linear transformations and matrices
... Definition 3.5 (Range). Let T : V → W be a linear transformation. The range of T , denoted R(T ), is defined as R(T ) := {T (v) : v ∈ V }. Remark 3.6. Note that R(T ) is a subspace of W , so its dimension can be defined. Definition 3.7 (Rank). Let V, W be vector spaces over a field F. Let T : V → W ...
... Definition 3.5 (Range). Let T : V → W be a linear transformation. The range of T , denoted R(T ), is defined as R(T ) := {T (v) : v ∈ V }. Remark 3.6. Note that R(T ) is a subspace of W , so its dimension can be defined. Definition 3.7 (Rank). Let V, W be vector spaces over a field F. Let T : V → W ...
Flux Splitting: A Notion on Stability
... Asymptotic Preserving and IMEX schemes. Both schemes for hyperbolic equations rely on a splitting of the convective flux into stiff and nonstiff parts. This choice is not arbitrary and has an influence on both the stability and the accuracy of the resulting methods. In this work, we consider a first ...
... Asymptotic Preserving and IMEX schemes. Both schemes for hyperbolic equations rely on a splitting of the convective flux into stiff and nonstiff parts. This choice is not arbitrary and has an influence on both the stability and the accuracy of the resulting methods. In this work, we consider a first ...
THE PROBABILITY OF CHOOSING PRIMITIVE
... Given any integer matrix B of full row rank, there exists a unimodular matrix U such that BU is in Hermite normal form (see, e.g., [5]; U will not, in general, be unique). This fact, together with the following lemma, gives a convenient characterization of when S is a primitive set. Lemma 5. Let {s1 ...
... Given any integer matrix B of full row rank, there exists a unimodular matrix U such that BU is in Hermite normal form (see, e.g., [5]; U will not, in general, be unique). This fact, together with the following lemma, gives a convenient characterization of when S is a primitive set. Lemma 5. Let {s1 ...
Aurifeuillian factorizations - American Mathematical Society
... found similar identities for every composite exponent n not divisible by 8. He showed that if n = N, 2N or 4N , with N odd, and if d is any squarefree divisor of N (where d is allowed to be negative when n = 4N ), then there exist polynomials Un,d (x), Vn,d (x) ∈ Z[x] such that ϕN (x) = UN,d (x)2 − ...
... found similar identities for every composite exponent n not divisible by 8. He showed that if n = N, 2N or 4N , with N odd, and if d is any squarefree divisor of N (where d is allowed to be negative when n = 4N ), then there exist polynomials Un,d (x), Vn,d (x) ∈ Z[x] such that ϕN (x) = UN,d (x)2 − ...
Sufficient conditions for the spectrality of self
... with s1 − s2 ∈ Zn such that the exponential functions es1 (x), es2 (x) are orthogonal in L2 (µM,D ), then there are infinite families of orthogonal exponentials E(Λ) in L2 (µM,D ) with Λ ⊆ Zn . In fact, let l = s1 − s2 ∈ Z(µ̂M,D ) ∩ Zn . Then there exists a positive integer k := k(l) such that mD (M ...
... with s1 − s2 ∈ Zn such that the exponential functions es1 (x), es2 (x) are orthogonal in L2 (µM,D ), then there are infinite families of orthogonal exponentials E(Λ) in L2 (µM,D ) with Λ ⊆ Zn . In fact, let l = s1 − s2 ∈ Z(µ̂M,D ) ∩ Zn . Then there exists a positive integer k := k(l) such that mD (M ...
Learning mixtures of product distributions over
... Such learning problems have been well studied in the past, as we now describe. 1.2 Related work. In [18] Kearns et al. gave efficient algorithms for learning mixtures of Hamming balls; these are product distributions over {0, 1}n in which all the coordinate means E[Xij ] must be either p or 1 − p fo ...
... Such learning problems have been well studied in the past, as we now describe. 1.2 Related work. In [18] Kearns et al. gave efficient algorithms for learning mixtures of Hamming balls; these are product distributions over {0, 1}n in which all the coordinate means E[Xij ] must be either p or 1 − p fo ...
Van Der Vaart, H.R.; (1966)An elementary deprivation of the Jordan normal form with an appendix on linear spaces. A didactical report."
... literature a complete, somewhat leisurely expositionl which in all its phases is essentially based on nothing more than the concepts of linear space and sUbspace, basis and 'Clirect sum, dimension, and the fundamental idea of mapping. ...
... literature a complete, somewhat leisurely expositionl which in all its phases is essentially based on nothing more than the concepts of linear space and sUbspace, basis and 'Clirect sum, dimension, and the fundamental idea of mapping. ...
Non-negative matrix factorization
NMF redirects here. For the bridge convention, see new minor forcing.Non-negative matrix factorization (NMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.NMF finds applications in such fields as computer vision, document clustering, chemometrics, audio signal processing and recommender systems.