The ring of evenly weighted points on the projective line
... In [37] Jakub Witaszek studies the multigraded Poincaré-Hilbert series of the homogeneous coordinate ring of the Plücker embedding of the Grassmannian G(2, n) for a certain Nn -grading. We obtain a closed formula for the multigraded Hilbert function and for the Poincaré-Hilbert series, see Remark ...
... In [37] Jakub Witaszek studies the multigraded Poincaré-Hilbert series of the homogeneous coordinate ring of the Plücker embedding of the Grassmannian G(2, n) for a certain Nn -grading. We obtain a closed formula for the multigraded Hilbert function and for the Poincaré-Hilbert series, see Remark ...
a basis for free lie rings and higher commutators in free groups
... It will be necessary for us to show that the above process terminates. From the definition of the standard monomials it is clear that expressions in standard form are left unaltered by the canonical process, and that expressions left unaltered are in standard form. If min (degree y, degree z)>n/3, t ...
... It will be necessary for us to show that the above process terminates. From the definition of the standard monomials it is clear that expressions in standard form are left unaltered by the canonical process, and that expressions left unaltered are in standard form. If min (degree y, degree z)>n/3, t ...
COMPLETELY RANK-NONINCREASING LINEAR MAPS Don
... skew-compressions. A simple counterexample is based on the following elementary fact: There do not exist nets {eλ } and {fλ } in C2 such that, for every 2 × 2 matrix A, (Aeλ , fλ ) → tr(A), where tr denotes the normalized trace on M2 . This follows from the fact that the above assertion is equivalen ...
... skew-compressions. A simple counterexample is based on the following elementary fact: There do not exist nets {eλ } and {fλ } in C2 such that, for every 2 × 2 matrix A, (Aeλ , fλ ) → tr(A), where tr denotes the normalized trace on M2 . This follows from the fact that the above assertion is equivalen ...
On variants of the Johnson-Lindenstrauss lemma
... calculation is similar in spirit to that of Achlioptas, but slightly simpler and divided into several independent conceptual steps, which should hopefully make it easier to grasp. After a preliminary version of the present article was written, I learned about a preprint of Indyk and Naor [9]. It con ...
... calculation is similar in spirit to that of Achlioptas, but slightly simpler and divided into several independent conceptual steps, which should hopefully make it easier to grasp. After a preliminary version of the present article was written, I learned about a preprint of Indyk and Naor [9]. It con ...
POLYHEDRAL POLARITIES
... Polar Cones is in the foundations of the solution of Linear Inequalities Systems and Farkas’s result about polar cones equivalent to the important theorems of Alternatives and to the Strong Duality Theorem in Linear Programming and was used to prove the Strong Duality Theorem. Although some extensio ...
... Polar Cones is in the foundations of the solution of Linear Inequalities Systems and Farkas’s result about polar cones equivalent to the important theorems of Alternatives and to the Strong Duality Theorem in Linear Programming and was used to prove the Strong Duality Theorem. Although some extensio ...
Lecture notes up to 08 Mar 2017
... Example 2.4 (Homogenous G-spaces). Let H ⊂ G be a closed subgroup. Then we equip G/H with the quotient topology (the finest topology for which the quotient map G → G/H is continuous), and the natural G-space structure (a(g1 , g2 H) = g1 g2 H). Remark 2.5. Let X be a G-space. Assume that the action o ...
... Example 2.4 (Homogenous G-spaces). Let H ⊂ G be a closed subgroup. Then we equip G/H with the quotient topology (the finest topology for which the quotient map G → G/H is continuous), and the natural G-space structure (a(g1 , g2 H) = g1 g2 H). Remark 2.5. Let X be a G-space. Assume that the action o ...
Composition followed by differentiation between weighted Bergman-Nevanlinna spaces
... where X ≍ Y means that there is a positive constant C such that C −1 X ≤ Y ≤ CX. See [3] for more about weighted Bergman spaces and weighted Bergman-Nevanlinna spaces. By the subharmonicity of log(1 + |f (z)|), we have ||f ||A0λ (D) , z∈D ...
... where X ≍ Y means that there is a positive constant C such that C −1 X ≤ Y ≤ CX. See [3] for more about weighted Bergman spaces and weighted Bergman-Nevanlinna spaces. By the subharmonicity of log(1 + |f (z)|), we have ||f ||A0λ (D) , z∈D ...
Groups
... Example 24. Let X be ANY finite-dimensional linear/vector space dim(X ) = n < ∞ over complex or real numbers (or other field F ) Then (L (X ), circ) is isomorphic to the well-known specific binary structure (Mn (F), ⋅) of n × n matrices over F with the binary operation of usual matrix multiplication ...
... Example 24. Let X be ANY finite-dimensional linear/vector space dim(X ) = n < ∞ over complex or real numbers (or other field F ) Then (L (X ), circ) is isomorphic to the well-known specific binary structure (Mn (F), ⋅) of n × n matrices over F with the binary operation of usual matrix multiplication ...
Three Dimensional Geometry
... skew lines and distance of a point from a plane. Most of the above results are obtained in vector form. Nevertheless, we shall also translate these results in the Cartesian form which, at times, presents a more clear geometric and analytic picture of the situation. ...
... skew lines and distance of a point from a plane. Most of the above results are obtained in vector form. Nevertheless, we shall also translate these results in the Cartesian form which, at times, presents a more clear geometric and analytic picture of the situation. ...
A Ramsey space of infinite polyhedra and the random polyhedron
... A polyhedron is a geometric object built up through a finite or countable number of suitable amalgamations of convex hulls of finite sets; polyhedra are generated in this way by simplexes. Simplicial morphisms are locally linear maps that preserve vertices. An ordered polyhedron is a polyhedron for ...
... A polyhedron is a geometric object built up through a finite or countable number of suitable amalgamations of convex hulls of finite sets; polyhedra are generated in this way by simplexes. Simplicial morphisms are locally linear maps that preserve vertices. An ordered polyhedron is a polyhedron for ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.