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8d2758e160578ea43317078ef1e7cb586d4ee339 | subsection | 3 | 18 | Body | The trace of L^n on \Gamma _s equalsIf s^{\prime } is sufficiently close to s we may consecutively replace the contours \Gamma _s for the \eta _i by \Gamma _{s^{\prime }}, obtaining the trace of L^n on \Gamma _{s^{\prime }}. So {\rm tr}\,L^n is a locally constant function of s and the usual argument shows that it is co... | {
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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682bc5932920a61d700134978fa132cb633a9700 | subsection | 4 | 18 | Body | The kernel K_1(\eta ,\,\eta ^{\prime }) extends analytically to \eta inside \gamma and to \eta ^{\prime } inside \gamma while K_2(\eta ,\,\eta ^{\prime }) extends analytically to \eta inside \gamma . Hence by Proposition 2 the determinant of K_2 acting on \gamma equals the determinant of K_2-K_1.
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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f54776970dee9c06d9bac2366f22ec243f2b6974 | subsection | 5 | 18 | Body | The computation of traces avoids that issue.\BoxDenote by R(\eta ,\,\eta ^{\prime };\,\lambda ) the resolvent kernel of K_1, the kernel of \lambda \,(I-\lambda K_1)^{-1}\,K_1. This is analytic everywhere except for \lambda =\tau ^{-k},\ k\ge 0. We defineProposition 5. Assume that \Gamma is star-shaped with 1 inside and... | {
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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4ab819f06a5ce361c071ba21bf7c38f969d9618e | subsection | 6 | 18 | Body | Observe thatBy Taylor's theorem with integral remainder G(\eta ,\,\eta ^{\prime },\tau ^{n+1}) is equal toWe multiply this by \varphi _\infty (\tau \eta )/\varphi _\infty (\eta ) times \lambda ^n and sum over n to get R(\eta ,\,\eta ^{\prime }\,;\lambda ). We obtain the statement of the proposition for \lambda sufficie... | {
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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99104b1f7c5cbaa280dc722100fb2da122b04ecb | subsection | 7 | 18 | Body | This gives the first estimate, \Vert K_2\Vert _1=O(e^{-\delta t}).Next, G^{(m)}(\eta ,\,\eta ^{\prime },u) is O(t^m) times the exponential ofand when |u|\le \tau ^{2}, as it is in (REF ), the real part of this when \eta \in \Gamma is at mostand the expression in brackets is negative. Thus the integrand in the integral ... | {
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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f17d4ea7afc85a55aa34bea2fed2d49320e1aa6c | subsection | 8 | 18 | Body | If we use \varphi (\eta )\,\varphi _\infty (\tau \eta )=\varphi _\infty (\eta ) again we see that the kernel of K_2\,R_1\,K_2\,R_2 is e^{-t} times a linear combination ofUsing (REF ) again we see that G^{(m)}(\zeta ,\,\eta ^{\prime },u) is O(t^m) times the exponential ofAs before the maximum real part of the expression... | {
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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7beed7f5f12936b8082764e254a10e6fce69056b | subsection | 9 | 18 | Body | Thus (REF ) equals e^{-t} times a polynomial of degree 2k-x in t with top coefficientIn particular the main contribution to the sum in (REF ) comes from the summand k=m-1, and if we recall the minus sign in (REF ) we get the statement of Theorem 1.\BoxRemark. As mentioned in the introduction, we can also show that
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b5767929605f37f5f7f9a5ff371ac8b3959178d6 | subsection | 10 | 18 | Body | This, finally, is the contour \Gamma in this section: the circle symmetric about the real line and meeting it at \eta =-t^{-1/2} and \eta =1.From the identityand the fact that the {\det }_2 is bounded on \Vert \,\cdot \,\Vert _2-bounded sets, we see that is suffices to prove thatWe shall prove more, namelyWe begin by o... | {
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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eacba6781b635722e3386ff75318801367a515b5 | subsection | 11 | 18 | Body | Using this and (REF ) we see that \Vert K_2\Vert _2=O(1).Next, K_2\,R.When x is given by (REF ) we find thatFrom this and the fact that u\eta is bounded away from 1 when \eta \in \Gamma and u\le \tau ^2 we find that eachis bounded by a linear combination of productsSince G(\eta ,\,\eta ^{\prime },\,0)=\varphi _\infty (... | {
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d7eff72f41198013bcafd6a0af8696effff42159 | subsection | 12 | 18 | Body | We define a kernel J(\eta ,\,\eta ^{\prime }) acting on a circle with center zero and radius r\in (0,\,1) bywhere the integral is taken over a circle with center zero and radius in the interval (1,\,r/\tau ).Lemma 4. With \lambda given by (REF ) we haveProof. Our operators K_1 and K_2 may be taken to act on a circle wi... | {
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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b6e180679d2eadd8f1001ca304a6c6a280c17a40 | subsection | 13 | 18 | Body | The only occurrence of \tau in \mu \,J(\eta ,\,\eta ^{\prime }) is in \mu \,f(\zeta /\eta ^{\prime }) and as \tau \rightarrow 0 this tends to \zeta /(\eta ^{\prime }-\zeta ). Since the probabilities \mathbb {P}(x_m(t)\le x) are continuous in p at p=0This follows, for example, from formula (2) of . the integral fomula w... | {
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00ff7589c8220987bfccc69851ec895994c9c076 | subsection | 14 | 18 | Body | Next we want to expand the \zeta -contour outward to \Gamma _o, but in the process we might encounter a singularity of f(\mu ,\,\zeta /\eta ^{\prime }), and this causes a problem. It will happen if a ray from zero meets \Gamma _i at a point \eta and \Gamma _0 at \zeta and \eta /\zeta \le \tau . This will not happen if ... | {
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} | 10.1007/s00220-009-0761-0 | 0807.1713 | Asymptotics in ASEP with Step Initial Condition | [
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bb274341e5053f84b1d8e701e6162a370861be67 | subsection | 15 | 18 | Body | We have Re(\psi _1)<0 inside C_i, Re(\psi _1)>0 between C_i and C_m, Re(\psi _1)>0 between C_m and C_o, and Re(\psi _1)>0 outside C_o. (All these may be seen by taking appropriate points in the regions and using the fact that they are connected.) Our curves \Gamma _\eta and \Gamma _\zeta will be very close to C_m, the ... | {
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d126ed8e8078802a5cfa85c2e4d3e34bed065b29 | subsection | 16 | 18 | Body | The curve \Gamma _\zeta is described as follows: it goes from \xi -t^{-1/3} in the direction -2\pi /3 until \eta _{-\varepsilon }^-, then it takes a left turn and goes counterclockwise around C^{(-\varepsilon )} (it will be very close to C_m the while) until \eta _{-\varepsilon }^+, and then it goes backwards along ray... | {
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4ff6bb74ea4936d29057cd32444f38111e16d21b | subsection | 17 | 18 | Body | In the rescaled kernels the factor 1/(\zeta -\eta ) in A(\zeta ,\,\eta ) remains the same. Because near z=1,the factor \mu \,f(\mu ,\zeta /\eta )/\eta in B(\eta ,\,\zeta ) becomesafter the rescaling. (The \mu and \eta appearing as they do is very nice.)As for the factors e^{\psi (\zeta )} and e^{-\psi (\eta )} we see f... | {
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691a3f1799d48b3e9acf7617a7510d22d0d238bf | abstract | 0 | 40 | Abstract | We give a simplified derivation of the expression of instanton numbers and of
mirror map in terms of Frobenius map on p-adic cohomology and use this
expression to prove integrality theorems. Modifying this proof we verify that
the Aganagic-Vafa formulas for the number of holomorphic disks can be expressed
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731a214ad64e360d1343958ec15a16707357ed8c | subsection | 1 | 40 | Introduction | The original goal of this paper was to prove
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disks given in , always gives an integer (as expected from physical considerations).
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d2096402118efd087fab116e7d22b216f06c82b7 | subsection | 2 | 40 | Introduction | The inclusion of the cohomology over \mathbb {Z}
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72e4062a3dbf1189b0ada27ca61db5335b01190c | subsection | 3 | 40 | Introduction | (In our definition we use algebraically integral sections instead of topologically integral sections. However, we should require that algebraically integral sections we are using are related to topologically rational sections.)Proving p-integrality we will always consider
a family of Calabi-Yau manifolds in a neighborh... | {
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7e5307d558b8590841bab939ba550f1535af4602 | subsection | 4 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | Let us consider a family of Calabi-Yau smooth threefolds
X_z over a base \mathcal {M}. We assume that the
family is
miniversal. ( This means that the family contains all complex deformations and the base has minimal possible dimension.) We will say that the base is the moduli space of complex structures. This terminolo... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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a92e3aaa405fa7f6849451fdb6f64fa79b1ae3a2 | subsection | 5 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | We assume that \Omega
depends holomorphically on complex structure;
it is defined up to a factor that is holomorphic on \mathcal {M}. The quotientst^a=\frac{X^a}{X^0}can be considered as coordinates on the moduli space \mathcal {M}; they are called special
coordinates. (These coordinates are multivalued in general, bu... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 808,
"openalex_id": "",
"raw": "Bershadsky, M., Cecotti, S., Ooguri, H., and Vafa, C., Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitude, Comm. Math. Phys., 165 (1994), 211-408.",
"source_ref_i... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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17011ec24084c28b90d64de920f4a3827065a7a5 | subsection | 6 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | Recall, that the coordinates in the neighborhood of the boundary point are denoted by z^1,...,z^r; the points on the boundary divisor obey z^1\cdots z^r=0.One classifies flat sections according their
behavior at the boundary point z=0. Our
assumption that the point is maximally unipotent
means that there exists one (up... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1152,
"openalex_id": "",
"raw": "Batyrev, V., Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. Journal, 69 (1993) 349- 409.",
"source_ref_id": "1db74a258cd2ca41b014ec26164fe2aaa11... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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5dd7dbefbee9907e50706ae520331bbbcd33b514 | subsection | 7 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | (The canonical
coordinate t^a does not depend on the choice of g^0, however, we have a freedom to
add a multiple of g^0 to g^a; hence
t^a is defined up an additive constant.)
We will assume that q^a is expressed in terms of
z^a as a series with rational coefficients. (It is sufficient to assume that the terms linear wi... | {
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... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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8cd06e0966d64b2ddc3197d4e3b2e9effc68ddd2 | subsection | 8 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | If we are interested in p-integrality we should work with \mathbb {Z}_p-cohomology; the above assumption takes the form : C_{abc}(0) reduced mod p is a matrix of rank r
(we consider the pair a,b as one index). See Appendix for more mathematical form of this assumption.We see that the basis
e^0,e^a, e_a, e_0 consists of... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1669,
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"raw": "Faltings, G., Crystalline cohomology and p-adic Galois representations, Algebraic Analysis, Geometry and Number Theory, (J. Igusa, ed) 1989.",
"source_ref_id": "e25f23423bad06716c672229e76b25... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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38058bcadd17716455c8a84c8ece79e9fe1957af | subsection | 9 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | This map (relative Frobenius map) is compatible with Gauss-Manin connection and with the scalar product on cohomology:\nabla _a{\rm Fr}=p{\rm Fr}\nabla _a ,<{\rm Fr}x,{\rm Fr}y>=p^3{\rm Fr}<x,y>.Where we denote by \nabla _a the covariant derivative on the bundle,
corresponding to the logarithmic derivative
\delta _a=z^... | {
"cite_spans": [
{
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"end": 1929,
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"raw": "A. Schwarz, I. Shapiro, Supergeometry and Arithmetic Geometry, hep-th/0605119, p-adic superspaces and Frobenius, math.NT/0605310",
"source_ref_id": "1fb2362402d71623fbdf64a635c314b7a8d884a3",... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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2abd1cd0b403ec1c04f6f3a997211df9180f46b6 | subsection | 10 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | In particular, this is true in canonical coordinates.Let us consider the simplest situation when
the three-dimensional Betti number is equal
to four; then the moduli space of complex
structures on Calabi-Yau manifold is one-dimensional (r=1). We will regard z as a coordinate on a punctured disk belonging to the moduli ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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-0... |
95fdacdfa533bbcf415e7ea739f99bba0cd7058e | subsection | 11 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | The entries of
the matrix of scalar products of g^0, g^1,g_1,g_0 with \Omega _b (period matrix) are obtained from periods by
means of differentiation: if g is a flat section then<g,\Omega _b>=\delta ^b <g, \Omega >.We have seen thate^0=g^0,\\\\
e^1=g^1+tg^0,
e_1=g_1+f^{\prime \prime }_0g^1-(f^{\prime }_0-tf^{\prime \pr... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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8ca21c113c56f4e7b402d776ff0d5495079d7353 | subsection | 12 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | (This follows from the assumption that the basis at hand is symplectic.)Let us denote by m_{a,b} the entries
of the matrix of Frobenius map in the basis
e^0,e^1,e_1,e_0.{\rm Fr}e^0=m_{1,1}e^0,
{\rm Fr}e^1=m_{2,2}e^1+m_{1,2}e^0,
{\rm Fr}e_1=m_{3,3}e_1+m_{2,3}e^1+m_{1,3}e^0,
{\rm Fr}e_0=m_{4,4}e_0+m_{3,4}e_1+m_{2,4}e^1+m... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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-0.... |
162f74d4ffdde001ec2d37834fd1d286defa94d0 | subsection | 13 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | Solving differential equations
we can find the dependence of matrix entries of z.In canonical coordinates (i.e. when Y_1=Y_3=1) the solutions
have the following form:m_{1,2}=\alpha ,m_{2,3}=({\rm Fr} f^{\prime \prime }_0p-f^{\prime \prime }_0p^2)+p^2 c_0\alpham_{1,3}=-({\rm Fr} f^{\prime }_0-p^2 f^{\prime }_0)+\frac{1}... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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76be75a5a153be344d3dad10e74ac76909d4f02c | subsection | 14 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | Recall, that the third derivative of f_0 is
Yukawa coupling Y=\sum _{k\ge 0} c_kq^k, hence we can takef_0=c_0\frac{t^3}{3!}+{\tilde{f}}_0where{\tilde{f}}_0=\sum _{k>0}\frac{c_k}{k^3}q^k.There exists a more direct way to obtain these formulas using the basis consisting of flat connections.We denote by \mu _{a,b} the ent... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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c67ac16e7079ea857323c227ed486fb717c3a8ef | subsection | 15 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | ( In some expressions containing
f_0 nonholomorphic terms cancel , hence
there is no necessity to replace f_0 with its holomorphic part.)The calculation of m_{i,j}(0) (of the behavior of entries of Frobenius matrix at the maximally unipotent boundary point) is much more difficult.
It is based on relation between flat s... | {
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"do... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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796eb8b9e057abd52310a0113c8ee05f6d4049bb | subsection | 16 | 40 | Frobenius map on cohomology of Calabi-Yau threefolds | This factor is a root of unity, p-adic logarithm of it is equal to zero, hence it does not contribute to the expression for m_{1,2}(0).The entry m_{1,2}(0) of Frobenius matrix in the coordinate u can be expressed in terms of c as \pm \log c ^{1-p} where c should be understood as a p-adic number.In canonical coordinates... | {
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} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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e7286587d39e42e6f00bc179217911cc1f6056be | subsection | 17 | 40 | Instanton numbers | In the case when there is one Kaehler parameter
(in terms of A-model) or the moduli space of complex structures on Calabi-Yau manifold is one-dimensional (in mirror B-model) the instanton numbers n_k are related to the genus zero free energy f_0
in the following way:{\tilde{f}}_0(q)=\sum _{k=1}\sum _{d=1}d^{-3}n_kq^{dk... | {
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"source_ref_id": "d0b38d730a0b35ec092bd64ad2785c... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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a1f99868eda36d9841d1948f9eb45f051e9b72a3 | subsection | 18 | 40 | Instanton numbers | (The equivalence of (REF ) and (REF )
follows from the remark that \delta ((f)\ast (g))=
\delta (f)\ast \delta (g).)Using the new form of (REF ) we can express
the instanton numbers in terms of Yukawa coupling:{n_kk^3}=\mu \ast (Y).Where \mu (k) stands for the Moebius function
that is equal to (-1)^s if k is represente... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
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"hep-th"
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475f1805128d3921e4b217a7a70a9a1a38948bae | subsection | 19 | 40 | Instanton numbers | \sum v(k)q^k=\sum _{r=1}g(r){\rm Li}_s(q^r))
and
m=(p^{-s}{\rm Fr}v-v), theng(p^at)=-\sum _{d|t}\frac{1}{d^s}\mu ({d})m\big (\frac{p^at}{d}\big ).Here v and g are arithmetic functions,
t is a natural number that is not divisible by p.Lemma 3 follows from Lemma 2 applied to
\delta ^sv=\lbrace 1\rbrace \ast \delta ^s g a... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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83dc1eeb0ee54ae42534b1a8bcafc2c0fdd58a6d | subsection | 20 | 40 | Instanton numbers | If m=(p^{-s}{\rm Fr}v-v) has
the form \sum m({\bf k})q^{\bf k} where m({\bf k}) are p-adic integers, then v=\sum _{r=1}g(\mathbf { r}){\rm Li}_s(q^{\mathbf { r}}) where
the coefficients g(\mathbf { r}) also are p-adic integers. Conversely, the integrality of g(\mathbf { r}) implies the integrality of m({\bf k}).Using L... | {
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"Albert Schwarz",
"Vadim Vologodsky"
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"hep-th"
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-0.0... |
99e2966cf0c427215e13dd59e81bf39e51c38c07 | subsection | 21 | 40 | Instanton numbers | 2 and Appendix for more detailed discussion of our assumptions about Yukawa couplings C_ {abc}(0)=
Y_{abc}(0) in the case r>1.) For p>3 one can prove the p-adic integrality of the basis e^0,e^a,e_a,e_0 without
this additional assumption; the proof is based on theorem by Faltings stating that inner product is a perfect ... | {
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"raw": "Faltings, G., Crystalline cohomology and p-adic Galois representations, Algebraic Analysis, Geometry and Number Theory, (J. Igusa, ed) 1989.",
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"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
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454124b9dbb1645cf9723cb978d19f345a575ac8 | subsection | 22 | 40 | Integrality of mirror map | Let us consider the Hodge filtration F^3\subset F^2\subset F^1\subset F^0 on the cohomology
with complex coefficients. Recall that
F^p is a direct sum of all groups H^{k,l}
where k\ge p. We restrict ourselves to the middle-dimensional cohomology H^3, hence
\dim F^3=h^{3,0}=1, \dim F^2/F^3=h^{2,1}=r,
\dim F^1/F^2=h^{1,2... | {
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"Albert Schwarz",
"Vadim Vologodsky"
] | [
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d10127c0a9a367a66c3e7cda98a2e7f605d838e3 | subsection | 23 | 40 | Integrality of mirror map | To construct such a basis we notice that both
W_0
and W_1/W_0 are trivial bundles; more precisely, they can be characterized
as trivial Hodge structures \mathbb {Z} and \mathbb {Z}(1)^r. We define e^0 as an algebraically (and topologically) integral generator
of W_0 and obtain e^a by lifting an algebraically integral b... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
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72a223f48edcddb383d46447eddee6e9439507e4 | subsection | 24 | 40 | Integrality of mirror map | Then for the infinite productQ^i(z)=z_i\Pi _{\bf k}E_p(z^{\bf k})^{-r_{\bf k}^i},where r_{\bf k}^i are coefficients of the power expansion of r^i(z) we have{\rm Fr}\nu _j^i -\nu _j^i=\delta _j r^iwhere
\nu _j^i =\delta _j \log Q^i . If r^i(z)\in \mathbb {Z}_p[[z_1,...,z_r]] then Q^i(z)\in \mathbb {Z} _p [[z_1,...,z_r]]... | {
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} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
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0997d404852bb078b3b4bde97757709a5e72b2e0 | subsection | 25 | 40 | Integrality of mirror map | (This form of f^a follows from the remark that the monodromy preserves topological integrality.) We will assume that \det m^a_b=\pm 1; this equation was called small monodromy condition in and Morrison integrality conjecture in . If it is satisfied then without loss of generality one can assume that m^a_b=\delta ^a_b: ... | {
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"doi"... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
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9e2d0c240554988ca258c530a486af821ce9c283 | subsection | 26 | 40 | Counting holomorphic disks. | If we consider an A-model on a Calabi-Yau threefold we can try to calculate the number of holomorphic disks with boundary on a Lagrangian submanifold. This problem is ambiguous from mathematical viewpoint (one needs additional information called framing) , but the
corresponding mirror problem is well defined
, . In the... | {
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... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
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8d4077d26e2373da10e21f3fdc36b68a04ecace8 | subsection | 27 | 40 | Counting holomorphic disks. | Of course, the normalized potential depends on
the choice of relative cohomology class paired with e_0; this means that it is defined up to addition of pairing of e_0 with absolute cohomology class in X (i.e. up to addition of solution to the Picard-Fuchs equation). We will use this freedom to get rid of omitted terms ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
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31b80a53c22cb1c44dfa2a060eb50b662e07f4c6 | subsection | 28 | 40 | Counting holomorphic disks. | (If X is a projective variety a homologically trivial subvariety Y cannot be irreducible.)
In this case
the group H^2(Y)/Im H^2(X) is one-dimensional, hence
\dim H^3(X,Y)=\dim H^3(X)+1. Notice that (\ref {e}) and (\ref {ee}) remain valid for any group of coefficients; they are valid also for de Rham cohomology over any... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
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822740a7e4286879de257b549d3089b9f4344df2 | subsection | 29 | 40 | Counting holomorphic disks. | If k=2n the first lattice can be obtained from the second one by means of multiplication by (2\pi i)^{n}.)}}
}Of course, there is some freedom in the choice of B, however, we can impose conditions that eliminate this freedom. Namely, we will prove in Appendix that there is a unique lifting B of b obeying B\in F^2 and
\... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
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f6ed6130b78673df5c522b3db159fce3cdb37207 | subsection | 30 | 40 | Counting holomorphic disks. | From the other side calculating the pairing between T and {\tilde{e}}_0 we see that \alpha _0 can be interpreted as
an integral of e_0 over a non-trivial relative cycle. This means that it can be considered as the normalized potential \tilde{\mathcal {}{W}, therefore the number of holomorphic disks can be expressed in ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
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7210a244b10273ba100e034a557a3c3cb4d61973 | subsection | 31 | 40 | Counting holomorphic disks. | Using this expression and the fact that \mu _0+\gamma is divisible by p^2
we see that\alpha _0=\sum _{\bf k}\sum _{d\in \mathbb {N}}\frac{1}{d^2}N_{\bf k}{\tilde{q}}^{d{\bf k}}=
\sum _{\bf k}N_{\bf k}{\rm Li}_2({\tilde{q}}^{\bf k}),where N_{\bf k} are integers. These integers represent the numbers of holomorphic disks.... | {
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... | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
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"hep-th"
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80d8ee95d095adfab948babe57eb9ba66cff1c8d | subsection | 32 | 40 | Counting holomorphic disks. | For the dual basis
e^0, e^a,e^{\alpha },... the Gauss-Manin connection obeys\nabla _ae^0=0,\nabla _ae^b=Y^b _a e^0.Notice that in the situation when B-branes correspond to one-dimensional subvarieties the dual cohomology bundle can be interpreted as
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"Albert Schwarz",
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286770be3371a60989eb1a8c727c7590d96e61db | subsection | 33 | 40 | Counting holomorphic disks. | (Notice that we do not separate closed and open string moduli, therefore the above expression contains not only the numbers of holomorphic disks, but also some combinations of instanton numbers.)Let us consider p-adic reductions of the cohomology bundle, of the bases in fibers of this bundle and of Gauss-Manin connecti... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
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2ddd7c6a044753c255fd72680c8c5d48f28eebfe | subsection | 34 | 40 | Counting holomorphic disks. | In canonical coordinates we obtain\delta _a m^0_0=\delta _a m^b_c=\delta _am^{\alpha }_{\beta }=0,\delta _a m^b_0 +m^0_0\delta ^b_a=pm^b_a,\delta _a m^{\alpha }_0+ m^b_0c^{\alpha }_{ab}=pm^{\alpha }_a,\delta _b m^{\alpha }_a+m^k_ac^{\alpha }_{bk}=pm^{\alpha }_{\beta }{\rm Fr}(c^{\beta }_{ab}.Here \delta _a stands for l... | {
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} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
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78e173d70d792103dcc109c3e3e20afcdc450ffe | subsection | 35 | 40 | Counting holomorphic disks. | We assume that this family is defined over complex numbers and can be extended to (non-smooth) semistable family One says that the family is semistable if the singular fibers have only mild singularities. For example,if \mathcal {M} is one-dimensional a singular fiber should be a normal crossing divisor in the total sp... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
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16409a45f8257735e228c9e12dc32797698c8e78 | subsection | 36 | 40 | Counting holomorphic disks. | (Recall that N_i comes from Gauss-Manin connection, hence it follows from Griffiths transversality that N_i acts from F^k into F^{k-1}.) It follows from the above statements that \mathcal {F} is a direct sum of intersections
W_k\bigcap F^k=\mathcal {F}_{3-k} (this means that the mixed Hodge structure on the special fib... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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05edfffde81f6aa9287dee55dd85e7c0e749ca2e | subsection | 37 | 40 | Counting holomorphic disks. | These numbers are related by the formula
\lambda _k=p\lambda _{k-1}, hence \lambda _3=p^3 \lambda _0.
From the second relation we obtain
\lambda _0 \lambda _3=p^3,
hence\lambda _0 =\pm 1.We will assume that \lambda _0=1; the modifications necessary in the case \lambda _0=-1 are obvious.One can prove that {\rm Fr}x for ... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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cf398f0731edf1c94196f873311de79d00b2117e | subsection | 38 | 40 | Counting holomorphic disks. | We can apply the above consideration to {\rm Fr}/p
to construct for an arbitrary element
of W_1/W_0 a unique flat section of
\mathcal {H}/W_0 that contains this element.
These sections specify an extension of W_1/W_0; lifting them (non-uniquely) to \mathcal {H} we obtain an extension of W_1 (again denoted by the same s... | {
"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
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"hep-th"
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3d6f33ee16edb15e685e0e54ae9e8c284c57984d | subsection | 39 | 40 | Counting holomorphic disks. | (This means that skipping some terms of weight filtration we obtain a Hodge-Tate structure on the special fiber.) It follows that the natural map of V_3\bigcap F^2
onto V_3/V_2 is an isomorphism. The generator b of I corresponds to a generator
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"cite_spans": []
} | 10.1016/j.nuclphysb.2009.05.014 | 0807.1714 | Integrality theorems in the theory of topological strings | [
"Albert Schwarz",
"Vadim Vologodsky"
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72676e57242eb75cf3aa4ff1c6bd325f17d298b3 | abstract | 0 | 21 | Abstract | We prove that evolution families on complex complete hyperbolic manifolds are
in one to one correspondence with certain semicomplete non-autonomous
holomorphic vector fields, providing the solution to a very general Loewner
type differential equation on manifolds. | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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f0f31084dc3c1155b6211091ea411d5284dcceaa | subsection | 1 | 21 | Introduction | In Loewner developed a tool to embed univalent
functions into particular families of univalent functions,
nowadays known as Loewner chains. Such a tool has been
studied and extended by many mathematicians in the past years
and has been proved to be very effective in the solution of
various problems and conjectures. For... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 140,
"openalex_id": "",
"raw": "K. Loewner, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, Math. Ann. 89 (1923), 103–121.",
"source_ref_id": "56eb3201bb5dcf84857a4aa39445112faaa8708b",
"start":... | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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98c1c6491f974629469dc7e066abb518a98561bf | subsection | 2 | 21 | Introduction | \varphi _{s,t}=\varphi _{u,t}\circ \varphi _{s,u} for all 0\le s\le u\le t<+\infty .
\varphi _{s,t}:M\rightarrow M is holomorphic for all 0\le s\le t<+\infty .
For any 0\le s<+\infty and for any compact set
K\subset \subset M the function [s,+\infty )\ni t\mapsto \varphi _{s,t}(z) is locally Lipschitz continuous unif... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
-0.07039372622966766,
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0.... | |
b50cd16f20d5ebc7b2d37834aa2684949178037b | subsection | 3 | 21 | Weak holomorphic vector fields, Herglotz vector
fields and evolution families | Let M be a complex manifold of complex dimension n. Let
TM denote the complex tangent bundle of M and let ||\cdot || be a Hermitian metric along the fibers of TM. We denote
by d(\cdot , \cdot ) the distance induced on M by ||\cdot ||. Also, we let k_M denote the Kobayashi pseudo-distance on
M. For definitions and prope... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 421,
"openalex_id": "",
"raw": "M. Abate, Iteration theory of holomorphic maps on taut manifolds, Mediterranean Press, Rende, Cosenza, 1989.",
"source_ref_id": "78f12bcd6ac28235cf9983ff0282686ac77ee057",
"start": 235
... | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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c735830ad3cae38c9c06796408ce0a2eb960a2c4 | subsection | 4 | 21 | Weak holomorphic vector fields, Herglotz vector
fields and evolution families | \end{split}By the Cauchy formula and
(REF ) (taking into account that the Hermitian metric
||\cdot || is equivalent to the metric | \cdot | of n
on U) we have\begin{split}
|G(z_1,\ldots , z_n,t)-G(z_1,\ldots ,z_{n-1}, w_n,t)|&\le \frac{1}{2\pi }\int _{|\xi |=\tilde{r}_n}\frac{|G(z_1,\ldots ,z_{n-1},\xi ,t)||z_n-w_n|}{|... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1110,
"openalex_id": "",
"raw": "E.A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955.",
"source_ref_id": "20785d2ecfe3b5754967180ac5755d73e334e3d1",
"start": 513
... | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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b1ca2fe731c858ead461a6d0c9c2f3ee3d1bb7d4 | subsection | 5 | 21 | Weak holomorphic vector fields, Herglotz vector
fields and evolution families | A Herglotz vector field of order
\infty is simply said a Herglotz vector field.Remark 2.6 According to if M=D is a strongly convex domain in
n with smooth boundary then G is a Herglotz vector field
if and only if it is a weak holomorphic vector field such that
for almost every t\in [0,+\infty ) the function D\ni z\maps... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 465,
"openalex_id": "",
"raw": "F. Bracci, M. D. Contreras, S. Diaz-Madrigal, Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains. J. Eur. Math. Soc., to appear.",
"... | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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026eaeb47eaed255e064bedb35f42c45ebf83cfc | subsection | 6 | 21 | From Herglotz vector fields to evolution families | In this section we prove that to each Herglotz vector field of
order d\ge 1 there corresponds a unique evolution family of
order d\ge 1.Proposition 3.1
Let M be a complete hyperbolic manifold with Kobayashi
distance k_M. Assume that k_M\in C^1(M\times M\setminus \hbox{Diag}). Let d\ge 1 and let G(z,t) be a Herglotz
ve... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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0.0028828284703195095,... | |
353238235cd4533d5ea98ff46fa6efa550bd30b3 | subsection | 7 | 21 | From Herglotz vector fields to evolution families | Differentiating we get\begin{split}
\overset{\bullet }{h}(t)&=\frac{\partial }{\partial t}k_M(\varphi _{0,t}(z),
\varphi _{0,t}(w))\\&=(dk_M)_{(\varphi _{0,t}(z),
\varphi _{0,t}(w))}(G(\varphi _{0,t}(z),t),(\varphi _{0,t}(w),t))\le 0\quad \hbox{a.e.\ }t\in [0,+\infty ).
\end{split}Hence h is decreasing in t and therefo... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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ef36854481eec1e8171e86f9d18916f9eebadd22 | subsection | 8 | 21 | From Herglotz vector fields to evolution families | If this is true and I<+\infty , then
letting I-\delta <s<I it follows that I(s,z)>I contradicting
Step 2, and thus I=+\infty .To prove the existence of \delta >0 as before, let U be a
local chart of M which trivializes TM and such that z_0
has coordinates O. With no loss of generality we can assume
that U contains a cl... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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8bc7e41a18f8a1de0742a02d04c3cf255f717b0e | subsection | 9 | 21 | From Herglotz vector fields to evolution families | Assuming that
|x^s_{n-1}(t)|\le r for all t\in [s,s+\delta ], by
(REF ) and (REF ) we have|x^s_n(t)|\le \int _s^t |G(x_{n-1}^s(\tau ),\tau )|d\tau \le \int _s^t C(\tau )d\tau \le r, \quad t\in [s,s+\delta ]which, by induction, implies that x_n^s(t) is well defined
for all n\in \mathbb {N} and t\in [s,s+\delta ] and |x^... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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... | |
fffafdc03e9af697848de0d29dd757f7d4a5289f | subsection | 10 | 21 | From Herglotz vector fields to evolution families | The absolutely continuous
(compact) curve [s,t]\ni \eta \mapsto \varphi _{\eta ,t}(z_0) is
covered by a finite number of coordinates charts, so that we
can find a partition s=t_0<t_1<\ldots <t_m=t such that each
curve [t_j,t_{j+1}]\ni \eta \mapsto \varphi _{\eta ,t_{j+1}}(z_0) is
contained in a coordinates chart. By Pr... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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dbc74b16b34f8cf4f6f1840b5d440d11442b97a3 | subsection | 11 | 21 | From Herglotz vector fields to evolution families | We will prove that\lim _{h\in |h|\rightarrow 0}\frac{\varphi _{s,t}(hv)-\varphi _{s,t}(0)}{h}=H^{-1}(t)\cdot v,showing that z\mapsto \varphi _{s,t}(z) is holomorphic.Since the topology induced by k_M coincides with the one of
M, the previous argument based on (REF ) shows that
there exists \delta >0 such that\sup \lbra... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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b8178c1652cb8c00f8c45c57ce639961d62d12a3 | subsection | 12 | 21 | From Herglotz vector fields to evolution families | In what follows we assume that s,j(hv)js,(O) for all 0<h<<1, j=1,..., n, and leave
to the reader the (obvious) changes in case
s,j(hv)js,(O). Then, for almost
every [s,t], it follows
\begin{equation*}
\begin{split}
\frac{d f_h}{d \eta }(\eta )&=\frac{\frac{\partial \varphi _{s,\eta }(hv)}{\partial \eta }-\frac{\partial... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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... | |
d1903eedb9de6f774793949d005ef52718e078c4 | subsection | 13 | 21 | From Herglotz vector fields to evolution families | Thus, multiplying by H(),
H(\eta )\frac{d f_h}{d \eta }(\eta )-H(\eta )A(\eta ) f_h(\eta ) =
H(\eta )\sum _{j=1}^n L^h_j(\eta )\quad \hbox{a.e. } \eta \in [s,t],
and, by the very definition of H, it holds
\frac{d}{d \eta }(H(\eta )f_h(\eta ))=H(\eta )\sum _{j=1}^n
L^h_j(\eta )\quad \hbox{a.e. } \eta \in [s,t].
Inte... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1619,
"openalex_id": "",
"raw": "H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969",
"source_ref_id": "834b6151fa9552259c9470b1101d... | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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df0acb492bcdcb4cc4f8b33f2d27cbaeadb7b49c | subsection | 14 | 21 | From Herglotz vector fields to evolution families | Uniqueness follows at once from the uniqueness of
solutions of ODE's.Remark 3.3 One can drop the hypothesis of regularity of k_M from the
statement of Proposition REF by giving a meaning in
the sense of currents and distributions to the inequality
dk_M(G,G)\le 0, similarly to what has been done by M. Abate
in for infin... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 397,
"openalex_id": "",
"raw": "M. Abate,The infinitesimal generators of semigroups of holomorphic maps. Ann. Mat. Pura Appl. (4) 161 (1992), 167–180.",
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"sta... | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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03afd7631c5ebdfb4990a167b5cac8387f677bef | subsection | 15 | 21 | From evolution families to
Herglotz vector fields | Proposition 4.1
Let M be a taut manifold with Kobayashi distance k_M.
Assume that k_M\in C^1(M\times M\setminus \hbox{Diag}). Then
for any evolution family (\varphi _{s,t}) over M there exists a
Herglotz vector field G\in \mathcal {H}_\infty (M)
which verifies (REF ).
Moreover, if H
is another weak holomorphic vector ... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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fe0b7e74271bf40fd8624c36603b36f17feb8eb5 | subsection | 16 | 21 | From evolution families to
Herglotz vector fields | For all t\in [0,+\infty ) there exists a
sequence m_j(t)\rightarrow 0^+ such that G(z,t)=\lim _{j\rightarrow \infty }G_{m_j(t)}(z,t) has the property that U^{\prime }\ni z\mapsto G(z,t) is holomorphic for all fixed t and [0,+\infty )\ni t\mapsto G(z,t) is measurable for all fixed z\in U^{\prime }. Let
\lbrace K_r\rbrac... | {
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"raw": "H. Luschgy, Measurable selectiones of limits points. Arch. Math. 45, (1985), 350-353.",
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} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
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82b35491331dc1f9be2feba18ca1953db65cb649 | subsection | 17 | 21 | From evolution families to
Herglotz vector fields | Now, by (REF ) and for almost every t\in [0,+\infty ) (and for s\le t so that \varphi _{s,\eta }(z)\in U^{\prime }
for \eta \in [s,t]) we have\begin{split}
\frac{\partial \varphi _{s,t}(z)}{\partial t}&=\lim _{h\rightarrow 0}\frac{\varphi _{s,t+h}(z)-\varphi _{s,t}(z)}{h}=\lim _{h\rightarrow 0}\frac{\varphi _{t,t+h}(\v... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
-0.030587520450353622,
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0.01... | |
3878289138ca7b910f60b98bf11b91b3d172a13a | subsection | 18 | 21 | From evolution families to
Herglotz vector fields | Uniqueness of G(z,t) on U^{\prime }. We are going to
prove that if H(z,t) is another weak holomorphic vector field
over U^{\prime } such that\frac{\partial \varphi _{s,t}(z)}{\partial t}=H(\varphi _{s,t}(z),t)for almost every t\in [0,+\infty ) (and for s\le t so that
\varphi _{s,\eta }(z)\in U^{\prime } for \eta \in [s... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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... | |
c306f6a8c915b0c49d1bdb9b52e466236394ac22 | subsection | 19 | 21 | From evolution families to
Herglotz vector fields | The map M\ni z\mapsto \varphi _{s,t}(z) is holomorphic, thus by Property (2) of
Definition REF\begin{split}
k_M(\varphi _{s,t+h}(z),\varphi _{s,t+h}(w))&\le k_M(\varphi _{t,t+h}(\varphi _{s,t}(z)),\varphi _{t,t+h}(\varphi _{s,t}(w)))\\&\le k_M(\varphi _{s,t}(z),\varphi _{s,t}(w))
\end{split}for all h\ge 0.Let Z(s) be t... | {
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"raw": "F. Bracci, M.D. Contreras, and S. Díaz-Madrigal, Evolution families and the Loewner equation I: the unit disc. Preprint 2008, available at arXiv:0807.1594.",
"source_ref_id": "44b6432e2ff249d... | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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0.... | |
c75fe8b2c4c5fe9280ae09a2e68dc81b573d0ddb | subsection | 20 | 21 | Univalence of evolution families | Proposition 5.1
Let (\varphi _{s,t}) be an evolution family on a taut manifold M.
Then for every 0\le s\le t<+\infty the map M\ni z \mapsto \varphi _{s,t}(z)\in M is univalent.Fix 0\le s\le t<+\infty . We show that
\varphi _{s,t} is injective on M. By contradiction, assume
\varphi _{s,t}(z_0)=\varphi _{s,t}(z_1) with ... | {
"cite_spans": []
} | 0807.1715 | Evolution Families and the Loewner Equation II: complex hyperbolic
manifolds | [
"Filippo Bracci",
"Manuel D. Contreras",
"S. Diaz-Madrigal"
] | [
"math.CV",
"math.CA"
] | 2,008 | en | Mathematics | [
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df5496a167183f1e3ba466b89766991722aa0827 | abstract | 0 | 9 | Abstract | Near Earth Objects (NEOs) are fragments of remnant primitive bodies that date
from the era of Solar System formation. At present, the physical properties and
origins of NEOs are poorly understood. We have measured thermal emission from
three NEOs -- (6037) 1988 EG, 1993 GD, and 2005 GL -- with Spitzer's IRAC
instrument... | {
"cite_spans": []
} | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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00a690ebc5ccab9c31a89bacd222a4914d7c51e1 | subsection | 1 | 9 | Introduction | Near Earth Objects (NEOs)
are
bodies whose orbits pass within a few tenths of an AU of the Earth's orbit. As of this
writing, there are around 5000 NEOs known. The Pan-STARRS program is
likely to increase the number of known NEOs to \sim 10,000 or more by 2013.
These bodies are of critical interest to both the scientif... | {
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"start": 1007
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{
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"doi": "",
... | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0.010... |
6c4f81c8b63e57b90d3105531cca79bafe577006 | subsection | 2 | 9 | Observations and data reduction | We observed three NEOs ((6037) 1988 EG;
1993 GD; and 2005 GL) at 3.6, 4.5, 5.8, and 8.0 \mu m
with Spitzer's InfraRed Array Camera using the moving cluster
observing mode, tracking according to the standard NAIF ephemeris.
Table gives
the observing log and observing geometries.
These objects were chosen to be visible ... | {
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{
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"doi": "",
"end": 1423,
... | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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... |
ce8af9349ef609659ada6d3d3601b519f74bbe83 | subsection | 3 | 9 | Model results and uncertainties | Thermal fluxes were measured in all four IRAC bands for
(6037) 1988 EG and 1993 GD.
For each target, the four-band data were fit using the NEATM
by varying diameter D, albedo p_V, and \eta until \chi ^2 was minimized. D and p_V are related through the optical magnitude H: p_V = 10^{-H/2.5} \left( 1329~\textrm {km}/D\ri... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 329,
"openalex_id": "",
"raw": "Fowler, J.W. & Chillemi, J.R. 1992, in The IRAS Minor Planet Survey, ed. E.D. Tedesco (Hanscom Air Force Base, Massachusetts), 17",
"source_ref_id": "293f7f68e573e3da061ec3afab2a497a94c1cdbf",... | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0... |
e83c97ce328d293a8a0fe512f3068bcf0784a915 | subsection | 4 | 9 | Model results and uncertainties | The formal errors
on these fixed \eta solutions are derived
directly from the measurement errors:
because any acceptable fit must pass within
the measurement error bars, the percent
error on diameter is equal to the percent
error on the best measurement utilized in the fit,
divided by two (since flux is proportional to... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1431,
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"start": 1058
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{
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"doi": "",
"en... | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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ef8c10cd3b12e074ec89aafec1ad00e6f260489e | subsection | 5 | 9 | Discussion | All three objects have diameters less than 500 meters, making them
among the smallest NEOs with known albedos and diameters, and
among the smallest individual objects studied with the Spitzer
Space Telescope. All three objects also have albedos close
to 0.3,
in agreement with the idea that the NEO
population is dominat... | {
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"end": 513,
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... | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
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c9bbc0e5f31d865965904217626516f0401e9b8f | subsection | 6 | 9 | The effect of unknown lightcurve variations on diameter uncertainties | Uncertainties in diameter can arise from the rotational flux variability (lightcurve)
of an observed asteroid.
To first order, the projected area A of an asteroid with
a double-peaked lightcurve varies as
A (\phi )/A_0 = 1+(10^{\frac{\Delta m/2}{2.5}}-1)\sin 2\phi ,
where \phi is rotational phase, A_0 is the average ar... | {
"cite_spans": []
} | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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431d1aecb19602987bac34ce863c2e4d9dde166d | subsection | 7 | 9 | The effect of unknown lightcurve variations on diameter uncertainties | Therefore, T is likely to be \gtrsim 0.4~P,
making \sigma _{D}\langle t\rangle small.lcccccccc|rrrr|c
Observing log
0pt
Target
AOR
UT
H
r
\Delta
\alpha
t_{{\rm frame}}
t_{{\rm exp}}
F_{3.6}
F_{4.5}
F_{5.8}
F_{8.0}
CommentDate (mag) (AU) (AU) (deg) (sec) (sec) (\mu Jy) (\mu Jy) (\mu Jy) (\mu Jy)
(6037) ... | {
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]
} | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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fdf0513149fc67a095a82b20dc6fece1f8133d79 | subsection | 8 | 9 | The effect of unknown lightcurve variations on diameter uncertainties | (3) H magnitude uncertainty around 0.3.lcccc|cccl
Physical properties of NEOs
0pt
Target
4cThermal fluxes (\mu Jy)
diameter
albedo
\eta
Model3.6 \mu m
4.5 \mu m
5.8 \mu m
8.0 \mu m
(m)
(6037) 1988 EG 10 142 806 2970 435 (23) 0.31 (0.03) 1.64 (0.11) Floating \eta806 2970 374 (34) 0.43 (0.08) 1.30 (0.18) Floa... | {
"cite_spans": []
} | 10.1086/591668 | 0807.1717 | Diameters and albedos of three sub-kilometer Near Earth Objects derived
from Spitzer observations | [
"D. E. Trilling",
"M. Mueller",
"J. L. Hora",
"G. Fazio",
"T. Spahr",
"J. A. Stansberry",
"H. A. Smith",
"S. R. Chesley",
"A. K. Mainzer"
] | [
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66d8a2247d58fdb743fb04c6336efb78ecae38e6 | abstract | 0 | 21 | Abstract | When used with coherent light, optical imaging systems, even
diffraction-limited, are inherently unable to reproduce both the amplitude and
the phase of a two-dimensional field distribution because their impulse
response function varies slowly from point to point (a property known as
non-isoplanatism). For sufficiently... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
"A. Kuhn"
] | [
"physics.optics"
] | 2,008 | en | Physics | [
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40ce5274c7967e673c895a1b27abe3163870b633 | subsection | 1 | 21 | Introduction | Current technology, especially the ability to
manufacture aspherical surfaces, allows lenses and mirrors to be
designed that minimize the most important geometrical aberrations.
Such optical elements are nearly ideal instruments obeying the laws
of Gaussian optics even for far off-axis points and non-paraxial
rays. Sel... | {
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... | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
"A. Kuhn"
] | [
"physics.optics"
] | 2,008 | en | Physics | [
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8af8f9d3959f0f17528293e1a6b97aee90887f3d | subsection | 2 | 21 | Non-isoplanatism of a pinhole camera | The problem of non-isoplanatism can be best
understood by considering the simple example of coherent image
formation through a pinhole camera (Fig. REF ).
[Figure: Scheme for coherent image formation by a pinhole camera. The pointsP^{\prime }_1 and P^{\prime }_2 in the image plane \mathcal {I} are thegeometrical images... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
"A. Kuhn"
] | [
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0.007740501780062914,
0.02... |
af2fa50a5e41055b6b564b358cba85dc43eac23f | subsection | 3 | 21 | Non-isoplanatism of a pinhole camera | Because of this
interference, the image points may be unresolved even
if their relative positions fulfill the Rayleigh criterion.The preceding example shows that different intensity distributions must be expected from
identical object patterns depending
on their positions in the object plane. This is due to an incorrec... | {
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} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
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... |
43a0ad03d106899e9dc9e5be73d879357286a2a9 | subsection | 4 | 21 | Non-isoplanatism of a pinhole camera | The integration domain
\mathcal {A} is the pinhole opening disc, whiler&=&\sqrt{x_{o}^{2}+y_{o}^{2}+z_{o}^{2}},\\
s&=&\sqrt{x_{i}^{2}+y_{i}^{2}+z_{i}^{2}}are the distances from the point source and the observation point to
the pinhole center, andf_{x}&=&\frac{1}{\lambda }\left(\frac{x_{i}}{s}+\frac{x_{o}}{r}\right), \\... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
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0.022... |
9e47b39326f9b6bd51cae3fa154eed29b1110ff9 | subsection | 5 | 21 | Non-isoplanatism of a pinhole camera | If the phase
factor were not there, the impulse function
corresponding to any point (x_o,y_o) lying in the object plane
would be the same as the impulse function of the origin
(0,0), but translated to the geometrical image point (M
x_o,M y_o); in that case, the system would
be space-invariant, or isoplanatic, and the
i... | {
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} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
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... |
054d735acf6f8458e02f8916daae210f6f01b8b7 | subsection | 6 | 21 | Non-isoplanatism of a pinhole camera | REF ), where \mathcal {S}_o
(\mathcal {S}_i) is a spherical surface of radius z_o (z_i)
centered on the pinhole, and having its vertex on \mathcal {O}
(\mathcal {I}). The spherical field distributions on
\mathcal {S}_o and \mathcal {S}_i are described
by the amplitudes U_{S_o}(x_o,y_o) and U_{S_i}(x_i,y_i),
respectivel... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
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0... |
b46734a3feea557be4566f85716b25dcce054c5f | subsection | 7 | 21 | Non-isoplanatism of a thin lens | The scenario exhibited in the simple example of the pinhole camera
actually occurs in all optical imaging systems. When imaging with
lenses or mirrors, however, the diffraction effects are less
dramatic than with a pinhole camera. The problem of phase
distortion remains though, and may sometimes induce
unwanted intensi... | {
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... | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
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... |
08b5979b77f639da392366bb1a6c0938a7f198b3 | subsection | 8 | 21 | Non-isoplanatism of a thin lens | (REF ) varies by less than \pi
when the point corresponding to coordinates
(x_o,y_o) explores the Airy pattern \delta _{\frac{a}{\lambda z_o}}(x_{i}/M- x_{o},y_{i}/M-y_{o}) in the object plane. If this is
the case, the phase factor \exp [i2\pi r/\lambda ] can
be taken out of the integral in
Eq. (REF ). Since the faste... | {
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} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
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] | [
"physics.optics"
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0.030... |
2df541e5954577fa6b4f2c8a1e1d820f977de6c3 | subsection | 9 | 21 | Non-isoplanatism and large field imaging | According to the discussion in Sec. ,
non-isoplanatism has no effect on the intensity detected in the
image plane if condition (REF ) is
satisfied. In that case, Eq. (REF ) can be used instead of
Eq. (REF ). In practical applications
of coherent imaging, the assumption is usually made that
the field mapping from the ob... | {
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} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
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] | [
"physics.optics"
] | 2,008 | en | Physics | [
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... |
a473d10d966aef35950f2a00be2758b9a3433027 | subsection | 10 | 21 | Non-isoplanatism and large field imaging | The scale ofthe figures is expressed in microns. The intensity scale is such that 1corresponds to the expected uniform intensity in the centerof the square in the limit of an infinite lens.]To give a deeper insight of the effect of the phase
factor \exp (i2\pi r/\lambda ), we use Eq. (REF )
instead of Eq. (REF ) to com... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
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"physics.optics"
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... |
003de3fb819b9d2ccff52b08c41f7222969b131e | subsection | 11 | 21 | Non-isoplanatism and large field imaging | (REF ) becomes|U_i(x_i,y_i)|^2 \ &=& \ \frac{1}{|M|^2} \ \left|U_o\left(\frac{x_i}{M},\frac{y_i}{M}\right)\right|^2 \\
&\phantom{=}& \ \times \ \mathrm {circ}^2\left(\frac{x_i}{Ma},\frac{y_i}{Ma}\right).In Eqs. (REF ) and (REF ),\mathrm {circ}(x,y)=
\left\lbrace \begin{array}{ll}{1 & if \ \sqrt{x^2+y^2}<1, \cr 0 & if \... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
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] | [
"physics.optics"
] | 2,008 | en | Physics | [
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-0.004931955598294735,
0.002660509664565325,
... |
8681c17e00451415911267176d0ac051a330e890 | subsection | 12 | 21 | Non-isoplanatism and large field imaging | REF , the percentage of
transmitted energy is, from left to right and top to bottom, 31.1%,
56.1%, 83.4%, 96.6%, 99.2%, 99.8%, and nearly 100% for the last three images.The discussion leading to Eqs. (REF ) to (REF ) only
concerned object fields U_o(x_o,y_o) that are slowly varying on
the length scale e_o=1.22\ \lambda... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
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] | [
"physics.optics"
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2b40579b536d39f529c8a0d5808386917d9e6e0a | subsection | 13 | 21 | Non-isoplanatism and large field imaging | REF show that
the intensity distribution in the image plane exhibit two distinct pseudo-periods
(in both x and y directions): the small-scale pseudo-period Md due to the grid
structure of the object and the large-scale pseudo-period X associated with the ring
patterns due to non-isoplanatism. Strictly speaking the imag... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
"A. Kuhn"
] | [
"physics.optics"
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... |
a8bf6c06c7f112854edc8c07628bf4c8d9d1c507 | subsection | 14 | 21 | Non-isoplanatism and large field imaging | Replacing the sum over n by a sum over
the values of n^{\prime }, we obtain|U_i&&(x_i+X,y_i)|^2 =\frac{1}{|M|^2} \times \\
&&\left|\sum _{n^{\prime },m} ^{i \phi _{n^{\prime }m} }
\delta _{\frac{a}{\lambda z_o}}(\frac{x_{i}}{M}- n^{\prime }
d,\frac{y_{i}}{M}-m d)\right|^2.where\phi _{n^{\prime }m}=\frac{\pi }{\lambda z... | {
"cite_spans": []
} | 10.1016/j.optcom.2008.10.058 | 0807.1718 | Coherent imaging of extended objects | [
"E. Brainis",
"C. Muldoon",
"L. Brandt",
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] | [
"physics.optics"
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... |
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