Lower Bounds in Communication Complexity: A Survey
... 2. Find an equivalent formulation of G in terms of a maximization problem. This will of course not always be possible, as in the case of approximate rank. This can be done, however, for rank and for a broad class of optimization problems over convex functions. 3. Prove lower bounds on G by exhibitin ...
... 2. Find an equivalent formulation of G in terms of a maximization problem. This will of course not always be possible, as in the case of approximate rank. This can be done, however, for rank and for a broad class of optimization problems over convex functions. 3. Prove lower bounds on G by exhibitin ...
CLUSTER ALGEBRAS II: FINITE TYPE CLASSIFICATION
... In order to understand a cluster algebra of finite type, one needs to study the combinatorial structure behind it, which is captured by its cluster complex. Roughly speaking, it is defined as follows. The cluster variables for a given cluster algebra are not given from the outset but are obtained fr ...
... In order to understand a cluster algebra of finite type, one needs to study the combinatorial structure behind it, which is captured by its cluster complex. Roughly speaking, it is defined as follows. The cluster variables for a given cluster algebra are not given from the outset but are obtained fr ...
NONLINEAR RANK-ONE MODIFICATION OF THE SYMMETRIC
... for k = 0, 1, . . . , n, where d0 = −∞ and dn+1 = +∞. Theorem 2.3. Let s(λ) be a positive decreasing function on Ik , i.e., s(λ) > 0 and s0 (λ) ≤ 0 for λ ∈ Ik , then (a) If k = 0, there is no eigenvalue of (2.3) on I0 . (b) If 1 ≤ k ≤ n, there is a simple eigenvalue of (2.3) on Ik . Proof. The state ...
... for k = 0, 1, . . . , n, where d0 = −∞ and dn+1 = +∞. Theorem 2.3. Let s(λ) be a positive decreasing function on Ik , i.e., s(λ) > 0 and s0 (λ) ≤ 0 for λ ∈ Ik , then (a) If k = 0, there is no eigenvalue of (2.3) on I0 . (b) If 1 ≤ k ≤ n, there is a simple eigenvalue of (2.3) on Ik . Proof. The state ...
Implementing a Toolkit for Ring
... probability theory is advisable. The presented toolkit uses several concepts from algebraic number theory, mainly specialized on the case of cyclotomic number fields. All necessary notions and facts are briefly introduced in Chapter 1. However, it might be helpful to have some background in algebrai ...
... probability theory is advisable. The presented toolkit uses several concepts from algebraic number theory, mainly specialized on the case of cyclotomic number fields. All necessary notions and facts are briefly introduced in Chapter 1. However, it might be helpful to have some background in algebrai ...
Subfield-Compatible Polynomials over Finite Fields - Rose
... polynomial g such that deg (g) < pn and the coefficients of g are sums of coefficients of terms in f that have degree greater than or equal to pn and coefficients of terms in f that have degree n strictly less than pn under the assumption that xp = x. By construction, deg (g) < pn and f (α) = g(α) f ...
... polynomial g such that deg (g) < pn and the coefficients of g are sums of coefficients of terms in f that have degree greater than or equal to pn and coefficients of terms in f that have degree n strictly less than pn under the assumption that xp = x. By construction, deg (g) < pn and f (α) = g(α) f ...
Chapter 2 Determinants
... Step 1: Prove the theorem under the assumption that column 1 is swapped with column k, for k>1. If this is proved, then swapping column k and column k’ will be equivalent to performing three swaps: first swapping column 1 and column k, then swapping column 1 and column k’, and finally swapping colum ...
... Step 1: Prove the theorem under the assumption that column 1 is swapped with column k, for k>1. If this is proved, then swapping column k and column k’ will be equivalent to performing three swaps: first swapping column 1 and column k, then swapping column 1 and column k’, and finally swapping colum ...
Answers to exercises LINEAR ALGEBRA - Joshua
... Thus, given y we can compute the associated x. Taking y = 0 gives the solution (c/a, 0), and since the second equation ax + dy = e is supposed to have the same solution set, substituting into it gives that a(c/a) + d · 0 = e, so c = e. Taking y = 1 in (∗) gives a((c − b)/a) + d · 1 = e, and so b = d ...
... Thus, given y we can compute the associated x. Taking y = 0 gives the solution (c/a, 0), and since the second equation ax + dy = e is supposed to have the same solution set, substituting into it gives that a(c/a) + d · 0 = e, so c = e. Taking y = 1 in (∗) gives a((c − b)/a) + d · 1 = e, and so b = d ...
Analysis of cumulants
... notwithstanding — assurances based, in part, on their fantastically complex mathematical models for measuring the risk in their various portfolios. There are many such models, but by far the most widely used is called VaR — Value at Risk. Built around statistical ideas and probability theories that ...
... notwithstanding — assurances based, in part, on their fantastically complex mathematical models for measuring the risk in their various portfolios. There are many such models, but by far the most widely used is called VaR — Value at Risk. Built around statistical ideas and probability theories that ...
Linear Algebra - UC Davis Mathematics
... of functions. In linear algebra, functions will again be the focus of your attention, but functions of a very special type. In precalculus you were perhaps encouraged to think of a function as a machine f into which one may feed a real number. For each input x this machine outputs a single real numb ...
... of functions. In linear algebra, functions will again be the focus of your attention, but functions of a very special type. In precalculus you were perhaps encouraged to think of a function as a machine f into which one may feed a real number. For each input x this machine outputs a single real numb ...
linear old
... of whom may go on to major in mathematics. It was motivated by the lack of a book that taught students basic structures of linear algebra without overdoing mathematical rigor or becoming a mindless exercise in crunching recipes at the cost of fundamental understanding. In particular we wanted a book ...
... of whom may go on to major in mathematics. It was motivated by the lack of a book that taught students basic structures of linear algebra without overdoing mathematical rigor or becoming a mindless exercise in crunching recipes at the cost of fundamental understanding. In particular we wanted a book ...
Extraneous Factors in the Dixon Resultant
... identi ed by Hong between the resultants of the two systems turns out to be exactly the same as the relationship identi ed by our results between their projection operators. This shows that the generalized Dixon method is optimal under such transformations because if the projection operator of the s ...
... identi ed by Hong between the resultants of the two systems turns out to be exactly the same as the relationship identi ed by our results between their projection operators. This shows that the generalized Dixon method is optimal under such transformations because if the projection operator of the s ...
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.