ÉTALE MORPHISMS OF SCHEMES Contents 1. Introduction 1 2 ...

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exposé - matière potentielle : for general schemes
exposé

ETALE MORPHISMS OF SCHEMES Contents 1. Introduction 1 2. Conventions 2 3. Unramified morphisms 2 4. Three other characterizations of unramified morphisms 4 5. The functorial characterization of unramified morphisms 5 6. Topological properties of unramified morphisms 6 7. Universally injective, unramified morphisms 7 8. Examples of unramified morphisms 8 9. Flat morphisms 9 10. Topological properties of flat morphisms 11 11. Etale morphisms 11 12. The structure theorem 13 13.

unramified morphisms
etale morphisms of schemes contents
unramified homomorphism of local rings
differentials
etale morphisms of schemes
ramification locus
projective modules
ring map
module

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edwui

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Lecture 7

M.J. Flynn

EE 486 lecture 7: Integer Multiplication

M. J. Flynn

slides prepared by Albert Liddicoat and Hossam Fahmy

Computer Architectur e & Arithmetic Group1

Stanford University

Multiplication Add-and-Shift Algorithm

Multiplicand“X”61 1 0 Multiplier“Y”0 1x5x 1 1 1 0 Partial Products“PP”0 0 0 1 1 0 Result“S”301 1 1 1 0

Y X Shift 0

ALU ShiftP

Computer Architectur e & Arithmetic Group2

Stanford University

Parallel Multiplication Generation of PP’s Using ROMs 1Simultaneous generation of partial products 4b x 4b Multiply Using 2Parallel reduction of partial products 256x8-bit ROM 8b x 8b Multiply Using 3Carry Propagate Addition (CPA) 256x8-bit ROMs X R Parallel generation of partial productsY X X X XX X X X 7 6 5 43 2 1 0 Y Y Y YY Y Y Y 7 6 5 43 2 1 0 x Using AND gate as 1x1-bit multiplierDOT representation X X X X *Y Y Y Y 3 2 1 03 2 1 0 X X XX X X 2 1 02 1 0 X2Y1X1Y1. Y Y Y x2 1YX X X Y Yx2 1 0X7* Y3Y2Y1Y0 06 5 4 X YX YX Y 2 01 00 0 X X X X *Y Y Y Y X Y 3 2 1 07 6 5 4 0 1 ...... X YX YX Y 2 21 20 2Y Y Y YX X X X * 7 6 5 47 6 5 4 R RR RR R 5 43 21 0 Computer Architectur e & Arithmetic Group3Stanford UniversityComputer Architectur e & Arithmetic Group4Stanford University

Generation of PP’s Using ROMs

8 x 8 bit Multiply using 256x8b ROMs a b c d

4, 8, and 16 bit Multiply using 256x8bit ROMs 16x16 bit 8x8 bit 4x4 bit

n hCSA Delay 4 10 b 8 31 d a 16 74 c 32 156 64 318 Computer Architectur e & Arithmetic Group5Stanford University

Booth’s Algorithm Observation:We can replace a string of 1’s in the multiplier by +1 and -1. . . . 0 1 1 1 1 1 0 . . . Example: . . . 0 1 1 1 1 1 0 . . . +1 - 1 . . . 1 0 0 0 0 0 0 . . . - 1 . . . 1 0 0 0 0-1 0 . . . X*(0 1 1 1 1 1)X*(1 0 0 0 0-1) Y Y X 1-X -1 X 10 0 X 10 0 X 10 0 X 10 0 0 0X 1

Computer Architectur e & Arithmetic Group6

Stanford University

EE 486

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