122:
219:
245:
25:
2899:(in this case, Daubechies 4). This implementation uses periodization to handle the problem of finite length signals. Other, more sophisticated methods are available, but often it is not necessary to use these as it only affects the very ends of the transformed signal. The periodization is accomplished in the forward transform directly in MATLAB vector notation, and the inverse transform by using the
318:
3680:
allowing the researchers to identify patterns, anomalies, and characteristic signatures within the signals associated with different engine conditions. This detailed spectral analysis aids in enhancing the accuracy of diagnostic assessments, enabling a more nuanced understanding of the vibrational and acoustic characteristics indicative of engine health or potential issues.
2158:
283:
behaviour or information in a signal. For example, D2, with one vanishing moment, easily encodes polynomials of one coefficient, or constant signal components. D4 encodes polynomials with two coefficients, i.e. constant and linear signal components; and D6 encodes 3-polynomials, i.e. constant, linear
3691:
The brachistochrone problem can be formulated and expressed as a variational problem, emphasizing the importance of finding the optimal curve that minimizes the time of descent. By introducing
Daubechies wavelets into the mathematical framework, scaling functions associated with these wavelets can
3673:
The elastohydrodynamic lubrication problem involves the study of lubrication regimes in which the deformation of the contacting surfaces significantly influences the lubricating film. Daubechies wavelets can address the challenges associated with accurately modeling and simulating such intricate
3685:
In practical terms, the
Daubechies wavelets facilitate a finely tuned examination of the temporal and spatial characteristics of dynamic waves within elastic materials. This approach enables a more nuanced understanding of how elastic solids respond to varying dynamic conditions over time. The
3679:
Daubechies
Wavelet can extract intricate details and features from the vibroacoustic signals, offering a comprehensive diagnostic approach for evaluating the condition and performance of diesel engines in combine harvesters. The Daubechies Wavelet spectrum serves as a powerful analytical tool,
3661:
Daubechies wavelet cepstral coefficients can be useful in the context of
Parkinson's disease detection. Daubechies wavelets, known for their efficient multi-resolution analysis, are utilized to extract cepstral features from vocal signal data. These wavelet-based coefficients can act as
1955:
3649:
The application of
Daubechies wavelet transform as a watermarking scheme has been proved effective. This approach operates in a proficient multi-resolution frequency domain, enabling the incorporation of an encrypted digital logo in the format of QR
3692:
construct an approximation of the optimal curve. Daubechies wavelets, with their ability to capture both high and low-frequency components of a function, prove instrumental in achieving a detailed representation of the brachistochrone curve.
188:
Among the 2 possible solutions of the algebraic equations for the moment and orthogonality conditions, the one is chosen whose scaling filter has extremal phase. The wavelet transform is also easy to put into practice using the
3667:
When it comes to analysis and detection of
Community Acquired Pneumonia (CAP), Complex Daubechies wavelets can be used to identify intricate details of the CAP affected areas in infected lungs to produce accurate results.
3655:
Daubechies wavelet approximation can be used to analyze
Griffith crack behavior in nonlocal magneto-elastic horizontally shear (SH) wave propagation within a finite-thickness, infinitely long homogeneous isotropic strip.
1796:
1305:
3628:(binomial QMF) is identical to the Daubechies wavelet filter, and its performance was ranked among known subspace solutions from a discrete-time signal processing perspective. It was an extension of the prior work on
1894:
631:
267:
Note that the spectra shown here are not the frequency response of the high and low pass filters, but rather the amplitudes of the continuous
Fourier transforms of the scaling (blue) and wavelet (red) functions.
2153:{\displaystyle c_{0}={\frac {1+{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{1}={\frac {3+{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{2}={\frac {3-{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{3}={\frac {1-{\sqrt {3}}}{4{\sqrt {2}}}}.}
1522:
296:(for example) signal components are treated differently by the transform depending on whether the points align with even- or odd-numbered locations in the sequence. The lack of the important property of
745:
3686:
integration of
Daubechies wavelets into the finite wavelet domain method likely contributes to a more versatile and robust analytical framework for studying transient dynamic waves in elastic solids.
819:
1145:
1398:
969:
1656:
233:
226:
259:
252:
3640:
filters are the unique maximally flat functions in a two-band perfect reconstruction QMF (PR-QMF) design formulation that is related to the wavelet regularity in the continuous domain.
492:
3784:
R.A. Haddad and A.N. Akansu, "A New
Orthogonal Transform for Signal Coding," IEEE Transactions on Acoustics, Speech and Signal Processing, vol.36, no.9, pp. 1404-1411, September 1988.
2261:
1038:
878:
397:
292:, and the lack of shift-invariance, which arise from the discrete shifting operation (below) during application of the transform. Sub-sequences which represent linear,
2190:
1569:
3674:
lubrication phenomena. Daubechies wavelets allows for a more detailed and refined exploration of the interactions between the lubricant and the contacting surfaces.
1947:
279:
equal to half the number of coefficients. For example, D2 has one vanishing moment, D4 has two, etc. A vanishing moment limits the wavelets ability to represent
375:
for details of this construction) will here be normalized to have sum equal 2 and sum of squares equal 2. In some applications, they are normalised to have sum
3920:
3662:
discriminative features for accurately identifying patterns indicative of Parkinson's disease, offering a novel approach to diagnostic methodologies.
1679:
1174:
200:
The Daubechies wavelets are not defined in terms of the resulting scaling and wavelet functions; in fact, they are not possible to write down in
2915:, a column vector with an even number of elements, has been pre-defined as the signal to be analyzed. Note that the D4 coefficients are /4.
1807:
527:
4071:
2880:
1413:
3852:
2168:
Below are the coefficients for the scaling functions for D2-20. The wavelet coefficients are derived by reversing the order of the
89:
1949:
using the quadrature mirror filter property results in the following solution for the coefficient values for filter of order 4.
642:
4058:, Proc. SPIE Video Communications and PACS for Medical Applications (Invited Paper), pp. 330-341, vol. 1977, Berlin, Oct. 1993.
61:
756:
4046:
402:
Using the general representation for a scaling sequence of an orthogonal discrete wavelet transform with approximation order
193:. Daubechies wavelets are widely used in solving a broad range of problems, e.g. self-similarity properties of a signal or
68:
4104:
3940:"Research of Daubechies Wavelet spectrum of vibroacoustic signals for diagnostic of diesel engines of combine harvesters"
1044:
42:
1323:
894:
1584:
4114:
358:
301:
108:
3636:
that led to the development of the Modified Hermite Transformation (MHT) in 1987. The magnitude square functions of
75:
4055:
173:
of vanishing moments, (this does not imply the best smoothness) for given support width (number of coefficients) 2
3775:
A.N. Akansu, Statistical Adaptive Transform Coding of Speech Signals. Ph.D. Thesis. Polytechnic University, 1987.
3750:
399:, so that both sequences and all shifts of them by an even number of coefficients are orthonormal to each other.
412:
3973:"Multiresolution Daubechies finite wavelet domain method for transient dynamic wave analysis in elastic solids"
1907:), thus one obtains 2 possible solutions. For extremal phase one chooses the one that has all complex roots of
57:
46:
4087:
3921:"Daubechies wavelet based numerical method for the solution of grease elastohydrodynamic lubrication problem"
2195:
4168:
1527:
shall be nonnegative on the interval translates into a set of linear restrictions on the coefficients of
4082:
4077:
3939:
3763:
980:
4096:
3900:
Natzina, Juanita S.R.F; Nadine, Suzanne S.R.F; Shojaa, Ayed Aljasar; Yubin, Xu; Muhammad, Saqib (2020).
1918:
For Daubechies wavelet transform, a pair of linear filters is used. Each filter of the pair should be a
142:
3902:"Analysis and Detection of Community Acquired Pneumonia Using PSPNET with Complex Daubechies Wavelets"
3822:"A new watermarking scheme based on Daubechies wavelet and chaotic map for quick response code images"
3766:, Proc. SPIE Visual Communications and Image Processing, pp. 609–618, vol. 1360, Lausanne, Sept. 1990.
3730:
1899:
that can be factored into two linear factors. One can assign either one of the two linear factors to
271:
Daubechies orthogonal wavelets D2–D20 resp. db1–db10 are commonly used. Each wavelet has a number of
1919:
827:
336:
297:
158:
3879:
378:
340:
35:
82:
3821:
3794:
3702:
1665:
one uses a technique called spectral factorization resp. Fejér-Riesz-algorithm. The polynomial
201:
190:
2175:
1538:
288:
signal components. This ability to encode signals is nonetheless subject to the phenomenon of
371:
Both the scaling sequence (low-pass filter) and the wavelet sequence (band-pass filter) (see
150:
3972:
3795:
A Generalized Parametric PR-QMF Design Technique Based on Bernstein Polynomial Approximation
185:
referring to the number of vanishing moments. So D4 and db2 are the same wavelet transform.
3629:
1925:
293:
285:
146:
208:, a numeric technique consisting of inverse-transforming an appropriate number of times.
8:
3956:
3633:
328:
372:
138:
121:
4110:
4042:
3901:
3853:"Griffith crack analysis in nonlocal magneto-elastic strip using Daubechies wavelets"
3753:(Binomial-QMF Daubechies Wavelets), Proc. 1st NJIT Symposium on Wavelets, April 1990.
205:
134:
3984:
3951:
3860:
3833:
3864:
2192:{−0.1830127, −0.3169873, 1.1830127, −0.6830127}). Mathematically, this looks like
2169:
4003:
2172:
coefficients and then reversing the sign of every second one, (i.e., D4 wavelet
153:. With each wavelet type of this class, there is a scaling function (called the
4138:
4134:
3837:
3807:
4120:
4004:"Solving brachistochrone problem via scaling functions of Daubechies wavelets"
4162:
3880:"Daubechies Wavelet Cepstral Coefficients for Parkinson's Disease Detection"
2895:
supports Daubechies wavelets directly a basic implementation is possible in
3637:
3625:
2292:
1791:{\displaystyle P(X)=(X-\mu _{1})\cdots (X-\mu _{N}),\qquad N=A+1+2\deg(R).}
1300:{\displaystyle P_{A}(X)=\sum _{k=0}^{A-1}{\binom {A+k-1}{A-1}}2^{-k}X^{k}.}
1161:
4032:
Proc. 1st NJIT Symposium on Wavelets, Subbands and Transforms, April 1990.
3878:
Soumaya, Zayrit; Taoufiq, Belhoussine Drissi; Abdelkrim, Ammoumou (2020).
4150:
2892:
232:
225:
169:
In general the Daubechies wavelets are chosen to have the highest number
4039:
Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets
1575:
results in a linear program with infinitely many inequality conditions.
258:
251:
218:
3621:
280:
244:
3919:
S.C., Shiralashetti; S.I., Hanaji; Sharada, S.Naregal (28 July 2020).
3820:
Umer, Aziz Waqas; Majid, Khan; Syeda, Iram Batool (18 December 2019).
3810:, IEEE Trans. Circuit Theory, vol CT-18, no. 3, pp. 411–413, May 1971.
2163:
1889:{\displaystyle X(Z)-\mu =-{\frac {1}{2}}Z+1-\mu -{\frac {1}{2}}Z^{-1}}
626:{\displaystyle a(Z)a\left(Z^{-1}\right)+a(-Z)a\left(-Z^{-1}\right)=4,}
4031:
3988:
24:
2879:
Parts of the construction are also used to derive the biorthogonal
4152:
Asymptotics of Daubechies Filters, Scaling Functions, and Wavelets
3925:
International Conference on Mathematical Sciences and Applications
3808:
On the Approximation Problem in Nonrecursive Digital Filter Design
4056:
Filter Banks and Wavelets in Signal Processing: A Critical Review
3938:
L L, Titova; Yu, M Chernik; Yu O, Gumenyuk; M M, Korobko (2020).
194:
333:. In particular, there is undefined math symbols (e.g. a, p, P).
300:, has led to the development of several different versions of a
4061:
3971:
Christos, V.Nastos; Dimitris, A. Saravanos (7 September 2021).
3851:
Jyotirmoy, Mouley; Nantu, Sarkar; Soumen, De (5 January 2023).
2896:
1517:{\displaystyle P(X)=P_{A}(X)+X^{A}(X-1)R\left((X-1)^{2}\right)}
2285:
Orthogonal Daubechies coefficients (normalized to have sum 2)
1160:, which can be obtained by division in the ring of truncated
521: − 1, one can write the orthogonality condition as
3721:
I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992, p. 194.
240:
Amplitudes of the frequency spectra of the above functions
3977:
International Journal for Numerical Methods in Engineering
3906:
Indian Journal of Computer Science and Engineering (IJCSE)
740:{\displaystyle (2-X)^{A}P(X)+X^{A}P(2-X)=2^{A}\qquad (*),}
3899:
3797:, IEEE Trans. Signal Process., pp. 2314–2321, July 1993.
1313:
The homogeneous equation for (*) is antisymmetric about
814:{\displaystyle X:={\frac {1}{2}}\left(2-Z-Z^{-1}\right)}
3877:
3944:
IOP Conference Series: Earth and Environmental Science
3731:
Daubechies Wavelet in Mathematica. Note that in there
3764:
Perfect Reconstruction Binomial QMF-Wavelet Transform
2198:
2178:
1958:
1928:
1810:
1682:
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1541:
1416:
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some polynomial with real coefficients. That the sum
1326:
1177:
1047:
983:
897:
830:
759:
645:
530:
415:
381:
3850:
1801:Each linear factor represents a Laurent-polynomial
4102:
2164:
The scaling sequences of lowest approximation order
1140:{\displaystyle p(e^{iw})p(e^{-iw})=|p(e^{iw})|^{2}}
145:and characterized by a maximal number of vanishing
125:
Daubechies 20 2-d wavelet (Wavelet Fn X Scaling Fn)
49:. Unsourced material may be challenged and removed.
2255:
2184:
2152:
1941:
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1393:{\displaystyle X^{A}(X-1)R\left((X-1)^{2}\right),}
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1299:
1139:
1032:
963:
872:
813:
739:
625:
486:
391:
4066:Journal of Mathematical Analysis and Applications
1915:) inside or on the unit circle and is thus real.
1265:
1230:
964:{\displaystyle P(X(Z))=p(Z)p\left(Z^{-1}\right).}
4160:
4008:Computational Methods for Differential Equations
3970:
1651:{\displaystyle P(X(Z))=p(Z)p\left(Z^{-1}\right)}
3918:
1922:. Solving the coefficient of the linear filter
1156:Equation (*) has one minimal solution for each
888:) stands for the symmetric Laurent-polynomial
3819:
1535:on the interval are bounded by some quantity
1310:Obviously, this has positive values on (0,2).
2271:is a coefficient of the wavelet sequence and
4036:
3937:
339:. There might be a discussion about this on
302:shift-invariant (discrete) wavelet transform
4143:Applied and Computational Harmonic Analysis
4037:Akansu, Ali N.; Haddad, Richard A. (1992),
204:. The graphs below are generated using the
1153:takes nonnegative values on the segment .
487:{\displaystyle a(Z)=2^{1-A}(1+Z)^{A}p(Z),}
181:using the length or number of taps, and db
4002:Azad, Kasnazani; Amjad, AliPanah (2021).
4001:
3955:
359:Learn how and when to remove this message
109:Learn how and when to remove this message
3762:A.N. Akansu, R.A. Haddad and H. Caglar,
3282:
177:. There are two naming schemes in use, D
120:
2275:a coefficient of the scaling sequence.
2256:{\displaystyle b_{k}=(-1)^{k}a_{N-1-k}}
1317:= 1 and has thus the general solution
824:generating all symmetric sequences and
197:problems, signal discontinuities, etc.
4161:
4109:. Berlin: Springer. pp. 157–160.
3626:binomial quadrature mirror filter bank
2279:is the wavelet index, i.e., 2 for D2.
311:
47:adding citations to reliable sources
18:
4097:The Daubechies D4 Wavelet Transform
4072:Hardware implementation of wavelets
1033:{\displaystyle X(e^{iw})=1-\cos(w)}
13:
3751:An Efficient QMF-Wavelet Structure
2881:Cohen–Daubechies–Feauveau wavelets
1234:
14:
4180:
4064:: "Generalized Self-similarity",
4018:
3857:Waves in Random and Complex Media
3826:Multimedia Tools and Applications
2886:
2906:
316:
257:
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243:
231:
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157:) which generates an orthogonal
23:
3995:
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3615:
2102:
2054:
2006:
1748:
724:
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34:needs additional citations for
4103:Jensen; la Cour-Harbo (2001).
4062:Carlos Cabrelli, Ursula Molter
4041:, Boston, MA: Academic Press,
3957:10.1088/1755-1315/548/3/032030
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1673:) splits into linear factors
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214:Scaling and wavelet functions
1:
3865:10.1080/17455030.2022.2163060
3708:
873:{\displaystyle X(-Z)=2-X(Z).}
750:with the Laurent-polynomial
164:
7:
4083:Encyclopedia of Mathematics
3793:H. Caglar and A.N. Akansu,
3696:
2826:−1.64709006 × 10
2802:−9.69947840 × 10
2799:−3.56329759 × 10
2772:−1.66137261 × 10
2725:−6.05496058 × 10
2722:−5.54004549 × 10
2699:−6.67962023 × 10
2696:−6.88771926 × 10
2693:−2.54790472 × 10
2664:−1.52353381 × 10
392:{\displaystyle {\sqrt {2}}}
10:
4185:
3838:10.1007/s11042-019-08570-5
2849:1.32354367 × 10
2778:2.81768659 × 10
2775:3.25814671 × 10
2753:1.97332536 × 10
2750:2.61296728 × 10
2747:9.55229711 × 10
2719:5.00226853 × 10
2702:5.10043697 × 10
2667:6.07514995 × 10
2646:3.54892813 × 10
2637:6.75606236 × 10
2610:7.83251152 × 10
2607:4.71742793 × 10
2531:6.68194092 × 10
2267:is the coefficient index,
509:having real coefficients,
143:discrete wavelet transform
2872:−1.875841 × 10
2823:5.5645514 × 10
4026:Ten Lectures on Wavelets
3286:
2917:
2185:{\displaystyle \approx }
1920:quadrature mirror filter
1564:{\displaystyle 4^{A-r},}
159:multiresolution analysis
2891:While software such as
133:, based on the work of
4135:Jianhong (Jackie) Shen
4106:Ripples in Mathematics
3703:Fast wavelet transform
2257:
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191:fast wavelet transform
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4078:"Daubechies wavelets"
4068:, 230: 251–260, 1999.
3283:Inverse transform, D4
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2155:
1944:
1942:{\displaystyle c_{i}}
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3630:binomial coefficient
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379:
329:confusing or unclear
58:"Daubechies wavelet"
43:improve this article
4169:Orthogonal wavelets
4024:Ingrid Daubechies:
3832:(9–10): 6891–6914.
3634:Hermite polynomials
2911:It is assumed that
2287:
337:clarify the section
139:orthogonal wavelets
131:Daubechies wavelets
16:Orthogonal wavelets
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373:orthogonal wavelet
137:, are a family of
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4048:978-0-12-047141-6
3983:(23): 7078–7100.
3739:/2 from the text.
3624:in 1990 that the
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2001:
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1986:
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774:
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277:vanishing moments
265:
264:
206:cascade algorithm
135:Ingrid Daubechies
119:
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93:
4176:
4131:
4129:
4128:
4119:. Archived from
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3989:10.1002/nme.6822
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3620:It was shown by
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3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2902:
2288:
2282:
2262:
2260:
2259:
2254:
2252:
2251:
2230:
2229:
2208:
2207:
2191:
2189:
2188:
2183:
2170:scaling function
2159:
2157:
2156:
2151:
2146:
2144:
2143:
2138:
2132:
2131:
2126:
2117:
2112:
2111:
2098:
2096:
2095:
2090:
2084:
2083:
2078:
2069:
2064:
2063:
2050:
2048:
2047:
2042:
2036:
2035:
2030:
2021:
2016:
2015:
2002:
2000:
1999:
1994:
1988:
1987:
1982:
1973:
1968:
1967:
1948:
1946:
1945:
1940:
1938:
1937:
1895:
1893:
1892:
1887:
1885:
1884:
1872:
1864:
1844:
1836:
1797:
1795:
1794:
1789:
1741:
1740:
1716:
1715:
1657:
1655:
1654:
1649:
1647:
1643:
1642:
1570:
1568:
1567:
1562:
1557:
1556:
1531:. The values of
1523:
1521:
1520:
1515:
1513:
1509:
1508:
1507:
1463:
1462:
1441:
1440:
1399:
1397:
1396:
1391:
1386:
1382:
1381:
1380:
1336:
1335:
1306:
1304:
1303:
1298:
1293:
1292:
1283:
1282:
1270:
1269:
1268:
1262:
1251:
1233:
1225:
1214:
1187:
1186:
1146:
1144:
1143:
1138:
1136:
1135:
1130:
1121:
1120:
1102:
1091:
1090:
1066:
1065:
1039:
1037:
1036:
1031:
1002:
1001:
970:
968:
967:
962:
957:
953:
952:
879:
877:
876:
871:
820:
818:
817:
812:
810:
806:
805:
804:
775:
767:
746:
744:
743:
738:
723:
722:
692:
691:
667:
666:
632:
630:
629:
624:
613:
609:
608:
607:
566:
562:
561:
513:(1) = 1 and deg(
493:
491:
490:
485:
468:
467:
446:
445:
398:
396:
395:
390:
388:
383:
364:
357:
353:
350:
344:
320:
319:
312:
298:shift-invariance
261:
254:
247:
235:
228:
221:
211:
210:
114:
107:
103:
100:
94:
92:
51:
27:
19:
4184:
4183:
4179:
4178:
4177:
4175:
4174:
4173:
4159:
4158:
4126:
4124:
4117:
4076:
4049:
4021:
4016:
4015:
4000:
3996:
3969:
3965:
3936:
3932:
3917:
3913:
3898:
3894:
3887:Complex Systems
3882:
3876:
3872:
3849:
3845:
3818:
3814:
3805:
3801:
3792:
3788:
3783:
3779:
3774:
3770:
3761:
3757:
3748:
3744:
3729:
3725:
3720:
3716:
3711:
3699:
3646:
3618:
3613:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3280:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2909:
2900:
2889:
2877:
2235:
2231:
2225:
2221:
2203:
2199:
2197:
2194:
2193:
2177:
2174:
2173:
2166:
2137:
2133:
2125:
2118:
2116:
2107:
2103:
2089:
2085:
2077:
2070:
2068:
2059:
2055:
2041:
2037:
2029:
2022:
2020:
2011:
2007:
1993:
1989:
1981:
1974:
1972:
1963:
1959:
1957:
1954:
1953:
1933:
1929:
1927:
1924:
1923:
1877:
1873:
1863:
1835:
1809:
1806:
1805:
1736:
1732:
1711:
1707:
1681:
1678:
1677:
1635:
1631:
1627:
1586:
1583:
1582:
1546:
1542:
1540:
1537:
1536:
1503:
1499:
1486:
1482:
1458:
1454:
1436:
1432:
1415:
1412:
1411:
1376:
1372:
1359:
1355:
1331:
1327:
1325:
1322:
1321:
1288:
1284:
1275:
1271:
1264:
1252:
1235:
1229:
1228:
1227:
1215:
1204:
1182:
1178:
1176:
1173:
1172:
1131:
1126:
1125:
1113:
1109:
1098:
1080:
1076:
1058:
1054:
1046:
1043:
1042:
994:
990:
982:
979:
978:
945:
941:
937:
896:
893:
892:
829:
826:
825:
797:
793:
780:
776:
766:
758:
755:
754:
718:
714:
687:
683:
662:
658:
644:
641:
640:
636:or equally as
600:
596:
592:
588:
554:
550:
546:
529:
526:
525:
463:
459:
435:
431:
414:
411:
410:
382:
380:
377:
376:
365:
354:
348:
345:
334:
321:
317:
310:
167:
149:for some given
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4182:
4172:
4171:
4157:
4156:
4139:Gilbert Strang
4132:
4115:
4100:
4093:
4074:
4069:
4059:
4052:
4047:
4034:
4029:
4020:
4019:External links
4017:
4014:
4013:
3994:
3963:
3930:
3911:
3892:
3870:
3843:
3812:
3799:
3786:
3777:
3768:
3755:
3742:
3723:
3713:
3712:
3710:
3707:
3706:
3705:
3698:
3695:
3694:
3693:
3688:
3687:
3682:
3681:
3676:
3675:
3670:
3669:
3664:
3663:
3658:
3657:
3652:
3651:
3645:
3642:
3617:
3614:
3287:
3284:
3281:
2918:
2908:
2905:
2888:
2887:Implementation
2885:
2874:
2873:
2870:
2868:
2866:
2864:
2862:
2860:
2858:
2856:
2854:
2851:
2850:
2847:
2845:
2843:
2841:
2839:
2837:
2835:
2833:
2831:
2828:
2827:
2824:
2821:
2819:
2817:
2815:
2813:
2811:
2809:
2807:
2804:
2803:
2800:
2797:
2795:
2793:
2791:
2789:
2787:
2785:
2783:
2780:
2779:
2776:
2773:
2770:
2768:
2766:
2764:
2762:
2760:
2758:
2755:
2754:
2751:
2748:
2745:
2743:
2741:
2739:
2737:
2735:
2733:
2730:
2729:
2726:
2723:
2720:
2717:
2715:
2713:
2711:
2709:
2707:
2704:
2703:
2700:
2697:
2694:
2691:
2689:
2687:
2685:
2683:
2681:
2678:
2677:
2674:
2671:
2668:
2665:
2662:
2660:
2658:
2656:
2654:
2651:
2650:
2647:
2644:
2641:
2638:
2635:
2633:
2631:
2629:
2627:
2624:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2603:
2601:
2599:
2596:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2575:
2573:
2571:
2568:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2544:
2542:
2539:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2515:
2513:
2510:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2483:
2480:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2453:
2450:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2356:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2324:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2281:
2250:
2247:
2244:
2241:
2238:
2234:
2228:
2224:
2220:
2217:
2214:
2211:
2206:
2202:
2181:
2165:
2162:
2161:
2160:
2149:
2141:
2136:
2129:
2124:
2121:
2115:
2110:
2106:
2101:
2093:
2088:
2081:
2076:
2073:
2067:
2062:
2058:
2053:
2045:
2040:
2033:
2028:
2025:
2019:
2014:
2010:
2005:
1997:
1992:
1985:
1980:
1977:
1971:
1966:
1962:
1936:
1932:
1897:
1896:
1883:
1880:
1876:
1870:
1867:
1862:
1859:
1856:
1853:
1850:
1847:
1842:
1839:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1799:
1798:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1747:
1744:
1739:
1735:
1731:
1728:
1725:
1722:
1719:
1714:
1710:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1659:
1658:
1646:
1641:
1638:
1634:
1630:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1560:
1555:
1552:
1549:
1545:
1525:
1524:
1512:
1506:
1502:
1498:
1495:
1492:
1489:
1485:
1481:
1478:
1475:
1472:
1469:
1466:
1461:
1457:
1453:
1450:
1447:
1444:
1439:
1435:
1431:
1428:
1425:
1422:
1419:
1401:
1400:
1389:
1385:
1379:
1375:
1371:
1368:
1365:
1362:
1358:
1354:
1351:
1348:
1345:
1342:
1339:
1334:
1330:
1308:
1307:
1296:
1291:
1287:
1281:
1278:
1274:
1267:
1261:
1258:
1255:
1250:
1247:
1244:
1241:
1238:
1232:
1224:
1221:
1218:
1213:
1210:
1207:
1203:
1199:
1196:
1193:
1190:
1185:
1181:
1148:
1147:
1134:
1129:
1124:
1119:
1116:
1112:
1108:
1105:
1101:
1097:
1094:
1089:
1086:
1083:
1079:
1075:
1072:
1069:
1064:
1061:
1057:
1053:
1050:
1040:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1000:
997:
993:
989:
986:
972:
971:
960:
956:
951:
948:
944:
940:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
822:
821:
809:
803:
800:
796:
792:
789:
786:
783:
779:
773:
770:
765:
762:
748:
747:
736:
733:
730:
727:
721:
717:
713:
710:
707:
704:
701:
698:
695:
690:
686:
682:
679:
676:
673:
670:
665:
661:
657:
654:
651:
648:
634:
633:
622:
619:
616:
612:
606:
603:
599:
595:
591:
587:
584:
581:
578:
575:
572:
569:
565:
560:
557:
553:
549:
545:
542:
539:
536:
533:
517:) =
495:
494:
483:
480:
477:
474:
471:
466:
462:
458:
455:
452:
449:
444:
441:
438:
434:
430:
427:
424:
421:
418:
386:
367:
366:
349:September 2019
324:
322:
315:
309:
306:
263:
262:
255:
248:
241:
237:
236:
229:
222:
215:
166:
163:
155:father wavelet
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4181:
4170:
4167:
4166:
4164:
4154:
4153:
4148:
4144:
4140:
4136:
4133:
4123:on 2008-12-02
4122:
4118:
4116:3-540-41662-5
4112:
4108:
4107:
4101:
4098:
4094:
4089:
4085:
4084:
4079:
4075:
4073:
4070:
4067:
4063:
4060:
4057:
4054:A.N. Akansu,
4053:
4050:
4044:
4040:
4035:
4033:
4030:
4027:
4023:
4022:
4009:
4005:
3998:
3990:
3986:
3982:
3978:
3974:
3967:
3958:
3953:
3949:
3945:
3941:
3934:
3926:
3922:
3915:
3907:
3903:
3896:
3888:
3881:
3874:
3866:
3862:
3858:
3854:
3847:
3839:
3835:
3831:
3827:
3823:
3816:
3809:
3806:O. Herrmann,
3803:
3796:
3790:
3781:
3772:
3765:
3759:
3752:
3749:A.N. Akansu,
3746:
3740:
3738:
3734:
3727:
3718:
3714:
3704:
3701:
3700:
3690:
3689:
3684:
3683:
3678:
3677:
3672:
3671:
3666:
3665:
3660:
3659:
3654:
3653:
3648:
3647:
3641:
3639:
3635:
3631:
3627:
3623:
2916:
2914:
2907:Transform, D4
2904:
2898:
2894:
2884:
2882:
2871:
2869:
2867:
2865:
2863:
2861:
2859:
2857:
2855:
2853:
2852:
2848:
2846:
2844:
2842:
2840:
2838:
2836:
2834:
2832:
2830:
2829:
2825:
2822:
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2017:
2012:
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1995:
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1103:
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1024:
1018:
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984:
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954:
949:
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928:
922:
919:
910:
904:
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883:
867:
861:
855:
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849:
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837:
831:
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798:
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790:
787:
784:
781:
777:
771:
768:
763:
760:
753:
752:
751:
734:
728:
719:
715:
711:
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674:
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663:
655:
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639:
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620:
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614:
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579:
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531:
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481:
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469:
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428:
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409:
408:
407:
405:
400:
384:
374:
363:
360:
352:
342:
341:the talk page
338:
332:
330:
325:This section
323:
314:
313:
305:
303:
299:
295:
291:
290:scale leakage
287:
282:
278:
274:
269:
260:
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113:
110:
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91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
4151:
4146:
4142:
4125:. Retrieved
4121:the original
4105:
4081:
4065:
4038:
4028:, SIAM 1992.
4025:
4007:
3997:
3980:
3976:
3966:
3947:
3943:
3933:
3924:
3914:
3905:
3895:
3886:
3873:
3856:
3846:
3829:
3825:
3815:
3802:
3789:
3780:
3771:
3758:
3745:
3736:
3732:
3726:
3717:
3644:Applications
3638:Binomial-QMF
3619:
3616:Binomial-QMF
2912:
2910:
2890:
2878:
2728:−0.01517900
2649:−0.04165925
2622:−0.10096657
2619:−0.09564726
2616:−0.06235021
2613:−0.02343994
2591:0.043452675
2588:−0.02456390
2585:−0.05378245
2582:−0.04466375
2579:−0.01779187
2551:−0.00882680
2548:−0.01498699
2537:−0.27710988
2534:−0.13695355
2508:−0.35333620
2505:−0.41475176
2502:−0.40165863
2499:−0.31683501
2496:−0.18351806
2493:−0.04560113
2472:−0.02238574
2469:−0.20351382
2466:−0.31998660
2463:−0.34265671
2460:−0.26450717
2457:−0.12083221
2430:−0.03957503
2427:−0.19093442
2284:
2276:
2272:
2268:
2264:
2167:
1917:
1912:
1908:
1904:
1900:
1898:
1800:
1670:
1666:
1662:
1660:
1577:
1572:
1532:
1528:
1526:
1404:
1402:
1314:
1312:
1309:
1165:
1162:power series
1157:
1155:
1150:
1149:
973:
885:
881:
823:
749:
635:
518:
514:
510:
506:
502:
498:
496:
403:
401:
370:
355:
346:
335:Please help
326:
308:Construction
289:
276:
273:zero moments
272:
270:
266:
199:
187:
182:
178:
174:
170:
168:
154:
130:
128:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
4095:I. Kaplan,
2901:circshift()
2893:Mathematica
2676:0.04696981
2673:0.03162417
2670:0.01236884
2643:0.01977216
2640:0.01774979
2594:0.13160299
2566:0.18012745
2563:0.21006834
2560:0.18207636
2557:0.11400345
2554:0.03892321
2525:0.13788809
2522:0.10970265
2478:0.39763774
2475:0.18836955
2448:0.97362811
2445:0.92954571
2442:0.82781653
2439:0.66437248
2436:0.44583132
2433:0.19576696
2424:−0.1830127
2417:0.74557507
2414:0.85534906
2411:0.95548615
2408:1.03114849
2405:1.06226376
2402:1.02432694
2399:0.89220014
2386:0.26612218
2383:0.34483430
2380:0.44246725
2377:0.56079128
2374:0.69950381
2371:0.85394354
2368:1.01094572
2365:1.14111692
2354:0.03771716
2351:0.05385035
2348:0.07695562
2345:0.11009943
2342:0.15774243
2339:0.22641898
2336:0.32580343
2333:0.47046721
1571:maximizing
202:closed form
141:defining a
99:August 2009
4127:2008-12-10
3950:: 032030.
3709:References
3622:Ali Akansu
2903:function:
2528:0.1008467
2519:0.0465036
2490:0.0436163
2487:0.0498175
2393:0.3169873
2362:1.1830127
2330:0.6830127
1578:To solve
331:to readers
281:polynomial
165:Properties
69:newspapers
4088:EMS Press
3517:circshift
3403:circshift
2396:0.650365
2246:−
2240:−
2216:−
2180:≈
2123:−
2075:−
1879:−
1861:−
1858:μ
1855:−
1833:−
1827:μ
1824:−
1774:
1734:μ
1730:−
1721:⋯
1709:μ
1705:−
1637:−
1551:−
1494:−
1471:−
1367:−
1344:−
1277:−
1257:−
1246:−
1220:−
1202:∑
1082:−
1019:
1013:−
947:−
880:Further,
853:−
838:−
799:−
791:−
785:−
729:∗
703:−
653:−
602:−
594:−
577:−
556:−
440:−
294:quadratic
286:quadratic
4163:Category
3859:: 1–19.
3697:See also
2883:(CDFs).
4090:, 2001
974:Since
327:may be
195:fractal
151:support
147:moments
83:scholar
4113:
4045:
3650:codes.
3211:s_even
3070:s_even
2992:s_even
2926:length
2897:MATLAB
2263:where
85:
78:
71:
64:
56:
4149:(3),
3883:(PDF)
3202:sqrt3
3184:s_odd
3175:sqrt3
3148:sqrt3
3133:sqrt3
3106:sqrt3
3085:sqrt3
3061:sqrt3
3046:s_odd
3031:sqrt3
2956:s_odd
2938:sqrt3
1403:with
497:with
90:JSTOR
76:books
4137:and
4111:ISBN
4043:ISBN
3632:and
3574:sqrt
3487:sqrt
3457:sqrt
3379:sqrt
3355:sqrt
3328:sqrt
3304:sqrt
3268:sqrt
3235:sqrt
2944:sqrt
2322:D20
2319:D18
2316:D16
2313:D14
2310:D12
2307:D10
2293:Haar
2291:D2 (
1661:for
284:and
129:The
62:news
3985:doi
3981:122
3952:doi
3948:548
3861:doi
3834:doi
3735:is
3388:));
3337:));
3277:));
3244:));
2304:D8
2301:D6
2298:D4
1771:deg
1164:in
1016:cos
501:= 2
406:,
275:or
45:by
4165::
4145:,
4141:,
4086:,
4080:,
4006:.
3979:.
3975:.
3946:.
3942:.
3923:.
3904:.
3885:.
3855:.
3830:79
3828:.
3824:.
3610:);
3568:s1
3532:);
3523:s1
3478:s1
3451:d1
3421:);
3409:d1
3397:s2
3391:s1
3352:((
3340:s2
3301:((
3289:d1
3019:);
2989:);
2953:);
2935:);
2359:1
2327:1
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1168:,
764::=
505:,
304:.
161:.
4155:.
4147:5
4130:.
4099:.
4092:.
4010:.
3991:.
3987::
3960:.
3954::
3927:.
3908:.
3889:.
3867:.
3863::
3840:.
3836::
3737:n
3733:n
3607:N
3604::
3601:2
3598::
3595:2
3592:(
3589:S
3586:*
3583:)
3580:3
3577:(
3571:-
3565:=
3562:)
3559:1
3556:-
3553:N
3550::
3547:2
3544::
3541:1
3538:(
3535:S
3529:1
3526:,
3520:(
3514:*
3511:4
3508:/
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3502:2
3499:-
3496:)
3493:3
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3484:(
3481:+
3475:*
3472:4
3469:/
3466:)
3463:3
3460:(
3454:+
3448:=
3445:)
3442:N
3439::
3436:2
3433::
3430:2
3427:(
3424:S
3418:1
3415:-
3412:,
3406:(
3400:+
3394:=
3385:2
3382:(
3376:/
3373:)
3370:1
3367:+
3364:)
3361:3
3358:(
3349:*
3346:s
3343:=
3334:2
3331:(
3325:/
3322:)
3319:1
3316:-
3313:)
3310:3
3307:(
3298:*
3295:d
3292:=
3274:2
3271:(
3265:*
3262:4
3259:(
3256:/
3253:d
3250:=
3247:d
3241:2
3238:(
3232:*
3229:4
3226:(
3223:/
3220:s
3217:=
3214:s
3208:*
3205:)
3199:-
3196:1
3193:-
3190:(
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3172:+
3169:3
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3163:+
3160:*
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3154:3
3151:-
3145:(
3142:+
3139:*
3136:)
3130:-
3127:1
3124:(
3121:=
3118:d
3115:;
3112:*
3109:)
3103:-
3100:1
3097:(
3094:+
3091:*
3088:)
3082:-
3079:3
3076:(
3073:+
3067:*
3064:)
3058:+
3055:3
3052:(
3049:+
3043:*
3040:)
3037:1
3034:+
3028:(
3025:=
3022:s
3016:N
3013::
3010:2
3007::
3004:2
3001:(
2998:S
2995:=
2986:1
2983:-
2980:N
2977::
2974:2
2971::
2968:1
2965:(
2962:S
2959:=
2950:3
2947:(
2941:=
2932:S
2929:(
2923:=
2920:N
2913:S
2277:N
2273:a
2269:b
2265:k
2249:k
2243:1
2237:N
2233:a
2227:k
2223:)
2219:1
2213:(
2210:=
2205:k
2201:b
2148:.
2140:2
2135:4
2128:3
2120:1
2114:=
2109:3
2105:c
2100:,
2092:2
2087:4
2080:3
2072:3
2066:=
2061:2
2057:c
2052:,
2044:2
2039:4
2032:3
2027:+
2024:3
2018:=
2013:1
2009:c
2004:,
1996:2
1991:4
1984:3
1979:+
1976:1
1970:=
1965:0
1961:c
1935:i
1931:c
1913:Z
1911:(
1909:p
1905:Z
1903:(
1901:p
1882:1
1875:Z
1869:2
1866:1
1852:1
1849:+
1846:Z
1841:2
1838:1
1830:=
1821:)
1818:Z
1815:(
1812:X
1786:.
1783:)
1780:R
1777:(
1768:2
1765:+
1762:1
1759:+
1756:A
1753:=
1750:N
1746:,
1743:)
1738:N
1727:X
1724:(
1718:)
1713:1
1702:X
1699:(
1696:=
1693:)
1690:X
1687:(
1684:P
1671:X
1669:(
1667:P
1663:p
1645:)
1640:1
1633:Z
1629:(
1625:p
1622:)
1619:Z
1616:(
1613:p
1610:=
1607:)
1604:)
1601:Z
1598:(
1595:X
1592:(
1589:P
1573:r
1559:,
1554:r
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1511:)
1505:2
1501:)
1497:1
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1484:(
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1477:)
1474:1
1468:X
1465:(
1460:A
1456:X
1452:+
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1446:X
1443:(
1438:A
1434:P
1430:=
1427:)
1424:X
1421:(
1418:P
1405:R
1388:,
1384:)
1378:2
1374:)
1370:1
1364:X
1361:(
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1338:(
1333:A
1329:X
1315:X
1295:.
1290:k
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1260:1
1254:A
1249:1
1243:k
1240:+
1237:A
1231:(
1223:1
1217:A
1212:0
1209:=
1206:k
1198:=
1195:)
1192:X
1189:(
1184:A
1180:P
1166:X
1158:A
1151:P
1133:2
1128:|
1123:)
1118:w
1115:i
1111:e
1107:(
1104:p
1100:|
1096:=
1093:)
1088:w
1085:i
1078:e
1074:(
1071:p
1068:)
1063:w
1060:i
1056:e
1052:(
1049:p
1028:)
1025:w
1022:(
1010:1
1007:=
1004:)
999:w
996:i
992:e
988:(
985:X
959:.
955:)
950:1
943:Z
939:(
935:p
932:)
929:Z
926:(
923:p
920:=
917:)
914:)
911:Z
908:(
905:X
902:(
899:P
886:X
884:(
882:P
868:.
865:)
862:Z
859:(
856:X
850:2
847:=
844:)
841:Z
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832:X
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802:1
795:Z
788:Z
782:2
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772:2
769:1
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726:(
720:A
716:2
712:=
709:)
706:X
700:2
697:(
694:P
689:A
685:X
681:+
678:)
675:X
672:(
669:P
664:A
660:)
656:X
650:2
647:(
621:,
618:4
615:=
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605:1
598:Z
590:(
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583:)
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559:1
552:Z
548:(
544:a
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538:Z
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515:p
511:p
507:p
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499:N
482:,
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437:1
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429:=
426:)
423:Z
420:(
417:a
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385:2
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356:(
351:)
347:(
343:.
183:A
179:N
175:A
171:A
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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