Bertini irreducibility theorems over finite fields
... Let X be a geometrically irreducible variety of dimension m ≥ 2 over a field k. Let F be a finite set of closed points in X . Then there exists a geometrically irreducible variety of dimension m − 1 Y ⊂ X containing F . Used in a similar form by Duncan-Reichstein, as well as Panin, who raised the qu ...
... Let X be a geometrically irreducible variety of dimension m ≥ 2 over a field k. Let F be a finite set of closed points in X . Then there exists a geometrically irreducible variety of dimension m − 1 Y ⊂ X containing F . Used in a similar form by Duncan-Reichstein, as well as Panin, who raised the qu ...
Rotation matrices - CS HUJI Home Page
... transforms ~0 = (0, 0) to a point other than ~0 = (0, 0). However, if we want to rotate around an arbitrary rotation center c, we can shift the plane by −c such that the rotation center will be 0, then perform the rotation around (0, 0) and shift the plane back by +c: Rc,α (x) = Rα (x − c) + c = Rα ...
... transforms ~0 = (0, 0) to a point other than ~0 = (0, 0). However, if we want to rotate around an arbitrary rotation center c, we can shift the plane by −c such that the rotation center will be 0, then perform the rotation around (0, 0) and shift the plane back by +c: Rc,α (x) = Rα (x − c) + c = Rα ...
Solutions to Assignment 8
... has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss. Again, we know that rank(A) + dim(Nul(A)) = 10. If the system is consistent for all possible ...
... has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss. Again, we know that rank(A) + dim(Nul(A)) = 10. If the system is consistent for all possible ...
Mathematics for Economic Analysis I
... The function y = f (x) is often called a ‘single valued function’ because there is a unique ‘y’ in the range for each specified x. A function whose domain and range are sets of real number is called a real valued function of a real variable. ...
... The function y = f (x) is often called a ‘single valued function’ because there is a unique ‘y’ in the range for each specified x. A function whose domain and range are sets of real number is called a real valued function of a real variable. ...
Homework 2. Solutions 1 a) Show that (x, y) = x1y1 + x2y2 + x3y3
... Hint: For any two given vectors x, y consider the quadratic polynomial At2 + 2Bt + C where A = (x, x), B = (x, y), C = (y, y). Show that this polynomial has at most one real root and consider its discriminant. Pn Pn Consider quadratic polynomial P (t) = i=1 (txi + y i )2 = At2 +2Bt +C, where A = i=1 ...
... Hint: For any two given vectors x, y consider the quadratic polynomial At2 + 2Bt + C where A = (x, x), B = (x, y), C = (y, y). Show that this polynomial has at most one real root and consider its discriminant. Pn Pn Consider quadratic polynomial P (t) = i=1 (txi + y i )2 = At2 +2Bt +C, where A = i=1 ...
A fast algorithm for approximate polynomial gcd based on structured
... algorithm [1], [2], [12], [17], optimization methods [15], SVD and factorization of resultant matrices [5], [4], [23], Padé approximation [3], [18], root grouping [18]. Some of them have been implemented inside numerical/symbolic packages like the algorithm of Zeng [23] in MatlabTM and the algorith ...
... algorithm [1], [2], [12], [17], optimization methods [15], SVD and factorization of resultant matrices [5], [4], [23], Padé approximation [3], [18], root grouping [18]. Some of them have been implemented inside numerical/symbolic packages like the algorithm of Zeng [23] in MatlabTM and the algorith ...