41. Feedback--invariant optimal control theory and differential
... We use definitions and notations from [4, Sec. 1]; see also Appendix for the definition and main properties of the Maslov-type index indΠ . Let (u, λ) be a Lagrangian point of the mapping f : U → M and let N be a germ of a submanifold in U at u, hence (u, λ) is a Lagrangian point of f |N . If N is f ...
... We use definitions and notations from [4, Sec. 1]; see also Appendix for the definition and main properties of the Maslov-type index indΠ . Let (u, λ) be a Lagrangian point of the mapping f : U → M and let N be a germ of a submanifold in U at u, hence (u, λ) is a Lagrangian point of f |N . If N is f ...
Core Mathematics 2
... Know that the gradient of a curve at a point is given by the gradient of the tangent at the point. Know that the gradient of the tangent is given by the limit of the gradient of a ...
... Know that the gradient of a curve at a point is given by the gradient of the tangent at the point. Know that the gradient of the tangent is given by the limit of the gradient of a ...
GEOMETRIC PROOFS OF SOME RESULTS OF MORITA
... Remark 8. (i) Since H1 (Γng , Z) is torsion free when g ≥ 3, there is no torsion in H 2 (Γng , Z) when g ≥ 3. So if a relation between integral cohomology classes holds in H 2 (Γng , Q), it holds in H 2 (Γng , Z). (ii) The (8g + 4)λ appears frequently in identities involving divisor classes on Mg , ...
... Remark 8. (i) Since H1 (Γng , Z) is torsion free when g ≥ 3, there is no torsion in H 2 (Γng , Z) when g ≥ 3. So if a relation between integral cohomology classes holds in H 2 (Γng , Q), it holds in H 2 (Γng , Z). (ii) The (8g + 4)λ appears frequently in identities involving divisor classes on Mg , ...
NOTES hist geometry
... when considered in the complex projective plane, leave the circular points at infinity fixed. These are the points (1,i,0) and (1,–i,0) which belong to every circle in the complex projective plane.) It seems that as a consequence of this, in the real plane there is a unique similarity transformation ...
... when considered in the complex projective plane, leave the circular points at infinity fixed. These are the points (1,i,0) and (1,–i,0) which belong to every circle in the complex projective plane.) It seems that as a consequence of this, in the real plane there is a unique similarity transformation ...
(pdf)
... The reason why we study these two objects is their usefulness for understanding algebraic number fields. In cases when elementary methods cannot reveal more information about some field k, for example when its Galois closure is a non-abelian extension, looking at the localizations of its adele ring ...
... The reason why we study these two objects is their usefulness for understanding algebraic number fields. In cases when elementary methods cannot reveal more information about some field k, for example when its Galois closure is a non-abelian extension, looking at the localizations of its adele ring ...
Isotriviality and the Space of Morphisms on Projective Varieties
... is reductive. Then φ is isotrivial if and only if φ has potential good reduction at all places v of K. Proof. The only if direction is clear. For the if direction, we imitate the proof in [12] to create a morphism from the complete curve C to the affine variety Md (X, L), which must be constant. Spe ...
... is reductive. Then φ is isotrivial if and only if φ has potential good reduction at all places v of K. Proof. The only if direction is clear. For the if direction, we imitate the proof in [12] to create a morphism from the complete curve C to the affine variety Md (X, L), which must be constant. Spe ...
Topology of Open Surfaces around a landmark result of C. P.
... algebraic variety. Recall the uniformization theorem for Riemann surfaces viz., any simply connected Riemann surface is bi-holomorphic to either the unit disc E in C, the complex plane C or the extended complex plane Ĉ. Ramanujam’s theorem is a 2-dimensional analogue of this classical 1-dimensional ...
... algebraic variety. Recall the uniformization theorem for Riemann surfaces viz., any simply connected Riemann surface is bi-holomorphic to either the unit disc E in C, the complex plane C or the extended complex plane Ĉ. Ramanujam’s theorem is a 2-dimensional analogue of this classical 1-dimensional ...
Computing in Picard groups of projective curves over finite fields
... by its values at these points), so that the multiplication maps can be computed pointwise. Example. Let X be a modular curve, say for concreteness X0 (n) or X1 (n) with n ≥ 5. Then a suitable line bundle L is the line bundle ω 2 of modular forms of weight 2 on X. The spaces Γ(X, Li ) can be computed ...
... by its values at these points), so that the multiplication maps can be computed pointwise. Example. Let X be a modular curve, say for concreteness X0 (n) or X1 (n) with n ≥ 5. Then a suitable line bundle L is the line bundle ω 2 of modular forms of weight 2 on X. The spaces Γ(X, Li ) can be computed ...
Chapter 1 PLANE CURVES
... If p = (x0 , x1 , x2 ) is a point of P2 , and if x0 6= 0, we may normalize the first entry to 1 without changing the point: (x0 , x1 , x2 ) ∼ (1, u1 , u2 ), where ui = xi /x0 . We did this for P1 above. The representative vector (1, u1 , u2 ) is uniquely determined by p, so points with x0 6= 0 corre ...
... If p = (x0 , x1 , x2 ) is a point of P2 , and if x0 6= 0, we may normalize the first entry to 1 without changing the point: (x0 , x1 , x2 ) ∼ (1, u1 , u2 ), where ui = xi /x0 . We did this for P1 above. The representative vector (1, u1 , u2 ) is uniquely determined by p, so points with x0 6= 0 corre ...
A practical Differential Power Analysis Attack against the Miller
... G1 × G2 → G3 We will consider pairings defined over an elliptic curve E over a finite field Fq , for q a prime number, or a power of a prime number different from 2 and 3. In characteristic 2 [11] and 3, the same scheme can be applied, but the equations are a little different. The equation of the el ...
... G1 × G2 → G3 We will consider pairings defined over an elliptic curve E over a finite field Fq , for q a prime number, or a power of a prime number different from 2 and 3. In characteristic 2 [11] and 3, the same scheme can be applied, but the equations are a little different. The equation of the el ...
1 Sample Paper – 2009 Class – XII Subject – Mathematics
... radius 5 3 is 500 cm3 An open box with a square base is to be made out of a given quantity of cardboard of area a2 square units. Find the dimensions of the box so that the volume of the box is maximum. Also find the maximum volume. Prove that the volume of the largest cone that can be inscribed in ...
... radius 5 3 is 500 cm3 An open box with a square base is to be made out of a given quantity of cardboard of area a2 square units. Find the dimensions of the box so that the volume of the box is maximum. Also find the maximum volume. Prove that the volume of the largest cone that can be inscribed in ...
2 - arXiv
... Proof of 1.3. Firstly we can replace X by a small crepant Q-factorialisation by [2, 1.6]. Furthermore we may extend the base field to assume it is uncountable. Lemma 4.1. Let X be a normal projective variety of dimension n over an uncountable field with nef Q-Cartier divisor D such that κ(X, D) = n ...
... Proof of 1.3. Firstly we can replace X by a small crepant Q-factorialisation by [2, 1.6]. Furthermore we may extend the base field to assume it is uncountable. Lemma 4.1. Let X be a normal projective variety of dimension n over an uncountable field with nef Q-Cartier divisor D such that κ(X, D) = n ...
The Picard group
... down to the study of X(Fp ), especially when p is a prime of good reduction for X. Varieties over finite fields have many advantages – in particular, they have only finitely many points which can therefore be listed! ...
... down to the study of X(Fp ), especially when p is a prime of good reduction for X. Varieties over finite fields have many advantages – in particular, they have only finitely many points which can therefore be listed! ...
WHICH ARE THE SIMPLEST ALGEBRAIC VARIETIES? Contents 1
... conic gives a one–to–one correspondence between the points on the conic and the points of a line. The inverse is given by quotients of polynomials of degree 2. The coefficients of these polynomials are in the same field as the coefficients of q(x, y) and the coordinates of P . ...
... conic gives a one–to–one correspondence between the points on the conic and the points of a line. The inverse is given by quotients of polynomials of degree 2. The coefficients of these polynomials are in the same field as the coefficients of q(x, y) and the coordinates of P . ...
CHAPTER 36 FUNCTIONS AND THEIR CURVES
... Equating the coefficient of the highest power of x term to zero gives 1 = 0 which is not an equation of a line. Hence there is no asymptote parallel with the x-axis Equating the coefficient of the highest power of y term to zero gives –x = 0 from which, x = 0 Hence, x = 0, y = x and y = –x are asymp ...
... Equating the coefficient of the highest power of x term to zero gives 1 = 0 which is not an equation of a line. Hence there is no asymptote parallel with the x-axis Equating the coefficient of the highest power of y term to zero gives –x = 0 from which, x = 0 Hence, x = 0, y = x and y = –x are asymp ...
1736 - RIMS, Kyoto University
... when dim X = 1. The following is fundamental for (*) in this case: Theorem 1.2 (Tango [T1]). Let D be an effective divisor on a smooth algebraic curve X. Then the kernel of the Frobenius map (1) is isomorphic to the space of exact differentials df of rational functions f on X with (df ) ≥ pD. The foll ...
... when dim X = 1. The following is fundamental for (*) in this case: Theorem 1.2 (Tango [T1]). Let D be an effective divisor on a smooth algebraic curve X. Then the kernel of the Frobenius map (1) is isomorphic to the space of exact differentials df of rational functions f on X with (df ) ≥ pD. The foll ...
PDF on arxiv.org - at www.arxiv.org.
... recall that (Fp f )00 is a finite-dimensional Fp -vector space provided with a natural structure of Gal(f )-module. Let C = Cf,p be the smooth projective model of the smooth affine K-curve y p = f (x). So C is a smooth projective curve defined over K. The rational function x ∈ K(C) defines a finite ...
... recall that (Fp f )00 is a finite-dimensional Fp -vector space provided with a natural structure of Gal(f )-module. Let C = Cf,p be the smooth projective model of the smooth affine K-curve y p = f (x). So C is a smooth projective curve defined over K. The rational function x ∈ K(C) defines a finite ...
Revised version
... that v ◦ v 0 ∈ V as well. The connections among v3 , (1.10) and v9 are equivalent to the equation v3 ◦ v3 = v9 ; it turns out that v3 ◦ v4 = v4 ◦ v3 = v12 . The homogeneous version of composition (which applies to infinite solutions as well) is given as follows. If v = (p : q : r) and v 0 = (p0 : q ...
... that v ◦ v 0 ∈ V as well. The connections among v3 , (1.10) and v9 are equivalent to the equation v3 ◦ v3 = v9 ; it turns out that v3 ◦ v4 = v4 ◦ v3 = v12 . The homogeneous version of composition (which applies to infinite solutions as well) is given as follows. If v = (p : q : r) and v 0 = (p0 : q ...
Most rank two finite groups act freely on a homotopy product of two
... two p-groups and all rank two finite simple groups except P SL3(Fp) for p an odd prime. We will be focused on rank two groups today and will verify the conjecture for most rank two groups. A result of A. Heller states that if h(G) ≤ 2, then rk(G) ≤ 2. To verify the conjecture for rank two groups, we ...
... two p-groups and all rank two finite simple groups except P SL3(Fp) for p an odd prime. We will be focused on rank two groups today and will verify the conjecture for most rank two groups. A result of A. Heller states that if h(G) ≤ 2, then rk(G) ≤ 2. To verify the conjecture for rank two groups, we ...
Solution 7 - WUSTL Math
... This is zero. R (3) Calculate the line integral xdy−ydx x2 +y 2 along the following path. Start from (1, 0), go along the x-axis to (a, 0) where a > 0, then go counterclockwise along the semicircle with radius a, ending at (−a, 0) and go along the x-axis to (−1, 0). π. H (4) If C is any closed loop ...
... This is zero. R (3) Calculate the line integral xdy−ydx x2 +y 2 along the following path. Start from (1, 0), go along the x-axis to (a, 0) where a > 0, then go counterclockwise along the semicircle with radius a, ending at (−a, 0) and go along the x-axis to (−1, 0). π. H (4) If C is any closed loop ...
WHAT IS A GLOBAL FIELD? A global field K is either • a finite
... field), C is a smooth projective curve over k = Fq or C, one may use some extra tools available when doing geometry over k. Example 1. The Riemanniann field case. Let π : X → C be a surjective morphism from a smooth projective surface X /C to a smooth projective curve C/C with generic fiber X/K. Let ...
... field), C is a smooth projective curve over k = Fq or C, one may use some extra tools available when doing geometry over k. Example 1. The Riemanniann field case. Let π : X → C be a surjective morphism from a smooth projective surface X /C to a smooth projective curve C/C with generic fiber X/K. Let ...
Notes
... classes, that form the fundamental group of M.) Free homotopy defines an equivalence relation on the free loop space ΛM. The equivalence classes are the path components of ΛM. They are called free homotopy classes, and we denote the set of free homotopy classes by F (M). The class of contractible cu ...
... classes, that form the fundamental group of M.) Free homotopy defines an equivalence relation on the free loop space ΛM. The equivalence classes are the path components of ΛM. They are called free homotopy classes, and we denote the set of free homotopy classes by F (M). The class of contractible cu ...
A COUNTER EXAMPLE TO MALLE`S CONJECTURE ON THE
... Example 1. Let G = Cℓ ≀ C2 for an odd prime ℓ. Then L = K := Q( ±ℓ) ⊆ Q(ζℓ ) has the wanted property. We remark that for ℓ > 3 we are only able to prove Z(Q, Cℓ ≀ C2 ; x) = O(x3/(2ℓ) ) since we do not know good estimates for the ℓ-rank of the class group of quadratic fields in these cases. 3. Commen ...
... Example 1. Let G = Cℓ ≀ C2 for an odd prime ℓ. Then L = K := Q( ±ℓ) ⊆ Q(ζℓ ) has the wanted property. We remark that for ℓ > 3 we are only able to prove Z(Q, Cℓ ≀ C2 ; x) = O(x3/(2ℓ) ) since we do not know good estimates for the ℓ-rank of the class group of quadratic fields in these cases. 3. Commen ...
Simultaneous equations : two equations, two variables (e
... (ii) e(x+1) (iii) e3x (iv) 5 ex (v) 6 e2x - 1 (vi) 1/ e3-x ...
... (ii) e(x+1) (iii) e3x (iv) 5 ex (v) 6 e2x - 1 (vi) 1/ e3-x ...
automorphisms of the field of complex numbers
... Waerden are, however, not needed in their full generality as far as our problem is concerned. Accordingly, in § 4, the argument is formulated in a more concrete way; this is less revealing than the general approach, but it is quite elementary and, except for an appeal to a few basic results in the t ...
... Waerden are, however, not needed in their full generality as far as our problem is concerned. Accordingly, in § 4, the argument is formulated in a more concrete way; this is less revealing than the general approach, but it is quite elementary and, except for an appeal to a few basic results in the t ...