55:. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the
87:
counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown
2040:
can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can be used to solve dynamic equations on time scales. If the time scale is the non-negative integers then the transform is equal to a modified
2680:
1501:
131:. For example, they can model insect populations that evolve continuously while in season, die out in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non-overlapping population.
2450:
2141:
2538:
401:
2265:
467:
3105:
Davis, John M.; Gravagne, Ian A.; Marks, Robert J. II (2010). "Bilateral
Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series".
2026:
713:
1232:
301:
2566:
1363:
1580:
83:
Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their
1722:
550:
1838:
1934:
1688:
290:
1315:
746:
928:
1767:
1286:
888:
250:
2334:
1615:
848:
608:
221:
196:
174:
703:
808:
3140:
Bastos, Nuno R. O.; Mozyrska, Dorota; Torres, Delfim F. M. (2011). "Fractional
Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform".
1802:
1650:
643:
2288:
2312:
1077:
1044:
1007:
974:
1883:
494:
1544:
1524:
1355:
1335:
1182:
1161:
1139:
1118:
766:
665:
573:
334:
2048:
92:
is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the
2346:
17:
2742:
68:
2936:
2465:
342:
2197:
2722:
408:
2780:
1942:
2675:{\displaystyle \delta _{a}^{\mathbb {H} }(t)={\begin{cases}{\dfrac {1}{\mu (a)}},&t=a\\0,&t\neq a\end{cases}}}
1496:{\displaystyle \left|f(\sigma (t))-f(s)-f^{\Delta }(t)(\sigma (t)-s)\right|\leq \varepsilon \left|\sigma (t)-s\right|}
1198:
2176:
3001:
2337:
1552:
1697:
72:
48:
3282:
2150:
2456:
502:
1807:
2154:
1888:
1841:
261:
3272:
2179:
and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.
1294:
723:
898:
2604:
1731:
1655:
1255:
2914:
858:
230:
3064:
Davis, John M.; Gravagne, Ian A.; Jackson, Billy J.; Marks, Robert J. II; Ramos, Alice A. (2007).
2317:
1585:
818:
578:
204:
179:
157:
2992:
2712:
670:
778:
2909:
1772:
1620:
613:
316:
operators represent the closest point in the time scale on the right and left of a given point
89:
71:. However, similar ideas have been used before and go back at least to the introduction of the
2273:
2940:
2188:
116:
40:
3225:
3194:
3159:
3077:
2930:
2850:
2809:
2702:
2297:
2165:
1055:
1022:
985:
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1859:
479:
8:
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1095:
128:
84:
36:
3198:
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3149:
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1529:
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319:
120:
56:
3207:
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2976:
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2797:
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2037:
93:
3126:
3050:
3202:
3114:
3085:
3038:
2971:
2858:
2817:
2291:
1098:
of a time scale is redefined as equivalent to density. A time scale is said to be
712:
124:
304:
The forward jump, backward jump, and graininess operators on a discrete time scale
2553:
127:. Dynamic equations on a time scale have a potential for applications such as in
2944:
496:
is the distance from a point to the closest point on the right and is given by:
3257:
3090:
3065:
3042:
2879:
201:
The two most commonly encountered examples of time scales are the real numbers
3118:
3019:(2012). "On the connection between the Hilger and Radon–Nikodym derivatives".
2863:
2838:
3266:
3016:
2754:
2445:{\displaystyle \int _{r}^{s}f(t)\Delta t=\int _{[r,s)}f(t)d\mu ^{\Delta }(t)}
224:
148:
52:
300:
1242:
1856:
is defined as the antiderivative with respect to the delta derivative. If
2549:
2042:
97:
28:
3236:
Dynamic
Equations on Time Scales: Qualitative Analysis and Applications
3215:
3181:
Agarwal, Ravi; Bohner, Martin; O’Regan, Donal; Peterson, Allan (2002).
2904:
Bohner, M; Guseinov, GS (2005). "Multiple integration on time scales".
2706:
2136:{\displaystyle {\mathcal {Z}}'\{x\}={\frac {{\mathcal {Z}}\{x\}}{z+1}}}
105:
67:
Time-scale calculus was introduced in 1988 by the German mathematician
3235:
152:
1691:
112:
44:
3154:
3033:
2770:
1725:
101:
2747:
Ein MaĂźkettenkalkĂĽl mit
Anwendung auf Zentrumsmannigfaltigkeiten
2533:{\displaystyle f^{\Delta }(t)={\frac {df}{d\mu ^{\Delta }}}(t).}
2171:
716:
Several points on a time scale with different classifications
3252:
3180:
2693:
on time scales is treated in Bastos, Mozyrska, and Torres.
2668:
396:{\displaystyle \sigma (t)=\inf\{s\in \mathbb {T} :s>t\}}
2260:{\displaystyle \mu ^{\Delta }(A)=\lambda (\rho ^{-1}(A)),}
3063:
2157:
are unified as partial dynamic equations on time scales.
462:{\displaystyle \rho (t)=\sup\{s\in \mathbb {T} :s<t\}}
3142:
International
Journal of Mathematics & Computation
2021:{\displaystyle \int _{r}^{s}f(t)\Delta (t)=F(s)-F(r).}
2685:
2608:
2569:
2468:
2349:
2320:
2300:
2276:
2200:
2051:
1945:
1891:
1862:
1810:
1775:
1734:
1700:
1658:
1623:
1588:
1555:
1532:
1512:
1366:
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1323:
1297:
1258:
1201:
1170:
1149:
1127:
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1058:
1025:
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955:
901:
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821:
781:
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673:
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616:
581:
561:
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482:
411:
345:
322:
264:
233:
207:
182:
160:
3139:
2031:
3104:
2674:
2532:
2444:
2328:
2306:
2282:
2259:
2135:
2020:
1928:
1877:
1832:
1796:
1761:
1716:
1682:
1644:
1609:
1574:
1538:
1518:
1495:
1349:
1329:
1309:
1280:
1226:
1176:
1155:
1133:
1112:
1071:
1038:
1001:
968:
922:
882:
842:
802:
760:
740:
697:
659:
637:
602:
567:
544:
488:
461:
395:
328:
284:
244:
215:
190:
176:. The common notation for a general time scale is
168:
3264:
3220:Journal of Computational and Applied Mathematics
3187:Journal of Computational and Applied Mathematics
3066:"The Laplace transform on time scales revisited"
2843:Journal of Computational and Applied Mathematics
2802:Journal of Computational and Applied Mathematics
2796:Ahlbrandt, Calvin D.; Morian, Christina (2002).
2182:
1245:, but is set to the real line for simplicity).
427:
361:
2798:"Partial differential equations on time scales"
2795:
2543:
2336:. The delta integral turns out to be the usual
1227:{\displaystyle f:\mathbb {T} \to \mathbb {R} ,}
3014:
2903:
2877:
2187:Associated with every time scale is a natural
255:A single point in a time scale is defined as:
2937:Missouri University of Science and Technology
2455:and the delta derivative turns out to be the
3183:"Dynamic equations on time scales: a survey"
2168:on time scales is treated in Bohner (2005).
2116:
2095:
2079:
2064:
456:
430:
390:
364:
295:
2771:Martin Bohner & Allan Peterson (2001).
2172:Stochastic dynamic equations on time scales
707:
2839:"Partial dynamic equations on time scales"
2314:is the backward shift operator defined on
2145:
1575:{\displaystyle \mathbb {T} =\mathbb {R} .}
941:if both left scattered and right scattered
104:but to more general time scales such as a
3206:
3153:
3089:
3032:
2975:
2913:
2862:
2821:
2581:
2322:
1717:{\displaystyle \mathbb {T} =\mathbb {Z} }
1710:
1702:
1565:
1557:
1217:
1209:
1141:. Similarly, a time scale is said to be
734:
440:
374:
278:
238:
209:
184:
162:
2957:
2880:"Partial differentiation on time scales"
711:
299:
3253:The Baylor University Time Scales Group
3107:Circuits, Systems and Signal Processing
2836:
2160:
945:As illustrated by the figure at right:
14:
3265:
2928:
2741:
3240:Nonlinear Dynamics And Systems Theory
2990:
2735:
1690:; is the derivative used in standard
545:{\displaystyle \mu (t)=\sigma (t)-t.}
134:
43:, unifying integral and differential
2878:Bohner, M.; Guseinov, G. S. (2004).
2766:
2764:
2749:(PhD thesis). Universität Würzburg.
1833:{\displaystyle f^{\Delta }=\Delta f}
78:
75:, which unifies sums and integrals.
51:, offering a formalism for studying
1929:{\displaystyle f(t)=F^{\Delta }(t)}
285:{\displaystyle t:t\in \mathbb {T} }
111:The three most popular examples of
24:
3226:Dynamic Equations And Applications
3174:
2723:Krylov–Bogoliubov averaging method
2686:Fractional calculus on time scales
2556:are unified on time scales as the
2510:
2474:
2428:
2377:
2206:
2090:
2055:
1973:
1912:
1824:
1816:
1664:
1416:
1264:
935:if both left dense and right dense
35:is a unification of the theory of
25:
3294:
3246:
2761:
2177:Stochastic differential equations
2032:Laplace transform and z-transform
1310:{\displaystyle \varepsilon >0}
741:{\displaystyle t\in \mathbb {T} }
3230:Advances in Difference Equations
3216:Dynamic Equations on Time Scales
2906:Dynamic Systems and Applications
2887:Dynamic Systems and Applications
2773:Dynamic Equations on Time Scales
18:Dynamic equations on time scales
3133:
3098:
3057:
3008:
923:{\displaystyle \sigma (t)>t}
2984:
2951:
2922:
2897:
2871:
2830:
2789:
2623:
2617:
2593:
2587:
2524:
2518:
2485:
2479:
2439:
2433:
2417:
2411:
2403:
2391:
2374:
2368:
2251:
2248:
2242:
2226:
2217:
2211:
2151:Partial differential equations
2113:
2101:
2076:
2070:
2012:
2006:
1997:
1991:
1982:
1976:
1970:
1964:
1923:
1917:
1901:
1895:
1872:
1866:
1847:
1844:used in difference equations.
1785:
1779:
1762:{\displaystyle \sigma (t)=t+1}
1744:
1738:
1683:{\displaystyle f^{\Delta }=f'}
1633:
1627:
1598:
1592:
1479:
1473:
1451:
1442:
1436:
1430:
1427:
1421:
1405:
1399:
1390:
1387:
1381:
1375:
1281:{\displaystyle f^{\Delta }(t)}
1275:
1269:
1213:
1121:if it is right dense at point
911:
905:
871:
865:
831:
825:
791:
785:
683:
677:
626:
620:
591:
585:
530:
524:
515:
509:
469:(backward shift/jump operator)
421:
415:
355:
349:
49:calculus of finite differences
13:
1:
3208:10.1016/S0377-0427(01)00432-0
3002:Ä°zmir Institute of Technology
2994:Measure theory on time scales
2977:10.1016/S0022-247X(03)00361-5
2823:10.1016/S0377-0427(01)00434-4
2728:
2459:with respect to this measure
2340:with respect to this measure
2183:Measure theory on time scales
1187:
1164:if it is left dense at point
1090:
883:{\displaystyle \rho (t)<t}
403:(forward shift/jump operator)
245:{\displaystyle h\mathbb {Z} }
2960:"Integration on time scales"
2932:Stochastic Dynamic Equations
2544:Distributions on time scales
2329:{\displaystyle \mathbb {R} }
2155:partial difference equations
1885:has a continuous derivative
1610:{\displaystyle \sigma (t)=t}
1317:there exists a neighborhood
843:{\displaystyle \sigma (t)=t}
603:{\displaystyle \sigma (t)=t}
216:{\displaystyle \mathbb {R} }
191:{\displaystyle \mathbb {T} }
169:{\displaystyle \mathbb {R} }
7:
2705:for dynamic equations on a
2696:
2338:Lebesgue–Stieltjes integral
1842:forward difference operator
698:{\displaystyle \rho (t)=t.}
336:, respectively. Formally:
10:
3299:
3091:10.1016/j.jmaa.2006.10.089
3043:10.1016/j.jmaa.2011.07.041
1100:right-continuous at point
803:{\displaystyle \rho (t)=t}
73:Riemann–Stieltjes integral
62:
3119:10.1007/s00034-010-9196-2
2864:10.1016/j.cam.2005.02.011
1797:{\displaystyle \mu (t)=1}
1645:{\displaystyle \mu (t)=0}
1252:(also Hilger derivative)
1143:left-continuous at point
638:{\displaystyle \mu (t)=0}
296:Operations on time scales
2958:Guseinov, G. S. (2003).
2457:Radon–Nikodym derivative
2283:{\displaystyle \lambda }
708:Classification of points
2713:Multiple-scale analysis
2146:Partial differentiation
1288:exists if and only if:
2929:Sanyal, Suman (2008).
2676:
2534:
2446:
2330:
2308:
2284:
2261:
2137:
2022:
1930:
1879:
1834:
1798:
1763:
1718:
1684:
1646:
1611:
1576:
1540:
1520:
1497:
1351:
1331:
1311:
1282:
1228:
1178:
1157:
1135:
1114:
1073:
1040:
1003:
970:
924:
884:
844:
804:
762:
742:
717:
699:
661:
639:
604:
569:
546:
490:
463:
397:
330:
305:
286:
246:
217:
192:
170:
41:differential equations
2677:
2535:
2447:
2331:
2309:
2307:{\displaystyle \rho }
2285:
2262:
2138:
2023:
1931:
1880:
1835:
1799:
1764:
1719:
1685:
1647:
1612:
1577:
1541:
1521:
1498:
1352:
1332:
1312:
1283:
1229:
1179:
1158:
1136:
1115:
1074:
1072:{\displaystyle t_{4}}
1041:
1039:{\displaystyle t_{3}}
1004:
1002:{\displaystyle t_{2}}
971:
969:{\displaystyle t_{1}}
925:
885:
845:
805:
763:
743:
715:
700:
662:
640:
605:
570:
547:
491:
464:
398:
331:
303:
287:
247:
218:
193:
171:
117:differential calculus
3283:Recurrence relations
2837:Jackson, B. (2006).
2703:Analysis on fractals
2567:
2466:
2347:
2318:
2298:
2274:
2198:
2166:Multiple integration
2161:Multiple integration
2049:
1943:
1889:
1878:{\displaystyle F(t)}
1860:
1808:
1773:
1732:
1698:
1656:
1621:
1586:
1553:
1530:
1510:
1364:
1341:
1321:
1295:
1256:
1199:
1168:
1147:
1125:
1104:
1056:
1023:
986:
953:
899:
859:
819:
779:
752:
724:
671:
651:
614:
579:
559:
503:
489:{\displaystyle \mu }
480:
409:
343:
320:
262:
231:
205:
180:
158:
37:difference equations
3199:2002JCoAM.141....1A
3164:2010arXiv1012.1555B
3082:2007JMAA..332.1291D
3070:J. Math. Anal. Appl
3021:J. Math. Anal. Appl
3000:(Master's thesis).
2964:J. Math. Anal. Appl
2855:2006JCoAM.186..391J
2814:2002JCoAM.141...35A
2718:Method of averaging
2691:Fractional calculus
2586:
2364:
1960:
129:population dynamics
121:difference calculus
115:on time scales are
33:time-scale calculus
2991:Deniz, A. (2007).
2672:
2667:
2628:
2570:
2530:
2442:
2350:
2326:
2304:
2280:
2257:
2133:
2018:
1946:
1926:
1875:
1830:
1794:
1759:
1714:
1680:
1642:
1607:
1572:
1536:
1516:
1493:
1347:
1327:
1307:
1278:
1224:
1174:
1153:
1131:
1110:
1069:
1036:
999:
966:
920:
880:
840:
800:
758:
738:
718:
695:
657:
635:
600:
565:
555:For a right-dense
542:
486:
459:
393:
326:
306:
282:
242:
213:
188:
166:
135:Formal definitions
57:forward difference
3273:Dynamical systems
3258:Timescalewiki.org
3238:Special issue of
3228:Special Issue of
3218:Special issue of
2782:978-0-8176-4225-9
2627:
2516:
2131:
2038:Laplace transform
1539:{\displaystyle U}
1519:{\displaystyle s}
1350:{\displaystyle t}
1330:{\displaystyle U}
1192:Take a function:
1177:{\displaystyle t}
1156:{\displaystyle t}
1134:{\displaystyle t}
1113:{\displaystyle t}
761:{\displaystyle t}
660:{\displaystyle t}
647:For a left-dense
568:{\displaystyle t}
329:{\displaystyle t}
79:Dynamic equations
16:(Redirected from
3290:
3212:
3210:
3168:
3167:
3157:
3137:
3131:
3130:
3113:(6): 1141–1165.
3102:
3096:
3095:
3093:
3076:(2): 1291–1307.
3061:
3055:
3054:
3036:
3027:(2): 1184–1189.
3012:
3006:
3005:
2999:
2988:
2982:
2981:
2979:
2955:
2949:
2948:
2926:
2920:
2919:
2917:
2901:
2895:
2894:
2884:
2875:
2869:
2868:
2866:
2834:
2828:
2827:
2825:
2793:
2787:
2786:
2768:
2759:
2758:
2739:
2681:
2679:
2678:
2673:
2671:
2670:
2629:
2626:
2609:
2585:
2584:
2578:
2539:
2537:
2536:
2531:
2517:
2515:
2514:
2513:
2500:
2492:
2478:
2477:
2451:
2449:
2448:
2443:
2432:
2431:
2407:
2406:
2363:
2358:
2335:
2333:
2332:
2327:
2325:
2313:
2311:
2310:
2305:
2292:Lebesgue measure
2289:
2287:
2286:
2281:
2266:
2264:
2263:
2258:
2241:
2240:
2210:
2209:
2142:
2140:
2139:
2134:
2132:
2130:
2119:
2094:
2093:
2086:
2063:
2059:
2058:
2027:
2025:
2024:
2019:
1959:
1954:
1935:
1933:
1932:
1927:
1916:
1915:
1884:
1882:
1881:
1876:
1839:
1837:
1836:
1831:
1820:
1819:
1803:
1801:
1800:
1795:
1768:
1766:
1765:
1760:
1723:
1721:
1720:
1715:
1713:
1705:
1689:
1687:
1686:
1681:
1679:
1668:
1667:
1651:
1649:
1648:
1643:
1616:
1614:
1613:
1608:
1581:
1579:
1578:
1573:
1568:
1560:
1545:
1543:
1542:
1537:
1525:
1523:
1522:
1517:
1502:
1500:
1499:
1494:
1492:
1488:
1458:
1454:
1420:
1419:
1356:
1354:
1353:
1348:
1336:
1334:
1333:
1328:
1316:
1314:
1313:
1308:
1287:
1285:
1284:
1279:
1268:
1267:
1250:delta derivative
1248:Definition: The
1233:
1231:
1230:
1225:
1220:
1212:
1183:
1181:
1180:
1175:
1162:
1160:
1159:
1154:
1140:
1138:
1137:
1132:
1119:
1117:
1116:
1111:
1078:
1076:
1075:
1070:
1068:
1067:
1045:
1043:
1042:
1037:
1035:
1034:
1008:
1006:
1005:
1000:
998:
997:
975:
973:
972:
967:
965:
964:
929:
927:
926:
921:
889:
887:
886:
881:
849:
847:
846:
841:
809:
807:
806:
801:
767:
765:
764:
759:
747:
745:
744:
739:
737:
704:
702:
701:
696:
666:
664:
663:
658:
644:
642:
641:
636:
609:
607:
606:
601:
574:
572:
571:
566:
551:
549:
548:
543:
495:
493:
492:
487:
468:
466:
465:
460:
443:
402:
400:
399:
394:
377:
335:
333:
332:
327:
291:
289:
288:
283:
281:
251:
249:
248:
243:
241:
222:
220:
219:
214:
212:
197:
195:
194:
189:
187:
175:
173:
172:
167:
165:
125:quantum calculus
21:
3298:
3297:
3293:
3292:
3291:
3289:
3288:
3287:
3263:
3262:
3249:
3177:
3175:Further reading
3172:
3171:
3138:
3134:
3103:
3099:
3062:
3058:
3013:
3009:
2997:
2989:
2985:
2956:
2952:
2927:
2923:
2902:
2898:
2882:
2876:
2872:
2835:
2831:
2794:
2790:
2783:
2769:
2762:
2740:
2736:
2731:
2699:
2688:
2666:
2665:
2654:
2645:
2644:
2633:
2613:
2607:
2600:
2599:
2580:
2579:
2574:
2568:
2565:
2564:
2554:Kronecker delta
2546:
2509:
2505:
2501:
2493:
2491:
2473:
2469:
2467:
2464:
2463:
2427:
2423:
2390:
2386:
2359:
2354:
2348:
2345:
2344:
2321:
2319:
2316:
2315:
2299:
2296:
2295:
2275:
2272:
2271:
2233:
2229:
2205:
2201:
2199:
2196:
2195:
2185:
2174:
2163:
2148:
2120:
2089:
2088:
2087:
2085:
2054:
2053:
2052:
2050:
2047:
2046:
2034:
1955:
1950:
1944:
1941:
1940:
1911:
1907:
1890:
1887:
1886:
1861:
1858:
1857:
1850:
1815:
1811:
1809:
1806:
1805:
1774:
1771:
1770:
1733:
1730:
1729:
1709:
1701:
1699:
1696:
1695:
1672:
1663:
1659:
1657:
1654:
1653:
1622:
1619:
1618:
1587:
1584:
1583:
1564:
1556:
1554:
1551:
1550:
1531:
1528:
1527:
1511:
1508:
1507:
1469:
1465:
1415:
1411:
1371:
1367:
1365:
1362:
1361:
1342:
1339:
1338:
1322:
1319:
1318:
1296:
1293:
1292:
1263:
1259:
1257:
1254:
1253:
1216:
1208:
1200:
1197:
1196:
1190:
1169:
1166:
1165:
1148:
1145:
1144:
1126:
1123:
1122:
1105:
1102:
1101:
1093:
1063:
1059:
1057:
1054:
1053:
1030:
1026:
1024:
1021:
1020:
1015:right scattered
993:
989:
987:
984:
983:
960:
956:
954:
951:
950:
900:
897:
896:
893:right scattered
860:
857:
856:
820:
817:
816:
780:
777:
776:
753:
750:
749:
733:
725:
722:
721:
710:
672:
669:
668:
652:
649:
648:
646:
615:
612:
611:
580:
577:
576:
560:
557:
556:
504:
501:
500:
481:
478:
477:
439:
410:
407:
406:
373:
344:
341:
340:
321:
318:
317:
298:
277:
263:
260:
259:
237:
232:
229:
228:
208:
206:
203:
202:
183:
181:
178:
177:
161:
159:
156:
155:
137:
81:
65:
23:
22:
15:
12:
11:
5:
3296:
3286:
3285:
3280:
3275:
3261:
3260:
3255:
3248:
3247:External links
3245:
3244:
3243:
3233:
3223:
3213:
3176:
3173:
3170:
3169:
3132:
3097:
3056:
3015:Eckhardt, J.;
3007:
2983:
2950:
2935:(PhD thesis).
2921:
2915:10.1.1.79.8824
2896:
2870:
2849:(2): 391–415.
2829:
2808:(1–2): 35–55.
2788:
2781:
2775:. Birkhäuser.
2760:
2743:Hilger, Stefan
2733:
2732:
2730:
2727:
2726:
2725:
2720:
2715:
2710:
2698:
2695:
2687:
2684:
2683:
2682:
2669:
2664:
2661:
2658:
2655:
2653:
2650:
2647:
2646:
2643:
2640:
2637:
2634:
2632:
2625:
2622:
2619:
2616:
2612:
2606:
2605:
2603:
2598:
2595:
2592:
2589:
2583:
2577:
2573:
2545:
2542:
2541:
2540:
2529:
2526:
2523:
2520:
2512:
2508:
2504:
2499:
2496:
2490:
2487:
2484:
2481:
2476:
2472:
2453:
2452:
2441:
2438:
2435:
2430:
2426:
2422:
2419:
2416:
2413:
2410:
2405:
2402:
2399:
2396:
2393:
2389:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2362:
2357:
2353:
2324:
2303:
2279:
2268:
2267:
2256:
2253:
2250:
2247:
2244:
2239:
2236:
2232:
2228:
2225:
2222:
2219:
2216:
2213:
2208:
2204:
2184:
2181:
2173:
2170:
2162:
2159:
2147:
2144:
2129:
2126:
2123:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2092:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2062:
2057:
2033:
2030:
2029:
2028:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1958:
1953:
1949:
1925:
1922:
1919:
1914:
1910:
1906:
1903:
1900:
1897:
1894:
1874:
1871:
1868:
1865:
1854:delta integral
1849:
1846:
1829:
1826:
1823:
1818:
1814:
1793:
1790:
1787:
1784:
1781:
1778:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1712:
1708:
1704:
1678:
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1666:
1662:
1641:
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1635:
1632:
1629:
1626:
1606:
1603:
1600:
1597:
1594:
1591:
1571:
1567:
1563:
1559:
1535:
1515:
1504:
1503:
1491:
1487:
1484:
1481:
1478:
1475:
1472:
1468:
1464:
1461:
1457:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1418:
1414:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1370:
1346:
1326:
1306:
1303:
1300:
1277:
1274:
1271:
1266:
1262:
1235:
1234:
1223:
1219:
1215:
1211:
1207:
1204:
1189:
1186:
1173:
1152:
1130:
1109:
1092:
1089:
1088:
1087:
1081:left scattered
1066:
1062:
1050:
1033:
1029:
1017:
996:
992:
980:
963:
959:
943:
942:
936:
930:
919:
916:
913:
910:
907:
904:
890:
879:
876:
873:
870:
867:
864:
853:left scattered
850:
839:
836:
833:
830:
827:
824:
810:
799:
796:
793:
790:
787:
784:
757:
736:
732:
729:
709:
706:
694:
691:
688:
685:
682:
679:
676:
656:
634:
631:
628:
625:
622:
619:
599:
596:
593:
590:
587:
584:
564:
553:
552:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
485:
471:
470:
458:
455:
452:
449:
446:
442:
438:
435:
432:
429:
426:
423:
420:
417:
414:
404:
392:
389:
386:
383:
380:
376:
372:
369:
366:
363:
360:
357:
354:
351:
348:
325:
297:
294:
293:
292:
280:
276:
273:
270:
267:
240:
236:
211:
186:
164:
136:
133:
80:
77:
64:
61:
53:hybrid systems
9:
6:
4:
3:
2:
3295:
3284:
3281:
3279:
3276:
3274:
3271:
3270:
3268:
3259:
3256:
3254:
3251:
3250:
3241:
3237:
3234:
3231:
3227:
3224:
3221:
3217:
3214:
3209:
3204:
3200:
3196:
3193:(1–2): 1–26.
3192:
3188:
3184:
3179:
3178:
3165:
3161:
3156:
3151:
3147:
3143:
3136:
3128:
3124:
3120:
3116:
3112:
3108:
3101:
3092:
3087:
3083:
3079:
3075:
3071:
3067:
3060:
3052:
3048:
3044:
3040:
3035:
3030:
3026:
3022:
3018:
3011:
3003:
2996:
2995:
2987:
2978:
2973:
2969:
2965:
2961:
2954:
2946:
2942:
2938:
2934:
2933:
2925:
2916:
2911:
2907:
2900:
2892:
2888:
2881:
2874:
2865:
2860:
2856:
2852:
2848:
2844:
2840:
2833:
2824:
2819:
2815:
2811:
2807:
2803:
2799:
2792:
2784:
2778:
2774:
2767:
2765:
2756:
2752:
2748:
2744:
2738:
2734:
2724:
2721:
2719:
2716:
2714:
2711:
2708:
2704:
2701:
2700:
2694:
2692:
2662:
2659:
2656:
2651:
2648:
2641:
2638:
2635:
2630:
2620:
2614:
2610:
2601:
2596:
2590:
2575:
2571:
2563:
2562:
2561:
2559:
2555:
2551:
2527:
2521:
2506:
2502:
2497:
2494:
2488:
2482:
2470:
2462:
2461:
2460:
2458:
2436:
2424:
2420:
2414:
2408:
2400:
2397:
2394:
2387:
2383:
2380:
2371:
2365:
2360:
2355:
2351:
2343:
2342:
2341:
2339:
2301:
2293:
2277:
2254:
2245:
2237:
2234:
2230:
2223:
2220:
2214:
2202:
2194:
2193:
2192:
2190:
2180:
2178:
2169:
2167:
2158:
2156:
2152:
2143:
2127:
2124:
2121:
2110:
2107:
2104:
2098:
2082:
2073:
2067:
2060:
2044:
2039:
2015:
2009:
2003:
2000:
1994:
1988:
1985:
1979:
1967:
1961:
1956:
1951:
1947:
1939:
1938:
1937:
1920:
1908:
1904:
1898:
1892:
1869:
1863:
1855:
1845:
1843:
1827:
1821:
1812:
1791:
1788:
1782:
1776:
1756:
1753:
1750:
1747:
1741:
1735:
1727:
1706:
1693:
1676:
1673:
1669:
1660:
1639:
1636:
1630:
1624:
1604:
1601:
1595:
1589:
1569:
1561:
1547:
1533:
1513:
1489:
1485:
1482:
1476:
1470:
1466:
1462:
1459:
1455:
1448:
1445:
1439:
1433:
1424:
1412:
1408:
1402:
1396:
1393:
1384:
1378:
1372:
1368:
1360:
1359:
1358:
1344:
1324:
1304:
1301:
1298:
1289:
1272:
1260:
1251:
1246:
1244:
1241:could be any
1240:
1221:
1205:
1202:
1195:
1194:
1193:
1185:
1171:
1163:
1150:
1128:
1120:
1107:
1097:
1086:
1082:
1064:
1060:
1051:
1049:
1031:
1027:
1018:
1016:
1012:
994:
990:
981:
979:
961:
957:
948:
947:
946:
940:
937:
934:
931:
917:
914:
908:
902:
894:
891:
877:
874:
868:
862:
854:
851:
837:
834:
828:
822:
814:
811:
797:
794:
788:
782:
774:
771:
770:
769:
755:
730:
727:
714:
705:
692:
689:
686:
680:
674:
654:
632:
629:
623:
617:
597:
594:
588:
582:
562:
539:
536:
533:
527:
521:
518:
512:
506:
499:
498:
497:
483:
476:
453:
450:
447:
444:
436:
433:
424:
418:
412:
405:
387:
384:
381:
378:
370:
367:
358:
352:
346:
339:
338:
337:
323:
315:
314:backward jump
311:
302:
274:
271:
268:
265:
258:
257:
256:
253:
234:
226:
225:discrete time
199:
154:
150:
149:closed subset
146:
145:measure chain
142:
132:
130:
126:
122:
118:
114:
109:
107:
103:
99:
95:
91:
86:
76:
74:
70:
69:Stefan Hilger
60:
58:
54:
50:
46:
42:
39:with that of
38:
34:
30:
19:
3239:
3229:
3219:
3190:
3186:
3148:(J11): 1–9.
3145:
3141:
3135:
3110:
3106:
3100:
3073:
3069:
3059:
3024:
3020:
3010:
2993:
2986:
2967:
2963:
2953:
2931:
2924:
2905:
2899:
2890:
2886:
2873:
2846:
2842:
2832:
2805:
2801:
2791:
2772:
2746:
2737:
2689:
2558:Hilger delta
2557:
2547:
2454:
2269:
2191:defined via
2186:
2175:
2164:
2149:
2035:
1853:
1851:
1548:
1505:
1290:
1249:
1247:
1243:Banach space
1238:
1236:
1191:
1142:
1099:
1094:
1084:
1080:
1047:
1014:
1010:
977:
944:
938:
932:
892:
852:
812:
772:
719:
554:
474:
472:
313:
310:forward jump
309:
307:
254:
200:
144:
140:
138:
110:
98:real numbers
82:
66:
32:
26:
2970:: 107–127.
2550:Dirac delta
2043:Z-transform
1848:Integration
1357:such that:
1085:right dense
813:right dense
29:mathematics
3267:Categories
3017:Teschl, G.
2893:: 351–379.
2729:References
2707:Cantor set
1291:For every
1188:Derivative
1096:Continuity
1091:Continuity
1011:left dense
773:left dense
475:graininess
141:time scale
106:Cantor set
100:or set of
85:continuous
59:operator.
3155:1012.1555
3034:1102.2511
2945:304364901
2910:CiteSeerX
2755:246538565
2660:≠
2615:μ
2572:δ
2511:Δ
2507:μ
2475:Δ
2429:Δ
2425:μ
2388:∫
2378:Δ
2352:∫
2302:ρ
2278:λ
2235:−
2231:ρ
2224:λ
2207:Δ
2203:μ
2001:−
1974:Δ
1948:∫
1936:one sets
1913:Δ
1825:Δ
1817:Δ
1777:μ
1736:σ
1665:Δ
1625:μ
1590:σ
1483:−
1471:σ
1463:ε
1460:≤
1446:−
1434:σ
1417:Δ
1409:−
1394:−
1379:σ
1299:ε
1265:Δ
1214:→
903:σ
863:ρ
823:σ
783:ρ
731:∈
675:ρ
618:μ
583:σ
534:−
522:σ
507:μ
484:μ
437:∈
413:ρ
371:∈
347:σ
275:∈
153:real line
47:with the
3278:Calculus
3127:16404013
3051:17178288
2941:ProQuest
2745:(1989).
2697:See also
2290:denotes
2061:′
1726:integers
1692:calculus
1677:′
1506:for all
1048:isolated
939:isolated
720:For any
223:and the
113:calculus
102:integers
90:function
45:calculus
3195:Bibcode
3160:Bibcode
3078:Bibcode
2851:Bibcode
2810:Bibcode
2189:measure
1840:is the
1237:(where
151:of the
147:) is a
63:History
3242:(2009)
3232:(2006)
3222:(2002)
3125:
3049:
2943:
2912:
2779:
2753:
2270:where
1052:Point
1019:Point
982:Point
949:Point
227:scale
123:, and
3150:arXiv
3123:S2CID
3047:S2CID
3029:arXiv
2998:(PDF)
2883:(PDF)
1724:(the
1694:. If
1582:Then
1549:Take
978:dense
933:dense
2777:ISBN
2751:OCLC
2552:and
2548:The
2294:and
2153:and
1852:The
1302:>
1083:and
1013:and
915:>
875:<
768:is:
610:and
473:The
451:<
385:>
312:and
308:The
143:(or
3203:doi
3191:141
3115:doi
3086:doi
3074:332
3039:doi
3025:385
2972:doi
2968:285
2859:doi
2847:186
2818:doi
2806:141
1728:),
1526:in
1337:of
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