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6a76273e3a16f2a5dba9393bb1bb05dcbb26537e | subsection | 19 | 43 | Nonequivalence between the local Lagrangian bias and the
local Eulerian bias | In our local Lagrangian biasing scheme, bias parameters are given by a
set of parameters, \lbrace \langle F^{(n)} \rangle \rbrace . In the local
Eulerian biasing scheme, on the other hand, bias parameters are given
by a set of parameters \lbrace b_n\rbrace which are coefficients of a Taylor
expansion,\delta _{\rm obj} ... | {
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distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
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aeb0f2cf16500af8c51d2fcb3b86771562750ac7 | subsection | 20 | 43 | Nonequivalence between the local Lagrangian bias and the
local Eulerian bias | (16) has a common factor of b_1^2, while the
factor (1 + \langle F^{\prime } \rangle )^2 cannot be factorized out in our
Eq. (REF ) with any reparameterization of \langle F^{\prime \prime }
\rangle . | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
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] | 2,008 | en | Physics | [
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e43baea5e05116f603db478ab3a4709c80eee81b | subsection | 21 | 43 | The halo approach as a local Lagrangian biasing scheme | The nonlinear power spectrum derived above depends on the local
Lagrangian bias only through \langle F^{(n)} \rangle defined by
Eq. (REF ). Up to one-loop corrections, we only need two
numbers, \langle F^{\prime } \rangle and \langle F^{\prime \prime } \rangle . For a general
local Lagrangian bias, these numbers could ... | {
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... | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
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] | 2,008 | en | Physics | [
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cfd697cee378ebe9d4d5aa4c1705f42c42bac61c | subsection | 22 | 43 | The halo approach as a local Lagrangian biasing scheme | In literature, a multiplicity function f(\tilde{\nu }) defined
by g(\nu ) = \tilde{\nu } f(\tilde{\nu })/2 and \tilde{\nu } = \nu ^2
is frequently introduced. The PS mass function is improved by Sheth
and Tormen (ST) to give a better fit to that of haloes in
numerical simulations of CDM-type cosmologies. The ST mass fu... | {
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{
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"doi": "",
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"raw": "R. K. Sheth and G. Tormen, Mon. Not. R. Astron. Soc., 308, 119 (1999).",
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"start": 159
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]
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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7fe0386d446b78d3ac1038b287ed70f8739b2a58 | subsection | 23 | 43 | The halo approach as a local Lagrangian biasing scheme | Defining
\nu _1 =\delta _1/\sigma _1, we have\frac{d\nu _{10}/\nu _{10}}{d\nu _1/\nu _1} =
\frac{\sigma _1^2}{\sigma _1^2 - \sigma _0^2},where \nu _1, \nu _{10}, \sigma _1, \sigma _0 are considered as
functions of halo mass M_1. Combining Eqs. (REF ),
(REF ), (REF ), and (REF ), the halo bias
function reduces toF_{M_1}... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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ea1b094d6c46d6ad8201d310d43ba03b5a498794 | subsection | 24 | 43 | The halo approach as a local Lagrangian biasing scheme | (REF )–(REF ), we have\left\langle F_{M_1}^{(n)} \right\rangle =
\frac{(-1)^n}{g(\nu _1)}
\left(\frac{\partial }{\partial \delta _1}\right)^n g(\nu _1) =
\frac{(-1)^n {\nu _1}^n}{{\delta _1}^n\, g(\nu _1)}
\frac{\partial ^n g(\nu _1)}{\partial {\nu _1}^n}.With this expression, the following consistency relation is
stra... | {
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... | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
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] | 2,008 | en | Physics | [
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3adb205cffdab195d6f0ac57819be5377a42b729 | subsection | 25 | 43 | The halo approach as a local Lagrangian biasing scheme | The above derivation is similarly applied
in this case. As a result, we have&
\left\langle F^{(n)} \right\rangle =
\frac{(-1)^n}{{\delta _{\rm c}}^n(z)}
\frac{
\displaystyle \int _{M_1}^{M_2} \nu ^n
\frac{d^n g}{d{\nu }^n} \frac{d\ln \sigma }{dM} \frac{dM}{M}
}{
\displaystyle \int _{M_1}^{M_2}
g(\nu ) \frac{d\ln \sigma... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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ffff9ac7ec6167d48d8d76d0f56182632de7b71c | subsection | 26 | 43 | Scale-dependence of the bias in quasi-linear regime | Applying Eqs. (REF ) and () to our results of
Eq. (REF ) or Eq. (REF ), a power spectrum of haloes
with one-loop corrections, either in real space or in redshift space,
can be evaluated. To demonstrate the effects of biasing and
redshift-space distortions, we show examples of power spectra and
correlation functions in ... | {
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"end"... | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
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] | 2,008 | en | Physics | [
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3a06b3200b26a17eb1b35ea47a9474a9563acb3c | subsection | 27 | 43 | Scale-dependence of the bias in quasi-linear regime | In the usual halo approach,
the scale dependence of the bias arises only from galaxy/dark matter
clustering within haloes, which is not considered in this paper. The
scale dependence shown in our results purely originates in clustering
of haloes themselves. Linear theory predicts constancy of halo bias on
large scales.... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 737,
"openalex_id": "",
"raw": "J. C. Jackson, Mon. Not. R. Astron. Soc., 156, 1P (1972); W. L. W. Sargent and E. L. Turner, Astrophys. J. Letters, 212, L3 (1977).",
"source_ref_id": "853561a50e6311571cd9667038a57711fd53b837... | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
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] | 2,008 | en | Physics | [
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dd96f78326891efaf0bcbb745071000bd544338c | subsection | 28 | 43 | Scale-dependence of the bias in quasi-linear regime | In lower panels, residual values relative to normalized
correlation functions of dark matter are plotted.Nonlinear effects degrade the signature of the BAO peak because of
random displacements of haloes, and the degradation is larger in
redshift space because of additional displacements along the lines of
sight. Halo b... | {
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"start": 381
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} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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e48af470e19fbe5d47cc62fbcf43c96011700052 | subsection | 29 | 43 | Summary | In this paper, we show that the nonlinear perturbation theory via the
Lagrangian picture is naturally incorporated with the local Lagrangian
bias on top of the redshift-space distortions. Explicit results of
one-loop power spectrum with a halo bias both in real space and in
redshift space are given. Our approach does n... | {
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"raw": "R. Angulo, C. M. Baugh, C. S. Frenk, R. G. Bower, A. Jenkins, & S. L. Morris, Mon. Not. R. Astron. Soc., 362, L25 (2005).",
"source_ref_id": "f46eee25bc06653f6c039ab1ee85a7fdc8b75496",
... | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
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] | 2,008 | en | Physics | [
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a703ed89b4b775387d5d527d47c121469903ad27 | subsection | 30 | 43 | Summary | It is straightforward to extend our formalism in
Sec. to include a nonlocal bias.Although our formalism contains a partial resummation of higher-order
Eulerian perturbations, Lagrangian perturbations are truncated at the
one-loop level. Recent developments of the renormalized perturbation
theory and its variants show ... | {
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"doi": "",
"end": 425,
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"raw": "R. Scoccimarro, Ann. N.Y. Acad. Sci. 927, 13 (2001); M. Crocce and R. Scoccimarro, , 73, 063519 (2006); M. Crocce and R. Scoccimarro, , 77, 023533 (2008); P. Valageas, Astron. Astrophys., 465, 725 (... | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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d4553dccd86ba24971a01544428330b18e96e600 | subsection | 31 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | In this Appendix, we outline a derivation of one-loop corrections to
the power spectrum with the local Lagrangian bias. Our goal is to
evaluate the cumulants of Eq. (REF ) by the perturbation
theory, and obtain a perturbative expansion of Eq. (REF ).
Most of the necessary techniques have been developed in paper I
.For ... | {
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"end": ... | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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d63113bfccaa99d3391dc85407b4839fc850f277 | subsection | 32 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | In real space, expressions of the kernels
{L}^{(n)} up to third order are given by
, & {L}^{(1)}({p}_1) = \frac{{k}}{k^2},
\\
& {L}^{(2)}({p}_1,{p}_2)
=\frac{3}{7} \frac{{k}}{k^2}
\left(1 - \mu _{1,2}^2\right),
\\
& {L}^{\rm (3a)}({p}_1,{p}_2,{p}_3)
= \frac{5}{7} \frac{{k}}{k^2}
\left(1 - \mu _{1,2}^2\right)
\left(1 - ... | {
"cite_spans": [
{
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"source_ref_id": "0c32cbecc19023adbed89ae2f9207616a902c46b",
... | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
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] | 2,008 | en | Physics | [
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a895964d06dfbb17cbaa83fe1105695f6d773f6d | subsection | 33 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | In a matrix
notation, the kernels in redshift space is given by{L}^{{\rm s}(n)} = R^{(n)}{L}^{(n)}.It is useful to define the following mixed polyspectra of linear
density field and displacement field:\left\langle \tilde{\delta }_{\rm L}({k}_1) \cdots \tilde{\delta }_{\rm L}({k}_l)
\tilde{\Psi }_{i_1}({p}_1) \cdots \ti... | {
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} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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458b74092c5a000e45c759b6b5c05ea26d8a9869 | subsection | 34 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | \\
\left.
+ \langle F^{\prime } \rangle \langle F^{\prime \prime } \rangle \, a_{11}({k})
+ \langle F^{\prime \prime } \rangle ^2\, a_{02}({k})
\right],wherea_{00}({k}) =
-\, k_i k_j C_{ij}({k})
- k_i k_j k_k \int \frac{d^3p}{(2\pi )^3}
C_{ijk}({k},-{p},{p}-{k})
\\
+ \frac{1}{2} k_i k_j k_k k_l
\int \frac{d^3p}{(2\pi )... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
-0.007209157571196556,
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0.033078208565711975,
-0.03252893686294556,... |
5a3b33b74dd17129fe2b15bcca26d3febdadc665 | subsection | 35 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | When the bias functions of these objects are F_1 and F_2,
the cross power spectrum is given by substitutions\langle F^{\prime } \rangle &\rightarrow \frac{1}{2}
\left(
\langle F_1^{\prime } \rangle + \langle F_2^{\prime } \rangle \right),
\\
\langle F^{\prime \prime } \rangle &\rightarrow \frac{1}{2}
\left(
\langle F_1... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
-0.035131845623254776,
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0.0024818924721330404,
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0... |
0700ac536d64b217256ebf9b223d6aa4f43fe345 | subsection | 36 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | (REF ) are given by sums of the polyspectra
of each order. In particular,&C_i({p}) = C^{(1)}_i({p}) + C^{(3)}_i({p}) + \cdots ,
\\
&C_{ij}({p}) = C^{(11)}_{ij}({p}) + C^{(22)}_{ij}({p})
+ C^{(13)}_{ij}({p}) + C^{(31)}_{ij}({p}) + \cdots ,
\\
&C_i({p}_1,{p}_2;{p}_3) =
C^{(2)}_i({p}_1,{p}_2;{p}_3) + \cdots ,
\\
&C_{ij}({... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
-0.04301109537482262,
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-0.010012518614530563,... |
e2f4d83dba003d6d726197966f6cd4c53083b572 | subsection | 37 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | (REF ), the mixed polyspectra of each order are given by& C({p}) = P_{\rm L}(p),
\\
& C^{(1)}_i({p})
= L^{(1)}_i({p}) P_{\rm L}(p),
\\
& C^{(11)}_{ij}({p})
= - L^{(1)}_i({p}) L^{(1)}_j({p}) P_{\rm L}(p),& C^{(3)}_i({p})
= \frac{1}{2} P_{\rm L}(p) \int \frac{d^3p^{\prime }}{(2\pi )^3}
L^{(3)}_i({p},-{p}^{\prime },{p}^{\... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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800807e955b532992cb5d0bc02bb4d4892f423ef | subsection | 38 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | With Feynman rules of
Fig. REF and appropriate statistical factors,
Eqs. (REF )–() are diagrammatically
represented in Fig. REF .
[Figure: Feynman rules for diagrammaticrepresentations.][Figure: Diagrammatic representations ofpolyspectra.]Substituting Eqs. (REF )–(REF ) and
(REF )–() into Eq. (REF ), we obtain a
length... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0.03785035386681557,
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... |
746dda8be00ea4d5bfd99d894f2851d50a12cef2 | subsection | 39 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | To present the results, we first define the following
integrals:Q_n(k) &= \frac{k^3}{4\pi ^2}
\int _0^\infty dr\, P_{\rm L}(kr)
\\
& \qquad \times \int _{-1}^1 dx
P_{\rm L}[k(1 + r^2 - 2rx)^{1/2}]\,
\tilde{Q}_n(r,x),
\\
R_n(k) &=
\frac{k^3}{4\pi ^2} P_{\rm L}(k)
\int _0^\infty dr\, P_{\rm L}(kr)\,\tilde{R}_n(r),where\t... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0.02535678818821907,
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0.... |
3b594a40909782bf412300b199fcbd774f94f76e | subsection | 40 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | (REF ) and (REF ). In deriving the above
equations, the transverse part of Eq. () does not
contribute at all, because the rotational covariance implies\int \frac{d^3p}{(2\pi )^3}
g({k},{p}) {T}({k},-{p},{p}) \propto {k},where g is a scalar function.Applying the transformation of Eq. (REF ), the corresponding
integrals ... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0.022598013281822205,
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0.0... |
b0473fa2b9b40311d932d3af740459371d949ade | subsection | 41 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | The final result is represented
as&P^{\rm (s)}_{\rm obj}({k}) =
\exp \left\lbrace -\left[1 + f(f+2)\mu ^2\right]
\left(k/k_{\rm NL}\right)^2 \right\rbrace
\\
&\quad \times \left[
\left( 1 + \langle F^{\prime } \rangle + f\mu ^2 \right)^2
P_{\rm L}(k)
+ \sum _{n,m} \mu ^{2n} f^m E_{nm}(k)
\right],wherek_{\rm NL}
= \lef... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
-0.0283824373036623,
0.006969719659537077,
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0.04489308223128319,
0.006801866460591555,
0.05288899317383766,
-0.021103715524077415,
0.02... |
072aa56876a9fdd92d0d2e291ec4d65262a28b55 | subsection | 42 | 43 | One-loop corrections to the biased power spectrum via the Lagrangian
perturbation theory | (REF )–(REF ), are complete set of equations
to give the general one-loop power spectrum with effects of local
Lagrangian bias and redshift-space distortions. Although the number of
terms are large, they are all given by simple integrals of Q_n(k)
and R_n(k) of Eqs. (REF ) and (), which are
numerically easy to evaluate... | {
"cite_spans": []
} | 10.1103/PhysRevD.78.083519 10.1103/PhysRevD.78.109901 | 0807.1733 | Nonlinear perturbation theory with halo bias and redshift-space
distortions via the Lagrangian picture | [
"Takahiko Matsubara"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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0.0... |
8c8db5b00ecc6f3f9f92908baa9a1f761a7aeba3 | abstract | 0 | 25 | Abstract | The Dynamic Time Warping (DTW) is a popular similarity measure between time
series. The DTW fails to satisfy the triangle inequality and its computation
requires quadratic time. Hence, to find closest neighbors quickly, we use
bounding techniques. We can avoid most DTW computations with an inexpensive
lower bound (LB_K... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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-0.00037384676397778094,
0.013756034895777702,... | |
944a6688b57dc6429b84aff2ec00b23273d3383e | subsection | 1 | 25 | Introduction | Dynamic Time Warping (DTW) was initially introduced
to recognize spoken words , but
it has since been applied to a wide range of information
retrieval and database problems:
handwriting recognition , ,
signature recognition , , image de-interlacing ,
appearance matching for security purposes ,
whale vocalization classi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 467,
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"raw": "H. Sakoe, S. Chiba, Dynamic programming algorithm optimization for spoken word recognition, IEEE Transactions on Acoustics, Speech, and Signal Processing 26 (1) (1978) 43–49.",
"source_ref_id"... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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0.018069440498948097,
... | |
bd435384278eb3c759e8e156032b6a60281c0164 | subsection | 2 | 25 | Conventions | Time series are arrays of values measured at certain times.
For simplicity, we assume a regular
sampling rate so that time series are generic arrays of floating-point values.
A time series x has length \vert x\vert .
Time series have length n and
are indexed from 1 to n.
The l_p norm of x
is \Vert x \Vert _p= (\sum _i ... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
-0.061785988509655,
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-0.025050021708011627,
-0... | |
68aeccc47e4ecfb03425ca1457f7d16360ce9ec9 | subsection | 3 | 25 | Related Works | Beside DTW, several similarity metrics have been
proposed including the
directed and general Hausdorff distance, Pearson's correlation,
nonlinear elastic matching distance ,
Edit distance with Real Penalty (ERP) ,
Needleman-Wunsch similarity ,
Smith-Waterman similarity ,
and SimilB .Dimensionality reduction, such as pi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 284,
"openalex_id": "",
"raw": "R. C. Veltkamp, Shape matching: similarity measures and algorithms, in: Shape Modeling and Applications, 2001, pp. 188–197.",
"source_ref_id": "b4166b155ec586b744125a51dcf3335812f5d97f",
... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
-0.04722749814391136,
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0.024956131353974342,
0... | |
a78b5af821fcc3141b3e7be06c1b96172969336a | subsection | 4 | 25 | Dynamic Time Warping | A many-to-many matching between the data points in time series x and the data point
in time series y matches every data point x_i in x with at least one data point y_j in y,
and every data point in y with at least a data point in x. The set of
matches (i,j) forms a warping path \Gamma .
We define the DTW as the minimiz... | {
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"arxiv_id": "",
"doi": "",
"end": 791,
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"raw": "H. Sakoe, S. Chiba, Dynamic programming algorithm optimization for spoken word recognition, IEEE Transactions on Acoustics, Speech, and Signal Processing 26 (1) (1978) 43–49.",
"source_ref_id"... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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-0.011087032966315746,
-0.013368534855544567,... | |
0cfd99dea214767c25cf9cf123e94e86ae707e46 | subsection | 5 | 25 | Dynamic Time Warping | Write q_{i,j}= \text{DTW}_p(x_{(i)},y_{(j)})^p so that \text{DTW}_p(x,y)=\@root p \of {q_{1,1}},
thenq_{i,j}= {\left\lbrace \begin{array}{ll}0 & \text{if $\vert x_{(i)}\vert = \vert (y_{(j)}\vert =0$}\\
\infty & \begin{matrix}\text{if $\vert x_{(i)}\vert = 0 \oplus \vert y_{(j)}\vert =0$}\\
\text{or $\vert i- j \vert >... | {
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"raw": "J. Colannino, M. Damian, F. Hurtado, S. Langerman, H. Meijer, S. Ramaswami, D. Souvaine, G. Toussaint, Efficient many-to-many point matching in one dimension, Graph. Comb. 23 (1) (2007) 169–178.",
... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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3d149c26b35740ae322c839ab3930f344db2a4a5 | subsection | 6 | 25 | Some Properties of Dynamic Time Warping | The DTW distance can be counterintuitive.
As an example, if x,y,z are three time series such that
x \le y \le z pointwise,
then it does not follow that \text{DTW}_p(x,z) \ge \text{DTW}_p(z,y).
Indeed, choose x=7,0,1,0, y=7,0,5,0, and z=7,7,7,0,
then \text{DTW}_\infty (z,y)=5 and \text{DTW}_\infty (z,x)=1.
Hence, we rev... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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... | |
9fe6482677ee0c246361d42e99f04a4002d70979 | subsection | 7 | 25 | Some Properties of Dynamic Time Warping | However, the DTW is translation invariant:
\textrm {DTW}_p(x,z)=\textrm {DTW}_p(x+b,z+b)
and \textrm {NDTW}_p(x,z)=\textrm {NDTW}_p(x+b,z+b) for any scalar b and 0<p\le \infty .The \text{DTW}_1 has the property that if
the time series are value-separated, then the DTW is the l_1 norm
as the next proposition shows.Propo... | {
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"start": 1128
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} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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0... | |
2fd6a32ee49946ca1e266b7e3e8aa215890997f0 | subsection | 8 | 25 | Some Properties of Dynamic Time Warping | As a consequence of Proposition REF ,
we have \vert x^{\prime }\vert =\vert y^{\prime }\vert \le 2n. From classical analysis, we
have \vert x^{\prime }\vert ^{1/p-1/q} \Vert x^{\prime }-y^{\prime }\Vert _q \ge \Vert x^{\prime }-y^{\prime } \Vert _p,
hence \vert 2n\vert ^{1/p-1/q} \Vert x^{\prime }-y^{\prime }\Vert _q \... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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157790faa2675513846fd6044dd893e5847d09b6 | subsection | 9 | 25 | The Triangle Inequality | The DTW is commonly used as a similarity measure: x and y are similar
if \text{DTW}_p(x,y) is small. Similarity measures
often define equivalence relations: A\sim A for all A (reflexivity), A\sim B \Rightarrow B \sim C (symmetry) and A\sim B \wedge B \sim C \Rightarrow A \sim C (transitivity).The DTW is reflexive and s... | {
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"arxiv_id": "",
"doi": "",
"end": 1436,
"openalex_id": "",
"raw": "F. Casacuberta, E. Vidal, H. Rulot, On the metric properties of dynamic time warping, IEEE Transactions on Acoustics, Speech, and Signal Processing 35 (11) (1987) 1631–1633.",
"source_ref_id"... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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... | |
f9682eade9a9356c6c920c27a9a741f1bf65d9fd | subsection | 10 | 25 | The Triangle Inequality | However, for the random-walk time series, we found that 20% and 15% of the triples violated the triangle inequality
for \text{DTW}_1 and \text{DTW}_2.The DTW satisfies a weak triangle inequality as the next theorem shows.Theorem 1 Given any 3 same-length time series x,y,z and 1\le p\le \infty , we have\text{DTW}_p(x,y)... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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5b9fe531fac1d7672b89905287e2f5afef364097 | subsection | 11 | 25 | The Triangle Inequality | By the triangle inequality in l_p, we have\min (2w+1,n)^{1/p}(\text{DTW}_p(x,y)+\text{DTW}_p(y,z)) & \ge \Vert x^{\prime \prime }-y^{\prime \prime } \Vert _p+\Vert y^{\prime \prime }-z^{\prime \prime } \Vert _p\\
&\ge \Vert x^{\prime \prime }-z^{\prime \prime } \Vert _p\ge \text{DTW}_p(x,z).For p=\infty , \max _{(i,j,k... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1452,
"openalex_id": "",
"raw": "T. Eiter, H. Mannila, Computing discrete frechet distance, Tech. Rep. CD-TR 94/64, Christian Doppler Laboratory for Expert Systems (1994).",
"source_ref_id": "95a01b9277bf23366e57933844c9952f... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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d9ce0f91756c0e3bfecc8864ba916923e64933a1 | subsection | 12 | 25 | Which is the Best Distance Measure? | The DTW can be seen as the minimization of the l_p distance
under warping. Which p should we choose?
Legrand et al. reported best results
for chromosome classification using \text{DTW}_1
as opposed to using \text{DTW}_2. However, they did not quantify the
benefits of \text{DTW}_1.
Morse and Patel reported similar resu... | {
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{
"arxiv_id": "",
"doi": "",
"end": 221,
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"raw": "B. Legrand, C. S. Chang, S. H. Ong, S. Y. Neo, N. Palanisamy, Chromosome classification using dynamic time warping, Pattern Recognition Letters 29 (3) (2007) 215–222.",
"source_ref_id": "bed81... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
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] | 2,008 | en | Computer Science | [
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c3d5538d595d2c0c1eed9856d4a9c8276c609aa3 | subsection | 13 | 25 | Computing Lower Bounds on the DTW | Given a time series x, define U(x)_i = \max _k \lbrace x_k | \,\vert k-i \vert \le w \rbrace
and L(x)_i = \min _k \lbrace x_k |\, \vert k-i \vert \le w \rbrace for i=1, \ldots , n. The pair U(x) and L(x) forms
the warping envelope of x (see Fig. REF ). We leave the time constraint w implicit.
[Figure: Warping envelope... | {
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"raw": "E. Keogh, C. A. Ratanamahatana, Exact indexing of dynamic time warping, Knowledge and Information Systems 7 (3) (2005) 358–386.",
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... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
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] | 2,008 | en | Computer Science | [
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aa3337e15268e743d73e9c0e3666311bb799b10c | subsection | 14 | 25 | Computing Lower Bounds on the DTW | Hence, we have that \text{NDTW}_p(x,y)^p \ge \sum _{(i,j) \in \Gamma } \vert x_i-h_i \vert ^p + \vert h_i - y_j \vert ^p
\ge \Vert x-h \Vert _p^p + \sum _{(i,j) \in \Gamma } \vert h_i - y_j \vert ^p.
This proves the result since \sum _{(i,j) \in \Gamma } \vert h_i - y_j \vert \ge \text{NDTW}_p(h,y).
For p=\infty , we h... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
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] | 2,008 | en | Computer Science | [
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81c10aa333d58563bb6454d8f46ff0d2cb6f8d9f | subsection | 15 | 25 | Computing Lower Bounds on the DTW | Hence, the approximation error is exactly \Vert h-y \Vert _1 in such instances. | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
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adc85edaaec30bb1e868f1562eda8f4ba7716ff5 | subsection | 16 | 25 | Warping Envelopes | The computation of the warping envelope U(x),L(x) requires O(nw) time using the naive approach
of repeatedly computing the maximum and the minimum over windows.
Instead, we compute the envelope using at most 3n comparisons between data-point values
using Algorithm .input a time series a indexed from 1 to n
input some ... | {
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{
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"doi": "",
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"raw": "D. Lemire, Streaming maximum-minimum filter using no more than three comparisons per element, Nordic Journal of Computing 13 (4) (2006) 328–339.",
"source_ref_id": "551d608124f908ae89dda081f7... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
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0fc03dbb879be3f0aa9d1d300c81247e9d1a3bf7 | subsection | 17 | 25 | Warping Envelopes | The second
result (L(U(h)) \ge h) follows similarly.Whereas L(U(h)) is greater or equal than h, the next lemma shows that U(L(U(h))) is
equal to U(h).Corollary 2 We have U(h)=U(L(U(h))) and L(h) = L(U(L(h))) for any h.By Lemma REF , we have L(U(h)) \ge h, hence U(L(U(h))) \ge U(h).
Again by Lemma REF , we have U(L(h^{\... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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f36f1ce3723c710cf5788590f45c6c17fdb55e94 | subsection | 18 | 25 | LB_Keogh | Let H(x,y) be the projection of x on y defined asH(x,y)_{i}= {\left\lbrace \begin{array}{ll}U(y)_i & \text{if $x_i\ge U(y)_i$}\\
L(y)_i & \text{if $x_i\le L(y)_i$} \\
x_i & \text{otherwise,}
\end{array}\right.}for i=1,2,\ldots ,n.
We have that H(x,y) is in the envelope of y. By Theorem REF and setting h= H(x,y),
we hav... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
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] | 2,008 | en | Computer Science | [
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82d26e25bc3a88fbfe982ac094a18748245e0039 | subsection | 19 | 25 | LB_Improved | Write \text{LB\_Improved}_p(x,y)^p=\text{LB\_Keogh}_p(x,y)^p+\text{LB\_Keogh}_p(y,H(x,y))^p
for 1 \le p<\infty . By definition, we have \text{LB\_Improved}_p(x,y)\ge \text{LB\_Keogh}_p(x,y).
Intuitively, whereas \text{LB\_Keogh}_p(x,y) measures the distance between x and the envelope of y,
\text{LB\_Keogh}_p(y,H(x,y)) ... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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a4ad98a2079f1e74df73ac52f40239ec69532175 | subsection | 20 | 25 | LB_Improved | If \alpha is the fraction of candidates pruned by LB_Keogh, at most
(2N+3)n+5(1-\alpha )Nn comparisons between data points are required to process a database
containing N time series.[1]
input a time series a indexed from 1 to n
input a set S of candidate time series
the nearest neighbor B to a in S under \text{DTW}_1
... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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fc028980f3b1c3b07095c4be8bacdb5aff886962 | subsection | 21 | 25 | Comparing LB_Keogh and LB_Improved | In this section, we benchmark Algorithms and .
We know that the LB_Improved approach has at least the pruning power of the LB_Keogh-based approach,
but does more pruning translate into a faster nearest-neighbor retrieval under the DTW distance?We implemented the algorithms in C++ and called the functions from Python s... | {
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"raw": "D. Lemire, Fast nearest-neighbor retrieval under the dynamic time warping, online: http://code.google.com/p/lbimproved/ (2008).",
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... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
"cs.DB"
] | 2,008 | en | Computer Science | [
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ad45b7a4bf2ba6452fb94db64cab5d8c48f233dd | subsection | 22 | 25 | Synthetic data sets | We tested our algorithms using the Cylinder-Bell-Funnel and
Control Charts data sets, as well as over a database of random walks.
We generated 1 000-sample random-walk time series using the formula x_i = x_{i-1}+N(0,1) and x_1=0.
Results for the Waveform and Wave+Noise data sets are similar and omitted.For each data ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 131,
"openalex_id": "",
"raw": "N. Saito, Local feature extraction and its applications using a library of bases, Ph.D. thesis, Yale University, New Haven, CT, USA (1994).",
"source_ref_id": "edd8357623b65089eda7e5023de06bf1... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
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] | 2,008 | en | Computer Science | [
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cbe3940d5bcde13768b7d297107049be3d0b210a | subsection | 23 | 25 | Shape data sets | For the rest of the section, we considered a large collection of time-series derived from shapes , .
The first data set is made of heterogeneous shapes which resulted in 5 844 1 024-sample times series.
The second data set is an arrow-head data set with of 15 000 251-sample time series.
We shuffled randomly each data s... | {
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"raw": "E. Keogh, L. Wei, X. Xi, S. H. Lee, M. Vlachos, LB_Keogh supports exact indexing of shapes under rotation invariance with arbitrary representations and distance measures, VLDB 2006 (2006) 882–893.",... | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
] | [
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5e847f72012b800a494ffb216551ec9a6ecf3e1c | subsection | 24 | 25 | Conclusion | We have shown that a two-pass pruning technique can improve the retrieval speed
by up to three times in several time-series databases. We do not use more memory.We expect to be able to significantly accelerate the retrieval with
parallelization. Several instances of Algo. can
run in parallel as long as they can commun... | {
"cite_spans": []
} | 0807.1734 | Faster Sequential Search with a Two-Pass Dynamic-Time-Warping Lower
Bound | [
"Daniel Lemire"
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d33671b541dae21ba814ec568d9238afef2019c6 | abstract | 0 | 28 | Abstract | Using a sample of 43 bright (V<16.1, distance <13 kpc) RR Lyrae stars (RRLS)
from the QUEST survey with spectroscopic radial velocities and metallicities,
we find that several separate halo substructures contribute to the Virgo
overdensity (VOD). While there is little evidence for halo substructure in the
spatial distr... | {
"cite_spans": []
} | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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32e23e0d319f3e1f87bf6fdd7db297d0f5f54002 | subsection | 1 | 28 | INTRODUCTION | Over the past several decades, a large number of investigations of the
stars and the globular clusters in the Galactic halo have concluded
that the halo formed chaotically and over a long period of time
through the accretion and disruption of dwarf galaxies (e.g., Searle
& Zinn 1978, Norris 1986; Carney et al. 1990; Ma... | {
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... | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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dd25f1e810fc2be9e1fde9031e9e52bb3f0044d3 | subsection | 2 | 28 | INTRODUCTION | Also using the SDSS photometric catalogue, Juric et al (2005,
2008) later identified a very large
overdensity of main sequence stars toward Virgo (the Virgo
Overdensity (VOD) ). On the sky, the VOD encompasses both the 12.4
hr clump and S297+63-20.5. But, according to , the VOD
is spread along the line of sight from \s... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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b8eba77857302efbc1f79d82664074696a0b0af7 | subsection | 3 | 28 | The Sample of RR Lyrae Stars | According to , the VOD occupies \sim 1000 sq. degrees
of the sky, and its rough limits are shown in
Figure REF . Notice that the SDSS does not extend south of
b\sim 58 at l\sim 300 . Thus, it is possible that the VOD extends farther in
that direction. The region surveyed by QUEST is a 23-wide band
centered at \delta =-... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
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2d89505a21a9f3833c86a0df283607e4e98fca82 | subsection | 4 | 28 | Spectroscopy | We obtained a total of 88 spectra of the 44 RRLS with 3 different
telescopes between 2001 and 2007. The instrumental setups of the
different observing runs are summarized in Table ,
and the individual observations are detailed in
Table . Most of the spectra were obtained with
the SMARTS 1.5m telescope at Cerro Tololo I... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
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] | [
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ba2b676438fd875aa80672637b855b3194272199 | subsection | 5 | 28 | Improving the Ephemerides | Since good ephemerides are essential for fitting the radial velocity
curves, it is important to improve them as much as possible. The
QUEST survey has a large number of epochs (an average 32 for our
targets), which should yield precise ephemerides. In ,
the time of maximum light (HJD_0) for the RRc stars was set by the... | {
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} | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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] | 2,008 | en | Physics | [
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c29ae41603b72ac11eeca4725726c9b104500b22 | subsection | 6 | 28 | Measurements of [Fe/H] | The metallicities of the RRLS have been measured from the spectrograms
following the methodology and calibration of , which
is based on Freeman & Rodgers' (1975) modification
of the \Delta S method. The method, which involves
plotting the pseudo-equivalent width of the Ca II K line (W(K)),
corrected for interstellar ab... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
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] | [
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78c5df3bc95aa4ef6de952051895b99efb389708 | subsection | 7 | 28 | Measurements of [Fe/H] | The 8 type c variables yield \langle [Fe/H]\rangle = -1.66
(std. dev. = 0.17), and the 5 that have spectrograms with 41 > S/N
> 17 yield \langle [Fe/H]\rangle = -1.63 (std. dev. = 0.19).
The high S/N samples differ in \langle [Fe/H]\rangle by 0.09 \pm 0.11,
which suggests that the offset, if any, between the type ab an... | {
"cite_spans": []
} | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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a86224ffd24b067fcaf74ed7c18defc859fa056d | subsection | 8 | 28 | Radial Velocities | Radial velocities for all blue spectra were obtained by Fourier
cross-correlation (IRAF’s fxcor task) with radial velocity
standard stars of A and F spectral type, in the spectral range 3800 -
5200 Å. The radial velocity standards were separated in two groups
according to their effective temperature. Each target star w... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
"astro-ph"
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6648fe5590e6976ed334c0763ffa037ba400de5a | subsection | 9 | 28 | Radial Velocities | In our low resolution spectrograms, the weak lines
produce poorly determined peaks in the cross-correlations, whereas
well-defined ones are produced by the Balmer lines and other strong
features. Furthermore, we could not obtain sufficiently large numbers
of spectra to determine the velocity curve for each star. For th... | {
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"o... | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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a095d0a0aae490dbafb72d505b184334362bd403 | subsection | 10 | 28 | Radial Velocities | Finally, we calculated the radial velocity
measured by an observer at the Sun who is at rest with the galactic
center, V_{gsr} byV_{gsr}=V_{\gamma }+10.0\,\cos {b}\,\cos {l}+225.2\,\cos {b}\,\sin {l}+7.2\,\sin {b};Our results are presented in Table , which lists:
ID, coordinates \alpha and \delta , the extinction corre... | {
"cite_spans": []
} | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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eb5e6928e60ad3afcddc1ee375a73f5e1a61e686 | subsection | 11 | 28 | DISTRIBUTION OF RADIAL VELOCITIES: SUBSTRUCTURES IN VELOCITY SPACE | As noted above, the VOD encompasses the entire range in \alpha and
perhaps a larger range of distance than our sample of RRLS. The
spatial distribution of our sample of RRLS (see the box in
Figure REF ) does not reveal any regions of exceptionally
high density that are strong candidates for halo substructure. There
is ... | {
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toward Virgo | [
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"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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8e6fe9c1531c6fe4463c2747e59f0a720711b5fa | subsection | 12 | 28 | DISTRIBUTION OF RADIAL VELOCITIES: SUBSTRUCTURES IN VELOCITY SPACE | In total, there are eight
RRLS, or 19% of our sample that have V_{gsr} > +180 {\rm km~s}^{-1}, which is very unlikely to be random occurrence (1 in
10,000, if \sigma _{\rm los}=101.6 {\rm km~s}^{-1}
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toward Virgo | [
"A. Katherina Vivas",
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"Robert Zinn",
"Rebeccah Winnick",
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ad935d3c248224b61307abf2384fde42cb48e1f9 | subsection | 13 | 28 | Identification of Groups | To identify possible groups of stars, we implemented with only a few
modifications the algorithm described by (the
Stellar Pair Search). In this technique, the distances of each star
from all others in the sample are calculated from the cartesian
Galactic coordinates x, y and z. Pairs of
stars are ones that are separat... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
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fe3601ea179c816878b3c886e5c4a5aa4a866472 | subsection | 14 | 28 | Identification of Groups | We restricted our sample to the 33 stars that have
D\ge 6.5 kpc since the incompleteness that is caused by the
different saturation levels of the CCDs of the QUEST camera
may effect the interpretation of results at closer
distances.Two significant groups were found among our RRLS
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toward Virgo | [
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"Robert Zinn",
"Rebeccah Winnick",
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bd22e18ee0c74dbfbafc8c5baa67f825f048f3ff | subsection | 15 | 28 | Comparison with Blue Horizontal Branch Stars | If any of the features described above are real substructures, then they should
also appear in the
distributions of the other types of stars that make up very old stellar populations.
While BHB stars and RRLS are, of course, samples of horizontal branch stars that
differ primarily in
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
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9e5486a6a157c44975f78dc1a7659ecbb205dce6 | subsection | 16 | 28 | Comparison with Blue Horizontal Branch Stars | A comparison, after making the
necessary transformations between photometric systems, of the
relationship to the one employed by
reveals that the difference in distance modulus is near zero at the
blue edge of the instability strip and again small at the highest
temperatures, but is 0.33 mag (a factor of 1.16 in dist... | {
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toward Virgo | [
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"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
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66156234a18d930945606d67800742187cf731f1 | subsection | 17 | 28 | Comparison with Blue Horizontal Branch Stars | The group, which was detected with d_{crit}=1 kpc and
v_{crit}=25 {\rm km~s}^{-1}, contains only 3 members, all of which are BHB
stars located in the very narrow intervals of D=11.1 to 11.3 kpc
and \alpha =205 to 210. The Monte Carlo simulations found
simialr groups in \sim 4\% of the cases (a 2.5\sigma detection).
Thi... | {
"cite_spans": []
} | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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d37e1a179816c018600e9bebdcefb9e30e3a455c | subsection | 18 | 28 | Comparison with main-sequence stars | The recent investigation of the VOD region by has
provided radial velocity measurements for many stars with the colors
of F type stars in two SEGUE fields, at (l,b) = (288,
62) and (300, 55), one of which overlaps with the
region we are considering here (see Figure REF ).
make the very reasonable assumption that the v... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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b6d9e9b10ca3a1ca6b3c8030e17ca3cfe8b0c1db | subsection | 19 | 28 | Group at | This group is probably the most remarkable feature seen in our data
since very few stars with such high velocities are expected in a
sample of halo stars (1 or 2 in our sample of RRLS). The 10 RRLS and
BHB stars in this group have a mean velocity of \langle {V_{\rm gsr}}\rangle = +215 {\rm km~s}^{-1}, and a standard de... | {
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... | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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1d8da2118d2adca6eb108b2051e54a8d6197806a | subsection | 20 | 28 | Group at | We are observing more RRLS in the range
12 < D < 16 kpc, with the expectation that more velocity and
metallicity measurements will reveal the origin of this substructure.This is the most populated group that we find in our data, containing
9 RRLS and 5 BHB stars with velocities in the range -80 < V_{gsr} <
-10 {\rm km~... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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863cfa562b3ee8f005a490c87b9f94f0ef2a1343 | subsection | 21 | 28 | Does the VSS extends to closer distances? | The F star data show a very prominent peak at 130 {\rm km~s}^{-1} in both
fields, and consider this to be the velocity of
S297+63-20.5 feature. On the basis of the spatial coincidence between
S297+63-20.5 and the VSS and the near coincidence in V_{gsr}, they
conclude that these features are probably the same or physica... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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] | 2,008 | en | Physics | [
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653a160c60fd288cbc093c3a2d4b6f9fa9a072b6 | subsection | 22 | 28 | THE RELATIONSHIP TO THE SGR STREAMS | Soon after the first detections of over-densities of stars in the
direction of Virgo, several authors
, speculated that they may be
debris from the disruption of the Sgr dSph galaxy. Even then there
was some evidence to contrary, for had noted that
the color of the turnoff in S297+63-20.5 was redder than the color of
t... | {
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toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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deae9fe3f46b4048ba90194e2cbb3047081fe5af | subsection | 23 | 28 | THE RELATIONSHIP TO THE SGR STREAMS | Also, none of the
BHB stars in the region have highly negative velocities (see
Figure REF ). In the prolate model, the leading stream
crosses the galactic plane far from the Sun, which explains why it
predicts relatively few stars in the diagrams at D < 20 kpc. We
conclude on the basis of these comparisons that none of... | {
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... | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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23d49dc0c184479acdfcf81ccfdbf14347f0b1b1 | subsection | 24 | 28 | DISCUSSION | Kinematical information is crucial to detect substructures in regions
where no obvious spatial overdensities are present. Taken together,
our RRLS sample, the BHB star sample of , and the F
star sample of paint a consistent picture that
several separate halo substructures exist at D \lesssim 13 kpc in
the part of the V... | {
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... | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
] | [
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16707b1280124eb8457c2f37fafdc5756fbf36b2 | subsection | 25 | 28 | DISCUSSION | We are indebted to the service mode observers Sergio Gonzalez, Alberto Pasten, Claudio
Aguilera and Alberto Miranda for their fine operation of the telescope and the spectrograph.
We also thank the anonymous referee for helpful suggestions
to the manuscript.
[Figure: Map in Galactic coordinates showing the strip (solid... | {
"cite_spans": []
} | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
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a06ecc56a682773f4244e59b20033353b9fa4d51 | subsection | 26 | 28 | DISCUSSION | The two significant groups found in our data are indicated bysolid symbols, one of them have stars with very high positive velocities and the otherone has stars with velocities around -60 {\rm km~s}^{-1}.][Figure: Histogram of V_{gsr} of the BHB stars in the same 3 distance bins as Figure .The dashed line indicatesthe ... | {
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} | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
"A. Katherina Vivas",
"Yara Jaffe",
"Robert Zinn",
"Rebeccah Winnick",
"Sonia Duffau",
"Cecilia Mateu"
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8e1c73faae79f5fff291f37e033a0d7731770796 | subsection | 27 | 28 | DISCUSSION | This measurement
was not taken into account for calculation of the mean [Fe/H] unless it was the only
available measurement for the star.ccccccccccccccc
15
0pc
Position, Distance, Velocity and Metallicity of the 43 RRLSa
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} | 10.1088/0004-6256/136/4/1645 | 0807.1735 | Spectroscopy of Bright QUEST RR Lyrae Stars: Velocity Substructures
toward Virgo | [
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7baa57912374262d1b0310c81cc9b06d475dd485 | abstract | 0 | 41 | Abstract | We show that the Mobius function mu(n) is strongly asymptotically orthogonal
to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected
nilpotent Lie group with a discrete and cocompact subgroup L (so G/L is a
nilmanifold), g : Z -> G is a polynomial sequence and F: G/L -> R is a
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} | 10.1103/PhysRevA.79.035801 | 0807.1736 | The Mobius function is strongly orthogonal to nilsequences | [
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d73b7076e8913ab8d2e046393a853a81a0af9ef8 | subsection | 1 | 41 | Introduction | Important remark. This paper is intimately tied to, and is intended to be read in conjunction with, the longer companion paper , which proves results about the distribution of finite polynomial orbits on nilmanifolds. In particular, we shall make heavy use of the notation and lemmas from that paper.The aim of this pape... | {
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4d820dde788b1220329feeefd280e6c05b8c1395 | subsection | 2 | 41 | Introduction | In fact, we shall prove the marginally stronger result that the Möbius function is asymptotically strongly orthogonal to any polynomial nilsequence (F(g(n)\Gamma ))_{n \in \mathbb {Z}}.Theorem 1 (Main Theorem)
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18f5d13e07b499b56e90039bb45b7c1e214727fa | subsection | 3 | 41 | Introduction | In particular, this would be the case if one assumed GRH. In fact, in that case it is not difficult to see from modifying the arguments below that we can replace the logarithmic decay \log ^{-A} N by polynomial decay N^{-c} for some c > 0 depending only on d and m.The authors learnt in that it is in many ways more natu... | {
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c1ce4355fb8460e0405d615ffe70a3cce7e3ee33 | subsection | 4 | 41 | Reducing to the equidistributed case | To prove Theorem REF , we will apply to decompose g as a product \varepsilon g^{\prime } \gamma where \varepsilon is “smooth”, \gamma is “rational” and g^{\prime } is highly equidistributed in some closed subgroup G^{\prime } \subseteq G. We will recall the precise statement shortly.In this section we shall show how th... | {
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441cf5bf28bb36f6a75b895896a7a453f48b7d0b | subsection | 5 | 41 | Reducing to the equidistributed case | This is the same method that one might use to tackle the “minor arcs” case of (REF ), where \alpha is not close to a rational with small denominator. We will describe it in detail in the next section. Our task for the remainder of this section is to reduce Theorem REF to Proposition REF .Proof that Proposition REF impl... | {
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2cd7d41a8ffa49198dea206eff44e49a178c6692 | subsection | 6 | 41 | Reducing to the equidistributed case | By (with M_0 := \log N) we can find an integer M,\log N \leqslant M \ll \log ^{O_B(1)} N,a rational subgroup G^{\prime } \subseteq G, a Mal'cev basis \mathcal {X}^{\prime } for G^{\prime }/\Gamma ^{\prime } (where \Gamma ^{\prime } := G \cap \Gamma ) in which each element is an M-rational combination (see ) of the elem... | {
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26210c500a6b81f49ea0aeb14acf71df331e368b | subsection | 7 | 41 | Reducing to the equidistributed case | Since \gamma _j = \gamma (j)\eta for some \eta with integer coordinates, it follows from that the coordinates \psi _{\mathcal {X}}(\gamma _j) are rationals with height \ll M^{O(1)}.We now take advantage of the periodicity of \gamma (n) \Gamma to split the right-hand side of (REF ) as\sum _{j = 0}^{q-1} \mathbb {E}_{n \... | {
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50f3673757f805c00b6bdd30f5bb3b23973f7ed2 | subsection | 8 | 41 | Reducing to the equidistributed case | Hence if we split each progression n \equiv j(\operatorname{mod}\, q) into further progressions P_{j,k} for k = O( M \log ^A N ), each having diameter at most \frac{N}{M\log ^A N}, we see that (REF ) is equal to\sum _{j,k} \mathbb {E}_{n \in [N]} \mu (n) 1_{P_{j,k}}(n) F(a_{j,k} g^{\prime }(n) \gamma _j \Gamma ) + O(\l... | {
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6849ed34e26fcb1539142c7b9714abab4ee5816a | subsection | 9 | 41 | Reducing to the equidistributed case | It is clear that each g_j is a polynomial sequence with coefficients in the filtration (H_j)_{\bullet } := \gamma _j^{-1} G^{\prime }_{\bullet } \gamma _j.Set \Lambda _j := \Gamma \cap H_j and define functionsF_{j,k} : H_j/\Lambda _j \rightarrow [-1,1]by the formulaF_{j,k}(x\Lambda _j) := F(a_{j,k}\gamma _j x\Gamma ).T... | {
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4177acb070e0ddd96307a9c734cb631eab4fd5ab | subsection | 10 | 41 | Reducing to the equidistributed case | Now the map x \mapsto F(a_{j,k}\gamma _j x \Gamma ) on G/\Gamma has Lipschitz constant at most M^{O(1)} by and the bounds |\psi _{\mathcal {X}}(a_{j,k})|, |\psi _{\mathcal {X}}(\gamma _j)| \leqslant M^{O(1)}. The last statement of the claim follows from . Finally, the statement about the quantitative equidistribution o... | {
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550c29ef989bbec524999a1a1edffce117addb4e | subsection | 11 | 41 | The equidistributed case: Type I and II sums | In this section we establish Proposition REF using Vinogradov's method of Type I and II sums in the form due to Vaughan . More precisely, we will use the following proposition.Proposition 1 (Method of Type I/II sums)
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448c46ca38e909ddc1cb7e2c8149d0b79abe040d | subsection | 12 | 41 | The equidistributed case: Type I and II sums | The abelian issues that one must then deal with are of a very similar nature to those involved in dealing with exponential sums such as \mathbb {E}_{n \in [N]} \mu (n) e(p(n)), where p : \mathbb {R}\rightarrow \mathbb {R}/\mathbb {Z} is an ordinary polynomial. Rather than quote results from the existing literature on t... | {
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89f70df81cd0010f618e577fb45e40fead9abc45 | subsection | 13 | 41 | The equidistributed case: Type I and II sums | It follows from that g_k,\tilde{g}_k \in \operatorname{poly}(\mathbb {Z},G_{\bullet }). Now (REF ) implies that (\tilde{g}_k(n)\Gamma )_{n \in [N_k]} fails to be \delta ^{O(c_1)}-equidistributed in G/\Gamma , where N_k \sim N/kl.It follows from that there is a nontrivial horizontal character \psi _k : G \rightarrow \ma... | {
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80fb21c7a2b1accdedad7ad1ba67a4a7223c6ba9 | subsection | 14 | 41 | The equidistributed case: Type I and II sums | Then there is some q \geqslant 1, q = O(1), such that \Vert q\beta _j \Vert _{\mathbb {R}/\mathbb {Z}} \ll N^{-j}\Vert p \Vert _{C^{\infty }[N]} for j = 1,\dots , d.Proof. Consider the representation (REF ) which is used to define the C^{\infty }[N]-norm. Observing that \beta _j can be written as a linear combination o... | {
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c4163fb4208ca1dbbbc20be988517ea5fe23f636 | subsection | 15 | 41 | The equidistributed case: Type I and II sums | Then by the Cauchy-Schwarz inequality and the preceding remarks we have\alpha ^{2t} K^{2t} = (\sum _n r(n))^2 \leqslant |tX| \sum _n r(n)^2 \ll _j |tX| K^{2t - j},which implies the result.By (REF ) and Lemma REF it follows that\Vert q l \beta _j \Vert _{\mathbb {R}/\mathbb {Z}} \ll \delta ^{-O(c_1)} (K/N)^jfor \gg \del... | {
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3af969ef6ef84cda72ca9ece745dc7816b5a19a6 | subsection | 16 | 41 | The equidistributed case: Type I and II sums | The conclusion is that there is some q^{\prime }, 1 \leqslant q^{\prime } \ll \delta ^{-O(c_1)}, such that\Vert qq^{\prime } \beta _i \Vert _{\mathbb {R}/\mathbb {Z}} \ll \delta ^{-O(c_1)} N^{-i}.Writing \tilde{\psi }:= qq^{\prime } \psi , it follows from (REF ) and (REF ) that for any n we have the bound\Vert \tilde{\... | {
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c10ccbc01d5cd9af5b1f95afa4741884ef054128 | subsection | 17 | 41 | The equidistributed case: Type I and II sums | Writing the left-hand side here as\mathbb {E}_{K < k,k^{\prime } \leqslant 2K} |\mathbb {E}_{W < w \leqslant 2W} f(kw)f(k^{\prime }w)|^2,we see that there are \gg \delta ^{O(c_1)} K^2 pairs (k,k^{\prime }) \in (K,2K]^2 such that|\mathbb {E}_{W < w \leqslant 2W} f(kw) f(k^{\prime }w)| \gg \delta ^{O(c_1)}.Written out in... | {
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78188b86a43ff82e5acd8287744298e13648158e | subsection | 19 | 41 | The equidistributed case: Type I and II sums | If\psi _1 \circ g(n) = \beta _d n^d + \dots + \beta _0and\psi _2 \circ g(n) = \beta ^{\prime }_d n^d + \dots + \beta ^{\prime }_0then\psi \circ g_{k,k^{\prime }}(n) = (\beta _d k^d + \beta ^{\prime }_d k^{\prime d})n^d + \dots + (\beta _0 + \beta ^{\prime }_0),By Lemma REF and (REF ) there is some 1 \leqslant q \ll \de... | {
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7a25823b00ce70b067a3ceedf661de91e4b63413 | subsection | 20 | 41 | Remarks on a nilpotent Hardy-Littlewood method | It may be of interest to interpret our method in terms of the “major and minor arcs” terminology of the Hardy-Littlewood method. Recall that to prove Davenport's estimate|\mathbb {E}_{n \in [N]} \mu (n) e(\alpha n)| \ll _A \log ^{-A} None divides into two cases: the major arcs where \alpha is close to a rational with s... | {
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cc7cadac7babfb4975614af6fc3af35f35ae4b25 | subsection | 21 | 41 | On bracket polynomials | By a bracket polynomial we mean an object formed from the scalar field \mathbb {R} and the indeterminate n using finitely many instances of the standard arithmetic operations +, \times together with the integer part operation \lfloor \; \rfloor and the fractional part operation \lbrace \; \rbrace . The following are al... | {
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7d7e3d9fe204f4cc2609936f96bbcaef99708cd6 | subsection | 22 | 41 | On bracket polynomials | Indeed the results of Leibman on the distribution of (g(n)\Gamma )_{n \in \mathbb {Z}} imply that the sequence (F(g(n)\Gamma ))_{n \in \mathbb {Z}} cannot have isolated values, yet there are bracket polynomials which do. A simple example is \lfloor 1 - \lbrace n \sqrt{2}\rbrace \rfloor , which is zero except when n = 0... | {
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"start": 487
},
{
... | 10.1103/PhysRevA.79.035801 | 0807.1736 | The Mobius function is strongly orthogonal to nilsequences | [
"Ben Green",
"Terence Tao"
] | [
"math.NT"
] | 2,008 | en | Mathematics | [
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