text: tree CP-nets) for any m, using only membership queries. (cid:48) For a CP-net N, a conﬂict pair w.r.t. v is a pair (x, x ) of swaps such that (cid:48) (cid:48) i (cid:48) (i) V (x) = V (x ) = v , (ii) x.1 and x .1 agree on v , (iii) x.2 and x .2 agree i (cid:48) i on v , and (iv) N entails one of the swaps x, x , but not the other. If v has i i(cid:48) a conﬂict pair, then v has a parent variable v whose values in x and x are i j diﬀerent. Such a variable v can be found with log(n) membership queries j by binary search (each query halves the number of candidate variables with (cid:48) diﬀerent values in x and x ) [27]. We use this binary search to learn tree-structured CP-nets from member- ship queries, by exploiting the following fact: if a variable v in a tree CP-net i has a parent, then a conﬂict pair w.r.t. v exists and can be detected by ask- i ing membership queries to sort m “test sets” for v . Let (vi, . . . , vi ) be an i ∈ { 1 m} arbitrary but ﬁxed permutation of D . Then, for all j 1, . . . , m , a test { vi − | ≤ ≤ } set I for v is deﬁned by I = (v1, . . . , vi 1, vi, vi+1, . . . , vn) 1 r m . i,j i i,j j j r j j Since v has no more than one parent, determining preference orders over m i such test sets of size m is suﬃcient for revealing conﬂict pairs, rather than O having to test all possible contexts in \{ }. V vi { } Example 7. Consider the set of variables V = A, B, C where D = { (cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48)} { (cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48)} { (cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48)} A a, a , a , a , D = b, b , b , b , and D = c, c , c , c . The following B C is one possible collection of test sets for the variable A: { (cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48) } I = abc, a bc, a bc, a bc A,1 { (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48) (cid:48) (cid:48)(cid:48)(cid:48) (cid:48) (cid:48)} I = ab c , a b c , a b c , a b c A,2 { (cid:48)(cid:48) (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48)} I = ab c , a b c , a b c , a b c A,3 { (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48)} I = ab c , a b c , a b c , a b c A,4 (cid:0) (cid:1) Clearly, a complete target CP-net imposes a total order on every I , i,j which can be revealed by posing enough membership queries selected from the m swaps over I ; a total of O(m log(m)) comparisons suﬃce to de- 2 i,j termine the order over I . This yields a simple algorithm for learning tree i,j CP-nets with membership queries: Algorithm 1. For every variable v , determine P a(v ) and CPT(v ) as i i i follows: (cid:0) (cid:1) ∈ { } 1. For every value j 1, . . . , m , ask O(m log(m)) membership queries from the m swaps over I to obtain an order over I . 2 i,j i,j ∈ { } 2. If for all j , j 1, . . . , m the obtained order over I imposes the 1 2 i,j1 same order on D as the obtained order over I does, i.e., there is vi ∅ i,j2 no conﬂict pair for v , then P a(v ) = . In this case, CPT(v ) is fully i i i 30

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text: (cid:31) { } vi vi which states that the preference over v with D = vi, vi, . . . , vi j k ⊆ \i { } vi 1 2∈ m is conditioned upon the assignment of U V v where vi, vi D . In ∈ O i j k vi particular, when U has the value u , vi is preferred to vi as a value U j k of v ceteris paribus (all other things being equal). That is, for any two i (cid:48) ∈ O (cid:48) outcomes o, o where o[v ] = vi and o [v ] = vi the preference holds (cid:48) i j (cid:48) i k \ ∪ { } when i) o[U] = o [U] = u and ii) o[Z] = o [Z] for all Z = V U v . In (cid:48) i such case, we say o is preferred to o ceteris paribus. Clearly, there could be (cid:48) exponentially many pairs of outcomes (o,o ) that are aﬀected by one such statement. O CP-nets provide a compact representation of preferences over by pro- ∈ viding such statements for every variable. For every v V , the decision ⊆ \{ } i maker2 chooses a set P a(v ) V v of parent variables that inﬂuence the i ∈i O preference order of v . For any u , the decision maker speciﬁes (cid:31) i P(cid:31)a(vi) an ordering vui over Dvi. We refer to vui as the conditional preference statement of v in the context of u. A Conditional Preference Table for i {(cid:31) (cid:31) } vi, CPT(vi), is a set of conditional preference statementsO vui1, . . . , vuik . CPT(v ) is said to be complete, if, for every element in , CPT(v ) i Pa(vi) i contains a statement that imposes a total order on D ; otherwise CPT(v ) vi i is incomplete. ∈ Deﬁnition 1 (CP-net [1]). Given, V , P a(v), and CPT(v) for v V , a ∈ ∈ CP-net is a directed graph (V, E), where, for any v , v V , (v , v ) E iﬀ ∈ i j i j v P a(v ). i j Analogously, a CP-net N is said to be complete if every CPT it poses is complete; otherwise it is incomplete. A CP-net is acyclic if it contains no cycles. It is separable if the edge set of its graph is the empty set and it is a tree if it is acyclic with indegree at most one. Lastly, we assume CP-nets are deﬁned in their minimal form, i.e., there is no dummy parent in any CPT that actually does not aﬀect the preference relation. { } Example 1. Figure 1a shows a CP-net over V = A, B, C with D = { } { } { } A a, a¯ , D = b,¯b , D = c, c¯ . Each variable is annotated with its CPT. B C For variable A, the user prefers a to a¯ unconditionally. For C, the preference { } depends on the values of B, i.e., P a(C) = B . For instance, in the context of ¯b, c¯ is preferred over c. ∈ O Two outcomes o, oˆ are swap outcomes (‘swaps’ for short) if they diﬀer in the value of exactly one variable v ; then v is called the swapped i i variable [1]. 2This can be any entity in charge of constructing the preference network, i.e., a com- puter agent, a person, a group of people, etc. 4 ##LTLine##

label: paragraph

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text: is

text: incomplete.

text: A

text: CP-net

text: is

text: acyclic

text: if

text: it

text: contains

text: no

text: cycles.

text: It

text: is

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text: edge

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text: the

text: empty

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text: it

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text: a

text: tree

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text: acyclic

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text: Lastly,

text: we

text: assume

text: CP-nets

text: are

text: deﬁned

text: in

text: their

text: minimal

text: form,

text: i.e.,

text: there

text: is

text: no

text: dummy

text: parent

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text: any

text: CPT

text: that

text: actually

text: does

text: not

text: aﬀect

text: the

text: preference

text: relation.

text: Example

text: Figure

text: shows

text: a

text: CP-net

text: over

text: V

text: A,

text: B,

text: C

text: with

text: D

text: A

text: a,

text: a¯

text: D

text: b,¯b

text: D

text: c,

text: c¯

text: Each

text: variable

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text: annotated

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text: its

text: CPT.

text: B

text: C

text: For

text: variable

text: A,

text: the

text: user

text: prefers

text: a

text: to

text: a¯

text: unconditionally.

text: For

text: C,

text: the

text: preference

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text: on

text: the

text: values

text: of

text: B,

text: i.e.,

text: P

text: a(C)

text: B

text: For

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text: in

text: the

text: context

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text: ¯b,

text: c¯

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text: c.

text: O

text: Two

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text: swap

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text: if

text: they

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text: in

text: the

text: value

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text: exactly

text: one

text: variable

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text: v

text: is

text: called

text: the

text: swapped

text: i

text: i

text: variable

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text: be

text: any

text: entity

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text: charge

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text: constructing

text: the

text: preference

text: network,

text: i.e.,

text: a

text: com-

text: puter

text: agent,

text: a

text: person,

text: a

text: group

text: of

text: people,

text: etc.

text: ##LTLine##


text: includes both complete and incomplete CP-nets, whereas our study focused on complete CP-nets.) Koriche and Zanuttini [2, Theorem 6] gave a lower C∗ bound on VCD( ,k,e) over the swap instance space. In particular, setting (cid:98) (cid:99) (cid:98)ac −(cid:99) C∗ u = ke and r = log nku , they claimed that VCD( ac,k,e) is lower-bounded by 1 , if k = 0 , LB = u(r + 1) , if k = 1 , − − u(2k + k(r 1) 1) , if k > 1 . We claim that this bound is not generally correct for large values of k and (cid:0) (cid:1) e. C∗ For any given k, it is easy to see that there is a target concept in ,k,e − ac whose graph has e = k +(n k)k edges. We can always construct such max 2 a graph G as follows: Let V and V be a partition over the n vertices of | | − | 1 | 2 (cid:0) (cid:1) G where V = n k and V = k. Add an edge from each node in V to 1 2 − 2 each node in V . This results in (n k)k edges and G is clearly acyclic with 1 indegree exactly k for every element in V . For the remaining k edges, let 1 2 < be any total order on V ; now the edge (v, w) is added to G if and only if 2 v < w. Next, we evaluate the lower bound LB for k > 1 when the target concept has e edges in its graph: max − − u(2k + k(r 1) 1) − (cid:98) (cid:99) (cid:98)(cid:0)e(cid:1)(cid:99) (cid:98) n e (cid:99) − − = (2k + k( log( k ) 1) 1) (setting e = e ) max k k k + (n − k)k (cid:98) n − (k2)+(n−k)k (cid:99) − − = 2 (2k + k( log( k ) 1) 1) k k 1 − − (cid:98) k + 1 (cid:99) − − = (2n k 1)(2k + k( log( ) 1) 1) 2 2k 1 − − − − = (2n k 1)(2k 2k 1) 2 − − − When k = n 1, this term evaluates to n2n 2 n2 + n which becomes − 2 − larger than 2n 1 for n > 6. More generally, given k = n c for some − constant 0 < c < n 1, the term equals − − − − − − − (2c 1)(n + (c 1)) (n + (c 1))2n (c+1) n2 (c 1)n + 2 44 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: includes

text: both

text: complete

text: and

text: incomplete

text: CP-nets,

text: whereas

text: our

text: study

text: focused

text: on

text: complete

text: CP-nets.)

text: Koriche

text: and

text: Zanuttini

text: Theorem

text: gave

text: a

text: lower

text: C∗

text: bound

text: on

text: VCD(

text: ,k,e)

text: over

text: the

text: swap

text: instance

text: space.

text: In

text: particular,

text: setting

text: C∗

text: u

text: ke

text: and

text: r

text: log

text: nku

text: they

text: claimed

text: that

text: VCD(

text: ac,k,e)

text: is

text: lower-bounded

text: by

text: 1

text: if

text: k

text: LB

text: u(r

text: if

text: k

text: k(r

text: if

text: k

text: We

text: claim

text: that

text: this

text: bound

text: is

text: not

text: generally

text: correct

text: for

text: large

text: values

text: of

text: k

text: and

text: e.

text: C∗

text: For

text: any

text: given

text: k,

text: it

text: is

text: easy

text: to

text: see

text: that

text: there

text: is

text: a

text: target

text: concept

text: in

text: ,k,e

text: ac

text: whose

text: graph

text: has

text: e

text: k

text: +(n

text: k)k

text: edges.

text: We

text: can

text: always

text: construct

text: such

text: max

text: a

text: graph

text: G

text: as

text: follows:

text: Let

text: V

text: and

text: V

text: be

text: a

text: partition

text: over

text: the

text: n

text: vertices

text: of

text: G

text: where

text: V

text: n

text: k

text: and

text: V

text: k.

text: Add

text: an

text: edge

text: from

text: each

text: node

text: in

text: V

text: to

text: each

text: node

text: in

text: V

text: This

text: results

text: in

text: (n

text: k)k

text: edges

text: and

text: G

text: is

text: clearly

text: acyclic

text: with

text: indegree

text: exactly

text: k

text: for

text: every

text: element

text: in

text: V

text: For

text: the

text: remaining

text: k

text: edges,

text: let

text: be

text: any

text: total

text: order

text: on

text: V

text: now

text: the

text: edge

text: (v,

text: w)

text: is

text: added

text: to

text: G

text: if

text: and

text: only

text: if

text: v

text: w.

text: Next,

text: we

text: evaluate

text: the

text: lower

text: bound

text: LB

text: for

text: k

text: when

text: the

text: target

text: concept

text: has

text: e

text: edges

text: in

text: its

text: graph:

text: max

text: k(r

text: n

text: e

text: k(

text: log(

text: k

text: (setting

text: e

text: e

text: max

text: k

text: k

text: k

text: (n

text: k)k

text: n

text: k(

text: log(

text: k

text: k

text: k

text: k

text: k

text: k(

text: log(

text: k

text: When

text: k

text: n

text: this

text: term

text: evaluates

text: to

text: n

text: which

text: becomes

text: larger

text: than

text: for

text: n

text: More

text: generally,

text: given

text: k

text: n

text: c

text: for

text: some

text: constant

text: c

text: n

text: the

text: term

text: equals

text: (c

text: (n

text: (c

text: (c

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: a¯¯bc¯ (cid:31) a a¯ a¯bc¯ a¯bc¯ a¯¯bc A (cid:31) ac : b (cid:31) ¯b (cid:31) abc¯ a¯bc a¯bc ac¯: b ¯b b : c c¯ (cid:31) (cid:31) a¯c : ¯b b ¯b : c¯ c a¯c¯: b (cid:31) ¯b abc B C (b) The induced preference (a) The network. graph. Figure 3: An example of an inconsistent cyclic CP-net. learning algorithm needs to identify a CP-net. Examples of such complexity notions will be introduced below. A speciﬁc complexity notion corresponds to a speciﬁc formal model of machine learning. Each such learning model assumes that there is an infor- mation source that supplies the learning algorithm with information about ∗ a hidden target concept c . The latter is a member of a concept class, which is simply the class of potential target concepts, and, in the context of this paper, also the class of hypotheses that the learning algorithm can formulate ∗ in the attempt to identify the target concept c . X Formally, one ﬁxes a ﬁnite set , called instance space, which contains all possible instances (i.e., elements of) an underlying domain. A concept c X { } is then deﬁned as a mapping from to 0, 1 . Equivalently, c can be seen as { ∈ X | } the set c = x c(x) = 1 , i.e., a subset of the instance space. A concept C class is a set of concepts. Within the scope of our study, the information source (sometimes called oracle), supplies the learning algorithm in some ∗ ∈ way or another with a set of labelled examples for the target concept c C. ∗ ∈ X × { } ∈ X A labelled example for c is a pair (x, b) 0, 1 where x and b = c(x). Under the set interpretation of concepts, this means that b = 1 if ∗ and only if the instance x belongs to the concept c . A concept c is consistent ⊆ × { } with a set S X 0, 1 of labelled examples, if and only if c(x) = b for ∈ all (x, b) S, i.e., if every element of S is an example for c. In practice, a concept is usually encoded by a representation σ(c) deﬁned R based on a representation class [18]. Thus, one usually has some ﬁxed R representation class in mind, with a one-to-one correspondence between the concept class C and its representation class R. We will assume in what follows that the representation class is chosen in a way that minimizes the worst case size of the representation of any concept in C. Generally, there may be various interpretations of the term “size;” since we will focus on 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: a¯¯bc¯

text: a

text: a¯

text: a¯bc¯

text: a¯bc¯

text: a¯¯bc

text: A

text: ac

text: b

text: ¯b

text: abc¯

text: a¯bc

text: a¯bc

text: ac¯:

text: b

text: ¯b

text: b

text: c

text: c¯

text: a¯c

text: ¯b

text: b

text: ¯b

text: c¯

text: c

text: a¯c¯:

text: b

text: ¯b

text: abc

text: B

text: C

text: (b)

text: The

label: caption

text: induced

text: preference

text: (a)

text: The

text: network.

text: graph.

text: Figure

text: An

text: example

text: of

text: an

text: inconsistent

text: cyclic

text: CP-net.

text: learning

text: algorithm

text: needs

text: to

text: identify

text: a

text: CP-net.

text: Examples

text: of

text: such

text: complexity

text: notions

text: will

text: be

text: introduced

text: below.

text: A

text: speciﬁc

text: complexity

text: notion

text: corresponds

text: to

text: a

text: speciﬁc

text: formal

text: model

text: of

text: machine

text: learning.

text: Each

text: such

text: learning

text: model

text: assumes

text: that

text: there

text: is

text: an

text: infor-

text: mation

text: source

text: that

text: supplies

text: the

text: learning

text: algorithm

text: with

text: information

text: about

text: a

text: hidden

text: target

text: concept

text: c

text: The

text: latter

text: is

text: a

text: member

text: of

text: a

text: concept

text: class,

text: which

text: is

text: simply

text: the

text: class

text: of

text: potential

text: target

text: concepts,

text: and,

text: in

text: the

text: context

text: of

text: this

text: paper,

text: also

text: the

text: class

text: of

text: hypotheses

text: that

text: the

text: learning

text: algorithm

text: can

text: formulate

text: in

text: the

text: attempt

text: to

text: identify

text: the

text: target

text: concept

text: c

text: X

text: Formally,

text: one

text: ﬁxes

text: a

text: ﬁnite

text: set

text: called

text: instance

text: space,

text: which

text: contains

text: all

text: possible

text: instances

text: (i.e.,

text: elements

text: of)

text: an

text: underlying

text: domain.

text: A

text: concept

text: c

text: X

text: is

text: then

text: deﬁned

text: as

text: a

text: mapping

text: from

text: to

text: Equivalently,

text: c

text: can

text: be

text: seen

text: as

text: X

text: the

text: set

text: c

text: x

text: c(x)

text: i.e.,

text: a

text: subset

text: of

text: the

text: instance

text: space.

text: A

text: concept

text: C

text: class

text: is

text: a

text: set

text: of

text: concepts.

text: Within

text: the

text: scope

text: of

text: our

text: study,

text: the

text: information

text: source

text: (sometimes

text: called

text: oracle),

text: supplies

text: the

text: learning

text: algorithm

text: in

text: some

text: way

text: or

text: another

text: with

text: a

text: set

text: of

text: labelled

text: examples

text: for

text: the

text: target

text: concept

text: c

text: C.

text: X

text: X

text: A

text: labelled

text: example

text: for

text: c

text: is

text: a

text: pair

text: (x,

text: b)

text: where

text: x

text: and

text: b

text: c(x).

text: Under

text: the

text: set

text: interpretation

text: of

text: concepts,

text: this

text: means

text: that

text: b

text: if

text: and

text: only

text: if

text: the

text: instance

text: x

text: belongs

text: to

text: the

text: concept

text: c

text: A

text: concept

text: c

text: is

text: consistent

text: with

text: a

text: set

text: S

text: X

text: of

text: labelled

text: examples,

text: if

text: and

text: only

text: if

text: c(x)

text: b

text: for

text: all

text: (x,

text: b)

text: S,

text: i.e.,

text: if

text: every

text: element

text: of

text: S

text: is

text: an

text: example

text: for

text: c.

text: In

text: practice,

text: a

text: concept

text: is

text: usually

text: encoded

text: by

text: a

text: representation

text: σ(c)

text: deﬁned

text: R

text: based

text: on

text: a

text: representation

text: class

text: Thus,

text: one

text: usually

text: has

text: some

text: ﬁxed

text: R

text: representation

text: class

text: in

text: mind,

text: with

text: a

text: one-to-one

text: correspondence

text: between

text: the

text: concept

text: class

text: C

text: and

text: its

text: representation

text: class

text: R.

text: We

text: will

text: assume

text: in

text: what

text: follows

text: that

text: the

text: representation

text: class

text: is

text: chosen

text: in

text: a

text: way

text: that

text: minimizes

text: the

text: worst

text: case

text: size

text: of

text: the

text: representation

text: of

text: any

text: concept

text: in

text: C.

text: Generally,

text: there

text: may

text: be

text: various

text: interpretations

text: of

text: the

text: term

text: “size;”

text: since

text: we

text: will

text: focus

text: on

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 DISCRETE HODGE STAR OPERATOR ON 3-MANIFOLDS [10] U. Pachner. P.L. homeomorphic manifolds are equivalent by elementary shellings. European J. Combin., 12(2):129–145, 1991. [11] K. Prasanna and A. Venkatesh. Automorphic cohomology, motivic cohomology, and the adjoint L-function. ArXiv e-prints, September 2016. [12] D Sullivan. Inﬁnitesimal computations in topology. Inst. Hautes E´tudes Sci. Publ. Math., (47):269– 331 (1978), 1977. [13] V. G. Turaev. Quantum invariants of knots and 3-manifolds, volume 18 of De Gruyter Studies in Mathematics. De Gruyter, Berlin, 2016. Third edition [of MR1292673]. [14] V. G. Turaev and O. Ya. Viro. State sum invariants of 3-manifolds and quantum 6j-symbols. Topology, 31(4):865–902, 1992. [15] A. Venkatesh. Cohomology of arithmetic groups and periods of automorphic forms. Jpn. J. Math., 12(1):1–32, 2017. [16] H. Whitney. Geometric integration theory. Princeton University Press, Princeton, N. J., 1957. [17] S. O. Wilson. Cochain algebra on manifolds and convergence under reﬁnement. Topology Appl., 154(9):1898–1920, 2007.

text: DISCRETE

text: HODGE

text: STAR

text: OPERATOR

text: ON

label: reference

text: U.

text: Pachner.

text: P.L.

text: homeomorphic

text: manifolds

text: are

text: equivalent

text: by

text: elementary

text: shellings.

text: European

text: J.

text: Combin.,

text: K.

text: Prasanna

text: and

text: A.

text: Venkatesh.

text: Automorphic

text: cohomology,

text: motivic

text: cohomology,

text: and

text: the

text: adjoint

text: L-function.

text: ArXiv

text: e-prints,

text: September

text: D

text: Sullivan.

text: Inﬁnitesimal

text: computations

text: in

text: topology.

text: Inst.

text: Hautes

text: E´tudes

text: Sci.

text: Publ.

text: Math.,

text: V.

text: G.

text: Turaev.

text: Quantum

text: invariants

text: of

text: knots

text: and

text: volume

text: of

text: De

text: Gruyter

text: Studies

text: in

text: Mathematics.

text: De

text: Gruyter,

text: Berlin,

text: Third

text: edition

text: [of

text: V.

text: G.

text: Turaev

text: and

text: O.

text: Ya.

text: Viro.

text: State

text: sum

text: invariants

text: of

text: and

text: quantum

text: Topology,

text: A.

text: Venkatesh.

text: Cohomology

text: of

text: arithmetic

text: groups

text: and

text: periods

text: of

text: automorphic

text: forms.

text: Jpn.

text: J.

text: Math.,

text: H.

text: Whitney.

text: Geometric

text: integration

text: theory.

text: Princeton

text: University

text: Press,

text: Princeton,

text: N.

text: J.,

text: S.

text: O.

text: Wilson.

text: Cochain

text: algebra

text: on

text: manifolds

text: and

text: convergence

text: under

text: reﬁnement.

text: Topology

text: Appl.,


text: DISCRETE HODGE STAR OPERATOR ON 3-MANIFOLDS 3 to the cuspidal subspaces be the discrete Hodge star operator. A general fact about (cid:70) (cid:70) combinatorial Hodge star operators shows that is an isomorphism. A priori, depends on K, although the dependence is suppressed in the notation. The following is our main result. Theorem (Theorem 2.2.1). Let M be a compact oriented 3-manifold. For i = 1, 2, the discrete Hodge star operator is independent of the choice of a triangulation K of M. (cid:70) Furthermore, commutes with pull-backs and transfers along ﬁnite coverings. We will prove the above theorem, as mentioned earlier, by verifying the invariance (cid:70) of with respect to the Pachner moves and the elementary shellings. The compatibility with ﬁnite coverings is a consequence of the invariance. (cid:70) ∈ Q Here is an application. Using , we deﬁne the pairing for x, y Hi (M, ) as cusp ∪ (cid:70) (x, y) := (x y) ([M]) w where [M] is the orientation class. Unraveling various deﬁnitions, we will prove:- − − Q Theorem (Theorem 5.1.1). The pairing ( , ) on Hi (M, ) is symmetric and pos- cusp itive deﬁnite for i = 1, 2. We move on to the application to the conjecture of Prasanna and Venkatesh [11]. Since the conjecture involves a few aspects of the Langlands Program [1], it will take some time for us to connect our main result to the conjecture. A more accessible account on the conjecture is provided by Venkatesh’s own exposition [15]. To accelerate our discussion, we will limit ourselves in the introduction to the special √ case of Bianchi groups. In addition, we will keep the classical point of view rather than Q − the adelic one. Let F = ( D) be an imaginary quadratic ﬁeld with a square-free ⊂ C positive integer D. We always ﬁx an embedding F . In particular, through the ⊂ C C inclusion PSL (F ) PSL ( ) and the natural action of PSL ( ) on the upper half- 2 2 2 H H space , any subgroup of PSL (F ) acts on . Let O be the ring of integers of F , with 2 F the associated Bianchi group PSL (O ). 2 F ⊂ The cohomology of a torsion-free ﬁnite index subgroup Γ PSL (O ) is identiﬁed 2 F with the cohomology of the orbit space \H Y := Γ , which is an oriented 3-manifold. We will always take its Borel-Serre compactiﬁcation ⊂ Y M which has the eﬀect of adding a torus to each cusp of Γ. The number of cusps of PSL (O ) equals the class number of O , which is, in particular, a positive integer. 2 F F Therefore the boundary of M is always nonempty for any Γ. The cuspidal cohomology groups • Q • Q • Q → • Q H (Γ, ) = H (M, ) = Ker(H (M, ) H (∂M, )) cusp cusp are to which our main theorem applies. A characteristic property of arithmetic groups is their rich theory of Hecke operators. ⊂ ∈ In the case of Γ PSL (O ), each element g PSL (F ) gives rise to a Hecke operator 2 F 2 Q → Q T i : Hi (M, ) Hi (M, ), g cusp cusp ##LTLine##

text: DISCRETE

text: HODGE

text: STAR

text: OPERATOR

text: ON

text: to

text: the

text: cuspidal

text: subspaces

text: be

text: the

text: discrete

text: Hodge

text: star

text: operator.

text: A

text: general

text: fact

text: about

text: combinatorial

text: Hodge

text: star

text: operators

text: shows

text: that

text: is

text: an

text: isomorphism.

text: A

text: priori,

text: depends

text: on

text: K,

text: although

text: the

text: dependence

text: is

text: suppressed

text: in

text: the

text: notation.

text: The

text: following

text: is

text: our

text: main

text: result.

text: Theorem

text: (Theorem

text: Let

text: M

text: be

text: a

text: compact

text: oriented

text: For

text: i

text: the

text: discrete

text: Hodge

text: star

text: operator

text: is

text: independent

text: of

text: the

text: choice

text: of

text: a

text: triangulation

text: K

text: of

text: M.

text: Furthermore,

text: commutes

text: with

text: pull-backs

text: and

text: transfers

text: along

text: ﬁnite

text: coverings.

text: We

text: will

text: prove

text: the

text: above

text: theorem,

text: as

text: mentioned

text: earlier,

text: by

text: verifying

text: the

text: invariance

text: of

text: with

text: respect

text: to

text: the

text: Pachner

text: moves

text: and

text: the

text: elementary

text: shellings.

text: The

text: compatibility

text: with

text: ﬁnite

text: coverings

text: is

text: a

text: consequence

text: of

text: the

text: invariance.

text: Q

text: Here

text: is

text: an

text: application.

text: Using

text: we

text: deﬁne

text: the

text: pairing

text: for

text: x,

text: y

text: Hi

text: (M,

text: as

text: cusp

text: (x,

text: y)

text: (x

text: y)

text: ([M])

text: w

text: where

text: [M]

text: is

text: the

text: orientation

text: class.

text: Unraveling

text: various

text: deﬁnitions,

text: we

text: will

text: prove:-

text: Q

text: Theorem

text: (Theorem

text: The

text: pairing

text: on

text: Hi

text: (M,

text: is

text: symmetric

text: and

text: pos-

text: cusp

text: itive

text: deﬁnite

text: for

text: i

text: We

text: move

text: on

text: to

text: the

text: application

text: to

text: the

text: conjecture

text: of

text: Prasanna

text: and

text: Venkatesh

text: Since

text: the

text: conjecture

text: involves

text: a

text: few

text: aspects

text: of

text: the

text: Langlands

text: Program

text: it

text: will

text: take

text: some

text: time

text: for

text: us

text: to

text: connect

text: our

text: main

text: result

text: to

text: the

text: conjecture.

text: A

text: more

text: accessible

text: account

text: on

text: the

text: conjecture

text: is

text: provided

text: by

text: Venkatesh’s

text: own

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text: To

text: accelerate

text: our

text: discussion,

text: we

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text: In

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text: PSL

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text: The

text: cuspidal

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text: groups

text: Q

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text: H

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text: H

text: (M,

text: Ker(H

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text: H

text: (∂M,

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text: to

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text: main

text: theorem

text: applies.

text: A

text: characteristic

text: property

text: of

text: arithmetic

text: groups

text: is

text: their

text: rich

text: theory

text: of

text: Hecke

text: operators.

text: In

text: the

text: case

text: of

text: Γ

text: PSL

text: (O

text: each

text: element

text: g

text: PSL

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text: gives

text: rise

text: to

text: a

text: Hecke

text: operator

text: F

text: Q

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text: Hi

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text: g

text: cusp

text: cusp

text: ##LTLine##


text: 6 DISCRETE HODGE STAR OPERATOR ON 3-MANIFOLDS ∈ (cid:70) which is determined by any triangulation K τ(M). Furthermore, commutes with pull-backs and transfers along ﬁnite coverings. (cid:70) 3. Construction of ∈ Fix a triangulation K τ(M) of a compact oriented n-manifold M. Our aim in this section is to construct an operator (cid:70) Q → − Q : Hi(K, ) Hn i(K, ) g0 · · · for i = 0, 1, 2, , n, and show that it induces an isomorphism (cid:70) Q −∼→ − Q : Hi (K, ) Hn i (K, ) cusp cusp between the cuspidal subspaces. We closely follow Wilson [17]. Let K be a triangulation of a manifold. The Whitney product is a (super-)commutative, but non-associative, product ∪ • Q × • Q → • Q : C (K, ) C (K, ) C (K, ) w of degree zero such that it induces the usual commutative and associative Alexander- Whitney cup product at the level of cohomology. Let us assign an arbitrary total ordering on vertices. It determines an orientation of B { · · · } every simplex of K. It determines a basis = b , b , for C•(K), called a standard 1 2 basis whose members are the oriented simplexes. If we diﬀerently order the vertices, B(cid:48) { (cid:48) (cid:48) · · · } (cid:48) then we obtain a diﬀerent basis = b , b , . They diﬀer by sign, in that b = u b ± 1 2 i i i with u = 1 for every i. i B∨ { ∨ ∨ · · · } • Since K is ﬁnite, we may take a basis = b , b , of C (K) to be the dual to B 1 2 | | the standard basis . The degree of b will be denoted b . We will sometimes denote i i the set of all vertices of a simplex b by the same b. For example, by the condition ∪ b b = b i j k we mean that every vertex of b is contained in either b or b . Similarly, the condition k i j ∩ #(b b ) = 1 i j means that b and b have exactly one vertex in common. i j (cid:40) ∪ Deﬁne the Whitney product to be w | | | | ∨ ∩ ∪ ∨ ∪ ∨ u | |bi |! bj| ! b if #(b b ) = 1 and b b = b , (3.1.1) bi w bj = ij ( bi + bj +1)! k i j i j k 0 otherwise, ∈ {± } where the sign u 1 is uniquely determined by ij · · · ∪ · · · · · · · · · [p p p ] [p p p ] = [p p p p p ] 0 1 r w r r+1 n 0 1 r r+1 n ∪ · · · together with the bilinearity of and [q q q ] being the characteristic function on w{ · · · 0 1} s the simplex with ordered vertices q , , q . Formulas for u , which are unimportant 0 s ij for us, can be found in Theorem 5.2 of [17] or (2.37) of [2]. ##LTLine##

text: DISCRETE

text: HODGE

text: STAR

text: OPERATOR

text: ON

text: which

text: is

text: determined

text: by

text: any

text: triangulation

text: K

text: τ(M).

text: Furthermore,

text: commutes

text: with

text: pull-backs

text: and

text: transfers

text: along

text: ﬁnite

text: coverings.

label: section

text: Construction

text: of

text: Fix

text: a

text: triangulation

text: K

text: τ(M)

text: of

text: a

text: compact

text: oriented

text: n-manifold

text: M.

text: Our

text: aim

text: in

text: this

text: section

text: is

text: to

text: construct

text: an

text: operator

text: Q

text: Q

text: Hi(K,

text: Hn

text: i(K,

text: for

text: i

text: n,

text: and

text: show

text: that

text: it

text: induces

text: an

text: isomorphism

text: Q

text: Q

text: Hi

text: (K,

text: Hn

text: i

text: (K,

text: cusp

text: cusp

text: between

text: the

text: cuspidal

text: subspaces.

text: We

text: closely

text: follow

text: Wilson

text: Let

text: K

text: be

text: a

text: triangulation

text: of

text: a

text: manifold.

text: The

text: Whitney

text: product

text: is

text: a

text: (super-)commutative,

text: but

text: non-associative,

text: product

text: Q

text: Q

text: Q

text: C

text: (K,

text: C

text: (K,

text: C

text: (K,

text: w

text: of

text: degree

text: zero

text: such

text: that

text: it

text: induces

text: the

text: usual

text: commutative

text: and

text: associative

text: Alexander-

text: Whitney

text: cup

text: product

text: at

text: the

text: level

text: of

text: cohomology.

text: Let

text: us

text: assign

text: an

text: arbitrary

text: total

text: ordering

text: on

text: vertices.

text: It

text: determines

text: an

text: orientation

text: of

text: B

text: every

text: simplex

text: of

text: K.

text: It

text: determines

text: a

text: basis

text: b

text: b

text: for

text: C•(K),

text: called

text: a

text: standard

text: basis

text: whose

text: members

text: are

text: the

text: oriented

text: simplexes.

text: If

text: we

text: diﬀerently

text: order

text: the

text: vertices,

text: then

text: we

text: obtain

text: a

text: diﬀerent

text: basis

text: b

text: b

text: They

text: diﬀer

text: by

text: sign,

text: in

text: that

text: b

text: u

text: b

text: i

text: i

text: i

text: with

text: u

text: for

text: every

text: i.

text: i

text: B∨

text: Since

text: K

text: is

text: ﬁnite,

text: we

text: may

text: take

text: a

text: basis

text: b

text: b

text: of

text: C

text: (K)

text: to

text: be

text: the

text: dual

text: to

text: B

text: the

text: standard

text: basis

text: The

text: degree

text: of

text: b

text: will

text: be

text: denoted

text: b

text: We

text: will

text: sometimes

text: denote

text: i

text: i

text: the

text: set

text: of

text: all

text: vertices

text: of

text: a

text: simplex

text: b

text: by

text: the

text: same

text: b.

text: For

text: example,

text: by

text: the

text: condition

text: b

text: b

text: b

text: i

text: j

text: k

text: we

text: mean

text: that

text: every

text: vertex

text: of

text: b

text: is

text: contained

text: in

text: either

text: b

text: or

text: b

text: Similarly,

text: the

text: condition

text: k

text: i

text: j

text: #(b

text: b

text: i

text: j

text: means

text: that

text: b

text: and

text: b

text: have

text: exactly

text: one

text: vertex

text: in

text: common.

text: i

text: j

text: Deﬁne

text: the

text: Whitney

text: product

text: to

text: be

text: w

text: u

text: |bi

text: bj|

text: b

text: if

text: #(b

text: b

text: and

text: b

text: b

text: b

text: bi

text: w

text: bj

text: ij

text: bi

text: bj

text: k

text: i

text: j

text: i

text: j

text: k

text: otherwise,

text: where

text: the

text: sign

text: u

text: is

text: uniquely

text: determined

text: by

text: ij

text: [p

text: p

text: p

text: [p

text: p

text: p

text: [p

text: p

text: p

text: p

text: p

text: r

text: w

text: r

text: n

text: r

text: n

text: together

text: with

text: the

text: bilinearity

text: of

text: and

text: [q

text: q

text: q

text: being

text: the

text: characteristic

text: function

text: on

text: w{

text: s

text: the

text: simplex

text: with

text: ordered

text: vertices

text: q

text: q

text: Formulas

text: for

text: u

text: which

text: are

text: unimportant

text: s

text: ij

text: for

text: us,

text: can

text: be

text: found

text: in

text: Theorem

text: of

text: or

text: of

text: ##LTLine##


text: 14 D´ıaz et al. 00..2205 BISHARPS 0.15 0.10 0.05 0.20 00..1126 FWHMHARPS 0.08 0.04 00..2205 SHARPS 0.15 0.10 0.05 0.10 00..0068 HαHARPS 0.04 0.02 0.10 00..0068 HeI HARPS 0.04 0.02 00..45 SPFS 0.3 0.2 0.1 00..45 SHIRES 0.3 0.2 0.1 0.20 0.16 SCOMBINED 0.12 0.08 0.04 1 10 100 1000 10000 Period (days) Fig. 7.— Generalized Lomb-Scargle periodogram of the activity indicators available from the diﬀerent spectrographs. From top to bottom: BISHARPS, CCF FWHMHARPS, SHARPS, HαHARPS, He iHARPS, SPFS, SHIRES and SCOMBINED. The arrows mark the position of the signal found at 42.37 days in the radial velocity series and the reported stellar rotation period of 37.1 days from Saar & Osten (1997). The dotted lines show the 0.1% signiﬁcance level, determined by 1,000 bootstrap resamplings. There are no statistically signiﬁcant power in the activity indicators matching the radial velocity period. rewoP rewoP rewoP rewoP rewoP rewoP rewoP rewoP ##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## ##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##

text: D´ıaz

text: et

text: al.

font: default

text: BISHARPS

text: FWHMHARPS

text: SHARPS

text: HαHARPS

text: HeI

text: HARPS

text: SPFS

text: SHIRES

text: SCOMBINED

text: Period

text: (days)

text: Fig.

text: Generalized

text: Lomb-Scargle

text: periodogram

text: of

text: the

text: activity

text: indicators

text: available

text: from

text: the

text: diﬀerent

text: spectrographs.

text: From

text: top

text: to

text: bottom:

text: BISHARPS,

text: CCF

text: FWHMHARPS,

text: SHARPS,

text: HαHARPS,

text: He

text: iHARPS,

text: SPFS,

text: SHIRES

text: and

text: SCOMBINED.

text: The

text: arrows

text: mark

text: the

text: position

text: of

text: the

text: signal

text: found

text: at

text: days

text: in

text: the

text: radial

text: velocity

text: series

text: and

text: the

text: reported

text: stellar

text: rotation

text: period

text: of

text: days

text: from

text: Saar

text: Osten

text: The

text: dotted

text: lines

text: show

text: the

text: signiﬁcance

text: level,

text: determined

text: by

text: bootstrap

text: resamplings.

text: There

text: are

text: no

text: statistically

text: signiﬁcant

text: power

text: in

text: the

text: activity

text: indicators

text: matching

text: the

text: radial

text: velocity

text: period.

text: rewoP

text: rewoP

text: rewoP

text: rewoP

text: rewoP

text: rewoP

text: rewoP

text: rewoP

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text: 8. ASAS PHOTOMETRY the GLS for the ASAS photometry. We sample the pe- riod space starting at a minimum period of 1 day and To complement the analysis we gathered photomet- up to 10,000 days, performing 80,000 period samples. ric data available from the All-Sky Automated Survey Considering just the data before JD=2452300 tends to (ASAS) Catalog (Pojmanski 1997) to investigate if any favor peaks with higher power towards high frequencies periodic signal could be seen in light curves, particularly (periods ∼1 day) but without any signiﬁcant period (or the rotational period of the star. As mentioned above, an integer multiple) near the period associated with the the literature values were reported by Noyes et al. (1984) 42 day signal of the reported planetary candidate. We and Saar & Osten (1997), where they found P ∼37.10 rot also ran the periodogram analysis on the full photometric days for this old star. We show the ASAS photometric dataset with no signiﬁcant periods found. Following the measurements in Figure 9. From the ﬁve diﬀerent aper- tures available, we selected aperture 1 as its MAD9 value relations in Hatzes (2002), we found that a ﬁlling factor of f=0.15 would be required to induce the RV amplitude of 0.219 mag was the smallest. The mean uncertainty − of 1.6 m s 1 of the signal found in the combined data. in the V-band photometry is σ = 0.036 mag. Grey ASAS If we consider the spots on the surface of the star to be circles correspond to the entire photometry set of 568 opaque, for the sake of simplicity, the ratio between the useful points acquired from 2000 to 2009. However, we stellar ﬂux and the ﬂux considering spots covering 0.15% excluded the data with poor quality (those not marked of the surface of the star would be 0.9985. This means, “A” or “B” in the catalog) and also those measurements the loss of light due to spots on the stellar surface can be that deviated more than 3-σ with respect to the mean translated into a ∆ m= 1.64 mmag. Given the precision value of the time series. The highest quality data (316 Draft version May 30, 2020 of the ASAS photometry for this star, we conclude it is points) are shown as black circles in the top panel of Preprint typeset using LATEX style emulateapj v. 12/16/11 insuﬃcient to be informative. Figure 9. The bottom panel in Figure 9 again shows THE TEST CASE OF HD26965: DIFFICULTIES DISENTANGLING WEAK DOPPLER SIGNALS FROM STELLAR ACTIVITY7 Mat´ıas R. D´ıaz1,2,(cid:63), James S. Jenkins1,3, Mikko Tuomi4, R. Paul Butler5, Maritza G. Soto1, Johanna K. Teske2,5, Fabo Feng4, Stephen A. Shectman2, Pamela Arriagada5, Jeffrey D. Crane2, Ian B. Thompson2 & Steven S. Vogt6 Subject headings: stars: individual HD26965 — techniques: spectroscopic, radial velocities — methods: statistical 9. MOUNT WILSON HK MEASUREMENTS data are publicly available from the NSO archive10 and include more than 2,000 stars observed from 1966 to Given that we ﬁnd some moderate correlations be- 1995. There are 1,155 HK observations for HD26965 tween the spectral activity indicators and the radial ve- from JD=2439787.8 to 2449771.7. The processed data locities, we supplemented our activity analysis by study- ing the original Ca ii H&K data from the Mount Wilson do not include associated uncertainties to the calibrated S-values. According to Duncan et al. (1991), the uncer- Observatory HK Project (Wilson 1978). The Project tainties in the Mount Wilson S-values can be calculated using the weights, W, included in the data that are de- 9 Median Absolute Deviation = median(|xi − median(x)|) rived from the photon counts of the measurements. The uncertainty in the S-index measurements is simply de- √ − 1 Departamento de Astronom´ıa, Universidad de Chile, Camino ﬁned as σ = S ( W) 1. We applied this formula to the S El Observatorio 1515, Las Condes, Santiago, Chile. reported weights to provide proper uncertainties for the 2 The Observatories, Carnegie Institution for Science, 813 Santa measurements of this star. All HK values for HD26965 Barbara Street, Pasadena, CA 91101, USA. 3 Centro de Astrof´ısica y Tecnolog´ıas Aﬁnes (CATA), Casilla 36- can be found in Table 13. D, Santiago, Chile. We proceeded to run the same periodogram analysis as 4 Center for Astrophysics Research, School of Physics, As- for the radial velocities and the photometric time-series tronomy and Mathematics, University of Hertfordshire, College from ASAS. A clear long-term variability of ∼4,100 days Lane, Hatﬁeld AL109AB, UK. 5 Department of Terrestrial Magnetism, Carnegie Institution of can be identiﬁed (see top and second top panels in Fig- Washington, 5124 Broad Branch Road, Washington, DC 20015- ure 10), providing evidence for a long-period magnetic 1305, USA. activity cycle, similar to the long-period solar cycle. Af- 6 UCO/Lick Observatory, Department of Astronomy and Astro- ter removing this signal by modeling it with a sinusoidal physics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA. function, a second period of ∼715 days is found in the 7 Some of the data presented herein were obtained at the W.M. periodogram (second bottom panel in the ﬁgure), likely Keck Observatory, which is operated as a scientiﬁc partnership representing another, shorter period magnetic cycle. Fi- among the California Institute of Technology, the University of California and the National Aeronautics and Space Administra- nally, the bottom panel in the ﬁgure shows the peri- tion. The Observatory was made possible by the generous ﬁnan- odogram of the Mount Wilson S-values after removing cial support of the W.M. Keck Foundation. the 4,100 and 715 day period signals from the data. In (cid:63)matias.diaz.m@ug.uchile.cl, Carnegie-Chile Graduate Fellow 2015-2016. 10 http://www.nso.edu/node/1335 ##LTLine##

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text: Keck

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text: from

text: the

text: data.

text: In

text: Carnegie-Chile

text: Graduate

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text: ##LTLine##


text: 22 D´ıaz et al. TABLE 11 HARPS NEW Radial Velocities of HD26965 BJD RV− σ R−V BI−S FWH−M S Hα He i (m s 1) (m s 1) (m s 1) (m s 1) (dex) (dex) (dex) 2457274.86039 -0.1517 0.1828 18.5315 5951.1924 0.1777 0.1170 0.5055 2457274.86304 -0.0462 0.1852 18.4779 5951.7017 0.1768 0.1171 0.5059 2457274.86566 0.4185 0.2041 18.6366 5951.8066 0.1777 0.1174 0.5059 2457274.86845 0.3169 0.2026 18.1923 5951.7324 0.1778 0.1173 0.5051 2457277.84809 0.4107 0.2189 18.9974 5950.3081 0.1763 0.1179 0.5056 2457277.85019 0.1997 0.2209 18.6706 5950.1743 0.1777 0.1176 0.5049 ... ... ... ... ... ... ... ... Note. — This table is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content. TABLE 12 ASAS Photometry of HD26965 HJD -2450000 MAG 4 MAG 0 MAG 1 MAG 2 MAG 3 MER 4 MER 0 MER 1 MER 2 MER 3 GRADE FRAME 1953.56936 29.999 29.999 29.999 29.999 29.999 0.028 0.063 0.050 0.034 0.029 C 9642 2172.77457 5.375 5.451 5.324 5.361 5.374 0.033 0.054 0.035 0.027 0.029 A 32848 2206.76432 29.999 29.999 29.999 29.999 29.999 0.030 0.084 0.057 0.037 0.031 C 37541 2227.69007 5.274 4.831 4.866 5.055 5.187 0.030 0.046 0.041 0.031 0.032 A 39837 2230.68849 5.235 4.866 4.942 5.091 5.179 0.038 0.107 0.090 0.063 0.050 D 40329 2234.67705 5.263 4.766 4.821 5.023 5.175 0.032 0.046 0.040 0.032 0.035 A 40805 2236.67316 29.999 29.999 29.999 29.999 29.999 0.034 0.046 0.039 0.031 0.036 C 41118 2501.90399 29.999 29.999 29.999 29.999 29.999 0.033 0.044 0.044 0.035 0.038 C 16126 2529.79004 4.598 3.968 4.145 4.378 4.523 0.044 0.060 0.052 0.043 0.046 B 19144 2549.77699 4.867 5.224 4.976 4.923 4.875 0.028 0.060 0.056 0.040 0.033 B 20837 2553.76674 4.727 4.377 4.445 4.585 4.681 0.035 0.049 0.052 0.043 0.041 B 21379 2558.77118 29.999 29.999 29.999 29.999 29.999 0.028 0.052 0.039 0.028 0.027 C 22103 2655.59412 4.934 6.289 5.861 5.440 5.139 0.029 0.036 0.037 0.027 0.029 A 35213 2954.75032 4.916 5.227 5.076 4.981 4.927 0.032 0.032 0.037 0.028 0.029 A 79986 ... ... ... ... ... ... ... ... ... ... ... ... ... Note. — This table is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content. TABLE 13 Mount Wilson HK Project measurements of HD26965 SMW JD - 2444000 W 0.223 -523.2 15956.8 0.207 -522.2 16877.2 0.234 -457.2 7264.7 0.206 -219.2 17620.2 0.218 -175.2 17201.9 0.213 -172.2 17475.1 0.219 -107.2 34054.6 0.205 172.8 8716.2 0.227 190.8 17363.2 0.221 200.8 10277.2 0.214 224.8 8639.2 0.205 260.8 8593.6 0.217 278.8 8700.7 0.194 482.9939 3409.7 ... ... ... Note. — This table is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: D´ıaz

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text: TABLE

text: HARPS

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text: 0.30 Disentangling Doppler signals and stellar activity 7 0.25 0.20 0.15 0.10 0.05 60 40 20 1 10 100 1000 10000 Period (days) Fig. 2.— Top: Generalized Lomb-Scargle periodogram of the unbinned combined velocities. The highest power is found at a period of 42.43 days. The vertical red arrows mark the position of the stellar rotation period and the period found in the time series. The dotted line shows the 0.1% signiﬁcance level, determined by 1,000 bootstrap resamplings. Bottom: Periodogram of sampling (window function) for the combined data. Lane, Hatﬁeld AL109AB, UK. 5 Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5124 Broad Branch Road, Washington, DC 20015- 1305, USA. 6 UCO/Lick Observatory, Department of Astronomy and Astro- physics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA. 7 Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientiﬁc partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administra- tion. The Observatory was made possible by the generous ﬁnan- cial support of the W.M. Keck Foundation. (cid:63)matias.diaz.m@ug.uchile.cl, Carnegie-Chile Graduate Fellow 2015-2016. rewoP rewoP ##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##

text: Disentangling

text: Doppler

text: signals

text: and

text: stellar

text: activity

text: Period

text: (days)

text: Fig.

text: Top:

text: Generalized

text: Lomb-Scargle

text: periodogram

text: of

text: the

text: unbinned

text: combined

text: velocities.

text: The

text: highest

text: power

text: is

text: found

text: at

text: a

text: period

text: of

text: days.

text: The

text: vertical

text: red

text: arrows

text: mark

text: the

text: position

text: of

text: the

text: stellar

text: rotation

text: period

text: and

text: the

text: period

text: found

text: in

text: the

text: time

text: series.

text: The

text: dotted

text: line

text: shows

text: the

text: signiﬁcance

text: level,

text: determined

text: by

text: bootstrap

text: resamplings.

text: Bottom:

text: Periodogram

text: of

text: sampling

text: (window

text: function)

text: for

text: the

text: combined

text: data.

text: Lane,

text: Hatﬁeld

text: UK.

text: Department

text: of

text: Terrestrial

text: Magnetism,

text: Carnegie

text: Institution

text: of

text: Washington,

text: Broad

text: Branch

text: Road,

text: Washington,

text: DC

text: USA.

text: UCO/Lick

text: Observatory,

text: Department

text: of

text: Astronomy

text: and

text: Astro-

text: physics,

text: University

text: of

text: California

text: at

text: Santa

text: Cruz,

text: Santa

text: Cruz,

text: CA

text: USA.

text: Some

text: of

text: the

text: data

text: presented

text: herein

text: were

text: obtained

text: at

text: the

text: W.M.

text: Keck

text: Observatory,

text: which

text: is

text: operated

text: as

text: a

text: scientiﬁc

text: partnership

text: among

text: the

text: California

text: Institute

text: of

text: Technology,

text: the

text: University

text: of

text: California

text: and

text: the

text: National

text: Aeronautics

text: and

text: Space

text: Administra-

text: tion.

text: The

text: Observatory

text: was

text: made

text: possible

text: by

text: the

text: generous

text: ﬁnan-

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text: support

text: of

text: the

text: W.M.

text: Keck

text: Foundation.

text: Carnegie-Chile

text: Graduate

text: Fellow

text: rewoP

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