text: I having served my country for 18 years and 2 wars, I am very upset that a spouse can get DIC benefits from the death of her late (First) husband who died of complications 35 years ago(vietnam) she has has been divorced and remarried 5 going on 6 times and is about to reach the age of 57 under the new VA if she is divorced or single prior to reaching her 57th birthday she is entitled to reapply for the 6 time and receive the DIC benefit for life. So she has convinced her present husband (#6) to divorce her and then she promised to remarry him once her July 28 birthday making her 57 years old and she qualify's under the new VA rule to obtain the benefit now remember she promised to remarry him after her birthday and qualification. To me this is fraud to the fullest extent and she should be denied the moneys and benefits. But for the love of my country and god that I serve, this should not be allowed, she is now boasting that after her birthday she will get her $1145.00/mo check for the rest of her life and can now keep it wither married or single or divorced. I think there are many woman out there that deserve this benefit, not some looser scammer, what can be done about this? Does any one have a contact or suggestion (VA Congressman) that can review this case? Its a discrase to her late husband whom died for his country and for all those in uniform today. I think Congress should address or at least the VA. If any one can help stop this stuff let me know as I have the information on this person. Cause I can see the future husband number 7 coming and that nice DIC check in her bank account every month. Which she never earned her late husband she was married to for 4 year 2 of which he spent in Nam and now 6 husband later awaits her check.


text:
I love escaping to the mountains. My parents have a cabin in the Swiss alps and it's definitely one of my family's happy places. I thought it would be nice to bring a little bit of alpine spirit to Jasmine's nursery, so I designed this sweet little mountain shelf. I had already created a pine forest wall with these adorable decals and then realised a little mountain shelf in the forest would be so cute! They are perfect for her little stuffed animals.
You will need:
MDF – I used small offcuts from our wardrobe we had fitted
Screws – they will need to be long enough to go through one layer of MDF (this will depend on how thick your MDF is) and then a further 2cm
White MDF primer
Water-based eggshell or gloss paint in white
Water-based eggshell or gloss paint in grey
Access to a jigsaw
Drilling tips
I recommend choosing a drill bit that is a fraction smaller than the screw, so that the screw tightens into the holes. If the drilled holes are too big, the screw will just spin around. Make sure you use a vice to safely secure the MDF before you start drilling.
Step 1. Firstly, decide on the size you'd like your shelf to be. Mine is approximately 25cm tall by 20cm wide. The little shelf at the bottom is about 12cm deep. On paper, draw out a template for the mountain shape and the shelf. Then take these along with the MDF to a carpenter (or a local wood shop with a jigsaw) and ask them to cut the shapes out. I asked them to curve the corners of the shelves so that they weren't sharp. Of course, if you have a jigsaw at home you can definitely do this part yourself! Once the shapes are cut out, sand off the edges so they are smooth.
Step 2. Prepare the two pieces for painting with a white MDF primer. Once dry, give both pieces a coating with either eggshell or gloss paint.
Step 3. Wait until the paint has dried, and mark two drill points at the bottom of the mountain shape, approximately 4cm in from the edge on either side. Drill holes all the way through the MDF. Then, to work out the correct place for the holes on the shelf part, place the shelf in line with the mountain base and mark with a pencil through the holes.
Step 4. Drill into the back edge of the shelf at these points, approximately 1.5cm deep.
Step 5. With a screwdriver, screw the shelf onto the mountain through the holes you have drilled. Tighten the screws as much as possible to ensure that the shelf is properly secured.
Step 6. With pencil, draw on the snow capped mountain line at the top of the mountains.
Step 7. Paint the shelf and mountain grey below the line you've just drawn. This will probably require a couple of coats.
Step 8. Attach a picture hook to the back of the shelf so that you can hang it on the wall. Alternatively, use superglue strength adhesive strips – just make sure the shelves are super secure. You don't want them falling off the wall and landing on anyone!
I hope you enjoyed my Mountain Shelf tutorial! As DIY projects go they are simple to make and really add an extra bit of cuteness to a child's bedroom. I hope Jasmine will love these for years to come.
Fancy saving this tutorial for later? Make sure you pin the image below to one of your Pinterest boards so you can always find it. Have fun DIYing everyone!

subset: commoncrawl


text:
Risks Involved for a High Maturity Organization to work with a Low Maturity Organization
This paper highlights some of the practical scenarios encountered in having to deal with differing process maturity in interplay. Requirements originated from the lower maturity organization and were often either dictated by the marketing department or were not very stable. Requirements management within the context of the lower maturity organization was always non existent. It was left to the higher maturity organization to document such changes and accordingly set the implementation goals. In such a scenario, the following appears to be the risks involved.
Maturity Model, Configuration Management, Cost Performance Index, Process,
Requirements Management.
For a high maturity organization to conduct business with a lower maturity organization, the risks involved are enormous. The image of the high maturity organization will be the first at risk for being damaged. Apart from the standards getting diluted in order to accommodate the requirements of the lower maturity organization, quality of the end product will suffer immensely due to time constraints within which the product will have to be developed while accommodating the process overheads. In the paper Software Process Improvement at Hughes Aircraft [1], Humphrey, Snyder and Willis discuss that productivity and quality increase with reduced risk levels as the maturity level increases from 1 to 5. This assessment however, is highly dependent upon the maturity of clients with whom the organization will interact. Given the fact that, of the 284 assessments performed in 1994, 75% of the assessments indicate level 1 capability, 16% of the assessments indicate level 2 capability and about 8% of the assessments indicate level 3 capability, a good 99% of the assessed software industry is below level 3 maturity [2]. This situation has not changed drastically at the time of this writing in 1999. Even though organizational processes have matured, the operational processes addressing the business needs of an organization are most definitely at a lower maturity because of the scenarios identified in this paper. Process management and adherence to it in this context is not only a matter of a tremendous overhead for a high maturity company but may also be a tremendous risk when it comes to relating its processes with other lower maturity processes. A few scenarios are discussed in what follows.
Changing requirements from the lower maturity organization.
requirements and frequent changes to requirements without a modified
delivery schedule is a very common feature.
Even if this means a lot of extra work for the higher maturity
organization to cope up with, such as, modifying the end of phase
deliverables from all the phases and then having to modify the code, it
might not meet the schedule deadlines.
This is especially so, if the lower maturity organization cannot
make up its mind on whether the change is required or can be postponed for
a future release. Most of
this uncertainty is due to the fact that requirements originate from the
marketing department that has to face a very competitive end user market.
The concept in such a work environment is "if the technology or
the software is late, it need not be there in the first place".
What is often targeted for is the look and feel of the software in
order to gain a market penetration. So,
a quick and dirty prototype is developed to fulfill this goal and then
software is built using this prototype as reference.
In a mature software development organization where the emphasis is
on end of phase deliverables and making sure that reviews and previews are
effectively conducted, a software development effort involving a quick and
dirty prototype is a potential challenge and a big risk.
Though prototype code is considered to be a throwaway code, if the
prototype is received well by the customer, it becomes the basis for
further development.
This risk is further compounded if the lower maturity organization does not keep track of all the changes to requirements it requested. This situation can be alleviated to an extent by the higher maturity organization by periodically updating the lower maturity organization of its changes to the requirements. This is a major overhead for the higher maturity organization because it will have to dedicate personnel to monitor the changes and keep the customer appraised about them. Firstly, it is difficult to find software specialists or software configuration managers who are willing to do such a job and secondly, it requires a software specialist to make these changes to the requirements because the end of phase deliverables from the analysis and design phases require such skills. So, this is the second risk for such projects.
situation can go out of control, if the customer is geographically
separated and does not acknowledge changes to requirements in a timely
fashion. In spite of having a
common development environment and configuration control mechanism, the
situation is bound to deteriorate over time.
This is because code development will require the discipline to
communicate changes on a daily basis for both teams to synchronize.
This might mean that code releases to either side happens on a
daily basis. Finding the
resources to do it effectively is a third risk.
Some of the resources, which are likely to be unavailable on a
consistently reliable basis, are -
With so many changes happening to the hardware and software platforms and vendor support for operating systems changing, there is an organization need to move to newer environments calling for a tremendous amount of latent learning curves. To cite an example, Domain Software Engineering Environment (DSEE) is a configuration management system in the Domain Operating System. With a change in focus of the software industry from this OS, a new configuration management system will have to be instituted which comes with a lot of added learning and idiosyncrasies that requires time to understand.
legacy systems from the lower maturity organization for feature updates
and for future maintenance.
This is by
far the most serious by way of "degree of risk".
Such legacy systems may not have a great deal of documentation to
support the process that was adopted to develop the code.
It requires highly skilled software specialists from the higher
maturity organization to read the code and understand what is going on.
Often, such an understanding comes after a lot of time is expended.
For the higher maturity organization to get an understanding of how
the feature changes will be handled, it must first do an impact analysis
[3]. This requires a schedule
to be determined after an assessment of the existing legacy system.
The legacy system may have the following risks.
configuration management.
documentation except code.
that is cryptic and is not governed by any coding standards.
enhancements are to be based on software tools that have not stood the
test of time.
In such a
situation, the onus and the consequent risks on the higher maturity
organization are tremendous. The
amount of time expended for creating the deliverables, which will help to
design the enhancements, may be a challenge.
If the design of the older legacy system is arbitrary and defective
with a guiding principle quite different from the best solution, then the
feature enhancements to the software will also be handicapped.
This is a risk that will have to be accepted if the legacy system
has to be reused.
As for point
4, technology transfer and tool induction is very poor in lower maturity
organizations. Tools are
introduced and experimented on live projects based on no evaluation but
purely on individual opinion. Further,
these decisions may become binding on the higher maturity organization to
use on the feature enhancements. If
tool evaluation is not on the higher maturity organization's project
schedule, it could become a major risk for schedule slippage and
associated costs. Usually,
the capabilities of the tool are superficially understood when they are
recommended and this could be a major risk. In depth understanding of the tool comes with experience with
the tool and not by merely experimenting.
With a lack of tool evaluation capabilities and its subsequent
induction into the organization, a lower maturity organization often
passes the buck to the higher maturity organization to determine its
applicability. This is cause
for a major software risk if the higher maturity organization does not
have the resources at the moment or is pre-occupied with meeting customer
expectations for valid business reasons.
requirements specification due to an uncertain customer.
No project
has begun in the author's experience wherein the requirements were all
specified which could be said to be correct, complete and consistent. So, to expect a software solution that would have the above
three characteristics besides being testable, verified and validated is a
major target for being prone to risk.
If the cause-effect analysis has to be conducted by the higher
maturity organization to determine whether the incompleteness of the
requirement can be resolved; given the limited resources at the hands of
the higher maturity organization, it could soon turn into a major
technical risk. Often the risk abatement procedure is to ask the customer for
a solution, which could get into a loop.
Management wisdom and technical expertise is always at a premium
and is seldom available when needed.
If this situation has to be mended, a separate group within the two
organizations must evolve alternative solutions in a proactive manner. If the resources to commit to such an activity are not
available, then this lack of resources could be a major risk.
seriousness of this risk can be mellowed to an extent by having a domain
expert or a systems engineer who will be the sole customer point of
contact. The domain expert
from the higher maturity organization must identify a counterpart in the
lower maturity organization and resolve ambiguities and incomplete
requirements. When about 75% of these requirements are stable, it is then
appropriate to begin with the analysis.
Partitioning the unstable requirements and the stable requirements
is a very important first step toward managing the system requirements.
To cite an example, the user interface code in the windowing
environment can be developed independent of the application code if the
user interface specifications are unclear.
It is then, only a matter of designing the application code
"interfaces" with the user interface code.
Testing these two independently before integrating them has the
potential advantage of increasing the quality of the software.
Customer who cares less for the process but is interested only in the final product.
The customer
is often interested only in his product.
The customer cares less as to what process the higher maturity
organization is going to adopt as long as it meets the schedules and
customer expectations of the software product.
In such a situation, involving the customer in the reviews and
previews of the end of phase deliverables is a potential problem that
could grow to become a risk if differences of opinion begin to surface in
the coding phase. There is
very little time left to amend the solution and trace these changes back
to the end of phase deliverables of the prior phases.
This situation can soon turn into a chaotic situation.
adherence to the process may be a matter of prime importance to the higher
maturity organization, it must not overlook the importance of creating
defect free software. If a
mature software production process cannot ensure defect free software with
deliverables on schedule, then it is like saying, "the operation was
successful, but the patient died".
Micro management of the process and the product development effort
by the management is a very useful step to reduce the magnitude of the
Customer who
believes that a process is applicable only in a stable environment such as
an automobile or a chemical industry but is seldom applicable to software.
It is not
very difficult to relate to this scenario!
The common feeling is that software soon becomes a legacy system,
whether inherited or created by an organization. And the analogous reasoning, which gets a good amount of
consideration, is, "if the input to an automobile manufacturing process
is a totally beaten up machine, then the tinkered product is going to be
as defective as defective software".
Giving this reasoning some thought, it is quite obvious that, the
creation of software processes just because an automobile manufacturing
industry or a chemical industry has a process, may not be appropriate.
A process in the context of software is to bring or apply a few
controls in an otherwise chaotic situation.
But, the challenges and the risks are an unavoidable outcome of the
very nature of software development activity, it being a creative science
goaded to action by the principle of - "to create a technology which
is new and ahead of its time and every one else".
This is further complicated by the lack of experienced software
testing capability in the industry. Though
testing is a recognized activity even in a level 2 process, there is a
lack of effective CASE support in this area.
addition from the use of a process comes about only if there is an effort
toward automating the transitions across the more formal phases of the
software development life cycle [4].
If the use of a process model does not reduce cycle time and
improve quality, then there must be an effort in evaluating the process
transitions with a view to automate as many of these transitions as
possible [5]. To cite an
example, "code standardization" is a very formal transition after
implementation. This
transition can be easily automated in order to improve the virtues of the
baseline code. It introduces
uniformity of appearance to the baseline code with consistency and a
reported cycle time reduction of about 1303X [6].
The trivial
solution available for a high maturity organization when faced with the
scenarios discussed in this paper, is to not do business with lower
maturity organization. This
is seldom the best choice considering the very high percentage of the
assessed software projects to be at levels 1 and 2 on the CMM.
There are also valid business reasons to work on projects from
highly time critical, market driven software projects of a low maturity
organization. Most market
driven software development activities will require the organization to
make intelligent initial guesses and informed decisions in the face of
uncertainty. If the initial guesses are quite different from what is
expected, it will also require a lot of rework.
The cost to rework in such a situation cannot be computed before
hand and often the cost-performance index that is defined as the ratio of
budgeted cost of work performed to the actual cost of work performed is
far from the ideal measure of 1.
To summarize,
even if the assessed capability is at a level 5 maturity on the CMM, given
the fabric of the current maturity levels in the software industry, a
level 5 organizations will have operational processes much below a level 5
because of its interactions with lower process maturity levels. The major risk for the higher maturity organization is to
stake its image and its processes in order to stay in business.
organizations that reach higher mature levels have a very important
responsibility to perform. After
reaching say a maturity of level 5 (Optimizing) on the CMM, software
automation along with reuse and external process education should become a
very important organization focus. Hence, these may need to be added as
KPAs to the existing Defect Prevention, Process Change Management and
Technology Change Management Key Process Areas at Level 5.
The paper
discusses first hand experience of the author with projects having the
characteristics described in the various scenarios.
The author has intentionally concealed the project details to
maintain confidentiality and proprietary nature of the information and to
focus only on the real issues that can be categorized as risks.
[1] Watts S.
Humphrey, Terry R. Snyder and Ronald R. Willis, "Software Process
Improvement at Hughes Aircraft". IEEE Software July 1991.
[2] Herbsleb,
J., et al., "Results of software process improvement efforts : Work in
progress.", Proceedings of the 1994 SEPG National Meeting.
Pittsburgh : Software Engineering Institute, Carnegie Mellon
University. 1994
R.S. Nandyal, "An Approach to Rework Projects", Software
Engineering Symposium, Motorola. Phoenix, Arizona. 1993.
R.S. Nandyal, "Cycle Time Reduction:A Survey of Solutions",
Software Engineering Symposium, Motorola, Chicago, Illinois.
R.S. Nandyal, "Automation of the product Life Cycle - A Solution
to Cycle Time Reduction", Software Engineering Symposium, Motorola,
Chicago, Illinois. 1994.
R.S. Nandyal, "Integrated Software Engineering Tool:ISET, Process
Automation Software", Motorola Tools Fair, Software Engineering
Symposium, Motorola, Chicago, Illinois.
Raghav S. Nandyal is a General Manager
heading Methods, Tools and Products Division for Intelligroup Asia Pvt.
For ease of navigation, use the selection box below ...


text:
Zurich – A liveable city
by Gabby Delauney • September 18, 2017 • Zurich
I exited the train station tired, dirty and disorientated at 8am that Friday morning. After trying to sleep on a train hurtling through Germany I wasn't 100% ready for my next adventure, but here I am. I make my way through town along the river, passing the bicycles and trams carrying commuters to work;...Read more →


text:
Inscribed slabs in Bihar Museum.
Photographer: Simpson, Sir Benjamin
Medium: Photographic print
Date: 1870
Photograph of the inscribed slabs in Bihar Museum taken in the 1870s and is part of the Archaeological Survey of India Collections: Indian Museum Series (volume 4: Bengal 161-210c). It was credited to 'Dr Simpson', but was possibly taken by Joseph David Beglar. The view shows a group of slabs, consisting of two Armenian tombstones, and five Islamic inscriptions. A note by Bloch reads, 'The sculptures photographed while exhibited in the Bihar Museum were collected from various places in Bihar, and are now in the Indian Museum. A paper by Mr Broadley, dealing with this collection, was published in Journal of the Asiatic Society of Bengal, vol. XLI, part I, 1872, pp. 209-311.


text:
Lecture notes on the Stefan problem
- Jemima Reynolds
- 3 years ago
- Views:
1 Lecture notes on the Stefan problem Daniele Andreucci Dipartimento di Metodi e Modelli Matematici Università di Roma La Sapienza via Antonio Scarpa Roma, Italy
3 Introduction These notes cover a portion of a course of the Doctorate Modelli e metodi matematici per la tecnologia e la società, given in 21 in Rome. The title of the course was Evolution equations and free boundary problems and its topics included, essentially, an introduction to Stefan and Hele-Shaw problems. Here only the material concerning the Stefan problem is partially reproduced. The present notes assume the reader has some knowledge of the elementary theory of L p and Sobolev spaces, as well as of the basic results of existence and regularity of solutions to smooth parabolic equations. The bibliography is minimal; only books and articles quoted in the text are referenced. See [12], [18] and [2] for further references. I thank the audience of the course for many stimulating comments and questions, and prof. R. Ricci for interesting discussions on the subject of these notes. Rome, January 22 Revised version: Rome, January 24 iii
5 Notation The notation employed here is essentially standard. Appendix D contains a list of the main symbols used in the text. Constants denoted by γ, C, C δ,...may change fom line to line. v
7 Contents Introduction Notation iii v Chapter 1. The classical formulation 1 1. The Stefan condition 1 2. The free boundary problem 4 3. Explicit examples of solutions 5 4. Basic estimates 7 5. Existence and uniqueness of the solution 8 6. Qualitative behaviour of the solution Regularity of the free boundary 15 Chapter 2. Weak formulation of the Stefan problem From the energy balance to the weak formulation Comparing the weak and the classical formulations 2 3. Definition of weak solution Uniqueness of the weak solution Existence of weak solutions A comparison result 37 Appendix A. Maximum principles for parabolic equations The weak maximum principle The strong maximum principle Hopf s lemma (the boundary point principle) Maximum principle for weak solutions 48 Appendix B. A theorem by Kruzhkov Mollifying kernels The main estimate 53 m+α m+α, Appendix C. The spaces H 2 (Q T ) Comments 55 Appendix D. Symbols used in text 57 Bibliography 59 vii
9 CHAPTER 1 The classical formulation In this chapter we consider the formulation of the Stefan problem as a classical initial boundary value problem for a parabolic partial differential equation. A portion of the boundary of the domain is a priori unknown (the free boundary), and therefore two boundary conditions must be prescribed on it, instead than only one, to obtain a well posed problem. 1. The Stefan condition The Stefan problem ([17]) is probably the simplest mathematical model of a phenomenon of change of phase. When a change of phase takes place, a latent heat is either absorbed or released, while the temperature of the material changing its phase remains constant. In the following we denote by L > the latent heat per unit of volume (p.u.v.), and neglect for the sake of simplicity any volume change in the material undergoing the change of phase. We also assume the critical temperature of change of phase to be a constant, θ. To be specific, consider at time t = t a domain A divided by the plane x 1 = s into two subdomains. At time t = t the sub-domain A 1 = A {x 1 < s } is filled by water, while A 2 = A {x 1 > s } is filled by ice. In the terminology of problems of change of phase, A 1 is the liquid phase and A 2 is the solid phase. The surface separating the two phases is referred to as the interface. Assume also the setting is plane symmetric, that is the temperature θ is a function of x 1 only, besides the time t, and the interface is a plane at all times. Denote by x 1 = s(t) the position of the interface at time t. Note that, due to the natural assumption that temperature is continuous, θ(s(t)+, t) = θ(s(t), t) = θ, for all t. (1.1) Assume ice is changing its phase, that is the interface is advancing into the solid phase. Due to the symmetry assumption we stipulate, we may confine ourselves to consider any portion D, say a disk, of the interface at time t. At a later time t 1 > t the interface occupies a position s(t 1 ) > s(t ) = s. The cylinder D (s(t 1 ), s(t )) has been melted over the time interval (t, t 1 ) (see Figure 1). The change of phase has therefore absorbed a quantity of heat volume of the melted cylinder latent heat p.u.v. = area(d)(s(t 1 ) s(t ))L. (1.2) The heat must be provided by diffusion, as we assume that no heat source or sink is present. We adopt for heat diffusion Fourier s law heat flux = k i Dθ, (1.3) where k 1 > is the diffusivity coefficient in water, and k 2 > is the diffusivity coefficient in ice (in principle k 1 k 2 ). Thus, the quantity of heat in (1.2) must 1
10 2 DANIELE ANDREUCCI water ice s(t ) s(t 1 ) e 1 Figure 1. Melting ice equal t 1 [ k 1 Dθ(s(t), t) e 1 k 2 Dθ(s(t)+, t) ( e 1 )] dx 2 dx 3 dt = t D(t) area(d) t 1 t [ k 1 θ x1 (s(t), t) + k 2 θ x1 (s(t)+, t)] dt. (1.4) Equating the two quantities, dividing the equation by t 1 t and letting t 1 t, we finally find k 1 θ x1 (s(t), t) + k 2 θ x1 (s(t)+, t) = Lṡ(t), (1.5) where we have substituted t with the general time t, as the same procedure can be obviously carried out at any time. This is called the Stefan condition on the free boundary. We stress the fact that the Stefan condition is merely a law of energetical balance. Several remarks are in order. Remark 1.1. Note that, although we did not assume anything on the values of θ(x 1, t) inside each one of the two phases, on physical grounds we should expect θ θ, in water, i.e., in A 1 ; θ θ, in ice, i.e., in A 2. (1.6) The equality θ θ in either one of the two phases (or in both) can not be ruled out in the model. Rather, it corresponds to the case when a whole phase is at critical temperature. Diffusion of heat, according to (1.3), can not take place in that phase, as θ x1 there. Thus (1.5) reduces to, e.g., if the solid phase is at constant temperature, k 1 θ x1 (s(t), t) = Lṡ(t). (1.7) Note that if θ > θ in water, (1.7) predicts that ṡ(t) >. In other words, melting of ice is predicted by the model, instead of solidification of water. This is consistent with obvious physical considerations. Problems where one of the two phases is everywhere at the critical temperature
11 1. THE STEFAN CONDITION 3 are usually referred to (somehow misleadingly) as one phase problems, while the general case where (1.5) is prescribed is the two phases problem. In the latter case, the sign of ṡ(t), and therefore the physical behaviour of the system water/ice predicted by the mathematical model, depends on the relative magnitude of the two heat fluxes at the interface. Remark 1.2. On the interface, which is also known as the free boundary two conditions are therefore prescribed: (1.1) and (1.5). In the case of a one phase problem, this fact has the following meaningful interpretation in terms of the general theory of parabolic PDE: The boundary of the domain where the heat equation (a parabolic equation based on (1.3)) is posed, contains an a priori unknown portion, corresponding to the interface separating the liquid phase from the solid one. Clearly, if only the Dirichlet boundary condition (1.1) was imposed on it, we could choose arbitrarily this part of the boundary, and solve the corresponding initial value boundary problem. Apparently the solution would not, in general, satisfy (1.7). A similar remark applies to solutions found imposing just (1.7) (where now the functional form of the arbitrarily given boundary x 1 = s(t) is explicitly taken into account). It is therefore evident that on the free boundary both conditions (1.1) and (1.7) should be prescribed in order to have a well posed problem. (Or, anyway, in more general free boundary problems, two different boundary conditions are required.) Incidentally, this circle of ideas provides the basic ingredient of a possible proof of the existence of solutions: we assign arbitrarily a candidate free boundary s and consider the solution θ to the problem, say, corresponding to the data (1.1). Then we define a transformed boundary s exploiting (1.7), i.e., k 1 θ x1 (s (t), t) = Lṡ (t). A fixed point of this transform corresponds to a solution of the complete problem. Remark 1.3. The meaning of conditions (1.1) and (1.5) in the context of two phases problems is probably better understood in terms of the weak formulation of the Stefan problem, which is discussed below in Chapter 2. We remark here that, actually, one phase problems are just two phases problems with one phase at constant temperature, so that the discussion in Chapter 2 applies to them too. Remark 1.4. Problems where the temperature restriction (1.6) is not fulfilled, are sometimes called undercooled Stefan problems. We do not treat them here, albeit their mathematical and physical interest (see however Subsection 2.4 of Chapter 2); let us only recall that they are, in some sense, ill posed Exercises Write the analogs of Stefan condition (1.5) in the cases of cylindrical and spherical symmetry in R Note that if we assume, in (1.5), θ < θ in A 1 and θ θ in A 2, we find ṡ(t) <. This appears to be inconsistent with physical intuition: a phase of ice at sub critical temperature should grow into a phase of water at identically critical temperature. Indeed, (1.5) is not a suitable model for the physical setting considered here. In other words, the Stefan condition in the form given above keeps memory of which side of the interface is occupied by which phase. Find out where we implicitly took into account this piece of information and write the Stefan condition when ice and water switch places.
12 4 DANIELE ANDREUCCI 1.3. Prove that Stefan condition (1.5) does not change its form if bounded volumetric heat sources are present (i.e., if the heat equation is not homogeneous). 2. The free boundary problem Keeping the plane symmetry setting considered above, we may of course assume the problem is one dimensional. Denoting by x the space variable, the complete two phases problem can be written as c 1 θ t k 1 θ xx =, in Q 1, (2.1) c 2 θ t k 2 θ xx =, in Q 2, (2.2) k 1 θ x (, t) = h 1 (t), < t < T, (2.3) k 2 θ x (d, t) = h 2 (t), < t < T, (2.4) θ(x, ) = Θ(x), < x < d, (2.5) k 1 θ x (s(t), t) + k 2 θ x (s(t)+, t) = Lṡ(t), < t < T, (2.6) θ(s(t), t) = θ(s(t)+, t) = θ, < t < T, (2.7) s() = b. (2.8) Here < b < d, T and c 1, c 2, k 1, k 2 are given positive numbers. The c i represent the thermal capacities in the two phases. The liquid phase occupies at the initial time t = the interval (, b), while the solid phase occupies (b, d). The problem is posed in the time interval (, T). Moreover we have set Q 1 = {(x, t) < x < s(t), < t < T }, Q 2 = {(x, t) s(t) < x < d, < t < T }. We are assuming that < s(t) < d for all < t < T. If the free boundary hits one of the two fixed boundaries x = and x = d, say at time t, of course the formulation above should be changed. In practice, one of the two phases disappears at t = t. We leave to the reader the simple task of writing the mathematical model for t > t. One could impose other types of boundary data, instead of (2.3), (2.4), e.g., Dirichlet data. If we are to attach the physical meaning of a change of phase model to the problem above, the data must satisfy suitable conditions. At any rate Θ(x) θ, < x < b; Θ(x) θ, b < x < d. Essentially, we need θ θ in Q 1 and θ θ in Q 2. Actually, we will deal mainly with the one phase version of (2.1) (2.8) where the solid phase is at constant temperature. Namely, after adimensionalization, we look at u t u xx =, in Q s,t, (2.9) u x (, t) = h(t) >, < t < T, (2.1) u(x, ) = u (x), < x < b, (2.11) u x (s(t), t) = ṡ(t), < t < T, (2.12) u(s(t), t) =, < t < T, (2.13) s() = b. (2.14)
13 3. EXPLICIT EXAMPLES OF SOLUTIONS 5 (We have kept the old names for all variables excepting the unknown u.) Here we denote for each positive function s C([, T]), such s() = b, Q s,t = {(x, t) < x < s(t), < t < T }. We regard the rescaled temperature u as a function defined in Q s,t. The solid phase therefore does not appear explicitly in the problem. As a matter of fact we assume it to be unbounded in the positive x direction (i.e., d = + ), so that no upper limit has to be imposed on the growth of the free boundary s. The sign restrictions in (2.1) and in (2.11) are imposed so that u > in Q s,t, see Proposition 4.1 below. Definition 2.1. A solution to problem (2.9) (2.14) is a pair (u, s) with s C 1 ((, T]) C([, T]), s() = b, s(t) >, t T ; u C(Q s,t ) C 2,1 (Q s,t ), u x C(Q s,t {t = }), and such that (2.9) (2.14) are satisfied in a classical pointwise sense. We prove a theorem of existence and uniqueness of solutions to (2.9) (2.14), under the assumptions h C([, T]), h(t) >, t T ; (2.15) u C([, b]), u (x) H(b x), x b. (2.16) We also study some qualitative behaviour of the solution. The adimensionalization of the problem does not play a substantial role in the mathematical theory we develop here. For further reading on the one-phase Stefan problem, we refer the reader to [5], [3]; we employ in this chapter the techniques found there, with some changes. Remark 2.1. The free boundary problem (2.9) (2.14) is strongly non linear, in spite of the linearity of the PDE and of the boundary conditions there. Indeed, recall that the free boundary s itself is an unknown of the problem; its dependence on the data is not linear (as, e.g., the explicit examples of Section 3 show) Exercises Prove that the change of variables θ αu + θ, x βξ, t γτ, allows one to write the one phase problem in the adimensionalized form (2.9) (2.14), for a suitable choice of the constants α, β, γ. Also note that adimensionalizing the complete two phases problem similarly is in general impossible. 3. Explicit examples of solutions Example 1. An explicit solution of the heat equation (2.9) is given by ( ) x v(x, t) = erf 2 = 2 x/2 t π t v(x, ) = 1, x >. e z2 dz, x,t >,
14 6 DANIELE ANDREUCCI Here erf denotes the well known error function. Fix C > arbitrarily, and set, for a α > to be chosen presently, { ( )} x u(x, t) = C erf α erf 2. t Define also s(t) = 2α t; note that s() =. Thus u > in Q s,t, and (2.9) as well as (2.13) are satisfied. By direct calculation u x (x, t) = C πt e x2 4t. Hence, u x (s(t), t) = u x (2α t, t) = C πt e α2 = ṡ(t) = α t, if and only if C = παe α2. Note that on the fixed boundary x = we may select either one of the conditions u x (, t) = C πt, or u(, t) = C erf α. We have proven that, when α is chosen as above, u(x, t) = 2αe α2 α x/2 t e z2 dz (3.1) solves the problem sketched in Figure 2. Note however that u is not continuous at (, ); the notion of solution in this connection should be suitably redefined. t s(t) = 2α t u x = C/ πt or u = C erf α u t u xx = u = u x = ṡ(t) = α/ t x Figure 2. The Stefan problem solved by u in (3.1) Example 2. It is obvious by direct inspection that the function u(x, t) = e t x 1, (3.2) solves the Stefan problem in Figure 3, corresponding to the free boundary s(t) = t. Note that we are forced to prescribe an exponentially increasing flux on x = in order to obtain a linear growth for s(t).
15 4. BASIC ESTIMATES 7 t s(t) = t u x = e t or u = e t 1 u t u xx = u = u x = ṡ(t) = 1 x Figure 3. The Stefan problem solved by u in (3.2) 3.1. Exercises Convince yourself that the solutions corresponding to u x (, t) = 2C/ πt, in the case of Example 1, and to u x (, t) = 2e t in the case of Example 2, can not be obtained by linearity from the ones given above. 4. Basic estimates Proposition 4.1. If (u, s) is a solution to (2.9) (2.14), then u(x, t) >, in Q s,t ; (4.1) ṡ(t) >, for all t >. (4.2) Proof. By virtue of the weak maximum principle, u must attain its maximum on the parabolic boundary of Q s,t, i.e., on Q s,t {(x, t) t = T, < x < s(t)}. The data h being positive, the maximum is attained on t = or on x = s(t). Therefore u (remember that u ). If we had u( x, t) = in some ( x, t) Q s,t, invoking the strong maximum principle we would obtain u in Q s,t {t < t}. This is again inconsistent with h >. Thus (4.1) is proven. Then, the value u = attained on the free boundary is a minimum for u. Recalling the parabolic version of Hopf s lemma, we infer ṡ(t) = u x (s(t), t) >, for all t >. Remark 4.1. The proof of estimate (4.1) does not make use of the Stefan condition (2.12). In the same spirit, we consider in the following results solutions to the initial value boundary problem obtained removing Stefan condition from the formulation. The rationale for this approach is that we want to apply those results to approximating solutions constructed according to the ideas of Remark 1.2. Lemma 4.1. Let u be a solution of (2.9), (2.1), (2.11), (2.13), where s is assumed to be a positive non decreasing Lipschitz continuous function in [, T], such that s() = b. Let (2.15) and (2.16) be in force. Then u x is continuous up to all the points of the boundary of Q s,t of the form (, t), (s(t), t), with T t >.
16 8 DANIELE ANDREUCCI The regularity of u x up to x = is classical; the proof of this result will be completed in Section 7, see Lemma 7.1. Proposition 4.2. Let u, s be as in Lemma 4.1. Then where M = max( h, H). < u(x, t) M(s(t) x), in Q s,t, (4.3) Proof. Define v(x, t) = M(s(t) x). It follows immediately v t v xx = Mṡ(t), in Q s,t ; v(s(t), t) = u(s(t), t) =, t T, v x (, t) = M h = u x (, t), < t < T, v(x, ) = M(b x) u (x) = u(x, ), x b. Therefore, taking into account the results of Appendix A, v(x, t) u(x, t), in Q s,t. Corollary 4.1. If u, s are as in Proposition 4.2, then > u x (s(t), t) M, < t < T, (4.4) where M is the constant defined in Proposition 4.2. Proof. A trivial consequence of Proposition 4.1 and of Proposition 4.2, as well as of Lemma 4.1. We also keep in mind Remark 4.1, and of course make use of Hopf s lemma for the strict inequality in (4.4). Remark 4.2. We stress the fact that the barrier construction of Proposition 4.2 is made possible by the fact that ṡ. In turn, this is for solutions of the Stefan problem a consequence of the positivity of u. Thus, such a barrier function, and in general any similar barrier function, does not exist in the case of the undercooled Stefan problem Exercises Note that the function v defined in the proof of Proposition 4.2 is just Lipschitz continuous in t. In spite of this fact, one may apply to v u the weak maximum principle in the form given in Section 4 of Appendix A. Carry out the proof in detail. We prove here 5. Existence and uniqueness of the solution Theorem 5.1. Assume (2.15), (2.16). Then there exists a unique solution to (2.9) (2.14). Let u, s be as in Proposition 4.2. Moreover assume that s Σ := {σ Lip([, T]) σ M, σ() = b}. Here M is the constant defined in Proposition 4.2. The set Σ is a convex compact subset of the Banach space C([, T]), equipped with the max norm. A useful property of all s Σ is b + Mt s(t) b, t T.
17 Define the transform T (s) by 5. EXISTENCE AND UNIQUENESS OF THE SOLUTION 9 t T (s)(t) = b u x (s(τ), τ) dτ, T t. Note that, as a consequence of Lemma 4.1 and of Corollary 4.1, and T (s) Lip([, T]) C 1 ((, T]), M d dt T (s)(t) = u x(s(t), t), t >. Then T : Σ Σ. Also note that a fixed point of T corresponds to a solution of our Stefan problem. We use the divergence theorem to transform the boundary flux integral defining T (s). Namely we write = t s(τ) (u τ u xx ) dxdτ = b t u (x) dx u(s(τ), τ)ṡ(τ) dτ + s(t) u(x, t) dx t t u x (s(τ), τ) dτ h(τ) dτ. Therefore t T (s)(t) = b b b + u x (s(τ), τ) dτ = t u (x) dx + h(τ) dτ s(t) u(x, t) dx =: F(t) s(t) u(x, t) dx. (5.1) Note that this equality allows us to express T (s) in terms of more regular functions than the flux u x (s(t), t), which appeared in its original definition. We are now in a position to prove that T is continuous in the max norm. Let s 1, s 2 Σ. Let us define α(t) = min(s 1 (t), s 2 (t)), β(t) = max(s 1 (t), s 2 (t)), i = 1, if β(t) = s 1 (t), i = 2, otherwise. (The number i is a function of time; this will not have any specific relevance.) Let us also define v(x, t) = u 1 (x, t) u 2 (x, t). Then v satisfies v t v xx =, in Q α,t, (5.2) v x (, t) =, < t < T, (5.3) v(x, ) =, < x < b, (5.4) v(α(t), t) = u i (α(t), t) M(β(t) α(t)), < t < T. (5.5)
18 1 DANIELE ANDREUCCI Therefore, we may invoke the maximum principle to obtain where we also denote On the other hand, we have v,t := max Q α,t v M s 1 s 2,t, (5.6) s 1 s 2,t = max τ t s 1(τ) s 2 (τ). T (s 1 )(t) T (s 2 )(t) = Therefore s 2 (t) s 1 (t) u 2 (x, t) dx u 1 (x, t) dx = α(t) v(x, t) dx + ( 1) i β(t) α(t) u i (x, t) dx. T (s 1 )(t) T (s 2 )(t) α(t) v,t + M(β(t) α(t)) 2 (b + MT)M s 1 s 2,t + M s 1 s 2 2,t, (5.7) and the continuity of T : Σ Σ is proven. By Schauder s theorem, it follows that a fixed point of T exists, and thus existence of a solution. Uniqueness might be proven invoking the monotone dependence result given below (see Theorem 6.1). However, mainly with the purpose of elucidating the role played in the theory of free boundary problems by local integral estimates, we proceed to give a direct proof of uniqueness of solutions. More explicitly, we prove the contractive character (for small t) of T, thereby obtaining existence and uniqueness of a fixed point Local estimates vs the maximum principle. Estimates like (5.6), obtained through the maximum principle, have the advantage of providing an immediate sup estimate of the solution in the whole domain of definition. However, the bound they give may be too rough, at least in some regions of the domain. Consider for example, in the setting above, a point (b/2, ε), with ε 1. The maximum principle predicts for v(b/2, ε) a bound of order M s 1 s 2,ε, that is, obviously, the same bound satisfied by the boundary data for v. On the other hand, taking into account (5.3), (5.4) one might expect v(b/2, ε) to be much smaller than v(α(ε), ε). This is indeed the case, as we show below. In order to do so, we exploit local integral estimates of the solution, that is, estimates involving only values of v in the region of interest, in this case, away from the boundary. We aim at proving that T is a contraction, that is, T (s 1 ) T (s 2 ),t d s 1 s 2,t, (5.8) with d < 1 for small enough t. A quick glance at (5.7) shows that (5.8) does not, indeed, follow from there (unless bm < 1). This failure is due only to the term
19 5. EXISTENCE AND UNIQUENESS OF THE SOLUTION 11 originating from the estimate of α(t) v(x, t) dx. Thus, a better estimate of this integral is needed. A key step in any local estimation is a good choice of cut off functions. These are, typically, non negative smooth functions equal to 1 in the region we want to single out, and identically vanishing away from it. Let b/2 > δ >, and define the cut off function ζ(x), such that ζ(x) = 1, x b 2δ, ζ(x) =, b δ x, 2 δ ζ x(x). Multiply (5.2) by vζ 2 and integrate by parts in Q b,t = [, b] [, t]. We get = (v τ vζ 2 v xx vζ 2 ) dxdτ Q b,t = 1 2 b v(x, t) 2 ζ(x) 2 dx + vxζ 2 2 dxdτ + 2 Q b,t whence (using the inequality 2ab εa 2 + b 2 /ε, ε > ) Q b,t vv x ζζ x dxdτ, (5.9) 1 b v(x, t) 2 ζ(x) 2 dx + v 2 xζ 2 2 dxdτ 2 v x v ζ x ζ dxdτ Q b,t Q b,t 2 v 2 ζ 2 1 x dxdτ + v 2 xζ 2 2 dxdτ. Q b,t Q b,t By absorbing the last integral into the left hand side, we find b v(x, t) 2 ζ(x) 2 dx + vxζ 2 2 dxdτ 4 Q b,t Let us now go back to t = 4 T (s 1 )(t) T (s 2 )(t) = = α(t) b 2δ b δ b 2δ Q b,t v 2 ζ 2 x dxdτ v 2 ζ 2 x dxdτ 16 δ 2 tδ v 2,t = 16 δ t v 2,t. (5.1) v(x, t) dx + ( 1) i v(x, t) dx + α(t) b 2δ β(t) α(t) u i (x, t) dx v(x, t) dx + ( 1) i β(t) α(t) u i (x, t) dx.
20 12 DANIELE ANDREUCCI Use Hölder s inequality and (5.1) to bound the integral over (, b 2δ) (recall that ζ 1 there), and find T (s 1 )(t) T (s 2 )(t) ( b 2δ v(x, t) 2 dx) 1/2 b + (α(t) b + 2δ) v,t + M(β(t) α(t)) 2 b 4 δ t v,t + (Mt + 2δ) v,t + M(β(t) α(t)) 2. Take now δ = t (t is fixed in this argument), and apply again the sup estimate (5.6), finally obtaining T (s 1 ) T (s 2 ),t { 4M b 4 t + M(Mt + 2 t) + M 2 t } s 1 s 2,t. Clearly, for t = t (M, b), T is a contractive mapping Exercises In Subsection 5.1 we proved existence and uniqueness of a fixed point of T in a small time interval (locally in time). Show how the argument can be completed to give existence and uniqueness of a fixed point in any time interval (global existence) Why uniqueness of solutions (u, s) with s Σ is equivalent to uniqueness of solutions in the class of Definition 2.1, without further restrictions? 5.3. Give an interpretation of (5.1) as an energetical balance Prove that T has the property (see also Figure 4) s 1 (t) < s 2 (t), < t < T = T (s 1 )(t) > T (s 2 )(t), < t < T Extend the proof of existence and uniqueness of solutions to the case when h. What happens if h, u? 5.6. To carry out rigorously the calculations in (5.9) actually we need an approximation procedure: i.e., we need first perform integration in a smaller 2- dimensional domain, bounded away from the boundaries x =, t =. Recognize the need of this approach, and go over the (easy) details. 6. Qualitative behaviour of the solution Theorem 6.1. (Monotone dependence) Let (u i, s i ) be solutions of (2.9) (2.14), i = 1, 2, respectively corresponding to data h = h i, b = b i, u = u i. Assume both sets of data satisfy (2.15), (2.16). If h 1 (t) h 2 (t), < t < T ; b 1 b 2 ; u 1 (x) u 2 (x), < x < b 1 ; (6.1) then s 1 (t) s 2 (t), < t < T. (6.2) Proof. 1) Let us assume first b 1 < b 2. Reasoning by contradiction, assume t = inf{t s 1 (t) = s 2 (t)} (, T). Then the function v = u 2 u 1 is strictly positive in { < t < t, < x < s 1 (t)},
21 6. QUALITATIVE BEHAVIOUR OF THE SOLUTION 13 t T (s) = s x = σ(t) x = T (σ)(t) b x Figure 4. Behaviour of the transform T by virtue of the strong maximum principle and Hopf s lemma. Indeed, v(s 1 (t), t) >, < t < t. Then v attains a minimum at (s 1 ( t), t) = (s 2 ( t), t), where Thus, due to Hopf s lemma, But we compute v(s 1 ( t), t) =. v x (s 1 ( t), t) <. v x (s 1 ( t), t) = u 2x (s 2 ( t), t) u 1x (s 1 ( t), t) = ṡ 2 ( t) + ṡ 1 ( t). Hence ṡ 2 ( t) > ṡ 1 ( t), which is not consistent with the definition of t. 2) Assume now b 1 = b 2. Let us extend the data u 2 to zero over (b, b + δ), where < δ < 1 is arbitrary. Let (u δ, s δ ) be the solution of problem (2.9) (2.14) corresponding to the data h = h 2, b = b 2 + δ, u = u 2. Then, by the first part of the proof, s 2 < s δ, s 1 < s δ, and for all < t < T s δ (t) s 2 (t) = δ s 2 (t) [u δ u 2 ](x, t) dx s δ (t) s 2 (t) u δ (x, t) dx δ. Therefore s δ s 2 + δ, so that s 1 < s δ s 2 + δ. On letting δ we recover s 1 s 2. Let us investigate the behaviour of the solution of the Stefan problem for large times. In doing so, we of course assume that T =. Note that the result of existence and uniqueness applies over each finite time interval; a standard extension technique allows us to prove existence and uniqueness of a solution defined for all positive times. Owing to Proposition 4.2, we have s(t) S, < t < = u(x, t) MS, in Q s,. (6.3)
22 14 DANIELE ANDREUCCI Moreover, s being monotonic, certainly there exists s = lim t s(t). (6.4) Theorem 6.2. Let (u, s) be the solution of Theorem 5.1. Then Proof. 1) Assume first b s = lim s(t) = b + t u (x) dx + h(t) dt. (6.5) h(t) dt = +. (6.6) We only need show s is unbounded. Let us recall that, for all t >, b s(t) = b + t u (x) dx + h(τ) dτ s(t) u(x, t) dx. (6.7) From (6.3), it follows that, if s is bounded over (, ), then u is also bounded over (, ). This is clearly inconsistent with (6.7), when we keep in mind (6.6). 2) Assume h(t) dt < +. (6.8) The balance law (6.7), together with u >, immediately yields b s(t) < b + u (x) dx + h(t) dt < +. (6.9) Then we have s(t) s < ; it is only left to identify s as the quantity indicated above. Owing to (6.7) again, we only need show lim t s(t) u(x, t) dx =. (6.1) On multiplying (2.9) by u and integrating by parts in Q s,t, we get 1 2 s(t) u(x, t) 2 dx + u 2 x dxdτ = Q s,t 1 b t u (x) 2 dx + u(, τ)h(τ) dτ. (6.11) 2 Recalling (6.3) and (6.9), the last integral in (6.11) is majorised by Thus a consequence of (6.11) is Ms h(τ) dτ <. Q s, u x (x, t) 2 dxdt < +. (6.12)
23 Elementary calculus then shows that u(x, t) 2 dxdt = Q s, 7. REGULARITY OF THE FREE BOUNDARY 15 Q s, s Q s, [ s(t) x s(t) x u ξ (ξ, t) dξ] 2 dxdt u ξ (ξ, t) 2 dξ dxdt s 2 Then there exists a sequence {t n }, t n, such that But the function s(t n ) t u(x, t n ) 2 dx. s(t) u(x, t) 2 dx, Q s, u ξ (ξ, t) 2 dξ dt <. when we take into account (6.11), is easily seen to have limit as t, so that this limit is. By Hölder s inequality, s(t) u(x, t) dx [ s(t) 1/2 s u(x, t) dx] 2, t, completing the proof of (6.1) Exercises Find conditions ensuring that the inequality in (6.2) is strict Discuss the necessity of assumptions (2.15), (2.16) in Theorem Regularity of the free boundary The approach in this Section is taken from [16], and provides an example of bootstrap argument, i.e., of an inductive proof where any given smoothness of the solution allows us to prove even more regularity for it. Our first result will become the first step in the induction procedure, and is however required to prove Lemma 4.1. Lemma 7.1. Let u C 2,1 (Q s,t ) C(Q s,t ), where s Lip([, T]), and s(t) > for t T. Assume u fulfils u t u xx =, in Q s,t, (7.1) u(s(t), t) =, t T. (7.2) Then for each small enough ε >, u x is continuous in P ε, where P ε = {(x, t) ε < x < s(t), ε < t < T }.
24 16 DANIELE ANDREUCCI Proof. The proof is based on standard local regularity estimates for solutions of parabolic equations. Introduce the following change of variables { y = x s(t), v(y, τ) = u(ys(τ), τ). τ = t ; The set Q s,t is mapped onto R = (, 1) (, T), where v solves v τ 1 s(τ) 2v yy ṡ(τ) s(τ) yv y =, in R, (7.3) v(1, τ) =, τ T. (7.4) More explicitly, (7.3) is solved a.e. in R, as v is locally a Sobolev function in R. Classical results, see [11] Chapter IV, Section 1, imply that for any fixed ε >, v τ, v y, v yy L q ((ε, 1) (ε, T)), for all q > 1. Then we use the embedding Lemma 3.3 of [11] Chapter II, to infer that, for q > 3, v y H α, α 2 ([ε, 1] [ε, T]), (7.5) where α = 1 3/q (see also Remark 11.2 of [11], p. 218; the space H α, α 2 is defined in Appendix C). Since u x (x, t) = 1 s(t) v ( x y s(t), t), the result follows. Our next result implies that the free boundary in the Stefan problem (2.9) (2.14) is of class C (, T). Theorem 7.1. Assume u and s are as in Lemma 7.1, and moreover u x (s(t), t) = cṡ(t), < t < T, (7.6) where c is a given constant. Then s C (, T). Proof. For v defined as in the proof of Lemma 7.1, we rewrite (7.6) as ṡ(τ) = 1 cs(τ) v y(1, τ), < τ < T. (7.7) Choose α (, 1). Then (7.5) and (7.7) yield at once ṡ H α, α 2 ([ε, 1] [ε, T]), for each fixed ε >. (7.8) Next we make use of the following classical result: If the coefficients in (7.3) (i.e., ṡ), are of class m+α m+α, H 2 ([ε, 1] [ε, T]), then v is of class H 2+m+α,2+m+α 2 ([2ε, 1] [2ε, T]). Here m is any integer. We prove by induction that for all m (7.9) m+α m+α, ṡ H 2 ([ε, 1] [ε, T]), for each fixed ε >. (7.1) We already know this is the case when m =, from (7.8). Assume then (7.1) is in force for a given m. Then, owing to (7.9), v H 2+m+α,2+m+α 2 ([ε, 1] [ε, T]), for each fixed ε >. (7.11)
The Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)More information
Systems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +More information
Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,More information
Numerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a numberMore information
The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same ifMore information
THE DIFFUSION EQUATION R. E. SHOWALTER 1. Heat Conduction in an Interval We shall describe the diffusion of heat energy through a long thin rod G with uniform cross section S. As before, we identify GMore information
1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is orMore information
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,More information
Metric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergenceMore information
Notes on weak convergence (MAT Spring 2006)
Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 pMore information
Continued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents IntroductionMore information
DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmoothMore information
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistableMore information
Course 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............More information
MATH 461: Fourier Series and Boundary Value Problems
MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 firstname.lastname@example.com MATH 461 ChapterMore information
1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dxMore information
The Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentionedMore information
6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are pointsMore information
Adaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650More information
Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of aMore information
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linearMore information
Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2More information
FIELDS-MITACS Conference. on the Mathematics of Medical Imaging. Photoacoustic and Thermoacoustic Tomography with a variable sound speed
FIELDS-MITACS Conference on the Mathematics of Medical Imaging Photoacoustic and Thermoacoustic Tomography with a variable sound speed Gunther Uhlmann UC Irvine & University of Washington Toronto, Canada,More information
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.More information
MA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki CounterexamplesMore information
Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +
Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.More information
Compactness in metric spaces
MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally theMore information
Metric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give someMore information
1. R In this and the next section we are going to study the properties of sequences of real numbers.
+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of realMore information
NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004
NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................More information
ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. Gacovska-Barandovska Smirnov (1949) derived fourMore information
Ideal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal idealsMore information
An Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Andrew J. Bernoff LECTURE 2 Cooling of a Hot Bar: The Diffusion Equation 2.1. Outline of Lecture An Introduction to Heat Flow Derivation of the DiffusionMore information
Course 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................More information
Chapter 4 Parabolic Equations
161 Chapter 4 Parabolic Equations Partial differential equations occur in abundance in a variety of areas from engineering, mathematical biology and physics. In this chapter we will concentrate upon theMore information
No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial resultsMore information
By W.E. Diewert. July, Linear programming problems are important for a number of reasons:
APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)More information
20 Applications of Fourier transform to differential equations
20 Applications of Fourier transform to differential equations Now I did all the preparatory work to be able to apply the Fourier transform to differential equations. The key property that is at use hereMore information
FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is isMore information
Class Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(xMore information
2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditionsMore information
PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS
PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,More information
The Relation between Two Present Value Formulae
James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 61-74. The Relation between Two Present Value Formulae James E. Ciecka,More information
x if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about theMore information
Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: firstname.lastname@example.com Narvik 6 PART I Task. Consider two-pointMore information
2.2. Instantaneous Velocity
2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematicalMore information
2.5 Gaussian Elimination
page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3More information
Tangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpointsMore information
Taylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we willMore information
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainlyMore information
ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochaticMore information
Quasi-static evolution and congested transport
Quasi-static evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposedMore information
5. Convergence of sequences of random variables
5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,More information
4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complex-valued function of a real variable
4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variableMore information
Math 317 HW #5 Solutions
Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to showMore information
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, butMore information
Numerical Solutions to Differential Equations
Numerical Solutions to Differential Equations Lecture Notes The Finite Element Method #2 Peter Blomgren, firstname.lastname@example.com Department of Mathematics and Statistics Dynamical Systems Group ComputationalMore information
Extremal equilibria for reaction diffusion equations in bounded domains and applications.
Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal Rodríguez-Bernal Alejandro Vidal-López Departamento de Matemática Aplicada Universidad Complutense de Madrid,More information
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operatorMore information
Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues suchMore information
The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line
The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,More information
OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of MathematicsMore information
Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be writtenMore information
Riesz-Fredhölm Theory
Riesz-Fredhölm Theory T. Muthukumar firstname.lastname@example.com Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed SpacesMore information
Double Sequences and Double Series
Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: email@example.com Abstract This research considers two traditional important questions,More information
Min-cost flow problems and network simplex algorithm
Min-cost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.More information
max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the studyMore information
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largelyMore information
Properties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,andMore information
MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π
MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that fMore information
Parametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final pointMore information
Analysis via Uniform Error Bounds Hermann Karcher, Bonn. Version May 2001
Analysis via Uniform Error Bounds Hermann Karcher, Bonn. Version May 2001 The standard analysis route proceeds as follows: 1.) Convergent sequences and series, completeness of the reals, 2.) Continuity,More information
1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.More information
Section 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure toMore information
1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)More information
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b aMore information
Inner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, andMore information
) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.
Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on FourierMore information
Mathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which areMore information
Lecture 12 Basic Lyapunov theory
EE363 Winter 2008-09 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12More information
16 Convergence of Fourier Series
16 Convergence of Fourier Series 16.1 Pointwise convergence of Fourier series Definition: Piecewise smooth functions For f defined on interval [a, b], f is piecewise smooth on [a, b] if there is a partitionMore information
Chapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values ofMore information
Cost Minimization and the Cost Function
Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This isMore information
Some stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discussMore information
MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a columnMore information
Weak topologies. David Lecomte. May 23, 2006
Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I suchMore information
1 Limiting distribution for a Markov chain
Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-checkMore information
Chapter 7. Induction and Recursion. Part 1. Mathematical Induction
Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simplyMore information
An optimal transportation problem with import/export taxes on the boundary
An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... JointMore information
Chapter 2 Limits Functions and Sequences sequence sequence Example
Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that studentsMore information
Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS
Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collectionMore information
The General, Linear Equation
Chapter 2 The General, Linear Equation Let a 1 (x), a 2 (x),..., a n (x) be continuous functions defined on the interval a x b, and suppose u (x) is n-times differentiable there. We form from u the functionMore information
since by using a computer we are limited to the use of elementary arithmetic operations
> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operationsMore information
Some Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limitedMore information
Math 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.More information
24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that noMore information
Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricingMore information
Understanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and otherMore information
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. TheseMore information
OPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2More information


text:
Entrainment of Air in Flowing Water: A Symposium: Closed Conduit Flowby A. A. Kalinske,
James M. Robertson,
Serial Information: Transactions of the American Society of Civil Engineers, 1943, Vol. 108, Issue 1, Pg. 1435-1447
Document Type: Journal Paper
Abstract: DataNotAvailable
Subject Headings: Entrainment | Air flow | Pipe flow
Services: Buy this book/Buy this article
Return to search


text:
An Unbiased View of reputation repair company
Abundant Agnew and his crew at Laptop or computer Geek are fantastic to work with. We switched internet hosting companies a handful of weeks back and contacted Prosperous about transferring all our files to The brand new internet hosting company. The transfer was effective and sleek. I believed i...[Read More]
These reviews are demonstrated further down the site s that users You should not always have to depart the website.
To model this factor, it is necessary to specify and recognize a far more refined classification of reputation.
The T25 applications are quick and sweet at 25 minutes with a couple more excess minutes of stretching When you entire the exercise session.
Customer support now ignores our calls and requests to critique our pending refund. We strongly discourage any one from managing Build.com"
You select no matter if that seems like coincidence. Owing to Citibank for staying on top of their game, nipping the fraud while in the bud and cancelling the card.
However, speaking to reviewers needs to be approached with treatment; Net messaging is really a blunt Resource and at times superior intentions run into badly. We have place together some illustrations that will help you get this equilibrium proper.
This is vital as it tends to make pages much more beneficial to customers, as well as improves the probability of position extremely from the SERPs. Boost rankings for 'products title' + evaluate
FleishmanHillard's Majlis workforce displays the wide geographic diversity with the Muslim population and may handle crucial benchmarks, significantly with the food stuff, beauty, pharmaceutical and money sectors.
Econsultancy on a regular basis hosts fascinating and enlightening situations for multi-disciplines everywhere in the world.
When you have damaging reviews exhibiting on the 1st site of Google for click here your personal company identify NOW is the time to try and do a little something over it. It could be here costing you 1000s of bucks each day that it continues read more to be on the main webpage.
Inside the context of brand extension techniques, quite a few firms depend upon reputation transfer as a method of transferring the good reputation of the company and its existings solutions to new marketplaces and new items. Buyers who will be currently accustomed to other merchandise of a longtime manufacturer, exhibiting consumer fulfillment and loyalty, will more easily take new items of the identical brand name. In contrast to model extension, the overall principle of reputation transfer also involves the transfer of the company's values and identification to The brand new merchandise and/or providers as well as the connected manufacturers when getting into new marketplaces. It is necessary, even so, to concentrate on the impression healthy involving preexisting and new makes, for this component has actually been confirmed for being vital to the good results of name extensions.
Cross web site scripting (XSS) is when criminal hackers use weak spot's in an internet site application to inject destructive code into that Site which will cause the website to behave in a way it wasn't built to do. The web site weakness which will allow ...[Go through Extra]
TripAdvisor is this type of effectively set up of A lot of people's on the internet journey analysis that websites must just think buyers will check with it just before scheduling.


text:
Release Year: 2016
Genres: anal, bigcocks, condoms,gay porn,bareback
Video language: English
Description:Rich can't get over the fact that his Brian is about to become a dad, and wants to do what he can to deflate his macho ego a little. He ties him naked to the cold, steel milking table to extract a sample and get a look at his seed. Rich takes his time, working up a nice big load, unaware of Brian's feelings of objectification and frustration.
Read moreSlow Teasing Handjobs – Brian, Rich; Daddy's Sperm Sample


text:
"One way to get the most out of life is to look upon it as an adventure."
--William Feather
This site is a listing of quotations I have found as inspiring, intriguing, intuitive, unique, insightful, and thought provoking. Hopefully, you will find them the same.
Search by Author or word or topic in search box at the top of the page, or by categories below. Also, if you want quotes with PICTURES, just put the term 'picture' in the search box.
The Profit Prophet


text: My story begins the 18th of September when I reserved the red groom bandeau and the groom porte round with 866 vuitton. I also reserve the silver miroir pouchette because I want to think overnight about a big purchase like the silver speedy. I keep a record of when I called, what I put myself on the waitlist, and who I talked to. 19th of September I call again and after thinking over the night, I reserve a red groom agenda and a miroir speedy. The SA assures me that I am on the list. I call again in a week to an SA to see if I am on the waiting list on the groom and for the miroir. SA won't even tell me how long the list is or whatever, etc. I call about a week ago to see why people are getting their stuff first and see when I will get my agenda. Turns out I was never put on the list. I suddenly freak out, not about the agenda per se, and ask "OMG AM I NOT ON THE LIST FOR THE MIROIR SPEEDY???????!!!" the one that is about a mile long and I reserved for on the 19th of September??? Apparently I am not. So the SA bumped me up on the list to be the first to be considered for a red agenda and replaces me on the list for the miroir speedy and gives me an extra 2 week bump on the list. ________________________________ Last night I stalk eluxury and finally get a groom cles to pop up. I buy a cancun and the cles and even print out the order confirmation stuff. The next morning I check and there is no order placed. I called eluxury and I find out that my order got deleted by the computer in some mishap. I still wanted the cancun so they gave me free shipping in condolences. So I am starting to get really pissed off with LV in general and call 866 vuitton just to make sure one more time I am on the list for the red agenda and the miroir speedy. The third SA who promised to correct my error and give me a 2 week bump NEVER PUT ME ON THE LIST EITHER. _____________________________ REMIND ME AGAIN WHY I LIKE VUITTON?!?!?!


text:
Window Tinting You Can Rely On For Cooler, More Comfortable Living
When shopping for window tint near Wentworth-nord, visit a dealer you can trust to complete the project on time and on budget. Whether you're looking to tint the windows of your home, car, or business, you'll find a variety of great tint shades when you visit your nearby location. Our window tint is custom cut for your home, business, or vehicle windows to assure a perfect fit—every time. Plus, it is manufactured to last, so you don't need to worry about the discoloration and bubbling that occurs with lower-grade window film. In addition, when you choose to have your window film installed by our expert technicians, you could get the extra protection of a lifetime limited warranty. In the very unlikely event that discoloration or bubbling occurs, simply give us a call and we'll remove and replace your window tint for free. Now that's exceptional service!
Enjoy a cooler ride with car window tinting
Over the lifetime of your car, continued exposure to light and heat can cause the interior to crack, fade, and peel. LLumar's window tints provide today's drivers with great protection and value for their money. Tint your car windows to help shield the interior, and your passengers, from the sun's destructive heat, rays, and glare. Ranging from clear to dark charcoal, your tinting company offers a full line of car window tint shades for you to choose from. From a legal perspective, know that your state may have laws that limit the level of darkness you can choose for your car window tint. Visit or call today to pick out an option that's within your state's legal guidelines!
Home window tint: Save money & protect your furnishings
When installed by one of our factory-trained technicians, your home window tint helps provides your family with protection from the sun's harmful rays, plus sometimes privacy from the prying eyes of strangers. Residential window tinting could help your flooring and treasured artwork and furnishings stay in like-new condition!It helps keep out up to 99% of the sun's destructive ultraviolet rays that lead to discoloration and fading. In warmer months, it could help keep your house cooler and more comfortable. What's more, once their home window tint was installed, many customers reported seeing their monthly energy bills go down by as much as 15%! Because LLumar is LEED-certified, you could be eligible to save even more with incentives and tax credits for the environmentally-aware!
Commercial window tint: Conserve energy & convey style
When you choose custom window tinting for your business, you can help limit your facility's energy expenses while achieving a look that's professional, polished, and unique to your company. Ranging from crystal clear to deep charcoal, there's a tint shade to match the needs of your business. When installed by your LLumar dealer, non-reflective window tint provides a uniform look that blends seamlessly with the design of your building. A special dual-reflective tint is also available for companies in need of a shiny exterior to encourage solar heat rejection, but a less reflective interior for privacy in the evening. What's more, custom office window tint is for more than the exterior of a building. Inside your facility, it can be installed as an additional design feature that offers style and privacy in stairwells, meeting rooms, and on interior office windows. Call or visit your LLumar window tint dealer near Wentworth-nord to get started on your commercial tinting project.
At your LLumar tinting location, get expert care & installation
Window tint installation is effortlessly smooth when you visit your nearby LLumar location. In fact, many custom tint projects can be finished within a few days, depending on the size of your home, car, or business. In general, installation takes about 15 minutes per window. For unparallelled window tinting, visit a LLumar tinting company. From commercial or home window tinting that requires opaque, translucent, mirrored, reflective, or color-tinted films, your LLumar specialist is ready to help you find the right one for your needs. Visit or call your LLumar dealer to learn more about window tint near Wentworth-nord, get a free estimate, and to schedule your installation today!It's time to improve your view of the world.
Find Nearby LLumar Dealers
332 Rue Dubois
St-Eustache, QC, J7P 4W9
- Hours:
- Mon-Fri 7:00 am - 5:00 pm
63 rue du Chaumont
Blainville, QC, J7B 6B4
- Hours:
- Mon-Fri 9:00 am - 5:00 pm
Sat 9:00 am - 1:00 pm
1490 Boul Cure Labelle
Laval, QC, H7V 2W1
- Hours:
- Mon-Thu 8:00 am - 5:30 pm
Fri 8:00 am - 5:00 pm
1439 Youville Dr
Unit 11 & 12
Orleans, ON, K1C 4M8
- Hours:
- Mon-Fri 8:00 am - 6:00 pm
Sat 8:00 am - 4:00 pm
1506 Michael St
Ottawa, ON, K1B 3S1
- Hours:
- Mon-Fri 8:00 am - 5:30 pm
Find Dealers in Nearby Cities
- Sainte-lucie-des-laurentides
- Ste-marguerite-du-lac-masson
- Barkmere
- Notre-dame-de-la-merci
- Huberdeau
- Lac-superieur
- Saint-faustin-lac-carre
- Lac-des-plages
- Amherst
- Val-morin
- Saint-remi-d'amherst
- La Conception
- Saint-adolphe-d'howard
- Lantier
- Brebeuf
- Esterel
- Val-david
- Arundel
- Lac-des-seize-iles
- Montcalm
- Val-des-lacs
- Entrelacs
Window Tint for Cars, Offices & Homes Near Wentworth-nord
LLumar dealers close to Wentworth-nord install high quality tint. Live stylishly, help lower your energy costs, and drive more comfortably with commercial, residential, and car window tinting. You can trust LLumar window tints; for more than 55 years, our tints have provided superior style and protection from the sun's destructive heat and ultraviolet rays. We help people live comfortably, wherever they work, drive, or play!Rest assured, your nearby window tint installer has been exclusively selected and certified to ensure that you have the finest service experience. Drop by a LLumar dealer near you in Wentworth-nord for a free and fast window tinting estimate today!


text:
Ty and I are both sick. I'm not sure if it's the same thing, but I highly suspect he brought me home a gift from the daycare kids. He is happy as a clam, runny nose, coughing and all. Is immune suppressed Mommy, is absolutely miserable. I even slept upstairs last night. Partly so I wouldn't wake the two of them up, and partly cause I knew I needed some sleep. I slept like a rock, but woke up feeling worse than ever. My head may explode and I'm more than half way through a large box of tissues I only opened last night right before bed. Me neck is all swelled up and my throat is really sore, and I took Sudafed, so now I'm tired again.
Ty is napping. When he wakes up I am going to drag myself over to the farm to help with harvesting the winter squash. Mike and Ty are going to pick more green/yellow and Roman beans for freezing. We're also going to stock up on swiss chard, spicy greens, and Mike's getting hot pepper to make a sodium free pepper sauce. I don't know how long I'll hold up...I guess we'll see.
Tonight we need to finish the apples, blanch and freeze the beans, blanch and freeze some corn on the cob (we've only frozen corn cut off the cob so far) and cook up some batches of the card to freeze. I am not sure what to do with spicy greens, but I found a recipe online for a mustard green soup, and it sounds pretty yummy, so we might try making that with the spicy greens, and then freezing it. I'm trying to fill up our freezer because it takes less energy to keep the freezer cold when everything inside it is frozen. Need to put more meat away too though...wish we could afford that 1/2 cow or 1/2 pig now...maybe in the spring...
Saturday, October 4, 2008
Runny Noses and Veggies
Posted by Callahan Crew at 12:50 PM
2 comments:
- Robin said...
Good God woman - stay in bed and get some rest! You're like a frontier woman. I may have to reconsider living on the compound with you - you'll likely work me to death... Seriously though, get better, m'kay? You've got a cute baby to enjoy.
- October 4, 2008 at 10:11 PM
Must be going around New England. Gaby and I were sick all weekend too.
- October 6, 2008 at 2:42 PM
Subscribe to: Post Comments (Atom)