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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
557cb85d228af5d0f4a7edd60c2b39d7283bd8f5 | subsection | 58 | 117 | Proof of Theorem | For each i \in [1:k] and
j \in [1:l], there exists[leftmargin=*]
a DRV
V_{i,j} : \left\lbrace \begin{array}{c} |\mathcal {V}_{i,j}| \le 2n^3\log _{2}|\mathcal {Y}| \\ V_{i,j} \ll (\mathbf {X},M_j) \\
h_{V_{i,j}}(v_{0,i,j}) > \frac{n^2-1}{n} \log _{2}\frac{n}{8} \end{array} \right\rbrace ,
positive real number
\tilde{... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05171293765306473,
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-0.016337627544999123,
-0.002133730798959732... | |
1cad16eb2b5da2b12925f67e7bad4d5af5c66e57 | subsection | 59 | 117 | Proof of Theorem | Second,|\mathcal {V}| \le ( 2n^3\log _{2}|\mathcal {Y}| )^{lk}since |\mathcal {V}_{i,j}| \le 2n^3\log _{2}|\mathcal {Y}|, while |\mathcal {V}| is
at most \prod _{i \in [1:k], j \in [1:l]} |\mathcal {V}_{i,j} |.Going forth, it will be important to note that V \gg V_{i,j}, thus
to every v \in \mathcal {V}, i \in [1:k] an... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.009055434726178646,
-0.00852141622453928,
0... | |
0ead45d45bb62b272be59b73302172e470ea7649 | subsection | 60 | 117 | Proof of Theorem | Hence there exists a
\delta = O(-\sqrt{\alpha } \log _{2}\alpha ) such that\hspace{-7.0pt}\Pr \left(\left| h(\mathbf {Y}_{i}|U,M_j) -\mathbb {H}(\mathbf {Y}_i|U,M_j) \right| >n \delta + 3 h(U) |U = u\right) & \\
&\hspace{-70.0pt} < 4 \cdot 2^{-n\alpha },for all i \in [1:k], j \in [1:l] and u \in \mathcal {U}_{i|j} due ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.006528121419250965,
-0.004082460422068834,
... | |
79bf816106f9e62daef5d36507f2f68abdeb03c7 | subsection | 61 | 117 | Proof of Theorem | Then identify the
following sets:\mathcal {U}_{j,\scriptscriptstyle {(\text{stable})}} &= \mathcal {U}_{j,*} \cap \left\lbrace u \in \mathcal {U} : q_j(u) <
\psi \right\rbrace \\
\mathcal {U}_{j,\scriptscriptstyle {(\text{saturate})}} &= \mathcal {U}_{j,*} \cap \left\lbrace u \in \mathcal {U} : q_j(u) =
\psi \right\rbr... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.008558728732168674,
-0... | |
29d98440c22d3251841faaa4be025d820faa11e6 | subsection | 62 | 117 | Proof of Theorem | To that end, observe first that&\hspace*{-10.0pt}
\Pr \left( | h(M_j|U) - Q_j| > \tilde{\rho }+ 1 | U = u \right) \\
&\le \Pr \left( | h(M_j|U) - h(M_j) | > \tilde{\rho }| U = u \right) \\
& \hspace{10.0pt} + \Pr \left( | h(M_j) - Q_j | > 1 | U = u \right) \\
&< 2^{-\rho }where the first probability in (REF ) is upper
... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.011857491917908192,
0.... | |
0c355a6bb0a9425d83620217151efed6baa4b400 | subsection | 63 | 117 | Proof of Theorem | Thus, by re-defining \mathcal {U}_{j,\scriptscriptstyle {(\text{stable})}} =
\mathcal {U}_{j,*} and \mathcal {U}_{j,*} = \emptyset ,
p_U\left(\mathcal {U}_{j,\scriptscriptstyle {(\text{stable})}} \right) \ge 1- 2^{-\rho }
and\Pr \left( |h(M_j|U) - \log _{2}|\mathcal {M}_j| | > \tilde{\rho }| U=u \right) < 2^{-\rho }for... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.03070514276623726,
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0.... | |
5ee25ef5b894fe630f4517eba94bfad2981d24dc | subsection | 64 | 117 | Proof of Theorem | For every w \in \mathcal {P}(\mathcal {Y}|\mathcal {X}), there exists a
\tilde{w} \in \mathcal {P}^{(\epsilon )}(\mathcal {Y}|\mathcal {X}) such that\sup _{{\hat{p} \in \mathcal {P}(\mathcal {X})\\ \hat{w} \in \mathcal {P}(\mathcal {Y}|\mathcal {X})} } \mathbb {D}_{\hat{w}}(w||\tilde{w}|\hat{p})
\le \epsilon - \frac{2|... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0... | |
4e54567b6ef6c1499e2092e497f980fed9709d60 | subsection | 65 | 117 | Proof of Theorem | \end{array}\right.}Next, form\mathcal {P}^{(\epsilon )}(\mathcal {Y}) &\triangleq \bigcup _{y \in \mathcal {Y}} \mathcal {P}^{(\epsilon )}(\mathcal {Y};y).Finally, we can define the approximating set as& \hspace{-10.0pt} \mathcal {P}^{(\epsilon )}(\mathcal {Y}|\mathcal {X})
\\
&\triangleq \left\lbrace w \in \mathcal {P... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0... | |
5c4adf238ce08dfe035fc8f772ac941420bda0e6 | subsection | 66 | 117 | Proof of Theorem | Observe that for every w \in \mathcal {P}(\mathcal {Y}|\mathcal {X}) there
exists a \tilde{w} \in \mathcal {P}^{(\epsilon )}(\mathcal {Y}|\mathcal {X}) that
has the following properties:\tilde{w}(y|x) \ge w(y|x) for all (y,x) \ne (y_x,x), and
\tilde{w}(y_x|x) \ge w(y_x|x) - |\mathcal {Y}|\tilde{\epsilon } for
all x \i... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.009229189716279507,
0.... | |
0ba27b377132316348c168617eab179f4b858c01 | subsection | 67 | 117 | Information Spectrum Slicing | Here we build a few results that relate to information (or entropy)
spectrum introduced by Han . Keep in mind that the
overarching goal of this work is to create quasi-images with nearly
uniform distribution. The entropy spectrum provides a language with
which we can succinctly discuss these variations.Definition 19
F... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-662-12066-8",
"end": 96,
"openalex_id": "https://openalex.org/W4251984466",
"raw": "T. S. Han, Information-Spectrum Methods in Information Theory. Applications of mathematics, Springer, 2003.",
"source_ref_id": "bbbc6b97... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04587678238749504,
0.020908579230308533,
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0.03675026446580887,
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0.02919570356607437,
-0.01993182860314846,
0.0... | |
ee730434fa9b8a66932b44a8751ac8ebd2804788 | subsection | 68 | 117 | Information Spectrum Slicing | In addition,t \lambda - \varrho (t) \le \log _{2}|\mathcal {S}_{Y}(t;\lambda ,t) | < (t+1)\lambdaif t> \lambda ^{-1} \log _{2}|\mathcal {Y}|.For s \in [0:t-1], (REF ) implies2^{-(s+1)\lambda } |\mathcal {S}_{Y}(s;\lambda ,t)| < \sum _{y \in \mathcal {S}_{Y}(s;\lambda ,t)} p_{Y}(y) \le 2^{-s\lambda }
|\mathcal {S}_{Y}(s... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.020021673291921616,
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0.05383876711130142,
-0.04343115910887718,
0.0... | |
fc142f8bf0fb5256502c0bb36d455ee39eaf8abf | subsection | 69 | 117 | Supporting Lemmas | Next, we begin to employ information spectrum slicing to derive a
number of lemmas which help to simplify and streamline the proof of
Theorem REF . All lemmas here are under the assumption that
(\emptyset ,\mathbf {X},\mathbf {Y}) form a regular collection of
DRVs. Furthermore, throughout the proofs we let \mathcal {A}... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.014203931204974651,
-0.0223357193171978,
0.037... | |
a38b6be889298cc959e9e21db55e939bd25a787c | subsection | 70 | 117 | Supporting Lemmas | If\mathcal {A}^{\prime } \triangleq \left\lbrace \mathbf {x} \in \mathcal {A} : p_{Y|X}^n (\mathcal {B}|\mathbf {x}) \ge \epsilon \right\rbrace ,thenp_{\mathbf {X}}(\mathcal {A}^{\prime } ) \ge \frac{\alpha - \epsilon }{1 - \epsilon } ,and\log _{2}g^{n}_{ Y|X }\left( \mathcal {A}^{\prime },1-\beta \right) \le \log _{2}... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.01123178843408823,
0.023775987327098846,
-0.... | |
dacf0123d7463bec73d2a2f118172c7491f81739 | subsection | 71 | 117 | Supporting Lemmas | Furthermore as \mathcal {B} is a minimum
\alpha -quasi image of \mathbf {X} by p_{Y|X}, we have\alpha &\le p_{\mathbf {Y}}(\mathcal {B})
\le \sum _{\mathbf {x} \in \mathcal {A}} p_{Y|X}^n (\mathcal {B}|\mathbf {x}) p_{\mathbf {X}}(\mathbf {x})
\\
&= \sum _{\mathbf {x} \in \mathcal {A}^{\prime } } p_{Y|X}^n (\mathcal {B... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.008010649122297764,
0... | |
9be34935a3934a0e006cf2237249e441f1a6c902 | subsection | 72 | 117 | Supporting Lemmas | Under this assumption, it follows that|\mathcal {S}_{\mathbf {Y}}(s)| \le \bar{g}^{n}_{ Y|X }\left( \mathbf {X},\eta _s \right) = \sum _{i=0}^s | \mathcal {S}_{\mathbf {Y}}(i)|,and thus (REF ) and (REF ) follow by
Corollary REF .Therefore what remains to be proven is that
\bigcup _{i=0}^{s} \mathcal {S}_{\mathbf {Y}}(i... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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-0.004084954038262367,
... | |
e46a2d3411401d6b41e9c2f2b851f9ceb6e8fd1c | subsection | 73 | 117 | Supporting Lemmas | On the other hand, for
s \in [0:t-1],& \hspace{-10.0pt} p_{\mathbf {Y}}(\mathcal { B}) \\
&= p_{\mathbf {Y}} \left(\bigcup _{i=0}^{s} \mathcal {S}_{\mathbf {Y}}(i) \right) -
p_{\mathbf {Y}} \left(\bigcup _{i=0}^{s} \mathcal {S}_{\mathbf {Y}}(i)\setminus \mathcal { B} \right)
\\ &\hspace{20.0pt}
+ p_{\mathbf {Y}} \left(... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.042514897882938385,
0.003383948002010584,
-0.... | |
c57fb9ad655b91e349434f5fb98b35ab3f3a7e2d | subsection | 74 | 117 | Supporting Lemmas | This is done in two parts.Lemma 25
Fix any U: \lbrace U \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {X} \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {Y}\rbrace and any u \in \mathcal {U}. Let \mathcal {A}_u... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.029487406834959984,
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0.03868601471185684,
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0.00030128107755444944,
-0.03166883438825607,
0.010663061402738094,
0.014438608661293983,
0.... | |
90ddf8ee49245c285d8bfa84893dfbbfa2f8289c | subsection | 75 | 117 | Supporting Lemmas | In other
words,\bar{g}^{n}_{ Y|X }\left( \mathbf {X}_{u},1-\beta \right)\le g^{n}_{ Y|X }\left( \mathcal {A}_u,1-\beta \right) \le 2^c,which follows from the assumption that
U \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {X} \operatorname{ \begin{}(1,1) \put (0,.2... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.00897471234202385,
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0.02... | |
66a18e883026f86b372a8ed95e04364ea3d633cc | subsection | 76 | 117 | Supporting lemmas | The first lemma repeatedly applies Theorem REF in order to
construct a DRV V which provides stability when conditioned
upon. That is, we apply Theorem REF to \mathcal {A} obtaining
a subset which gives stability. This subset is then removed from
\mathcal {A}, and Theorem REF is applied to the remaining set
again to obt... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.01846432499587536,
0.028428956866264343,
-0.01908997632563114,
0.011330381967127323,
-0.014939318411052227,
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0.02746759168803692,
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-0.005913162138313055,
-0.0337851382791996,
-0.014557823538780212,
-0.04052995890378952,
0.... | |
1298b902345981d955fa659c24626d923949bf25 | subsection | 77 | 117 | Supporting lemmas | The random variable V is then an index to the stable
subset that \mathbf {X} belongs.Lemma 27
Given any regular collection (\emptyset , \mathbf {X},\mathbf {Y}), positive
real number
\alpha \in \left( \frac{\log _{2}n}{n}, \frac{1}{8 \ln 2} \right), and
\zeta \in \mathbb {N}_+, there exists:[leftmargin=*]
a DRV V : \l... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05505209043622017,
-0.023482585325837135,
-0.04577501863241196,
0.02317741885781288,
-0.0030898137483745813,
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0.004051089286804199,
0.016967274248600006,
0.03469746559858322,
-0.0027045407332479954,
-0.020156266167759895,
0.036681048572063446,
-0.02206355892121792,
... | |
f97949ba19570e15da97940b8176ad31763a6b4e | subsection | 78 | 117 | Supporting lemmas | Now for each i \in [2:\zeta -1], given the
recursively defined
\mathbf {X}_i \triangleq \mathbf {X} | \lbrace \mathbf {X} \notin \bigcup _{j=1}^{i-1}
\mathcal {A}_j^\dagger \rbrace there exists a
\mathcal {A}_i^\dagger \subseteq \mathcal {A}\setminus \bigcup _{j=1}^{i-1}
\mathcal {A}_j^\dagger such thatp_{\mathbf {X}_i... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04119846597313881,
-0.011367724277079105,
-0.050719887018203735,
-0.020263541489839554,
-0.01158134639263153,
0.04248019680380821,
0.013595493510365486,
0.024642786011099815,
-0.00113295775372535,
0.0023383942898362875,
-0.0374753437936306,
0.021575788035988808,
-0.007064773701131344,
0... | |
eb2b76fe98b73ee6c242d7920bd0a613646fff7f | subsection | 79 | 117 | Supporting lemmas | Hence,p_{V}(0)
&=
\Pr \left( \mathbf {X} \in \mathcal {A} \setminus \bigcup _{i=1}^{\zeta -1}
\mathcal {A}^\dagger _i \right) \\
&=
\prod _{j=1}^{\zeta -1} \Pr \left(\mathbf {X}\in \mathcal {A} \setminus \bigcup _{i=1}^{j} \mathcal {A}^\dagger _i | \mathbf {X} \in \mathcal {A}
\setminus \bigcup _{i=1}^{j-1} \mathcal {A... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04681657254695892,
0.003929357044398785,
-0.054995737969875336,
0.0038778556045144796,
-0.000591311021707952,
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0.0016184754204005003,
0.026200799271464348,
-0.024522239342331886,
0.00852250773459673,
-0.007641264237463474,
... | |
d8505d86ebd7513a266ae4c0a2c91f5813d4dfbe | subsection | 80 | 117 | Supporting lemmas | Hence it also
follows that3 \cdot 2^{-n\alpha }& > \Pr \left( \left| h(\mathbf {Y}|U) - r_i \right| > n \delta _i + h(U) | U=u \right) \\
&\ge \Pr \left( \left| h(\mathbf {Y}|U) - r(V) \right| > n \delta + h(U) | U=u \right),establishing (REF ).Notice that Lemma REF only applies to regular collections
of the form (\emp... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04461587592959404,
-0.01037578471004963,
-0.026595577597618103,
0.008834675885736942,
-0.01571626216173172,
0.017593057826161385,
-0.008270110934972763,
0.021636562421917915,
0.027892550453543663,
-0.00012159122707089409,
-0.03298889100551605,
0.056975264102220535,
-0.0011338978074491024,... | |
a9b3c49eb38d0cebb441dd793d7e923106582671 | subsection | 81 | 117 | Supporting lemmas | This generalization is achieved by
leveraging Lemma REF with the fact that
(M,\mathbf {X},\mathbf {Y})|\lbrace M = m\rbrace is a regular collection for all
m \in \mathcal {M}.Corollary 28
Given any regular collection (M, \mathbf {X},\mathbf {Y}), positive real
number
\alpha \in \left( \frac{\log _{2}n}{n}, \frac{1}{8 ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0366608127951622,
-0.022939717397093773,
-0.0424606092274189,
-0.006349249742925167,
0.004628389608114958,
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0.008852319791913033,
0.04777200147509575,
-0.005696773063391447,
-0.035348229110240936,
0.017613062635064125,
-0.020879264920949936,
0... | |
2065481821e6dcdb0976be254295c94ff1f9dfd6 | subsection | 82 | 117 | Supporting lemmas | For
each m \in \mathcal {M}, let (\mathbf {X}_{(m)},\mathbf {Y}_{(m)}) be DRVs
defined by setting their distributions according top_{\mathbf {X}_{(m)},\mathbf {Y}_{(m)}}(\mathbf {x},\mathbf {y}) = p_{\mathbf {X},\mathbf {Y}|M}(\mathbf {x},\mathbf {y}|m).It is clear then that the set
(\emptyset , \mathbf {X}_{(m)},\math... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0032103597186505795,
-0.04397983103990555,
-0.026522187516093254,
0.013367915526032448,
-0.03399967402219772,
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0.03702118992805481,
0.0033992044627666473,
-0.013810460455715656,
-0.0005226610810495913,
-0.012490455061197281... | |
f36b74f1d6520f15cdc6ab5769f23dfbae1a4ae2 | subsection | 83 | 117 | Supporting lemmas | Thus for each m \in \mathcal {M}, there exists:[leftmargin=*]
a DRV
V_{(m)} : \left\lbrace \begin{array}{c} \mathcal {V}_{(m)} = [0:\zeta -1] \\
V_{(m)} \ll \mathbf {X}_{(m)} \\ h_{V_{(m)}}(0) > \frac{\zeta -1}{n}
\log _{2}\frac{n}{8} \end{array} \right\rbrace ,
a positive real number \delta _{(m)} = O(-\sqrt{\alpha ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.00931540783494711,
-0.0275876447558403,
-0.026351694017648697,
-0.012634164653718472,
-0.00242612580768764,
0.04382285103201866,
0.002435662318021059,
0.03564421087503433,
0.04235801845788956,
-0.0004234276129864156,
-0.014007443562150002,
0.02996799349784851,
-0.011917008087038994,
0.0... | |
7929d39f829164ce2434ca823a283126a1d4b478 | subsection | 84 | 117 | Supporting lemmas | Next,
that V is a deterministic function of (\mathbf {X},M) follows
from (REF ) and V_{(m)} being a deterministic
function of \mathbf {X}_{(m)} for each m \in \mathcal {M}. Finally,p_{V}(0) &= \sum _{m \in \mathcal {M}} p_{M}(m) p_{V|M}(0|m) = \sum _{m \in \mathcal {M}} p_{M}(m) p_{V_{(m)}}(0)
\\
&< \sum _{m \in \mathc... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.003521721577271819,
-0.013735858723521233,
-0.036720529198646545,
-0.011354976333677769,
-0.02831112965941429,
0.03986451402306557,
-0.0018295400077477098,
0.025686055421829224,
0.010225583799183369,
0.01930651254951954,
0.0016988585703074932,
-0.00020079154637642205,
0.005700381007045507... | |
6dda8ba250246bec76df7dd2766f5634f55ded68 | subsection | 85 | 117 | Supporting lemmas | Notice that U_{(m)} \gg V_{(m)} and
U_{(m)} \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {X}_{(m)} \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {Y}_{(m)} and
p_{V_{(m)}|U_{(m)}}(0|u) = 0 clearly follow
from (... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03470909595489502,
-0.005777217913419008,
-0.07143297046422958,
-0.02790159545838833,
0.01002045813947916,
0.0356249064207077,
-0.03757862746715546,
0.01782771572470665,
0.003008808707818389,
-0.002982097677886486,
-0.0387997031211853,
0.01469870749861002,
0.014927659183740616,
0.011501... | |
adc669f339791af6679705a439fdd6d3d8511a5e | subsection | 86 | 117 | Supporting lemmas | Hence it follows that&\Pr \left( |h(\mathbf {Y}|U,M) \!\! -\!\! r(V,M)| \!\! >\!\! n \delta + h(U|M) | (U,M) = (u,m) \right) \\
&=\sum _{{\mathbf {y},v: \\ | h_{\mathbf {Y}|U,M} (\mathbf {y}|u,m) - r(v,m)| \\
~~~\ge n \delta + h_{U|M}(u|m)}} p_{\mathbf {Y},V|U,M}(\mathbf {y},v|u,m) \\
&=\sum _{{\mathbf {y},v: \\ | h_{\... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04000413790345192,
-0.01867470145225525,
-0.018201731145381927,
-0.03481672331690788,
-0.00001126403731177561,
0.012236201204359531,
-0.024777544662356377,
0.027539080008864403,
0.0330468975007534,
0.008177810348570347,
-0.035976260900497437,
0.006617770995944738,
0.0021779523231089115,
... | |
bde7b036c1ddfb2f11c455ecb20d8f34503ecbd0 | subsection | 87 | 117 | Supporting lemmas | Now define the
following:DRV
\hat{V}= \left\lfloor \tilde{r}(\tilde{V},M) \right\rfloor ,
DRV
V = {\left\lbrace \begin{array}{ll}(\hat{V},\tilde{V}) &\text{ if } \tilde{V} \ne 0 \\ v_0 &\text{ o.w.} \end{array}\right.},
constant
\delta = \tilde{\delta }+ \alpha + \frac{1}{n} = O(-\sqrt{\alpha }
\log _{2}\alpha )
(... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.005972701590508223,
-0.04116053506731987,
-0.012959923595190048,
0.018734298646450043,
-0.01972593367099762,
0.026591109111905098,
-0.019802214577794075,
0.03133570775389671,
0.016430649906396866,
0.01498896349221468,
0.007570761721581221,
-0.014477889984846115,
-0.018413923680782318,
0... | |
f2607979063fff65dbf0f1f38fe1fed6566cc20b | subsection | 88 | 117 | Supporting lemmas | Finally,
p_{V}(v_0) = p_{\tilde{V}}(0) < 2^{-\frac{\zeta -1}{n} \log _{2}\frac{n}{8}}.Now, to prove (REF ) fix any DRV
U : \left\lbrace \begin{array}{c} U \gg V \\ (U,M) \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {X} \operatorname{ \begin{}(1,1) \put (0,.22){\li... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.032067906111478806,
-0.020946355536580086,
-0.012494584545493126,
-0.02942863665521145,
-0.01478296983987093,
0.05495176091790199,
0.030664363875985146,
0.028375979512929916,
0.021190449595451355,
0.03188483789563179,
-0.03176278993487358,
0.022960133850574493,
-0.008367862552404404,
0.... | |
06aa97d2d2a73b34fd2e658baad10bd9d351ebf0 | subsection | 89 | 117 | Supporting lemmas | \cdot \!2^{-n\alpha } ,\\
\Pr \left( h(U|M) > h(U) + n\alpha | U = u \right) &\le 2^{-n\alpha } , \\
\Pr \left( |r(V) - \tilde{r}(\tilde{V},M)| > 1 | U = u \right) &=0 .We finish the proof by
confirming (REF )–(). First,
(REF ) is a property directly obtained
from (REF ) of Corollary REF . | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05536884069442749,
0.0053071980364620686,
0.011148549616336823,
-0.011911625973880291,
-0.0038249215576797724,
0.01967211626470089,
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0.0471581369638443,
0.014292425476014614,
0.013155440799891949,
-0.038611676543951035,
0.02357906848192215,
-0.003155321814119816,
0... | |
87feb3869e69275093a51eff8730a5f4ab03eb49 | subsection | 90 | 117 | Supporting lemmas | Next,
() can be derived as follows:&\hspace{-15.0pt}\Pr \left( h(U|M) > h(U) + n\alpha | U =u \right) \\
&= \sum _{m: p(u|m) < 2^{-n\alpha } p(u) } p(m|u) \\
&= \sum _{m: p(u|m) < 2^{-n\alpha } p(u) } \frac{p(u|m) p(m)}{p(u)} \\
&< \sum _{m} 2^{-n\alpha } p(m) = 2^{-n\alpha }.Finally, for proving (),| r(V) - \tilde{r}(... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
0.0008475161157548428,
0.011713559739291668,
-0.018863791599869728,
-0.018818005919456482,
-0.011232808232307434,
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0.0337594710290432,
0.03513304889202118,
0.02371709793806076,
-0.010958092287182808,
0.021900923922657967,
-0.01616242341697216,
0.0... | |
04ab3eab3b21c7d25da84a6c1f4f063140b960ec | subsection | 91 | 117 | Supporting lemmas | \operatornamewithlimits{arg\,max}_{{ (\hat{w},\hat{p}): \\ \hat{p} \in \mathcal {P}_n(\mathcal {X})
\\ \hat{w} \in \mathcal {P}_n(\mathcal {Y}|\hat{p}) }} \zeta (\hat{w},\hat{p},\mathbf {y}) 2^{-n \mathbb {D}(\hat{w}||w|\hat{p})}.We will prove\log _{2}\frac{w_n(\mathbf {y}|u)}{\tilde{w}_n(\mathbf {y}|u)} \!
&\le \!
n \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1561/9781933019543",
"end": 1284,
"openalex_id": "https://openalex.org/W2055309977",
"raw": "I. Csiszár, P. C. Shields, et al., “Information theory and statistics: A tutorial,” Foundations and Trends® in Communications and Information Th... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
0.0006079397862777114,
0.006683045998215675,
-0.02461130917072296,
-0.0032385308295488358,
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0.023695822805166245,
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0.022673530504107475,
0.02030852623283863,
0.009429503232240677,
0.02085781656205654,
-0.01812661811709404,
... | |
4261b73ffcc317782fe538d0a8703c4518dd5f5d | subsection | 92 | 117 | Supporting lemmas | Further notice
that (REF ) and () implyw_n(\mathbf {y}|u)
\le |\mathcal {P}_n(\mathcal {Y},\mathcal {X})| \zeta (\hat{w}^{(\mathbf {y})},\hat{p}^{(\mathbf {y})},\mathbf {y}) 2^{-n \mathbb {D}(\hat{w}^{(\mathbf {y})}||w|\hat{p}^{(\mathbf {y})})}and\tilde{w}_n(\mathbf {y}|u)
\ge \zeta (\hat{w}^{(\mathbf {y})},\hat{p}^{(\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 631,
"openalex_id": "https://openalex.org/W1549664537",
"raw": "I. Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems. Cambridge University Press, 2nd ed., 2011.",
"source_ref_id": "d9... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02473948709666729,
0.005776616744697094,
-0.00997363030910492,
-0.011087746359407902,
-0.006860209628939629,
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0.0552174374461174,
-0.005784247536212206,
0.027593456208705902,
-0.008905299939215183,
... | |
c4b3604103ba8f6d155485fcf1935a906725d2de | subsection | 93 | 117 | Supporting lemmas | 2^{-n(\alpha -\epsilon )}) \left[
\log _{2}|\mathcal {Y}| \!- \!\frac{2}{n}\log _{2}( 2^{-n\epsilon }
\!+\!2^{-n(\alpha - \epsilon )}) \right],for all w \in \mathcal {P}(\mathcal {Y}|\mathcal {X}) such that\sup _{{\hat{p} \in \mathcal {P}(\mathcal {X}) \\ \hat{w} \in \mathcal {P}(\mathcal {Y}|\mathcal {X})}} \mathbb {D... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0029691956005990505,
-0.0023456073831766844,
-0.002162535674870014,
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0.01297521311789751,
0.010732583701610565,
0.0022616994101554155,
0.04149627313017845,
-0.03643128648400307,
... | |
c0b1a66db1a1f526c6e2c5d8f4646dcbe9fd169b | subsection | 94 | 117 | Supporting lemmas | Because of (REF ), it must also
follow that|h_{\mathbf {Y}_{w}|M}(\mathbf {y}|m) - \mathbb {H}(\mathbf {Y}_{\tilde{w}}|M) | \le n\delta + n \epsilon ,for all \mathbf {y} \notin \mathcal {B}^*(m) \cup \mathcal {B}^-(m), where\mathcal {B}^*(m) &\triangleq \left\lbrace \mathbf {y} : |h_{\mathbf {Y}_{\tilde{w}}|M}(\mathbf ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04979430511593819,
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0.01638452336192131,
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0.0019565317779779434,
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0.024256417527794838,
0.03887128829956055,
0.03810850903391838,
-0.04857385531067848,
0.03508789837360382,
0.014218227006494999,
0.0... | |
ada004eaab337c2158390f691cade623c0b8e779 | subsection | 95 | 117 | Supporting lemmas | First,
for (REF ),& \hspace{-10.0pt} \Pr \left(\mathbf {Y}_{w} \in \mathcal {B}^*(M) \right) \\
&= \sum _{m} p_{M}(m) \sum _{\mathbf {y}\in \mathcal {B}^*(m) } w_n(\mathbf {y}|m)
\\
&\le \sum _{m} p_{M}(m) \sum _{\mathbf {y}\in \mathcal {B}^*(m) } \tilde{w}_n(\mathbf {y}|m) 2^{n\epsilon }
\\
&\le 2^{n \epsilon } \Pr \... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02477249875664711,
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0.06367890536785126,
-0.026314102113246918,
0... | |
5f7903938fd31127ba9f85d72c0bdda1c29e6332 | subsection | 96 | 117 | Supporting Lemma | We streamline the arguments of the proof by first considering the
following lemma:Lemma 30
For any DRVs U, V, and \alpha \in \mathbb {R}_+,\Pr \left( \left| h(V|U) - h(V) \right| > \alpha \right) <
(|\mathcal {U}|+1)2^{-\alpha } .By the union bound, it is clear that\Pr \left(\left| h_{V|U}(V|U) - h(V) \right| > \alpha... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.03409361466765404,
0.012178470380604267,
0.... | |
d041a36b38fa3deb7a69519f3b8526fab8925854 | subsection | 97 | 117 | Concluding remarks | Our contribution is simply the construction of a DRV that provides information stability, and the examples demonstrating how such a result can be applied. This work is a self contained collection and refinement of our previous works , , , . Perhaps more accurately, our current work is to our past work as Theseus' final... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/isit.2014.6875052",
"end": 240,
"openalex_id": "https://openalex.org/W2963878611",
"raw": "E. Graves and T. F. Wong, “Equating the achievable exponent region to the achievable entropy region by partitioning the source,” in Proc. 201... | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.026515213772654533,
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-0.03960500285029411,
0.... | |
9b5a62884c4450ff62f7fd5306f179a2bbdb69e4 | subsection | 98 | 117 | Proof of Lemma | Let
\mathcal {B}
= \left\lbrace (y,u) \in \mathcal {Y} \times \mathcal {U} : |h_{Y|U }(y|u) - c| >
\epsilon \right\rbrace
and Q = \mathfrak {1}\left({(Y,U) \in \mathcal {B}}\right) . From the hypothesis of the lemma,
we have p_Q(1) < \mu < \frac{1}{2}. The conditional entropy
\mathbb {H} (Y|U) can always be expanded i... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.02517237700521946,
-0.... | |
7bd1b542f91616eaed9e1b49c878d8ef635fc3fd | subsection | 99 | 117 | Proof of Lemma | On the other hand,0 \le \zeta _{Q=1}
&= p_{Q}(1) \mathbb {H}(Y |U,Q=1 )
\\
& \hspace*{10.0pt} -
\sum _{u} p_{Q|U}(1|u)p_{U}(u)
\log _{2}p_{Q|U}(1|u) \\
&\le p_{Q}(1) \mathbb {H}(Y |U,Q=1 ) + \mathbb {H}(Q|U) \\
&\le \mu \log _{2}\frac{|\mathcal {Y}|}{\mu ^2}where (REF ) results from the bound
\mathbb {H}(Y|U,Q=1 ) \le ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.018656987696886063,
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0.014240636490285397,
-0.001594158005900681,
0... | |
fcbec1350f096e41d89f5e76621be7f85f356445 | subsection | 100 | 117 | Proof of Corollary | Let us be given a regular collection (M_{[1:l]},\mathbf {X},\mathbf {Y}_{\mathcal {P}(\mathcal {Y}|\mathcal {X})}) and DRV T: \left\lbrace (T,M_{[1:l]}) \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put (.5,.22){\circle {.3}} \end{}}\mathbf {X} \operatorname{ \begin{}(1,1) \put (0,.22){\line (1,0){1}} \put... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.02781299687922001,
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-0.0015628599794581532,
-0.013265715911984444,
0... | |
4c7c51294197d700084d1c768827e0c5a725ec48 | subsection | 101 | 117 | Proof of Corollary | By Theorem REF there exists a set \tilde{P}\subseteq \mathcal {P}(\mathcal {Y}|\mathcal {X}), where|\mathcal {\tilde{P}}| \le \tilde{k} \triangleq \left( |\mathcal {Y}|\left( 1 + \left\lfloor \frac{4|\mathcal {Y}|^2}{\varepsilon _n} \right\rfloor \right) \right)^{|\mathcal {X}||\mathcal {Y}|}which has the following pro... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.04346207156777382,
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0.023333393037319183,
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0.02296713925898075,
-0.014879044145345688,... | |
611f1609e5758b70c01f3a210066d6e135a80959 | subsection | 102 | 117 | Proof of Corollary | 2^{-n(\hat{\alpha }- \varepsilon _n)}) ).Now for (M_{[1:l]},\mathbf {X},\mathbf {Y}_{\tilde{P}}), there exists:[leftmargin=*]
a DRV V : \left\lbrace \begin{array}{c} |\mathcal {V}| \le (2n^3\log _{2}|\mathcal {Y}| )^{l \tilde{k}} \\ V \ll (\mathbf {X},M_{[1:l]}) \end{array} \right\rbrace ,
a real number \tilde{\delta... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.007520289625972509,
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0.019703082740306854,
-0.029394373297691345,
0.... | |
1c31340d90fbf2dd8b3cc04d55f2cec18f5cd7da | subsection | 103 | 117 | Proof of Corollary | The existence of a \delta = O(-\sqrt{\varepsilon _n} \log _{2}\varepsilon _n) so that\Pr \left( |h(\mathbf {Y}_w|M_j,U) - \mathbb {H}_{U}(\mathbf {Y}_w|M_j)| > n\delta + 7 h(U) | U=u\right) &\\
&\hspace{-100.0pt}< 5\cdot 2^{-n\varepsilon _n}for each w \in \mathcal {P}(\mathcal {Y}|\mathcal {X}) and u \in \mathcal {U}_{... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0699264407157898,
0.004225485026836395,
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0.0005038738599978387,
-0.00663568964228034,
0.02372068352997303,
-0.0036019592080265284,
... | |
a84fb282736a158d8900ad30d45eef96b2783370 | subsection | 104 | 117 | Proof of Corollary | First, U \gg T is a direct consequence of the definition of U. Second,\log _{2}|\mathcal {U}| &= \log _{2}|\mathcal {V}||\mathcal {Q}||\mathcal {T}| \\&\le \log _{2}|\mathcal {T}| + 3l\tilde{k} \log _{2}n \\
&\hspace{10.0pt} + l \log _{2}(n^2+1) + (l\tilde{k}) \log _{2}(2 \log _{2}|\mathcal {Y}|) \\
&\le \log _{2}|\mat... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.026382949203252792,
0.03323427587747574,
-0.05737414211034775,
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0.008110200986266136,
0.00439080037176609,
-0.005954856052994728,
0.013710283674299717,
-0.0031757960096001625,
... | |
b1e7abac33f3038eed53f1436fd769cf3f8b9e20 | subsection | 105 | 117 | Proof of Corollary | For this reason let \mathcal {\check{U}} be the set of all u \in \mathcal {U} such thath(u)< n\varepsilon _n + \log _{2}|\mathcal {U}|and for each j \in [1:l] let \mathcal {\hat{U}}_j be the set of all u \in \mathcal {U} such that\Pr \left( |h(M_j|U) - h(M_j)| > 2 n\varepsilon _n + \log _{2}(|\mathcal {U}|+1) | U = u \... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.034068528562784195,
0.015027090907096863,
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0.0000235513780353358,
-0.002629168564453721,
0.02962679974734783,
-0.02352132834494114,... | |
0521655349db5ab5433290835cb56f57469b529b | subsection | 106 | 117 | Proof of Corollary | \begin{array}{l r l}
&|h(\mathbf {Y}_w|M_j,U) - \mathbb {H}_{U}(\mathbf {Y}_w|M_j)| &> n \nu _n \\
\text{or } &|h(M_j|U) - \mathbb {H}_{U}(M_j)| &> n \nu _n \\
\text{or } & |h(M_j) - h(M_j|U)| &> n \nu _n \\
\end{array} | U = u \right) \\
&\hspace{150.0pt} < 8\cdot 2^{-n\varepsilon _n} ,where\nu _n &= \max \left( \delt... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.031901177018880844,
0.012752839364111423,
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0.03815930336713791,
-0.0047928085550665855,
0... | |
13479a2be4d799a0a7b5ac1353a4d04f023f0a60 | subsection | 107 | 117 | Proof of Corollary | Furthermore\nu _n = O(n^{-1} \log _{2}|\mathcal {T}| -l \sqrt{\varepsilon _n} \log _{2}\varepsilon _n)as detailed in Appendix REF , and\Pr \left( (\mathbf {Y}_w,M_{[1:l]}) \notin \mathcal {D}_{\scriptscriptstyle {(\text{stable})},(M_j)}(U,w; \nu _n) | U = u \right) &< 8 \cdot 2^{-n\varepsilon _n}in the first case, whil... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03790571168065071,
0.03409072384238243,
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-0.008797360584139824,
0.0... | |
594c55245e7ac3e72b7f340c066c067f1fafd07b | subsection | 108 | 117 | Order terms | Before deriving the order terms explicitly, a few inequalities useful inequalities need to be derived for \alpha \in \left(\frac{\log _{2}n}{n},\frac{1}{8 \ln 2}\right), and n\ge 27. | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02134593203663826,
0.019743843004107475,
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-0.026091167703270912,
-0.057797275483608246,
... | |
e8e65db8e1ecb7f61ab1f301491ebd68a3bedd23 | subsection | 109 | 117 | Order terms | Letting \alpha _- = \frac{\log _{2}n}{n} and \alpha _+ = \frac{1}{8 \ln 2} these inequalities are as follows:2^{-n\alpha } &\le 2^{-n \alpha _-} \le \frac{1}{n} \\
\frac{1}{n} &\le -\sqrt{\alpha } \log _{2}(\alpha ) \cdot \frac{1}{-n\sqrt{\alpha _-} \log _{2}(\alpha _- )} \\
&\le -\sqrt{\alpha } \log _{2}\alpha \frac{1... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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0.021047774702310562,
-0.05864071473479271... | |
1b0a663a2ebc0a4dfebb76ff5b40c80ebd87da75 | subsection | 110 | 117 | Equation ( | In specific the combination of Equations (REF ) and () provides\Pr \left( |h(\mathbf {Y}|U) - s^* \lambda _n | < n \tilde{\delta }+ \lambda + h(U) | u \right) < 3\cdot 2^{-n\alpha } .Hence we need to show that \tilde{\delta }+ n^{-1}\lambda = O(-\sqrt{\alpha } \log _{2}\alpha ). Towards this goal, let B be a Bernoulli ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.05332469940185547,
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0.05857475474476814,
-0.01355247013270855,
... | |
bdf61d917f8e12bd87d545444fa5bf12f2128da3 | subsection | 111 | 117 | Equation ( | Combining Equations (REF ) and (REF ) with Equations ()–() gives\tilde{\delta }+ n^{-1}\lambda &\le - \mu \sqrt{\alpha } \log _{2}\alphawhere\mu &\triangleq \sqrt{2 \ln 2} + \frac{\sqrt{2 \ln 2} \log _{2}|\mathcal {Y}|}{\log _{2}( 8 \ln 2)} \\
&\hspace{20.0pt} + \frac{1 }{2\sqrt{2 \ln 2} \log _{2}(8 \ln 2)} + \frac{11.... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.028506778180599213,
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0.027972659096121788,
-0.033359646797180176,
0.04874232038855553,
-0.011346247047185898,
... | |
b2c3e25073089239985894d4dfdee6156f816adf | subsection | 112 | 117 | Equation ( | To finish the proof note that \delta = O(-\sqrt{\alpha } \log _{2}\alpha ) since \delta _{i,j} = O(-\sqrt{\alpha } \log _{2}\alpha ) for all i,j, and \frac{1}{n} = O(-\sqrt{\alpha } \log _{2}\alpha ) by Equation ().Given Equation () ,\Pr \left( |\mathbb {H}(M_j|U) - h(M_j|U) | > \tilde{\beta }| U = u \right) < 2^{-\tau... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.03350337594747543,
0.028407689183950424,
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0.025585228577256203,
0.018429908901453018,
0.004176479764282703,
-0.01368512213230133,
0.02192365750670433,
-0.007307885680347681,
0... | |
73bde013b34f6cdd08cb02414659ce7d5c86eaa9 | subsection | 113 | 117 | Equation ( | But \rho \ge 1 and |\mathcal {U}| \ge 1, and hence\tilde{\beta }< 8 \rho + 3\log _{2}|\mathcal {U}| + 6 + 2^{-\rho } \psi &\le 14 \rho + 2^{-\rho } \psi + 3 \log _{2}|\mathcal {U}|.Thus\Pr \left( |\mathbb {H}(M_j|U) - h(M_j|U) | > \beta + 3\log _{2}|\mathcal {U}| | U = u \right) < 2^{-\rho }where \beta = 14 \rho + 2^{-... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
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fc39a7c70dfaaf8c94d41e4a825fd4f02d076ab6 | subsection | 114 | 117 | Equation ( | Thus\Pr \left( |h(\mathbf {Y}_{w}|M_j,U) - \mathbb {H}_{U}(\mathbf {Y}_{w}|M_j)| > n \delta + 7 \log _{2}|\mathcal {U}| | U =u\right) &\\
&\hspace{-80.0pt} \le 5\cdot 2^{-n\varepsilon _n},for\delta = \frac{7}{3} \tilde{\delta }+ 3 \varepsilon _n + 5 \cdot 2^{-n\varepsilon _n} \log _{2}|\mathcal {Y}|.Furthermore \delta ... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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ca354736a0d1571004004956bc0013901ecc9bb9 | subsection | 115 | 117 | Equation ( | If we show that each of these terms is O(n^{-1} \log _{2}|\mathcal {T}| -l \sqrt{\varepsilon _n}\log _{2}\varepsilon _n) then it must also follow that \nu _n = O(n^{-1} \log _{2}|\mathcal {T}| -l \sqrt{\varepsilon _n}\log _{2}\varepsilon _n). | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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5162227845da34b5d366d38a4000ed5c0c6bbe10 | subsection | 116 | 117 | Equation ( | First&\delta + 7 \varepsilon _n + 7 \frac{\log _{2}|\mathcal {U}|}{n} \\
&= O(-\sqrt{\varepsilon _n} \log _{2}\varepsilon _n) + O( n^{-1} \log _{2}|\mathcal {T}| - l \varepsilon _n \log _{2}\varepsilon _n) \\
&\le O(n^{-1} \log _{2}|\mathcal {T}| - l \sqrt{\varepsilon _n} \log _{2}\varepsilon _n)by Equations () and (),... | {
"cite_spans": []
} | 1806.05589 | Inducing information stability and applications thereof to obtaining
information theoretic necessary conditions directly from operational
requirements | [
"Eric Graves",
"Tan F. Wong"
] | [
"cs.IT",
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] | 2,018 | en | Computer Science | [
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281bac0527990baff3774fa6e22b2181e05229db | abstract | 0 | 18 | Abstract | Khovanov-Floer theories are a class of homological link invariants which
admit spectral sequences from Khovanov homology. They include Khovanov
homology, Szab{\'o}'s geometric link homology, singular instanton homology, and
various Floer theories applied to branched double covers. In this short note we
show that certai... | {
"cite_spans": []
} | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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084bb7c1a0d4b5137a33239b5f2a35c0c08c915e | subsection | 1 | 18 | Introduction | Let L \subset S^3 be a link. Suppose that there is a sphere in S^3 which
intersects L transversely in four points, splitting the link into two tangles,
each in a three-ball. Mutation is the operation of regluing the
three-balls by some orientation-preserving involution of the sphere. Mutation can also be
represented di... | {
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{
"arxiv_id": "",
"doi": "10.1007/bf01098418",
"end": 1206,
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"raw": "O. Ya. Viro. Nonprojecting isotopies and knots with homeomorphic coverings. Journal of Soviet Mathematics, 12(1):86–96, Jul 1979.",
"source_r... | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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3f66b4c8abfb93db9beeb27cce3d02836da00d0e | subsection | 2 | 18 | Introduction | The corollary was conjectured by Peter
Lambert-Cole in (and earlier by Seed, see below). He defined a particular class of
Khovanov-Floer theories, so-called extended theories, characterized by
the fact that their reduced versions satisfy a Künneth theorem for connected sums.Theorem (Lambert-Cole, )
Extended Khovanov-... | {
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{
"arxiv_id": "",
"doi": "10.1016/j.aim.2019.106734",
"end": 89,
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"raw": "P. Lambert-Cole. On Conway mutation and link homology. ArXiv e-prints, January 2017.",
"source_ref_id": "75f0496fdcaf11c252a5a0f68faa7d5... | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
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756d93c2c3638b29ac2b54576698afc632463738 | subsection | 3 | 18 | The structure of Szabó homology | Szabó homology is a link homology theory which interpolates between the
combinatorial and analytic Khovanov-Floer theories. To a link diagram
\operatorname{\mathcal {D}} it assigns a filtered chain complex \operatorname{CSz}(\operatorname{\mathcal {D}}) whose underlying vector
space is identical to the Khovanov chain g... | {
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{
"arxiv_id": "",
"doi": "10.1112/jtopol/jtv027",
"end": 1246,
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"raw": "Zoltán Szabó. A geometric spectral sequence in Khovanov homology. Journal of Topology, 8(4):1017 – 1044, 2015.",
"source_ref_id": "adfdb1a... | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
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025d25bf00f24225e05fad4c9819ec43f02c2027 | subsection | 4 | 18 | The structure of Szabó homology | Szabó homology is isomorphic to mirror Szabó homology.We prove the first and second of Seed's conjectures in the course of proving Theorem
REF , and of course the third is the main subject of this
paper.The first conjectured property is part of Lambert-Cole's definition
of an extended Khovanov-Floer theory and therefor... | {
"cite_spans": []
} | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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e3b97385b3ed31e80328f8f22d40f73fc804f6ec | subsection | 5 | 18 | Strong Khovanov-Floer theories | The following definition is from , following .Definition 2.1
A strong Khovanov-Floer theory is a rule which assigns to a link
diagram \operatorname{\mathcal {D}} and some auxiliary data A a filtered chain complex
\mathcal {K}(\operatorname{\mathcal {D}},A) so thatFor any two collections of auxiliary data A_\alpha , A_... | {
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"source_ref_id": "c1958967f2a6822b4869e69f0bd09715a2ab9066... | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
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57397d9cb24a9ca24ca466719be82c59e405936c | subsection | 6 | 18 | Strong Khovanov-Floer theories | If \operatorname{\mathcal {D}}^{\prime } is obtained from \operatorname{\mathcal {D}} by a planar isotopy,
then \mathcal {K}(\operatorname{\mathcal {D}}) is filtered chain homotopy equivalent to
\mathcal {K}(\operatorname{\mathcal {D}}^{\prime }).
Let \operatorname{\mathcal {D}} be the disjoint union of two diagrams \... | {
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"source_ref_id": "c1958967f2a6822b4869e69f0bd09715a2ab90... | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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65ab5b9a1c64a622ea4a3c954d27239646d513d0 | subsection | 7 | 18 | Strong Khovanov-Floer theories | \mathcal {K}(-) factors through \llbracket - \rrbracket as a functor.More precisely, recall that \llbracket - \rrbracket is a functor from a category of diagrams and cobordisms to the category \operatorname{{Mat}(\operatorname{{Cob}})}. There is a
functor F_\mathcal {K} from a certain subcategory of \operatorname{{Mat}... | {
"cite_spans": []
} | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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390456bbddc3e1f45f2463cb4820321f99bab604 | subsection | 8 | 18 | The basepoint action and dotted cobordisms | By a basepoint of a link diagram \operatorname{\mathcal {D}} we mean a point which is not a double point.Definition 3.1
Let \mathcal {K} be a strong Khovanov-Floer theory over the Frobenius algebra R = \mathbb {F}[X]/(r(X)) for some polynomial r. Let U be a crossingless diagram of the unknot. Fix a chain homotopy equi... | {
"cite_spans": []
} | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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6701d3e01769f58ae128420980dea6f2b233c4e1 | subsection | 9 | 18 | The basepoint action and dotted cobordisms | Therefore r(\chi _p) is null-homotopic on \mathcal {K}(\circ ).Observe that \chi _p \circ \chi _p is chain homotopic to the following composition:\mathcal {K}(\operatorname{\mathcal {D}}) &\mathrel {\mathop {\longrightarrow }_{Z}} \mathcal {K}(\operatorname{\mathcal {D}}) \otimes \mathcal {K}(U) \mathrel {\mathop {\lon... | {
"cite_spans": []
} | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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c0252b369b0d3021c930ce9e8c57f9b6e9a55116 | subsection | 10 | 18 | Dotted cobordisms | Over \mathbb {F}, the map \operatorname{Id}\otimes X in Definition REF does
not have an obvious cobordism-theoretic interpretation.Over a ring
without 2-torsion, multiplication by X can be represented as attaching a
tube. In , multiplication by X is represented by dots on
a cobordism. In this section we check carefully... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 286,
"openalex_id": "",
"raw": "Dror Bar-Natan. Khovanov's homology for tangles and cobordisms. Geometry & Topology, 9(3):1443–1499, 2005.",
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"start": 223
... | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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60340f57a5909c1812ee80e00652424114ad9d67 | subsection | 11 | 18 | Dotted cobordisms | So by the first point, \mathcal {K}(W_p)
\simeq \mathcal {K}(W_q).Suppose that p lies at the critical point of a zero-handle attachment
and that q is a generic dot which lies just next to it. Then \mathcal {K}(W_p)
\simeq \mathcal {K}(W_q) using Condition 8 of Definition REF . A similar argument applies to two-handle a... | {
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"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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29a365e4e0e50d4614005058871abe80e8bb7246 | subsection | 12 | 18 | Reduced theories | Definition 3.5
Let \mathcal {K} be a strong Khovanov-Floer theory over \mathbb {F}[X]/(X^2). The complex \mathrm {Im}(X_p) is called \mathcal {K} reduced at p or the reduced version of \mathcal {K}. We write often use the notation \widetilde{\mathcal {K}}_p or just \widetilde{\mathcal {K}}.Now we can prove Seed's firs... | {
"cite_spans": []
} | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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4a7f0e23a5760769c5335cf2e052e35f296b7c4c | subsection | 13 | 18 | Reduced theories | The 4Tu relation states thatN_{12} + N_{23} + N_{34} + N_{41} = 0,taking the cobordisms to be morphisms in the linear cobordism category \operatorname{{Mat}(\operatorname{{Cob}})}. Observe that N_{12} + N_{34} is a summand of N \circ \mathfrak {h}_{ k+1 }, that N_{41} is a summand of \mathfrak {h}_{ k+1 } \circ N, and ... | {
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"doi": "10.4064/fm225-1-16",
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"source_ref_id": "1320ad37383bddde411... | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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183389120919c88d7d66a03586e5d680746cb770 | subsection | 14 | 18 | Reduced theories | So we will show that \widetilde{\operatorname{CSz}}(\operatorname{\mathcal {D}},p) = \mathrm {Im}(X_p).The action of X_p on \operatorname{CSz}(\operatorname{\mathcal {D}}) can be writtenX_p = X_{p,0} + X_{p,1} + \cdots + X_{p,c}where C is the number of crossings in \operatorname{\mathcal {D}}. Here X_{p,k} is
homologic... | {
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"source_ref_id": "adfdb1ab... | 1806.05595 | Mutation-invariance of Khovanov-Floer theories | [
"Adam Saltz"
] | [
"math.GT"
] | 2,018 | en | Mathematics | [
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57dfd725742b5e66c89e7de312921fba2a4cb476 | subsection | 15 | 18 | Reduced theories | Proving
this statement for strong Khovanov-Floer theories would allow us to adapt
Wehrli's proof of mutation-invariance.Seed's second conjecture follows from the fact that the Leray spectral sequence
for a direct sum of complexes is isomorphic to the direct sum of the individual
spectral sequences and from the fact tha... | {
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16d5b4770a4fe581ebbe1e9a3fe5d62d31b9831d | subsection | 16 | 18 | Connected sums and mutation | Proposition 4.1
Let \operatorname{\mathcal {D}} and \operatorname{\mathcal {D}}^{\prime } be link diagrams and write \operatorname{\mathcal {D}}\# \operatorname{\mathcal {D}}^{\prime } for their connected sum. Let \mathcal {K} be a conic, strong Khovanov-Floer theory over \mathbb {F}[X]/(X^2). Then\widetilde{\mathcal ... | {
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7b8e7a590bd3147348c70714dfd44a03b7697db6 | subsection | 17 | 18 | Connected sums and mutation | \tilde{\mathcal {A}}(L,p) \cong \tilde{\mathcal {A}}(L,q).Theorem ()
Extended Khovanov-Floer theories are invariant under Conway mutation.Now we can prove Theorem REF .[Proof of Theorem REF ]
In we constructed a Khovanov-Floer theory for every strong Khovanov-Floer theory – use the Leray spectral sequence given by th... | {
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d83a1402dc69bab58ee42c81978e19f152ee2513 | abstract | 0 | 34 | Abstract | The current revolution in collaborating distributed things is seen as the
first phase of IoT to develop various services. Such collaboration is
threatened by the fragmentation found in the industry nowadays as it brings
challenges stemming from the difficulty to integrate diverse technologies in
system. Diverse network... | {
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} | 1808.07355 | Is Fragmentation a Threat to the Success of the Internet of Things? | [
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28801ab5adfb4a848ec461c21905a02e52dba0db | subsection | 1 | 34 | Introduction | Internet of Things (IoT) is a new technology paradigm that is gaining power due to the huge advancements in the electronic and wireless communication technologies fields . In the coming years, the IoT is expected to bridge diverse Internet collaborative technologies to enable new services and applications by connecting... | {
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4b0da1ddbc5d1c67dc8c324631638f12fc45ce05 | subsection | 2 | 34 | Background on the Internet of Things | There is no exact definition of IoT yet since it is still in the forming process and is subject to the perspectives taken , , . It was first introduced to the community as a “dynamic global network infrastructure with self-configuring capabilities based on standards and interoperable communication protocols”. In IoT, p... | {
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d1fbc8b05c12c6c356ac612a0ecad172c121bf80 | subsection | 3 | 34 | Background on the Internet of Things | Connected entities are now integrated to the Internet, allowing us to process and consume feeds of data collected from the field in real time, and interact with each other in virtual environments.In the industrial sectors nowadays, machines are equipped with sensors to help monitor their health and communicate importan... | {
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} | 1808.07355 | Is Fragmentation a Threat to the Success of the Internet of Things? | [
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9a188b13e7f1e91b458992acaf80cb3027a47331 | subsection | 4 | 34 | Architectures for the Internet of Things | The highly competitive nature of the IoT makes interoperability between different things a difficult task to achieve. A crucial need of an IoT ecosystem is that things found within the network must be interconnected to exchange information. System architecture should be able to guarantee the practical operations of the... | {
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afc9d91f5f7b79c1bd64cb19ad477bb385709174 | subsection | 5 | 34 | Prior and Related Works | Most popular technologies in IoT, e.g., sensor network technologies, have been the subject of other surveys. Contributions in regards to IoT protocol standardization has been summarized by Sheng et al. . However, this previous work focused only on some specific technologies (e.g., the IEEE 802.15.4 standard as well as ... | {
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edfc227261b09140ebe7187911ed3f1a8a3b38bf | subsection | 6 | 34 | Study Methodology | The following subsections present our proposed methodology to perform the Systematic Literature Review (SLR), and the outcomes of the SLR: | {
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e14df99a6d8a1976136f82e1e94af203489953a7 | subsection | 7 | 34 | Data Sources | Literature collection was done by making a comprehensive systematic search on the major indexing databases following the guidelines given by . We used ACM digital library, ScienceDirect, Springer, IEEE Xplore, Engineering Village, Web of Science and Google Scholar and did an electronically-based search considering the ... | {
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3c1ea9e22b9868ba56e08c3904c0b6dd242b1616 | subsection | 8 | 34 | Search and Selection Processes | Relevant studies from the aforementioned data sources are organized in three rounds as described in Figure 2.Round 1: We perform electronic search and we narrow our scope review to identify and categorize the preliminary studies related to our subject, i.e., integration and interoperability. Then, we read and select th... | {
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f2d68ee43c4f6ab195b4cf48fbbc38fd38c431ec | subsection | 9 | 34 | Quality of the Selected Papers | We apply various inclusion and exclusion criteria on the remaining set of studies that resulted from the second and third rounds. These selection criteria help to decide whether to include a paper for further investigation. Below are the criteria used in our SLR.-
Documents in form of abstracts, powerpoint presentation... | {
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} | 1808.07355 | Is Fragmentation a Threat to the Success of the Internet of Things? | [
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225564824547bc93656e900c66e55c8ae49d99b6 | subsection | 10 | 34 | Organization of the SLR | The following subparagraphs describe the motivation behind the following tackled parts in this SLR:Part 1: Integration and Interoperability challenges in IoT: this part describes the potential interoperability issues that can affect the IoT paradigm. Such study aims to provide a comprehensive overview of common integra... | {
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} | 1808.07355 | Is Fragmentation a Threat to the Success of the Internet of Things? | [
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0dfdd4cb432c67897e1cc842ea266b4f95a7954e | subsection | 11 | 34 | Integration and Interoperability Challenges | IoT integration has become a problem due to of its expensiveness . Not only that, but also it is hard to keep IoT physical parts up-to-date as all devices depend on an integration to provide access or information, hence, can span a wide diversity of technologies, locations, operations and sensitivity levels. The data t... | {
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7b16be5061a95a7e5506abdee0216b7cd473d659 | subsection | 12 | 34 | Network-layer Interoperability | Power constrained devices require efficient networking standards and protocols. Conventionally, the paradigm is scattered between a number of different power networking protocols (e.g., ZigBee, Bluetooth), traditional networking protocols, like Ethernet, WiFi, as well as hardwired connections. Such protocols are sugges... | {
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f23b4cf7aad36be6148317f11ef4a44f1f88c53d | subsection | 13 | 34 | Messaging-protocol Interoperability | In newly developed IoT applications, a number of application level protocols, see Section , are proposed by different enterprises to become the de-facto standards to help the provisioning of communication interoperability . Each protocol possesses specific messaging architecture and unique characteristics that are help... | {
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01eecf08b83987d93d0963b57dbc4fedee19d632 | subsection | 14 | 34 | Data Annotation-level Interoperability | Conventional IoT service model provides raw data captured from the heterogeneous collaborated things found within the system. Such data do not contain intellectual annotation that needs extensive manual efforts to build practical usable applications. Because of the proprietary approaches employed by the IoT providers, ... | {
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} | 1808.07355 | Is Fragmentation a Threat to the Success of the Internet of Things? | [
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5f95f00479bb768cafcf6f3a6eec0c3fc8dfee9a | subsection | 15 | 34 | Proposed Solutions to Integration and Interoperability Issues | To address the above challenges, the research community and industry have been working on the development of standard and implementation practices that would allow a better communication between the services provided by different providers and help ease the pain of their integration.In the following, we discuss some ke... | {
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"Yann-Gaël Guéhéneuc",
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003f9a039a4f4434e0916801f5bce63ee8781190 | subsection | 16 | 34 | Standards and Technologies | IoT requires a number of different technologies. Specially, communication technologies are considered to be a fundamental framework to realize various IoT services. Standards, on one hand, help both developers and users to determine the best technical protocol for dynamic services and applications in IoT. On the other ... | {
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1a3289dbe4547a17e14d3a1e10bc76494090d3e4 | subsection | 17 | 34 | Standards and Technologies | As a consequence, constrained protocols have been suggested for an IP network as well as the application layers.Message Queue Telemetry Transport (MQTT) is a M2M/IoT connectivity protocol that is extremely lightweight publish/subscribe messaging transport; it is useful for connections with remote locations where a smal... | {
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17f0444f0085f5227510b8e1c9328e052ae895ab | subsection | 18 | 34 | Standards and Technologies | The protocol uses a dispatch field that is found in the first part of the packet to recognize a type of the packet and defines two types of dispatches as well, first and subsequent fragments, for carrying an IP datagram. Conventionally, the first fragment is used to carry a compressed IPv6 header information, a transpo... | {
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797120ac411651c9c61e26cd062c792fb26d6fd9 | subsection | 19 | 34 | Body | IoT embedded devices use a communication facility to connect to the Internet. A well-known network standard, IEEE 802.3 (the Ethernet) , , is used to provision such functionality. Some of the IoT devices do employ such standard to connect to a network when the participating devices are fixed in a facility because power... | {
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"raw": "D. Law, D. Dove, J. D'Ambrosia, M. Hajduczenia, M. Laubach, and S. Carlson, “Evolution of ethernet standards in the ieee 802.3 working group,” IEEE Communications Magazine, vol. 51, no. 8, pp. 88–96... | 1808.07355 | Is Fragmentation a Threat to the Success of the Internet of Things? | [
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304fb105fd3df343e7b44c8eecf3b8328efe2a85 | subsection | 20 | 34 | Body | One of the advantages of HART is that it has a backward compatibility to traditional HART instruments.
- Thread is an IPv6 based mesh topology network protocol that provides Thread networking stack on top of the IEEE 802.15.4; this will allow each Thread end device to connect to the Internet through native IP protocols... | {
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"raw": "TSN, “Ieee: Ieee 802.1 time-sensitive networking task group,” Available at http://www.ieee802.org/1/pages/tsn.html, accessed (2018/01/22), 2012.",
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bfdb2e56ce7f19021398dacbaca3c281c2b85eb1 | subsection | 21 | 34 | Body | The IEEE 802.15.4e is a standard for MAC layer mechanisms; it is used to realize a low power intermittent operations. It is used alongside with IEEE 802.15.4g as it does not define any physical layer in its configuration.
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"raw": "3GPP, “3rd generation partnership project; technical specification group radio access network; evolved universal terrestrial radio access (e-utra); user equipment (ue) radio transmission and recepti... | 1808.07355 | Is Fragmentation a Threat to the Success of the Internet of Things? | [
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cf0d6fcfba99f1decacc4cce04fe8a4ab8440286 | subsection | 22 | 34 | Body | GSMA/eSIM assume Embedded Universal Integrated Circuit Card (eUICC) as a new embedded SIM function for smart devices; the eUICC identification information (eICCiD) associated to the eUICC allows both the over the air `OTA' provisioning of an initial operator subscription as well as changing of subscription from one ope... | {
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