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# American Institute of Mathematical Sciences

March  2015, 14(2): 407-419. doi: 10.3934/cpaa.2015.14.407

## Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift

 1 Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia 2 Dipartimento di Matematica E. De Giorgi, Università del Salento, 73100, Lecce 3 Dipartimento di Matematica "Ennio De Giorgi”, Università del Salento, C.P. 193, Lecce, I-73100 4 Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe

Received  December 2013 Revised  July 2014 Published  December 2014

We study second order elliptic operators whose diffusion coefficients degenerate at the boundary in first order and whose drift term strongly points outward. It is shown that these operators generate analytic semigroups in $L^2$ where they are equipped with their natural domain without boundary conditions. Hence, the corresponding parabolic problem can be solved with optimal regularity. In a previous work we had treated the case of inward pointing drift terms.
Citation: Simona Fornaro, Giorgio Metafune, Diego Pallara, Roland Schnaubelt. Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift. Communications on Pure & Applied Analysis, 2015, 14 (2) : 407-419. doi: 10.3934/cpaa.2015.14.407
##### References:
 [1] M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups,, Arch. Math., 70 (1998), 377.  doi: 10.1007/s000130050210.  Google Scholar [2] P. Daskalopoulos and P. M. N. Feehan, Existence, uniqueness and global regularity for degenerate elliptic obstacle problems in mathematical finance,, preprint, ().   Google Scholar [3] P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation,, J. Amer. Math. Soc., 11 (1998), 899.  doi: 10.1090/S0894-0347-98-00277-X.  Google Scholar [4] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000).   Google Scholar [5] C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology,, Ann. Math. Stud. 185, (2013).   Google Scholar [6] P. M. N. Feehan and C. Pop, Degenerate elliptic operators in mathematical finance and Hölder continuity for solutions to variational equations and inequalities,, preprint, ().   Google Scholar [7] P. M. N. Feehan and C. Pop, Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations,, J. Differential Equations, 256 (2014), 895.  doi: 10.1016/j.jde.2013.08.012.  Google Scholar [8] W. Feller, The parabolic differential equations and the associated semi-groups of transformations,, Ann. of Math., 55 (1952), 468.   Google Scholar [9] W. Feller, Diffusion processes in one dimension,, Trans. Amer. Math. Soc., 97 (1954), 1.   Google Scholar [10] S. Fornaro, G. Metafune, D. Pallara and J. Prüss, $L^p$-theory for some elliptic and parabolic problems with first order degeneracy at the boundary,, J. Math. Pures Appl., 87 (2007), 367.  doi: 10.1016/j.matpur.2007.02.001.  Google Scholar [11] S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Degenerate operators of Tricomi type in $L^p$-spaces and in spaces of continuous functions,, J. Differential Equations, 252 (2012), 1182.  doi: 10.1016/j.jde.2011.09.017.  Google Scholar [12] S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, One-dimensional degenerate operators in $L^p$-spaces,, J. Math. Anal. Appl., 402 (2013), 308.  doi: 10.1016/j.jmaa.2013.01.030.  Google Scholar [13] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar [14] C. Kienzler, Flat fronts and stability for the porous medium equation,, preprint, ().   Google Scholar [15] J. U. Kim, An $L^p$ a priori estimate for the Tricomi equation in the upper half-space,, Trans. Amer. Math. Soc., 351 (1999), 4611.  doi: 10.1090/S0002-9947-99-02349-1.  Google Scholar [16] K.-H. Kim, Sobolev space theory of parabolic equations degenerating on the boundary of $C^1$ domains,, Comm. Partial Differential Equations, 32 (2007), 1261.  doi: 10.1080/03605300600910449.  Google Scholar [17] H. Koch, Non-euclidean singular integrals and the porous medium equation,, Habilitation thesis (1999), (1999).   Google Scholar [18] J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of first order,, Comm. Pure Appl. Math., 20 (1967), 797.   Google Scholar [19] P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus,, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, (2004), 65.  doi: 10.1007/978-3-540-44653-8_2.  Google Scholar [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser Verlag, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar [21] A. Lunardi, Interpolation Theory,, Scuola Normale Superiore Pisa, (2009).   Google Scholar [22] G. Metafune, Analyticity for some degenerate one-dimensional evolution equation,, Studia Math., 127 (1998), 251.   Google Scholar [23] O. A. Oleinik and E. V. Radkevic, Second Order Equations with Non Negative Characteristic Form., Plenum Press, (1973).   Google Scholar [24] E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Princeton University Press, (2005).   Google Scholar [25] N. Shimakura, Partial Differential Operators of Elliptic Type,, Amer. Math. Soc., (1992).   Google Scholar [26] K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations,, Osaka J. Math., 32 (1995), 275.   Google Scholar [27] H. Tanabe, Functional Analytic Methods for Partial Diffeeential Equations,, Marcel Dekker, (1997).   Google Scholar [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978).   Google Scholar

show all references

##### References:
 [1] M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups,, Arch. Math., 70 (1998), 377.  doi: 10.1007/s000130050210.  Google Scholar [2] P. Daskalopoulos and P. M. N. Feehan, Existence, uniqueness and global regularity for degenerate elliptic obstacle problems in mathematical finance,, preprint, ().   Google Scholar [3] P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation,, J. Amer. Math. Soc., 11 (1998), 899.  doi: 10.1090/S0894-0347-98-00277-X.  Google Scholar [4] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000).   Google Scholar [5] C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology,, Ann. Math. Stud. 185, (2013).   Google Scholar [6] P. M. N. Feehan and C. Pop, Degenerate elliptic operators in mathematical finance and Hölder continuity for solutions to variational equations and inequalities,, preprint, ().   Google Scholar [7] P. M. N. Feehan and C. Pop, Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations,, J. Differential Equations, 256 (2014), 895.  doi: 10.1016/j.jde.2013.08.012.  Google Scholar [8] W. Feller, The parabolic differential equations and the associated semi-groups of transformations,, Ann. of Math., 55 (1952), 468.   Google Scholar [9] W. Feller, Diffusion processes in one dimension,, Trans. Amer. Math. Soc., 97 (1954), 1.   Google Scholar [10] S. Fornaro, G. Metafune, D. Pallara and J. Prüss, $L^p$-theory for some elliptic and parabolic problems with first order degeneracy at the boundary,, J. Math. Pures Appl., 87 (2007), 367.  doi: 10.1016/j.matpur.2007.02.001.  Google Scholar [11] S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Degenerate operators of Tricomi type in $L^p$-spaces and in spaces of continuous functions,, J. Differential Equations, 252 (2012), 1182.  doi: 10.1016/j.jde.2011.09.017.  Google Scholar [12] S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, One-dimensional degenerate operators in $L^p$-spaces,, J. Math. Anal. Appl., 402 (2013), 308.  doi: 10.1016/j.jmaa.2013.01.030.  Google Scholar [13] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar [14] C. Kienzler, Flat fronts and stability for the porous medium equation,, preprint, ().   Google Scholar [15] J. U. Kim, An $L^p$ a priori estimate for the Tricomi equation in the upper half-space,, Trans. Amer. Math. Soc., 351 (1999), 4611.  doi: 10.1090/S0002-9947-99-02349-1.  Google Scholar [16] K.-H. Kim, Sobolev space theory of parabolic equations degenerating on the boundary of $C^1$ domains,, Comm. Partial Differential Equations, 32 (2007), 1261.  doi: 10.1080/03605300600910449.  Google Scholar [17] H. Koch, Non-euclidean singular integrals and the porous medium equation,, Habilitation thesis (1999), (1999).   Google Scholar [18] J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of first order,, Comm. Pure Appl. Math., 20 (1967), 797.   Google Scholar [19] P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus,, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, (2004), 65.  doi: 10.1007/978-3-540-44653-8_2.  Google Scholar [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser Verlag, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar [21] A. Lunardi, Interpolation Theory,, Scuola Normale Superiore Pisa, (2009).   Google Scholar [22] G. Metafune, Analyticity for some degenerate one-dimensional evolution equation,, Studia Math., 127 (1998), 251.   Google Scholar [23] O. A. Oleinik and E. V. Radkevic, Second Order Equations with Non Negative Characteristic Form., Plenum Press, (1973).   Google Scholar [24] E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Princeton University Press, (2005).   Google Scholar [25] N. Shimakura, Partial Differential Operators of Elliptic Type,, Amer. Math. Soc., (1992).   Google Scholar [26] K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations,, Osaka J. Math., 32 (1995), 275.   Google Scholar [27] H. Tanabe, Functional Analytic Methods for Partial Diffeeential Equations,, Marcel Dekker, (1997).   Google Scholar [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978).   Google Scholar
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# Understanding the concept | Finite Element Method | Advantages of FEA | Disadvantages of FEA | Applications of FEA | Finite Element Analysis

Finite element analysis is a great method to solve complex engineering problems.
Before we start with the concept of Finite element analysis (FEA), let us understand FEA practically.

## Finite Element Analysis (FEA) - Practical Approach:

→ Let us assume that you are a Design Engineer working in a car company.

→ Your role is to design a car.

→ While designing a car you should not only cover up the outer body shape and geometry, i.e., Aesthetics.

Designing (for car component) means to derive the dimensions (parameters) require to generate a car component with given constraints and based on affecting forces/loads and/or moments.

→ So, you start by deriving force and moments acting on different components of the car while designing.

→ But, to set the limit or capacity to withstand specific load/moment for the car depends on the material of the components of the car.

→ Materials have their own Ultimate tensile strengths (UTS) and Yield strengths (exceeding UTS after some limit results in failure).

→ So, important step in designing is the selection of material.

→ After you calculate the parameters based on forces and moments based on materials, you need to test its design by doing testing of the components.

Testing is the crucial process to check design of the components after its final manufacturing or in-between fabrication by applying same or more loads/moments on the components as per design.

→ This process of first time trail of a component after designing and manufacturing is known as Prototyping.

→ This will upon testing gives you the result as the valid proof of designed component upon applying pre-assumed loads/moments. You will find the result as feedback for your design. You may get success or failure based on application of loads/moments on the component.

→ After getting results, you can update design and can again go for prototyping.

→ All this process are like a loop - to get optimized, efficient and cost-effective product. This in simple words known as Research and Development (R&D) for/of the product.

→ Research and Development is a time consuming and costly process as to get optimized results you have to improve the design and product based on prototyping again and again.

→ To get this simplified we start using computers for iterations in design calculations.

→ With the development in the industrial sectors, man develops great simulation softwares like Ansys., PV-Elite, Solid works simulation, Solid Edge simulation, and many others.

Simulation is the process of generating required environment (means to help designer to constraint the designed component along with application of loads and/or moments) for testing the designed component.

→ This process of simulation of designed components in computers helps in very magical way for prototyping by reducing time as well as cost of prototyping.

→ Now, the point to explain the whole concept this way practically is because all these softwares uses the method known as Finite Element Method (FEM).

→ Let us understand the method which became revolution in the field of Designing (impacting prototyping).

## Concept of Finite Element Analysis (FEA):

→ For simple engineering problems we always use analytical methods, which uses mathematical expressions that gives us unknown quantity at any location in a body/structure.

→ As a limitation of such methods, it is valid for infinite number of locations in body/structure.

→ But, for many complex engineering problems, it becomes very difficult or impossible to obtain analytical solutions. In such cases, the method known as Finite Element Method (FEM) is used.

## Finite Element Method (FEM):

→ In Finite Element Method, the body/structure is divided into finite number of smaller units known as elements.

→ This process of dividing whole body into finite number of elements is known as discretization or meshing.

Fig-1: Discretization or Meshing

→ As shown in fig-1 above, we have stepped shaft at the left side and upon discretization (meshing) we get something like on right side.

→ The assemblage of these elements then represents the original body or structure.

→ These elements are considered to be interconnected at joints which are know as nodes.

Fig-2: Illustrating nodes in elements

→ As we can see in the fig-2, which is the detailed view of mesh; we have triangular element mesh and all the elements are connected by dot like junctions known as nodes.

→ In FEM, solutions are formulated for each element and then combined to obtain the solution for the original body/structure.

→ This approach is known as "going from part to whole."

→ In order to make analysis more simpler, it is only at nodes where the continuity or equilibrium equations are required to be satisfied.

→ In FEM, the amount of data to be handled is dependent upon the number of elements into which original body/structure is discretized.

→ For large number of elements, it is formidable task to handle the volume of data manually, and hence use of computer is evitable in such cases.

→ As we are satisfying equilibrium/continuity equations at the nodes of the elements, the accuracy of the solution increases with the increase in the number of elements (fine mesh).

→ However, more number of elements results in increased computation.

## Boundary Conditions:

→ In boundary value problems, a solution is sought in the region of the body/structure, while the values of certain variables are prescribed on the boundaries of the region.

→ The values of variables prescribed on the boundaries of the region are called as boundary conditions.

### Types of Boundary Conditions:

1. Geometric (Essential) Boundary Conditions

2. Force (Natural) Boundary Conditions

Let us discuss them in detail with examples.

### 1. Geometric (Essential) Boundary Conditions :

• In structural mechanics problems, the geometric boundary conditions includes prescribed displacements and slopes.

• These are also known as kinematic boundary conditions

Fig-3: Geometric boundary conditions

• As we can see in the above figure, the stepped shaft is fixed from the one end.
• This is the boundary condition provided to shaft to constraint its position in the space.
• This kind of boundary conditions which is related to geometry, are geometric boundary conditions.

### 2. Force (Natural) Boundary Conditions :

• In structural mechanics problems, the force boundary conditions includes prescribed forces and moments.

• These are also known as static boundary conditions

Fig-3: Force boundary conditions

• As we can see in the above figure, load of 150N is applied to one end of the fixed shaft.
• This is the boundary condition provided to shaft simulating the load acting on the shaft.
• This kind of boundary conditions which is related to loadings, are force boundary conditions.

## Advantages of Finite Element Analysis (FEA):

→ The physical problems which were so far insolvable and complex for any closed bound solutions can be analyzed by using this method.

→ This method can be efficiently applied to bodies (or structures) with irregular geometry.

→ It can deal with any kind of boundary conditions.

→ This method can deal with material anisotropy and non-homogeneity very easily.

## Disadvantages of Finite Element Analysis (FEA):

→ The accuracy of results from this method highly depends upon the degree of discretization.

→ Manual judgement is essential in discretization process.

→ Finite element analysis requires large computer memory and time, and hence cost involved is high.

→ In this method, errors in the input data may go undetected and erroneous solution obtained therefrom may appear acceptable.

→ This method is complicated and so not useful for simple problems.

## Applications of Finite Element Analysis (FEA):

→ This method was originated for the solution of structural problems (structural analysis).

→ However, the theory on which it is based has made possible its successful application for the solutions of complex problems in other fields of engineering such as:
Mechanical design, mechanical vibrations, aeronautical engineering, heat transfer, fluid flow, soil and rock mechanics, bio-engineering, etc.

→ The different analysis that can be carried out using finite element analysis are:

\begin{aligned} &1) \ \text{Static-linear analysis} \\ &2) \text{ Static non-linear analysis}\\ &3) \text{ Dynamic linear analysis} \\ &4)\text{ Dynamic non-linear analysis}\\ &5) \text{ Thermal analysis} \\ &6) \ \text{Fluid flow analysis} \\ \end{aligned}

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## Intermediate Algebra for College Students (7th Edition)

$True$
$\frac{14}{19}+\frac{5}{19}=\frac{19}{19}=1$, and 1 is an element of the natural numbers, so that means the first statement is true. $12\times(50)^{2}\times(50+10)-12\times(50+10)\times(50)^{2}=0$ $30000\times60-720\times2500=0$ $1800000-1800000=0$ $0=0$, the second statement is also true. Since both of the statements are true, the answer is $true$.


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Biomart Vs Amigo
2
8
Entering edit mode
12.2 years ago
Mike Dewar ★ 1.6k

Say I'm interested in the molecular function of Ywhaz. On AmiGO, which seems to be the basic web-tool immediately accessible from the GO website, I get 3 associations about protein binding.

On Biomart I get 4 results, the extra one being a "transcription factor binding" (which has caused me some trouble).

I was wondering:

1. What are the differences between these two services?
2. In general, whether there was a canonical source I should consult, or if I should become a lot more careful source material when querying online services
biomart gene • 2.5k views
1
Entering edit mode

I use the Ensemble Genes 58 (SANGER UK) database, selecting mouse genes. My Amigo query is mosue too. I have no rational way to pick between databases, so I use Sanger because I met a girl once who worked there. She was nice.

0
Entering edit mode

Which Biomart server and database did you query? When I query biomart.org using MGI as the database, I get exactly the same three molecular function terms as you get on AmiGO, which is just how it should be since I can see that AmiGO used MGI as the source of annotations.

0
Entering edit mode

For mouse genes I would recommend using MGI - it is the model organism database for mouse and thus presumably has the best annotation of mouse genes.

0
Entering edit mode

I think Lars is right. I would choose the data source where the provenance of the annotation is most clearly indicated.

5
Entering edit mode
12.2 years ago

The human annotation contains a Molecular Function annotation for Transcription Factor Binding (last item here)

The mouse does not.

There are frequently differences of this kind between the different species annotations; I typically find the Human has more annotations than the mouse.

0
Entering edit mode

Thanks! Though I'm aware of the differences between mouse and human annotations in general, this is bound to be the source of the discrepancy, no? This response doesn't answer the question about the differences between Amigo and Biomart (which, seemingly, might actually be about the Sanger DB and whatever amiGO points at), though, so I'll wait to see if any other answers are forthcoming before ticking...

5
Entering edit mode
12.2 years ago

The EnsEMBL annotation for Ywhaz (ENSMUSG00000022285) has 3 transcripts annotated to GO:0008134 (transcription factor binding) with evidence code IEA. The MGI annotation does not contain this. As the EnsEMBL gene has been merged with a Vega manually-annotated gene, it is possible that the extra term came from there... No, that hunch was wrong; the Vega versions have fewer GO terms annotated which suggests to me that the extra one is from the automatic gene build.


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# American Institute of Mathematical Sciences

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June 2017, 37(6): 3285-3299. doi: 10.3934/dcds.2017139

## Local well-posedness of the Camassa-Holm equation on the real line

 1 Department of Mathematics, The Graduate Center, City University of New York, New York, NY 10016, USA 2 Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY 11210, USA

Received  December 2016 Revised  January 2017 Published  February 2017

In this paper we prove the local well-posedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms with an appropriate decaying condition. This work was motivated by G. Misiolek who proved the same result for the Camassa-Holm equation on the periodic domain. We use the Lagrangian approach and rewrite the equation as an ODE on the Banach space. Then by using the standard ODE technique, we prove existence and uniqueness. Finally, we show the continuous dependence of the solution on the initial data by using the topological group property of the diffeomorphism group.

Citation: Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
##### References:
 [1] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics Springer, New York, 1998. [2] R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Advances in Mathematics, 154 (2000), 229-257. doi: 10.1006/aima.1999.1883. [3] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Archive for Rational Mechanics and Analysis, 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [5] R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Advances in Applied Mechanics, 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [6] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, Journal of Differential Equations, 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [7] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, Journal of Functional Analysis, 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. [8] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Annales de l'institut Fourier, 50 (2000), 321-362. doi: 10.5802/aif.1757. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. [10] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Communications on Pure and Applied Mathematics, 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [11] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Mathematische Zeitschrift, 233 (2000), 75-91. doi: 10.1007/PL00004793. [12] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, Journal of Physics A: Mathematical and General, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201. [13] A. Constantin and H. McKean, A shallow water equation on the circle, Communications on Pure and Applied Mathematics, 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [14] A. Constantin and W. A. Strauss, Stability of peakons, Communications on Pure and Applied Mathematics, 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [15] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, Journal of Nonlinear Science, 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x. [16] R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988. [17] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Mathematics, 92 (1970), 102-163. doi: 10.2307/1970699. [18] B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D: Nonlinear Phenomena, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [19] A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic CH equation, Communications in Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746. [20] H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms American Mathematical Society 226 (2003), vi+60 pp. doi: 10.1090/S0065-9266-2013-00676-4. [21] B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Advances in Mathematics, 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4. [22] B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 365 (2007), 2333-2357. doi: 10.1098/rsta.2007.2012. [23] S. Lang, Differential Manifolds Second edition. Springer-Verlag, New York, 1985. [24] J. Lenells, A variational approach to the stability of periodic peakons, Journal of Nonlinear Mathematical Physics, 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2. [25] J. Lenells, Stability for the periodic Camassa-Holm equation, Mathematica Scandinavica, 97 (2005), 188-200. doi: 10.7146/math.scand.a-14971. [26] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, Journal of Differential Equations, 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007. [27] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, Journal of Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. [28] F. Linares, G. Ponce and T. Sideris, Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons, arXiv: 1609.06212. [29] H. McKean, Breakdown of a shallow water equation, Asian Journal of Mathematics, 2 (1998), 867-874. doi: 10.4310/AJM.1998.v2.n4.a10. [30] H. McKean, Shallow water & the diffeomorphism group, Applied and Industrial Mathematics, Venice2 (2000), 135-143. [31] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, Journal of Geometry and Physics, 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [32] G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geometric & Functional Analysis GAFA, 12 (2002), 1080-1104. doi: 10.1007/PL00012648. [33] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Analysis: Theory, Methods} & Applications, 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X.

show all references

##### References:
 [1] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics Springer, New York, 1998. [2] R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Advances in Mathematics, 154 (2000), 229-257. doi: 10.1006/aima.1999.1883. [3] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Archive for Rational Mechanics and Analysis, 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [5] R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Advances in Applied Mechanics, 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [6] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, Journal of Differential Equations, 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [7] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, Journal of Functional Analysis, 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. [8] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Annales de l'institut Fourier, 50 (2000), 321-362. doi: 10.5802/aif.1757. [9] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586. [10] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Communications on Pure and Applied Mathematics, 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [11] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Mathematische Zeitschrift, 233 (2000), 75-91. doi: 10.1007/PL00004793. [12] A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, Journal of Physics A: Mathematical and General, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201. [13] A. Constantin and H. McKean, A shallow water equation on the circle, Communications on Pure and Applied Mathematics, 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [14] A. Constantin and W. A. Strauss, Stability of peakons, Communications on Pure and Applied Mathematics, 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [15] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, Journal of Nonlinear Science, 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x. [16] R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988. [17] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Mathematics, 92 (1970), 102-163. doi: 10.2307/1970699. [18] B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D: Nonlinear Phenomena, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [19] A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic CH equation, Communications in Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746. [20] H. Inci, T. Kappeler and P. Topalov, On the regularity of the composition of diffeomorphisms American Mathematical Society 226 (2003), vi+60 pp. doi: 10.1090/S0065-9266-2013-00676-4. [21] B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Advances in Mathematics, 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4. [22] B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 365 (2007), 2333-2357. doi: 10.1098/rsta.2007.2012. [23] S. Lang, Differential Manifolds Second edition. Springer-Verlag, New York, 1985. [24] J. Lenells, A variational approach to the stability of periodic peakons, Journal of Nonlinear Mathematical Physics, 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2. [25] J. Lenells, Stability for the periodic Camassa-Holm equation, Mathematica Scandinavica, 97 (2005), 188-200. doi: 10.7146/math.scand.a-14971. [26] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, Journal of Differential Equations, 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007. [27] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, Journal of Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. [28] F. Linares, G. Ponce and T. Sideris, Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons, arXiv: 1609.06212. [29] H. McKean, Breakdown of a shallow water equation, Asian Journal of Mathematics, 2 (1998), 867-874. doi: 10.4310/AJM.1998.v2.n4.a10. [30] H. McKean, Shallow water & the diffeomorphism group, Applied and Industrial Mathematics, Venice2 (2000), 135-143. [31] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, Journal of Geometry and Physics, 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [32] G. Misiolek, Classical solutions of the periodic Camassa-Holm equation, Geometric & Functional Analysis GAFA, 12 (2002), 1080-1104. doi: 10.1007/PL00012648. [33] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Analysis: Theory, Methods} & Applications, 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X.
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Article Contents
Article Contents

# Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

• For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\}$. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0$. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.
Mathematics Subject Classification: 35J20, 35Q55, 35B40.

 Citation:

•  [1] A. Ambrosetti, A. Malchiodi and W.-M Ni, Singularly perturves elliptic equation with symmetry: existence of solutions concentrating on spheres. I, Comm. Math. Phys., 235 (2003), 427-466.doi: 10.1007/s00220-003-0811-y. [2] A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. d'Anayse Math., 98 (2006), 317-348.doi: 10.1007/BF02790279. [3] A. Ambrosetti, A. Malchiodi and A. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Rational Mech. Anal., 159 (2001), 253-271.doi: 10.1007/s002050100152. [4] A. Ambrosetti and D. Ruiz, Radial solutions concentrating on sphere of nonlinear Schrödinger equations with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 889-907.doi: 10.1017/S0308210500004789. [5] M. Badiale. and T. D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation, Nonlinear Anal., 49 (2002), 947-985.doi: 10.1016/S0362-546X(01)00717-9. [6] V. Benci. and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Diff. Equat., 184 (2002), 109-138.doi: 10.1006/jdeq.2001.4138. [7] J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential, Nonlinear Anal., 50 (2002), 1135-1151.doi: 10.1016/S0362-546X(01)00805-7. [8] J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 1877-1904.doi: 10.1081/PDE-200040205. [9] J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 138A (2008), 975-987.doi: 10.1017/S0308210507000236. [10] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 165 (2002), 295-316.doi: 10.1007/s00205-002-0225-6. [11] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II, Calc. Var. Partial Differential Equations, 18 (2003), 207-219.doi: 10.1007/s00526-002-0191-8. [12] D. Cao, E. S. Noussair and S. Yan, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations, to appear in Trans. of AMS. doi: 10.1090/S0002-9947-08-04348-1. [13] D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency, Math. Ann, 336 (2006), 925-948.doi: 10.1007/s00208-006-0021-y. [14] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1998), 120-156.doi: 10.1016/0022-0396(88)90021-6. [15] E. N. Dancer and S. Yan, On the existence of multipeak solutions for nonlinear field equations on $R^N$, Discrete Contin. Dynam. Systems, 6 (2000), 39-50. [16] E. N. Dancer, K. Y. Lam and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations, Abstr. Apply. Anal., 3 (1998), 293-318.doi: 10.1155/S1085337598000578. [17] M. del Pino and P. L. Felmer, Local mountaion passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.doi: 10.1007/BF01189950. [18] M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.doi: 10.1006/jfan.1996.3085. [19] M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, 15 (1998), 127-149.doi: 10.1016/S0294-1449(97)89296-7. [20] M. del Pino and P. L. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.doi: 10.1007/s002080200327. [21] M. del Pino, P. L. Felmer and O. H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential, Nonlinear Anal. TMA, 34 (1998), 979-989.doi: 10.1016/S0362-546X(97)00593-2. [22] M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pur. App. Math., LX (2007), 113-146.doi: 10.1002/cpa.20135. [23] A. Floer and A. Weinstein, Non spreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.doi: 10.1016/0022-1236(86)90096-0. [24] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Berlin, Heidelberg, New York and Tokyo: Springer. Grundlehren 224. (1983).doi: 10.1007/978-3-642-61798-0. [26] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820.doi: 10.1080/03605309608821208. [27] Y. Kabeya and K. Tanaka, Uniqueness of positive solutions of semilinear elliptic equations in $R^N$ and Séré's non-degeneracy condition, Comm. Partial Differential Equations, 24 (1999), 563-598.doi: 10.1080/03605309908821434. [28] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928. [29] T. Kato, Perturbation Theory for Linear Operators, second ed., Grundlehren Math. Wiss., Band 132, Springer, Berlin-New York, (1976). [30] O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity, Proc. Roy. Soc. Edinburgh Sect. A, 139A (2009), 833-852.doi: 10.1017/S0308210508000309. [31] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations, 16 (1996), 585-615.doi: 10.1080/03605309108820770. [32] Y. Y. Li, On a singular perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980. [33] F. Mahmoudi and A. Malchiodi, Concentration on minimal submainifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525.doi: 10.1016/j.aim.2006.05.014. [34] F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold, Geom. Funct. Anal., 16 (2006), 924-958.doi: 10.1007/s00039-006-0566-7. [35] F. Mahmoudi, F. S. Sánchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture for general submanifolds, J. Differential Equations, 258 (2015), 243-280.doi: 10.1016/j.jde.2014.09.010. [36] A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, G.A.F.A., 15-16 (2005), 1162-1222.doi: 10.1007/s00039-005-0542-7. [37] A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 15 (2002), 1507-1568.doi: 10.1002/cpa.10049. [38] A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.doi: 10.1215/S0012-7094-04-12414-5. [39] S. Minakshisundaram and and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math., 1 (1949), 242-256. [40] Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equations, 13 (1988), 1499-1519.doi: 10.1080/03605308808820585. [41] Y. G. Oh, Correction to: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equations, 14 (1989), 833-834.doi: 10.1080/03605308908820631. [42] Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multile well potential, Comm. Math. Phys., 131 (1990), 223-253. [43] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.doi: 10.1007/BF00946631. [44] Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency, Calc. Var. Partial Differential Equations, 29 (2007), 365-395.doi: 10.1007/s00526-006-0070-9. [45] B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $R^N$, Ann. Mat. Pura Appl., 184 (2002), 73-83.doi: 10.1007/s102310200029. [46] W. Strauss, Existence of solitary waves in higher demensions, Comm. Math. Phys., 55 (1977), 149-162. [47] J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238 (2007), 201-219.doi: 10.1016/j.jde.2007.03.018. [48] L. P. Wang, J. Wei and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture for general hypersurfaces, Comm. Partial Differential Equations, 36 (2011), 2117-2161.doi: 10.1080/03605302.2011.580033. [49] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. [50] Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations, J. Differential Equations, 159 (1999), 102-137.doi: 10.1006/jdeq.1999.3650.


text:
ganeti

Ganeti.Storage.Lvm.LVParser

Description

Logical Volumer information parser

This module holds the definition of the parser that extracts status information about the logical volumes (LVs) of the system from the output of the lvs command.

Synopsis

# Parser implementation

lvCommand :: String Source #

The command providing the data, in the format the parser expects

lvParams :: [String] Source #

The parameters for getting the data in the format the parser expects

lvParser :: Parser [LVInfo] Source #

The parser for a whole diskstatus file.


text:
# heavytailed

## October 6, 2014

### Kernel Random Effects

Filed under: Uncategorized — heavytailed @ 10:57 pm

I ran into a couple of grad students studying for a big AI exam yesterday. I barged in wondering what they were doing with all the pretty inner products and norms on the board, precipitating a moment of intense awkwardness. One of them asked how much I knew about Mercer’s theorem, and if I could explain it. The best I could do was recall the gist of Mercer’s theorem: which is that if you have a Kernel $K: \mathcal{X}\times\mathcal{X} \rightarrow \mathbb{R}$ that obeys certain conditions (positive-definiteness and boundedness), then you can think of it as an inner product in a larger feature space. In other words (letting $\mu$ be a measure on $\mathcal{X}$):

$(1) \;\;\; \sum_{i,j=1}^n a_i a_j K(x_i,x_j) \geq 0 \; \; \forall x_i \in \mathcal{X}, a_i \in \mathbb{R}$

$(2) \;\;\;\; \int_X \int_X K(x,z)^2 d\mu(x)d\mu(z) < \infty$

$(1),(2) \Rightarrow K(x,z) = \sum_{k=1}^\infty \lambda_k\phi_k(x)\phi_k(z)$

Where the convergence is uniform and absolute. This is, basically, obvious, and is the extension of the linear algebraic notion that positive semidefinite matrices can be written as an inner product, and the point is that so long as you can construct a positive-definite K for which condition (2) holds, you can solve certain optimization functions over a Hilbert space without ever directly calculating the image of your data in that space. For support vector machines (the typical application of Kernels) this basically means if you magically pick a high-dimensional representation in which there is a good separating hyperplane, you can find a really good hypothesis.

Okay well what does it mean for there to be a good separating hyperplane in a large (possibly infinite) dimensional space? It means that there’s a $(w,b)$ such that

$w^\dagger x_i + b < 0 \;\; \mathrm{if} \; \; y_i = 0 \;\;\; w^\dagger x_i + b > 0 \; \mathrm{otherwise}$

Note that for Kernel methods, the inner product is replaced by the kernel, that is $K(w,x) + b < 0$ (etc). One way of interpreting this is as showing that there is a high-dimensional inner product space characterized by K on which the classification function is linear; another way is to interpret this as saying that in that high-dimensional space, the data cluster on either side of the dividing line. Perhaps there are many clusters, but on either side of that line: the explanatory data is somehow better aligned with the responses.

(Note that this is different from the Manifold hypotheses articulated in previous posts. The weak MH says that the data are distributed on a low-dimensional manifold; the Strong MH says that the hypothesis class has a simple representation in that very same manifold. The Kernel-MH says the hypothesis class has a simple representation on some manifold of dimensionality up to that of the data, in a potentially infinite ambient space. This is why Kernel methods don’t magically escape the curse of dimensionality.)

Obviously, a better alignment of features with response makes all manner of machine learning methods work more appropriately. In particular, in the regression setting, one would like to identify features better correlated with the response. The only issue is that while the predictors can fly into the infinite-dimensional Kernel space, the response can’t. And even if you could, all you get from $K(x,z)$ is the Kernel matrix $K_{ij} = K(x_i,x_j)$, with no access to $\phi(x_i)$ and $\phi(x_j)$.

But. For Random effects (see for instance here), $K$ is all you need. This is one of those (excellent, but few) non-trivial cases where you’re forced to use random effects. This blog post will provide some background and demonstrate some results. There’s not all that much to develop — merely pointing out that K is a matrix of appropriate dimensions is enough for anyone to take this forward and run with it — except to point out relationships between the current literature on Kernels and assumptions about random effects models (and these mostly have to do with normalization).

From the Annals of Bad Examples: Why Use Kernels

“Doing a good thing for a bad reason”

The reason to use a Kernel is to project the data into a space in which it is linearly separable. What this is really doing is choosing a space in which the image of the separating criterion function is linear, enabling linear classification in the image of the data, which corresponds to nonlinear classification of the unprojected data. This works really well for many types of hypotheses, but one type on which it tends to fail miserably are disjoint intervals. Yet somehow the canonical example is as follows:

Got some red points, some blue points (the y-axis is just 0 or 1 depending on the class, it’s a 1-dimensional classification) and they’re obviously not linearly separable since they’re disjoint intervals. Now there is a transformation $C: \mathbb{R}\rightarrow\mathbb{R}$ that will wind up separating the red points from the blue points, it’s the cheating kernel, where you simply multiply the endpoints together to form a polynomial, i.e. for endpoints $z_i \in Z$:

$C(x) = \prod_{i=1}^{|Z|}(x-z_i) = (x_i-3)(x_i-2)(x_i+4)(x_i+1)(x_i-6)(x_i-8)(x_i+11)(x^i-12)$

and “magically” by constructing a polynomial with zeros on the edges of the intervals the two classes are linearly separable!

Therefore the Kernel trick works! My beef is: this is really a kernel dirty trick: the Kernel function itself encodes the hypothesis, and so if we didn’t know it ahead of time, we’d’ve been screwed. Before we move on to realistic examples using “real” kernels, it’s worth asking what the Kernel Matrix looks like for this, and we can choose either to normalize within the kernel feature space or not (the normalization sets $\tilde{K}(x,y) = \frac{K(x,y)}{\sqrt{K(x,x)K(y,y)}}$)

For a reason to be determined, normalizing the Kernel (right) reveals the huge structure in the data (the fuzziness is due to a small constant I added to deal with 0 valued denominators), and very obviously the eigenspaces of these two kernel matrices will be different. The one on the right should be very easy to classify, the latter maybe less so. Looking at the projections, it’s clear there’s a much larger margin in the latter case

The latter case clearly indicates that a random effects model (or fixed effect using PCs) could easily pull signal out of this. This immediately leads to means of extending the Kernel trick to general regression problems. In part this is already well-known; for the general regression problem of solving:

$\mathrm{min}_w L(X^\dagger w)$

where L is a loss function (such as $L = (y-y_\mathrm{pred})^2$) we can rewrite as

$\mathrm{min}_v L(X^\dagger Xv)$

simply by letting $w = Xv$. For L2-penalized regression the transformation works as well (the penalty term is $+ \lambda v^\dagger X^\dagger Xv$). The story here is a little bit more restrictive, but at the same time a bit cooler: by being a bit more explicit about an underlying probability model (e.g. gaussian, probit, etc), the kernel matrix K is taken as a hypothesis about the covariance of observations. If you’ve found a “good” representation of your data – not just a space where it’s separable but where distances mean something, in terms of “close points are correlated, far points are not,” the Kernel matrix should have the property of explaining a large proportion of your data.

Let’s not over-trivialize: this is a pretty radical re-interpretation of the Kernel matrix and what it’s trying to accomplish. Usually we only care about separability: the fact that the magnitude of the positive points (in the first figure) is far smaller than that of the negative points, and that overall there’s a vast range not associated at all with the response, is in general ignored. It’s separable: let’s move on. By contrast, considering the Kernel matrix as though it were part of a variance components model means caring about the structure of the data, beyond smearing it out enough that the class is more or less separable. It may even be worth making normative statements: good Kernels provide both improved separability and discover/retain covariance in the predictors that translates into covariance in the response. (This is, at its core, a statement about scale).

Digression: Kernel Normalization

Before moving to the empirical section (which can be summarized by: blindly sticking K into generalized linear models sometimes works), it’s worth linking the way scale relates to linear models to the way the Kernel represents a map into a latent high-dimensional feature space.

One thing to notice is that Kernel normalization restricts the Kernel feature space to a unit sphere. This follows from Mercer’s theorem: we can just rewrite

$K(x,y)/\sqrt{K(x,x)K(y,y)} = \frac{\phi(x)}{||\phi(x)||}^\dagger \frac{\phi(y)}{||\phi(y)||} = \cos \theta_{\phi_x\phi_y}$

for suitably defined inner product (it may be an integral). Clearly you lose a dimension from this, and so if K corresponded to an N-dimensional space, then the normalized version is the (N-1)-dimensional sphere. This precisely explains the behavior observed above: restricting the one-dimensional polynomial transform $C(x)$ above to the 0-dimensional sphere (the points {1,-1}) effectively makes $K(x,y)/\sqrt{K(x,x)K(y,y)} = \mathrm{sgn}(C(x))\mathrm{sgn}(C(y))$: ergo the constant strobes in the normalized Kernel matrix.

Conceptually at least, normalization is a statement that the original kernel maps the data into a space where the angles (suitably defined) are predictive, but the position of the data in absolute terms is irrelevant. In some sense, there’s less smearing, and this can be amenable to methods not just looking for a separating hyperplane, but looking for a space where distance between points has some kind of meaning.

One way to examine the impact of normalization is by its effect on the canonical feature map. That is, given a kernel $K(x,y): X \times X \rightarrow \mathbb{R}$ it induces a number of objects. One such object is the functional $L_K: \ell^2(X) \rightarrow \ell^2(X)$ defined as $g(y) = L(f) = \int_X f(x)K(x,y)dx$. This can be thought of as a change of bases much like a Laplace transform; and the canonical feature map is the application of this functional to the delta-representation of the datapoint, i.e. $\phi_K(x) = K_x(y) = L_K(\delta(x)) = \int_X K(z,y)\delta(z-x)dz$.

There are many features consistent with a Kernel, not all of which draw from Mercer’s theorem like the canonical map does. However the exact feature depends on how you define your inner product, for instance if $\langle f(x),g(y)\rangle = \int\int f(x)K(x,y)g(y) dxdy$ then the above representation is good to go. By contrast you might want to factor this as $\int\int f(x)\sqrt{K(x,y)}\sqrt{K(x,y)}g(y)dxdy$, which is perfectly fine given that K is positive definite, in this case you might want the canonical feature to be $\int_X \sqrt{K(z,y)}\delta(z-x)dz$

So what does the canonical feature look like for a polynomial kernel? On the left is a standard quadratic kernel, and on the right, the same kernel but normalized. Remember, these are canonical features, in this case $\phi_z(x,y) = \int_x\int_y \sqrt{K(x,y)}\delta(x-z_x)\delta(y-z_y)$ is parametrized by the 2-vector $z = (z_x,z_y) = (1,-1)$, the black dot in the pictures.

Increasing the polynomial to degree six:

So Kernel normalization here appears to make the canonical features look more like radial functions. What are inner products of radial functions? Effectively similarities: radial functions whose centers are close by (in comparison to the function volume, say) have a large inner product, those with centers far apart have a small inner product. This is going to link us back to random effects, but it’s worth addressing (in our own poor fashion) what’s going on geometrically when one normalizes a Kernel.

Some Geometry of Kernel Normalization

So normalizing a kernel is basically rescaling so that the diagonal is one. Note that the standard RBF $\exp(-||x-y||^2)$ is automatically normalized, which is of course related to the intuitive connection between inner products of radial functions and similarities. But what really does it mean if $K(x,y) = (x^\dagger y + c)^d$, when normalized, results in a better classification that $K(x,y) = \exp(-||x-y||^2)$? (These are .gifs by the way, if they’re not animating click on them to view).

Since these kernels are just inner products, they should be rotation invariant; that is invariant under unitary operations on L2 (the space of the canonical representation). In the case of kernels that are functions of a norm or inner product (i.e. $K(x,y) = K(x^\dagger y)$ or $K(x,y) = K(||x-y||)$), rotations of feature space induce a unitary operator in Kernel space – and therefore the Kernel matrix will be invariant under such rotations.

But one big difference between the exponential and polynomial kernel is that the former is translation invariant, and the latter is not.

However, unlike the polynomial kernel, the exponential kernel is invariant under translations, while the polynomial kernel has a “minimum” (0) where $x^\dagger y = -c$. In some sense, while pure radial kernels encode “locality”, other kernels encode biases about the ambient feature space. The polynomial kernel treats points as similar if they are (1) far away from the minimum c, and (2) collinear (or nearly so) with each other and c. The ANOVA kernel (where rotations of feature space do not induce unitary transformations in Kernel space) encodes for axis-parallel similarities, and a wavelet kernel (which is admissible though not positive definite) encodes for a patterned similarity:

So basically, one kernel performing “better” than another means that the canonical feature of the “superior” kernel function encodes something about the classification geometry in the ambient feature space that is closer to the truth. So for instance, if the normalized polynomial kernel outperforms the standard RBF, it indicates that similarities are not well modeled by gaussian-scaled distances, but instead by distances that are biased by overall distance from (and angle to) the reference point c.

But what does it mean if normalization improves performance? What’s the difference between the normalized kernel and the unnormalized one? Well if a kernel is a generalized covariance, then a normalized kernel is some kind of a generalized correlation. That is, we go from some unknown space $\mathcal{D}$ with an inner product onto a sphere in that space: $\{x \in \mathcal{D}: \langle x, x \rangle = 0_D\}$ where $0_D$ is the 0 element of that space. This is a kind of poor man’s projective space: we’ve made our destination manifold scale-invariant (with respect to rescaling the data). Indeed for SVMs, this kind of normalization is recommended as theoretically it improves the margin.

Would we want to do this for a random effect model? Should we do this for Kernel-PCA? There are really two answers here: for RBF Kernels, there is no actual question, as these are automatically normalized. However for things like the polynomial kernel, the question is one of whether we want to equalize scales across the features in our Hilbert space. From a physical point of view, data has units associated with it; so if x is impressions/day then x^2 is impressions^2/day^2. This automatically places features on different scales, so it seems to me from this perspective normalization is appropriate. On the other hand, if the data is normalized to be unitless, then features in the Kernel space may be “physical” in some sense.

Ultimately, there’s nothing stopping a Kernel from being used in an old-school linear model. However, there are connections between Kernel machines and Linear Mixed models


text:
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###### \${selected_topic_name}
• Notes

If $P(A)=0.3, P(B)=0.2,$ and $P(A \cap B)=0.1$

$P(A)=0.3$
$P(B)=0.2$
$P(A \cap B)=0.1$

$P\left(A^{\prime}\right)=1-P(A)=1-0.3=0.7$

b) $P(A \cup B)=P(A)+P(B)-P(A \cap B)$

$=0.3+0.2-0.1=0.4$

c) $P\left(A^{\prime} \cap B\right)$

$P(B)=P(A \cap B)+P(A^{\prime} \cap B)$

$0.2=0.1+P(A^{\prime} \cap B)$

$\Rightarrow P(A^{\prime} \cap B) = 0.1$

d) $P\left(A \cap B^{\prime}\right)$

$P(A)=P(A \cap B)+P(A \cap B^{\prime})$

$0.3=0.1+P(A \cap B^{\prime})$

$P(A \cap B^{\prime})=0.2$

e) $P\left[(A \cup B)^{\prime}\right]$

$=1-P(A \cup B)=0.6$

f) $P\left(A^{\prime} \cup B\right)$

$=P\left(A^{\prime}\right)+P(B)-P\left(A^{\prime} \cap B\right)$

$=0.7+0.2-0.1=0.8$


text:
CSE 11 Programming Assignment 1

Learning Goals

• Set up and test your Java development environment
• Write functions that take integer parameters and return integer results.
• Practice using design recipes

FAQs

Permission denied issue while trying to run lecture code (Mac/Linux)

If you are working on Mac or Linux machines and you want to run the lecture code (or PA1) by using the run script, you might encounter an error that says

-bash: ./run: Permission denied

To solve this issue, run the following command first:

chmod +x run

before running the command ./run FirstExample.

Collaboration

Different assignments in this course have different collaboration policies.

On this assignment, you can collaborate with anyone in the course, including sharing code. Feel free to offer help to other students, converse about the *assignment, and so on.

In your submission, give credit to all students and course staff who helped you with this assignment by noting their name and how you used their ideas or work. Note that using someone’s work without giving credit to them is a violation of academic integrity.

Submission Checklist

Here is a list of items to check for before submitting PA1.

• DesignRecipeExamples.java
• Perimeter Method
• At least 2 tests, each with different parameters and a comment (at least 2) with the expected value of the test
• BorderArea Method
• At least 2 tests, each with different parameters and a comment (at least 2) with the expected value of the test
• Converter Method
• At least 2 tests, each with different parameters and a comment (At least 2) with the expected value of the test
• 1 comment describing what the converter does, what parameters it takes, and any relevant outside sources
• Combiner Method
• At least 2 tests, each with different parameters and a comment (at least 2) with the expected value of the test
• At least 1 test with parameters that produce an incorrect output and a comment explaining why the output is incorrect OR 1 comment describing why all parameters will produce the correct output
• 1 comment describing what the combiner does, what parameters it takes, and any relevant sources.
• transcript.txt
• Converter method output using the run command with at least 2 tests
• Combiner method output using the run command with at least 2 tests

Part 1 - Setting Up

• On your own computer: General instructions for setting up your environment can be found in this document. There are specific video tutorials for Windows and Mac.
• In the CSE Labs: Watch and follow the instructions from this video: this video. In text form, the instructions are:
• If the lab computer is showing a Windows deskto or login, restart it and choose CentOS (first option) on the screen you get when it starts up.
• Locate your CSE11 account, which you can find using the ACMS account lookup tool: https://sdacs.ucsd.edu/~icc/index.php. The username starts with “cs11s222”. You will need to do an initial password reset, which you only need to do once. It may take up to 20 minutes for the new password to be effective.
• Once you are logged in, you can follow the instructions for using Visual Studio Code to get fully set up

If you get stuck at any point, do any one of the following:

• Take a screenshot and ask on Piazza! (Not sure how to take a screenshot? Try Googling it for your particular platform, like “screenshot microsoft windows” or “screenshot chromebook” or “screenshot os x”, or ask how on Piazza and we can help.)

Running Code

Unzip the directory you downloaded, and open it in Visual Studio Code.

You will be creating a simple program from scratch by following the following steps.

1. Create a new file named FirstExample.java.

2. Create a new class named FirstExample

3. Create a new field of type int named theNumberFive and assign it the value of 5

After completing these steps, you should have something similar to the following piece of code.

class FirstExample {
int theNumberFive = 5;
}

Open the terminal, then run your program using ./run FirstExample (Mac, Lab) or .\run.bat FirstExample (Windows). You should see output similar to the following.

Let’s explore this program a little. We see that the code is encased inside class FirstExample { ... }. This means that we have created a new class using the class keyword and named it FirstExample. The { and } denote the beginning and end of the FirstExample class. Code we write will be contined inside classes. Classes in Java can contain many different types of information, two of which we will be focusing on in Part 2. They are Fields and Methods. The program we just wrote contains just one field.

Fields are named variables that belong to a class and are written outside of Methods. The class FirstExample contains a single field of type int named theNumberFive with an assigned value of 5. Notice the line int theNumberFive = 5; and how it takes the form of [Type] [Variable Name] = [Assigned Value];. Additionally notice that it belongs to the class FirstExample and not to any method.

By using the run command, we are able to confirm several aspects of our program. We are able to identify our class via the line new FirstExample:1( ... ). The FirstExample class contains 1 field named theNumberFive and its value is 5. We can use the run command to check that our program is working as you expect it to.

Part 2 - Writing Methods

• You will write your code in DesignRecipeExamples.java. You can run the code to test it yourself with:

Mac and Lab: ./run DesignRecipeExamples

Windows: .\run.bat DesignRecipeExamples

• For problem 3 and 4, you will use the following design recipe:
2. Write a short comment documenting the method (including assumptions, like if a parameter should take values only in a certain range)
3. Write examples of calling the method
4. Write the method body

Using this recipe for problems 1 and 2 is optional, but highly encouraged.

• For problems 1 - 4, you will also write tests to ensure the correctness of your code.

To write a test, you will call the written method and assign it to a field that you create. When running your program using the run command, check to see if the actual output matches what you expect from the console.

For problems 3 and 4, you will be creating a new file called transcript.txt and copy/pasting test output from the run command into it. See problems 3 and 4 for more details.

(Several of these examples are borrowed from How to Design Programs, and its Supplemental Material)

Problem 1: Perimeter

Develop a method named perimeter that when given an integer width and integer height of a rectangle will return its perimeter.

The rubric we will use to grade this method is:

• The method has a correct header (parameters and return type)
• The method has a correct method body
• There are at least two examples of calling the method with different arguments, commented with their expected values

Problem 2: Border Area

Develop a method named borderArea that given 2 integers, width and height, describing a rectangle, and another 2 integers, width and height, describing a rectangle cut out of the center of the other, returns the area of the region between them. For example, the blue area in this shape:

• The method has a correct header (parameters and return type)
• The method has a correct method body
• There are at least two examples of calling the method with different arguments, commented with their expected values

Problem 3: Converter

Implementation

Develop a method (with a name of your choice!) that takes one int parameter, converts it to another int value, and returns the new int value. For example, your function could convert:

• Fahrenheit to Celsius
• Between currencies

Write a comment for your method, describing what it does and what parameter it takes. If you use any outside sources for the conversion, be sure to cite those in the comment as well.

Testing

Call your method on at least 2 different examples. Compare each result against a reference converter (Many search engines have built-in converters, or you can use a calculator) and add a comment above the function call to discuss differences (if any) that you get between your program and the reference.

• The method has a correct header (parameters and return type)
• The method has a method body that matches the examples you chose
• There are at least two examples of calling the method with different arguments, commented with their expected values
• In the file transcript.txt, show your program running and the expected values are produced by copy/pasting from the terminal.
• You can justify the output based on an external source or your own design description

Problem 4: Combiner

Implementation

Develop a method (with a name of your choice!) that takes at least two int parameters, combines them to another int value, and returns the new int value. For example, your function could combine:

• Feet and inches into total in inches
• Starting hour and minute and end hour and minute into total number of minutes

Write a comment for your method, describing what it does and what parameters it takes. If you use any outside sources, be sure to cite those in the comment as well.

Testing

Call your method on at least 2 different examples. Can you call your method with arguments that run, but produce an incorrect output? Call your method with at least one such pair of arguments, and add a comment above it explaining why the output is incorrect. If you cannot find such a pair of arguments, write a comment explaining why you believe no such input exists.

• The method has a correct header (parameters and return type)
• The method has a method body that matches the examples you chose
• There are at least two examples of calling the method with different arguments, commented with their expected values
• In the file transcript.txt, show your program running and the expected values are produced by copy/pasting from the terminal.
• You can justify the output based on an external source or your own design description

Submission

Your submission will include the following:

1. A code submission of your program with the methods you wrote for part two, at PA1 gradescope assignment. You should select and upload the files DesignRecipeExamples.java and transcript.txt

A more detailed list of requirements can be found here


text:
# The Stacks Project

## Tag 039X

Lemma 34.3.8. Let $R \to A$ be a faithfully flat ring map, and let $R \to R'$ be faithfully flat. Set $A' = R' \otimes_R A$. If all descent data for $R' \to A'$ are effective, then so are all descent data for $R \to A$.

Proof. Let $(N, \varphi)$ be a descent datum for $R \to A$. Set $N' = R' \otimes_R N = A' \otimes_A N$, and denote $\varphi' = \text{id}_{R'} \otimes \varphi$ the base change of the descent datum $\varphi$. Then $(N', \varphi')$ is a descent datum for $R' \to A'$ and $H^0(s(N'_\bullet)) = R' \otimes_R H^0(s(N_\bullet))$. Moreover, the map $A' \otimes_{R'} H^0(s(N'_\bullet)) \to N'$ is identified with the base change of the $A$-module map $A \otimes_R H^0(s(N)) \to N$ via the faithfully flat map $A \to A'$. Hence we conclude by Lemma 34.3.7. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 615–620 (see updates for more information).

\begin{lemma}
\label{lemma-descent-descends}
Let $R \to A$ be a faithfully flat ring map, and let $R \to R'$
be faithfully flat. Set $A' = R' \otimes_R A$. If all descent data
for $R' \to A'$ are effective, then so are all descent data for $R \to A$.
\end{lemma}

\begin{proof}
Let $(N, \varphi)$ be a descent datum for $R \to A$.
Set $N' = R' \otimes_R N = A' \otimes_A N$, and denote
$\varphi' = \text{id}_{R'} \otimes \varphi$ the base change
of the descent datum $\varphi$. Then $(N', \varphi')$ is
a descent datum for $R' \to A'$ and
$H^0(s(N'_\bullet)) = R' \otimes_R H^0(s(N_\bullet))$.
Moreover, the map
$A' \otimes_{R'} H^0(s(N'_\bullet)) \to N'$ is identified
with the base change of the $A$-module map
$A \otimes_R H^0(s(N)) \to N$ via the faithfully flat map
$A \to A'$. Hence we conclude by Lemma \ref{lemma-recognize-effective}.
\end{proof}

Comment #2874 by Ko Aoki on October 5, 2017 a 3:57 pm UTC

Possibly wrong statement: The proof uses 039W, but it holds under an additional condition that $R \to A$ is faithfully flat. So I think faithfully flatness of $R \to A$ is needed. The same for 039Y.

Comment #2936 by Johan (site) on October 10, 2017 a 2:15 am UTC

Thanks, fixed here.

Comment #2952 by Ko Aoki on October 11, 2017 a 12:50 pm UTC

Thank you for fixing. (1) of 039Y contains the same error.

There are also 2 comments on Section 34.3: Descent.

## Add a comment on tag 039X

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text:
Journal ArticleDOI

Block Designs Robust Against the Presence of an Aberration in a Treatment-Control Setup

20 Jan 2012-Communications in Statistics-theory and Methods (Taylor & Francis Group)-Vol. 41, Iss: 5, pp 920-933

TL;DR: In this article, the authors considered the problem of finding designs insensitive to the presence of an outlier in a treatment-control block design setup for estimating the set of elementary contrasts between the effects of each test treatment and a control treatment.

AbstractThe author considers the problem of finding designs insensitive to the presence of an outlier in a treatment-control block design setup for estimating the set of elementary contrasts between the effects of each test treatment and a control treatment. The criterion of robustness suggested by Mandal (1989) in block design setup for estimating a full set of orthonormal treatment contrasts is adapted. A new class viz. partially balanced treatment incomplete block designs (PBTIBD) is introduced and it is shown that balanced treatment incomplete block designs (BTIBD) and PBTIB designs, under certain conditions, are robust in the previous sense. Such designs are important in the sense that the inference on the treatment contrasts under consideration remain unaffected by the presence of an outlier.

Citations
More filters
Journal ArticleDOI

20 citations

Journal ArticleDOI
TL;DR: In this article, the robustness of incomplete block designs against non-normality in the observations has been investigated and a new measure of robustness is introduced for further study, which is called robustness measure.
Abstract: In this article, designs are found for which the F-test of analysis of variance is insensitive to violation of normality assumption. Atiqullah (1962) proved that the F-test for treatments adjusting for blocks in the intra-block analysis of a balanced incomplete block design is robust against non-normality in the observations. Here an attempt has been made to identify other designs robust in this sense. In particular, it is observed that for testing relevant hypothesis, a partially balanced incomplete block design in block design setup, under certain conditions, is robust. Robustness of a balanced treatment incomplete block design and a partially balanced treatment incomplete block design (Biswas, 2012), in treatment-control design setup, is also studied. Moreover, a new measure of robustness is introduced for further study. The performance of the F-test in presence of non-normality in the observations for a quadratically balanced design is also examined.

4 citations

Cites background or methods from "Block Designs Robust Against the Pr..."

• ...However, a separable S-type BTIB design (Biswas, 2012) for which t = 0 (t + 1 and t being the number of times the control treatment occurs in s and the remaining b − s blocks respectively) can be considered a nearly robust design among competing designs as it can be easily proved that for such a…...

[...]

• ...Generalizing GDT designs, a new general class of partially balanced treatment incomplete block designs (PBTIB designs) was introduced in Biswas (2012)....

[...]

• ...Robustness of a balanced treatment incomplete block design and a partially balanced treatment incomplete block design (Biswas, 2012), in treatment-control design setup, is also studied....

[...]

• ...Further, examples of such designs are given in Biswas (2012)....

[...]

• ...…obtained robust designs against the presence of an outlier in block design setup for estimating a full set of orthonormal treatment contrasts and Biswas (2012) extended it to the treatment-control setup for the estimation of a set of elementary contrasts between each test treatment and a…...

[...]

Journal ArticleDOI
TL;DR: In this article, the authors find that complete and partial diallel cross designs are robust to the presence of an outlier in a block design setup for estimating a complete set of orthonormal contrasts among the effects of the general combining abilities of a set of parental lines.
Abstract: In this article, we find designs insensitive to the presence of an outlier in a diallel cross design setup for estimating a complete set of orthonormal contrasts among the effects of the general combining abilities of a set of parental lines. The criterion of robustness, suggested by Mandal (1989) in block design setup and used by Biswas (2012) in treatment-control setup, is adapted here. Complete diallel cross designs, suggested by Gupta and Kageyama (1994), and partial diallel cross designs, suggested by Gupta et al. (1995) and Mukerjee (1997), are found to be robust under certain conditions.

1 citations

Cites background or methods from "Block Designs Robust Against the Pr..."

• ...The criterion of robustness, suggested by Mandal (1989) in block design setup and used by Biswas (2012) in treatment-control setup, is adapted here. Complete diallel cross designs, suggested by Gupta and Kageyama (1994), and partial diallel cross designs, suggested by Gupta et al....

[...]

• ...Without going into the details of the methods for deriving the criterion, given in Biswas (2012), only the definition is given below....

[...]

• ...Box and Draper (1975) were the first to investigate, in this context, the problem of predicting the observed response vector in a response surface model....

[...]

• ...The criterion of robustness, suggested by Mandal (1989) in block design setup and used by Biswas (2012) in treatment-control setup, is adapted here. Complete diallel cross designs, suggested by Gupta and Kageyama (1994), and partial diallel cross designs, suggested by Gupta et al. (1995) and Mukerjee (1997), are found to be robust under certain conditions....

[...]

• ...Box and Draper (1975) were the first to investigate, in this context, the problem of predicting the observed response vector in a response surface model. Later, Gopalan and Dey (1976), Ghosh and Kipngeno (1985), Mandal (1989), Mandal and Shah (1993), Biswas (2012), and some others studied robustness of designs against the presence of outliers under different contexts....

[...]

Journal ArticleDOI
Ganesh Dutta
TL;DR: In this paper, the authors show that the presence of one or more outliers in the observations affect inference procedure in statistical analysis and propose robust designs robust against presence of a single outlier.
Abstract: Presence of one or more outliers in the observations affect inference procedure in statistical analysis. Designs robust against presence of a single outlier can be found in the literature for both ...
Journal ArticleDOI
Ganesh Dutta
01 Dec 2020
TL;DR: In this article, an attempt has been made to find robust designs in a block design set-up with covariates, when there is one single wild observation in the study variable.
Abstract: Presence of one or more aberration in the observations affects inference procedure in statistical analysis. Block designs robust against presence of aberrations can be found in the literature for both regression and block design set-ups. In this paper, an attempt has been made to find robust designs in a block design set-up with covariates, when there is one single wild observation in the study variable. Specifically, effects on the estimation of a full set of orthonormal treatment contrasts and that on the estimation of covariate parameters have been considered.

More filters
Book
01 Jan 1971

622 citations

"Block Designs Robust Against the Pr..." refers background or methods in this paper

• ...Examples of designs, with the parametric combinations given in the list of Raghavarao (1971) for which the value of t satisfying condition (3.9) comes out as zero, are given in Appendix 6.1....

[...]

• ...…contains c test treatments from each group, where k∗1 = cm, c being any positive integer and m being the number of groups of test treatments (cf. Raghavarao, 1971). c) All triangular PBIB designs in the test treatments satisfying r + n− 4 1 − n− 3 2 = 0, there being v = n n− 1 /2 test…...

[...]

• ...Here every block contains 2k∗1/n test treatments from each row (or column), 2k ∗ 1/n being an integer (cf. Raghavarao, 1971). d) All Latin-square type designs in the test treatments with two constraints L2 satisfying r + s − 2 1 − s − 1 2 = 0, there being v = s2 test treatments, where each block is…...

[...]

• ...Here every block contains k∗1/s test treatments from each row (or column; cf. Raghavarao, 1971)....

[...]

• ...The BTIB designs are taken from the list of designs in Raghavarao (1971)....

[...]

Journal ArticleDOI
A. S. Hedayat
TL;DR: In this article, existing knowledge on optimal designs for comparing test treatments with controls under 0-, 1-, and 2-way elimination of heterogeneity models is presented. But the results are motivated through numerical examples.
Abstract: This article outlines existing knowledge on optimal designs for comparing test treatments with controls under 0-, 1- and 2-way elimination of heterogeneity models. The results are motivated through numerical examples.

98 citations

Journal ArticleDOI
TL;DR: In this article, a theory of optimal incomplete block designs for comparing several treatments with a control is developed, and a procedure for making exact joint confidence statements for this multiple comparisons problem is described.
Abstract: In this paper we develop a theory of optimal incomplete block designs for comparing several treatments with a control. This class of designs is appropriate for comparing simultaneously p ≥ 2 test treatments with a control treatment (the so-called multiple comparisons with a control (MCC) problem) when the observations are taken in incomplete blocks of common size k < p + 1. For this problem we propose a new general class of incomplete block designs that are balanced with respect to (wrt) test treatments. We shall use the abbreviation BTIB to refer to such designs. We study their structure and give some methods of construction. A procedure for making exact joint confidence statements for this multiple comparisons problem is described. By using a new concept of admissibility of designs, it is shown how “inferior” designs can be eliminated from consideration, and attention limited to a small class of BTIB designs that can be constructed from so-called generator designs in the minimal complete class of such d...

98 citations

"Block Designs Robust Against the Pr..." refers background or methods in this paper

• ...For a definition and other properties of such designs see Bechhofer and Tamhane (1981) and Hedayat et al. (1988)....

[...]

• ...In the class of incomplete treatment-control block designs, Bechhofer and Tamhane (1981) introduced balanced treatment incomplete block designs (BTIBD)....

[...]

Journal ArticleDOI
TL;DR: In this article, the problem of finding optimal incomplete blocks designs for comparing $p$ test treatments with a control is studied, and conditions for a design to be φ-optimal are found for a large class of functions.
Abstract: The problem of finding optimal incomplete blocks designs for comparing $p$ test treatments with a control is studied. B.I.B. designs are found to be $D$-optimal. $A$- and $E$-optimal designs are also obtained. For a large class of functions $\phi$, conditions for a design to be $\phi$-optimal are found. Most of the optimal designs are certain types of B.T.I.B. designs, introduced by Bechhofer and Tamhane (1981), which are binary in test treatments.

82 citations

"Block Designs Robust Against the Pr..." refers background in this paper

• ...Such designs are also optimal as can be seen in Majumdar and Notz (1983)....

[...]

• ...Such designs are also optimal as can be seen in Majumdar and Notz (1983). In the class of PBTIB designs introduced by us, those designs are found to be robust for which the number of first associates of any test treatment in any block is a constant....

[...]

Book
01 Jan 1986
TL;DR: Introduction Some Mathematical and Statistical Results General Properties of Block Designs Balanced Incomplete Block Designs - Properties and Analysis Construction and Existence of Balanced In complete Block Designs Partially Balanced In Complete Block Designs Additional Topics in Block Designs Index.
Abstract: Introduction Some Mathematical and Statistical Results General Properties of Block Designs Balanced Incomplete Block Designs - Properties and Analysis Construction and Existence of Balanced Incomplete Block Designs Partially Balanced Incomplete Block Designs Additional Topics in Block Designs Index.

78 citations