MTH232 - National Open University of Nigeria

... Definition: A real or complex valued function y = ø (x) defined on an interval I is called a solution or an integral of the differential equation g(x, y, y’,…., y(n)) = 0 if ø (x) is n time differentiable and if x, ø (x), ø ’(x), ….., ø (n) (x) satisfy this equation for all x in I. For example, the ...

... Definition: A real or complex valued function y = ø (x) defined on an interval I is called a solution or an integral of the differential equation g(x, y, y’,…., y(n)) = 0 if ø (x) is n time differentiable and if x, ø (x), ø ’(x), ….., ø (n) (x) satisfy this equation for all x in I. For example, the ...

On Brauer Groups of Lubin

... pbq As an associative ring spectrum, the Morava K-theory A cannot be recovered from the Lubin-Tate spectrum E. To elaborate on pbq, it is useful to introduce some terminology. Definition 1.0.2. Let E be a Lubin-Tate spectrum, so that R “ π0 E is a complete regular local ring with maximal ideal m Ď R ...

... pbq As an associative ring spectrum, the Morava K-theory A cannot be recovered from the Lubin-Tate spectrum E. To elaborate on pbq, it is useful to introduce some terminology. Definition 1.0.2. Let E be a Lubin-Tate spectrum, so that R “ π0 E is a complete regular local ring with maximal ideal m Ď R ...

Dilation Theory, Commutant Lifting and Semicrossed Products

... Shilov modules as a primary building block. Muhly and Solel adopt this view, but focus more on a somewhat stronger property of orthoprojective modules. They may have gone further, as we do, had they known what we do today. We will argue that these are more central to the theory. Another important in ...

... Shilov modules as a primary building block. Muhly and Solel adopt this view, but focus more on a somewhat stronger property of orthoprojective modules. They may have gone further, as we do, had they known what we do today. We will argue that these are more central to the theory. Another important in ...

abstract algebra: a study guide for beginners - IME-USP

... the smallest positive solution of the congruence ax ≡ 0 (mod n). 37. (a) Compute the last digit in the decimal expansion of 4100 . (b) Is 4100 divisible by 3? 38. Find all integers n for which 13 | 4(n2 + 1). 39. Prove that 10n+1 + 4 · 10n + 4 is divisible by 9, for all positive integers n. 40. Prov ...

... the smallest positive solution of the congruence ax ≡ 0 (mod n). 37. (a) Compute the last digit in the decimal expansion of 4100 . (b) Is 4100 divisible by 3? 38. Find all integers n for which 13 | 4(n2 + 1). 39. Prove that 10n+1 + 4 · 10n + 4 is divisible by 9, for all positive integers n. 40. Prov ...

On topological centre problems and SIN quantum groups

... whenever (aα ) and (bβ ) are nets in A converging to m and n, respectively, in σ (A∗∗ , A∗ ). The algebra A is said to be Arens regular if and ♦ coincide on A∗∗ . Every operator algebra, in particular, every C ∗ -algebra, is Arens regular. However, Banach algebras studied in abstract harmonic anal ...

... whenever (aα ) and (bβ ) are nets in A converging to m and n, respectively, in σ (A∗∗ , A∗ ). The algebra A is said to be Arens regular if and ♦ coincide on A∗∗ . Every operator algebra, in particular, every C ∗ -algebra, is Arens regular. However, Banach algebras studied in abstract harmonic anal ...

# Linear algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.