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# zbMATH — the first resource for mathematics

## Journal of $$K$$-Theory

### $$K$$-Theory and its Applications in Algebra, Geometry, Analysis & Topology

 Short Title: J. $$K$$-Theory Publisher: Cambridge University Press, Cambridge ISSN: 1865-2433; 1865-5394/e Online: http://journals.cambridge.org/KAG Comments: No longer indexed
 Documents Indexed: 276 Publications (2008–2014) References Indexed: 262 Publications with 6,086 References.
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#### Latest Issues

 14, No. 3 (2014) 14, No. 2 (2014) 14, No. 1 (2014) 13, No. 3 (2014) 13, No. 2 (2014) 13, No. 1 (2014) 12, No. 3 (2013) 12, No. 2 (2013) 12, No. 1 (2013) 11, No. 3 (2013) 11, No. 2 (2013) 11, No. 1 (2013) 10, No. 3 (2012) 10, No. 2 (2012) 10, No. 1 (2012) 9, No. 3 (2012) 9, No. 2 (2012) 9, No. 1 (2012) 8, No. 3 (2011) 8, No. 2 (2011) 8, No. 1 (2011) 7, No. 3 (2011) 7, No. 2 (2011) 7, No. 1 (2011) 6, No. 3 (2010) 6, No. 2 (2010) 6, No. 1 (2010) 5, No. 3 (2010) 5, No. 2 (2010) 5, No. 1 (2010) 4, No. 3 (2009) 4, No. 2 (2009) 4, No. 1 (2009) 3, No. 3 (2009) 3, No. 2 (2009) 3, No. 1 (2009) 2, No. 3 (2008) 2, No. 2 (2008) 2, No. 1 (2008) 1, No. 3 (2008) 1, No. 2 (2008) 1, No. 1 (2008)
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#### Fields

 161 $$K$$-theory (19-XX) 91 Algebraic geometry (14-XX) 61 Algebraic topology (55-XX) 58 Category theory; homological algebra (18-XX) 48 Functional analysis (46-XX) 36 Associative rings and algebras (16-XX) 33 Number theory (11-XX) 27 Global analysis, analysis on manifolds (58-XX) 24 Group theory and generalizations (20-XX) 20 Manifolds and cell complexes (57-XX) 15 Commutative algebra (13-XX) 10 History and biography (01-XX) 9 Nonassociative rings and algebras (17-XX) 8 Differential geometry (53-XX) 7 Topological groups, Lie groups (22-XX) 4 Quantum theory (81-XX) 3 Field theory and polynomials (12-XX) 3 Abstract harmonic analysis (43-XX) 3 Operator theory (47-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Geometry (51-XX) 1 General and overarching topics; collections (00-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 General topology (54-XX) 1 Mechanics of particles and systems (70-XX)

#### Citations contained in zbMATH Open

220 Publications have been cited 1,400 times in 1,219 Documents Cited by Year
Dimensions of triangulated categories. Zbl 1165.18008
Rouquier, Raphaël
2008
Weight structures vs. $$t$$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general). Zbl 1303.18019
Bondarko, M. V.
2010
Under Spec $$\mathbb Z$$. (Au-dessous de Spec $$\mathbb Z$$.) Zbl 1177.14022
Toën, Bertrand; Vaquié, Michel
2009
Bak’s work on the $$K$$-theory of rings. Zbl 1183.19001
Hazrat, Roozbeh; Vavilov, Nikolai
2009
Symmetric monoidal structure on non-commutative motives. Zbl 1256.19002
Cisinski, Denis-Charles; Tabuada, Gonçalo
2012
Hermitian $$K$$-theory of exact categories. Zbl 1328.19009
Schlichting, Marco
2010
Bimonads and Hopf monads on categories. Zbl 1239.18002
Mesablishvili, Bachuki; Wisbauer, Robert
2011
On the Lie enveloping algebra of a pre-Lie algebra. Zbl 1178.17011
Oudom, J.-M.; Guin, D.
2008
Convergence of the motivic Adams spectral sequence. Zbl 1309.14018
Hu, P.; Kriz, I.; Ormsby, K.
2011
Remarks on motivic homotopy theory over algebraically closed fields. Zbl 1248.14026
Hu, Po; Kriz, Igor; Ormsby, Kyle
2011
Recollement of homotopy categories and Cohen-Macaulay modules. Zbl 1251.18008
Iyama, Osamu; Kato, Kiriko; Miyachi, Jun-Ichi
2011
Vector bundles and Gromov-Hausdorff distance. Zbl 1189.53041
Rieffel, Marc A.
2010
Homotopy invariance of non-stable $$K_{1}$$-functors. Zbl 1314.19002
Stavrova, A.
2014
Relative subgroups in Chevalley groups. Zbl 1204.20066
Hazrat, R.; Petrov, V.; Vavilov, N.
2010
The noncommutative geometry of $$k$$-graph $$C^*$$-algebras. Zbl 1167.46036
2008
Report on locally finite triangulated categories. Zbl 1252.18028
Krause, Henning
2012
A Thom-Porteous formula for connective $$K$$-theory using algebraic cobordism. Zbl 1314.14015
Hudson, Thomas
2014
Stratifications and Mackey functors. II: Globally defined Mackey functors. Zbl 1243.20014
Webb, Peter
2010
Thom isomorphism and push-forward map in twisted K-theory. Zbl 1204.55006
Carey, Alan L.; Wang, Bai-Ling
2008
$$\mathbb A^1$$-homotopy of Chevalley groups. Zbl 1200.14039
Wendt, Matthias
2010
The analytic index of elliptic pseudodifferential operators on a singular foliation. Zbl 1237.57030
Androulidakis, Iakovos; Skandalis, Georges
2011
Invariance of the $$K$$-theory for derived equivalences. (Invariance de la $$K$$-théorie par équivalences dérivées.) Zbl 1230.19002
Cisinski, Denis-Charles
2010
Fiberwise $$KK$$-equivalence of continuous fields of $$C^*$$-algebras. Zbl 1173.46050
2009
Versal deformations of Leibniz algebras. Zbl 1191.17001
Fialowski, Alice; Mandal, Ashis; Mukherjee, Goutam
2009
Motives over simplicial schemes. Zbl 1194.14029
2010
The Farrell-Jones isomorphism conjecture for 3-manifold groups. Zbl 1160.57018
Roushon, S. K.
2008
The Guillemin-Sternberg conjecture for noncompact groups and spaces. Zbl 1159.19004
Hochs, P.; Landsman, N. P.
2008
Separable functors and formal smoothness. Zbl 1167.16032
Ardizzoni, Alessandro
2008
On the homology of unitary groups with polynomial coefficients. (Sur l’homologie des groupes unitaires à coefficients polynomiaux.) Zbl 1281.19004
Djament, Aurélien
2012
Auslander-Reiten triangles in subcategories. Zbl 1201.16021
Jørgensen, Peter
2009
Elementary symplectic orbits and improved $$K_{1}$$-stability. Zbl 1218.19001
Chattopadhyay, Pratyusha; Rao, Ravi A.
2011
An index theorem of Callias type for pseudodifferential operators. Zbl 1248.58012
Kottke, Chris
2011
Bounded and unbounded Fredholm modules for quantum projective spaces. Zbl 1207.58008
D’Andrea, Francesco; Landi, Giovanni
2010
The additivity of traces in monoidal derivators. Zbl 1349.18015
Groth, Moritz; Ponto, Kate; Shulman, Michael
2014
A fundamental property of Suslin matrices. Zbl 1200.13019
Jose, Selby; Rao, Ravi A.
2010
Geometry of Reidemeister classes and twisted Burnside theorem. Zbl 1215.22003
Fel’shtyn, Alexander; Troitsky, Evgenij
2008
Motivic connective $$k$$-theories and the cohomology of $$A(1)$$. Zbl 1266.14015
Isaksen, Daniel C.; Shkembi, Armira
2011
$$KK$$-theory and spectral flow in von Neumann algebras. Zbl 1260.19001
Kaad, J.; Nest, R.; Rennie, A.
2012
Cyclic multicategories, multivariable adjunctions and mates. Zbl 1326.18005
Cheng, Eugenia; Gurski, Nick; Riehl, Emily
2014
Simplicial radditive functors. Zbl 1194.55021
2010
Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra $$O_{n}$$. Zbl 1220.19004
Carey, A. L.; Phillips, J.; Rennie, A.
2010
Analysis on arithmetic schemes. II. Zbl 1225.14019
Fesenko, Ivan
2010
Motivic decomposition of anisotropic varieties of type $$F_{4}$$ into generalized Rost motives. Zbl 1163.14014
Nikolenko, S.; Semenov, N.; Zainoulline, K.
2009
The projective bundle theorem for $$I^j$$-cohomology. Zbl 1272.14019
Fasel, Jean
2013
Comparison of secondary invariants of algebraic $$K$$-theory. Zbl 1241.19001
2011
A Bloch-Wigner complex for $$\mathrm{SL}_{2}$$. Zbl 1284.19003
Hutchinson, Kevin
2013
Reduction of filtered $$K$$-theory and a characterization of Cuntz-Krieger algebras. Zbl 1330.46058
Arklint, Sara E.; Bentmann, Rasmus; Katsura, Takeshi
2014
Higher quasi-categories vs higher Rezk spaces. Zbl 1322.18002
Ara, Dimitri
2014
Perfect complexes on Deligne-Mumford stacks and applications. Zbl 1189.19003
Krishna, Amalendu
2009
Higher homotopy operations and cohomology. Zbl 1207.55012
Blanc, David; Johnson, Mark W.; Turner, James M.
2010
Rationally trivial torsors in $$\mathbb A^1$$-homotopy theory. Zbl 1228.19002
Wendt, Matthias
2011
The fundamental theorem via derived Morita invariance, localization, and $$\mathbb A^{1}$$-homotopy invariance. Zbl 1257.18007
2012
Local left invertibility for operator tuples and noncommutative localizations. Zbl 1185.46048
Dosiev, Anar
2009
Cycles of codimension 2 and unramified $$H^3$$ for varieties over finite fields. (Cycles de codimension 2 et $$H^3$$ non ramifié pour les variétés sur les corps finis.) Zbl 1263.19004
Colliot-Thélène, Jean-Louis; Kahn, Bruno
2013
Prime ideals of mixed Artin-Tate motives. Zbl 1275.14021
Peter, Tobias J.
2013
Connective algebraic $$K$$-theory. Zbl 1326.19001
Dai, Shouxin; Levine, Marc
2014
Flat bundles, von Neumann algebras and $$K$$-theory with $$\mathbb{R}/\mathbb{Z}$$-coefficients. Zbl 1315.46077
Antonini, Paolo; Azzali, Sara; Skandalis, Georges
2014
Index maps in the $$K$$-theory of graph algebras. Zbl 1260.46051
Carlsen, Toke Meier; Eilers, Søren; Tomforde, Mark
2012
Twisted homology of quantum $$SL(2)$$. II. Zbl 1201.58003
Hadfield, Tom; Krähmer, Ulrich
2010
Characterizing C$$^*$$-algebras of compact operators by generic categorical properties of Hilbert C$$^*$$-modules. Zbl 1163.46039
Frank, Michael
2008
Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds. Zbl 1183.58005
Lescure, Jean-Marie
2009
Index theory and non-commutative geometry. II: Dirac operators and index bundles. Zbl 1155.58009
Benameur, Moulay-Tahar; Heitsch, James L.
2008
Cancellation problem for projective modules over affine algebras. Zbl 1180.13012
Keshari, Manoj Kumar
2009
On the functoriality of the slice filtration. Zbl 1319.14029
Pelaez, Pablo
2013
On a localisation sequence for the K-theory of skew power series rings. Zbl 1270.16035
Witte, Malte
2013
Injective stability for unitary $$K_{1}$$. Revisited. Zbl 1270.19003
Sinchuk, S.
2013
On (co)homology of Frobenius Poisson algebras. Zbl 1309.17012
Zhu, Can; Van Oystaeyen, Fred; Zhang, Yinhuo
2014
Slices of motivic Landweber spectra. Zbl 1249.14008
Spitzweck, Markus
2012
Hochster’s theta pairing and numerical equivalence. Zbl 1326.13013
Dao, Hailong; Kurano, Kazuhiko
2014
Compactly supported analytic indices for Lie groupoids. Zbl 1188.19004
Rouse, Paulo Carrillo
2009
Geometric $$K$$-homology with coefficient. I: $$\mathbb Z / k\mathbb Z$$-cycles and Bockstein sequence. Zbl 1256.19005
Deeley, Robin J.
2012
Relative Zariski open objects. Zbl 1266.14005
Marty, Florian
2012
Derived moduli of schemes and sheaves. Zbl 1257.14008
Pridham, J. P.
2012
Representability of derived stacks. Zbl 1257.14009
Pridham, J. P.
2012
An elementary differential extension of odd $$K$$-theory. Zbl 1327.19014
Tradler, Thomas; Wilson, Scott O.; Zeinalian, Mahmoud
2013
On the Witt groups of projective bundles and split quadrics: geometric reasoning. Zbl 1223.19005
Nenashev, Alexander
2009
Module categories for group algebras over commutative rings. Zbl 1291.20010
Benson, Dave; Iyengar, Srikanth B.; Krause, Henning
2013
Cyclic homology, Serre’s local factors and $$\lambda$$-operations. Zbl 1327.14109
Connes, Alain; Consani, Caterina
2014
Twisted $$K$$-theory constructions in the case of a decomposable Dixmier-Douady class. Zbl 1360.19014
Harju, Antti J.; Mickelsson, Jouko
2014
Cohomological obstruction theory for Brauer classes and the period-index problem. Zbl 1236.14023
Antieau, Benjamin
2011
Reduced Steenrod operations and resolution of singularities. Zbl 1238.14006
Haution, Olivier
2012
Nisnevich descent for $$K$$-theory of Deligne-Mumford stacks. Zbl 1284.19005
Krishna, Amalendu; Østvær, Paul Arne
2012
Norm index formula for the Tate kernels and applications. Zbl 1254.11105
Assim, J.; Movahhedi, A.
2012
On the Lie-formality of Poisson manifolds. Zbl 1149.17013
Sharygin, G.; Talalaev, D.
2008
Simplicial homotopy in semi-abelian categories. Zbl 1190.18007
Van der Linden, Tim
2009
Bounded rigidity of manifolds and asymptotic dimension growth. Zbl 1202.53048
Chang, Stanley S.; Ferry, Steven; Yu, Guoliang
2008
Orderings and signatures of higher level on multirings and hyperfields. Zbl 1270.12003
Gładki, Paweł; Marshall, Murray
2012
Higher spectral flow and an entire bivariant JLO cocycle. Zbl 1276.58002
Benameur, Moulay-Tahar; Carey, Alan L.
2013
On the algebraic $$K$$-theory of truncated polynomial algebras in several variables. Zbl 1317.19006
Angeltveit, Vigleik; Gerhardt, Teena; Hill, Michael A.; Lindenstrauss, Ayelet
2014
Tame kernels and second regulators of number fields and their subfields. Zbl 1286.11199
Browkin, Jerzy; Gangl, Herbert
2013
A Quillen model structure for Gray-categories. Zbl 1257.18013
Lack, Stephen
2011
Finitely summable Fredholm modules over higher rank groups and lattices. Zbl 1236.19006
Puschnigg, Michael
2011
Determinant functors on triangulated categories. Zbl 1243.18023
Breuning, Manuel
2011
Brown representability often fails for homotopy categories of complexes. Zbl 1250.18015
Modoi, George Ciprian; Šťovíček, Jan
2012
$$K$$-theory, reality, and duality. Zbl 1325.55003
Heard, Drew; Stojanoska, Vesna
2014
Tannakization in derived algebraic geometry. Zbl 1325.18001
Iwanari, Isamu
2014
Relative pairing in cyclic cohomology and divisor flows. Zbl 1175.46066
Lesch, Matthias; Moscovici, Henri; Pflaum, Markus J.
2009
Noncommutative geometry of foliations. Zbl 1155.58004
Kordyukov, Yuri A.
2008
Comparing homotopy categories. Zbl 1155.55007
Blanc, David
2008
Grothendieck groups for categories of complexes. Zbl 1163.13008
Foxby, Hans-Bjørn; Halvorsen, Esben Bistrup
2009
Homotopy invariance of non-stable $$K_{1}$$-functors. Zbl 1314.19002
Stavrova, A.
2014
A Thom-Porteous formula for connective $$K$$-theory using algebraic cobordism. Zbl 1314.14015
Hudson, Thomas
2014
The additivity of traces in monoidal derivators. Zbl 1349.18015
Groth, Moritz; Ponto, Kate; Shulman, Michael
2014
Cyclic multicategories, multivariable adjunctions and mates. Zbl 1326.18005
Cheng, Eugenia; Gurski, Nick; Riehl, Emily
2014
Reduction of filtered $$K$$-theory and a characterization of Cuntz-Krieger algebras. Zbl 1330.46058
Arklint, Sara E.; Bentmann, Rasmus; Katsura, Takeshi
2014
Higher quasi-categories vs higher Rezk spaces. Zbl 1322.18002
Ara, Dimitri
2014
Connective algebraic $$K$$-theory. Zbl 1326.19001
Dai, Shouxin; Levine, Marc
2014
Flat bundles, von Neumann algebras and $$K$$-theory with $$\mathbb{R}/\mathbb{Z}$$-coefficients. Zbl 1315.46077
Antonini, Paolo; Azzali, Sara; Skandalis, Georges
2014
On (co)homology of Frobenius Poisson algebras. Zbl 1309.17012
Zhu, Can; Van Oystaeyen, Fred; Zhang, Yinhuo
2014
Hochster’s theta pairing and numerical equivalence. Zbl 1326.13013
Dao, Hailong; Kurano, Kazuhiko
2014
Cyclic homology, Serre’s local factors and $$\lambda$$-operations. Zbl 1327.14109
Connes, Alain; Consani, Caterina
2014
Twisted $$K$$-theory constructions in the case of a decomposable Dixmier-Douady class. Zbl 1360.19014
Harju, Antti J.; Mickelsson, Jouko
2014
On the algebraic $$K$$-theory of truncated polynomial algebras in several variables. Zbl 1317.19006
Angeltveit, Vigleik; Gerhardt, Teena; Hill, Michael A.; Lindenstrauss, Ayelet
2014
$$K$$-theory, reality, and duality. Zbl 1325.55003
Heard, Drew; Stojanoska, Vesna
2014
Tannakization in derived algebraic geometry. Zbl 1325.18001
Iwanari, Isamu
2014
The triangulated category of $$\mathrm K$$-motives $$DK_{-}^{\mathrm{eff}}(k)$$. Zbl 1327.14106
Garkusha, Grigory; Panin, Ivan
2014
Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras. Zbl 1311.16003
Ara, P.; Pardo, E.
2014
The noncommutative infinitesimal equivariant index formula. Zbl 1348.58015
Wang, Yong
2014
Dendroidal sets as models for connective spectra. Zbl 1323.55009
Bašić, Matija; Nikolaus, Thomas
2014
Twisted homological stability for extensions and automorphism groups of free nilpotent groups. Zbl 1308.19001
Szymik, Markus
2014
Morava $$K$$-theory rings for the groups $$G_{38}, \ldots, G_{41}$$ of order 32. Zbl 1304.55004
2014
An interesting example for spectral invariants. Zbl 1300.58009
Benameur, Moulay-Tahar; Heitsch, James L.; Wahl, Charlotte
2014
An additive variant of Somekawa’s $$K$$-groups and Kähler differentials. Zbl 1317.19005
Hiranouchi, Toshiro
2014
Reciprocity laws on algebraic surfaces via iterated integrals. Zbl 1317.19008
Horozov, Ivan; Kerr, Matt
2014
$$K$$-groups of reciprocity functors for $$\mathbb{G}_a$$ and abelian varieties. Zbl 1326.19002
Rülling, Kay; Yamazaki, Takao
2014
Algebraic $$K$$-theory of $$\infty$$-operads. Zbl 1325.55009
Nikolaus, Thomas
2014
Twisted $$K$$-theory and obstructions against positive scalar curvature metrics. Zbl 1305.19004
Pennig, Ulrich
2014
Twisted Witt groups of flag varieties. Zbl 1305.19002
Zibrowius, Marcus
2014
Equivariant $$\mathrm{KK}$$-theory for generalised actions and Thom isomorphism in groupoid twisted $$K$$-theory. Zbl 1311.19003
Moutuou, El-Kaïoum M.
2014
Cyclic cohomology and Baaj-Skandalis duality. Zbl 1325.19002
Voigt, Christian
2014
$$E$$-theory for $$C[0, 1]$$-algebras with finitely many singular points. Zbl 1318.46040
2014
Milnor $$K$$-theory and the graded representation ring. Zbl 1345.12005
Guillot, Pierre; Mináč, Ján
2014
On the motive of the group of units of a division algebra. Zbl 1325.14037
Shinder, Evgeny
2014
On the rational $$K_2$$ of a curve of $$\mathrm{GL}_{2}$$ type over a global field of positive characteristic. Zbl 1322.19002
Chida, Masataka; Kondo, Satoshi; Yamauchi, Takuya
2014
The projective bundle theorem for $$I^j$$-cohomology. Zbl 1272.14019
Fasel, Jean
2013
A Bloch-Wigner complex for $$\mathrm{SL}_{2}$$. Zbl 1284.19003
Hutchinson, Kevin
2013
Cycles of codimension 2 and unramified $$H^3$$ for varieties over finite fields. (Cycles de codimension 2 et $$H^3$$ non ramifié pour les variétés sur les corps finis.) Zbl 1263.19004
Colliot-Thélène, Jean-Louis; Kahn, Bruno
2013
Prime ideals of mixed Artin-Tate motives. Zbl 1275.14021
Peter, Tobias J.
2013
On the functoriality of the slice filtration. Zbl 1319.14029
Pelaez, Pablo
2013
On a localisation sequence for the K-theory of skew power series rings. Zbl 1270.16035
Witte, Malte
2013
Injective stability for unitary $$K_{1}$$. Revisited. Zbl 1270.19003
Sinchuk, S.
2013
An elementary differential extension of odd $$K$$-theory. Zbl 1327.19014
Tradler, Thomas; Wilson, Scott O.; Zeinalian, Mahmoud
2013
Module categories for group algebras over commutative rings. Zbl 1291.20010
Benson, Dave; Iyengar, Srikanth B.; Krause, Henning
2013
Higher spectral flow and an entire bivariant JLO cocycle. Zbl 1276.58002
Benameur, Moulay-Tahar; Carey, Alan L.
2013
Tame kernels and second regulators of number fields and their subfields. Zbl 1286.11199
Browkin, Jerzy; Gangl, Herbert
2013
Beilinson’s Hodge conjecture for smooth varieties. Zbl 1300.14009
de Jeu, Rob; Lewis, James D.
2013
Milnor descent for cohesive dg-categories. Zbl 1296.18005
Ben-Bassat, Oren; Block, Jonathan
2013
Self-dual blended extensions. (Extensions panachées autoduales.) Zbl 1272.18007
Bertrand, Daniel
2013
Algebraic $$K$$-theory and cyclic homology. Zbl 1286.19004
Loday, Jean-Louis
2013
On a local to global principle in étale $$K$$-groups of curves. Zbl 1291.19005
Banaszak, Grzegorz; Krasoń, Piotr
2013
On additive invariants of actions of additive and multiplicative groups. Zbl 1328.14074
Hu, Wenchuan
2013
Sheaf theory and Paschke duality. Zbl 1368.46066
Roe, John; Siegel, Paul
2013
Geometric $$K$$-homology with coefficients. II: The analytic theory and isomorphism. Zbl 1297.19006
Deeley, Robin J.
2013
A residue formula for the fundamental Hochschild class on the Podleś sphere. Zbl 1293.58009
Krähmer, Ulrich; Wagner, Elmar
2013
Modular invariants and twisted equivariant $$K$$-theory. II: Dynkin diagram symmetries. Zbl 1310.81137
Evans, David E.; Gannon, Terry
2013
Algebraic cobordism theory attached to algebraic equivalence. Zbl 1273.14045
Krishna, Amalendu; Park, Jinhyun
2013
Central simple algebras of prime exponent and divided power operations. Zbl 1275.16020
Sivatski, A. S.
2013
Quillen model categories. Zbl 1285.18019
Hovey, Mark
2013
Spectrum of group cohomology and support varieties. Zbl 1297.20055
Friedlander, Eric M.
2013
Quillen’s solution of Serre’s problem. Zbl 1282.13001
Suslin, Andrei
2013
Remarks on Murre’s conjecture on Chow groups. Zbl 1284.14011
Xu, Kejian; Xu, Ze
2013
$$K$$-cycles for twisted $$K$$-homology. Zbl 1300.19003
Baum, Paul; Carey, Alan; Wang, Bai-Ling
2013
On $$\lambda-$$invariants of number fields and étale cohomology. Zbl 1348.11085
Kolster, Manfred; Movahhedi, Abbas
2013
Smoothness in relative geometry. Zbl 1296.18003
Marty, Florian
2013
Equivariant semi-topological $$K$$-homology and a theorem of Thomason. Zbl 1297.19004
Heller, Jeremiah; Hornbostel, Jens
2013
A candidate for the abelian category of mixed elliptic motives. Zbl 1295.14010
Patashnick, Owen
2013
Remarks on 1-motivic sheaves. Zbl 1317.14011
Bertapelle, Alessandra
2013
On lambda operations on mixed motives. Zbl 1290.14005
Biglari, Shahram
2013
Godeaux-Serre varieties and the étale index. Zbl 1272.14018
Antieau, Benjamin; Williams, Ben
2013
Symmetric monoidal structure on non-commutative motives. Zbl 1256.19002
Cisinski, Denis-Charles; Tabuada, Gonçalo
2012
Report on locally finite triangulated categories. Zbl 1252.18028
Krause, Henning
2012
On the homology of unitary groups with polynomial coefficients. (Sur l’homologie des groupes unitaires à coefficients polynomiaux.) Zbl 1281.19004
Djament, Aurélien
2012
$$KK$$-theory and spectral flow in von Neumann algebras. Zbl 1260.19001
Kaad, J.; Nest, R.; Rennie, A.
2012
The fundamental theorem via derived Morita invariance, localization, and $$\mathbb A^{1}$$-homotopy invariance. Zbl 1257.18007
2012
Index maps in the $$K$$-theory of graph algebras. Zbl 1260.46051
Carlsen, Toke Meier; Eilers, Søren; Tomforde, Mark
2012
Slices of motivic Landweber spectra. Zbl 1249.14008
Spitzweck, Markus
2012
Geometric $$K$$-homology with coefficient. I: $$\mathbb Z / k\mathbb Z$$-cycles and Bockstein sequence. Zbl 1256.19005
Deeley, Robin J.
2012
Relative Zariski open objects. Zbl 1266.14005
Marty, Florian
2012
Derived moduli of schemes and sheaves. Zbl 1257.14008
Pridham, J. P.
2012
Representability of derived stacks. Zbl 1257.14009
Pridham, J. P.
2012
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# How exactly do logic and mathematics interact and what happens when we change the logic?

[Note: this question turned out to be pretty huge, so if you think it would be better to split it up into smaller questions, please comment. The questions here are quite conceptually intertwined and dependent on each other, so I thought it was best to leave them in one piece, since I strongly suspect there might be a short answer that answers all of them at once. (But I really do hope the finale below sufficiently motivates the length of this question.)]

I've been thinking a bit about the foundations of mathematics and decided it's time to clear a few things up. I think it will be easiest to explain what bothers me on some concrete examples, so here goes. (Since, despite always having been fascinated with the subject, I have not yet found the time to study logic deeply, it is very possible that I shall say some things that are just terribly wrong. In this case comments are most welcome, since one of the main purposes of this question is clearing up my misconceptions.)

Suppose I am a mathematician and I want to study vector spaces. Then, as I understand it, what I want to do is the following. First, I define a formal theory $T$, describing the basic properties of vectors. This theory $T$ is the union $V\cup L$ of non-logical axioms (about formal properties of vectors) and logical axioms (these are just the axioms of predicate calculus). Its models are precisely the vector spaces. A model is something that interprets my theory, for example, the models in $\mathrm{ZFC}$ of the theory $T$ are precisely those sets that interpret our theory so it becomes true in this setting. (As you can see from this phrasing, I am already a bit confused by this, so here comes my first question.)

What is the precise role of $\mathrm{ZFC}$ here? Is it just the metatheory or is it something else? Does it even make sense to want to be able to interpret your theory in the metatheory?

As you see, I mention metatheory here. This is the next step. A mathematician always wants to have a good metatheory to be able to talk about the theory. As I understand it, the metatheory is mostly taken to be some sufficiently powerful theory, such as $\mathrm{ZFC}$ or maybe $\mathrm{IZF}$ if you like intuitionism and so on (category theory, perhaps?) What is interesting here is that the theory itself is of almost no use to mathematicians without some kind of metatheory. For example, most of linear algebra doesn't really happen in the theory of the vector space, it happens in this metatheory. The theory alone can't even express a statement like "every vector space has a basis" let alone prove it. (If I have confused every single bit of terminology out there, please let me know. Actually after thinking a bit more, I am becoming convinced that by "metatheory" here I probably mean the metatheory of the theory in which we interpret $T$, in this case the metatheory of $\mathrm{ZFC}$.)

But now comes the interesting thing. The properties we can prove will depend on the theory in which we interpret $T$. For example every $\mathrm{ZFC}$-model of a vector space will have a basis of a uniquely determined cardinality. And I believe I have read somewhere it is possible to construct in some intuitionistic set theory a vector space without a basis. It can also have two bases of different cardinalities.

This sounds as interesting as it confuses me.

Are these actually the models of the same theory $T$ or do we have to change the logical axioms of $T$ also? I imagine if one wants to be able to really interpret $T$ in an intuitionistic set theory, one might want to replace the predicate calculus $L$ with some sort of an intuitionistic variant of it, $L_I$? Or is this completely unnecessary? (In this case it probably is, since the axioms $V$ deal only with finitely many vectors at a time, right? But what about if we were to interpret $\mathrm{ZFC}$ in $\mathrm{IZF}$?)

I am completely fascinated that one can get such different-looking vector spaces by just changing the logic. So I was also wondering how much logic can we replace with something else in order to get interesting results. This, I suspect, would only mean interpreting the theory $T$ in some theory with a different logic. This seems to me to open up many interesting possibilities:

Is it possible to go the other way around and interpret $\mathrm{ZFC}$ in $T$ or something bizarre like that? Is there a model of $T$ in some weird theory that has in some sense two finite bases of different cardinalities?

And for the finale, which was actually my main motivation for these considerations:

How much logic (in any sensible sense of the word) do we have to "throw away" in order to make it possible to construct a three-dimensional associative division algebra over the reals?

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There is an area called "reverse mathematics" that tries to find minimal sets of axiom to prove a certain theorem – Thomas Andrews Feb 22 '12 at 6:43
I think I will never understand the obsession with finding a three-dimensional associative division algebra... and as for vector spaces without the invariant basis number property, one does not have to retreat to IZF to get a problem. Already there are models of ZF in which vector spaces fail to have invariant basis number. – Zhen Lin Feb 22 '12 at 7:38
@ZhenLin: Well, it is not really an obsession. The question could also be "how much logic do we have to throw away in order to make it possible to construct a non-cyclic group with three elements?" The emphasis is not on this particular problem, I am more interested about what unimaginable things are possible when we are willing to radically change our logic. – Dejan Govc Feb 22 '12 at 13:41
@Dejan Govc: it requires almost nothing to prove that a group with three elements must be cyclic, and certainly does not require us to assume ZFC. For example if we just take the group axioms and add an axiom $(\forall x)(x = e \lor x = a \lor x = b) \land (e \not = a) \land (e \not = b) \land (a \not = b)$ which says that the group has exactly three elements $\{e, a, b\}$ then we can prove from these axioms that $(a\cdot a = b \lor b \cdot b = a)$. – Carl Mummert Feb 22 '12 at 14:02
@Dejan Govc: the axioms of a group are not typically called "logical axioms"; that refers to the axioms of the logic itself, not to axioms of the object theory. If you change the group axioms, it does seem harder to claim that your new axioms are still axioms for "groups". – Carl Mummert Feb 22 '12 at 14:43

The role of ZFC in studying vector spaces, among other things, is that we don't just look at the vector space itself, we look at sub_sets_ of the vector space to see if they are subspaces, and we want to look at functions (which are also represented as sets) from one vector space to another.

The first-order theory of vector spaces is good for studying some properties of a particular vector space, but it is not able to quantify over, or construct, subsets of the vector space or functions between different vector spaces. So to really study vector spaces in the way that mathematicians do, we need to use a system that allows us to work with sets and functions. There are many possible systems that we could use, but ZFC is convenient because (1) we know it pretty well and (2) in practice ZFC is strong enough to do the things that we want to do with vector spaces.

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but this kind of question are solved in model theory via concepts of morphism of structures and substructures, am I wrong? – Giorgio Mossa Feb 22 '12 at 14:31
@Ineff: a structure is also represented by a set, so to work with structures or quantify over them you have to be able to work with sets in some form, which cannot be done in the first-order theory of vector spaces. – Carl Mummert Feb 22 '12 at 14:43
ah ok, now I see what you mean, thanks for the clarification. :) – Giorgio Mossa Feb 22 '12 at 14:47
Thanks, this is a very helpful perspective. So until we find an interesting property "of vector spaces" that ZFC cannot interpret, we can reasonably work with ZFC. Then we just switch to something more powerful. Or in case we want to study some very weak properties of vector spaces, we could just interpret them in some very weak theory? – Dejan Govc Feb 22 '12 at 15:06
@Dejan Gove: yes, that is exactly what I was saying. – Carl Mummert Feb 22 '12 at 16:37

I think you're confusing the idea of a foundational theory with the idea of a metatheory.

ZFC is a commonly used foundational theory in mathematics. For example, it is possible to construct the real number system out of ZFC.

A metatheory is a theory that describes another theory. Here's an example of a theory, which I'll call FOO:

Propositions in FOO consist of strings of the characters A and B.

Axiom 1: B is a theorem.

Axiom 2: If P is a theorem, then PA is a theorem.

If I start proving theorems in FOO, I will prove B, BA, BAA, BAAA, ... All of these are theorems. I start to see a pattern. I say, "Hey, the only theorems I can prove in FOO consist of B followed by zero or more A's." This is a theorem in the metatheory of FOO.

You asked, "What is the precise role of ZFC here? Is it just the metatheory or is it something else?" I think the answer is that it is not just the metatheory -- and it's not even a metatheory at all. In this situation ZFC is a theory that you're using to construct models of T. For example, suppose I'm worried that there is some horrible contradiction lurking at the heart of the theory of vector spaces. If I can construct models of vector spaces using ZFC, then I can prove that any contradiction in the theory of vector spaces also leads to a contradiction in ZFC. This is comforting, because I'm pretty sure that ZFC is consistent.

As far as I can tell, the term "metatheory" is not formally defined. It seems to have originated with Hilbert, which means it predates Godel, model theory, and ZFC. It's a vague, philosophical term. But the general idea seems to be that a metatheory is an informal thing you construct in order to describe some more formal theory. In the example above, FOO is formal, but my observation about the kinds of theorems that can be proved in FOO is informal.

By contrast, a foundational theory like ZFC is formal.

It may help to observe that the relationship of model to theory can be symmetric. Let R be a theory of the real numbers. It has axioms like x+y=y+x, etc. Let E be a theory of Euclidean geometry. It has axioms like the parallel postulate. Both R and E can be formalized, so let's assume they have been. We can make a model of E in R, which is what Descartes did by modeling the plane as ordered pairs of real numbers. But we can also make a model of R in E, which is essentially what Euclid did in the Elements, where he discusses, e.g., the distributive law x(y+z)=xy+xz as a geometrical theorem.

The relationship of a metatheory to the theory is describes is not symmetric in this way.

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FOO also known as "sheep mathematics" – Seamus Feb 22 '12 at 12:32
@BenCrowell I disagree with you on some points: there's a rigorous definition of metatheory and a metatheory doesn't need to be a intuitive theory. – Giorgio Mossa Feb 22 '12 at 12:56
Thanks, this certainly helps. "Foundational theory" might be just the term that I needed. So, if I understand correctly, the foundational theory can be any formal theory in which we interpret our original formal theory? So in the case of Descartes and Euclid, $R$ can be seen as a foundational theory for $E$ and vice versa? Is this standard terminology? – Dejan Govc Feb 22 '12 at 13:55
I agree with @ineff. A metatheory can and should be formalizable, otherwise it does not belong to mathematics. The point is that it is formalizable in higher-order logic in general. That's because it deals with quantization of formulae/statements in lower-order logic (often first-order logic). The metatheorem mentioned above is an example of this. You can also look in wikipedia for "metatheory". – magma Feb 23 '12 at 16:52

I'll try to answer some of your questions. I think your main problem is some misunderstanding of the concept of metatheory and definition, I'll try to explain here to you.

A definition can be formally "defined" as a predicate of the form "$x$ is an $A$ iff $P(x)$ holds" for some formula $P$ in a language you use. So in order to make definitions we need a language.

Now suppose we want to study a theory with formal tools, in order to do so we have to define in formal way the objects we want to study: thus to do this we need a language in which we can formalize/define the objects of our theory (formulas, theorems, theory, interpretation and so on) and a theory which states facts from which we can prove our theorems (metatheorems) about the theory, this data (language+theory) are our metatheory. We can choose many different metatheories, the only thing we require is that this theory is rich enough to allows to formalize (aka define) both the syntax and semantic for the theory that we want to study. Clearly you can use ZFC, or an intuitive set theory, but also an intuitionistic set theory or ETCS (a categorical set theory) or any other theory which satisfies the previous requirements.

Now we have to focus to the concepts of theory and model of a theory. The first one is just a set of formulae of the language (which are objects of our metatheory) the second is an interpretation of the said theory which satisfy the formulae of the theory. Now we have two kinds of definition that we can give in a theory: an explicit definition and an axiomatic definition. By explicit definition I mean a definition as I explained above. An axiomatic definition is a way of saying that an object (of my metatheory) is of a given kind if its a model of a certain theory (theory described in my metatheory). So you can see any axiomatic definition as explicit definition in the metatheory and many times you can reverse this construction turning an explicit definition into an axiomatic one: for instance in the case of vector space you can both define a vector space as a set which have certain operations satisfying some axioms, but you can also shortly say that a vector space is a model of the theory of the said axioms, these things being the same thing.

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Algebraic $\overline{\mathbb{Q}}$-Groups as Abstract Groups

Olivier Frécon Laboratoire de Mathématiques et Applications, Université de Poitiers, Poitiers, France
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AMS Member Price: $70.20 Click above image for expanded view Algebraic$\overline{\mathbb{Q}}$-Groups as Abstract Groups Olivier Frécon Laboratoire de Mathématiques et Applications, Université de Poitiers, Poitiers, France Available Formats:  Softcover ISBN: 978-1-4704-2923-2 Product Code: MEMO/255/1219  List Price:$78.00 MAA Member Price: $70.20 AMS Member Price:$46.80
 Electronic ISBN: 978-1-4704-4815-8 Product Code: MEMO/255/1219.E
 List Price: $78.00 MAA Member Price:$70.20 AMS Member Price: $46.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$117.00 MAA Member Price: $105.30 AMS Member Price:$70.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 2552018; 99 pp
MSC: Primary 20; Secondary 03; 14;

The author analyzes the abstract structure of algebraic groups over an algebraically closed field $K$.

For $K$ of characteristic zero and $G$ a given connected affine algebraic $\overline{\mathbb Q}$-group, the main theorem describes all the affine algebraic $\overline{\mathbb Q}$-groups $H$ such that the groups $H(K)$ and $G(K)$ are isomorphic as abstract groups. In the same time, it is shown that for any two connected algebraic $\overline{\mathbb Q}$-groups $G$ and $H$, the elementary equivalence of the pure groups $G(K)$ and $H(K)$ implies that they are abstractly isomorphic.

In the final section, the author applies his results to characterize the connected algebraic groups, all of whose abstract automorphisms are standard, when $K$ is either $\overline {\mathbb Q}$ or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited.

• Chapters
• 1. Introduction
• 2. Background material
• 3. Expanded pure groups
• 4. Unipotent groups over $\bar {\mathbb {Q}}$ and definable linearity
• 5. Definably affine groups
• 6. Tori in expanded pure groups
• 7. The definably linear quotients of an $ACF$-group
• 8. The group $D_G$ and the Main Theorem for $K=\bar {\mathbb {Q}}$
• 9. The Main Theorem for $K\neq \bar {\mathbb {Q}}$
• 10. Bi-interpretability and standard isomorphisms
• Acknowledgements
• Index of notations

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Volume: 2552018; 99 pp
MSC: Primary 20; Secondary 03; 14;

The author analyzes the abstract structure of algebraic groups over an algebraically closed field $K$.

For $K$ of characteristic zero and $G$ a given connected affine algebraic $\overline{\mathbb Q}$-group, the main theorem describes all the affine algebraic $\overline{\mathbb Q}$-groups $H$ such that the groups $H(K)$ and $G(K)$ are isomorphic as abstract groups. In the same time, it is shown that for any two connected algebraic $\overline{\mathbb Q}$-groups $G$ and $H$, the elementary equivalence of the pure groups $G(K)$ and $H(K)$ implies that they are abstractly isomorphic.

In the final section, the author applies his results to characterize the connected algebraic groups, all of whose abstract automorphisms are standard, when $K$ is either $\overline {\mathbb Q}$ or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited.

• Chapters
• 1. Introduction
• 2. Background material
• 3. Expanded pure groups
• 4. Unipotent groups over $\bar {\mathbb {Q}}$ and definable linearity
• 5. Definably affine groups
• 6. Tori in expanded pure groups
• 7. The definably linear quotients of an $ACF$-group
• 8. The group $D_G$ and the Main Theorem for $K=\bar {\mathbb {Q}}$
• 9. The Main Theorem for $K\neq \bar {\mathbb {Q}}$
• 10. Bi-interpretability and standard isomorphisms
• Acknowledgements
• Index of notations
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## Elementary Algebra

$3x(x+1)(x-1)$
Since $3x$ is common to all the terms in the expression, we obtain: $3x^{3}-3x$ =$3x(x^{2}-1)$ Now, we use the relation $a^{2}-b^{2}=(a+b)(a-b)$ to simplify: $3x(x^{2}-1)$ =$3x(x^{2}-1^{2})$ =$3x(x+1)(x-1)$


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# Not every metric comes from a norm

1. Nov 14, 2009

### littleHilbert

Hello!

It is said that not every metric comes from a norm.

Consider for example a metric defined on all sequences of real numbers with the metric:

$$d(x,y):=\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|}$$

I can't grasp how can that be.

There is a proof, so could you please give me hints so that I can try and do the proof on my own.

At the moment I can only see that the whole thing is bounded by 1. What does it actually mean when we say "it does not come from a norm"?

2. Nov 14, 2009

### Office_Shredder

Staff Emeritus
3. Nov 14, 2009

### littleHilbert

4. Nov 14, 2009

### rasmhop

Re: Metric

On the site linked by Office_Shredder:
Thus for a metric not to come from a norm means that there exists no norm that induces that metric. Thus you must show that d is a metric, but that if p(v) is any norm, then $p(x-y) \not= d(x,y)$ for some pair x,y.

5. Nov 16, 2009

### littleHilbert

Re: Metric

this was easier than I thought. In fact take x=(1,1,...), then 2||x|| is not equal to ||2x||!

First I tried to see whether the triangle inequality is not satisfied, but it did not to work, because both the norm and the metric seem somehow to be "conform" in this respect.


text:
×

# The Robbery

There was a robbery in which a lot of goods were stolen. The robber(s) left in a truck. It is known that : (1) Nobody else could have been involved other than A, B and C. (2) C never commits a crime without A's participation. (3) B does not know how to drive. So, is A innocent or guilty?

Note by Vishwathiga Jayasankar
3 years ago

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since " b " doesn't know how to drive so it can't be the robber which means it's either " a " or " c " ... now " c " does not commit a crime without " a " .. which means that if " c " is guilty then " a " is guilty as well .. and if " c " is not .. that leaves " a " only the answer is " a "

- 2 years, 9 months ago

great

- 2 years, 9 months ago


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# Simulate Infinity mirror effect

I'm trying to model an infinity mirror but can't seem to get the reflections working correctly. I have a simple node setup of a glossy and transparent into a mixed node and the geometry node connected using the backfacing node to the FAC of the mix shader. I only have one "layer" of reflection ... how can I get the mirroring effect of infinity.


text:
Opened 10 years ago

# Slave's ShellCommand and msys are broken

Reported by: Owned by: even even major 0.9.+ 0.7.10 windows fqueze

### Description

While using buildbot in msys, new bugs appeared somewhere between 0.7.9 and 0.7.10p1.

Those are related to the changes in the ShellCommand? object even if, right now, I can't tell what exactly caused this. What I know is, when I call pwd using buildbot, it used to return a path in msys path format (full UNIX like, so it was looking alike /c/some/folder) when it returns a path in windows format now (c:path ofolder). Using some tricks with environment variables before launching buildbot I achieved to get UNIX compatible path using the windows format (like c:\path\to\folder) but it stills cause problems on some executables like make that completely fails.

Here is what the error looks like :

make -f client.mk build
in dir c:uildbotxp-compilwin32uild (timeout 1200 secs)
watching logfiles {}
argv: ['make', '-f', 'client.mk', 'build']
environment:
!EXITCODE=00000001
ALLUSERSPROFILE=C:Documents and SettingsAll Users
COMMONPROGRAMFILES=C:Program FilesFichiers communs
COMPUTERNAME=XP-32-COMPIL
COMSPEC=C:WINDOWSsystem32cmd.exe
CVS_RSH=ssh
DEVENVDIR=C:Program FilesMicrosoft Visual Studio 8Common7IDE
EDITOR=xemacs.exe
FP_NO_HOST_CHECK=NO
FRAMEWORKDIR=C:WINDOWSMicrosoft.NETFramework
FRAMEWORKSDKDIR=C:Program FilesMicrosoft Visual Studio 8SDKv2.0
FRAMEWORKVERSION=v2.0.50727
HISTFILE=C:/mozilla-build/msys/home/buildbot/.bash_history
HOME=c:/Documents and Settings/buildbot
HOMEDRIVE=C:
HOMEPATH=
HOSTNAME=XP-32-COMPIL
HOSTTYPE=i686
INCLUDE=C:Program FilesMicrosoft SDKsWindowsv6.0\include;C:Program FilesMicrosoft SDKsWindowsv6.0\includeatl;C:Program FilesMicrosoft Visual Studio 8VCATLMFCINCLUDE;C:Program FilesMicrosoft Visual Studio 8VCINCLUDE;C:Program FilesMicrosoft Visual Studio 8VCPlatformSDKinclude;C:Program FilesMicrosoft Visual Studio 8SDKv2.0include;
INPUTRC=C:/mozilla-build/msys/etc/inputrc
LIB=C:Program FilesMicrosoft SDKsWindowsv6.0\lib;C:Program FilesMicrosoft Visual Studio 8VCATLMFCLIB;C:Program FilesMicrosoft Visual Studio 8VCLIB;C:Program FilesMicrosoft Visual Studio 8VCPlatformSDKlib;C:Program FilesMicrosoft Visual Studio 8SDKv2.0lib;;C:mozilla-buildatlthunk_compat
LIBPATH=C:WINDOWSMicrosoft.NETFrameworkv2.0.50727;C:Program FilesMicrosoft Visual Studio 8VCATLMFCLIB
LOGNAME=buildbot
LOGONSERVER=\XP-32-COMPIL
MACHTYPE=i686-pc-msys
MAKEFLAGS= --unix
MAKE_MODE=unix
MFLAGS=- --unix
MOZBUILDDIR=C:mozilla-build
MOZCONFIG=c:uildbotxp-compilwin32uild/tools/buildbot-configs/win32/mozconfig
MOZILLABUILD=C:mozilla-build
MOZ_MSVCVERSION=8
MOZ_TOOLS=C:mozilla-buildmoztools
MSVC6KEY=HKLMSOFTWAREMicrosoftVisualStudio6.0SetupMicrosoft Visual C++
MSVC71KEY=HKLMSOFTWAREMicrosoftVisualStudio7.1SetupVC
MSVC8EXPRESSKEY=HKLMSOFTWAREMicrosoftVCExpress8.0SetupVC
MSVC8KEY=HKLMSOFTWAREMicrosoftVisualStudio8.0SetupVC
MSVC9EXPRESSKEY=HKLMSOFTWAREMicrosoftVCExpress9.0SetupVC
MSVC9KEY=HKLMSOFTWAREMicrosoftVisualStudio9.0SetupVC
MSVCROOTKEY=HKLMSOFTWAREMicrosoftVisualStudio
MSYSTEM=MINGW32
NUMBER_OF_PROCESSORS=1
OLDPWD=c:/mozilla-build/python25/lib/site-packages/win32
OLD_PATH=C:Program FilesWindows Resource KitsTools;C:WINDOWSsystem32;C:WINDOWS;C:WINDOWSSystem32Wbem;c:Program FilesMicrosoft SQL Server90Toolsinn;C:mozilla-buildPython25;C:mozilla-buildPython25Scripts
OS=Windows_NT
OSTYPE=msys
PATH=C:mozilla-buildmsyslocalin;c:mozilla-buildwget;c:mozilla-build7zip;c:mozilla-buildlat261full;c:mozilla-buildpython25;c:mozilla-buildsvn-win32-1.4.2in;c:mozilla-buildupx203w;c:mozilla-buildxemacsXEmacs-21.4.19i586-pc-win32;c:mozilla-buildinfo-zip;c:mozilla-build
sis-2.22;c:mozilla-build
sis-2.33u;c:mozilla-buildhg;c:mozilla-buildpython25Scripts;.;C:mozilla-buildmsyslocalin;C:mozilla-buildmsysmingwin;C:mozilla-buildmsysin;c:Program FilesMicrosoft SDKsWindowsv6.0in;c:Program FilesMicrosoft Visual Studio 8Common7IDE;c:Program FilesMicrosoft Visual Studio 8VCBIN;c:Program FilesMicrosoft Visual Studio 8Common7Tools;c:Program FilesMicrosoft Visual Studio 8Common7Toolsin;c:Program FilesMicrosoft Visual Studio 8VCPlatformSDKin;c:Program FilesMicrosoft Visual Studio 8SDKv2.0in;c:WINDOWSMicrosoft.NETFrameworkv2.0.50727;c:Program FilesMicrosoft Visual Studio 8VCVCPackages;c:WINDOWSSystem32;c:WINDOWS;c:WINDOWSSystem32Wbem;c:mozilla-buildmoztoolsin
PATHEXT=.COM;.EXE;.BAT;.CMD;.VBS;.VBE;.JS;.JSE;.WSF;.WSH;.PY
PROCESSOR_ARCHITECTURE=x86
PROCESSOR_IDENTIFIER=x86 Family 6 Model 23 Stepping 7, GenuineIntel
PROCESSOR_LEVEL=6
PROCESSOR_REVISION=1707
PROGRAMFILES=C:Program Files
PROMPT=$P$G
PS1=[33]0;$MSYSTEM:w07 33[32m]u@h [33[33mw33[0m]$
PWD=c:uildbotxp-compilwin32uild
PYTHON=python
SDK2003SP1KEY=HKLMSOFTWAREMicrosoftMicrosoftSDKInstalledSDKs8F9E5EF3-A9A5-491B-A889-C58EFFECE8B3
SDK2003SP2KEY=HKLMSOFTWAREMicrosoftMicrosoftSDKInstalledSDKsD2FF9F89-8AA2-4373-8A31-C838BF4DBBE1
SDK6KEY=HKLMSOFTWAREMicrosoftMicrosoft SDKsWindowsv6.0WinSDKBuild
SDKDIR=C:Program FilesMicrosoft SDKsWindowsv6.0
SDKMINORVER=0
SDKROOTKEY=HKLMSOFTWAREMicrosoftMicrosoftSDKInstalledSDKs
SDKVER=6
SHELL=C:/mozilla-build/msys/bin/sh
SHLVL=1
SSH_AGENT_PID=3232
SSH_AUTH_SOCK=C:/DOCUME~1/buildbot/LOCALS~1/Temp/ssh-vHocOE3172/agent.3172
SYSTEMDRIVE=C:
SYSTEMROOT=C:WINDOWS
TEMP=C:/DOCUME~1/buildbot/LOCALS~1/Temp
TERM=cygwin
TMP=C:/DOCUME~1/buildbot/LOCALS~1/Temp
USERDOMAIN=XP-32-COMPIL
USERPROFILE=C:Documents and Settingsuildbot
USESDK=1
VC8DIR=C:Program FilesMicrosoft Visual Studio 8VC
VCINSTALLDIR=C:Program FilesMicrosoft Visual Studio 8VC
VS80COMNTOOLS=C:Program FilesMicrosoft Visual Studio 8Common7Tools
VSINSTALLDIR=C:Program FilesMicrosoft Visual Studio 8
WINDIR=C:WINDOWS
_=c:/mozilla-build/python25/python
closing stdin
using PTY: False
/bin/sh: c:buildbotxp-compilwin32build/client.mk: No such file or directory
client.mk:118: *** This source tree appears to have Windows-style line endings. To convert it to Unix-style line endings, run "python mozilla/build/win32/mozilla-dos2unix.py".  Stop.
program finished with exit code 2
elapsedTime=0.563000

While looking at the main differences with the previous version I used (0.7.8) that was working, the call to cmd.exe that previously preceded each calls disappeared. That might be the reason, but also it might be something else.

What I know for sure is that I noticed the version of ShellCommand? I can find on this trac is not mine and if I replace mine with it (just adding a usePTY argument to the constructor to get it working), everything works again.

I intend to look further on this bug but don't have time right now. If someone happens to find the exact reason of this and to produce a patch quickly (meaning before I find more time to look about that), I would really appreciate it.

### comment:2 Changed 10 years ago by even

• Milestone changed from undecided to 0.7.11
• Owner set to even
• Status changed from new to assigned

I just added a patch correcting the bug on msys. It seems that changing the PWD variable is making everything bug on msys. It's better not to do it in these conditions.

If you ABSOLUTELY want to set PWD, I might try another patch that convert the path in msys format before setting it.

Since my patch does not affect other OS and seems to work, I add the review flag and set the milestone to next version.

Correct the bug.

### comment:4 Changed 10 years ago by blackett

I had the same problem with buildslaves in Windows Server 2008 that use cygwin

It seems that once the PWD was set with a windows like path, cygwin maintained this form, whereas I had scripts that ran assuming the cygwin form, which is what had been defined up to that point.

My workaround was to unset PWD in the bat file that starts my cygwin processes for buildbot:

  SET PWD=

### comment:5 Changed 10 years ago by dustin

• Resolution set to fixed
• Status changed from assigned to closed

### comment:6 Changed 9 years ago by hushp1pt

• Resolution fixed deleted
• Status changed from closed to reopened

I had the same problem with Cygwin-1.7 on a Windows 2003 server.

However, I too had the same pwd problem (Windows path when cygwin-style path was expected). I also tried to unset PWD, but I never got that to change anything.

I eventually commented out the

self.environPWD? = os.path.abspath(self.workdir)

statement in runprocess.py, and everything was great.

Can you add a test for cygwin as well as msys to the patch that fixed this bug for msys?

### comment:7 follow-up: ↓ 8 Changed 9 years ago by dustin

• Keywords review sprint removed

how can I identify a cygwin system in Python?

### comment:8 in reply to: ↑ 7 Changed 9 years ago by hushp1pt

how can I identify a cygwin system in Python?

I was wondering, in the earlier patch here, how did MACHTYPE know it was an MSYS system?

self.environ.get('MACHTYPE', None) == 'i686-pc-msys'

Do you need to start the buildslave from Msys for MACHTIME to say "msys"? If you started the buildslave from Cygwin, would MACHTYPE say "cygwin"?

In fact, I do not start our Windows buildslave from Cygwin. I used the run-buildbot-as-service contrib. But I still want the BuildCommand? commands to run under Cygwin.

If you don't have MACHTYPE that says Cygwin, there probably is no foolproof way to tell if a ShellCommand? buildstep wanted to run Cygwin. However, I bet if the first word of the first command in the build step was "bash", "bash.exe", or a full path to bash, and buildslave is Windows, then the odds would be pretty good that the command(s) were really meant for Cygwin bash- or for Msys bash. Either way, the patch would help.

Those who prefer to have their ShellCommand? start a BAT file to run Cygwin commands, I guess they can still set PWD= before they call bash or whatever.

### comment:9 follow-up: ↓ 10 Changed 9 years ago by dustin

• Milestone changed from 0.7.11 to 0.8.3

I'm worried about playing whack-a-mole here, or getting too specific in analyzing the components of a buildstep. Could we add a ShellCommand? option that would be passed to the slave to enable this particular behavior?

### comment:10 in reply to: ↑ 9 Changed 9 years ago by hushp1pt

I'm worried about playing whack-a-mole here, or getting too specific in analyzing the components of a buildstep. Could we add a ShellCommand? option that would be passed to the slave to enable this particular behavior?

Thanks, I think that would be ideal. You could call the option with_cygwin_path. If value == None, no change- same as regular Windows process. If != None, skip the PWD reset in runprocess.py.

For extra credit, if value looks like a Cygwin path string, then override the PATH that bash (or whatever) sees when it lands. In the example http://buildbot.net/trac/wiki/CygwinShellCommands, I passed Cygwin's PATH through Shell Command's envPATH?. Implementing some kind of short cut for this override would be helpful, because envPATH? was the only value in env that had to be different between Windows and Linux builds and I had serious python and/or buildbot language troubles in my build factory, trying to define that env dictionary the right way at the right time. Your call. Thanks again.

### comment:11 follow-up: ↓ 12 Changed 9 years ago by ayust

• Milestone changed from 0.8.3 to 0.8.+

### comment:12 in reply to: ↑ 11 Changed 7 years ago by hushp1pt

For extra credit, ...

Hi, please forget all about that "for extra credit" idea- my mistake.

Please reconsider the simple bug fix dustin proposed,

Could we add a ShellCommand option that would be passed to the slave to enable this particular behavior?

I suggest the new ShellCommand option be named with_cygwin_path. If with_cygwin_path == None, no change- same as regular Windows process. If != None, skip the PWD reset in runprocess.py.

Thanks.

### comment:13 Changed 7 years ago by tom.prince

I wonder if it actually makes sense to be setting PWD anywhere? If the command being run is a shell script, then the shell should be setting PWD. Otherwise, I'd guess that things should be using getcwd(3) to find the current directory, not a environment variable. Perhaps we should just unset it?

### comment:14 Changed 5 years ago by dustin

• Milestone changed from 0.8.+ to 0.9.+

Ticket retargeted after milestone closed

Note: See TracTickets for help on using tickets.


text:
### Archive

Archive for August, 2007

## eclipse (a transit by any other name)

Image Source: APOD.

Last night, the alarm went off at 2:45 AM, just prior to the start of the full lunar eclipse. Remarkably, the fog had stayed away. The air was slightly warm, and the town was absolutely quiet. The shadow of the Earth was covering nearly the entire lunar surface, with just a small oblique portion of the lower right hemisphere still in sunlight. A few minutes later, the whole moon was glowing a dull orange-red against the easily visible stars of the ecliptic. It was creepy, weird. Definitely worth getting out of bed for.

The Sun and the Moon occupy nearly the same angular size in Earth’s sky. This means that to good approximation, the patch of sky covered by the moon during a central lunar eclipse contains stars that can see the Earth in transit across the face of the Sun.

And during a lunar eclipse, they see a double transit.

The famous “tooth” in the HST light curve for TrES-1 is generally attributed to the planet passing over starspots, but for those who prefer not to shave with Occam’s razor, it can be equally well modeled by a double transit.

Last night, during the eclipse, the Moon (at RA 22h 26 min, Dec= -09 deg 57 min) was only a few degrees away from the planet bearing stars GJ 876, GJ 849, HD 21707, and HD 219449.

Categories: non-technical Tags:

## Æ’r3$h R4Ã14Â£ V3Â£0(1713$

Image Source.

Eugenio has finished combing through this summer’s literature, and has added twenty newly published radial velocity data sets to both the systemic backend and to the current version of the downloadable systemic console. As a result of his efforts, new or augmented data is now available for the following stars: Cha Ha 8, GJ 317, HD3651, HD5319, HD11506, HD17156, HD37605, HD43691, HD75898, HD80606, HD89744, HD125612, HD132406, HD170469, HD171028, HD231701, NGC2423, NGC4349, HAT-P-3, and TrES-4. As always, the published literature citations for the velocities are contained in the “vels_list.txt” file that comes bundled with the systemic console download. The vels_list.txt file can be indispensible if you want to publish results that use the systemic package as a research tool — indeed, we’re quite excited that researchers are starting to adopt the console in the course of carrying out state-of-the-art research (see, e.g. here.)

There’s quite a bit to explore with these new data sets. Eugenio has had a first look, and included in his recommendations are:

GJ 317: This system (discovered by John Johnson and the California-Carnegie planet search team, preprint here) is only the third red dwarf that’s been found to harbor a Jovian-mass companion. The data shows clear evidence for one planet “b”, with at least 1.2 Jupiter masses and a 693-day orbit, and there’s a strong hint of a second planet in the radial velocity variations. Check it out with the console!

HD 17156: This data comes from a recent paper by the California-Carnegie team. There are radial velocities from both the Keck and the Subaru telescopes, and the signal-to-noise of the orbit is very high.

The data show a ~3 Jupiter-mass planet on a 21.2 day orbit. The orbit is remarkably eccentric for a planet on such a short period, leading to a 25-fold variation in the amount of light received during each trip around the star.

It’ll be interesting to get a weather forecast for this world, and it’s also important to point out that the orientation of the orbit is very well suited for the possibility of observing transits. Periastron is reasonably close to being aligned with the line of sight to Earth, leading to an a-priori transit probability of more than 10%. In the discovery paper, a preliminary transit search is reported, but only about 1/4th of the transit window was ruled out. With a Dec of +71 degrees and a nice situation in the winter sky, this is definitely one for Transitesearch.org’s Finland contingent.

Categories: Tags:

## Countdown

August 20th, 2007 1 comment

Image Source.

August 1st marked the most recent ‘606 day, which came and went without wide remark. Perhaps this was because in late Summer, HD 80606 rises and sets in near-synch the Sun, and is thus lost from the Earth’s night skies.

At the moment, HD 80606b is headed back out toward apastron.

The global storms and shockwaves that were unleashed at the beginning of August are dissipating rapidly, and the flux of heat from the planet is likely fading back down to the sullen baseline glow that arises from tidal heating.

HD 80606’s next periastron passage occurs on November 20th, and the Spitzer Space Telescope is scheduled to observe the whole event (details here). It’s going to be a big deal. Spitzer can only observe HD 80606 during two three-week windows each year, and fortunately, the Nov. 20th Periastron passage occurs during one of these windows. It’s literally the only opportunity to catch HD 80606 b’s big swing before Spitzer’s cryogen runs out in 2009.

The orbital geometry of the periastron passage looks like this:

Each marker of the orbit is separated by one hour. The prediction for the pseudo-synchronous rotation of the planet is also indicated. The planet should be spinning with a period of 36.8 hours. Jonathan Langton’s hydrodynamics code predicts what the temperature distribution on the planet should look like at each moment from Spitzer’s viewpoint in our solar system:

Transitsearch.org observers have covered a number of the HD 80606 b transit opportunities, and it seems pretty certain that the planet doesn’t transit. This isn’t surprising. The geometry of the orbit is such that when the planet crosses the plane containing the line of sight to the Earth, it’s quite a distance away from the star. Not so, however, for the secondary transit. There’s a very respectable 15% chance that Spitzer will detect a secondary transit centered two hours prior to the periastron passage.

Even if the planet doesn’t transit, we should be able to get a good sense of the orbital inclination from the shape of the light curve. If the orbit is nearly in the plane of the sky, then we should see a steady rise followed by a plateau in the 8-micron flux coming from the planet. For more nearly edge-on configurations, the flux peak should be clearly discernable. The observations are scheduled to start 20 hours prior to periastron and end 10 hours after.

Categories: Tags:

## Vorticity

Vorticity can be thought of as the tendency of a paddlewheel to spin if placed in the flow. High vorticity is a large counter-clockwise spin, zero vorticity is no spin, and a large negative vorticity is a tendency to spin clockwise. The climate models of short-period extrasolar planets that Jonathan Langton and I have developed show a remarkable variety of vorticity patterns on their surfaces, in keeping with the incredibly stormy and complex nature of their atmospheres. Here’s a gallery of Mercator-projection vorticity maps for the known strongly irradiated Jovian planets that have significant eccentricities. The red arrows indicate the wind speeds and directions across the planetary surfaces. These figures are all from a paper that’s currently under review at the Astrophysical Journal (see here for an overview of the numerical method that we’re using). Also, a shout-out is due to Edward Tufte for advocating the strong graphic-design effect of small spots of saturated color on a gray-scaled backdrop.

HAT-P-2b
:

Here are 1.1 MB North Pole, South Pole and Mercator Projection animations of the HAT-P2b vorticity evolution.

HD 80606 b
:

1.1 MB Mercator animation here.

HD 185269 b:

1.1 MB Mercator animation here.

HD 108147 b

1.1 MB Mercator animation here.

HD 118203 b
:

1.1 MB Mercator animation here. The animations above are hosted on the Oklo Corporation’s servers.

It’s interesting to compare the vorticity maps with the temperature distributions on the planetary surfaces (shown in the same order as above):

Categories: worlds Tags:

## Gigantic

Image Source.

The TrES survey announced the discovery of a new transiting planet today, raising the number of known transits to twenty (including Mercury and Venus). The new planet, “TrES-4″, has a mass of order 84% that of Jupiter, and with a radius of 1.67 Rjup, it’s pumped to nearly five times Jupiter’s volume:

The false color image of Jupiter was produced from near-infrared data obtained with the Gemini telescope. The even more luridly false-color representation of TrES-4 is based on a vorticity map from one of Jonathan Langton’s recent simulations.

In order for TrES-4 to be swollen to its current size, it needs to be experiencing heating of order 6×10^27 ergs per second. One way to do this is to have a significant perturbing companion which drives large time-averaged variations in TrES-4’s orbital eccentricity. So far, there are only four published radial velocities for TrES-4, so the orbit could easily be non-circular. More provocatively, if strong orbital forcing is indeed occurring, then there’s a reasonable chance that the perturber might also be observable in transit. I recommend that Transitsearch.org observers keep this bad boy under constant supervision.

Categories: worlds Tags:

## Whorls

Image Source.

HAT-P-2b. The name doesn’t exactly ring of grandeur, but this planet — a product of GÃ¡spÃ¡r Bakos’ HAT Net transit survey — is poised to give the Spitzer Space Telescope its most dramatic glimpse to date of a hot Jupiter.

HAT-P-2b’s orbit is remarkably eccentric for a planet with an orbital period of only 5.6 days, and by a stroke of luck, periastron is located almost exactly midway between the primary and the secondary transits (as viewed from Earth). The strength of the stellar insolation at periastron is nine times as strong as at apastron, which more than guarantees that the planet will have disaster-movie-ready weather.

On June 6th, Josh Winn and his collaborators used the Keck telescope to obtain 97 radial velocities for HAT-P-2. The observations were timed to occur before, during, and after primary transit, and the Rossiter-McLaughlin effect is clearly visible in their data (preprint here):

The symmetry of the Rossitered points indicates that the angular momentum vector of the planetary orbit is aligned with the spin pole of the star:

This state of affairs also holds true for the other transiting planets — HD 209458b, HD 149026b, HD 189733b — for which the effect has been measured. The observed alignments are evidence in favor of disk migration as the mechanism for producing hot Jupiters.

With its apparent magnitude of V=8.7, the HAT-P-2b parent star is roughly ten times brighter than the average planet-bearing star discovered in a wide-field transit survey. The star is bright enough, in fact, to have earned an entry in both the Henry Draper Catalog (HD 147506) and the Hipparcos Database (HIP 80076), but with its surface temperature of 6300K (F8 spectral type) it was too hot to have been a sure-fire “add” to the ongoing radial velocity surveys. Prior to this May, it had been entirely ignored in the astronomical literature (save a brief mention in this paper from 1969).

HAT-P-2’s intrisic brightness and its planet’s orbital geometry mean that in a relatively compact 34-hour observation, Spitzer can collect on the most interesting features of the orbit with high signal-to-noise. In particular, there is an excellent opportunity to measure the rate at which the day-side atmosphere heats up during the close approach to the star. The planet, in fact, presents such a remarkable situation that a block of Director’s Discretionary time was awarded so that the observations can be made during the current GO-4 cycle. They’ll be occurring soon.

Both HAT-P-2b and HD 80606 b will provide a crucial ground truth for extrasolar planetary climate simulations. Jonathan Langton’s current model, for example, predicts that that the temperatures on HAT-P-2b will range over more than 1000K. At the four times shown in the above orbital diagram, the hemisphere facing Earth is predicted to show the following appearances:

Spitzer, of course, can’t resolve the planetary disk. It measures the total amount of light coming from the planet in chosen passband. At 8-microns, the planet’s light curve should look like this:

The temperature maps only hint at the complex dynamics of the surface flow. A better indication is given by the distribution of vorticity,

which we’ll pick up in the next post…

Categories: Tags:


text:
Why is the Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure:

Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued $\Sigma-$measurable functions, i.e. for every $k\ge 1$ and every $x$ in $X$, $$0 \le f_k(x) \le f_{k+1}(x).$$ Next, set the pointwise limit of the sequence ${f_n}$ to be $f$. That is, for every $x$ in $X$, $$f(x) = \lim_{k\to \infty}f_k(x).$$ Then $f$ is $\Sigma-$measurable and $$\lim_{k\to \infty}\int f_k d\mu = \int f d\mu.$$

I've noticed that when it comes to monotone convergence theorem (either Lebesgue or general measure), usually its definition restricts the monotone function sequences to be nonnegative. I'm not sure why the 'nonnegative' is necessary.

• Think of $-|x|/n$ on $\mathbb R$. The sequence converges to 0 but the integral doesn't even though it is increasing. – Matt Samuel Feb 9 '16 at 4:46
• @MattSamuel: You are right. so negative function can monotonely converge to zero while its integral may not follow? – Bear and bunny Feb 9 '16 at 4:54
• it's just like a huge positive function converging to 0. It is symmetric, you just have to flip your perspective. – Matt Samuel Feb 9 '16 at 14:53
• @MattSamuel I cannot see how your example work, the integral is $- \frac{|x|x}{n}$ which converge also to $0$! Could you explain to me, please? – The_lost Jul 6 '18 at 11:04
• @the The integral doesn't converge to 0 because it is always $-\infty$. – Matt Samuel Jul 6 '18 at 11:36

Well, if $f_k$ could be negative, then its integral might not even be defined. For instance, if $X=\mathbb{R}$ with Lebesgue measure and $f_k(x)=x$ for some $k$, there is no good way to define $\int f_k$ (it should morally be "$\infty-\infty$"). On the other hand, the integral of a nonnegative measurable function can always be defined (though it might be $\infty$).
Even if you require $\int f_k$ to be defined for all $k$, if $\int f_k$ is allowed to be $-\infty$, the result can be false. For instance, let $X=\mathbb{N}$ with counting measure and let $f_k(n)=-1$ if $n>k$ and $0$ if $n\leq k$. Then the $f_k$ are monotone increasing and converge pointwise to the constant function $0$, but $\int f_k=-\infty$ for all $k$.
On the other hand, if you require $\int f_k$ to be defined and $>-\infty$ for all $k$, the result is true. Indeed, you can just replace each $f_k$ by $f_k-f_1$ and use the usual version of the theorem, since all these functions are nonnegative (and the equation $\int f_k=\int f_1+\int (f_k-f_1)$ is guaranteed to make sense and be true since $\int f_1>-\infty$).
• @Eric Wofsey In the last case, if you require $\int f_{k}$ to be defined (not necessarily integrable) and $>-\infty$ for all $k$. Why does make sense consider the $\int f_{k} = \int f_{1} + \int (f_{k} -f_{1})$? since $f_{1} - f_{k} \geq 0$ we can have that $\int f_{1} - f_{k} = +\infty$. If we have too that $\int f_{1} = +\infty$. so, $\int f_{k} = +\infty -\infty$. – Alladin Sep 7 '16 at 13:52
• @Alladin: $f_k-f_1$ is nonnegative, not nonpositive. – Eric Wofsey Sep 7 '16 at 18:55
If they could be negative, then the statement would also have to be true for any sequence $f_n$ where $f_n(x)\geq f_{n+1}(x)$ by just reflecting across $0$. But then consider when $f_n:[0,\infty]\rightarrow R, f_n(x)=\frac{x}{n}$.
In this case, for each $x$, $\lim_n f_n(x)=0$, and thus $f_n \rightarrow f=0$ pointwise, but the integrals are infinite for each finite $n$.


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## Before You Watch

This video follows directly on from Linear Equations. It shows how to take two linear equations and solve them together to get a single value for the two variables (which are usually x and y, but could be any two letters). When we think about this process graphically, we are finding the point where the two lines intercept. This is called solving simultaneous equations. Keep in mind that in Australia this topic is known as simultaneous equations, but in other countries it is often referred to as “solving a system of equations”.

If you haven't watched Linear Equations yet, take a look at that video first, then come back.

...

## Now What?

This video showed the simplest way of solving two linear equations simultaneously. In general, any two equations with the same two unknowns can be solved simultaneously. Bear in mind, however, that depending on the equations, that’s sometimes not possible. Let's think about it graphically: each equation is a line, and if the two lines cross, you can solve them. If they don’t cross, you won’t be able to find a real solution.

Moving beyond two equations, what if there are three, or four equations? If you have more equations, you can find more unknowns.

Once you are comfortable with solving two equations simultaneously, consider giving three equations a go at:

You should also consider having a look at how to solve other equations, such as Quadratic Equations or equations with Exponentials.

## But When I am Going to Use This

Linear equations are very commonly used in everyday life to model situations. So when two things are both modelled by a linear equation, we can solve them together (simultaneously) to find out important information.

A common example of this is in finance and the analysis of revenue and costs. The revenue for a company from a product (say R) is simply the price of the product (let's say $30) multiplied by how many they sell (n), so R = 30n This relationship can be represented by a straight line. Similarly, the cost of that product (C) is a fixed starting amount (the development cost, staff wages and so on) plus whatever the cost of making the product is multiplied by how many they sell. Let's say there is a fixed monthly cost of$1500 and the product costs \$10 to make. That means the costs per month are:

C = 1500 + 10n

This can also be represented as a straight line. So when revenue is equal to costs (R = C), the company is breaking even. By solving these two equations simultaneously:

C = 30n

C = 1500 + 10n

we can determine how many products, n, must be sold per month for the company to break even. See if you can work it out, the answer is n = 75. A graph of the functions is given below.

This is just a single example. Some other examples can be found at:


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# Checking Solutions to Equations

## Substitute values for variables to check equation solutions

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Checking Solutions to Equations

### Checking Solutions: Equations

You will often need to check solutions to equations in order to check your work. In a math class, verifying that you arrived at the correct solution is very good practice. We check a solution to an equation by replacing the variable in the equation with the value of the solution. A solution should result in a true statement when simplified.

#### Checking Solutions to Equations

1. Check that the given number is a solution to the equation: \begin{align*}y = -1; \ 3y + 5 = -2y\end{align*}

Replace the variable \begin{align*}y\end{align*} in the equation with the given value, -1.

\begin{align} 3(y)+5&=-2(y)\\ 3(-1) + 5 &= -2(-1)\\ -3 + 5 &= 2\\ 2 &= 2\\ \end{align}

2 = 2 is a true statement. This means that \begin{align*}y = -1\end{align*} is a solution to \begin{align*}3y + 5 = -2y\end{align*}.

2. Check that the given number is a solution to the equation: \begin{align*}z =3; \ z^2 + 2z = 8\end{align*}

Replace the variable \begin{align*}z\end{align*} in the equation with the given value, 3.

\begin{align} (z)^2+2(z)&=8\\ (3)^2 + 2(3) &= 8\\ 9 + 6 &= 8\\ 15 &= 8\\ \end{align}

15 = 8 is not a true statement. This means that \begin{align*}z = 3\end{align*} is not a solution to \begin{align*}z^2 + 2z = 8\end{align*} .

Let’s use what we have learned about defining variables, writing equations and writing inequalities to solve some real-world problems.

#### Real-World Applications

Tomatoes cost $0.50 each and avocados cost$2.00 each. Anne buys six more tomatoes than avocados. Her total bill is 8. How many tomatoes and how many avocados did Anne buy? Check your answer! Define Let \begin{align*}a\end{align*} equal the number of avocados Anne buys. Translate Anne buys six more tomatoes than avocados. This means that \begin{align*}a + 6\end{align*} is equal to the number of tomatoes. Tomatoes cost0.50 each and avocados cost $2.00 each. Her total bill is$8. This means that .50 times the number of tomatoes plus 2 times the number of avocados equals 8.

\begin{align} 0.5(a + 6) + 2a &= 8\\ 0.5a + 0.5 \cdot 6 + 2a &= 8\\ 2.5a + 3 &= 8\\ 2.5a &= 5\\ a &= 2\\ \end{align}

Remember that \begin{align*}a\end{align*} represents the number of avocados. Since \begin{align*}a=2\end{align*}, Anne buys two avocados. The number of tomatoes is \begin{align*}(a) + 6 = (2) + 6 = 8\end{align*}.

Anne bought 2 avocados and 8 tomatoes.

Check

If Anne bought two avocados and eight tomatoes, the total cost would be: \begin{align*}(2 \times 2) + (8 \times 0.5) = 4 + 4 = 8\end{align*}. The answer checks out.

### Example

#### Example 1

Check that the given number is a solution to the equation: \begin{align*}x = -\frac{1}{2}; \ 3x + 1 = x\end{align*}

\begin{align} 3(x)+1&=x\\ 3 \left ( - \frac{1}{2} \right ) + 1 &= - \frac{1}{2}\\ \left ( - \frac{3}{2} \right ) + 1 &= - \frac{1}{2}\\ - \frac{1}{2} &= - \frac{1}{2}\\ \end{align}

\begin{align*}-\frac{1}{2}=-\frac{1}{2}\end{align*} is a true statement. This means that \begin{align*}x = - \frac{1}{2}\end{align*} is a solution to \begin{align*}3x + 1 = x\end{align*}.

### Review

For 1-9, check whether the given number is a solution to the corresponding equation.

1. \begin{align*}a = -3; \ 4a + 3 = -9\end{align*}
2. \begin{align*}x = \frac{4}{3}; \ \frac{3}{4}x + \frac{1}{2} = \frac{3}{2}\end{align*}
3. \begin{align*}y = 2; \ 2.5y - 10.0 = -5.0\end{align*}
4. \begin{align*}z = -5; \ 2(5 - 2z) = 20 - 2(z - 1)\end{align*}
5. \begin{align*}a = 10; \ 5a-7=43\end{align*}
6. \begin{align*}x = \frac{2}{3}; 3x+5=7 \end{align*}
7. \begin{align*}x = -9; \ \frac{x}{3}\cdot 10 = -30\end{align*}
8. \begin{align*}z = \frac{1}{2}; \ 2z=1.5-z\end{align*}
9. \begin{align*}z = 0.5; \ z(1 - 2z) = 6 + 6(4z - 3)\end{align*}
10. The cost of a Ford Focus is 27% of the price of a Lexus GS 450h. If the price of the Ford is \$15000, what is the price of the Lexus?

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
inequality An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.
solution A solution to an equation or inequality should result in a true statement when substituted for the variable in the equation or inequality.

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affine regular implies noetherian theorem

Let $X$ be a scheme, and possibly some arrows, satisfying the following conditions:

• $X$ regular, affine
Then
• $X$ noetherian

References

• Stacks project, https://stacks.math.columbia.edu/tag/02IU (Fabio)


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# Tag Info

1

That's fairly small for an object. It wouldn't have any significant gravitational effect on the moon or the earth. Tidal effects go as the cube of the distance. So the sun has about half the tidal effects of the moon. If this object were in low earth orbit (400km altitude), then the relative tidal effects on the surface when it is overhead would be about ...

1

Now perfect balance between the centrifugal force of orbital rotation and sun's gravity is impossible so the earth's orbit should either be slowly decaying inwards or expanding outwards due to difference in magnitude of those opposing forces. This assumption is incorrect. We could make the same argument about a weight suspended from a spring. ...

2

I am certain that mathematical analysis of tidal locking has been done many times but I have failed to find such an analysis where the mechanism for the transfer of angular momentum to spin angular momentum is included which is the question which has been asked. Perhaps someone is able to produce a reference or an analysis? Having experienced on a number ...

3

This is actually a famous theorem known as the Einstein Equivalence Postulate (sometimes Equivalence Principle). It's true that since Earth is spinning, acceleration in a spacecraft isn't quite the same situation we experience daily, but in general, yes, gravity is indistinguishable from uniform acceleration. Specifically, if you are in a box with no windows ...

1

Gravitational fields aren't homogeneous. Here on the Earth, a clock on the floor runs more slowly than a clock on the table, and we have clocks precise enough to measure such small differences due to the gravitational gradient. But doesn't a clock in an accelerating spaceship run at the same rate no matter where in the ship you put it? See page ...

Top 50 recent answers are included


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## anonymous551 2 years ago Estimating slope of a graph without a grid? Slope of (0,0) to (3,6) ? Slope of (0,2) to (6,3) ? Slope of (1,0) to (6,6) ?

1. naynaylala

first number the sides with ruler(centimeters/inches depending on problem)and you plot in coordinates and remember that x horizontal

2. naynaylala

goes first

3. asnaseer

you do not need to graph anything in order to find the slope between two given points. the slope is just the change in y divided by the change in x. e.g. the slope of a line between (x1,y1) and (x2,y2) would be: slope = (y2 - y1)/(x2 - x1)

4. precal

|dw:1351727020793:dw|

5. precal

use the slope formula, as told by asnaseer

6. anonymous551

|dw:1351727856210:dw|

7. satellite73

right 3 units, up six units, slope is $$\frac{6}{3}=2$$

8. precal

satellite is doing rise over run, another way to do slopes.


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# Complex dynamics and quasi-conformal geometry.

23-25 October 2017
Université d'Angers
Europe/Paris timezone

## The Milnor-Thurston determinant and the Ruelle transfer operator.

23 Oct 2017, 14:30
55m
L003 (Université d'Angers)

### L003

#### Université d'Angers

2 Boulevard Lavoisier 49000 Angers

### Speaker

Hans Henrik Rugh (Université de Paris-Sud, Orsay)

### Description

The topological entropy $\htop$ of a continuous piecewise monotone interval map measures the exponential growth in the number of monotonicity intervals for iteratesof the map. Milnor and Thurston showed that $\exp(-\htop)$ is the smallest zero of an analytic function, now coined the Milnor-Thurston determinant, that keeps track of relative positions of forward orbits of critical points. On the other hand $\exp(\htop)$ equals the spectral radius of a Ruelle transfer operator $L$, associated with the map. Iterates of $L$ keep track of inverse orbits of the map. For no obvious reason, a Fredholm determinant for the transfer operator has not only the same leading zero as the M-T determinant but all peripheral (those lying in the unit disk) zeros are the same. In the talk I will show that on a suitable function space, the dual of the Ruelle transfer operator has a regularized determinant, identical to the Milnor-Thurston determinant, hereby providing a natural explanation for the above puzzle. This work was inspired by a collaboration with Tan Lei in 2014.

### Primary author

Hans Henrik Rugh (Université de Paris-Sud, Orsay)

### Presentation Materials

 Transparents


text:
3 + 2 log (base 2) X = log (base 2) y

1. Feb 5, 2005

Gughanath

3 + 2 log (base 2) X = log (base 2) y

2. Feb 5, 2005

arildno

Rewrite this equality as follows:
$$2^{(3+2log_{2}x)}=2^{log_{2}y}$$

Now, use properties of exponents and logarithms to derive the result!

3. Feb 5, 2005

Gughanath

oh...I SEE....thanx...wasnt that hard :s

4. Feb 5, 2005

Gughanath

how did u write the numbers (powers) above the normal number btw???

5. Feb 5, 2005

arildno

Click on the image to see the LATEX code behind it.


text: Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 17 (2007), No. 3, 449--468Copyright Heldermann Verlag 2007 Derivations from the Even Parts into the Odd Parts for Lie Superalgebras W and S Wende Liu Dept. of Mathematics, Harbin Normal University, Harbin 150080, P. R. China wendeliu@hrbnu.edu.cn Baoling Guan (1) Dept. of Mathematics, Harbin Normal University, Harbin 150080, P. R. China (2) Dept. of Mathematics, Qiqihar University, Qiqihar 161006, P. R. China [Abstract-pdf] \def\Z{{\mathbb Z}\,} \def\Der{\mathop{\rm Der\,}\nolimits} Let $\cal W$ and $\cal S$ denote the even parts of the generalized Witt superalgebra $W$ and the special superalgebra $S$ over a field of characteristic $p>3$, respectively. In this note, using the method of reduction on $\Z$-gradations, we determine the derivation space $\Der( {\cal W}, W_{\overline1})$ from $\cal W$ into $W_{\overline1}$ and the derivation space $\Der({\cal S}, W_{\overline1})$ from $\cal S$ into $W_{\overline1}$. In particular, the derivation space $\Der({\cal S}, S_{\overline1})$ is determined. Keywords: Generalized Witt superalgebra, special superalgebra, derivation space, canonical torus. MSC: 17B50, 17B40 [ Fulltext-pdf  (234  KB)] for subscribers only.


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# Trigonometric Proof using Compound Angle Formula

There are many areas to apply the compound angle formulas, and trigonometric proof using the compound angle formula is one of them.  \begin{aligned} \require{color}\sin (x + y) &= \sin x \cos y + \sin y \cos x &\color{green} (1) \\\sin (x – y) &= \sin x \cos y – \sin y \cos x &\color{green} […]