text:
## Väth, Martin

Compute Distance To:
 Author ID: vath.martin Published as: Väth, Martin; Väth, M. External Links: MGP · Wikidata · GND · IdRef
 Documents Indexed: 92 Publications since 1993, including 6 Books Co-Authors: 25 Co-Authors with 45 Joint Publications 420 Co-Co-Authors
all top 5

### Co-Authors

 47 single-authored 14 Appell, Jürgen M. 6 Eisner, Jan 6 Kučera, Milan 4 Kim, In-Sook 3 Andres, Jan 3 Chen, Churjen 3 Giorgieri, Elena 2 Benedetti, Irene 2 D’Aniello, Emma 2 Erzakova, Nina A. 2 Ricker, Werner Joseph 2 Santucci, Paola 2 Tseng, Shiojenn 2 Vignoli, Alfonso 2 Zabreĭko, Pëtr Petrovich 1 Baltaev, Jamol I. 1 Biberdorf, Èlina A. 1 Dirr, Gunter 1 Falcón Santana, Sergio 1 Guanda, Nathaly 1 Gurevich, Pavel L. 1 Kim, Yunho 1 Luxemburg, Wilhelmus Anthonius Josephus 1 Salem, Hussein A. H. 1 Taddei, Valentina
all top 5

### Serials

 13 Zeitschrift für Analysis und ihre Anwendungen 8 Topological Methods in Nonlinear Analysis 6 Journal of Integral Equations and Applications 5 Journal of Mathematical Analysis and Applications 5 Annali di Matematica Pura ed Applicata. Serie Quarta 5 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 4 Journal of Differential Equations 3 Nonlinear Functional Analysis and Applications 3 Fixed Point Theory and Applications 2 Ricerche di Matematica 2 Topology and its Applications 2 Applications of Mathematics 2 Indagationes Mathematicae. New Series 2 Positivity 2 Nonlinear Analysis. Theory, Methods & Applications 1 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 1 Collectanea Mathematica 1 Glasgow Mathematical Journal 1 Journal of Functional Analysis 1 Proceedings of the American Mathematical Society 1 Rendiconti di Matematica e delle sue Applicazioni. Serie VII 1 Progress of Mathematics (Varanasi) 1 Differential and Integral Equations 1 Mathematica Bohemica 1 Irish Mathematical Society Bulletin 1 Journal of Dynamics and Differential Equations 1 Functional Differential Equations 1 Analysis (München) 1 Nonlinear Analysis. Real World Applications 1 Advanced Nonlinear Studies 1 Journal of Function Spaces and Applications 1 Fixed Point Theory 1 Cubo 1 Dynamics of Partial Differential Equations 1 De Gruyter Series in Nonlinear Analysis and Applications 1 Lecture Notes in Mathematics 1 Pure and Applied Mathematics, Marcel Dekker
all top 5

### Fields

 75 Operator theory (47-XX) 26 Functional analysis (46-XX) 24 Partial differential equations (35-XX) 14 Integral equations (45-XX) 12 Measure and integration (28-XX) 12 Ordinary differential equations (34-XX) 12 General topology (54-XX) 7 Algebraic topology (55-XX) 6 Real functions (26-XX) 3 Mathematical logic and foundations (03-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Global analysis, analysis on manifolds (58-XX) 1 Special functions (33-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Optics, electromagnetic theory (78-XX)

### Citations contained in zbMATH Open

76 Publications have been cited 449 times in 229 Documents Cited by Year
Fixed point theorems and fixed point index for countably condensing maps. Zbl 0964.47025
Väth, Martin
1999
Volterra and integral equations of vector functions. Zbl 0940.45002
Väth, Martin
2000
Ideal spaces. Zbl 0896.46018
Väth, Martin
1997
Global bifurcation of the $$p$$-Laplacian and related operators. Zbl 1073.35027
Väth, Martin
2005
Degree and global bifurcation for elliptic equations with multivalued unilateral conditions. Zbl 1111.47055
Eisner, Jan; Kučera, Milan; Väth, Martin
2006
Some remarks on small sets. Zbl 1102.28300
Appell, Jürgen; D’Aniello, Emma; Väth, Martin
2001
Global solution branches and a topological implicit function theorem. Zbl 1150.58006
Väth, Martin
2007
Global bifurcation for a reaction-diffusion system with inclusions. Zbl 1182.35025
Eisner, Jan; Kučera, Milan; Väth, Martin
2009
Nonlinear spectral theory for homogeneous operators. Zbl 1045.47053
Giorgieri, Elena; Appell, Jürgen; Väth, Martin
2002
Integration theory. A second course. Zbl 1043.28001
Väth, Martin
2002
An axiomatic approach to a coincidence index for noncompact function pairs. Zbl 0992.47028
Väth, Martin
2000
An abstract Gronwall lemma and applications to global existence results for functional differential and integral equations of fractional order. Zbl 1080.45006
Salem, Hussein A. H.; Väth, Martin
2004
On some Banach space constants arising in nonlinear fixed point and eigenvalue theory. Zbl 1074.47025
Appell, Jürgen; Erzakova, Nina A.; Santana, Sergio Falcon; Väth, Martin
2004
Coincidence points of function pairs based on compactness properties. Zbl 1008.47059
Väth, M.
2002
On the definition of eigenvalues for nonlinear operators. Zbl 0956.47038
Santucci, Paola; Väth, Martin
2000
Calculation of Lefschetz and Nielsen numbers in hyperspaces for fractals and dynamical systems. Zbl 1173.55001
Andres, J.; Väth, M.
2007
Bifurcation for a reaction-diffusion system with unilateral and Neumann boundary conditions. Zbl 1237.35013
Kučera, Milan; Väth, Martin
2012
Evolution problems with nonlinear nonlocal boundary conditions. Zbl 1281.34108
Benedetti, Irene; Taddei, Valentina; Väth, Martin
2013
On the minimal displacement problem of $$\gamma$$-Lipschitz maps and $$\gamma$$-Lipschitz retractions onto the sphere. Zbl 1038.46015
Väth, M.
2002
On the connection of degree theory and 0-epi maps. Zbl 1021.47028
Väth, Martin
2001
Bifurcation for a reaction-diffusion system with unilateral obstacles with pointwise and integral conditions. Zbl 1225.35023
Väth, Martin
2011
Topological analysis. From the basics to the triple degree for nonlinear Fredholm inclusions. Zbl 1248.47001
Väth, Martin
2012
Continuity of single- and multivalued superposition operators in generalized ideal spaces of measurable vector functions. Zbl 1119.47060
Väth, Martin
2006
New beams of global bifurcation points for a reaction-diffusion system with inequalities or inclusions. Zbl 1188.35019
Väth, Martin
2009
A disc-cutting theorem and two-dimensional bifurcation of a reaction-diffusion system with inclusions. Zbl 1169.35352
Väth, Martin
2008
The existence of non-trivial bounded functionals implies the Hahn-Banach extension theorem. Zbl 0995.46007
Luxemburg, W. A. J.; Väth, M.
2001
A characterization of 0-epi maps with a degree. Zbl 1021.47029
Giorgieri, Elena; Väth, Martin
2001
Nonstandard analysis. Zbl 1108.03061
Väth, Martin
2007
A compactness criterion of mixed Krasnoselskiĭ-Riesz type in regular ideal spaces of vector functions. Zbl 0941.46016
Väth, M.
1999
Coincidence index for noncompact mappings on nonconvex sets. Zbl 1045.47050
Andres, Jan; Väth, Martin
2002
Bifurcation points for a reaction-diffusion system with two inequalities. Zbl 1185.35074
Eisner, Jan; Kučera, Milan; Väth, Martin
2010
Some remarks on measures of noncompactness and retractions onto spheres. Zbl 1137.47039
Kim, In-Sook; Väth, Martin
2007
A general theorem on continuity and compactness of the Uryson operator. Zbl 0872.45010
Väth, Martin
1996
Global nontrivial bifurcation of homogeneous operators with an application to the $$p$$-Laplacian. Zbl 1142.35033
Väth, Martin
2006
$$\mathcal F$$-epi maps. Zbl 1008.47060
Appell, Jürgen; Väth, Martin; Vignoli, Alfonso
2001
Grasping the phantom – a new approach to nonlinear spectral theory. Zbl 1150.47042
Santucci, P.; Väth, M.
2001
Abstract Volterra equations of the second kind. Zbl 0917.45011
Väth, Martin
1998
Complete continuity of some nonlinear Volterra operator in Banach spaces. Zbl 0894.47055
Väth, M.
1998
Continuity, compactness, and degree theory for operators in systems involving $$p$$-Laplacians and inclusions. Zbl 1165.47060
Väth, Martin
2008
Weakly singular Hammerstein-Volterra operators in Orlicz and Hölder spaces. Zbl 0802.45003
Appell, Jürgen; Väth, M.
1993
Global solution branches for equations involving nonhomogeneous operators of $$p$$-Laplace type. Zbl 1211.35116
Kim, Yun-Ho; Väth, Martin
2011
Fixed point free maps of a closed ball with small measures of noncompactness. Zbl 0996.47054
Väth, Martin
2001
Linear and nonlinear abstract Volterra equations. Zbl 1046.45008
Väth, M.
1998
Function spaces with the Matkowski property and degeneracy phenomena for composition operators. Zbl 1253.47038
Appell, Jürgen; Guanda, Nathaly; Väth, Martin
2011
Location of bifurcation points for a reaction-diffusion system with Neumann-Signorini conditions. Zbl 1258.35020
Eisner, Jan; Väth, Martin
2011
Coepi maps and generalizations of the Hopf extension theorem. Zbl 1034.47026
Väth, Martin
2003
A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions. Zbl 1389.35047
Eisner, Jan; Kučera, Milan; Väth, Martin
2016
Semilinear inclusion with nonlocal conditions without compactness in non-reflexive spaces. Zbl 1365.34106
Benedetti, Irene; Väth, Martin
2016
Two topological definitions of a Nielsen number for coincidences of noncompact maps. Zbl 1079.55004
Andres, Jan; Väth, Martin
2004
The dual space of $$L_\infty$$ is $$L_1$$. Zbl 0922.46066
Väth, Martin
1998
Approximation, complete continuity, and uniform measurability of Uryson operators on general measure spaces. Zbl 0931.47049
Väth, M.
1998
A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions. Zbl 1249.35020
Baltaev, Jamol I.; Kučera, Milan; Väth, Martin
2012
Some measurability results and applications to spaces with mixed family-norm. Zbl 1120.46013
Väth, Martin
2006
Continuity and differentiability of multivalued superposition operators with atoms and parameters. I. Zbl 1237.47065
Väth, Martin
2012
Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities. Zbl 1340.35145
Väth, Martin
2014
Continuity of near-duality maps and characterizations of ideal spaces of measurable functions. Zbl 0953.46019
Dirr, Gunter; Väth, Martin
2000
Generalized ideal spaces and applications to the superposition operator. Zbl 1128.47054
Väth, Martin
2006
The Furi-Martelli-Vignoli spectrum vs. the phantom. Zbl 1042.47533
Väth, M.
2001
Convergence theorems and measures of noncompactness for noncompact Urysohn operators in ideal spaces. Zbl 1092.47052
Väth, Martin
2004
On a class of maps related to the Furi-Martelli-Vignoli spectrum. Zbl 1221.47104
Appell, J.; Giorgieri, E.; Väth, M.
2001
Compactness and existence results for ordinary differential equations in Banach spaces. Zbl 0964.34048
Appell, J.; Väth, M.; Vignoli, A.
1999
Merging of degree and index theory. Zbl 1118.54025
Väth, Martin
2006
Continuity and differentiability of multivalued superposition operators with atoms and parameters. II. Zbl 1241.47049
Väth, Martin
2012
Degree and index theories for noncompact function triples. Zbl 1143.47041
Väth, Martin
2007
Compactness estimates for integral operators of vector functions with nonmeasurable kernels. Zbl 1137.45005
Väth, Martin
2006
Degree, instability and bifurcation of reaction-diffusion systems with obstacles near certain hyperbolas. Zbl 1386.35160
Eisner, Jan; Väth, Martin
2016
Some Crandall-Rabinowitz type results and applications to reaction-diffusion systems. Zbl 1378.35027
Väth, Martin
2018
Bifurcation of obstacle problems with inclusions follow from degree results for variational inequalities. Zbl 1313.35020
Väth, Martin
2013
Calculation of degree and global bifurcation for variational inequalities with nonsymmetric operators in Banach spaces. Zbl 1290.47066
Väth, Martin
2014
Stability for semilinear parabolic problems in $$L_2$$ and $$W^{1,2}$$. Zbl 1375.35266
Gurevich, Pavel; Väth, Martin
2016
Spaces of complex functions and vector measures in incomplete spaces. Zbl 1067.28010
Ricker, Werner; Väth, Martin
2004
Some topological methods in nonlinear analysis. (Einige topologische Methoden in der nichtlinearen Analysis.) Zbl 1066.47060
Väth, Martin
2004
On the spectrum of orthomorphisms and Barbashin operators. Zbl 0947.45002
Biberdorf, E. A.; Väth, M.
1999
A continuation and existence result for a boundary value problem on an unbounded domain arising for the electrical potential in a cylindrical double layer. Zbl 1160.34012
Appell, Jürgen; Chen, Chur-Jen; Tseng, Shiojenn; Väth, Martin
2007
The Krasnosel’skiĭ-Quittner formula and instability of a reaction-diffusion system with unilateral obstacles. Zbl 1308.35031
Kim, In-Sook; Väth, Martin
2014
Iterative approximation for a boundary value problem arising for the electrical potential on a cylindrical double layer. Zbl 1165.35049
Appell, Jürgen; Chen, Chur-Jen; Tseng, Shiojenn; Väth, Martin
2008
Some Crandall-Rabinowitz type results and applications to reaction-diffusion systems. Zbl 1378.35027
Väth, Martin
2018
A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions. Zbl 1389.35047
Eisner, Jan; Kučera, Milan; Väth, Martin
2016
Semilinear inclusion with nonlocal conditions without compactness in non-reflexive spaces. Zbl 1365.34106
Benedetti, Irene; Väth, Martin
2016
Degree, instability and bifurcation of reaction-diffusion systems with obstacles near certain hyperbolas. Zbl 1386.35160
Eisner, Jan; Väth, Martin
2016
Stability for semilinear parabolic problems in $$L_2$$ and $$W^{1,2}$$. Zbl 1375.35266
Gurevich, Pavel; Väth, Martin
2016
Instability of Turing type for a reaction-diffusion system with unilateral obstacles modeled by variational inequalities. Zbl 1340.35145
Väth, Martin
2014
Calculation of degree and global bifurcation for variational inequalities with nonsymmetric operators in Banach spaces. Zbl 1290.47066
Väth, Martin
2014
The Krasnosel’skiĭ-Quittner formula and instability of a reaction-diffusion system with unilateral obstacles. Zbl 1308.35031
Kim, In-Sook; Väth, Martin
2014
Evolution problems with nonlinear nonlocal boundary conditions. Zbl 1281.34108
Benedetti, Irene; Taddei, Valentina; Väth, Martin
2013
Bifurcation of obstacle problems with inclusions follow from degree results for variational inequalities. Zbl 1313.35020
Väth, Martin
2013
Bifurcation for a reaction-diffusion system with unilateral and Neumann boundary conditions. Zbl 1237.35013
Kučera, Milan; Väth, Martin
2012
Topological analysis. From the basics to the triple degree for nonlinear Fredholm inclusions. Zbl 1248.47001
Väth, Martin
2012
A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions. Zbl 1249.35020
Baltaev, Jamol I.; Kučera, Milan; Väth, Martin
2012
Continuity and differentiability of multivalued superposition operators with atoms and parameters. I. Zbl 1237.47065
Väth, Martin
2012
Continuity and differentiability of multivalued superposition operators with atoms and parameters. II. Zbl 1241.47049
Väth, Martin
2012
Bifurcation for a reaction-diffusion system with unilateral obstacles with pointwise and integral conditions. Zbl 1225.35023
Väth, Martin
2011
Global solution branches for equations involving nonhomogeneous operators of $$p$$-Laplace type. Zbl 1211.35116
Kim, Yun-Ho; Väth, Martin
2011
Function spaces with the Matkowski property and degeneracy phenomena for composition operators. Zbl 1253.47038
Appell, Jürgen; Guanda, Nathaly; Väth, Martin
2011
Location of bifurcation points for a reaction-diffusion system with Neumann-Signorini conditions. Zbl 1258.35020
Eisner, Jan; Väth, Martin
2011
Bifurcation points for a reaction-diffusion system with two inequalities. Zbl 1185.35074
Eisner, Jan; Kučera, Milan; Väth, Martin
2010
Global bifurcation for a reaction-diffusion system with inclusions. Zbl 1182.35025
Eisner, Jan; Kučera, Milan; Väth, Martin
2009
New beams of global bifurcation points for a reaction-diffusion system with inequalities or inclusions. Zbl 1188.35019
Väth, Martin
2009
A disc-cutting theorem and two-dimensional bifurcation of a reaction-diffusion system with inclusions. Zbl 1169.35352
Väth, Martin
2008
Continuity, compactness, and degree theory for operators in systems involving $$p$$-Laplacians and inclusions. Zbl 1165.47060
Väth, Martin
2008
Iterative approximation for a boundary value problem arising for the electrical potential on a cylindrical double layer. Zbl 1165.35049
Appell, Jürgen; Chen, Chur-Jen; Tseng, Shiojenn; Väth, Martin
2008
Global solution branches and a topological implicit function theorem. Zbl 1150.58006
Väth, Martin
2007
Calculation of Lefschetz and Nielsen numbers in hyperspaces for fractals and dynamical systems. Zbl 1173.55001
Andres, J.; Väth, M.
2007
Nonstandard analysis. Zbl 1108.03061
Väth, Martin
2007
Some remarks on measures of noncompactness and retractions onto spheres. Zbl 1137.47039
Kim, In-Sook; Väth, Martin
2007
Degree and index theories for noncompact function triples. Zbl 1143.47041
Väth, Martin
2007
A continuation and existence result for a boundary value problem on an unbounded domain arising for the electrical potential in a cylindrical double layer. Zbl 1160.34012
Appell, Jürgen; Chen, Chur-Jen; Tseng, Shiojenn; Väth, Martin
2007
Degree and global bifurcation for elliptic equations with multivalued unilateral conditions. Zbl 1111.47055
Eisner, Jan; Kučera, Milan; Väth, Martin
2006
Continuity of single- and multivalued superposition operators in generalized ideal spaces of measurable vector functions. Zbl 1119.47060
Väth, Martin
2006
Global nontrivial bifurcation of homogeneous operators with an application to the $$p$$-Laplacian. Zbl 1142.35033
Väth, Martin
2006
Some measurability results and applications to spaces with mixed family-norm. Zbl 1120.46013
Väth, Martin
2006
Generalized ideal spaces and applications to the superposition operator. Zbl 1128.47054
Väth, Martin
2006
Merging of degree and index theory. Zbl 1118.54025
Väth, Martin
2006
Compactness estimates for integral operators of vector functions with nonmeasurable kernels. Zbl 1137.45005
Väth, Martin
2006
Global bifurcation of the $$p$$-Laplacian and related operators. Zbl 1073.35027
Väth, Martin
2005
An abstract Gronwall lemma and applications to global existence results for functional differential and integral equations of fractional order. Zbl 1080.45006
Salem, Hussein A. H.; Väth, Martin
2004
On some Banach space constants arising in nonlinear fixed point and eigenvalue theory. Zbl 1074.47025
Appell, Jürgen; Erzakova, Nina A.; Santana, Sergio Falcon; Väth, Martin
2004
Two topological definitions of a Nielsen number for coincidences of noncompact maps. Zbl 1079.55004
Andres, Jan; Väth, Martin
2004
Convergence theorems and measures of noncompactness for noncompact Urysohn operators in ideal spaces. Zbl 1092.47052
Väth, Martin
2004
Spaces of complex functions and vector measures in incomplete spaces. Zbl 1067.28010
Ricker, Werner; Väth, Martin
2004
Some topological methods in nonlinear analysis. (Einige topologische Methoden in der nichtlinearen Analysis.) Zbl 1066.47060
Väth, Martin
2004
Coepi maps and generalizations of the Hopf extension theorem. Zbl 1034.47026
Väth, Martin
2003
Nonlinear spectral theory for homogeneous operators. Zbl 1045.47053
Giorgieri, Elena; Appell, Jürgen; Väth, Martin
2002
Integration theory. A second course. Zbl 1043.28001
Väth, Martin
2002
Coincidence points of function pairs based on compactness properties. Zbl 1008.47059
Väth, M.
2002
On the minimal displacement problem of $$\gamma$$-Lipschitz maps and $$\gamma$$-Lipschitz retractions onto the sphere. Zbl 1038.46015
Väth, M.
2002
Coincidence index for noncompact mappings on nonconvex sets. Zbl 1045.47050
Andres, Jan; Väth, Martin
2002
Some remarks on small sets. Zbl 1102.28300
Appell, Jürgen; D’Aniello, Emma; Väth, Martin
2001
On the connection of degree theory and 0-epi maps. Zbl 1021.47028
Väth, Martin
2001
The existence of non-trivial bounded functionals implies the Hahn-Banach extension theorem. Zbl 0995.46007
Luxemburg, W. A. J.; Väth, M.
2001
A characterization of 0-epi maps with a degree. Zbl 1021.47029
Giorgieri, Elena; Väth, Martin
2001
$$\mathcal F$$-epi maps. Zbl 1008.47060
Appell, Jürgen; Väth, Martin; Vignoli, Alfonso
2001
Grasping the phantom – a new approach to nonlinear spectral theory. Zbl 1150.47042
Santucci, P.; Väth, M.
2001
Fixed point free maps of a closed ball with small measures of noncompactness. Zbl 0996.47054
Väth, Martin
2001
The Furi-Martelli-Vignoli spectrum vs. the phantom. Zbl 1042.47533
Väth, M.
2001
On a class of maps related to the Furi-Martelli-Vignoli spectrum. Zbl 1221.47104
Appell, J.; Giorgieri, E.; Väth, M.
2001
Volterra and integral equations of vector functions. Zbl 0940.45002
Väth, Martin
2000
An axiomatic approach to a coincidence index for noncompact function pairs. Zbl 0992.47028
Väth, Martin
2000
On the definition of eigenvalues for nonlinear operators. Zbl 0956.47038
Santucci, Paola; Väth, Martin
2000
Continuity of near-duality maps and characterizations of ideal spaces of measurable functions. Zbl 0953.46019
Dirr, Gunter; Väth, Martin
2000
Fixed point theorems and fixed point index for countably condensing maps. Zbl 0964.47025
Väth, Martin
1999
A compactness criterion of mixed Krasnoselskiĭ-Riesz type in regular ideal spaces of vector functions. Zbl 0941.46016
Väth, M.
1999
Compactness and existence results for ordinary differential equations in Banach spaces. Zbl 0964.34048
Appell, J.; Väth, M.; Vignoli, A.
1999
On the spectrum of orthomorphisms and Barbashin operators. Zbl 0947.45002
Biberdorf, E. A.; Väth, M.
1999
Abstract Volterra equations of the second kind. Zbl 0917.45011
Väth, Martin
1998
Complete continuity of some nonlinear Volterra operator in Banach spaces. Zbl 0894.47055
Väth, M.
1998
Linear and nonlinear abstract Volterra equations. Zbl 1046.45008
Väth, M.
1998
The dual space of $$L_\infty$$ is $$L_1$$. Zbl 0922.46066
Väth, Martin
1998
Approximation, complete continuity, and uniform measurability of Uryson operators on general measure spaces. Zbl 0931.47049
Väth, M.
1998
Ideal spaces. Zbl 0896.46018
Väth, Martin
1997
A general theorem on continuity and compactness of the Uryson operator. Zbl 0872.45010
Väth, Martin
1996
Weakly singular Hammerstein-Volterra operators in Orlicz and Hölder spaces. Zbl 0802.45003
Appell, Jürgen; Väth, M.
1993
all top 5

all top 5

### Cited in 82 Serials

 25 Journal of Mathematical Analysis and Applications 24 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 13 Zeitschrift für Analysis und ihre Anwendungen 8 Applied Mathematics and Computation 8 Journal of Integral Equations and Applications 7 Topology and its Applications 7 Applied Mathematics Letters 6 Annali di Matematica Pura ed Applicata. Serie Quarta 6 Journal of Differential Equations 6 Mathematical and Computer Modelling 5 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 5 Fixed Point Theory and Applications 5 Boundary Value Problems 4 Indagationes Mathematicae. New Series 4 Discrete and Continuous Dynamical Systems 3 Computers & Mathematics with Applications 3 Mathematica Slovaca 3 Acta Mathematica Hungarica 3 Applications of Mathematics 3 Nonlinear Analysis. Real World Applications 3 Central European Journal of Mathematics 3 Journal of Function Spaces 2 Mathematische Semesterberichte 2 Fuzzy Sets and Systems 2 Journal of Functional Analysis 2 Numerical Functional Analysis and Optimization 2 Proceedings of the American Mathematical Society 2 Quaestiones Mathematicae 2 Computational Optimization and Applications 2 Topological Methods in Nonlinear Analysis 2 Journal of Inequalities and Applications 2 Fractional Calculus & Applied Analysis 2 Mediterranean Journal of Mathematics 2 Journal of Fixed Point Theory and Applications 2 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 2 Nonlinear Analysis. Theory, Methods & Applications 2 British Journal for the History of Mathematics 1 Applicable Analysis 1 Bulletin of the Australian Mathematical Society 1 Lithuanian Mathematical Journal 1 Rocky Mountain Journal of Mathematics 1 Demonstratio Mathematica 1 Journal of the London Mathematical Society. Second Series 1 Manuscripta Mathematica 1 Real Analysis Exchange 1 Results in Mathematics 1 Revista de la Unión Matemática Argentina 1 Ricerche di Matematica 1 Synthese 1 Transactions of the American Mathematical Society 1 Physica D 1 Optimization 1 Journal of Global Optimization 1 Computational Mathematics and Mathematical Physics 1 Elemente der Mathematik 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Journal of Dynamics and Differential Equations 1 Russian Mathematics 1 Tatra Mountains Mathematical Publications 1 Journal of Mathematical Sciences (New York) 1 Mathematical Logic Quarterly (MLQ) 1 Georgian Mathematical Journal 1 Opuscula Mathematica 1 Sbornik: Mathematics 1 Abstract and Applied Analysis 1 Positivity 1 Communications in Contemporary Mathematics 1 Acta et Commentationes Universitatis Tartuensis de Mathematica 1 Nonlinear Analysis. Modelling and Control 1 Journal of Dynamical Systems and Geometric Theories 1 Milan Journal of Mathematics 1 Journal of Function Spaces and Applications 1 Fixed Point Theory 1 Proceedings of the Steklov Institute of Mathematics 1 Discrete and Continuous Dynamical Systems. Series S 1 Differential Equations and Applications 1 Set-Valued and Variational Analysis 1 Applied General Topology 1 Journal of Pseudo-Differential Operators and Applications 1 Annals of Functional Analysis 1 Electronic Journal of Mathematical Analysis and Applications EJMAA 1 Bulletin of Computational Applied Mathematics
all top 5

### Cited in 31 Fields

 138 Operator theory (47-XX) 45 Integral equations (45-XX) 43 Partial differential equations (35-XX) 39 Functional analysis (46-XX) 35 General topology (54-XX) 32 Ordinary differential equations (34-XX) 19 Real functions (26-XX) 18 Measure and integration (28-XX) 11 Algebraic topology (55-XX) 10 Mathematical logic and foundations (03-XX) 7 Calculus of variations and optimal control; optimization (49-XX) 5 Dynamical systems and ergodic theory (37-XX) 5 Global analysis, analysis on manifolds (58-XX) 4 History and biography (01-XX) 4 Probability theory and stochastic processes (60-XX) 3 Operations research, mathematical programming (90-XX) 2 Difference and functional equations (39-XX) 2 Numerical analysis (65-XX) 1 General and overarching topics; collections (00-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Topological groups, Lie groups (22-XX) 1 Sequences, series, summability (40-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Convex and discrete geometry (52-XX) 1 Manifolds and cell complexes (57-XX) 1 Statistics (62-XX) 1 Mechanics of deformable solids (74-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Biology and other natural sciences (92-XX) 1 Systems theory; control (93-XX) 1 Mathematics education (97-XX)

### Wikidata Timeline

The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.


text:
How to use braces at left and top of array in equation mode?

I tried to put 2 braces, one at the top and one at the left, but I got an ugly effect. How to fit both braces with the array?

\documentclass[12pt]{report}
\begin{document}
$$lines \overbrace{\left\{\begin{array}{|r|r|r|r|r|} \hline 4 & 6 & 5 & 7 & 9 \\ \hline 3 & 5 & 7 & 8 & 10 \\ \hline 1 & 3 & 9 & 10 & 12 \\ \hline 2 & 4 & 3 & 9 & 11 \\ \hline \end{array}\right.}^{columns}$$
\end{document}

• Instead \documentclass$12pt${report} is correct \documentclass[12pt]{report}. Correcting this I obtain nice braces, only over on also consider left one ... Aug 12 '16 at 19:59
• Thank you for the correction @Zarko. Did you get nice braces with this code? the top brace takes the left brace. How to solve this problem? Aug 12 '16 at 20:04
• this looks like what you want: How can I have a brace on the left and the top of a matrix without overlapping? Aug 12 '16 at 20:05

An rude solution, similar to solution which Barbara Beeton pointed in her comment:

\documentclass[12pt]{report}
\begin{document}
$$lines \left\{\vphantom{\begin{array}{c} \hline 4 \\ \hline 3 \\ \hline 1 \\ \hline 2 \\ \hline \end{array}}\right.% \overbrace{\begin{array}{|r|r|r|r|r|} \hline 4 & 6 & 5 & 7 & 9 \\ \hline 3 & 5 & 7 & 8 & 10 \\ \hline 1 & 3 & 9 & 10 & 12 \\ \hline 2 & 4 & 3 & 9 & 11 \\ \hline \end{array}}^{columns}$$
\end{document}

• As you pointed, similar. But in fact, better, because make use of all first column with \vphantom. Aug 12 '16 at 20:23

Here is another solution, based on blkarray:

\documentclass[12pt]{report}
\usepackage[utf8]{inputenc}%
\usepackage{blkarray, amsmath}

\begin{document}

$$\renewcommand\arraystretch{1.6} \overbrace{\mkern-8mu\begin{blockarray}{rrrrr}% \hphantom{10}& \hphantom{10} &\hphantom{10} & & \\[-4ex] \begin{block}{\Left{\footnotesize lines\,}{\{\hspace{0.8em}}|r|r|r|r|r|} \BAhline 4 & 6 & 5 & 7 & 9 \\ \BAhline 3 & 5 & 7 & 8 & 10 \\ \BAhline 1 & 3 & 9 & 10 & 12 \\ \BAhline 2 & 4 & 3 & 9 & 11 \\ \BAhline \end{block} \end{blockarray}\mkern-8mu}^\text{columns}%$$

\end{document}

format: plain


text:
# Kunneth formula for motivic cohomology

I was wondering when the Kunneth formula holds for motivic cohomology: $$H^p(X,A(\alpha)) = \bigoplus_{i+j=p;\beta+\gamma = \alpha} H^j(X,A(\beta)) \otimes H^i(X,A(\gamma))$$ where $H^p(X,A(\alpha))$ is defined as you wish: by higher Chow groups, Hom groups in $DM(X)$, etc... The case I'm most interested in is $A= \mathbb{Q}$ and $M(X) \in DM(\mathbb{Q})_{\mathbb{Q}}$ in the thick sub-triangagulated category generated by the $\mathbb{Q}(n)$, $n\in \mathbb{Z}$.

-

I now remember a nice argument, why there's no Kunneth formula for Chow groups of $X \times X$ unless $X$ has a Tate motive. Let $X$ be smooth projective of dimension $d$. We start with a decomposition of a diagonal: $$[\Delta] = \sum_{i,j} \alpha^i_j \beta^{d-i}_j \in \oplus_i CH^i(X) \otimes CH^{d-i}(X)$$ We can assume $\alpha^i_j$ are linearly independent. In this case we can show that $\alpha^i_j$ form a basis of Chow groups and $\beta^{d-i}_j$ is the dual basis.

Indeed, as a correspondence $[\Delta]$ acts as identity on Chow groups, so for any class c, $$c = [\Delta]c = \sum_{i,j} \alpha^i_j deg(\beta^{d-i}_j \cup c),$$ and the claim follows if we substitute $c = \alpha^i_j$.

Now $CH_i(X) = Hom(\mathbb Z(i)[2i], M(X))$ and we can consider the set of $\alpha^i_j$ as a morphism of motives $$\oplus_{i,j}\mathbb Z(i)[2i] \to M(X).$$ A simple computation shows that it is an isomorphism with the inverse given by $\beta^i_j$.

And of course, on the other hand, if $X$ has a Tate motive, then Kunneth formula for Chow groups follows (it doesn't answer the question, since I only consider smooth projective varieties).

-
That's really nice thanks. –  YBL Jan 26 '10 at 1:22

I don't think so; for example if X is Spec(Q) then this doesn't seem to be true. If you define, for any $Y$ in this thick subcategory, $\mathbb{H}(Y)$ to be the bigraded ring $\oplus H^s(Y, \mathbb{Q}(t))$, then I'd expect there to be a spectral sequence of the form $${\rm Tor}^{{\mathbb H}({\rm Spec} \mathbb{Q}))}(\mathbb{H}(X),\mathbb{H}(X)) \Rightarrow \mathbb{H}(X \times X)$$ instead, i.e. you should take the module structure over the motivic cohomology of a point into account on the righthand side of the formula you were proposing.

-

Correct me if I'm wrong. I think here is how it goes. For any $X$ and any Tate motive $M$, there is an isomorphism of modules over $H^{\*,\*}(Spec F)$: $$H^{\*,\*}(M(X) \otimes M) = H^{\*,\*}(X) \otimes_{H^{\*,\*}(Spec F)} H^{\*,\*}(M).$$

Remarks.

1. Right hand side makes sense, since $H^{\*,\*}(M)$ is indeed a module over $H^{\*,\*}(Spec F)$

2. When $M = M(\mathbf P^n)$, this is the the statement of the projective bundle theorem.

3. It's unclear to me how to write down the individual terms $H^{p,q}(M(X) \otimes M)$ it terms of cohomology $M(X)$ and $M$.

To prove the theorem in this form it is sufficient to consider the case $M = \mathbf Q(n)$. Note that $H^{\*,\*}(\mathbf Q(n))$ is $H^{\*,\*}(Spec F)$ with bidegree shifted by (0,n), therefore tensoring with this module is the same as shifting the bidegree by (0,n). And the same thing is in the left-hand side, according to Voevodsky's Cancellation Theorem.

-

There is a paper by Dugger & Isaksen called Motivic Cell Structures in which they establish a Künneth formula for various cohomology theories that are represented in the $\mathbb{A}^1$-homotopy category, provided the object $X$ satisfies some sort of cellularity condition which is similar to the requirement of having a Tate motive.

Of course, as Tyler suggested this Künneth formula is of the form of a spectral sequence over the motivic cohomology of the ground field, to wit $\mathrm{Tor}_{H(\mathrm{spec}\; k)} (H(X) , H(Y)) \Rightarrow H(X \times Y)$.

In general, this spectral sequence fails, as can probably be seen by all the counterexamples already given, and certainly can be seen by considering $\mathrm{spec}\; \mathbb{C} \times_\mathbb{R} \mathrm{spec} \; \mathbb{C}$ with $\mathbb{Z}/2$-coefficients, where the motivic cohomology rings are known in their entirities (These calculations also appear in papers of Dugger & Isaksen on the motivic Adams spectral sequence).

-

Very briefly, I believe the following is true: Motivic cohomology does not satisfy a Kunneth formula on the level of cohomology groups, but it does satisfy a kind of Kunneth formula on the level of some suitable derived category of sheaves. This should hold true in general for any Bloch-Ogus cohomology theory, I think.

-

Did you mean $H^p(X \times X, A(\alpha) = \dots$?

I 'm pretty sure Kunneth formula doesn't hold for motivic cohomology in general. I think you'll get the wrong statement even when you specialize to the case of Picard group of a product of two curves of genus > 0.

However, for Tate motives over $\mathbf Q$ it probaby does hold. Seems like it's sufficient to check the case of $\mathbf{P^n}$ for which motivic cohomology "coincide" with the usual cohomology.

-
This is exactly what I had in mind. In the case of a mixed Tate motives, it feels like this is equivalent to Ext^2(A(0),A(n)) = 0 for the base field but I don't know how to turn this intuition into a proof. Any thoughts? –  YBL Nov 25 '09 at 21:09
I had a wrong impression, that $H^{p,q}(Spec(F), \mathbf Q)$ is nonzero only for $p=q=0$. However, since it is not the case I agree with Tyler Lawson, that the formula must have tensor product over $H^{*,*}(Spec F)$ or something like that. Otherwise even H^{,}(X \times Spec(F)) doesn't satisfy the decomposition. –  Evgeny Shinder Nov 25 '09 at 21:39


text:
Thread: 0^0 = 1? View Single Post
 Quote by CRGreathouse I don't think that it's undefined; I think that it's 1. 0^0 is an empty product, and empty products are necessarily equal to 1. As Galileo points out above, we need 0^0=1 for series to have compact formulas. Everyone agrees that $$x^0=1$$ for $$x\neq0$$, but there's no reason to think that it should be different at 0 -- $$0^x$$ is only 0 for x > 0, since it's not defined for negative x. There's no problem accepting 0^0 as 1, and there are many good reasons to think it shouldn't be undefined or 0.
$$x^0 = \frac{x}{x}$$.


text:
# Mandated Curriculum vs. Developmentally Appropriate Practices

You are a parent who is concerned about the new state-mandated, textbook-based curriculum for kindergarten, and you are writing a letter to the school board in protest of this new policy. In your letter, define DAP, describe the benefits of DAP, and explain why you feel the use of a textbook-based curriculum may be inappropriate. Describe potential negative effects of this curriculum and how it is developmentally inappropriate to literacy development. Recommend an alternative approach to developing literacy skills. In your letter, use the kindergarten standards from my state and the guidelines for DAP described in the course text as well as in NAEYC resources.

The paper should be two to three pages in addition to the title page and the reference page. Use at least one scholarly reference in addition to your text.

my state ALABAMA

$10.00 ANSWER Tutor has posted answer for$10.00. See answer's preview


text:
# Prevent a page break across two different environment

I need to tie two different environments, i.e., prevent a page break from happening. I tried \nopagebreak, \nobreak, \begin{samepage}-\end{samepage}, but nothing worked.

\begin{enumerate}
\item ....
\item týden
Pondělí:
\begin{enumerate}
\begin{multicols}{2}
\item $\frac{\frac{1}{a-1}+\frac{2}{a-1}+\frac{3}{a-1}}{\frac{1}{a^{2}-1}}$
\vysl{$6(a+1)$ PODM: $a\neq\pm1$}
\item $(y^{2}-2)^{3}$
\vysl{$y^{6}-6y^{4}+12y^{2}-8$}
\end{multicols}
\item $A-B=?;A=\{x\in\R;-3\leq x\leq 10\};B=\{x\in\R;-4<x<8\}$
\vysl{$A-B=\{x\in\R;8 \leq x\leq 10\}$}
\end{enumerate}

Page breaks somewhere between word "Pondělí" and environment multicols, but I need it together.

Thanks for all answers and excuse my English, I hope everything is understandable.

• Welcome to TeX.SE. Have you looked into the needspace package? – Mico Sep 28 '15 at 19:42

You should try the needspace package. Since it looks like 5 to 6 lines are occupied by the material that needs to be kept together, you might try:

\documentclass{article}
\usepackage{multicol,needspace}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
%% how is the "\vysl" macro defined?

\begin{document}

\begin{enumerate}
\item \dots
\item týden
\needspace{6\baselineskip}
Pondělí:
\begin{enumerate}
\begin{multicols}{2}
\item $\frac{\frac{1}{a-1}+\frac{2}{a-1}+\frac{3}{a-1}}{\frac{1}{a^{2}-1}}$
\vysl{$6(a+1)$ PODM: $a\neq\pm1$}
\item $(y^{2}-2)^{3}$
\vysl{$y^{6}-6y^{4}+12y^{2}-8$}
\end{multicols}
\item $A-B=?;A=\{x\in\R;-3\leq x\leq 10\};B=\{x\in\R;-4<x<8\}$
\vysl{$A-B=\{x\in\R;8 \leq x\leq 10\}$}
\end{enumerate}

\end{document}

(I'm afraid I can't post a screenshot as I don't know how the \vsyl macro is defined.)


text:
# Find the long-term mean number of cars in line at the toll booth.

Cars arrive at a toll booth according to a Poisson process at the rate of two cars per minute. The time taken by the attendant to collect the toll is exponentially distributed with mean $$20$$ seconds.

(a) Find the long-term mean number of cars in line at the toll booth.

(b) Find the long-term probability that there are more than three cars at the toll booth.

If the arrival is $$2$$ cars per minute, then the inter-arrival times between each successive car is $$\text{exp}(1/2)$$ distributed. So cars arrive with rate $$\text{exp}(1/2)$$ and they leave the booth with rate $$\text{exp}(1/3)$$.

Let the statespace be $$S=\{0,1,2,...,n\}$$ denote the number of cars at the booth at different times. Starting at state $$0$$, the first car to arrive is the car with time equal to

$$\text{min}\left(\text{exp}(1/2),...n \ \text{terms}...,\text{exp}(1/2)\right)=\text{exp}\left(\sum_{k=1}^n\frac{1}{2}\right)=\text{exp}\left(\frac{n}{2}\right).$$

The rate for the second car can be found summing $$n-1$$ terms in similar fashion. That is $$\text{exp}\left(\frac{n-1}{2}\right)$$ and so on for the third etc. At the same time, the cars leave the queue as the attendant is finished, an average each $$20$$ seconds. So the transition rate diagram is

$$0\xrightarrow{\frac{n}{2}}1\xrightarrow{\frac{n-1}{2}}2\xrightarrow{\frac{n-2}{2}}3...n-2\xrightarrow{1}n-1\xrightarrow{\frac{1}{2}}n.$$

(OBS: there are also arrows going to the left from each state with rate $$1/3$$, did not know how to latex that.)

This is where it gets tedious. I need to find the generator matrix $$Q$$ and then find the limiting distribution $$\vec{\pi}$$ using $$\vec{\pi}\ Q=0$$. Once I have it, I can simply multiply each entry in the limiting distribution vector with each state and sum all the factors obtained. How do I get there?

EDIT as per EDZ's suggestion using Little's rule. I'd be happy if someone could verify the below!

The problem is an M/M/1 process, which is a birth and death process with constant birth and death rates, that is $$\lambda_i=\lambda$$ and $$\mu_i=\mu.$$ Thus, the stationary distribution is given by

$$\pi_k=\pi_0\prod_{i=1}^k\frac{\lambda}{\mu}=\pi_0\left(\frac{\lambda}{\mu}\right)^k$$

and

$$\pi_0=\left(\sum_{k=0}^{\infty}\prod_{i=1}^{k}\frac{\lambda}{\mu}\right)^{-1}=1-\frac{\lambda}{\mu},$$

so

$$\pi_k=\left(1-\frac{\lambda}{\mu}\right)\left(\frac{\lambda}{\mu}\right)^k,$$

which is a geometric distribution with parameter $$1-\lambda/\mu.$$ In our case, $$\lambda=2$$ and $$\mu=3$$ since each minute, $$3$$ cars are given service and $$2$$ new cars arrive. So our distribution is of the number of cars $$N$$ in the queue is

$$N\sim\text{Geometric}\left(\frac{1}{3}\right).$$

For (a), the long term mean number of cars in line is the mean of the geometric distribution, which is

$$E[N]=\frac{1-1/3}{1/3}=2.$$

For (b), since we have the distribution, we can compute the desired probability as follows:

$$P(N>3)=1-P(N=0)-P(N=1)-P(N=2)=1-\frac{1}{3}-\frac{2}{9}-\frac{4}{27}\approx0.2963$$

• have you looked at this as a queue where you can use Little's rules as well as other established formulas? en.wikipedia.org/wiki/M/M/1_queue – EDZ Jan 10 at 23:38
• No I have not because I don't really know how to know when to use Little's rule or not. Some very similar assignments involving queue-theory uses it and some don't. However, I'm gonna try this now and edit in my attempt in the question above. Thanks! – Parseval Jan 11 at 0:33
• @EDZ - I've added my attempt as per your suggestion. Can you please check if there are any mistakes? My book only has answers to odd numbered questions and this is an even numbered unfortunately. – Parseval Jan 11 at 1:20


text:
## Elementary Algebra

Published by Cengage Learning

# Chapter 1 - Some Basic Concepts of Arithmetic and Algebra - 1.5 - Use of Properties - Problem Set 1.5 - Page 33: 7

#### Answer

Additive inverse property

#### Work Step by Step

RECALL: The additive inverse property states that for any real number $a$: $a+(-a)=0$, so $-a+a=0$ The given statement is justified by the additive inverse property.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
## Distinct values of a proto-logarithmic function

Published on Saturday, 19th January 2019, 10:00 pm; Solved by 113;
Difficulty rating: 70%

### Problem 652

Consider the values of $\log_2(8)$, $\text{log}_4(64)$ and $\text{log}_3(27)$. All three are equal to $3$.

Generally, the function $f(m,n)=\text{log}_m(n)$ over integers $m,n \ge 2$ has the property that
$f(m_1,n_1)=f(m_2,n_2)$ if

1. $\, m_1=a^e, n_1=a^f, m_2=b^e,n_2=b^f \,$ for some integers $a,b,e,f \, \,$ or
2. $\, m_1=a^e, n_1=b^e, m_2=a^f,n_2=b^f \,$ for some integers $a,b,e,f \,$

We call a function $g(m,n)$ over integers $m,n \ge 2$ proto-logarithmic if

• $\quad \, \, \, \, g(m_1,n_1)=g(m_2,n_2)$ if any integers $a,b,e,f$ fulfilling 1. or 2. can be found
• and $\, g(m_1,n_1) \ne g(m_2,n_2)$ if no integers $a,b,e,f$ fulfilling 1. or 2. can be found

Let $D(N)$ be the number of distinct values that any proto-logarithmic function $g(m,n)$ attains over $2\le m, n\le N$.
For example, $D(5)=13$, $D(10)=69$, $D(100)=9607$ and $D(10000)=99959605$.

Find $D(10^{18})$, and give the last 9 digits as answer.

Note: According to the four exponentials conjecture the function $\text{log}_m(n)$ is proto-logarithmic.
While this conjecture is yet unproven in general, $\text{log}_m(n)$ can be used to calculate $D(N)$ for small values of $N$.


text:
# Is it possible for a VHDL component to have multiple architectures?

Just a thought I had: is it possible for a VHDL component to have multiple architectures if outputs are not modified by both? If so how can we select the one to use at synthesis time (like the C preprocessor)? What happens if multiple architectures can coexist and outputs conflict?

An entity defines an interface to a box; an architecture defines what's inside. If you don't have the same interface, you don't have the same entity. If you have the same port names, but each architecture uses them for different purposes, well, that's legal, but maybe not advisable - it depends on the specifics.

If you want to use multiple architectures simultaneously, an alternative to configurations is just to use direct instantiation:

U1 : entity my_lib.my_comp(arch1)
...

U2 : entity my_lib.my_comp(arch2)
...

If you want to use only one at a time, some sort of configuration is probably the way to go.

A VHDL entity can have different VHDL architectures. You can select the correct binding between 'entity' and 'achitecture' with the 'configuration'. The entity is describing the inputs and outputs. So, they have to stay the same.


text:
# What is the slope and intercept of 2x + y = -2?

Nov 14, 2015

Slope $= - 2$,
$y$-intercept $= - 2$,
$x$-intercept $= - 1$.

#### Explanation:

$2 x + y = - 2$.

The equation of a straight line is usually written as:
$y = m x + c$, where m is the gradient and $c$ the $y$-intercept. So, we will put it in that format:
$2 x + y = - 2$,
$y = - 2 x - 2$. So, we know that $m = - 2$ and that the $y$-intercept is at $- 2$.

If we want to know the $x$-intercept, we can simply substitute $y$ by 0 and solve for $x$:
$0 = - 2 x - 2$,
$2 x = - 2$,
$x = - 1$.

Hope it Helps! :D .


text:
# On analysing a sample of water, a chemist finds that 100cm^3 of it contains 0.007g of calcium carbonate. What is the concentration of calcium carbonate in ppm?

Jan 11, 2017

$\text{1 ppm"-=} 1 \cdot m g \cdot {L}^{-} 1$

#### Explanation:

And thus $\text{concentration}$ $=$ $\text{Mass of solute"/"Volume}$.

$= \frac{7 \cdot m g}{0.100 \cdot L} = 70 \cdot m g \cdot {L}^{-} 1$

And thus $0.070 \cdot g \cdot {L}^{-} 1 \equiv 70 \cdot m g \cdot {L}^{-} 1 \equiv \text{70 ppm}$.

At these concentrations, we don't really have to worry too much about the density of the solution. $\text{70 ppm}$ denotes a concentration of $70 \cdot m g \cdot {L}^{-} 1$ with respect to $\text{calcium carbonate}$. Of course this will be a GREATER concentration than the concentration with respect to the metal ion. Why should this be so?

I think you can see why a concentration of $1 \cdot m g \cdot {L}^{-} 1$ is referred to as a $\text{part per million}$. A litre of water has a mass of $1000 \cdot g$, and each of these $\text{grams}$ is composed of a $1000 \cdot m g$ (i.e. $1 \cdot m g = {10}^{-} 3 g$. $1000 \times 1000 \equiv \text{million}$, hence $1 \cdot m g$ $\equiv$ $\text{1 part per million}$.


text:
# Why exactly is the Guassian filter a low-pass filter?

In image processing, a Guassian filter is used to blur an image. A low-pass filter is a filter that attenuates the high frequencies, preserving only smooth variations in the provided image. Is a Guassian filter a low-pass filter because it smooths the edges (or, in general, the parts of the image where there is a high variation of the intensity)? Why exactly is the Guassian filter a low-pass filter?

• @nbro, whether it's $$h(t) = e^{-\alpha t^2}$$ or $$H(f) = e^{-\beta f^2}$$ as $|t|$ or $|f|$ increase, the $h(t)$ or $H(f)$ get very small. – robert bristow-johnson Mar 28 at 19:35


text:
SMS scnews item created by John Enyang at Tue 6 Nov 2012 1134
Type: Seminar
Distribution: World
Expiry: 10 Nov 2012
Calendar1: 9 Nov 2012 1205-1255
CalLoc1: Carslaw 175
Auth: enyang@penyang.pc (assumed)

# The Littlewood-Richardson rule and the hive model

### Kim

###### Speaker:

Sangjib Kim (University of Queensland)

###### Title:

The Littlewood-Richardson rule and the hive model.

###### Abstract:

The Littlewood-Richardson rule provides a combinatorial description of the multiplicities in the decomposition of tensor products of irreducible representations of $$\mathrm{GL}(n,\mathbb{C})$$. We will discuss the hive model of the Littlewood-Richardson rule and its relation to other combinatorial results in representation theory.

---------------------------------------------------------------------------------------

After the seminar we will take the speaker to lunch.

See the Algebra Seminar web page for information about other seminars in the series.

John Enyang John.Enyang@sydney.edu.au

If you are registered you may mark the scnews item as read.
School members may try to .


text:
Example of the equality of an inequality

This question is related to Daniel Fischer's answer here. Suppose $f$ is a real $C^{1}$ function on $[0, 1]$ such that $f(0) = 0$ and $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$. Then (essentially by Cauchy-Schwarz), we have $\left|\int_{0}^{1}f(x)\, dx\right| \leq 2/3$ as Daniel Fischer stated in his answer.

My question is: Is there an example of a function such that we have equality in $\left|\int_{0}^{1}f(x)\, dx\right| \leq 2/3$? I was thinking of maybe a smoothed version of a function which takes the value $0$ on $[0, 1/2)$ and $4/3$ on $(1/2, 1]$ but that seems complicated to construct, moreover, the derivative in some neighbourhood of $1/2$ might be hard to control.

-
I think the absolute values are not needed since we always have $\int_0^1 |f(x)| \, dx \geq \left|\int_0^1f(x)\, dx\right|$ while $f'(x)^2$ is unchanged (except at the cusps). So we can just focus on positive functions. Also, instead of investigating if we can reach $\frac{2}{3}$ now that $1$ was proved to be impossible, I think it would be more interesting to find out the maximum possible value of $\int_0^1 f(x) \, dx$ (and the corresponding function) with the given constraints $f(0) = 0$ and $\int_0^1 f'(x)^2 \, dx \leq 1$. –  Pratyush Sarkar Aug 12 '13 at 3:04

No, we can't ever reach $\frac23$. When applying the Cauchy-Schwarz inequality

$$\int_0^x 1\cdot f'(t)\, dt \leqslant \left(\int_0^x 1^2\,dt\right)^{1/2}\left(\int_0^x f'(t)^2\,dt\right)^{1/2} = \sqrt{x}\left(\int_0^x f'(t)^2\,dt\right)^{1/2},$$

we have two factors that ensure that the inequality $f(x) < \sqrt{x}$ is strict.

1. For small $x$, the integral $\int_0^x f'(t)^2\,dt$ is small, much smaller than $1$.
2. We only have equality in the Cauchy-Schwarz inequality if the two functions differ only by a constant factor.

The first defect is minimised if the bulk of the weight of $f'$ is concentrated near $0$, but that increases the difference in the CS inequality.

We can get $\int_0^1 f(x)\,dx = \frac{\sqrt{3}}{3}$ by choosing $f'(x) = \sqrt{3}(1-x)$, and that is the maximal possible value:

\begin{align} \int_0^1 f(x)\,dx &= \int_0^1 \int_0^x f'(t)\,dt\,dx\\ &= \iint_{0 \leqslant t \leqslant x\leqslant 1} f'(t)\,dt\,dx\\ &= \int_0^1 \int_t^1 f'(t)\,dx\,dt\\ &= \int_0^1 (1-t)f'(t)\,dt\\ &\leqslant \left(\int_0^1 (1-t)^2\,dt\right)^{1/2}\left(\int_0^1 f'(t)^2\,dt\right)^{1/2}\\ &= \sqrt{\frac13} = \frac{\sqrt{3}}{3}. \end{align}

-
Woah, clever. I like this solution. –  ADF Aug 12 '13 at 15:05


text:
# RD Sharma Solutions For Class 7 Maths Chapter 23 Data Handling - II (Central Values) Exercise 23.1

Get free PDF of RD Sharma Solutions for Class 7 Maths Exercise 23.1 of Chapter 23 Data Handling – II from the provided links. These PDF’s can be easily downloaded by the students. Our experts have uniquely formulated these questions to help the students gain competence. RD Sharma Solutions for Class 7 is one of the best study material designed for CBSE students. This exercise deals with the measures of central values.

## Download the PDF of RD Sharma Solutions For Class 7 Maths Chapter 23 – Data Handling – II (Central Values) Exercise 23.1

### Access answers to Maths RD Sharma Solutions For Class 7 Chapter 23 – Data Handling – II (Central values) Exercise 23.1

1. Ashish studies for 4 hours, 5 hours and 3 hours on three consecutive days. How many hours does he study daily on an average?

Solution:

Given Ashish studies for 4 hours, 5 hours and 3 hours on three consecutive days

Average number of study hours = sum of hours/ number of days

Average number of study hours = (4 + 5 + 3) ÷ 3

= 12 ÷ 3

= 4 hours

Thus, Ashish studies for 4 hours on an average.

2. A cricketer scores the following runs in 8 innings: 58, 76, 40, 35, 48, 45, 0, 100.

Find the mean score.

Solution:

Given runs in 8 innings: 58, 76, 40, 35, 48, 45, 0, 100

Mean score = total sum of runs/number of innings

The mean score = (58 + 76 + 40 + 35 + 48 + 45 + 0 + 100) ÷ 8

= 402 ÷ 8

= 50.25 runs.

3. The marks (out of 100) obtained by a group of students in science test are 85, 76, 90, 84, 39, 48, 56, 95, 81 and 75. Find the

(i) Highest and the lowest marks obtained by the students.

(ii) Range of marks obtained.

(iii) Mean marks obtained by the group.

Solution:

In order to find the highest and lowest marks, we have to arrange the marks in ascending order as follows:

39, 48, 56, 75, 76, 81, 84, 85, 90, 95

(i) Clearly, the highest mark is 95 and the lowest is 39.

(ii) The range of the marks obtained is: (95 – 39) = 56.

(iii) From the following data, we have

Mean marks = Sum of the marks/ Total number of students

Mean marks = (39 + 48 + 56 + 75 + 76 + 81 + 84 + 85 + 90 + 95) ÷ 10

= 729 ÷ 10

= 72.9.

Hence, the mean mark of the students is 72.9.

4. The enrolment of a school during six consecutive years was as follows:

1555, 1670, 1750, 2019, 2540, 2820

Find the mean enrolment of the school for this period.

Solution:

Given enrolment of a school during six consecutive years as follows

1555, 1670, 1750, 2019, 2540, 2820

The mean enrolment = Sum of the enrolments in each year/ Total number of years

The mean enrolment = (1555 + 1670 + 1750 + 2019 + 2540 + 2820) ÷ 6

= 12354 ÷ 6

= 2059.

Thus, the mean enrolment of the school for the given period is 2059.

5. The rainfall (in mm) in a city on 7 days of a certain week was recorded as follows:

 Day Mon Tue Wed Thu Fri Sat Sun Rainfall (in mm) 0.0 12.2 2.1 0.0 20.5 5.3 1.0

(i) Find the range of the rainfall from the above data.

(ii) Find the mean rainfall for the week.

(iii) On how many days was the rainfall less than the mean rainfall.

Solution:

(i) The range of the rainfall = Maximum rainfall – Minimum rainfall

= 20.5 – 0.0

= 20.5 mm.

(ii) The mean rainfall = (0.0 + 12.2 + 2.1 + 0.0 + 20.5 + 5.3 + 1.0) ÷ 7

= 41.1 ÷ 7

= 5.87 mm.

(iii) Clearly, there are 5 days (Mon, Wed, Thu, Sat and Sun), when the rainfall was less than the mean, i.e., 5.87 mm.

6. If the heights of 5 persons are 140 cm, 150 cm, 152 cm, 158 cm and 161 cm respectively, find the mean height.

Solution:

The mean height = Sum of the heights /Total number of persons

= (140 + 150 + 152 + 158 + 161) ÷ 5

= 761 ÷ 5

= 152.2 cm.

7. Find the mean of 994, 996, 998, 1002 and 1000.

Solution:

Mean = Sum of the given numbers/Total number of given numbers

Mean = (994 + 996 + 998 + 1002 + 1000) ÷ 5

= 4990 ÷ 5

= 998.

8. Find the mean of first five natural numbers.

Solution:

We know that first five natural numbers = 1, 2, 3, 4 and 5

Mean of first five natural numbers = (1 + 2 + 3 + 4 + 5) ÷ 5

= 15 ÷ 5

= 3

9. Find the mean of all factors of 10.

Solution:

We know that factors of 10 are 1, 2, 5 and 10

Arithmetic mean of all factors of 10 = (1 + 2 + 5 + 10) ÷ 4

= 18 ÷ 4

= 4.5

10. Find the mean of first 10 even natural numbers.

Solution:

The first 10 even natural numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20.

Mean of first 10 even natural numbers = (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20) ÷ 10

= 110 ÷ 10

= 11

11. Find the mean of x, x + 2, x + 4, x + 6, x + 8

Solution:

Mean = Sum of observations ÷ Number of observations

Mean = (x + x + 2 + x + 4 + x + 6 + x + 8) ÷ 5

Mean = (5x + 20) ÷ 5

Mean = 5 (x + 4) ÷ 5

Mean = x + 4

12. Find the mean of first five multiples of 3.

Solution:

The first five multiples of 3 are 3, 6, 9, 12 and 15.

Mean of first five multiples of 3 are = (3 + 6 + 9 + 12 + 15) ÷ 5

= 45 ÷ 5

= 9

13. Following are the weights (in kg) of 10 new born babies in a hospital on a particular day: 3.4, 3.6, 4.2, 4.5, 3.9, 4.1, 3.8, 4.5, 4.4, 3.6 Find the mean $${\overline{X}}$$

Solution:

We know that

$${\overline{X}}$$ = sum of observations/ number of observations

$${\overline{X}}$$ = sum of weights of babies/ number of babies

$${\overline{X}}$$ = (3.4 + 3.6 + 4.2 + 4.5 + 3.9 + 4.1 + 3.8 + 4.5 + 4.4 + 3.6) ÷ 10

$${\overline{X}}$$ = (40) ÷ 10

$${\overline{X}}$$ = 4 kg

14. The percentage of marks obtained by students of a class in mathematics are:

64, 36, 47, 23, 0, 19, 81, 93, 72, 35, 3, 1 Find their mean.

Solution:

Mean = sum of the marks obtained/ total number of students

= (64 + 36 + 47 + 23 + 0 + 19 + 81 + 93 + 72 + 35 + 3 + 1) ÷ 12

= 474 ÷ 12

= 39.5%

15. The numbers of children in 10 families of a locality are:

2, 4, 3, 4, 2, 3, 5, 1, 1, 5 Find the mean number of children per family.

Solution:

Mean number of children per family = sum of total number of children / total number of families

= (2 + 4 + 3 + 4 + 2 + 3 + 5 + 1 + 1 + 5) ÷ 10

= 30 ÷ 10

= 3

Thus, on an average there are 3 children per family in the locality.

16. The mean of marks scored by 100 students was found to be 40. Later on it was discovered that a score of 53 was misread as 83. Find the correct mean.

Solution:

Given n = the number of observations = 100, Mean = 40

Mean = sum of observations/total number of observations

40 = sum of the observations/ 100

Sum of the observations = 40 x 100

Thus, the incorrect sum of the observations = 40 x 100 = 4000.

Now,

The correct sum of the observations = Incorrect sum of the observations – Incorrect observation + Correct observation

The correct sum of the observations = 4000 – 83 + 53

The correct sum of the observations = 4000 – 30 = 3970

Correct mean = correct sum of the observations/ number of observations

= 3970/100

= 39.7

17. The mean of five numbers is 27. If one number is excluded, their mean is 25. Find the excluded number.

Solution:

We know that

Mean = sum of five numbers/5 = 27

So, sum of the five numbers = 5 x 27 = 135.

Now,

The mean of four numbers = sum of the four numbers/4 = 25

So, sum of the four numbers = 4 x 25 = 100.

Therefore, the excluded number = Sum of the five number – Sum of the four numbers

The excluded number = 135 – 100

= 35.

18. The mean weight per student in a group of 7 students is 55 kg. The individual weights of 6 of them (in kg) are 52, 54, 55, 53, 56 and 54. Find the weight of the seventh student.

Solution:

We know that

Mean = sum of weights of students/ number of students

Let the weight of the seventh student be x kg.

Mean = (52 + 54 + 55 + 53 + 56 + 54 + x)/ 7

55 = (52 + 54 + 55 + 53 + 56 + 54 + x)/ 7

55 x 7 = 324 + x

385 = 324 + x

x = 385 – 324

x = 61 kg.

Therefore weight of seventh student is 61kg.

19. The mean weight of 8 numbers is 15 kg. If each number is multiplied by 2, what will be the new mean?

Solution:

Let x1, x2, x3…x8 be the eight numbers whose mean is 15 kg. Then,

15 = x1 + x2 + x3+……+ x8 /8

x1 + x2 + x3 + …+ x8 = 15 × 8

x1 + x2 + x3 +…+ x8 = 120.

Let the new numbers be 2x1, 2x2, 2x3 …2x8.

Let M be the arithmetic mean of the new numbers.

Then,

M = 2x1 + 2x2 + 2x3 +…+ 2x8/8

M = 2 (x1 + x2 + x3 + …+ x8)/8

M = (2 × 120)/8

= 30

20. The mean of 5 numbers is 18. If one number is excluded, their mean is 16. Find the excluded number.

Solution:

Let x1, x2, x3, x4 and x5 be five numbers whose mean is 18. Then,

18 = Sum of five numbers ÷ 5

Hence, sum of five numbers = 18 × 5 = 90

Now, if one number is excluded, then their mean is 16.

So,

16 = Sum of four numbers ÷ 4

Therefore sum of four numbers = 16 × 4 = 64.

The excluded number = Sum of five observations – Sum of four observations

The excluded number = 90 – 64

Therefore The excluded number = 26.

21. The mean of 200 items was 50. Later on, it was discovered that the two items were misread as 92 and 8 instead of 192 and 88. Find the correct mean.

Solution:

Given n = Number of observations = 200

Mean = sum of observations/ number of observations

50 = sum of observations/ 200

Sum of the observations = 50 x 200 = 10,000.

Thus, the incorrect sum of the observations = 50 x 200

Now,

The correct sum of the observations = Incorrect sum of the observations – Incorrect observations + Correct observations

Correct sum of the observations = 10,000 – (92 + 8) + (192 + 88)

Correct sum of the observations = 10,000 – 100 + 280

Correct sum of the observations = 9900 + 280

Correct sum of the observations = 10,180.

Therefore correct mean = correct sum of the observations/ number of observations

= 10180/200

= 50.9

22. The mean of 5 numbers is 27. If one more number is included, then the mean is 25. Find the included number.

Solution:

Given Mean = Sum of five numbers ÷ 5

Sum of the five numbers = 27 × 5 = 135.

Now, New mean = 25

25 = Sum of six numbers ÷ 6

Sum of the six numbers = 25 × 6 = 150.

The included number = Sum of the six numbers – Sum of the five numbers

The included number = 150 – 135

Therefore the included number = 15.

23. The mean of 75 numbers is 35. If each number is multiplied by 4, find the new mean.

Solution:

Let x1, x2, x3…x75 be 75 numbers with their mean equal to 35. Then,

35 = x1 + x2 + x3 + …..+ x75 /75

x1 + x2 + x3 + …..+ x75 = 35 × 75

x1 + x2 + x3 +…+ x75 = 2625

The new numbers are 4 x 1, 4 x 2, 4 x 3…4 x 75

Let M be the arithmetic mean of the new numbers. Then,

M = 4x1 + 4x2 + 4x3 +…+ 4x75/75

M = 4 (x1 + x2 + x3 + …+ x75)/75

M = (4 × 2625)/75

= 140


text:
# 🆕 Check the dashboard

Find debug info on the dashboard

The dashboard’s debug page will indicate you basic debug info like if one of Reachy’s motor or force sensor is disconnected or if you forgot to turn on Reachy’s motors before booting its computer.

You can also check the status of reachy_sdk_server.service in the dashboard’s services page to get error messages.

## Access the dashboard#

From the robot:
Access the dashboard at 127.0.0.1:3972

From any other device on the same network as the robot:
Access the dashboard at <robot-ip>:3972


text:
# Ac analysis with dependent source

#### craka

1. Homework Statement
Determine I1

independent AC voltage source 10 cos 10^5t
inductor 60mH
capacitor 5nF
dependent voltage controlled voltage source 3Vx

2. Homework Equations
KVL & KCL

3. The Attempt at a Solution

$$\begin{array}{l} - 7.071 + 6000I_1 - j6000I_3 = 0 \\ I_3 = I_1 - I_2 \\ - 7.071 + 6000I_1 - j6000(I_1 - I_2 ) = 0 \\ j6000(I_1 - I_2 ) + 3v_x - j2000 = 0 \\ v_x = 6000I_1 \\ - 7.071 + 6000I_1 - j6000(I_1 - I_2 ) = 0 \\ j6000(I_1 - I_2 ) + 3(6000I_1 ) - j2000 = 0 \\ 24000I_1 = 7.071 + j2000 \\ I_1 = (2.94625 \times 10^{ - 4} ) + j0.083333 \\ I_1 = 0.0833\angle 89.79^o \\ \end{array}$$

The answer for I1 is given as 0.33mA @ 171.9 degrees

Could someone help me here please, gone over this for a couple hours still stumped.
Cheers

#### Attachments

• 3.4 KB Views: 250
Related Engineering and Comp Sci Homework Help News on Phys.org

#### Kruum

Quickly going through your attempt, all I can spot is that your complex value for the independent supply is incorrect. cos(x)=sin(x+pi/2)...

One sure-fire way to get it is to use mesh method.

#### craka

If I convert to the frequency domain and have my voltage as my zero phase reference it shouldn't matter should it? ie voltage 7.071 V @ 0degrees

I thought what I tried was mesh loop analysis.

#### Kruum

Yes, of course you can calculate it as cos as the reference. But in that case you need to start playing with the signs a bit more due to the definition of inductor and capacitor.

And by mesh method I meant using the matrix form. You only get 2x2 matrix, which is easy to deal with.

Edit: Going more in-depth to your calculations, looks like your sign convention differs from mine. Can you explain the method how you got rows 1 and 4? Especially in row 4 why do you only use the impedance of the capcitor or is it a typo? And is the voltage source given as 10cos(10^5t), because in the picture the frequency is 15.9kHz, which would give w=31831pi?

Last edited:

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving


text:
# Aerodynamics, mathematical problems of

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problems involved in the solution of fundamental equations in aerodynamics, which are an accurate description of the laws of motion of a gas-like medium and its interaction forces with solids moving in this medium. Turbulence, for which no satisfactory mathematical model has yet been established, forms an exception to this rule. Processes without turbulence or in which the turbulence can be neglected may, in principle, be sufficiently completely investigated by a mathematical solution of the relevant equations. The real possibilities of solving such equations differ from the theoretical; accordingly, the orders of the individual terms in the equations are estimated, and on this basis simplified mathematical models are established for different classes of aerodynamic processes, in which one or more "small" terms in the complete set of aerodynamic equations are discarded. Even for such simplified models analytic solutions can be obtained only for the simplest cases, and numerical methods are widely employed in practical work in aerodynamics. These methods have become prevalent with the introduction of electronic computers.

As regards the nature of the equations involved in aerodynamical problems, these can be subdivided into four large classes.

1) Problems in the aerodynamics of an ideal (non-viscous and thermally non-conducting) fluid, the word "fluid" being used in this context to denote both liquids and gases.

2) Problems in the aerodynamics of viscous fluids.

3) Problems in the aerodynamics of radiant gases.

4) Problems in the aerodynamics of rarefied gases.

The equations of an ideal fluid are obtained from the general equations by making the Reynolds number — which describes the order of the ratio between the inertial and the viscous forces — grow without limit.

In most problems of applied aerodynamics of aircraft and rockets, and in seacraft hydrodynamics, the Reynolds numbers are very large ( $10 ^ {6}$– $10 ^ {8}$), so that the model of an ideal fluid yields an accurate description of flow processes around solid bodies under these conditions, except for the boundary layer zone. The system of aerodynamic equations of an ideal fluid are:

$$\rho \frac{d \mathbf V }{dt} = \rho \mathbf g - \nabla p$$

(equation of motion);

$$\frac{d \rho }{dt} + \rho \mathop{\rm div} \mathbf V = 0$$

(continuity equation);

$$\rho \frac{d}{dt} \left ( \mathbf V ^ \frac{2}{2} + U \right ) = \rho ( \mathbf g , \mathbf V ) - \mathop{\rm div} ( p \mathbf V )$$

(equation of conservation of energy). Here $\mathbf V$ is the velocity vector, $\rho$ is the density, $p$ is the pressure, $\mathbf g$ is the acceleration due to gravity and $U$ is the internal energy. It is a system of equations of the first order with respect to the functions sought. This is also true for a mixture of chemically interacting gases: The equations of the transfer of the mixture components are added to the above equations, while the heat generation due to chemical reactions is added to the energy equation.

A number of simplified models can be derived from the model of an ideal fluid. The most important ones are the model of a weight-less fluid and the model of an incompressible fluid. The former model is the limiting case of the general model as the Froude number increases without limit; the Froude number describes the order of the ratio of the inertial to the gravity forces. This model is very accurate when applied to the aerodynamics of aircraft and rockets (with Froude numbers of order $10 ^ {2}$– $10 ^ {3}$). However, the force of gravity becomes significant in seacraft hydrodynamics and especially so in meteorological applications (with Froude number of order $\leq 1$).

The equation of an incompressible fluid is obtained from the general system by putting $\rho= \textrm{ const }$. This form of the equation of motion is obtained as the Mach number tends to zero (which is the relationship of the characteristic velocity of sound in a fluid medium). In the general case this is not true; it is only valid for Froude numbers $\geq 1$( i.e. if $\textrm{ Ma } \ll \textrm{ Fr }$), and also for quasi-stationary processes, in which the Strouhal number (the criterion of similitude of non-stationary motions of the liquid or the gas) is of order $\leq 1$.

The simplest model is that of an incompressible fluid. In this model the entire system of equations is reduced to the single Laplace equation for the velocity potential $\Phi$( $\mathop{\rm grad} \Phi = \mathbf V$). However, it is only in the case of plane-parallel motions of a weight-less fluid and axi-symmetric motion around solids of revolution that the solution of aerodynamic problems is reduced to problems in classical potential theory.

The problem of flow around a given shape is solved by classical methods of the theory of conformal mapping. The flow around solids of revolution at zero angle of incidence is determined by the classical solution of the exterior Neumann problem for the Laplace equation. In view of the practical importance of these problems (determination of the characteristics of wing profiles and fuselages) many methods for their solutions have been developed that achieve an acceptable accuracy by means of relatively simple calculations. If electronic computers are employed, there is no point in using these methods, since accurate numerical methods for solving these problems belong to the simplest problems of computational mathematics.

In the spatial case the formal solution of the Neumann problem for the velocity potential will correspond to the physical picture of the flow only in very exceptional cases. In a viscous fluid a solid is followed by a turbulent trail. As the Reynolds number increases, this trail becomes thinner (for a continuous flow) and in the limit it becomes an infinitely thin turbulent surface, the intensity of the turbulent layer of which vanishes only in exceptional cases (e.g. in the case of plane-parallel motion). Accordingly, the problems involving discontinuous velocity potentials in the region of the flow zone represent the real spatial problems of an incompressible fluid. The location of this discontinuity surface is unknown, and for this reason the exact problem of flow around solid bodies in the presence of a discontinuous potential surface is a very complicated non-linear problem. It is only in the linear approximation, i.e. on the assumption that the solid around which the fluid is flowing disturbs the basic mainstream only to a small extent (a thin wing at a small angle of incidence) that the problem could be solved. In such a case the surface of turbulence can be considered as horizontal, and it may be assumed that the potential along the stream has a constant discontinuity. The normal derivatives of the velocity potential are given on the projection plane of the solid. Analytic solutions have been obtained on these assumptions (for a circular and an elliptic wing). Numerical methods of solving this problem for other shapes of wings are available and simplify the task to a considerable extent (cf. Wing theory).

The principal difficulty in problems involving a heavy fluid is that the boundary conditions for the potential on the free surface are non-linear. Exact solutions have been found for planar problems of the theory of waves only. Little progress has been made on the spatial problem of waves with a finite amplitude. On the other hand, the linearized problem (small wave amplitudes) has been largely solved, and the calculation of the wave resistance to the motion of a ship is actually based on the linear theory.

The possibility of aircraft travelling at near-sonic and supersonic speeds stimulated the search for solutions of aerodynamic problems of compressible fluids. The problems involving subsonic, transsonic and supersonic velocities are all treated in substantially different manners. In the subsonic range the aerodynamic equations remain elliptic and the solutions qualitatively resemble those obtained for incompressible fluids.

In the first period of research on the aerodynamics of compressible fluids much research was done on the effect of compressibility on the aerodynamic characteristics of the solids around which the flow takes place. All these methods involve some sort of linearization of the velocity of the unperturbed flow. Khristianovich's method is an exception, permitting as it does to obtain accurate solutions of the equations of subsonic aerodynamics, though for a solid whose shape is unknown in advance.

In practice, the difficulties involved in the problems of subsonic aerodynamics are the same as those to be dealt with when investigating incompressible fluids: problems involving plane-parallel and axi-symmetric flows may be fairly easily solved by numerical methods (cf. Integral-relation method; Adjustment method). In the case of spatial flows with discontinuous velocity potential, solutions can only be obtained for linearized equations. These equations can be converted, by an elementary transformation, into equations for incompressible fluids in which the boundary conditions and the conditions on the surface of discontinuity are the same as those for incompressible fluids.

The zone of transsonic aerodynamics is, from the point of view of mathematics, including numerical mathematics, the most difficult to study. The presence of local supersonic (parabolic) zones which almost always terminate in shock waves precludes any analytic approximation. Moreover, these problems display the principal drawback of equations of elliptic type — the propagation of the effect of any particular disturbance throughout the space. The most suitable method for solving the problems of transsonic aerodynamics is the adjustment method, in which a non-stationary aerodynamic problem is solved (starting from an arbitrary initial state), and the solution of the stationary problem of transsonic flow around a solid is obtained as a limit solution of the non-stationary problem as the time tends to infinity. This method is very laborious, but is quite feasible if electronic computers are employed.

As regards mathematical methods of study, supersonic aerodynamics may be subdivided into three fields.

1) Pure supersonic flow.

2) Mixed flow, with formation of local subsonic zones.

3) High-temperature hypersonic flow producing chemical reactions in the gas.

At higher velocities and, consequently, at higher temperatures ionization takes place and radiation processes become significant. This part of hypersonic aerodynamics may be said to form a separate field, since the mathematical problems involved are quite different from those arising in the case of "transparent" gases.

Problems in pure supersonic flow have been most thoroughly investigated. This included the development of numerical methods of characteristics, finite difference methods and methods of semi-characteristics, by which relatively simple solutions may be obtained not only for plane-parallel or axi-symmetric but also for spatial flows. The linear theory of supersonic flow has also been exhaustively studied. Analytic solutions can be obtained of numerous practical problems. The latter may be somewhat complicated by the occurrence of weak shock waves, but even these difficulties are of a computational and not of a principle character.

Practically speaking, these problems are limited to flows around sharply pointed solids and to the problems of internal aerodynamics (calculations of jet nozzles). However, if the supersonic velocities are very high, sharp-edged solids are not employed (charring of the edges), while a local subsonic flow always occurs at the blunt end. Having computed this region, the remaining (pure supersonic) flows are calculated by the methods of supersonic aerodynamics.

As compared with the aerodynamics of transsonic flow, the advantage of the problems of aerodynamics of supersonic flow with local subsonic zones (even though such zones are mixed) consists in the fact that such zones usually represent bounded narrow domains in the vicinity of the blunt ends. This is why it was possible to develop effective numerical methods for the computation of local subsonic zones (the method of integral relations, the method of inverse problems and the adjustment method). It should be noted, however, that up to the time of writing (1977) no rigorous mathematical investigations of these problems have ever been performed. There is also no proof that their solutions do exists and are unique. Accordingly, numerical methods are developed on the assumption that the conditions of continuous velocities and accelerations during the transition from the subsonic to the supersonic zone ensure the uniqueness of a solutions, which is physically realistic. This hypothesis is confirmed by all the results of numerical calculations.

Up till here, it has been said that the computation of the local subsonic zone (including the supersonic part of the region of influence) is followed by calculations of the flow by methods of purely supersonic aerodynamics.

There is another type of problems, which are connected with hypersonic flow around thin solids (prior to the initiation of chemical reactions). Hypersonic flows around thin solids (excepting the blunt-shaped regions) are distinguished by small changes in the velocity component along the main stream. For this reason the equations may be simplified so that the problem of flow around a thin body of a given shape (in the planar and axi-symmetric cases) becomes identical to the one-dimensional stationary problem. Many important qualitative characteristics of hypersonic flows have been obtained in this way and approximate similitude laws have been established; these are also extensively employed in the analysis of the results of numerical calculations, which are then reduced to very compact relations in a very wide range of Mach numbers and geometric parameters of the solid.

Aerodynamic problems concerned with gases undergoing chemical reactions involve the simultaneous solution of the equations of motion and the equations of chemical kinetics. While the resulting systems of equations are more complicated, the numerical methods employed in their solution are essentially the same as those employed in the aerodynamics of an ideal gas, except that they are much more laborious. Such calculations are extensively carried out in practical work, and since modelling natural conditions in aerodynamic tubes cannot be realized in this temperature range, the aerodynamic characteristics of hypersonic instruments are mainly obtained by numerical methods.

There are two principal trends in the theory of viscous fluids — the theory of the complete equations of a viscous fluid (Navier–Stokes equations), and boundary-layer theory. The boundary-layer equations represent the principal term of the asymptotic expansion of the Navier–Stokes equations in the vicinity of the boundary of the solid to which the fluid particles adhere, completely or in part. It is this fact which radically disturbs the solutions obtained for ideal fluids near the surface of the solid around which the flow takes place. The order of the error involved in the boundary-layer equations is $1 / \sqrt { \mathop{\rm Re}}$, which means that the theory is valid for large Reynolds numbers only, and even then only in the regions of smooth continuous flow. Despite the fact that the boundary-layer equations contain all the main terms of viscous stresses, their mathematical structure is much simpler. While the complete equations are of elliptic type, the boundary-layer equations are parabolic with characteristics directed along the normal to the surface of the solid. It is therefore possible to conduct the computations "layer by layer" , i.e. to pass from one section of the boundary layer to another, irrespective of the conditions prevailing outside the range between the sections.

One consequence of the practical importance of the theory of boundary layers (computations of the resistance, of the surface temperatures, of the rate of destroyment of the surface under hypersonic flight conditions) was the development of numerous approximate methods for performing such calculations (the Pohlhausen method, the one-parameter Kochin–Loitsyanskii method, etc.). However, if electronic computers are employed, all these approximate methods become superfluous, since an accurate numerical solution of the boundary-layer equations presents no difficulties, even in the difficult case of a hot gas in which chemical reactions are taking place. The numerical routines may be so constructed that at each step the system of equations is split, with respect to the $x$- variable (along the tangent to the surface of the solid), into a number of separate second-order differential equations which, from the computational point of view, is highly advantageous.

What was said above applies to plane-parallel and to axi-symmetric flows. The structure of three-dimensional boundary layers is more complicated. The equations of the three-dimensional layers themselves vary significantly with the geometry of the solid around which the flow takes place. The development of methods of calculation of three-dimensional boundary layers is much less advanced, partly because calculations on plane sections are often used in practice; this, despite the lack of a theoretical basis, often yields sufficiently accurate results. There are no principal difficulties in solving the exact equations of a three-dimensional layer.

Except for a few special cases, in which analytic solutions could be obtained, the development of methods for solving the complete system of equations for a viscous liquid is due to the introduction of electronic computers. The principal difficulty in solving such equations is their high order and the fact that the region of influence is unbounded. In the simplest case of stationary motion of a viscous incompressible fluid with a constant viscosity coefficient one obtains a system of two equations of the second order with respect to the stream function $\psi$ and vortex velocity $\omega$. They may be written in the dimension-less form as follows:

$$\Delta \omega = 2 \left ( \frac{\partial \psi }{\partial y } \frac{\partial \omega }{\partial x } - \frac{\partial \psi }{\partial x } \frac{\partial \omega }{\partial y } \right ) ,$$

$$\Delta \psi = \omega .$$

The boundary conditions on solid boundaries are only for the stream function $\psi = \textrm{ const }$, $\partial \psi / \partial n = 0$. The solution of this non-linear system of the fourth order can only be obtained by some kind of iterative process. The iterative process is so constructed that separate equations for the vortex velocity and for the flow function are obtained at each iteration step. Such a separation must take place not only in the equations, but in the boundary conditions as well. Thus, by choosing the solution of the ideal fluid equations as the first approximation, one obtains the following set of iterations:

$$\Delta \omega _ {n+1} - 2 \frac{\partial \omega _ {n+1} }{\partial x } = 2 \left ( \frac{\partial \phi _ {n} }{\partial y } \frac{\partial \omega _ {n} }{\partial x } - \frac{\partial \phi _ {n} }{\partial x } \frac{\partial \omega _ {n} }{\partial y } \right ) ,$$

$$\Delta \psi _ {n+1} = \omega _ {n+1} ,$$

with the following boundary conditions on the solid boundaries:

$$\psi _ {n+1} = {\textrm{ const } } ,\ \ \omega _ {n+1} = \alpha \frac{\partial \psi _ {n} }{\partial n } + \omega _ {n} ,$$

where $\phi = \psi - y$, which means that a separate set of equations for the vortex flow and the flow function is thus obtained in the ( $n + 1$)- st approximation. The solution may be obtained in a similar manner by the adjustment method, but it is not necessary in such a case to take a real system of non-stationary equations; rather, one may consider the system of parabolic equations:

$$\frac{\partial \omega }{\partial t } = \Delta \omega - 2 \left ( \frac{\partial \psi }{\partial y } \frac{\partial \omega }{\partial x } - \frac{\partial \psi }{\partial x } \frac{\partial \omega }{\partial y } \right ) ,$$

$$\frac{\partial \psi }{\partial t } = \Delta \psi - \omega ,$$

which is more convenient to handle in numerical calculations.

The problem of flow around a body by a viscous fluid involves not merely the development of computation methods, but also research on fundamental questions. Thus, it is unclear whether a stationary solution exists at large Reynolds numbers (e.g. for the case of flow around a cylinder). Numerical methods yield acceptable solutions up to Reynolds numbers of several hundreds, but above these values of the Reynolds number the numerical calculations no longer converge.

Problems in the aerodynamics of radiant gases and of rarefied gases have been solved only for the simplest cases. However, in view of the practical importance of these problems (in both these fields), mathematical models of the relevant phenomena have been proposed. Such models very often are no limiting cases of the complete equations (model of gray radiation, model of diffusive radiation in the theory of radiant gases, Crook's model in the theory of rarefied gases), but merely qualitative reflections of the relationships described in the complete equations. The volume of the computations involved in solving these problems is very large, and success in this field will depend on the possibilities for increasing the effectiveness of electronic computers.

#### References

 [1] N.E. Kochin, I.A. Kibel', N.V. Roze, "Theoretical hydrodynamics" , Interscience (1964) (Translated from Russian) [2] S.A. Khristianovich, "Gas flow around bodies with large subsonic speed" , Moscow (1940) (In Russian) [3] L.J. [L.I. Sedov] Sedov, "Two-dimensional problems in hydrodynamics and aerodynamics" , Acad. Press (1965) (Translated from Russian) MR0210373 Zbl 0131.40901 [4] W. Hayes, "Hypersonic flow theory" , Acad. Press (1966) MR0101731 Zbl 0148.21502 [5] G.G. Chernyi, "Gas flow with large supersonic speed" , Moscow (1959) (In Russian) [6] O.M. Belotserkovskii, P.I. Chushkin, "The numerical solution of problems in gas dynamics" , Basic developments in fluid dynamics , 1 , Acad. Press (1965) pp. 1–126 MR0204017 [7] P.I. Chushkin, "The method of characteristics for spatial supersonic flows" , Moscow (1968) (In Russian) [8] , Flow around blunted bodies of a supersonic gas flow , Moscow (1967) (In Russian) [9] , Numerical methods in fluid mechanics , Wroclaw (1969) (In Polish)

In the equation of the conservation of energy $( \mathbf g , \mathbf V )$ denotes the scalar product of two vectors.


text:
Discover The Ground Plan For HGTV Smart Dwelling 2017

Patrons want to see the property structure, to visualise the movement of the rooms, how they fit in relation to others rooms and picture their furniture inside the house. You may as well use a floor plan to speak with contractors and distributors about an upcoming transforming venture. Ground plans might also include details of fixtures like sinks, water heaters, furnaces, and many others. All of our normal modular home plans usually embrace a flooring plan and an exterior elevation.

Whereas portray partitions or designing furnishings has its own problems, building a staircase generally is a daunting task to even essentially the most skilled builder. RoomSketcher Pro subscribers can select from an array of 3D Flooring Plan profile choices – select your 3D perspective, wall-prime coloration, textures and materials.

Constructing stairs not solely requires a builder’s talent but additionally a watch for arithmetic and of course, a plan. In case you’re about to try and increase the house’s plan then try to assume ahead to however you’d just like the home to feel. Much like a map the orientation of the view is downward from above, but not like a conventional map , a plan is drawn at a selected vertical place (generally at about 4 feet above the floor).

However while it’s easy to search out plans of some of the most well-known homes, plenty of the other house plans do not exist except in books. The world of the lounge within the flooring plan is $$1\frac78 \textual content in\instances 2 \frac12 \textual content in = \frac7516 \text in^2.$$ Since the lengths scale by $80$, the realm scales by $80^2$.

Whenever you discover a few modular house plans that appeal to you, imagine living in each home. You too can take one in every of our modular house plans and make it into a completely different kind. Clear strains and natural colors ar different fashionable fashion choices which can in all probability be in your modern house plans.