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Example 2. Let \( M \) be the oriented three-dimensional euclidean space \( {\mathbb{R}}^{3} \) . Then every 1 -form on \( M \) corresponds to some vector field \( \mathbf{A}\left( {{\omega }^{1} = {\omega }_{\mathbf{A}}^{1}}\right) \), where | \[ {\omega }_{\mathbf{A}}^{1}\left( \xi \right) = \left( {\mathbf{A},\xi }\right) \] The integral of \( {\omega }_{\mathbf{A}}^{\mathbf{1}} \) on a chain \( {c}_{1} \) representing a curve \( l \) is called the circulation of the field \( \mathbf{A} \) over the curve \( l \) : \[ {\int }_{{c}_{1}}{\omega }_{\mathbf{A}}... | Yes |
Problem 14. Suppose that, in the \( {2n} \)-dimensional space \( {\mathbb{R}}^{n} = \left\{ \left( {{p}_{1},\ldots ,{p}_{n};{q}_{1},\ldots ,{q}_{n}}\right) \right\} \), we are given a 2-chain \( {c}_{2} \) representing a two-dimensional oriented surface \( S \) with boundary \( l \). Find \[ {\int }_{{c}_{2}}d{p}_{1} \... | Answer. The sum of the oriented areas of the projection of \( S \) on the two-dimensional coordinate planes \( {p}_{i},{q}_{i} \). | No |
Show that\n\n\[ \operatorname{div}\left\lbrack {\mathbf{A},\mathbf{B}}\right\rbrack = \left( {\operatorname{curl}\mathbf{A},\mathbf{B}}\right) - \left( {\operatorname{curl}\mathbf{B},\mathbf{A}}\right) ,\] | Hint. By the formula for differentiating the product of forms,\n\n\[ d\left( {\omega }_{\left\lbrack \mathbf{A},\mathbf{B}\right\rbrack }^{2}\right) = d\left( {{\omega }_{\mathbf{A}}^{1} \land {\omega }_{\mathbf{B}}^{1}}\right) = d{\omega }_{\mathbf{A}}^{1} \land {\omega }_{\mathbf{B}}^{1} - {\omega }_{\mathbf{A}}^{1} ... | No |
Problem 7. Show that curl grad \( = \) div curl \( = 0 \) . | Hint. \( {dd} = 0 \) . | No |
Problem 8. Given the components of a vector field \( \mathbf{A} = {A}_{1}{\mathbf{e}}_{1} + {A}_{2}{\mathbf{e}}_{2} + {A}_{3}{\mathbf{e}}_{3} \), find the components of its curl. | Solution. According to Section 34E\n\n\[ \n{\omega }_{\mathbf{A}}^{1} = {A}_{1}\sqrt{{E}_{1}}d{x}_{1} + {A}_{2}\sqrt{{E}_{2}}d{x}_{2} + {A}_{3}\sqrt{{E}_{3}}d{x}_{3}. \n\]\n\nTherefore,\n\n\[ \nd{\omega }_{\mathbf{A}}^{1} = \left( {\frac{\partial {A}_{3}\sqrt{{E}_{3}}}{\partial {x}_{2}} - \frac{\partial {A}_{2}\sqrt{{E... | Yes |
Problem 9. Find the divergence of the field \( \mathbf{A} = {A}_{1}{\mathbf{e}}_{1} + {A}_{2}{\mathbf{e}}_{2} + {A}_{3}{\mathbf{e}}_{3} \) . | Solution. \( {\omega }_{A}^{2} = {A}_{1}\sqrt{{E}_{2}{E}_{3}}d{x}_{2} \land d{x}_{3} + \cdots \) . Therefore,\n\n\[ d{\omega }_{A}^{2} = \frac{\partial }{\partial {x}_{1}}\left( {{A}_{1}\sqrt{{E}_{2}{E}_{3}}}\right) d{x}_{1} \land d{x}_{2} \land d{x}_{3} + \cdots . \]\n\nBy the definition of divergence,\n\n\[ d{\omega ... | Yes |
Problem 10. The Laplace operator on \( M \) is the operator \( \Lambda = \) div grad. Find its expression in the coordinates \( {x}_{i} \) . | \[ {\Delta f} = \frac{1}{\sqrt{{E}_{1}{E}_{2}{E}_{3}}}\left\lbrack {\frac{\partial }{\partial {x}_{1}}\left( {\sqrt{\frac{{E}_{2}{E}_{3}}{{E}_{1}}}\frac{\partial f}{\partial {x}_{1}}}\right) + \cdots }\right\rbrack . \] | Yes |
Problem 12. Prove Poincaré's lemma for 1-forms. | Hint. Consider \( {\int }_{{x}_{0}}^{{x}_{1}}{\omega }^{1} = \varphi \left( {x}_{1}\right) \) . | No |
Show that in a vector space the integral of a closed form over any cycle is zero. | Hint. Construct a \( \\left( {k + 1}\\right) \) -chain whose boundary is the given cycle (Figure 164).\n\n\n\nFigure 164 Cone over a cycle\n\n\n\n## 7: Differential forms\n\nNamely, for any chain \( c \) consider the... | No |
Lemma 1. The mixed partial derivative \( {\partial }^{2}\Delta /\partial s\partial t \) at 0 is equal to the commutator of differentiation in the directions \( \mathbf{A} \) and \( \mathbf{B} \) : | Proof. By the definition of \( {L}_{A} \) , \[ {\left. \frac{\partial }{\partial t}\right| }_{t = 0}\varphi \left( {{A}^{t}{B}^{s}x}\right) = \left( {{L}_{\mathbf{A}}\varphi }\right) \left( {{B}^{s}x}\right) . \] If we denote the function \( {L}_{\mathrm{A}}\varphi \) by \( \psi \), then by the definition of \( {L}_{\m... | Yes |
Lemma 2. The operator \( {L}_{\mathrm{B}}{L}_{\mathrm{A}} - {L}_{\mathrm{A}}{L}_{\mathrm{B}} \) is a first-order linear differential operator. | Proof. Let \( \left( {{A}_{1},\ldots ,{A}_{n}}\right) \) and \( \left( {{B}_{1},\ldots ,{B}_{n}}\right) \) be the components of the fields \( \mathbf{A} \) and \( \mathbf{B} \) in the local coordinate system \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) on \( M \) . Then\n\n\[ \n{L}_{\mathbf{B}}{L}_{\mathbf{A}}\varphi ... | Yes |
Corollary 2. The Poisson bracket of the functions \( F \) and \( H \) is equal to the value of the 1 -form \( {dF} \) on the velocity vector \( {IdH} \) of the phase flow with hamiltonian function \( H \) : | \[ \left( {F, H}\right) = {dF}\left( {IdH}\right) \] | Yes |
Problem 1. Compute the Poisson bracket of two functions \( F \) and \( H \) in the canonical coordinate space \( {\mathbb{R}}^{2n} = \{ \left( {\mathbf{p},\mathbf{q}}\right) \} ,{\omega }^{2}\left( {\mathbf{\xi },\mathbf{\eta }}\right) = \left( {I\mathbf{\xi },\mathbf{\eta }}\right) \) . | Solution. By Corollary 3 we have\n\n\[\n\left( {F, H}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}\frac{\partial H}{\partial {p}_{i}}\frac{\partial F}{\partial {q}_{i}} - \frac{\partial H}{\partial {q}_{i}}\frac{\partial F}{\partial {p}_{i}}\n\]\n\n(we use the fact that \( I \) is symplectic and has the form\n\n\[\nI =... | Yes |
Problem 2. Compute the Poisson brackets of the basic functions \( {p}_{i} \) and \( {q}_{j} \) . | Solution. The gradients of the basic functions form a \ | No |
Show that the map \( A : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) sending \( \left( {\mathbf{p},\mathbf{q}}\right) \rightarrow \left( {\mathbf{P}\left( {\mathbf{p},\mathbf{q}}\right) ,\mathbf{Q}\left( {\mathbf{p},\mathbf{q}}\right) }\right) \) is canonical if and only if the Poisson brackets of any two functi... | Solution. Let \( A \) be canonical. Then the symplectic structures \( d\mathbf{p} \land d\mathbf{q} \) and \( d\mathbf{P} \land d\mathbf{Q} \) coincide. But the definition of the Poisson bracket \( \left( {F, H}\right) \) was given invariantly in terms of the symplectic structure; it did not involve the coordinates. Th... | Yes |
Corollary 5. Let \( \mathbf{B} \) and \( \mathbf{C} \) be hamiltonian fields with hamiltonian functions \( B \) and \( C \) . Consider the Poisson bracket \( \left\lbrack {\mathbf{B},\mathbf{C}}\right\rbrack \) of the vector fields. This vector field is hamiltonian, and its hamiltonian function is equal to the Poisson ... | Proof. Set \( \left( {B, C}\right) = D \) . The Jacobi identity can be rewritten in the form\n\n\[ \left( {A, D}\right) = \left( {\left( {A, B}\right), C}\right) - \left( {\left( {A, C}\right), B}\right) ,\]\n\n\[ {L}_{\mathbf{D}} = {L}_{\mathbf{C}}{L}_{\mathbf{B}} - {L}_{\mathbf{B}}{L}_{\mathbf{C}}\;{L}_{\mathbf{D}} =... | Yes |
Corollary 8. The map of the Lie algebra of functions onto the Lie algebra of hamiltonian fields is an algebra homomorphism. Its kernel consists of the locally constant functions. If \( {M}^{2n} \) is connected, the kernel is one-dimensional and consists of constants. | Proof. Our map is linear. Corollary 5 says that our map carries the Poisson bracket of functions into the Poisson bracket of vector fields. The kernel consists of functions \( H \) for which \( {IdH} \equiv 0 \) . Since \( I \) is an isomorphism, \( {dH} \equiv 0 \) and \( H = \) const. | Yes |
Corollary 9. The phase flows with hamiltonian functions \( {H}_{1} \) and \( {H}_{2} \) commute if and only if the Poisson bracket of the functions \( {H}_{1} \) and \( {H}_{2} \) is (locally) constant. | Proof. By the theorem in Section 39, \( E \), it is necessary and sufficient that \( \left\lbrack {{\mathbf{H}}_{1},{\mathbf{H}}_{2}}\right\rbrack \equiv 0 \), and by Corollary 8 this condition is equivalent to \( d\left( {{H}_{1},{H}_{2}}\right) \) \( \equiv 0 \) . | Yes |
Corollary 1. The phase flow preserves the integral of the form \( \mathbf{p}d\mathbf{q} = \) \( {p}_{1}d{q}_{1} + \cdots + {p}_{n}d{q}_{n} \) on closed curves. | Proof. Let \( {g}_{{t}_{0}}^{{t}_{1}} : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) be the transformation of the phase space \( \left( {\mathbf{p},\mathbf{q}}\right) \) realized by the phase flow from time \( {t}_{0} \) to \( {t}_{1} \) (i.e., \( {g}_{{t}_{0}}^{t}\left( {{\mathbf{p}}_{0},{\mathbf{q}}_{0}}\right)... | Yes |
Corollary 2. The phase flow preserves the sum of the oriented areas of the projections of a surface onto the \( n \) coordinate planes \( \left( {{p}_{i},{q}_{i}}\right) \) : | \[ {\iint }_{\sigma }d\mathbf{p} \land d\mathbf{q} = {\iint }_{{g}_{{t}_{0}}^{{t}_{1}}\sigma }d\mathbf{p} \land d\mathbf{q} \] In other words, the 2 -form \( {\omega }^{2} = d\mathbf{p} \land d\mathbf{q} \) is an absolute integral invariant of the phase flow. | Yes |
Corollary 1. If, in a canonical system with two degrees of freedom, a first integral \( F \) is known which does not depend on the hamiltonian \( H \), then the system is integrable by quadratures; a compact connected two-dimensional submanifold of the phase space \( H = h, F = f \) is an invariant torus, and motion on... | Proof. \( F \) and \( H \) are in involution since \( F \) is a first integral of a system with hamiltonian function \( H \) . | No |
Lemma 1. On the n-dimensional manifold \( {M}_{\mathrm{f}} \) there exist \( n \) tangent vector fields which commute with one another and which are linearly independent at every point. | Proof. The symplectic structure of phase space defines an operator \( I \) taking 1 -forms to vector fields. This operator \( I \) carries the 1 -form \( d{F}_{i} \) to the field \( {Id}{F}_{i} \) of phase velocities of the system with hamiltonian function \( {F}_{i} \) . We will show that the \( n \) fields \( {Id}{F}... | Yes |
Lemma 2. Let \( {M}^{n} \) be a compact connected differentiable \( n \) -dimensional manifold, on which we are given \( n \) pairwise commutative and linearly independent at each point vector fields. Then \( {M}^{n} \) is diffeomorphic to an \( n \) -dimensional torus. | Proof. We denote by \( {g}_{i}^{t}, i = 1,\ldots, n \), the one-parameter groups of diffeo-morphisms of \( M \) corresponding to the \( n \) given vector fields. Since the fields commute, the groups \( {g}_{i}^{t} \) and \( {g}_{j}^{s} \) commute. Therefore, we can define an action \( g \) of the commutative group \( {... | Yes |
Show that the map \( g \) (Figure 211) of a sufficiently small neighborhood \( V \) of the point \( 0 \in {\mathbb{R}}^{n} \) gives a chart in a neighborhood of \( {x}_{0} \) : every point \( {x}_{0} \in M \) has a neighborhood \( U\left( {{x}_{0} \in U \subset M}\right) \) such that \( g \) maps \( V \) diffeomorphica... | Hint. Apply the implicit function theorem and use the linear independence of the fields at \( {x}_{0} \). | No |
Problem 3. Show that \( \Gamma \) is a subgroup of the group \( {\mathbb{R}}^{n} \), independent of the point \( {x}_{0} \) . | Solution. If \( {g}^{\$ }{x}_{0} = {x}_{0} \) and \( {g}^{t}{x}_{0} = {x}_{0} \), then \( {g}^{\$ + t}{x}_{0} = {g}^{\$ }{g}^{t}{x}_{0} = {g}^{\$ }{x}_{0} = {x}_{0} \) and \( {g}^{-t}{x}_{0} = \) \( {g}^{-t}{g}^{t}{x}_{0} = {x}_{0} \) . Therefore, \( \Gamma \) is a subgroup of \( {\mathbb{R}}^{n} \) . If \( x = {g}^{t}... | No |
Lemma 3. Let \( \Gamma \) be a discrete subgroup of \( {\mathbb{R}}^{n} \) . Then there exist \( k\left( {0 \leq k \leq n}\right) \) linearly independent vectors \( {\mathbf{e}}_{1},\ldots ,{\mathbf{e}}_{k} \in \Gamma \) such that \( \Gamma \) is exactly the set of all their integral linear combinations. | Proof. We will consider \( {\mathbb{R}}^{n} \) with some euclidean structure. We always have \( 0 \in \Gamma \) . If \( \Gamma = \{ 0\} \) the lemma is proved. If not, there is a point \( {\mathbf{e}}_{0} \in \Gamma \) , \( {\mathbf{e}}_{0} \neq 0 \) (Figure 215). Consider the line \( {\mathbb{{Re}}}_{0} \) . We will s... | Yes |
Show that there are no points of \( \Gamma \) on the plane \( {\mathbb{{Re}}}_{1} + {\mathbb{{Re}}}_{2} \) other than integral linear combinations of \( {\mathbf{e}}_{1} \) and \( {\mathbf{e}}_{2} \) . | Hint. Partition the plane into parallelograms (Figure 216) \( \Delta = \left\{ {{\lambda }_{1}{\mathbf{e}}_{1} + {\lambda }_{2}{\mathbf{e}}_{2}}\right\} \) , \( {m}_{i} \leq {\lambda }_{i} < {m}_{i} + 1 \) . If there were an \( \mathbf{e} \in \Delta \) with \( \mathbf{e} \neq {m}_{1}{\mathbf{e}}_{1} + {m}_{2}{\mathbf{e... | No |
Problem 8. Show that this closest point always exists. | Hint. Take the closest of the finite number of points in a \ | No |
Problem 10. Show that the map of charts \( A : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) gives a diffeomorphism \( \widetilde{A} : {T}^{k} \times {\mathbb{R}}^{n - k} \rightarrow {M}_{\mathrm{f}}, \) | But, since the manifold \( {M}_{\mathbf{f}} \) is compact by hypothesis, \( k = n \) and \( {M}_{\mathbf{f}} \) is an \( n \) -dimensional torus. Lemma 2 is proved. | No |
Problem 11. Show that under the action of the phase flow with hamiltonian \( H \) the angular coordinates \( \mathbf{\varphi } \) vary uniformly with time | \[ {\dot{\varphi }}_{i} = {\omega }_{i}\;{\omega }_{i} = {\omega }_{i}\left( \mathbf{f}\right) \;\mathbf{\varphi }\left( t\right) = \mathbf{\varphi }\left( 0\right) + \mathbf{\omega }t. \] | Yes |
Problem. Find the action-angle variables in the case of the simple harmonic oscillator \( H = \frac{1}{2}{p}^{2} + \frac{1}{2}{q}^{2}. \) | Solution. If \( r,\varphi \) are polar coordinates, then \( {dp} \land {dq} = {rdr} \land {d\varphi } = d\left( {{r}^{2}/2}\right) \land {d\varphi } \) . Therefore, \( I = H = \left( {{p}^{2} + {q}^{2}}\right) /2 \) . \n\n\( {}^{88} \) It is not hard to see that I has the dimensions of action. \n\nIn order to construct... | No |
Corollary 1. If the frequencies are independent, then every trajectory \( \{ \mathbf{\varphi }\left( t\right) \} \) is dense on the torus \( {T}^{n} \) . | Proof. Assume the contrary. Then in some neighborhood \( D \) of some point of the torus, there is no point of the trajectory \( \varphi \left( t\right) \) . It is easy to construct a continuous function \( f \) equal to zero outside \( D \) and with space average equal to 1 . The time average \( {f}^{ * }\left( {\varp... | Yes |
Corollary 2. If the frequencies are independent, then every trajectory is uniformly distributed on the torus \( {T}^{n} \) . | Proof. We apply the theorem to the characteristic function \( f \) of the set \( D \) ( \( f \) is Riemann integrable since \( D \) is Jordan measurable). Then \( {\int }_{0}^{T}f\left( {\varphi \left( t\right) }\right) {dt} = \) \( {\tau }_{D}\left( T\right) \), and \( \bar{f} = {\left( 2\pi \right) }^{-n} \) mes \( D... | Yes |
Lemma 1. The theorem is true for exponentials \( f = {e}^{i\left( {\mathbf{k},\varphi }\right) },\mathbf{k} \in {\mathbb{Z}}^{n} \) . | Proof. If \( \mathbf{k} = 0 \), then \( \bar{f} = f = {f}^{ * } = 1 \) and the theorem is obvious. If \( \mathbf{k} \neq 0 \) , then \( \bar{f} = 0 \) . On the other hand,\n\n\[{\int }_{0}^{T}{e}^{i\left( {\mathbf{k},{\varphi }_{0} + {\omega t}}\right) }{dt} = {e}^{i\left( {\mathbf{k},{\varphi }_{0}}\right) }\frac{{e}^... | Yes |
Lemma 2. The theorem is true for trigonometric polynomials\n\n\[ f = \mathop{\sum }\limits_{{\left| \mathbf{k}\right| < N}}{f}_{\mathbf{k}}{e}^{i\left( {\mathbf{k},\varphi }\right) } \] | Proof. Both the time and space averages depend linearly on \( f \), and therefore agree by Lemma 1. | No |
Lemma 3. Let \( f \) be a real continuous (or at least Riemann integrable) function. Then, for any \( \varepsilon > 0 \), there exist two trigonometric polynomials \( {P}_{1} \) and \( {P}_{2} \) such that \( {P}_{1} < f < {P}_{2} \) and \( \left( {1/{\left( 2\pi \right) }^{n}}\right) {\int }_{{T}^{n}}\left( {{P}_{2} -... | Proof. Suppose first that \( f \) is continuous. By the Weierstrass theorem, we can approximate \( f \) by a trigonometric polynomial \( P \) with \( \left| {f - P}\right| < \frac{1}{2}\varepsilon \) . The polynomials \( {P}_{1} = P - \frac{1}{2}\varepsilon \) and \( {P}_{2} = P + \frac{1}{2}\varepsilon \) are the ones... | Yes |
Here we show the adiabatic invariance of the action variable in a system with one degree of freedom. | Let \( \mathbf{I},\varphi \) be action-angle variables in an integrable (\ | No |
Problem 2. Show that the curvature tensor can be expressed in terms of covariant differentiation in the following way:\n\n\[ \Omega \left( {{\xi }_{0},{\eta }_{0}}\right) {\zeta }_{0} = - {\nabla }_{\xi }{\nabla }_{\eta }\zeta + {\nabla }_{\eta }{\nabla }_{\xi }\zeta + {\nabla }_{\left\lbrack \eta ,\xi \right\rbrack }\... | where \( \xi ,\eta ,\zeta \) are any vector fields whose values at the point under consideration are \( {\xi }_{0},{\eta }_{0} \), and \( {\zeta }_{0} \) . | No |
Problem 4. Suppose that the riemannian metric is given in local coordinates \( {x}_{1},\ldots ,{x}_{n} \) by the symmetric matrix \( {g}_{ij} \) :\n\n\[ d{s}^{2} = \sum {g}_{ij}d{x}_{i}d{x}_{j} \]\n\nDenote by \( {e}_{1},\ldots ,{e}_{n} \) the coordinate vector fields (so that differentiation in the direction \( {e}_{i... | By using the expression for the curvature tensor in terms of the connection in Problem 2, we also obtain an explicit formula for the curvature. The numbers \( {R}_{ijkl} = \left\langle {\Omega \left( {{e}_{i},{e}_{j}}\right) {e}_{k},{e}_{l}}\right\rangle \) are called the components of the curvature tensor. | No |
Theorem 1. The vector of angular momentum relative to space is preserved under motion: | \[ \frac{d{M}_{s}}{dt} = 0 \] | No |
Theorem 2. The vector of angular momentum relative to the body satisfies\n\nEuler's equation\n\n\[ \frac{d{M}_{c}}{dt} = \left\{ {{\omega }_{c},{M}_{c}}\right\} \] | These theorems are proved for a generalized rigid body in the same way as for an ordinary rigid body. | No |
Theorem 3. The orbits of the co-adjoint representation of a group in the dual space to the algebra are invariant manifolds for the flow in this space given by Euler's equation. | Proof. \( {M}_{c}\left( t\right) \) is obtained from \( {M}_{s}\left( t\right) \) by the action of the co-adjoint representation, and \( {M}_{s}\left( t\right) \) remains fixed. | No |
Theorem 4. On every orbit \( V \) of the co-adjoint representation, Euler’s equation is hamiltonian with hamiltonian function \( H \) . | Proof. Every vector \( \xi \) tangent to \( V \) at a point \( M \) has the form \( \xi = \{ f, M\} \) . where \( f \in \mathfrak{g} \) . In particular, the vector field on the right side of Euler's equation can be written in the form \( X = \{ {dT}, M\} \) (here the differential of the function \( T \) at a point \( M... | Yes |
Theorem 5. The motion of the vector of angular velocity in the body is determined by the initial position of this vector and does not depend on the initial position of the body. The vector of angular velocity in the body satisfies an equation with quadratic right-hand side: | \[ {\dot{\omega }}_{c} = B\left( {{\omega }_{c},{\omega }_{c}}\right) \] We will call this equation Euler's equation for angular velocity. We notice that, under the action of the operator \( {A}^{-1} : {\mathfrak{g}}^{ * } \rightarrow \mathfrak{g} \), the orbits of the co-adjoint representation are carried to invariant... | No |
Theorem 6. Euler's equations (for momentum and angular velocity) have a quadratic first integral, whose value is equal to the kinetic energy | \[ T = \frac{1}{2}\left( {{M}_{c},{A}^{-1}{M}_{c}}\right) = \frac{1}{2}\left( {A{\omega }_{c},{\omega }_{c}}\right) . \] | No |
Theorem 7. The angular momentum (respectively, angular velocity) of a stationary rotation with respect to the body is a critical point of the energy on the orbit of the co-adjoint representation (respectively on the image of the orbit under the action of the operator \( {A}^{-1} \) ). Conversely, every critical point o... | The proof is a straightforward computation or application of Theorem 4. | No |
Theorem 8. Suppose that a regular point \( M \) of the space of angular momenta is a critical point of the energy on an orbit of the co-adjoint representation, and that the second differential of the energy \( {d}^{2}H \) at this point is a (positive-or negative-)definite form. Then \( M \) is a (Liapunov) stable equil... | Proof. It follows from the regularity of the orbits near this point that on every neighboring orbit there exists near \( M \) a point which is a conditional maximum or minimum of energy. | No |
Theorem 10. The curvature of a group in the direction determined by an orthonormal pair of vectors \( \xi ,\eta \) in the algebra is given by the formula\n\n\[ \n{K}_{\xi ,\eta } = \langle \delta ,\delta \rangle + 2\langle \alpha ,\beta \rangle - 3\langle \alpha ,\alpha \rangle - 4\left\langle {{B}_{\xi },{B}_{\eta }}\... | The proof is a tedious but straightforward calculation. It is based on the easily verified formula for covariant derivative\n\n\[ \n{\left( {\nabla }_{\xi }\eta \right) }_{e} = \frac{1}{2}\left( {\left\lbrack {\xi ,\eta }\right\rbrack - B\left( {\xi ,\eta }\right) - B\left( {\eta ,\xi }\right) }\right) ,\]\n\nwhere \( ... | Yes |
Theorem 11. Assume that the region \( D \) is bounded by a compact analytic surface, and that the field of velocities is analytic and not everywhere collinear with its curl. Then the region of the flow can be partitioned by an analytic sub-manifold into a finite number of cells, in each of which the flow is constructed... | To prove this theorem we look at the \ | No |
Theorem 13. Suppose that the stream function of a stationary flow, \( \psi = \psi \left( {x, y}\right) \) , in a region \( D \) is a function of the vorticity function (i.e., of the function \( {\Delta \psi } \) ) not only locally, but globally. Suppose that the derivative of the stream function with respect to the vor... | This theorem implies the stability of a stationary flow in the case of a positive-definite quadratic form\n\n\[ {\iint }_{D}{\left( \nabla \varphi \right) }^{2} + \frac{\nabla \psi }{\nabla {\Delta \psi }}{\left( \Delta \varphi \right) }^{2}{dxdy} \]\n\nwith respect to \( \nabla \varphi \) (where \( \varphi \) is a con... | Yes |
Theorem 14. The explicit formulas for the scalar product, commutator, operation \( B \), connection, and curvature of a right-invariant metric on the group \( {S}_{0} \) Diff \( {T}^{2} \) have the following form:\n\n\[ \left\langle {{e}_{k},{e}_{l}}\right\rangle = 0\;\text{ for }k + l \neq 0, \]\n\n\[ \left\langle {{e... | The proof of this theorem is in the first article listed in the introduction to this appendix. | No |
Theorem 15. The curvature of the group \( {S}_{0} \) Diff \( {T}^{2} \) in any two-dimensional plane containing the direction \( \xi \) is non-positive. Namely,\n\n\[ \langle \Omega \left( {\xi ,\eta }\right) \rangle = - \frac{S}{4}\mathop{\sum }\limits_{l}{a}_{k, l}^{2}{\left| {x}_{l} + {x}_{l + {2k}}\right| }^{2}. \] | From this formula it follows, in particular, that\n\n1. The curvature is equal to zero only for those two-dimensional planes which consist of parallel flows in the same direction as \( \xi \), so that \( \left\lbrack {\xi ,\eta }\right\rbrack = 0 \) ;\n\n2. The curvature in the plane defined by the flow functions \( \x... | Yes |
Problem 1. Calculate the symplectic structure \( \Omega \) in the affine chart \( w = {z}_{1} : {z}_{0} \) of the projective line \( \mathbb{C}{P}^{1} \) . | Answer. \( \Omega = \left( {1/\pi }\right) \left( {{dx} \land {dy}}\right) /{\left( 1 + {x}^{2} + {y}^{2}\right) }^{2} \), where \( w = x + {iy} \) . The coefficient in the definition of the form \( \Omega \) is chosen to obtain the usual orientation of the complex line \( \left( {{dx} \land {dy}}\right) \) and so that... | Yes |
The set of all planes tangent to the graph of a function \( f = \varphi \left( x\right) \) in an \( \left( {n + 1}\right) \) -dimensional euclidean space with coordinates \( \left( {{x}_{1},\ldots ,{x}_{n};f}\right) \) is a Legendre submanifold of the \( \left( {{2n} + 1}\right) \) -dimensional space of all nonvertical... | The Legendre transformation can be described in these terms in the following way.\n\nConsider a second \( \left( {{2n} + 1}\right) \) -dimensional contact space with coordinates \( \left( {P, X, F}\right) \) and contact structure given by the form\n\n\[ \Omega = {PdX} - {dF} \]\n\nThe Legendre involution is the map tak... | Yes |
Corollary 2. An axially symmetric rigid body fixed at a point on the axis of symmetry, has at least two stationary rotations (for every value of the angular momentum with respect to the axis of symmetry). | Both corollaries follow from the fact that a function on the sphere has at least two critical points. | No |
Theorem 1. The set of ellipsoids of revolution is a finite union of smooth sub-manifolds of codimension 2 and higher in the manifold of all ellipsoids. | Proof. We first consider an ellipsoid in \( n \) -dimensional space which has two equal axes, and whose other axes are distinct. Such an ellipsoid is defined by the directions of the distinct axes, which gives\n\n\[ \left( {n - 1}\right) + \left( {n - 2}\right) + \cdots + 2 = \frac{\left( {n + 1}\right) \left( {n - 2}\... | Yes |
Corollary 1. The form with pole singularity\n\n\\[ \n\\frac{h\\left( {x, y, z}\\right) {dx} \\land {dy} \\land {dz}}{f\\left( {x, y, z}\\right) },\\;h\\left( 0\\right) \\neq 0, \n\\]\n\nwhere \\( f \\) is one of the polynomials \\( A, D, E \\), may be reduced to the form \\( {dx} \\land {dy} \\land {dz}/{fby} \\) a hol... | In exactly the same way for any \\( n \\geq 3 \\), a factor \\( h\\left( {{x}_{1},\\ldots ,{x}_{n}}\\right) \\) which does not vanish at the origin can be converted to unity. | No |
Corollary 5. For \( l \leq 6 \), in generic \( l \) -parameter families of forms \( {dx} \land {dy} \land {dz}/ \) \( f\left( {x, y, z}\right) \), one finds only forms which in the neighborhood of each point are locally equivalent to one of the following 24 types: | \[ \frac{{dx} \land {dy} \land {dz}}{1},\;\frac{{dx} \land {dy} \land {dz}}{x},\;\frac{{dx} \land {dy} \land {dz}}{{x}^{2} + {y}^{2} \pm {z}^{2}},\;\frac{{dx} \land {dy} \land {dz}}{{x}^{3} + {y}^{2} \pm {z}^{2}}, \] \[ \frac{{dx} \land {dy} \land {dz}}{{x}^{4} \pm {y}^{2} \pm {z}^{2}},\;\frac{{dx} \land {dy} \land {dz... | Yes |
Corollary 6. Let \( f \) be a nondegenerate quasi-homogeneous polynomial of weight 1 with argument weights \( {w}_{1},{w}_{2} \). Then the form \[ \frac{h\left( {x, y}\right) {dx} \land {dy}}{f\left( {x, y}\right) }, \; h\left( {0,0}\right) \neq 0, \] where \( h \) is a smooth (holomorphic) function in a neighborhood o... | Correspondingly, bivector fields and Poisson structures may be locally reduced to the form \[ \frac{f\left( {x, y}\right) \left( {{\partial }_{x} \land {\partial }_{y}}\right) }{\pm 1 + \phi \left( {x, y}\right) }, \; \{ x, y\} = \frac{f\left( {x, y}\right) }{\pm 1 + \phi \left( {x, y}\right) }.\] | Yes |
Theorem 1 (Jacobi). Through each point of an n-dimensional euclidean space there pass \( n \) quadrics confocal to a given ellipsoid. Smooth confocal quadrics intersect at right angles. | Proof. Each point other than 0 in our space corresponds to an affine hyperplane in the dual space, consisting of those linear functionals whose value is 1 at the given point. In terms of the dual space, Theorem 1 means that every hyperplane not passing through 0 in an \( n \) -dimensional euclidean space is tangent to ... | Yes |
Theorem 2 (Chasles). Given a family of confocal quadrics in n-dimensional euclidean space, a line in general position is tangent to \( n - 1 \) different quadrics in the family, and the planes tangent to the quadrics at the points of tangency are pairwise orthogonal. | Proof. We project the quadrics in the confocal family along a pencil of parallel lines onto the hyperplane perpendicular to the pencil. Each quadric defines an apparent contour (the set of critical values of the projection of the quadric). For a projection whose direction is in general position, the apparent contour is... | "No" |
(1.9) If \( \left( {X, A}\right) \) is a relative CW-complex and \( C \) is a compact subset of \( X \), then there is a finite subcomplex \( \left( {{X}^{\prime },{A}^{\prime }}\right) \) such that \( C \subset {X}^{\prime } \) . | It follows from (6) of (1.6) that \( C \subset {X}_{n} \) for some \( n \) . Therefore it suffices to prove, by induction on \( n \), that, if \( C \) is any compact subset of \( \left( {{X}_{n}, A}\right) \), then \( C \subset {X}^{\prime } \) for some finite subcomplex \( \left( {{X}^{\prime },{A}^{\prime }}\right) \... | Yes |
We claim: Orientations \( {e}^{n} \) of the cells \( {\mathbf{E}}_{ + }^{n} \) can be found so that\n\n(7.1)\n\n\[ \partial {e}^{n} = \left\{ \begin{array}{ll} \left( {1 - \tau }\right) {e}^{n - 1} & \left( {n\text{ odd }}\right) , \\ \left( {1 + \tau }\right) {e}^{n - 1} & \left( {n\text{ even }}\right) . \end{array}\... | In fact, let \( {e}^{0} \) be the homology class of the point \( {\mathbf{E}}_{ + }^{0} \) ; then \( \partial : {H}_{1}\left( {{\mathbf{E}}_{ + }^{1},{\mathbf{S}}^{0}}\right) \approx \) \( {\widetilde{H}}_{0}\left( {\mathbf{S}}^{0}\right) \), and the homology class of the cycle \( {e}^{0} - \tau {e}^{0} = \left( {1 - \... | Yes |
Recall from \( §2 \), Chapter I that we may regard \( {\mathbf{S}}^{\infty } \) as the unit sphere in complex Euclidean space \( {\mathbf{C}}^{\infty } \), and \( {\mathbf{S}}^{{2n} + 1} \) as the unit sphere in \( {\mathbf{C}}^{n + 1} \) . The operation of scalar multiplication by complex numbers of absolute value 1 d... | Let \( {\mathbf{E}}_{ * }^{2n} \) be the set of all points \( \left( {{z}_{0},\ldots ,{z}_{n}}\right) \in {\mathbf{S}}^{{2n} + 1} \) such that \( {z}_{n} \) is real and non-negative. Then \( {\mathbf{E}}_{ * }^{2n} \) is a cell with boundary \( {\mathbf{S}}^{{2n} - 1} \), and \( p : \left( {{\mathbf{E}}_{ * }^{2n},{\ma... | Yes |
Theorem If \( Y \) is an \( \mathrm{H} \) -space, then \( {\pi }_{1}\left( Y\right) \) operates trivially on \( \left\lbrack {X, Y}\right\rbrack \) for any space \( X \) . | For let \( u : \left( {\mathbf{I},\mathbf{I}}\right) \rightarrow \left( {Y, e}\right) \) and \( f : \left( {X, * }\right) \rightarrow \left( {Y, e}\right) \) be maps. Then the map \( g : \mathbf{I} \times X \rightarrow Y \) defined by\n\n\[ g\left( {t, x}\right) = u\left( t\right) \cdot f\left( x\right) \]\n\nis a free... | Yes |
Lemma 1 Every finitely generated subgroup of \( G \) is conjugate to a subgroup of H. | For if \( F \) is a finite subset of \( G \), there exist integers \( m, n \) such that each \( \sigma \in F \) fixes all integers not belonging to the closed interval \( \left\lbrack {-m, n}\right\rbrack \) . If \( m \leq 0 \) , \( F \subset H \) . Suppose \( m > 0 \) . Then there is a permutation \( \tau \) such that... | Yes |
Let \( f : \Pi \rightarrow G \) be a homomorphism of abelian groups. If \( X \) is a CW-complex, composition with \( f \) is a chain map of the cochain complex \( \operatorname{Hom}\left( {\Gamma \left( X\right) ,\Pi }\right) \) into \( \operatorname{Hom}\left( {\Gamma \left( X\right), G}\right) \), and therefore induc... | and it is clear that \( {f}_{ * } \) is a cohomology operation of type \( \left( {n, n;\Pi, G}\right) \), the coefficient group homomorphism induced by \( f \) . | No |
Example 1 (Coefficient group homomorphisms). Let \( \\Pi, G \) be abelian groups, \( f : \\Pi \\rightarrow G \) a homomorphism. Then the coefficient group homomorphism\n\n\\[ \n{f}_{ * } : {H}^{n}\\left( {\\;;\\Pi }\\right) \\rightarrow {H}^{n}\\left( {\\;;G}\\right) ,\n\\]\n\nas we saw in Example 1 of \( §8 \), Chapte... | Clearly \( {f}_{ * } \) commutes with the coboundary operator of the Mayer-Vietoris sequence of a proper triad, and hence the \( {f}_{ * } \), for each value of \( n \), are the components of a stable operation \( {f}_{ * } \) . | No |
Let \( X \) be an \( \left( {m - 1}\right) \) -connected space \( \left( {m \geq 2}\right) \), and let \( \left\{ {{\Pi }_{n},{X}^{n - 1},{p}_{n}}\right\} \) be a fibred Postnikov system for \( X \) ,(which may as well begin with \( n = m - 1 \) ). Then \( \left\{ {{\Pi }_{n + 1},\Omega {X}^{n - 1},\Omega {p}_{n}}\righ... | \[ {\Delta }_{ * } : {\pi }_{n + 1}\left( X\right) \rightarrow {\pi }_{n}\left( {\Omega X}\right) \] of the homotopy sequence of the fibration \( p : {\mathbf{P}}^{\prime }\left( X\right) \rightarrow X \) . It follows from Exercise 5 and from Exercise 7 of Chapter VI that the Postnikov invariants of \( X \) and of \( {... | No |
Example 4 (Suspensions). Let \( X \) be an \( \left( {m - 1}\right) \) -connected space \( \left( {m \geq 2}\right) \), and let \( \left\{ {{X}^{n},{f}_{n}}\right\} \) be a homotopy resolution of \( X \) . Then \( \left( {{X}^{q}, X}\right) \) is \( \left( {q + 1}\right) \) -connected; it follows from the Relative Hure... | Since suspension is an isomorphism in homology, \( {H}_{r}\left( {\mathbf{S}{X}^{q},\mathbf{S}X}\right) = 0 \) for \( r \leq q + 2 \) . Since \( \mathbf{S}X \) is 1-connected, we can apply the converse of the Relative Hurewicz Theorem to deduce that \( \left( {\mathbf{S}{X}^{q},\mathbf{S}X}\right) \) is \( \left( {q + ... | Yes |
Let \( D \) be one of the standard division algebras, viz. R, C, Q, K, and let \( d \) be the dimension of \( D \) over \( R \) . Let \( f : {\mathbf{S}}^{d - 1} \times {\mathbf{S}}^{d - 1} \rightarrow {\mathbf{S}}^{d - 1} \) be the map given by\n\n\[ f\left( {x, y}\right) = {x}^{-1}y \]\n\nthen \( g : {\mathbf{S}}^{{2... | For we may represent \( {\mathbf{S}}^{{2d} - 1} \) as the set of all pairs \( \left( {x, y}\right) \in D \times D \) such that \( \parallel x{\parallel }^{2} + \parallel y{\parallel }^{2} = 1 \) ; in this representation the sets \( {\mathbf{S}}^{d - 1} \times {\mathbf{E}}^{d} \) and \( {\mathbf{E}}^{d} \times {\mathbf{... | Yes |
Let \( f : B \rightarrow A,\;g : B \rightarrow C \) be homomorphisms. Then \( g \circ {f}^{-1} : A \rightsquigarrow C \), and | \[ \operatorname{Dom}\left( {g \circ {f}^{-1}}\right) = \operatorname{Im}f \] \[ \operatorname{Ker}\left( {g \circ {f}^{-1}}\right) = f\left( {\operatorname{Ker}g}\right) , \] \[ \operatorname{Im}\left( {g \circ {f}^{-1}}\right) = \operatorname{Im}g \] \[ \operatorname{Ind}\left( {g \circ {f}^{-1}}\right) = g\left( {\o... | Yes |
Let \( f : A \rightarrow B,\;g : C \rightarrow B \) be homomorphisms. Then \( {g}^{-1} \circ f : A \rightsquigarrow C \), and | \[ \operatorname{Dom}\left( {{g}^{-1} \circ f}\right) = {f}^{-1}\left( {\operatorname{Im}g}\right) \] \[ \operatorname{Ker}\left( {{g}^{-1} \circ f}\right) = \operatorname{Ker}f \] \[ \operatorname{Im}\left( {{g}^{-1} \circ f}\right) = {g}^{-1}\left( {\operatorname{Im}f}\right) \] \[ \operatorname{Ind}\left( {{g}^{-1} ... | Yes |
Excision Theorem | 57, 580, 598 | No |
2.3.1. Theorem. Every infinite cyclic group is isomorphic to the group \( \mathbf{Z} \), and every cyclic group of finite order is isomorphic to some group \( {\mathbf{Z}}_{n} \) . | Proof. Let \( \langle a\rangle \) be an infinite cyclic group. Define a mapping \( \phi : \mathbf{Z} \rightarrow \langle a\rangle \) by \( {n\phi } = {a}^{n} \) . It is one-to-one: if for \( m > n \) we had \( {m\phi } = {n\phi } \), i.e. \( {a}^{m - n} = e \), then the group \( \langle a\rangle \) would turn out to be... | Yes |
Theorem 2. A graph is bipartite iff it does not contain an odd cycle. | Proof. Suppose \( G \) is bipartite with vertex classes \( {V}_{1} \) and \( {V}_{2} \) . Let \( {x}_{1}{x}_{2}\ldots {x}_{l} \) be a cycle in \( G \) . We may assume that \( {x}_{1} \in {V}_{1} \) . Then \( {x}_{2} \in {V}_{2},{x}_{3} \in {V}_{1} \), and so on: \( {x}_{i} \in {V}_{1} \) iff \( i \) is odd. Since \( {x... | Yes |
Theorem 3. A graph is a forest iff for every pair \( \{ x, y\} \) of distinct vertices it contains at most one \( x - y \) path. | Proof. If \( {x}_{1}{x}_{2}\ldots {x}_{l} \) is a cycle in a graph \( G \) then \( {x}_{1}{x}_{2}\ldots {x}_{l} \) and \( {x}_{1}{x}_{l} \) are two \( {x}_{1} - {x}_{l} \) paths in \( G \) .\n\nConversely, let \( {P}_{1} = {x}_{0}{x}_{1}\ldots {x}_{l} \) and \( {P}_{2} = {x}_{0}{y}_{1}{y}_{2}\ldots {y}_{k}{x}_{l} \) be... | Yes |
Corollary 5. Every connected graph contains a spanning tree, that is a tree containing every vertex of the graph. | Proof. Take a minimal connected spanning subgraph. There are several simple constructions of a spanning tree of a graph \( G \) ; we present two of them. Pick a vertex \( x \) and put \( {V}_{i} = \{ y \in G : d\left( {x, y}\right) = i\}, i = 0,1,\ldots \) Note that if \( {y}_{i} \in {V}_{i}, i > 0 \), and \( x{z}_{1}{... | Yes |
Corollary 7. A tree of order at least 2 contains at least 2 vertices of degree 1. | Proof. Let \( {d}_{1} \leq {d}_{2} \leq \cdots \leq {d}_{n} \) be the degree sequence of a tree \( T \) of order \( n \geq 2 \) . Since \( T \) is connected, \( \delta \left( T\right) = {d}_{1} \geq 1 \) . Hence if \( T \) had at most one vertex of degree 1, by (1) and Corollary 5 we would have\n\n\[ \n{2e}\left( T\rig... | Yes |
Theorem 8. Each of the four methods described above produces an economical spanning tree. If no two edges have the same cost then there is a unique economical spanning tree. | Proof. Choose an economical spanning tree \( T \) of \( G \) that has as many edges in common with \( {T}_{1} \) as possible. ( \( {T}_{1} \) is the spanning tree constructed by the first method.)\n\nSuppose that \( E\left( {T}_{1}\right) \neq E\left( T\right) \) . The edges of \( {T}_{1} \) have been selected one by o... | No |
Theorem 9. For \( n \geq 3 \) the complete graph \( {K}^{n} \) is decomposable into edge disjoint Hamilton cycles iff \( n \) is odd. For \( n \geq 2 \) the complete graph \( {K}^{n} \) is decomposable into edge disjoint Hamilton paths iff \( n \) is even. | The result above shows that if \( n \geq 3 \) is odd, then we can string together \( \frac{1}{2}\left( {n - 1}\right) \) edge disjoint cycles in \( {K}^{n} \) to obtain a circuit containing all the edges of \( {K}^{n} \). In general a circuit in a graph \( G \) containing all the edges is said to be and Euler circuit o... | No |
Theorem 10. A non-trivial connected graph has an Euler circuit iff each vertex has even degree. | Proof. The conditions are clearly necessary. For example, if \( {x}_{1}{x}_{2}\ldots {x}_{m} \) is an Euler circuit in \( G \) and \( x \) occurs \( k \) times in the sequence \( {x}_{1},{x}_{2},\ldots ,{x}_{m} \) , then \( d\left( x\right) = {2k} \) .\n\nWe prove the sufficiency of the first condition by induction on ... | Yes |
Theorem 11. If a connected plane graph \( G \) has \( n \) vertices, \( m \) edges and \( f \) faces, then\n\n\[ n - m + f = 2\text{.} \] | Proof. Let us apply induction on the number of faces. If \( f = 1 \) then \( G \) does not contain a cycle so it is a tree and the result holds.\n\nSuppose now that \( f > 1 \) and the result holds for smaller values of \( f \) . Let \( {ab} \) be an edge in a cycle of \( G \) . Since a cycle separates the plane, the e... | Yes |
Theorem 12. A planar graph of order \( n \geq 3 \) has at most \( {3n} - 6 \) edges. Furthermore, a planar graph of order \( n \) and girth at least \( g,3 \leq g < \infty \), has size at most\n\n\[ \max \left\{ {\frac{g}{g - 2}\left( {n - 2}\right), n - 1}\right\} \text{.} \] | Proof. The first assertion is the case \( g = 3 \) of the second, so it suffices to prove the second assertion. Let \( G \) be a planar graph of order \( n \), size \( m \) and girth at least \( g \) . If \( n \leq g - 1 \) then \( G \) is acyclic so \( m \leq n - 1 \) . Assume now that \( n \geq g \) and the assertion... | Yes |
Theorem 14. Let \( R \) be a commutative ring and let \( {A}_{1},{A}_{2},\ldots ,{A}_{2k} \in {M}_{k}\left( R\right) \) . Then \( \left\lbrack {{A}_{1},{A}_{2},\ldots ,{A}_{2k}}\right\rbrack = 0 \) . | Proof. We shall deduce the result from a lemma about Euler trails in directed multigraphs. Let \( \overrightarrow{G} \) be a directed multigraph of order \( n \) with edges \( {e}_{1},{e}_{2},\ldots \) , \( {e}_{m} \) . Thus each \( {e}_{i} \) is an ordered pair of not necessarily distinct vertices. Every (directed) Eu... | No |
Lemma 15. If \( m \geq {2n} \) then \( \varepsilon \left( {\overrightarrow{G};x, y}\right) = 0 \) . | Proof of Lemma 15. We may clearly assume that \( \overrightarrow{G} \) has no isolated vertices. Let \( {\overrightarrow{G}}^{\prime } \) be obtained from \( \overrightarrow{G} \) by adding to it a vertex \( {x}^{\prime } \), a path of length \( m + 1 - {2n} \) from \( {x}^{\prime } \) to \( x \) and an edge from \( y ... | Yes |
Theorem 1. Given an edge \( {ab} \), denote by \( N\left( {s, a, b, t}\right) \) the number of spanning trees of \( G \) in which the (unique) path from \( s \) to \( t \) contains a and \( b \), in this order. Define \( N\left( {s, b, a, t}\right) \) analogously and write \( N \) for the total number of spanning trees... | Proof. For each spanning tree \( T \) there is exactly one neighbour \( {x}_{T} \) of \( s \) that is on the \( s - t \) path \( {P}_{T} \) contained in \( T \) . Hence \( \mathop{\sum }\limits_{{b \in \Gamma \left( s\right) }}N\left( {s, s, b, t}\right) = N \) , where \( \Gamma \left( s\right) \) is the set of neighbo... | Yes |
Theorem 6. \( B{B}^{t} = D - A \), where \( {B}^{t} \) denotes the transpose of \( B \) and \( D \) is the \( n \times n \) diagonal matrix in which \( {\left( D\right) }_{ii} \) is \( d\left( {v}_{i}\right) \), the degree of the vertex \( {v}_{i} \) in \( G \) . | Proof. What is \( {\left( B{B}^{t}\right) }_{ij} \) ? It is \( \mathop{\sum }\limits_{{l = 1}}^{m}{b}_{il}{b}_{jl} \), which is \( d\left( {v}_{i}\right) \) if \( i = j, - 1 \) if \( {v}_{i}{v}_{j} \) is an edge (if \( {e}_{l} = {v}_{i}{v}_{j} \) is directed from \( {v}_{i} \) to \( {v}_{j} \), then \( {b}_{il}{b}_{jl}... | Yes |
Theorem 7. The electric current \( \mathbf{w} \) satisfying \( \mathbf{p} = D\mathbf{w} + \mathbf{g} \) is given by \( \mathbf{w} = \) \( - C{\left( {C}^{t}DC\right) }^{-1}{C}^{t}g \) . | Proof. Equation (1) implies that \( {B}_{T}{\mathbf{w}}_{T} + {B}_{N}{\mathbf{w}}_{N} = \mathbf{0} \), so \( {\mathbf{w}}_{T} = - {B}_{T}^{-1}{B}_{N}{\mathbf{w}}_{N} \) \( = {C}_{T}{\mathbf{w}}_{N} \) . Hence \( \mathbf{w} = C{\mathbf{w}}_{N} \) . Combining (2) and (3) we find that \( {C}^{t}D\mathbf{w} + \) \( {C}^{t}... | Yes |
Theorem 1. (Max-flow min-cut theorem.) The maximal flow value from s to \( t \) is equal to the minimum of the capacities of cuts separating \( s \) from \( t \) . | Proof. We have remarked already that there is a flow \( f \) with maximal value, say \( v \), and the capacity of every cut is at least \( v \) . Thus, in order to prove the theorem we have to show that there is a cut with capacity \( v \) . We shall, in fact, do considerably more than this: we shall give a very simple... | Yes |
Theorem 3. The maximum of the flow value from a set of sources to a set of sinks is equal to the minimum of the capacity of cuts separating the sources from the sinks. | Let us assume now that we have capacity restrictions on the vertices, except the source and the sink. Thus we are given a function \( c : V - \{ s, t\} \rightarrow \) \( {\mathbb{R}}^{ + } \) and every flow \( f \) from \( s \) to \( t \) has to satisfy the following inequality:\n\n\[ \mathop{\sum }\limits_{{y \in {\Ga... | Yes |
Theorem 7. A bipartite graph \( G \) with vertex sets \( {V}_{1} \) and \( {V}_{2} \) contains a complete matching from \( {V}_{1} \) to \( {V}_{2} \) iff\n\n\[ \left| {\Gamma \left( S\right) }\right| \geq \left| S\right| \;\text{ for every }S \subset {V}_{1}. \] | 1st Proof. Both Menger’s theorem (applied to the sets \( {V}_{1} \) and \( {V}_{2} \), as at the end of \( §2 \) ) and the max-flow min-cut theorem (applied to the directed graph obtained from \( G \) by sending each edge from \( {V}_{1} \) to \( {V}_{2} \), in which each vertex has capacity 1) imply the following. If ... | Yes |
Corollary 8. Suppose the bipartite graph \( G = {G}_{2}\left( {m, n}\right) \) with vertex sets \( {V}_{1},{V}_{2} \) satisfies the following condition:\n\n\[ \left| {\Gamma \left( S\right) }\right| \geq \left| S\right| - d\;\text{ for every }S \subset {V}_{1}. \]\n\nThen \( G \) contains \( m - d \) independent edges. | Proof. Add \( d \) vertices to \( {V}_{2} \) and join them to each vertex in \( {V}_{1} \) . The new graph \( {G}^{ * } \) satisfies the condition for a complete matching. At least \( m - d \) of the edges in this matching belong to \( G \) . | Yes |
Corollary 9. Let \( G \) be a bipartite graph with vertex classes \( {V}_{1} = \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) and \( {V}_{2} = \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) . Then \( G \) contains a subgraph \( H \) such that \( {d}_{H}\left( {x}_{i}\right) = {d}_{i} \) and \( 0 \leq {d}_{H}\left( {y}_{i}\... | Proof. Replace each vertex \( {x}_{i} \) by \( {d}_{i} \) vertices joined to each vertex in \( \Gamma \left( {x}_{i}\right) \) . Then there is a subgraph \( H \) iff the new graph has a matching from the new first vertex class to \( {V}_{2} \) . The result follows from Theorem 7. | Yes |
Theorem 10. Let \( G = {G}_{2}\left( {m, n}\right) \) be a bipartite graph with vertex classes \( {V}_{1} = \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) and \( {V}_{2} = \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) . For \( S \subset {V}_{1} \) and \( 1 \leq j \leq n \) denote by \( {S}_{j} \) the number of edges from ... | Proof. Turn \( G \) into a directed graph \( \overrightarrow{G} \) by sending each edge from \( {V}_{1} \) to \( {V}_{2} \) . Give each edge capacity 1, a vertex \( {x}_{i} \) capacity \( {d}_{i} \) and a vertex \( {y}_{j} \) capacity \( {e}_{j} \) . Then there is a subgraph \( H \) with required properties iff in \( \... | Yes |
Theorem 11. If every antichain in a (finite) partially ordered set \( P \) has at most m elements then \( P \) is the union of \( m \) chains. | Proof. Let us apply induction on \( \\left| P\\right| \) . If \( P = \\varnothing \) there is nothing to prove so we suppose that \( \\left| P\\right| > 0 \) and the theorem holds for sets with fewer elements.\n\nLet \( C \) be a maximal chain in \( P \) . (Thus if \( x \\notin C \) then \( C \\cup \\{ x\\} \) is no lo... | Yes |
Corollary 13. A graph \( G \) contains a set of independent edges covering all but at most \( d \) of the vertices iff\n\n\[ q\left( {G - S}\right) \leq \left| S\right| + d\;\text{ for every }S \subset V\left( G\right) . \] | Proof. Since the number of vertices not covered by a set of independent edges is congruent to \( \left| G\right| \) modulo 2, we may assume that\n\n\[ d \equiv \left| G\right| \;\left( {\;\operatorname{mod}\;2}\right) . \]\n\nPut \( H = G + {K}^{d} \), that is let \( H \) be obtained from \( G \) by adding to it a set ... | Yes |
Theorem 1. For \( g \geq 3 \) and \( \delta \geq 3 \) put\n\n\[ \n{n}_{0}\left( {g,\delta }\right) = \left\{ \begin{array}{ll} 1 + \frac{\delta }{\delta - 2}\left\{ {{\left( \delta - 1\right) }^{\left( {g - 1}\right) /2} - 1}\right\} & \text{ if }g\text{ is odd,} \\ \frac{2}{\delta - 2}\left\{ {{\left( \delta - 1\right... | Proof. Suppose first that \( g \) is odd, say \( g = {2d} + 1, d \geq 1 \) . Pick a vertex \( x \) . There is no vertex \( z \) for which \( G \) contains two distinct \( z - x \) paths of length at most \( d \), since otherwise \( G \) has a cycle of length at most \( {2d} \) . Consequently there are at least \( \delt... | Yes |
Theorem 2. Let \( G \) be a connected graph of order \( n \geq 3 \) such that for any two non-adjacent vertices \( x \) and \( y \) we have\n\n\[ d\left( x\right) + d\left( y\right) \geq k \]\n\nIf \( k = n \) then \( G \) is Hamiltonian and if \( k < n \) then \( G \) contains a path of length \( k \) . and a cycle of... | Proof. Assume that \( G \) is not Hamiltonian and let \( P = {x}_{1}{x}_{2}\ldots {x}_{l} \) be a longest path in \( G \) . The maximality of \( P \) implies that the neighbours of \( {x}_{1} \) and \( {x}_{l} \) are vertices of \( P \) . As \( G \) does not contain a cycle of length \( l,{x}_{1} \) is not adjacent to ... | Yes |
Theorem 3. Let \( G \) be a graph of order \( n \) without a path of length \( k\left( { \geq 1}\right) \) . Then\n\n\[ e\left( G\right) \leq \frac{k - 1}{2}n \]\n\nA graph is an extremal graph (that is equality holds for it) iff all its components are complete graphs of order \( k \) . | Proof. We fix \( k \) and apply induction on \( n \) . The assertion is clearly true if \( n \leq k \) . Assume now that \( n > k \) and the assertion holds for smaller values of \( n \) .\n\nIf \( G \) is disconnected, the induction hypothesis implies the result. Now if \( G \) is connected, it contains no \( {K}^{k} ... | Yes |
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