-empty set of real numbers that is bounded above, there exists a least upper bound (supremum) in the real numbers.