{wiki=Schatten_class_operator}

Schatten class operators, denoted as \\( \\mathcal\{S\}_p \\) for \\( p \\geq 1 \\), are a generalization of compact operators on a Hilbert space. They are defined in terms of the singular values of the operators.


Schatten class operator

{wiki=Schröder–Bernstein_theorems_for_operator_algebras}

The Schröder–Bernstein theorem, traditionally framed in set theory, states that if there are injective (one-to-one) functions \\( f: A \\to B \\) and \\( g: B \\to A \\) between two sets \\( A \\) and \\( B \\), then there exists a bijection (one-to-one and onto function) between \\( A \\) and \\( B \\).


Schröder–Bernstein theorems for operator algebras

{wiki=Sectorial_operator}

A sectorial operator is a type of linear operator in functional analysis that generalizes the concept of self-adjoint operators. Sectorial operators arise in the study of partial differential equations and the theory of semigroups of operators. They are particularly important in the context of evolution equations and their solutions. An operator \\( A \\) on a Banach space \\( X \\) is said to be sectorial if it has a sector in the complex plane where its spectrum lies.


Sectorial operator

{wiki=Singular_integral_operators_of_convolution_type}

Singular integral operators of convolution type are a particular class of linear operators that arise in the study of functional analysis, partial differential equations, and harmonic analysis. These operators are defined through convolution with a kernel (a function that describes the behavior of the operator) which typically has certain singular properties.


Singular integral operators of convolution type

{wiki=Singular_integral_operators_on_closed_curves}

Singular integral operators are a class of mathematical operators that arise in various areas of analysis, particularly in the study of partial differential equations, harmonic analysis, and complex analysis. When we talk about singular integral operators on closed curves, we are often considering how these operators act on functions defined on the plane or in higher-dimensional spaces, particularly in relation to their behavior around singularities or points of discontinuity.


Singular integral operators on closed curves

{wiki=Stein–Strömberg_theorem}

The Stein–Strömberg theorem is a result in the field of harmonic analysis and complex analysis, particularly concerning the behavior of functions defined on certain sets and their Fourier transforms. It provides bounds on the integral of the exponential of a function, specifically concerning the Plancherel measure associated with it. In essence, the theorem states conditions under which the Fourier transform of a function within a specific space will be contained in another function space, highlighting the interplay between various functional spaces.


Stein–Strömberg theorem

{wiki=Beltrami_identity}

The Beltrami identity is a mathematical result related to the calculus of variations, particularly in the context of classical mechanics and fluid dynamics. It is named after the Italian mathematician Ernesto Beltrami. In the calculus of variations, the Beltrami identity provides a necessary condition for a functional to be extremized.


Beltrami identity

{wiki=Caratheodory-π_solution}

The Carathéodory-π (pi) solution is a concept found in the field of differential equations, particularly in the study of differential inclusions and differential equations with certain types of discontinuities. The traditional concept of a solution for ordinary differential equations typically involves classical solutions, which are functions that are continuously differentiable and satisfy the equation pointwise.


Caratheodory-π solution

{wiki=Costate_equation}

In optimal control theory, the costate equations are derived from the Pontryagin's Maximum Principle, which is a method for solving optimal control problems. The principle provides necessary conditions for optimality when determining control strategies that minimize or maximize a certain objective (or cost) function subject to dynamic constraints.


Costate equation

{wiki=Covector_mapping_principle}

The Covector Mapping Principle is a concept in differential geometry and mathematical physics that relates to the study of vector spaces and their duals. To understand the principle, let's break down the key components: 1. **Vectors and Covectors**: - In a vector space \\( V \\), a **vector** can be thought of as an element that can represent a point or a direction in that space.


Covector mapping principle

{wiki=Double-setpoint_control}

Double-setpoint control is a control strategy often used in industrial automation and process control systems. It involves maintaining a process variable (such as temperature, pressure, or flow rate) within a specified range defined by two setpoints: an upper setpoint and a lower setpoint.


Double-setpoint control

{wiki=GPOPS-II}

GPOPS-II (General Purpose Optimal Control Software) is a software package designed for solving optimal control problems using direct collocation methods. It offers a robust framework for formulating and solving problems in which the goal is to determine control inputs that will optimize a certain performance criterion, subject to dynamic constraints and boundary conditions.


GPOPS-II

{wiki=Hamiltonian_(control_theory)}

In control theory, the Hamiltonian is a function that is central to optimal control problems. It is used in the formulation of the Hamiltonian control methods, particularly in dynamic programming and optimal control strategies, such as the Pontryagin's Maximum Principle. \#\#\# Definition of the Hamiltonian The Hamiltonian \\( H \\) is typically defined for a control system described by: - A set of state variables \\( x(t) \\) that represent the system's configuration at time \\( t \\).


Hamiltonian (control theory)

{wiki=Hydrological_optimization}

Hydrological optimization refers to a set of methods and techniques used to manage water resources effectively in a given watershed or water system. It involves the analysis and optimization of the hydrological cycle, which includes precipitation, evaporation, infiltration, runoff, and groundwater recharge. The goal is to enhance the efficiency of water use, improve water quality, and maximize the benefits derived from water resources while minimizing negative environmental impacts.


Hydrological optimization

{wiki=Legendre–Clebsch_condition}

The Legendre–Clebsch condition is a criterion in the calculus of variations that helps determine whether a given differential equation can be derived from a variational principle, typically in the context of optimal control or mechanics. More specifically, it relates to the conditions under which a function can be considered a Hamiltonian function in a variational formulation.


Legendre–Clebsch condition

{wiki=Meixner_polynomials}

Meixner polynomials are a class of orthogonal polynomials that arise in the context of probability theory and various applications in mathematical physics. They are associated with the Meixner distribution, which is a natural generalization of the Poisson distribution and is used to model various types of counting processes.


Meixner polynomials

{wiki=Category:Optica_(society)_academic_journals}

Optica is a professional society focused on advancing the field of optics and photonics. It was established to promote research, education, and applications related to optics, and it is involved in organizing conferences, publishing research, and fostering collaboration among professionals in the field. In terms of academic journals, Optica publishes several prestigious journals that cover a wide range of topics within optics and photonics.


Optica (society) academic journals

{wiki=Category:Presidents_of_Optica_(society)}

The Presidents of Optica refer to the individuals who have led the Optica society, which is an international professional organization dedicated to promoting optics and photonics. Formerly known as the Optical Society (OSA), Optica focuses on advancing the study and application of light science and technology. The role of the president typically includes guiding the society's strategic vision, overseeing initiatives related to education, policy, and research in optics, and representing the organization in various professional settings.


Presidents of Optica (society)

{wiki=Joseph_Rosen_(professor)}

Joseph Rosen is a professor known for his work in the field of mathematics, particularly in areas such as mathematical logic and the foundations of mathematics. He has made contributions to various topics, such as model theory and set theory. However, it is important to verify the specific context in which you are asking about him, as there may be multiple individuals with that name in academia.


Joseph Rosen (professor)

{wiki=Optics_Classification_and_Indexing_Scheme}

The Optics Classification and Indexing Scheme is a systematic framework used for categorizing and indexing literature and research in the field of optics. This scheme helps researchers, scholars, and professionals organize and retrieve information pertaining to various areas of optics, such as theoretical, experimental, and applied optics. \#\#\# Key Aspects of the Scheme: 1. **Classification**: The scheme provides a hierarchical structure that classifies different topics within optics.


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