"Discoveries" by Peter Kolény is a publication that explores various topics, often related to science, technology, or innovation. However, specific details about the content and focus of the book or work may be limited and can vary.

"Discoveries" by Pierre Méchain refers to the significant contributions he made to the field of geodesy, particularly in the late 18th century. Méchain was a French astronomer and geodesist who is best known for his work on the metric system and the measurement of the meridian arc in France.

R. J. Mitchell, an astronomer, is known for his contributions to the field of astronomy and his work in discovering various astronomical phenomena. However, specific details about his discoveries might need to be clarified, as there may be various contributions in different areas of astronomy. Mitchell's notable work often involves studies of celestial objects, contributions to theoretical astronomy, or advancements in observational techniques.

"Discoveries" by R. Scott Dunbar is a book that explores various themes related to science, innovation, and the impact of human curiosity on our understanding of the world.

"Discoveries" by Seiji Ueda is a concept that revolves around exploring and delivering insights or findings in various fields. However, as of my last update in October 2023, there isn't a widely recognized work or publication titled "Discoveries" by Seiji Ueda that fits a particular context or subject.

"Discoveries" by Richard Martin West is a book that explores various scientific and technological breakthroughs, delving into how these discoveries have shaped our understanding of the world. West examines the interplay between innovation and society, highlighting key figures and moments in the history of science that have led to significant advancements.

"Discoveries" by Richard Schorr refers to a book that presents a collection of insights and ideas related to the natural world, science, and the process of discovery itself. Richard Schorr is known for his engaging writing style and ability to make complex scientific concepts accessible to a broader audience. The book likely explores themes such as curiosity, exploration, and the importance of scientific inquiry, offering readers a chance to reflect on the wonders of discovery in various fields.

"Discoveries" by Robert H. McNaught is a book that provides insights and narratives about various scientific discoveries and the individuals behind them. McNaught, an astronomer known for his work in comet discovery and research, typically focuses on the methods and stories that lead to significant advancements in science. The book might explore themes such as the challenges faced by scientists, the impact of their work on society, and the excitement of scientific exploration.

"Discoveries" is a work by Royal Harwood Frost, who was an American poet, essayist, and editor. This piece reflects themes of exploration and introspection, often delving into the complexities of human experience and the world around us. Frost's writing is characterized by its lyrical quality and profound insights into nature and life.

"Discoveries" by Schelte J. Bus is a book that explores the field of astronomy, particularly focusing on the history and significance of astronomical discoveries. The author, Schelte J. Bus, is known for his work in planetary science and asteroid research. In "Discoveries," he likely discusses various key astronomical observations, the impact of these discoveries on our understanding of the universe, and the evolution of astronomical techniques and technologies.

The order of a algebraic structure is just its cardinality.

Sometimes, especially in the case of structures with an infinite number of elements, it is often more convenient to talk in terms of some parameter that characterizes the structure, and that parameter is usually called the degree.

The degree of some algebraic structure is some parameter that describes the structure. There is no universal definition valid for all structures, it is a per structure type thing.

This is particularly useful when talking about structures with an infinite number of elements, but it is sometimes also used for finite structures.

Examples:

Examples:

The term is not very clear, as it could either mean:

A linear map is a function $f:V_1(F)\to V_2(F)$ where $V_1(F)$ and $V_2(F)$ are two vector spaces over underlying fields $F$ such that:

A common case is $F=\mathbb{R}$, $V_1={\mathbb{R}}_m$ and $V_2={\mathbb{R}}_n$.

One thing that makes such functions particularly simple is that they can be fully specified by specifyin how they act on all possible combinations of input basis vectors: they are therefore specified by only a finite number of elements of $F$.

Every linear map in finite dimension can be represented by a matrix, the points of the domain being represented as vectors.

As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.

The prototypical building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters, e.g.

For example, the left side of the time-independent Schrödinger equation is a linear map. And the time-independent Schrödinger equation can be seen as a eigenvalue problem.

A form is a function from a vector space to elements of the underlying field of the vector space.

Examples:

A Linear map where the image is the underlying field of the vector space, e.g. ${\mathbb{R}}^n\to \mathbb{R}$.

The set of all linear forms over a vector space forms another vector space called the dual space.

For the typical case of a linear form over ${\mathbb{R}}^n$, the form can be seen just as a row vector with n elements, the full form being specified by the value of each of the basis vectors.

The dual space of a vector space $V$, sometimes denoted $V^{\ast }$, is the vector space of all linear forms over $V$ with the obvious addition and scalar multiplication operations defined.

Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the $V$, and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.

Infinite dimension seems more interesting however, see: en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case

One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space $V$.

Dual vectors are the members of a dual space.

In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis:The dual basis vectors $e^i$ are defined to "pick the corresponding coordinate" out of elements of V. E.g.:By expanding into the basis, we can put this more succinctly with the Kronecker delta as:

Note that in Einstein notation, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector $f$ as:

In the context of quantum mechanics, the bra notation is also used for dual vectors.