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0704.0012.txt |@word enseignement:1 4ac:1 results:4 proposition:4 follow:1 primes:21 whose:2 4np:1 equivalence:1 γv:4 tm:6 aut:1 elliptic:1 norm:1 conjecture:6 g2:1 regional:1 every:12 ramified:1 bxy:1 school:1 level:1 t0:2 t1:4 large:4 math:9 upper:2 occurrence:1 zero:1 odd:12 theory:3 even:1 appear:2 sum:1 new:2 congruences:2 method:1 satisfy:3 respectively:1 here:5 modular:23 objects:1 address:1 obtained:1 remark:2 study:5 studied:2 contradiction:2 fourier:9 classification:1 f1:5 apply:2 use:1 prove:5 positive:17 negative:4 next:1 noting:1 m2:7 conductor:1 tℓ:1 berlin:2 normalized:2 particular:1 known:3 central:1 springer:2 ma:1 recall:1 mk:1 values:3 following:13 root:1 example:1 proof:17 claim:2 give:3 application:1 nonnegative:1 dk:1 gcd:12 criterion:1 ordinary:2 verlag:1 kias:1 divides:3 gordon:1 fp:6 product:1 mar:1 npj:1 may:1 applications:3 plane:2 fd:5 natural:1 so:2 postech:1 coprime:1 counted:1 nα2:3 soc:1 mathematicians:1 paper:2 london:1 boylan:1 satisfies:4 its:1 sλ:14 derive:2 singular:11 interesting:1 gtm:1 forms:14 choose:1 mail:1 e2πiz:1 serre:1 cusp:8 non:2 bernoulli:1 newman:5 they:1 half:13 now:1 term:2 classified:1 2z:3 sufficiently:4 2g:1 each:6 fψ:3 2m:8 repeating:1 traces:9 referred:1 square:2 dong:1 weight:17 series:3 24h:3 related:1 society:1 eisenstein:1 combining:1 our:2 ep:3 special:2 furthermore:1 since:4 denotes:5 formula:4 mλ:6 plete:1 inert:1 free:2 york:2 e4:3 generate:1 definition:1 dirichlet:4 r3:2 phrases:1 modularity:1 times:1 cbms:1 4p2:4 first:1 already:1 ono:7 omitted:1 number:17 ℓj:1 ℓi:3 another:1 given:1 introduction:3 least:2 their:1 zagier:2 convention:1 fonctions:1 gives:5 holomorphic:11 part:3 completes:1 coefficients:17 finitely:2 2f1:1 overpartition:2 was:4 imply:1 classes:1 kth:1 were:1 cohen:4 curves:1 say:1 2t1:2 need:1 relatively:1 ax2:1 k3:1 k2:6 ideal:1 concerns:1 take:2 begin:1 trace:3 rankin:3 algebraic:1 involving:1 class:9 χq:1 χp:1 ki:2 partition:3 kr:1 proceedings:1 bracket:3 kv:6 section:2 show:2 12h:2 notations:1 spaces:1 analogue:3 trivial:1 q3:1 q2:13 divisibility:2 proportion:5 lemma:9 only:2 do:2 overpartitions:2 de:1 db:1 dz:1 words:1 prime:23 fields:1 arxiv:1 factorization:1 qk:2 qn:4 coefficient:1 moduli:11 set:2 modulo:18 ser:1 exists:10 bruinier:3 com2:1 see:10 result:1 recently:2 subject:1 αq:5 satisfying:1 exponent:1 definite:1 wiss:1 αf:3 invariant:1 meromorphic:1 korea:1 terms:2 hecke:8 extend:2 web:1 jm:2 key:1 certaines:1 amer:2 both:1 many:4 eta:1 otherwise:2 16n:1 cz:2 cy:1 moreover:3 16v:1 polylogarithms:1 ps:1 pp:1 discriminant:3 16p2:3 compos:1 sl2:2 int:2 motives:1 pe:2 pj:1 ann:1 define:5 generating:1 hurwitz:5 θp:8 p2:11 p3:1 hughes:1 main:2 ǫd:1 heidelberg:1 suppose:9 lect:1 property:12 k1:5 cont:1 overlined:1 vol:1 vi:1 thus:9 vp:3 conf:1 binary:1 ahlgren:4 complex:4 arguments:1 purpose:1 characters:1 weakly:3 k21:1 well:2 stating:1 usual:5 obtain:1 integer:21 somerville:1 note:12 infinitely:1 hℓ:1 character:4 except:1 k0:3 theorem:40 remaining:2 real:2 eigenform:1 sums:1 press:2 using:6 unique:1 mod:66 follows:1 pole:1 arithmetic:2 multiplicative:1 4mn:2 mathematical:1 deal:1 sequence:1 some:3 b2:1 proofs:5 nonzero:1 hodge:1 does:1 slash:1 bm:2 equivalent:1 q2k:2 expansion:1 bk:2 imaginary:1 nℓ:1 properties:4 chapter:3 integers:8 ok:2 choi:7 theorems:1 o:2 2k:1 22m:1 h1:8 integral:17 references:1 operator:6 ek:2 there:12 quadratic:4 sections:1 function:6 complete:2 form:18 gauss:1 logx:6 hm:1 4z:6 true:2 2πi:4 multiplicity:1 up:1 mth:3 critical:1 distribution:14 similar:1 ωf:4 detailed:1 ac:4 proved:7 ad:1 ag:7 af:10 defined:4 certain:1 abstract:1 congruence:1 denote:3 twist:4 az:2 ni:10 functions:3 no:5 when:3 vanishes:1 field:1 mathematics:2 nt:4 nv:1 15v:2 polynomials:1 koblitz:1 lang:1 implies:5 assume:4 4n:9 n1:9 n2:3 |
0704.0013.txt |@word consider:4 gp:5 results:3 invent:1 proposition:16 primes:2 gl:1 compact:1 analytic:1 tr:4 γt:7 tm:2 th:1 under:1 kθ:1 g7:2 mpb:3 orthogonal:3 far:1 bilinear:1 g8:3 every:19 lecture:3 associated:1 dim:11 school:1 m2g:1 riemann:1 level:2 notes:3 t1:2 large:3 vector:3 math:17 anal:1 upper:2 zn:1 des:1 sturm:2 japanese:1 zero:11 further:2 reine:1 odd:6 canonical:2 residue:6 3x:1 appear:2 sum:1 version:1 new:2 congruences:11 method:1 coefficient:1 respectively:2 b2r:1 here:9 modular:58 objects:3 groups:2 address:2 112b:3 divisibility:1 obtained:3 remark:6 divisor:1 52b:1 institute:1 study:1 studied:4 allows:1 products:1 fourier:25 classification:1 ls:2 texts:1 9z:1 apply:2 prove:4 remains:1 positive:27 negative:3 next:1 call:1 therefore:2 m2:1 c4n:5 more:4 atf:9 elliptic:2 definition:4 thus:3 berlin:3 dirichlet:1 known:1 iii:1 springer:4 isomorphism:5 ma:1 recall:2 work:1 det:4 mh:5 mk:1 mt:2 values:3 ρt:5 mr:6 following:12 trans:1 root:1 example:2 proof:24 claim:3 lτ:2 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0704.0014.txt |@word simplexes:3 consider:1 chain:5 invent:2 eds:1 proposition:16 transversally:3 getzler:1 research:1 reparameterization:2 gh:2 compact:5 whose:2 preserves:3 γs:10 equivalence:2 send:1 γt:5 finally:2 tk:4 th:1 under:1 rham:3 include:1 p2:1 string:4 induce:1 level:10 evaluation:1 t1:4 vector:7 neighborhood:4 chen:17 spectral:4 filtration:6 section:13 abs:1 above:1 international:1 satisfy:2 homology:3 led:1 degree:5 respectively:2 here:7 groups:3 path:4 along:4 bk:2 equations:2 obtained:3 study:2 spheres:1 haller:1 proof:24 makes:3 via:2 projection:2 diffeomorphisms:1 devoted:1 apr:1 limp:4 rπ:3 total:1 would:2 distinct:2 ψp:3 quasi:3 therefore:12 totality:3 associated:4 type:3 auser:1 duality:1 formality:1 becomes:4 surjection:1 normalized:1 projective:1 cases:1 isomorphism:25 work:2 morgan:2 following:22 trans:1 deg:3 figure:1 give:6 organized:1 professor:1 numbers:1 ahler:4 verify:10 hom:90 maps:6 those:3 fp:33 fq:6 product:23 algebra:17 diagram:2 dα:1 st:1 filtrations:3 sh:5 so:3 realizes:1 fundamental:4 sc:13 representation:1 sa:3 acts:1 indeed:1 paper:4 through:1 satisfies:2 its:7 τl:1 group:7 thank:1 τi:6 fix:1 lm:87 forms:6 connes:2 fig:1 τp:1 τq:4 non:2 introduce:1 they:2 sσ:1 now:1 definedness:1 2t:3 easily:2 term:1 finite:4 equality:5 isle:1 2d:1 iff:1 2g:2 each:4 found:1 associative:6 differential:20 ιm:2 dωi:1 map:27 series:1 tubular:6 ep:2 er:1 category:1 shown:2 space:9 restricted:1 since:3 induces:2 denotes:1 contained:1 formula:1 induced:5 suffices:5 math:8 rk:1 fiber:1 theory:4 diagonal:2 free:8 standard:1 reason:1 org:1 e1:3 e2:5 categorical:1 bϕ2:1 kohno:1 βg:1 δij:1 onto:1 βi:2 βj:9 first:2 subcomplex:2 describes:1 conjugacy:2 number:1 ring:2 given:11 similarly:1 introduction:1 construct:5 their:3 homologies:1 normalization:1 gives:2 shifted:1 progr:1 deligne:2 completes:1 coefficients:1 showed:5 cyclic:6 classes:5 powers:1 projections:1 cohen:2 curves:3 gentle:1 seen:1 any:10 tierney:1 lie:11 constructions:1 lim:5 εp:1 εr:5 oriented:5 ideal:3 εi:4 take:8 εk:2 coincide:2 regard:1 asserts:7 opposite:1 goldman:5 birkh:1 complexes:2 connected:6 class:1 indeterminates:1 constructed:1 especially:1 neighborhoods:6 calculated:1 vertices:2 points:7 sum:1 fact:1 show:3 spaces:9 relation:2 trivial:1 pontryagin:1 completion:2 based:3 basel:1 merkulov:2 lemma:25 generators:1 preliminaries:1 dj:1 intersect:3 viewpoint:1 df:7 de:3 db:4 dc:2 bar:8 covering:1 arxiv:2 coproduct:1 qz:1 da0:1 fixed:1 τk:1 set:7 surjectivity:1 respect:1 see:3 result:1 techniques:1 review:2 outside:1 closed:7 between:2 αi:3 invariant:1 boundary:4 αj:8 notice:2 terms:1 wr:2 τj:29 amer:2 ker:3 topology:7 σk:2 σj:1 σi:54 cor:1 last:2 σn:1 according:5 homotopy:2 thom:1 connection:1 faces:1 genus:1 otherwise:2 point:1 sullivan:5 ca:1 cohomology:4 corresponds:1 dimensional:2 simply:6 sequences:2 cr:1 cq:14 pr:2 pp:2 commutative:3 pt:1 basis:2 pz:3 due:1 ai:3 much:1 pk:2 algebras:12 ann:1 define:19 surjective:3 direction:2 aj:1 commutes:2 cotangent:1 subspace:1 noncommutative:3 bull:1 c1:6 c0:10 manifolds:4 case:8 fujii:1 p1:1 chas:3 near:2 loop:15 vol:1 endowed:1 them:1 surface:5 dual:3 verified:3 author:1 same:3 poincar:1 complex:14 proved:1 φk:2 subalgebra:6 composition:1 http:1 flows:1 hand:2 v0:4 v1:2 v2:2 purpose:1 identity:1 σp:31 acknowledgement:1 well:2 model:3 dimension:1 manifold:11 graded:13 composing:3 λx:1 obtain:18 converges:1 λm:21 griffiths:2 gt:1 preprint:1 generator:1 cup:5 regarded:1 discuss:1 theorem:28 coalgebras:1 unbased:1 real:2 isomorphisms:1 spanned:1 intersection:7 formal:2 like:1 follows:1 integrals:14 arbitrary:1 become:1 immersions:1 reduced:1 because:1 sequence:2 sends:1 some:5 b1:3 diff:2 integration:1 comments:1 does:1 decomposition:1 ddc:2 αg:1 considering:1 power:1 equivalent:2 bi:3 bj:1 tv2:2 simplex:3 properties:1 injective:1 algebraic:1 soc:2 inclusion:5 h0:28 h1:7 integral:1 isomorphic:11 three:1 references:1 operator:2 support:3 there:14 yan:2 way:3 notices:1 form:2 hs:3 hp:8 hq:11 koichi:1 ht:3 construction:2 hd:1 he:1 ωy:1 ωx:10 ωp:46 inside:3 ωr:27 smooth:11 up:5 us:2 ωi:39 ωk:37 ωj:2 ωm:4 ωl:8 iterated:15 ωn:32 called:1 constant:3 defined:11 ak:2 describe:3 denote:14 aq:4 ap:1 dη:1 dγ:1 bracket:9 ni:3 functions:1 no:3 when:1 field:1 other:2 mathematics:1 briefly:2 elements:4 bundle:1 helpful:1 a1:9 a0:5 generated:1 a2:7 augmentation:5 structure:6 jones:5 dω:1 assume:6 curve:1 dϕ:1 having:2 |
0704.0015.txt |@word replacing:1 limited:1 gs:1 consider:5 pointing:1 gp:1 mnp:1 2s12:1 mnr:1 iib:2 desirable:2 results:4 integration:4 leads:4 scalar:6 γmnp:5 computation:3 edu:1 consists:3 mna:1 expressions:17 contributions:6 issues:2 gh:10 γz:5 γy:5 γx:5 const:1 calculate:1 superspace:13 γs:3 γq:3 technique:1 tr:44 δhr:2 containing:1 presents:1 γi:2 program:1 δhk:1 those:4 th:18 under:4 γc:1 γb:6 γa:9 must:4 worth:1 γe:1 eliminate:1 conservation:1 case:9 supported:1 amplitudes:28 string:3 straightforward:4 returning:1 very:3 rise:1 choice:4 4c2:1 witten:1 mpr:5 stays:1 application:2 affect:2 extending:1 simplify:5 matrices:7 condition:1 chapel:1 wa:2 four:51 poitiers:1 6a1:1 subsets:1 immediately:1 level:3 turns:3 xyz:4 t5:1 list:2 solution:2 t1:20 stemming:1 solved:7 vector:7 manifestly:1 gauge:10 dαn:1 exhibits:1 equation:7 multiplies:2 jhep:1 mn:24 slower:1 zi:8 ten:9 dealing:2 sign:4 translating:2 elegant:1 zero:17 mnpq:11 displays:1 correlator:21 further:1 odd:16 canonical:4 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Stanford Triplets
Some measurable characteristics of the dataset:
- D — number of documents
- W — modality dictionary size (number of unique tokens)
- len D — average document length in modality tokens (number of tokens)
- len D uniq — average document length in unique modality tokens (number of unique tokens)
| D | @word W | @word len D | @word len D uniq | |
|---|---|---|---|---|
| value | 963564 | 3.55391e+09 | 3688.3 | 962.404 |
Information about document lengths in modality tokens:
| len_total@word | len_uniq@word | |
|---|---|---|
| mean | 3688.3 | 962.404 |
| std | 3482.8 | 458.434 |
| min | 0 | 0 |
| 25% | 1791 | 665 |
| 50% | 2835 | 878 |
| 75% | 4608 | 1184 |
| max | 754814 | 17874 |
Metadata: There are triplets that can be used to measure quality.
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