paper stringlengths 9 16 | proof stringlengths 0 131k |
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math/9903187 | This follows directly from REF . |
math/9903187 | The first assertion is clear because the eigenspaces of MATH are invariant subspaces under the action of MATH. Next we prove the second assertion. Let MATH and MATH be in a same fiber of MATH above MATH, and set MATH and MATH. Then REF holds for MATH, and also for MATH replaced by MATH. There exists MATH in MATH such t... |
math/9903187 | Let MATH be as in REF. Direct verification yields MATH. The lemma follows now from REF (with MATH replaced by MATH). |
math/9903187 | Let MATH be a large integer. For MATH in MATH, we consider the subset MATH of MATH consisting of all points MATH in MATH such that MATH and MATH. Note that MATH, because MATH. Thus MATH with MATH, since MATH is integrable on MATH. By the first assertion of REF and by REF below, for MATH in MATH large enough with respec... |
math/9903187 | Since REF is a direct consequence of assertion REF in the proof of REF , taking MATH, and REF follows from REF , it remains to prove REF . By the first assertion in the statement of REF , MATH is cylindrical at level MATH, taking MATH large enough. In order to prove REF , we may assume that MATH is a locally closed sub... |
math/9903187 | The first statement is a direct consequence of REF with MATH, MATH replaced by MATH, MATH. The second follows then, using the decomposition MATH and the fact that MATH. |
math/9903187 | We will use the functor MATH of CITE which to a variety MATH over MATH associates an object MATH of the homotopy category MATH of bounded complexes of objects in MATH, such that MATH is the NAME characteristic of MATH. Consider the functor MATH which to an object MATH associates the complex in MATH which is zero in non... |
math/9903187 | This is proved in exactly the same way as REF using REF . |
math/9903187 | Straightforward exercise, using REF . |
math/9903187 | We may assume that MATH is irreducible. Set MATH . We may assume there exists a closed subscheme MATH of MATH with MATH such that MATH is contained in MATH, because otherwise MATH and MATH have measure zero. Since MATH is measurable and MATH is contained in cylindrical subsets MATH of MATH with MATH arbitrary small, we... |
math/9903187 | CASE: First assume that MATH. Then, since MATH is cylindrical, we have MATH and MATH is contained in some MATH, with MATH a closed subscheme of MATH, with MATH. This yields REF when MATH. Now suppose that MATH. Take MATH in MATH large enough to insure that MATH. We may assume that MATH is contained in MATH. Now we choo... |
math/9903187 | Reasoning as in the proof of REF , we reduce to the case where MATH is cylindrical and satisfies MATH, with MATH . For this reduction we use the assumption that MATH is strongly measurable to insure that the cylinder MATH in REF is contained in MATH, so that the restriction of MATH to MATH is injective. Next we can red... |
math/9903189 | MATH . Notice that by REF , MATH. MATH since MATH and MATH link, clearly MATH. So MATH is well defined. We will distinguish the two cases: CASE: Suppose that MATH. Let us suppose that MATH. Set MATH. By the classical deformation lemma (see CITE) we have that for any MATH, there exists MATH such that MATH . But by the d... |
math/9903198 | Every NAME algebra that consists of point vector fields preserves the vertical distribution MATH, which is a one-dimensional subdistribution of the contact distribution MATH. Consequently, any reducible NAME algebra of contact vector fields also preserves a one-dimensional subdistribution of MATH. Conversely, let MATH ... |
math/9903198 | Let MATH be an irreducible NAME algebra of contact vector fields. For an arbitrary point MATH, we let MATH and define MATH and MATH. Then MATH is obviously an open subset in MATH. The NAME algebra MATH is transitive at point in general position if and only if MATH. Assume the contrary. Then the subspaces MATH form a co... |
math/9903198 | Fix a basis MATH in the space MATH. Then the action of the elements of MATH on MATH is given by the following matrices: MATH . Therefore, the NAME algebra MATH may be identified with MATH, and the MATH - module MATH with the natural MATH - module. Any irreducible subalgebra of MATH is conjugate to one and only one of t... |
math/9903198 | Suppose MATH for some MATH. For every MATH, consider the subspace MATH . It is easy to show that, MATH for all MATH. Thus, the subspace MATH may be identified with MATH, and since MATH, this identification is in agreement with the structure of the NAME algebras MATH and MATH. Hence, we have found an isomorphism of the ... |
quant-ph/9903014 | Let MATH. Since MATH is a normal matrix, MATH can be written as MATH where MATH is a unitary matrix and MATH is the diagonal matrix of eigenvalues with the MATH-th eigenvalue having the form MATH CITE. If all eigenvalues in MATH are rotations through rational fractions of MATH, that is, MATH is rational, then let MATH ... |
quant-ph/9903014 | The `if' direction follows from the fact that the transition function for a GFA is also a valid transition function for an MO-QFA that can accept the same language with certainty. For the `only if' direction, by contradiction, assume that there exists a language MATH that can be accepted by an MO-QFA with bounded error... |
quant-ph/9903014 | Let MATH where MATH, MATH, MATH, and MATH is defined by the transition matrices MATH where MATH is an irrational fraction of MATH. Since MATH is a rotation matrix and MATH is an irrational fraction of MATH, the orbit formed by applying MATH to MATH is dense in the circle, and there exists only one MATH, such that MATH,... |
quant-ph/9903014 | Construct a free group of rotation matrices drawn from the group MATH as discussed by CITE. Let MATH be a MATH-state MO-QFA where MATH such that MATH is equal to the sum of the number of rotation matrices and their inverses, MATH is defined by the rotation matrices and their inverses, and MATH. The MO-QFA will accept i... |
quant-ph/9903014 | The second result follows from REF because every GFA is also a PFA. Since we can bilinearize MATH, MATH is a generalized cut-point event (GCE)CITE. Since the class of GCEs is equal to the class of probabilistic cut-point events (PCEs)CITE, which are accepted by PFAs, there exists a PFA that can accept MATH with some cu... |
quant-ph/9903014 | Closure under complement follows from the fact that we can exchange the accept and reject states of the MM-QFA. This exchanges the probabilities of acceptance and rejection but does not affect the margin. Given an MM-QFA MATH and a homomorphism MATH we construct an MM-QFA MATH that accepts MATH. Let MATH and MATH. Assu... |
quant-ph/9903014 | Let MATH and define a homomorphism MATH to be MATH, MATH, and MATH. Since MATH can be accepted by an RFA, MATH CITE, but MATH, the result follows. |
quant-ph/9903014 | By contradiction, assume that MATH. Let MATH. Since the minimal DFA for MATH does not satisfy the partial order condition there exist states MATH and strings MATH as defined above and a distinguishing string MATH such that MATH if and only if MATH. Without loss of generality assume that MATH and MATH. Let MATH be the s... |
quant-ph/9903014 | Let MATH be a DFA and MATH be the corresponding minimal DFA. Assume by contradiction that MATH does not satisfy the partial order condition. Hence, MATH has two states that correspond to the equivalence classes MATH and MATH such that MATH and MATH. By the NAME theorem CITE, the equivalence classes partition the set of... |
quant-ph/9903014 | Let MATH be the minimal DFA accepting the language MATH and let MATH be the minimal DFA accepting the language MATH. We first construct an automaton MATH that accepts MATH by combining MATH and MATH using a direct product. Define MATH where MATH, MATH, MATH and MATH. We argue that if MATH and MATH satisfy the partial o... |
quant-ph/9903014 | By REF the intersection of two languages that satisfy the partial order condition is a language that satisfies the partial order condition. |
quant-ph/9903014 | We construct an MM-QFA MATH with MATH states that accepts MATH where MATH and MATH. For each link in the trigger chain we require a junk state and a non-halting state. We order the states to correspond with the description of the MATH matrices. Specifically, the first MATH states are the non-halting states, interleaved... |
quant-ph/9903014 | Let MATH and MATH be end-decisive MM-QFAs that accept MATH and MATH. Using these two MM-QFAs we construct an MM-QFA MATH that satisfies the above inequalities. Let MATH and MATH. The sets of halting states are defined as MATH and the transition function MATH is defined as MATH which is a tensor product of the transitio... |
quant-ph/9903014 | Let MATH, MATH, MATH, and MATH be the respective cut-points and margins of MM-QFAs MATH and MATH. Since MATH, MATH, and MATH, the result follows from REF . |
quant-ph/9903014 | Using REF to compose MATH copies of MATH yields the result. |
quant-ph/9903014 | Assume that MATH accepts words in MATH with probability at least MATH and accepts words not in MATH with probability at most MATH. Similarly, assume that MATH accepts words in MATH with probability at least MATH and accepts words not in MATH with probability at most MATH. Using REF let MATH be the MATH-th tensor power ... |
quant-ph/9903014 | Since MATH, the same argument as in REF applies. |
quant-ph/9903014 | Let MATH be an end-decisive MM-QFA that accepts MATH with bounded positive one-sided error. Since MATH rejects all strings not in MATH with certainty, for every computation of MATH on MATH zero amplitude is placed into the accepting states of MATH. Let MATH, let MATH and assume that MATH. We use MATH to construct an en... |
quant-ph/9903014 | Let MM-QFA MATH accept MATH with cut-point MATH, margin MATH, and maximum margin MATH, and let MM-QFA MATH accept MATH with cut-point MATH, margin MATH, and maximum margin MATH. First, consider the inequalities in REF that occur when we compose the MM-QFAs MATH and MATH using the tensor technique. Since MM-QFA MATH acc... |
quant-ph/9903014 | Let MATH be a piecewise testable set. We first rewrite it in canonical form: MATH . By REF we can construct end-decisive MM-QFAs that accept partial piecewise testable sets, MATH with bounded positive one-sided error. Using these constructions and REF , we can construct end-decisive MM-QFAs that accept languages MATH a... |
quant-ph/9903014 | Let MATH be an MO-QFA with left and right end-markers, effectively allowing MATH to start in any possible configuration. Define MATH from MATH. Let MATH be defined in terms of the transition matrices MATH. We define MATH from MATH in the following way: for every MATH let MATH and let MATH . Now consider what happens wh... |
quant-ph/9903014 | Let MATH be an MM-QFA that uses two end-markers and accepts MATH. Assume without loss of generality that MATH which facilitates a simpler description of MATH. We construct MATH that accepts MATH with only the right end-marker. Let MATH, MATH and MATH. Assume that MATH is defined in terms of transition matrices MATH. Th... |
quant-ph/9903035 | Both items are proved in a similar way which has two parts. The first part shows that computing a function in MATH can be reduced to computing the parity of MATH other queries to MATH. The second part then proceeds by showing that using binary search one can compute the parity of MATH-queries with MATH adaptive queries... |
quant-ph/9903035 | For simplicity we only describe what is happening to the states that get effected by the oracle query. Construct the following initial state: MATH . Next, make the only query to MATH depending on the value of the first bit. Note that MATH will thus be queried in superposition for both MATH and MATH. Applying MATH estab... |
quant-ph/9903035 | Without loss of generality we will assume that the queries are made to MATH, and that the predicate that is computable with MATH queries to MATH is MATH. Let MATH, MATH,, MATH be the queries that the computation of MATH makes. We will use the proof technique of REF (also called mind-change technique) which enables us t... |
quant-ph/9903035 | The proof is by induction on MATH. For MATH we have back the situation of REF . Let the predicate MATH be computable with MATH non-adaptive queries to MATH. As in the proof of REF we reduce the MATH queries MATH that MATH makes, to the calculation of the parity-bit MATH. Next, we construct MATH new formulae MATH accord... |
quant-ph/9903035 | Fix MATH, the input MATH of length MATH and let MATH be the function in MATH. Suppose that MATH with MATH for some MATH depending on MATH. The goal is to obtain the following state: MATH . Since with this state one application of MATH will give us MATH (see REF ). Similar to REF we can obtain this state if we had acces... |
quant-ph/9903035 | Let MATH be the following set: MATH. Using the the same approach as the proof of REF it is not hard to see that MATH can be computed relative MATH with only a single query. |
quant-ph/9903035 | Let MATH be the function we want to compute relative to MATH. Without loss of generality we assume that MATH for some MATH depending on MATH. As before we construct the following set: MATH . As in REF it follows that MATH is computable with one quantum query to MATH. Since MATH is closed under MATH-reductions and MATH,... |
chao-dyn/9904005 | The existence of the manifold follows quite directly from the Center Manifold Theorem CITE. We would now like to compute MATH. First, we introduce the matrix MATH satisfying MATH, which exists because MATH and MATH have no eigenvalues in common CITE. In fact, it is given by MATH . The change of variables MATH yields MA... |
chao-dyn/9904005 | The normal form of REF-jet of REF can be written as MATH with MATH and MATH. The assertion has been proved in CITE, see also CITE. In fact, the origin is an unstable NAME - NAME point if MATH or if MATH. If MATH, it is asymptotically stable if MATH and MATH are both negative, and unstable if one of them is positive. |
chao-dyn/9904005 | Let us consider MATH . The first two conditions of REF are satisfied if we choose MATH and MATH. To satisfy the third one, we have to choose MATH in such a way that MATH, which is possible under our assumptions. The last condition looks more difficult to satisfy, but in fact, we have MATH, where the constant depends on... |
chao-dyn/9904005 | REF is obtained by eliminating nonresonant terms and rescaling space and time. REF comes from the linear transformation MATH followed by a scaling. REF is obtained with the transformation MATH. |
chao-dyn/9904005 | REF can also be written as MATH . The transformation MATH yields a system whose linearization at the point MATH is the matrix MATH . The center manifold theorem implies the existence of a local invariant manifold MATH. Moreover, it is shown in REF , p. REF, that in some neighbourhood of the origin, any solution satisfi... |
chao-dyn/9904005 | Use integration by parts once. |
chao-dyn/9904005 | Notice that for small enough MATH, we have MATH. CASE: Simplification of the linear part. Consider the initial value problem MATH . Since MATH, its solution satisfies MATH. (In fact, one can prove that MATH). The transformation MATH yields MATH where MATH satisfies MATH, and MATH and MATH satisfy similar bounds as MATH... |
chao-dyn/9904005 | After subtracting MATH from REF and eliminating the term linear in MATH in the same way as in REF , we obtain MATH where MATH . Let MATH be a primitive of MATH. The solution of REF with initial condition MATH should satisfy MATH . Using the a priori estimate of REF and the bounds REF, we can apply REF to estimate the i... |
chao-dyn/9904005 | One can use a standard averaging result, see for instance REF . page REF. |
chao-dyn/9904005 | To simplify the notation, we consider the case MATH. CASE: Transformation of the equation. Let MATH be the positive solution of MATH. Then we have MATH . We decrease the order of the drift term MATH by the transformation MATH which implies that MATH, MATH and yields the system MATH where MATH . There is a time MATH suc... |
chao-dyn/9904005 | The proof is similar to the proof of REF , so we only outline the differences. M. enumREF CASE: We have to eliminate quadratic terms from the equation as well. In order to get a normal form similar to REF, we start by eliminating quadratic terms, then we remove the term linear in MATH, and then only the nonresonant cub... |
chao-dyn/9904005 | CASE: Hamiltonian system. Consider, as a special case of REF, the Hamiltonian system MATH . REF shows the existence of a particular solution MATH, MATH. If MATH, the dynamics of MATH is governed by a Hamiltonian of the form MATH where MATH and MATH are bounded functions. Using REF, one can show that for sufficiently sm... |
cs/9904019 | Set MATH. Consider the following algorithm: CASE: Apply exact search for MATH, each of which takes MATH queries. CASE: If no solution has been found, then conduct MATH searches, each with MATH queries. CASE: Output a solution if one has been found, otherwise output `no'. The query complexity of this algorithm is bounde... |
cs/9904019 | From REF we obtain the upper bound MATH . To prove a lower bound on MATH we distinguish two cases. CASE: MATH. By REF , we can achieve error MATH using MATH queries. Now (leaving out some constant factors): MATH . CASE: MATH. We can achieve error MATH using MATH queries, and then classically amplify this to error MATH ... |
cs/9904019 | Let MATH be the sensitivity of MATH: the maximum, over all MATH, of the number of variables that we can individually flip in MATH to change MATH. Let MATH be an input on which the sensitivity of MATH equals MATH. Assume without loss of generality that MATH. All sensitive variables must be REF in MATH, and setting one o... |
cs/9904019 | Run the algorithms MATH and MATH of REF side-by-side until one of them terminates with a certificate. This gives a certificate-finding quantum algorithm for MATH with expected number of queries MATH. Run this algorithm for twice its expected number of queries and answer `don't know' if it hasn't terminated after that t... |
cs/9904019 | A similar analysis as before shows MATH and MATH. For the quantum lower bound: note that if we set all variables to REF except for the MATH variables in the first subtree, then MATH becomes the OR of MATH variables. This is known to have zero-error complexity exactly MATH CITE, hence MATH. |
cs/9904019 | The quantum upper bound follows from REF and the CITE reduction. For the classical lower bound, suppose we have a classical zero-error protocol MATH for MATH with MATH bits of communication. We will show how we can use this to solve the Disjointness problem on MATH variables. (Given NAME 's input MATH and NAME 's MATH,... |
cs/9904019 | For the lower bound, let MATH denote the degree of the unique multilinear multivariate polynomial MATH that represents a function MATH (that is, MATH for all MATH). CITE proves that MATH for every MATH. CITE prove that MATH for all monotone graph properties MATH. Combining these two facts gives the lower bound. Let MAT... |
cs/9904019 | The quantum lower bound follows from MATH REF and MATH. Consider the property ``the graph contains a star", where a star is a node that has edges to all other nodes. This property corresponds to a REF-level tree, where the first level is an OR of MATH subtrees, and each subtree is an AND of MATH variables. The MATH var... |
cs/9904019 | For MATH: MATH. |
cs/9904019 | A polynomial MATH with MATH and MATH for all integers MATH must have degree MATH. Since MATH for our MATH, we have MATH. Now MATH has degree MATH and MATH for all integers MATH . Applying REF to MATH (which is bounded by REF at integer points) with MATH we obtain: MATH . Now we rescale MATH to MATH (that is, the domain... |
cs/9904019 | By induction on MATH. Base step. For MATH the bounds are trivial. Induction step (assume the lemma for MATH). Let MATH be the uniform MATH-level AND-OR tree on MATH variables. The root is an OR of MATH subtrees, each of which has MATH variables. We construct MATH as follows. First use multi-level NAME as in CITE to fin... |
hep-th/9904002 | Consider any weight vector MATH whose entries are only MATH's and MATH's. The scalar product of such a vector with a simple root MATH is positive only if the vector has a MATH at the MATH-th position and a MATH at the MATH-st position. Then and only then will MATH also be a weight, according to REF . The effect of subt... |
hep-th/9904062 | This is an immediate consequence of the fact that REF is an affine bundle modelled over the pull-back vector bundle MATH. |
hep-th/9904062 | Given an arbitrary connection MATH on the fiber bundle MATH, the corresponding Hamiltonian map REF defines the form MATH which is exactly the Hamiltonian form REF . Since MATH is a MATH-valued basic REF-form on MATH, MATH is a horizontal density on MATH. Then the result follows from REF . Note that MATH iff REF . |
hep-th/9904062 | If MATH is closed, there is a contractible neighbourhood MATH of a point MATH which belongs to a holonomic coordinate chart MATH and where the local form MATH is exact. We have MATH on MATH. It is readily observed that the second term in the right-hand side of this equality is also an exact form on MATH. By virtue of t... |
hep-th/9904062 | Given a Hamiltonian form MATH, let us consider the first order differential operator MATH on MATH where MATH is the canonical monomorphism REF . It is called the NAME operator associated with MATH. A glance at REF shows that this operator is an affine morphism over MATH of constant rank. It follows that its kernel is a... |
hep-th/9904062 | The proof is based on REF . |
hep-th/9904062 | The relation REF takes the coordinate form MATH . Substituting REF , we obtain the relation REF . |
hep-th/9904062 | Let MATH be a vertical vector field on the affine jet bundle MATH which takes its values into the kernel of the tangent map MATH to MATH. Then MATH. |
hep-th/9904062 | Given a vector MATH, the value MATH is the same for all Hamiltonian maps MATH satisfying the relation REF . Then the results follow from the relation REF . |
hep-th/9904062 | Acting by the exterior differential on the relation REF , we obtain the relation MATH which is equivalent to the system of equalities MATH . Using these equalities and REF , one can easily see that MATH where MATH is the NAME - NAME - NAME operator REF . Let MATH be a section of MATH which lives in the Lagrangian const... |
hep-th/9904062 | The NAME REF hold by virtue of REF . Substituting MATH in the NAME REF and using the relations REF , we come to the NAME REF for MATH as follows: MATH . |
hep-th/9904062 | Such a Hamiltonian form MATH defines the global section MATH of the fibred manifold REF . Due to the relation REF , MATH and the constrained NAME equations can be written as MATH . Note that they differ from the NAME REF restricted to MATH which read MATH where MATH is a section of MATH and MATH is an arbitrary vertica... |
hep-th/9904062 | In accordance with the relation REF , the projection REF yields a surjection of MATH onto MATH. Given a section MATH of the fibred manifold REF , we have the morphism MATH . By virtue of REF , this is a surjection such that MATH . Hence, MATH is a bundle isomorphism over MATH which is independent of the choice of a glo... |
hep-th/9904062 | The first of the relations REF is an immediate consequence of the relation REF . The latter follows from REF and the relation REF if we put MATH for some Hamiltonian form MATH associated with the almost regular Lagrangian MATH. |
hep-th/9904062 | Solutions MATH of the pointwise linear algebraic REF form an affine space modelled over the linear space of solutions of the linear algebraic equations MATH. At each point of MATH, these spaces are obviously connected. |
hep-th/9904062 | The map REF is a solution of the algebraic equations MATH . After pointwise diagonalization, the matrix MATH has some non-vanishing components MATH, MATH. Then a solution of REF takes the form MATH while the remaining components MATH, MATH, are arbitrary. In particular, there is a solution with MATH . It satisfies the ... |
hep-th/9904062 | The proof follows from a direct computation by means of the relations REF . |
hep-th/9904062 | By the very definitions of MATH and MATH, the Hamiltonian map REF satisfies REF . A direct computation shows that MATH. Then the relation REF also holds and, if MATH is a connection REF , the Hamiltonian form REF is associated with the Lagrangian REF . Let us write the corresponding NAME REF for a section MATH of the N... |
hep-th/9904062 | Due to the splitting REF , we have the corresponding splitting of the vertical tangent bundle MATH of the NAME bundle MATH. In particular, any vertical vector field MATH on MATH admits the decomposition MATH such that MATH is a vertical vector field on the Lagrangian constraint space MATH. Let us consider the equations... |
hep-th/9904062 | CASE: Let us consider the affine difference MATH over MATH. We have MATH iff MATH. CASE: In the proof of REF , we have shown that, given MATH, there exists a connection REF which fulfills the relation REF . Let us consider the affine difference MATH over MATH. This is a local section of the vector bundle MATH over MATH... |
hep-th/9904062 | Similar to the well-known isomorphism between the fiber bundles MATH and MATH,MATH the isomorphism MATH can be established by inspection of the transformation laws of the holonomic coordinates MATH on MATH and MATH on MATH. |
hep-th/9904062 | The proof follows from a direct computation. We have MATH . Components of this connection obey the NAME REF and the equations MATH . |
hep-th/9904136 | We show how to derive the first relation. Inserting the definition we find: MATH where we used the relation MATH on the last line. The second equation follows from a similar calculation. |
math-ph/9904013 | The function MATH is positive, for MATH odd. |
math-ph/9904013 | The idea is to write MATH as the sum of a function MATH that is positive and a function MATH that absorbs the asymptotic behavior at infinity. Since MATH for MATH small, with MATH there exists MATH such that MATH for all MATH . Let MATH to be chosen below, and let MATH be the NAME step function, that is, MATH for MATH ... |
math-ph/9904013 | In terms of the functions REF - REF we get that, for MATH and therefore MATH, where MATH and where MATH and MATH are as defined above. Since MATH it remains to be shown that MATH . Using the differential equations for MATH and MATH we find that MATH where MATH as defined above, where MATH and where MATH . The functions... |
math-ph/9904013 | From REF it follows that either MATH or MATH . In the first case we get using REF that MATH and therefore MATH and in the second case we get using REF that MATH and therefore MATH . |
math-ph/9904013 | REF follow by using the triangle inequality and the asymptotic properties of the functions MATH, MATH, MATH and MATH. |
math-ph/9904013 | The case MATH follows from REF . Let now MATH . Since MATH for all MATH we find that MATH . Furthermore, since MATH near MATH . Since the function MATH is bounded, it follows that MATH . MATH with MATH . For MATH we improve this bound using additional properties of the function MATH . Namely, since MATH we have that MA... |
math-ph/9904013 | Since MATH the solution of the integral equation will be dominated by MATH and, as we will see, MATH has been chosen such that MATH . The idea is therefore to show that, if MATH is large enough to make the nonlinear part of the map MATH small, and if MATH, then the map MATH contracts the ball MATH into itself, which by... |
math-ph/9904013 | We first prove that MATH is unique. Given a function MATH from MATH to MATH we define the function MATH. Assume that there are two values MATH such that the functions MATH and MATH are positive and satisfy MATH . We first show that the function MATH is positive for all MATH . Namely, if we assume the contrary, then bec... |
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