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19aaa53ff362d4519eceec3f6eacadddcea51bd3 | subsection | 10 | 16 | Comparison of the spectral functions for the SCA and C-T QMC methods | First we compare the one-particle spectral functions obtained from the SCA and CT QMC
methods with Pade approximation for analytical continuation. Since the process, in which G(i\omega ) calculated by the CT QMC
method changes into G(\omega ) with Pade approximation, introduces large
error, it is useful check to compar... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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7ebaa771b537e6e7e95ea3764a77627f435657a4 | subsection | 11 | 16 | The metal-insulator transition with a first-order transition | Here we present our results on the MIT due to geometrical frustration effect
obtained with the CT QMC method. In previous study of unfrustrated square lattice using DCA method,
it was shown that short-range AF correlations destroy the Fermi liquid
quasiparticle peak at finite temperature . According to
Ref. 25, the aut... | {
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"doi"... | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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f32b595a876e18e2de20007c2e6db3ddc6057137 | subsection | 12 | 16 | The metal-insulator transition with a first-order transition | U_c=7.2 for T=0.2, U_c=6.9 for T=0.1 andU_c=6.7 for T=0.05]Moreover, we find that the anomalous character in the metallic
state is unlike the results of the nearest-neighbor spin
correlation on the Kagome lattice. In the metallic state the spin correlation
is weak. This is the reason that geometrical
frustration is mor... | {
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} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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61172db52fcf9bc53f4033e8aa8a8c81784c6b53 | subsection | 13 | 16 | Comparison of Green's functions among the DCA, DF and DMFT methods | In this part we using the DMFT, DF and DCA methods with N_c=4 and N_c=16 to study
the non-local correlation effects and compare the on-site and
nearest-neighbor Green's functions in the Matsubara space.
[Figure: The imaginary part of on-site Green's function for \beta =4, (a)U=6 and (b) U=10. The real part of nearest-n... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
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b0871628f8ec8516bb6832c7d796c9e1c468c962 | subsection | 14 | 16 | The spin susceptibility using the DF method | In order to explore a magnetic instability we measure the spin
susceptibility using the DF method. The reason why we employ DF method for
the spin susceptibility is that the cluster-extension method of the DMFT takes
a large amount of time in order to obtain the two-particle properties. On the other
hand, because the D... | {
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} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
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eae21640de6ec38aa3ac69ebfab298a753906726 | subsection | 15 | 16 | CONCLUSIONS | In summary, we have investigated the Hubbard model on the triangular lattice
using the DCA and DF method. Using the DCA method we compared the spectral functions
obtained from SCA and CT QMC methods. We found a good agreement of both methods and the quasiparticle
peak and gap structure are presented in the weak and the... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.205117 | 0807.1683 | Study of the Hubbard model on the triangular lattice using dynamical
cluster approximation and dual fermion methods | [
"Hunpyo Lee",
"Gang Li",
"Hartmut Monien"
] | [
"cond-mat.str-el"
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3aa427eaf6ab6ed9bf329f2ea27b15a3dffd79a6 | abstract | 0 | 33 | Abstract | We consider the variational problem consisting of minimizing a polyconvex
integrand for maps between manifolds. We offer a simple and direct proof of the
existence of a minimizing map. The proof is based on Young measures. | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
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5358663014ef15e9354b551e98d886cc706ba8d0 | subsection | 1 | 33 | Introduction | Let \bar{N} be a compact Riemannian manifold
with boundary of dimension
n
and M be a complete Riemannian manifold of dimension m.
We denote by N the interior of \bar{N},
and by {\rm {d}}t the non-negative Borel measure on N associated
with the metric. It can be defined as the n-Hausdorff measure
associated with the Rie... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
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3338c53289276ea02153f26ddd32d990f584ebc5 | subsection | 2 | 33 | Introduction | Let us denote by J^1(N,M) the manifold of 1-jets of maps from
N to M. If N is an open subset of \mathbb {R}^n and M=\mathbb {R}^m,
thenJ^1(N,M)=N\times M\times \mathcal {L}(\mathbb {R}^n,\mathbb {R}^m).Regularity :
We say that L:J\longrightarrow \mathbb {R}\cup \lbrace +\infty \rbrace is a normal integrand if
it is a B... | {
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"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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0.0... |
27fed7fd6cb2f38e4892b0708f96c23bd60c8b65 | subsection | 3 | 33 | Introduction | Our goal is to provide a short and direct proof of the following result:Theorem 1
Let k\in \lbrace 1,\ldots ,\min (m,n)\rbrace be given
and let L:J\longrightarrow \mathbb {R}\cup \lbrace +\infty \rbrace be a k-convex and k-superlinear
normal integrand.
Let u_0\in \text{cart}_k( N,M) be given,
such that\int _N L(t,u_0(... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
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"raw": "J. Ball, V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rat. Mech. Anal. 90 (1985), 325–388.",
"source_ref_id": "1062a326... | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
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8c7f180682b50988cc385914059c0fcadb5b079f | subsection | 4 | 33 | Introduction | In section , we prove a compactness results
under boundary conditions.
In section , we briefly expose how the various tools
exposed in sections and
lead to a proof of Theorem REF .
We also collect various related remarks.
Finally, we recall in the Appendix some relevant facts on the
topology of some spaces of measures... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
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1c53a63adc5dbe05bf58d9d2e6507b7ce92ff7c2 | subsection | 5 | 33 | Some algebra | Let E and F be two Euclidean vector spaces
of dimension n and m. It will be convenient to set p:=\min (m,n).
We denote by \mathcal {L}(E,F) the set of linear maps
between E and F.
Recall that the vector space \wedge _l E is endowed with
a natural inner product (which is induced from the inner product
of E).
This inner ... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
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-... |
94835603be29919d95c6ffdd181c684e2e3890a4 | subsection | 6 | 33 | Some algebra | If e_j^* is the dual base of E^*, then we have
\Omega =e_1^*\wedge \ldots \wedge e_n^*.
If I is a subset of \lbrace 1,\ldots , n\rbrace ,
we denote by e_I the product
e_{\alpha _1}\wedge \ldots \wedge e_{\alpha _i},
where i is the cardinal of I, and \alpha _j, 1\leqslant j\leqslant i
are the elements of I in increasing... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
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51a4f190cd13467df945d8dac24e92b5dbcafd78 | subsection | 7 | 33 | Generalized maps | Let us first recall the definitions of the Sobolev space
W^{1,q}(N,M). We say that u belongs to this space if there
exists a v(t)\in \mathcal {L}(T_tN,T_{u(t)}M), depending measurably on t,
such that\int _N\Vert v(t)\Vert ^q_{t,u(t)} {\rm {d}}t<\inftyand{\rm {d}}(\chi \circ u)_t={\rm {d}}\chi _{u(t)}\circ v(t)in the se... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
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d0d18c51474e11b46f98e8e4090c1fb469e30773 | subsection | 8 | 33 | Young measures | Let us denote by J the set of
1-jets of maps u:N\longrightarrow M.
In many examples, N is an open subset of \mathbb {R}^n,
M is \mathbb {R}^m, and thenJ=N\times \mathbb {R}^m\times \mathcal {L}(\mathbb {R}^n,\mathbb {R}^m).We shall usually denote by (t,x,v) the points of J.
We define the functionr_k(t,x,v)=1+\Vert v\Ve... | {
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0cf9675de1f9ae7f9c1dfdd46ec240039a396269 | subsection | 9 | 33 | Young measures | It follows that the map \eta \longmapsto \int L{\rm {d}}\eta
is the uniform limit on \mathcal {Y}_k(N,M)
of the continuous maps \eta \longmapsto \int L_i{\rm {d}}\eta ,
and therefore it is continuous on \mathcal {Y}_k(N,M).In the general case,
we first write the integrand L(t,x,v)=r_k(t,x,v)g(t,x,v)
with a normal inte... | {
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a1826bed603eb941898cf423a9fc23bf522cc505 | subsection | 10 | 33 | Closed measures | It is a fundamental and well known observation that
there exists many null-Lagrangians, that is functions
F(t,x,v):J^1(N,M)\longrightarrow \mathbb {R} such that\int _N F(t,u(t),{\rm {d}}u_t){\rm {d}}t=0for all C^1 maps u:N\longrightarrow M.
We define \mathcal {N}_k(N,M) as the sets of continuous functions
F(t,x,v) such... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
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0.... |
a6df8b53099145e86b6037d2c9c2ca1ca2e7faea | subsection | 11 | 33 | Closed measures | If l=1, for example, U is a vector-field, and
\dot{U}=\text{div} U.Lemma 4
For each l\in 1,\ldots , k, each smooth (l-1)-form \chi
on M such that both \chi and {\rm {d}}\chi are bounded,
and each compactly supported smooth field U(t) of l-vectors on N, the functionF(t,x,v):=
\chi _x\circ \wedge _{l-1}v\cdot \dot{U}(t... | {
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} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
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] | 2,008 | en | Mathematics | [
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0.032559238374233246,
... |
2068e883474421ce2fb834cb85f28e609d6470d6 | subsection | 12 | 33 | Closed measures | We have0=\int _N(Id\times u)^*{\rm {d}}\xi =
\int _N{\rm {d}}\xi _{(t,u(t))}\circ \wedge _n(Id\times {\rm {d}}u_t)\cdot \lambda {\rm {d}}t=(-1)^{l+1}\int _N (i_{\dot{U}}\Omega \wedge \chi )
\circ \wedge _n(Id\times {\rm {d}}u_t)\cdot \lambda {\rm {d}}t
+ (-1)^{n-l}
\int _N (i_{ U}\Omega \wedge {\rm {d}}\chi )
\circ \we... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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4ebc87ec355a34ec41eaf26ad866bcc69307cbd1 | subsection | 13 | 33 | Generalized maps and Cartesian maps | The closed measure \eta \in \mathcal {C}_k is called a generalized map
if there exists a measurable map u:N\longmapsto M
such that the marginal of \eta on N\times M
is concentrated on the graph of u.
We then say that \eta is a generalized map over u.
We denote by
\mathcal {G}_k(N,M) the set of generalized maps.Definiti... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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0.... |
8be318431de57ed8388417e56b1e888ca969b749 | subsection | 14 | 33 | Generalized maps and Cartesian maps | Settingg_i(t):=\int _{\mathcal {L}(T_tN,T_{u(t)}M)} \wedge _j v \;d\Gamma _t(v),we have u\in W^{1,1}(N,M), g_1(t)= du_t and g_i(t)=\wedge _i g_1(t)
for almost all t.By Jensen's inequality, we immediately obtain:Corollary 6
If \eta is a generalized map over u, and if L is k-convex,
then\int _{J^1(N,M)}L \textrm {d}\eta... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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fa61d618a56a8cbc76df5e20dca5aa9e493d85e4 | subsection | 15 | 33 | Generalized maps and Cartesian maps | The maps g_l satisfy
the following equations:\int _N {\rm {d}}\chi _{u(t)}\circ g_1(t) \cdot U(t) {\rm {d}}t+
\int _N \chi (u(t))\cdot \dot{U}(t){\rm {d}}t=0for all smooth vector-field U on N supported in the interior
of N, and all smooth function \chi :M\longrightarrow \mathbb {R},
and\int _N {\rm {d}}\chi _{u(t)}\cir... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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0... |
8f26c8d3d0ef24790e0a87e400f98813a5f5b4e8 | subsection | 16 | 33 | Generalized maps and Cartesian maps | If f is smooth with bounded differential, h is a smooth diffeomorphism onto its image
h(\tilde{N})\subset N,
and u\in \text{cart}_k(N,M), then f\circ u\circ h\in \text{cart}_k(\tilde{N},\tilde{M}).
Moreover, the distributional minors \tilde{g}_i of f\circ u\circ h
are :\tilde{g}_i(\tilde{t})= \wedge _i {\rm {d}}f_{u(h(... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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dd4b75bf04c1ddfbcb5d9857d084c8fda4472987 | subsection | 17 | 33 | Generalized maps and Cartesian maps | This is true because (REF ) is equivalent to
(REF ), and (REF ) for C^1 maps follows because
\tilde{F} is a null Lagrangian, and thus it sends the C^1 map
f\circ u\circ h into zero; this amounts to (REF ) by the chain rule.In order to prove the equality between distributional minors,
we expand (REF ) to\int _{\tilde{N}... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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595b4aa6e255cfdbedd0e50a07d36736c77dc018 | subsection | 18 | 33 | Generalized maps and Cartesian maps | On the other hand, since the function u^{\infty }
is smooth, we know that its distributional minors
are \wedge _ldu^{\infty }(t), which here are
just the constant functions \wedge _l g_1(0).
Therefore, by uniqueness of the distributional minors,
we have proved that
\wedge _lg_1(0)=g_l(0).We have proved Theorem REF . We... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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7a2d88c13ccb76ca539e1f07e01f08df81d63c28 | subsection | 19 | 33 | Topology | The set \mathcal {G}_k of generalized maps is endowed with the topology of
\mathcal {Y}_k.Proposition 11
The set \mathcal {G}_k(N,M) of generalized maps is closed in \mathcal {Y}_k(N,M).Proof. Let \eta ^j be a sequence of generalized maps above u^j.
Let us assume that the sequence \eta ^j is converging to \eta
in \ma... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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1c36f1f07f9e60f466e528dca3d90b74277d636a | subsection | 20 | 33 | Boundary conditions and compactness | In most applications, the manifold M is not compact,
and it is necessary to introduce boundary conditions in order
to get compactness.
We fix, as explained in the introduction,
a map u_0\in \text{cart}_k(N,M).
We define the set \mathcal {C}_k(N,M;u_0)\subset \mathcal {C}_k(N,M)
of closed measures
with boundary u_0 as
t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1224,
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"raw": "M. Giaquinta, G. Modica, J. Soucek, Cartesian currents in the calculus of variations I, Springer, Berlin, 1998.",
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"start": 1... | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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80dd4ae476ea6b03b3080915111c13dfdfcaa109 | subsection | 21 | 33 | Boundary conditions and compactness | So it is enough to prove that it is relatively compact.
By the appendix, this follows if we can prove that it
is r_k-tight.
In other words, we have to show
that for each \epsilon >0 there exists a compact subset Z\in J
such that\int _{J\setminus Z(R)}r_k(t,x,v){\rm d}\eta (t,x,v)\leqslant 2\epsilonfor each measure \eta... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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fca4bc7f778b8b9cbedf4767a1cca6faeeeba840 | subsection | 22 | 33 | Boundary conditions and compactness | Indeed, taking R_0 such that (REF ) holds, and then
setting S=\max _{\tilde{Z}(R_0)}r_k, we get\int _{J\setminus Z(R)}r_k {\rm {d}}\eta \leqslant \int _{J\setminus \tilde{Z}(R_0)}r_k {\rm {d}}\eta +S\eta (J\setminus \hat{Z}(R))\leqslant \epsilon + S\epsilon (R).In order to prove (REF ), we consider, for each R>0, a
fun... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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0e7075324f970487a8744f9d37b0f299338a6a68 | subsection | 23 | 33 | Conclusion | We now collect the tools we have introduced to prove Theorem
REF . We also add some discussions and variations. | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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44ab58fc961e257391ee836eb7fc5efa861ba0ca | subsection | 24 | 33 | Proof of Theorem | By Propositions REF and REF ,
there exists a generalized map \eta over some u\in W^1(N,M) such that \eta minimizes \int L{\rm {d}}\eta on \mathcal {G}_k(N,M;u_0). We want to show that u minimizes in \text{cart}_k(N,M;u_0).If v\in \text{cart}_k(N,M;u_0) is another map,
we have\int _{J^1(N,M)} L(t,u(t),{\rm {d}}u_t){\rm ... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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edee1be5287a9b464ff02341e7219c3ade81ea35 | subsection | 25 | 33 | An example | We consider M=\mathbb {R}^2; N=B, the unit
open ball of
\mathbb {R}^2, and the LagrangianL(t,x,v)=\epsilon (|v|^p+|t|^4|v|^4 )+|\text{det } v|^2,with p\in ]1,2[.
We claim that\inf _{u\in W^{1,1}(N,M;Id)} \int _N L(t,u(t),{\rm {d}}u_t){\rm {d}}t
<
\min _{u\in \text{\rm cart}_2(N,M;Id)}\int _N L(t,u(t), {\rm {d}}u_t)dtwh... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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d93f31fc8678a501aa4382c4eccf2c0bf07e45a2 | subsection | 26 | 33 | An example | Since |{\rm {d}}f(u)|\leqslant 1 we have that |{\rm {d}}(f\circ u)|\leqslant |{\rm {d}}u| and
|\det {\rm {d}}(f\circ u)|\leqslant |\det {\rm {d}}u|; the first inequality is strict if |u|>1 and {\rm {d}}u\ne 0.
If we did not have u(t)\subset \bar{B} for almost every t, the action of f\circ u
would be strictly smaller th... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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1a7f2b106cf8df3efa9d9df1643d1251b0f67001 | subsection | 27 | 33 | An example | In order to prove (REF ),
we set u_i=(u^1_i,u^2_i) and u=(u^1,u^2), we call (x_1,x_2)
the coordinates on the target space \mathbb {R}^2 and we
assert that\int _B\det {\rm {d}}u\cdot \phi _r{\rm {d}}t=
\int _B
u^1(\partial _2u^2,-\partial _1u^2) \cdot \nabla \phi _r{\rm {d}}t .Indeed, this formula is just (REF ) with l=... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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d422d1ae90d33b015a42145ae985000361acf7d7 | subsection | 28 | 33 | Weak continuity of minors | Let us mention the following classical result
which follows from our tools (see , 3.3.1
or , 8.3):Proposition 15 Let N be a bounded disc in \mathbb {R}^n.
Let u_i be a sequence of maps in \text{cart}_k(N,\mathbb {R}^m),
and let u\in W^{1,1}(N,M) and
g_j(t)\in L^1(N,\mathcal {L}(\wedge _j\mathbb {R}^n,\wedge _j\mathbb {... | {
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... | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
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] | 2,008 | en | Mathematics | [
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e88b41ff0086d3a8650fc69b5eb33ddbce104237 | subsection | 29 | 33 | On Null-Lagrangians | It may seem unnatural in the definitions of the sets
\mathcal {N}_k(N,M) to require that the null-Lagrangians
F(t,x,v)
be k-affine functions of v.
Indeed, working with a larger set \mathcal {N}(N,M)
of null-Lagrangians would make the result stronger,
and may allow to relax somewhat the k-convexity hypothesis on L.
The ... | {
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... | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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e9b514d25321695a088dad9e78603b9a0c36058e | subsection | 30 | 33 | More general setting | The heart of the matters is the Jensen's inequality obtained
in Corollary REF .
This inequality is the result of an equilibrium between
the known properties of the measures \Gamma _t
appearing in the disintegration of generalized maps
and the convexity assumed on the integrand L.Other, but less explicit equilibria migh... | {
"cite_spans": []
} | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
"math.OC",
"math.AP"
] | 2,008 | en | Mathematics | [
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2f756e54a34928bb94e45d1827f9e083aba5a520 | subsection | 31 | 33 | More general setting | Let \mathsf {P}_{t,x} be the closure, in \mathcal {P}_{t,x}
of the measures of the form\Gamma =({\rm {d}}u)_{\sharp } ({\rm {d}}t_{|B})where u:T_tN\longrightarrow T_xM is a smooth map supported in B.
Note that if \Gamma \in \mathsf {P}_{t,x}, then
\int _{\mathcal {L}(T_tN,T_xM)} v {\rm {d}}\Gamma =0.
A last notation is... | {
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{
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"raw": "P. Pedregal, Parametrized Measures and Variational Principles, Progress in nonlinear Partial Differential Equations and their Applications Birkhauser (1997).",
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"Patrick Bernard",
"Ugo Bessi"
] | [
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"math.AP"
] | 2,008 | en | Mathematics | [
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31578dc7b5ae9c4b45a47ce4332f59d9ca735bfd | subsection | 32 | 33 | Kantorovich-Rubinstein space | Let us recall some standard facts on probability
measures, see , .
Let (X,d) be a complete and separable metric space,
and let r:X\longrightarrow [1,\infty ) be a continuous function.
Let \mathcal {P}_r(X) be the set of Borel probability measures
\mu on X
which satisfy\int _X r(x) d\mu (x)<\infty .Let us denote by C_r(... | {
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{
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"end": 66,
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"raw": "L. Ambrosio, N. Gigli and G. Savaré, Gradient flows, Lectures in Math. ETH Zürich, Birkhäuser (2005).",
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... | 10.4134/JKMS.2010.47.2.331 | 0807.1684 | Young measures, Cartesian maps, and polyconvexity | [
"Patrick Bernard",
"Ugo Bessi"
] | [
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d4fa93e105b6d796d783e8a19054a45183bedc66 | abstract | 0 | 14 | Abstract | We consider the model of Directed Polymers in an i.i.d. gaussian or bounded
Environment in the $L^2$ region. We prove the convergence of the law of the
environment seen by the particle. As a main technical step, we establish a
lower tail concentration inequality for the partition function for bounded
environments. Our ... | {
"cite_spans": []
} | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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... | |
71036849a8df691447804484f72e4d490f7528f3 | subsection | 1 | 14 | Introduction | We consider the following model of directed polymers in a random media: let d\ge 3 and let \lbrace \eta (t,x):\, t\in \mathbb {Z},\, x\in \mathbb {Z}^d\rbrace denotes a family of real variables. We call it the environment.
For y\in \mathbb {Z}^d, let P_y be the law of the simple symmetric nearest neighbor random walk o... | {
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"raw": "Imbrie, J.Z., Spencer, T. (1988) Diffusion of directed polymers in random environment, J. Stat. Ph. 52 3/4, 609-626.",
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"star... | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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4dbd267ada86244a47ea8f0c3f15d2297bec4e62 | subsection | 2 | 14 | Introduction | It turns out that the behavior of the polymer depends strongly on the
temperature. Here we will be concerned by a region of very high temperature, or equivalently, we will focus
on very small values of \beta . Let state this more precisely: we can define the normalized partition function,W_N = Z_N \exp \lbrace -N\lambd... | {
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"raw": "Comets, F., Shiga, T., Yoshida, N. (2004) Probabilistic Analysis of Directed Polymers in a Random Environment: a Review, Advanced Studies in Pure Mathematics 39, 115-142",
"source_ref_id": "17... | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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68f2eda09e2a37b394bba4507390b78c0afda084 | subsection | 3 | 14 | Introduction | We observe that
L_{+\infty }(\omega ,\tilde{\omega }) has a geometric distribution with parameter \pi _d, so that Q(W^2_N) is
uniformly bounded in N for\lambda (2\beta )-2\lambda (\beta )< \log (1/\pi _d),and uniform integrability follows. We call this the L^2 condition. In the general case, we can prove that, in fact,... | {
"cite_spans": [
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"start... | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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cf26c15ffb80f64b0dbbcf060cfdbd040e3837f5 | subsection | 4 | 14 | Introduction | We can state the principal result of this article:
let \overleftarrow{\eta }(t,x):= \eta (-t,x), \overleftarrow{W}_N(\eta ):= W_N(\overleftarrow{\eta })
and let \overleftarrow{W}_{+\infty } be the (almost sure) limit of \overleftarrow{W}_N when
N tends to infinity.Theorem 2
In the L^2 region,q_N:=\frac{dQ_N}{dQ} \long... | {
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"end": 1537,
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"raw": "Bolthausen,E., Sznitman, A.S. (2002) On the static and dynamic points of views for certain random walks in random environment, Methods and Applications of Analysis, 9, 3, 345-376.",
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Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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d5c3460c56b54b9d14a67ae9ce37523c443b1000 | subsection | 5 | 14 | Introduction | Then, there exists a unique law Q_{+\infty } on E such that for any A \in \mathcal {G}_N,Q_{+\infty }(A) = Q({\bf 1}_A \overleftarrow{W}_N e^{\beta \eta (0,0)-\lambda (\beta )}).The density of this law coincides with the limit in Theorem 2 but we emphasize
that our strategy of proof here is completely different.Let us ... | {
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{
"arxiv_id": "",
"doi": "",
"end": 428,
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"raw": "Comets, F., Yoshida, N.(2006) Directed polymers in random environment are diffusive at weak disorder Annals of Probability 34 1746–1770",
"source_ref_id": "cb29e8cf5f86de9e5d2ed2865a987a3ef6ec... | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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65157767be02a4a21189d6e76670f45fda3316b9 | subsection | 6 | 14 | Introduction | Again, this result is available in the L^2 region only.We will now introduce some obvious notation that will be useful in what follows:
For x\in \mathbb {Z}^d and M<N \in \mathbb {Z}, we writeW_{M,N}(x)=Z^x_{M,N}e^{-(N-M)\lambda (\beta )}.Similarly, we write\overleftarrow{W}_{M,N}(x)=e^{-(N-M)\lambda }P^x [\exp \lbrace... | {
"cite_spans": []
} | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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2258e067046310beb3499c7d47957cf44eda9447 | subsection | 7 | 14 | Proof of Theorems | In this section, we make no specific assumptions on the environment. We just need Proposition REF
and Theorem REF to hold, which is the case for gaussian or bounded environments in the L^2 region.
Actually, we will see that we don't need the whole strength of the invariance principle, but just an averaged version
of i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 806,
"openalex_id": "",
"raw": "Vargas, V. (2006) A local limit theorem for directed polymers in random media: the continuous and the discrete case, Ann. Inst. H. Poincaré Probab. Statist. 42, 5, 521–534.",
"source_ref_id": ... | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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... | |
9aa31d786c4f3ced7b441e7b0fb6d2b4b07ff6b7 | subsection | 8 | 14 | Proof of Theorems | \sum _{|x|<A N^{1/2}} \frac{W_{-N,-N+l_N}(x)}{W_{-N,0}(x)}\,p_N(x,0)\\
&+& \sum _{|x|<A N^{1/2}} \frac{R_{-N,0}(x,0)}{W_{-N,0}(x)}\,p_N(x,0).We again integrate the second term of the sum (REF ). We use Cauchy-Schwarz inequality and translation invariance:Q \sum _{|x|<A N^{1/2}} \frac{R_{-N,0}(x,0)}{W_{-N,0}(x)}\,p_N(x,... | {
"cite_spans": []
} | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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0.0... | |
e202fa7cc09d63da4754b9273fc8acd32674dd1c | subsection | 9 | 14 | Proof of Theorems | It is then enough to
prove that h_N tends to zero in L^1. Indeed, using translation invariance,
Cauchy-Schwarz inequality and the uniform boundedness of negative moments
of W_{-N,0},Q(|h_N|) &\le & \sum _{|x|<A N^{1/2}}Q \left( \left| \frac{W_{-N,-N+l_N}(x)}{W_{-N,0}(x)}-1 \right| \right) p_N(x,0)\\
&\le & Q \left( \le... | {
"cite_spans": []
} | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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0.... | |
d5a8397ae2b3bfc218a012f9d43c55655afc006f | subsection | 10 | 14 | Proof of Theorems | We can apply the central limit theorem toQ \sum _{|x|>AN^{1/2}}\mu ^x_{-N,M}(\omega _N=0)= Q \mu ^0_{0,N+M}\left(
|\omega _N|>AN^{1/2} \right).Now, by the Markov property and the local limit theorem, we have\sum _{|x|\le AN^{1/2}} \mu ^x_{-N,M}\left(\omega _N=0 \right)&=&
\sum _{|x|\le AN^{1/2}}\frac{W_{0,M}(0)\, W_{-N... | {
"cite_spans": []
} | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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d8759924c031dcd4e70ebb5d9bac37ef10679bfe | subsection | 11 | 14 | Concentration inequalities | The proof follows closely , Section 2 (see the proof of the lower bound of Theorem 1.1 therein).
Recall that we assumed that the environment is bounded by one.
In the L^2 region, it is known that QZ^2_N \le K (QZ_N)^2 (see (REF )). This implies thatQ \left( Z_N \ge \frac{1}{2}QZ_N \right) \ge \frac{1}{4} \frac{(QZ_N)^2... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 96,
"openalex_id": "",
"raw": "Talagrand, M. (1998) Rigorous results for the Hopfield model with many patterns, Probab. Th. Relat. Fields 110, 177-276.",
"source_ref_id": "390be7da7ab9eeda8a7b2f59c4d54663f083c245",
"st... | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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... | |
440ff5ceeb3b025687a4edae23470d1cdf65c27e | subsection | 12 | 14 | Concentration inequalities | Then for
every finite sequence (y^{(k)})^M_{k=1} \subset B and any sequence of non-negative number such that
\sum ^M_{k=1} \alpha _k=1, we have that|x_i - \sum ^M_{k=1} \alpha _k y^{(k)}_i| \le 2 \sum ^M_{k=1}\alpha _k h(x,y^{(k)})_i,for every i \in T_N. This yields||x- \sum ^M_{k=1} \alpha _k y^{(k)} || \le 2 \left( \... | {
"cite_spans": []
} | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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0.... | |
d0a43b888444186e605ce8d4e3a6bad129fd9e39 | subsection | 13 | 14 | Concentration inequalities | But
using successively the Cauchy-Schwarz inequality, (REF ) and the fact that \overline{x}\in A, we find that\left|\left\langle \sum ^N_{t=1} (\overline{z}(t,\omega _t)-\overline{x}(t,\omega _t)\right\rangle _{\overline{x}} \right|
&=& \left|\sum ^N_{t=1} \sum _a (\overline{z}(t,a)-\overline{x}(t,a) \left\langle {\bf ... | {
"cite_spans": []
} | 0807.1685 | Convergence of the law of the Environment Seen by the Particle for
Directed Polymers in Random Media in the $L^2$ region | [
"Gregorio Moreno Flores"
] | [
"math.PR"
] | 2,008 | en | Mathematics | [
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f8b654bc9a35912b62b18d42bd4b5915ed60a35e | abstract | 0 | 23 | Abstract | We use lattice QCD simulations, with MILC configurations (including vacuum
polarization from u, d, and s quarks), to update our previous determinations of
the QCD coupling constant. Our new analysis uses results from 6 different
lattice spacings and 12 different combinations of sea-quark masses to
significantly reduce ... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
] | 2,008 | en | Physics | [
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1ff2532613d3e1b555ec119e75528dc292bb410b | subsection | 1 | 23 | Introduction | An accurate value for the coupling constant \alpha _s in quantum
chromodynamics (QCD) is important both for QCD phenomenology, and as an
input for possible theories beyond the Standard Model. Some of the most
accurate values for the coupling constant come from numerical
simulations of QCD using lattice techniques, when... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 488,
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"raw": "Q. Mason et al. [HPQCD Collaboration], Phys. Rev. Lett. 95, 052002 (2005) [arXiv:hep-lat/0503005].",
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"start": 386
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... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
] | 2,008 | en | Physics | [
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2dced318c9911e7b381ddbf9742f05ec1fae3537 | subsection | 2 | 23 | Introduction | We know the values of the coefficients
c_n through order n=3 (next-to-next-to-leading order) for the
quantities we examine, yet unknown higher-order coefficients still have
an impact at the level of accuracy we seek. A main focus of this paper
is to address these complications, and quantify the uncertainties in
our det... | {
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]
} | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
] | 2,008 | en | Physics | [
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7dd290badd516c92522aac62d5f6f4a8a8d8d49d | subsection | 3 | 23 | Perturbation Theory | The simplest short-distance quantities to simulate are vacuum
expectation values of Wilson loop operators:W_{mn} \equiv \mbox{$\frac{1}{3}$}\,\langle 0|\,
\mathrm {Re\,Tr}\,\mathrm {P}\,\mathrm {
e}^{-ig\oint _{nm}
\!A\cdot dx}\, |0\rangle ,where {\rm P} denotes path ordering, A_\mu is the QCD vector
potential, and th... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1586,
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"raw": "Q. Mason and H. Trottier, in preparation (2005); Q. Mason, Cornell University Ph.D. thesis (2004).",
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"start": 1531
}
]... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
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d0bf6375ef54a2f0a506676bfd23cbcb8463495f | subsection | 4 | 23 | Perturbation Theory | The quarks in our
simulations are not exactly massless, but the masses are sufficiently
small that the difference is negligible, {\cal O}(\alpha _V^2 (am)^2), in
perturbation theory (but less so nonperturbatively, as we will
discuss).
[Table: Perturbative scale and coefficients for several small Wilsonloops W_{ij}, Cre... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1453,
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"raw": "G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250 (1993) [arXiv:hep-lat/9209022].",
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"start": 1350
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"... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
] | 2,008 | en | Physics | [
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6479c1b2eafa81456a05301b20a9191a2b21acf6 | subsection | 5 | 23 | Perturbation Theory | This is possible because the
coupling \alpha _V(d/a) changes value with different lattice
spacings a:q^2\,\frac{d\alpha _V(q)}{dq^2} = - \beta _0 \alpha _V^2 - \beta _1
\alpha _V^3
-\beta _2 \alpha _V^4 -\beta _3\alpha _V^5where the \beta _i are constants . In this
paper, we follow our earlier analysis by parameterizin... | {
"cite_spans": [
{
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"doi": "",
"end": 257,
"openalex_id": "",
"raw": "Coupling \\alpha _V(\\mu ) is defined in terms of \\alpha _{\\rm \\overline{MS}}(\\overline{\\mu }), where \\overline{\\mu }\\equiv \\mu \\exp (-5/6) , , using the static-quark potential. The potent... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
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8eac698d304202e7c1bc12fa4c9f8da7bb36216e | subsection | 6 | 23 | QCD Simulations | The gluon-configuration sets we use were created by the MILC
collaboration . The relevant simulation parameters are
listed in Table REF .The input parameters for a QCD simulation are the bare coupling
constant and bare quark masses. The coupling constant is specified
through the \beta parameter, listed in Table REF ,
... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 76,
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"raw": "C. Aubin et al, Phys. Rev. D70:094505, 2004; and MILC Collaboration, private communication.",
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"start": 0
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{
"a... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
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] | [
"hep-lat"
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e8659b8a1c51b33fc1f4379219c09f29845c4a99 | subsection | 7 | 23 | Systematic Errors | The goal of our analysis is to determine
\alpha _0\equiv \alpha _V(7.5\,\mathrm {GeV}).
The only relevant systematic errors, other than from the truncation of
perturbation theory, are from nonperturbative effects and from a^2
errors in our determination of the lattice spacings. Finite-volume
errors are no larger than o... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
] | 2,008 | en | Physics | [
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2e6db3d55407e6992bd3742db80f40e5a13f952e | subsection | 8 | 23 | Chiral Corrections | Wilson loops, being very short-distance, are almost independent of the
light-quark masses. The dependence in perturbation theory is
{\cal O}(\alpha _V^2(am_q)^2), which is negligible here given other
errors. There is a larger contribution, however, from nonperturbative
contributions that is important to our analysis. T... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
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-... |
d4d8506dd01032bfef443da4809f951bc664f0db | subsection | 9 | 23 | Chiral Corrections | We expect
w^{(1)}_m to be roughly independent of loop size since w^{(0)} is
approximately proportional to L/a (see
Section ).We can estimate the size of w^{(1)}_m from a simple argument. For
light-quark hadrons, hadronic quantities like meson decay constants or
baryon masses depend approximately linearly on the masses ... | {
"cite_spans": [
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"doi": "",
"end": 897,
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"raw": "This is quite obvious from lattice QCD simulations. It is perhaps because sea-quark contributions are suppressed by 1/N_c, where N_c=3 is the number of QCD colors, in a 1/N_c expansion of QCD.",
... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
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... |
95b876b5a627c0ed3e285d8630c6edcda22086d3 | subsection | 10 | 23 | Gluon Condensate | The leading gluonic nonperturbative contribution comes from the gluonic
condensate, \langle \alpha _s G^2/\pi \rangle . The contribution of the
condensate to a Wilson loop is easily calculated to leading order in
perturbation theory:\delta W_\mathrm {cond} = -\frac{\pi ^2}{36}\,
\left(\frac{A}{a^2}\right)^2
a^4 \langle... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 652,
"openalex_id": "",
"raw": "For a recent analysis of condensate values see B. L. Ioffe, Prog. Part. Nucl. Phys. 56, 232 (2006) [arXiv:hep-ph/0502148].",
"source_ref_id": "3edb27be770a28f9e88e64534c89b3b422c27932",
... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
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5e5e4abfb2366f3ce4bda0d9241c116a397ab1cf | subsection | 11 | 23 | Finite- | In our analysis, the scale for the couplings comes from the lattice
spacing, and the lattice spacing comes from measurements of r_1/a in
the simulations. As for any physical quantity, lattice QCD
measurements of r_1 have finite-a errors; and, using an analysis
similar to the one we outlined for Wilson loops, they shoul... | {
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"raw": "Rather than approximating \\alpha _V by a constant, we could put in the running coupling at a scale like 1/a. In this case, however, it makes little difference to the final fits. Our analysis is not... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
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7da6b2e11ebb70cec59970c472dce72170138214 | subsection | 12 | 23 | Analysis and Results | We have 22 different short-distance quantities in our analysis, each of
which produces a separate value for
\alpha _0\equiv \alpha _V(7.5\,\mathrm {GeV}). These consist of \log (W)s
for each of 8 Wilson loops, 6 independent Creutz ratios built from
these loops, 7 tadpole-improved \log (W)s, and the tadpole improved
bar... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
"J. Shigemitsu",
"H. Trottier"
] | [
"hep-lat"
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0... |
f24427ed8b55dd5b63d696b30de38ad9be35320d | subsection | 13 | 23 | Constrained Fits | We analyze each short-distance
quantity Y separately. We use a constrained fitting procedure, based
upon Bayesian ideas , to fit the values
Y_i\pm \sigma _{Y_i} coming from each of our configuration sets
(Table REF ) to a single formula. In this procedure we
minimize an augmented \chi ^2 function of the form\chi ^2 \eq... | {
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"source_ref_id": "7a... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
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"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
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84c4620b66cddc869fb75eb8dad6aa3f411efbca | subsection | 14 | 23 | Constrained Fits | For n\ge 4, we set \overline{c}_n=0
and\sigma _{c_n} = 2.5\, \mathrm {max}(|c_1|,|c_2|,|c_3|).Thus the c_ns in the fit are constrained by the values obtained from
our Feynman integrals where these are available (taking correct account
of the uncertainties in those values), while the others are allowed to
vary over a ra... | {
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"source_ref_id": "7a... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
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"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
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"hep-lat"
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86af736e6e3a2aa7947e07bd4b4bc0ec651d73b6 | subsection | 15 | 23 | Constrained Fits | For r_{1a}^{(2)}, the empirical Bayes
criterion suggests a width for the prior that is twice what we
anticipated in Section REF :\overline{r}_{1a}^{(2)} = 0
\quad \quad \sigma _{r_{1a}^{(2)}} = 2\alpha _s \approx 0.6.We use this more conservative prior in our fits. Higher-order
corrections are easily added but have no ... | {
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"raw": "A. Gray, I. Allison, C. T. H. Davies, E. Dalgic, G. P. Lepage, J. Shigemitsu and M. Wingate [HPQCD Collaboration], Phys. Rev. D 72, 094507 (2005) [arXiv:hep-lat/0507013]. The value for r_1 given in... | 10.1103/PhysRevD.78.114507 | 0807.1687 | Update: Accurate Determinations of alpha_s from Realistic Lattice QCD | [
"C. T. H. Davies",
"K. Hornbostel",
"I. D. Kendall",
"G. P. Lepage",
"C. McNeile",
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91ae128baf6bc0e5b0f4121aad0336e786dad98d | subsection | 16 | 23 | Results | The results from our 22 determinations of the coupling are listed and
shown in Figure REF . The gray band corresponds to
our final result of\alpha _{\rm \overline{MS}}(M_Z,n_f\!=\!5) = 0.1183\,(8)which was obtained from a weighted average of all of
22 determinations . Our error estimate here is that of a
typical entry ... | {
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68850430701e967a6a2183acccba56ccc2b4fd68 | subsection | 17 | 23 | Results | The
error estimate produced by our fitting code for a quantity like
\alpha _{\rm \overline{MS}} is approximately linear in all the variances \sigma ^2
that appear in the \chi ^2 function:\sigma _{\alpha _{\rm \overline{MS}}}^2 \approx &
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c16607b68a3613dae0aef068fb6b5991f2e8f6ee | subsection | 18 | 23 | Results | As noted in , the n=4
coefficients are large, particularly for \log (W)s where typically our
fits imply c_4/c_1\approx -4(2). As expected, perturbative
higher-order coefficients are smaller for other quantities: for
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94491446d71baeba6913df4432a33a8c78f9947d | subsection | 19 | 23 | Results | If we keep only the three
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nonperturbative effects, we get 0.1180(10) whether or not the gluon
condensates are included. We cannot fit all of our simulation data if
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e1115ed82d56db5d2cf5db509b2fd2fc3724c83e | subsection | 20 | 23 | Conclusions | Any high-precision determination of \alpha _s based upon lattice QCD
simulations has to address several key issues:Finite-Lattice-Spacing Errors: Errors due to the finite
lattice-spacing can enter in two ways. First they affect lattice
determinations of the physical quantity or quantities used to set the
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72329a491fe6fe72792d5c8d55e67da198689b75 | subsection | 21 | 23 | Conclusions | In the past we have used simulations with fewer than
three light-quarks and extrapolated to n_f=3 (1/\alpha _{\rm \overline{MS}}(M_Z)
appears to be reasonably linear in n_f) . Here
(and in our earlier paper ) contributions from all
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4a6b6fa97f99c98ceae79000dd8a83e814e9982e | subsection | 22 | 23 | Conclusions | While early concerns about the light-quark discretization
used here have been largely
addressed , , it remains important to
test the simulation technology of lattice QCD at increasing levels of
precision given the critical importance of lattice results for
phenomenology .We thank K. Maltman for discussions and comments... | {
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8f1deb3048666fbe16a163aae27eb531310ac8ee | abstract | 0 | 16 | Abstract | We exhibit a varying speed of light (VSL) theory that implements the recently
proposed decaying speed of sound mechanism for generating density fluctuations.
We avail ourselves of bimetric VSL theories, where the speed of gravity differs
from that of light. We first show that a Dirac-Born-Infeld (DBI) type of
$K$-essen... | {
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d2f85a8a405d03c389408189b293b7c8fd9fbf6b | subsection | 1 | 16 | Introduction | In a recent letter we proposed a mechanism for producing
scale-invariant density fluctuations of appropriate amplitude
based on a decaying speed of sound c_s.
We emphasized the mechanism's generality and how it could
be implemented using a variety of methods. The examples of
K-essence ,
and varying speed of light (VS... | {
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874c1e0155050efc364311e5841359d04abeaab4 | subsection | 2 | 16 | Introduction | A few non-perturbative features of bimetric theories are
not captured by this map; but after identifying them we're able
in Section to write down the full non-perturbative
bi-metric structure behind this model, which is simply
a cosmological constant with respect
to the matter metric.
We conclude with a discussion of ... | {
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1448ad54896dbc80eea616d36b13cd0c77efc6d9 | subsection | 3 | 16 | Bimetric VSL theories | Bimetric theories have been proposed as VSL
theories , , solving the horizon, flatness, and dark
matter problems. In the model proposed by Clayton and Moffat, for
example, there are two metrics: an “Einstein” metric
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d46bacc8551678f5bb0f3cb1e08a26e87f638de6 | subsection | 4 | 16 | Bimetric VSL theories | In the original formulation, due to Clayton
and Moffat, \phi has a Klein-Gordon Lagrangian
in the Einstein (g_{\mu \nu }) frame, that isS_\phi =\int d^4 x {\sqrt{-g}}
\,{\left(\frac{1}{2}g^{\mu \nu }\partial _\mu \phi \partial _\nu \phi -V\right)}\; .But we could also make it
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9287d19c9f3ea6db5316c555ea44e9451b155508 | subsection | 5 | 16 | Bimetric VSL theories | This is because, upon
performing variations in S, one finds\delta {\hat{g}}_{\mu \nu }=\delta g_{\mu \nu }+ 4B\partial _{(\mu } \phi \partial _{\nu )}\delta \phiso that a term related to the
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3d84ea4003f524d97ad0695d81f1423d1835d96a | subsection | 6 | 16 | The decaying speed of sound mechanism | We now review the mechanism for generating scale-invariant fluctuations
proposed in . The central result is that for any (constant)
equation of state w=p/\rho scale-invariance follows from a sound speed
c_s\propto \rho , if the fluctuations originate
from a vacuum state defined inside the (sound) horizon,
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93caf4ae154464514d11a1e61e5fc5cf9a397d16 | subsection | 7 | 16 | The decaying speed of sound mechanism | At low densities c_s must be constant so scale-invariance must
effectively require c_s=c_0(1+\rho /\rho _\star ), where c_s\approx c_0 at
low-energy and \rho _\star is the density that triggers its
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dd7528ecc08e052a2e7aa94eb2429d6c4dfaa918 | subsection | 8 | 16 | The (anti-)DBI model as a realization of scale-invariance | In we already exhibited a proof of concept realization
of the scenario just reviewed. Here we show that this realization
is the limiting case of a Dirac-Born-Infeld theory, with
a crucial change in the sign of one constant with respect to what is
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2ebebe95649417dd205fe81e36337c8a011749be | subsection | 9 | 16 | The (anti-)DBI model as a realization of scale-invariance | DBI inflation
has been studied in recent papers, where the model is tweaked so
as to reduce the speed of sound during inflation. However
the model could also be used to implement the varying speed of
sound mechanism, dispensing with inflation altogether. The DBI
model for a scalar field is based on the action:{\cal L}=... | {
"cite_spans": []
} | 10.1103/PhysRevD.79.043525 | 0807.1689 | Bimetric varying speed of light theories and primordial fluctuations | [
"Joao Magueijo"
] | [
"gr-qc",
"astro-ph",
"hep-th"
] | 2,008 | en | Physics | [
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0c9126daeb0dcb65a7e2c69a4326171e0a851fd4 | subsection | 10 | 16 | The mapping between | We now relate the two leitmotifs in this paper: bimetric
VSL theories and the varying speed of sound mechanism as implemented
by the DBI model. K-essence models can be seen as theories with an
emergent second metric . Although the authors of
distance their work from full bimetric theories, in the limit of
perturbative... | {
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"doi": "",... | 10.1103/PhysRevD.79.043525 | 0807.1689 | Bimetric varying speed of light theories and primordial fluctuations | [
"Joao Magueijo"
] | [
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84e0b35ede76fc7ed09e5e7c199d2be4415c8113 | subsection | 11 | 16 | The mapping between | Finally the subtleties in
defining the stress-energy tensor
(see Section ) are altogether ignored.
Therefore the mapping just described can only be used gingerly.A stronger connection is established in ,
considering a
background \phi =\phi _0 plus perturbations around it, a situation
well suited to cosmology. Leray's t... | {
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} | 10.1103/PhysRevD.79.043525 | 0807.1689 | Bimetric varying speed of light theories and primordial fluctuations | [
"Joao Magueijo"
] | [
"gr-qc",
"astro-ph",
"hep-th"
] | 2,008 | en | Physics | [
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5ac88336af65eecd59f8ffe08c04f78b64ba9878 | subsection | 12 | 16 | The mapping between | If we consider a minimal bimetric
theory where the bi-scalar is Klein-Gordon in the matter frame, then
in a perturbative scheme we should identify {\hat{g}}_{\mu }=M_{\mu \nu }.
Then we find that in the Einstein frame, g_{\mu \nu }, the theory
maps into a DBI model, i.e. precisely the K-essence model
that leads to scal... | {
"cite_spans": []
} | 10.1103/PhysRevD.79.043525 | 0807.1689 | Bimetric varying speed of light theories and primordial fluctuations | [
"Joao Magueijo"
] | [
"gr-qc",
"astro-ph",
"hep-th"
] | 2,008 | en | Physics | [
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5c04528f4dec94a6505f2624f23ba61d86f02e7c | subsection | 13 | 16 | The full non-perturbative structure | We are now prepared to identify the full non-perturbative bimetric
structure behind the various models presented in this paper. It
will be simpler and more elegant, and perhaps we should have
presented it upfront, bypassing the hardships of cosmology.
We stress that the work in is distinct from the bimetric
picture to... | {
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{
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"doi": "",... | 10.1103/PhysRevD.79.043525 | 0807.1689 | Bimetric varying speed of light theories and primordial fluctuations | [
"Joao Magueijo"
] | [
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8653df070a15a56b4996682630a45603f014d9ea | subsection | 14 | 16 | The full non-perturbative structure | The
Bianchi identities then force it to have a mass term
ensuring the appropriate background
scaling solutions at high energies. Its DBI behaviour in the
Einstein frame induces a varying speed of sound with the correct
profile to generate scale-invariant fluctuations with amplitude\frac{2^{3/2}(5+3w)^2}{w(1+w)}\frac{1}... | {
"cite_spans": [
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}
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} | 10.1103/PhysRevD.79.043525 | 0807.1689 | Bimetric varying speed of light theories and primordial fluctuations | [
"Joao Magueijo"
] | [
"gr-qc",
"astro-ph",
"hep-th"
] | 2,008 | en | Physics | [
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68a0b16c0d0a87172fbce93be3d4527976e11dd3 | subsection | 15 | 16 | Discussion | It is often stated that units can always be defined so that the speed of
light is a constant. This is true in the same way that units can
always be defined so that the Hubble “constant” is indeed a constant,
the acceleration of gravity is the same everywhere as it is on Earth,
the Universe is not expanding, etc, etc, e... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1508,
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"raw": "A. Guth, Phys.Rev. D23 347 (1981); A. Linde, Phys. Lett B 108, 1220 (1982).",
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"start": 1393
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"arxiv_id": "... | 10.1103/PhysRevD.79.043525 | 0807.1689 | Bimetric varying speed of light theories and primordial fluctuations | [
"Joao Magueijo"
] | [
"gr-qc",
"astro-ph",
"hep-th"
] | 2,008 | en | Physics | [
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e4a574a8355c4b1a63b32863f016982bc109bea2 | abstract | 0 | 58 | Abstract | We consider the infinite family of Feynman graphs known as the "banana
graphs" and compute explicitly the classes of the corresponding graph
hypersurfaces in the Grothendieck ring of varieties as well as their
Chern-Schwartz-MacPherson classes, using the classical Cremona transformation
and the dual graph, and a blowup... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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52a5bbc97fea193cde3e6796ab79db707f76423e | subsection | 1 | 58 | Introduction | Since the extensive study of revealed the systematic
appearance of multiple zeta values as the result of Feynman diagram
computations in perturbative quantum field theory, the question of
finding a direct relation between Feynman diagrams and periods of
motives has become a rich field of investigation. The formulation
... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 304,
"openalex_id": "",
"raw": "D. Broadhurst, D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B, Vol.393 (1997) 403–412.",
"source_ref_id": "d29d0ef6dc447d... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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... | |
442a72aafb327eb251e8fb8dcf4e2b615e36bf23 | subsection | 2 | 58 | Introduction | We first prove a
general pullback formula for these classes, which is
necessary in order to compute the contribution to the
CSM class of the complement of the algebraic simplex
in the graph hypersurface. The formula is then obtained
by assembling the contribution of the intersection with
the algebraic simplex and of it... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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bdf0597b0cffe469583932aac36399652bfe947e | subsection | 3 | 58 | Parametric Feynman integrals | We briefly recall some well known facts (cf. §6-2-3 of ,
§18 of , and §6 of ) about the parametric
form of Feynman integrals.Given a scalar field theory with Lagrangian written in Euclidean signature as{\mathcal {L}}(\phi )= \frac{1}{2} (\partial \phi )^2 + \frac{m^2}{2} \phi ^2 +
{\mathcal {L}}_{int}(\phi ),where the ... | {
"cite_spans": [
{
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... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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0.024520... | |
a21c816b5c63f1814158712c3977f10818f270ee | subsection | 4 | 58 | Parametric Feynman integrals | The coefficient s_C is a function of the external momenta attached
to the vertices in either one of the two componentss_C = \left(\sum _{v\in V(\Gamma _1)} P_v\right)^2 = \left(\sum _{v\in V(\Gamma _2)} P_v\right)^2,where the P_v are defined asP_v=\sum _{e\in E_{ext}(\Gamma ), t(e)=v} p_e,where the p_e are incoming ext... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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1529d4907493d2733215fdb39ff3fe566d306d26 | subsection | 5 | 58 | Feynman graphs, varieties, and periods | The graph polynomial \Psi _\Gamma (t) of (REF ) also admits
a description as determinant\Psi _\Gamma (t)=\det M_\Gamma (t)of an \ell \times \ell -matrix
M_\Gamma (t) associated to the graph (, §3 and
, §18), of the form(M_\Gamma )_{kr}(t)=\sum _{i=1}^n t_i \eta _{ik} \eta _{ir},where the n \times \ell -matrix \eta _{ik... | {
"cite_spans": [
{
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"start": 0
},
{
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... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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0... | |
393753ab99f6eb6327b6771c0f61c5b8561e98fb | subsection | 6 | 58 | Feynman graphs, varieties, and periods | In this sense, it can be viewed as the
evaluation of a period of the algebraic variety given by the
complement of the graph hypersurface. To understand the nature of this
period, one is faced with two main problems. One is
eliminating divergences (regularization and renormalization of Feynman
integrals), and the other ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 681,
"openalex_id": "",
"raw": "S. Bloch, E. Esnault, D. Kreimer, On motives associated to graph polynomials, Commun. Math. Phys., Vol.267 (2006) 181–225.",
"source_ref_id": "1ab9b50d98e4ce8c593ec98f03fa8e6aa2c3049d",
... | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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95a87079b9b1e6248755ae570a91d38af043c490 | subsection | 7 | 58 | Dual graphs and Cremona transformation | In the case of planar graphs, there is an interesting relation
between the hypersurface of the graph and the one of the dual graph.
This will be especially useful in the explicit calculation we perform
below in the special case of the banana graphs. We recall it here in
the general case of arbitrary planar graphs.The s... | {
"cite_spans": []
} | 0807.1690 | Feynman motives of banana graphs | [
"Paolo Aluffi",
"Matilde Marcolli"
] | [
"hep-th",
"math-ph",
"math.AG",
"math.MP"
] | 2,008 | en | Physics | [
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