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# zbMATH — the first resource for mathematics

## Kochubeĭ, Anatoliĭ Naumovych

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 Author ID: kochubei.anatoly-n Published as: Kochubei, A. N.; Kochubei, Anatolii N.; Kochubei, Anatoly; Kochubei, Anatoly N.; Kochubej, A. N.; Kochubej, Anatolij N.; Kochubeĭ, A. N. Homepage: https://www.imath.kiev.ua/people/profile.php?pid=32 External Links: MGP · Math-Net.Ru · Wikidata · dblp · GND
 Documents Indexed: 147 Publications since 1971, including 7 Books Reviewing Activity: 2,025 Reviews Biographic References: 1 Publication
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#### Co-Authors

 116 single-authored 7 Gorbachuk, Myroslav L’vovych 6 Kondrat’yev, Yuriĭ Grygorovych 5 Berezans’kyĭ, Yuriĭ Makarovych 4 Eĭdel’man, Samuïl Davydovych 4 Gorbachuk, Valentyna Ivanivna 3 Adamyan, Vadym Movsesovych 3 Antonyuk, Oleksandra Viktorivna 3 Khrennikov, Andreĭ Yur’evich 3 Sait-Ametov, Mustafa R. 2 da Silva, José Luís 2 Gohberg, Israel 2 Khruslov, Eugene Yakovlevich 2 Langer, Heinz 2 Luchko, Yuri 2 Popov, Gennadiĭ Yakovlevich 1 Anashin, Vladimir Sergeevich 1 Boĭchuk, Oleksandr Andriĭovych 1 Dragovich, Branko 1 Gerasimenko, Viktor Ivanovich 1 Gubreev, Gennadiĭ Mykhaĭlovych 1 Hutlyans’kyi, V. Ya. 1 Ivasyshen, Stepan Dmytrovych 1 Kaneko, Hiroshi 1 Korolyuk, Volodymyr Semenovych 1 Kozyrev, Sergeĭ Vladimirovich 1 Kushnir, Roman M. 1 Lukovsky, Ivan Oleksandrovych 1 Makarov, Volodymyr Leonidovich 1 Malamud, Mark M. 1 Marchenko, V. O. 1 Mykhajlets’, V. A. 1 Nikitin, A. H. 1 Nizhnik, Leonid Pavlovich 1 Parasyuk, Igor O. 1 Pastur, Leonid Andreevich 1 Perestyuk, Mykola Oleksiĭovych 1 Piskarëv, S. I. 1 Portenko, Mykola Ivanovych 1 Rofe-Beketov, Fedor Semenovich 1 Ronto, M. I. 1 Rybak, M. A. 1 Samoĭlenko, Anatoliĭ Mykhaĭlovych 1 Samoĭlenko, Yuriĭ Stefanovych 1 Sharkovskiĭ, Oleksandr Mykolaiovych 1 Shklyar, Alexander Ya. 1 Soskin, Daniel S. 1 Tkachenko, Victor Ivanovich 1 Trofimchuk, Sergei I. 1 Volovich, Igor’ Vasil’evich
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#### Serials

 12 Ukrainian Mathematical Journal 9 Ukraïns’kyĭ Matematychnyĭ Zhurnal 8 Differential Equations 7 Methods of Functional Analysis and Topology 5 Differentsial’nye Uravneniya 4 Siberian Mathematical Journal 4 Soviet Mathematics. Doklady 3 Mathematical Notes 3 Mathematics of the USSR. Izvestiya 3 Dopovidi Akademiï Nauk Ukraïns’koï RSR. Seriya A 3 Potential Analysis 3 Doklady Mathematics 3 Fractional Calculus & Applied Analysis 3 $$p$$-Adic Numbers, Ultrametric Analysis, and Applications 3 Operator Theory: Advances and Applications 2 Applicable Analysis 2 Journal of Mathematical Analysis and Applications 2 Journal of Mathematical Physics 2 Theoretical and Mathematical Physics 2 Functional Analysis and its Applications 2 Journal of Number Theory 2 Pacific Journal of Mathematics 2 Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 2 Russian Academy of Sciences. Izvestiya. Mathematics 2 Algebras and Representation Theory 2 Infinite Dimensional Analysis, Quantum Probability and Related Topics 2 Dopovidi Natsional’noï Akademiï Nauk Ukraïny. Matematyka, Pryrodoznavstvo, Tekhnichni Nauky 2 Handbook of Fractional Calculus with Applications 1 Letters in Mathematical Physics 1 Teoriya Veroyatnosteĭ i eë Primeneniya 1 Theory of Probability and its Applications 1 Funktsional’nyĭ Analiz i ego Prilozheniya 1 Integral Equations and Operator Theory 1 Journal of Algebra 1 Journal of Differential Equations 1 Sibirskiĭ Matematicheskiĭ Zhurnal 1 Tohoku Mathematical Journal. Second Series 1 Advances in Applied Mathematics 1 Soviet Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences 1 Journal of Theoretical Probability 1 Izvestiya Akademii Nauk Armyanskoi SSR. Matematika 1 Journal of Physics A: Mathematical and General 1 Expositiones Mathematicae 1 Teoriya Funktsiĭ, Funktsional’nyĭ Analiz i ikh Prilozheniya 1 Funktsional’nyĭ Analiz 1 Bulletin of the Belgian Mathematical Society - Simon Stevin 1 Finite Fields and their Applications 1 Izvestiya: Mathematics 1 The Journal of Fourier Analysis and Applications 1 European Mathematical Society Newsletter 1 Analysis (München) 1 Stochastics and Dynamics 1 International Journal of Number Theory 1 Cambridge Tracts in Mathematics 1 Proceedings of the Steklov Institute of Mathematics 1 Pure and Applied Mathematics, Marcel Dekker 1 Translations. Series 2. American Mathematical Society 1 Journal of Physics A: Mathematical and Theoretical 1 Zbirnyk Prats’ Instytutu Matematyky NAN Ukraïny 1 Kinetic and Related Models 1 Journal of Pseudo-Differential Operators and Applications
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#### Fields

 59 Operator theory (47-XX) 51 Partial differential equations (35-XX) 43 Ordinary differential equations (34-XX) 33 Number theory (11-XX) 24 Functional analysis (46-XX) 21 Probability theory and stochastic processes (60-XX) 17 Real functions (26-XX) 10 Quantum theory (81-XX) 9 Field theory and polynomials (12-XX) 7 Functions of a complex variable (30-XX) 6 General and overarching topics; collections (00-XX) 6 Associative rings and algebras (16-XX) 6 Special functions (33-XX) 5 Abstract harmonic analysis (43-XX) 4 History and biography (01-XX) 3 Combinatorics (05-XX) 3 Measure and integration (28-XX) 3 Fluid mechanics (76-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Approximations and expansions (41-XX) 2 Differential geometry (53-XX) 2 Global analysis, analysis on manifolds (58-XX) 1 Group theory and generalizations (20-XX) 1 Topological groups, Lie groups (22-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Sequences, series, summability (40-XX) 1 Integral transforms, operational calculus (44-XX) 1 Mechanics of particles and systems (70-XX) 1 Mechanics of deformable solids (74-XX)

#### Citations contained in zbMATH

75 Publications have been cited 1,011 times in 707 Documents Cited by Year
Cauchy problem for fractional diffusion equations. Zbl 1068.35037
Eidelman, Samuil D.; Kochubei, Anatoly N.
2004
Distributed order calculus and equations of ultraslow diffusion. Zbl 1149.26014
Kochubei, Anatoly N.
2008
Pseudo-differential equations and stochastics over non-Archimedean fields. Zbl 0984.11063
Kochubei, Anatoly N.
2001
Fractional-order diffusion. Zbl 0729.35064
Kochubei, A. N.
1990
Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. Zbl 1062.35003
Eidelman, Samuil D.; Ivasyshen, Stepan D.; Kochubei, Anatoly N.
2004
General fractional calculus, evolution equations, and renewal processes. Zbl 1250.26006
Kochubei, Anatoly N.
2011
A Cauchy problem for evolution equations of fractional order. Zbl 0696.34047
Kochubej, A. N.
1989
Cauchy problem for evolution equations of a fractional order. Zbl 1129.35427
Kochubei, A. N.; Èidelman, S. D.
2004
Extensions of symmetric operators and symmetric binary relations. Zbl 0322.47006
Kochubej, A. N.
1975
Fractional-parabolic systems. Zbl 1259.35218
Kochubei, Anatoly N.
2012
A non-Archimedean wave equation. Zbl 1190.35235
Kochubei, Anatoly N.
2008
Parabolic pseudodifferential equations, hypersingular integrals, and Markov processes. Zbl 0682.35113
Kochubej, A. N.
1989
Cauchy problem for fractional diffusion-wave equations with variable coefficients. Zbl 1297.35274
Kochubei, Anatoly N.
2014
Extension theory for symmetric operators and boundary value problems for differential equations. Zbl 0706.47001
Gorbachuk, V. I.; Gorbachuk, M. L.; Kochubej, A. N.
1989
Asymptotic properties of solutions of the fractional diffusion-wave equation. Zbl 1323.35197
Kochubei, Anatoly N.
2014
Distributed order derivatives and relaxation patterns. Zbl 1191.34005
Kochubei, Anatoly N.
2009
Fundamental solutions of pseudodifferential equations connected with $$p$$-adic quadratic forms. Zbl 0934.35218
Kochubei, A. N.
1998
Stochastic integrals and stochastic differential equations over the field of $$p$$-adic numbers. Zbl 0874.60047
Kochubei, Anatoly N.
1997
$$p$$-adic analogue of the porous medium equation. Zbl 1401.35165
Khrennikov, Andrei Yu.; Kochubei, Anatoly N.
2018
Interaction measures on the space of distributions over the field of $$p$$-adic numbers. Zbl 1053.81067
Kochubei, Anatoly N.; Sait-Ametov, Mustafa R.
2003
One-dimensional point interactions. Zbl 0703.34086
Kochubej, A. N.
1989
Elliptic operators with boundary conditions on a subset of measure zero. Zbl 0506.35029
Kochubej, A. N.
1982
Weak solutions of stochastic differential equations over the field of $$p$$-adic numbers. Zbl 1136.60039
Kaneko, Hiroshi; Kochubei, Anatoly N.
2007
Limit theorems for sums of $$p$$-adic random variables. Zbl 0954.60001
Kochubei, Anatoly N.
1998
A Schrödinger-type equation over the field of $$p$$-adic numbers. Zbl 0780.35088
Kochubei, Anatolii N.
1993
Hausdorff measure for a stable-like process over an infinite extension of a local field. Zbl 1018.60038
Kochubei, Anatoly N.
2002
Harmonic oscillator in characteristic $$p$$. Zbl 1036.11520
Kochubei, Anatoly N.
1998
$$p$$-adic commutation relations. Zbl 0905.46051
Kochubei, Anatoly N.
1996
Distributed-order calculus: An operator-theoretic interpretation. Zbl 1164.26009
Kochubei, A. N.
2008
Differential equations for $${\mathbb F}_q$$-linear functions. Zbl 1010.11066
Kochubei, Anatoly N.
2000
Diffusion of fractional order. Zbl 0712.35049
Kochubej, A. N.
1990
Fractional-hyperbolic systems. Zbl 1312.35183
Kochubei, Anatoly
2013
Analysis in positive characteristic. Zbl 1171.12005
Kochubei, Anatoly N.
2009
On $$p$$-adic Green’s functions. Zbl 0843.35147
Kochubei, A. N.
1993
Parabolic equations on a p-adic number field. Zbl 0765.35002
Kochubej, A. N.
1991
Schrödinger-type operator over $$p$$-adic number field. Zbl 0809.47060
Kochubej, A. N.
1991
On characteristic functions of symmetric operators and their extensions. Zbl 0444.47023
Kochubej, A. N.
1980
Analysis and probability over infinite extensions of a local field. Zbl 1023.60501
Kochubei, Anatoly N.
1999
$$\mathbb{F}_q$$-linear calculus over function fields. Zbl 1029.11019
Kochubei, Anatoly N.
1999
Additive and multiplicative fractional differentiations over the field of $$p$$-adic numbers. Zbl 0882.26015
Kochubei, Anatoly N.
1997
One-dimensional point interactions. Zbl 0693.34018
Kochubej, A. N.
1989
Fractional kinetic hierarchies and intermittency. Zbl 1359.82014
Kochubei, Anatoly N.; Kondratiev, Yuri
2017
Radial solutions of non-Archimedean pseudodifferential equations. Zbl 1396.35070
Kochubei, Anatoly N.
2014
On some classes of non-Archimedean operator algebras. Zbl 1335.47047
Kochubei, Anatoly N.
2013
Differential equations for $$\mathbb F_q$$-linear functions. II: Regular singularity. Zbl 1081.11075
Kochubei, Anatoly N.
2003
Symmetric operators and nonclassical spectral problems. Zbl 0454.47023
Kochubej, A. N.
1979
Multidimensional nonlinear pseudo-differential evolution equation with $$p$$-adic spatial variables. Zbl 07193644
Antoniouk, Alexandra V.; Khrennikov, Andrei Yu.; Kochubei, Anatoly N.
2020
Linear and nonlinear heat equations on a $$p$$-adic ball. Zbl 1439.35537
Kochubei, A. N.
2018
On the compactness and the uniform continuity of a resolvent family for a fractional differential equation. Zbl 1313.34001
Antonyuk, O. V.; Kochubei, A. N.; Piskarev, S. I.
2014
Fractional differential equations: $$\alpha$$-entire solutions, regular and irregular singularities. Zbl 1178.26007
Kochubei, Anatoly N.
2009
Umbral calculus in positive characteristic. Zbl 1062.05020
Kochubei, Anatoly N.
2005
Equations of one-dimensional fractal diffusion. Zbl 1037.35117
Kochubeĭ, A. N.; Èĭdel’man, S. D.
2003
Heat equation in a $$p$$-adic ball. Zbl 0926.60091
Kochubej, Anatolij N.
1996
Parabolic equations over the field of $$p$$-adic numbers. Zbl 0778.35001
Kochubej, A. N.
1991
On extensions of a positive-definite symmetric operator. Zbl 0555.47029
Kochubej, A. N.
1984
On best approximation in normed modules. Zbl 0262.41035
Kochubej, A. N.
1973
Dissipative extensions of differential operators in a space of vector- functions. Zbl 0261.47026
Gorbachuk, M. L.; Kochubej, A. N.; Rybak, M. A.
1972
Basic theory. Zbl 1410.26003
Kochubei, Anatoly (ed.); Luchko, Yuri (ed.)
2019
Fractional approximation of solutions of evolution equations. Zbl 1338.34109
Kochubei, Anatoly N.; Kondratiev, Yuri G.
2016
Non-Archimedean group algebras with Baer reductions. Zbl 1320.47074
Kochubei, Anatoly N.
2014
Non-Archimedean normal operators. Zbl 1309.47089
Kochubei, Anatoly N.
2010
$$p$$-adic spherical coordinates and their applications. Zbl 1187.60039
Kochubei, A. N.
2009
On a $$p$$-adic wave equation. Zbl 1191.47090
Kochubei, Anatoly N.
2009
Dwork-Carlitz exponential and overconvergence for additive functions in positive characteristic. Zbl 1160.11033
Kochubei, Anatoly N.
2008
Evolution equations and functions of hypergeometric type over fields of positive characteristics. Zbl 1197.12004
Kochubei, Anatoly N.
2007
Hypergeometric functions and Carlitz differential equations over function fields. Zbl 1211.12008
Kochubei, Anatoly N.
2007
Construction of interaction measures on the space of distributions over the field of $$p$$-adic numbers. Zbl 1098.81060
Kochubei, A. N.; Sait-Ametov, M. R.
2004
The differentiation operator on subsets of the field of $$p$$-adic numbers. Zbl 0784.47057
Kochubej, A. N.
1992
Self-adjoint extensions of a Schrödinger operator with singular potential. Zbl 0737.34061
Kochubej, A. N.
1991
Self-adjoint extensions of Schrödinger operators with singular potentials. Zbl 0725.47006
Kochubei, A. N.
1990
Stabilization of solutions of dissipative hyperbolic equations with almost-periodic coefficients. Zbl 0697.35082
Kochubej, A. N.
1987
Stabilization of solutions of dissipative hyperbolic equations. Zbl 0625.35051
Kochubej, A. N.
1986
On extensions of a positive definite symmetric operator. Zbl 0454.47022
Kochubej, A. N.
1979
The extensions of a nondensely defined symmetric operator. Zbl 0409.47014
Kochubej, A. N.
1977
On extensions of a nondensely defined symmetric operator. Zbl 0362.47008
Kochubej, A. N.
1977
Multidimensional nonlinear pseudo-differential evolution equation with $$p$$-adic spatial variables. Zbl 07193644
Antoniouk, Alexandra V.; Khrennikov, Andrei Yu.; Kochubei, Anatoly N.
2020
Basic theory. Zbl 1410.26003
Kochubei, Anatoly (ed.); Luchko, Yuri (ed.)
2019
$$p$$-adic analogue of the porous medium equation. Zbl 1401.35165
Khrennikov, Andrei Yu.; Kochubei, Anatoly N.
2018
Linear and nonlinear heat equations on a $$p$$-adic ball. Zbl 1439.35537
Kochubei, A. N.
2018
Fractional kinetic hierarchies and intermittency. Zbl 1359.82014
Kochubei, Anatoly N.; Kondratiev, Yuri
2017
Fractional approximation of solutions of evolution equations. Zbl 1338.34109
Kochubei, Anatoly N.; Kondratiev, Yuri G.
2016
Cauchy problem for fractional diffusion-wave equations with variable coefficients. Zbl 1297.35274
Kochubei, Anatoly N.
2014
Asymptotic properties of solutions of the fractional diffusion-wave equation. Zbl 1323.35197
Kochubei, Anatoly N.
2014
Radial solutions of non-Archimedean pseudodifferential equations. Zbl 1396.35070
Kochubei, Anatoly N.
2014
On the compactness and the uniform continuity of a resolvent family for a fractional differential equation. Zbl 1313.34001
Antonyuk, O. V.; Kochubei, A. N.; Piskarev, S. I.
2014
Non-Archimedean group algebras with Baer reductions. Zbl 1320.47074
Kochubei, Anatoly N.
2014
Fractional-hyperbolic systems. Zbl 1312.35183
Kochubei, Anatoly
2013
On some classes of non-Archimedean operator algebras. Zbl 1335.47047
Kochubei, Anatoly N.
2013
Fractional-parabolic systems. Zbl 1259.35218
Kochubei, Anatoly N.
2012
General fractional calculus, evolution equations, and renewal processes. Zbl 1250.26006
Kochubei, Anatoly N.
2011
Non-Archimedean normal operators. Zbl 1309.47089
Kochubei, Anatoly N.
2010
Distributed order derivatives and relaxation patterns. Zbl 1191.34005
Kochubei, Anatoly N.
2009
Analysis in positive characteristic. Zbl 1171.12005
Kochubei, Anatoly N.
2009
Fractional differential equations: $$\alpha$$-entire solutions, regular and irregular singularities. Zbl 1178.26007
Kochubei, Anatoly N.
2009
$$p$$-adic spherical coordinates and their applications. Zbl 1187.60039
Kochubei, A. N.
2009
On a $$p$$-adic wave equation. Zbl 1191.47090
Kochubei, Anatoly N.
2009
Distributed order calculus and equations of ultraslow diffusion. Zbl 1149.26014
Kochubei, Anatoly N.
2008
A non-Archimedean wave equation. Zbl 1190.35235
Kochubei, Anatoly N.
2008
Distributed-order calculus: An operator-theoretic interpretation. Zbl 1164.26009
Kochubei, A. N.
2008
Dwork-Carlitz exponential and overconvergence for additive functions in positive characteristic. Zbl 1160.11033
Kochubei, Anatoly N.
2008
Weak solutions of stochastic differential equations over the field of $$p$$-adic numbers. Zbl 1136.60039
Kaneko, Hiroshi; Kochubei, Anatoly N.
2007
Evolution equations and functions of hypergeometric type over fields of positive characteristics. Zbl 1197.12004
Kochubei, Anatoly N.
2007
Hypergeometric functions and Carlitz differential equations over function fields. Zbl 1211.12008
Kochubei, Anatoly N.
2007
Umbral calculus in positive characteristic. Zbl 1062.05020
Kochubei, Anatoly N.
2005
Cauchy problem for fractional diffusion equations. Zbl 1068.35037
Eidelman, Samuil D.; Kochubei, Anatoly N.
2004
Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. Zbl 1062.35003
Eidelman, Samuil D.; Ivasyshen, Stepan D.; Kochubei, Anatoly N.
2004
Cauchy problem for evolution equations of a fractional order. Zbl 1129.35427
Kochubei, A. N.; Èidelman, S. D.
2004
Construction of interaction measures on the space of distributions over the field of $$p$$-adic numbers. Zbl 1098.81060
Kochubei, A. N.; Sait-Ametov, M. R.
2004
Interaction measures on the space of distributions over the field of $$p$$-adic numbers. Zbl 1053.81067
Kochubei, Anatoly N.; Sait-Ametov, Mustafa R.
2003
Differential equations for $$\mathbb F_q$$-linear functions. II: Regular singularity. Zbl 1081.11075
Kochubei, Anatoly N.
2003
Equations of one-dimensional fractal diffusion. Zbl 1037.35117
Kochubeĭ, A. N.; Èĭdel’man, S. D.
2003
Hausdorff measure for a stable-like process over an infinite extension of a local field. Zbl 1018.60038
Kochubei, Anatoly N.
2002
Pseudo-differential equations and stochastics over non-Archimedean fields. Zbl 0984.11063
Kochubei, Anatoly N.
2001
Differential equations for $${\mathbb F}_q$$-linear functions. Zbl 1010.11066
Kochubei, Anatoly N.
2000
Analysis and probability over infinite extensions of a local field. Zbl 1023.60501
Kochubei, Anatoly N.
1999
$$\mathbb{F}_q$$-linear calculus over function fields. Zbl 1029.11019
Kochubei, Anatoly N.
1999
Fundamental solutions of pseudodifferential equations connected with $$p$$-adic quadratic forms. Zbl 0934.35218
Kochubei, A. N.
1998
Limit theorems for sums of $$p$$-adic random variables. Zbl 0954.60001
Kochubei, Anatoly N.
1998
Harmonic oscillator in characteristic $$p$$. Zbl 1036.11520
Kochubei, Anatoly N.
1998
Stochastic integrals and stochastic differential equations over the field of $$p$$-adic numbers. Zbl 0874.60047
Kochubei, Anatoly N.
1997
Additive and multiplicative fractional differentiations over the field of $$p$$-adic numbers. Zbl 0882.26015
Kochubei, Anatoly N.
1997
$$p$$-adic commutation relations. Zbl 0905.46051
Kochubei, Anatoly N.
1996
Heat equation in a $$p$$-adic ball. Zbl 0926.60091
Kochubej, Anatolij N.
1996
A Schrödinger-type equation over the field of $$p$$-adic numbers. Zbl 0780.35088
Kochubei, Anatolii N.
1993
On $$p$$-adic Green’s functions. Zbl 0843.35147
Kochubei, A. N.
1993
The differentiation operator on subsets of the field of $$p$$-adic numbers. Zbl 0784.47057
Kochubej, A. N.
1992
Parabolic equations on a p-adic number field. Zbl 0765.35002
Kochubej, A. N.
1991
Schrödinger-type operator over $$p$$-adic number field. Zbl 0809.47060
Kochubej, A. N.
1991
Parabolic equations over the field of $$p$$-adic numbers. Zbl 0778.35001
Kochubej, A. N.
1991
Self-adjoint extensions of a Schrödinger operator with singular potential. Zbl 0737.34061
Kochubej, A. N.
1991
Fractional-order diffusion. Zbl 0729.35064
Kochubei, A. N.
1990
Diffusion of fractional order. Zbl 0712.35049
Kochubej, A. N.
1990
Self-adjoint extensions of Schrödinger operators with singular potentials. Zbl 0725.47006
Kochubei, A. N.
1990
A Cauchy problem for evolution equations of fractional order. Zbl 0696.34047
Kochubej, A. N.
1989
Parabolic pseudodifferential equations, hypersingular integrals, and Markov processes. Zbl 0682.35113
Kochubej, A. N.
1989
Extension theory for symmetric operators and boundary value problems for differential equations. Zbl 0706.47001
Gorbachuk, V. I.; Gorbachuk, M. L.; Kochubej, A. N.
1989
One-dimensional point interactions. Zbl 0703.34086
Kochubej, A. N.
1989
One-dimensional point interactions. Zbl 0693.34018
Kochubej, A. N.
1989
Stabilization of solutions of dissipative hyperbolic equations with almost-periodic coefficients. Zbl 0697.35082
Kochubej, A. N.
1987
Stabilization of solutions of dissipative hyperbolic equations. Zbl 0625.35051
Kochubej, A. N.
1986
On extensions of a positive-definite symmetric operator. Zbl 0555.47029
Kochubej, A. N.
1984
Elliptic operators with boundary conditions on a subset of measure zero. Zbl 0506.35029
Kochubej, A. N.
1982
On characteristic functions of symmetric operators and their extensions. Zbl 0444.47023
Kochubej, A. N.
1980
Symmetric operators and nonclassical spectral problems. Zbl 0454.47023
Kochubej, A. N.
1979
On extensions of a positive definite symmetric operator. Zbl 0454.47022
Kochubej, A. N.
1979
The extensions of a nondensely defined symmetric operator. Zbl 0409.47014
Kochubej, A. N.
1977
On extensions of a nondensely defined symmetric operator. Zbl 0362.47008
Kochubej, A. N.
1977
Extensions of symmetric operators and symmetric binary relations. Zbl 0322.47006
Kochubej, A. N.
1975
On best approximation in normed modules. Zbl 0262.41035
Kochubej, A. N.
1973
Dissipative extensions of differential operators in a space of vector- functions. Zbl 0261.47026
Gorbachuk, M. L.; Kochubej, A. N.; Rybak, M. A.
1972
all top 5

#### Cited by 765 Authors

 30 Kochubeĭ, Anatoliĭ Naumovych 25 Khrennikov, Andreĭ Yur’evich 24 Zúñiga-Galindo, Wilson A. 14 Cuevas, Claudio 12 Albeverio, Sergio A. 12 Lopushans’ka, Galyna Petrovna 12 Luchko, Yuri 12 Shelkovich, Vladimir M. 11 Kozyrev, Sergeĭ Vladimirovich 11 Mainardi, Francesco 11 Nane, Erkan 10 Yamamoto, Masahiro 9 Lopushans’kyi, Andriy Olegovich 9 Malamud, Mark M. 9 Peng, Jigen 9 Zacher, Rico 8 Kaneko, Hiroshi 8 Meerschaert, Mark M. 7 Allahverdiev, Bilender Paşaoğlu 7 Bazhlekova, Emilia G. 7 Chen, Pengyu 7 D’Ovidio, Mirko 7 Li, Yongxiang 7 Lizama, Carlos 7 Protsakh, Nataliya P. 6 Chen, Zhen-Qing 6 Kemppainen, Jukka T. 6 Kolokoltsov, Vassili N. 6 Krasnoshchok, Mykola Valeriĭovych 6 Leonenko, Nikolai N. 6 Sandev, Trifce 6 Vasylyeva, Nataliya V. 6 Wang, Rongnian 5 Agarwal, Ravi P. 5 Ahmad, Bashir 5 Behrndt, Jussi 5 Beshtokov, Murat Khamidbievich 5 Bikulov, Al’bert Khakimovich 5 Fu, Zunwei 5 Gao, Jinghuai 5 Gorenflo, Rudolf 5 Kirane, Mokhtar 5 Li, Kexue 5 Li, Zhiyuan 5 Mamchuev, Murat Osmanovich 5 Mukhamedov, Farruh Maksutovich 5 Osypchuk, Mykhailo M. 5 Povstenko, Yuriy Z. 5 Toaldo, Bruno 5 Vergara, Vicente 5 Wu, Qingyan 5 Zhou, Zhi 4 Anh, Vo V. 4 Atanackovic, Teodor M. 4 Bendikov, Alexander D. 4 Casas-Sánchez, Oscar Francisco 4 Chacón-Cortes, Leonardo Fabio 4 Chechkin, Aleksei V. 4 de Andrade, Bruno 4 Estala-Arias, Samuel 4 Feng, Binhua 4 Galeano-Peñaloza, Jeanneth 4 Ivasyshen, Stepan Dmytrovych 4 Jin, Bangti 4 Kim, Panki 4 Li, Yaning 4 Liu, Fawang 4 Luchko, Yurii F. 4 Mei, Zhandong 4 Mijena, Jebessa B. 4 Murugesu, R. 4 Oleschko, Klaudia 4 Piskarëv, S. I. 4 Torresblanca-Badillo, Anselmo 4 Umarov, Sabir R. 4 Wu, Bo 4 Zhang, Quanguo 4 Zhang, Shuqin 4 Zhang, Zufeng 4 Zhou, Yong 3 Abbaszadeh, Mostafa 3 Al-Refai, Mohammed 3 Al-saedi, Ahmed Eid Salem 3 Ansari, Alireza 3 Băleanu, Dumitru I. 3 Bazzaev, Aleksandr Kazbekovich 3 Cao, Junfei 3 Cygan, Wojciech 3 Derkach, Vladimir Alexandrovich 3 Digernes, Trond 3 Dos Santos, José Paulo Carvalho 3 Fëdorov, Vladimir Evgen’evich 3 Gao, Guanghua 3 Garrappa, Roberto 3 Gesztesy, Fritz 3 Geyler, Vladimir A. 3 Giusti, Andrea 3 Grubb, Gerd 3 Gu, Haibo 3 Jia, Junxiong ...and 665 more Authors
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#### Cited in 195 Serials

 44 Fractional Calculus & Applied Analysis 33 $$p$$-Adic Numbers, Ultrametric Analysis, and Applications 30 Journal of Mathematical Analysis and Applications 21 Ukrainian Mathematical Journal 20 Computers & Mathematics with Applications 18 Journal of Mathematical Sciences (New York) 16 Journal of Differential Equations 14 Applied Mathematics and Computation 14 Stochastic Processes and their Applications 13 Journal of Functional Analysis 12 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 12 Abstract and Applied Analysis 11 Advances in Difference Equations 11 Carpathian Mathematical Publications 10 Journal of Computational and Applied Mathematics 10 The Journal of Fourier Analysis and Applications 9 Journal of Mathematical Physics 9 Differential Equations 9 Proceedings of the Steklov Institute of Mathematics 8 Mathematical Methods in the Applied Sciences 8 Doklady Mathematics 7 Chaos, Solitons and Fractals 7 Applied Mathematics Letters 7 Potential Analysis 7 Integral Transforms and Special Functions 6 Nonlinear Dynamics 6 Communications in Nonlinear Science and Numerical Simulation 6 Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki 6 Journal of Pseudo-Differential Operators and Applications 5 Applicable Analysis 5 Theoretical and Mathematical Physics 5 Integral Equations and Operator Theory 5 Proceedings of the American Mathematical Society 5 Siberian Mathematical Journal 5 Journal of Evolution Equations 5 Fractional Differential Calculus 4 Mathematical Notes 4 Functional Analysis and its Applications 4 Journal of Number Theory 4 Mathematische Nachrichten 4 Publications of the Research Institute for Mathematical Sciences, Kyoto University 4 Tohoku Mathematical Journal. Second Series 4 Statistics & Probability Letters 4 Numerical Algorithms 4 Computational and Applied Mathematics 4 Stochastics and Dynamics 4 Ufimskiĭ Matematicheskiĭ Zhurnal 4 Evolution Equations and Control Theory 4 Mathematics 4 Vestnik KRAUNTS. Fiziko-Matematicheskie Nauki 3 Reports on Mathematical Physics 3 ZAMP. Zeitschrift für angewandte Mathematik und Physik 3 Reviews in Mathematical Physics 3 Advances in Mathematics 3 The Annals of Probability 3 Mathematische Annalen 3 Applied Numerical Mathematics 3 Journal of Theoretical Probability 3 Computational Mathematics and Mathematical Physics 3 Applied Mathematical Modelling 3 Applied and Computational Harmonic Analysis 3 Electronic Journal of Differential Equations (EJDE) 3 Discrete and Continuous Dynamical Systems 3 Matematychni Metody ta Fizyko-Mekhanichni Polya 3 Matematychni Studiï 3 Infinite Dimensional Analysis, Quantum Probability and Related Topics 3 Nonlinear Analysis. Real World Applications 3 Discrete and Continuous Dynamical Systems. Series B 3 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 3 Communications on Pure and Applied Analysis 3 Journal of Applied Mathematics and Computing 3 Mediterranean Journal of Mathematics 2 Communications in Mathematical Physics 2 International Journal of Theoretical Physics 2 Inverse Problems 2 Nonlinearity 2 Nuclear Physics. B 2 Mathematics of Computation 2 Journal of Pure and Applied Algebra 2 Mathematische Zeitschrift 2 Meccanica 2 Numerical Functional Analysis and Optimization 2 Rendiconti del Seminario Matematico della Università di Padova 2 Results in Mathematics 2 Transactions of the Moscow Mathematical Society 2 Acta Applicandae Mathematicae 2 Probability Theory and Related Fields 2 Journal of Scientific Computing 2 Forum Mathematicum 2 Journal of Integral Equations and Applications 2 Journal de Mathématiques Pures et Appliquées. Neuvième Série 2 Expositiones Mathematicae 2 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 2 Indagationes Mathematicae. New Series 2 Theory of Probability and Mathematical Statistics 2 Journal of Inverse and Ill-Posed Problems 2 Mathematical Problems in Engineering 2 Algebras and Representation Theory 2 Matematicheskie Trudy 2 Journal of Applied Mathematics ...and 95 more Serials
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#### Cited in 52 Fields

 381 Partial differential equations (35-XX) 181 Operator theory (47-XX) 147 Probability theory and stochastic processes (60-XX) 131 Ordinary differential equations (34-XX) 117 Real functions (26-XX) 64 Number theory (11-XX) 62 Numerical analysis (65-XX) 55 Integral equations (45-XX) 45 Quantum theory (81-XX) 44 Functional analysis (46-XX) 36 Harmonic analysis on Euclidean spaces (42-XX) 35 Special functions (33-XX) 23 Statistical mechanics, structure of matter (82-XX) 18 Systems theory; control (93-XX) 16 Integral transforms, operational calculus (44-XX) 14 Dynamical systems and ergodic theory (37-XX) 13 Abstract harmonic analysis (43-XX) 13 Fluid mechanics (76-XX) 9 Field theory and polynomials (12-XX) 9 Mechanics of deformable solids (74-XX) 9 Biology and other natural sciences (92-XX) 8 Potential theory (31-XX) 8 Global analysis, analysis on manifolds (58-XX) 7 Associative rings and algebras (16-XX) 7 Measure and integration (28-XX) 7 Classical thermodynamics, heat transfer (80-XX) 6 Functions of a complex variable (30-XX) 6 Difference and functional equations (39-XX) 5 Combinatorics (05-XX) 5 Calculus of variations and optimal control; optimization (49-XX) 5 Relativity and gravitational theory (83-XX) 5 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 4 Commutative algebra (13-XX) 4 Group theory and generalizations (20-XX) 4 Topological groups, Lie groups (22-XX) 3 Optics, electromagnetic theory (78-XX) 3 Geophysics (86-XX) 3 Information and communication theory, circuits (94-XX) 2 General and overarching topics; collections (00-XX) 2 Order, lattices, ordered algebraic structures (06-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Approximations and expansions (41-XX) 2 Statistics (62-XX) 1 History and biography (01-XX) 1 Algebraic geometry (14-XX) 1 Nonassociative rings and algebras (17-XX) 1 Sequences, series, summability (40-XX) 1 General topology (54-XX) 1 Computer science (68-XX) 1 Mechanics of particles and systems (70-XX) 1 Astronomy and astrophysics (85-XX)

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text:
# Homework Help: Problem with basic doppler effect question

1. Aug 13, 2013

### PsychonautQQ

1. The problem statement, all variables and given/known data
The source of a 1 kilohertz sound is getting closer to the listener at a speed of .9 times the speed of sound. What frequency does the listener hear?

2. Relevant equations
fl = ((v-vs) / (v + vl))fs

3. The attempt at a solution
((343+343(.9)) / 343)*1000=1900 Hz
this is the wrong answer, what did I do wrong? it's supposed to be 10,000 Hz

2. Aug 13, 2013

### besulzbach

Correct formula:
$f = \frac{c\ +\ v_{r}}{c\ +\ v_{s}} \cdot f_{0}$

In your problem the source is getting closer, so $v_{s}$ is negative.

As you didn't miss that, your error is in the formula. Use the one above.

You also SUMMED values that you should have SUBTRACTED.

Remember:

$v_{r}$ is the velocity of the receiver relative to the medium; positive if the receiver is moving towards the source;

$v_{s}$ is the velocity of the source relative to the medium; positive if the source is moving away from the receiver;

3. Aug 13, 2013

### Zondrina

Use this equation : $f = f_0( \frac{v}{v ± v_s} )$

Where $f$ is the stationary frequency. $f_2$ is the apparent frequency detected. $v$ is the speed of sound in air and $v_s$ is the speed of the source.

Since the source is moving closer to your listener, you want to use $v - v_s$ ( Doppler effect ). Don't forget to convert kHz to Hz otherwise this wont work!

I got $f = 10000 Hz$.

format: plain


text:
# Assigning per-sample genotypes (HaplotypeCaller)

edited January 8

This document describes the procedure used by HaplotypeCaller to assign genotypes to individual samples based on the allele likelihoods calculated in the previous step. For more context information on how this fits into the overall HaplotypeCaller method, please see the more general HaplotypeCaller documentation. See also the documentation on the QUAL score as well as the one on PL and GQ.

This procedure is NOT applied by Mutect2 for somatic short variant discovery. See this article for a direct comparison between HaplotypeCaller and Mutect2.

#### Contents

1. Overview
2. Preliminary assumptions / limitations
3. Calculating genotype likelihoods using Bayes' Theorem
4. Selecting a genotype and emitting the call record

## 1. Overview

The previous step produced a table of per-read allele likelihoods for each candidate variant site under consideration. Now, all that remains to do is to evaluate those likelihoods in aggregate to determine what is the most likely genotype of the sample at each site. This is done by applying Bayes' theorem to calculate the likelihoods of each possible genotype, and selecting the most likely. This produces a genotype call as well as the calculation of various metrics that will be annotated in the output VCF if a variant call is emitted.

Note that this describes the regular mode of HaplotypeCaller, which does not emit an estimate of reference confidence. For details on how the reference confidence model works and is applied in GVCF modes (-ERC GVCF and -ERC BP_RESOLUTION) please see the reference confidence model documentation.

## 2. Preliminary assumptions / limitations

### Quality

Keep in mind that we are trying to infer the genotype of each sample given the observed sequence data, so the degree of confidence we can have in a genotype depends on both the quality and the quantity of the available data. By definition, low coverage and low quality will both lead to lower confidence calls. The GATK only uses reads that satisfy certain mapping quality thresholds, and only uses “good” bases that satisfy certain base quality thresholds (see documentation for default values).

### Ploidy

Both the HaplotypeCaller and GenotypeGVCFs assume that the organism of study is diploid by default, but the desired ploidy can be set using the -ploidy argument. The ploidy is taken into account in the mathematical development of the Bayesian calculation using a generalized form of the genotyping algorithm that can handle ploidies other than 2. Note that using ploidy for pooled experiments is subject to some practical limitations due to the number of possible combinations resulting from the interaction between ploidy and the number of alternate alleles that are considered. There are some arguments that aim to mitigate those limitations but they are not fully documented yet.

Reads that are mates in the same pair are not handled together in the reassembly, but if they overlap, there is some special handling to ensure they are not counted as independent observations.

### Single-sample vs multi-sample

We apply different genotyping models when genotyping a single sample as opposed to multiple samples together (as done by HaplotypeCaller on multiple inputs or GenotypeGVCFs on multiple GVCFs). The multi-sample case is not currently documented for the public but is an extension of previous work by Heng Li and others.

## 3. Calculating genotype likelihoods using Bayes' Theorem

We use the approach described in Li 2011 to calculate the posterior probabilities of non-reference alleles (Methods 2.3.5 and 2.3.6) extended to handle multi-allelic variation.

The basic formula we use for all types of variation under consideration (SNPs, insertions and deletions) is:

$$P(G|D) = \frac{ P(G) P(D|G) }{ \sum_{i} P(G_i) P(D|G_i) }$$

If that is meaningless to you, please don't freak out -- we're going to break it down and go through all the components one by one. First of all, the term on the left:

$$P(G|D)$$

is the quantity we are trying to calculate for each possible genotype: the conditional probability of the genotype G given the observed data D.

Now let's break down the term on the right:

$$\frac{ P(G) P(D|G) }{ \sum_{i} P(G_i) P(D|G_i) }$$

We can ignore the denominator (bottom of the fraction) because it ends up being the same for all the genotypes, and the point of calculating this likelihood is to determine the most likely genotype. The important part is the numerator (top of the fraction):

$$P(G) P(D|G)$$

which is composed of two things: the prior probability of the genotype and the conditional probability of the data given the genotype.

The first one is the easiest to understand. The prior probability of the genotype G:

$$P(G)$$

represents how probably we expect to see this genotype based on previous observations, studies of the population, and so on. By default, the GATK tools use a flat prior (always the same value) but you can input your own set of priors if you have information about the frequency of certain genotypes in the population you're studying.

The second one is a little trickier to understand if you're not familiar with Bayesian statistics. It is called the conditional probability of the data given the genotype, but what does that mean? Assuming that the genotype G is the true genotype,

$$P(D|G)$$

is the probability of observing the sequence data that we have in hand. That is, how likely would we be to pull out a read with a particular sequence from an individual that has this particular genotype? We don't have that number yet, so this requires a little more calculation, using the following formula:

$$P(D|G) = \prod{j} \left( \frac{P(D_j | H_1)}{2} + \frac{P(D_j | H_2)}{2} \right)$$

You'll notice that this is where the diploid assumption comes into play, since here we decomposed the genotype G into:

$$G = H_1H_2$$

which allows for exactly two possible haplotypes. In future versions we'll have a generalized form of this that will allow for any number of haplotypes.

Now, back to our calculation, what's left to figure out is this:

$$P(D_j|H_n)$$

which as it turns out is the conditional probability of the data given a particular haplotype (or specifically, a particular allele), aggregated over all supporting reads. Conveniently, that is exactly what we calculated in Step 3 of the HaplotypeCaller process, when we used the PairHMM to produce the likelihoods of each read against each haplotype, and then marginalized them to find the likelihoods of each read for each allele under consideration. So all we have to do at this point is plug the values from that table into the equation above, and we can work our way back up to obtain:

$$P(G|D)$$

for the genotype G.

## 4. Selecting a genotype and emitting the call record

We go through the process of calculating a likelihood for each possible genotype based on the alleles that were observed at the site, considering every possible combination of alleles. For example, if we see an A and a T at a site, the possible genotypes are AA, AT and TT, and we end up with 3 corresponding probabilities. We pick the largest one, which corresponds to the most likely genotype, and assign that to the sample.

Note that depending on the variant calling options specified in the command-line, we may only emit records for actual variant sites (where at least one sample has a genotype other than homozygous-reference) or we may also emit records for reference sites. The latter is discussed in the reference confidence model documentation.

Assuming that we have a non-ref genotype, all that remains is to calculate the various site-level and genotype-level metrics that will be emitted as annotations in the variant record, including QUAL as well as PL and GQ. For more information on how the other variant context metrics are calculated, please see the corresponding variant annotations documentation.

Post edited by Sheila on
Tagged:

• Member ✭✭

In P(G|D)=P(G)P(D|G)/P(D), if P(G) is a flat prior and P(D|G) is computed using each sample's own reads only; that is, no cohort info is used in computing P(G|D). Then how joint variant calling can have higher sensitivity than individual variant calling?

are the cohort reads used in the identification of potential variant sites?

Hey @blueskypy,
I'm just helping out our new forum support person today and I'm noticing that you've posted four similar questions across different threads for the last few days. Would it be possible for you to consolidate all of your questions into a single new post?

Going forward, it would help us tremendously if you could not ask duplicate questions and to group your related questions into the same thread. Thank you!

• Member

Dear GATK4 experts,

Using HaplotypeCaller (GATK4.0.4.0). I am contacting you about the following warning message that appear when I used HC in the -ERC mode:

13:53:50.764 WARN DepthPerSampleHC - Annotation will not be calculated, genotype is not called or alleleLikelihoodMap is null
13:53:50.764 WARN StrandBiasBySample - Annotation will not be calculated, genotype is not called or alleleLikelihoodMap is null

Using my regular pre-processed BAM files, I applied HC in two different modes:

1) without the -ERC GCVF mode (to bootstrap the BQSR)

and 2) with the -ERC GCVF mode (after recalibration of the base quality scores to produce the .g.vcf file)

The error files generated during both analysis are very different and I don't understand why.

In the first case (HC without the -ERC mode), reads are processed "normally" without any warnings.
In the second case (HC with the -ERC mode), many many warnings appear in the .error file generated during the analysis.
I have to add that despite these many warn messages, the ProgressMeter reach the end of BAM file.

Since both HC analysis are performed with the same original BAM file, could you explain me what these warnings mean and if they affect the quality of the .g.vcf file that is created and that I will use for downstream analysis (JoinGenotyping)?


text:
# Global solar radiation prediction using hybrid online sequential extreme learning machine model

### Abstract

Accurate global solar radiation prediction is highly essential for related research on renewable energy sources. The cost implication and measurement expertise of global solar radiation emphasize that intelligence prediction models need to be applied. On the basis of long-term measured daily solar radiation data, this study uses a novel regularized online sequential extreme learning machine, integrated with variable forgetting factor (FOS-ELM), to predict global solar radiation at Bur Dedougou, in the Burkina Faso region. Bayesian Information Criterion (BIC) is applied to build the seven input combinations based on speed (Wspeed), maximum and minimum temperature ($T_{max}$ and $T_{min}$), maximum and minimum humidity ($H_{max}$ and $H_{min}$), evaporation ($E_{o}$) and vapor pressure deficiency ($V_{PD}$). For the difference input parameters magnitudes, seven models were developed and evaluated for the optimal input combination. Various statistical indicators were computed for the prediction accuracy examination. The experimental results of the applied FOS-ELM model demonstrated a reliable prediction accuracy against the classical extreme learning machine (ELM) model for daily global solar radiation simulation. In fact, compared to classical ELM, the FOS-ELM model reported an enhancement in the root mean square error (RMSE) and mean absolute error (MAE) by (68.8–79.8%). In summary, the results clearly confirm the effectiveness of the FOS-ELM model, owing to the fixed internal tuning parameters.

Publication
Energies


text:
# How to scale imported .gltf model?

I imported a gltf model in Blender and it's way too big. If I convert the same model to .obj before, I can import it in Blender and set the Clamp-Size to 1.00 and the scaling is alright. Unfortunately, the gltf importer in Blender 2.8 doesn't have such an option.

I already tried selecting everything and scaling it down in the Transform tab to 0.01 (seems to be the lowest I can set it to) but it's still way too big (still so big that Blender is cutting the view in half because one end is "too far" away).

Is there any way to resize it? Im importing an ArchiCAD model which unfortunately doesn't give me any options on export.

Thanks

• Once you've loaded a model into Blender it doesn't matter what the original file format was, scaling works the same with any mesh. Are you sure you can't scale the mesh less than 0.01? Can you share a .blend file? – Don McCurdy Apr 22 '19 at 5:29

As Don mentioned in a comment, you should be able to scale the object smaller in a variety of ways, regardless of import format. You can use the S key and drag the mouse towards the center, which can be repeated multiple times, or you can directly enter numbers (smaller than 0.01) on the Transform panel.

But before you do that, here's something else to consider:

If this is an architectural or other genuinely large model, scaling it down is likely the wrong solution. You want to configure Blender to be able to preview a larger volume of space.

Open the Numeric panel (if it's not already open, press N in the 3D panel), and look for the clip settings:

The rules here are a little strange, but I'll try to explain. Roughly, you don't want the Clip End value to be more than about 10,000 to 100,000 times larger than the Clip Start value. The more digits you put between these two numbers, the more strain you're placing on something called the "depth buffer" which is how a 3D graphics card figures out which polygons are in front.

For example, if you set Start to 100m and End to 10000000m, you should be able to preview a very large model, cleanly. But if you only set the End value super large, and leave the Start value at its default, you may start to see flickering (Z-fighting) between some of the polygons, depending on how close they are to overlapping. The good news is, when you're viewing something that large, you don't typically need to get the camera to zoom in on the tiniest details, so the Clip Start can move further away from the camera.

So my advice is, start by adding zeros onto the back of the Clip End number, until the whole back of the model comes into range. Then, if you see problems, start adding zeros to the Clip Start value, one digit at a time, until the problems subside.

• thanks, I didn't know that changing the clip values was the proper way to that! – tobias Apr 22 '19 at 14:41


text:
# Category of Ordered Sets has Enough Constants

## Theorem

Let $\mathbf{OrdSet}$ be the category of ordered sets.

Then $\mathbf{OrdSet}$ has enough constants.

## Proof

By Singleton Ordered Set is Terminal Object, we have that any ordered set with a singleton as underlying set is terminal in $\mathbf{OrdSet}$.

Let $1$ be such a singleton ordered set.

To show that $\mathbf{OrdSet}$ has enough constants, it is to be shown that if:

$f: P \to Q \ne g: P \to Q$

then there exists an $x: 1 \to P$ such that $f \circ x \ne g \circ x$.

Now by definition of $\mathbf{OrdSet}$, $f$ and $g$ are increasing mappings.

In particular, they are mappings, and by Equality of Mappings, there must be a $p \in P$ such that:

$\map f p \ne \map g p$

Define $\bar p: 1 \to P$ by $\map {\bar p} * = p$ (where $*$ is the unique element of $1$).

If this is a morphism in $\mathbf{OrdSet}$, we are evidently done.

This follows from Mapping is Constant iff Increasing and Decreasing.

Hence $\mathbf{OrdSet}$ has enough constants.

$\blacksquare$


text:
# 2.2.4.16.6 GetGraphPoints::GetPoints

## Description

This method initiates the point picking process. A maximum number of point sto pick can be specified along with the specific GraphLayer to pick from.

When called, further OriginC code will pause while the point picking process is taking place. Once either the maximum number of points has been picked or the processed cancelled via the ESC key or deactivation of the target graph, this method will return.

Hint: To pick a seemingly unlimited number of points nPts parameter could be set to something very high such as 1E6.

## Syntax

BOOL GetPoints(int nPts, GraphLayer &gl = NULL, LPCSTR lpcszLTvarName = NULL)

## Parameters

nPts
[input] Maximum number of points to get.
gl
[input] GraphLayer object, Points are to be selected from a plot or plots this Layer.
lpcszLTvarName
[input] An optional LabTalk variable that can be used for interaction with a dialog. Not used except in certain cases.

## Return

Returns non-zero if successful.

## Examples

For an example of using this method, please see the Example Section in GetGraphPoints.


text:
# Vertex-Facet Assignments For Polytopes

For polytopes in R^d with at least as many facets as vertices, we prove that vertices can be mapped injectively to non-incident facets when 1≤ d≤ 5 and give counterexamples for d≥ 7.

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## Authors

• 2 publications
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## 1. Background and notation

Polytopes are present in many fields of mathematics such as poset theory, discrete mathematics, linear optimization, and convex geometry. Unsurprisingly, problems involving polytopes can be addressed by a variety of methods, and different aspects of such problems can be emphasized. A motivation for the contents of the present paper is given by a metrical notion called reducedness introduced for general convex sets in [5]. While the existence of reduced polytopes is clear in the Euclidean plane, the only known examples of reduced polytopes in higher-dimensional Euclidean spaces are those given in [3]. This existence problem for a polytope with a certain metrical property has a combinatorial feature at its core: The authors of [1, Theorem 4] show that, in particular, for any reduced polytope, each vertex of the polytope is (in some sense) opposite to a facet to which it is not incident, and this assignment is injective. Thus, are there combinatorial types of polytopes in for which their vertices cannot be mapped injectively to non-incident facets and therefore cannot have any reduced realizations? In the following, we proof that the answer is negative for and affirmative for .

Let us recall some standard notation. For this, we refer to the books [4, 7]. A polytope in is the convex hull of finitely many points , i.e.

 P=conv{v1,...,vn}:={n∑i=1λivi\medspace∣∣ ∣∣\medspaceλ1,...,λn≥0,n∑i=1λi=1}.

By its dimension, we mean the dimension of its affine hull, or if is empty. A face of is a subset with the property, that for all and , the condition already implies . The faces of include the empty set and itself. Each face is also a polytope and therefore has a dimension. The family of -dimensional faces of is finite and shall be denoted by . Any two faces of which are not disjoint are said to be incident to each other. Throughout this paper, we use the standing assumption that the polytope is of dimension , and that this dimension is (we exclude the empty and singleton polytope). In this case, its -dimensional faces are called vertices, edges and facets when , and , respectively. The set consists of singletons and we will not distinguish between the point and the singleton . In particular, we will write . A polytope is spanned by its vertices: . A face of a face of and the intersection of faces of is again a face of . In particular, any face of is spanned by some vertices in . These will be called the vertices of the face. Our central problem now reads as follows:

###### Problem 1.1.

Given a polytope with , is there an injective mapping for which and are non-incident for all ?

Obviously, the assumption cannot be dropped as it is necessary for the existence of injective mappings without any extra condition. Polytope duality, which interchanges the roles of -dimensional faces and -dimensional faces, carries over any proof or counterexample to or from the following equivalent problem.

###### Problem 1.2.

Given a polytope with , is there an injective mapping for which and are non-incident for all ?

The injective maps mentioned in Problems 1.1 and 1.2 shall be called vertex-facet assignments and facet-vertex assignments, respectively.

## 2. Graph-theoretic formulation

Problems involving injective maps between finite sets can be stated conveniently in the language of graph theory, using matchings of bipartite graphs. We refer the reader to [2, Chapter 2]. In order to avoid confusion with the language of polytopes, we refer to the nodes of a graph, instead of its vertices. A bipartite graph is then a graph for which there is a decomposition of the set of nodes into non-empty disjoint sets called partition classes, such that edges (the elements of ) do not join nodes of the same partition class. For each node , denote by the neighborhood of , i.e. the set of to adjacent nodes in . Further, set . A matching in a bipartite graph is a -regular subgraph of . The matching of is said to be -covering (resp. -covering) if every node (resp. ) is a node in . In this sense, -covering matchings of correspond to injective single-valued selections of the set-valued mapping given by , that is, -covering matchings of are injective mappings which map nodes to adjacent ones.

For our purposes, a bipartite graph can be assigned to any polytope as follows.

###### Definition 2.1.

The vertex-facet graph of the polytope is the bipartite graph whose partition classes are (disjoint copies of) the vertices and the facets of the polytope. A vertex and a facet are adjacent in if and only of they are non-incident in .

With this terminology in place, the vertex-facet assignments (resp. facet-vertex assignments) are just -covering (resp. -covering) matchings in the vertex-facet graph.

Hall’s marriage theorem [2, Theorem 2.1.2] gives a necessary and sufficient condition for the existence of -covering matchings of bipartite graphs .

###### Theorem 2.2.

The bipartite graph possesses a -covering matching if and only if for all .

The condition for all appearing in Theorem 2.2 is called Hall condition. While one direction of the theorem is almost trivial, the other one gives the power to construct the desired matchings (provided they exist).

## 3. Existence in low dimensions

The answers to Problems 1.1 and 1.2 are trivially affirmative for . Trial and error for polytopes in with low facet or vertex count does not yield any counterexamples. Using Hall’s marriage theorem (Theorem 2.2) for the vertex-facet graph (see Definition 2.1), we are able to show that there are no counterexamples up to dimension . For the proof, we will take the dual point of view, i.e. we address Problem 1.2 which requires finding facet-vertex assignments for polytopes . This is motivated by the following geometric interpretation of the non-neighborhood of a set of nodes in the vertex-facet graph .

###### Proposition 3.1.

Let be the vertex-facet graph of the polytope . For , the set is the set of vertices of .

###### Proof.

Note that a vertex is in if and only if there is at least one facet to which it is non-incident (since then they are adjacent in ). Hence, a vertex lies in the complement if and only if it incident to all the facets , i.e. it lies in the intersection . ∎

In the sequel, we denote by and the vertex count and the facet count of the polytope under consideration. Also, we denote by the vertex-facet graph of . As a corollary of Theorem 2.2 and Proposition 3.1, we obtain the following statement.

###### Corollary 3.2.

A polytope possesses a facet-vertex assignment if and only if for all and facets , we have

 (1) |F0(f1∩⋯∩fk)|≤n−k.

In order to construct a counterexample to Problem 1.2, it is required to find a polytope and facets such that . The possible existence of counterexamples is constrained by the affine dimension of .

###### Theorem 3.3.

Let , be a polytope with and . If , then condition 1 holds.

###### Proof.

We deal with each case for separately.

Case : This means that is a vertex of . Assume that condition 1 is violated, i.e. . This implies , and we conclude , i.e. . Thus the intersection uses all facets of . But no vertex of is incident to all facets of , a contradiction.

Case : This means that is an edge of . Assume that condition 1 is violated, i.e. . This implies , and we conclude , i.e. . The affine hull of the edge

is a straight line which is contained in the hyperplanes obtained as the affine hulls of the facets

. Two additional hyperplanes (or facets of ) are required to “cut out” the edge from its affine hull. However, since , there is at most one facet of left which is not one of . This is a contradiction.

Case : This means that is a polygon with, say, vertices and edges. Assume that condition 1 is violated, i.e. . This implies , and we conclude , i.e. . Thus the intersection uses all but at most facets of . However, in order to “cut out” the edges of the polygon, we need more facets not already used in the intersection, a contradiction.

Case : This means that is a -face of . As such, it is incident to exactly two facets of , see [7, Theorem 2.7]. Hence . Assume that condition 1 is violated, i.e. . Thus the two facets and must intersect in all vertices, or all but one. Since the intersection is -dimensional, the convex hull of the union of and the (possibly) remaining vertex of is at most -dimensional. This contradicts the assumption of being -dimensional.

Case : This means that the intersection consists of a single facet, i.e. . Assume that condition 1 is violated, i.e. . Hence, the facet contains all vertices, which contradicts the assumption of being -dimensional. ∎

An immediate consequence of Theorem 3.3 is the existence of facet-vertex assignments for polytopes up to dimension .

###### Corollary 3.4.

Every polytope of dimension possesses a facet-vertex assignment.

Therefore, the smallest possible counterexample to Problem 1.2 might be of dimension , where some facets intersect in a face of dimension .

## 4. Counterexamples in high dimensions

In this section, we construct for any polytopes which do not possess vertex-facet assignments. This leaves open only the case of dimension . Our counterexamples are based on a standard construction for polytopes. Given two polytopes , their free join

is defined as the convex hull of these polytopes after embedding them into skew affine subspaces of

. If the vertices of the two polytopes are given as , then the free join can be defined as

Most importantly, if the polytopes have vertices and facets, then their free join has vertices and facets. This is quite obvious for the vertices. For the facets note the following: each facet of is spanned either by and a facet of , or by and a facet of . For details regarding the free join and its properties see [6, Corollary 2].

###### Theorem 4.1.

For , let be a polytope with vertices and facets. Assume that and , and that . Then does not possess any vertex-facet assignment.

###### Proof.

Note that , i.e.  has more facets than vertices, and this is precisely the minimal requirement for the existence of vertex-facet assignments. Let be the facets of which contain (there are many because one for each facet of ). Then

The claim now follows from Theorem 2.2 and Corollary 3.2. ∎

A more direct reasoning (without using the detour over the matching formulation) might go like this: In a vertex-facet assignment, each vertex originally from must be assigned to a unique non-incident facet. The only facets not incident to are the ones spanned by and a (non-incident) facet of . However, since , there are not enough facets of to provide such non-incident facets in for all vertices of .

Note that the condition is not used in the proof but follows from and anyway. Furthermore, the conditions and cannot be satisfied for polytopes in dimension . This is the reason why this construction only yields counterexamples for Problem 1.1 in dimensions .

###### Example 4.2.

We list some concrete counterexamples for .

1. [label=(),leftmargin=*,align=left,noitemsep]

2. For odd dimensions

, let be any polytope with more vertices than facets, and let be the dual of . The free join then is a self-dual -dimensional polytope, see [6, Corollary 2]. By Theorem 4.1, the polytope does not possess vertex-facet assignments, and by self-duality, it does not possess facet-vertex assignments either.

3. We can construct counterexamples for Problem 1.1 from any polytope with and . Take to be a bipyramid with vertices. Then and . The free join then satisfies the assumptions of Theorem 4.1, and thus is a polytope without vertex-facet assignments in dimension .

## 5. Open problems

###### Question 5.1.

Does every -dimensional polytope possess a vertex-facet assignment, or are there any counterexamples?

As noted in Section 3, any possible counterexample in six dimensions must have facets that intersect in a -dimensional face with more than vertices.
Acknowledgements. The second author gratefully acknowledges the support by the funding of the European Union and the Free State of Saxony (ESF).

## References

• [1] G. Averkov and H. Martini, On reduced polytopes and antipodality, Adv. Geom. 8 (2008), no. 4, pp. 615–626, doi: 10.1515/ADVGEOM.2008.037.
• [2] R. Diestel, Graph Theory, 5th ed., Graduate Texts in Mathematics, vol. 173, Springer, Berlin, 2017, doi: 10.1007/978-3-662-53622-3.
• [3] B. González Merino, T. Jahn, A. Polyanskii, and G. Wachsmuth, Hunting for reduced polytopes, Discrete Comput. Geom. 60 (2018), no. 3, pp. 801–808, doi: 10.1007/s00454-018-9982-3.
• [4] B. Grünbaum, Convex Polytopes, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer, New York, NY, 2003, doi: 10.1007/978-1-4613-0019-9.
• [5] E. Heil, Kleinste konvexe Körper gegebener Dicke, Preprint 453, TU Darmstadt, 1978.
• [6] H. R. Tiwary and K. Elbassioni, Self-duality of polytopes and its relations to vertex enumeration and graph isomorphism, Graphs Combin. 30 (2014), no. 3, pp. 729–742, doi: 10.1007/s00373-013-1299-7.
• [7] G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, New York, NY, 1995, doi: 10.1007/978-1-4613-8431-1.


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## 1.4 Keeping Track of Decimal Places

Consider the numbers 3.7, 37, and 370. Using our rules for logarithms, we see that each power of ten adds “one” to the value of the logarithm:

$\begin{eqnarray*} \log 3.7 &=& \log (3.7\times 1) &=& \log 3.7 + \log 1 &=& \log 3.7 + 0 \\ \log 37 &=& \log (3.7\times 10) &=& \log 3.7 + \log 10 &=& \log 3.7 + 1 \\ \log 370 &=& \log (3.7\times 100) &=& \log 3.7 + \log 100 &=& \log 3.7 + 2 \\ \end{eqnarray*}$

We can see a similar pattern when looking back at our table of logarithms above. For example, notice that $$\log 6$$ = 0.7782, while $$\log 60$$ = 1.7782. By keeping track of powers of 10, one only needs the values of the logarithms for numbers between 1 and 10 to be able to determine the logarithm of any other number outside that range.

As an illustration, recall the plot we made earlier using Base 2. The pattern between each integer value of $$p$$ repeats; the corresponding point in each repeat section is just “2” times larger than the point in the previous section of the plot:

Hence, if we want to know the value of $$2^{7.25}$$, we only need to know the value of $$2^{0.25}$$ and then multiply this answer by $$2^7$$ (= 128).

We can use any number as our base, and the same general rules will apply; for our Base 10 system of common logarithms, multiplying by factors of 10 simply means adding zeroes or moving decimal points. Hence, by keeping track of powers of ten, detailed tables of common logarithms of numbers between 1 and 10, which themselves will have values between 0 and 1, are sufficient to perform a fairly accurate general multiplication or division calculation, as we shall soon see.

For example, the table below gives the 3-place logarithms for numbers between 1 and 10 in increments of 0.1. For finer increments and for further accuracy of the logarithms, tables can take many pages of text. This led to the publication of books of significant length that contained detailed tables of logarithms.

x Log(x) x Log(x) x Log(x) x Log(x)
1 0 3.3 0.519 5.6 0.748 7.9 0.898
1.1 0.041 3.4 0.531 5.7 0.756 8 0.903
1.2 0.079 3.5 0.544 5.8 0.763 8.1 0.908
1.3 0.114 3.6 0.556 5.9 0.771 8.2 0.914
1.4 0.146 3.7 0.568 6 0.778 8.3 0.919
1.5 0.176 3.8 0.58 6.1 0.785 8.4 0.924
1.6 0.204 3.9 0.591 6.2 0.792 8.5 0.929
1.7 0.23 4 0.602 6.3 0.799 8.6 0.934
1.8 0.255 4.1 0.613 6.4 0.806 8.7 0.94
1.9 0.279 4.2 0.623 6.5 0.813 8.8 0.944
2 0.301 4.3 0.633 6.6 0.82 8.9 0.949
2.1 0.322 4.4 0.643 6.7 0.826 9 0.954
2.2 0.342 4.5 0.653 6.8 0.833 9.1 0.959
2.3 0.362 4.6 0.663 6.9 0.839 9.2 0.964
2.4 0.38 4.7 0.672 7 0.845 9.3 0.968
2.5 0.398 4.8 0.681 7.1 0.851 9.4 0.973
2.6 0.415 4.9 0.69 7.2 0.857 9.5 0.978
2.7 0.431 5 0.699 7.3 0.863 9.6 0.982
2.8 0.447 5.1 0.708 7.4 0.869 9.7 0.987
2.9 0.462 5.2 0.716 7.5 0.875 9.8 0.991
3 0.477 5.3 0.724 7.6 0.881 9.9 0.996
3.1 0.491 5.4 0.732 7.7 0.886 10 1
3.2 0.505 5.5 0.74 7.8 0.892

One can immediately see how logarithms can be used to multiply numbers. For example, notice from our table that the logarithm of 1.5 is 0.176 and the logarithm of 3.6 is 0.556. The sum of the logarithms is 0.732, which we can see from the table is the logarithm of 5.4 = $$1.5\times$$ 3.6. Clearly many other examples can be found just from this simple table. Now suppose we wanted to multiply $$150\times$$ 3.6. We don’t need to have the logarithm of 150 in our table – the logarithm of 150 will just be 2 + the logarithm of 1.5, or 2.176. The result will just be $$10^2$$ times the result for $$1.5\times$$ 3.6, or 540. By keeping track of powers of ten, calculations can be performed by just using logarithms of numbers between 1 and 10.