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42f25e2284027a32c679c473edecd37c50dc8b3d | subsection | 1 | 60 | Introduction | Let A be a bounded and translation-invariant operator on \mathsf {L}^2(\mathbb {R}^d). In other words, consider a Fourier multiplierA=A(\sigma )=\mathcal {F}^\ast \sigma \mathcal {F}with a bounded, complex-valued symbol \sigma \in \mathsf {L}^\infty (\mathbb {R}^d). Here, the Fourier transform \mathcal {F} is chosen to... | {
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6206d623967ce16cf1a6846504f81e8ea63cb969 | subsection | 2 | 60 | Introduction | While\mathcal {B}_d=\frac{|\Omega |}{(2\pi )^d}\int \limits _{\mathbb {R}^d}d\xi \,(h\circ \sigma )(\xi )only depends on \Omega through its volume |\Omega |, the coefficients \mathcal {B}_j for j\le d-1 contain geometric information on the boundary \partial \Omega : \mathcal {B}_{d-1} arises from a hyperplane approxima... | {
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f9943e2b32fa91368d794d0cb2834693eb96d04a | subsection | 3 | 60 | Introduction | In particular, one observes that the edges (or if d=2 the corners) of \Omega do not enter the trace asymptotics up to order L^{d-1}. In the special case of cubes \Omega , actually implies complete asymptotics for (REF ), consisting of d+1 terms. However, the latter result is established in the more general framework of... | {
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4bb35a3d59a4f454fe8661cc75616d0e9d69b2da | subsection | 4 | 60 | Introduction | Yet, in the polygon case c_0 includes additional terms due to the presence of non-parallel edges. Furthermore, we compute c_0 explicitly as a function of the polygon's interior angles for radially symmetric symbols \sigma and quadratic test functions h, see Theorem REF . As a consequence, one can compare c_0 with the c... | {
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744fd4d3bacdcd3926e45d0623f320a8e9ef86b2 | subsection | 5 | 60 | Results | Let h:\mathbb {C}\rightarrow \mathbb {C} be an entire function with h(0)=0 and consider a symbol \sigma \in \mathsf {W}^{\infty ,1}(\mathbb {R}^2), see (REF ) for the definition. These assumptions will be sufficient to obtain the asymptotic trace formula (REF ) with well-defined coefficients c_j=c_j(P,h,\sigma ). In or... | {
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polygons | [
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321b52a2789d81d9ff590ebab7761f1d154a2dd4 | subsection | 6 | 60 | Notation for the polygon | Let \Xi (P)\subset \mathbb {R}^2 denote the set of vertices of P and \mathcal {E}(P) the set of edges of P. In the following we specify the contribution of each edge E\in \mathcal {E}(P) and each corner at X\in \Xi (P) to the asymptotics (REF ).First, fix an edge E\in \mathcal {E}(P). Let \nu _E be its inward pointing ... | {
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polygons | [
"Bernhard Pfirsch"
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891e9ae30fbc409b3db20cc0375de63724912980 | subsection | 7 | 60 | Notation for the polygon | Clearly, also the operator M(x\cdot \nu _E)\big [h_1(A_{H_E})-h_1(A)\big ] is invariant with respect to translations along the edge E.Fix now a vertex X\in \Xi (P). Its adjacent edges are named E^{(1)}(X) and E^{(2)}(X), where the enumeration is again chosen according to the orientation of \partial P. Corresponding to ... | {
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189e82c3fc4a1efb32c036c1ee991bbdaac8443f | subsection | 8 | 60 | Main result | Our first and main theorem provides a complete asymptotic expansion of \operatorname{tr}h(A_{P_L}) and contains formulas for all the coefficients in (REF ).Theorem 2.1
Assume that \sigma \in \mathsf {W}^{\infty ,1}(\mathbb {R}^2), see (REF ), and let h:\mathbb {C}\rightarrow \mathbb {C} be an entire function with h(0)... | {
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5a793ed2fd8ec86f463ede218ea6ed63e65c04c0 | subsection | 9 | 60 | Main result | As anticipated, the formula (REF ) for c_1 reduces the corresponding formula from the smooth boundary case, see .Theorem 2.3
Let \sigma \in \mathsf {W}^{\infty ,1}(\mathbb {R}^2) and let h:\mathbb {C}\rightarrow \mathbb {C} be an entire function with h(0)=0. Define, for E\in \mathcal {E}(P) and t\in \mathbb {R}, the f... | {
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b2fed0b941771874b21c6a39b3a70a05c9188215 | subsection | 10 | 60 | Main result | Referring to the same proposition, one similarly gets thata_0(\nu _E)=&-\frac{1}{64\pi ^2}\int \limits _\mathbb {R}dt\int \limits _{\mathbb {R}} d\xi \, h^{\prime \prime }(\sigma _{E,t}(\xi ))\sigma ^{\prime }_{E,t}(\xi )^2 \\
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a396323cf02184a98cb87bd3141a67b47fcd2734 | subsection | 11 | 60 | The radially symmetric case | In contrast to the above, the coefficients b_0(X), see (REF ) and (REF ), can naturally not be transformed into integrals over traces of one-dimensional fibre operators since they incorporate the truly two-dimensional sector operators h(A_{C(X)}). This makes their explicit calculation rather involved. However, we manag... | {
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4532d3f66059887d125a1f01b5c1b7a9b416ad51 | subsection | 12 | 60 | The radially symmetric case | Notice that, due to the radial symmetry of \sigma , the dependence of the coefficients c_j, j=0,1,2, on the geometry of P separates from their dependence on the symbol \sigma .
Interestingly, the contribution of convex corners and concave corners to c_0 are obtained via the same formula, in contrast to the two distinc... | {
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46503a2e117c25808d352e707904551598de4fa4 | subsection | 13 | 60 | The radially symmetric case | As a simple but representative example consider a disc \Omega and let \lbrace P_n \rbrace be a sequence of inscribed regular n-gons, approximating \Omega . As the function f, see (REF ), vanishes to second order at \gamma =\pi , one easily checks thatc_0(P_n,h,\sigma )\rightarrow 0=\mathcal {B}_0(\Omega ,h,\sigma ),as ... | {
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c7bd9fff6203eed7301247e2d9bda73921b2ccfa | subsection | 14 | 60 | Strategy of the proofs | Let us comment on the basic ideas for the proofs of Theorems REF , REF , and REF .The strategy of the proof of Theorem REF is as follows. The leading order term in the asymptotics originates from approximating the operator h(A_{P_L}) by its bulk approximation \chi _{P_L} h(A)\chi _{P_L}, which is a very familiar idea. ... | {
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polygons | [
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817da7323199c95465a38e7b266b89f065c5c46c | subsection | 15 | 60 | Strategy of the proofs | \begin{}[!t]
\caption {The sector C(X), the one-sided boundary neighbourhood \mathcal {V}, and the corner neighbourhood \mathcal {N}(X)}
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642b6a24a358812ab90b3de0df6e40f6700edae1 | subsection | 16 | 60 | Strategy of the proofs | In view of the translation-invariance of A we may assume that X=0, hence the sector C(X) models the corner at X\in \Xi (P), see (\ref {eq:defC(X)}) and Figure \ref {fig:1}. Again the locality of the operator A implies that one can replace the operator h_1(A_{P_L}) in (\ref {eq:cornertrace}) by the L-independent sector ... | {
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acd5d478346fe121db25f184ed26879e1c105b4b | subsection | 17 | 60 | Trace norm estimates | In this section we collect the trace norm estimates that will be sufficient to prove Theorems REF and REF . | {
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43f05a65364ef374be93a6be6a661907d166032c | subsection | 18 | 60 | Schatten-von Neumann classes | We introduce the standard notation for Schatten-von Neumann classes \mathfrak {S}_p for p>0, see e.g. , . A compact operator T is an element of \mathfrak {S}_p iff its singular values \lbrace s_k(T)\rbrace _{k=1}^\infty are p-summable, i.e.\Vert T\Vert _p^p:=\sum \limits _{k=1}^\infty s_k(T)^p <\infty .We shall often m... | {
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aa93b975024030ef0232e0c60d240d95e76a7edf | subsection | 19 | 60 | Finite volume truncations of the operator | We recall the notationA=A(\sigma )=\mathcal {F}^\ast \sigma \mathcal {F},andA_\Omega =A_\Omega (\sigma )=\chi _\Omega \mathcal {F}^\ast \sigma \mathcal {F}\chi _\Omega ,where \Omega \subseteq \mathbb {R}^d is a measurable subset and \sigma :\mathbb {R}^d\rightarrow \mathbb {C} is the symbol of the operator A, acting on... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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0... | |
447d3990e48175587f1c511eb30070ad67eae43b | subsection | 20 | 60 | Finite volume truncations of the operator | We have that\chi _\Lambda A\chi _\Omega = B_1B_2,where B_1 and B_2 are the operators on \mathsf {L}^2(\mathbb {R}^d) with kernelsB_1(x,\xi )&:=(2\pi )^{-d/2}\chi _\Lambda (x)e^{ix\cdot \xi }\sqrt{\sigma (\xi )} \\
B_2(\xi ,y)&:=(2\pi )^{-d/2}\sqrt{\sigma (\xi )}e^{-iy\cdot \xi }\chi _\Omega (y).Hence, (REF ) yields\Ver... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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0.... | |
418bd45bcf1d13234134d159212dffefd34ed239 | subsection | 21 | 60 | Symbol estimates | Introduce, for N\ge 0, the Sobolev spaces\mathsf {W}^{N,1}(\mathbb {R}^d):=\lbrace f\in \mathsf {L}^1(\mathbb {R}^d): \ \partial ^\alpha f\in \mathsf {L}^1(\mathbb {R}^d)\ \text{for all} \ \alpha \in \mathbb {N}_0^d, \ |\alpha |\le N\rbrace ,with corresponding norms\Vert f\Vert _N:=\sum \limits _{|\alpha |\le N} \Vert ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-1-4471-2807-6",
"end": 798,
"openalex_id": "https://openalex.org/W617215881",
"raw": "F. Demengel and G. Demengel, Functional spaces for the theory of elliptic partial differential equations. Universitext, Springer, London; EDP ... | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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54b5444ef5f0c88fc607fc58074ec25223ab0ae0 | subsection | 22 | 60 | Symbol estimates | Moreover, for every N\in \mathbb {N}_0, it holds that (t\mapsto \sigma _t)\in \mathsf {L}^1\big (\mathbb {R}, \mathsf {W}^{N,1}(\mathbb {R}^{d-1})\big )\cap \mathsf {C}\big (\mathbb {R}, \mathsf {W}^{N,1}(\mathbb {R}^{d-1})\big ).Using the fact that \sigma \in \mathsf {W}^{\infty ,1}(\mathbb {R}^d) and integrating by p... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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376c18ce10884b9427f25289faf221a3e3e76eae | subsection | 23 | 60 | Symbol estimates | Moreover, the fact that \sigma \in \mathsf {L}^1(\mathbb {R}^d)\cap \mathsf {C}(\mathbb {R}^d) and the uniform bound (REF ) ensure that (t\mapsto \sigma _t)\in \mathsf {L}^1\big (\mathbb {R},\mathsf {L}^1(\mathbb {R}^{d-1})\big )\cap \mathsf {C}\big (\mathbb {R},\mathsf {L}^1(\mathbb {R}^{d-1})\big ). The analogous sta... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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7c79c812c0e89ac6a913ae3c7a28f09810fdcf70 | subsection | 24 | 60 | Localisation estimates | Throughout this subsection, let \sigma \in \mathsf {W}^{\infty ,1}(\mathbb {R}^d) and let h:\mathbb {C}\rightarrow \mathbb {C} be an entire function that vanishes to second order at z=0. One of the main tools for proving Theorem REF is the next proposition. It is of similar spirit as , which was recently established in... | {
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{
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"doi": "10.1007/s00220-018-3161-5",
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"raw": "A. Dietlein, Full Szegő-Type Trace Asymptotics for Ergodic Operators on Large Boxes. Comm. Math. Phys. 2018. doi:10.1007/s00220-018-3161-5.",... | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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d662f47159291dab411b7b6bfb47d4723cdbaaee | subsection | 25 | 60 | Localisation estimates | This assumption is sufficient to prove Theorem REF : we will exclusively apply Proposition REF to the function h_1, see (REF ).Proposition REF follows from approximation of the test function h by polynomials and the next lemma.Lemma 3.6
Let a,b\in \mathbb {R}^d. Then, for all N\in \mathbb {N}, there exists constants c... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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067471a8b6a51c0b475244afd91250f4751c7b70 | subsection | 26 | 60 | Localisation estimates | \Vert \chi _{Q_a} A\chi _{Q_{y_1}}\Vert _2 \Vert \chi _{Q_{y_1}} A\chi _{Q_{y_2}}\Vert \cdot \dots \cdot \Vert \chi _{Q_{k-2}} A\chi _{Q_{y_{k-1}}}\Vert \Vert \chi _{Q_{y_{k-1}}} A\chi _{Q_{b}}\Vert _2\\
&\le \tilde{c}_{M,\sigma }^k\!\!\!\!\sum \limits _{y_1,\dots ,y_{k-1}\in \mathbb {Z}^d}\!\! \langle a-y_1 \rangle ^{... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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0... | |
c966b7a602e78a6e38f3fb9984836c8f22f1ca06 | subsection | 27 | 60 | Localisation estimates | As we may interpolate with (REF ), it suffices to show that\Vert \chi _{Q_a\cap \Lambda }\big [h(A_\Lambda )-h(A_\Omega )\big ]\chi _{Q_b}\Vert _1 \le C_{h,\sigma ,N}^{\prime \prime }\langle \operatorname{dist}(a,\Omega \setminus \Lambda ) \rangle ^{-N}\langle \operatorname{dist}(b,\Omega \setminus \Lambda ) \rangle ^{... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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b33731933389cbf7c8c6e099dfa571c1d2df045e | subsection | 28 | 60 | Localisation estimates | Defining for m,n\in \mathbb {N}_0 the operators\tau _{mn}:=\chi _{Q_a}[A_\Lambda ]^m\chi _\Lambda A\chi _{\Omega \setminus \Lambda }[A_\Omega ]^n\chi _{Q_b},one gets that\chi _{Q_a\cap \Lambda }\big ([A_\Lambda ]^k-[A_\Omega ]^k\big )\chi _{Q_b}&=\sum \limits _{l=0}^{k-1}\chi _{Q_a\cap \Lambda }[A_\Lambda ]^{k-l-1}(A_\... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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... | |
533e2e5726a2afe447cf3596bcb11e87c2388829 | subsection | 29 | 60 | Localisation estimates | Similarly, we estimate for n\ge 1,\Vert \tau _{0n}\Vert _1&\le \sum \limits _{y\in (\Omega \setminus \Lambda )_+}\Vert \chi _{Q_a}A\chi _{Q_y}\Vert _2\Vert \chi _{Q_y}[A_\Omega ]^n\chi _{Q_b}\Vert _2\\
&\le \sum \limits _{y\in (\Omega \setminus \Lambda )_+} \big [c_{M}\Vert \sigma \Vert _{M^{\prime }}\big ]^{n+1}\langl... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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8b9e582824d657de8add4e788175620d99f296bb | subsection | 30 | 60 | Localisation estimates | If Q_b\cap \Omega \setminus \Lambda \ne \emptyset , we estimate\Vert \tau _{m0}\Vert _1&\le \Vert \chi _{Q_a}[A_\Lambda ]^m\chi _\Lambda A\chi _{Q_b}\Vert _1\\
&\le \sum \limits _{x\in \Lambda _+}\Vert \chi _{Q_a} [A_\Lambda ]^m\chi _{Q_x}\Vert _2\Vert \chi _{Q_x}A\chi _{Q_b}\Vert _2 \\
&\le \sum \limits _{x\in \Lambda... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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4503f4a74ee360300fce453b2aceafb0725cb53e | subsection | 31 | 60 | Localisation estimates | Applying Proposition REF for N=d+\beta +1 and the assumption \mathsf {M}\subseteq \Lambda , one gets that\big \Vert \chi _{\mathsf {M}}\big [h(A_\Lambda )-h(A_\Omega )\big ]\big \Vert _1&\le \sum \limits _{{a\in \mathsf {M}_+\\ b\in \mathbb {Z}^d}}\big \Vert \chi _{Q_a\cap \Lambda }\big [h(A_\Lambda )-h(A_\Omega )\big ... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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c008713b4403d897c69ebca9b9e08cd8cce4cbd6 | subsection | 32 | 60 | Localisation estimates | Estimate (REF ) holds.Then one has that\big \Vert \chi _\mathsf {M} \big [h(A_\Lambda )-h(A_\Omega )\big ]\big \Vert _1=\mathcal {O}(L^{-\infty }),as L\rightarrow \infty .As in the proof of Corollary REF , we may assume that \Lambda \subseteq \Omega . Moreover, similarly as in (REF ), an application of Proposition REF ... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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c1f957f90cd1420ff2c445b3399b431d05c6ab84 | subsection | 33 | 60 | Proof of Theorem | Fix \epsilon >0 to be chosen later and recall the definition () of \mathcal {V}=\mathcal {V}^{(\epsilon )}, the one-sided \epsilon –neighbourhood of \partial P. As indicated in Subsection REF , we split \mathcal {V} into (almost) disjoint sets \mathcal {N}(X), X\in \Xi (P), such that \mathcal {N}(X) contains the part o... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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7808fc9921c410d15f3d7e03d313fd9adcfc1980 | subsection | 34 | 60 | Partition of | Fix a vertex X\in \Xi (P) and recall that its adjacent edges are named E^{(1)}(X) and E^{(2)}(X), see Subsection REF . It will be convenient to introduce the following two choices for the unit normal and the unit tangent vector at X:\begin{aligned}
(\tau _X^{(1)},\nu _X^{(1)})&:=(-\tau _{E^{(1)}(X)},\nu _{E^{(1)}(X)}),... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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6b402806f1e32555eeae8121eed4e572a82da715 | subsection | 35 | 60 | Reduction to individual corner contributions | As in the formulation of Theorem REF , let \sigma \in \mathsf {W}^{\infty ,1}(\mathbb {R}^2) and assume that h is an entire function with h(0)=0. Notice that due to Lemma REF the operators h(A_{P_L}) and \chi _{P_L}h(A)\chi _{P_L} are trace class, the trace of the latter operator being computed in (REF ). This gives us... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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9f18d9e8e282c219f29543c863d6fce0b8c0824c | subsection | 36 | 60 | Reduction to individual corner contributions | Moreover, it is not difficult to see that\operatorname{dist}\big (\mathsf {N}_L(X)\,,\,C(X)\triangle (P-X)_L\big )\gtrsim L.Hence, Corollary REF with Assumption (REF ) yields that\operatorname{tr}\big (\chi _{\mathsf {N}_L(X)}\big [h(A_{(P-X)_L})-h(A)\big ]\big )=\operatorname{tr}\big (\chi _{\mathsf {N}_L(X)}\big [h(A... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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431bb25e6c45b09290a72658084b3c370202f659 | subsection | 37 | 60 | The | In the smooth boundary case, the sub-leading order term in the asymptotics (REF ) is (at least morally) obtained via approximation of the operator h(A_{\Omega _L}) by half-space operators: around x\in \partial \Omega _L, the operator h(A_{\Omega _L}) is replaced by h(A_{H_x}) where H_x is the half-space approximation o... | {
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"raw": "B. H. Thorsen, An N-dimensional analogue of Szegő's limit theorem. J. Math. Anal. Appl. 198(1): 137–165, 1996.",
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"start": 491... | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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0.0023977786768227816,
0.017244169488549... | |
b3a23f40f90fe40af8a2eb0c39418507a7bfed66 | subsection | 38 | 60 | The | Then it follows from Corollary REF that\operatorname{tr}\big (\chi _{T_L}\big [h(A_{H})-h(A)\big ]\big )=\operatorname{tr}\big (\chi _{L\tfrac{|E|}{2}\cdot S}\big [h(A_H)-h(A)\big ]\big )+\mathcal {O}(L^{-\infty }).Here, the trace on the right-hand side is well-defined due to Corollary REF . Also, the invariance of the... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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8fafcf4e417273f2cba394af8bd64903453c163d | subsection | 39 | 60 | Regularisation of sector operators | The key to finding the constant order term in the asymptotics of (REF ) is a trace-class regularisation of the sector operator h(A_{C}) with the help of the half-space operators h(A_{H^{(j)}}), j=1,2, and the full-space operator h(A). This regularisation is given in the next proposition. For its proof we consider spati... | {
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"raw": "A. Dietlein, Full Szegő-Type Trace Asymptotics for Ergodic Operators on Large Boxes. Comm. Math. Phys. 2018. doi:10.1007/s00220-018-3161-5.",... | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
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] | 2,018 | en | Mathematics | [
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732d7339c8184732c0700b6072bcc761400fba68 | subsection | 40 | 60 | Regularisation of sector operators | Then one can writeZ&=\chi _{C_l}\big [ h(A_C)-h(A_{H^{(1)}})\big ]+\chi _{C_r}\big [ h(A_C)-h(A_{H^{(2)}})\big ]\\
&\ \ \ +\chi _{C_l}\big [ h(A)-h(A_{H^{(2)}})\big ]+\chi _{C_r}\big [ h(A)-h(A_{H^{(1)}})\big ].Thus, Corollary REF implies that the operator Z is trace class since the estimate (REF ) with \beta =1 is eas... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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c17395d2c7b3a0681615ca7524a6959a64f83ee9 | subsection | 41 | 60 | Contributions from non-right-angled corners | In the next subsection we will apply the regularisation for the sector operator h(A_{C}) from Proposition REF to find the asymptotics of the trace (REF ). As it turns out during this process, non-perpendicular edges E^{(1)} and E^{(2)} generate an extra term of constant order. Technically, this relies on the fact that ... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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525fe03f306b40467fa79a3497574ca56df19222 | subsection | 42 | 60 | Contributions from non-right-angled corners | The other cases can be reduced to this one via a symmetry argument. After a suitable rotation we may also assume that H=\mathbb {R}\times [0,\infty ), \Gamma =\lbrace (x_1,x_2)\in H: 0\le x_1\le \cot (\gamma )x_2\rbrace , and S=[0,1]\times [0,\infty ). Splitting the strip S into unit cubes, one easily gets from Proposi... | {
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{
"arxiv_id": "",
"doi": "10.1090/s0002-9939-1988-0929421-x",
"end": 901,
"openalex_id": "https://openalex.org/W2013275976",
"raw": "C. Brislawn, Kernels of Trace Class Operators. Proc. Amer. Math. Soc. 104(4): 1181–1190, 1988.",
"source_ref_id": "ec6075b58249... | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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07fddc7c8472586ec28f31af06cec652a0f8af47 | subsection | 43 | 60 | Complete asymptotics | Equipped with Proposition REF and Lemmas REF and REF , we are now ready to extract the asymptotics from (REF ). As the regularisation for the sector operators in Proposition REF depends on the type of the sector, we naturally have to distinguish convex and concave corners of the polygon P_L. Propositions REF and REF co... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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0... | |
59180d7eb560f4f47fc105f6606cedb38ed01cf4 | subsection | 44 | 60 | Complete asymptotics | Thus it remains to find the asymptotics for\operatorname{tr}\big (\chi _{\mathsf {N}_L}\big [h(A_{H^{(j)}})-h(A)\big ]\big ), \ j=1,2.Recall the definition (REF ) of the sectors \Gamma ^{(j)}
and define its finite sections\Gamma ^{(j)}[r]:=\lbrace y \in \Gamma ^{(j)}:y\cdot \nu ^{(j)}\le r\rbrace ,\ j=1,2,\ r\ge 0.Appl... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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0... | |
6d440f2cef60ab93f58157c81777494553e1d903 | subsection | 45 | 60 | Complete asymptotics | Then we have that\operatorname{tr}\big (\chi _{\mathsf {N}_L}&[h(A_{C})-h(A)]\big )=L\,\sum \limits _{j=1}^2\tfrac{|E^{(j)}|}{2}\operatorname{tr}\big (\chi _{S^{(j)}}\big [h(A_{H^{(j)}})-h(A)\big ]\big )\\
&+\operatorname{tr}\big (\chi _{H^{(1)}\cap H^{(2)}}\big [h(A_{C})-h(A)\big ]\big )\\[2ex]
&+\operatorname{tr}\big... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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... | |
f060ba07aaf5235f6eb499d2c88a7dad1dc0e4b7 | subsection | 46 | 60 | Complete asymptotics | We write\operatorname{tr}\big (\chi _{\mathsf {N}_L}[h(A_{C})-h(A)]\big )=&\eta _1(L)+\eta _2(L),with\eta _1(L)&:=\operatorname{tr}\big (\chi _{\mathsf {N}_L\cap H^{(1)}\cap H^{(2)}}\big [h(A_C)-h(A)\big ]\big )+\operatorname{tr}\big (\chi _{\mathsf {N}_L\cap C\setminus H^{(1)}}\big [h(A_C)-h(A_{H^{(2)}})\big ]\big )\\... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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0.... | |
05e52c06118c0a83aa04c3337354a660366618ac | subsection | 47 | 60 | Complete asymptotics | Alternatively, one easily gets that, for instance,\operatorname{tr}\big (\chi _{\mathsf {N}_L\cap C\setminus H^{(1)}}\big [h(A_{H^{(2)}})&-h(A)\big ]\big )=\operatorname{tr}\big (\chi _{T_L^{(2)}}\big [h(A_{H^{(2)}})-h(A)\big ]\big )\\
&+\operatorname{sgn}(\gamma -\tfrac{3\pi }{2})\operatorname{tr}\big (\chi _{\mathsf ... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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e8265d4e8c6c3a33be0fbaf4e607b4d0df41698c | subsection | 48 | 60 | Proof of Theorem | It suffices to prove the theorem for test functions h of the form h(z)=\sum \limits _{k=2}^\infty a_kz^k since both sides of (REF ) and () vanish for linear functions h. Moreover, we may assume after a suitable rotation that H_E=H=\mathbb {R}\times [0,\infty ), i.e. S_E=S=[0,1]\times [0,\infty ). Thus, we have that\sig... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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ad0d9870474e5323527681887bb53651dbf3322e | subsection | 49 | 60 | Proof of Theorem | In particular, the right-hand sides of (REF ) and () are well-defined under our assumptions on h and \sigma .Introduce the unitary (identification) mapJ:\mathsf {L}^2(\mathbb {R}^2)\rightarrow \mathsf {L}^2\big (\mathbb {R},\mathsf {L}^2(\mathbb {R})\big ),\ (Jf)(t):=f(t,\,\cdot \,).Moreover, define the partial Fourier... | {
"cite_spans": [
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"raw": "M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.",
"source_ref_id": "db89... | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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-... | |
a8a32c343907520516ec51f0f581313e2d91aaef | subsection | 50 | 60 | Proof of Theorem | To verify (REF ), notice that\mathcal {F}_1\chi _H =\chi _H\mathcal {F}_1,henceA_H=\chi _H\mathcal {F}_1^\ast \mathcal {F}_2^\ast \sigma \mathcal {F}_2\mathcal {F}_1\chi _H=\mathcal {F}_1^\ast \chi _H\mathcal {F}_2^\ast \sigma \mathcal {F}_2\chi _H\mathcal {F}_1.Moreover, the definition of J yields that\chi _H\mathcal ... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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8d78f26a3c67c426bf6981acb7b7c3c329b92b82 | subsection | 51 | 60 | Proof of Theorem | Namely, for \phi ,\psi \in \mathsf {L}^2(\mathbb {R}), we have that\big \langle \phi \otimes \psi ,\tilde{B}_\alpha (\phi \otimes \psi )\big \rangle _{\mathsf {L}^2(\mathbb {R}^2)}&=\big \langle J((\mathcal {F}\phi )\otimes \psi ), B_\alpha J ((\mathcal {F}\phi )\otimes \psi )\big \rangle _{\mathsf {L}^2(\mathbb {R},\m... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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f3410593a21ecc2ebcf8d5c016c55a713cd5df8c | subsection | 52 | 60 | Proof of Theorem | Then (REF ) implies that\operatorname{tr}\big (\chi _S \tilde{B}_\alpha \chi _S\big )&=\sum \limits _{n,m\in \mathbb {N}} \langle \psi _n\otimes \psi _m,\chi _S\tilde{B}_\alpha \chi _S \psi _n\otimes \psi _m\rangle _{\mathsf {L}^2(\mathbb {R}^2)}\\
&=\sum \limits _{n,m\in \mathbb {N}}\int \limits _\mathbb {R}dt\, |\mat... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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0.0040... | |
e4183f3d0fb238ce28c389440d56466a6b582768 | subsection | 53 | 60 | Radially symmetric symbols – Proof of Theorem | As in the statement of Theorem REF assume that the symbol \sigma is radially symmetric and the test function h is a quadratic polynomial, i.e. h(z)=z^2+bz for some b\in \mathbb {C}. The coefficient c_2=c_2(P,h,\sigma ) is easily computed from Theorem REF . Recall also that the linear part of h does not contribute to th... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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f0fe88ae8c23afa81056dbb60b53a02bc0547b65 | subsection | 54 | 60 | Radially symmetric symbols – Proof of Theorem | \int \limits _x^\infty dy\,\check{\sigma }_t(y)\check{\sigma }_t(-y)\\
&=\int \limits _0^\infty dy\,\check{\sigma }_t(y) \check{\sigma }_t(-y)\int \limits _0^y dx\,x^\alpha \\
&=\frac{1}{2}\int \limits _{-\infty }^\infty dy\, \frac{|y|^{\alpha +1}}{\alpha +1}\check{\sigma }_t(y)\check{\sigma }_t(-y).Parseval's identity... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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8e56a468fac054cc45654ea703d044a96037b233 | subsection | 55 | 60 | Radially symmetric symbols – Proof of Theorem | This calculation is performed in the next lemma.Lemma 7.2
Let h(z)=z^2 and assume that the symbol \sigma \in \mathsf {W}^{\infty ,1}(\mathbb {R}^2) is radially symmetric.Then for every X\in \Xi (P) the formulab_0(X)=\frac{1-\gamma _X\cot (\gamma _X)}{2}\int \limits _0^\infty dr\, r^3\check{\sigma }(r)^2holds.Fix X\in ... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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764726cda3ca53e596b19e907a1477ab80350abf | subsection | 56 | 60 | Radially symmetric symbols – Proof of Theorem | Then we get from (REF ) thatb_0&=\operatorname{tr}\big [\chi _{H^{(1)}\cap H^{(2)}}\big ([A_{C}]^2-A^2\big )\big ]+\operatorname{tr}\big [\chi _{C\setminus H^{(1)}}\big ([A_{C}]^2-[A_{H^{(2)}}]^2\big )\big ]\\[2ex]
&\ \ \ +\operatorname{tr}\big [\chi _{C\setminus H^{(2)}}\big ([A_{C}]^2-[A_{H^{(1)}}]^2\big )\big ]\\
&=... | {
"cite_spans": []
} | 1807.04714 | A Szeg\H{o} limit theorem for translation-invariant operators on
polygons | [
"Bernhard Pfirsch"
] | [
"math.SP",
"math-ph",
"math.MP"
] | 2,018 | en | Mathematics | [
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7a377a48b4ed5fca82bf7ef87fb966ee016174a1 | subsection | 57 | 60 | Appendix | The purpose of this appendix is to provide a proof of the following result.Lemma A.1
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ba4e7f6d145d912412b890db9855424eaa8b74bc | abstract | 0 | 42 | Abstract | HIV RNA viral load (VL) is an important outcome variable in studies of HIV
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67d9861650ee2a87f9295a69cd75008fa9d6a09f | subsection | 1 | 42 | Abstract | HIV RNA viral load (VL) is an important outcome variable in studies of HIV infected persons. There exists only a handful of methods which classify patients by viral load patterns. Most methods place limits on the use of viral load measurements, are often specific to a particular study design, and do not account for com... | {
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586cd8a66c77be0610928792c13527e12818de99 | subsection | 2 | 42 | Introduction | The primary clinical goal of HIV treatment and patient engagement is suppression of the HIV viral load (VL), as measured by low or undetectable circulating HIV RNA levels. However, VL most often fluctuates over repeated measurements, with a range that spans 8 orders of magnitude from 0 (undetectable) - 10^7 copies/mL. ... | {
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bba6c81371d14b3e8972c62769f2dd0aec12709d | subsection | 3 | 42 | Introduction | These methods begin by assigning a set of features to a patient (e.g. demographics, laboratory measurements, therapies) and then performing computational clustering to identify similar classes of patients. Several studies have applied machine learning methods in HIV research to predict HIV viral load responses or CD4 T... | {
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73b298b1e9528124538b5acfccd88fc072367d39 | subsection | 4 | 42 | Human Subjects Protection | This proposal was reviewed and approved by the University of Rochester Human Subjects Review Board (protocol number RSRB00068884). The analysis in this paper is presented in compliance with Center for Medicare Services (CMS) current cell size suppression policy. Data were coded such that patients could not be identifie... | {
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30831acc840a8595c6ca5d0180af5bf80d4fd4b7 | subsection | 5 | 42 | Study Data | We obtained medical encounter data from all patients with an HIV diagnosis in the University of Rochester Medical Center's electronic medical record system (EMR) between 2011 and 2016. For each patient we have information on their age, gender, race, ethnicity, zip code, and coded procedures with associated ICD9 and ICD... | {
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9bd1434a89885941439e6db5d1f974bc10ca09b9 | subsection | 6 | 42 | Data Availability | The data is provided in | {
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d3a06861bb4eb5f8cb5341854028cf64c7790934 | subsection | 7 | 42 | Hardware and Software Specifications | Analyses were performed on a Windows 8 server with Intel(R) Xeon(R) CPUs E5-2620 v2 @ 2.10GHz and 256GB of RAM. Python 2.7 was used for most data mining and machine learning under Spyder v.3 installed from Aanaconda2 (64-bit). The default packages available in Anaconda were used for analysis, including, but not limited... | {
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f9ba2f3e5ab42ec913db47295e02522006dcb2aa | subsection | 8 | 42 | Viral Load Analysis Methods | Mathematical notations for this work are described in Table REF .
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30a998f7a1671ab47dbe7accd260b025749eddb4 | subsection | 9 | 42 | Feature Vector Definition | We next designed a feature vector to capture characteristics that would allow us to distinguish between viral load patterns. Since viral load data is asynchronous and noisy, with variable numbers of data points for each subject, we argue that one or two viral load measurements are too few to accurately judge viral load... | {
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7723190089e02900976dba8f9ee4e5d26eabdf14 | subsection | 10 | 42 | Feature Vector Definition | More specifically, the weight function follows an inverse square root function (f(x) = \frac{1}{\sqrt{x}}) rather than an inverse function (g(x) = \frac{1}{x}). This has the advantage of avoiding rapid convergence of g(x) to zero when time is measured in units of days (Eq REF ). Weighted recency is then calculated as t... | {
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d8fce110f33e5978183c7389d666d69f59ae92d9 | subsection | 11 | 42 | Feature Vector Definition | \textrm {grnd}(x) =
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900b3431096287789e40d26edd1301d80da6c768 | subsection | 12 | 42 | Analytic Terminology | Here we formally define keywords which will appear in the analysis: Let Feature extraction be the process of determining the values \dot{A}, wRR, \check{D}, and IQR from a set of patients (using their viral load patterns) with the formulations given above. Then a feature vector (\vec{F}_p) contains the values \dot{A}_p... | {
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54b8d928fb02d8846e2542395c572c4fe30a209c | subsection | 13 | 42 | Feature Extraction and Normalization | We began by transforming viral load data by min-max normalization to equally weight the temporal features of the VL series (Eq REF ). That is, we normalize the features, F, to a range between [0, 1] using equation REF where F^* = f(F) (sfig:outlineB.1).F^* = f(F) = \frac{F - \min F}{\max F - \min F}Next, we examined ea... | {
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49bd6b66b7026ac40e69ed6bde9df324d424a710 | subsection | 14 | 42 | Hierarchical Clustering | We then performed hierarchical clustering of the individual subjects using a Euclidean distance metric and the Ward's criterion to minimize the total within-cluster variance, revealing a clear separation into 5 distinct patient groups (sfig:outlineC).From Fig REF and Fig REF , we find that the bluish green cluster (n=4... | {
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ceabdb92136eaeff5c2fcf8376ee55baed841251 | subsection | 15 | 42 | Comparison with Existing Categorization Methods | Visually, we find that the SLVL group detected by our method is very similar to the LLVR group defined by Greub et. al. (Fig REF ). Furthermore, it appears that the methods trying to capture SHVL, viral rebound, and viral failure patients did not succeed as well as the identification of SHVL and RVL patients in our met... | {
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0731c05997170374be46cec9e085fbea973a1163 | subsection | 16 | 42 | Classification Stability | To address the potential issue of misclassification, and to assess the changing nature of viral load pattern membership assignment, we performed a time-varying classification sensitivity analysis. First, we train a supervised machine learning method on the results from our original classification. Then, to determine th... | {
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2c16d77fc56d84a33a86ae0908735cc093af524d | subsection | 17 | 42 | Centroid Summarization | Prior to performing partial viral load membership assignment, we initially consider several common supervised learning methods for learning on our initial classification results: k-nearest neighbors (k=5,7,9) , support vector machine (SVM), decision tree, AdaBoost, and random forests. We tested the performance of these... | {
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9b80145a5ee7775145dd0ae13d2c656110e251c0 | subsection | 18 | 42 | Centroid Summarization | BIRCH defines its centroid as the average of all the points (multidimensional mean) which scored lower than four other centroid methods. From amongst these CM, the polyhedral CMuRN scored the highest in terms of average F_1, and although this method did not outperform the more intricate supervised machine learning meth... | {
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7466e58c830e8b2c6eb78a4b026c98bddf45947e | subsection | 19 | 42 | Membership Assignment on Partially Retained VL Patterns | To determine the classification stability, we iteratively assign VL pattern membership using only partial viral load data for every patient. First, we extract the feature vector {F} from the partially retained data (sfig:outlineG.2). We then transform {F} to the same normalized space the polyhedral CM was trained on. N... | {
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0e00dbb9fc6fcca4bd1ce4ae7125a885f3dd8d95 | subsection | 20 | 42 | Membership Assignment on Partially Retained VL Patterns | This type of progression is likely because the viral load suppression class is time dependent and thus VL time series taken early, or later, with respect to treatment initiation would be expected to differ based on the full treatment response. Similarly we find the RVL group's true viral load membership assignment incr... | {
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87035e7fe4a029ecec7ebb363a815e666dad3291 | subsection | 21 | 42 | Projected Hyperplane Normalization | We were initially perplexed to find the smallest disk scoring low in Table REF because this method attempts to find the center by minimizing the radius, hence one may speculate that it should score the highest from amongst the centroid methods using RN. However, the result led us to recognize that one potential drawbac... | {
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f2ee0f49cc22b7b2c1a8f30ae3147e634a060ff0 | subsection | 22 | 42 | Discussion | Researchers have previously performed HIV population case studies using differing schema to classify VL patterns , , , . Our work is unique as it suggests a method for standardizing the classification of VL patterns using a set of optimally segregating features. These features have been specifically engineered to optim... | {
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751cc48a45c61af4bd8a2b225e266de9cc7a95aa | subsection | 23 | 42 | Discussion | In addition, our feature vector was designed specifically to suit the literature rather than objectively clustering the data using a time-series based clustering method , , . Also, some of our features are slightly collinear - with the greatest correlation coefficient being between IQR and wRR (-0.717). However, while ... | {
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0257f570a75c29479a08b1e2eed00d576d18aca3 | subsection | 24 | 42 | Conclusion | We have proposed a set of four unambiguous features which have been successfully used in segregating five different types of viral load patterns: durably suppressed viral load (DSVL), sustained low viral load (SLVL), sustained high viral load (SHVL), high viral load suppression (HVLS), and rebounding viral load (RVL). ... | {
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b2a15ca86ad3a4e00dddbe782b2856b248bb1703 | subsection | 25 | 42 | Fig S1 | Viral load distribution. For each pair of viral load measurements, we calculate the change in days and the change in viral load counts for all patients and plot it as a scatter. The horizontal line of dots which appears between 0 and 2 are an artifact of using 20 and 48 in data to replace the “Pos <20" and “Pos <48" va... | {
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e2e621accafb8d6dd8474ed094e60cd2bedf2213 | subsection | 26 | 42 | Fig S2 | Paper Outline. The light green box indicates the training step and the light red box indicates the testing step. (A) Feature extraction (B) Normalization of extracted features. Each feature has a unique linear transformation for normalization. (C) Unsupervised hierarchical clustering into clouds (the six different VL p... | {
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e79b0d665f0a4444a8cf5582dcf60aa49338dfc1 | subsection | 27 | 42 | Fig S3 | Patient feature extraction. Feature extraction on 1576 patients displayed as 2D splicing of the 4 dimensional feature space. Each splice plots a dimension versus another in the form of a scatter plot. | {
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52adbaf71260ce2c1d589f0d96f5bf31fbd93c12 | subsection | 28 | 42 | Fig S4 | Decision Tree. While some useful rules may be pruned, the tree is otherwise complicated and difficult to draw useful conclusions from. | {
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172b31227d568cc223a452101d89ecafe1fd5d98 | subsection | 29 | 42 | Fig S5 | Decision Tree Classification Stability. Classification stability results using decision tree learned on 100% retained data. The title of each plot corresponds to which category of patients are being tested. The dashed white line is the 80% membership assignment probability line, included for reference. The solid white ... | {
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a4d436d041aecee38440c08c4cb8dcdfc2780af7 | subsection | 30 | 42 | Fig S6 | SVM Classification Stability. Classification stability results using SVM learned on 100% retained data. The title of each plot corresponds to which category of patients are being tested. The dashed white line is the 80% membership assignment probability line, included for reference. The solid white line represents the ... | {
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6c8d68d845d72a71b449e39f70268c38058e2377 | subsection | 31 | 42 | Fig S7 | k-Nearest Neighbor Classification Stability. Classification stability results using k-Nearest Neighbor (k = 5) learned on 100% retained data. The title of each plot corresponds to which category of patients are being tested. The dashed white line is the 80% membership assignment probability line, included for reference... | {
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be764f85935c49ddd882e40bdd0ddf3668d54565 | subsection | 32 | 42 | Fig S8 | Polyhedral CMuPHN Classification Stability. Classification stability results using the polyhedral method for detecting center and projected hyperplane normalization method for classification. The title of each plot corresponds to which category of patients are being tested. The dashed white line is the 80% membership a... | {
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e021e568ef20800bdf45df9eb5767261a9dc9223 | subsection | 33 | 42 | Fig S9 | Centroid Methods. Gives a visual of how the seven methods work on an example point set. The green target signifies the exact center which is found according to the different methods in our algorithm. | {
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0a431ab75ccf62910ffa5ec136c238bec6bdc7fe | subsection | 34 | 42 | Data S1 | Viral load data. The data set used for this study is provided in a completely deidentified format. The data is in a csv format where the first column represents a unique subject, with a random identifier. The subsequent values are as t_{i,j}, VL_{i,j}, where t_{i,j} is the time from a universal T_0 for the VL measureme... | {
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48af0f4945881d53d63f00dcd86ba494445f96ec | subsection | 35 | 42 | Greub et. al. LLVR | Greub et. al. were particularly focused on detecting low level viral rebound (LLVR) in patients . The following procedure was used to categorize the patients of their study:If the patient has two consecutive viral load measurements (VLM) less than 50, within a 24 week period, and they have two VLM after this consecutiv... | {
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f1edb84d3926b7ba821a7ae9533fddfb7de1c619 | subsection | 36 | 42 | Rose et. al. SMVL/RMVL | The focus of Rose et. al. was to investigate the use of several frameworks in categorizing suppressed versus not-suppressed viral load . First they omitted the patients from their study whom were virally suppressed at baseline, where they define viral suppression as < 200 copies/mL because they were found to have no su... | {
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f9c30aafbf34eb17c420966ab21f81b4a52f5730 | subsection | 37 | 42 | Terzian et. al. SHVL | The objective of Terzian et. al. was to develop a method of categorizing a patient as DSVL or SHVL for the purpose of monitoring successful ART uptake . Their procedure for categorizing patients is as follows:If the maximum viral load of the patient is \le 400 copies/mL then the patient is labeled as `DSVL'. If the pat... | {
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1744731a3addc9d2a85965698f86507256bcbc8f | subsection | 38 | 42 | Phillips et. al. Viral Rebound | The aim of Phillips et. al. was to characterize virological response to ART . While the statistical methods proposed by Phillips et. al. went beyond categorizing patients, they composed a method to identify two populations of HIV patients (Viral Failure and Viral Rebound):Only patients who have at least one VLM within ... | {
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4fecd2cd0caaa372a6195f48edf4b34df0507d2b | subsection | 39 | 42 | Considered but Removed Features | There were several features which were thought to have significance in segregating viral load patterns but did not make it into our feature vector. We had to be careful of extracting features which may be collinear as it would cause a shift in the weighting of features. These collinear features are too many to list her... | {
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f5a7dd3a2738d8a8b80c7db5f717926dffb5ce05 | subsection | 40 | 42 | Centroid Detection Methodologies | Centroid detection is a problem which several machine learning algorithms attempt to solve, such as Support Vector Machine (SVM), Bayesian Point Machines (BPM), Analytic Centre Machines, k-Means, BIRCH, among others , . The center of a cloud of samples is generally considered the average , but is still a matter of inte... | {
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