2218:
22:
1996:
The detailed mechanism of many real physico-chemical complex systems includes both reversible and irreversible reactions. Such mechanisms are typical in homogeneous combustion, heterogeneous catalytic oxidation, and complex enzyme reactions. The classical thermodynamics of perfect systems is defined
232:
Yablonsky, together with Lazman, developed the general form of steady-state kinetic description (the kinetic polynomial’), which is a non-linear generalization of many theoretical expressions proposed previously (the
Langmuir –Hinshelwood and Hougen–Watson equations). Yablonsky also created a theory
1987:
This quasithermodynamic property of the systems without interaction of different components is important for the analysis of the dynamics of catalytic reactions: nonlinear steps with two (or more) different intermediate reagents are responsible for nontrivial dynamical effects like multiplicity of
71:
2011:
requirement, these restrictions should be formulated as follows: A system with some irreversible reactions should be at the limit of the systems with all reversible reactions and the detailed balance conditions. Such systems have been completely described in 2011. The
1471:
629:), Z is the "adsorption place" on the surface of the solid catalyst (for example, Pt), AZ and BZ are the intermediates on the surface (adatoms, adsorbed molecules, or radicals), and (BZ) is an intermediate that does not participate in the main reaction.
2248:
During his career, Yablonsky has organized many conferences and workshops at national and international levels. Yablonsky frequently participates in interdisciplinary dialogues involving mathematicians, chemists, physicists, and chemical engineers.
1979:
2031:: The principle of detailed balance is valid for the reversible part. This means that for the set of all reversible reactions, there exists a positive equilibrium where all the elementary reactions are equilibrated by their reverse reactions.
1352:
2001:
gives the possibility to write the chemical kinetic equations for any combination of reversible and irreversible reactions. Without additional restrictions, this class of equations is extremely wide and can approximate any
2006:
with preservation of positivity of concentrations and the linear conservation laws. (This general approximation theorem was proved in 1986.) The model of real systems should satisfy some restrictions. Under the standard
2128:
632:
Let the concentration of the gaseous components be constant. Then the law of mass action gives for this reaction mechanism a system of three ordinary differential equations that describe kinetics on the surface.
1081:
1831:
736:
2286:
Yablonsky has numerous international designations as an honorary professor, fellow, doctor, and member of prestigious science academies and universities in
Belgium, India, China, Russia, and Ukraine.
248:(University of Leicester, UK), obtained new results on coincidences and intersections in kinetic dependences and found a new type of symmetry relation between the observable and initial kinetic data.
2016:
is the characteristic property of all systems that obey the generalized mass action law and is the limit of systems with detailed balance when some of the reaction rate constants tend to zero (the
599:
1557:
2185:
834:
1730:
1166:
are the stoichiometric coefficients (usually they are integer numbers). (The components that are present in excess and the components with almost constant concentrations are not included.)
1164:
530:
373:
1673:
464:
916:
2245:. Since 2007, Yablonsky has been an associate professor at Saint Louis University's Parks College of Engineering, Aviation, and Technology as well as in the department of chemistry.
271:
A simple scheme for the nonlinear kinetic oscillations in heterogeneous catalytic reactions has been proposed by Bykov, Yablonsky, and Kim in 1978. The authors have started with the
1364:
2265:
Lifetime
Achievement Award, in recognition of outstanding contributions to the research field of chemical kinetics, Mathematics in Chemical Kinetics and Engineering, MaCKiE, 2013
1839:
207:
and non-steady-state catalytic reactions is widely used by research teams in many countries of the world (the USA, UK, Belgium, Germany, France, Norway, and
Thailand).
1988:
steady states, oscillations, or bifurcations. Without interaction between different components, the kinetic curves converge into a simple norm, even for open systems.
2357:
Estathiou, A.M.; G.S. Yablonsky; Gleaves, J. T. (2012). "Transient
Techniques: Temporal Analysis of Products and Steady State Isotopic Transient Kinetic Analysis".
1757:
1614:
1587:
1111:
2046:
of the stoichiometric vectors of the reversible reactions. (Physically, this means that the irreversible reactions cannot be included in oriented cyclic pathways.)
214:
169:
1179:
3005:
2870:
272:
2891:
2206:
2205:
of the stoichiometric vectors of the reversible reactions, and the reaction rate constants of the remaining reversible reactions should satisfy the
200:, particularly in catalytic technology of complete and selective oxidation, which is one of the main driving forces of sustainable development.
2975:
2970:
2965:
2732:
Bykov, V. I.; Elokhin, V. I.; Yablonskii, G. S. (1976). "The simplest catalytic mechanism permitting several steady states of the surface".
2017:
2053:
1002:
1173:
of CO oxidation on PT provides a simple example of such a reaction mechanism without interaction of different components on the surface:
2241:
From 1997 to 2007, Yablonsky was a research associate professor in the department of energy, environmental, and chemical engineering at
2405:
Grigoriy, Yablonsky; M. Olea; G. Marin (May–June 2003). "Temporal
Analysis of Products: Basic Principles, Applications, and Theory".
2190:(It may be useful to recall the formal convention: the linear span of an empty set is 0; the convex hull of an empty set is empty.)
2291:
121:
2715:
M.G. Slin'ko, V.I. Bykov, G.S. Yablonskii, T.A. Akramov, "Multiplicity of the Steady State in
Heterogeneous Catalytic Reactions",
2698:
V.I. Bykov, G.S. Yablonskii, V.F. Kim, "On the simple model of kinetic self-oscillations in catalytic reaction of CO oxidation",
2426:
Yablonsky, Grigoriy; I.M.Y. Mareels; M. Lazman (November 2003). "The principle of critical simplification in chemical kinetics".
1762:
641:
2347:
2960:
1490:
560:
2384:
Gleaves, J.T.; G.S. Yablonsky; P. Phanawadee; Y. Schuurman (14 October 1997). "TAP-2. Interrogative
Kinetics Approach".
2137:
2359:
Transient
Techniques: Temporal Analysis of Products (TAP) and Steady-State Isotopic Transient Kinetic Analysis (SSITKA)
2242:
32:
759:
2374:
1678:
549:
2794:, Part 1, 1975, Computing Center USSR Academy of Sciences, Siberian Branch, Novosibirsk (in Russian), pp. 199–206.
2482:
1120:
487:
287:
2309:
980:
This three-dimensional system includes seven parameters. The detailed analysis shows that there are 23 different
1619:
3000:
2995:
1466:{\displaystyle \alpha _{r}m_{i_{r}}=\sum _{j}\beta _{rj}m_{j}{\text{ for some }}m_{j}>0{\text{ and all }}r,}
857:
396:
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V.I. Bykov, T.A. Akramov, and G.S. Yablonskii, "Studies of the dynamic properties of catalytic systems", In
2628:
234:
2895:
2845:
2667:
2234:
2230:
2198:
2193:
The extended principle of detailed balance gives an ultimate and complete answer to the following problem:
2039:
252:
245:
2766:
A.I. Khibnik, V.I. Bykov, and G.S. Yablonski, "23 phase portraits of the simplest catalytic oscillator",
2303:
2256:, which recognizes a distinguished individual for outstanding achievement in engineering or technology.
2195:
how to throw away some reverse reactions without violating thermodynamics and microscopic reversibility?
1616:. Then the dynamic of the kinetic system is very simple: the steady states are stable and all solutions
2980:
2940:
2574:
2557:
2540:
2519:
2297:
2008:
1170:
2805:"Thermodynamic function analogue for reactions proceeding without interaction of various substances"
2277:
Honorary Fellow of the
Australian Institute of High Energetic Materials, Gladstone, Australia (2011)
1974:{\displaystyle \|\mathbf {c} ^{(1)}-\mathbf {c} ^{(2)}\|=\sum _{i}m_{i}|c_{i}^{(1)}-{c}_{i}^{(2)}|,}
953:
is the concentration of the free places of adsorption on the surface ("per one adsorption center"),
2990:
154:(2013), Lifetime Achievement Award (Mathematics in Chemical Kinetics and Engineering, MaCKiE-2013)
256:
2955:
2042:
of the stoichiometric vectors of the irreversible reactions has an empty intersection with the
211:
2804:
2221:
The programm committee of the Russian Conference "Mathematical Methods in Chemical Kinetics",
984:
for this system, including oscillations, multiplicity of steady states, and various types of
39:
2985:
2916:
2435:
1735:
1592:
1565:
1089:
197:
126:
2201:
of the stoichiometric vectors of the irreversible reactions should not intersect with the
1997:
for reversible kinetics and has no limit for irreversible reactions. On the contrary, the
8:
2253:
2217:
2439:
2268:
James B. Eads Award, Academy of Science of St. Louis Outstanding Scientist Award (2013).
233:
of precise catalyst characterization for the advanced worldwide experimental technique (
2749:
2683:
2648:
2607:
985:
151:
105:
2418:
2397:
2274:
Chevron Chair Professorship at the Indian Institute of Technology (IIT), Madras (2011)
2846:"Saint Louis University Professors Receive Honors for Their Contributions to Science"
2819:
2503:
2370:
2343:
1347:{\displaystyle {\ce {2Pt(+O2)<=>2Pt;\;\;{PtO}+CO<=>{Pt}+CO2\!\uparrow }}}
193:
131:
117:
2941:
Yablonsky's faculty profile at Parks College of Engineering, Aviation and Technology
2753:
2815:
2741:
2675:
2640:
2603:
2599:
2464:
2443:
2414:
2393:
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2226:
2003:
260:
241:
185:
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1998:
1481:
2679:
2501:
2366:
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981:
2644:
2468:
2447:
2383:
2949:
1477:
276:
2558:"Coincidences in Chemical Kinetics: Surprising News about Simple Reactions"
2455:
Feres, R.; G.S. Yablonsky (2004). "Knudsen Diffusion and Random Billiards".
991:
129:(physical chemistry) (Boreskov Institute of Catalysis), Novosibirsk, USSR),
2222:
204:
2324:
Wiley-VCH (2011) (together with Guy B. Marin), and more than 200 papers.
2202:
2043:
2745:
2302:
2011–present: Member of the Scientific Council on Catalysis at the
97:
2871:"Academy of Science Honors Yablonsky with Outstanding Scientist Award"
2590:
G.S. Yablonsky, A.N. Gorban, D. Constales, V. Galvita and G.B. Marin,
237:) developed by John T. Gleaves at Washington University in St. Louis.
2356:
2123:{\displaystyle \sum _{i}\alpha _{i}A_{i}\to \sum _{j}\beta _{j}A_{j}}
969:, correspondingly (also normalized "per one adsorption center"). and
122:
National Technical University of Ukraine (Kiev Polytechnic Institute)
2575:"Equilibrium relationships for non-equilibrium chemical dependences"
1076:{\displaystyle \alpha _{r}A_{i_{r}}\to \sum _{j}\beta _{rj}A_{j}\,,}
210:
Yablonsky serves as an associate research professor of chemistry at
2629:"Extended detailed balance for systems with irreversible reactions"
2520:"Overall Reaction Rate Equation of Single Route Catalytic Reaction"
2271:
Honorary Doctor Degree from the University of Ghent, Belgium (2010)
2687:
2652:
2611:
2023:
The extended principle of detailed balance consists of two parts:
220:
Since 2006, Yablonsky has been an editor of the Russian-American
2849:
2502:
Yablonsky, G.S.; V.I. Bykov; A.N. Gorban'; V.I. Elokhin (1991).
2425:
2329:
Yablonsky, G.S.; V.I. Bykov; A.N. Gorban'; V.I. Elokhin (1991).
1991:
266:
70:
2281:
240:
In 2008–2011, Yablonsky, together with Constales and Marin (
1826:{\displaystyle \mathbf {c} ^{(1)}(t),\mathbf {c} ^{(2)}(t)}
101:
731:{\displaystyle {\dot {x}}=2q_{1}z^{2}-2q_{5}x^{2}-q_{3}xy}
1336:
1208:
992:
Reactions without the interaction of different components
303:
2668:"Thermodynamics in the limit of irreversible reactions"
2539:
S.O. Shekhtman, G.S. Yablonsky, S. Chen, J.T. Gleaves,
2404:
2731:
2623:
2621:
2619:
2320:
Yablonsky is the author of seven books, most recently
1299:
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433:
338:
2140:
2056:
1842:
1765:
1738:
1681:
1622:
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1367:
1358:
Let the reaction mechanism have the conservation law
1182:
1123:
1092:
1005:
860:
762:
644:
563:
490:
399:
290:
215:
Parks College of Engineering, Aviation and Technology
170:
Parks College of Engineering, Aviation and Technology
2662:
2660:
594:{\displaystyle {\ce {{B}+Z\rightleftharpoons (BZ)}}}
134:(Boreskov Institute of Catalysis, Novosibirsk, USSR)
2616:
2454:
2337:
2322:
Kinetics of Chemical Reactions: Decoding Complexity
1552:{\displaystyle W_{r}=k_{r}c_{i_{r}}^{\alpha _{r}},}
2541:" Thin-Zone TAP-Reactor – Theory and Application "
2340:Kinetics of Complex Reactions. Decoding Complexity
2180:{\displaystyle \gamma _{i}=\beta _{i}-\alpha _{i}}
2179:
2122:
1973:
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1724:
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1608:
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367:
2657:
1339:
1307:
1306:
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1244:
1243:
1226:
1225:
996:Let the reaction mechanism consist of reactions.
441:
440:
423:
422:
346:
345:
328:
327:
2947:
829:{\displaystyle {\dot {y}}=q_{2}z-q_{6}y-q_{3}xy}
275:(1976), the simplest catalytic reaction without
142:Theory of complex non-linear catalytic reactions
1725:{\displaystyle m(\mathbf {c} )=\sum m_{i}c_{i}}
973:is the concentration of the buffer component (
2592:"Reciprocal Relations Between Kinetic Curves"
1159:{\displaystyle \alpha _{r},\beta _{rj}\geq 0}
1117:is the number of the elementary reaction and
525:{\displaystyle {\ce {{AZ}+BZ -> {AB}+2Z}}}
368:{\displaystyle {\ce {{A2}+2Z <=> 2AZ}}}
2508:. Amsterdam–Oxford–New York–Tokyo: Elsevier.
2333:. Amsterdam–Oxford–New York–Tokyo: Elsevier.
1885:
1843:
1675:with the same value of the conservation law
2917:"Dr. Yablonsky Receives Honorary Doctorate"
2773:(1987), 1388–1390 . English translation in
2361:. Vol. 1 & 2. pp. 1013–1073.
548:Then they have supplemented this classical
279:that allows multiplicity of steady states.
227:
3006:Washington University in St. Louis faculty
2831:A.N. Gorban, V.I. Bykov, G.S. Yablonskii,
2803:A.N. Gorban, V.I. Bykov, G.S. Yablonskii,
2666:A.N. Gorban, E.M. Mirkes, G.S. Yablonsky,
2050:The stoichiometric vector of the reaction
1992:The extended principle of detailed balance
1759:norm: the distance between such solutions
1668:{\displaystyle \mathbf {c} (t)=(c_{i}(t))}
1266:
1265:
552:of catalytic oxidation by a "buffer" step
459:{\displaystyle {\ce {{B}+Z <=> BZ}}}
267:Catalytic trigger and catalytic oscillator
69:
2892:"Outstanding St. Louis Scientists Awards"
2573:G. S. Yablonsky, D. Constales, G. Marin,
2556:G. S. Yablonsky, D. Constales, G. Marin,
2282:Professional memberships and associations
1258:
1187:
1069:
911:{\displaystyle {\dot {s}}=q_{4}z-kq_{4}s}
517:
360:
313:
2848:. Saint Louis University. Archived from
2292:American Institute of Chemical Engineers
2216:
263:in the limit of irreversible reactions.
2734:Reaction Kinetics and Catalysis Letters
2315:
2252:Yablonsky was selected in 2013 for the
1732:monotonically converge in the weighted
1282:
1219:
416:
321:
2948:
2014:extended principle of detailed balance
621:, B, and AB are gases (for example, O
2976:Saint Louis University mathematicians
2775:Russian Journal of Physical Chemistry
2505:Kinetic Models of Catalytic Reactions
2331:Kinetic Models of Catalytic Reactions
2971:21st-century American mathematicians
2966:20th-century American mathematicians
2894:. Academy of Science. Archived from
2338:Marin, G.B.; G.S. Yablonsky (2011).
2259:
851:
753:
635:
554:
481:
390:
281:
255:, Yablonsky developed the theory of
15:
13:
2627:A. N. Gorban and G. S. Yablonsky,
2243:Washington University in St. Louis
939: = 1 − (
217:and college of arts and sciences.
14:
3017:
2934:
1984:monotonically decreases in time.
42:and remove advice or instruction.
2792:Mathematicheskie problemy khimii
2577:, Chemical Engineering Science,
2560:, Chemical Engineering Science,
2543:, Chemical Engineering Science,
2524:Advances in Chemical Engineering
2483:Chemical reaction network theory
1869:
1848:
1798:
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1624:
20:
2909:
2884:
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2797:
2784:
2760:
2725:
2709:
2310:Academy of Science of St. Louis
2692:
2584:
2567:
2550:
2533:
2518:M. Lazman and G.S. Yablonsky,
2512:
2495:
2237:, G.S. Yablonsky, A.N.Ivanova.
2087:
1964:
1958:
1952:
1932:
1926:
1912:
1880:
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1859:
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1820:
1814:
1809:
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1628:
1340:
1309:
1284:
1246:
1221:
1211:
1192:
1033:
586:
580:
503:
443:
418:
348:
323:
192:) is an expert in the area of
1:
2833:Essays on Chemical Relaxation
2488:
2419:10.1016/S0021-9517(02)00109-4
2398:10.1016/S0926-860X(97)00124-5
235:temporal analysis of products
2820:10.1016/0009-2509(86)80005-7
2809:Chemical Engineering Science
2633:Chemical Engineering Science
2457:Chemical Engineering Science
2428:Chemical Engineering Science
2386:Applied Catalysis A: General
2229:, 1980. From left to right:
190:Григорий Семенович Яблонский
150:James B. Eads Award via the
7:
2961:American chemical engineers
2835:, Nauka, Novosibirsk, 1986.
2680:10.1016/j.physa.2012.10.009
2476:
2304:Russian Academy of Sciences
1113:are symbols of components,
924:
842:
744:
607:
538:
472:
381:
10:
3022:
2604:10.1209/0295-5075/93/20004
2367:10.1002/9783527645329.ch22
2342:. Wiley-VCH. p. 428.
2308:2013–present: Fellow,
961:are the concentrations of
2645:10.1016/j.ces.2011.07.054
2469:10.1016/j.ces.2004.01.016
2448:10.1016/j.ces.2003.08.004
2298:American Chemical Society
2212:
2009:microscopic reversibility
189:
175:
165:
158:
146:
138:
113:
80:
68:
61:
2814:(11) (1986), 2739–2745;
2134:vector with coordinates
2018:Gorban-Yablonsky theorem
1589:is the concentration of
228:Scientific contributions
212:Saint Louis University's
75:Grigoriy Yablonsky, 2004
2594:, Europhysics Letters,
2197:The answer is that the
257:chemical thermodynamics
100:, Stavropol'skij kraj,
2674:392 (2013) 1318–1335;
2238:
2207:Wegscheider identities
2181:
2124:
1975:
1827:
1753:
1726:
1669:
1610:
1583:
1553:
1467:
1348:
1160:
1107:
1077:
912:
830:
732:
595:
526:
460:
369:
203:His theory of complex
3001:Soviet mathematicians
2996:20th-century chemists
2717:Dokl. Akad. Nauk SSSR
2220:
2182:
2125:
1976:
1828:
1754:
1752:{\displaystyle l_{1}}
1727:
1670:
1611:
1609:{\displaystyle A_{i}}
1584:
1582:{\displaystyle c_{i}}
1554:
1468:
1349:
1171:Eley–Rideal mechanism
1161:
1108:
1106:{\displaystyle A_{i}}
1078:
913:
831:
733:
596:
527:
461:
370:
2706:(3) (1978), 637–639.
2407:Journal of Catalysis
2316:Notable publications
2138:
2054:
2036:structural condition
1840:
1763:
1736:
1679:
1620:
1593:
1566:
1491:
1433: for some
1365:
1180:
1121:
1090:
1003:
858:
760:
642:
561:
550:adsorption mechanism
488:
397:
288:
198:chemical engineering
40:rewrite this article
2440:2003ChEnS..58.4833Y
2254:James B. Eads Award
2029:algebraic condition
1962:
1936:
1545:
1454: and all
1338:
1295:
1232:
1210:
429:
334:
305:
2746:10.1007/BF02061998
2639::5388–5399, 2011;
2598::2004–2007, 2011;
2239:
2177:
2120:
2099:
2066:
1971:
1940:
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1900:
1823:
1749:
1722:
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1606:
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1344:
1326:
1314:
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1103:
1073:
1045:
908:
826:
728:
591:
522:
456:
448:
365:
353:
293:
182:Grigoriy Yablonsky
152:Academy of Science
106:Russian Federation
63:Grigoriy Yablonsky
2981:Jewish scientists
2434:(21): 4833–4842.
2349:978-3-527-31763-9
2260:Honors and awards
2090:
2057:
1891:
1455:
1434:
1398:
1329:
1321:
1316:
1277:
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389:
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363:
355:
316:
296:
273:catalytic trigger
194:chemical kinetics
184:(or Yablonskii) (
179:
178:
160:Scientific career
91:September 7, 1940
57:
56:
33:a manual or guide
3013:
2928:
2927:
2925:
2924:
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2903:
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2842:
2836:
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2823:
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2795:
2788:
2782:
2781:(1987), 722–723.
2764:
2758:
2757:
2729:
2723:
2722:(4) (1976), 876.
2713:
2707:
2696:
2690:
2664:
2655:
2625:
2614:
2588:
2582:
2571:
2565:
2564::6065–6076, 2010
2554:
2548:
2547::4371–7378, 1999
2537:
2531:
2516:
2510:
2509:
2499:
2472:
2463:(7): 1541–1556.
2451:
2422:
2413:(1–2): 120–134.
2401:
2380:
2353:
2334:
2227:Krasnoyarsk Krai
2186:
2184:
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2178:
2176:
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2118:
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2086:
2085:
2076:
2075:
2065:
2004:dynamical system
1980:
1978:
1977:
1972:
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1961:
1950:
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1884:
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261:detailed balance
253:Alexander Gorban
246:Alexander Gorban
244:, Belgium), and
242:Ghent University
191:
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88:
73:
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52:
49:
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31:is written like
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2991:Soviet chemists
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2919:. Parks College
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2873:. Parks College
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2132:gain minus loss
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1999:mass action law
1994:
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2935:External links
2933:
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2740:(2): 191–198.
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139:Known for
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55:
54:
28:
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2956:Living people
2954:
2953:
2951:
2942:
2939:
2938:
2918:
2912:
2898:on 2013-05-03
2897:
2893:
2887:
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2852:on 2015-01-19
2851:
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2768:J. Fiz. Khim.
2763:
2755:
2751:
2747:
2743:
2739:
2735:
2728:
2721:
2718:
2712:
2705:
2702:(Chemistry)
2701:
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2530::47–102, 2008
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120:(chemistry),
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107:
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99:
93:(age 84)
83:
79:
72:
67:
60:
51:
41:
36:
34:
29:This article
27:
18:
17:
2921:. Retrieved
2911:
2900:. Retrieved
2896:the original
2886:
2875:. Retrieved
2865:
2854:. Retrieved
2850:the original
2840:
2832:
2827:
2811:
2808:
2799:
2791:
2786:
2778:
2774:
2770:
2767:
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2727:
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2703:
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2527:
2523:
2514:
2504:
2497:
2460:
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2410:
2406:
2392:(1): 55–88.
2389:
2385:
2358:
2339:
2330:
2321:
2319:
2285:
2251:
2247:
2240:
2223:Shushenskoye
2194:
2192:
2189:
2131:
2049:
2038:is that the
2035:
2028:
2022:
2013:
1995:
1986:
1983:
1561:
1480:satisfy the
1476:and let the
1475:
1357:
1168:
1114:
1085:
995:
986:bifurcations
979:
974:
970:
966:
962:
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954:
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944:
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936:
933:
922:
840:
742:
631:
625:, CO, and CO
616:
605:
547:
536:
470:
379:
270:
250:
239:
231:
221:
219:
209:
205:steady-state
202:
181:
180:
166:Institutions
159:
45:
38:Please help
30:
2986:1940 births
2235:A.N. Gorban
2203:linear span
2199:convex hull
2044:linear span
2040:convex hull
222:Middle West
2950:Categories
2923:2013-04-25
2902:2013-04-25
2877:2013-04-25
2856:2013-04-25
2489:References
98:Yessentuki
87:1940-09-07
48:March 2023
2777:, Volume
2688:1207.2507
2672:Physica A
2653:1101.5280
2612:1008.1056
2169:α
2165:−
2156:β
2143:γ
2102:β
2092:∑
2088:→
2069:α
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1938:−
1893:∑
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1865:−
1844:‖
1700:∑
1536:α
1410:β
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1247:⇀
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