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Du Bois-Reymond, Paul

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Author ID: du-bois-reymond.paul Recent zbMATH articles by "Du Bois-Reymond, Paul"
Published as: du Bois-Reymond, P.; Du Bois-Reymond, P.; Bois-Reymond, P. du; du Bois-Reymond, Paul; Du Bois-Reymond, Paul; Du Bois Reymond, P.
Further Spellings: Du Bois-Reymond, David Paul Gustave
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Documents Indexed: 69 Publications since 1865, including 7 Books
Biographic References: 4 Publications

Co-Authors

65 single-authored
4 Jourdain, Philip Edward Bertrand

Publications by Year

Citations contained in zbMATH Open

33 Publications have been cited 112 times in 93 Documents Cited by Year
An attempt at a classification of arbitrary functions of a real argument according to their variation on the smallest intervals. (Versuch einer Classification der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen.) JFM 06.0241.01
Bois-Reymond, P. du
20
1874
On asymptotic values, infinitary approximations and infinitary solutions of equations. Addendum. (Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichungen. Nachträge.) JFM 07.0249.01
du Bois-Reymond, P.
11
1875
A new theory of convergence and divergence of series with positive terms. (Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern.) JFM 05.0128.01
du Bois-Reymond, P.
10
1873
On the relative size of the infinities of functions. (Sur la grandeur relative des infinis des fonctions.) JFM 03.0197.01
du Bois-Reymond, P.
9
1870
The general theory of functions. Part I. Metaphysics and the theory of fundamental concepts of mathematics: quantity, limit, argument and function. (Die allgemeine Functionentheorie. I. Teil. Metaphysik und Theorie der mathematischen Grundbegriffe: Grösse, Grenze, Argument und Function.) JFM 14.0309.01
du Bois-Reymond, P.
8
1882
On Fourier series. (Ueber die Fourier’schen Reihen.) JFM 05.0145.01
du Bois-Reymond, P.
7
1873
Convergenz und Divergenz der Fourier’schen Darstellungsformeln. JFM 08.0128.02
du Bois-Reymond, P.
5
1876
Proof of the fundamental theorem of integral calculus: \(\int_a^b F'(x) dx = F(b) F(a).\). (Der Beweis des Fundamentalsatzes der Integralrechnung: \(\int_a^b F'(x) dx = F(b) - F(a).\).) JFM 12.0217.02
Du Bois-Reymond, P.
4
1880
On the paradoxes of the infinitary calculus. (Ueber die Paradoxen des Infinitär-Calcüls.) JFM 09.0297.01
du Bois-Reymond, P.
4
1877
On the domain of validity of Taylor series expansions. (Ueber den Gültigkeitsbereich der Taylor’schen Reihenentwickelung.) JFM 15.0183.01
Du Bois-Reymond, P.
3
1882
Investigations on the convergence and divergence of Fourier expansions. (Untersuchungen über die Convergenz und Divergenz der Fourier’schen Darstellungsformeln.) JFM 09.0175.02
du Bois-Reymond, Paul
3
1876
Investigations of the convergence and divergence of the Fourier representation formula. (Untersuchungen über die Convergenz und Divergenz der Fourier’schen Darstellungsformeln.) JFM 09.0298.01
du Bois-Reymond, Paul
3
1876
The general theory of functions, translated by G. Milhaud and A. Girot. Part I. Metaphysics and the theory of the fundamental concepts of mathematics: size, limit, argument, and function. (Théorie générale des fonctions, traduite par G. Milhaud et A. Girot. Partie I: Métaphysique et théorie des concepts mathématiques fondamentaux, grandeur, limite, argument et fonction.) JFM 20.0410.01
du Bois-Reymond, P.
2
1888
Proof that the coefficients of the trigonometric series \[ \quad f(x) = \varSigma (a_{p} \cos px + b_{p} \sin . px) \] have the values \[ a_{0} = \frac{1}{2\pi} \int_{ \pi}^{+\pi} d\alpha f(\alpha), \quad a_{p} = \frac{1}{\pi} \int_{ \pi}^{+\pi} d\alpha f(\alpha) \cos . p\alpha, \]
\[ b_{p} = \frac{1}{\pi} \int_{ \pi}^{+\pi} d\alpha f(\alpha) \sin . p \alpha, \] whenever these integrals are finite and definite. (Beweis, dass die Coefficienten der trigonometrischen Reihe \[ (1)\quad f(x) = \varSigma (a_{p} \cos px + b_{p} \sin . px) \] die Werthe \[ a_{0} = \frac{1}{2\pi} \int_{-\pi}^{+\pi} d\alpha f(\alpha), \quad a_{p} = \frac{1}{\pi} \int_{-\pi}^{+\pi} d\alpha f(\alpha) \cos . p\alpha, \]
\[ b_{p} = \frac{1}{\pi} \int_{-\pi}^{+\pi} d\alpha f(\alpha) \sin . p \alpha \] haben, jedesmal, wenn diese Integrale endlich und bestimmt sind.)
 JFM 08.0128.04
du Bois-Reymond, Paul
2
1876
A general theorem concerning the relative growth of the infinities of functions and of their derivatives. (Théorème général concernant la grandeur relative des infinis des fonctions et de leurs dérivées.) JFM 04.0196.01
du Bois-Reymond, P.
2
1872
A note on a theorem by Cauchy concerning the continuity of infinite series. (Notiz über einen Cauchy’schen Satz, die Stetigkeit von Summen unendlicher Reihen betreffend.) JFM 03.0099.01
du Bois-Reymond, P.
2
1871
Ueber die Grundlagen der Erkenntnis in den exacten Wissenschaften. JFM 22.0062.01
Du Bois-Reymond, Paul
1
1890
Ueber lineare partielle Differentialgleichungen zweiter Ordnung. JFM 21.0343.01
du Bois-Reymond, P.
1
1889
Remarks on \[ \varDelta z=\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}=0. \]. (Bemerkungen über \[ \varDelta z=\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}=0. \].) JFM 20.0384.01
du Bois-Reymond, P.
1
1888
Ueber den Convergenzgrad der variabeln Reihen und den Stetigkeitsgrad der Functionen zweier Argumente. JFM 18.0201.01
du Bois-Reymond, P.
1
1886
On representation formulas for functions. (Sur les formules de représentation des fonctions.) JFM 13.0327.01
du Bois-Reymond, P.
1
1881
On the integration and differentiation of infinitesimal relations. (Ueber Integration und Differentiation infinitärer Relationen.) JFM 11.0199.01
Du Bois-Reymond, P.
1
1879
Comments on the rudiments of the calculus of variations. (Erläuterungen zu den Anfangsgründen der Variationsrechnung.) JFM 11.0259.01
Du Bois-Reymond, P.
1
1879
On the theorem \[ \lim f'(x) = \lim \frac{f(x)}{x}. \] . (Ueber den Satz \[ \lim f'(x) = \lim \frac{f(x)}{x}. \].) JFM 12.0208.01
Du Bois-Reymond, P.
1
1880
Ein allgemeiner Satz der Integrirbarkeitslehre. JFM 12.0211.02
Du Bois-Reymond, P.
1
1880
Comments on the rudiments of the calculus of variations. (Erläuterungen zu den Anfangsgründen der Variationsrechnung.) JFM 12.0299.01
Du Bois-Reymond, P.
1
1879
Note on the convergence of integrals with non vanishing argument. (Notiz über Convergenz von Integralen mit nicht verschwindendem Argument.) JFM 10.0288.01
du Bois-Reymond, P.
1
1878
Two theorems on the limit values of functions of two variables. (Zwei Sätze über Grenzwerthe von Functionen zweier Veränderlichen.) JFM 09.0302.01
du Bois-Reymond, P.
1
1877
Notiz über infinitäre Gleichheiten. JFM 08.0128.03
du Bois-Reymond, P.
1
1876
On the jumps in the values of analytic functions. (Ueber die sprungweisen Werthänderungen analytischer Functionen.) JFM 06.0238.01
Bois-Reymond, P. du
1
1874
General theorems on the domain of validity of integral formulae which serve to define arbitrary functions. (Allgemeine Lehrsätze über den Gültigkeitsbereich der Integralformeln, die zur Darstellung willkürlicher Functionen dienen.) JFM 06.0244.01
Bois-Reymond, P. du
1
1874
Theory of Fourier integrals and formulas. (Die Theorie der Fourier’schen Integrale und Formeln.) JFM 03.0138.01
du Bois-Reymond, P.
1
1871
General theory of functions. Part One. Metaphysics and the theory of fundamental mathematical concepts: magnitude, limit, argument and functions. Transl. from the 1882 orig. by G. Milhaud and A. Girot. Reprint of the 1887 translation. (Théorie générale des fonctions. Première Partie. Métaphysique et théorie des concepts mathématiques fondamentaux: grandeur, limite, argument et fonction.) Zbl 0974.01504
Du Bois-Reymond, Paul
1
1995
General theory of functions. Part One. Metaphysics and the theory of fundamental mathematical concepts: magnitude, limit, argument and functions. Transl. from the 1882 orig. by G. Milhaud and A. Girot. Reprint of the 1887 translation. (Théorie générale des fonctions. Première Partie. Métaphysique et théorie des concepts mathématiques fondamentaux: grandeur, limite, argument et fonction.) Zbl 0974.01504
Du Bois-Reymond, Paul
1
1995
Ueber die Grundlagen der Erkenntnis in den exacten Wissenschaften. JFM 22.0062.01
Du Bois-Reymond, Paul
1
1890
Ueber lineare partielle Differentialgleichungen zweiter Ordnung. JFM 21.0343.01
du Bois-Reymond, P.
1
1889
The general theory of functions, translated by G. Milhaud and A. Girot. Part I. Metaphysics and the theory of the fundamental concepts of mathematics: size, limit, argument, and function. (Théorie générale des fonctions, traduite par G. Milhaud et A. Girot. Partie I: Métaphysique et théorie des concepts mathématiques fondamentaux, grandeur, limite, argument et fonction.) JFM 20.0410.01
du Bois-Reymond, P.
2
1888
Remarks on \[ \varDelta z=\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}=0. \]. (Bemerkungen über \[ \varDelta z=\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}=0. \].) JFM 20.0384.01
du Bois-Reymond, P.
1
1888
Ueber den Convergenzgrad der variabeln Reihen und den Stetigkeitsgrad der Functionen zweier Argumente. JFM 18.0201.01
du Bois-Reymond, P.
1
1886
The general theory of functions. Part I. Metaphysics and the theory of fundamental concepts of mathematics: quantity, limit, argument and function. (Die allgemeine Functionentheorie. I. Teil. Metaphysik und Theorie der mathematischen Grundbegriffe: Grösse, Grenze, Argument und Function.) JFM 14.0309.01
du Bois-Reymond, P.
8
1882
On the domain of validity of Taylor series expansions. (Ueber den Gültigkeitsbereich der Taylor’schen Reihenentwickelung.) JFM 15.0183.01
Du Bois-Reymond, P.
3
1882
On representation formulas for functions. (Sur les formules de représentation des fonctions.) JFM 13.0327.01
du Bois-Reymond, P.
1
1881
Proof of the fundamental theorem of integral calculus: \(\int_a^b F'(x) dx = F(b) F(a).\). (Der Beweis des Fundamentalsatzes der Integralrechnung: \(\int_a^b F'(x) dx = F(b) - F(a).\).) JFM 12.0217.02
Du Bois-Reymond, P.
4
1880
On the theorem \[ \lim f'(x) = \lim \frac{f(x)}{x}. \] . (Ueber den Satz \[ \lim f'(x) = \lim \frac{f(x)}{x}. \].) JFM 12.0208.01
Du Bois-Reymond, P.
1
1880
Ein allgemeiner Satz der Integrirbarkeitslehre. JFM 12.0211.02
Du Bois-Reymond, P.
1
1880
On the integration and differentiation of infinitesimal relations. (Ueber Integration und Differentiation infinitärer Relationen.) JFM 11.0199.01
Du Bois-Reymond, P.
1
1879
Comments on the rudiments of the calculus of variations. (Erläuterungen zu den Anfangsgründen der Variationsrechnung.) JFM 11.0259.01
Du Bois-Reymond, P.
1
1879
Comments on the rudiments of the calculus of variations. (Erläuterungen zu den Anfangsgründen der Variationsrechnung.) JFM 12.0299.01
Du Bois-Reymond, P.
1
1879
Note on the convergence of integrals with non vanishing argument. (Notiz über Convergenz von Integralen mit nicht verschwindendem Argument.) JFM 10.0288.01
du Bois-Reymond, P.
1
1878
On the paradoxes of the infinitary calculus. (Ueber die Paradoxen des Infinitär-Calcüls.) JFM 09.0297.01
du Bois-Reymond, P.
4
1877
Two theorems on the limit values of functions of two variables. (Zwei Sätze über Grenzwerthe von Functionen zweier Veränderlichen.) JFM 09.0302.01
du Bois-Reymond, P.
1
1877
Convergenz und Divergenz der Fourier’schen Darstellungsformeln. JFM 08.0128.02
du Bois-Reymond, P.
5
1876
Investigations on the convergence and divergence of Fourier expansions. (Untersuchungen über die Convergenz und Divergenz der Fourier’schen Darstellungsformeln.) JFM 09.0175.02
du Bois-Reymond, Paul
3
1876
Investigations of the convergence and divergence of the Fourier representation formula. (Untersuchungen über die Convergenz und Divergenz der Fourier’schen Darstellungsformeln.) JFM 09.0298.01
du Bois-Reymond, Paul
3
1876
Proof that the coefficients of the trigonometric series \[ \quad f(x) = \varSigma (a_{p} \cos px + b_{p} \sin . px) \] have the values \[ a_{0} = \frac{1}{2\pi} \int_{ \pi}^{+\pi} d\alpha f(\alpha), \quad a_{p} = \frac{1}{\pi} \int_{ \pi}^{+\pi} d\alpha f(\alpha) \cos . p\alpha, \]
\[ b_{p} = \frac{1}{\pi} \int_{ \pi}^{+\pi} d\alpha f(\alpha) \sin . p \alpha, \] whenever these integrals are finite and definite. (Beweis, dass die Coefficienten der trigonometrischen Reihe \[ (1)\quad f(x) = \varSigma (a_{p} \cos px + b_{p} \sin . px) \] die Werthe \[ a_{0} = \frac{1}{2\pi} \int_{-\pi}^{+\pi} d\alpha f(\alpha), \quad a_{p} = \frac{1}{\pi} \int_{-\pi}^{+\pi} d\alpha f(\alpha) \cos . p\alpha, \]
\[ b_{p} = \frac{1}{\pi} \int_{-\pi}^{+\pi} d\alpha f(\alpha) \sin . p \alpha \] haben, jedesmal, wenn diese Integrale endlich und bestimmt sind.)
 JFM 08.0128.04
du Bois-Reymond, Paul
2
1876
Notiz über infinitäre Gleichheiten. JFM 08.0128.03
du Bois-Reymond, P.
1
1876
On asymptotic values, infinitary approximations and infinitary solutions of equations. Addendum. (Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichungen. Nachträge.) JFM 07.0249.01
du Bois-Reymond, P.
11
1875
An attempt at a classification of arbitrary functions of a real argument according to their variation on the smallest intervals. (Versuch einer Classification der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen.) JFM 06.0241.01
Bois-Reymond, P. du
20
1874
On the jumps in the values of analytic functions. (Ueber die sprungweisen Werthänderungen analytischer Functionen.) JFM 06.0238.01
Bois-Reymond, P. du
1
1874
General theorems on the domain of validity of integral formulae which serve to define arbitrary functions. (Allgemeine Lehrsätze über den Gültigkeitsbereich der Integralformeln, die zur Darstellung willkürlicher Functionen dienen.) JFM 06.0244.01
Bois-Reymond, P. du
1
1874
A new theory of convergence and divergence of series with positive terms. (Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern.) JFM 05.0128.01
du Bois-Reymond, P.
10
1873
On Fourier series. (Ueber die Fourier’schen Reihen.) JFM 05.0145.01
du Bois-Reymond, P.
7
1873
A general theorem concerning the relative growth of the infinities of functions and of their derivatives. (Théorème général concernant la grandeur relative des infinis des fonctions et de leurs dérivées.) JFM 04.0196.01
du Bois-Reymond, P.
2
1872
A note on a theorem by Cauchy concerning the continuity of infinite series. (Notiz über einen Cauchy’schen Satz, die Stetigkeit von Summen unendlicher Reihen betreffend.) JFM 03.0099.01
du Bois-Reymond, P.
2
1871
Theory of Fourier integrals and formulas. (Die Theorie der Fourier’schen Integrale und Formeln.) JFM 03.0138.01
du Bois-Reymond, P.
1
1871
On the relative size of the infinities of functions. (Sur la grandeur relative des infinis des fonctions.) JFM 03.0197.01
du Bois-Reymond, P.
9
1870
all top 5

Cited by 110 Authors

3 Bernal-Gonzàlez, Luis
3 Lie, Victor Daniel
3 Neuenschwander, Erwin
3 Prodanov, Dimiter
2 Aviles Lopez, Antonio
2 Baglini, Lorenzo Luperi
2 Benci, Vieri
2 Dahmen, Wolfgang A.
2 Ehrlich, Philip
2 Hirschorn, James
2 Kanamori, Akihiro
2 Kragh Sørensen, Henrik
2 McCarty, David Charles
2 Singh, Avadhesh Narayan
2 Todorcevic, Stevo B.
2 Volkert, Klaus Thomas
1 Aschenbrenner, Matthias
1 Aubry, Jean-Marie
1 Avdispahić, Muharem
1 Bastin, Françoise
1 Bedürftig, Thomas
1 Berkovitz, Joseph
1 Berwald, Ludwig
1 Bhouri, Imen
1 Bini, Dario Andrea
1 Błaszczyk, Piotr
1 Boche, Holger
1 Borovik, Alexandre V.
1 Bråting, Kajsa
1 Brioschi, Alessio
1 Brive, Bruno
1 Bruneau, Michel
1 Buescu, Jorge
1 Burn, Robert Pemberton
1 Carty, D. C. Mc.
1 Chabert, Jean-Luc
1 De Amo, Enrique
1 Diaz Carrillo, Manuel
1 Duchamp, Gerard Henry Edmond
1 Dugac, Pierre
1 Eceizabarrena, Daniel
1 Enjalbert, Jean-Yves
1 Erdős, Pál
1 Fernández-Sánchez, Juan
1 Fila, Marlena
1 Finet, Catherine
1 Folland, Gerald B.
1 Fraser, Craig G.
1 Gát, György
1 Gmeiner, Joseph Anton
1 Görlich, Ernst
1 Gronwall, Thomas Hakon
1 Haidar, Nassar H. S.
1 Hannabuss, Keith C.
1 Hewitt, Edwin
1 Hewitt, Robert E.
1 Hoang Ngoc Minh, Vincel
1 Itatsu, Seiichi
1 Kahane, Jean-Pierre
1 Kairies, Hans-Heinrich
1 Kanovei, Vladimir G.
1 Katz, Mikhail G.
1 Khanmohamadi, Omid
1 Khanmohammadi, Ehssan
1 Lacan, Francis
1 Laugwitz, Detlef
1 Lukashenko, S. Yu.
1 Lyubetsky, Vassily Aleksandrovich
1 Ma, Shiguang
1 Massei, Stefano
1 McNeal, Jeffery D.
1 Meini, Beatrice
1 Metzler, Wolfgang
1 Mouze, Augustin
1 Müller, Max
1 Nestoridis, Vassili
1 Nicolay, Samuel
1 Niculescu, Constantin P.
1 Obermaier, Josef
1 Pejlare, Johanna
1 Pellegrino, Daniel Marinho
1 Pinkus, Allan M.
1 Ploščica, Miroslav
1 Pohl, Volker
1 Popovici, Florin
1 Proietti, Carlo
1 Qing, Jie
1 Radu, Mircea
1 Robol, Leonardo
1 Rogers, Claude Ambrose
1 Rosenlicht, Maxwell
1 Rosso, Riccardo
1 Seoane-Sepúlveda, Juan Benigno
1 Serpa, Cristina
1 Shackell, John R.
1 Simons, Laurent
1 Smith, Donald R.
1 Steuding, Jörn
1 Taylor, Samuel James
1 Tazzioli, Rossana
...and 10 more Authors
all top 5

Cited in 55 Serials

8 Archive for History of Exact Sciences
6 Historia Mathematica
5 Journal of Mathematical Analysis and Applications
5 Journal of Approximation Theory
3 Mathematische Annalen
3 The Bulletin of Symbolic Logic
2 Mathematische Semesterberichte
2 Chaos, Solitons and Fractals
2 The Mathematical Intelligencer
2 Advances in Mathematics
2 Mathematische Zeitschrift
2 Proceedings of the American Mathematical Society
2 Transactions of the American Mathematical Society
2 History and Philosophy of Logic
2 Constructive Approximation
2 Bulletin of the American Mathematical Society. New Series
2 Rendiconti del Circolo Matematico di Palermo
1 Analysis Mathematica
1 Centaurus
1 Discrete Mathematics
1 Indian Journal of Pure & Applied Mathematics
1 Journal d’Analyse Mathématique
1 Mathematics Magazine
1 Acta Mathematica
1 Applied Mathematics and Computation
1 Bulletin de la Société Mathématique de France
1 Publications Mathématiques
1 Journal of Algebra
1 The Journal of Symbolic Logic
1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods
1 Notre Dame Journal of Formal Logic
1 Proceedings of the Japan Academy. Series A
1 Rendiconti del Circolo Matemàtico di Palermo. Serie II
1 Results in Mathematics
1 Synthese
1 Bulletin of the Section of Logic
1 Annals of Pure and Applied Logic
1 Forum Mathematicum
1 Aequationes Mathematicae
1 SIAM Journal on Scientific Computing
1 Calculus of Variations and Partial Differential Equations
1 Fractional Calculus & Applied Analysis
1 NTM. Neue Serie
1 Annals of Mathematics. Second Series
1 Erkenntnis
1 Journal of the European Mathematical Society (JEMS)
1 Communications in Nonlinear Science and Numerical Simulation
1 Foundations of Science
1 La Gaceta de la Real Sociedad Matemática Española
1 Journal of the Australian Mathematical Society
1 Mediterranean Journal of Mathematics
1 Monatshefte für Mathematik und Physik
1 Discrete and Continuous Dynamical Systems. Series S
1 Bulletin of Mathematical Analysis and Applications
1 Arabian Journal of Mathematics
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Cited in 38 Fields

30 Real functions (26-XX)
29 History and biography (01-XX)
18 Harmonic analysis on Euclidean spaces (42-XX)
17 Mathematical logic and foundations (03-XX)
8 Functional analysis (46-XX)
7 Sequences, series, summability (40-XX)
7 Approximations and expansions (41-XX)
6 General and overarching topics; collections (00-XX)
6 Measure and integration (28-XX)
5 Field theory and polynomials (12-XX)
5 Functions of a complex variable (30-XX)
4 Combinatorics (05-XX)
3 Ordinary differential equations (34-XX)
3 Partial differential equations (35-XX)
3 Probability theory and stochastic processes (60-XX)
3 Computer science (68-XX)
2 Number theory (11-XX)
2 Linear and multilinear algebra; matrix theory (15-XX)
2 Special functions (33-XX)
2 Difference and functional equations (39-XX)
2 Abstract harmonic analysis (43-XX)
2 Operator theory (47-XX)
2 Calculus of variations and optimal control; optimization (49-XX)
2 General topology (54-XX)
2 Mathematics education (97-XX)
1 Order, lattices, ordered algebraic structures (06-XX)
1 General algebraic systems (08-XX)
1 Algebraic geometry (14-XX)
1 Associative rings and algebras (16-XX)
1 Potential theory (31-XX)
1 Several complex variables and analytic spaces (32-XX)
1 Dynamical systems and ergodic theory (37-XX)
1 Integral equations (45-XX)
1 Numerical analysis (65-XX)
1 Classical thermodynamics, heat transfer (80-XX)
1 Quantum theory (81-XX)
1 Operations research, mathematical programming (90-XX)
1 Information and communication theory, circuits (94-XX)

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