The Adjacency Matrices of Complete and Nutful Graphs
... For a graph G = Γ(G), the vertices can be partitioned into G–core and G–coreforbidden vertices for G. A vertex of G corresponding to a non–zero entry in some kernel eigenvector x is said to be a G–core vertex (G–CV) for the adjacency matrix G associated with G. A G–core-forbidden vertex (G–CFV), for ...
... For a graph G = Γ(G), the vertices can be partitioned into G–core and G–coreforbidden vertices for G. A vertex of G corresponding to a non–zero entry in some kernel eigenvector x is said to be a G–core vertex (G–CV) for the adjacency matrix G associated with G. A G–core-forbidden vertex (G–CFV), for ...
Title and Abstracts - Chi-Kwong Li
... (2000)], this problem have been studied extensively and many interesting partial results have been reported. However, it remained unknown for a long time when two general quantum operations are perfectly distinguishable. In 2009 we obtained a feasible necessary and sufficient condition for the perfe ...
... (2000)], this problem have been studied extensively and many interesting partial results have been reported. However, it remained unknown for a long time when two general quantum operations are perfectly distinguishable. In 2009 we obtained a feasible necessary and sufficient condition for the perfe ...
pivot position
... Steps 1–3 require no work for this submatrix, and we have reached an echelon form of the full matrix. We perform one more step to obtain the reduced echelon form. STEP 5: Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, ma ...
... Steps 1–3 require no work for this submatrix, and we have reached an echelon form of the full matrix. We perform one more step to obtain the reduced echelon form. STEP 5: Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, ma ...
The Power of Depth 2 Circuits over Algebras
... we know an F-basis {e1 , . . . , ek } of R and we also know how ei e j can be expressed in terms of the basis elements, for all i and j. Since elements of a finite dimensional algebra, given in basis form, can be expressed as matrices over F, the problem at hand is the following. P ROBLEM 2. Given a ...
... we know an F-basis {e1 , . . . , ek } of R and we also know how ei e j can be expressed in terms of the basis elements, for all i and j. Since elements of a finite dimensional algebra, given in basis form, can be expressed as matrices over F, the problem at hand is the following. P ROBLEM 2. Given a ...
Shiftless Decomposition and Polynomial
... It is well known that the size of R may be exponentially large as a function of the size of F . Consider the case K = . The algorithm we present here is distinguished from previous algorithms because it always returns a compact representation of R in time polynomial in the input size. For example, i ...
... It is well known that the size of R may be exponentially large as a function of the size of F . Consider the case K = . The algorithm we present here is distinguished from previous algorithms because it always returns a compact representation of R in time polynomial in the input size. For example, i ...
Geometric Means - College of William and Mary
... The geometric mean of two positive semi-definite matrices arises naturally in several areas, and it has many of the properties of the geometric mean of two positive scalars. Researchers have tried to define a geometric mean on three or more positive definite matrices, but there is still no satisfact ...
... The geometric mean of two positive semi-definite matrices arises naturally in several areas, and it has many of the properties of the geometric mean of two positive scalars. Researchers have tried to define a geometric mean on three or more positive definite matrices, but there is still no satisfact ...
Lecture 12 Semidefinite Duality
... {ai }ni=1 such that Aij = E[ai aj ]. Similarly, let Bij = E[bi bj ] for r.v.s {bi }. Moreover, we can take the a’s to be independent of the b’s. So if we define the random variables ci = ai bi , then ...
... {ai }ni=1 such that Aij = E[ai aj ]. Similarly, let Bij = E[bi bj ] for r.v.s {bi }. Moreover, we can take the a’s to be independent of the b’s. So if we define the random variables ci = ai bi , then ...
Full Text - J
... relation between Stokes matrices of (1) and monodromy data of (3) may have important applications in this case, such as the understanding of the monodromy data associated to special solutions of the sixth Painlevé equation, including the solution associated to the quantum cohomology of CP 2 . Monod ...
... relation between Stokes matrices of (1) and monodromy data of (3) may have important applications in this case, such as the understanding of the monodromy data associated to special solutions of the sixth Painlevé equation, including the solution associated to the quantum cohomology of CP 2 . Monod ...
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 ...
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.