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cs/0005010 | Note that MATH is anti-monotonic in its first argument, that is, MATH implies MATH, and monotonic in its second argument. Fix a program MATH, a stable model MATH of MATH, and a set of literals MATH such that MATH agrees with MATH. Define MATH and MATH . Let MATH be the least fixed point of MATH. Since MATH agrees with ... |
cs/0005010 | Assume that MATH covers MATH and that MATH. Then, MATH and MATH for every MATH, since MATH. Thus, MATH is the least fixed point of MATH from which we infer, by REF , that MATH is a stable model of MATH. |
cs/0005010 | Note that for every atom MATH, the list MATH is traversed at most twice and the lists MATH and MATH are traversed at most once. For every rule MATH, the list MATH is traversed at most twice. To be precise, the list MATH is only traversed in the procedure MATH when MATH or MATH is being set and this can only happen once... |
cs/0005010 | Note that for every atom MATH, the list MATH is traversed at most twice and the lists MATH is traversed at most once. |
cs/0005010 | Let MATH be the set of atoms that appear as not-atoms in some cycles or that appear in the heads of some choice rules of MATH. Define MATH and MATH . If MATH, then MATH does not appear in the head of a choice rule nor does it appear as a not-atom in MATH . Hence, MATH for any MATH. Now, for any MATH and for any MATH, M... |
cs/0005010 | Let MATH. Assume that MATH. Then, there exists an atom MATH and a rule MATH that is not a choice rule such that MATH . Since otherwise MATH, MATH, and consequently MATH by REF . As the atoms in MATH can not appear as not-atoms in the body of MATH, MATH for MATH and MATH. Hence, MATH since MATH is monotonic in its secon... |
cs/0005010 | We show that completeness is not lost when backjumping is introduced. Since backtracking is chronological, and therefore exhaustive, for a depth of recursion smaller than MATH, we only have to consider the case MATH. Assume that there is no stable model agreeing with MATH and that when MATH returns false, MATH returns ... |
cs/0005010 | Given MATH we can check whether MATH using the oracle MATH a linear number of time. Namely, the language MATH is decided by the function CASE: MATH each atom MATH in MATH in order, most significant first there exists a stable model of MATH that agrees with MATH the least significant atom is in MATH return true return f... |
cs/0005010 | We begin by showing that the problem is in MATH. Assume without loss of generality that all weights are positive. If MATH is the minimize statement MATH then define MATH . Furthermore, if MATH is a set of atoms, then define MATH . Since the oracle MATH is in NP, the procedure MATH in REF shows that the problem of findi... |
cs/0005010 | Let MATH. Then, MATH . Consequently, MATH . In addition, MATH implies MATH . As MATH implies MATH (we can assume MATH), MATH . It follows that MATH . Hence, MATH and therefore MATH . Thus, MATH . Let MATH. Then, MATH . Hence, MATH is a fixed point of MATH and MATH . Now, MATH as MATH and we have proved that the well-fo... |
cs/0005010 | Consider the program MATH and assume that MATH is a modular mapping. Then, MATH is unsatisfiable as MATH has no stable models. It follows that also MATH is unsatisfiable. But this implies that MATH has no stable models, which is clearly not the case. Hence, MATH is not modular. |
cs/0005010 | Define MATH . Then, MATH implies MATH, which in turn implies MATH by the monotonicity of MATH. Hence, MATH, and consequently MATH . Now, MATH implies MATH, which by the definition of MATH implies MATH. Thus, MATH. Moreover, for any fixed point MATH, MATH and hence MATH by definition. |
cs/0005012 | CASE: Let MATH stem from MATH. This implies MATH and MATH. For each MATH and MATH, MATH, this implies MATH and hence MATH and MATH are isomorphic. CASE: REF hold by construction. Obviously, MATH stems from MATH and from REF it follows that each interpretation MATH stemming from MATH is isomorphic to MATH, hence REF hol... |
cs/0005012 | For the only if-direction let MATH be an interpretation with MATH and MATH. From REF it follows that the canonical witness MATH is a witness for MATH that is admissible with respect to MATH. For the if-direction let MATH be an witness for MATH that is admissible with respect to MATH. This implies that there is an inter... |
cs/0005012 | MATH is a witness, hence there is an interpretation MATH stemming from MATH. From REF and the fact that MATH satisfies the properties stated in REF it follows that, for each MATH, MATH . Hence, MATH and MATH is admissible with respect to MATH. |
cs/0005012 | The if-direction follows from the definition of ``correct absorption". For the only if-direction, let MATH be a concept and MATH a witness for MATH that is admissible with respect to MATH. This implies the existence of an interpretation MATH stemming from MATH such that MATH and MATH. Since MATH we have MATH and hence ... |
cs/0005012 | Trivially, MATH holds. Given an unfolded witness MATH, we have to show that there is an interpretation MATH stemming from MATH with MATH. We fix an arbitrary linearisation MATH of the ``uses" partial order on the atomic concept names appearing on the left-hand sides of axioms in MATH such that, if MATH uses MATH, then ... |
cs/0005012 | In both cases, MATH holds trivially. CASE: Let MATH be a concept and MATH be an unfolded witness for MATH with respect to the absorption MATH. This implies that MATH is unfolded with respect to the (smaller) absorption MATH. Since MATH is a correct absorption, there is an interpretation MATH stemming from MATH with MAT... |
cs/0005018 | By the fact that MATH is MATH-closed and REF , it is sufficient to prove that for all positive literals MATH, all NAME of MATH are finite. Let us consider an atom MATH. If MATH is defined in MATH, then the thesis trivially holds by hypothesis. If MATH is defined in MATH, MATH is bounded with respect to MATH by hypothes... |
cs/0005018 | Let MATH be a query strongly bounded with respect to MATH and MATH. We prove the theorem by induction on MATH. Base. Let MATH. This case follows immediately by REF , where MATH, MATH is empty and MATH is the class of strongly bounded queries with respect to MATH and MATH, and the fact that a strongly bounded atom is al... |
cs/0005018 | It is sufficient to extend the proof in CITE by showing that if a query MATH is well-moded and MATH is ground then both MATH and MATH are well-moded. This follows immediately by definition of well-modedness. If MATH is non-ground then the query above has no descendant. |
cs/0005018 | Similarly to the case of well-moded programs, to extend the result to general programs it is sufficient to show that if a query MATH is well-typed then both MATH and MATH are well-typed. In fact, by REF , MATH is well-typed and by REF , if the first literal in a well-typed query is negative, then it is not used to dedu... |
cs/0005018 | Let MATH be the class of well-moded queries of MATH. By REF , MATH is MATH-closed. Moreover CASE: MATH is acceptable with respect to a moded level mapping MATH and MATH, by hypothesis; CASE: for all well-moded queries MATH, all NAME of MATH are finite, by hypothesis; CASE: for all well-moded atoms MATH, if MATH is defi... |
cs/0005018 | Let MATH be the class of well-typed queries of MATH. By REF , MATH is MATH-closed. Moreover CASE: MATH is acceptable with respect to a level mapping MATH and MATH, by hypothesis; CASE: for all well-typed queries MATH, all NAME of MATH are finite, by hypothesis; CASE: for all well-typed atoms MATH, if MATH is defined in... |
cs/0005018 | It is a consequence of REF and (the proof of) REF. |
cs/0005018 | Immediate by the definitions of semi-acceptability and strongly boundedness, since we are considering a finest decomposition. |
cs/0005018 | Since we are considering a finest decomposition of MATH, by REF , MATH is acceptable with respect to MATH, while MATH is acceptable with respect to MATH such that if MATH is defined in MATH then MATH else MATH. By REF all NAME of MATH are bounded with respect to MATH and MATH. By definition of boundedness, for all NAME... |
cs/0005026 | Suppose we have a MATH-bit length message and a perfect (non predictable) random encryption key with the same length. It is easy to see that we can establish a bijective relation between the bits of the message and the bits of the key. Let us suppose that the MATH-th bit of the encryption key could change the state of ... |
cs/0005026 | We have a MATH-bit length ciphertext MATH obtained using an exclusive-or bitwise operator as described in subsection REFEF (here MATH where MATH denotes the MATH-th bit of the message and MATH labels the MATH-th bit of the encryption key). It is possible to hide an arbitrary message MATH if we define the key as: MATH w... |
cs/0005026 | Suppose we have a ciphertext MATH provided with the message as digital signature. These ciphertext hides a codeword (a random bitmap) MATH and a message field digest MATH that depends on the codeword itself. It is easy to see that we have MATH possible codewords and a message field digest for each codeword field. Apply... |
gr-qc/0005031 | Let MATH. REF gives that MATH is an open neighbourhood of MATH in MATH. Hence, by the definition of MATH, there exists a smooth timelike curve MATH of MATH having a future endpoint at MATH in MATH. One has MATH by REF . Since MATH is a smooth timelike curve of MATH with a future endpoint at MATH one therefore has MATH ... |
gr-qc/0005031 | The time reverse of REF gives that MATH cannot be contained entirely in MATH. Let MATH and let MATH. One then has MATH. Let MATH. The set MATH is compact in MATH by REF and MATH is compact in MATH. So by REF the set MATH is a slice of MATH. The set MATH is contained in MATH which does not intersect MATH. Hence MATH is ... |
gr-qc/0005031 | One may assume MATH otherwise one may redefine MATH as MATH. The Lemma then gives that there exists a slice MATH of MATH such that MATH. |
gr-qc/0005031 | It suffices to assume MATH since one may otherwise redefine MATH as MATH. The set MATH is open in MATH and MATH is a slice of MATH. By the time reverse of REF one has that MATH is compact in MATH and such that MATH. The time reverse of REF gives MATH whereby one has MATH. Hence one has MATH. Let MATH be the compact set... |
gr-qc/0005031 | Let MATH be the associated ASE space-time and MATH an ASE asymptote of MATH. If MATH was a timelike curve of MATH from MATH to MATH then, because MATH is a null hypersurface of MATH, one would have MATH so there would exist a timelike curve MATH of MATH such that MATH. But then MATH would be a timelike curve of MATH fr... |
gr-qc/0005031 | It suffices to assume MATH since one may otherwise redefine MATH as MATH. Let MATH be a slice of MATH and let MATH be a slice of MATH lying strictly to the past of MATH along the null geodesic generators of MATH. Let MATH and MATH. By REF there exists a slice MATH of MATH such that MATH and MATH for MATH. REF gives MAT... |
gr-qc/0005031 | One may, by passing to a subset of MATH if necessary, assume MATH. Suppose, for the purpose of obtaining a contradiction, that MATH is non-empty. Then MATH is non-empty and so is MATH. By REF there exists a slice MATH of MATH such that MATH. Since MATH is a non-empty proper subset of MATH it follows that MATH is a non-... |
gr-qc/0005050 | CASE: Suppose MATH is a photon surface. Let MATH and let MATH be null. There exists an affine null geodesic MATH of MATH such that MATH. One has MATH along MATH. At MATH this gives MATH. CASE: Let MATH. By REF one has MATH null MATH. Let MATH be an orthonormal basis for MATH with MATH timelike and MATH, MATH spacelike.... |
gr-qc/0005050 | Let MATH be the induced Lorentzian MATH-metric on MATH and, for each MATH, let MATH be the induced Riemannian MATH-metric on MATH. The expansion of MATH in MATH is given by MATH where the covariant derivative is that of MATH. Since MATH is both shear-free and vorticity-free in MATH, the NAME equation for MATH in MATH a... |
gr-qc/0005050 | By spherical symmetry, and since MATH is unit timelike, the vector field MATH must be proportional to MATH. Hence it suffices to show that MATH is a photon surface iff along MATH one has MATH or equivalently MATH . Construct for MATH a local orthonormal basis field of the form MATH. With respect to this basis field the... |
gr-qc/0005050 | Note that the unit future-directed timelike tangent field MATH along MATH in REF is proportional to the restriction to MATH of the Killing field MATH. Suppose first that there exists a MATH-invariant MATH-sphere MATH such that REF holds. The quantities MATH, MATH and MATH remain constant as MATH is mapped along the flo... |
gr-qc/0005050 | Since MATH is both spherically symmetric and static, the surface MATH is of the form MATH . The unit spacelike normal to MATH is therefore given by MATH for MATH . The second fundamental form of MATH is given by MATH . The vector fields MATH form an orthonormal frame field along MATH, with MATH and MATH unit spacelike ... |
gr-qc/0005050 | Fix MATH and let MATH be the outward future-directed null normal field along each MATH, MATH, normalized such that MATH, where MATH is the outward radial unit tangent to MATH . Since MATH is parallelly propagated along each of the geodesic integral curves of MATH, one has that MATH is a well-defined, nowhere-zero null ... |
gr-qc/0005050 | By REF with REF one has MATH. By REF one therefore has MATH. The left side of REF is thus bounded from below by MATH. This is positive by REF . The left side of REF is therefore non-vanishing for all MATH. |
gr-qc/0005050 | Let MATH be the left side of REF . Since MATH and MATH are MATH, piecewise MATH functions of MATH, one has by REF that MATH is a MATH, piecewise MATH function of MATH. The function MATH is then a piecewise MATH function of MATH which, by means of of the NAME REF , is given by MATH . For MATH such that MATH REF reduces ... |
hep-th/0005002 | Let MATH be a state of ghost number MATH and MATH-invariant (MATH). Assuming that the filtration is bounded, we write MATH where MATH. Then, MATH . Each parenthesis vanishes separately since they carry different MATH. So, MATH. The MATH-cohomology is trivial by assumption, thus MATH. But then MATH, which is cohomologou... |
hep-th/0005002 | FGZ's filtration is originally given for the MATH flat spacetime as MATH . The filtration itself does not require MATH; this filtration can be naturally used for MATH, replacing MATH with MATH. Then, the modified filtration assigns the following degrees to the operators: MATH . The operator MATH satisfies REF and the d... |
hep-th/0005002 | The number of states of the NAME space MATH and that of the NAME module MATH are the same for a given level MATH. Thus, the NAME module furnishes a basis of the NAME space if all the states in a highest weight representation, MATH are linearly independent, where MATH. This can be shown using the NAME determinant. Consi... |
hep-th/0005002 | Using REF , a state MATH can be written as MATH where MATH and MATH. Note that the states in MATH all have nonpositive ghost number: MATH. We define a new filtration degree MATH as MATH which corresponds to MATH . The algebra then determines MATH (for MATH) from the assignment. The operator MATH satisfies REF and the d... |
hep-th/0005002 | At each mass level, states MATH in MATH are classified into two kinds of representations: BRST singlets MATH and BRST doublets MATH, where MATH. The ghost number of MATH is the ghost number of MATH plus REF. Therefore, MATH causes these pairs of states to cancel in the index and only the singlets contribute: MATH . We ... |
hep-th/0005002 | At a given mass level, the matrix of inner products among MATH takes the form MATH . We have used MATH, MATH and MATH. If MATH were degenerate, there would be a state MATH which is orthogonal to all states in MATH. Thus, the matrix MATH should be nondegenerate. (Similarly, the matrix MATH should be nondegenerate as wel... |
hep-th/0005002 | We prove REF by explicitly calculating the both sides. In order to calculate the left-hand side of REF , take an orthonormal basis of definite MATH, MATH [the basis REF ], MATH and an orthonormal basis of MATH. Then, MATH. Similarly, for the right-hand side, take an orthonormal basis of definite MATH, MATH, MATH and an... |
hep-th/0005023 | The cohomology groups with coefficient in the constant sheaf MATH on homotopic paracompact topological spaces are isomorphic CITE. If MATH is a vector bundle, its base MATH is a strong deformation retract of the infinite-order jet space MATH. To show this, let us consider the map MATH . A glance at the transition funct... |
hep-th/0005023 | Local exactness of a vertical complex on a coordinate chart MATH, MATH on MATH follows from a version of the NAME lemma with parameters (see, for example, CITE). We have the the corresponding homotopy operator MATH where MATH. Since MATH is a vector bundle, it is readily observed that this homotopy operator is globally... |
hep-th/0005057 | MATH being a bounded NAME set, the same is true of its closure MATH, so that, due to compactness and convexity of MATH, there exists a number MATH for which the relation MATH is satisfied, where MATH denotes the sum MATH with MATH terms. The spectrum condition then entails: MATH . Note, that in the derivation of this r... |
hep-th/0005057 | CASE: By partition of unity (compare CITE), applied to elements of MATH which have arbitrary energy-momentum transfer in MATH, any MATH can be written as a finite sum MATH where the MATH belong to MATH and the operators MATH have energy-momentum transfer in compact and convex subsets MATH of MATH. Since MATH we infer M... |
hep-th/0005057 | For the MATH-seminorms on MATH the assertion follows from the order relation for operators in MATH. Let MATH belong to the left ideal MATH, then MATH which by REF has the consequence MATH . This relation extends by continuity of the seminorms to all of MATH. In case of the MATH-topologies, note that for any NAME set MA... |
hep-th/0005057 | CASE: Note, that we can define a linear subspace MATH of MATH consisting of all those operators MATH which fulfill MATH for any bounded NAME set MATH. On this space the mappings MATH act as seminorms whose restrictions to MATH coincide with MATH (compare the Remark following REF ). Now let MATH be arbitrary. Given a bo... |
hep-th/0005057 | CASE: For any MATH and arbitrary MATH the relation MATH leads to the estimate MATH for any MATH and thus, by REF and the notation of the proof of REF , to MATH . This shows that MATH belongs to MATH and at the same time that the seminorm MATH (on MATH) can be replaced by MATH to yield REF. CASE: Let MATH be a normal fu... |
hep-th/0005057 | CASE: According to REF , we have for any MATH whereas the reverse inequality is a consequence of REF . This proves the assertion. CASE: Note, that MATH is invariant under the operation of taking adjoints defined by MATH with MATH, MATH, for any linear functional MATH on MATH. Thus MATH for any MATH (compare the proof o... |
hep-th/0005057 | MATH as well as its intersection MATH with the positive cone MATH are invariant under the mapping MATH defined by MATH for any unitary operator MATH and any linear functional MATH on MATH. CASE: Now, MATH for any MATH. This implies MATH for any MATH and any MATH, henceforth MATH . Therefore the introductory remark in c... |
hep-th/0005057 | CASE: Note, that continuity of the mapping MATH with respect to the locally convex space MATH is equivalent to its continuity with respect to each of the topologizing seminorms MATH. Let the NAME subset MATH of MATH be arbitrary but fixed. We shall first consider the special point MATH and restrict attention to an oper... |
hep-th/0005057 | CASE: By REF is an integrable majorizing function for the integrand of REF, so MATH exists as a NAME integral in MATH. The same holds true for the integrals constructed by use of an approximating net MATH for the almost local operator MATH: MATH . Due to compactness of MATH, these operators belong to the local algebras... |
hep-th/0005057 | By translation invariance of the norm MATH as well as of the seminorms MATH (compare REF ) the (measurable) integrand on the right-hand side of REF is majorized by the functions MATH and MATH for any bounded NAME set MATH. These are NAME and therefore MATH exists as a unique element of MATH, satisfying the claimed esti... |
hep-th/0005057 | First we consider the special case of two elements MATH and MATH in MATH having energy-momentum transfer in compact and convex subsets MATH and MATH of MATH, respectively, with the additional property that MATH and MATH lie in the complement of MATH, too. According to REF MATH and we are left with the task to investiga... |
hep-th/0005057 | Due to the assumed continuity of MATH, the assertions follow from REF in connection with REF . |
hep-th/0005057 | Due to the assumed continuity of the functional MATH, there exists a bounded NAME set MATH such that, according to REF in connection with REF , for any MATH there holds the inequality MATH . Therefore the linear operator MATH turns out to be continuous, so that the assertion follows by an application of REF from the re... |
hep-th/0005057 | REF state that MATH exist in the complete locally convex spaces MATH and MATH, respectively. Now, the functional MATH, which lies in MATH according to the remark following REF , is linear and continuous with respect to MATH and, by REF , also with respect to both MATH. Therefore it commutes with the locally convex inte... |
hep-th/0005057 | Let MATH be an arbitrary compact subset of MATH and note that MATH . Thus, according to the construction of MATH, MATH belongs to the algebra of counters and exists furthermore as an integral in the locally convex space MATH. Therefore the functional MATH can be interchanged with the integral CITE to give MATH . Applic... |
hep-th/0005057 | First, we re-write the argument MATH, commuting the operators MATH and MATH, to get MATH . This implies MATH where the first term on the right-hand side is evidently integrable over MATH, due to almost locality of the operators encompassed by the commutator. For MATH we have the estimate MATH . The second term can be e... |
hep-th/0005057 | As implied by REF , MATH is a continuous mapping on MATH with respect to the MATH-topology, hence it is uniformly continuous on any compact set MATH. This means that to MATH there exists MATH such that MATH and MATH imply MATH where MATH denotes the MATH-dimensional volume of MATH. Consequently, under the above assumpt... |
hep-th/0005057 | If a function MATH belongs to the space MATH, then the operator MATH lies in MATH, according to REF , and has energy-momentum transfer in MATH, the support of the NAME transform of MATH. If this happens to satisfy MATH, we infer MATH and henceforth, by REF , MATH. Since MATH is assumed to belong to MATH, REF results in... |
hep-th/0005057 | Taking into account the fact that the NAME measure on MATH is invariant under translations, one can express MATH for any finite time MATH and any given MATH by the following integral MATH . Next, we want to evaluate MATH which, according to the respective limits of MATH-integration, can be split into a sum of three int... |
hep-th/0005057 | Consider the functional MATH at finite time MATH. Applying to the absolute value of its defining REF the NAME inequality with respect to the inner product (MATH large enough) MATH of square-integrable functions MATH and MATH depending on the time variable MATH, one gets in the special case of MATH the estimate MATH . N... |
hep-th/0005057 | CASE: The proof of the various properties stated in the Theorem is readily carried out, once the NAME has been realized. CASE: Since a particle weight satisfies the NAME inequality its null space MATH turns out to be a left ideal in MATH (and hence in MATH). The defining sesquilinear form endows the quotient space of M... |
hep-th/0005057 | CASE: Due to continuity of the particle weight MATH with respect to NAME transformations as claimed in REF , the integrand of REF can be estimated with respect to the seminorm MATH induced on MATH, which gives the NAME function MATH. Therefore the integral in question indeed exists in the completion of the locally conv... |
hep-th/0005057 | The energy-momentum transfer of an operator MATH can be stated in terms of the support properties of the NAME transform of the mapping MATH considered as an operator-valued distribution (compare the remark following REF ). For the operator MATH this has the consequence that MATH if MATH is any NAME function with MATH. ... |
hep-th/0005057 | To establish this result we follow in the main the strategy of the proof of REF . Applied to the problem at hand in terms of MATH, this yields initially the estimate MATH for any MATH. The first term on the right-hand side turns out to be majorized by MATH in view of the fact that the particle weight is invariant under... |
hep-th/0005057 | Let MATH denote the NAME of the particle weight MATH with associated spectral measure MATH for the generator MATH of the intrinsic space-time translations. For the time being, suppose that MATH is an open bounded NAME set in MATH. Let furthermore MATH be an arbitrary element of MATH and MATH. We are interested in an es... |
hep-th/0005057 | With the definitions MATH and MATH, MATH, where the latter obviously leaves invariant MATH and is such that the corresponding spectral measure turns out to be MATH for any NAME set MATH, all features of the restricted MATH-particle weight are readily checked on the grounds of REF . |
hep-th/0005057 | The presuppositions of this theorem meet the requirements for an application of CITE. This supplies us with CASE: a standard NAME space MATH, CASE: a bounded positive measure MATH on MATH, CASE: a MATH-measurable field MATH on MATH consisting of irreducible representations MATH of the MATH-algebra MATH on the NAME spac... |
hep-th/0005057 | CASE: MATH-boundedness of the particle weight MATH means, according to REF , that to any bounded NAME set MATH there exist another such set MATH containing MATH and an appropriate positive constant MATH, so that the estimate MATH holds for any MATH. Then a finite cover of MATH by sets of diameter less than a given MATH... |
hep-th/0005057 | Select a dense sequence MATH in the norm-separable MATH-algebra MATH and consider the countable set of compact balls MATH of radius MATH in MATH. The corresponding operators MATH are decomposable according to REF : MATH and CITE tells us that the respective norms are related in compliance with the equation MATH . With ... |
hep-th/0005057 | CASE: Let MATH be a bounded NAME set and suppose that MATH is a normal functional on MATH. Then the same applies to the functional MATH, and therefore the mapping MATH is continuous with respect to the relative MATH-weak topology of MATH. Now, according to the Compactness Condition, MATH maps the unit ball MATH of the ... |
hep-th/0005057 | In a first step it will be shown that, setting MATH for any compact subset MATH of MATH, the following estimate is in force for arbitrary bounded NAME sets MATH: MATH with a suitable constant MATH and an appropriate bounded NAME set MATH. If MATH denotes a state on MATH which is induced by a vector MATH we can immediat... |
hep-th/0005057 | We have to calculate the integral on the right-hand side of REF and, to do so, it is split into two parts according to MATH or MATH with an abitrary radius MATH which is held fixed for the moment. For large MATH we use the estimate REF for the integrand and get in terms of the norm MATH: MATH . Accordingly, the respect... |
hep-th/0005057 | Let MATH denote the closed MATH-ball, MATH, in MATH with respect to MATH. By REF , the condition MATH, MATH, implies MATH, stating a property of uniform approximation. This means that, given MATH, there exists a radius MATH, take e. CASE: MATH, such that to any MATH we can find a local operator MATH with MATH . Again a... |
hep-th/0005057 | Let MATH and MATH be arbitrary distinct points in MATH. We shall assume MATH and want to show that MATH. Define MATH and consider one of the seminorms MATH topologizing MATH. There are two possibilities: MATH . Depending on the actual situation we define an interval MATH, choosing MATH, MATH in REF and MATH, MATH in RE... |
hep-th/0005057 | By CITE MATH is a well-defined MATH-valued mapping on the compact interval MATH. For MATH and MATH satisfying MATH we have MATH hence MATH . Now by assumption, MATH is continuous on the compact interval MATH of integration for any of the defining seminorms MATH of MATH, and, according to CITE, one has for any MATH the ... |
hep-th/0005057 | Given MATH and MATH as above we define two mappings MATH and MATH on the compact interval MATH to MATH respectively MATH through MATH . From REF we infer MATH for any MATH. This implies, according to REF , that the mapping MATH is constant on the interval MATH (Note, that MATH as well as MATH are continuous.). Hence MA... |
hep-th/0005057 | CASE: If MATH is continuously differentiable the mappings MATH are continuous for any MATH; furthermore REF correspond for each MATH exactly to REF setting MATH, so that the first part of the statement is almost trivial. CASE: Let all the partial derivatives of MATH exist as continuous mappings MATH, then, for small MA... |
hep-th/0005057 | For MATH and sufficiently small MATH consider the following expression which involves two increments of MATH: MATH . By assumption on the existence and continuity of the mixed derivatives we can apply the Mean Value REF twice to the above expression: One can consider the increments with respect to MATH and apply the Me... |
hep-th/0005057 | CASE: To prove the non-trivial part, suppose that MATH is arbitrary but fixed. Then MATH, MATH, is a local chart around MATH with MATH. According to the definition of MATH we have MATH and, since the automorphisms are norm-preserving, the assumed differentiability of the mapping MATH at MATH carries over to MATH which ... |
hep-th/0005057 | By assumption on MATH (relations REF) in connection with linearity of MATH, the increment of MATH at MATH allows for the representation MATH where MATH for any seminorm MATH, MATH. But, due to continuity of MATH, there exist to any seminorm MATH on MATH a finite number of seminorms MATH on MATH, MATH, and a positive co... |
hep-th/0005057 | Let MATH be a dense sequence of non-zero vectors in MATH and let MATH denote the NAME algebra generated by MATH. According to NAME 's Density Theorem, MATH coincides with the strong closure MATH of the algebra MATH, which by assumption acts non-degenerately on MATH (compare CITE, CITE). First we assume the existence of... |
hep-th/0005103 | CASE: ` REF ': Since MATH is a group homomorphism this is trivial. CASE: ` REF ': Assume that MATH is free but MATH. Thus MATH, in contradiction to the NAME isomorphism REF . |
hep-th/0005273 | Assume first that MATH is real, and consider the functional MATH for MATH. Then MATH where MATH and MATH was used. Hence MATH is invariant as well, and REF follows using uniqueness of the integral (up to normalization). For MATH, we define MATH with the star structure REF . Using MATH and MATH, an analogous calculation... |
hep-th/0005273 | For MATH, we have MATH and hermiticity is immediate. For MATH, consider MATH . NAME follows using REF : MATH . Using REF for MATH, it is not difficult to see that they are also positive - definite. |
hep-th/0005273 | Using MATH (which follows from REF ) and MATH, it is easy to check that MATH for all MATH, and REF follows. REF follows immediately from MATH, and To see REF , one needs the well - known relation MATH, as well as MATH; the latter follows from the quasitriangularity of MATH. The commutation relations among the MATH are ... |
math-ph/0005010 | Though MATH-modules MATH fail to be MATH-modules CITE, one can use the fact that the sheaves MATH are projections MATH of sheaves of MATH-modules. Let MATH be a locally finite open covering of MATH and MATH the associated partition of unity. For any open subset MATH and any section MATH of the sheaf MATH over MATH, let... |
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