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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
95a11529d1c381eb6606e6b79c457898e3bb8e05 | subsection | 84 | 108 | Conclusion | After defining adaptive convolutions (for other types of adaptive convolutions
see Appendix ) and analyzing their theoretical
properties, we have derived a formula for the adaptation function
\mu _f, which allows automatic adjustment of the local smoothing of a function f.
The requirements for such a formula were reaso... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1805.01729",
"end": 2192,
"openalex_id": "https://openalex.org/W2802336905",
"raw": "I. Klebanov. Axiomatic approach to variable kernel density estimation. ArXiv e-prints, 2018.",
"source_ref_id": "5122feebb607cdd56ab37... | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.018886689096689224,
0.01844426989555359,
-0.03545449301600456,
-0.012494505383074284,
0.025706034153699875,
-0.014340458437800407,
-0.008772089146077633,
-0.026560358703136444,
0.062121644616127014,
0.042045000940561295,
-0.020366501063108444,
0.025004267692565918,
-0.015835527330636978,
... | |
3a9dbefd64de9cb50e49c3a8aa6514302b2ec77c | subsection | 85 | 108 | Other Types of Adaptive Convolutions | In the case of the common convolution f\ast g, the contribution of f(y) to
(f\ast g) (x) depends, roughly speaking, on the distance between x and y.
In the following, we will introduce two further types of adaptive
convolutions, for which the contribution of f(y) to the convolution evaluated
in x depends on the distanc... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01740972138941288,
-0.0380847193300724,
-0.018493061885237694,
0.00703407684341073,
0.006965414620935917,
-0.03494150936603546,
-0.03210346773266792,
-0.00647333450615406,
0.035887524485588074,
-0.01747075468301773,
-0.07421638071537018,
-0.008620940148830414,
-0.004001489374786615,
0.0... | |
bd7b455d07153f8115a02a2965bacadce6fb5c96 | subsection | 86 | 108 | Other Types of Adaptive Convolutions | \end{}
\end{align*}}\begin{}[Young^{\prime }s inequality]
Under the conditions of Definition \ref {def\mathrel {\mathop }weightedAdapt}, we have\mathrel {\mathop }
\left\Vert f\ast ^p [g\, |\, h]\right\Vert _p\le \left\Vert f\right\Vert _1\left\Vert g\right\Vert _p
\qquad \text{and}\qquad \left\Vert f\ast ^p [g_1,g_2... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.016754046082496643,
-0.037078626453876495,
-0.018081553280353546,
-0.037353284657001495,
-0.006061516236513853,
-0.02659590356051922,
-0.007739972788840532,
0.05572475120425224,
0.04052709415555,
0.026244955137372017,
-0.02664167992770672,
-0.009170475415885448,
-0.007492018863558769,
0... | |
b5b95131f48f34b79a707f20cc7ec746d95762a3 | subsection | 87 | 108 | Other Types of Adaptive Convolutions | Let f\in L^q\left(\mathbb {R}^d\right), g_1=g_2=\mathrel {\mathop }g\in L^1\cap L^p\left(\mathbb {R}^n\right) and h\colon \mathbb {R}^d\rightarrow \mathbb {R}^n be a measurable function such that0 < \left\Vert g(h({\cdot )-z)_p < \infty for almost all
z\in \mathbb {R}^d. Then \tilde{G}_p is symmetric and
\left\Vert f\... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.035248056054115295,
0.018356453627347946,
0.01188668143004179,
-0.029434414580464363,
0.0064468844793736935,
-0.01976027339696884,
-0.026230046525597572,
0.00875097792595625,
0.038696564733982086,
0.0035362497437745333,
-0.05172766372561455,
-0.019226212054491043,
0.001713764970190823,
... | |
d4211e7d6ff0561a3601904ee03811753053dcbc | subsection | 88 | 108 | Other Types of Adaptive Convolutions | \item If
g_1\in L^1\left(\mathbb {R}^d\times \mathbb {R}^n\right),\ g_2,\, g\in L^p\left(\mathbb {R}^d\times \mathbb {R}^n\right)
depend on an additional parameter in \mathbb {R}^d and
\begin{}
\item \displaystyle \int _{\mathbb {R}^d} |g_1(y,z-h(y))|\, \mathrm {d}y\le \Gamma _1 for
some constant \Gamma _1>0 (independe... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03345469385385513,
0.017292048782110214,
-0.007631089072674513,
-0.009722007438540459,
-0.004075001459568739,
-0.0006686741835437715,
0.006471163593232632,
0.02269485965371132,
0.04114683344960213,
0.01467458438128233,
-0.05399758741259575,
-0.045969683676958084,
-0.00911152083426714,
0... | |
a3fba8520a729c929f1f35dfa0ac7079c0b49635 | subsection | 89 | 108 | Other Types of Adaptive Convolutions | More precisely, in this case the
linearization of h at y, h(x)-h(y)\approx Dh(y)\, (x-y), is meaningful and yields
\begin{align*}
G_p(x,y)
&=
\left\Vert g\right\Vert _p
\frac{g\left(h(x)-h(y)\right)}{\left\Vert g(h({\cdot )-h(y))_p}
\approx \left\Vert g\right\Vert _p \frac{g\left(Dh(y)\, (x-y)\right)}{\left\Vert g(Dh(y... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01635691523551941,
-0.019698485732078552,
-0.006397048942744732,
-0.02656472846865654,
-0.008903227746486664,
-0.014289412647485733,
-0.02654946967959404,
0.00981872621923685,
0.04626321420073509,
-0.00469193235039711,
-0.029082350432872772,
-0.01888979598879814,
0.008964261040091515,
0... | |
226de2d4a08fcae3172e0ed3dd36513b7f6707c6 | subsection | 90 | 108 | Other Types of Adaptive Convolutions | For
example, we can `let a value f(y) contribute strongly to f\ast ^p[g\,
|\, h](x), even though x is far away from y (without contributing strongly to most values in
between)^{\prime } by choosing h such that h(y)\approx h(x), see Figure
\ref {fig\mathrel {\mathop }weighted_quadratic}.
}
\section {Proofs}
\right.\beg... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01829780451953411,
0.003637813962996006,
0.006733866408467293,
-0.03394021466374397,
-0.021807808429002762,
-0.03610726073384285,
0.04379874840378761,
0.05423719435930252,
0.02537885680794716,
0.010179012082517147,
-0.05036092922091484,
0.014574147760868073,
-0.03772491589188576,
-0.006... | |
68528f798fed6fdf9d89ab1ef3ad07f963ce8e3c | subsection | 91 | 108 | Other Types of Adaptive Convolutions | Hölder^{\prime }s inequality yields
\begin{align*}
|f \bar{\ast }G|(x)
&\le \int _{\mathbb {R}^d} |f(y)|^{\frac{1}{p^{\prime }}}\, |f(y)|^{\frac{1}{p}}\,
|G(x,y)|\, \mathrm {d}y
\le \left\Vert |f|^{\frac{1}{p^{\prime }}}\right\Vert _{p^{\prime }}\, \left\Vert |f|^{\frac{1}{p}}\,
|G(x,{\cdot )|_{p},
}
which implies
\beg... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.007306648883968592,
0.01929108053445816,
-0.019489485770463943,
-0.029241858050227165,
-0.019809985533356667,
-0.002937921555712819,
0.007165476214140654,
0.03708649054169655,
0.029714977368712425,
-0.014071499928832054,
-0.0776527002453804,
-0.0002345329412491992,
-0.028326142579317093,
... | |
9a43c4c5bb400915f88be7d3023f8fd06704e1cc | subsection | 92 | 108 | Other Types of Adaptive Convolutions | \end{align*}\end{}}\right.}\end{align*}\begin{}[Proof of Corollary \ref {cor\mathrel {\mathop }young}]
First note that for p<\infty and y\in \mathbb {R}^d the change of variables formula
implies\mathrel {\mathop }
\begin{equation}
\left\Vert g_{\mu ,p}({\cdot ,y)_p^p
=
\int _{\mathbb {R}^d}|\det \mu (y)|\, \left| g\bi... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.06435072422027588,
-0.005735243204981089,
-0.0025356414262205362,
-0.034434352070093155,
-0.03452593460679054,
-0.011737596243619919,
-0.012867092154920101,
0.044660866260528564,
0.005510107148438692,
-0.018835103139281273,
-0.0012134446296840906,
-0.01438580546528101,
-0.0032034174073487... | |
f6425b443202784f7f3140c0794eb58eedd4a35b | subsection | 93 | 108 | Other Types of Adaptive Convolutions | Since
\frac{r-q}{qr} + \frac{r-p}{pr} + \frac{1}{r}
=
\frac{1}{q} - \frac{1}{r} + \frac{1}{p} - \frac{1}{r} + \frac{1}{r}
=
\frac{1}{q} + \frac{1}{p} - \frac{1}{r}
=
1,
the generalized Hölder's inequality yields
|f \bar{\ast }G|(x)
&\le \int _{\mathbb {R}^d} |f(y)|^{1-\frac{q}{r}}\, |f(y)|^{\frac{q}{r}}\,
|G(x,y)|^{... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.051741477102041245,
0.021093428134918213,
-0.019597657024860382,
-0.01784241572022438,
0.0030297001358121634,
0.0026710203383117914,
-0.0007268990739248693,
0.02225341461598873,
0.01205774862319231,
-0.006643209140747786,
-0.055099330842494965,
0.004208764992654324,
-0.025244956836104393,... | |
39483bd60a4f51f32d981b8dd28df56ff03072db | subsection | 94 | 108 | Other Types of Adaptive Convolutions | \end{align*}
\right.
}\right.}
[Proof of Proposition REF ]
For all j=1,\dots ,d and \alpha \in \mathbb {N}^d with |\alpha |<m, we have by
induction:
\partial _{x_j}\partial ^\alpha \left(f\ast _{\mu }^p g\right)(x)
&=
\partial _{x_j}\left(\int _{\mathbb {R}^d} f(y)\, |\det (\mu (y))|^{1/p}\,
\alpha (\mu (y))\, D^{|\al... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.00034682522527873516,
0.007345543708652258,
-0.03786425665020943,
-0.026697199791669846,
-0.013668966479599476,
0.009328764863312244,
0.013821521773934364,
0.006102217361330986,
0.0364302359521389,
0.0036250983830541372,
-0.018474461510777473,
-0.0026906963903456926,
-0.006914574652910232... | |
bbad90835d89d0fdce538a2533996289341c10fa | subsection | 95 | 108 | Other Types of Adaptive Convolutions | The observations
\begin{}
\begin{align*}
\left({\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]
\right)_{k=1}^d j_t(y)
&=
\sum _{k=1}^d {\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]\, j_{k,t}(y)
=
{\rm tr}\left[\mu _t^{-1}(y)\sum _{k=1}^d\partial _{k}\mu _t(y)\, j_{k,t}(y)\right]
\\
&=
{\rm tr}\lef... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.00036994070978835225,
0.028954533860087395,
-0.03228018805384636,
-0.01919114962220192,
-0.014095884747803211,
0.010213414207100868,
0.02962576597929001,
0.02478983998298645,
0.02382875792682171,
0.046559132635593414,
-0.02880198135972023,
0.003932049963623285,
-0.003943491727113724,
-0.... | |
60376110a7b368d2e22e7b5962d74a7509cdf6c8 | subsection | 96 | 108 | Other Types of Adaptive Convolutions | \end{pmatrix}
lead to
\begin{align*}
&\left(\nabla _y\Big [\delta _t(y) \, g_{\mu _t}(x,y)\Big ]\right)^{\intercal }j_t(y)
\\
&=
\delta _t(y) \left(
g_{\mu _t}(x,y) \left({\rm tr}\left[\mu _t^{-1}(y)\partial _{k}\mu _t(y)\right]
\right)_{k=1}^d + (\nabla g)_{\mu _t}(x,y)^{\intercal }
\left[M_t(x,y)-\mu _t(y)\right] \ri... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.0151362968608737,
0.012786508537828922,
-0.005977463908493519,
-0.03228669613599777,
-0.017928577959537506,
0.038786761462688446,
0.013343438506126404,
0.02511526644229889,
0.0034064296633005142,
0.03222566470503807,
-0.008926142938435078,
0.0026072729378938675,
-0.008956659585237503,
0... | |
b826f881dbbbd42821b985193bb8729fecb2e868 | subsection | 97 | 108 | Other Types of Adaptive Convolutions | \end{align*}
Combining these two, we get\mathrel {\mathop }
\begin{align*}
\partial _t \rho _{g,t}(x)
&=
\int _{\mathbb {R}^d} \underbrace{\partial _t\rho _t(y)}_{= -\operatorname{div}j_t(y)}
\delta _t(y) \, g_{\mu _t}(x,y)
+ \partial _t\Big [\delta _t(y) \, g_{\mu _t}(x,y)\Big ]\, \rho _t(y)
\, \mathrm {d}y
\\
&=
\int... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.015744362026453018,
0.011442123912274837,
0.0023551704362034798,
-0.017559845000505447,
-0.002164468402042985,
-0.008520567789673805,
0.00038569490425288677,
0.02915453165769577,
-0.0009239514474757016,
0.01267024502158165,
-0.002175910398364067,
-0.010175861418247223,
-0.0237996168434619... | |
298b43791b684b7502e826924071c94ded3f1872 | subsection | 98 | 108 | Other Types of Adaptive Convolutions | We have:
&(f({\cdot \, - a)\ast ^p_{\mu _{f({\cdot \, - a)}} g) (x)
=
\int f(y-a) \left|\det \left(\mu _{f}(y-a)\right)\right|
g\left(\mu _{f}(y-a)(x-y)\right)\, \mathrm {d}y
\\
& \hspace{28.45274pt} =
\int f(y) \left|\det \left(\mu _{f}(y)\right)\right|
g\left(\mu _{f}(y)(x-a-y)\right)\, \mathrm {d}y
=
(f\ast ^p_{\mu... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005618390627205372,
0.003526274347677827,
-0.021588657051324844,
-0.05312487855553627,
-0.010733299888670444,
0.009733966551721096,
-0.01843045838177204,
0.019681531935930252,
0.02998000755906105,
0.006201970856636763,
-0.030468231067061424,
0.03019360452890396,
-0.010618872940540314,
0... | |
b34a42695bcb1e72de656777575eebdd654a8b2f | subsection | 99 | 108 | Other Types of Adaptive Convolutions | \end{align*}
\right.
}\right.\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muFourier}]
The proof is analogous to the one of Proposition
\ref {prop\mathrel {\mathop }muWigner2}.
\end{}
\right.\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muFBI}]
The proof is analogous to the one of Proposition
... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.044177960604429245,
-0.00014349065895657986,
-0.025017905980348587,
-0.003083380637690425,
-0.023324621841311455,
-0.021188946440815926,
0.019160054624080658,
0.03499456122517586,
0.06748732924461365,
0.04579497128725052,
-0.03325551003217697,
-0.012043680995702744,
0.00414549745619297,
... | |
fa548c3472d374d80675a2b79490a1bdcef0e5d8 | subsection | 100 | 108 | Other Types of Adaptive Convolutions | Then we have for x,y\in \mathbb {R}^d
\big (\nabla \tilde{f}\nabla \tilde{f}^{\intercal } - \tilde{f}\, D^2 \tilde{f}\big )(x)
=
A^\intercal \, \big (\nabla f\nabla f^{\intercal } - f\, D^2 f\big )(Ax)\, A,
\qquad G_{\mu _{\tilde{f}}^{-2}(x)}(y)
=
*{\det A}\, G_{\mu _f^{-2}(Ax)}(Ay),
and, since for any functions \phi... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.014687075279653072,
0.009872766211628914,
0.009514173492789268,
-0.0009026829502545297,
-0.006878129206597805,
-0.002988915191963315,
0.008552837185561657,
-0.010902768932282925,
0.07165767252445221,
0.024826880544424057,
-0.033021122217178345,
-0.024414878338575363,
0.0002694220165722072,... | |
5d47501f3997524c208c2f569aaf6f22b5ffe2a6 | subsection | 101 | 108 | Other Types of Adaptive Convolutions | Therefore,
\mu _{f^{(t)}}(x+a_k^{(t)})
\stackrel{(A1)}{=}
\mu _{\tilde{f}^{(t)}}(x)
\xrightarrow{}
\mu _{f_k}(x).
}
}}}\begin{}[Proof of Proposition \ref {prop\mathrel {\mathop }muWigner2}]
As the Wigner transform is real-valued, we get for real-valued functions~f\mathrel {\mathop }
\begin{align*}
W f(x,-\xi )
&=
(2\... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005172934383153915,
-0.004856301937252283,
-0.011620028875768185,
-0.014893168583512306,
-0.01591554842889309,
0.0057489764876663685,
0.015656137838959694,
0.03415052220225334,
0.046693746000528336,
0.026490308344364166,
0.0038415524177253246,
0.014999983832240105,
0.0035401794593781233,
... | |
c4f2879055e7d1aa9e4a2c54e89b5f62be36663a | subsection | 102 | 108 | Other Types of Adaptive Convolutions | For the covariance matrix, we use the transformation
z_1 = y_1-y_2,\qquad z_2 = y_1+y_2
and the function
F(z_1,z_2) = f\left(x+\frac{z_2 + z_1}{4}\right)f\left(x-\frac{z_2 +
z_1}{4}\right)f\left(x+\frac{z_2 - z_1}{4}\right)f\left(x-\frac{z_2 -
z_1}{4}\right)
to compute\mathrel {\mathop }
\begin{}
\begin{align*}
\in... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.005715889856219292,
-0.0007206866866908967,
0.0014432811876758933,
0.03531793877482414,
0.008554755710065365,
0.0049985419027507305,
0.03183804079890251,
0.022527778521180153,
-0.011828609742224216,
0.023825110867619514,
-0.049848053604364395,
-0.01061522401869297,
0.002384800463914871,
... | |
f61082c0e7230c28c36f0f7269e396092adb3bd3 | subsection | 103 | 108 | Other Types of Adaptive Convolutions | \end{align*}
\end{}
Taking the quotient proves the formula for the covariance matrix.
\end{}
\right.}\end{}
[Proof of Theorem REF ]
Adaptation Axiom REF (A1) follows from
\left(f({\cdot -a)\ast g({\cdot -b)(x)
}}\right.&=
\int f(y-a)\, g(x-y-b)\, \mathrm {d}y
=
\int f(y)\, g(x-(a+b)-y)\, \mathrm {d}y
\\
&=
(f\ast g)(x... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.0013511285651475191,
-0.00341167114675045,
-0.029612313956022263,
-0.011846451088786125,
0.0258440300822258,
0.0018088150536641479,
0.03987974673509598,
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0.020611146464943886,
-0.006117742508649826,
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0.0018316993955522776,
-0.021862156689167023,
... | |
808f065bd0107ec84bf7f7566d50afca8e425ce2 | subsection | 104 | 108 | Other Types of Adaptive Convolutions | The claim follows from the definitions (REF ) of
\mu _f^{(d)} and (REF ) of
\mu _f^{(e)}.
[Proof of Proposition ]
For the h-adaptive convolution of type two the property \left\Vert
G({\cdot ,y)_p = \left\Vert g\right\Vert _p is straightforward for all
y\in \mathbb {R}^d, 1\le p \le \infty and Theorem \ref {theorem\mat... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03579450771212578,
0.008384092710912228,
-0.010909237898886204,
-0.02456485852599144,
-0.020018070936203003,
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0.006740078330039978,
0.03478750213980675,
0.03912068158388138,
0.005679669789969921,
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-0.005008331965655088,
-0.0037705532740801573,
... | |
352c3da9918e9b05a4830c8cb63059942206dc44 | subsection | 105 | 108 | Other Types of Adaptive Convolutions | Hölder^{\prime }s inequality yields
\begin{align*}
|\tilde{G}(x,y)|
&\le \left\Vert g_2\right\Vert _p \int |g_1(z-h(y))|^{1/p^{\prime }}\, |g_1(z-h(y))|^{1/p}
\gamma (x,z)\, \mathrm {d}z
\\
&\le \left\Vert g_2\right\Vert _p \left\Vert g_1({\cdot -h(y))^{1/p^{\prime }}_{p^{\prime }}
\left\Vert g_1({\cdot -h(y))^{1/p}\, ... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.023305442184209824,
0.013987843878567219,
-0.02129082754254341,
-0.02271021530032158,
-0.027670443058013916,
0.0026613534428179264,
0.020527714863419533,
0.03063131868839264,
0.01092776469886303,
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0.00428105890750885,
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-0... | |
a9ab97aae47b9ffa68b1c631b988addfcda80539 | subsection | 106 | 108 | Other Types of Adaptive Convolutions | \end{align*}
\end{}
Therefore \left\Vert \tilde{G}({\cdot ,y)_p \le \left\Vert g_1\right\Vert _1
\left\Vert g_2\right\Vert _p (for all y\in \mathbb {R}^d and 1\le p\le \infty ) also holds for
type three and again Theorem \ref {theorem\mathrel {\mathop }young1} proves the claim.
}
\right.}\right.}\right.\begin{}[Proof o... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.000177927955519408,
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... | |
30f0452c817821d01e4abbd5a1ce8442a0c9cd08 | subsection | 107 | 108 | Other Types of Adaptive Convolutions | Springer, 1997.
R. A. Horn and C. R. Johnson.
Matrix analysis.
Cambridge University Press, 2012.
I. Klebanov.
Axiomatic approach to variable kernel density estimation.
ArXiv e-prints, 2018.
W. Young.
On the multiplication of successions of fourier constants.
Proceedings of the Royal Society of London. Series A, Contai... | {
"cite_spans": []
} | 1805.00703 | Adaptive Convolutions | [
"Ilja Klebanov"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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... | |
77809e708c3379e7ab4537fa384e443523a2879d | abstract | 0 | 51 | Abstract | An analysis of the transmission of `scalar' phonons across partially unzipped
square and triangular lattice tubes, assuming nearest neighbor interactions
between particles, is presented. The phonon transport is assumed to involve the
out-of-plane phonons in the unzipped portion and the radial phonons in the
tubular por... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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6eee9c56247fec6b33d989409f972ddf22a06406 | subsection | 1 | 51 | Introduction | The rise of nanostructures in technological applications has invigorated a slurry of questions concerning the nature of thermal transport , , .
The reduced physical dimensions overcome the hurdles of the phonon mean free path, while new physical processes become crucial in the transport problem , .
For instance, the th... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
e0ecb8e962a31326fff3fdddde8ec485475cee5e | subsection | 2 | 51 | Square lattice model | Consider a partly unzipped two-dimensional tube of square lattice, denoted by {{\mathfrak {S}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ }}}}}, with {\mathtt {N}} number of rows containing infinite number of particles with unit mass,
as shown schematically in Fig. REF (a). The `in-plane' nearest neighbors, ... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.012342878617346287,
-0.011564773507416248,
-0.0090... | |
8e37f3e0e2f33315e316118969103f89646a9f7b | subsection | 3 | 51 | Square lattice model | The total displacement field {\mathtt {u}}^{{t}}, a sum of the incident wave field {\mathtt {u}}^{i} and the scattered wave field {\mathtt {u}}^{{s}}, of an arbitrary particle in the lattice {{\mathfrak {S}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ }}}}} satisfies the discrete Helmholtz equation for all ({... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.04406338930130005,
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0.036... | |
b36c7814c6b027f5150cddcd8a42a9a892ae3ee0 | subsection | 4 | 51 | Triangular lattice model | Similar to the square lattice, {\mathtt {N}} number of rows of particles with unit mass, are arranged in the form of one-dimensional lattices, but now these rows form a two-dimensional strip of triangular lattice {{\mathfrak {T}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ }}}}}, as shown schematically in Fig... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.020374301820993423,
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... | |
673e0539c7f735da3267dc79e2c2f47140b5de74 | subsection | 5 | 51 | Triangular lattice model | Without any loss of generality, it is assumed that
a(i)y=-a(i)N-y-1, yZ0N,
as the definition of the incident wave mode on the replica {{\mathfrak {T}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ }}}}}{{^\mathrm {R}}} in terms of that on {{\mathfrak {T}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ ... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.0... | |
b8c3e70d2e4cd07c90aad4044b45efaf520b2f72 | subsection | 6 | 51 | Wiener–Hopf formulation | The boundaries are such that {{\mathtt {y}}}=-1 and {{\mathtt {y}}}={{\mathtt {N}}}-1 are identified as same (shown schematically in Fig. REF ), while the unzipped portion lies along the broken bonds between {\mathtt {y}}=0 and {\mathtt {y}}=-1\simeq {{\mathtt {N}}}-1. | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.0... | |
6f6e0a92e84693fec263a542dd44fb57a449c4a4 | subsection | 7 | 51 | Square lattice tube | The equation of motion at {\mathtt {y}}=0, {\mathtt {y}}={\mathtt {N}}-1\simeq -1, is
2utx, y
=utx+1, y+utx-1, y+utx, y1NH(x)+utx, y1
-(3+H(x))utx, y, xZ,
where the letter {{\mathit {H}}} stands for the Heaviside function: {{\mathit {H}}}(m)=0, m<0 and {{\mathit {H}}}(m)=1, m\ge 0.
Suppose the discrete Fourier trans... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
fd468d2a493b7cbac2850e528ad20f76fcb0483d | subsection | 8 | 51 | Square lattice tube | Introducing
vx=ux, 0-ux, -1, vix=uix, 0-uix, -1,
with {\mathrm {v}}^F={\mathtt {u}}^F_{0}-{\mathtt {u}}^F_{-1},
it is found that {\mathtt {u}}^F_{0}={{\mathrm {v}}}^F{{\mathtt {U}}_{{\mathtt {N}}-1}}/({{\mathtt {U}}_{{\mathtt {N}}-1}-{\mathtt {U}}_{{\mathtt {N}}-2}-1}),
as well as in the context of (REF ), -{\mathtt ... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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75d8b778177f322e4c035ec7f8a016a59dd0b768 | subsection | 9 | 51 | Square lattice tube | Finally, using (), for {{z}}\in {{{A}}}{{A}} is an annulus containing the unit circle {\mathbb {T}} in the complex plane that makes the Wiener–Hopf problem
well-defined.
v+(z)+L(z)v-(z)=(1-L(z))Av(i)D-(zzP-1),
with
v(i):=a(i)0-a(i)N-1,
is obtained as a Wiener–Hopf equation, using (REF ), with the kernel \operatorna... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.0144... | |
9303d6c3772cac5a8449d2e3791d9814f1d2f569 | subsection | 10 | 51 | Square lattice tube | Additionally, using the even/odd mode based factorizations given by , from (REF ),L=
2TN2(+1)UN-1, for {\mathtt {N}}=2{\mathrm {N}}VNWN, for {\mathtt {N}}=2{\mathrm {N}}+1.This completes the Wiener–Hopf formulation for {\mathfrak {S}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ }}}} assuming incidence of wave... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
bfd2c4fc23dc7b4a27ce787c6a430fa67c27816a | subsection | 11 | 51 | Triangular lattice tube | In terms of a more convenient labelling shown in Fig. REF (b), it is required that {\mathtt {u}}^F_{{\mathrm {N}}}={\mathtt {u}}^F_{-{\mathrm {N}}}. Due to the manufactured symmetry on {\mathfrak {R}}{\mathbin {{{\circledcirc }\\{\circ }}}}, i.e., odd symmetry (REF ) in the choice of the incident wave mode (REF ),
usin... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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e4b14b0b3eda74ca28b9d148c7214d3e2ba404ff | subsection | 12 | 51 | Triangular lattice tube | The details of aforementioned `even' and `odd' modes has been provided by .The Wiener–Hopf
kernel (REF ) can be simplified to (compare with (REF ))
L(z)
=112(z+z-1)12(z+z-1)WN-WN-12(+1)UN-1=:ND,
where {\vartheta }, the argument of the Chebyshev polynomials, is given by (REF ) using the definition of {{Q}} for triang... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
1ade62096ce9865e365570c46b736eada499637f | subsection | 13 | 51 | Triangular lattice tube | Indeed, {{N}}(-{z})=-{\frac{1}{2}}({z}+{z}^{-1})({{\mathtt {U}}}_{{\mathrm {N}}}({\vartheta }(-{z}))+{{\mathtt {U}}}_{{\mathrm {N}}-1}({\vartheta }(-{z})))-({{\mathtt {U}}}_{{\mathrm {N}}-1}({\vartheta }(-{z}))+{{\mathtt {U}}}_{{\mathrm {N}}-2}({\vartheta }(-{z})))=(-1)^{{\mathrm {N}}}({\frac{1}{2}}({z}+{z}^{-1}){{\mat... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.0005... | |
ec6185795cd67a9c0dd41593f183ac3de9f18ca2 | subsection | 14 | 51 | Incidence from unzipped portion | In contrast to §REF and §REF discussed so far in this paper, consider the case when a wave mode is incident from the unzipped portion.
The scattering of such a wave (Fig. REF ) occurs due to the `new' bonds placed between the `through-crack'. It is assumed that the incident wave has the form (REF ),
with {}_x such that... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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-0.0525362566113472,
-0.01... | |
d9525dbf30e3965fc4a6d889f20fa734b23efc33 | subsection | 15 | 51 | Exact Solution of Wiener–Hopf equation | The multiplicative factors for the Wiener–Hopf kernels (REF ), and (REF ), can be constructed easily (see Appendix
for the details).
Using \operatorname{{L}}_{{}}=\operatorname{{L}}_{{}+}\operatorname{{L}}_{{}-}, Wiener–Hopf equations (REF ) and (REF ) (resp. (REF )), can be expressed as {{\operatorname{{L}}}_{{}+}^{-... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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b352ad34b80db4f439b2c150dcfad054e37fcdcc | subsection | 16 | 51 | Square lattice structure | Using (REF ) for {{\mathfrak {s}}}={{\mathcal {R}}} and (REF ) for {{\mathfrak {s}}}={{\mathcal {L}}},C(z)=(1L+(z)-L-(z))Av(i)
(D-(z zP-1)s,R-D+(z zP-1)s,L),
zA, ,
leading to
C(z)=Av(i)(1L+(zP)-L1(z))D-(z zP-1)s,R
Av(i)(L-(zP)-L1(z))D+(z zP-1)s,R, zA.
Note that {{{\mathrm {v}}}_{({{{\kappa }}^{i}})}} denotes the ex... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0bea6041d6785560f2d2a69233b8791f89fefb91 | subsection | 17 | 51 | Square lattice structure | Thus, it is required that ahead and behind the edge of unzipped portion
ux, yAa=1NRAaa(a)yzax, x+,
ux, y-Aa(a)yzax+Ab=1NLAba(b)yzbx, x-.
Comparing (REF ) with the expression of {\mathtt {v}} ({\mathtt {v}}_{{\mathtt {x}}}={\mathtt {u}}_{{\mathtt {x}}, 0}-{\mathtt {u}}_{{\mathtt {x}}, -1}) based on (REF ), the coeffi... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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52db579aa86af445d1a68873a1e8ef476f65d919 | subsection | 18 | 51 | Square lattice structure | (REF ), and the numerical solution on a finite grid is shown in the same figures in (c), (d), and (e), where the displacement of particles located at lattice sites forming a discrete rectangle (shown as white bubbles in top of part (a)) with two sides aligned with `boundary' of the strips is plotted.
Note that the far ... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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500960afb97726244f6140976fa52b6d8940f3b0 | subsection | 19 | 51 | Square lattice structure | The expression for the reflectance and the transmittance (using the far-field expansion (REF ) for a wave incident from the tubular portion) is given by.\hspace{-32.52127pt}
{{R}}_{{{\mathcal {L}}}\leftarrow {{\mathcal {R}}}}&=&\frac{-{{\mathtt {V}_g}}({}_{\tilde{{\mathsf {a}}}})^{-1}}{|{\operatorname{{L}}}_{{}+}({{z}}... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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c4675fe675ecaeeca82fa805c7ddc99b8ec5ca4b | subsection | 20 | 51 | Square lattice structure | And for the unzipped portion, {{{\mathrm {v}}}_{({{\mathsf {b}}})}}=2/\sqrt{{\mathtt {N}}}\cos {{\frac{1}{2}}}{{}} for an appropriate {} .
After various manipulations, analagous to those applied for the bifurcated waveguides of square lattice , it is found that the reflectance {{R}}_{{{\mathcal {L}}}\leftarrow {{\mathc... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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420a1f37af7746171443d59f038578c7c079edf9 | subsection | 21 | 51 | Square lattice structure | N in the plot label refers to N_{\text{grid}}.][Figure: Illustration for triangular lattice structure with details same as those in Fig. except for the incidence from the unzipped side.] | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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e23451f9bb3f5fc10914d9907906fd1e25ce2533 | subsection | 22 | 51 | Triangular lattice structure | Using (REF ) for {{\mathfrak {s}}}={{\mathcal {R}}} and (REF ) for {{\mathfrak {s}}}={{\mathcal {L}}}, in the context of the first paragraph of this section §,The odd symmetry of the incident wave on {{\mathfrak {T}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ }}}}} is assumed (i.e., - sign in (REF )) since t... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
83e170aedb9def9f83cef03d19d620514801c71c | subsection | 23 | 51 | Triangular lattice structure | Note that \sum \nolimits _{{\mathtt {x}}\in {\mathbb {Z}}}({\mathtt {u}}_{{\mathtt {x}}, 0}-{\mathtt {u}}_{{\mathtt {x}}-1, -1}){{z}}^{-{\mathtt {x}}}
=(1+{{z}}^{-1}){\mathtt {u}}_{{0}}^F({{z}}),
and therefore, using
(REF ),
(1+z-1)u0;(z)u-1, 0=v.
Note that the Fourier transform of the second type of slant bondlengt... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.... | |
e55aec56b770eff1d3c774cc3366fe261569ea2b | subsection | 24 | 51 | Triangular lattice structure | As a result the following asymptotic expression holds,
vx
Aa(a)0(1+za-1)D+(za)N+(za)a=1NR
1za-zaN+(za)D'+(za)zax,vxAa(a)0(1+za-1)(-zax
+D+(za)N+(za)b=1NL
1zb-zaD-(zb)N'-(zb)zbx),
as {\mathtt {x}}\rightarrow \pm \infty , respectively,
where the values of {z}, corresponding to the outgoing wave modes, that appear in t... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.... | |
27633d30434b42b793d778e056a8ca86756c5cb8 | subsection | 25 | 51 | Triangular lattice structure | Further, on the lattice {\mathfrak {T}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ }}}} where {\mathtt {x}}\in 2{\mathbb {Z}} at {\mathtt {y}}=0,
the total displacement field in the far-field is asymptotically given by (with {{a}}_{({{{\kappa }}^{i}})}={{a}}_{(\tilde{{\mathsf {a}}})}\delta _{{{\mathfrak {s}}... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
40cb224ff245348dfa5222692e79c704c93b3f06 | subsection | 26 | 51 | Triangular lattice structure | (REF ), and the numerical solution on a finite grid is shown in the same figures in (c), (d), and (e), where the displacement of particles located at lattice sites forming a discrete rectangle (big white dots) with two sides aligned with `boundary' of the strips is plotted.
Note that the far field displacement, for any... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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688ba0ec276fd80b27cb5121f699a363b9ee9b30 | subsection | 27 | 51 | Triangular lattice structure | The dispersion relations for the odd modes in the portion ahead of the unzipped portion and behind are almost overlapping.Using (REF ), for incidence from the tubular portion, the reflectance and the transmittance, respectively, are given by\hspace{-54.2025pt}&&{{R}}_{{{\mathcal {L}}}\leftarrow {{\mathcal {R}}}}=\hspac... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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5be4d957bbf04c5032c1731f5767ef9e346b2459 | subsection | 28 | 51 | Triangular lattice structure | The expression {\mathtt {V}_g}({}) denotes the group velocity of the propagating wave with wave number {} in the appropriate lattice strip.
[Figure: Choice of {}_{\tilde{{\mathsf {a}}}} ({}_{\tilde{{\mathsf {b}}}}), so that the group velocity is positive (resp. negative), for wave incident from the tubular (resp. unzip... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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a7efaa70020a14c31a636bdefae286c795582095 | subsection | 29 | 51 | Triangular lattice structure | On the other hand, (while observing that {z}\ne \pm 1, {{N}}({z})=0, using (REF )) it is found that
{{N}}^{\prime }({{z}})={\frac{1}{2}}{z}^{-1}({z}-{z}^{-1}){{\mathtt {W}}}_{{\mathrm {N}}}+({\vartheta }-1)^{-1}{\vartheta }^{\prime }(\cos {}({\mathrm {N}}+{\frac{1}{2}})(1-\alpha ){\mathtt {U}}_{{\mathrm {N}}}-({\mathrm... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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296a6cfb8b52bcaab57705dda3d6a331207ab736 | subsection | 30 | 51 | Triangular lattice structure | Then, (REF ) implies\hspace{-25.29494pt}{{R}}_{{{\mathcal {L}}}\leftarrow {{\mathcal {R}}}}
&=&-\frac{{\mathtt {V}_g}({}_{\tilde{{\mathsf {a}}}})^{-1}}{|{\operatorname{{L}}}_{{}+}({{z}}_{\tilde{{\mathsf {a}}}})|^2{\mathrm {N}}}\sum \limits _{{\mathsf {a}}=1}^{N^{{{\mathcal {R}}}}}\frac{\cos ^2{\frac{1}{2}}{}_{\tilde{{\... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0b6795e646acc16abbdb96f1cb0490f434a8c5da | subsection | 31 | 51 | Triangular lattice structure | Also
the identity {{\mathtt {U}}}_{n}^2=1+{{\mathtt {U}}}_{n-1}{{\mathtt {U}}}_{n+1} implies {{\mathtt {U}}}_{{\mathtt {N}}}^2=1 when {{\mathtt {U}}}_{{\mathtt {N}}-1}=0, so that {{\mathtt {T}}}_{{\mathtt {N}}}={{\mathtt {U}}}_{{\mathtt {N}}}=\pm 1. But {{\mathtt {T}}}_{{\mathtt {N}}}\ne 1 when {{D}}\ne 0 which leaves ... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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99317e5f4fd263f52c9852327d4d96128acc5de9 | subsection | 32 | 51 | Triangular lattice structure | Finally, the expected, and elegant, form of the expression (REF ) results.Similarly, (using (1+\alpha ^2-2{\vartheta }\alpha ){\mathtt {U}}^2_{{\mathrm {N}}}=1) it is found that{{T}}_{{{\mathcal {L}}}\leftarrow {{\mathcal {R}}}}&=&{2}i\frac{\sin ^2{\frac{1}{2}}{{}_{\tilde{{\mathsf {a}}}}}\cos ^2{\frac{1}{2}}{}_{\tilde{... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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81da55f1e8e7b40f175daf5a9a4c7e11230fb480 | subsection | 33 | 51 | Triangular lattice structure | In fact, by inspection it is clear that the relations (REF ) and (REF ) are same as those for the square lattice structure;
moreover, for the incidence from the unzipped portion, the expressions retains the same form (recall (REF ) and statement preceding it).It is worth a non-trivial note that the final expression for... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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6de095f564075328bdcf738bd6dca332e492fcdb | subsection | 34 | 51 | Triangular lattice structure | In particular, with {{z}}_{{P}}={{z}}_{\tilde{{\mathsf {a}}}}\delta _{{{\mathfrak {s}}}, {{\mathcal {R}}}}+{{z}}_{\tilde{{\mathsf {b}}}}\delta _{{{\mathfrak {s}}}, {{\mathcal {L}}}}, in the first two equations,\hspace{-25.29494pt}{{R}}\delta _{{{\mathfrak {s}}}, {{\mathcal {R}}}}+{{T}}\delta _{{{\mathfrak {s}}}, {{\mat... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
e1cc384374ef7472c3b151767674a3786604f09b | subsection | 35 | 51 | Triangular lattice structure | REF with {}_{\tilde{{\mathsf {a}}}} and {}_{\tilde{{\mathsf {b}}}}, respectively, as a variable on horizontal axis.
[Figure: Reflectance for the partly unzipped tube with (a) {\mathtt {N}}=6 (b) {\mathtt {N}}=8. The critical point associated with local maximum of the dispersion curves in the interval (0, {\frac{1}{2}}\... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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7efea196d896ccf4dc888b454e8aa3c8b6d5f952 | subsection | 36 | 51 | Scattering matrix | Under the limit {}_2=\Im {}\rightarrow 0 and |{\mathtt {x}}|\rightarrow \infty , it is easy to see that
the far-field can be determined (suitably) in terms of the propagating waves associated with the two different portions of the lattice strip.
Suppose that the symbol {\mathbb {T}} denotes the unit circle (as a counte... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.0... | |
339bf56846c56ebc3a704568a03770466d337695 | subsection | 37 | 51 | Scattering matrix | Thus, () and () can be written as, respectively,
{{\mathcal {Z}}}^+_{{{\mathcal {R}}}}=\lbrace {z}_{{\mathsf {a}}}\rbrace _{{\mathsf {a}}=1}^{N^{{{\mathcal {R}}}}}, {{\mathcal {Z}}}^{-}_{{{\mathcal {L}}}}=\lbrace {z}_{{\mathsf {b}}}\rbrace _{{\mathsf {b}}=1}^{N^{{{\mathcal {L}}}}},
and
{{\mathcal {Z}}}^{-}_{{{\mathcal ... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.0088... | |
890396b85e6353737caf036709be6ac634ffc771 | subsection | 38 | 51 | Scattering matrix | The asymptotic form of solution to equation of motion is thus
u{{ll}
\widetilde{{\mathsf {I}}}_{{{\mathcal {R}}}}{\mathtt {u}}_{{\tilde{{\mathsf {a}}}}}+\widetilde{{\mathsf {I}}}_{{{\mathcal {R}}}}\sum \limits _{{{\mathsf {a}}}=1}^{N^{{{\mathcal {R}}}}}{{\tau }}^{{{\mathcal {R}}}\tilde{{{\mathcal {R}}}}}_{{\mathsf {a}... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
d36d548254b36c5cd0874c275e1d48cffe7b9e5d | subsection | 39 | 51 | Scattering matrix | Suppose
OR|Vga|OR=IRRRaaVga+ILRLabVgb
IRRRaa+ILRLab,
and
OL|Vgb|OL=IRLRbaVga+ILLLbbVgb
IRLRba+ILLLbb,
where {{\mathsf {O}}}_{{{\mathcal {R}}},{{\mathcal {L}}}} \equiv \sqrt{|{{\mathtt {V}_g}}_{{\mathsf {a}},{\mathsf {b}},{\mathsf {b}}}|}{\widetilde{{\mathsf {O}}}}_{{{\mathcal {R}}},{{\mathcal {L}}}} and {{\mathsf... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.... | |
d74147a72a003cf3e18f533345b29c2730526521 | subsection | 40 | 51 | Scattering matrix | With
{{\mathsf {O}}}=[{{\mathsf {O}}}^1_{{{\mathcal {R}}}}~
\cdots ~
{{\mathsf {O}}}^{N^{{{\mathcal {R}}}}}_{{{\mathcal {R}}}}~
{{\mathsf {O}}}^1_{{{\mathcal {L}}}}~
\cdots ~
{{\mathsf {O}}}^{N^{{{\mathcal {L}}}}}_{{{\mathcal {L}}}}]^T, {{\mathsf {I}}}=[{{\mathsf {I}}}^1_{{{\mathcal {R}}}}~
\cdots ~
{{\mathsf {I}}}^{N^... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0.012... | |
0d28a593e18dd890b8aebf5701f278f4932b90dc | subsection | 41 | 51 | Scattering matrix | As a necessary consequence of the unitarity of \mathbf {S} it follows that
\sum \nolimits _{{{\mathsf {a}}}=1}^{N^{{{\mathcal {R}}}}}|{{\tau }}^{{{\mathcal {R}}}\tilde{{{\mathcal {R}}}}}_{{\mathsf {a}}\tilde{{\mathsf {a}}}}|^2+\sum \nolimits _{{{\mathsf {b}}}=1}^{N^{{{\mathcal {L}}}}}|{{\tau }}^{{{\mathcal {L}}}\tilde{... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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... | |
b25fcab524e8390d3f36426a7764bd2e8f367e9d | subsection | 42 | 51 | Conductance | Finally, the conductance for the transmission from right to left can be expressed at a given frequency {} as
GLR=a=1NRTLR
which equals that for transmission from left to right given by {{G}}_{{{{\mathcal {L}}}}\rightarrow {{{\mathcal {R}}}}}=\sum _{\tilde{{\mathsf {b}}}=1}^{N^{{\mathcal {L}}}}{{T}}_{{{\mathcal {L}}}\r... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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7af0afc834a9bdb00ddf8edcf716c04a10984773 | subsection | 43 | 51 | Concluding remarks | A closed form expression has been provided for the conductance in partly unzipped tubes of square and triangular lattice, and in fact the same form of expression holds. An exact solution of the wave propagation problem has been harnessed for this purpose.
A provision of the reflection and transmission coefficients usin... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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a36c0c48570d71b5e4760e089802c10a120e3363 | subsection | 44 | 51 | Body | The discrete Fourier transform {\mathtt {u}}_{{\mathtt {y}}}^F: {\mathbb {C}}\rightarrow {\mathbb {C}} of \lbrace {\mathtt {u}}_{{\mathtt {x}}, {\mathtt {y}}}\rbrace _{{\mathtt {x}}\in {\mathbb {Z}}} (along the {\mathtt {x}} axis) is defined by
{\mathtt {u}}_{{\mathtt {y}}}^F={\mathtt {u}}_{{\mathtt {y}};+}+{\mathtt {u... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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cebf43ae9ef67ecab8cab12da7b9b296e0ba1836 | subsection | 45 | 51 | Body | Using the definition of Chebyshev polynomial of the second kind, for 0<n\in {\mathbb {Z}}, it follows that
{\lambda }^{-n}-{\lambda }^{n}=({\lambda }^{-1}-{\lambda }){\mathtt {U}}_{n-1}({\frac{1}{2}}({\lambda }+{\lambda }^{-1})).Due to their frequent appearance in the rest of the paper,
it is also useful to define
zP... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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4e7aef24a1ab27bee8cd8d99a892fd134884a259 | subsection | 46 | 51 | Body | Since J({{z}}) is bounded on the complex plane and tends to zero as {{z}} tends to 0, it follows that J\equiv 0.Recall that the incident wave is {{\mathrm {A}}}{{a}}_{({{{\kappa }}^{i}}){{\mathtt {y}}}}e^{i{}_x {{\mathtt {x}}}} on the physical sub-lattice, while the incident wave is -{{\mathrm {A}}}{{a}}_{({{{\kappa }}... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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f549b2c087896bd66f8d6e5c4b788cf58d8bfefa | subsection | 47 | 51 | Body | Similarly, for incidence from the unzipped portion,
-qF(z)=x=0+ z-x(uix, 0-uix-1, -1)+x=-1+ z-x(uix, 0-uix+1, -1)=Aa(i)0(1+z)(1zP-1)D+(zzP-1).The simplification of (1+{{z}}^{-1})({\mathtt {u}}_+({{z}})+{\mathtt {u}}_-({{z}})), stated in (REF ), is detailed as follows
(1+z-1)uF(z)=
-12Aa(i)0zP-1(L+(z)z(1L+(z)-l+0)+1-(... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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decefc1f64596d683fc20af7697df36c8486b0b7 | subsection | 48 | 51 | Wiener–Hopf factorization: | Following , assuming that |{{z}}_{{F}}|<1, {{z}}_{{F}}\in {\mathbb {C}}, let
{{F}}({{z}}; {{z}}_{{F}}){:=}{{z}}_{{F}}^{-1}(1-{{z}}_{{F}}{{z}})(1-{{z}}_{{F}}{{z}}^{-1}),
and {{F}}_\pm ({{z}}; {{z}}_{{F}})={{z}}_{{F}}^{-{\frac{1}{2}}}(1-{{z}}_{{F}}{{z}}^{\mp 1}). The Wiener–Hopf kernel (REF ) can be expressed as \operato... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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7f6a6c7a9a22ba7b97af0e941b90a8bc9ddaf665 | subsection | 49 | 51 | Wiener–Hopf factorization: | For example, in case of the periodic boundary, i.e. {\mathfrak {T}\hspace{-1.69997pt}}{\mathbin {{{\circledcirc }\\{\circ }}}}, the expressions of {{N}} and {{D}} (REF ) can be found to be
N(z)=j=1N(6-(z2+z-2)-322-2(z+z-1)jN+1)
-2j=1N-1(6-(z2+z-2)-322-2(z+z-1)jN)D(z)=j=1N(6-(z2+z-2)-322-2(z+z-1)2j2N+1),
which indica... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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419e2d43484a20933f79f3232de61c2ad08b7681 | subsection | 50 | 51 | Chebyshev polynomials | The Chebyshev polynomials, following Appendix A of , are:
first kind {\mathtt {T}}_n=
{\frac{1}{2}}(({\vartheta }+\sqrt{{\vartheta }^2-1})^n+({\vartheta }-\sqrt{{\vartheta }^2-1})^n),
second kind
{\mathtt {U}}_n={\frac{1}{2}}(({\vartheta }+\sqrt{{\vartheta }^2-1})^n-({\vartheta }-\sqrt{{\vartheta }^2-1})^n)/\sqrt{{\va... | {
"cite_spans": []
} | 1808.01873 | Kinematically restricted phonon transmission in partly-unzipped tubes of
square and triangular lattices | [
"Basant Lal Sharma"
] | [
"cond-mat.mes-hall",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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2f8774125d402aa817046e3b0ba6971e7a77a8e5 | abstract | 0 | 50 | Abstract | We classify those rational maps $f: \mathbb{P}^1 \to \mathbb{P}^1$ for which
there exists a contravariant tensor $q$ which is parallel, i.e. such that $f^*q
// q$, by proving that such maps preserve a parabolic orbifold. | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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c79aaa199b86966159f992285710faf6d9ed5972 | subsection | 1 | 50 | Introduction | A holomorphic dynamical system on the Riemann sphere \mathbb {P}^1
is the data of a rational map f:\mathbb {P}^1\rightarrow \mathbb {P}^1.
From the viewpoint of Dynamics, the principal object of interest
is the study of the space of orbits \mathbb {P}^1/f under the equivalence
relation generated by f, namely z \sim w i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf02685881",
"end": 1123,
"openalex_id": "https://openalex.org/W2057221822",
"raw": "P. Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math., (1974), pp. 5–77.",
"source_ref_id": "22a4d6ba3e06effe44d3e4553335a2... | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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98c79570d54a5539ce1f5673066a16843bb25bd7 | subsection | 2 | 50 | Introduction | This led us to ask for which maps
f:\mathbb {P}^1\rightarrow \mathbb {P}^1 does there exist a non-zero meromorphic global section q of \Omega _{\mathbb {P}^1}^{\otimes k} and a constant \lambda \in * such that:f^*q=\lambda qwherein we employ the standard convention, , of identifying
A_f of a differential with its image... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf02685881",
"end": 321,
"openalex_id": "https://openalex.org/W2057221822",
"raw": "P. Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math., (1974), pp. 5–77.",
"source_ref_id": "22a4d6ba3e06effe44d3e4553335a22... | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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f2d6d64e0031e4179a55bdc7388fa058e75dbbae | subsection | 3 | 50 | A simple case | The space Rat(d) of all rational maps f:\mathbb {P}^1\rightarrow \mathbb {P}^1
with deg(f)=d, is never a group unless d=1, i.e. the group of automorphisms of the
complex projective line PGL_2(.The subgroup generated by f \in PGL_2( is clearly isomorphic
to \mathbb {Z}, and it acts on \mathbb {P}^1 through[row sep=0.6pc... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.060525063425302505,
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... | |
0f5d5e16ba9a8d807fe9dfcc7955ac127a769e88 | subsection | 4 | 50 | A simple case | It follows that \alpha is a primitive n-th root of unity,
and as f^*q=\lambda q we see easily that \lambda =\alpha ^j for some j<n.Note that the action of f: \mathbb {G}_m \rightarrow \mathbb {G}_m is a free action,
so the quotient map p: \mathbb {G}_m \rightarrow \mathbb {G}_m/f is canonically a \mu _n-torsor.DefineQ(... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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0.012020477093756199,
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0... | |
bb1e1290ec3c1ada02142012e410b13da7a1e815 | subsection | 5 | 50 | A simple case | Since f is an automorphism, for any x \in \mathbb {P}^1 we have
ord_x(q)=ord_{f(x)}(q), hence for any k \in \mathbb {Z} the sets
S_k=\lbrace x\in \mathbb {P}^1: ord_x(q)=k\rbrace are completely invariant for the dynamics,
i.e. f^{-1}(S_k)=S_k.From REF we deduce easily that in Case 1)
the only finite set which is comple... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.04454679414629936,
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0... | |
8d4e043fa160e678976504a077a080e2e6393f92 | subsection | 6 | 50 | A simple case | Q \in H^0(\mathbb {G}_m/f,\Omega ^{\otimes k}\otimes L_{\lambda }^{-1}) where L_{\lambda } denotes the sheaf on \mathbb {G}_m/f given by the action
on the trivial sheaf A_f(f^*1)=\lambda .Thus “multiplication by Q” yields an f-invariant
isomorphism of sheaves on \mathbb {G}_m\mathcal {O}\mathop {\longrightarrow }\limit... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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... | |
e762d5acd2076a5cc72f36ab8dc5c07c9c7066ee | subsection | 7 | 50 | Dynamical systems on the Riemann sphere with a parallel tensor | In the holomorphic category, a non-unit endomorphism of \mathbb {P}^1
is a rational map f: \mathbb {P}^1\rightarrow \mathbb {P}^1 of degree d>1.
We denote by \Omega _{\mathbb {P}^1} the sheaf of holomorphic
differential forms on \mathbb {P}^1, given by the canonical
action dz \rightarrow f^{\prime }(z)\,dz, and by \Ome... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.041957564651966095,
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0.006144875660538673,
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0.00927643571048975,
-0.014326600357890129,
-0.00... | |
7ebfb96bdf53daaa8bdfeb35a1a86c2a12c3e524 | subsection | 8 | 50 | Dynamical systems on the Riemann sphere with a parallel tensor | Within this notation we have f^*q=q(s^n)\,(ns^{n-1}ds)^k and
it follows easily thatord_y(f^*q)=deg_y(f)(ord_x(q) +k) -k.Now Assumption REF clearly implies that ord_y(f^*q)=ord_y(q), so we obtain\forall x \in \mathbb {P}^1, \qquad ord_x(q)=deg_x(f)(ord_{f(x)}(q) +k) -k | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.025737281888723373,
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0... | |
81d8fc0198818ab49c6fb92bb2997615c27e1adf | subsection | 9 | 50 | Considerations on zeroes and poles | In this section we are going to show that (REF ) constrains the number of zeroes and poles of q.Lemma 3.2
Let f: \mathbb {P}^1\rightarrow \mathbb {P}^1 be a rational map of degree d>1 which satisfies Assumption REF .Then q has no zeroes, that is the set Z:=\lbrace x \in \mathbb {P}^1: ord_x(q) >0 \rbrace is the empty ... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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0.019664525985717773,
-0... | |
ea9772994b665536645622fbc67301b55e34ed89 | subsection | 10 | 50 | Considerations on zeroes and poles | Given a zero x \in Z of q with ord_x(q)=e>0, we see from (REF ) that for any
y \in f^{-1}(x), setting n:=deg_y(f), we have ord_y(q)=ne + k(n-1)>0, i.e. y \in Z.Observe now that, from the definition of f^*\mathcal {Z}, we haveord_y(f^*q)=ord_y(f^*\mathcal {Z}) + k(n-1) \ge ord_y(f^*\mathcal {Z})Thus summing (REF ) over ... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.041768670082092285,
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0.03... | |
002ada786f2f19c517a552fbf6e83fd00af4ea91 | subsection | 11 | 50 | Considerations on zeroes and poles | Moreover we have -ord_y(q) -k= deg_y(f)(-ord_x(q)-k) =
ord_y(f^* \mathcal {P}_k), so that,deg(\mathcal {P}_k)= \sum \limits _{x \in P_k} (-ord_x(q) -k)
\ge & \sum \limits _{y \in f^{-1}(P_k) } (-ord_y(q) -k) = \\
\sum \limits _{y \in f^{-1}(P_k) } ord_y(f^* \mathcal {P}_k)=
deg(f^* \mathcal {P}_k)= & d \cdot deg(\mathc... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.014942570589482784,
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0.029244419187307358,
-0.009854926727712154,
0.... | |
60aa0c5d9c9342735086c6e6e5dec958c5061e92 | subsection | 12 | 50 | Considerations on zeroes and poles | From deg(f^*\mathcal {P}_m)= d\, deg(\mathcal {P}_m) and d>1,
we deduce that deg(\mathcal {P}_m)=0, which implies (REF ).In this section we are going to show that (REF ) constrains the number of zeroes and poles of q.Lemma 3.2
Let f: \mathbb {P}^1\rightarrow \mathbb {P}^1 be a rational map of degree d>1 which satisfie... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.06740158796310425,
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... | |
4f15f174bf78319349c1f5247411bc8ac51a8ffb | subsection | 13 | 50 | Considerations on zeroes and poles | Given a zero x \in Z of q with ord_x(q)=e>0, we see from (REF ) that for any
y \in f^{-1}(x), setting n:=deg_y(f), we have ord_y(q)=ne + k(n-1)>0, i.e. y \in Z.Observe now that, from the definition of f^*\mathcal {Z}, we haveord_y(f^*q)=ord_y(f^*\mathcal {Z}) + k(n-1) \ge ord_y(f^*\mathcal {Z})Thus summing (REF ) over ... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.041768670082092285,
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0.03... | |
bfe2acbab219e47e2eaba3bf7b5700d20deea4a4 | subsection | 14 | 50 | Considerations on zeroes and poles | Moreover we have -ord_y(q) -k= deg_y(f)(-ord_x(q)-k) =
ord_y(f^* \mathcal {P}_k), so that,deg(\mathcal {P}_k)= \sum \limits _{x \in P_k} (-ord_x(q) -k)
\ge & \sum \limits _{y \in f^{-1}(P_k) } (-ord_y(q) -k) = \\
\sum \limits _{y \in f^{-1}(P_k) } ord_y(f^* \mathcal {P}_k)=
deg(f^* \mathcal {P}_k)= & d \cdot deg(\mathc... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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0.0... | |
f4f4edb3ae78cc3621aad19eb934df836cc964da | subsection | 15 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | In this section we discuss the main consequence
of Assumption REF ,
i.e. the existence of an orbifold or eventually an orbifold with boundary
that is preserved by f.
We refer to and to
for a formal definition of an orbifold.
Observe that under our hypothesis, given any ramification point
x\in Ram_f, its image f(x) is ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 226,
"openalex_id": "",
"raw": "W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, vol. 35 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.",
"source_ref... | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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0.009206565096974373,
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0.00601058267056942,
-0.0035049021244049072,
0.... | |
579174d5bb42a34ab86db52518d2393c22e5acaa | subsection | 16 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | Then f preserves one of the following
parabolic orbifolds (resp. orbifold with boundary):(\infty ,\infty )
(2,2,\infty )
(2,2,2,2)
(3,3,3)
(2,4,4)
(2,3,6)It is a fact that, for q a meromorphic global section
of the sheaf \Omega _{\mathbb {P}^1}^{\otimes k}, we have\sum \limits _{x \in \mathbb {P}^1} ord_x(q)= deg(... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.024299971759319305,
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0... | |
68724600fec0334b79b405e6a2adae34b2c66ebf | subsection | 17 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | \end{array}\right.}It follows from the discussion above that \nu _f
satisfies condition (REF ), so f
preserves an orbifold with boundary of type (ii).Our aim is now to show that any solution of (REF )
with n_k=0 corresponds to one of the orbifold (iii)-(vi) listed above.
Note that we must have n_i\le 4, \;\forall i and... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.028636060655117035,
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0.018002424389123917,
-0.014478221535682678,
0.0... | |
03bb3130f1b4590b9facc25b48f3dc77520405e0 | subsection | 18 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | In fact, note that if n_i=2 then there is only another nonzero n_j solving (REF ), hence
n_j \alpha _j=2(1-\alpha _i) and consequently, if \alpha _i > \frac{3}{4},
we should have n_j \alpha _j < \frac{1}{2} which is impossible.
Note that \alpha _i\ne \frac{2}{3}, otherwise we would have n_i=3.We claim that there are no... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.04710285738110542,
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0.005735223181545734,
0.018972408026456833,
... | |
fb4c2a20015fe3840df0d89db235754082d9b82e | subsection | 19 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | Thus we have \nu _f(y)deg_y(f)=4 in each case, so we conclude that f
preserves an orbifold of type (v).Finally we suppose that n_i\le 1,\;\forall i.
It is clear that in this case we must have \#\lbrace i:n_i\ne 0\rbrace =3,
so we can rewrite equation (REF )
in the form \alpha + \beta + \gamma =2,
with \alpha ,\beta ,\g... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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0.0150142852216959,
-0.021239720284938812,
0.01910... | |
0825e2869ede135b10d700e0dd4447198f2a187b | subsection | 20 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | It follows that \nu _f(y)deg_y(f)=6 in each case, so \nu _f satisfies
condition (REF ), i.e.
f preserves an orbifold of type (vi).\textbf {Remark} 3.6
We have shown that every solution of (REF )
with \alpha _i \ne 1 is such that
\alpha _i=\left(1-\frac{1}{n}\right) for some
n=2,3,4 \mbox{ or } 6, i.e. q may only have... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1116,
"openalex_id": "",
"raw": "W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, vol. 35 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.",
"source_re... | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.005244331434369087,
0.01638519950211048,
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0.014073879458010197,
-0.013127991929650307,
0... | |
eadc19c2dbdd8d0deab26c1ebbc835899c85e477 | subsection | 21 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | Also, we denote by \mathcal {O}=(\mathbb {P}^1, \nu _f) the orbifold
preserved by f and we shall refer to it, for brevity's sake,
through the string (\nu _f(x)\mathop {)}_{x \in P_f}.By definition if \nu _f takes the value \infty
we will say that \mathcal {O} is an orbifold with boundary, which
we will denote by (\mat... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.03894321620464325,
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0.012383393943309784,
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0.011... | |
fa5e7ac0491c92e69656b8fdc9165aec286167f7 | subsection | 22 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | We proved in REF that the set of poles of q of order
k is a complete invariant set for the dynamics,
hence in the first case if we define\nu _f(x)=
{\left\lbrace \begin{array}{ll}
\infty , & \mbox{if } ord_x(q)=k;\\
1, & \mbox{otherwise}.
\end{array}\right.}it is clear that \nu _f satisfies condition (REF ),
i.e f pres... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
-0.0362800695002079,
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0.009893870912492275,
-0.018338369205594063,
0.03... | |
c20cc1d5d903fba1404adfaa6cf4003fe6b12e28 | subsection | 23 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | Therefore n_i=3 for some i if and only if \alpha _i=\frac{2}{3}
(note that it makes sense only if k\equiv 0\;(mod\, 3)).
In this case let P=\lbrace p_1,p_2,p_3\rbrace be the set of poles of q,
each of order \frac{2}{3} k and let us define\nu _f(x)=
{\left\lbrace \begin{array}{ll}
3, & \mbox{if } x \in P; \\
1, & \mbox{... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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03f96dc6e411b319f7426d7cade56ebf2d799364 | subsection | 24 | 50 | Main Lemma: the dynamical system preserves a parabolic orbifold | By the Riemann-Hurwitz formula, the ramification of a rational map of degree
d has order 2d-2, hence we obtain 2d-2=(e_1-1)+\dots +(e_r-1)=d-(r+s),
which is absurd since d>1.Therefore we conclude that n_i=2 for some
i \ne k, \frac{k}{2} if and only if \alpha _i=\frac{3}{4}
(note that it makes sense only if k \equiv 0 ... | {
"cite_spans": []
} | 1806.10056 | Dynamical systems with a parallel tensor | [
"Jacopo Garofali"
] | [
"math.DS"
] | 2,018 | en | Mathematics | [
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