Infinitesimal refers to a quantity that is extremely small, approaching zero but never actually reaching it. In mathematics, infinitesimals are used in calculus, particularly in the formulation of derivatives and integrals. In the context of non-standard analysis, developed by mathematician Abraham Robinson in the 1960s, infinitesimals can be rigorously defined and treated like real numbers, allowing for a formal approach to concepts that describe quantities that are smaller than any positive real number.

"Nova Methodus pro Maximis et Minimis" is a work by the mathematician and philosopher Gottfried Wilhelm Leibniz, published in 1684. The title translates to "A New Method for Maxima and Minima," and it is significant for its contributions to the field of calculus and optimization. In this work, Leibniz explores methods for finding the maxima and minima of functions, which are critical concepts in calculus.

The reflection formula typically refers to a specific mathematical property involving special functions, particularly in the context of the gamma function and trigonometric functions. One of the most common reflection formulas is for the gamma function, which states: \[ \Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin(\pi z)} \] for \( z \) not an integer.

In complex analysis, theorems provide important results and tools for working with complex functions and their properties. Here are some fundamental theorems in complex analysis: 1. **Cauchy's Integral Theorem**: This theorem states that if a function is analytic (holomorphic) on and within a closed curve in the complex plane, then the integral of that function over the curve is zero.

Bicoherence is a statistical measure used in signal processing and time series analysis to assess the degree of non-linearity and the presence of interactions between different frequency components of a signal. It is a higher-order spectral analysis technique that extends the concept of coherence, which is primarily used in linear systems. The bicoherence is particularly useful in identifying and quantifying non-linear relationships between signals in the frequency domain.

In the context of topology, continuous functions on a compact Hausdorff space play a crucial role in various areas of mathematics, particularly in analysis and algebraic topology.

In the context of mathematics and dynamical systems, an "escaping set" typically refers to a set of points in the complex plane (or other spaces) that escape to infinity under the iteration of a particular function. The concept is frequently encountered in the study of complex dynamics, particularly in relation to Julia sets and the Mandelbrot set. **Key Concepts:** 1.

Fuchs' relation is a concept from condensed matter physics, particularly in the context of quantum mechanics and statistical mechanics. It describes a specific relationship among different correlation functions of a many-body quantum system, especially in the context of systems exhibiting long-range order or critical phenomena. In statistical mechanics, Fuchs' relation is often applied to systems exhibiting phase transitions, providing insights into the fluctuations and parameters that characterize the behavior of the system near critical points.

A Specker sequence is a type of sequence that is associated with the study of the theory of computation and constructible sets. More specifically, the most famous Specker sequence is a sequence constructed by Ernst Specker in the context of the study of the limitations of certain types of computational sequences, particularly in relation to concepts like non-reducibility and the foundations of mathematics.

The term "regular part" can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **In Mathematics (Topology)**: The regular part of a measure or function might refer to a subset that behaves nicely according to certain criteria, such as being continuous or differentiable. For example, in the context of measures, the "regular part" of a measure could refer to the portion that can be approximated by more regular sets.

The Stefan Bergman Prize is an award given for outstanding contributions in the field of complex analysis, especially in areas related to the theory of functions of several complex variables. Established in honor of the mathematician Stefan Bergman, who made significant contributions to several complex variables and other areas of mathematics, the prize aims to recognize individuals whose work exhibits the same level of excellence and innovation. The prize is typically awarded every two years by the American Mathematical Society (AMS) or other mathematics organizations associated with the field.

The Anderson function is a mathematical concept frequently encountered in various fields, especially in physics, mathematics, and materials science. In its most common context, it relates to the study of disordered systems and electron localization, particularly in solid-state physics. The function is often associated with the Anderson localization phenomenon, which is the absence of diffusion of waves in a disordered medium. The original paper by Philip W.

Biholomorphism is a concept from complex analysis, specifically in the study of several complex variables and complex manifolds. It refers to a certain type of mapping between complex manifolds.

An injective function, also known as a one-to-one function, is a type of function in mathematics that preserves distinctness: if two inputs to the function are different, then their outputs will also be different.

The Kolmogorov–Arnold representation theorem, also known as the Kolmogorov–Arnold function representation theorem, is a result in the theory of multivariate functions that provides a way to express any continuous multivariate function as a superposition of continuous functions of fewer variables.

The Squeeze Theorem, also known as the Sandwich Theorem or Pinching Theorem, is a fundamental concept in calculus, specifically in the context of limits. It helps to determine the limit of a function by comparing it with two other functions that "squeeze" it in a defined manner.

The constant associated with Somos' quadratic recurrence, often denoted as \( c \), is given by the formula: \[ c = \frac{1 + \sqrt{5}}{2} \] This is known as the golden ratio, commonly denoted by \( \phi \). In the context of the recurrence, the sequence defines terms using the previous terms in relation to this constant.

The Behnke-Stein theorem is a significant result in several complex variables and complex analysis. It describes the holomorphicity of certain types of functions under certain conditions related to domains in complex manifolds.

The Cauchy–Euler operator, also known as the Cauchy–Euler differential operator, refers to a specific type of differential operator that is commonly used in the analysis of differential equations of the form: \[ a x^n \frac{d^n y}{dx^n} + a x^{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1 x \frac{dy}{dx