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## Kainen, Paul C.

Compute Distance To:
 Author ID: kainen.paul-c Published as: Kainen, P. C.; Kainen, Paul; Kainen, Paul C.
 Documents Indexed: 74 Publications since 1968, including 1 Book
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#### Co-Authors

 39 single-authored 17 Kůrková, Věra 6 Harary, Frank 6 Vogt, Andrew 5 Hammack, Richard H. 4 Sanguineti, Marcello 2 Hechler, Stephen Herman 2 Schwenk, Allen J. 2 White, Arthur T. 1 Bernhart, Frank R. 1 Erdős, Pál 1 Hunter, Rachel 1 Kreinovich, Vladik Yakovlevich 1 Overbay, Shannon 1 Riskin, Adrian 1 Saaty, Thomas L. 1 Sirisaengtaksin, Ongard
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#### Serials

 5 Bulletin of the Institute of Combinatorics and its Applications 5 Geombinatorics 4 Journal of Approximation Theory 4 Journal of Combinatorial Theory. Series B 4 Applied Mathematics Letters 3 Cahiers de Topologie et Géométrie Différentielle Catégoriques 2 Proceedings of the American Mathematical Society 2 Missouri Journal of Mathematical Sciences 2 The Art of Discrete and Applied Mathematics 1 Discrete Applied Mathematics 1 Discrete Mathematics 1 IEEE Transactions on Information Theory 1 Israel Journal of Mathematics 1 Journal of the Franklin Institute 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Bulletin of the Calcutta Mathematical Society 1 1 Journal of Graph Theory 1 Journal of the London Mathematical Society. Second Series 1 Mathematische Nachrichten 1 Mathematica Scandinavica 1 Mathematische Zeitschrift 1 Nanta Mathematica 1 Pacific Journal of Mathematics 1 Quaestiones Mathematicae 1 Graphs and Combinatorics 1 Journal of Complexity 1 Neural Networks 1 Annals of Operations Research 1 Neural Computation 1 SIAM Journal on Optimization 1 Neural, Parallel & Scientific Computations 1 Diagrammes 1 JMMA. Journal of Mathematical Modelling and Algorithms 1 Journal of Combinatorial Theory 1 Ars Mathematica Contemporanea 1 Bulletin of the Hellenic Mathematical Society
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#### Fields

 51 Combinatorics (05-XX) 12 Computer science (68-XX) 11 Approximations and expansions (41-XX) 9 Category theory; homological algebra (18-XX) 9 Manifolds and cell complexes (57-XX) 5 Biology and other natural sciences (92-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Operations research, mathematical programming (90-XX) 2 History and biography (01-XX) 2 Numerical analysis (65-XX) 1 General and overarching topics; collections (00-XX) 1 Group theory and generalizations (20-XX) 1 Real functions (26-XX) 1 Measure and integration (28-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Integral transforms, operational calculus (44-XX) 1 Functional analysis (46-XX) 1 Geometry (51-XX) 1 Convex and discrete geometry (52-XX) 1 General topology (54-XX) 1 Algebraic topology (55-XX) 1 Probability theory and stochastic processes (60-XX)

#### Citations contained in zbMATH

49 Publications have been cited 353 times in 279 Documents Cited by Year
The book thickness of a graph. Zbl 0427.05028
Bernhart, Frank; Kainen, Paul C.
1979
The four-color problem. Assaults and conquest. Zbl 0463.05041
Saaty, Thomas L.; Kainen, Paul C.
1977
Toroidal graphs with arbitrarily high crossing numbers. Zbl 0285.05104
Harary, Frank; Kainen, Paul C.; Schwenk, Allen J.
1973
Complexity of Gaussian-radial-basis networks approximating smooth functions. Zbl 1162.65006
Kainen, Paul C.; Kůrková, Věra; Sanguineti, Marcello
2009
An integral upper bound for neural network approximation. Zbl 1186.68370
Kainen, Paul C.; Kůrková, Věra
2009
A lower bound for crossing numbers of graphs with application to $$K_n$$,$$K_{pq}$$g, and $$Q(d)$$. Zbl 0237.05104
Kainen, Paul C.
1972
The book thickness of a graph. II. Zbl 0749.05030
Kainen, Paul C.
1990
Thickness and coarseness of graphs. Zbl 0264.05108
Kainen, Paul C.
1973
Kainen, P. C.
1971
A Sobolev-type upper bound for rates of approximation by linear combinations of Heaviside plane waves. Zbl 1119.41023
Kainen, Paul C.; Kurková, Věra; Vogt, Andrew
2007
Minimization of error functionals over variable-basis functions. Zbl 1061.49020
Kainen, Paul C.; Kurková, Vera; Sanguineti, Marcello
2003
On a problem of P. Erdős. Zbl 0182.57901
Kainen, Paul C.
1968
Dependence of computational models on input dimension: tractability of approximation and optimization tasks. Zbl 1365.68373
Kainen, Paul C.; Kůrková, Věra; Sanguineti, Marcello
2012
Geometry and topology of continuous best and near best approximations. Zbl 0969.41020
Kainen, Paul C.; Kůrková, Věra; Vogt, Andrew
2000
On stable crossing numbers. Zbl 0401.05039
Kainen, Paul C.; White, Arthur T.
1978
Quasiorthogonal dimension of Euclidean spaces. Zbl 0783.05030
Kainen, Paul C.; Kůrková, Věra
1993
A generalization of the 5-color theorem. Zbl 0293.05111
Kainen, Paul C.
1974
Extension of a theorem of Whitney. Zbl 1131.05035
Kainen, Paul C.; Overbay, Shannon
2007
Best approximation by linear combinations of characteristic functions of half-spaces. Zbl 1038.41011
Kainen, Paul C.; Kůrková, Věra; Vogt, Andrew
2003
On robust cycle bases. Zbl 1075.05555
Kainen, Paul C.
2002
Continuity of approximation by neural networks in $$L_p$$ spaces. Zbl 1055.41020
Kainen, Paul C.; Kurková, Vera; Vogt, Andrew
2001
Chromatic number and skewness. Zbl 0298.05116
Kainen, Paul C.
1975
Some recent results in topological graph theory. Zbl 0319.05103
Kainen, Paul C.
1974
Universal coefficient theorems for generalized homology and stable cohomotopy. Zbl 0211.32704
Kainen, P. C.
1971
Cycle construction and geodesic cycles with application to the hypercube. Zbl 1329.05162
Kainen, Paul C.
2015
Replacing points by compacta in neural network approximation. Zbl 1075.54013
Kainen, Paul C.
2004
A maximal toroidal graph which is not a triangulation. Zbl 0271.05108
Harary, Frank; Kainen, P. C.; Schwenk, A. J.; White, A. T.
1973
On the stable crossing number of cubes. Zbl 0253.05118
Kainen, Paul C.
1972
Graph bases and diagram commutativity. Zbl 1395.05092
Hammack, Richard H.; Kainen, Paul C.
2018
Integral combinations of Heavisides. Zbl 1225.26029
Kainen, Paul C.; Kůrková, Věra; Vogt, Andrew
2010
Singularities of finite scaling functions. Zbl 0852.42020
Kůrková, V.; Kainen, P. C.
1996
Every commutative square of Abelian groups can be realized by a homotopy commutative square of Moore, or co-Moore, spaces. Zbl 0243.55007
Kainen, Paul C.
1972
Embeddings and orientations of graphs. Zbl 0247.05109
Kainen, Paul C.
1970
On 2-skeleta of Platonic polytopes. Zbl 1425.57002
Hammack, Richard; Kainen, Paul C.
2018
Robust cycle bases do not exist for $$K_{n, n}$$ if $$n \geq 8$$. Zbl 1375.05150
Hammack, Richard H.; Kainen, Paul C.
2018
Comparing fixed and variable-width Gaussian networks. Zbl 1343.68200
Kurková, Věra; Kainen, Paul C.
2014
Graph cycles and diagram commutativity. Zbl 1338.05135
Kainen, Paul C.
2012
Rates of minimization of error functionals over Boolean variable-basis functions. Zbl 1080.49020
Kainen, P. C.; Kurková, V.; Sanguineti, M.
2005
Isolated squares in hypercubes and robustness of commutativity. Zbl 1042.18007
Kainen, Paul C.
2002
Every graph of cyclic bandwidth 3 is toroidal. Zbl 0938.05025
Harary, Frank; Kainen, Paul C.; Riskin, Adrian
1999
Utilizing geometric anomalies of high dimension: When complexity makes computation easier. Zbl 0906.68126
Kainen, Paul C.
1997
Uniqueness of network parametrization and faster learning. Zbl 0938.68801
Kainen, Paul C.; Kůrková, Věra; Kreinovich, Vladik; Sirisaengtaksin, Ongard
1994
Is the four color theorem true? Zbl 0854.05045
Kainen, Paul C.
1993
The cube of a path is maximal planar. Zbl 0804.05046
Harary, Frank; Kainen, Paul C.
1993
Upper bound for linear arboricity. Zbl 0732.05020
Kainen, Paul C.
1991
Arboricity and edge partitions. Zbl 0763.05028
Kainen, Paul C.
1991
On triangular colorings of a planar graph. Zbl 0416.05041
Harary, Frank; Kainen, Paul C.
1977
Relative colorings of graphs. Zbl 0258.05108
Kainen, Paul C.
1973
Trivalent descriptions of graphs. Zbl 0288.05103
Kainen, Paul C.
1971
Graph bases and diagram commutativity. Zbl 1395.05092
Hammack, Richard H.; Kainen, Paul C.
2018
On 2-skeleta of Platonic polytopes. Zbl 1425.57002
Hammack, Richard; Kainen, Paul C.
2018
Robust cycle bases do not exist for $$K_{n, n}$$ if $$n \geq 8$$. Zbl 1375.05150
Hammack, Richard H.; Kainen, Paul C.
2018
Cycle construction and geodesic cycles with application to the hypercube. Zbl 1329.05162
Kainen, Paul C.
2015
Comparing fixed and variable-width Gaussian networks. Zbl 1343.68200
Kurková, Věra; Kainen, Paul C.
2014
Dependence of computational models on input dimension: tractability of approximation and optimization tasks. Zbl 1365.68373
Kainen, Paul C.; Kůrková, Věra; Sanguineti, Marcello
2012
Graph cycles and diagram commutativity. Zbl 1338.05135
Kainen, Paul C.
2012
Integral combinations of Heavisides. Zbl 1225.26029
Kainen, Paul C.; Kůrková, Věra; Vogt, Andrew
2010
Complexity of Gaussian-radial-basis networks approximating smooth functions. Zbl 1162.65006
Kainen, Paul C.; Kůrková, Věra; Sanguineti, Marcello
2009
An integral upper bound for neural network approximation. Zbl 1186.68370
Kainen, Paul C.; Kůrková, Věra
2009
A Sobolev-type upper bound for rates of approximation by linear combinations of Heaviside plane waves. Zbl 1119.41023
Kainen, Paul C.; Kurková, Věra; Vogt, Andrew
2007
Extension of a theorem of Whitney. Zbl 1131.05035
Kainen, Paul C.; Overbay, Shannon
2007
Rates of minimization of error functionals over Boolean variable-basis functions. Zbl 1080.49020
Kainen, P. C.; Kurková, V.; Sanguineti, M.
2005
Replacing points by compacta in neural network approximation. Zbl 1075.54013
Kainen, Paul C.
2004
Minimization of error functionals over variable-basis functions. Zbl 1061.49020
Kainen, Paul C.; Kurková, Vera; Sanguineti, Marcello
2003
Best approximation by linear combinations of characteristic functions of half-spaces. Zbl 1038.41011
Kainen, Paul C.; Kůrková, Věra; Vogt, Andrew
2003
On robust cycle bases. Zbl 1075.05555
Kainen, Paul C.
2002
Isolated squares in hypercubes and robustness of commutativity. Zbl 1042.18007
Kainen, Paul C.
2002
Continuity of approximation by neural networks in $$L_p$$ spaces. Zbl 1055.41020
Kainen, Paul C.; Kurková, Vera; Vogt, Andrew
2001
Geometry and topology of continuous best and near best approximations. Zbl 0969.41020
Kainen, Paul C.; Kůrková, Věra; Vogt, Andrew
2000
Every graph of cyclic bandwidth 3 is toroidal. Zbl 0938.05025
Harary, Frank; Kainen, Paul C.; Riskin, Adrian
1999
Utilizing geometric anomalies of high dimension: When complexity makes computation easier. Zbl 0906.68126
Kainen, Paul C.
1997
Singularities of finite scaling functions. Zbl 0852.42020
Kůrková, V.; Kainen, P. C.
1996
Uniqueness of network parametrization and faster learning. Zbl 0938.68801
Kainen, Paul C.; Kůrková, Věra; Kreinovich, Vladik; Sirisaengtaksin, Ongard
1994
Quasiorthogonal dimension of Euclidean spaces. Zbl 0783.05030
Kainen, Paul C.; Kůrková, Věra
1993
Is the four color theorem true? Zbl 0854.05045
Kainen, Paul C.
1993
The cube of a path is maximal planar. Zbl 0804.05046
Harary, Frank; Kainen, Paul C.
1993
Upper bound for linear arboricity. Zbl 0732.05020
Kainen, Paul C.
1991
Arboricity and edge partitions. Zbl 0763.05028
Kainen, Paul C.
1991
The book thickness of a graph. II. Zbl 0749.05030
Kainen, Paul C.
1990
The book thickness of a graph. Zbl 0427.05028
Bernhart, Frank; Kainen, Paul C.
1979
On stable crossing numbers. Zbl 0401.05039
Kainen, Paul C.; White, Arthur T.
1978
The four-color problem. Assaults and conquest. Zbl 0463.05041
Saaty, Thomas L.; Kainen, Paul C.
1977
On triangular colorings of a planar graph. Zbl 0416.05041
Harary, Frank; Kainen, Paul C.
1977
Chromatic number and skewness. Zbl 0298.05116
Kainen, Paul C.
1975
A generalization of the 5-color theorem. Zbl 0293.05111
Kainen, Paul C.
1974
Some recent results in topological graph theory. Zbl 0319.05103
Kainen, Paul C.
1974
Toroidal graphs with arbitrarily high crossing numbers. Zbl 0285.05104
Harary, Frank; Kainen, Paul C.; Schwenk, Allen J.
1973
Thickness and coarseness of graphs. Zbl 0264.05108
Kainen, Paul C.
1973
A maximal toroidal graph which is not a triangulation. Zbl 0271.05108
Harary, Frank; Kainen, P. C.; Schwenk, A. J.; White, A. T.
1973
Relative colorings of graphs. Zbl 0258.05108
Kainen, Paul C.
1973
A lower bound for crossing numbers of graphs with application to $$K_n$$,$$K_{pq}$$g, and $$Q(d)$$. Zbl 0237.05104
Kainen, Paul C.
1972
On the stable crossing number of cubes. Zbl 0253.05118
Kainen, Paul C.
1972
Every commutative square of Abelian groups can be realized by a homotopy commutative square of Moore, or co-Moore, spaces. Zbl 0243.55007
Kainen, Paul C.
1972
Kainen, P. C.
1971
Universal coefficient theorems for generalized homology and stable cohomotopy. Zbl 0211.32704
Kainen, P. C.
1971
Trivalent descriptions of graphs. Zbl 0288.05103
Kainen, Paul C.
1971
Embeddings and orientations of graphs. Zbl 0247.05109
Kainen, Paul C.
1970
On a problem of P. Erdős. Zbl 0182.57901
Kainen, Paul C.
1968
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#### Cited by 412 Authors

 22 Kainen, Paul C. 15 Kůrková, Věra 15 Sanguineti, Marcello 10 Gnecco, Giorgio 10 Liotta, Giuseppe 9 Costarelli, Danilo 9 Jendrol’, Stanislav 6 Czap, Július 6 Di Giacomo, Emilio 5 Klešč, Marián 5 Salazar, Gelasio 5 Székely, László A. 5 Vinti, Gianluca 5 Wood, David Ronald 4 Bekos, Michael A. 4 Hammack, Richard H. 4 Hartung, Elizabeth J. 4 Sýkora, Ondrej 4 Vogt, Andrew 4 Vrto, Imrich 4 Wismath, Stephen K. 3 Alessandri, Angelo 3 Auer, Christopher 3 Beineke, Lowell W. 3 Binucci, Carla 3 Brandenburg, Franz-Josef 3 Cimikowski, Robert J. 3 Fabrici, Igor 3 Gleißner, Andreas 3 Hasunuma, Toru 3 Hong, Seok-Hee 3 Kaufmann, Michael 3 Kobourov, Stephen G. 3 Montecchiani, Fabrizio 3 Raftopoulou, Chrysanthi N. 3 Širáň, Jozef 3 Spigler, Renato 3 Tóth, Csaba D. 3 Voigt, Margit 2 Ábrego, Bernardo Manuel 2 Albertson, Michael O. 2 Alfaro Montúfar, Carlos Alejandro 2 Arroyo, Alan 2 Bachmaier, Christian 2 Bernhart, Frank R. 2 Bruckdorfer, Till 2 Cao, Feilong 2 Chia, Gek-Ling 2 Czumaj, Artur 2 Derňár, Marek 2 Didimo, Walter 2 Dujmović, Vida 2 Duncan, Christian A. 2 Eliahou, Shalom 2 Enomoto, Hikoe 2 Fernández-Merchant, Silvia 2 Fulek, Radoslav 2 Gaggero, Mauro 2 Garcia, Alfredo Daniel 2 Gibbons, Alan M. 2 Gorban, Alexander N. 2 Graver, Jack E. 2 Gronemann, Martin 2 Harary, Frank 2 Haythorpe, Michael 2 He, Meng 2 Heath, Lenwood S. 2 Hurtado, Ferran 2 Kauffman, Louis Hirsch 2 Korman, Matias 2 Kozyrev, V. P. 2 Kwiatkowska, Anna Beata 2 Meijer, Henk G. 2 Meng, Jixiang 2 Mohar, Bojan 2 Nakamoto, Atsuhiro 2 Nešetřil, Jaroslav 2 Newcombe, Alex 2 Nowakowski, Richard J. 2 Nozawa, Takayuki 2 Ota, Katsuhiro 2 Overbay, Shannon 2 Richter, Robert Bruce 2 Saleur, Hubert 2 Schäfer, Marcus 2 Shahrokhi, Farhad 2 Shibata, Yukio 2 Stadler, Peter F. 2 Stöhr, Elena A. 2 Syslo, Maciej M. 2 Thomassen, Carsten 2 Tyukin, Ivan Yu. 2 Vrbjarová, Michaela 2 Whitesides, Sue H. 2 Yannakakis, Mihalis 2 Zhao, Bin 1 Abbasi, Naser 1 Adamsson, Jay 1 Aichholzer, Oswin 1 Alam, Jawaherul Md. ...and 312 more Authors
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#### Cited in 96 Serials

 29 Discrete Mathematics 25 Discrete Applied Mathematics 16 Journal of Combinatorial Theory. Series B 13 Computational Geometry 10 Algorithmica 9 Neural Networks 7 Information Processing Letters 7 Journal of Optimization Theory and Applications 7 Theoretical Computer Science 6 Journal of Approximation Theory 5 Journal of Graph Algorithms and Applications 4 Journal of Combinatorial Theory. Series A 4 Graphs and Combinatorics 4 Applied Mathematics Letters 4 SIAM Journal on Discrete Mathematics 3 Bulletin of Mathematical Biology 3 Information Sciences 3 Journal of Soviet Mathematics 3 Mathematische Nachrichten 3 European Journal of Combinatorics 3 Journal of Mathematical Chemistry 3 Journal of Discrete Algorithms 3 The Art of Discrete and Applied Mathematics 2 Communications in Mathematical Physics 2 Journal of Mathematical Analysis and Applications 2 Periodica Mathematica Hungarica 2 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2 Applied Mathematics and Computation 2 Journal of Graph Theory 2 Journal of Pure and Applied Algebra 2 Proceedings of the American Mathematical Society 2 Quaestiones Mathematicae 2 Rendiconti del Seminario Matemàtico e Fisico di Milano 2 Results in Mathematics 2 Topology and its Applications 2 SIAM Journal on Algebraic and Discrete Methods 2 Discrete & Computational Geometry 2 Neural Computation 2 Journal of Knot Theory and its Ramifications 2 The Electronic Journal of Combinatorics 1 American Mathematical Monthly 1 Bulletin of the Australian Mathematical Society 1 Computers & Mathematics with Applications 1 Journal of the Franklin Institute 1 Chaos, Solitons and Fractals 1 The Mathematical Intelligencer 1 BIT 1 Fuzzy Sets and Systems 1 Glasgow Mathematical Journal 1 Integral Equations and Operator Theory 1 Journal of Computational and Applied Mathematics 1 Journal of Computer and System Sciences 1 Kybernetika 1 Mathematica Slovaca 1 Mathematical Systems Theory 1 Mathematische Zeitschrift 1 Osaka Journal of Mathematics 1 Ricerche di Matematica 1 Combinatorica 1 Chinese Annals of Mathematics. Series B 1 Order 1 Information and Computation 1 Journal of Parallel and Distributed Computing 1 Machine Learning 1 International Journal of Computational Geometry & Applications 1 International Journal of Algebra and Computation 1 MSCS. Mathematical Structures in Computer Science 1 European Journal of Operational Research 1 Historia Mathematica 1 Expositiones Mathematicae 1 The Australasian Journal of Combinatorics 1 Computational Optimization and Applications 1 Applied Categorical Structures 1 Applied and Computational Harmonic Analysis 1 Opuscula Mathematica 1 Theory and Applications of Categories 1 Discussiones Mathematicae. Graph Theory 1 Journal of Heuristics 1 INFORMS Journal on Computing 1 Optimization Methods & Software 1 Chicago Journal of Theoretical Computer Science 1 Journal of Discrete Mathematical Sciences & Cryptography 1 Acta Mathematica Sinica. English Series 1 Far East Journal of Mathematical Sciences 1 Algebraic & Geometric Topology 1 Miscelánea Matemática 1 Journal of Applied Mathematics 1 Journal of Machine Learning Research (JMLR) 1 JMMA. Journal of Mathematical Modelling and Algorithms 1 Cahiers de Topologie et Géométrie Différentielle Catégoriques 1 Journal of Statistical Mechanics: Theory and Experiment 1 Optimization Letters 1 Journal of Homotopy and Related Structures 1 Afrika Matematika 1 Computer Science Review 1 Bulletin of the Hellenic Mathematical Society
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#### Cited in 40 Fields

 191 Combinatorics (05-XX) 71 Computer science (68-XX) 26 Approximations and expansions (41-XX) 21 Biology and other natural sciences (92-XX) 18 Operations research, mathematical programming (90-XX) 13 Category theory; homological algebra (18-XX) 12 Manifolds and cell complexes (57-XX) 9 Information and communication theory, circuits (94-XX) 7 Operator theory (47-XX) 7 Statistics (62-XX) 4 Order, lattices, ordered algebraic structures (06-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 4 General topology (54-XX) 4 Algebraic topology (55-XX) 4 Numerical analysis (65-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Mathematical logic and foundations (03-XX) 3 Group theory and generalizations (20-XX) 3 Integral transforms, operational calculus (44-XX) 3 Convex and discrete geometry (52-XX) 3 Probability theory and stochastic processes (60-XX) 2 History and biography (01-XX) 2 Number theory (11-XX) 2 Nonassociative rings and algebras (17-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 1 General and overarching topics; collections (00-XX) 1 Commutative algebra (13-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Associative rings and algebras (16-XX) 1 $$K$$-theory (19-XX) 1 Real functions (26-XX) 1 Measure and integration (28-XX) 1 Partial differential equations (35-XX) 1 Integral equations (45-XX) 1 Functional analysis (46-XX) 1 Geometry (51-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Quantum theory (81-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Systems theory; control (93-XX)