paper stringlengths 9 16 | proof stringlengths 0 131k |
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hep-th/0010227 | In the case MATH the parity behaviour of the vacuum state can be obtained from an indirect argument. We know that the ground states in absence of potential term are holomorphic sections of the line bundle MATH with NAME class MATH (see Ref. CITE for a review and references therein). Any holomorphic section of MATH can ... |
math-ph/0010005 | Let MATH be a minimizing sequence for MATH under the normalization condition MATH, with corresponding densities MATH. Using the one-dimensional NAME inequality CITE we have MATH . Moreover (compare REF ), MATH . Since the potential energy is relatively bounded with respect to the kinetic energy (compare REF), the right... |
math-ph/0010005 | Observe that MATH where we set MATH. Using the positive definiteness of the NAME kernel and the fact that MATH implies MATH for all MATH, we immediately get the desired result. |
math-ph/0010005 | We proceed essentially as in CITE. For any MATH . In particular, for all MATH, MATH . Now if there exists a MATH with MATH and MATH we can choose MATH small enough to conclude that MATH which contradicts the fact that MATH minimizes MATH. |
math-ph/0010005 | For fixed MATH, MATH and MATH define MATH as above. Let MATH be the infimum of MATH under the constraint MATH. For MATH let MATH be the corresponding minimizers. We have, with MATH, MATH . Dividing by MATH and taking the limit MATH followed by MATH, we conclude that MATH . From REF we infer that MATH for all MATH, with... |
math-ph/0010005 | With MATH the two-dimensional Laplacian, one easily computes that MATH for MATH. Multiplying REF with MATH and integrating over MATH, we therefore get, for any density matrix MATH (recall REF ), MATH where we used partial integration for the second step, and the fact that MATH for MATH. To treat the last term in REF , ... |
math-ph/0010005 | This follows immediately from REF , noting that MATH for all MATH. |
math-ph/0010005 | After an appropriate scaling, this is a direct consequence of REF , using the estimates MATH and MATH . |
math-ph/0010005 | The lower bound is quite easy, using the results of CITE. As shown in REF, we have MATH where we have used the scaling properties of MATH. It is shown in CITE that the right hand side of REF divided by MATH converges to MATH. For the upper bound we assume MATH for the moment. We use as trial density matrices MATH where... |
math-ph/0010005 | The convergence of the densities in REF follows from the convergence of the energies in a standard way by considering perturbations of the external potential (compare for example, CITE). Moreover, since the convergence in REF is uniform in MATH, REF holds for any function MATH, so we can conclude that REF holds uniform... |
math-ph/0010005 | We first treat the case MATH. From REF and the fact that MATH is convex in MATH, we get MATH . Suppose that MATH is not the ground state energy of some MATH. Then MATH, because the second lowest eigenvalue of MATH is equal to the ground state energy of the three-dimensional operator MATH . This follows because MATH is ... |
math-ph/0010005 | Using the definition of MATH it can be shown (compare CITE) that MATH for some coefficients MATH that fulfill MATH. Since MATH for all MATH (CITEb) we get MATH where we used the fact that MATH if MATH (CITEg). Moreover, using convexity of MATH (CITEi), we arrive at MATH . REF follows if we can show that MATH . This is ... |
math-ph/0010005 | Let MATH denote the eigenvectors of MATH, that is, MATH . Multiplying REF with MATH and integrating we get MATH where we used that MATH. Since MATH is convex and MATH as MATH we have MATH. Using this and partial integration we can estimate the last term in REF by MATH . Summing over all MATH and MATH (and, according to... |
math-ph/0010013 | The assumptions of REF imply those of CITE. |
math-ph/0010013 | See for example CITE. |
math-ph/0010013 | See REF. |
math-ph/0010013 | We may assume MATH, because the general case MATH follows therefrom. Let MATH be a monotone increasing sequence, MATH if MATH, of positive simple functions approximating MATH. More precisely, these functions are assumed to be of the form MATH with suitable constants MATH and bounded NAME sets MATH which are pairwise di... |
math-ph/0010013 | See REF. |
math-ph/0010013 | If MATH for MATH-almost all MATH, then MATH using REF . Conversely, for every MATH, normalized in the sense MATH, there exists a bounded open cube MATH compatible with MATH such that MATH and therefore MATH . Taking the probabilistic expectation on both sides and using REF we arrive at the sandwiching estimate MATH by ... |
math-ph/0010013 | NAME 's inequality CITE, a subsequent shift in the MATH-integration, and an enlargement of its domain show that MATH where the cube MATH is the arithmetic difference of the unit cubes MATH and MATH. Using NAME 's inequality again, we thus arrive at MATH . Since MATH is contained in the cube centered at the origin and c... |
math-ph/0010013 | We first define the following one-parameter family MATH of smooth NAME probability densities on MATH which approximates the NAME measure MATH on MATH supported at MATH as MATH. Moreover, let MATH denote the convolution of MATH and MATH. The fundamental theorem of calculus yields MATH and hence MATH . The supremum on th... |
math-ph/0010013 | REF : REF follows from REF and the finiteness MATH. Moreover, for every MATH one has MATH by partial integration. Vague convergence of MATH to MATH is now a consequence of the dominated-convergence theorem. It is applicable since REF implies the existence of a locally bounded function dominating all but finitely many o... |
math-ph/0010013 | To show that MATH is a positive NAME measure on MATH, it suffices that MATH for any compact energy interval MATH, MATH, MATH. This follows from the elementary inequality MATH, the spectral theorem applied to MATH and the functional calculus. REF below and REF ensure that the right-hand side of REF is indeed finite. To ... |
math-ph/0010013 | Since we have already established the vague convergence of the density-of-states measures in REF , it remains to verify relation REF for the corresponding random distribution functions MATH for MATH-almost all MATH. To this end, we employ the elementary inequality MATH valid for all MATH, MATH with MATH defined in REF ... |
math-ph/0010013 | See CITE where the appropriate NAME formula for the infinite-volume and the NAME semigroup is employed; see also CITE. |
math-ph/0010013 | Thanks to REF below and REF , the integrals MATH and (analogously) MATH are finite for all MATH and MATH-almost all MATH such that REF yields MATH . Here the quantity MATH, which depends on MATH and MATH, was introduced in REF and vanishes for MATH. We further estimate the first term with the help of REF choosing MATH ... |
math-ph/0010013 | Thanks to REF , the integrals MATH and (analogously) MATH are finite for all MATH. Moreover, MATH for all MATH by REF again. This implies vague convergence by REF . |
math-ph/0010013 | The proof consists of an approximation argument. To this end, we recall REF of the truncated random potential MATH. Since MATH is bounded and MATH enjoys REF , we may apply CITE (or CITE together with REF ) which gives MATH for all MATH, all MATH and all MATH. Using the triangle inequality we estimate MATH . The proof ... |
math-ph/0010013 | CASE: Let MATH if MATH and zero otherwise stand for the signum function of a complex-valued function MATH. The polar decompositions MATH and MATH together with NAME 's inequality CITE and CITE show that MATH . Let MATH denote an orthonormal eigenbasis associated with MATH and MATH the eigenvalue corresponding to MATH. ... |
math-ph/0010013 | CASE: The claim follows from the chain of REF . Here the first inequality is a consequence of the polar decomposition MATH. The second one is a special case of CITE. For the third one we used the diamagnetic inequality CITE in the version MATH for any MATH, together with CITE. The fourth inequality eventually follows f... |
math-ph/0010013 | The proof is split into the following three parts: CASE: Proof of REF for MATH, CASE: Proof of REF for MATH, CASE: Approximation argument for the validity of REF . Throughout the proof we use the abbreviations MATH in agreement with REF . As to REF. Let MATH. We may then apply the (second) resolvent equation CITE MATH ... |
math-ph/0010013 | Throughout, we assume that MATH is MATH-ergodic. The proof is split into three parts. CASE: Proof of REF , CASE: Proof of REF with MATH replaced by MATH with MATH arbitrary, CASE: Approximation argument for the validity of REF with MATH replaced by MATH and of REF . We use the abbreviations MATH for the resolvents of M... |
math-ph/0010013 | We consider the product of the probability spaces MATH and MATH. The latter corresponds to a uniform distribution on the open unit cube MATH. On this enlarged space we define the random potential MATH . It is MATH-ergodic by construction CITE and enjoys REF . The latter assertion is proven by tracing the claimed proper... |
math-ph/0010016 | Recall that MATH is the solution of MATH with MATH . Thus MATH is analytic in MATH with at worst algebraic singularities at MATH and MATH, that is, the boundaries of the stability interval. But, as was determined before, both MATH have identical analytic properties, and by the definition of MATH and MATH, MATH so we ar... |
math-ph/0010016 | Suppose that MATH for all MATH, and hence all MATH by analyticity. We know then that for every MATH and MATH, the NAME solution MATH is MATH (for all MATH); that is, MATH is exponentially decaying in this region of the upper half plane. Thus MATH is the NAME solution for the perturbed equation MATH. We may therefore ca... |
math-ph/0010016 | Since MATH, we have that MATH, MATH, and MATH from MATH and MATH. Thus MATH is invertible. For the rest of the proof we will drop the fixed parameter MATH. First note that expressing the solutions MATH and MATH in terms of the NAME solutions MATH yields MATH . Clearly, MATH and using that MATH are the NAME solutions we... |
math-ph/0010016 | CASE: By (MATH), MATH uniformly in MATH and MATH. Thus by dominated convergence, MATH, if MATH. CASE: We have that MATH where the last inequality follows as MATH for all MATH by REF and (MATH). |
math-ph/0010016 | CASE: We first show that MATH weakly as MATH. Note that in general if MATH weakly and MATH weakly, then MATH weakly (for the definition of the convolution, see REF). By MATH we have that MATH weakly and an application of NAME implies that every subsequence of MATH contains a weakly convergent subsequence. Using this, t... |
math-ph/0010016 | One calculates that MATH . The result follows by letting MATH and noting both REF of the above Corollary. |
math-ph/0010016 | REF confirms that both assumptions of REF hold uniformly with respect to MATH. That this implies the above estimates, again uniformly with respect to MATH, follows from observing that the proofs of REF yield uniform results under uniform assumptions. |
math-ph/0010016 | REF follows as in V. REF using that MATH . REF follows as in REF using REF as done therein. |
math-ph/0010016 | This lemma can be proved in exactly the same way REF is proved in CITE. For the reader's convenience, we sketch the argument briefly. By our choice of the interval MATH and the results from REF, in particular REF , we have MATH. Using the inequality MATH and NAME 's inequality, one shows as in CITE that for every MATH ... |
math-ph/0010016 | We will follow the same strategy as NAME in their proof of CITE, that is, we will use NAME continuity of the integrated density of states to derive the estimate REF. The only difficulty that arises is that the cutoff of eigenfunctions as performed by NAME in the discrete case may produce elements outside the domain of ... |
math-ph/0010016 | We closely follow the proof of REF and make the necessary modifications. Let MATH be as above and for each odd MATH, we set MATH with some MATH to be chosen later. For every MATH and MATH, we define the events MATH . Let MATH with MATH and MATH from REF . Then MATH and REF immediately implies MATH provided MATH is larg... |
math-ph/0010016 | This follows from REF. Note that in the case MATH the assumptions required there are equivalent to MATH for some MATH and that MATH is non-compact and strongly irreducible. In particular, non-compactness is equivalent to the contractivity required in CITE; see CITE. Integrability of MATH with respect to MATH for all MA... |
math-ph/0010016 | This follows from REF, whose assumptions, that is, MATH strongly irreducible and MATH for some MATH, are satisfied. |
math-ph/0010016 | We closely follow the proof of REF. Define MATH by MATH . Then MATH . Define MATH by MATH . For MATH, choose representatives MATH and MATH such that MATH and the angle between MATH and MATH is at most MATH. Then, by REF, MATH recall that MATH. This implies, using notation from REF, MATH . The definition of MATH shows M... |
math-ph/0010016 | From REF we get for all MATH, MATH with probability at least MATH. This yields the assertion. |
math-ph/0010016 | Let MATH be the solutions of MATH with NAME boundary conditions at MATH, that is, MATH, MATH. Then the NAME 's function MATH (that is, the kernel of MATH) is given by MATH where the Wronskian MATH is constant in MATH. Setting MATH, we get MATH where MATH denotes the transfer matrix from MATH to MATH. By stationarity we... |
math-ph/0010016 | With probability MATH we have that both the event in REF and the complementary event in REF hold. Thus, by assumption we have that for every MATH, we have MATH and moreover, by the resolvent equation, MATH . Thus for these MATH's, the cube MATH is MATH-good. |
math-ph/0010016 | Fix an arbitrary compact interval MATH. It follows from REF that we have both a NAME estimate and a fixed energy initial length scale estimate for every MATH. REF shows that these two results imply a variable-energy initial length scale estimate for a ball MATH of explicit radius around MATH . The variable-energy multi... |
math-ph/0010016 | It essentially follows from CITE that the variable-energy resolvent decay estimates, as given by the output of the the variable-energy multiscale analysis, imply strong dynamical localization in the sense of REF . For the curious reader we briefly sketch the argument, referring him to CITE for necessary notation. Given... |
math-ph/0010016 | For MATH one has MATH that is, MATH, which implies the lemma. |
math-ph/0010016 | Without restriction let MATH. The solutions MATH and MATH satisfy MATH and thus MATH . NAME 's lemma, for example, CITE, yields MATH . By REF we have for all MATH that MATH . Inserting this into REF yields the result. |
math-ph/0010016 | By REF there are constants MATH only depending on MATH such that for all MATH . With MATH and MATH we get MATH . It now follows from elementary geometric considerations, for example, CITE, that MATH contains an interval of length MATH on which MATH. This yields REF . |
math-ph/0010016 | See REF , and REF. |
math/0010019 | Assume then that MATH is bounded. To check the continuity, let MATH be a net weakly convergent to MATH. Then for all MATH we have MATH as MATH is dense in MATH and MATH is bounded by assumption, it follows that MATH weakly. It remains to show that MATH is bounded if MATH is norm closed. We shall show that MATH is closa... |
math/0010019 | Assuming in MATH that MATH is contained in the ball of radius MATH, we shall show that the same is true for MATH, that is, MATH holds. Let MATH. By NAME density theorem CITE, there exists a net of operators MATH strongly convergent to MATH. Since MATH, we may assume, possibly restricting to a subnet, that MATH weakly c... |
math/0010019 | If MATH is pure, then MATH, hence, if the boundedness condition holds, MATH is bounded. As MATH, it follows that MATH is bounded, thus MATH is semibounded. The converse is obvious. |
math/0010019 | Immediate by REF . |
math/0010019 | Clearly MATH for all MATH. If MATH is MATH-holomorphic on MATH and MATH, then MATH is boundary value of a function in MATH, thus by REF MATH. If moreover the bound REF holds, then MATH thus MATH is MATH-bounded. Conversely, if MATH is MATH-bounded, then MATH and the same computation done above yields, by REF , that MAT... |
math/0010019 | Immediate by REF . |
math/0010019 | As the map MATH satisfies MATH, REF holds with MATH (see REF), where MATH namely MATH . Assuming first that MATH is also separating, that is, MATH ,if MATH and MATH we then have MATH . Since MATH implements automorphims of MATH and MATH, by the modular theory MATH and MATH commute, thus there exists a strongly dense su... |
math/0010019 | By REF applied to the MATH-fold tensor product, we have MATH. By the modular theory MATH and MATH commute, thus a simple application of the NAME theorem implies MATH for all MATH, thus MATH. Taking the limit as MATH we obtain MATH . As MATH and MATH, the above inequality also entails that MATH . Taking logarithms, as M... |
math/0010019 | Let MATH be the NAME operator and MATH be the spectral projection of MATH corresponding to the interval MATH. Then MATH commutes with the (real, unbounded) projection MATH onto MATH (see REF ) because it commutes with MATH: indeed MATH commutes both with MATH and with MATH (being a real even function of MATH). Denoting... |
math/0010019 | As is well known, MATH, where MATH is the NAME 's operator associated with MATH and MATH. With MATH, let MATH the operator given by MATH, MATH, and define analogously MATH with respect to MATH. Then MATH and MATH and MATH, thus MATH and MATH. |
math/0010019 | Since the kernel of MATH is invariant under MATH and MATH, one can decompose MATH as a direct sum of two components, one corresponding to the kernel of MATH, and one to its orthogonal complement, thus REF can be proved for each component separately. Since the inequality is obviously satisfied on the kernel of MATH, we ... |
math/0010019 | Set MATH as in REF . Note first that MATH is equal to MATH where MATH is the NAME operator. As MATH commutes both with MATH and MATH, the same is true for MATH and thus MATH commutes with MATH. It follows that MATH . We then have for all MATH . Clearly the same is true if we replace the NAME algebra MATH by MATH (infin... |
math/0010019 | MATH . If MATH satisfies the KMS condition at inverse temperature MATH then it is MATH-holomorphic with constant MATH. It is also immediate by REF that MATH is not completely MATH-holomorphic if MATH. For the same reason it is not completely MATH-holomorphic if MATH. If MATH is a ground state, then MATH is obviously co... |
math/0010019 | Let MATH be the family of projections associated with MATH by the spectral theorem, namely MATH. Then MATH if and only if MATH, that is, MATH, where MATH is the finite NAME spectral measure associated with MATH. By the next NAME (with a change of sign of MATH) this holds iff MATH is the boundary value of a function hol... |
math/0010019 | If MATH belongs to MATH, then also MATH belongs to MATH for all MATH and MATH defines a function in the strip MATH, that can be easily seen to belong to MATH. Conversely suppose MATH to be the boundary value of a function holomorphic in MATH and continuous in MATH. Decompose MATH as MATH, where the first term is suppor... |
math/0010019 | By replacing MATH with MATH, we may assume that MATH is non-negative. Let MATH be a vector orthogonal to MATH. We have to show that MATH. If MATH is a function in the NAME space MATH, we have MATH for all MATH, where MATH the NAME anti-transform of MATH. If MATH is a bounded NAME function with compact support, we may c... |
math/0010019 | Note first that, as a consequence of the commutation relations REF , we have MATH for all MATH CITE. By the the criterion given in REF below, it will suffice to construct a core MATH for MATH such that MATH for all MATH. The set MATH is contained in the domain of MATH, and clearly MATH . If moreover MATH is Ad-MATH-inv... |
math/0010019 | Since MATH implements automorphisms of MATH, it commutes with the modular operator MATH associated with MATH, thus MATH is a one-parameter group of unitaries. We denote by MATH its self-adjoint generator. By the modular theory MATH also commutes with the modular conjugation MATH associated with MATH, thus MATH. We then... |
math/0010019 | Let MATH commute with MATH and MATH. Then MATH and MATH for some MATH. As MATH is separating for MATH, then MATH and this entails the thesis. |
math/0010019 | Let MATH be the weak closure of MATH. Clearly MATH, hence, by REF, see also CITE, MATH. By the mean ergodic theorem, the one dimensional projection MATH onto MATH belongs to NAME algebra generated by MATH, hence to MATH. Then MATH by REF . |
math/0010019 | Set MATH and apply REF . |
math/0010019 | In this proof we drop the subscript MATH on the operators associated with MATH. MATH: As MATH, then MATH for all MATH, which immediately implies MATH. MATH is obvious. MATH: We only need to show the spectrum condition. By NAME covariance it is sufficient to show that the positivity of the generator of a one-parameter g... |
math/0010019 | MATH: By the split property the set MATH is compact for MATH CITE and metrically nuclear for MATH CITE. By the KMS property MATH for all MATH and MATH (see CITE); we omit the suffix MATH on MATH and MATH. MATH: Assume first that the underlying NAME MATH space is separable. Let MATH and choose a sequence MATH norm dense... |
math/0010019 | The proof is similar to the one of REF . We only notice that in this case condition MATH refers to a bounded interval MATH. This is possible because MATH and, by scaling the interval, this gives MATH for any MATH, thus the boundedness condition holds for MATH. |
math/0010019 | MATH. The inclusion MATH is half-sisd modular, therefore the translations (with positive generator) can be constructed as in CITE, compare also PropositionREFA. REF. MATH. First we extend the net MATH to all intervals in MATH setting MATH, MATH. Clearly MATH and MATH act as translations and dilations on the net, theref... |
math/0010020 | That MATH acts faithfully on MATH is well-known and easy to prove. If MATH are represented by sections MATH, MATH, then there is a fiberwise translation in the part of MATH that is smooth over MATH which sends MATH to MATH. As recalled above, this translation extends as an automorphism MATH of MATH. Then MATH fixes the... |
math/0010020 | Let MATH. So for every MATH we have MATH. It follows that for every reducible fiber MATH, MATH preserves the root subsystem MATH of MATH generated by MATH. So MATH normalizes the associated affine NAME group MATH. Choose a section MATH. Then MATH sends its class MATH to an element of the form MATH with MATH, where MATH... |
math/0010020 | Let MATH be the locus of pairs of distinct points of MATH with the same image in MATH. For the first assertion it is enough to show that MATH is of dimension MATH. A point of MATH for which MATH are mutually distinct can be represented by a triple MATH with MATH so that MATH is the divisor defined by MATH. Notice that ... |
math/0010020 | In view of REF we must show that the MATH-orbit space of MATH is rational. Generically MATH is fibered in lines over the product of projective spaces MATH. Let MATH be the locus where MATH acts freely. Then MATH is open-dense in MATH, and the orbit space MATH is a rational curve. So if MATH denotes the preimage of MATH... |
math/0010020 | We begin with proving the first part of REF . Suppose that MATH has a singular point MATH over MATH. Denote the closures of the connected components of MATH by MATH and MATH. Then on MATH we have a NAME fiber of type MATH for some MATH. The NAME characteristic of such a fiber is MATH and hence the degree of MATH on MAT... |
math/0010020 | The first statement follows in a straightforward manner from the fact that MATH, our computation of MATH, and the MATH-linearity of MATH. The second follows from the first if we bear in mind the Formulae REF for MATH. |
math/0010020 | From the definitions we find that MATH and MATH. We know a priori that MATH and MATH are covering transformations, hence it is enough to show that these elements act on MATH as respectively, MATH and MATH. This is verified in a straightforward manner using REF . |
math/0010020 | Choose an affine equation for MATH as before. First note that MATH lies in MATH. At a point of multiplicity MATH, a local equation of MATH is MATH. A straightforward calculation shows that the pull-back of MATH under normalization has in each the preimage of this singularity a zero of order MATH for MATH. Any other ele... |
math/0010020 | We use our fixed two dimensional vector space MATH equipped with a generator MATH of MATH. Given a semistable MATH, regard MATH as a homogeneous function on MATH. Then MATH defines a degree MATH covering of MATH. It is an affine surface with good MATH-action (so that MATH has weight MATH) whose curve at infinity is a M... |
math/0010020 | This is clear for the group of covering transformations. Any such automorphism that is not a covering transformation must permute the ramification points nontrivially. It is easy to see that such an automorphism acts nontrivially on MATH. |
math/0010020 | Let MATH be a stable effective degree MATH divisor in MATH (so all multiplicities MATH). Given a neighborhood MATH of MATH in the space of effective degree MATH divisors, denote by MATH the divisors that are reduced. Then MATH defines a locally constant sheaf of MATH-modules. If MATH has multiplicities MATH (so that MA... |
math/0010020 | Since MATH is MATH-equivariant, it is enough to prove that MATH is an isomorphism. To this end, one first shows that MATH is a local isomorphism in codimension one (this is based on simple type of local NAME theorem) and has discrete fibers. This implies that MATH has no ramification. So MATH is a local isomorphism eve... |
math/0010020 | The group MATH may be identified with the orbifold fundamental group of MATH. Via the orbifold isomorphism MATH, we then get a MATH-covering. This covering factorizes over a covering of MATH with the kernel of MATH as covering group. Since MATH is trivial in MATH, a simple loop around a deleted hyperplane has monodromy... |
math/0010020 | It is clear that the projection MATH is a MATH-covering. There is no ramification outside the discriminant divisor MATH since MATH is there locally liftable to a morphism to MATH. The remaining statements follow easily. |
math/0010020 | The first assertion follows from the fact that MATH is MATH-equivariant and of degree MATH and the second from the observation that MATH. The last clause requires more work. In view of the connectedness of MATH, it is enough to prove that assertion for one particular rational elliptic surface. We take the case studied ... |
math/0010020 | Consider the case when the closed fiber represents a general point of MATH, MATH, MATH or MATH. The image of such a point in MATH is a semistable orbit of a degree MATH divisor on MATH of type MATH, MATH, MATH, MATH respectively. So its image under the period isomorphism is going to be perpendicular to a (primitive) su... |
math/0010020 | Let MATH be the maximal integer for which MATH is nonempty. So MATH is an isomorphism, but MATH is not. So MATH is locally the intersection of MATH members of MATH in general position. From this it follows that the blowup of MATH factorizes over MATH. The pull-back of MATH to MATH is up to a twist with a principal idea... |
math/0010020 | Consider the set MATH of pairs of MATH-vectors MATH in MATH with MATH. The mod MATH reduction of a pair MATH is pair of vectors MATH in MATH with symplectic product MATH. The number of such pairs of vectors is MATH. The MATH-vectors mapping to MATH are the elements of the MATH-orbit of MATH and likewise for MATH. So th... |
math/0010020 | Let MATH be a MATH-vector in MATH with MATH. It follows from the preceding that MATH is MATH-equivalent to MATH, with MATH and MATH a vector mentioned in one of the cases MATH. In these cases the exponent MATH is determined by the inner product of MATH with MATH. The last part of the corollary is straightforward. |
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