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77eca990c859c577485c33e306c222f13a22cd64 | subsection | 7 | 17 | Low-rankness on tensor factors | We propose a new definition on low-rank tensor, which gives the low-rankness on the decomposition factors of a tensor, for TR decomposition, the low-rank model is formulated as:\begin{aligned}\min \limits _{\lbrace {\mathcal {G}}_n\rbrace _{n=1}^N,{\mathcal {X}}} \ \sum _{n=1}^N \text{Rank}({\mathcal {G}}_n) + \frac{\l... | {
"cite_spans": []
} | 1805.08468 | Rank Minimization on Tensor Ring: A New Paradigm in Scalable Tensor
Decomposition and Completion | [
"Longhao Yuan",
"Chao Li",
"Danilo Mandic",
"Jianting Cao",
"Qibin Zhao"
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f4a662296a43bd3c9d68ca88566a985789696e67 | subsection | 8 | 17 | Low-rankness on tensor factors | More specifically, our tensor ring overlapped low-rank factor (TR-OLRF) model is formulated as follows:\begin{aligned}\min \limits _{\lbrace {\mathcal {G}}_n\rbrace _{n=1}^N,{\mathcal {X}}} \ \sum _{n=1}^N\sum _{i=3}^N \Vert \Gamma _i(\mathbf {G}_n) \Vert _* + \frac{\lambda }{2}\Vert {\mathcal {X}}- {\mathcal {Z}}(\lbr... | {
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Decomposition and Completion | [
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26bc9c70eced00a50a1945d6fee2dccfe282942a | subsection | 9 | 17 | TR-OLRF | To solve the equations (REF ) and (REF ), we apply the augmented Lagrangian multiplier method (ADMM) which is efficient and widely used. Because the variables of TR-OLRF are interdependent, we adopt alternative variables, and the augmented Lagrangian function of TR-OLRF model is:\begin{aligned}&{\mathcal {L}} \left( \l... | {
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Decomposition and Completion | [
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e96c461096ae1efcaa19f0d0ab19279cebe06c1e | subsection | 10 | 17 | TR-OLRF | \end{aligned}For n=1, \ldots , N, the kth iteration update scheme of alternating direction method of multipliers (ADMM) of TR-OLRF model is listed below:\left\lbrace
\begin{array}{lr}
{\mathcal {G}}_n^{k+1}=\bar{\Gamma }_2((\lambda \Delta _n(\mathbf {X}^k)(\mathbf {Q}_n^k)^T+\mu \sum _{i=1}^{3}\Gamma _2(\mathbf {M}_{n... | {
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Decomposition and Completion | [
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638eecdf3228a5c728aebac0853734966de8d664 | subsection | 11 | 17 | TR-LLRF | Similarly, the augmented Lagrangian function of TR-LLRF model can be written as:\begin{aligned}&{\mathcal {L}}\left(\lbrace {\mathcal {G}}_n\rbrace _{n=1}^N,{\mathcal {X}},\lbrace {\mathcal {W}}_{ni}\rbrace _{n=1,i=1}^{N,3},\lbrace {\mathcal {Y}}_n\rbrace _{n=1}^N\right)= \sum _{n=1}^N\sum _{i=1}^3\Vert \Gamma _i(\math... | {
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} | 1805.08468 | Rank Minimization on Tensor Ring: A New Paradigm in Scalable Tensor
Decomposition and Completion | [
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"Chao Li",
"Danilo Mandic",
"Jianting Cao",
"Qibin Zhao"
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352e18507d7fc62b06154d581db319cc9a175ed4 | subsection | 12 | 17 | TR-LLRF | \end{aligned}The corresponding update scheme of TR-LLRF model is listed below:\left\lbrace
\begin{array}{lr}
{\mathcal {G}}_n^{k+1}=\bar{\Gamma }_2((\lambda \Delta _n(\mathbf {X}^k)(\mathbf {Q}_n^k)^T+\mu \sum _{i=1}^3 \Gamma _2(\mathbf {W}_{ni}^k)+\Gamma _2(\mathbf {Y}_n^k))(\lambda \mathbf {Q}_n^k(\mathbf {Q}_n^k)^T... | {
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} | 1805.08468 | Rank Minimization on Tensor Ring: A New Paradigm in Scalable Tensor
Decomposition and Completion | [
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d4e31f757f437f3c8a7ce93a9d8b68c15e9fa4eb | subsection | 13 | 17 | Computational complexity | We next compared the computational complexity of our TR-OLRF and TR-LLRF to the state-of-the-art algorithms TR-ALS , SiLRTC-TT , SiLRTC and FBCP . The comparative algorithms are state-of-the-art algorithms and are similar to our algorithms. The complexities are summarized in Tab. REF , where we denote the dimension of ... | {
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Decomposition and Completion | [
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144aa104657e74ccf885ac16e389b7edfa97dea8 | subsection | 14 | 17 | Synthetic data | To verifying the performance of our two proposed algorithms, we test two tensors of size 10\times 10\times 10\times 10 and 4 \times 4\times 4\times 6\times 6\times 6. The tensors were generated by TR factors of TR-ranks \lbrace 4,5,4,5\rbrace and \lbrace 4,4,4,4,4,4\rbrace respectively. The values of the TR factors wer... | {
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} | 1805.08468 | Rank Minimization on Tensor Ring: A New Paradigm in Scalable Tensor
Decomposition and Completion | [
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900d7403032cfb28e888121dc24b2fad783f2a3c | subsection | 15 | 17 | Hyperspectral image | A hyperspectral image of size 200\times \ 200\times 80 was next considered. This was an image of urban landscape collected by a satellite. We compare our TR-OLRF and TR-LLRF to TR-ALS , TT-SiLRTC , SiLRTC and BCPF. We examined order-three, order-five, order-seven and order-eight tensors respectively. The missing rate i... | {
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Decomposition and Completion | [
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9230008f46e24f7dbdf93054d67c1297f7ac4f9e | subsection | 16 | 17 | Conclusion | In order to solve the large-scale SVD calculation and rank selection problem that most tensor completion methods have. We proposed two algorithms which impose low-rank assumption on tensor factors. Based on tensor ring decomposition, we proposed two optimization models named as TR-OLRF and TR-LLRF. The two models can b... | {
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} | 1805.08468 | Rank Minimization on Tensor Ring: A New Paradigm in Scalable Tensor
Decomposition and Completion | [
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"Chao Li",
"Danilo Mandic",
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a80be8eaf3e30128c5c3d14074c200c97d14dba8 | abstract | 0 | 62 | Abstract | We study the injectivity and surjectivity of the Borel map in three
instances: in Roumieu-Carleman ultraholomorphic classes in unbounded sectors of
the Riemann surface of the logarithm, and in classes of functions admitting,
uniform or nonuniform, asymptotic expansion at the corresponding vertex. These
classes are defi... | {
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} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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269dc9657211a678baa5807618d15243f2ed9c40 | subsection | 1 | 62 | Introduction | In 1886, H. Poincaré boosted the mathematical interest in formal (usually divergent) power series by introducing the notion of asymptotic expansion, which is a kind of
Taylor expansion which provides successive approximations: a complex function f, holomorphic on a sector
S=\lbrace z\in \mathbb {C}: 0<|z|<r,\ a<\arg (z... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
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4363ad2aa9f39b302177e952ec37bb5d2cc92fc3 | subsection | 2 | 62 | Introduction | The sequence \mathbb {M}_{1/k}=(p!^{1/k})_{p\in \mathbb {N}_0} is the Gevrey sequence of order 1/k, f is said to be 1/k-Gevrey asymptotic to \widehat{f} (denoted by f\in \widetilde{\mathcal {A}}_{\mathbb {M}_{1/k}}(S)), and \widehat{f}, because of the estimates satisfied by its coefficients, is said to be a 1/k-Gevrey ... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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aac86adc322e94c1d1265ed8e6e9e6adda6909a8 | subsection | 3 | 62 | Introduction | M_p} <\infty .In order to guarantee some stability properties for these classes, and to avoid trivial situations, we will always assume that
\mathbb {M} is a weight sequence, that is, a logarithmically convex sequence such that its sequence of quotients of consecutive terms, {m}=(m_p=M_{p+1}/M_p)_{p\in \mathbb {N}_0}, ... | {
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eb0adeccc544d2d20bc87d8b6a5a9cac4d074196 | subsection | 4 | 62 | Introduction | Indeed, it was observed that, for the previous arguments to work, d_{\mathbb {M}} need not be a nonzero proximate order, but rather be close enough to one such order (we say
\mathbb {M} admits a nonzero proximate order, see Theorem REF ). It is then natural to ask oneself whether every strongly regular sequence admits ... | {
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0c7105e81ac65656f0d19dd9aeebd54ce0faa01a | subsection | 5 | 62 | Introduction | Finally, in the second author generalized the Borel–Ritt–Gevrey theorem for strongly regular sequences such that the auxiliary function d_{\mathbb {M}} is a proximate order (or, less demanding, sequences admitting a nonzero proximate order): the Borel map \widetilde{\mathcal {B}}: \widetilde{\mathcal {A}}_{\mathbb {M}... | {
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87bdd70a248656bf8dd0664d38fd717783988076 | subsection | 6 | 62 | Introduction | Without any other assumption on \mathbb {M}, no result stating the surjectivity of the Borel map is available, but we may give some information on the maximal possible opening for which surjectivity could occur by resting on results by Schmets and Valdivia and on the use of suitable Borel-like integral transforms, see... | {
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141f15d12869f028d571bb1cc676943a93e905ae | subsection | 7 | 62 | Notation | We set \mathbb {N}:=\lbrace 1,2,...\rbrace , \mathbb {N}_{0}:=\mathbb {N}\cup \lbrace 0\rbrace .
\mathcal {R} stands for the Riemann surface of the logarithm, and
\mathbb {C}[[z]] is the space of formal power series in z with complex coefficients.For \gamma >0, we consider unbounded sectors bisected by direction 0,S_{\... | {
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"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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c26b5f1ae3276f5638eae05236469396fee90e0a | subsection | 8 | 62 | Sequences and associated functions | In what follows, \mathbb {M}=(M_p)_{p\in \mathbb {N}_0} will always stand for a sequence of
positive real numbers, and we will always assume that M_0=1. The following properties for such a sequence will play a role in this paper.Definition 2.1 We say that:(i)
\mathbb {M} is logarithmically convex (for short, (lc)) if
M... | {
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"source_ref_id": "a... | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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] | 2,018 | en | Mathematics | [
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332631a61dc56585a8116310e60145c9b6d10a77 | subsection | 9 | 62 | Sequences and associated functions | It is immediate that if \mathbb {M} is (lc) and (snq), then \mathbb {M} is a weight sequence.Example 2.4
We mention some interesting examples. In particular, those in (i) and (iii) appear in the applications of summability theory to the study of formal power series solutions for different kinds of equations.(i)
The se... | {
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... | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
] | [
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64072418457385b412e2bd2a135e28ad7c82f7b8 | subsection | 10 | 62 | Sequences and associated functions | \end{array}\right.}One may also consider the function\omega _{\mathbb {M}}(t):=\sup _{p\in \mathbb {N}_{0}}\log \big (\frac{t^p}{M_{p}}\big )= -\log \big (h_{\mathbb {M}}(1/t)\big ),\quad t>0;\qquad \omega _{\mathbb {M}}(0)=0,which is a nondecreasing continuous map in [0,\infty ) with \lim _{t\rightarrow \infty }\omega... | {
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"start": 348
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
] | [
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01e0f8a0deb78e9891d71c188aa8c5c7aea38d50 | subsection | 11 | 62 | Asymptotic expansions, ultraholomorphic classes and the asymptotic Borel map | In this paragraph G is a sectorial region and \mathbb {M} a sequence. We start recalling the concept of asymptotic expansion.We say a holomorphic function f in G admits the formal power series \widehat{f}=\sum _{p=0}^{\infty }a_{p}z^{p}\in \mathbb {C}[[z]] as its \mathbb {M}-asymptotic expansion in G (when the variable... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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] | 2,018 | en | Mathematics | [
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03ddea29ad976bc2305060839cf5c2c82e57cd86 | subsection | 12 | 62 | Asymptotic expansions, ultraholomorphic classes and the asymptotic Borel map | Note that, taking p=0 in (REF ), we deduce that every function in \widetilde{\mathcal {A}}^u_{\mathbb {M}}(G) is a bounded function.Finally, we define for every A>0 the class \mathcal {A}_{\mathbb {M},A}(G) consisting of the functions holomorphic in G such that\left\Vert f\right\Vert _{\mathbb {M},A}:=\sup _{z\in G,n\i... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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2106f54ba3957d679adf1efb7efaad1717ba2fec | subsection | 13 | 62 | Asymptotic expansions, ultraholomorphic classes and the asymptotic Borel map | Consequently, we have that
\mathcal {A}_{\mathbb {M}}(S)\subseteq \widetilde{\mathcal {A}}^u_{\mathbb {M}}(S) \subseteq \widetilde{\mathcal {A}}_{\mathbb {M}}(S).
f\in \widetilde{\mathcal {A}}_{\mathbb {M}}(G) if and only if for every T\ll G there exists A_T>0 such that f|_T\in \mathcal {A}_{\mathbb {M},A_T}(T).
In ca... | {
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} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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27533585d54674bb6281dd64ca5e248f46d6dd0a | subsection | 14 | 62 | Asymptotic expansions, ultraholomorphic classes and the asymptotic Borel map | If f\in \widetilde{\mathcal {A}}_{\mathbb {M}}(G), its \mathbb {M}-asymptotic expansion \widehat{f} is unique.One may accordingly define classes of formal power series\mathbb {C}[[z]]_{\mathbb {M},A}=\Big \lbrace \widehat{f}=\sum _{n=0}^\infty a_nz^n\in \mathbb {C}[[z]]:\, \left|\,{a} \,\right|_{\mathbb {M},A}:=\sup _{... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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122140674070f24da24cb6e3481c4a41a55861ae | subsection | 15 | 62 | Asymptotic expansions, ultraholomorphic classes and the asymptotic Borel map | Finally, note that if \mathbb {M}\approx \mathbb {L}, then \mathbb {C}[[z]]_{\mathbb {M}}=\mathbb {C}[[z]]_{\mathbb {L}}.A fundamental role in the discussion about the injectivity and surjectivity of the asymptotic Borel map will be played by the flat functions.Definition 2.10 A function f in any of the previous classe... | {
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"source_ref_... | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
] | [
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2b14c41e746336f339493dc71faf984c362064ea | subsection | 16 | 62 | Injectivity and surjectivity intervals for the asymptotic Borel map | By using a simple rotation, we see that the injectivity and the surjectivity of the Borel map in any of the previously considered classes do not depend on the bisecting direction d of the sectorial region G, so we limit ourselves to the case d=0. Moreover, in this paper we will restrict our study to the unbounded secto... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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9a80a12e235bc0d9ddce8a68bd3fdbb84ccf55cb | subsection | 17 | 62 | Injectivity and surjectivity intervals for the asymptotic Borel map | Hence, I_{\mathbb {M}}, \widetilde{I}^u_{\mathbb {M}} and \widetilde{I}_{\mathbb {M}} are either empty or unbounded intervals contained in (0,\infty ), which we call quasianalyticity or injectivity intervals.Similarly, we defineS_{\mathbb {M}}:=&\lbrace \gamma >0; \quad \widetilde{\mathcal {B}}:\mathcal {A}_{\mathbb {M... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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0a28153ae2cb944f969d5a0b90dd1c9807d3937d | subsection | 18 | 62 | Injectivity and surjectivity intervals for the asymptotic Borel map | Hence, by Proposition REF , if G is any sectorial region and f\in \widetilde{\mathcal {A}}_{\mathbb {M}}(G) is flat, we have that f(t)=0 for every t\in (0,A] which, by the identity principle, implies that f(z) identically vanishes in G. Consequently, the Borel map is always injective.On the other hand, in the same situ... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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0d9dce4351a374e861d2028b8823df59dbab03fa | subsection | 19 | 62 | Injectivity intervals: known results, and complete solution of the problem | The quasianalyticity intervals \widetilde{I}^u_{\mathbb {M}} and I_{\mathbb {M}} were determined in the literature in the 1950's.
The first case is basically answered by the following result of S. Mandelbrojt in 1952.Theorem 3.1 (, Section 2.4.III)
Let \mathbb {M} be a weight sequence, \gamma >0, b\ge 0 andH_b=\lbrace... | {
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... | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
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9f5bfee2884adf45aa2a4c83a96a2fa228a6d322 | subsection | 20 | 62 | Injectivity intervals: known results, and complete solution of the problem | Then, one has\lambda _{(c_p)}=\limsup _{p\rightarrow \infty }\frac{\log (p)}{\log (c_p)}.We consider now the closely related growth index (introduced in , see also ) for weight sequences \mathbb {M},\omega (\mathbb {M}):= \displaystyle \liminf _{p\rightarrow \infty } \frac{\log (m_{p})}{\log (p)}\in [0,\infty ],and we ... | {
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"source... | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
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df3df2ba3200b826f2d67f7b6743c731b29c6d13 | subsection | 21 | 62 | Injectivity intervals: known results, and complete solution of the problem | (iii)
Either \gamma >\omega (\mathbb {M}), or \gamma =\omega (\mathbb {M}) and \sum _{p=0}^{\infty } ((p+1)m_{p})^{-1/(\omega (\mathbb {M})+1)}=\infty .From Theorem REF one may deduce the following partial generalization of Watson's Lemma for nonuniform asymptotics, included in ; although in that paper strongly regular... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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f8d17136fca581840b08289b64e637789cc1e227 | subsection | 22 | 62 | Injectivity intervals: known results, and complete solution of the problem | If \omega (\mathbb {M})\in (0,\infty ), we have the situation described in Table REF , where \sum _{p=0}^{\infty } \sigma _p denotes the series \sum _{p=0}^{\infty } \left((p+1)m_{p}\right)^{-1/(\omega (\mathbb {M})+1)} and \sum _{p=0}^{\infty } \left(m_{p} \right)^{-1/\omega (\mathbb {M})} is abbreviated to \sum _{p=0... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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a4aacc28df81e45a12b3036f08bf269ba82209e6 | subsection | 23 | 62 | Injectivity intervals: known results, and complete solution of the problem | Hence, Table REF contains all the information about the injectivity intervals deduced from the classical results for the sequences \mathbb {M}_{\alpha ,\beta }.
[Table: Injectivity intervals for the sequence \mathbb {M}_{\alpha ,\beta } with \alpha >0, \beta \in \mathbb {R}.]Note that even if the Gevrey case \mathbb {M... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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0e8af148583b13efd35e794f570abc8a6b70c319 | subsection | 24 | 62 | Injectivity intervals: known results, and complete solution of the problem | In the works of K. V. Trunov and R. S. Yulmukhametov , a characterization is given, for a convex bounded region containing 0 in its boundary, in terms of the sequence \mathbb {M} and also of the way the boundary approaches 0. In particular, for bounded sectors, if \gamma \le 1, d\in \mathbb {R} and r>0, it turns out th... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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66406848413c1e37a48215b2e444fbf56614ef25 | subsection | 25 | 62 | Injectivity intervals: known results, and complete solution of the problem | \overline{V(z)}=V(\overline{z}) for every z \in S_\gamma (where, for z=(|z|,\arg (z)), we put \overline{z}=(|z|,-\arg (z))).
V(t) is positive in (0,\infty ), strictly increasing and \lim _{t\rightarrow 0}V(t)=0.
The function t\in \mathbb {R}\rightarrow V(e^t) is strictly convex (i.e. V is strictly convex relative to ... | {
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"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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0dfbe05b128bb48405978cc99829fdac676983b2 | subsection | 26 | 62 | Injectivity intervals: known results, and complete solution of the problem | 2.1)
Let \omega :(a,\infty )\rightarrow (0,\infty ) be a nonnegative, nondecreasing continuous function with \rho [\omega ]:=\limsup _{t\rightarrow \infty } \log (\omega (t))/\log (t)<\infty . Then, there exists a proximate order \rho (t) with \lim _{t \rightarrow \infty } \rho (t)=\rho [\omega ] such that\limsup _{t\... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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bb01dafbb6304b90f7b71c12a95fe7ecdc520da8 | subsection | 27 | 62 | Injectivity intervals: known results, and complete solution of the problem | By property (i) in Theorem REF we have\lim _{t\rightarrow \infty }\frac{V(t/c)}{V(t)}=\left(\frac{1}{c}\right)^{1/\omega } <\frac{b}{A_2},so that there exists R_1>0 such thatbV(t)> A_2V(t/c),\quad t\ge R_1.Let R_2:=\max (R_0,R_1,ct_2) and r:=R_2^{-1}. Then, using (REF ), (REF ) and (REF ), for z\in S(0,\beta ,r) we hav... | {
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} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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b434c2fb892a362b546cd4b0eb896a15d85eb925 | subsection | 28 | 62 | Injectivity intervals: known results, and complete solution of the problem | Suppose \widetilde{\mathcal {B}}:\widetilde{\mathcal {A}}_{\mathbb {M}}(S_\gamma )\longrightarrow \mathbb {C}[[z]]_{\mathbb {M}} is surjective. Since it is clear that the series \sum _{n=0}^{\infty } z^n belongs to \mathbb {C}[[z]]_{\mathbb {M}}, there exists f\in \widetilde{\mathcal {A}}_{\mathbb {M}}(S_\gamma ) such ... | {
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} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
] | [
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114a8f427ecf8c18b27554d2c2b4278ce6076575 | subsection | 29 | 62 | Injectivity intervals: known results, and complete solution of the problem | Furthermore, g(z)\sim _{\mathbb {M}} \widehat{0} uniformly in S(0,\gamma ,1), so there exist C,A>0 such that
for every z\in S(0,\gamma ,1) one has|g(z)|
\le CA^pM_p|z|^p,\qquad p\in \mathbb {N}_0.Hence, the holomorphic function \psi :\lbrace z\in \mathbb {C}:\Re (z)>0\rbrace \rightarrow \mathbb {C}, defined by \psi (u)... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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dc9e36b2bc6b0bbafe16309332c09bb1e91523fd | subsection | 30 | 62 | Injectivity intervals: known results, and complete solution of the problem | We proceed now to estimate |g(z)|. Firstly, parameterizing we have that|g(z)|&\le \left|\int _0^\infty e^{-re^{i\varphi }z}f(re^{i\varphi })e^{i\varphi }\,dr-
\int _0^\infty e^{-re^{i\varphi }z}re^{i\varphi }e^{i\varphi }\,dr\right|\\
&\le \int _0^\infty e^{-r\Re (e^{i\varphi }z)}|f(re^{i\varphi })|\,dr+
\left| \int _0... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
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8c8dfc4baf76b591719132c61a0fcac99cf877e9 | subsection | 31 | 62 | Injectivity intervals: known results, and complete solution of the problem | Note that the estimates in (REF ) imply for \Re (w)>1 (and so |w|>1) that|h(w)|\le \frac{C}{\Re (e^{i\varphi }w^{\gamma +1})}+\frac{1}{|w^{\gamma +1}|\Re (e^{i\varphi }w^{\gamma +1})}\le \frac{2C}{\Re (e^{i\varphi }w^{\gamma +1})}.These last estimates and the ones in (REF ) and (REF ) can now be summed up for h as|h(w)... | {
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} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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90162f3eab76d537142bd34dead6e4d9bfb45d30 | subsection | 32 | 62 | Injectivity intervals: known results, and complete solution of the problem | Now, observe that in this case
0<\frac{\pi }{2}-|\arg (w)|\le (\gamma +1)(\frac{\pi }{2}-|\arg (w)|)\le \frac{\pi }{2},
and so
|w|\cos \left((\gamma +1)|\arg (w)|-\frac{\gamma \pi }{2}\right)&=|w|\sin \left((\gamma +1)\left(\frac{\pi }{2}-|\arg (w)|\right)\right)\\
&\ge |w|\sin \left(\frac{\pi }{2}-|\arg (w)|\right)... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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f53543d360468ffbbdba58f0b39f66a11ecfdac6 | subsection | 33 | 62 | Surjectivity intervals for the Borel map | In the study of the surjectivity intervals a new index for the sequence \mathbb {M}, introduced in this regard by V. Thilliez , will play a central role.Definition 4.1
Let \mathbb {M}=(M_{p})_{p\in \mathbb {N}_{0}} be a strongly regular sequence and \gamma >0. We say \mathbb {M} satisfies property \left(P_{\gamma }\ri... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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9517804a2d4504aa9ebf24d3b19d09aa57650609 | subsection | 34 | 62 | Surjectivity intervals for the Borel map | Valdivia , we can obtain (see , ) an alternative expression of the index:
\gamma (\mathbb {M})=\sup \lbrace \beta >0; \,\, {m}\,\, \text{satisfies} \,\, (\gamma _{\beta }) \rbrace .In and , the connections between the indices \gamma (\mathbb {M}) and \omega (\mathbb {M}), the growth properties usually imposed on weig... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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51197871418fc4888136c9933c3e77596bffff2b | subsection | 35 | 62 | Weight sequences | Our first result is based on a theorem by H.-J. Petzsche in the ultradifferentiable setting and we need to consider the following space.Definition 4.3 We say that f\in \mathcal {E}_{\mathbb {M}}([-1,1]) if f\in \mathcal {C}^{\infty }([-1,1]) and there exists a constant A>0 for which\sup _{p\in \mathbb {N}_0,\, x\in [-1... | {
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45f869beb2da2078a6445bfb67007b9ccc789cd2 | subsection | 36 | 62 | Weight sequences | A suitable rotation shows that also \widetilde{\mathcal {B}}:\widetilde{\mathcal {A}}_{\mathbb {M}}(S(\pi ,\gamma ))\longrightarrow \mathbb {C}[[z]]_{\mathbb {M}} is surjective and so there exists a function f_2\in \widetilde{\mathcal {A}}_{\mathbb {M}}(S(\pi ,\gamma )) such that \widetilde{\mathcal {B}}(f_2)=\widehat{... | {
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ultraholomorphic classes | [
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2fe9eeae7a594a36d569e13ecc9fe90e35c96ade | subsection | 37 | 62 | Weight sequences | Although we do not treat this case here, some of their proofs can be adapted to, or suitably modified for, our Roumieu-like spaces.While the aforementioned authors impose condition (dc) on the sequence \mathbb {M}, i.e., there exists A>0 such that M_{p+1}\le A^p M_p for every p\in \mathbb {N}_0, we will show that, in s... | {
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ultraholomorphic classes | [
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7753836d1b3881978a2ceb4bcc4bc09b186ab317 | subsection | 38 | 62 | Weight sequences | As it was pointed out in , a simple computation leads to\mathbb {P}_{1,\mathbb {M}}=\mathbb {M},
P_{kr}=M_k for every k\in \mathbb {N}_0,
p_{kr+j}=(m_k)^{1/r} for all k\in \mathbb {N}_0 and j\in \lbrace 0,\dots ,r-1\rbrace ,
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527d4af456403e4abf2faa8725ff5974a21331f6 | subsection | 39 | 62 | Weight sequences | Now, for every \beta \in (1, 3/2) and any t\in S_{(\beta -1)/2}, we define\phi _{\beta ,t}:=\beta +2\arg (t)/\pi \in ((\beta +1)/2,(3\beta -1)/2)\subseteq (1,7/4).Hence, the change of variables u=t/w maps \gamma _{\phi _{\beta ,t}} into \delta _{\beta } which is a path consisting of a segment from the origin to a point... | {
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-0.013922407291829586,
0.010375053621828556,
0.02018558233976364,
-0.013243450783193111,
0.012022856622934341,
-0.02287089079618454,
0.... |
b322d8e0feb486e073dce5bf80b86c9262c61557 | subsection | 40 | 62 | Weight sequences | Firstly, for {=(M_p/p!)_{p\in \mathbb {N}_0} we will prove that the restriction map \mathcal {B}_{r}:\mathcal {E}_{r,{}([0,1])\longrightarrow \mathbb {C}[[z]]_{{} is surjective.
Since r\notin \mathbb {N}, we may choose two numbers \beta _1,\beta _2 with
1<\beta _1<\beta _2<\min \lbrace \frac{\alpha }{r},\frac{3}{2}\rb... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
-0.024304049089550972,
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0.032466426491737366,
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-0.017407985404133797,
0.03408364579081535,
-0... |
32c5985fd23388365fb389d4e0ee24ce44f5b07a | subsection | 41 | 62 | Weight sequences | \end{equation}
We consider now a path
\delta _{\beta _1} in S(0,{\beta _2},R) like the ones used in the classical Borel transform, made up of a segment \delta _1 from the origin to a point u_0 with |u_0|=R_0<R and \arg (u_0)=\pi \beta _1/2, then the circular arc \delta _2, traversed clockwise on the circumference |u|=R... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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0.031557388603687286,
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0.023637522011995316,
0.021165423095226288,
-0.0... |
8436de5fa0a7304ded3211c86ac89bd393b90990 | subsection | 42 | 62 | Weight sequences | \end{equation}
So,
if we choose R_0=|t|/p<R, we may apply (\ref {equaRemainderPhiProofDCbis}) and see that
\begin{equation}
\left|\int _{\delta _2}e^{t/u}
\left( \varphi (u)-\sum _{k=0}^{p-1}b_ku^{kr}\right)\, \frac{du}{u}\right|
\le \pi \beta _1 e^pCA^pM_p\left(\frac{|t|}{p}\right)^{pr}.
\end{equation}
On the other ha... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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0.0592021681368351,
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0.010413783602416515,
0.021193765103816986,
-0.02879... |
726d077627f52b48aaa2d09c39167ba3dc978fb9 | subsection | 43 | 62 | Weight sequences | \end{align*}
According to
(\ref {equaRemainderProofDCbis}), (\ref {equacotadelta2}) and (\ref {equacotadelta13}), and using Stirling^{\prime }s formula, we find that there exist constants C_2,A_2>0 such that for every p\in \mathbb {N} and t\in S(0,(\beta _1-1)/2,R) one has
\begin{equation}
\left|f(t)-\sum _{k=0}^{p-1}b... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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-0.00018433862715028226,
0.0019268514588475227,... |
b0f3ed7187186f49d65a0e618318a46092d0195a | subsection | 44 | 62 | Weight sequences | Then, Cauchy^{\prime }s integral formula together with~(\ref {equaAsympExpanfProofDCbis}) allow us to deduce that for every p\in \mathbb {N}_0,
\begin{align*}
|F^{(pr)}(t)|&=\left|\left(f(t)- \sum _{k=1}^{p-1}b_k\frac{t^{kr}}{(kr)!}\right)^{(pr)}\right|\le (pr)!\left(\frac{1+\varepsilon }{\varepsilon }\right)^{pr} \fra... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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0.020887188613414764,
0.0003156345628667623,
-0... |
f8af17ed3ffe46267244b5f8c4444ec472a2b53d | subsection | 45 | 62 | Weight sequences | By Proposition~\ref {prop.51.Lr.SchmetsValdivia}, we conclude that {m} satisfies (\gamma _r), what amounts to \gamma (\mathbb {M})>r=\lfloor \alpha \rfloor .
}}(ii) It is an immediate consequence of~(i).
}
}}}Corollary 4.11
Whenever \mathbb {M} is a weight sequence, if \gamma (\mathbb {M})<\infty one always has\wideti... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
-0.016527121886610985,
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0.019914953038096428,
-0.05225195363163948,
0.0... |
023b89c23eeb4da5b2e7c8b0562c479537b32ad6 | subsection | 46 | 62 | Weight sequences | The conclusion easily follows.Remark 4.12
Summing up, for a weight sequence \mathbb {M} and taking into account () and Theorem REF we see that:if \gamma (\mathbb {M})=0 (equivalently, if \mathbb {M} has not (snq)) then S_{\mathbb {M}}=\widetilde{S}^u_{\mathbb {M}}=\widetilde{S}_{\mathbb {M}}=\emptyset .
if \gamma (\m... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
-0.04096085578203201,
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0.01774868741631508,
-0.02437201514840126,
-0.0071... |
63c7248b049891f011a902ae79fcfb4572b461df | subsection | 47 | 62 | Weight sequences satisfying derivation closedness condition | As it has been pointed out in Remark REF , Corollary REF provides also information about \widetilde{S}^u_{\mathbb {M}}. In order to slightly improve it, one needs to impose (dc), which is a natural condition on the sequence \mathbb {M}, in the sense that it guarantees that the ultraholomorphic classes under considerati... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4064/sm-143-3-221-250",
"end": 496,
"openalex_id": "https://openalex.org/W869244473",
"raw": "J. Schmets, M. Valdivia, Extension maps in ultradifferentiable and ultraholomorphic function spaces, Studia Math. 143 (3) (2000), 221–250.",
... | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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0.0... |
0fa7578714cd5f9a46e563d2764908c3947c3895 | subsection | 48 | 62 | Weight sequences satisfying derivation closedness condition | Note that in this case no use has been made of (dc).Suppose now that \alpha \ge 1 and put r=\lfloor \alpha \rfloor , a positive natural number (note that, by Theorem REF , we only would need to consider the case \alpha =r\in \mathbb {N} but the proof works anyway).
Our aim is to show that \mathcal {B}_r:\mathcal {N}_{r... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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... |
57a5183da6e67e9d826d2f2d9a5f80795ea0af02 | subsection | 49 | 62 | Weight sequences satisfying derivation closedness condition | By the classical Hankel formula~(\ref {eq.Hankel.formula}) for the reciprocal Gamma function, for every natural number p\ge 2 and every t\in \mathbb {R} we may write
\begin{equation}
f(t)-\sum _{k=1}^{p-1}(-1)^{kr}b_k\frac{t^{kr}}{(kr)!}=
\frac{1}{2\pi i}\int _{1-\infty \,i}^{1+\infty \,i}e^{tu}
\left(\frac{\varphi (u)... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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0.01435878686606884,
-0.004318317398428917,
-... |
9f438ede900d3e4faff8090fd4a1e0b135534e80 | subsection | 50 | 62 | Weight sequences satisfying derivation closedness condition | Obviously, F\in \mathcal {C}^{\infty }([0,\infty )) and
F^{(pr)}(0)=b_p, p\in \mathbb {N}_0; F^{(m)}(0)=0 otherwise.
In order to conclude, we estimate the derivatives of F of order pr for some p\in \mathbb {N}_0. For p=0 and t\ge 0, we take into account~(\ref {equaBoundsbpProofDC}) and (\ref {equaBoundsVarphiProofDC}) ... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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-0.00915008969604969,
-0... |
bcf5be17e828eb9932b1a02e4fcac1ab3215ba31 | subsection | 51 | 62 | Weight sequences satisfying derivation closedness condition | \end{equation}
From (\ref {equaBoundFtProofDC}) and~(\ref {equaBoundDerivFtProofDC}), and since \mathbb {M} satisfies (dc), we deduce that there exist C_1,A_1>0 such that for every p\in \mathbb {N}_0 one has
|F^{(pr)}(t)|\le C_1A_1^pM_p=C_1A_1^p p!{,\quad t\ge 0,
and so F\in \mathcal {N}_{r,{}([0,\infty )) and \mathc... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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-... |
600893a0db4287b71ebf27c4f810106ee1e1cb2a | subsection | 52 | 62 | Weight sequences satisfying derivation closedness condition | Then \gamma (\mathbb {M})\in (0,\infty ], and we have the situation described in Table REF , with the corresponding conventions if \gamma (\mathbb {M})=\infty or \omega (\mathbb {M})=\infty . With the same assumptions, one might be able to show at least that \widetilde{S}_{\mathbb {M}} \subseteq \widetilde{S}^u_{\mathb... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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0.007637774106115103,
-0.03... |
1a9244b1140f34b8a69b1acb945dc3997b244b5f | subsection | 53 | 62 | Strongly regular sequences | We need to impose more conditions on the sequence \mathbb {M} in order to get extra information about surjectivity. We recall that \mathbb {M} is said to be strongly regular if is (lc), (snq) and (mg). As commented before, the first two conditions are natural in this context, and moderate growth, which is stronger than... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
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-0.... |
edce67db65b6ead95f7d2254276fc243ed8d190a | subsection | 54 | 62 | Strongly regular sequences | The following assertions are equivalent:(i)
r<\gamma (\mathbb {M}),
(ii)
there exists d\ge 1 such that for every A>0 there is a linear continuous operator
T_{\mathbb {M},A,r}:\mathbb {C}[[z]]_{\mathbb {M},A} \rightarrow \mathcal {A}_{\mathbb {M},dA}(S_{r})
such that \widetilde{\mathcal {B}}\circ T_{\mathbb {M},A,\gam... | {
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"source_ref_id": "a... | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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2ac6d9461ec3eff6b93761fe2863bd256b7c3fe0 | subsection | 55 | 62 | Strongly regular sequences | Let us define a new formal power series \widehat{g}=\sum _{j=0}^\infty b_jz^j with coefficientsb_{qj}=a_j,\ j\in \mathbb {N}_0;\quad b_m=0\text{ otherwise.}The log-convexity of \mathbb {M} implies that M_j^q\le M_{qj} for every j, so we have that|b_{qj}|\le CA^jM_j^q\le C(A^{1/q})^{qj}M_{qj},and consequently, \widehat{... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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85dc934895a2c5e994a6427c041912562bbc8d31 | subsection | 56 | 62 | Strongly regular sequences | Then, S_{\mathbb {M}}=\widetilde{S}^u_{\mathbb {M}}=(0,\gamma (\mathbb {M})).By Theorem REF and (), we have (0,\gamma (\mathbb {M}))\subseteq S_{\mathbb {M}}\subseteq \widetilde{S}^u_{\mathbb {M}}, while (iii)\Rightarrow (i) in Theorem REF ensures that, \gamma (\mathbb {M}) being rational, it cannot be the case that \g... | {
"cite_spans": []
} | 10.1016/j.jmaa.2018.09.011 | 1805.01153 | Injectivity and surjectivity of the asymptotic Borel map in Carleman
ultraholomorphic classes | [
"Javier Jiménez-Garrido",
"Javier Sanz",
"Gerhard Schindl"
] | [
"math.CV",
"math.FA"
] | 2,018 | en | Mathematics | [
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8d6b9f407ca50347e42d503f3a2a07d23e4f6342 | subsection | 57 | 62 | Strongly regular sequences | For every q\in \mathbb {N} we have that \zeta =\xi q\notin \mathbb {N}, we will show that \widetilde{\mathcal {B}}:\widetilde{\mathcal {A}}_{{\mathbb {M}}^{q}}(S_{\zeta })\rightarrow \mathbb {C}[[z]]_{\mathbb {M}^{q}} is surjective so, by Theorem REF .(i), we see that \lfloor \zeta \rfloor <\gamma (\mathbb {M}^{q}). Th... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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94f30796f4d743dd0c92d9079eecb0205df5d997 | subsection | 58 | 62 | Strongly regular sequences | This means that for opening \alpha \pi with \alpha in the interval (\gamma ,\omega ), the Borel map is neither injective nor surjective and the corresponding injectivity and surjectivity intervals for this sequence are either [\omega ,\infty ) or (\omega ,\infty ) and (0,\gamma ) or (0,\gamma ], respectively. | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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b1f143734fd71ed4514d43a5598d88920a1fe134 | subsection | 59 | 62 | Sequences admitting a nonzero proximate order | In this final subsection, taking into account that the Borel map is never bijective, Theorem REF , we will deduce more information regarding the surjectivity intervals. In order to be able to infer from that result whether or not \gamma (\mathbb {M}) belongs to S_{\mathbb {M}} and \widetilde{S}^u_{\mathbb {M}}, strongl... | {
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ultraholomorphic classes | [
"Javier Jiménez-Garrido",
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"Gerhard Schindl"
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c6d79386a681bb8edbe2a14c7099130c973f5534 | subsection | 60 | 62 | Sequences admitting a nonzero proximate order | For nonuniform asymptotics, he proved that \widetilde{S}_{\mathbb {M}}=(0,\gamma (\mathbb {M})] employing the truncated Laplace transform technique, where the classical exponential kernel was replaced by a function which is constructed using proximate orders and Maergoiz's functions.
The weight sequences \mathbb {M} fo... | {
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ultraholomorphic classes | [
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f3d9c30d90a9657ffadff1d81ec59466ac3de753 | subsection | 61 | 62 | Sequences admitting a nonzero proximate order | The following statements are equivalent:(i)
\gamma \le \omega (\mathbb {M})=\gamma (\mathbb {M}),
(ii)
For every \widehat{f}=\sum _{p\in \mathbb {N}_0} a_p z^p\in \mathbb {C}[[z]]_{\mathbb {M}} there exists a function f\in \widetilde{\mathcal {A}}_{\mathbb {M}}(S_{\gamma }) such that
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ultraholomorphic classes | [
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"Gerhard Schindl"
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af4447df74c8d729342ce5fd2382e508801d2706 | abstract | 0 | 21 | Abstract | In this paper, the wavelet analysis is used to study the ECG signal. We show
that the high-frequency wavelet components of the ECG signal contain
information on the functioning of the heart and can be used in diagnosis. We
describe the automated classification system that separates the ECG of sick and
healthy persons u... | {
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} | 1807.09964 | Wavelet analysis in problems of classification of ECG signals | [
"N. K. Smolentsev",
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800222ac0d894a299c8ce48d2a4eb349a2dd38ec | subsection | 1 | 21 | Introduction | Heartbeat represents a complex electrochemical process. It is registered in the form of electrocardiogram by skin electrodes placed in certain places of body surface. One heartbeat cycle recorded on ECG usually consists of several bursts: P wave, then QRS complex, T wave and U wave (Fig. 2). After a while this complex ... | {
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717e7132fdbba16f10da360823eda57408095e1d | subsection | 2 | 21 | Introduction | The choice of the 4th decomposing level is explained by the fact that the first four high-frequency components represent high ECG frequencies from 30 to 350 Hz, and low-frequency component represents the undistorted smoothed ECG signal cleared of high-frequency oscillations. In case of more deep signal expansion the fo... | {
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4a3e136a6c6959f7e64321289630f9db9aeab16d | subsection | 3 | 21 | Materials | For classification system creation and its testing ECG data sets of two groups are studied: healthy persons with normal ECG data and patients who recently came through myocardial infarction (MI). For the analysis digitized 30 seconds long cardiosignals made on the high-resolution cardiograph (1028 counts per second) "C... | {
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53a7b0aa58deec3a502bdc62089fa3c861238ffa | subsection | 4 | 21 | Wavelet-decomposition | The key elements of wavelet-analysis are two functions: the scaling function \varphi (t) and the wavelet function \psi (t), satisfying equations\varphi (t)=\sqrt{2}\sum _{k\in \mathbb {Z}} h_k \varphi (2t-k),\psi (t)=\sqrt{2}\sum _{k\in \mathbb {Z}} g_k \varphi (2t-k),where, h_k and g_k are the low and high-pass filter... | {
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1d47b718a1187bf740c6b1757dcc40f0b37af890 | subsection | 5 | 21 | Wavelet-decomposition | This allows us to take its representation A_j(f) in the scale V_j instead of the function f(t) for a sufficiently large j = j_0. Instead of the values of the function f(t), we can consider its approximation coefficients a_{j,k} (for a sufficiently large j = j_0).For every j\in \mathbb {Z} we have V_{j-1}\subset V_j.
Le... | {
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9f3a2fa857468a8ba5fe96b3d60f1825d9567a97 | subsection | 6 | 21 | Wavelet-decomposition | Then we obtain the decomposition of V_j into an orthogonal direct sumV_j=V_{j-N}\oplus W_{j-N}\oplus W_{j-N+1}\oplus \dots \oplus W_{j-1}.If initially the signal was represented by the coefficients a_{j,k}, now we have obtained the approximation coefficients a_{j-N,k} on a smaller scale V_{j-N} and the set of high-freq... | {
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4c208e07c3e2583c919667fa7e5221a62c1c6e8f | subsection | 7 | 21 | Wavelet-decomposition | 2).
[Figure: Wavelet components of ECG signal during decomposition to the 4th level (across is signal counting from 1 to 3400)]For example, RecD_2 is the signal component, reconstructed on the following set of wavelet coefficients \lbrace 0, D_2, 0, \dots , 0\rbrace , where 0 means the array from zeros. Similarly, low-... | {
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4c6399a5968c117ff3aae8ee1ca0413e9036660b | subsection | 8 | 21 | Feature space | For every high-frequency decomposition components RecD_1, RecD_2, ..., RecD_N many various statistical, frequency and stochastic characteristics can be calculated:maximum absolute value;
dispersion;
L_1- and L_2-energy;
relative L_2-energy;
maximum value of the power spectrum;
frequency, where maximum value of the... | {
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} | 1807.09964 | Wavelet analysis in problems of classification of ECG signals | [
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471d26a7ef520b11da003d70e6afbd772c754878 | subsection | 9 | 21 | Feature space | Then Hurst exponent H can be found from the ratio:R/S = (N/2)^H,where S is a standard deviation with selective average m_X, and R is the so-called range, accumulated deviation from average:R=\max _{1\le n \le N}\sum _{k=1}^n\, (x_k-m_X) -{\rm min}_{1\le n \le N}\sum _{k=1}^n\, (x_k-m_X).In MATLAB there is wfbmesti func... | {
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401ef029f9b1f7f56f6f6334072fe26b93bb1303 | subsection | 10 | 21 | Reduction of the feature space | Let us suppose that wavelet decomposition is done and feature vectors Y = [y_1, y_2, \dots , y_n] are made for some set of ECG records. Features space can have too big n dimension that complicates creation of classifiers, as it assumes working with high order matrixes.
It is desirable to somehow reduce its dimension wi... | {
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c1409c19d4e0e142268c6e3f919d5493c4d340f1 | subsection | 11 | 21 | Reduction of the feature space | In practice these values are approximated by selective estimates:M_k=\frac{1}{N_k}\sum _{j=1}^{N_k}Y_j^{(k)}, \,
\Sigma _k =\frac{1}{N_k}\sum _{j=1}^{N_k} (Y_j^{(k)} -M_k)(Y_j^{(k)} -M_k)^T, \,
P_k=\frac{N_k}{N}, \, k = 1, \dots , c.Scatter matrix S_b between classes shows vectors dispersion of expected values around a... | {
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ebb14fa4fba1072cc95cc81d8fbd7ee6a0f29a59 | subsection | 12 | 21 | Reduction of the feature space | Therefore, vectors Z are also divided into c subsets \lbrace Z_i^{(k)} = A^T Y_i^{(k)}, i = 1, \dots , N_k, k = 1, \dots , c\rbrace . Now the task is in creation of classifiers, i.e. of functions which divide all these subsets. | {
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5d4baf4a6dbc2ad9d2dbe3a50b782533bb5260cd | subsection | 13 | 21 | Linear classifiers | The algorithm of classification assumes the division of patients into two groups (healthy and those who had myocardial infarction (MI)). Therefore it is possible to use linear classifiers. Let us remind their construction , ch. 4, 10. The linear classifier has the form of linear heterogeneous function h(Z) = V^TZ + v_0... | {
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9c3ab2f96fae560d09bb3015ef4035038fb602df | subsection | 14 | 21 | Results and discussion | For creation of the classifying system ECG records of two groups of patients were used. The first group of ECG records of healthy persons contains 96 ECG fragments for four persons aged from 21 to 27. The second group of ECG records of patients who have recently came through myocardial infarction (subacute period) cont... | {
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7d5d7f774f155e719277e22e0b653b61b972cec0 | subsection | 15 | 21 | Wavelet selecting | In the wok the orthogonal Meyer wavelet dmey is used, which is derived from Meyer wavelet of infinite impulse response by truncation of its filter to 102 members. It has the carrier on the interval [0,101] and central frequency Fr = 0.6634 Hz. The choice of this wavelet is explained by well localization of frequency sp... | {
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"N. K. Smolentsev",
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9d355cfc3398d27b442162ed4f82f9c60b5e46c5 | subsection | 16 | 21 | Wavelet decomposition | Decomposing of ECG signal to the 4th level is made: S\mapsto \lbrace D_1, D_2, D_3, D_4, A_4\rbrace (Fig. 2). For signal components we have:S = RecD_1 + RecD_2 + RecD_3 + RecD_4 + RecA_4.Frequency spectrum of power of the first component RecD_1 is concentrated within the limits of 220 to 350 Hz, for the second componen... | {
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} | 1807.09964 | Wavelet analysis in problems of classification of ECG signals | [
"N. K. Smolentsev",
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a9ec4d0fe6ca5e9e61f7a576550108c7887bd707 | subsection | 17 | 21 | Feature space and its reduction | As it was noted in section Feature space, for every high-frequency ECG decomposing component RecD_1, RecD_2, RecD_3, RecD_4 it can be calculated to 10 various statistical, frequency and stochastic characteristics. In total there are 40 features for ECG. In the course of work those features which influence was very litt... | {
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1afec62ed0212681611c4118119a8320fd6de323 | subsection | 18 | 21 | Linear classifier construction | As it was determined in the previous section, the reduced feature space is one-dimensional and can be derived by projection Z = \Psi _1^T Y in one-dimensional space generated by vector \Psi _1. The reduced vector of features Z is a scalar. That is why the linear classifier h(Z) =V^TZ +v_0 takes the form of h(Z) = VZ + ... | {
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} | 1807.09964 | Wavelet analysis in problems of classification of ECG signals | [
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b8f705142803cb358996a489ec5602457c1326f5 | subsection | 19 | 21 | Testing of the classifying system | For testing 96 ECG fragments of four healthy persons aged from 21 to 56 and 120 ECG fragments of five sick persons aged from 45 to 57 who recently came through myocardial infarction (subacute period) are used. For these groups features vectors are created and space reduction of features to one-dimensional space is made... | {
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d2463b8cd5beef6d8c43c94132963a4694b572ee | subsection | 20 | 21 | Conclusion | Based on the wavelet analysis, a classification system has been constructed that reliably separates groups of healthy and sick patients.
Positive results of testing show that high-frequency ECG wavelet-components carry essential diagnostic information concerning ECG.
Such a system can be used as a complement to classif... | {
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} | 1807.09964 | Wavelet analysis in problems of classification of ECG signals | [
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e3c9871621bf64aa59d61f758bf04b9b7caf40ba | abstract | 0 | 37 | Abstract | It is a neat result from functional programming that libraries of parser
combinators can support rapid construction of decoders for quite a range of
formats. With a little more work, the same combinator program can denote both a
decoder and an encoder. Unfortunately, the real world is full of gnarly
formats, as with th... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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c6296f9e721fd8f4226a64cabffa17988d92fbfb | subsection | 1 | 37 | Introduction | Decoders and encoders are vital components of any software that
communicates with the outside world, and accordingly functions that
process untrusted data represent a key attack surface for malicious
actors. Failures to produce or interpret standard formats routinely
result in data loss, privacy violations, and service... | {
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"raw": "2015. CVE-2015-0618. Available from MITRE, CVE-ID CVE-2015-0618.. (Feb. 2015). https://cve.mitre.org/cgi-bin/cvename.cgi?name=CVE-2015-0618",
"source_ref_id": "40f48241e38f1fca413ac847e05fa13c... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
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4d52b1c6109364707d49a9b4fa312c2a4a4b858c | subsection | 2 | 37 | Getting started | Our first format is extremely simple:1.1: User input
1.2: Encoder
1.3: DecoderAll user input is contained in [esh:sensor0.1]box REF .1. sensor_msg is a record
type with two fields; the Coq Record command defines accessor
functions for these two fields. format specifies how instances
of this record are serialized usin... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
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0.01635... | |
7bb45c91c2040716421471a70d1c70718c44fbcc | subsection | 3 | 37 | Underspecification | We now consider a twist: to align
data on a 16-bit boundary, we introduce 8 bits of
padding after stationID; these bits will be reserved for future use:2.1: User input
2.2: Encoder
2.3: DecoderThese eight underspecified bits introduce an asymmetry: the encoder always writes 0x00, but the decoder
accepts any value. Th... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
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f29288a3daed8a96cee2a37ae31240a0c2bc972c | subsection | 4 | 37 | Constants and enums | Our next enhancements are to add a version number to our format and to tag each measurement with a kind, "TEMP" or "HUMIDITY". To save space, we allocate 2 bits for the tag and 14 bits for the measurement:3.1: User input
3.2: Encoder
3.3: DecoderThe use of format_const in the specification forces conforming
encoders ... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
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4d6f9ba50d564fbbe3a95dbe9f0394aab758af59 | subsection | 5 | 37 | Lists and dependencies | Our penultimate example illustrates data dependencies and input
restrictions. To do so, we replace our single data point with a list
of measurements (for conciseness, we remove tags and use 16-bit
words):4.1: User inputThe format_list combinator encodes a value by simply
applying its argument combinator in sequence to ... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
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4b7a291aca83a328cb8e1515206858c65dde190b | subsection | 6 | 37 | User-defined formats | Our final example illustrates a key benefit of the combinator-based
approach: integration of user-defined formats and decoders. The
advantage here is that Narcissus does not sacrifice correctness
for extensibility: every derived function must be correct. This
example uses a custom type for sensor readings, reading. To
... | {
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"Clément Pit--Claudel",
"Qianchuan Ye",
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