text:
Having problems with your PC Desktops ?
BIOS Update, need BIOS-Flash software
\015 My computer is from 2002 and I would like to update my BIOS chip. Model: Shuttle (Spacewalker)MS51/MS51N. \015\012BIOS ID: et-MS51-C-00\015\012BIOS Type: Award Software Inc.\015\012BIOS Date: 02/05/2002\015\012Chipset ID: Chip (SiS650/961)\015\012OEM Info: MS51S004 02/05/2002\015\012I am waiting for your answer.\015\012\015\012e-mail: [email protected]\015\012\015\012Thanks,\015\012\015\012Carl Nacsin\015
Answers :
Typically their should be instructions specific to your motherboard model at the manufacturers website. You will need to have downloaded the new bios.rom file and a small application that does the flash itself.
Usually you wont ever need a bios update unless dealing with enthusiast hardware which is brand new hardware that has just been sold and needs updates regularly.
Usually you wont ever need a bios update unless dealing with enthusiast hardware which is brand new hardware that has just been sold and needs updates regularly.
Repair Help & Product Troubleshooting for PC Desktops
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Suggested Questions/Answers :
Question regarding ur answer to my original problem...
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BIOS Update, need BIOS-Flash software
Typically their should be instructions specific to your motherboard model at the manufacturers website. You will need to have downloaded the new bios.rom file and a small application that does the flash itself.Usually you wont ever need a ... PC Desktops
Can turn computer off and on and I will see the windows screen then the screen goes blue with a blinking cursor. I also have a dell.
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Dell inspiration 1150 laptop
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Need bios not just the update,
Find out what make and bios version you have i.e. pheonix labs etc go to the bios website and download the latest version in exe file. here is link with full instructionshttp://www.chat11.com/Bios_Upgrade_Instructions_For_Sony_Vaio_PCG_TR ... Sony PCG-TR3A Notebook
Manually flashing a Dell XPS 1330..
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Vertical Dotted Lines on my Toshiba Tecra
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Frezes at verifying DMI settings. Already changed MB and Power supply. Lan connection only flashes green once in a while should be constant. Tried on a working terminal and does the same.
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I have the same "freezing" problem!
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The monitor turns off and on repeatedly.
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Updated firmware for cd/dvd drive SN324F now the drive is not working and unavailable on my Samsung v25 laptop pc?
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BIOS Update Procedure
BIOS Update Procedure\015\012\015\012All latest Motherboards today, 486/ Pentium / Pentium Pro etc.,ensure that upgrades are easily obtained by incorporating the system BIOS in a FLASH Memory component. With FLASH BIOS, there is no need to repl ... PC Desktops
Updating BIOS I upgraded my computers RAM memory and was told I had to update my BIOS to make it work. How do I update BIOS? I am totally clueless. Thanks! Hi, I just changed the ram in my acer aspire 5630 from 2GB (pre-installed) to 4GB (supposed Max.) but the System still shows only the 2GB. I re-booted twice but it will not show up. What else do I have to do?
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Armada m700 no post
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Toshiba 5205-s503 won't start up
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Wrong bios NEED HELP How and can I make a flask BIOS CD to get my BIOS working again MY DV9000 LAPTOP windows vista was not reading the full amount of how much memory that is on the computer this started after HP sent me a recovery disc the computer kept rebooting so I was told to update the bios so I did but I downloaded the wrong bios now the computer screen does not work. I can hear the computer boot up lights turn on I tried using the restore disc I can hear the cd driver readin
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Monitor won't turn on -- flashes on then off again
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"bootlock" When I turn the computer I get a message
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Easy Note W3332 BIOS shows NEC DVD + RW ND-6650A; does not play DVD's
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How to flash my BIOS - Toshiba Satellite M30 Notebook
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My laptop will not turn on or charge - Acer Aspire 3680 Laptop
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USB not recognized
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text:
BIOLOGY | A World of Experience
Biology students at Baker are accepted into dental school at a 100 percent rate and accepted into medical school at a 87 percent rate. Why? Between Baker's excellent faculty members, personalized learning, and incredible resources, our students are limited only by the goals they set for themselves.
Biology majors have a choice of four tracks: molecular bioscience, vertebrate structure and function, ecology and evolution, and certification to teach biology at the secondary level.
All students have the opportunity to do original research with a faculty member and present their research at regional and national conferences. Some have even traveled internationally.
Baker maintains the Baker Wetlands and the Ivan L. Boyd Arboretum for in-depth research and observation.
Students can gain experience during summer research programs at major research institutions such as the Stowers Institute for Medical Research, University of Kansas, University of Missouri, and Quintiles.
Students preparing for a career in health care participate in shadowing experiences with health-care professionals.
The Baker Wetlands and Discovery Center is paradise for our student biologists, ecologists, biochemists, and prehealth professionals. But our students aren't the only beneficiaries of the stunning 927-acre natural habitat. Stargazers, bird watchers, and exercise enthusiasts are always bustling around the area, which is open to the public. Located just 10 miles north of the Baldwin City campus, the Baker Wetlands is a true regional distinction for the university. Learn more >
JAMISON | Biology Major, Class of 2016
After serving as a featured presenter at Baker's Dialogos research symposium and internships with several pharmacies, Jamison was accepted into the University of Kansas School of Pharmacy before walking across the stage to get his diploma. "The professors in Baker's Biology Department offer an expanse of knowledge and altruism, insurmountable in comparison to others. Their entirety exceptionally prepares all incoming students to perform beyond their perceived limits. This expertise not only forges a strong foundation in the core principles of biology, but furthermore establishes each student as a formidable competitor for post-undergraduate work, schooling, and research."
Our biology majors go on to land these jobs:
- Science teachers
- Junior scientists in research labs and for the Environmental Protection Agency
- Wildlife biologists for Kansas Department of Wildlife and Parks
- Park rangers
- Army Corps of Engineers
of graduates are employed full time or enrolled in graduate school within six months of receiving their diploma.
Ecology and Evolution
Students in Baker University's Ecology and Evolution class tagged monarchs last fall and one of their tagged butterflies was recovered in March on its wintering grounds in El Rosario, Michoacan, Mexico.
"Each fall our introductory ecology students tag monarchs in one of our labs, where they investigate migratory behavior, navigation, and conservation of a globally vulnerable, but very charismatic, species," said Dr. Scott Kimball, assistant professor of biology. "The monarchs we catch are all tagged, weighed, and measured by our students before being sent on their way. It is one of their favorite activities."
Course Descriptions
R: course can be repeated for credit; P/NC: course graded on a pass/no credit basis
Courses required for these programs are listed in the current catalog.
BI 120 – Human Ecology
This course is intended for the non-science major. It presents the terminology, methodology, and worldview of biological science through a consideration of the impact of modern technology on human ecology. (This course may not be counted towards either the major or minor in Biology.) (3 credit hours)
BI 125 – Human Genetics
This course is intended for the non-science major. It will focus on the role of genetics in health, medicine, society, ethics, and evolution. Students will discover how biology and genetics affects them directly. (This course may not be counted towards either the major or minor in Biology.) (3 credit hours)
BI 151 – Molecular and Cellular Biology
This course is an entry-level course for Biology majors. The course will address the relationship between structure and function of the major biological molecules (proteins, nucleic acids, carbohydrates, and fats) as well as the relationship between structure and function of cells and organelles. The students will also be introduced to the basic principles of molecular genetics and evolution. Co-requisite: BI 151L. (3 credit hours)
BI 151L – Molecular and Cellular Biology Laboratory
The laboratory that accompanies BI 151. Co-requisite: BI 151. (1 credit hour)
BI 152 – Genetics
Genetics is the study of heredity at the population, organismal, cellular, and molecular levels. This course will focus on the Mendelian rules of inheritance in individuals and in populations. The molecular mechanisms that control cell division and gene expression will also be discussed. There will be three lecture/discussion periods per week, and one laboratory session per week where students will learn techniques used to study inheritance patterns. Prerequisite: BI 151. (4 credit hours)
BI 246 – Human Anatomy and Physiology I
This course is the first of a two-semester sequence in which human anatomy and physiology are studied using a body systems approach, with emphasis on the interrelationships between form and function at the gross and microscopic levels of organization. Human Anatomy and Physiology I is required for students in Exercise Science and Physical Education, and for pre-professional students in Nursing and other allied health sciences. This course is not recommended for pre-medical and pre-dental students. The course covers the basic anatomical and directional terminology; homeostasis; cell biology; histology; skeletal, muscle, nervous, cardiovascular, and respiratory systems; and the digestive system and metabolism. There are three lecture and discussion sessions and one laboratory session each week. This course does not count towards a major or minor in Biology. Prerequisite: Sophomore status or instructor permission. (4 credit hours)
BI 247 – Human Anatomy and Physiology II
Human Anatomy and Physiology II is a continuation of Human Anatomy and Physiology I, and is required for pre-professional students in many of the allied health sciences. This course is not recommended for pre-medical and pre-dental students. The course covers the integumentary system; neural integration; general and special senses; the endocrine system; blood; the lymphatic system and immunity; the urinary system, fluid/electrolyte and acid/base balance; and the reproductive system and human genetics. Homeostasis is emphasized as a unifying theme throughout both semesters. There are three lecture and discussion sessions and one laboratory session each week. This course does not count towards a major or minor in Biology. Prerequisite: BI 246 with a grade of "C" or higher. (4 credit hours)
BI 251 – Ecology and Evolution
This course is an entry-level course for Biology majors. The course will work from an evolutionary perspective to address the basic ecological levels of structure and function including populations, communities, and ecosystems. Three lecture/discussion and one laboratory session will occur each week. (4 credit hours)
BI 254 – Organismal Diversity
This course is an entry-level course for Biology majors. The course is a survey of prokaryotic and eukaryotic animal diversity. The evolutionary relationships, taxonomy, and characteristics of major organismal groups, especially plants and animals, will be emphasized. There are three lecture/discussion periods and one laboratory session each week. Prerequisite: BI 251. (4 credit hours)
BI 262 – Microbiology
This course is an introduction to microbiology. Topics include a survey of microbes including viruses, bacteria, archaea, fungi, protozoa, and algae. In addition to consideration of the taxonomy, genetics, anatomy, and physiology of these organisms, we will also consider how they interact with their environments. For pathogenic organisms, this will entail an introduction to the field of immunology. Co-requisite: BI 262L or QS 211. (3 credit hours)
BI 262L – Microbiology Laboratory
The laboratory that accompanies BI 262. Co-requisite: BI 262. (1 credit hour)
BI 298 – Experimental Design Workshop
Students who have completed BI 151, BI 152, and BI 251 with a minimum grade of "C" and are interested in pursuing a major in Biology will enroll in this course in the spring of their sophomore experience. Students will design an original research project with the help of the faculty. Prerequisite: BI 151, 152, and 251, all with grade of "C" or higher, and sophomore status. (1 credit hour)
BI 342 – Plant Physiology
This course covers the biology of plants from an evolutionary perspective at the population, organism, cell, and molecular levels of organization. It also emphasizes how plant biology can be used to better understand aspects of ecology, agriculture, and medicine. There are three lecture and discussion sessions and one laboratory session each week. Prerequisite: BI 251 (4 credit hours)
BI 344 – Forest Ecology
This course will provide a basic introduction to the ecology of forests, with special consideration given to the relationships between plants and animals adapted to eastern deciduous forests found in Northeastern Kansas. The course will include lab- and field-based exercises as well as assigned readings from texts and primary literature to provide a fundamental understanding of ecological principles and field techniques that are unique to forest ecology. Prerequisite: BI 254. (4 credit hours)
BI 356 – Ornithology
This course covers the biology of birds, including their classification, physiology, behavior, ecology, evolution, and speciation. The field identification of local species is emphasized. This course is recommended as an elective for Biology majors, teachers, and anyone seriously interested in birds. There are three lecture sessions and one laboratory field session each week. Prerequisite: BI 254 (4 credit hours)
BI 360 – Comparative Vertebrate Anatomy
This course is a thorough exploration of anatomical similarities and differences among the vertebrate classes and an analysis of the anatomical evidence for evolutionary relationships. In the laboratory, students will gain experience dissecting a series of representative vertebrates to study anatomical diversity. This course is fundamental for Biology majors who plan careers in vertebrate biology and for pre-medical and pre-dental students. There are three lecture and discussion periods and one laboratory dissection period each week. Prerequisite: BI 246 or BI 254. (4 credit hours)
BI 361 – Developmental Biology
In this course students will address the fundamental questions: how does the fertilized egg give rise to the adult body, and how does the adult body produce gametes that can result in the fertilized egg? There are three lecture/discussion sessions and one laboratory session each week in which the students will learn how scientists are using information gleaned from cell/molecular biology, physiology, anatomy, cancer research, neurobiology, immunology, evolutionary biology, and ecology to answer these questions. The scope of this course makes it advisable to have a solid background in biology prior to attempting this study. Prerequisite: BI 152. (4 credit hours)
BI 363 – Virology
The world between the living (cellular organisms) and the non-living is occupied by a variety of microbes including viruses, viroids, virusoids, and prions. Though minute, these particles have a huge impact on human society. In proof, consider the fact that more humans died between the years 1917 and 1920 from the flu than in the battles of World War I. This course is designed to allow students who have completed the introductory microbiology course an opportunity to expand their knowledge concerning these subcellular microbes. Prerequisite: BI 152 or BI 262. (3 credit hours)
BI 375 – Evolution
This course is an examination of the theory of evolution, including its historical and social implications. It emphasizes the intellectual skills associated with the testing of evolutionary hypotheses. There are three lecture and discussion sessions each week. Prerequisite: BI 254. (3 credit hours)
BI 377 – Population and Community Ecology
This course covers the relationship between organisms and their environment. Field and laboratory techniques are covered in the laboratory sessions. This course is recommended for all Biology majors and other serious students of ecology. There are three lecture and discussion periods and one field or laboratory session each week. Prerequisite: BI 254. (4 credit hours)
BI 380 – Animal Behavior
This course explores the proximate and ultimate evolutionary explanations for the behavior of animals. The role of scientific process in producing research discoveries is emphasized. There are three lecture and discussion periods and one laboratory session each week. Prerequisite: BI 254. (4 credit hours)
BI 382 – Comparative Physiology
This course covers the comparative function of animal organisms from the molecular to the organismal level. The physical and chemical basis for the similarities and differences in function are analyzed. Mechanisms for meeting common problems, such as water and ion balance, feeding and digestion, gas exchange, internal transport, and nerve and muscle functions are studied. The theme of physiological adaptations to the environment will be emphasized. This course is recommended for junior and senior Biology majors and for pre-medical and pre-dental students. There are three lecture and discussion periods and one laboratory session each week. Prerequisite: BI 246 or BI 254. (4 credit hours)
BI 383 – Advanced Cell Biology
This course is designed to provide students who will become career scientists or health professionals a solid and deep understanding of the biology of the cell. We will introduce the students to important factual information, terminology, and methodology concerning modern cellular biology via reading assignments in the text and in the primary literature, lectures, and discussions. A secondary goal is to help students develop a clearer understanding of the ethical challenges involved in science. Prerequisite: Junior status and BI 152. Prerequisite or co-requisite: CH 251. (3 credit hours)
BI 385 – Advanced Molecular Biology
This course will allow students to expand on their previously acquired knowledge of genetics. Students will study classical and current experiments in molecular biology that are used to determine how the molecules involved in DNA replication, RNA transcription, and protein translation interact and function at the molecular level. Prerequisites: Junior status, BI 152, and CH 251. (3 credit hours)
BI 386 – Methods in Biochemistry and Molecular Biology
This intensive laboratory course will give students the opportunity to carry out modern molecular experimental techniques such as recombinant DNA, electrophoresis, protein purification, blotting, and DNA sequencing. Today these techniques are utilized to answer questions in cell biology, molecular biology, and biochemistry. Students should have completed most other biology and chemistry requirements prior to taking this course so that they are prepared to both understand and design molecular experiments. Prerequisites: Senior status and BI 383 or 385 or CH 370. (3 credit hours)
BI 388 – Immunology
This course focuses on the cellular and molecular structure and function of the human immune system. It will provide a conceptual background for understanding the development and function of the cells and mechanisms that produce innate and adaptive immunity. The role of pathogens in the immune response, abnormal immune responses, and experimental manipulation of the immune system will also be discussed. Prerequisite: BI 152. (3 credit hours)
BI 392 – Wetland and Prairie Ecology
This course examines the plants and animals of a variety of habitats classified within the context of wetlands and prairie. Both of these areas have been greatly reduced in size within the North American continent through agriculture and development. Several areas near Baldwin City provide rare opportunities for study of these areas. The upland community at the Ivan Boyd Prairie Preserve, prairie savannahs at the Ivan Boyd Woodlands, and a variety of wetland types at the 573-acre Baker University Wetlands Research and Natural Area will be the focal points of field study. This course is designed for upper-college Biology majors with an interest in field research. There are three lecture and discussion periods and one laboratory session each week. Prerequisite: BI 377. (4 credit hours)
BI 410 – Senior Seminar in Biology
This is the senior capstone course for all seniors graduating with a degree in Biology. The students will consult with the course instructor and project mentor in order to choose a topic that is relevant to their career interests and goals. Through research of the literature, each student will develop a significant paper that will demonstrate their ability to research, interpret, and write in their chosen field. Prerequisite: Senior status, Biology major. It is strongly suggested that students have completed at least two biology courses at the 300 level prior to taking this seminar. (2 credit hours)
BI 498 – Research in Biology
Students who have completed BI 298 with a minimum grade of "C" will enroll in two credit hours of BI 498 at some point during their junior experience in order to carry out the research project planned. Students may repeat this course in subsequent semesters to further their research; however, only the original term will count toward the major in Biology. Prerequisite: BI 298 with a "C" or higher. R (1-3 credit hours)
Do you want to teach biology?
Baker University's undergraduate education programs license teachers in biology (6-12). Students pursuing a degree in education and teacher licensure work closely with faculty advisors from the School of Education to fulfill the requirements for a degree from Baker University and teacher licensure in Kansas. Candidates are required to complete education course work and the required course work in at least one content area. Learn More >>
The Department of Biology and Chemistry gives these awards with financial prizes to be applied to the following year's tuition:
- Mildred Hunt Riddle Departmental Recognition Scholarship for Biology
- Mildred Hunt Riddle Departmental Recognition Scholarship for Chemistry
- Howard T. Bonnett Scholarship and Mrs. Ivan L. Boyd Memorial Scholarship
- Margaret E. Scanlon Endowed Memorial Scholarship
- E.J. Cragoe Scholarship
- Chemical Rubber Company Press Freshman Chemistry Achievement Award
- Sonia Browning Endowed Scholarship
- Doris Cink and Kathryn Zimney Endowed Scholarship
Academic Recognition
- Outstanding Senior in Biology
- Outstanding Senior in Chemistry
- Outstanding Junior in Biology
- Outstanding Sophomore in Biology
- Outstanding Freshman in Biology
- Dr. Darcy Russell
- Dr. Roger Boyd
- Danielle Hemingson
- Dr. Charmaine Henry
- Dr. Scott Kimball
- Dr. William Randy Miller
- Dr. Erin Morris
Dr. Darcy Russell
Professor of Biology & Duboc Chair, Chair of the Department of Biology & Chemistry | firstname.lastname@example.com
Curiosity is what drove Dr. Darcy Russell into the lab, and working with students is what has kept her there. To her, the best thing about the department is its laboratory and field experiences, which are tied into the curriculum. And when she's not studying how cells and viruses combat each other, she feeds her Jane Austen/George Eliot/Charlotte Bronte addiction.
B.S. Baker University, Ph.D. Kansas State University
Expertise: microbiology
Office: Boyd Science Center 229 | 785.594.8418
Dr. Roger Boyd
Education Coordinator for the Baker Wetlands & Professor Emeritus of Biology | email@example.com
Baker science history runs deep in the Boyd family. Dr. Roger Boyd has been able to both teach and develop the Baker University Wetlands into a unique learning tool—something he had been working on for 50 years that has come to fruition. His most memorable experience at Baker is teaching abroad at Harlaxton College, our partner college, and taking a group of students to Mexico 11 times as a part of the interterm experience.
Ph.D. Colorado State University
Expertise: ornithology, wetland and prairie ecosystems, neotropics
Baker Wetlands Office, 1365 N. 1250 Road | 785.594.4702
Danielle Hemingson
Visiting Assistant Professor of Biology | firstname.lastname@example.com
Office: Mulvane | 785.594.8752
Dr. Charmaine Henry
Associate Professor of Biology | firstname.lastname@example.com
B.S. University of the Virgin Islands; Ph.D. University of Medicine & Dentistry of New Jersey
Expertise: physiology, immunology, anatomy
Office: Boyd Science Center 228 | 785.594.4596
Dr. Scott Kimball
Assistant Professor of Biology | email@example.com
When asked what drew him to his field, Dr. Kimball's reaction was immediate: "Birds. I love birds." And with that love, he has been able to develop it into a career and land at a university that, according to him, appreciates the value of learning and understanding. He loves Baker's dedication to learning experiences outside the classroom as well, which enables majors to experience the state of the art in their focus area of biology or chemistry.
B.A. Baker University, M.S. Boise State University, Ph.D. The Ohio State University
Expertise: avian ecology and animal behavior
Office: Boyd Science Center 226 | 785.594.4563
Dr. William Randy Miller
Coordinator of Student Research in Biology | firstname.lastname@example.com
It was the opportunity to explore the unknown that drove Dr. William Randy Miller to the field of biology. Dr. Miller's program "Tardigrades and Wheel Chairs in the Canopy" Research Experience for Undergraduates, was awarded a National Science Foundation grant for $286,285 over three years. The program hosts eight students each summer as they study and collect tardigrades locally and nationally, from Baldwin City to Massachusetts.
B.S., M.A. University of Montana; Ph.D. University of New England
Office: Boyd Science Center 225 | 785.594.8379
Dr. Erin Morris
Associate Professor of Biology | email@example.com
"I always say if I were not a scientist I would be a historian. I love looking through old documents and understanding how past events have shaped the world today. I get to combine these two things when I teach a Genology Interterm. That class covers deep human history through genetics, and recent family history using ancestry databases."
B.A. Drury University, Ph.D. University of Missouri at Columbia
Office: Boyd Science Center 230 | 785.594.7881
Kathy Wright
Department Assistant
Biology & Chemistry
Math, Physics, & Computer Science
Office: Boyd Science Center 227

subset: commoncrawl


text: Divxturka.net is using Redbunker! Because Redbunker is better than Hotfile, Netload and Rapidshare ! Click here and buy REDBUNKER premium account! Plot: Amistad is the name of a slave ship traveling from Cuba to the U.S. in 1839. It is carrying a cargo of Africans who have been sold into slavery in Cuba, taken on board, and chained in the cargo hold of the ship. As the ship is crossing from Cuba to the U.S., Cinque, who was a tribal leader in Africa, leads a mutiny and takes over the ship. They continue to sail, hoping to find help when they land. Instead, when they reach the United States, they are imprisoned as runaway slaves. They don't speak a word of English, and it seems like they are doomed to die for killing their captors when an abolitionist lawyer decides to take their case, arguing that they were free citizens of another country and not slaves at all. The case finally gets to the Supreme Court, where John Quincy Adams makes an impassioned and eloquent plea for their release. Genre: Drama | History | Mystery IMDB Rating: 7.2/10 from 45,489 users Metascore: 63/100 Directed by: Steven Spielberg Starring: Djimon Hounsou, Matthew McConaughey, Anthony Hopkins Release Name: Amistad.1997.720p.BluRay.x264-HD4U Size: 6.55 GB Video: MKV | 1280×688 | 4554 Kbps Audio: English | DTS | 1509 Kbps Runtime: 155 mins Subtitles: English, Spanish, Portuguese, French Information: Code: http://www.imdb.com/title/tt0118607/ Download: Please use this LINK to REGISTER or RENEW your account on REDBUNKER to support us. Code: http://www.redbunker.net/ppuzb6av5fwj/Amistad.1997.720p.BluRay.x264-HD4U.mkv.html Download One Link Amistad (1997) 720p BluRay Screens: NFO: IF YOU LIKE MY UPLOADS PLEASE GET A REDBUNKER PREMIUM ACCOUNT, Thank You All for your Support


text:
Sobolev spaces for planar domains
In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.
- 1 Sobolev spaces with boundary conditions
- 2 Application to Dirichlet problem
- 3 Sobolev spaces without boundary condition
- 4 Regularity for Dirichlet problem
- 5 Application to smooth Riemann mapping theorem
- 6 Trace map
- 7 Abstract formulation of boundary value problems
- 8 Application to Neumann problem
- 9 Regularity for Neumann problem
- 10 Notes
- 11 References
Sobolev spaces with boundary conditions[edit]
Let Ω ⊂ R2 be a bounded domain with smooth boundary. Since Ω is contained in a large square in R2, it can be regarded as a domain in T2 by identifying opposite sides of the square. The theory of Sobolev spaces on T2 can be found in Bers, John & Schechter (1979), an account which is followed in several later textbooks such as Warner (1983) and Griffiths & Harris (1994).
For k an integer, the (restricted) Sobolev space Hk
0(Ω) is defined as the closure of C∞
c(Ω) in the standard Sobolev space Hk(T2).
- Vanishing properties on boundary: For k > 0 the elements of Hk
0(Ω) are referred to as "L2 functions on Ω which vanish with their first k − 1 derivatives on ∂Ω."[1] In fact if f ∈ Ck(Ω) agrees with a function in Hk
0(Ω), then g = ∂ αf is in C1. Let fn ∈ C∞
c(Ω) be such that fn → f in the Sobolev norm, and set gn = ∂ αfn . Thus gn → g in H1
0(Ω). Hence for h ∈ C∞(T2) and D = a∂x + b∂y,
- By Green's theorem this implies
- where
- with n the unit normal to the boundary. Since such k form a dense subspace of L2(Ω), it follows that g = 0 on ∂Ω.
- Support properties: Let Ωc be the complement of Ω and define restricted Sobolev spaces analogously for Ωc. Both sets of spaces have a natural pairing with C∞(T2). The Sobolev space for Ω is the annihilator in the Sobolev space for T2 of C∞
c(Ωc) and that for Ωc is the annihilator of C∞
c(Ω).[2] In fact this is proved by locally applying a small translation to move the domain inside itself and then smoothing by a smooth convolution operator.
- Suppose g in Hk(T2) annihilates C∞
c(Ωc). By compactness, there are finitely many opens U0, U1, ... , UN covering Ω such that the closure of U0 is disjoint from ∂Ω and each Ui is an open disc about a boundary point zi such that in Ui small translations in the direction of the normal vector ni carry Ω into Ω. Add an open UN+1 with closure in Ωc to produce a cover of T2 and let ψi be a partition of unity subordinate to this cover. If translation by n is denoted by λn, then the functions
- tend to g as t decreases to 0 and still lie in the annihilator, indeed they are in the annihilator for a larger domain than Ωc, the complement of which lies in Ω. Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in Ω. These are necessarily smooth functions of compact support in Ω.
- Further vanishing properties on the boundary: The characterization in terms of annihilators shows that f ∈ Ck(Ω) lies in H k
0(Ω) if (and only if) it and its derivatives of order less than k vanish on ∂Ω.[3] In fact f can be extended to T2 by setting it to be 0 on Ωc. This extension F defines an element in Hk(T2) using the formula for the norm
- Moreover F satisfies (F, g) = 0 for g in C∞
- Duality: For k ≥ 0, define H−k(Ω) to be the orthogonal complement of H−k
0(Ωc) in H−k(T2). Let Pk be the orthogonal projection onto H−k(Ω), so that Qk = I − Pk is the orthogonal projection onto H−k
0(Ωc). When k = 0, this just gives H0(Ω) = L2(Ω). If f ∈ Hk
0(Ωc) and g ∈ H−k(T2), then
- This implies that under the pairing between Hk(T2) and H−k(T2), Hk
0(Ωc) and H−k(Ω) are each other's duals.
- Approximation by smooth functions: The image of C∞
c(Ω) is dense in H−k(Ω) for k ≤ 0. This is obvious for k = 0 since the sum C∞
c(Ω) + C∞
c(Ωc) is dense in L2(T2). Density for k < 0 follows because the image of L2(T2) is dense in H−k(T2) and Pk annihilates C∞
- Canonical isometries: The operator (I + ∆)k gives an isometry of H 2k
0(Ω) into H0(Ω) and of H k
0(Ω) onto H−k(Ω). In fact the first statement follows because it is true on T2. That (I + ∆)k is an isometry on H k
0(Ω) follows using the density of C∞
c(Ω) in H−k(Ω): for f, g ∈ C∞
c(Ω) we have:
- Since the adjoint map between the duals can by identified with this map, it follows that (I + ∆)k is a unitary map.
Application to Dirichlet problem[edit]
Invertibility of ∆[edit]
The operator ∆ defines an isomorphism between H1
0(Ω) and H−1(Ω). In fact it is a Fredholm operator of index 0. The kernel of ∆ in H1(T2) consists of constant functions and none of these except zero vanish on the boundary of Ω. Hence the kernel of H1
0(Ω) is (0) and ∆ is invertible.
In particular the equation ∆f = g has a unique solution in H1
0(Ω) for g in H−1(Ω).
Eigenvalue problem[edit]
Let T be the operator on L2(Ω) defined by
where R0 is the inclusion of L2(Ω) in H−1(Ω) and R1 of H1
0(Ω) in L2(Ω), both compact operators by Rellich's theorem. The operator T is compact and self-adjoint with (Tf, f ) > 0 for all f. By the spectral theorem, there is a complete orthonormal set of eigenfunctions fn in L2(Ω) with
Since μn > 0, fn lies in H1
0(Ω). Setting λn = μ−n, the fn are eigenfunctions of the Laplacian:
Sobolev spaces without boundary condition[edit]
To determine the regularity properties of the eigenfunctions fn and solutions of
enlargements of the Sobolev spaces Hk
0(Ω) have to be considered. Let C∞(Ω−) be the space of smooth functions on Ω which with their derivatives extend continuously to Ω. By Borel's lemma, these are precisely the restrictions of smooth functions on T2. The Sobolev space Hk(Ω) is defined to the Hilbert space completion of this space for the norm
This norm agrees with the Sobolev norm on C∞
c(Ω) so that Hk
0(Ω) can be regarded as a closed subspace of Hk(Ω). Unlike Hk
0(Ω), Hk(Ω) is not naturally a subspace of Hk(T2), but the map restricting smooth functions from T2 to Ω is continuous for the Sobolev norm so extends by continuity to a map ρk : Hk(T2) → Hk(Ω).
- Invariance under diffeomorphism: Any diffeomorphism between the closures of two smooth domains induces an isomorphism between the Sobolev space. This is a simple consequence of the chain rule for derivatives.
- Extension theorem: The restriction of ρk to the orthogonal complement of its kernel defines an isomorphism onto Hk(Ω). The extension map Ek is defined to be the inverse of this map: it is an isomorphism (not necessarily norm preserving) of Hk(Ω) onto the orthogonal complement of Hk
0(Ωc) such that ρk ∘ Ek = I. On C∞
c(Ω), it agrees with the natural inclusion map. Bounded extension maps Ek of this kind from Hk(Ω) to Hk(T2) were constructed first constructed by Hestenes and Lions. For smooth curves the Seeley extension theorem provides an extension which is continuous in all the Sobolev norms. A version of the extension which applies in the case where the boundary is just a Lipschitz curve was constructed by Calderón using singular integral operators and generalized by Stein (1970).
- It is sufficient to construct an extension E for a neighbourhood of a closed annulus, since a collar around the boundary is diffeomorphic to an annulus I × T with I a closed interval in T. Taking a smooth bump function ψ with 0 ≤ ψ ≤ 1, equal to 1 near the boundary and 0 outside the collar, E(ψf ) + (1 − ψ) f will provide an extension on Ω. On the annulus, the problem reduces to finding an extension for Ck( I ) in Ck(T). Using a partition of unity the task of extending reduces to a neighbourhood of the end points of I. Assuming 0 is the left end point, an extension is given locally by
- Matching the first derivatives of order k or less at 0, gives
- This matrix equation is solvable because the determinant is non-zero by Vandermonde's formula. It is straightforward to check that the formula for E( f ), when appropriately modified with bump functions, leads to an extension which is continuous in the above Sobolev norm.[4]
- Restriction theorem: The restriction map ρk is surjective with ker ρk = Hk
0(Ωc). This is an immediate consequence of the extension theorem and the support properties for Sobolev spaces with boundary condition.
- Duality: Hk(Ω) is naturally the dual of H−k0(Ω). Again this is an immediate consequence of the restriction theorem. Thus the Sobolev spaces form a chain:
- The differentiation operators ∂x, ∂y carry each Sobolev space into the larger one with index 1 less.
- Sobolev embedding theorem: Hk+2(Ω) is contained in Ck(Ω−). This is an immediate consequence of the extension theorem and the Sobolev embedding theorem for Hk+2(T2).
- Characterization: Hk(Ω) consists of f in L2(Ω) = H0(Ω) such that all the derivatives ∂αf lie in L2(Ω) for |α| ≤ k.Here the derivatives are taken within the chain of Sobolev spaces above.[5] Since C∞
c(Ω) is weakly dense in Hk(Ω), this condition is equivalent to the existence of L2 functions fα such that
- To prove the characterization, note that if f is in Hk(Ω), then ∂αf lies in Hk−|α|(Ω) and hence in H0(Ω) = L2(Ω). Conversely the result is well known for the Sobolev spaces Hk(T2): the assumption implies that the (∂x − i∂y)k f is in L2(T2) and the corresponding condition on the Fourier coefficients of f shows that f lies in Hk(T2). Similarly the result an be proved directly for an annulus [−δ, δ] × T. In fact by the argument on T2 the restriction of f to any smaller annulus [−δ',δ'] × T lies in Hk: equivalently the restriction of the function fR (x, y) = f (Rx, y) lies in Hk for R > 1. On the other hand ∂α fR → ∂α f in L2 as R → 1, so that f must lie in Hk. The case for a general domain Ω reduces to these two cases since f can be written as f = ψf + (1 − ψ) f with ψ a bump function supported in Ω such that 1 − ψ is supported in a collar of the boundary.
- Regularity theorem: If f in L2(Ω) has both derivatives ∂x f and ∂y f in Hk(Ω) then f lies in Hk+1(Ω). This is an immediate consequence of the characterization of Hk(Ω) above. In fact if this is true even when satisfied at the level of distributions: if there are functions g, h in Hk(Ω) such that (g,φ) = (f, φx) and (h,φ) = (f,φy) for φ in C∞
c(Ω), then f is in Hk+1(Ω).
- Rotations on an annulus: For an annulus I × T, the extension map to T2 is by construction equivariant with respect to rotations in the second variable,
- On T2 it is known that if f is in Hk, then the difference quotient δh f = h−1(Rh f − f ) → ∂y f in Hk−1; if the difference quotients are bounded in Hk then ∂yf lies in Hk. Both assertions are consequences of the formula:
- These results on T2 imply analogous results on the annulus using the extension.
Regularity for Dirichlet problem[edit]
Regularity for dual Dirichlet problem[edit]
If ∆u = f with u in H1
0(Ω) and f in Hk−1(Ω) with k ≥ 0, then u lies in Hk+1(Ω).
Take a decomposition u = ψu + (1 − ψ)u with ψ supported in Ω and 1 − ψ supported in a collar of the boundary. Standard Sobolev theory for T2 can be applied to ψu: elliptic regularity implies that it lies in Hk+1(T2) and hence Hk+1(Ω). v = (1 − ψ)u lies in H1
0 of a collar, diffeomorphic to an annulus, so it suffices to prove the result with Ω a collar and ∆ replaced by
The proof[6] proceeds by induction on k, proving simultaneously the inequality
for some constant C depending only on k. It is straightforward to establish this inequality for k = 0, where by density u can be taken to be smooth of compact support in Ω:
The collar is diffeomorphic to an annulus. The rotational flow Rt on the annulus induces a flow St on the collar with corresponding vector field Y = r∂x + s∂y. Thus Y corresponds to the vector field ∂θ. The radial vector field on the annulus r∂r is a commuting vector field which on the collar gives a vector field Z = p∂x + q∂y proportional to the normal vector field. The vector fields Y and Z commute.
The difference quotients δhu can be formed for the flow St. The commutators [δh, ∆1] are second order differential operators from Hk+1(Ω) to Hk−1(Ω). Their operators norms are uniformly bounded for h near 0; for the computation can be carried out on the annulus where the commutator just replaces the coefficients of ∆1 by their difference quotients composed with Sh. On the other hand, v = δhu lies in H1
0(Ω), so the inequalities for u apply equally well for v:
The uniform boundedness of the difference quotients δhu implies that Yu lies in Hk+1(Ω) with
It follows that Vu lies in Hk+1(Ω) where V is the vector field
Moreover, Vu satisfies a similar inequality to Yu.
Let W be the orthogonal vector field
It can also be written as ξZ for some smooth nowhere vanishing function ξ on a neighbourhood of the collar.
It suffices to show that Wu lies in Hk+1(Ω). For then
so that ∂xu and ∂yu lie in Hk+1(Ω) and u must lie in Hk+2(Ω).
To check the result on Wu, it is enough to show that VWu and W2u lie in Hk(Ω). Note that
are vector fields. But then
with all terms on the right hand side in Hk(Ω). Moreover, the inequalities for Vu show that
Smoothness of eigenfunctions[edit]
It follows by induction from the regularity theorem for the dual Dirichlet problem that the eigenfunctions of ∆ in H1
0(Ω) lie in C∞(Ω−). Moreover, any solution of ∆u = f with f in C∞(Ω−) and u in H1
0(Ω) must have u in C∞(Ω−). In both cases by the vanishing properties, the eigenfunctions and u vanish on the boundary of Ω.
Solving the Dirichlet problem[edit]
The dual Dirichlet problem can be used to solve the Dirichlet problem:
By Borel's lemma g is the restriction of a function G in C∞(Ω−). Let F be the smooth solution of ∆F = ∆G with F = 0 on ∂Ω. Then f = G − F solves the Dirichlet problem. By the maximal principle, the solution is unique.[7]
Application to smooth Riemann mapping theorem[edit]
The solution to the Dirichlet problem can be used to prove a strong form of the Riemann mapping theorem for simply connected domains with smooth boundary. The method also applies to a region diffeomorphic to an annulus.[8] For multiply connected regions with smooth boundary Schiffer & Hawley (1962) have given a method for mapping the region onto a disc with circular holes. Their method involves solving the Dirichlet problem with a non-linear boundary condition. They construct a function g such that:
- g is harmonic in the interior of Ω;
- On ∂Ω we have: ∂ng = κ − KeG, where κ is the curvature of the boundary curve, ∂n is the derivative in the direction normal to ∂Ω and K is constant on each boundary component.
Taylor (2011) gives a proof of the Riemann mapping theorem for a simply connected domain Ω with smooth boundary. Translating if necessary, it can be assumed that 0 ∈ Ω. The solution of the Dirichlet problem shows that there is a unique smooth function U(z) on Ω which is harmonic in Ω and equals −log|z| on ∂Ω. Define the Green's function by G(z) = log|z| + U(z). It vanishes on ∂Ω and is harmonic on Ω away from 0. The harmonic conjugate V of U is the unique real function on Ω such that U + iV is holomorphic. As such it must satisfy the Cauchy-Riemann equations:
The solution is given by
where the integral is taken over any path in Ω. It is easily verified that Vx and Vy exist and are given by the corresponding derivatives of U. Thus V is a smooth function on Ω, vanishing at 0. By the Cauchy-Riemann f = U + iV is smooth on Ω, holomorphic on Ω and f (0) = 0. The function H = arg z + V(z) is only defined up to multiples of 2π, but the function
is a holomorphic on Ω and smooth on Ω. By construction, F(0) = 0 and |F(z)| = 1 for z ∈ ∂Ω. Since z has winding number 1, so too does F(z). On the other hand, F(z) = 0 only for z = 0 where there is a simple zero. So by the argument principle F assumes every value in the unit disc, D, exactly once and F′ does not vanish inside Ω. To check that the derivative on the boundary curve is non-zero amounts to computing the derivative of eiH, i.e. the derivative of H should not vanish on the boundary curve. By the Cauchy-Riemann equations these tangential derivative are up to a sign the directional derivative in the direction of the normal to the boundary. But G vanishes on the boundary and is strictly negative in Ω since |F| = eG. The Hopf lemma implies that the directional derivative of G in the direction of the outward normal is strictly positive. So on the boundary curve, F has nowhere vanishing derivative. Since the boundary curve has winding number one, F defines a diffeomorphism of the boundary curve onto the unit circle. Accordingly, F : Ω → D is a smooth diffeomorphism, which restricts to a holomorphic map Ω → D and a smooth diffeomorphism between the boundaries.
Similar arguments can be applied to prove the Riemann mapping theorem for a doubly connected domain Ω bounded by simple smooth curves Ci (the inner curve) and Co (the outer curve). By translating we can assume 1 lies on the outer boundary. Let u be the smooth solution of the Dirichlet problem with U = 0 on the outer curve and −1 on the inner curve. By the maximum principle 0 < u(z) < 1 for z in Ω and so by the Hopf lemma the normal derivatives of u are negative on the outer curve and positive on the inner curve. The integral of −uydx + uydx over the boundary is zero by Stoke's theorem so the contributions from the boundary curves cancel. On the other hand, on each boundary curve the contribution is the integral of the normal derivative along the boundary. So there is a constant c > 0 such that U = cu satisfies
on each boundary curve. The harmonic conjugate V of U can again be defined by
and is well-defined up to multiples of 2π. The function
is smooth on Ω and holomorphic in Ω. On the outer curve |F| = 1 and on the inner curve |F| = e−c = r < 1. The tangential derivatives on the outer curves are nowhere vanishing by the Cauchy-Riemann equations, since the normal derivatives are nowhere vanishing. The normalization of the integrals implies that F restricts to a diffeomorphism between the boundary curves and the two concentric circles. Since the images of outer and inner curve have winding number 1 and 0 about any point in the annulus, an application of the argument principle implies that F assumes every value within the annulus r < |z| < 1 exactly once; since that includes multiplicities, the complex derivative of F is nowhere vanishing in Ω. This F is a smooth diffeomorphism of Ω onto the closed annulus r ≤ |z| ≤ 1, restricting to a holomorphic map in the interior and a smooth diffeomorphism on both boundary curves.
Trace map[edit]
The restriction map τ : C∞(T2) → C∞(T) = C∞(1 × T) extends to a continuous map Hk(T2) → Hk − ½(T) for k ≥ 1.[9] In fact
so the Cauchy–Schwarz inequality yields
where, by the integral test,
The map τ is onto since a continuous extension map E can be constructed from Hk − ½(T) to Hk(T2).[10][11] In fact set
Thus ck < λn < Ck. If g is smooth, then by construction Eg restricts to g on 1 × T. Moreover, E is a bounded linear map since
It follows that there is a trace map τ of Hk(Ω) onto Hk − ½(∂Ω). Indeed, take a tubular neighbourhood of the boundary and a smooth function ψ supported in the collar and equal to 1 near the boundary. Multiplication by ψ carries functions into Hk of the collar, which can be identified with Hk of an annulus for which there is a trace map. The invariance under diffeomorphims (or coordinate change) of the half-integer Sobolev spaces on the circle follows from the fact that an equivalent norm on Hk + ½(T) is given by[12]
It is also a consequence of the properties of τ and E (the "trace theorem").[13] In fact any diffeomorphism f of T induces a diffeomorphism F of T2 by acting only on the second factor. Invariance of Hk(T2) under the induced map F* therefore implies invariance of Hk − ½(T) under f*, since f* = τ ∘ F* ∘ E.
Further consequences of the trace theorem are the two exact sequences[14][15]
where the last map takes f in H2(Ω) to f|∂Ω and ∂nf|∂Ω. There are generalizations of these sequences to Hk(Ω) involving higher powers of the normal derivative in the trace map:
The trace map to Hj − ½(∂Ω) takes f to ∂k − j
nf |∂Ω
Abstract formulation of boundary value problems[edit]
The Sobolev space approach to the Neumann problem cannot be phrased quite as directly as that for the Dirichlet problem. The main reason is that for a function f in H1(Ω), the normal derivative ∂nf |∂Ω cannot be a priori defined at the level of Sobolev spaces. Instead an alternative formulation of boundary value problems for the Laplacian Δ on a bounded region Ω in the plane is used. It employs Dirichlet forms, sesqulinear bilinear forms on H1(Ω), H1
0(Ω) or an intermediate closed subspace. Integration over the boundary is not involved in defining the Dirichlet form. Instead, if the Dirichlet form satisfies a certain positivity condition, termed coerciveness, solution can be shown to exist in a weak sense, so-called "weak solutions". A general regularity theorem than implies that the solutions of the boundary value problem must lie in H2(Ω), so that they are strong solutions and satisfy boundary conditions involving the restriction of a function and its normal derivative to the boundary. The Dirichlet problem can equally well be phrased in these terms, but because the trace map f |∂Ω is already defined on H1(Ω), Dirichlet forms do not need to be mentioned explicitly and the operator formulation is more direct. A unified discussion is given in Folland (1995) and briefly summarised below. It is explained how the Dirichlet problem, as discussed above, fits into this framework. Then a detailed treatment of the Neumann problem from this point of view is given following Taylor (2011).
The Hilbert space formulation of boundary value problems for the Laplacian Δ on a bounded region Ω in the plane proceeds from the following data:[16]
- A closed subspace H1
- A Dirichlet form for Δ given by a bounded Hermitian bilinear form D( f, g) defined for f, g ∈ H1(Ω) such that D( f, g) = (∆f, g) for f, g ∈ H1
- D is coercive, i.e. there is a positive constant C and a non-negative constant λ such that D( f, f ) ≥ C ( f, f )(1) − λ( f, f ).
A weak solution of the boundary value problem given initial data f in L2(Ω) is a function u satisfying
for all g.
For both the Dirichlet and Neumann problem
For the Dirichlet problem H = H1
0(Ω). In this case
By the trace theorem the solution satisfies u|Ω = 0 in H½(∂Ω).
For the Neumann problem H is taken to be H1(Ω).
Application to Neumann problem[edit]
The classical Neumann problem on Ω consists in solving the boundary value problem
Green's theorem implies that for u, v ∈ C∞(Ω−)
Thus if Δu = 0 in Ω and satisfies the Neumann boundary conditions, ux = uy = 0, and so u is constant in Ω.
Hence the Neumann problem has a unique solution up to adding constants.[17]
Consider the Hermitian form on H1(Ω) defined by
Since H1(Ω) is in duality with H−1
0(Ω), there is a unique element Lu in H−1
0(Ω) such that
The map I + L is an isometry of H1(Ω) onto H−1
0(Ω), so in particular L is bounded.
In fact
On the other hand, any f in H−1
0(Ω) defines a bounded conjugate-linear form on H1(Ω) sending v to ( f, v). By the Riesz–Fischer theorem, there exists u ∈ H1(Ω) such that
Hence (L + I)u = f and so L + I is surjective. Define a bounded linear operator T on L2(Ω) by
where R1 is the map H1(Ω) → L2(Ω), a compact operator, and R0 is the map L2(Ω) → H−1
0(Ω), its adjoint, so also compact.
The operator T has the following properties:
- T is a contraction since it is a composition of contractions
- T is compact, since R0 and R1 are compact by Rellich's theorem
- T is self-adjoint, since if f, g ∈ L2(Ω), they can be written f = (L + I)u, g = (L + I)v with u, v ∈ H1(Ω) so
- T has positive spectrum and kernel (0), for
- and Tf = 0 implies u = 0 and hence f = 0.
- There is a complete orthonormal basis fn of L2(Ω) consisting of eigenfunctions of T. Thus
- with 0 < μn ≤ 1 and μn decreasing to 0.
- The eigenfunctions all lie in H1(Ω) since the image of T lies in H1(Ω).
- The fn are eigenfunctions of L with
- Thus λn are non-negative and increase to ∞.
- The eigenvalue 0 occurs with multiplicity one and corresponds to the constant function. For if u ∈ H1(Ω) satisfies Lu = 0, then
- so u is constant.
Regularity for Neumann problem[edit]
Weak solutions are strong solutions[edit]
The first main regularity result shows that a weak solution expressed in terms of the operator L and the Dirichlet form D is a strong solution in the classical sense, expressed in terms of the Laplacian Δ and the Neumann boundary conditions. Thus if u = Tf with u ∈ H1(Ω), f ∈ L2(Ω), then u ∈ H2(Ω), satisfies Δu + u = f and ∂nu|∂Ω = 0. Moreover, for some constant C independent of u,
Note that
Take a decomposition u = ψu + (1 − ψ)u with ψ supported in Ω and 1 − ψ supported in a collar of the boundary.
The operator L is characterized by
so that
The function v = ψu and w = (1 − ψ)u are treated separately, v being essentially subject to usual elliptic regularity considerations for interior points while w requires special treatment near the boundary using difference quotients. Once the strong properties are established in terms of ∆ and the Neumann boundary conditions, the "bootstrap" regularity results can be proved exactly as for the Dirichlet problem.
Interior estimates[edit]
The function v = ψu lies in H1
0(Ω1) where Ω1 is a region with closure in Ω. If f ∈ C∞
c(Ω) and g ∈ C∞(Ω−)
By continuity the same holds with f replaced by v and hence Lv = ∆v. So
Hence regarding v as an element of H1(T2), ∆v ∈ L2(T2). Hence v ∈ H2(T2). Since v = φv for φ ∈ C∞
c(Ω), we have v ∈ H2
0(Ω). Moreover,
so that
Boundary estimates[edit]
The function w = (1 − ψ)u is supported in a collar contained in a tubular neighbourhood of the boundary. The difference quotients δhw can be formed for the flow St and lie in H1(Ω), so the first inequality is applicable:
The commutators [L, δh] are uniformly bounded as operators from H1(Ω) to H−1
0(Ω). This is equivalent to checking the inequality
for g, h smooth functions on a collar. This can be checked directly on an annulus, using invariance of Sobolev spaces under dffeomorphisms and the fact that for the annulus the commutator of δh with a differential operator is obtained by applying the difference operator to the coefficients after having applied Rh to the function:[18]
Hence the difference quotients δhw are uniformly bounded, and therefore Yw ∈ H1(Ω) with
Hence Vw ∈ H1(Ω) and Vw satisfies a similar inequality to Yw:
Let W be the orthogonal vector field. As for the Dirichlet problem, to show that w ∈ H2(Ω), it suffices to show that Ww ∈ H1(Ω).
To check this, it is enough to show that VWw, W 2u ∈ L2(Ω). As before
are vector fields. On the other hand, (Lw, φ) = (∆w, φ) for φ ∈ C∞
c(Ω), so that Lw and ∆w define the same distribution on Ω. Hence
Since the terms on the right hand side are pairings with functions in L2(Ω), the regularity criterion shows that Ww ∈ H2(Ω). Hence Lw = ∆w since both terms lie in L2(Ω) and have the same inner products with φ's.
Moreover, the inequalities for Vw show that
It follows that u = v + w ∈ H2(Ω). Moreover,
Neumann boundary conditions[edit]
Since u ∈ H2(Ω), Green's theorem is applicable by continuity. Thus for v ∈ H1(Ω),
Hence the Neumann boundary conditions are satisfied:
where the left hand side is regarded as an element of H½(∂Ω) and hence L2(∂Ω).
Regularity of strong solutions[edit]
The main result here states that if u ∈ Hk+1 (k ≥ 1), ∆u ∈ Hk and ∂nu|∂Ω = 0, then u ∈ Hk+2 and
for some constant independent of u.
Like the corresponding result for the Dirichlet problem, this is proved by induction on k ≥ 1. For k = 1, u is also a weak solution of the Neumann problem so satisfies the estimate above for k = 0. The Neumann boundary condition can be written
Since Z commutes with the vector field Y corresponding to the period flow St, the inductive method of proof used for the Dirichlet problem works equally well in this case: for the difference quotients δh preserve the boundary condition when expressed in terms of Z.[19]
Smoothness of eigenfunctions[edit]
It follows by induction from the regularity theorem for the Neumann problem that the eigenfunctions of D in H1(Ω) lie in C∞(Ω−). Moreover, any solution of Du = f with f in C∞(Ω−) and u in H1(Ω) must have u in C∞(Ω−). In both cases by the vanishing properties, the normal derivatives of the eigenfunctions and u vanish on ∂Ω.
Solving the associated Neumann problem[edit]
The method above can be used to solve the associated Neumann boundary value problem:
By Borel's lemma g is the restriction of a function G ∈ C∞(Ω−). Let F be a smooth function such that ∂nF = G near the boundary. Let u be the solution of ∆u = −∆F with ∂nu = 0. Then f = u + F solves the boundary value problem.[20]
- ^ Bers, John & Schechter 1979, pp. 192–193
- ^ Chazarain & Piriou 1982
- ^ Folland 1995, p. 226
- ^ Folland 1995
- ^ See:
- Agmon 2010
- Folland 1995, pp. 219–223
- Chazarain & Piriou 1982, p. 94
- ^ Taylor 2011
- ^ Folland 1995, p. 84
- ^ Taylor 2011, pp. 323–325
- ^ Chazarain & Piriou 1982
- ^ Taylor 2011, p. 275
- ^ Renardy, Rogers & 2004 214-218
- ^ Hörmander 1990, pp. 240–241
- ^ Renardy & Rogers 2004
- ^ Chazarain & Piriou 1982
- ^ Renardy & Rogers 2004
- ^ Folland 1995, pp. 231–248
- ^ Taylor 2011
- ^ Folland 1995, pp. 255–260
- ^ Taylor 2011, p. 348
- ^ Folland 1995, p. 85
- John, Fritz (1982), Partial differential equations, Applied Mathematical Sciences, 1 (4th ed.), Springer-Verlag, ISBN 0-387-90609-6
- Bers, Lipman; John, Fritz; Schechter, Martin (1979), Partial differential equations, with supplements by Lars Gȧrding and A. N. Milgram, Lectures in Applied Mathematics, 3A, American Mathematical Society, ISBN 0-8218-0049-3
- Agmon, Shmuel (2010), Lectures on Elliptic Boundary Value Problems, American Mathematical Society, ISBN 0-8218-4910-7
- Stein, Elias M. (1970), Singular Integrals and Differentiability Properties of Functions, Princeton University Press
- Greene, Robert E.; Krantz, Steven G. (2006), Function theory of one complex variable, Graduate Studies in Mathematics, 40 (3rd ed.), American Mathematical Society, ISBN 0-8218-3962-4
- Taylor, Michael E. (2011), Partial differential equations I. Basic theory, Applied Mathematical Sciences, 115 (2nd ed.), Springer, ISBN 978-1-4419-7054-1
- Zimmer, Robert J. (1990), Essential results of functional analysis, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 0-226-98337-4
- Folland, Gerald B. (1995), Introduction to partial differential equations (2nd ed.), Princeton University Press, ISBN 0-691-04361-2
- Chazarain, Jacques; Piriou, Alain (1982), Introduction to the Theory of Linear Partial Differential Equations, Studies in Mathematics and Its Applications, 14, Elsevier, ISBN 0-444-86452-0
- Bell, Steven R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 0-8493-8270-X
- Warner, Frank W. (1983), Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, 94, Springer, ISBN 0-387-90894-3
- Griffiths, Phillip; Harris, Joseph (1994), Principles of Algebraic Geometry, Wiley Interscience, ISBN 0-471-05059-8
- Courant, R. (1950), Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience
- Schiffer, M.; Hawley, N. S. (1962), "Connections and conformal mapping", Acta Math., 107: 175–274, doi:10.1007/bf02545790
- Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, ISBN 3-540-52343-X
- Renardy, Michael; Rogers, Robert C. (2004), An Introduction to Partial Differential Equations, Texts in Applied Mathematics, 13 (2nd ed.), Springer, ISBN 0-387-00444-0


text:
It's been a while since the last post was online. We have been very busy in working on one of the very fun and important features for our WPF 4 release, and I will post more details about the feature later.
While taking short breaks, I've noticed a few blog posts about finding managed memory leaks in WPF as well as Silverlight applications, some have very detailed steps for locating and debugging the memory leaks, some also introduced samples and tools, I found them quite useful and interesting:
- Finding Memory Leaks in WPF-based applications – by WPF Performance team (our PM Jossef Goldberg). This post provided detailed steps in finding the leaks with sample apps, cause analysises and suggested fixes
- Where's your leak at? [Using WinDbg, SOS, and GCRoot to diagnose a .NET memory leak] – by Silverlight dev David Anson, it has a few good links and a very interesting sample
- Tracking down managed memory leaks (how to find a GC leak) – by Rico Mariani, it offers very detailed steps to follow in diagnosing memory leaks
Techniques and Tools:
SOS: finding the method bound to an EventHandler with WinDbg – by Mark Smith
WinDbg, a core part of the Debugging Tools for Windows
Other useful resources:
Simulating Weak Delegates for EventHandler-like Delegates – by Ian Griffiths
Simulating "Weak Delegates" in the CLR – by Greg Schechter
Updated WPFPerf Performance Profiling Tools for WPF – by WPF Performance team


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By Michele Pedulla, DNP, RN, ARNP, CPNP-PC; Google+ ProfileFaculty Member,
Kaplan University School of Nursing
nation is facing a growing shortage of primary care physicians. Are nurse
practitioners the answer?
most people have an illness like a cold or a symptom such as a sore throat
that's gone on for several days, they usually take a trip to the doctor's
office. But, with an ever-increasing shortage of primary care physicians in the
United States—and demand for them increasing every year—it's getting harder these
days for patients to find a doctor nearby, let alone secure a same-day appointment.
So, what's a sick person to do? As it turns out, more and more folks are turning
to their neighborhood nurse practitioner to treat their medical needs instead.
to the National Nursing Centers ConsortiumConvenient Care Association, an arm
of the Public Health Management Corporation, approximately 20 million patients have sought the care of nurse practitioners in 2013.Nurse practitioners are registered nurses who hold a
master's or doctoral degree, with advanced training in a specialty area, such
as family medicine. "They're trained to do physical exams, diagnose, treat,
assess, and prescribe," explained Ken Miller co-president of the American Association of Nurse Practitioners
(AANP) and a professor at the Catholic University of America's School of
Nursing, to Kaplan University "They can do 85% to 90% of what a family practice
physician does, essentially."
AANP estimates that there are more than 189,000 nurse practitioners providing
care in the country today. That's up from 128,000 in 2008. The RAND Corporation, a nonprofit research organization, predicts the number will rise
to 244,000 by 2025. A majority of those in the field today work in primary
care. Many collaborate alongside doctors in private practices or in community
health centers, some lead nurse-managed clinics, and others provide services at
retail clinics, such as those in CVS and Walgreens.
an influx of people entering the nurse practitioner profession each year, some health
care experts say expanding the roles of nonphysician providers, such as nurse
practitioners and physician assistants, could potentially ease the expanding
primary care gap. "Nurse practitioners represent 189,000 possible solutions to
the country's current primary care crisis," Miller said. "Can we alone solve the
problem entirely? No—but we can help."
what kind of deficit are we looking at? Forecasts by the Association of American Medical Colleges say the nation faces a
shortage of 91,500 physicians across all specialties by 2020, including 45,000
primary care physicians. By 2025, the country is expected to be short 130,600 physicians.
factors are contributing to the widening gap. For one, baby boomers are aging.
Over the next decade, the U.S. Census Bureau
estimates a 36% growth in the number of Americans over age 65—the very
demographic with the greatest health care needs. The physician population is
getting older, too. Nearly one third of all doctors are projected to retire over
the next 10 years. Plus, millions of newly insured Americans under the Affordable
Care Act means that millions more people are expected to be seeking care.
the time of this writing, nearly 60 million Americans lacked adequate access to
primary health care due to a scarcity of providers in their communities. Perhaps
the most visible example of the shortfall's impact is the burgeoning VA scandal.
In May 2014, the Department of Veterans Affairs blamed an acute shortage of 400
primary care doctors for excessive treatment delays experienced by veterans across
the nation—which then reportedly led certain facilities to falsify records of
wait times.
veterans aren't the only ones affected by the shortage. Residents of rural and inner-city
areas have been feeling the strain for years. Often, those living in areas with
limited primary care resources are forced to travel long distances to see a
doctor. This can lead to them delaying or even forgoing care, and can result in
a patient's condition worsening or even a pricey trip to the ER—all things that
can be avoided with greater access to providers.
are the types of situations where nurse practitioners can make a great impact.
"Nurse practitioners are playing a vital role by making patient-centered care
available in rural and underserved communities," George Zangaro, director of
the National Center for Health Workforce Analysis in the Bureau of Health Workforce, told Kaplan
University. Zangaro, who supports a team-based approach to tackling the primary
care crisis, added, "One of the most important services being provided in these
areas is health promotion and disease prevention, which will decrease the
incidence rates of diseases and hospitalizations."
an effort to offset the shortage, 19 states plus the District of Columbia have
passed legislation increasing the scope of practice for nurse practitioners. These
laws allow nurse practitioners to provide services with less physician
oversight, practicing closer to the full extent of their education and
training. More states are considering moving in this direction as well. However,
some physician groups argue that substituting nurse practitioners for doctors isn't
the answer. The American Academy of Family Physicians has been vocal about its
opposition, citing questions of patient safety and quality of care as a concern.
practitioner advocates say there is no data to support those claims. In fact,
says AANP's Miller, studies over the years have found the quality of care
provided by nurse practitioners comparable to that of family physicians. The same
studies also found that nurse practitioners were given equal or even higher marks
in patient satisfaction, likely due to the fact that they are able to devote
more time to each appointment. Research published last year in the journal Health
Affairs has also shown that, while most patients still prefer to see a
doctor for their medical needs, they are happy to see a nurse practitioner or
physician assistant if a doctor is unavailable.
nurse practitioner Bessie Burk told Kaplan University that she likes to spend at
least 10 minutes with each of her patients. Burk works at the Sun Life Family
Health Center in Casa Grande, Arizona. Arizona is one state that grants nurse
practitioners full-practice authority. On any given day, Burk sees about 25 to
30 patients, some scheduled and some walk-ins, for issues ranging from minor colds
to chronic diseases. Compare this with the doctors at the Center, who see at
least 40 patients daily.
said the additional time allows her to build extra trust and a rapport with her
patients. She also said her work as a nurse has trained her to better educate patients
on how to take care of themselves. "Our patients like us. They can talk to us.
When we sit down together, they just have a better connection with the nurse
part of us," Burk explained. "We absolutely
make a difference in these communities."
to make a difference on a national level, said Miller, nurse practitioners in every state must be allowed to practice
with full authority. And, echoing Zangaro, he added, "We need to stop looking
at this as a single discipline issue. This crisis concerns all health care
providers. It's going to take all of us working together to solve it."
Pedulla is a full-time faculty member at Kaplan University. The views expressed
in this article are solely those of the author and do not represent the view of
Kaplan University.
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By DAN KARELL
For the first time all season, the New York Cosmos took a lead and never let it go.
Despite going down to ten-men late, a second-half header goal from Sebastian Guenzatti gave the Cosmos a 1-0 victory over Minnesota United on Saturday evening at James Shuart Stadium. Cosmos goalkeeper Kyle Reynish earned his second shutout of the season in the win.
With the 2-2 draw between the Carolina RailHawks and the Tampa Bay Rowdies on Saturday, the Cosmos moved two points clear at the top of the NASL table.
Cosmos head coach Giovanni Savarese made three changes to his starting lineup from the week before, bringing back Ayoze and Peri Marosevic, while starting Japanese youth international Satoru Kashiwase as the lone forward up top.
The continuity in the rest of the lineup seemed to make a difference for the hosts in the first half, as they possessed better than in previous matches. However, the visiting Minnesota United did very well, through on-loan midfielder Calum Mallace, to cut out the Cosmos rhythm in the middle of the park.
Without many good chances, the teams moved into the second half with an emphasis on using the wings. Minnesota created opportunities through Miguel Ibarra and Lucas Rodriguez, though Ibarra's best chance came to nothing after a heavy touch.
The Cosmos finally took advantage of the spaces out wide with the overlapping run of Hunter Freeman down the right side in the 78th minute.
Freeman whipped in a great cross into the middle of the box, and the 22-year-old Guenzatti met the ball with a flying header that went in for a goal, the forward's first of the season. The goal ended a streak of 258 minutes for the Cosmos without a goal from open play.
Two minutes later, the Cosmos were given a stunning blow. A studs up tackle from Freeman resulted in the right back being shown a straight red card, reducing the Cosmos to ten men. However, the hosts were able to defend well and outlast the late opportunities from Minnesota United.
Ayoze nearly gave the Cosmos a two-goal lead in stoppage time but his effort went wide of goal.
I was at the game. Brutal finishing by both sides, but ultimately worse by Minnesota. Fun time, though.
Remind me again why a second/third rate side like the Cosmos is getting any coverage on this site… Who a cares?
A lot of people want to read about them. A lot of people don't, no one is forcing you to click the link man.
It's not that complicated and it's been explained over and over on this blog. I'm sure you can figure it out.
I care. You don't. So go away.
You seem a bit jealous Marc about the coverage the Cosmos gets, are your team being overlooked? There is a simple reason the Cosmos catches the interrest of people. Working in the football industry and talking football all over Europe, I can tell you that most of the people talks about the Cosmos when they talk about football in the US. Galaxy and NYRB are far behind…. Especially in Europe Cosmos is what people think and talk about when it comes to football in the US.
Why should we care what Europe think and talk about? Cosmos are in a lower tier and will remain there for quite some time. Without strong TV revenue the team will perpetually spend more money trying to compete with MLS than they are capable of earning.
Well, you should care about what the European and for that sake the rest of the worlld thinks ad Garber wants MLS to develop into a league that is acknowledged as a good league world wide. When the strongest brand the nation have in football is not playing there, that isn't good for achieving that goal.
The thing is Garber and a lot of the MLS supporters hopes that NYCFC will help build the MLS brand, the problem with that is that that franchise is just looked upon as a reserve team for Man City in Europe, and if MLS want their brand to develop having a reserve team in it won't help.
Before MLS was looked at as a league were former stars ended their careers, and only a few of them were are formr world stars (Beckham, Henry and Nesta) Now it's looked at as a league were former stars end their career and that allows reserve teams to enter.
It's a pity that Garber were more after the entrance fee than looking at what a strong brand name could give to the league in future earnings. Unfortunately this is were Garber shows him self as a short sightet business man not an football man.
It's a pity with Cosmos in MLS, they would have had a lot more attention from the rest of the world. That would also made the league grow much more than having Man City move Gareth Barry or anyone else of their second rated players to NYCFC. Considering a former European Champion signed for the Cosmos and a former two times Champions League winner would have signed for them if not a Serie A Club came calling, the consider what players Cosmos could have attracted to MLS.
The league would have thrived on Cosmos entering even with a lower fee as the future revenue would have grown massively with them in it. Garber has done well expanding the league, but recent choice makes me worry that he isn't the right person to get MLS to the next stage as the NYCFC choice shows, after that: Do you seriously believ that Orlando, Oklahoma etc will bring more attention to MLS?
Thanks for making all of that up. It was fun to read.
You really should know that making fun of someones statements only shows that you can't make a sensible answer…It's the oldest trick in the book and most people have moved past it..
You make a lot of statements without any proof that they are true. Show me proof that Europeans think about Cosmos and not MLS or that Cosmos can attract bigger names than MLS (hello Henry, Beckham, LD, Dempsey, et al). Prove that Cosmos is the strongest brand. You don't know who NYCFC is going to target as players so you decide to assume the best they can hope for is Gareth Barry. Say something with truth behind it and I will respond in kind.
I written wat the truth is, you can like it or not. First of all: Dempsey and LD are stars in US yes, but saying they are world stars and comparing them to Beckham and Henry is way of. LD has one good loan stint in Everton and couldn't get regular playing time in Germany. Dempsey has done well in Fulham, but then didn't really make the step up at Tottenham…
Why Cosmos is the strongest brand, even if it was the last ownership and a makeshift team, tell me about one other MLS club that has been invited to play against Man Utd or any other of the Europan giants at the European giant's stadium.
And when the European giants goes to the state in the summers, who do they mainly play against? MLS all-stars and other European teams, but no MLS teams. They probably won't play against Cosmos either, but you can't rule out Man Utd doing it as they both have Nike as kit suppliers.
Cosmos has Fly Emirates as sponsors, tell me about a MLS team that has the same shirt sponsor as a European giant has?
Of course with the shit loads of money NYCFC got they probably can get better players than Barry, but then again who wants to play for Man Citys reserve team.
It seems to me like you think I'm against MLS, but the case is I just say what the toughts about MLS is in Europe. You proably read Blatters words about the lague a year or so ago. Ives blasted him in here so you must have read it. In my case I'm following the MLS, NASL and USL Pro, Streaming mathces from all leagues, but the way forward for football in US is not having NASL and MLS having a go at each other, they should join forces. That will improve football and the most important thing sometime in the future lift the USmen team into a WC contender.
BTW: wans't able to reply your post. So I replied here..
John O'Donnell
Once again who cares about the rest of the world.Plenty of people here like MLS just the way it is.
Might be that a lot of people like MLS the way its is, but do you honestly believe that Garber & Co don't want to make it bigger better and more respected world wide?
Yeah right.. And how much are you being paid by the Cosmos up create a veneer of legitimacy? I think all these comments are part of a well paid ultimately doomed to fail plan.
Clyde Frog
How much are you getting paid to make comments that drive hits on this blog entry?
"Well paid" ahahhahahaha
That's genius! Hahahahahaha
He said blog posters are
"Well paid" oh that's just classic!!!
Indulge me for a minute: by your standards, name 3 "first rate" sides…
Great win from the Cosmos, obviously still some sort of finishing problem but a win is a win. Our midfield played great, Ayoze is class and Senna is incredible on both sides of the ball. Amazing what that man is doing at his age.
Top of the table for now, hoping for a draw or loss by the Strikers tomorrow. If we can keep this up, preferably with some bigger scorelines, we'll be on our way to Soccer Bowl. But this short season is one hell of a tight race.
I'm amazed at how people get upset over the Cosmos being covered. Personally, I'm offended by the coverage Toronto gets. They are a terrible team with awful ownership, no clue and wannabe Euro fans. They offend me!
Very happy the Cosmos are doing so well in their opening season. Senna is amazing shots are on point and dangerous and he distributions are high class.
I just don't care about the Cosmo's, maybe if they join MLS
we don't care about dog either.
Heh. Minn U.
Jeff Fulton
I don't mind the Cosmos being covered, This is Ives' site. He can cover what ever he wants to cover. I think he is New Your based, so it makes sense that he cover a team that is successful and in his market. The center wheel on your mouse or "magic" thingy on your Apple mouse helps easily scroll by stories you don't want to read.


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Antique Edwardian 10K Gold, Glass Pearl & Ruby Sweetheart Pin/Brooch
Welcome to ROBINS NEST MIDWEST! May I offer….
AGE: Catch is marked Pat 5-17-10
ITEM: A lovely rose gold watch pin/sweetheart pin featuring a small bird at the top left, perched upon an interesting design. With a ton of glass split pearls (and if I'm wrong in my description, they are genuine), the bird has a cab garnet eye at the top far right, a faceted ruby. The pin is definitely a sweetheart piece, most likely a gift from the husband-to-be to his new affianced. Simple c-catch at the back to hold the pin in place.
CONDITION: Very nice with no major problems. One of the prongs holding the ruby appears to have been clipped but the remaining 5 claws are holding the stone quite tightly. The ruby, in the area of the missing claw, has a slightly rough edge - only visible with a loupe (yes, I know, I'm really picky about my descriptions).
METAL CONTENT: 10K Gold
HALLMARKS: Just the catch is marked as above - metal has been gently acid tested for content.
SIZE: 1-1/16" wide x 1" high
WEIGHT: 3.1 grams
FINAL COMMENTS: A lovely brooch just too cute for words!
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Round ornament with black glitter. Inserts of zebra print fabric and silver cording accent. Silver cap and loop for hanging. Made of foam.6"H X 6"W X 6"Dheight with loop: 9"H1 set of 4.
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Clinical Neurophysiologist, Division of Neurology
Epileptologist, Division of Neurology
Professor, UC Department of Pediatrics
Phone 513-636-4222
Fax 513-636-3980
Email firstname.lastname@example.com
Epilepsy; improvement of the quality of life for children with epilepsy; studies of brain and physiology to better understand brain function
Neuropharmacology; epilepsy surgery; central auditory processing; magnetoencephalography; functional cortical mapping; fMRI; SPECT; vagus nerve stimulator; ketogenic diet; language development and language disorders
MD: Case Western Reserve University, Cleveland, Ohio, 1977.
Residency: Rainbow Babies and Childrens, Cleveland, Ohio, 1979.
Fellowship: University Hospitals, Cleveland, Ohio, 1982.
Instructor:, Department of Neurology, University Hospitals, Cleveland, Ohio, 1983.
Fellowship: Epilepsy, National Institutes of Health, Bethesda, MD, 1983-1984.
Fellowship: Neurophysiology, National Institutes of Health, Bethesda, MD, 1984-1985.
Medical Staff Fellow: Clinical Epilepsy Section, National Institutes of Health, 1983-1984.
Medical Staff Fellow: EEG Laboratory, National Institutes of Health, 1984-1986.
Medical Staff Fellow: Unit on Neurophysiology, National Institutes of Health, 1986-1987.
Senior Staff Fellow: Unit on Neurophysiology, National Institutes of Health, 1987-1989.
Associate Professor: Department of Neurology, University of New Mexico, Albuquerque, NM, 1989-1992.
Director: Center for Magnetoencephalography, Albuquerque, NM, 1989-1992.
Associate Professor: Department of Pediatrics, University of Tennessee, Memphis, TN, 1992-2000.
Associate Professor: Department of Neurology, University of Tennessee, Memphis, TN, 1992-2000.
Adjunct Associate Professor: Memphis Speech and Hearing Center, Department of Audiology, University of Memphis, TN, 1998-present.
Division Chief: Child Neurology, Department of Pediatrics, University of Tennessee, 1994-2000.
Medical Director: Comprehensive Epilepsy Center, Le Bonheur Children's Medical, Center, Memphis, TN, 1992-2000.
Director: Cincinnati Children's Neurodiagnostic Laboratory, 2003 to present.
Director: Cincinnati Children's Clinical Neurophysiology Fellowship Program, 2001 to present.
Medical Director: Cincinnati Children's MEG Center, 2005 to present.
Certification: Pediatrics, 1984; Neurology with special competence in Child Neurology, 1984; Clinical Neurophysiology, 1986; Added Qualifications in Clinical Neurophysiology, 1994.
Practice Locations: Cincinnati Children's Hospital Medical Center Main Campus; Outpatient Services -- Anderson.
3333 Burnet Avenue, Cincinnati, Ohio 45229-3026
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