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d1ae00721bc16ff87f292f61ec86d1140cd5a45e | subsection | 22 | 61 | Almost inclusions | For q,r \in \mathbb {Q} with q<r, let A_{q,r} = \lbrace \alpha \in A \ \mid \ q,r \in I_{\alpha }\rbrace , and let A^{\prime } = \bigcup _{q<r \in \mathbb {Q}} A_{q,r}.
Since \operatorname{rad}M \in \mathrm {(qtame)}, for each q<r \in \mathbb {Q}, \vert A_{q,r}\vert < \infty .
Therefore A^{\prime } is countable.Further... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
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c6cd8ddf755b00faa03322d7a8665ff3ebe87fe6 | subsection | 23 | 61 | Enveloping distance | In this section, we define a non-symmetric distance between classes of persistence modules and calculate its value for most of the pairs in Figure REF .Definition 3.10
Let \mathcal {A} and \mathcal {B} be classes of persistence modules.
We define the enveloping distance from \mathcal {A} to \mathcal {B} as follows.E(\... | {
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6964b9a29348bdca938076742f0d61a508d1b45b | subsection | 24 | 61 | Enveloping distance | By our first observation, there is an A \in \mathcal {A} such that D and A are s-interleaved.
Since \mathcal {A} (almost) includes in \mathcal {B}, there is a B \in \mathcal {B} such that A and B are \varepsilon -interleaved.
Therefore by Remark REF , C and B are (s+2\varepsilon )-interleaved.
So for all C \in there is... | {
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} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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984cc595ff585af5cb70402aba42671de183aea1 | subsection | 25 | 61 | Enveloping distance | Thus d_I(M,(d,d+2z]) \ge d_I((d,d+2z],0) \ge z.
The other three cases follow from the same arguments.
\mathrm {(cfid)} to \mathrm {(fid)}: Consider [0,\infty ).
\mathrm {(qtame)} to \mathrm {(cfid)}: Consider \bigoplus _{k=1}^{\infty } [0,k).
\mathrm {(ffid)}\subset \mathrm {(fid)}: Consider [0,\infty ).
\ma... | {
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} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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fe7f17e61e76316a87c358dc63f15054b635e9d2 | subsection | 26 | 61 | Enveloping distance | We define the p-persistent submodule of M byM^{(p)}(a) = \operatorname{im}M(a-p \le a).For a \le b, there is an induced map between objects M^{(p)}(a) and M^{(p)}(b) given by M(a \le b).
Since M is a persistence module, so is M^{(p)}, and since M^{(p)}(a) is a sub-vector space of M(a) for all a, M^{(p)} is a submodule ... | {
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} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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6033740ce8b8d1636ff5dc9407dfda540892092e | subsection | 27 | 61 | Sets of persistence modules | Next we consider whether the classes defined above are sets or proper classes.
We will use the following notation.
Let \overline{\mathbb {R}}:=\mathbb {R}\cup \lbrace \pm \infty \rbrace and \overline{\mathbb {N}}:=\mathbb {N}\cup \lbrace \infty \rbrace .
Given a set X, let \mathcal {P}(X) denote its power set.
Let \mat... | {
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} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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] | [
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a51a3715d794c490170e745f9083f20a57b69917 | subsection | 28 | 61 | Sets of persistence modules | By Lemma REF , we can define the following map.\mathrm {(pfd)}\longrightarrow \mathcal {P}(\overline{\mathbb {R}}^2\times \lbrace 1,2,3,4\rbrace \times \mathbb {N})\bigoplus _{\alpha \in A} I_{\alpha } \longmapsto \bigcup _{\alpha \in A} \big [ \lbrace (\inf I_{\alpha },\sup I_{\alpha })\rbrace \times \lbrace f(I_{\alp... | {
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} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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1fb375c5ec673d9546168601f67d2ba41447dbac | subsection | 29 | 61 | Interval-decomposable persistence modules of arbitrary cardinality | Motivated by the desire to have a set of persistence modules that contains all of the sets of persistence modules in Section REF and the proofs of Proposition REF and REF , we make the following definition.Definition 3.28
Given a cardinal \kappa , let \operatorname{(\kappa -id}) denote the class of persistence modules... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
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aaa5a6691b44d6dea563fc2c13bc98a9badc4d8b | subsection | 30 | 61 | Topological properties | Since we are interested in studying topological spaces of persistence modules, we will for the most part restrict ourselves to the sets in Figure REF .
We will consider the basic topological properties of these sets with the topology induced by the interleaving metric.
[Figure: Sets of metric spaces, each with the topo... | {
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} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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] | [
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cc1517022d622767a3306bf818a6d0d5178ec008 | subsection | 31 | 61 | Open subsets | In this section we consider which of the inclusion maps in Figure REF are inclusions of open subsets.
Recall that in a pseudometric space X, a subset A\subset X is said to be open if for all a \in A, there exists \varepsilon >0 such that B_\varepsilon (a)\subset A.Proposition 4.1
Among the inclusion maps in Figure REF... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
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] | 2,018 | en | Mathematics | [
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02782a24beb52dd443841647ea3039df6442649a | subsection | 32 | 61 | Separation | Proposition 4.3 Any set of ephemeral persistence modules with the interleaving distance has the indiscrete topology.Let S be a set of ephemeral persistence modules.
By Lemma REF , each M \in \mathrm {(eph)} has d_I(M,0)=0. So for M,N \in S, by the triangle inequality, d_I(M,N)=0.
Thus for all M \in S and for all \varep... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
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] | 2,018 | en | Mathematics | [
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67ab22b839ecd1578be18b1953c8c2b06b2d03ce | subsection | 33 | 61 | Compactness | Let X be an extended pseudometric space. Then a subset S\subset X is totally bounded if and only if for each \varepsilon >0, there exists a finite subset F=\lbrace x_1,x_2,\ldots ,x_n \rbrace \subset X such that S\subset \cup _{i=1}^n B_\varepsilon (x_i).
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"Peter Bubenik",
"Tane Vergili"
] | [
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0f599b6195ddc2bbe98169e10ad97417127ebf52 | subsection | 34 | 61 | Compactness | It does not have a finite subcover, since there does not exist a persistence module N such that B_{\frac{\delta }{6}}(N) contains M_n and M_m for m \ne n.Corollary 4.9
All of the spaces in Figure REF are not locally compact.An open covering \mathcal {O}=\lbrace O_i\rbrace _{i \in I} of X is locally finite if every x\i... | {
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"Peter Bubenik",
"Tane Vergili"
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0566881be1377ddba43d0bbf5055053eabe4b026 | subsection | 35 | 61 | Path Connectedness | Lemma 4.11
Let S be an extended pseudometric space. Let a,b \in S with d(a,b)= \infty . Then there does not exist a path in S from a to b.Suppose there is a path \gamma from a to b in S.
Then \gamma has a compact image.
Therefore the cover \lbrace B_{1}(x) \ \mid \ x \in \gamma \rbrace should have a finite subcover, w... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
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e0377befc5af6ddbdf2cb62578ff03de152c6575 | subsection | 36 | 61 | Path Connectedness | So M^{(t)} is a continuous path from M to 0.Remark 4.15 It is not the case that the path component of the zero module in \mathrm {(cid)} is \mathrm {(cfid)}, since \mathrm {(cfid)} is not path connected.
Since infinite intervals have infinite distance from the zero module, the path component of the zero module in \math... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
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a527103486806d3e41456aa826a9f89cc295c7cd | subsection | 37 | 61 | Path Connectedness | Let d denote the product metric on S\times [0,1].
Whenever (N,s) \in S\times [0,1] satisfies d((M,t),(N,s))< \delta , d_I(M,N)< \delta and \vert t-s\vert <\delta .
Furthermored_I(M^{(t)},N^{(s)}) \le d_I(M^{(t)},M^{(s)}) + d_I(M^{(s)},N^{(s)}) \\
\le \vert t-s\vert d_I(M,0) + s d_I(M,N)
\le \delta d_I(M,0) + \delta = \... | {
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} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
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d745cdb0453f88b7d69d36e4ab07ca39bfeec6a2 | subsection | 38 | 61 | Separability | A topological space is said to be separable if it has a countable dense subset.Theorem 4.18
The spaces \mathrm {(fid)}, \mathrm {(ffid)} and \mathrm {(ffid^{[c,d]})} are separable.First we will show that \mathrm {(ffid)} is separable.
LetD_n =\big \lbrace \bigoplus _{i=1}^n (p_i,q_i) \in \mathrm {(fid)}\ | \ p_i, q_i ... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
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a1bc8b4b182c0b7d0f64719f508002e4fe9b243e | subsection | 39 | 61 | Separability | Then any dense subset of \mathrm {(cfid)}, \mathrm {(cid)}, \mathrm {(pfd)}, or \mathrm {(rid)}, contains a point in an open ball centered at each M_\alpha of radius \frac{1}{2} and thus cannot be countable.
The same is true
for the subspace of \mathrm {(cid)} with finite distance to 0, and
for \mathrm {(cid)}\cap \mat... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
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548449ef8f517c0cfc4a70da19b320a1f6d14f8c | subsection | 40 | 61 | Countability | A topological space is said to be a first countable if it has a countable basis at each of its points.Lemma 4.21
An extended pseudometric space is first countable.Let x be a point in the space.
Then the countable collection of open balls \lbrace B_{\frac{1}{n}}(x) \ | \ n\in \mathbb {N}\rbrace is the desired local bas... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
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c16555912b35225770c83fe86a2c8a1ad833e0a2 | subsection | 41 | 61 | Completeness | An extended pseudometric space is said to be complete if every Cauchy sequence converges (see the end of Section REF ).Theorem 4.24
The spaces \mathrm {(pfd)}, \mathrm {(fid)}, \mathrm {(ffid)} and \mathrm {(ffid^{[c,d]})} are not complete.For n \ge 0, let M_n = \bigoplus _{k=0}^n \left[-\frac{1}{2^k},\frac{1}{2^k}\ri... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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6af3f5f558a3013cb71606573a04bec9cfc086e0 | subsection | 42 | 61 | Completeness | By the definition of interleaving, there exist natural transformations \varphi _{k} : M_k \Rightarrow M_{k+1} T_{\frac{1}{2^{k}}} and \psi _{k} : M_{k+1} \Rightarrow M_k T_{\frac{1}{2^{k}}} such that the triangles corresponding to (REF ) commute.Now we define shifted versions of \varphi and \psi .
For k \ge 0,
let \alp... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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e80627d7a2a33c561dc77701768073d68de0e379 | subsection | 43 | 61 | Completeness | By the universal properties of the colimit and the limit, we have a map \theta _a: A(a) \rightarrow B(a), and\mu ^k_a \theta _a \lambda ^k_a = M_k(a-\textstyle \frac{1}{2^k} \le a+\textstyle \frac{1}{2^k}).Let M(a) denote the image of \theta _a.
Thus, \theta _a factors as follows.\begin{}[row sep=small]
A(a) [rr, "\the... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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7b93d8d6251777defd5b2f82ef2ae4832aad348c | subsection | 44 | 61 | Completeness | We obtain the commutative diagram in Figure REF .
[Figure: The bi-infinite sequence in (), its limit and colimit, and three induced maps.]By the universal properties of limit and colimit, we have the following commutative diagram.\begin{}
A(a) [d,"\theta _a"^{\prime }] [r,"A(a\le b)"] & A(b) [d,"\theta _b"] \\
B(a) [r,... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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7c991613b758316be2b1430d55451f8666344b02 | subsection | 45 | 61 | Completeness | For the left hand side of the first identity, we haveM(a) \xrightarrow{} A(a+\frac{1}{2^k}) \xrightarrow{} M(a+\frac{1}{2^k}) \xrightarrow{} B(a+\frac{1}{2^k}) \xrightarrow{} M(a+\frac{1}{2^{k-1}}) = M(b).Using (REF ) the composition of the inner two maps equals \theta _{a+\frac{1}{2^k}}.
Then using (REF ) we see that ... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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1b24f57b9635178157549d39ec31c8b50baa6da1 | subsection | 46 | 61 | Completeness | Thus \mathrm {(cid)}\cap \mathrm {(qtame)} is complete.Finally, assume that in addition, each M^{\prime }_n \in \mathrm {(cfid)}\cap \mathrm {(qtame)}.
Since M is \frac{1}{2^k}-interleaved with M_k, which does not contain any infinite intervals in its direct sum decomposition, neither does M.
Therefore \operatorname{ra... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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58f330380844b387f89bf2a3e1724dbeb6d38b16 | subsection | 47 | 61 | Completeness | Thus, M_{\infty } is a limit of the Cauchy sequence. | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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0a47a25c6267a85eea0b26cf4a8dc648c84e7c23 | subsection | 48 | 61 | Baire spaces | Let X be a topological space. A subspace A \subset X has empty interior in X if A does not contain an open set in X.
The space X is said to be a Baire space if for any countable collection of closed sets in X with empty interior in X, their union also has empty interior in X.Theorem 4.35 (Baire category theorem)
A com... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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61b8d7c575a438d681b64f920d0e538df52ac1e4 | subsection | 49 | 61 | Baire spaces | Also, for all n, the sequence x_n,x_{n+1},x_{n+2},\ldots in \overline{B_{s_n}(x_n)} converges to x, so x \in \overline{B_{s_n}(x_n)}. Thus x \notin A_n for all n.Corollary 4.36
Hence \mathrm {(cid)}\cap \mathrm {(qtame)} and \mathrm {(cfid)}\cap \mathrm {(qtame)} are Baire spaces. | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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a1a82d220c2c34f7906c11e41aa1bb38eb4c7ef1 | subsection | 50 | 61 | Topological dimension | Let X be a topological space.
A collection of subsets of X has order m if there is a point in X contained in m of the subsets, but no point of X is contained in m+1 of the subsets.
The topological dimension of X (also called the Lebesgue covering dimension) is the smallest number m such that every open cover of X has a... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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0195f60818df9b43a17f43df82c5deaceb23e548 | subsection | 51 | 61 | Open questions | We end with some unresolved questions.Are \mathrm {(cid)} and \mathrm {(cfid)} complete?
Can the results presented here be extended to multiparameter persistence modules and generalized persistence modules? | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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901b9eb6741861ef3133e11788e6f3a6d977f15c | subsection | 52 | 61 | The arithmetic of maps and interleavings of interval modules | In this appendix, we give some basic results on interval modules, maps of interval modules, interleavings of interval modules, and neighborhoods of interval modules. | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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25682455d770bb2fd85d9fb1bf2fa84ddae8e860 | subsection | 53 | 61 | Some relations between intervals | First we define some relations between intervals that will be useful in the following sections and describe some of their properties.Recall that I \subset \mathbb {R} is an interval if a, c \in I and a \le b \le c then b \in I. It follows that the intersection of two intervals is an interval.Definition 6.1
For A,B \su... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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16cb32e4c8956e7ca52a2c54dffc196806ba5808 | subsection | 54 | 61 | Some relations between intervals | Then
J \setminus (I \cap J) \prec (I \cap J), and
(I \cap J) \prec I \setminus (I \cap J).First note that if either A or B is empty then A \prec B.
Suppose j \in J \setminus (I \cap J) and i \in I \cap J with i < j.
Since J \le I, there is an i^{\prime } \in I with j \le i^{\prime }. Since I is an interval, j \in I, wh... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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57840bc7157ea23628bc331814563d335d76a58e | subsection | 55 | 61 | Nonzero maps of interval modules | In this section we characterize nonzero maps of interval modules.Proposition 6.7
Let I and J be nonempty intervals. There is a nonzero map of persistence modules f:I \rightarrow J if and only if J \le I and I \cap J \ne \emptyset .(\Rightarrow ) Assume f \ne 0. Then there is an a \in \mathbb {R} such that 0 \ne f_a:I(... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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6164a4310872335cb9b1353a13ec98aa9500eba6 | subsection | 56 | 61 | Nonzero maps of interval modules | Therefore a \notin I which implies that I(a) = 0 and thus the diagram commutes.Lemma 6.8
Assume there is a nonzero map f: I \rightarrow J of interval modules. Then (up to isomorphism) f_a = 1 if a \in I \cap J and f_a = 0 otherwise.Assume f \ne 0. The there is a b \in I \cap J such that f_b is nonzero. Without loss of... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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891313496934485aead31e3e2f759a5b012e9921 | subsection | 57 | 61 | Interleavings of interval modules | In this section we characterize interleavings of interval modules.Definition 6.10 Let I be an interval and \varepsilon \in \mathbb {R}. Define the shifted interval I[\varepsilon ] by x \in I[\varepsilon ] if and only if x+\varepsilon \in I.
For example, [a,b)[\varepsilon ] = [a-\varepsilon ,b-\varepsilon ).The next lem... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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504541d0b0c33367adc9cd9d9bd3d10038518f3a | subsection | 58 | 61 | Interleavings of interval modules | Since J is an interval, x \in J.Lemma 6.16
If K \le J \le I then I \cap K = (I \cap J) \cap (J \cap K).One direction is easy: (I \cap J) \cap (J \cap K) = I \cap J \cap K \subset I \cap K.
The other direction follows from Lemma REF .Proposition 6.17
Let I and J be interval modules and \varepsilon \ge 0.
If J[\varepsi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s41468-018-0012-6",
"end": 1224,
"openalex_id": "https://openalex.org/W2588688245",
"raw": "Amit Patel. Generalized persistence diagrams. Journal of Applied and Computational Topology, 1(3):397–419, Jun 2018.",
"source_ref_id"... | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
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4fd3aa02856782a06b1a21bf42ba8efdd57cf07c | subsection | 59 | 61 | Interleavings of interval modules | If either \varphi or \psi are zero, then from Definition REF , I^{(2\varepsilon )} and J^{(2\varepsilon )} are zero. It follows that I^{-\varepsilon } and J^{-\varepsilon } are both empty and the condition is satisfied. If both \varphi and \psi are nonzero, then by Proposition REF , J[\varepsilon ] \le I and I[\varepsi... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
"math.GN"
] | 2,018 | en | Mathematics | [
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3c2beb4922879c644c7f92d486b9b082ef653328 | subsection | 60 | 61 | Neighborhoods of interval modules | Using Theorem REF , one obtains a complete characterization of the interval modules within distance \varepsilon of an interval module.Example 6.21 Consider the interval module [a,b) and let \varepsilon \in [0,\frac{b-a}{2}).
Then an interval module I is \varepsilon -interleaved with [a,b) if and only if [a+\varepsilon ... | {
"cite_spans": []
} | 10.1007/s41468-018-0022-4 | 1802.08117 | Topological spaces of persistence modules and their properties | [
"Peter Bubenik",
"Tane Vergili"
] | [
"math.AT",
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930ee29357d07c6a4f284e910ea3a0a1be6f53ef | abstract | 0 | 35 | Abstract | In this paper, we give an estimate of sub-Laplacian of Riemannian distance
functions in pseudo-Hermitian geometry which plays a similar role as Laplacian
comparison theorem in Riemannian geometry, and deduce a prior horizontal
gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to
regular balls of... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
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dcd471fcd701200b7ea0b40b3050475727d1c526 | subsection | 1 | 35 | Introduction | Inspired by Eells-Sampson's theorem, one natural problem is to consider the existence of harmonic maps from complete noncompact Riemannian manifolds.
Usually some convexity conditions on the images will lead this existence (cf. , , ).
Based on elliptic theory, some existence theorems have been studied for generalized h... | {
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"source_... | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
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9a9457d38ab52333941255acb9a70cb1310024b0 | subsection | 2 | 35 | Introduction | If for some k, k_1 \ge 0,R_* \ge - k, \mbox{ and } |A|, | \mbox{div} A | \le k_1 , \quad \mbox{ on } B_R (x_0),where R_* is the pseudo-Hermitian Ricci curvature and A is the pseudo-Hermitian torsion,
then there exists C_{\ref *{cst-2}} = C_{\ref *{cst-2}} (m) such that\Delta _b r \le C_{\ref *{cst-2}} \left(\frac{1}{r... | {
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"raw": "H.I. Choi. On the Liouville theorem for harmonic maps. Proc. Amer. Math. Soc., 85(1):91–94, 1982.",
"source_ref_id": "6fbb095f5594e97bd... | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
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d5fe92ba33090ee950d1f9ffc62a71a793a568b7 | subsection | 3 | 35 | Introduction | Then there is no nontrivial pseudo-Hermitian map from M to any regular ball of N.Another application of Theorem REF is the global existence of pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to regular balls which is due to an exhaustion process combined with the Dirichlet existence of pseudo-h... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
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6068bb29f0bf71b1fde8bdac3c13351bf3577090 | subsection | 4 | 35 | Basic Notions | In this section, we present some basic notions of pseudo-Hermitian geometry and pseudo-harmonic maps.
For details, readers may refer to , , . Recall that a smooth manifold M of real dimension 2m+1 is said to be a CR manifold if
there exists a smooth rank n complex subbundle T_{1,0} M \subset TM \otimes \mathbb {C} such... | {
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To Regular Balls | [
"Tian Chong",
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"Yibin Ren",
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018bc3df0728d3de60ca92036cce6d705b290118 | subsection | 5 | 35 | Basic Notions | Henceforth it is always regarded as the canonical volume form in pseudo-Hermitian geometry.It is remarkable that (M, HM, G_\theta ) could also be viewed as a sub-Riemannian manifold which satisfies the strong bracket generating hypothesis (see Appendix for details).
The completeness of a sub-Riemannian manifold is well... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
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0a0cc47d3a100803cadcfeb28a64fed0c0643822 | subsection | 6 | 35 | Basic Notions | For example,R_{\bar{\alpha } \beta \lambda \mu } = 2 i (A_{\beta \mu } \delta _{\bar{\alpha } \lambda } - A_{\beta \lambda } \delta _{\bar{\alpha } \mu }), \quad R_{\bar{\alpha } \beta 0 \mu } = - A_{\beta \mu , \bar{\alpha }}, \quad R_{\bar{\alpha } \beta 0 \bar{\mu }} = A_{\bar{\alpha } \bar{\mu }, \beta }where A_{\b... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
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44c2dedff17b8463e7e31405b5523191a242b6bb | subsection | 7 | 35 | Basic Notions | The sub-Laplacian \Delta _b u of a smooth function u is defined by\Delta _b u = \mbox{trace}_{G_\theta } \nabla _b d_b u,which is viewed as the special case of \tau _H acting on functions.Lemma 2.2 (CR Bochner Formulas, cf. , , )
For any smooth map f: M \rightarrow N , we have\frac{1}{2} \Delta _b |d_b f|^2=& |\nabla ... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
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d44abc885f20f73975c6f1f6464c9f5d94b7008d | subsection | 8 | 35 | Basic Notions | The commutation relation (cf. , )f^i_{\alpha \bar{\beta }} - f^i_{\bar{\beta } \alpha } = 2 i f^i_0 \delta _{\alpha \bar{\beta }}shows that| \pi _{(1,1)} \nabla _b d_b f|^2 \ge & 2 \sum _{\alpha = 1}^m f^i_{\alpha \bar{\alpha }} f^i_{\bar{\alpha } \alpha } \\
= & \frac{1}{2} \sum _{\alpha =1}^m \big [ |f^i_{\alpha \bar... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
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97f55a935c665b1fc4ca4e242aad8ba0e9eaf653 | subsection | 9 | 35 | Basic Notions | In particular, if k =0 and k_1 =0, then C_{\ref *{cst-reebcrbochner}} = 0.For (REF ), due to (REF ), Cauchy inequality and the identityi (f^i_{\bar{\alpha }} f^i_{0 \alpha } - f^i_\alpha f^i_{0 \bar{\alpha }}) = -\langle \nabla _b f_0 , d_b f \circ J \rangle ,it suffice to prove thatf^i_{\bar{\alpha }} f^j_{\beta } f^k... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
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49ad54eeb66e694ea7ae5b9edc2ae78ad0f81ffb | subsection | 10 | 35 | Basic Notions | Let (M,\theta ) be a pseudo-Hermitian manifold and \Omega \Subset M. For any k \in \mathbb {N} and p>1, the Folland-Stein space S^p_k (\Omega ) is given byS_k^p (\Omega ) = \big \lbrace u \in L^p (\Omega ) \big | \: \nabla _b^l u \in L^p (\Omega ), l = 0, 1, \dots , k \big \rbracewhere \nabla ^l_b u is the horizontal r... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
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24bd7fe9a7b3111cece6ae521979dc09e41a6ffc | subsection | 11 | 35 | Basic Notions | A direct calculation shows that for any \sigma \in \Gamma ( \otimes ^k T^* M ) and X_1 , \cdots , X_k, X, Y \in \Gamma (HM) , we have& ( \nabla ^2 \sigma ) ( X_1 , \cdots , X_k ; X, Y ) - ( \nabla ^2 \sigma ) ( X_1 , \cdots , X_k ; Y, X ) \\
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"raw": "G.B. Folland and E.M. Stein. Estimates for the \\bar{\\partial }_b complex and analysis on the Heisenberg group. Comm. Pure Appl. Math., 27:429–522, 1974.",
"source_ref_id": "e16709426d37a1ad... | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
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535441845005a02f901e386f09badd55db22781c | subsection | 12 | 35 | Sub-Laplacian Comparison Theorem | In this section, we will deduce Theorem REF which plays a similar role as Laplacian comparison theorem in Riemannian geometry.Suppose that (M^{2m+1}, \theta ) is a complete noncompact pseudo-Hermitian manifold.
Let r be the Riemannian distance with respect to Webster metric g_\theta from a reference point x_0 \in M.
We... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
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8eb324c145503aedfef901f7f35d3877576582ac | subsection | 13 | 35 | Sub-Laplacian Comparison Theorem | Since \widehat{Hess} (r) (\nabla r, \cdot ) =0 , then\Delta _b r \big |_{\gamma (a)} = \sum _{B=1}^{2m} \widehat{Hess} (r) (e_B (a), e_B (a)) = \sum _{B=1}^{2m} \widehat{Hess} (r) (e_B^{\perp } (a), e_B^\perp (a))Using the Riemannian exponential map, we could extend e_B^{\perp } (a) as a Jacobi field U_B along \gamma w... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
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c55bfc37de49eaf2e7064097447018c633f2ca04 | subsection | 14 | 35 | Sub-Laplacian Comparison Theorem | Then there is a constant C_{\ref *{cst-1}} = C_{\ref *{cst-1}} (m) such that\Delta _b r \big |_{\gamma (a)} \le C_{\ref *{cst-1}} \left(\frac{1}{a} + \sqrt{1 + k_1 + k_1^2 + \hat{k}} \right).Due to (REF ), we have\hat{\nabla }_{\dot{\gamma }} e_B = - [ d \theta (\dot{\gamma }, e_B) + A (\dot{\gamma }, e_B) ] \xi + \the... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
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49777c95cfb6212a11e94a39c8a08c81fe468f2d | subsection | 15 | 35 | Sub-Laplacian Comparison Theorem | By the curvature assumption, the Index lemma and (REF ), we have\Delta _b r \big |_{\gamma (a)} & \le \sum _{B=1}^{2m} I_a (V_B, V_B) = \sum _{B=1}^{2m} \int _0^a \left( \big | \hat{\nabla }_{\dot{\gamma }} V_B \big |^2 - \langle \hat{R} (V_B, \nabla r) \nabla r , V_B \rangle \right) dt \\
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To Regular Balls | [
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401620206aae6ee17edfad322eb93722533c57a3 | subsection | 16 | 35 | Sub-Laplacian Comparison Theorem | Denote \eta _\alpha = \frac{1}{\sqrt{2}} (e_\alpha - i J e_\alpha ).Lemma 3.3
For X, Y \in TM, we have\sum _{B=1}^{2m} \langle \hat{R} (e_B, X) Y, e_B \rangle & = \sum _{B=1}^{2m} \langle R(e_B, X) Y, e_B \rangle - 3 \langle \pi _H X, \pi _H Y \rangle \\
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To Regular Balls | [
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... | |
b0bab050feb0045bfbe679dfe462ed9f5cac6686 | subsection | 17 | 35 | Sub-Laplacian Comparison Theorem | On the other hand,\langle L e_B, e_B \rangle = trace_{G_\theta } \tau + trace_{G_\theta } J = 0.Substituting () and (REF ) into (REF ), the result is\sum _{B=1}^{2m} \langle (L e_B \wedge L X) Y, e_B \rangle = \langle \tau X, \tau Y \rangle - \langle \pi _H X, \pi _H Y \rangle .For the third term in (), we have\sum _{B... | {
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"raw": "N. Tanaka. A differential geometric study on strongly pseudo-convex manifolds, volume 9 of Lectures in Mathematics, Department of Mathematics, Kyoto University. Kino... | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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a4f73669bbcb007400d97e66453b62365ad2f707 | subsection | 18 | 35 | Sub-Laplacian Comparison Theorem | One can prove it by applying Riemannian first Bianchi identity to ().Lemma 3.4
For any X, Y \in TM, we have\langle R_* X, Y \rangle =& \sum _{B=1}^{2m} \langle R (e_B, \pi _H X) \pi _H Y, e_B \rangle - 2 (m-1) A( X, J Y),Since J X is horizontal, we can use the first Bianchi identity (REF ) and obtain-i & \sum _{\alpha... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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c60da1a231d7af9b1bbdfbca7c7d489cd1ce1f55 | subsection | 19 | 35 | Horizontal Gradient Estimates | Suppose that (M^{2m+1}, \theta ) is a complete noncompact pseudo-Hermitian manifold.
Let r be the Riemannian distance function from x_0 \in M associated with the Webster metric g_\theta and B_R be the geodesic ball of radius R centered at x_0.
Assume thatR_* \ge - k, \mbox{ and } |A|, | \mbox{div} A | \le k_1, \quad \m... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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00da7ae44caec8f74b3c0f9b6f021350f3793e47 | subsection | 20 | 35 | Horizontal Gradient Estimates | Moreover, Hessian comparison theorem shows that\mbox{Hess } \psi \ge \cos ( \sqrt{\kappa } \rho ) \cdot h.Lemma 4.1
For any 0 < D < \frac{\pi }{2 \sqrt{\kappa }}, there exist \nu \in [1,2), b > \phi (D) and \delta >0 only depending on D such that\nu \frac{\cos (\sqrt{\kappa } t)}{b - \phi (t)} -2 \kappa > \delta , \qu... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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b7641e13804ef8aede6800cf344ff3f9111b5f93 | subsection | 21 | 35 | Horizontal Gradient Estimates | Then\nu \frac{\Delta _b \psi \circ f}{b - \psi \circ f} - 2 \kappa |d_b f|^2 \ge \delta |d_b f|^2To estimate |d_b f|^2, we consider the following auxiliary function\Phi _{\mu \chi } = |d_b f|^2 + \mu \chi |f_0|^2where \mu will be determined later.Lemma 4.3
Suppose \mu and \epsilon satisfyC_{\ref *{cst-reebcrbochner}} ... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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00c08bc3b22c7d3e3cbaed243bc0568afeac469b | subsection | 22 | 35 | Horizontal Gradient Estimates | The \epsilon in Lemma REF is chosen as\epsilon = \frac{1}{\nu } - \frac{1}{2} \le 1and \mu satisfyC_{\ref *{cst-reebcrbochner}} \mu \le \epsilon .Let x be a maximum point of \chi F_{\mu \chi } on B_{2 R} which is nonzero. Assume that r is smooth at x. Otherwise we can modify the distance function r as . | {
"cite_spans": [
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"doi": "10.1090/pspum/036/573431",
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"raw": "S.Y. Cheng. Liouville theorem for harmonic maps. In Proc. Sympos. Pure Math., volume 36, pages 147–151. Amer. Math. Soc., Providence, RI, 1980... | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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... | |
f4f5cfe916ad54b4b6aec2a3b25ccedf5d13a41e | subsection | 23 | 35 | Horizontal Gradient Estimates | Hence at x, we have0 = \nabla _b \ln (\chi F_{\mu \chi }) &= \frac{\nabla _b \chi }{\chi } + \frac{\nabla _b \Phi _{\mu \chi }}{\Phi _{\mu \chi }} + \nu \frac{\nabla _b (\psi \circ f)}{b - \psi \circ f} , \\
0 \ge \Delta _b \ln (\chi F_{\mu \chi }) & = \frac{\Delta _b \chi }{\chi } - \frac{|\nabla _b \chi |^2}{\chi ^2}... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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aba0b1fbb7ff1d127d838f66cae47f22b614f82d | subsection | 24 | 35 | Horizontal Gradient Estimates | By definition of \Phi _{\mu \chi },|f_0|^2 = \mu ^{-1} \chi ^{-1} (\Phi _{\mu \chi } - |d_b f|^2)which, together with (REF ), shows at x,0 \ge \frac{1}{\chi } \left( 2m \epsilon \mu ^{-1} -C_{\ref *{cst-reebcrbochner}} - \frac{ 2 C_{\nu }}{R} \right) + \bigg [ \delta \chi \Phi _{\mu \chi } - 2m \epsilon \mu ^{-1} - \le... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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d4eeccc6d786e4f583ab936a47da76c8d60bb64f | subsection | 25 | 35 | Global Existence Theorem | Jost and Xu studied the minimizing sequence of Dirichlet problem of subelliptic harmonic maps and obtained the existence theorem under some convexity condition.
Their results seem to depend on the global fields which satisfy the Hörmander condition and the noncharacteristic assumption of the boundary.
But the weak exis... | {
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"doi": "10.1090/s0002-9947-98-01992-8",
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"raw": "J. Jost and C.J. Xu. Subelliptic harmonic maps. Trans. Amer. Math. Soc., 350(11):4633–4649, 1998.",
"source_ref_id": "86eb47f3b8884... | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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ecd3f3a1574db9b03b84674828d8aa9b50e868b6 | subsection | 26 | 35 | Global Existence Theorem | If f is continuous inside M, then f \in C^\infty (M, N).Now let's come to prove Theorem REF .
[Proof of Theorem REF ]
Suppose that (M, \theta ) is a complete noncompact pseudo-Hermitian manifold and (N, h) is a Riemannian manifold with sectional curvature K^N \le \kappa for some \kappa \ge 0. Let B_D (p_0) \subset N be... | {
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{
"arxiv_id": "",
"doi": "10.1093/acprof:oso/9780198564959.001.0001",
"end": 2206,
"openalex_id": "https://openalex.org/W4238244177",
"raw": "C.P. Boyer and K. Galicki. “Sasakian geometry\". Oxford University Press, Oxford, 2008.",
"source_ref_id": "1d80b0adba... | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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e3b577084ab69833b3061ec8dded43ab1da30507 | subsection | 27 | 35 | Global Existence Theorem | Since the identity I of B^n_{\mathbb {C}} is a holomorphic map from B^n_{\mathbb {C}} to \mathbb {C}^n, then it is also a harmonic map from (B^n_{\mathbb {C}}, \omega ) to (\mathbb {C}^n, \omega _0).
The lift of I is denoted by \tilde{I} such that\tilde{I} = I \circ \pi : B^n_{\mathbb {C}} \times \mathbb {R} \rightarro... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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f5ee76ca6ef15b761e9ac36f779ab70564e48108 | subsection | 28 | 35 | Global Existence Theorem | On one hand, the relation (REF ) guarantees that\tau _H (\tilde{I}) &= \sum _{B=1}^{2n} (\nabla _{\tilde{e}_B} d \tilde{I}) (\tilde{e}_B) \\
& = \sum _{B=1}^{2n} \hat{\nabla }_{\tilde{e}_B} \left( d \tilde{I} (\tilde{e}_B) \right) - \sum _{B=1}^{2n} d \tilde{I} \left( \nabla _{\tilde{e}_B} \tilde{e_B} \right) \\
& = \s... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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a144a0d50a32d1a9a4049833313d60dcef4c5d0e | subsection | 29 | 35 | Appendix | This section will deduce Theorem REF and Theorem REF by the theory of subelliptic analysis.
Suppose that (M, \theta ) is a pseudo-Hermitian manifold of real dimension 2m +1.
Let \Omega be a coordinate neighborhood in M and \lbrace e_B \rbrace _{B=1}^{2m} be an orthonormal basis of HM \big |_\Omega with J e_i = e_{i +m}... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
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f64e676a46ca5f40384abb307d7c4e4456acd09c | subsection | 30 | 35 | Appendix | Assume that u, v \in L_{loc}^1 (\Omega ) and \Delta _b u = v in the distribution sense. For any \chi \in C^\infty _0 (\Omega ), if v \in S^p_k (\Omega ) with p >1 and k \in \mathbb {N}, then \chi u \in S^p_{k+2} (\Omega ) and||\chi u||_{S^p_{k+2} (\Omega )} \le C_{\chi } \left( ||u||_{L^p (\Omega )} + ||v||_{S^p_k (\Om... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
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f456dc9e11f232fd7baa2f94490a8c17d4ea5063 | subsection | 31 | 35 | Appendix | Consider the minimizing problem\lambda = \inf _{f \in \mathcal {S}} E_H (f) = \inf _{f \in \mathcal {S}} \int _M h_{ij} (f) \langle \nabla _b f^i, \nabla _b f^j \ranglewhere h_{ij} = h (\frac{\partial }{\partial z^i}, \frac{\partial }{\partial z^j}) .
Since \varphi \in \mathcal {S} , then \lambda is finite.
Let \lbrace... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
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"math.DG"
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2eb28354aaa275e9580e0319d982aa0e88bf9c27 | subsection | 32 | 35 | Appendix | The positivity of (h_{ij}) implies that0
& \le \sum _{i, j, A} h_{ij} (f_s) e_A (f^i_s - f^i) \: e_A (f^j_s - f^j) \\
& = \sum _{i, j, A} h_{ij} (f_s) e_A f^i_s \: e_A f^j_s
- \sum _{i, j, A} h_{ij} (f_s) e_A f^i \: e_A f^j
- 2 \sum _{i, j, A} h_{ij} (f_s) e_A f^i \: e_A (f^j_s - f^j) ,which yields that\sum _{i, j, A} ... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
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354f57f4263ee071cd6eacce5cc06b988523b2ad | subsection | 33 | 35 | Appendix | Hence\lim _{s \rightarrow \infty } \sum _{i, j, A} \int _K h_{ij} (f) e_A f^i \: e_A (f^j_s- f^j) = 0 .Using (REF ), (REF ) and (REF ), we find that\sum _{i, j, A} \int _K h_{ij} (f) e_A f^i \: e_A f^j \le \liminf _{s \rightarrow \infty } \sum _{i, j, A} \int _K h_{ij} (f_s) e_A f^i_s \: e_A f^j_s,which implies that\su... | {
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To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.... | |
306c65f48eb94e788d5fa5b2868691edd9e87d54 | subsection | 34 | 35 | Acknowledge | The authors would like to thank the referees for their valuable comments.Tian ChongSchool of Science, College of Arts and SciencesShanghai Polytechnic UniversityShanghai, 201209, P. R. Chinachongtian@sspu.edu.cnYuxin DongSchool of Mathematical SciencesFudan UniversityShanghai, 200433, P. R. Chinayxdong@fudan.edu.cnYibi... | {
"cite_spans": []
} | 1802.08034 | Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds
To Regular Balls | [
"Tian Chong",
"Yuxin Dong",
"Yibin Ren",
"Zhang Wei"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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dd6f33590f7aaa63534285602db71460626c940f | abstract | 0 | 6 | Abstract | The performance of the missing transverse momentum (E$_{T}^{miss}$)
reconstruction with the ATLAS detector is evaluated using data collected in
proton-proton collisions at the LHC at a center-of-mass energy of 13 TeV in
2015. To reconstruct E$_{T}^{miss}$, fully calibrated electrons, muons,
photons, hadronically decayi... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6288-9 | 1802.08168 | Performance of missing transverse momentum reconstruction with the ATLAS
detector using proton-proton collisions at $\sqrt{s}$ = 13 TeV | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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... |
409d6ffe1ba63fafec623f82f2b21bd5f63f64f4 | subsection | 1 | 6 | Introduction | The missing transverse momentum () is an important observable serving as an experimental proxy for the transverse momentum carried by
undetected particles produced in proton–proton () collisions measured with the detector at the Large Hadron Collider ().
It is reconstructed from the signals of detected particles in the... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6288-9 | 1802.08168 | Performance of missing transverse momentum reconstruction with the ATLAS
detector using proton-proton collisions at $\sqrt{s}$ = 13 TeV | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0... |
8f66d42d00cfa188e0795b4a6db7e9fec3010a86 | subsection | 2 | 6 | ATLAS detector | The ATLAS experiment at the LHC features a multi-purpose particle
detector with a forward–backward symmetric cylindrical geometry and a nearly full (4\pi )
coverage in solid angle.ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP)
in the centre of the detector and the z-ax... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6288-9 | 1802.08168 | Performance of missing transverse momentum reconstruction with the ATLAS
detector using proton-proton collisions at $\sqrt{s}$ = 13 TeV | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.03537306562066078,
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d078d035d1a6147c0f86ca37033fb91f5f7f1fec | subsection | 3 | 6 | reconstruction | The reconstructed in is characterised by two contributions.
The first one is from the hard-event signals comprising fully reconstructed and calibrated particles and jets (hard objects).
The reconstructed particles are electrons, photons, , and muons.
While muons are reconstructed from and tracks, electrons and are iden... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6288-9 | 1802.08168 | Performance of missing transverse momentum reconstruction with the ATLAS
detector using proton-proton collisions at $\sqrt{s}$ = 13 TeV | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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0.0415... |
20b98558c25301dfec95af0540e30e1c4a0b0f72 | subsection | 4 | 6 | basics | The missing transverse momentum reconstruction provides a set of observables constructed from the components of the transverse momentum vectors () of the various contributions.
The missing transverse momentum components serve as the basic input for most of these observables. They are given by&= - \sum _{i\in \lbrace \t... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6288-9 | 1802.08168 | Performance of missing transverse momentum reconstruction with the ATLAS
detector using proton-proton collisions at $\sqrt{s}$ = 13 TeV | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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... |
80fbe87f525be6cf558a3405650a6324cb62e697 | subsection | 5 | 6 | basics | A discussion of the treatment of isolated and non-isolated muons is given in subsec:muon-sel.Generally, jets are rejected if they overlap with accepted higher-priority particles.
To avoid signal losses for reconstruction in the case of partial or marginal overlap, and to suppress the accidental
inclusion of jets recons... | {
"cite_spans": []
} | 10.1140/epjc/s10052-018-6288-9 | 1802.08168 | Performance of missing transverse momentum reconstruction with the ATLAS
detector using proton-proton collisions at $\sqrt{s}$ = 13 TeV | [
"ATLAS Collaboration"
] | [
"hep-ex"
] | 2,018 | en | Physics | [
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... |
71580acdfb70ba5c4c558a5acdcfbb36061cf9fb | abstract | 0 | 60 | Abstract | We investigate the limiting behavior of sample central moments, examining the
special cases where the limiting (as the sample size tends to infinity)
distribution is degenerate. Parent (non-degenerate) distributions with this
property are called \emph{singular}, and we show in this article that the
singular distributio... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.03732490539550781,
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0.028260722756385803,
-0.014107472263276577,
-0.0... | |
030ab83035ab06ce60bb3589255ce5445a72fc59 | subsection | 1 | 60 | Introduction | Let X be a random variable with distribution function F
and finite moment of order k, for some positive integer k\ge 2.
Then, X has finite central moment of order k.Based on a random sample of size n from F,
a natural estimator of the kth central moment of X is
the kth sample central moment, and
the strong law of large... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-1-4614-1412-4_97",
"end": 535,
"openalex_id": "https://openalex.org/W2148451394",
"raw": "Lehmann, E.L. (1999). Elements of Large-Sample Theory. Springer, N.Y.",
"source_ref_id": "9c5a7f7b27e474ed552bd10e29dbb4adb5d3f761",... | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.015780800953507423,
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0.008523769676685333,
-0.02129034511744976,
... | |
fb5b53f00da4d3544dec9774f33ab02245671d62 | subsection | 2 | 60 | Notation and Terminology | Let X\sim F with \operatorname{{E}}|X|^{k}<\infty for some (fixed) k\in \lbrace 1,2,\ldots \rbrace ;
and let us consider a random sample X_1,\ldots ,X_n from F.
To avoid trivialities we further assume that X is non-degenerate,
that is, the set of points of increase of F,S_F\doteq \lbrace x\in {R}\colon F(x+\textrm {\sc... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
0.0027687707915902138,
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... | |
01c22d6da78686e588337d2d221e67fc3026e3eb | subsection | 3 | 60 | Notation and Terminology | In the sequel, we shall use the notation\textrm {\scalebox {.882}{Ė}}_j\doteq \operatorname{{E}}(X-\textrm {\scalebox {.882}{Ė}})^j, \quad j=0,\ldots ,k.The sample moments of the centered X_is arem_{j,n}\doteq \frac{1}{n}\sum _{i=1}^n (X_i-\textrm {\scalebox {.882}{Ė}})^j,
\quad j=1,\ldots ,k.The moment estimator of \t... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.0501580610871315,
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-0.00917581282556057,
-0.0539717823266983,
0.0... | |
466aab4213fb62b274972d4559e70605d81319f3 | subsection | 4 | 60 | Notation and Terminology | Therefore,
{M}_{k,n}={g}_k({m}_{k,n}), where {g}_k=(g_{1,k},\ldots ,g_{k,k})^{\prime }.Finally, let {X}_n be a sequence of random vectors.
The terminology {X}_n \surd {n}-converges in distribution
to a distribution, say F_{0}, means that there exists
{\textrm {\scalebox {.882}{Ė}}} such that \surd {n}({X}_n-{\textrm {\... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.07148998975753784,
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-0.013026623986661434,
-0.... | |
9da57bd5f1904b1909a811f35cb3029fb8ea7d8a | subsection | 5 | 60 | Motivation and our contributions | Based on the asymptotic distribution
of the vector of the sample skewness and kurtosis,
gave an asymptotic
result for testing normality.
establishes moment-based estimators
of the parameter vector of the characteristic quadratic polynomial
for both, integrated Pearson and cumulative Ord families of distributions,
and... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.jspi.2004.04.014",
"end": 378,
"openalex_id": "https://openalex.org/W2071301783",
"raw": "Pewsey, A. (2005). The large-sample distribution of the most fundamental of statistical summaries. J. Statist. Plann. Inference, 134, 434–44... | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.0034953865688294172,
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0.024606911465525627,
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0.023416992276906967,
-0.009984639473259449,... | |
58d27448ae9f4ad219ec66eecbf2c35cfc95a70a | subsection | 6 | 60 | The limiting distribution and a characterization of normality | Assume that k\ge 2 and \operatorname{{E}}|X|^{2k}<\infty .
The multivariate central limit theorem immediately yields
that\surd {n}({m}_{k,n}-{\textrm {\scalebox {.882}{Ė}}}_{k}){\mathrm {d}}N_k\mathopen {}\mathclose {\left({0}_k,\operatorname{\mathrel {}_k,
}\right.
where {0}_k=(0,\ldots ,0)^{\prime }\in {R}^k
and \ope... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.06616592407226562,
-0.0027410180773586035,
-0.025514166802167892,
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-0.031709592789411545,
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0.0026589971967041492,
-0.056094542145729065,... | |
fea4e63b7c47c75530b40e00b28252d5b58c9d02 | subsection | 7 | 60 | The limiting distribution and a characterization of normality | This result is presented in the following proposition.
}\begin{}
If \operatorname{{E}}|X|^{2k}<\infty , then
\begin{equation}
\surd {n}({M}_{k,n}-{\textrm {\scalebox {.882}{Ė}}}_{k}){\mathrm {d}}N_k({0}_k,\mathbf {V}_k),
\end{equation}
where the variance-covariance matrix \mathbf {V}_k=(v_{ij})\in {R}^{k\times k}
has... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.06856273859739304,
0.010615624487400055,
-0.01976597122848034,
-0.023093370720744133,
-0.020193343982100487,
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0.025688132271170616,
-0.012027479708194733,
-0.014774872921407223,
-0.01794963888823986,... | |
41d4ef35a0703a1ed0945b388fda1819e9e56be7 | subsection | 8 | 60 | The limiting distribution and a characterization of normality | Particular cases of the preceding result are contained in the next corollary.If k\ge 2 and \operatorname{{E}}|X|^{2k}<\infty , then\surd {n}(M_{k,n}-\textrm {\scalebox {.882}{Ė}}_k){\mathrm {d}}N\mathopen {}\mathclose {\left(0,v_k^2\right)};\surd {n}\mathopen {}\mathclose {\left({\bar{X}_n-\textrm {\scalebox {.882}{Ė}}... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.05492764711380005,
0.0020979309920221567,
-0.035519879311323166,
-0.007472448516637087,
-0.015776440501213074,
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0.036496371030807495,
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-0.030454328283667564,
0.005012147594243288,
... | |
23d32ee6a499e17a35ca684919ecd504f643e208 | subsection | 9 | 60 | The limiting distribution and a characterization of normality | Let k,r\in \lbrace 2,3,\ldots \rbrace be fixed.If \operatorname{{E}}|X|^k<\infty , then \operatorname{{E}}(M_{k,n})=\textrm {\scalebox {.882}{Ė}}_k+o(1/\surd {n});
If \operatorname{{E}}|X|^{k+1}<\infty , then
\mathsf {Cov}(\bar{X}_n, M_{k,n})=(\textrm {\scalebox {.882}{Ė}}_{k+1}-k\textrm {\scalebox {.883}{ě}}^2\textr... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.04492921382188797,
-0.019763359799981117,
-0.018344059586524963,
0.0001856391754699871,
-0.004021347500383854,
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0.007191878743469715,
0.006558535620570183,
-0.03354429826140404,
-0.018496673554182053,
... | |
855f84f0b9d4f979a816355fb5ca792ffe7e4f79 | subsection | 10 | 60 | The limiting distribution and a characterization of normality | Let
k\in \lbrace 2,3,\ldots \rbrace be fixed.The sample mean, \bar{X}_n, is called
asymptotically independent
of the sample central moment, M_{k,n}, if
there exist independent random variables W_1 and W_k such
that
\surd {n}
\mathopen {}\mathclose {\left({\bar{X}_n-\textrm {\scalebox {.882}{Ė}}\atop M_{k,n}-\textrm {\... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.03484925627708435,
-0.0067936209961771965,
-0.019133523106575012,
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0.009101392701268196,
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0.0254198145121336,
0.008323234505951405,
0.021910477429628372,
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-0.007308578584343195,
-0.023741435259580612,
... | |
5ee61b74f9513eed779b21677d3fa435a28f1ed5 | subsection | 11 | 60 | The limiting distribution and a characterization of normality | Observing the dispersion matrix in (), it becomes clear
that the first column – except of the first element, \textrm {\scalebox {.883}{ě}}^2 – vanish.
This is so because \textrm {\scalebox {.882}{Ė}}_k=0 for all odd k and
\textrm {\scalebox {.882}{Ė}}_{2r}=\textrm {\scalebox {.883}{ě}}^{2r}(2r)!/(2^r r!); thus,
for any... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.023458678275346756,
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0.005151141434907913,
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0.008623392321169376,
0.0040026274509727955,
0.02092508040368557,
0.0014633056707680225,
-0.009157584048807621,
-0.035745106637477875,
-0.04365115240216255... | |
4d8c7691f9c9a1e62311416ce57fddf46d3cfd55 | subsection | 12 | 60 | The limiting distribution and a characterization of normality | (),
(REF )) it follows that \bar{X}_n and M_{k,n} are
asymptotically uncorrelated if and only if
\textrm {\scalebox {.882}{Ė}}_{k+1}=k\textrm {\scalebox {.883}{ě}}^2\textrm {\scalebox {.882}{Ė}}_{k-1}. Since we have assumed that
this relation holds for all k\ge 2 it follows
that \textrm {\scalebox {.882}{Ė}}_1=\textrm ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/2291440",
"end": 713,
"openalex_id": "https://openalex.org/W2056099894",
"raw": "Billingsley, P. (1995). Probability and Measure (3rd ed.), John Wiley & Sons, New York.",
"source_ref_id": "0735b6093ee152737a021622ed7cbf27f67d5... | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.016650760546326637,
-0.022236624732613564,
-0.03510242700576782,
-0.038490574806928635,
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0.019520001485943794,
0.006097138859331608,
-0.028463490307331085,
0.0007201720727607608,
-0.025502676144242287... | |
b65bf77f1cdcb84b8a2c797f55a7c593ccad3e3b | subsection | 13 | 60 | The limiting distribution and a characterization of normality | It is well-known that for all m\ge k+2,
P_{m} is orthogonal to \lbrace 1,x,\ldots ,x^{k+1}\rbrace in the interval
[0,1], that is,\int _{0}^1 x^{j}P_{m}(x)\operatorname{d\textrm {{x}}}=0, \quad j=0,\ldots ,k+1.Since P_m is continuous on [0,1], it follows that
0<\max _{x\in [0,1]}|P_m(x)|\doteq a_m<\infty . Also,
\min _{... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.02684575319290161,
-0.023152362555265427,
-0.035010889172554016,
-0.026311585679650307,
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0.01138540729880333,
-0.07337940484285355,
0.014643831178545952,
-0.006215425208210945,
... | |
f22fd488b7301ae324779b21c46cd6c36424a7e3 | subsection | 14 | 60 | The singular distributions | First, center the rv X as U=X-\textrm {\scalebox {.882}{Ė}}
with \operatorname{{E}}(U^j)=\textrm {\scalebox {.882}{Ė}}_j for all j.
Assume that \operatorname{{E}}|X|^{2k}<\infty for some k=2,3,\ldots
and consider the random vector{U}_k=\mathopen {}\mathclose {\left(U,U^2,U^3-3\textrm {\scalebox {.883}{ě}}^2 U,U^4-4\te... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.009458571672439575,
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0.0177119392901659,
-0.004332636017352343,
0.... | |
daa0f9d4703ae6f49119682355d50bdb0e8d1901 | subsection | 15 | 60 | The singular distributions | For fixed k\ge 2, a non-degenerate random variable X, or its
distribution function F, is
called singular (of order k)
if \operatorname{{E}}|X|^{2k}<\infty and\surd {n}(M_{k,n}-\textrm {\scalebox {.882}{Ė}}_k){\mathrm {p}}0.The set of all singular random variables of order k will
be denoted by {F}_k; the subset of all s... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
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-0.002985761035233736,
-0.02339768223464489,
... | |
cc1f269c2f79e0fd0eba6db689586fce9cb9be4a | subsection | 16 | 60 | The singular distributions | Moreover, we shall show below
(lem.odd) that we can find a unique
value of p=p_{2k+1}\in (1/2,1),
for which the two-valued random variable W_{2k+1},
with\operatorname{{P}}\mathopen {}\mathclose {\left(W_{2k+1}=\sqrt{{(1-p)}/{p}}\right)}
=p
=1-\operatorname{{P}}\mathopen {}\mathclose {\left(W_{2k+1}=-\sqrt{{p}/{(1-p)}}\... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.0303998664021492,
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0.001818917691707611,
-0.029362119734287262,
-... | |
4a4ebf9e510c4db7bf8a99f6a53e31f6f1bc0a97 | subsection | 17 | 60 | The singular distributions | Hence,
\operatorname{{P}}(x_i-\textrm {\scalebox {.883}{ď}}<X\le x_i+\textrm {\scalebox {.883}{ď}})\le \operatorname{{P}}[(X-\textrm {\scalebox {.882}{Ė}})^k\ne \textrm {\scalebox {.882}{Ė}}_k+k\textrm {\scalebox {.882}{Ė}}_{k-1}(X-\textrm {\scalebox {.882}{Ė}})].
Since, however, x_i is a point of increase of F,
we hav... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.04737234115600586,
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-0.045907218009233475,
-0.019473940134048462,
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0.012980083003640175,
-0.054636914283037186,
-0.01475044060498476,
-0.05710931122303009,
... | |
0a8cc18fc3d2f3187939809b3a690a3ab2e6eddf | subsection | 18 | 60 | The singular distributions | Since \textrm {\scalebox {.882}{Ė}}=p and\textrm {\scalebox {.882}{Ė}}_k=p(1-p)\mathopen {}\mathclose {\left[(1-p)^{k-1}+(-1)^k p^{k-1}\right]},
\quad k=1,2,\ldots ,(REF ) shows that X\in {F}_k if and only
if(x-p)^k=\textrm {\scalebox {.882}{Ė}}_k+k\textrm {\scalebox {.882}{Ė}}_{k-1}(x-p) \quad \textrm {for } x=0 \text... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.03390945866703987,
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-0.04721686244010925,
-0.017382031306624413,
-0.007428186014294624,
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0.0038781464099884033,
-0.013139533810317516,
-0.01909123919904232,
-0.027103150263428688,
... | |
11ba0d8caccf4f44df1035daba1d4c0273c03d62 | subsection | 19 | 60 | The singular distributions | Therefore, there exist \textrm {\scalebox {.883}{Ě}}_1<\textrm {\scalebox {.883}{Ě}}_2, with 0<\textrm {\scalebox {.883}{Ě}}_1<1<k-2<\textrm {\scalebox {.883}{Ě}}_2<\infty ,
such that q_k(t)<0 for t in (0,\textrm {\scalebox {.883}{Ě}}_1)\cup (\textrm {\scalebox {.883}{Ě}}_2,\infty )
and q_k(t)>0 for t in (\textrm {\sca... | {
"cite_spans": []
} | 1806.02314 | On the limiting distribution of sample central moments | [
"Georgios Afendras",
"Nickos Papadatos",
"Violetta Piperigou"
] | [
"math.ST",
"stat.TH"
] | 2,018 | en | Mathematics | [
-0.07308017462491989,
0.00048627585056237876,
-0.016148461028933525,
-0.03132007643580437,
-0.0012057924177497625,
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0.0184684656560421,
-0.02240636944770813,
-0.011317659169435501,
-0.03898220136761665,
... |
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