text: 9 Scalar φ4 theory is not asymptotically free in four dimen- sions, but we can understand the mathematical features of the sOPE in more detail by studying the renormal- ization group equations for the simple example of scalar ﬁelds. VI. RENORMALIZATION GROUP EQUATIONS O FIG. 7. Two-loop diagrams that appear at (m2) and there- fore do not contribute to d . For details of the Feynman We consider the matrix elements of scalar operators ρ2 diagrams, see the caption of Fig. 2. coupled to N external, unsmeared scalar ﬁelds. We can derive renormalization group equations for the Wilson coeﬃcients of the OPE by considering the scale depen- dence of suitably chosen Green functions [53]. The Green 2. Next-to-leading-order function for N + 2 external scalar ﬁelds obeys the renor- ma(cid:20)lization group equ(cid:21)ation (cid:10) (cid:11) To determine the next-to-leading-order contribution to (cid:101) (cid:101) the connected Wilson coeﬃcient, we must incorporate the d | | Feynman diagrams of Fig. 4. The two-loop diagrams of µ + (N + 2) γ Ω φ(p ) . . . φ(p ) Ω = 0, (45) dµ 1 N+2 Fig. 7 do not contribute to the Wilson coeﬃcient d , O ρ2 because they appear at (m2). Looking at the Feynman diagrams of Fig. 4, and bear- while the Green function of the renormalized operator ing in mind our experience of the next-to-leading-order φ2(cid:20)(0) coupled to N (cid:21)external scalar ﬁelds satisﬁes R (cid:10) (cid:11) contribution to dI, we immediately observe that these di- (cid:101) (cid:101) agrams are simply the product of the one-loop Wilson co- d − | | eﬃcient, d(1), and the next-to-leading-order renormalized µ γ + Nγ Ω φ2 (0)φ(p ) . . . φ(p ) Ω = 0. ρ2 dµ m R 1 N four-point vertex. This contribution to the vertex is, of (46) course, nothing other than the next-to-leading-order con- tribution to the renormalized coupling constant. Thus, Here the renormalization group operator for scalar ﬁeld we write the Wilso(cid:20)n coeﬃcient q(cid:18)uite (cid:19)sim(cid:21) ply as theory is (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) d = 1 + λR γ − 1 + log x2 + O(λ3), (43) d ∂ ∂ − ∂ ρ2 32π2 E 8τ µdµ = µ∂µ + β ∂λ γmmR ∂m (47) λ,mR µ,mR R µ,λ where λR is the renormalized coupling. In the MS and the coeﬃcients are [59] scheme, the renormalized coupl(cid:18)ing is(cid:19)given by dλ 3λ2 O − 3λ2 µ2 O βMS(λ) = µ = + (λ3), (48) λ = λ log + (λ3). (44) dµ (16π2)2 R 2(16π2) m2 − µ d log(m ) γMS(λ) = R m 2 dµ For a calculation of this to ﬁve loops, see [59]. − λ 5λ2 O As we move beyond the leading-order Wilson coeﬃ- = + + (λ3), (49) 2(16π2) 12(16π2)2 cients, i.e. d(I0) and d(1), divergent radiative corrections ρ2 µ d log(Z ) λ2 O appear in our calculations, because all ﬁeld interaction γMS(λ) = φ = + (λ3). (50) vertices appear at zero ﬂow time. These divergences 2 dµ 12(16π2)2 can be removed by the renormalization parameters of the original theory and the renormalization scale dependence In general these coeﬃcients depend on the mass, the of the smeared operators is completely contained in the renormalization scale, and the renormalized coupling renormalization parameters of the original theory. This constant, but in a mass independent renormalization is to be expected: it follows from the fact that renormal- scheme, such as the MS scheme, they depend on the ized matrix elements at zero ﬂow time remain ﬁnite at renormalization scale only through the renormalized cou- nonzero ﬂow time and require no further renormalization pling constant. These functions are known to ﬁve loops [31, 35]. for the O(N)-symmetric theory, given by the N-multiplet { } Ultimately, for realistic calculations in lattice QCD, Φ = φ , . . . , φ [59, 60]. 1 N the perturbative calculation of smeared Wilson coeﬃ- Returning again to our example of the OPE for the cients must be combined with nonperturbative determi- two-point function, Eq. (2), we can derive an renor- nations of matrix elements at hadronic energy scales. malization group equation for the Wilson coeﬃcient c φ2 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Scalar

text: φ4

text: theory

text: is

text: not

text: asymptotically

text: free

text: in

text: four

text: dimen-

text: sions,

text: but

text: we

text: can

text: understand

text: the

text: mathematical

text: features

text: of

text: the

text: sOPE

text: in

text: more

text: detail

text: by

text: studying

text: the

text: renormal-

text: ization

text: group

text: equations

text: for

text: the

text: simple

text: example

text: of

text: scalar

text: ﬁelds.

text: VI.

text: RENORMALIZATION

text: GROUP

text: EQUATIONS

text: O

text: FIG.

text: Two-loop

text: diagrams

text: that

text: appear

text: at

text: and

text: there-

text: fore

text: do

text: not

text: contribute

text: to

text: d

text: For

text: details

text: of

text: the

text: Feynman

text: We

text: consider

text: the

text: matrix

text: elements

text: of

text: scalar

text: operators

text: ρ2

text: diagrams,

text: see

text: the

text: caption

text: of

text: Fig.

text: coupled

text: to

text: N

text: external,

text: unsmeared

text: scalar

text: ﬁelds.

text: We

text: can

text: derive

text: renormalization

text: group

text: equations

text: for

text: the

text: Wilson

text: coeﬃcients

text: of

text: the

text: OPE

text: by

text: considering

text: the

text: scale

text: depen-

text: dence

text: of

text: suitably

text: chosen

text: Green

text: functions

text: The

text: Green

text: Next-to-leading-order

text: function

text: for

text: N

text: external

text: scalar

text: ﬁelds

text: obeys

text: the

text: renor-

text: group

text: To

text: determine

text: the

text: next-to-leading-order

text: contribution

text: to

text: the

text: connected

text: Wilson

text: coeﬃcient,

text: we

text: must

text: incorporate

text: the

text: d

text: Feynman

text: diagrams

text: of

text: Fig.

text: The

text: two-loop

text: diagrams

text: of

text: µ

text: (N

text: γ

text: Ω

text: φ(p

text: φ(p

text: Ω

text: dµ

text: Fig.

text: do

text: not

text: contribute

text: to

text: the

text: Wilson

text: coeﬃcient

text: d

text: O

text: ρ2

text: because

text: they

text: appear

text: at

text: Looking

text: at

text: the

text: Feynman

text: diagrams

text: of

text: Fig.

text: and

text: bear-

text: while

text: the

text: Green

text: function

text: of

text: the

text: renormalized

text: operator

text: ing

text: in

text: mind

text: our

text: experience

text: of

text: the

text: next-to-leading-order

text: coupled

text: to

text: N

text: scalar

text: ﬁelds

text: satisﬁes

text: R

text: contribution

text: to

text: dI,

text: we

text: immediately

text: observe

text: that

text: these

text: di-

text: agrams

text: are

text: simply

text: the

text: product

text: of

text: the

text: one-loop

text: Wilson

text: co-

text: d

text: eﬃcient,

text: and

text: the

text: next-to-leading-order

text: renormalized

text: µ

text: γ

text: Nγ

text: Ω

text: φ2

text: φ(p

text: Ω

text: ρ2

text: dµ

text: m

text: R

text: N

text: four-point

text: vertex.

text: This

text: contribution

text: to

text: the

text: vertex

text: is,

text: of

text: course,

text: nothing

text: other

text: than

text: the

text: next-to-leading-order

text: con-

text: tribution

text: to

text: the

text: renormalized

text: coupling

text: constant.

text: Thus,

text: Here

text: the

text: renormalization

text: group

text: operator

text: for

text: scalar

text: ﬁeld

text: we

text: write

text: the

text: coeﬃcient

text: ply

text: as

text: theory

text: is

text: d

text: λR

text: γ

text: log

text: d

text: ρ2

text: E

text: 8τ

text: µdµ

text: µ∂µ

text: β

text: ∂λ

text: γmmR

text: ∂m

text: λ,mR

text: µ,mR

text: R

text: µ,λ

text: where

text: λR

text: is

text: the

text: renormalized

text: coupling.

text: In

text: the

text: MS

text: and

text: the

text: coeﬃcients

text: are

text: scheme,

text: the

text: renormalized

text: by

text: dλ

text: O

text: µ2

text: O

text: βMS(λ)

text: µ

text: λ

text: λ

text: log

text: dµ

text: R

text: µ

text: d

text: log(m

text: γMS(λ)

text: R

text: m

text: dµ

text: For

text: a

text: calculation

text: of

text: this

text: to

text: ﬁve

text: loops,

text: see

text: λ

text: O

text: As

text: we

text: move

text: beyond

text: the

text: leading-order

text: Wilson

text: coeﬃ-

text: cients,

text: i.e.

text: and

text: divergent

text: radiative

text: corrections

text: ρ2

text: µ

text: d

text: log(Z

text: λ2

text: O

text: appear

text: in

text: our

text: calculations,

text: because

text: all

text: ﬁeld

text: interaction

text: γMS(λ)

text: φ

text: vertices

text: appear

text: at

text: zero

text: ﬂow

text: time.

text: These

text: divergences

text: dµ

text: can

text: be

text: removed

text: by

text: the

text: renormalization

text: parameters

text: of

text: the

text: original

text: theory

text: and

text: the

text: renormalization

text: scale

text: dependence

text: In

text: general

text: these

text: coeﬃcients

text: depend

text: on

text: the

text: mass,

text: the

text: of

text: the

text: smeared

text: operators

text: is

text: completely

text: contained

text: in

text: the

text: renormalization

text: scale,

text: and

text: the

text: renormalized

text: coupling

text: renormalization

text: parameters

text: of

text: the

text: original

text: theory.

text: This

text: constant,

text: but

text: in

text: a

text: mass

text: independent

text: renormalization

text: is

text: to

text: be

text: expected:

text: it

text: follows

text: from

text: the

text: fact

text: that

text: renormal-

text: scheme,

text: such

text: as

text: the

text: MS

text: scheme,

text: they

text: depend

text: on

text: the

text: ized

text: matrix

text: elements

text: at

text: zero

text: ﬂow

text: time

text: remain

text: ﬁnite

text: at

text: renormalization

text: scale

text: only

text: through

text: the

text: renormalized

text: cou-

text: nonzero

text: ﬂow

text: time

text: and

text: require

text: no

text: further

text: renormalization

text: pling

text: constant.

text: These

text: functions

text: are

text: known

text: to

text: ﬁve

text: loops

text: for

text: the

text: O(N)-symmetric

text: theory,

text: given

text: by

text: the

text: N-multiplet

text: Ultimately,

text: for

text: realistic

text: calculations

text: in

text: lattice

text: QCD,

text: Φ

text: φ

text: φ

text: N

text: the

text: perturbative

text: calculation

text: of

text: smeared

text: Wilson

text: coeﬃ-

text: Returning

text: again

text: to

text: our

text: example

text: of

text: the

text: OPE

text: for

text: the

text: cients

text: must

text: be

text: combined

text: with

text: nonperturbative

text: determi-

text: two-point

text: function,

text: Eq.

text: we

text: can

text: derive

text: an

text: renor-

text: nations

text: of

text: matrix

text: elements

text: at

text: hadronic

text: energy

text: scales.

text: malization

text: group

text: equation

text: for

text: the

text: Wilson

text: coeﬃcient

text: c

text: φ2

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Rapid Sequence Identiﬁcation of Potential Pathogens Using Techniques from Sparse Linear Algebra Stephanie Dodson, Darrell O. Ricke, Jeremy Kepner, Nelson Chiu, and Anna Shcherbina MIT Lincoln Laboratory, Lexington, MA, U.S.A Abstract— The decreasing costs and increasing speed and Despite the sequencing, the source was not correctly iden- accuracy of DNA sample collection, preparation, and sequencing tiﬁed as sprouts until 41 days after the ﬁrst hospitalized case, has rapidly produced an enormous volume of genetic data. and in the mean time the media and critics ran wild with However, fast and accurate analysis of the samples remains the possibility that the bacteria was a genetically engineered a bottleneck. Here we present D4RAGenS, a genetic sequence bioattack [5]. Additionally, the misidentiﬁcation of the source identiﬁcation algorithm that exhibits the Big Data handling and as Spanish cucumbers caused European Union farmers weekly computational power of the Dynamic Distributed Dimensional losses up to $611 million [6]. Data Model (D4M). The method leverages linear algebra and statistical properties to increase computational performance while In 2001 and 2011 genetic sequencing machines existed, but retaining accuracy by subsampling the data. Two run modes, Fast the lack of widely available fast analysis techniques inhibited and Wise, yield speed and precision tradeoffs, with applications the identiﬁcation of the source of the E. coli and anthrax. Since in biodefense and medical diagnostics. The D4RAGenS analysis the anthrax attacks the United States has developed federal algorithm is tested over several datasets, including three utilized for the Defense Threat Reduction Agency (DTRA) metagenomic laboratories to counter biological terrorism, but fast detection algorithm contest. and characterization methods are still inadequate [7]. Sequencing machines continue to be upgraded. At a frac- tion of the previous cost, new Next Generation Sequencers (NGS) have the capability of sequencing an entire human I. INTRODUCTION genome in mere days, with smaller genomes in minutes to hours. These sequencers combined with fast analytic methods With advances in biological and chemical knowledge as will allow for fast detection methods with endless application well as increased world travel, bioterrorism and deadly disease in homeland protection as well as infectious disease, inherited outbreaks are real concerns. The mitigation of these threats lies disease, cancer genomics, and individualized medicine. in accurate source and organism identiﬁcation, making rapid genetic sequencing and organism identiﬁcation of particular In general, metagenomics algorithms seek to tackle four interest in homeland defense. analysis problems: (1) organism identiﬁcation, (2) correct classiﬁcation of sequences, (3) gene identiﬁcation, and (4) Bioterror attacks made headlines in 2001 with the mailing variant identiﬁcation. With the goal of ﬁnding an algorithm that of anthrax containing letters to notable media and political would allow DNA sequencing to become a rapid diagnostic ﬁgures that resulted in 17 cases and 5 deaths. The timing of medical tool, the Defense Treat Reduction Agency (DTRA) the ﬁrst letter, shortly after the September 11th terror attacks, sponsored a challenge to ﬁnd a fast and accurate method [8]. increased public anxiety of the possibility of a foreign attack Several metagenomic algorithms are currently available, some [1]. A nine-year F.B.I. investigation ensued that dismissed the as a result of the DTRA challenge. Each algorithm uses a idea of a foreign assailant when analysis linked the anthrax wide range of techniques, with trade-offs between time and strain and concentration to those previously developed by the accuracy. United States biowarfare defense program [2]. Finally, a U.S. Many algorithms utilize existing tools, such as the “gold- government scientist was indicted for the crimes [3]. standard” BLAST (Basic Local Alignment Search Tool), and An example of widespread food borne illness lies in the modify the results to achieve better identiﬁcation levels [9] 2011 virulent Germany E. coli outbreak. A slightly pathogenic [10] [11]. These algorithms, including MetaPhlyer [9] and organism, E. coli typically has minor gastrointestinal effects MetaPhlAn [10], can achieve faster speeds by running BLAST and little chance of lasting damage or death. The Germany searches on reduced datasets of marker genes. Other algorithms outbreak however was much more serious and over the course test sequence similarity through the use of computationally of almost three months, the bacteria swept through Germany expensive hidden-Markov-models (HMMs) and creation of affecting over 4,000 individuals and resulting in 53 fatalities phylogenetic trees [12] [13]. [4]. Genetic sequencing did play a role in the discovery of a To avoid cumbersome external programs, HMMs, and bacteriophage insertion carrying the Shiga toxin in the E. coli phylogenetic trees, [14] instead segments the sequences into strain O104:H4 that accounted for the severity [4]. short k-mers, and performs sequence comparisons by matching the k-mers in a sorted list. The methods MetaCV [15], LMAT This work is sponsored by the Assistant Secretary of Defense for Research [16], and Kraken [17] employ a similar k-mer segmenting & Engineering under Air Force Contract #FA8721-05-C-0002. Opinions, strategy in combination with phylogenetic trees to discover interpretations, recommendations and conclusions are those of the authors and are not necessarily endorsed by the United States Government. the least common ancestors. ##LTLine##

text: Rapid

label: title

text: Sequence

text: Identiﬁcation

text: of

text: Potential

text: Pathogens

text: Using

text: Techniques

text: from

text: Sparse

text: Linear

text: Algebra

text: Stephanie

label: author

text: Dodson,

text: Darrell

text: O.

text: Ricke,

text: Jeremy

text: Kepner,

text: Nelson

text: Chiu,

text: and

text: Anna

text: Shcherbina

text: MIT

label: paragraph

text: Lincoln

text: Laboratory,

text: Lexington,

text: MA,

text: U.S.A

text: Abstract—

text: The

label: abstract

text: decreasing

text: costs

text: and

text: increasing

text: speed

text: and

text: Despite

text: the

text: sequencing,

text: the

text: source

text: was

text: not

text: correctly

text: iden-

text: accuracy

text: of

text: DNA

text: sample

text: collection,

text: preparation,

text: and

text: sequencing

text: tiﬁed

text: as

text: sprouts

text: until

text: days

text: after

text: the

text: ﬁrst

text: hospitalized

text: case,

text: has

text: rapidly

text: produced

text: an

text: enormous

text: volume

text: of

text: genetic

text: data.

text: and

text: in

text: the

text: mean

text: time

text: the

text: media

text: and

text: critics

text: ran

text: wild

text: with

text: However,

text: fast

text: and

text: accurate

text: analysis

text: of

text: the

text: samples

text: remains

text: the

text: possibility

text: that

text: the

text: bacteria

text: was

text: a

text: genetically

text: engineered

text: a

text: bottleneck.

text: Here

text: we

text: present

text: a

text: genetic

text: sequence

text: bioattack

text: Additionally,

text: the

text: misidentiﬁcation

text: of

text: the

text: source

text: identiﬁcation

text: algorithm

text: that

text: exhibits

text: the

text: Big

text: Data

text: handling

text: and

text: as

text: Spanish

text: cucumbers

text: caused

text: European

text: Union

text: farmers

text: weekly

text: computational

text: power

text: of

text: the

text: Dynamic

text: Distributed

text: Dimensional

text: losses

text: up

text: to

text: million

text: Data

text: Model

text: The

text: method

text: leverages

text: linear

text: algebra

text: and

text: statistical

text: properties

text: to

text: increase

text: computational

text: performance

text: while

text: In

text: and

text: genetic

text: sequencing

text: machines

text: existed,

text: but

text: retaining

text: accuracy

text: by

text: subsampling

text: the

text: data.

text: Two

text: run

text: modes,

text: Fast

text: the

text: lack

text: of

text: widely

text: available

text: fast

text: analysis

text: techniques

text: inhibited

text: and

text: Wise,

text: yield

text: speed

text: and

text: precision

text: tradeoffs,

text: with

text: applications

text: the

text: identiﬁcation

text: of

text: the

text: source

text: of

text: the

text: E.

text: coli

text: and

text: anthrax.

text: Since

text: in

text: biodefense

text: and

text: medical

text: diagnostics.

text: The

text: analysis

text: the

text: anthrax

text: attacks

text: the

text: United

text: States

text: has

text: developed

text: federal

text: algorithm

text: is

text: tested

text: over

text: several

text: datasets,

text: including

text: three

text: utilized

text: for

text: the

text: Defense

text: Threat

text: Reduction

text: Agency

text: (DTRA)

text: metagenomic

text: laboratories

text: to

text: counter

text: biological

text: terrorism,

text: but

text: fast

text: detection

text: algorithm

text: contest.

text: and

text: characterization

text: methods

text: are

text: still

text: inadequate

text: Sequencing

text: machines

text: continue

text: to

text: be

text: upgraded.

text: At

text: a

text: frac-

text: tion

text: of

text: the

text: previous

text: cost,

text: new

text: Next

text: Generation

text: Sequencers

text: (NGS)

text: have

text: the

text: capability

text: of

text: sequencing

text: an

text: entire

text: human

text: I.

text: INTRODUCTION

text: genome

text: in

text: mere

text: days,

text: with

text: smaller

text: genomes

text: in

text: minutes

text: to

text: hours.

text: These

text: sequencers

text: combined

text: with

text: fast

text: analytic

text: methods

text: With

text: advances

text: in

text: biological

text: and

text: chemical

text: knowledge

text: as

text: will

text: allow

text: for

text: fast

text: detection

text: methods

text: with

text: endless

text: application

text: well

text: as

text: increased

text: world

text: travel,

text: bioterrorism

text: and

text: deadly

text: disease

text: in

text: homeland

text: protection

text: as

text: well

text: as

text: infectious

text: disease,

text: inherited

text: outbreaks

text: are

text: real

text: concerns.

text: The

text: mitigation

text: of

text: these

text: threats

text: lies

text: disease,

text: cancer

text: genomics,

text: and

text: individualized

text: medicine.

text: in

text: accurate

text: source

text: and

text: organism

text: identiﬁcation,

text: making

text: rapid

text: genetic

text: sequencing

text: and

text: organism

text: identiﬁcation

text: of

text: particular

text: In

text: general,

text: metagenomics

text: algorithms

text: seek

text: to

text: tackle

text: four

text: interest

text: in

text: homeland

text: defense.

text: analysis

text: problems:

text: organism

text: identiﬁcation,

text: correct

text: classiﬁcation

text: of

text: sequences,

text: gene

text: identiﬁcation,

text: and

text: Bioterror

text: attacks

text: made

text: headlines

text: in

text: with

text: the

text: mailing

text: variant

text: identiﬁcation.

text: With

text: the

text: goal

text: of

text: ﬁnding

text: an

text: algorithm

text: that

text: of

text: anthrax

text: containing

text: letters

text: to

text: notable

text: media

text: and

text: political

text: would

text: allow

text: DNA

text: sequencing

text: to

text: become

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text: diagnostic

text: ﬁgures

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text: deaths.

text: The

text: timing

text: of

text: medical

text: tool,

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text: Reduction

text: Agency

text: (DTRA)

text: the

text: ﬁrst

text: letter,

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text: September

text: terror

text: attacks,

text: sponsored

text: a

text: challenge

text: to

text: ﬁnd

text: a

text: fast

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text: accurate

text: method

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text: public

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text: of

text: the

text: possibility

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text: metagenomic

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text: currently

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text: some

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text: nine-year

text: F.B.I.

text: investigation

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text: dismissed

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text: as

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text: result

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text: DTRA

text: challenge.

text: Each

text: algorithm

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text: a

text: idea

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text: foreign

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text: analysis

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text: the

text: anthrax

text: wide

text: range

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text: techniques,

text: with

text: trade-offs

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text: time

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text: concentration

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text: those

text: previously

text: developed

text: by

text: the

text: accuracy.

text: United

text: States

text: biowarfare

text: defense

text: program

text: Finally,

text: a

text: U.S.

text: Many

text: algorithms

text: utilize

text: existing

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text: the

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text: government

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text: example

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text: widespread

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text: coli

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text: organism,

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text: coli

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text: minor

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text: effects

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text: lasting

text: damage

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text: death.

text: The

text: Germany

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text: datasets

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text: marker

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text: course

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text: the

text: use

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text: the

text: bacteria

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text: expensive

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text: (HMMs)

text: and

text: creation

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text: affecting

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text: resulting

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text: fatalities

text: phylogenetic

text: trees

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text: sequencing

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text: The

text: methods

text: MetaCV

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text: work

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text: sponsored

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text: the

text: Assistant

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text: Defense

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text: Research

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text: Kraken

text: employ

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text: Engineering

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text: of

text: the

text: authors

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text: the

text: least

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text: ancestors.

text: ##LTLine##


text: Virus Identification TABLE I. NUMBER OF READS CLASSIFIED AS EACH TRUTH ORGANISM FN to Species 0 TP to Genus BY THE SIX DETECTION ALGORITHMS THAT REPORT READ MAPPING. THE 11 LAST ROW GIVES THE RESULTS OF RUNNING WISE MODE ON A GDNA 6.6 8 REFERENCE DATABASE FOR SAMPLES A AND B. 13 5 Algorithm SampleA Sample B Sample C 20 Species Genus Species Genus Species Genus 0 61 120 180 2 FP to Genus TP to Species Truth 58,707 58,707 49,948 49,948 39,996 39,996 4.8 11 16 22 2.7 Wise 390 D4RAGenS 37,486 37,486 961 24,558 4,107 21,261 1.8 260 MetaScope 53,999 54,109 48,675 49,517 39,613 39,867 0.9 140 Kraken 0 v0.10.4-beta 49,158 51,995 9,287 49,948 9,444 40,184 FP to Species 17 Time LMAT v1.2.1 9,767 36,339 106 36,253 2,369 12,492 Kraken v 0.10.4-Mbeettaa S c o p e F a s t D ⁴ MRAetGaCenVS v 2 . 3 . 0 L M A T v 1 . 2.1 MetaPhyler 0 45 0 71 0 188 1.25 Fig. 4. Results of organism identiﬁcation on the 22 virus in silico Illumina MetaCV 0 75,749 0 49,936 0 5,847 test set. The Fast D4RAGenS algorithm scores perfect marks. (gdna) Wise 58,707 58,707 35,018 49,303 D4RAGenS V. CONCLUSION Along with NGS tools, D4RAGenS can aid in homeland smaller reference database built from RNA sequences. Since security by detecting bioterror threats and disease outbreaks. In RNA is only the coding portion of the genome (i.e. the genes), 2001 and 2011, faster analysis would have resulted in quicker and the bacterial samples contain the coding and non-coding implication of the source of the E. coli and anthrax, fewer portions, all of the unknown sequences will not be detected fatalities, reduced public anxiety, and less economic damage. using the RNA database. Considering that 10-15% of bacterial The analysis can also be performed on an individual basis at genomes are non-coding regions [26] and that many reads health facilities to determine the cause of an illness before span coding and non-coding regions, the Wise D4RAGenS read administering medications. mappings are very reasonable. The techniques applied in Wise D4RAGenS naturally lead To exhibit the read identiﬁcation power of the Wise mode, to strain-level identiﬁcation of pathogens. The pipeline can be the algorithm was run using the full genomic sequences (gdna) modiﬁed to use a larger, strain speciﬁc dataset after initial of the truth organisms as the reference database. The tallies of identiﬁcation of target organisms with Fast mode. Such an read mapping using the gdna are shown in the last row of enhanced dataset would allow for detection of the minute Table I. Over 93% of sample A reads are correctly classiﬁed differences between strains, leading to increased beneﬁts for to the species level, and 98% of sample B reads to the genus medical and threat response. Similar extensions can lead to level, bringing the results level with MetaScope. The low gene detection. false positives, reasonable run times and mapping abilities place MetaScope, D4RAGenS, and LMAT above the other Additionally, the simple string comparisons using matrix algorithms. multiplications and parallel capabilities can be exploited to create an ultra-fast multiple sequence aligner. Sequence align- C. Virus Identiﬁcation and Read Mapping ers show relatedness of multiple sequences and are useful in All seven algorithms were again run on the generated locating mutations between healthy and diseased cells (such virus dataset, and results analyzed for identiﬁcation and read as those associated with genetic disease and cancer). mapping. The virus RNA dataset is about 15 times smaller than The basic mathematical ﬁlters used in D4RAGenS simplify the bacterial reference. In order to account for the decreased the current algorithms while maintaining a high level of size, the percentage of data used in Fast mode is increased from accuracy. Results show the algorithm to report comparable 2 to 8%. Despite the increase in data usage, the computation results to MetaScope and the publicly available LMAT v1.2.1 times remain fast due to the smaller number of viral organisms. analysis tools. The high identiﬁcation rate along with low run Organism detection results displayed in Figure 4 show that time and number of false positives places D4RAGenS as a top competitor for the best genetic analysis algorithm. Fast D4RAGenS, MetaScope, and Kraken v0.10.4-beta suc- cessfully identify all 22 organisms without any false positives. VI. ACKNOWLEDGEMENTS LMAT v1.2.1 reported 2 false negatives and 3 false positives. MetaPhyler 1.25 and MetaPhlan v1.7.7 failed to run on the The authors are indebted to the following individuals for virus test set, and MetaCV v2.3.0 only identiﬁed bacterial their technical contributions to this work: Chansup Byun, matches. Ashley Conard, and Dylan Hutchinson. Read mapping results in Table II reveal the four algorithms REFERENCES that produced results have similar scores on this test set. [1] D. Canedy and A. Kuczynski, “A Nation Challenged: A Medical Mys- Compared to bacteria, viral DNA is gene dense, with fewer tery; Second Case of Anthrax Leads F.B.I to Inquiry,” The New York non-coding regions, allowing for high read identiﬁcation using Times, Oct. 9, 2001. only RNA. In this case, Kraken v0.10.4-beta is believed to have [2] W. Broad, “A Nation Challenged: The Spores; Terror Anthrax Resembles no false positives due to the absence of a human background. Type Made by U.S.,” The New York Times, Dec 3, 2001. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: A speed and departure time optimization algorithm for the PRP Kramer, R.; Maculan, N.; Subramanian, A.; Vidal, T. 3 The proposed speed and departure time optimization algorithm The fuel consumption per distance unit F f (v ) is a convex function. The speed value ∗ σiσi+1 σiσi+1 v that minimizes fuel costs is given in Equation (4). Similarly, for any arc (σ , σ ), assuming f i i+1 ∗ that there is no waiting time in the route after σ , the speed value v that minimizes fuel and i fd ∗ ∗ driver costs is expressed in Equation (5). Both values, v and v , are independent of the arc f fd (cid:18) (cid:19) under consideration. dF f ∗ ⇔ ∗ w 1/3 σiσi+1 (v ) = 0 v = 1 (4) f f dv 2w σiσi+1 4 (cid:18) (cid:19) v∗ = ωωffdc + w1 1/3 (5) fd 2w 4 Given a route σ, the speed and departure time optimization problem consists of ﬁnding the departure time from the ﬁrst location (i.e., the depot when dealing with vehicle routing problems) and the optimal speeds for each arc while respecting customers’ time-windows. In order to solve this problem, we propose a recursive algorithm that extends those presented in Demir et al. (2012); Kramer et al. (2015); Hvattum et al. (2013). The proposed approach can be generally viewed as a divide-and-conquer strategy, which iteratively solves a relaxed speed optimization problem obtained by ignoring time windows at intermediate destinations. If the resulting solution satisﬁes all constraints, then it is returned. Otherwise the customer p with maximum time-window violation is identiﬁed and its arrival time is set to its closest feasible value. Fixing this decision variable creates two sub-problems which are recursively solved (Alg. 1, Line 20-21). Algorithm 1 is ﬁrst applied on a given route σ by setting the “start” s to 1 and the | | “end” e to σ . We recall that the departure time and return time to the depot are con- sidered as decision variables. In this ﬁrst iteration, still, the departure time is set to the earliest possible value t = a (Alg. 1, Line 6). This decision will be revised later on. From this σ1 σ1 starting time, the arrival time at each customer is derived as in Kramer et al. (2015): at ﬁrst, ∗ the arrival time at the last customer when traveling at speed v is determined and, in case of fd violation, updated to its closest time-window bound (Alg. 1, Line 8). Such arrival time leads to a reference speed v on the route (Alg. 1, Line 11) which will be used to compute the arrival ref time at each customer as well as the maximum time-window violation (Alg. 1, Lines 12-17). In case of violation, two subproblems will be recursively solved. Any subproblem starting at the depot is now solved without ﬁxing the departure time to a (as opposed to the method σ1 presented in Kramer et al. (2015)). Indeed, the arrival time to the last customer of this sub- 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: speed

text: and

text: departure

text: time

text: optimization

text: algorithm

text: for

text: the

text: PRP

text: Kramer,

text: R.;

text: Maculan,

text: N.;

text: Subramanian,

text: A.;

text: Vidal,

text: T.

label: section

text: The

text: proposed

text: speed

text: and

text: departure

text: time

text: optimization

text: algorithm

text: The

text: fuel

text: consumption

text: per

text: distance

text: unit

text: F

text: f

text: (v

text: is

text: a

text: convex

text: function.

text: The

text: speed

text: value

text: v

text: that

text: minimizes

text: fuel

text: costs

text: is

text: given

text: in

text: Equation

text: Similarly,

text: for

text: any

text: arc

text: (σ

text: σ

text: assuming

text: f

text: i

text: that

text: there

text: is

text: no

text: waiting

text: time

text: in

text: the

text: route

text: after

text: σ

text: the

text: speed

text: value

text: v

text: that

text: minimizes

text: fuel

text: and

text: i

text: fd

text: driver

text: costs

text: is

text: expressed

text: in

text: Equation

text: Both

text: values,

text: v

text: and

text: v

text: are

text: independent

text: of

text: the

text: arc

text: f

text: fd

text: under

text: consideration.

text: dF

text: f

text: w

text: (v

text: v

text: f

text: f

text: dv

text: v∗

text: ωωffdc

text: fd

text: Given

text: a

text: route

text: σ,

text: the

text: speed

text: and

text: departure

text: time

text: optimization

text: problem

text: consists

text: of

text: ﬁnding

text: the

text: departure

text: time

text: from

text: the

text: ﬁrst

text: location

text: (i.e.,

text: the

text: depot

text: when

text: dealing

text: with

text: vehicle

text: routing

text: problems)

text: and

text: the

text: optimal

text: speeds

text: for

text: each

text: arc

text: while

text: respecting

text: customers’

text: time-windows.

text: In

text: order

text: to

text: solve

text: this

text: problem,

text: we

text: propose

text: a

text: recursive

text: algorithm

text: that

text: extends

text: those

text: presented

text: in

text: Demir

text: et

text: al.

text: Kramer

text: et

text: al.

text: Hvattum

text: et

text: al.

text: The

text: proposed

text: approach

text: can

text: be

text: generally

text: viewed

text: as

text: a

text: divide-and-conquer

text: strategy,

text: which

text: iteratively

text: solves

text: a

text: relaxed

text: speed

text: optimization

text: problem

text: obtained

text: by

text: ignoring

text: time

text: windows

text: at

text: intermediate

text: destinations.

text: If

text: the

text: resulting

text: solution

text: satisﬁes

text: all

text: constraints,

text: then

text: it

text: is

text: returned.

text: Otherwise

text: the

text: customer

text: p

text: with

text: maximum

text: time-window

text: violation

text: is

text: identiﬁed

text: and

text: its

text: arrival

text: time

text: is

text: set

text: to

text: its

text: closest

text: feasible

text: value.

text: Fixing

text: this

text: decision

text: variable

text: creates

text: two

text: sub-problems

text: which

text: are

text: recursively

text: solved

text: (Alg.

text: Line

text: Algorithm

text: is

text: ﬁrst

text: applied

text: on

text: a

text: given

text: route

text: σ

text: by

text: setting

text: the

text: “start”

text: s

text: to

text: and

text: the

text: “end”

text: e

text: to

text: σ

text: We

text: recall

text: that

text: the

text: departure

text: time

text: and

text: return

text: time

text: to

text: the

text: depot

text: are

text: con-

text: sidered

text: as

text: decision

text: variables.

text: In

text: this

text: ﬁrst

text: iteration,

text: still,

text: the

text: departure

text: time

text: is

text: set

text: to

text: the

text: earliest

text: possible

text: value

text: t

text: a

text: (Alg.

text: Line

text: This

text: decision

text: will

text: be

text: revised

text: later

text: on.

text: From

text: this

text: σ1

text: σ1

text: starting

text: time,

text: the

text: arrival

text: time

text: at

text: each

text: customer

text: is

text: derived

text: as

text: in

text: Kramer

text: et

text: al.

text: at

text: ﬁrst,

text: the

text: arrival

text: time

text: at

text: the

text: last

text: customer

text: when

text: traveling

text: at

text: speed

text: v

text: is

text: determined

text: and,

text: in

text: case

text: of

text: fd

text: violation,

text: updated

text: to

text: its

text: closest

text: time-window

text: bound

text: (Alg.

text: Line

text: Such

text: arrival

text: time

text: leads

text: to

text: a

text: reference

text: speed

text: v

text: on

text: the

text: route

text: (Alg.

text: Line

text: which

text: will

text: be

text: used

text: to

text: compute

text: the

text: arrival

text: ref

text: time

text: at

text: each

text: customer

text: as

text: well

text: as

text: the

text: maximum

text: time-window

text: violation

text: (Alg.

text: Lines

text: In

text: case

text: of

text: violation,

text: two

text: subproblems

text: will

text: be

text: recursively

text: solved.

text: Any

text: subproblem

text: starting

text: at

text: the

text: depot

text: is

text: now

text: solved

text: without

text: ﬁxing

text: the

text: departure

text: time

text: to

text: a

text: (as

text: opposed

text: to

text: the

text: method

text: σ1

text: presented

text: in

text: Kramer

text: et

text: al.

text: Indeed,

text: the

text: arrival

text: time

text: to

text: the

text: last

text: customer

text: of

text: this

text: sub-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A speed and departure time optimization algorithm for the PRP Kramer, R.; Maculan, N.; Subramanian, A.; Vidal, T. problem is already ﬁxed. It is thus possible to evaluate the reference speed “backwards”, deriving the best departure time at the depot (Alg. 1, Line 10), and the arrival time at each customer. The other sub-problems are solved in a similar manner. For these cases, the starting date – and possibly the arrival date to the last customer or depot – are already set. The recursion is repeated until no violation is found. | | Finally, once arrival times t are known for all customers i = 1, . . . , σ , the associated σi ∗ ∗ speeds are revised in such a way that any speed below v is replaced by v and a waiting time f f (Alg. 1, Lines 22-24). Figure 1 shows an execution example of the presented algorithm in a route involving seven nodes. The horizontal lines represent the customers and the brackets their corresponding time windows. Bullet points indicate the arrival times. The best departure and arrival times of this example are depicted in Figure 1.e. Algorithm 1 Speed and Departure Time Optimization Algorithm — SDTOA (cid:80) 1: Procedure SDTOA(σ, s, e) ← (cid:80) ← ← 2: p violation maxV iolation 0 ← − 3: D ← ei−=s1 dσi,σi+1 4: T ei=s1 τσi | | 5: if s = 1 and e = σ then 6: tσ1 =| a| σ1 7: if e = σ then { { ∗ } } 8: tσe = min max aσe, tσs + D/vfd + T , bσe 9: if s = 1 then { { − ∗ − } } 10: tσs←= min m−ax aσ−s, tσe D/vfd T , bσs 11: vref D/(tσe tσs T) 12: for i = s + 1 . . . e do 13: tσi = tσi−1 + τσi−{1 + dσ−i−1,σi/vre−f } 14: violation = max 0, tσi bσi, aσi tσi 15: if violation > maxV iolation then 16: maxV iolation = violation 17: p = i 18: if maxV iolation > 0 then { { } } 19: tσp = min max aσp, tσp , bσp 20: SDTOA(σ, s, p) 21: SDTOA(σ, p, e) | | 22: if s = 1 and e = σ then | | 23: for i = 2 . . . σ do { − − ∗} 24: vσi−1,σi = max dσi−1,σi/(tσi tσi−1 τσi−1)), vf 4 Computational Results The proposed algorithm was integrated into the ILS-SP-SOA matheuristic of Kramer et al. (2015), which makes use of an adaptive speed matrix through the search to keep track of speed decisions. Instead of SOA, the SDTOA is executed on each route associated to a local optimal solution found by the ILS, returning routes with optimized speeds and departure times. These routes are stored in a pool that are further combined together by means of integer programming over a set partitioning (SP) formulation. 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: speed

text: and

text: departure

text: time

text: optimization

text: algorithm

text: for

text: the

text: PRP

text: Kramer,

text: R.;

text: Maculan,

text: N.;

text: Subramanian,

text: A.;

text: Vidal,

text: T.

text: problem

text: is

text: already

text: ﬁxed.

text: It

text: is

text: thus

text: possible

text: to

text: evaluate

text: the

text: reference

text: speed

text: “backwards”,

text: deriving

text: the

text: best

text: departure

text: time

text: at

text: the

text: depot

text: (Alg.

text: Line

text: and

text: the

text: arrival

text: time

text: at

text: each

text: customer.

text: The

text: other

text: sub-problems

text: are

text: solved

text: in

text: a

text: similar

text: manner.

text: For

text: these

text: cases,

text: the

text: starting

text: date

text: and

text: possibly

text: the

text: arrival

text: date

text: to

text: the

text: last

text: customer

text: or

text: depot

text: are

text: already

text: set.

text: The

text: recursion

text: is

text: repeated

text: until

text: no

text: violation

text: is

text: found.

text: Finally,

text: once

text: arrival

text: times

text: t

text: are

text: known

text: for

text: all

text: customers

text: i

text: σ

text: the

text: associated

text: σi

text: speeds

text: are

text: revised

text: in

text: such

text: a

text: way

text: that

text: any

text: speed

text: below

text: v

text: is

text: replaced

text: by

text: v

text: and

text: a

text: waiting

text: time

text: f

text: f

text: (Alg.

text: Lines

text: Figure

text: shows

text: an

text: execution

text: example

text: of

text: the

text: presented

text: algorithm

text: in

text: a

text: route

text: involving

text: seven

text: nodes.

text: The

text: horizontal

text: lines

text: represent

text: the

text: customers

text: and

text: the

text: brackets

text: their

text: corresponding

text: time

text: windows.

text: Bullet

text: points

text: indicate

text: the

text: arrival

text: times.

text: The

text: best

text: departure

text: and

text: arrival

text: times

text: of

text: this

text: example

text: are

text: depicted

text: in

text: Figure

text: Algorithm

text: Speed

text: and

text: Departure

text: Time

text: Optimization

text: Algorithm

text: SDTOA

text: Procedure

text: SDTOA(σ,

text: s,

text: e)

text: p

text: violation

text: maxV

text: iolation

text: D

text: T

text: τσi

text: if

text: s

text: and

text: e

text: σ

text: then

text: a|

text: σ1

text: if

text: e

text: σ

text: then

text: tσe

text: min

text: max

text: aσe,

text: tσs

text: D/vfd

text: T

text: bσe

text: if

text: s

text: then

text: tσs←=

text: min

text: m−ax

text: aσ−s,

text: tσe

text: D/vfd

text: T

text: bσs

text: vref

text: D/(tσe

text: tσs

text: T)

text: for

text: i

text: s

text: e

text: do

text: tσi

text: τσi−{1

text: violation

text: max

text: tσi

text: bσi,

text: aσi

text: tσi

text: if

text: violation

text: maxV

text: iolation

text: then

text: maxV

text: iolation

text: violation

text: p

text: i

text: if

text: maxV

text: iolation

text: then

text: tσp

text: min

text: max

text: aσp,

text: tσp

text: bσp

text: SDTOA(σ,

text: s,

text: p)

text: SDTOA(σ,

text: p,

text: e)

text: if

text: s

text: and

text: e

text: σ

text: then

text: for

text: i

text: σ

text: do

text: max

text: vf

text: Computational

text: Results

text: The

text: proposed

text: algorithm

text: was

text: integrated

text: into

text: the

text: ILS-SP-SOA

text: matheuristic

text: of

text: Kramer

text: et

text: al.

text: which

text: makes

text: use

text: of

text: an

text: adaptive

text: speed

text: matrix

text: through

text: the

text: search

text: to

text: keep

text: track

text: of

text: speed

text: decisions.

text: Instead

text: of

text: SOA,

text: the

text: SDTOA

text: is

text: executed

text: on

text: each

text: route

text: associated

text: to

text: a

text: local

text: optimal

text: solution

text: found

text: by

text: the

text: ILS,

text: returning

text: routes

text: with

text: optimized

text: speeds

text: and

text: departure

text: times.

text: These

text: routes

text: are

text: stored

text: in

text: a

text: pool

text: that

text: are

text: further

text: combined

text: together

text: by

text: means

text: of

text: integer

text: programming

text: over

text: a

text: set

text: partitioning

text: (SP)

text: formulation.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 FIG. 6. Plots of the mean curvatures H of the individual MOTSs laid out on the (ρ, σ) coordinates of the six patches of the horizon ﬁnder at the time of osculation t = 1.97384M. Their mean curvatures now agree at the contact point, as required by the theory. This is mainly achieved through the distortion of the large MOTS. penetrated approximately half-way inside the larger one. At this stage, the puncture near the center of the small MOTS can be expected to have a signiﬁcant eﬀect on the larger one. This is evident in Fig. 7 from the sharp localized feature that has developed at the front of the large MOTS. As the simulation progresses further, the horizon ﬁnder is not able to locate the larger individual MOTS. This could be due either to the horizon ﬁnder breaking down for numerical reasons due to the puncture or simply due to the larger MOTS ceasing to exist. Further code development would be necessary to resolve this issue. The left and right panels of Fig. 8 give surface plots of the mean-curvature of the individual MOTSs at the same times t = 2.1785M and t = 2.538M as in Fig. 7. The left panel of Fig. 8 shows that the mean curvature at the front of the larger MOTS has continued to grow and now overshoots the mean curvature of the smaller MOTS. At the later time displayed in the right panel of Fig. 8, the mean curvature at the front of the larger MOTS has now fallen back below the mean curvature of the smaller MOTS and has developed some highly oscillatory behavior. Again, this could be the result of the puncture inside the smaller MOTS as it crosses the surface of the larger MOTS at t ≈ 2.45M. Figure 9 shows the time evolution of the mean curvatures at the (facing) front ends of the individual MOTSs as they approach and penetrate. These are the points around which their mutual distortion is greatest. The mean curvature at the front end of the smaller MOTS grows steadily from its initial time symmetric value H = 0 and then settles to a fairly constant value as its geometry adjusts to the ambient gauge. In contrast, the mean curvature H at the (large) front end of the larger MOTS shows a rapid increase as the smaller MOTS approaches and penetrates. At the time of osculation, t = 1.97384M, both mean curvatures agree, as required by (3.19). As the penetration continues, H (large) continues to rise until the puncture inside the small MOTS crosses it. H then ﬁrst drops discontinuously to a (large) much smaller value and then begins to rise again during the period when both MOTS overlap. Runs with diﬀerent spatial resolutions and diﬀerent time steps conﬁrm that this jump is not a numerical artifact of the evolution code. It could possibly result from a discontinuity in the MOTT trace out by the large MOTS similar to the jump that occurs in the formation of the outer common horizon. Or it could be an indication of a more complicated horizon structure which is diﬃcult for the horizon ﬁnder to fully describe. This is another issue that we defer to future work. This discontinuous drop in H has a counterpart in Fig. 10, which plots the time evolution of the areas A of (large) both individual MOTSs. The larger MOTS shows a discontinuous jump in area at the same time its mean curvature undergoes a discontinuous drop. The small MOTS shows very small growth in area during the entire run, with an approximate range 3.941 < A < 3.948. (Near the outset, A undergoes a small anomalous decrease which (small) (small) can be attributed to numerical error since this would otherwise be inconsistent with the strict stability of the small MOTS at this time.) At the end of the run, the ratio A /A has not undergone substantial change, which (large) (small) oﬀers no clue regarding their possible future coalescence. E. Convergence In order to verify the new geometrical features we have found, we have measured the convergence of the harmonic constraints Cµ, i.e. (2.3) with the gauging forcing Fµ = 0 used in our simulations. We monitor the constraints via the norm ##LTFigure## ##LTFigure##

text: FIG.

text: Plots

text: of

text: the

text: mean

text: curvatures

text: H

text: of

text: the

text: individual

text: MOTSs

text: laid

text: out

text: on

text: the

text: (ρ,

text: σ)

text: coordinates

text: of

text: the

text: six

text: patches

text: of

text: the

text: horizon

text: ﬁnder

text: at

text: the

text: time

text: of

text: osculation

text: t

text: Their

text: mean

text: curvatures

text: now

text: agree

text: at

text: the

text: contact

text: point,

text: as

text: required

text: by

text: the

text: theory.

text: This

text: is

text: mainly

text: achieved

text: through

text: the

text: distortion

text: of

text: the

text: large

text: MOTS.

text: penetrated

text: approximately

text: half-way

text: inside

text: the

text: larger

text: one.

text: At

text: this

text: stage,

text: the

text: puncture

text: near

text: the

text: center

text: of

text: the

text: small

text: MOTS

text: can

text: be

text: expected

text: to

text: have

text: a

text: signiﬁcant

text: eﬀect

text: on

text: the

text: larger

text: one.

text: This

text: is

text: evident

text: in

text: Fig.

text: from

text: the

text: sharp

text: localized

text: feature

text: that

text: has

text: developed

text: at

text: the

text: front

text: of

text: the

text: large

text: MOTS.

text: As

text: the

text: simulation

text: progresses

text: further,

text: the

text: horizon

text: ﬁnder

text: is

text: not

text: able

text: to

text: locate

text: the

text: larger

text: individual

text: MOTS.

text: This

text: could

text: be

text: due

text: either

text: to

text: the

text: horizon

text: ﬁnder

text: breaking

text: down

text: for

text: numerical

text: reasons

text: due

text: to

text: the

text: puncture

text: or

text: simply

text: due

text: to

text: the

text: larger

text: MOTS

text: ceasing

text: to

text: exist.

text: Further

text: code

text: development

text: would

text: be

text: necessary

text: to

text: resolve

text: this

text: issue.

text: The

text: left

text: and

text: right

text: panels

text: of

text: Fig.

text: give

text: surface

text: plots

text: of

text: the

text: mean-curvature

text: of

text: the

text: individual

text: MOTSs

text: at

text: the

text: same

text: times

text: t

text: and

text: t

text: as

text: in

text: Fig.

text: The

text: left

text: panel

text: of

text: Fig.

text: shows

text: that

text: the

text: mean

text: curvature

text: at

text: the

text: front

text: of

text: the

text: larger

text: MOTS

text: has

text: continued

text: to

text: grow

text: and

text: now

text: overshoots

text: the

text: mean

text: curvature

text: of

text: the

text: smaller

text: MOTS.

text: At

text: the

text: later

text: time

text: displayed

text: in

text: the

text: right

text: panel

text: of

text: Fig.

text: the

text: mean

text: curvature

text: at

text: the

text: front

text: of

text: the

text: larger

text: MOTS

text: has

text: now

text: fallen

text: back

text: below

text: the

text: mean

text: curvature

text: of

text: the

text: smaller

text: MOTS

text: and

text: has

text: developed

text: some

text: highly

text: oscillatory

text: behavior.

text: Again,

text: this

text: could

text: be

text: the

text: result

text: of

text: the

text: puncture

text: inside

text: the

text: smaller

text: MOTS

text: as

text: it

text: crosses

text: the

text: surface

text: of

text: the

text: larger

text: MOTS

text: at

text: t

text: Figure

text: shows

text: the

text: time

text: evolution

text: of

text: the

text: mean

text: curvatures

text: at

text: the

text: (facing)

text: front

text: ends

text: of

text: the

text: individual

text: MOTSs

text: as

text: they

text: approach

text: and

text: penetrate.

text: These

text: are

text: the

text: points

text: around

text: which

text: their

text: mutual

text: distortion

text: is

text: greatest.

text: The

text: mean

text: curvature

text: at

text: the

text: front

text: end

text: of

text: the

text: smaller

text: MOTS

text: grows

text: steadily

text: from

text: its

text: initial

text: time

text: symmetric

text: value

text: H

text: and

text: then

text: settles

text: to

text: a

text: fairly

text: constant

text: value

text: as

text: its

text: geometry

text: adjusts

text: to

text: the

text: ambient

text: gauge.

text: In

text: contrast,

text: the

text: mean

text: curvature

text: H

text: at

text: the

text: (large)

text: front

text: end

text: of

text: the

text: larger

text: MOTS

text: shows

text: a

text: rapid

text: increase

text: as

text: the

text: smaller

text: MOTS

text: approaches

text: and

text: penetrates.

text: At

text: the

text: time

text: of

text: osculation,

text: t

text: both

text: mean

text: curvatures

text: agree,

text: as

text: required

text: by

text: As

text: the

text: penetration

text: continues,

text: H

text: (large)

text: continues

text: to

text: rise

text: until

text: the

text: puncture

text: inside

text: the

text: small

text: MOTS

text: crosses

text: it.

text: H

text: then

text: ﬁrst

text: drops

text: discontinuously

text: to

text: a

text: (large)

text: much

text: smaller

text: value

text: and

text: then

text: begins

text: to

text: rise

text: again

text: during

text: the

text: period

text: when

text: both

text: MOTS

text: overlap.

text: Runs

text: with

text: diﬀerent

text: spatial

text: resolutions

text: and

text: diﬀerent

text: time

text: steps

text: conﬁrm

text: that

text: this

text: jump

text: is

text: not

text: a

text: numerical

text: artifact

text: of

text: the

text: evolution

text: code.

text: It

text: could

text: possibly

text: result

text: from

text: a

text: discontinuity

text: in

text: the

text: MOTT

text: trace

text: out

text: by

text: the

text: large

text: MOTS

text: similar

text: to

text: the

text: jump

text: that

text: occurs

text: in

text: the

text: formation

text: of

text: the

text: outer

text: common

text: horizon.

text: Or

text: it

text: could

text: be

text: an

text: indication

text: of

text: a

text: more

text: complicated

text: horizon

text: structure

text: which

text: is

text: diﬃcult

text: for

text: the

text: horizon

text: ﬁnder

text: to

text: fully

text: describe.

text: This

text: is

text: another

text: issue

text: that

text: we

text: defer

text: to

text: future

text: work.

text: This

text: discontinuous

text: drop

text: in

text: H

text: has

text: a

text: counterpart

text: in

text: Fig.

text: which

text: plots

text: the

text: time

text: evolution

text: of

text: the

text: areas

text: A

text: of

text: (large)

text: both

text: individual

text: MOTSs.

text: The

text: larger

text: MOTS

text: shows

text: a

text: discontinuous

text: jump

text: in

text: area

text: at

text: the

text: same

text: time

text: its

text: mean

text: curvature

text: undergoes

text: a

text: discontinuous

text: drop.

text: The

text: small

text: MOTS

text: shows

text: very

text: small

text: growth

text: in

text: area

text: during

text: the

text: entire

text: run,

text: with

text: an

text: approximate

text: range

text: A

text: (Near

text: the

text: outset,

text: A

text: undergoes

text: a

text: small

text: anomalous

text: decrease

text: which

text: (small)

text: (small)

text: can

text: be

text: attributed

text: to

text: numerical

text: error

text: since

text: this

text: would

text: otherwise

text: be

text: inconsistent

text: with

text: the

text: strict

text: stability

text: of

text: the

text: small

text: MOTS

text: at

text: this

text: time.)

text: At

text: the

text: end

text: of

text: the

text: run,

text: the

text: ratio

text: A

text: /A

text: has

text: not

text: undergone

text: substantial

text: change,

text: which

text: (large)

text: (small)

text: oﬀers

text: no

text: clue

text: regarding

text: their

text: possible

text: future

text: coalescence.

text: E.

text: Convergence

text: In

text: order

text: to

text: verify

text: the

text: new

text: geometrical

text: features

text: we

text: have

text: found,

text: we

text: have

text: measured

text: the

text: convergence

text: of

text: the

text: harmonic

text: constraints

text: Cµ,

text: i.e.

text: with

text: the

text: gauging

text: forcing

text: Fµ

text: used

text: in

text: our

text: simulations.

text: We

text: monitor

text: the

text: constraints

text: via

text: the

text: norm

text: ##LTFigure##

text: ##LTFigure##


text: 9 The operator L and the function W are given by the geometry of S and the space-time geometry of M , whereas τ¯ τ¯ f is a function determined by (3.16). If S is strictly stable then L is invertible. Thus the previous calculation can τ¯ be turned around to conclude the existence of the desired MOTT. The causal structure of the tube follows by the observation that the null energy condition implies W ≥ 0. From stability one then obtains f ≥ 0 via equation (3.18) as in [14, Lemma 3]. A similar argument was used in [13] to construct a MOTT through S in the case it is stable but not strictly stable (cid:83) τ¯ assuming W ≥ 0 and W (cid:54)≡ 0. A diﬀerent approach is to take the outermost MOTS Sout of all slices M and deﬁne the apparent horizon of the τ τ slicing as H = Sout. In generic space-times H is smooth up to a discrete set of outward jumps. See [13] for the τ τ details and the particular notion of genericity used therein. The same reference also provides the causal character of H, namely that H is achronal. Moreover, if Ω denotes the interior of Sout, then J+(Ω ) ∩ M ⊂ Ω for all τ < σ. τ τ τ σ σ Here J+(Ω ) denotes the causal future of Ω . In particular, if there is a MOTS at an initial time τ¯ then it will persist τ τ for all times. D. Approaching MOTSs Assume that the space-time and the slicing are completely regular. Then the constant δ, the bound for the area, and the length of the second fundamental form of Sout remains uniformly controlled. If in this situation two components of a MOTS are closer than δ, neither of these two components can be part of Sout. In particular, the evolution of Sout will be discontinuous at some stage of the evolution. By the causality of H, the jump is outward, and a new outermost MOTS has formed outside the tubes of the two original ones. Generically, after the jump time the jump target will split into two branches of MOTTs, a stable branch traveling outward and an unstable branch traveling inward. For a complete discussion of this jump in the outermost MOTS, we refer to [13]. An important fact to point out is that if two MOTSs are close to each other then there is no reason to expect that they be stable, in contrast to the stability of Sout. Note that it is not known whether the area of Sout is monotonic across the jumps. For MOTSs in general, including Sout, it is not even clear whether monotonicity of area should be expected along smooth pieces. However, if the MOTS has positive mean curvature, so that H = −P > 0, then its expansion in any outward spacelike direction Nµ + αnµ, where 0 ≤ α < 1, must be positive. As a consequence, the continuous portion of the MOTT traced out by a stable MOTS with H > 0 must have a monotonically increasing area. Such MOTSs will also be inner trapped, i.e. θ− = P − H < 0. E. Exterior osculation of MOTSs The following theoretical observations pertain to the collision between the MOTSs of a large and small black hole as they ﬁrst touch. Note that this situation is not prevented by the strong maximum principle. However, at the time of ﬁrst contact, a common horizon already has formed according to the previous section . In (3.6), the contribution to θ from P is common to both MOTSs, so that at a common point of osculation we + must have H = H . (3.19) (small) (large) Here the two mean curvatures are deﬁned with respect to the respective outer normals, which in this case have opposite orientations. Thus the two MOTSs do not share the same outgoing null direction (cid:96). As a result of (3.19), the mean extrinsic curvature radius of the large and small MOTSs must match at the point of osculation. This can be deceiving in terms of a coordinate picture of the MOTSs since the connection of the Cauchy slice enters into the mean curvature. We have H = sij∂ N − sijΓk N . (3.20) i j ij k At the point of osculation N = −N so that the second term has the same magnitude but opposite sign i(small) i(large) for the small and large MOTSs. Hence (3.19) implies sij∂ N = sij∂ N − 2sijΓk N (3.21) i j(small) i j(large) ij k(large)

text: The

text: operator

text: L

text: and

text: the

text: function

text: W

text: are

text: given

text: by

text: the

text: geometry

text: of

text: S

text: and

text: the

text: space-time

text: geometry

text: of

text: M

text: whereas

text: τ¯

text: τ¯

text: f

text: is

text: a

text: function

text: determined

text: by

text: If

text: S

text: is

text: strictly

text: stable

text: then

text: L

text: is

text: invertible.

text: Thus

text: the

text: previous

text: calculation

text: can

text: τ¯

text: be

text: turned

text: around

text: to

text: conclude

text: the

text: existence

text: of

text: the

text: desired

text: MOTT.

text: The

text: causal

text: structure

text: of

text: the

text: tube

text: follows

text: by

text: the

text: observation

text: that

text: the

text: energy

text: condition

text: implies

text: W

text: From

text: stability

text: one

text: then

text: obtains

text: f

text: via

text: equation

text: as

text: in

text: Lemma

text: A

text: similar

text: argument

text: was

text: used

text: in

text: to

text: construct

text: a

text: MOTT

text: through

text: S

text: in

text: the

text: case

text: it

text: is

text: stable

text: but

text: not

text: strictly

text: stable

text: τ¯

text: assuming

text: W

text: and

text: W

text: A

text: diﬀerent

text: approach

text: is

text: to

text: take

text: the

text: outermost

text: MOTS

text: Sout

text: of

text: all

text: slices

text: M

text: and

text: deﬁne

text: the

text: apparent

text: horizon

text: of

text: the

text: τ

text: τ

text: slicing

text: as

text: H

text: Sout.

text: In

text: generic

text: space-times

text: H

text: is

text: smooth

text: up

text: to

text: a

text: discrete

text: set

text: of

text: outward

text: jumps.

text: See

text: for

text: the

text: τ

text: τ

text: details

text: and

text: the

text: particular

text: notion

text: of

text: genericity

text: used

text: therein.

text: The

text: same

text: reference

text: also

text: provides

text: the

text: causal

text: character

text: of

text: H,

text: namely

text: that

text: H

text: is

text: achronal.

text: Moreover,

text: if

text: Ω

text: denotes

text: the

text: interior

text: of

text: Sout,

text: then

text: J+(Ω

text: M

text: Ω

text: for

text: all

text: τ

text: σ.

text: τ

text: τ

text: τ

text: σ

text: σ

text: Here

text: J+(Ω

text: denotes

text: the

text: causal

text: future

text: of

text: Ω

text: In

text: particular,

text: if

text: there

text: is

text: a

text: MOTS

text: at

text: an

text: initial

text: time

text: τ¯

text: then

text: it

text: will

text: persist

text: τ

text: τ

text: for

text: all

text: times.

text: D.

text: Approaching

text: MOTSs

text: Assume

text: that

text: the

text: space-time

text: and

text: the

text: slicing

text: are

text: completely

text: regular.

text: Then

text: the

text: constant

text: δ,

text: the

text: bound

text: for

text: the

text: area,

text: and

text: the

text: length

text: of

text: the

text: second

text: fundamental

text: form

text: of

text: Sout

text: remains

text: uniformly

text: controlled.

text: If

text: in

text: this

text: situation

text: two

text: components

text: of

text: a

text: MOTS

text: are

text: closer

text: than

text: δ,

text: neither

text: of

text: these

text: two

text: components

text: can

text: be

text: part

text: of

text: Sout.

text: In

text: particular,

text: the

text: evolution

text: of

text: Sout

text: will

text: be

text: discontinuous

text: at

text: some

text: stage

text: of

text: the

text: evolution.

text: By

text: the

text: causality

text: of

text: H,

text: the

text: jump

text: is

text: outward,

text: and

text: a

text: new

text: outermost

text: MOTS

text: has

text: formed

text: outside

text: the

text: tubes

text: of

text: the

text: two

text: original

text: ones.

text: Generically,

text: after

text: the

text: jump

text: time

text: the

text: jump

text: target

text: will

text: split

text: into

text: two

text: branches

text: of

text: MOTTs,

text: a

text: stable

text: branch

text: traveling

text: outward

text: and

text: an

text: unstable

text: branch

text: traveling

text: inward.

text: For

text: a

text: complete

text: discussion

text: of

text: this

text: jump

text: in

text: the

text: outermost

text: MOTS,

text: we

text: refer

text: to

text: An

text: important

text: fact

text: to

text: point

text: out

text: is

text: that

text: if

text: two

text: MOTSs

text: are

text: close

text: to

text: each

text: other

text: then

text: there

text: is

text: no

text: reason

text: to

text: expect

text: that

text: they

text: be

text: stable,

text: in

text: contrast

text: to

text: the

text: stability

text: of

text: Sout.

text: Note

text: that

text: it

text: is

text: not

text: known

text: whether

text: the

text: area

text: of

text: Sout

text: is

text: monotonic

text: across

text: the

text: jumps.

text: For

text: MOTSs

text: in

text: general,

text: including

text: Sout,

text: it

text: is

text: not

text: even

text: clear

text: whether

text: monotonicity

text: of

text: area

text: should

text: be

text: expected

text: along

text: smooth

text: pieces.

text: However,

text: if

text: the

text: MOTS

text: has

text: positive

text: mean

text: curvature,

text: so

text: that

text: H

text: −P

text: then

text: its

text: expansion

text: in

text: any

text: outward

text: spacelike

text: direction

text: Nµ

text: αnµ,

text: where

text: α

text: must

text: be

text: positive.

text: As

text: a

text: consequence,

text: the

text: continuous

text: portion

text: of

text: the

text: MOTT

text: traced

text: out

text: by

text: a

text: stable

text: MOTS

text: with

text: H

text: must

text: have

text: a

text: monotonically

text: increasing

text: area.

text: Such

text: MOTSs

text: will

text: also

text: be

text: inner

text: trapped,

text: i.e.

text: θ−

text: P

text: H

text: E.

text: Exterior

text: osculation

text: of

text: MOTSs

text: The

text: following

text: theoretical

text: observations

text: pertain

text: to

text: the

text: collision

text: between

text: the

text: MOTSs

text: of

text: a

text: large

text: and

text: small

text: black

text: hole

text: as

text: they

text: ﬁrst

text: touch.

text: Note

text: that

text: this

text: situation

text: is

text: not

text: prevented

text: by

text: the

text: strong

text: maximum

text: principle.

text: However,

text: at

text: the

text: time

text: of

text: ﬁrst

text: contact,

text: a

text: common

text: horizon

text: already

text: has

text: formed

text: according

text: to

text: the

text: previous

text: section

text: In

text: the

text: contribution

text: to

text: θ

text: from

text: P

text: is

text: common

text: to

text: both

text: MOTSs,

text: so

text: that

text: at

text: a

text: common

text: point

text: of

text: osculation

text: we

text: must

text: have

text: H

text: H

text: (small)

text: (large)

text: Here

text: the

text: two

text: mean

text: curvatures

text: are

text: deﬁned

text: with

text: respect

text: to

text: the

text: respective

text: outer

text: normals,

text: which

text: in

text: this

text: case

text: have

text: opposite

text: orientations.

text: Thus

text: the

text: two

text: MOTSs

text: do

text: not

text: share

text: the

text: same

text: outgoing

text: direction

text: As

text: a

text: result

text: of

text: the

text: mean

text: extrinsic

text: curvature

text: radius

text: of

text: the

text: large

text: and

text: small

text: MOTSs

text: must

text: match

text: at

text: the

text: point

text: of

text: osculation.

text: This

text: can

text: be

text: deceiving

text: in

text: terms

text: of

text: a

text: coordinate

text: picture

text: of

text: the

text: MOTSs

text: since

text: the

text: connection

text: of

text: the

text: Cauchy

text: slice

text: enters

text: into

text: the

text: mean

text: curvature.

text: We

text: have

text: H

text: sij∂

text: N

text: sijΓk

text: N

text: i

text: j

text: ij

text: k

text: At

text: the

text: point

text: of

text: osculation

text: N

text: −N

text: so

text: that

text: the

text: second

text: term

text: has

text: the

text: same

text: magnitude

text: but

text: opposite

text: sign

text: i(small)

text: i(large)

text: for

text: the

text: small

text: and

text: large

text: MOTSs.

text: Hence

text: implies

text: sij∂

text: N

text: sij∂

text: N

text: N

text: i

text: j(small)

text: i

text: j(large)

text: ij

text: k(large)


text: Bp f^ 2s/B1M KsB f^ 2s/B1M Bp f 2/1-M KsB f 2/1-M || sB || sB 12 4.0 1.0 243 ensemble, am = 0.005 243 ensemble, am = 0.005 243 ensemble, aml = 0.01 243 ensemble, aml = 0.01 3.0 333222333 eeennnssseeemmmbbbllleee,,, aaammmlll === 000...000000468 0.8 333222333 eeennnssseeemmmbbbllleee,,, aaammmlll === 000...000000468 l l continuum continuum 0.6 2.0 0.4 c 2/dof = 0.91, p = 52% c 2/dof = 1.39, p = 14% 1.0 0.2 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 ( Ep / M )2 ( Ep / M )2 Bs Bs 4.0 1.0 243 ensemble, am = 0.005 243 ensemble, am = 0.005 243 ensemble, aml = 0.01 243 ensemble, aml = 0.01 3.0 333222333 eeennnssseeemmmbbbllleee,,, aaammmlll === 000...000000468 0.8 333222333 eeennnssseeemmmbbbllleee,,, aaammmlll === 000...000000468 l l continuum continuum 0.6 2.0 0.4 c 2/dof = 0.74, p = 75% c 2/dof = 1.21, p = 23% 1.0 0.2 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 ( E / M )2 ( E / M )2 K Bs K Bs → → FIG. 6. Chiral-continuum extrapolation of the B π(cid:96)ν (upper plots) and B K(cid:96)ν (lower plots) form factors from correlated s ﬁts using NLO SU(2) hard-pion/kaon HMχPT. Fits of f⊥ are on the left and of f(cid:107) are on the right. In each plot, the colors ≈ distinguish between data points on the ﬁve diﬀerent ensembles: circles and squares correspond to the a 0.11 fm data and ≈ triangles and diamonds to a 0.086 fm data. The colored ﬁt curves show the interpolation/extrapolation in pion/kaon energy: the ﬁt function is evaluated at the unphysical sea-quark masses and nonzero lattice spacings on the diﬀerent ensembles, such that the curves should go through the data points of the same color. The continuum, physical-quark-mass form factors are shown as a function of pion/kaon energy by the black lines with gray error band. The vertical dashed line on the left-hand side of each plot shows the physical pion or kaon mass. For each form factor, we obtain the largest diﬀerence produce the experimental value of the B meson mass, s from our preferred ﬁt using the following variation: and carry out our analysis in terms of dimensionless ra- ≤ ≤ tios over M to remove all explicit dependence on the fBπ : analytic for 18.7 GeV2 q2 22.7 GeV2 Bs + lattice scale. We then obtain the form factors and mo- and omitting the M term elsewhere, mentum transfers in physical units by multiplying by the π f0Bπ : analytic, atapipnrtoiepsriiantethpeowfoerrmoffaMctBosrs=du5e.36to7(t4o) GtheeVe.xTphereimunencetar-l f+BsK : analytic, error on MBs are negligible. fBsK : omitting the a2 and M terms, We do, however, still need to consider the implicit de- 0 π pendence on the lattice spacing through the parameters We therefore use these ﬁts to obtain the q2-dependent of the b-quark action. We estimate the size of this depen- chiral-continuum extrapolation errors quoted in Table VI dence, by computing the form factors f⊥ and f(cid:107) for seven and shown in Fig. 7. sets of RHQ parameters. We then calculate the slopes with respect to the parameters – ∆f/∆m a, ∆f/∆c 0 P and ∆f/∆ξ – for all momenta used in the analysis. Next B. Lattice-scale uncertainty we multiply each slope by the uncertainty in the corre- sponding RHQ parameter due to the lattice spacing from × We tuned the parameters of the b-quark action to re- Table II, e.g. ∆f/∆m σa . Finally, we add the indi- 0 m0a ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Chiral-continuum

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text: (upper

text: plots)

text: and

text: B

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text: plots)

text: form

text: factors

text: from

text: correlated

text: s

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text: using

text: NLO

text: hard-pion/kaon

text: HMχPT.

text: Fits

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text: colors

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text: ensembles:

text: circles

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text: the

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text: fm

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text: and

text: triangles

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text: diamonds

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text: fm

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text: The

text: colored

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text: the

text: interpolation/extrapolation

text: in

text: pion/kaon

text: energy:

text: the

text: ﬁt

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text: evaluated

text: at

text: the

text: unphysical

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text: ensembles,

text: such

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text: the

text: same

text: color.

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text: pion/kaon

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text: the

text: black

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text: gray

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text: band.

text: The

text: vertical

text: dashed

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text: the

text: left-hand

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text: physical

text: pion

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text: kaon

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text: therefore

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text: obtain

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text: of

text: the

text: b-quark

text: action.

text: We

text: estimate

text: the

text: size

text: of

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text: chiral-continuum

text: extrapolation

text: errors

text: quoted

text: in

text: Table

text: VI

text: dence,

text: by

text: computing

text: the

text: form

text: factors

text: f⊥

text: and

text: for

text: seven

text: and

text: shown

text: in

text: Fig.

text: sets

text: of

text: RHQ

text: parameters.

text: We

text: then

text: calculate

text: the

text: slopes

text: with

text: respect

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text: parameters

text: ∆f/∆m

text: a,

text: ∆f/∆c

text: P

text: and

text: ∆f/∆ξ

text: for

text: all

text: momenta

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text: the

text: analysis.

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text: uncertainty

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text: multiply

text: each

text: slope

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text: parameters

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text: the

text: b-quark

text: action

text: to

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text: Table

text: II,

text: e.g.

text: ∆f/∆m

text: σa

text: Finally,

text: we

text: add

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