MONTREAL, May 09, May 09, 2018 (GLOBE NEWSWIRE via COMTEX) -- New Look Vision Group Inc. ("New Look Vision" or the "Company") (BCI), a leader in the Canadian retail optical business with 376 stores across Canada, reported financial results today for the 13 weeks ended March 31, 2018 ("Q1 2018"). This press release should be read in conjunction with the Company's management discussion and analysis (the "MD&A") and interim consolidated financial statements for Q1 2018, which are available on the Company's website at www.newlookvision.ca/investors and have been posted on SEDAR at www.sedar.com.
-- Revenues reached $69.8 million and adjusted EBITDAreached $11.3 million for Q1 2018, representing increases of 36.7% and 44.1% respectively over last year. The revenue increase was mainly due to a full quarter of ownership of Iris and comparable stores sales growth of 1.8% over last year. This revenue growth, along with improvement in the materials consumed ratio, resulted in the strong adjusted EBITDA performance. As a percentage of revenues, adjusted EBITDA in the first quarter increased 90 basis points to 16.3%. Depreciation and amortization expenses increased principally as a result of the acquisition of Iris and capital investments made on the Company's stores and laboratories.
-- Adjusted net earnings attributed to shareholders (defined as net earnings adjusted to remove the impact of depreciation, acquisition-related costs, equity-based compensation, and other non-comparable costs) for the first quarter increased by $2.3 million to $6.9 million. Adjusted net earnings per share for the quarter compared to Q1 2017 increased 33.3% to $0.44 per diluted share. The increase is mainly due to higher EBITDA offset by higher depreciation, financial expenses, and income taxes.
-- Net earnings attributed to shareholders were $2.2 million, compared to $1.3 million last year, the increase being mainly due to a higher EBITDA, offset by depreciation, financial expenses, and income taxes as mentioned above.
-- Cash flows from operating activities before income taxes paid and changes in working capital items were $10.2 million or $0.65 per diluted share in the first quarter of 2018, up significantly from $7.1 million or $0.51 per diluted share last year. Income tax instalments paid in the first quarter of 2018 were $2.8 million compared to $1.7 million for 2017.
-- When adjusted for acquisition-related costs and other non-comparable costs, adjusted cash flows from operating activities were $10.7 million or $0.69 per diluted share, an increase of $2.9 million, or 37.9% over last year.
Antoine Amiel, the President and CEO of New Look Vision, stated that: "The year is off to a strong start with solid Q1 results for the Group as a whole, primarily due to the inclusion of Iris and other acquisitions made in 2017, in addition to achieving our 15 consecutive quarter of comparable store sales growth. As we carry out our strategic growth plan, we continue to generate cost synergies as highlighted by our decreasing operating expense ratio. We maintain our focus on the integration of recent acquisitions and strengthening our position in the consolidating Canadian retail optical industry."
Pursuant to its previously announced dividend policy, the Board of Directors of New Look Vision declared a dividend of $0.15 per common share. The quarterly cash dividend will be paid on June 29, 2018 to the shareholders of record as of June 22, 2018. The dividend has been designated as an "eligible dividend", that is a dividend entitling shareholders who are Canadian resident individuals to a higher dividend tax credit.
As of April 30, 2018, New Look Vision had 15,491,520 Class A common shares issued and outstanding.
1) EBITDA, Adjusted EBITDA, Adjusted net earnings, Cash flows from operating activities before income taxes paid and changes in working capital items, and Adjusted cash flows from operating activities are not recognized measures under IFRS and may not be comparable to similar measures used by other entities. See Table C and Table D attached for a reconciliation of net earnings to these measures. See Table E for reconciliation of cash flows.
About New Look Vision Group Inc. New Look Vision is a leader in the eye care industry in Canada having a network of 376 stores mainly under the New Look Eyewear, Vogue Optical, Greiche & Scaff and Iris banners and laboratory facilities using state-of-the-art technologies. Tax information regarding payments to shareholders is available at www.newlookvision.ca in the Investors section.
a) Comparable stores are stores which have been operating for at least 12 months. Revenues are recognized at time of delivery of goods to customers, however management measures the comparable store performance on the basis of sales orders, whether delivered or not. b) EBITDA, adjusted EBITDA , adjusted net earnings, cash flows from operating activities before income taxes paid and changes in working capital items, and adjusted cash flows from operating activities are not recognized measures under IFRS and may not be comparable to similar measures used by other entities. Refer to Table C and Table D for a reconciliation of these measures to net earnings. Also, refer to Table E for reconciliation of cash flows. c) Adjusted net debt is not a recognized measures under IFRS and may not be comparable to similar measures used by other entities. Refer to Table F Adjusted net debt for the definition and reconciliation. d) The amounts of dividends shown in the table above refer to amounts declared in the periods. e) The increase in the number of stores in the last twelve months reflects the acquisitions of 152 stores, two openings, and five closures.
a) EBITDA and adjusted EBITDA are not recognized measures under IFRS and may not be comparable to similar measures used by other entities. New Look Vision believes that EBITDA and adjusted EBITDA are useful financial metrics as they assist in determining the ability to generate cash from operations. Investors should be cautioned that EBITDA and adjusted EBITDA should not be considered as an alternative to net earnings or cash flows as determined under IFRS. b) Equity-based compensation represents the fair value of New Look Vision stock options vested in the period. c) Acquisition-related costs are mainly comprised of legal and other fees related to the business acquisitions, whether completed or in progress.
a) Adjusted net earnings attributed to shareholders are not a recognized measure under IFRS and may not be comparable to similar measures used by other entities. New Look Vision believes that this disclosure provides useful information as it allows the comparison of net results excluding depreciation, amortization and loss on disposal of assets, acquisition-related costs, equity-based compensation, and other non-comparable costs which may vary significantly from quarter to quarter. Investors should be cautioned that adjusted net earnings should not be considered as an alternative to net earnings as determined under IFRS.
a) Adjusted cash flows from operating activities are not a recognized measure under IFRS and may not be comparable to similar measures used by other entities. New Look Vision believes that this disclosure provides useful information as it allows the comparison of net operating cash flows excl uding acquisition-related costs and other non-comparable costs, which may vary significantly from quarter to quarter. Investors should be cautioned that adjusted cash flows from operating activities should not be considered as an alternative to cash flows from operating activities as determined under IFRS.
a) Adjusted net debt calculated above is not a recognized measure under IFRS and is therefore unlikely to be comparable to similar measures used by other entities. Investors should be cautioned that adjusted net debt should not be considered as an alternative to total debt as determined under IFRS.


Korcula Island is a fantastic destination for a family holiday. Whether you're making the island your base for your full holiday or including it as part of a multiple destination trip, we can guarantee a fun time for all. To help take the stress out of planning your holiday to Korcula, we've done some prep work for you.
Depending on where you are travelling from, you will likely be making your way to Korcula Island by boat, bus or car. Our helpful Korcula FAQ page will give you full details about how to travel to Korcula from nearby destinations.
If using public transport we recommend that you book tickets online for boats and buses, so there is no risk of not getting tickets in time.
If you plan to take longer or more complicated journeys, hiring a car will make travelling much easier. Car rental agencies also have child and baby seats available for hire.
If you want to get somewhere quick and hassle free, but do not want to hire a car, a private transfer would be an ideal option (although this will be more expensive than public transport).
Where to stay on Korcula?
As the larger towns offer more entertainment, we would suggest you staying in one of these. Korcula Town, Vela Luka and Lumbarda would all be good choices. Lumbarda in particular is a popular destination for families to stay. There are many accommodation options, the town is compact and easy to navigate and the sandy beaches are ideal for small feet.
There are many good reasons to book an apartment rather than a hotel for your family holiday to Korcula and you'll find a wide selection of good value apartments.
You can self cater which is convenient for feeding hungry children quickly and to keep holiday costs down.
An apartment has more space including bedrooms for everyone, typically a living area/kitchen and one or two bathrooms.
Many apartments have an outside area, a perfect place for the children to play when you aren't at the beach.
Take a look at our favourite family friendly accommodation options on Korcula.
1. It goes without saying, making the most of the sea during your holiday is vital. The sea is amazingly clean and safe. Take the children kayaking across the channel and go snorkelling.
2. Book older children & teenagers a wind-surfing lesson.
3. Bounce around on the water inflatables located on Prvi Zal in Lumbarda & Trstenica in Orebic.
4. With several sandy beaches in Lumbarda, enjoy good old fashioned bat & ball games and sandcastle building.
5. Take a relaxing boat trip around the archipelago and jump off the boat into the sea to cool off.
6. You will find play parks in most villages and towns across the island, the perfect place for children to burn off energy. There are also trampolines in Korcula Town by Banje Beach.
7. Go for a bike ride to Lumbarda or Vrbovica Bay. You can hire bikes in various sizes as well as bike seats for smaller children from tourist agencies in Korcula.
8. Take a bus ride to Zrnovo and visit the Kocje nature park. The amazing rock structure provides a really fun place to explore and great for a treasure hunt.
9. Check out the events scheduled by the local tourist boards which are specific for families. One of our favourites is the donkey racing in Lumbarda.
10. Be in Korcula Town on 30 June and join in the Half New Year fun. Fancy dress is optional but make sure you watch the billiant costume parade and have a dance in the square.
Croatian people are generally very hospitable and welcoming to children, so most restaurants will be very happy to accommodate you. Don't expect child menus; however there's usually some more "child friendly" dishes on the main menu. Pizza, pasta, simple grilled meats and chips are common and of course you'll always find ice cream close by!
Two of our favourite family restaurants are Belin in Zrnovo and Torkul in Lumbarda. Belin serves traditional food and Torkul is a pizzeria. Both have high chairs (not so easy to find in restaurants) and areas where children can play. Torkul has a large garden and there's usually a selection of toys available. Belin has a large terrace just outside of the restaurant and they also have a small swing you can ask to use. Belin also arranges a culinary experience which is a fun family activity.
If you are travelling with a baby you can find powdered milk and baby food at many of the supermarkets (Konzum and Tommy are best for this) and also the health & beauty store Muller. You'll also find nappies and wet wipes here.
Sun tan lotion can be purchased from the supermarkets, Kozmo and the pharmacy.
Over the counter medicines, plasters and other minor first add equipment can be purchased from the pharmacy.
The town markets sell various beach equipment, such as buckets & spades, snorkel gear and plastic balls.
Swimming shoes are really useful for non sandy beaches, protecting feet from the pebbles and also the urchins, which common in the area.
You can find lots more planning tips and helpful advice on our Family Holiday Travel board on Pinterest.


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Tue, 17 Jan, 2023 - 07:00
Dirty old town: how do we stop littering in Cork city black-spots?
Where does the mentality of thinking it's acceptable to throw your rubbish around come from? So asks Kathriona Devereux in her weekly column
38th place out of 40 in a national litter survey is not an acceptable place for Cork city to be, says Kathriona Devereux
Kathriona Devereux
REPETITION seems to be a big part of parenting. Common refrains in our house are "close the door", "put on your slippers" and "put your bowl in the sink".
Despite daily reminders, there are just some instructions that are impossible for my children to retain and go in one ear and out the other, to be repeated the next day and the day after until…when?!
There are tones of an exasperated parent in the latest Irish Business Against Litter (IBAL) report about Cork's performance at keeping the place tidy and litter free.
Much of the city centre received A or B grades when it came to cleanliness but there are a few locations where litter problems were highlighted years ago.
The report says "many of the heavily littered sites have been this way for quite some time and have been noted in previous IBAL surveys".
"An obvious and heavy litter presence prevailed along this road" (emphasis added). And "no improvement at this site - it has been subject to dumping for quite some time", "no change from previous surveys", and on it goes.
Behind those measured sentences, I can hear a well intentioned report writer pulling their hair out, and screaming at the computer, "I've said all this before, why don't you listen and do something. I said clean up Kennedy Quay back in 2017. Go get some black bags and sort it out, for the love of God!"
Despite top marks for locations like Patrick Street, Anglesea Street, Shalom Park, The Lough and Bus Station - "A credit to the station - staff" - all three surveyed areas, Cork City Centre, Cork Mahon and Cork Northside, placed in the 'Moderately Littered', 'Littered' or 'Seriously Littered' categories.
Meanwhile, top of the pile locations like Naas, Kilkenny and Maynooth are 'Cleaner than European Norms'.
Considering people from Cork are known for being vocal about how great Cork is, you have to wonder if our excessive pride in our home place is misplaced, given a sizeable chunk of the population clearly have no respect for the city.
Maybe the street cleaners who keep the city centre clean (thank you!) should take a week off so we get a real sense of how dirty we are.
I am slow to call somebody out when I witness blatant littering. I passed a young fella recently who opened a packet of cigarettes as he walked along the road, shedding plastic and paper as he went. He was oblivious both to his dirty behaviour and the dirty look I was throwing at him.
I didn't want a confrontation so didn't say anything, but maybe I should have deployed one of my best mammy lines, "Who do you think is going to clean up after you?"
If the City Council didn't have to direct so much energy and resources at keeping on top of the day to day litter, maybe they'd be able to tackle the persistently bad areas that crop up year after year on the IBAL survey, dragging down our overall score and generating such bad publicity.
Areas like the North Ring Road and Kennedy Quay won't be cleaned up without proper engagement with the area and a new plan for managing the problems there.
City Council can't be responsible for everything though. Having pride in our own patch of place is important too - if there's a crisp packet and sodden discarded face mask outside your door, or rolling around on your road, just pick it up. Someone has to, so it might as well be you.
When I was a Northsider, I used to occasionally take part in Blarney Street's Saturday morning street clean-up. Community clean-ups are fantastic initiatives that deliver much more than clean streets. Opportunities to meet your neighbours and do something tangibly worthwhile together are very welcome these days.
I used to be appalled at the number of black bags we would fill over the course of two hours. Crisp and chocolate wrappers, plastic bottles, and cigarette butts were the dominant forms of litter. Items that could easily, and should have been stuffed in a pocket or held in hand until the next bin presented itself.
Some Saturday mornings, though, I'd be down-hearted and disgusted by the mess created by what must be, a sizeable minority.
Cleaning up is important, but stopping the source of the mess is too.
Where does the mentality of thinking it's acceptable to throw your rubbish around come from?
Lots of people have an opinion on where the fault lies and who is to blame, everything from privatised bin collection to an insufficient number of bins outside the city centre. Is the cost of living crisis leading to an increase in dumping as people try to avoid refuse charges? Or do some people think waste collection is simply something they shouldn't have to pay for and are willing to throw their waste wherever they want?
To avoid a repetition of these IBAL scores next year, we need to fully understand the varying factors at play and tackle them imaginatively, because 38th place out of 40 in a national litter survey is not an acceptable place for Cork city to be.
Yes, I've joined a gym... but it seems such a waste of energy
kathriona devereuxlitteringenvironment

redpajama_set_name: RedPajamaCommonCrawl


This Glass CO2 Bubble Counter is a visual tool to count the number of bubbles per second leaving the system and entering the aquarium. With this knowledge, you can make more informed decisions about the supply of CO2 to your planted aquarium.
No CO2 system is complete without one.
Hand crafted glass bubble counter.
Designed to be installed in-line with the CO2 tubing.
Harmonises perfectly with the aquarium.
Allows for an accurate visual measurement of the CO2 supply.


You are currently browsing the category archive for the 'polymath' category.
This is the tenth "research" thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
Most of the progress since the last thread has been on the numerical side, in which the various techniques to numerically establish zero-free regions to the equation have been streamlined, made faster, and extended to larger heights than were previously possible. The best bound for now depends on the height to which one is willing to assume the Riemann hypothesis. Using the conservative verification up to height (slightly larger than) , which has been confirmed by independent work of Platt et al. and Gourdon-Demichel, the best bound remains at . Using the verification up to height claimed by Gourdon-Demichel, this improves slightly to , and if one assumes the Riemann hypothesis up to height the bound improves to , contingent on a numerical computation that is still underway. (See the table below the fold for more data of this form.) This is broadly consistent with the expectation that the bound on should be inversely proportional to the logarithm of the height at which the Riemann hypothesis is verified.
As progress seems to have stabilised, it may be time to transition to the writing phase of the Polymath15 project. (There are still some interesting research questions to pursue, such as numerically investigating the zeroes of for negative values of , but the writeup does not necessarily have to contain every single direction pursued in the project. If enough additional interesting findings are unearthed then one could always consider writing a second paper, for instance.
Below the fold is the detailed progress report on the numerics by Rudolph Dwars and Kalpesh Muchhal.
Polymath15, ninth thread: going below 0.22?
This is the ninth "research" thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
We have now tentatively improved the upper bound of the de Bruijn-Newman constant to . Among the technical improvements in our approach, we now are able to use Taylor expansions to efficiently compute the approximation to for many values of in a given region, thus speeding up the computations in the barrier considerably. Also, by using the heuristic that behaves somewhat like the partial Euler product , we were able to find a good location to place the barrier in which is larger than average, hence easier to keep away from zero.
The main remaining bottleneck is that of computing the Euler mollifier bounds that keep bounded away from zero for larger values of beyond the barrier. In going below we are beginning to need quite complicated mollifiers with somewhat poor tail behavior; we may be reaching the point where none of our bounds will succeed in keeping bounded away from zero, so we may be close to the natural limits of our methods.
Participants are also welcome to add any further summaries of the situation in the comments below.
This is the seventh "research" thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
The most recent news is that we appear to have completed the verification that is free of zeroes when and , which implies that . For very large (for instance when the quantity is at least ) this can be done analytically; for medium values of (say when is between and ) this can be done by numerically evaluating a fast approximation to and using the argument principle in a rectangle; and most recently it appears that we can also handle small values of , in part due to some new, and significantly faster, numerical ways to evaluate in this range.
One obvious thing to do now is to experiment with lowering the parameters and and see what happens. However there are two other potential ways to bound which may also be numerically feasible. One approach is based on trying to exclude zeroes of in a region of the form , and for some moderately large (this acts as a "barrier" to prevent zeroes from flowing into the region at time , assuming that they were not already there at time ). This require significantly less numerical verification in the aspect, but more numerical verification in the aspect, so it is not yet clear whether this is a net win.
Another, rather different approach, is to study the evolution of statistics such as over time. One has fairly good control on such quantities at time zero, and their time derivative looks somewhat manageable, so one may be able to still have good control on this quantity at later times . However for this approach to work, one needs an effective version of the Riemann-von Mangoldt formula for , which at present is only available asymptotically (or at time ). This approach may be able to avoid almost all numerical computation, except for numerical verification of the Riemann hypothesis, for which we can appeal to existing literature.
This is the sixth "research" thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
We have explicit upper bounds on , , ; see this wiki page for details. They are tabulated in the range here. For , the upper bound for is monotone decreasing, and is in particular bounded by , while and are known to be bounded by and respectively (see here).
we can obtain a lower bound of for , which settles the test problem in this regime. One can get more efficient lower bounds by multiplying both Dirichlet series by a suitable Euler product mollifier; we have found for to be good choices to get a variety of further lower bounds depending only on , see this table and this wiki page. Comparing this against our tabulated upper bounds for the error terms we can handle the range .
In the range , we have been able to obtain a suitable lower bound (where exceeds the upper bound for ) by numerically evaluating at a mesh of points for each choice of , with the mesh spacing being adaptive and determined by and an upper bound for the derivative of ; the data is available here.
This leaves the final range (roughly corresponding to ). Here we can numerically evaluate to high accuracy at a fine mesh (see the data here), but to fill in the mesh we need good upper bounds on . It seems that we can get reasonable estimates using some contour shifting from the original definition of (see here). We are close to finishing off this remaining region and thus solving the toy problem.
Beyond this, we need to figure out how to show that for as well. General theory lets one do this for , leaving the region . The analytic theory that handles and should also handle this region; for presumably the argument principle will become relevant.
The full argument also needs to be streamlined and organised; right now it sprawls over many wiki pages and github code files. (A very preliminary writeup attempt has begun here). We should also see if there is much hope of extending the methods to push much beyond the bound of that we would get from the above calculations. This would also be a good time to start discussing whether to move to the writing phase of the project, or whether there are still fruitful research directions for the project to explore.
Polymath15, fifth thread: finishing off the test problem?
This is the fifth "research" thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
where the expressions are quite small in practice ( is typically about two orders of magnitude smaller than the main term once is moderately large, and the error terms are even smaller). See this page for details. In principle we could also obtain an effective upper bound for (the term would be replaced by something smaller).
The ratio takes the form of a difference of two Dirichlet series, where is a phase whose value is explicit but perhaps not terribly important, and the coefficients are explicit and relatively simple ( is , and is approximately ). To bound this away from zero, we have found it advantageous to mollify this difference by multiplying by an Euler product to cancel much of the initial oscillation; also one can take advantage of the fact that the are real and the are (approximately) real. See this page for details. The upshot is that we seem to be getting good lower bounds for the size of this difference of Dirichlet series starting from about or so. The error terms are already quite small by this stage, so we should soon be able to rigorously keep from vanishing at this point. We also have a scheme for lower bounding the difference of Dirichlet series below this range, though it is not clear at present how far we can continue this before the error terms become unmanageable. For very small we may have to explore some faster ways to compute the expression , which is still difficult to compute directly with high accuracy. One will also need to bound the somewhat unwieldy expressions by something more manageable. For instance, right now these quantities depend on the continuous variable ; it would be preferable to have a quantity that depends only on the parameter , as this could be computed numerically for all in the remaining range of interest quite quickly.
As before, any other mathematical discussion related to the project is also welcome here, for instance any summaries of previous discussion that was not covered in this post.
This is the fourth "research" thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/. Progress will be summarised at this Polymath wiki page.
for some explicit (but somewhat messy) error terms : see this wiki page for details. The original approximations can be considered deprecated at this point in favour of the (slightly more complicated) approximation ; the approximation is a simplified version of which is not quite as accurate but might be useful for testing purposes.
It is convenient to normalise everything by an explicit non-zero factor . Asymptotically, converges to 1; numerically, it appears that its magnitude (and also its real part) stays roughly between 0.4 and 3 in the range , and we seem to be able to keep it (or at least the toy counterpart ) away from zero starting from about (here it seems that there is a useful trick of multiplying by Euler-type factors like to cancel off some of the oscillation). Also, the bounds on the error seem to be of size about 0.1 or better in these ranges also. So we seem to be on track to be able to rigorously eliminate zeroes starting from about or so. We have not discussed too much what to do with the small values of ; at some point our effective error bounds will become unusable, and we may have to find some more faster ways to compute .
In addition to this main direction of inquiry, there have been additional discussions on the dynamics of zeroes, and some numerical investigations of the behaviour Lehmer pairs under heat flow. Contributors are welcome to summarise any findings from these discussions from previous threads (or on any other related topic, e.g. improvements in the code) in the comments below.
This is the third "research" thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this previous thread. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
which would be the first quantitative improvement over the de Bruijn bound of (or the Ki-Kim-Lee refinement of ). Of course we can try to lower the two parameters of later on in the project, but this seems as good a place to start as any. One could also potentially try to use finer analysis of dynamics of zeroes to improve the bound further, but this seems to be a less urgent task.
with and given by (1).
Analytically bounding for medium and large (with a bound that is better than the bound away from zero in the previous two tasks).
Note that tasks 2 and 3 do not directly require any further understanding of the function .
Below we will give a progress report on the numeric and analytic sides of these tasks.
There is some progress on the code side but not at the pace I was hoping. Here are a few things which happened (rather, mistakes which were taken care of).
We got rid of code which wasn't being used. For example, @dhjpolymath computed based on an old version but only realized it after the fact.
We implemented tests to catch human/numerical bugs before a computation starts. Again, we lost some numerical cycles but moving forward these can be avoided.
David got set up on GitHub and he is able to compare his output (in C) with the Python code. That is helping a lot.
Computing quantities like , , , etc. with the goal of understanding the zero free regions.
seems to be settling around range.
See the figure below. The top plot is on the complex plane and the bottom plot is the absolute value. The code to play with this is here.
Steepest descent heuristics then predict that , , and . For the purposes of this project, we will need effective error estimates here, with explicit error terms.
for smallish complex numbers , as these sorts of integrals appear in the form of . As of this time of writing, there are effective bounds for the case, and I am currently working on extending them to the case, which should give enough control to approximate and . The most complicated task will be that of upper bounding , but it also looks eventually doable.
This is the first official "research" thread of the Polymath15 project to upper bound the de Bruijn-Newman constant . Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.
Numerical computation of the entire functions , with the ultimate aim of establishing zero-free regions of the form for various .
Improved understanding of the dynamics of the zeroes of .
Establishing the zero-free nature of when and is sufficiently large depending on and .
The formula (2) is valid for any . Naively one would think that it would be simplest to take ; however, when and is large (with bounded), it seems asymptotically better to take closer to , in particular something like seems to be a reasonably good choice. This is because the integrand in (3) becomes significantly less oscillatory and also much lower in amplitude; the term in (3) now generates a factor roughly comparable to (which, as we will see below, is the main term in the decay asymptotics for ), while the term still exhibits a reasonable amount of decay as . We will use the representation (2) in the asymptotic analysis of below, but it may also be a useful representation to use for numerical purposes.


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Archive for the 'Success Stories' Category
Nicholas Rongione
Posted on: April 5th, 2017 by JaydeepM No Comments
Nicholas Rongione, a former FSGC Ambassador at the University of Miami was awarded a National Science Foundation Graduate Research Fellowship. He will be focusing on developing aerospace applications of nanotechnology, and in particular, thermoelectric materials development. As a participant in the FSGC Hybrid Motor Rocket Competition my freshman year in 2011-2012, he gained invaluable insight into project management and rocketry. Most importantly, the experience inspired him to delve deeper into aerospace engineering, and served as a springboard into additional projects over the following years as an undergraduate. By sophomore year, he became the President of the American Institute of Aeronautics and Astronautics, and the FSGC Student Ambassador at the University of Miami. With support from FSGC, he then completed the NASA Propulsion Academy in the summer of 2013. In his junior year he furthered his efforts and began a two year endeavor developing a satellite program at UM that culminated in 2nd place in the FSGC sponsored Florida University Satellite Design Competition (FUNSAT 2014) and, ultimately, his senior design project. During this time he also participated in the NASA Reduced Gravity Student Flight Opportunities Program. Finally, FSGC provided additional support in his efforts to create a balloon satellite program, and a satellite ground station, and even help address concerns regarding rocket testing facilities. Now as a graduate student, he is supported by FSGC in a grant to work at NASA Kennedy Space Center on antimicrobial materials development employed in the Advanced Nano Systems Laboratory.
Frank Monzon
2006 Florida Space Grant Scholar "The experience you need to succeed at the next level" One of my professors introduced me to the Florida Space Grant program in 2006 while I was attending the University of Florida. He graciously mentored me in research methods for astrodynamics in association with the Defense Advanced Research Projects Agency (DARPA). This research eventually became the basis for my presentation on cube and modular satellites at a regional conference of the American Institute of Aeronautics and Astronautics (AIAA).
The hands-on experience gained through internships and the aerospace elective class on satellite design involved not just setting up a team, but also living out the process of project development. This was more than just a learning environment that developed my abilities in research, scientific writing methods, and formal presentations…it served an extension into real world experiences that I rely upon to this day.
One of my goals was to earn a master's degree in both science and business to expand my professional options. The foundational skills I learned though the Florida Space Grant program concerning project management, communication, and writing have served me well since the topics and projects are always changing.
I now work in private sector aerospace and our company is currently developing a new 737 aircraft that offers various features to accommodate individuals with reduced mobility. I'm quite proud of it as it is not just an innovation, it will also tangibly improve the quality of life for those with physical disabilities.
The Florida Space Grant program greatly expanded the scope of my professional options because it did more than just look good on my resume; it provided concrete knowledge and skills that enabled me to secure my first job at Boeing.
The program will definitely take you out of your comfort zone, but it will also give you the experience you need to succeed at the next level.
B/E Aerospace
2008-2011 Florida Space Grant Fellow, "Gave me the toolset to do research" I am a first-generation college student and the only one in my family to pursue a doctorate degree. The Florida Space Grant program taught me to think about things differently and allowed me to graduate with a degree that my family could not afford to give me. I am very grateful for their assistance.
While at the University of Florida I participated in the Space Grant program from 2008 to 2011 and researched application-based control systems theory. Among other things, this led to working on satellite alignment clustering. More importantly however, it afforded me the opportunity to expand those applied theories into my dissertation. This essentially became a research platform that I used in real world applications.
Even before being accepted into the program, I knew that I wanted to pursue career in aerospace engineering, but the opportunities the Florida Space Grant provided really allowed me to explore my various interests more deeply through application-specific projects. Being involved with these projects enabled me to significantly narrow my focus and find areas where I could give valuable contributions to the field.
The Space Grant program helped me to investigate programs in a rich and thorough way by encouraging questions, individual investigations, conceptual exploration, and collaborative solutions that moved the "state of the art" forward. Although these things did not directly prepare me for later employment at NASA Armstrong, Boeing, and NASA Ames (where I am now), it was indirectly invaluable; it gave me the research tools to excel at each of those places because I learned both in theory and in the field. In a very real sense, I felt empowered to transform the conceptual into reality.
Possessing these tools has greatly benefited my current work as a flight test director for UAF (drones) traffic management. In conjunction with the FAA, we are developing technology to safely manage the anticipated increase in the use of these drones.
Ultimately, the Florida Space Grant program allowed me to pursue higher education instead of being forced to choose a career that was necessary to pay bills. I am a firm believer that you should pursue your interests and continually apply what you learn to other applications until you are employed in your field of interest.
NASA Ames Research Center
Research Aerospace Engineer
UAS Traffic Management Project
Jennifer Kissinger
2006-2008 Florida Space Grant, "Opened my eyes to the vast array of options available to me" While I was attending the Florida Institute of Technology, the Florida Space Grant program funded my internship at the NASA Academy program in California at NASA Ames Research Center. This was a completely immersive program that included not only research, but leadership training as well.
Although the program definitely provided a number of academic and professional benefits, there was also a personal benefit. While participating in this program, I was one of several people who met their future spouse.
My specific role in the 10-week program was to examine Martian soil samples and determine their morphologies [?]. Additionally, there were several trips and tours included in the program that allowed our team to network with top leaders in the space aeronautics industry.
Just being around the like-minded, intellectually curious, and gifted students who were part of the Florida Space Grant was inspiring and helped to expand and develop my own areas of strength. It was also a wonderful opportunity to hone my leadership and teamwork skills–which paid dividends as I progressed through the program.
While completing my graduate work at UCLA, I chose to focus on space physics which only served to reinforce my long-term goal to work at NASA Goddard Space Center. Our cohort used 10 to 15 years of data gathered from satellites to conduct statistical analysis concerning the interactions of solar influences.
Though it was rewarding work, I am currently a software engineer in the healthcare industry, largely because of my husband's profession, and am hopeful that I can eventually transition back into the space industry.
There is no question that the opportunities provided by the Florida Space Grant program opened my eyes to the vast array of options that were available to me. But more importantly, the program provided a network of connections and internships that enabled me to actually explore and pursue those options.
I was just very fortunate to benefit from a lot of great professional direction and advice and will always be grateful to the program.
Sofware Engineer
Jack Russell Software


text:
Q: Как в Unity прикрепить скрипт ко множеству объектов? У меня в иерархии множество обьектов, около тысячи, я их все выделил и перетащил на них свой скрипт но во вкладке Inspector/Script/Object  написано None(GameObject), естественно перетаскивать в это поле по одному больше тысячи обьектов очень сложное занятие, так вот как мне сделать что бы Unity сам подцеплял объект? Я пробывал так, запросить все объекты с тэгом Player:
public GameObject obj;

void Start() {
    obj = GameObject.FindWithTag("Player");
}
void Update() {
    Color32 color32 = new Color32 (4, 128, 192, 70);
    obj.GetComponent <Renderer> ().material.color = color32;
}

но почему то цвет менял лишь один объект.

A: Адекватных вариантов 2:
Для первого необходимо, чтобы у всех этих объектов был одинаковый tag и после этого делаешь выборку с помощью метода:
FindGameObjectsWithTag

Во втором варианте ты просто делаешь perfab
После чего добавляешь 1000 раз (или сколько там у тебя объектов) этот перфаб в окно Hierarchy. В итоге все эти объекты будут связаны с твоим перфабом и любое изменение в перфабе приведёт к изменению во всех 1000 добавленных объектах

redpajama_set_name: RedPajamaStackExchange


I purchased it with a texture that is good for the texture and antique.
This review is about item#38620951.
It is not a place that specially requires high specs, so it was completely usable without problems.
This review is about item#38621213.
There is no problem with the product. I am satisfied with the price and the delivery date.
This review is about item#38621143.
This review is about item#38620705.
This review is about item#38621073.
I was looking for a finish of black skin as it is a painting premise. The material of the black leather finish was very helpful because there is no place to be handled.
This review is about item#38621082.
PS threading is uneven and can not be screwed well. Is the outline of the thread before cutting uneven?
This review is about item#38620845.
It is a good product with a sense of stability. I would like to purchase it next time.
This review is about item#38621003.
Very cheap and fast delivery. Rust inside is slightly worrisome, but it is generally satisfactory.
There seems to be two kinds of texture of black, 1/2 size here is a pipe of a good atmosphere drifting vintage from the former. 3/4 size is different in surface treatment, it is shiny and difficult to say black color. Also, the black of the no-brand of cut-out is also gray rather than black. I do not know other sizes, but as for black it is good to have a texture of 1/2 size of Mr. Yoshiyuki, so it is best if it can be sold out by this manufacturer. Although it is different from the original use, pipes are also trends in the world of DIY, so I'd like you to listen to the voice of this hand as well.
This review is about item#38621021.


Home / Security And Privacy / Is Vivaldi Open Source?
Vivaldi is not made available under one unified open source license. It does contain the Chromium source code with changes made to allow the HTML/CSS/JS based UI to run. All changes to the Chromium source code are made available under a BSD license and can be read by anyone on vivaldi.com/source/. Details in this regard are explained in the the README and LICENSE files within the package.
In addition, our UI code is written in plain, accessible code for those who read HTML, CSS and JS. This means that for all practical purposes the Vivaldi source code is available for audit.


text:
Q: Are TIKZ and PGF the same thing or not? When learning how to use tikz I've seen many guides and example using commands that begin with pgfmath and they used the two package names interchangeably so I assumed they refer to the same thing. Since \usepackage{tikz} was enough to use pgfmath commands and I didn't need \usepackage{pgf} this reinforced my belief.
However more recently I've seen some comments here and there where users point out that command x is a feature of pgf not tikz and vice versa. This confused me. Are they aliases of a single package or two very closely associated packages?

A: TikZ is "only" a frontend to PGF.
Without PGF there's no TikZ.
PGFMath is used by both PGF and TikZ to parse arbitrary input but can also be used without PGF and TikZ.
Technically, PGFMath has nothing to do with PGF
(which stands for Portable Graphics Format)
but since it was developed with PGF,
it somehow inherited its prefix.
The examples might start with \pgfmathsetmacro\myValue{<eval>}
because that's one way to define a macro \myValue with the evaluation of <eval>.
But \newcommand*\myValue{<eval>} could have also been used,
especially if <eval> is just a number or a length.
PGF commands start with \pgf…,
except for those that start with \pgfmath,
those are macros that can also be used when only PGFMath is loaded but also outside of a pgfpicture or a tikzpicture.
(Some exceptions like \pgfmathanglebetweenpoints exist.)

The PGF/TikZ manual (which is just named pgfmanual.pdf to confuse you a bit more) has its own sections about

*

*the Design Principles of TikZ and

*the Design Principles of PGF:

*

*All commands and environments start with pgf.

*All commands that specify a point (a coordinate) start with \pgfpoint.

*All commands that extend the current path start with \pgfpath.

*All commands that set/change a graphics parameter start with \pgfset.

*All commands that use a previously declared object (like a path, image or shading) start with \pgfuse.

*All commands having to do with coordinate transformations start with \pgftransform.

*All commands having to do with arrow tips start with \pgfarrows.

*All commands for "quickly" extending or drawing a path start with \pgfpathq or \pgfusepathq.

*All commands having to do with matrices start with \pgfmatrix.

Notes:

*

*A tikzpicture internally creates a pgfpicture.
The commands of TikZ are

*

*\path (and its derivatives \draw, \fill, \filldraw, \shade, \shadedraw, \clip, \graph, \useasboundingbox, \node, \coordinate, \matrix, \pic),

*the environment scope and its \scoped command version,

*some like \datavisualization or \spy that are only available via a library and, of course,

*\tikzset to set options via the key-value syntax outside optional [] parameter of the mentioned commands (and \tikzgraphset and \tikzdatavisualizationset).

The \tikz macro provides a shortcut for a tikzpicture.

*In TikZ coordinates are enclosed in ( and ).

*In TikZ path operations are --, |-, rectangle, .. controls … .. and similar (but also to and edge which are special again).

*Graphics parameters are usually set via the key-value syntax (like colors and dash patterns).

*These is done by TikZ in the ; at the end of the line in combination with the keys draw, fill and so on.

*Transformations are also done via the key-value syntax (scale, shift, …).

*Arrows are specified via – you guessed it – the key-value syntax. Explicitily via the arrows key but TikZ has set up its PGFKeys namespace so that keys with a - are interpreted as an arrow tip specification.

*These quick commands are not available on the TikZ level.

*Besides \matrix most matrix settings are done via the key-value syntax again. (Though, the TeX counts \pgfmatrixcurrentrow and \pgfmatrixcurrentcolumn can often times be seen in a TikZ matrix.)
Speaking of PGFKeys, similar to PGFMath this is just another package that can be used standalone without PGF and TikZ. PGFKeys does not draw anything.


I have now added searchable categories for my Waite-Smith, Tarot de Marseille and Lenormand-related posts and am beginning to populate them. These posts aren't as easy to chase down as the Thoth group because they are often tied to broader commentary, so it will take me some time to capture them. As before, use the "Select Category" drop-down on the home page and look for "RWS Material," TdM Material" or "Lenormand Material." Once again, I'm not including spreads and example readings in the binning at this time.


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The Santa Fe Film Festival presents a night of celebration of Native Cinema at The Santa Fe Oxygen and Healing bar! We will be celebrating Native Cinema's poster and will be auctioning off the actual 4ft X 4ft oil painting by Kathy Whitman-Elk Woman. We will have a live and silent auction while serving delicious hors d'oeuvres and amazing mocktails while you enjoy an evening celebrating Native Cinema. Tickets are $30 and available! The first 25 people to sign up will get a Native Cinema signed poster by the artist!


text:
\section{Introduction}

Our aim in this article is to show the global existence of global weak solutions to the fractional strongly damped wave equation with nonlinear hyperbolic dynamic boundary conditions.
We establish the global existence of weak solutions under a balance condition imposed on the nonlinear terms.
This condition is motivated by \cite[Lemma 3.1]{RBT01}.
In the present article, both nonlinearities are allowed supercritical polynomial growth.
Special attention is given to obtaining the compact resolvent for the associated linear operator which contains (fractional) Wentzell-Laplacians.

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ with smooth boundary $\Gamma:=\partial\Omega$.
Throughout we assume $\theta\in[\frac{1}{2},1)$, $\omega\in(0,1]$ and $\alpha\in(0,1]$.
We consider the equations in the unknown $u=u(t,x)$,
\begin{align}
\partial^2_tu-\omega\Delta^\theta \partial_tu+\partial_tu-\Delta u+u+f(u)=0 & \quad \text{in} \quad (0,\infty)\times\Omega,  \label{pde1} \\
\partial^2_tu+\omega\partial_{\bf n}^\theta \partial_tu+\partial_{\bf n}u-\alpha\omega\Delta_\Gamma \partial_tu+\partial_tu-\Delta_\Gamma u+u+g(u)=0 & \quad \text{on} \quad (0,\infty)\times\Gamma.  \label{pde2}
\end{align}
Additionally, we impose the initial conditions
\begin{align}
u(0,x)=u_0(x) \quad \text{and} \quad \partial_tu(0,x)=u_1(x) \quad \text{at} \quad \{0\}\times\Omega,  \label{ic1}
\end{align}
and
\begin{align}
u_{\mid\Gamma}(0,x)=\gamma_{0}(x) \quad \text{and} \quad \partial_tu_{\mid\Gamma}(0,x)=\gamma_{1}(x) \quad \text{at}\ \{0\}\times\Gamma.  \label{ic2}
\end{align}
Above, $\Delta_\Gamma$ denotes the Laplace-Beltrami operator (cf. e.g. \cite{Ch-La-10}).

We assume $f\in C(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ satisfy the sign conditions
\begin{equation}
\liminf_{|s|\rightarrow\infty}\frac{f(s)}{s}> -M_1, \quad g'(s)\ge -M_2, \quad \forall s\in \mathbb{R},\label{sign}
\end{equation}
for some $M_1,M_2>0$, and the growth assumptions, for all $s\in \mathbb{R}$,
\begin{equation}
|f(s)|\le \ell_1(1+|s|^{r_1-1}), \quad |g(s)|\le \ell_2(1+|s|^{r_2-1}),  \label{growth}
\end{equation}
for some positive constants $\ell_1$ and $\ell_2$, and where $r_1,r_2\ge 2$.
In addition, we assume there exists $\varepsilon \in (0,\omega )$ so that
the following balance condition holds,
\begin{equation}
\liminf_{|s|\rightarrow \infty} \frac{f(s)s+\frac{|\Gamma|}{|\Omega|}
g(s)s - \frac{C_\Omega^2|\Gamma|^2}{4\varepsilon|\Omega
|^2}|g'(s)s + g(s)|^2}{|s|^{r_1}} > 0,  \label{balance}
\end{equation}
for $r_1\ge \max \{r_2,2(r_2-1)\}$, where $C_\Omega>0$ is the best Sobolev constant in the following Sobolev-Poincar\'{e} inequality
\begin{equation}
\|u-\langle u\rangle_\Gamma\|_{L^2(\Omega)}\le C_\Omega\|\nabla u\|
_{L^2(\Omega)}, \quad \langle u\rangle_\Gamma := \frac{1}{|\Gamma|}\int\limits_\Gamma tr_D(u) d\sigma,  \label{eq:Poincare}
\end{equation}
for all $u\in H^1(\Omega)$.

Let us provide further context for the balance condition (\ref{balance}) in our setting (also see \cite{RBT01} and \cite{Gal-Shomberg15-2} for other settings).
Suppose that for $|y|\rightarrow\infty,$ both the internal and boundary functions satisfy the following:
\begin{equation*}
\lim_{|y|\rightarrow \infty }\frac{f(y)}{|y|^{r_1-1}}=\left( r_1-1\right) c_f, \quad \lim_{|y|\rightarrow \infty }\frac{g'(y)}{|y|^{r_2-2}}=\left( r_2-1\right) c_g,
\end{equation*}
for some constants $c_f,c_g\in \mathbb{R}\setminus \{0\}$.
In particular, there holds
\begin{equation*}
f(y)y\sim c_f|y|^{r_1}, \quad g(y)y\sim c_g|y|^{r_2} \quad \text{as}\ |y|\rightarrow \infty .
\end{equation*}
For the case of bulk dissipation (i.e., $c_f>0$) and anti-dissipative behavior at the boundary $\Gamma $ (i.e., $c_g<0$), assumption (\ref{balance}) is automatically satisfied provided that $r_1>\max \{r_2,2(r_2-1)\}$.
Furthermore, if $2<r_2<2\left( r_2-1\right)=r_1$ and
\begin{equation*}
c_f>\frac{1}{4\varepsilon}\left( \frac{C_\Omega|\Gamma|c_gr_2}{|\Omega|}\right)^2,
\end{equation*}
for some $\varepsilon \in (0,\omega )$, then (\ref{balance}) is again satisfied.
In the case when $f$ and $g$ are sublinear (i.e., $r_1=r_2=2$ in (\ref{growth})), the condition (\ref{balance}) is also automatically satisfied provided that
\begin{equation*}
\left( c_f +\frac{|\Gamma|}{|\Omega|}c_g\right) >\frac{1}{\varepsilon}\left( \frac{C_\Omega |\Gamma| c_g}{|\Omega|}\right)^2
\end{equation*}
for some $\varepsilon \in \left( 0,\omega \right)$.

\textbf{Notation and conventions.} Let us introduce some notation and conventions that are used throughout the article.
Norms in the associated space are clearly denoted $\|\cdot\|_B$ where $B$ is the corresponding Banach space.
We use the notation $(\cdot,\cdot)_H$ to denote the inner-product on the Hilbert space $H$.
The dual product on $H^*\times H$ is denoted $\langle \cdot,\cdot \rangle_{H^*\times H}$.
The notation $\langle \cdot,\cdot \rangle$ is also used to denote the product on the phase space and various other vectorial function spaces.
Denote by $(u,v)^{tr}$ the vector-valued function $\binom{u}{v}.$
In many calculations, functional notation indicating dependence on the variable $t$ is dropped; for example, we will write $u$ in place of $u(t)$.
Throughout the article, $C>0$ will denote a \emph{generic} constant which may depend on various structural parameters such as $|\Omega|$, $|\Gamma|$, $M_1,$ $M_2$, etc, and these constants may even change from line to line.
Furthermore, $Q:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ will be a generic monotonically increasing function whose specific dependance on other parameters will be made explicit on occurrence.
All of these constants/quantities are {\em independent} of the perturbation parameters $\theta,$ $\alpha$ and $\omega.$

\textbf{Outline of the article.} In the next section we establish the variational formulation of Problem {\textbf{P} and define weak solutions.
A proof of the existence of global weak solutions is developed in Section \ref{s:wp}.
Because of the nature of the balance condition, a continuous dependence type estimate is not available.
The article continues with some remarks on this difficulty and plans for possible further research.
An appendix contains some explicit characterizations for the fractional Wentzell-Laplacian used throughout the article, as well as a certain compact embedding result that we need to draw upon.

\section{Formulation of the model problem}  \label{s:prelim}

In this section we first recall the Wentzell-Laplacian defined on vectorial Hilbert spaces.
(For this we largely refer to \cite[Section 2]{AreMetPalRom-2003} and \cite[Section 2 and Appendix]{Gal12-1}.)
Following this, we give the basic functional setup in order to formulate the model problem.
We also provide various results pertaining to the problem.

To begin, let $\lambda_\Omega>0$ denote the best constant satisfying the Sobolev inequality in $\Omega$
\begin{equation}  \label{Sobolev}
\lambda_\Omega \int_\Omega u^2 dx \le \int_\Omega (|\nabla u|^2+u^2) dx.
\end{equation}
We will also rely on the Laplace-Beltrami operator $-\Delta_\Gamma$ on the surface $\Gamma.$
This operator is positive definite and self-adjoint on $L^2(\Gamma)$ with domain $D(\Delta_\Gamma)$.
The Sobolev spaces $H^s(\Gamma)$, for $s\in\mathbb{R}$, may be defined as $H^s(\Gamma)=D((\Delta_\Gamma)^{s/2})$ when endowed with the norm whose square is given by, for all $u\in H^s(\Gamma)$,
\begin{equation}  \label{LB-norm}
\|u\|^2_{H^s(\Gamma)} := \|u\|^2_{L^2(\Gamma)} + \left\|(-\Delta_\Gamma)^{s/2}u\right\|^2_{L^2(\Gamma)}.
\end{equation}
On the boundary, let $\lambda_\Gamma>0$ denote the best constant satisfying the Sobolev inequality on $\Gamma$
\begin{equation}  \label{bndry-Sobolev}
\lambda_\Gamma \int_\Gamma u^2 d\sigma \le \int_\Gamma\left( |\nabla_\Gamma u|^2+u^2 \right) d\sigma.
\end{equation}

Next, recall that $\Omega$ is a bounded domain of $\mathbb{R}^3$ with boundary $\Gamma$, to which we now assume is of class $\mathcal{C}^2$.
To this end, consider the space $\mathbb{X}^2 = L^2(\overline{\Omega},d\mu),$ where $d\mu =dx_{\mid\Omega}\oplus d\sigma$ is such that $dx$ denotes the Lebesgue measure on $\Omega$ and $d\sigma$ denotes the natural surface measure on $\Gamma$.
Then $\mathbb{X}^2=L^2(\Omega,dx)\oplus L^2(\Gamma,d\sigma)$ may be identified by the natural norm
\begin{equation*}
\|u\|_{\mathbb{X}^2}^2=\int_\Omega|u(x)|^2dx + \int_\Gamma
|u(x)|^2d\sigma.
\end{equation*}
Moreover, if we identify every $u\in C(\overline{\Omega})$ with $U=(u_{\mid\Omega},u_{\mid\Gamma})^{tr} \in C(\Omega) \times C(\Gamma)$, we may also define $\mathbb{X}^2$ to be the completion of $C(\overline{\Omega})$ with respect to the norm $\|\cdot\|_{\mathbb{X}^2}$.
Thus, in general, any function $u\in \mathbb{X}^2$ will be of the form $u=\binom{u_1}{u_2}$ with $u_1\in L^2(\Omega,dx)$ and $u_2\in L^2(\Gamma,d\sigma)$.
It is important to note that there need not be any connection between $u_1$ and $u_2$.
From now on, the inner product in the Hilbert space $\mathbb{X}^2$
will be denoted by $\langle \cdot ,\cdot \rangle_{\mathbb{X}^2}.$
Now we recall that the Dirichlet trace map $tr_D:C^\infty(\overline{\Omega}) \rightarrow C^\infty(\Gamma),$ defined by $tr_D(u) =u_{\mid\Gamma}$ extends to a linear continuous operator $tr_D:H^r(\Omega) \rightarrow H^{r-1/2}(\Gamma),$
for all $r>1/2$, which is onto for $1/2<r<3/2.$
This map also possesses a bounded right inverse $tr_D^{-1}:H^{r-1/2}(\Gamma)
\rightarrow H^r(\Omega)$ such that $tr_D(tr_D^{-1}\psi) =\psi,$ for any $\psi \in H^{r-1/2}(\Gamma)$.
We can thus introduce the subspaces of $H^r(\Omega) \times H^{r-1/2}(\Gamma)$ and $H^r(\Omega) \times H^r(\Gamma)$, respectively, by
\begin{align}
\mathbb{V}_0^r &:= \{U=(u,\gamma) \in H^r(\Omega) \times H^{r-1/2}(\Gamma) : tr_D(u) =\gamma \},  \label{vvv} \\
\mathbb{V}^r &:= \{U=(u,\gamma) \in \mathbb{V}_0^r: tr_D(u) = \gamma \in H^r(\Gamma) \},  \notag
\end{align}
for every $r>1/2,$ and note that $\mathbb{V}_0^r,$ $\mathbb{V}^r$ are
not product spaces.
However, we do have the following dense and compact embeddings $\mathbb{V}_0^{r_1}\subset \mathbb{V}_0^{r_2},$ for any $r_1>r_2>1/2$ (by definition, this also true for the sequence of spaces $\mathbb{V}^{r_1}\subset \mathbb{V}^{r_2}$). Naturally, the norm on the spaces $\mathbb{V}_0^r,$ $\mathbb{V}^r$ are defined by
\begin{equation}
\|U\|_{\mathbb{V}_0^r}^2:=\|u\|_{H^r(\Omega)}^2+\|\gamma\|_{H^{r-1/2}(\Gamma)}^2, \quad \|U\|_{\mathbb{V}^r}^2:=\|u\|_{H^r(\Omega)}^2+\|\gamma\|_{H^r(\Gamma)}^2.
\label{Vr-norm}
\end{equation}

Here we consider the basic (linear) operator associated with the model problem \eqref{pde1}-\eqref{ic2}, the so-called Wentzell-Laplacian.
Let
\begin{align}
\Delta_W \binom{u_1}{u_2}:=\begin{pmatrix} \Delta u_1-u_1 \\ -\partial_{\bf n}u_1+\Delta_\Gamma u_2-u_2 \end{pmatrix},  \label{A_Wentzell1}
\end{align}
with
\begin{equation}
D(\Delta_W) := \left\{ U=\binom{u_{1}}{u_{2}}\in \mathbb{V}^1 : -\Delta u_1\in L^2(\Omega),\ \partial_{\bf n}u_1-\Delta_\Gamma u_2\in L^2(\Gamma) \right\}. \label{A_Wentzell}
\end{equation}
By, for example, \cite[see Appendix and in particular Theorem 5.3]{Gal12-1}, the operator $(\Delta_W,D(\Delta_W))$ is self-adjoint and strictly positive operator on $\mathbb{X}^2$, and the resolvent operator $(I+\Delta_W)^{-1}\in \mathcal{L}(\mathbb{X}^2)$ is compact.
Since $\Gamma$ is of class $\mathcal{C}^2,$ then $D(\Delta_W)=\mathbb{V}^2$. Indeed, the map $L:U\mapsto \Delta_WU,$ as a mapping from $\mathbb{V}^2$ into $\mathbb{X}^2=L^2(\Omega) \times L^2(\Gamma),$ is an isomorphism, and there exists a
positive constant $C_*$, independent of $U=(u,\gamma)^{tr}$, such that, for all $U\in \mathbb{V}^2$,
\begin{equation}
C_*^{-1}\|U\|_{\mathbb{V}^2}\le \|L(U)\|_{\mathbb{X}^2}\le C_*\|U\|_{\mathbb{V}^2},  \label{regularity-oper}
\end{equation}
(cf. Lemma \ref{t:pre-reg-op}, see also \cite{CFGGGOR09}).

The following basic elliptic estimate is taken from \cite[Lemma 2.2]
{Gal&Grasselli08}.

\begin{lemma}  \label{t:pre-reg-op}
Consider the linear boundary value problem,
\begin{equation}
\left\{ \begin{array}{rl} -\Delta u & =p_1 \quad \text{in} \quad \Omega, \\
-\Delta_\Gamma u + \partial_{\bf n}u + u & =p_2 \quad \text{on} \quad \Gamma.
\end{array}\right.  \label{pre-reg-BVP}
\end{equation}
If $(p_1,p_2) \in H^s(\Omega)\times H^s(\Gamma)$ for $s\ge0$ and $s+\frac{1}{2}\not\in \mathbb{N}$, then the following estimate holds for some constant $C>0$,
\begin{equation}
\|u\|_{H^{s+2}(\Omega)}+\|u\|_{H^{s+2}(\Gamma)}\le C\left( \|p_1\|_{H^s(\Omega)}+\|p_2\|_{H^s(\Gamma)}\right).  \label{H2-reg}
\end{equation}
\end{lemma}

We also recall the following basic inequality which gives interior control over some boundary terms (cf. \cite[Lemma A.2]{Gal12-2}).

\begin{lemma}  \label{t:pre-bnd-reg}
Let $s>1$ and $u\in H^1(\Omega)$.
Then, for every $\varepsilon>0$, there exists a positive constant $C_\varepsilon\sim\varepsilon^{-1}$ such that,
\begin{equation}
\|u\|_{L^s(\Gamma)}^s \le \varepsilon\|\nabla u\|_{L^2(\Omega)}^2 + C_\varepsilon(\|u\|_{L^\gamma(\Omega)}^\gamma + 1),  \label{pre-bnd-reg}
\end{equation}
where $\gamma=\max\{s,2(s-1)\}$.
\end{lemma}
We refer the reader to more details to e.g., \cite{CGGM10}, \cite{CFGGGOR09} and \cite{Gal&Warma10} and the references therein.

Finally, since the operator $\Delta_W$ with domain $D(\Delta_W)$ is positive and self-adjoint on $\mathbb{X}^2$, we may define fractional powers of $\Delta_W$ (see Appendix \ref{s:ap-1}).
Indeed, with $\theta\in[\frac{1}{2},1)$, $\alpha\in(0,1]$ and $\omega\in(0,1]$, we define
\begin{align}
\Delta_W^\theta \binom{u_1}{u_2} := \begin{pmatrix} \Delta^\theta u_1-u_1 \\ -\partial_{\bf n}^\theta u_1+\Delta_\Gamma u_2-u_2 \end{pmatrix}  \notag
\end{align}
and
\begin{align}
\Delta_W^{\theta,\alpha,\omega} \binom{u_1}{u_2} := \begin{pmatrix} \omega\Delta^\theta u_1-u_1 \\ -\omega\partial_{\bf n}^\theta u_1+\alpha\omega\Delta_\Gamma u_2-u_2 \end{pmatrix}  \notag
\end{align}
with domain
\begin{equation}
D(\Delta_W^{\theta,\alpha,\omega}) := \left\{ U=\binom{u_{1}}{u_{2}}\in \mathbb{V}^1 : -\omega\Delta^\theta u_1\in L^2(\Omega),\ \omega\partial_{\bf n}^\theta u_1-\alpha\omega\Delta_\Gamma u_2\in L^2(\Gamma) \right\}. \label{A_Wentzell2}
\end{equation}
Hence, $\Delta_W^{\theta,1,1}=\Delta_W^\theta$.
The fractional flux $\partial_{\bf n}^\theta$ are defined as follows.
Consider $\partial_{\bf n}u=\nabla u\cdot {\mathbf n}$, and recall $\partial_{\bf n}u\in L^2(\Gamma)$ whenever $u\in H^{3/2}(\Omega)$.
So we can define $\partial_{\bf n}^\theta u=\nabla_W^{\theta/2} u \cdot {\mathbf n}$ when $u\in H^{\frac{1}{2}+\theta}(\Omega)$ guaranteeing the fractional flux $\partial_{\bf n}^\theta u\in L^2(\Gamma).$
(These fractional flux operators are explicitly written in Appendix \ref{s:ap-1}.)
Moving toward the linear operator associated with the model problem \eqref{pde1}-\eqref{ic2}
Let $U=(u_1,u_2)\in \mathbb{V}^1$ and $V=(v_1,v_2)\in \mathbb{X}^2$, and let $\mathcal{X}=(U,V)$.
Motivated by \cite{Carvalho_Cholewa_02}, we define the unbounded linear operator $\mathcal{A}_{\theta,\alpha,\omega}$ written as
\[
\mathcal{A}_{\theta,\alpha,\omega}\mathcal{X} := \begin{pmatrix} 0 & I_{2\times2} \\ \Delta_W & \Delta_W^{\theta,\alpha,\omega} \end{pmatrix}\begin{pmatrix} U \\ V \end{pmatrix} = \begin{pmatrix} V \\ \Delta_W U+\Delta_W^{\theta,\alpha,\omega} V \end{pmatrix} = \begin{pmatrix} V \\ \Delta^{\theta,1,1}_W(\Delta_W^{1-\theta,1,1}U+\Delta_W^{0,\alpha,\omega}V )\end{pmatrix}
\]
with domain
\begin{align}
D(\mathcal{A}_{\theta,\alpha,\omega}) := & \left\{ \mathcal{X}=\binom{U}{V}\in \mathbb{V}^1\times\mathbb{X}^2 : \Delta_W^{1-\theta,1,1}U+\Delta_W^{0,\alpha,\beta}V\in D(\Delta_W^{\theta,1,1}) \right\}.  \notag
\end{align}
By \cite[Theorem 3.1 (a)]{Haraux-Otani-13}, the resolvent $(I_{4\times4}+\mathcal{A}_{\theta,\alpha,\omega})^{-1}\in\mathcal{L}(\mathbb{V}^1\times\mathbb{X}^2)$ is compact.
Hence, we can support the local existence of weak solutions (defined below) with a Galerkin method.

Next we define the nonlinear mapping on $\mathbb{V}^1\times\mathbb{X}^2$ given by
\begin{align}
F(U):=\binom{0}{-f(u)}, \quad G(U):=\binom{0}{-g(\gamma)},
\end{align}
and
\begin{align}
\mathcal{F}(\mathcal{X}) &:= \begin{pmatrix} F(U) \\ G(U) \end{pmatrix} = \begin{pmatrix} 0 \\ -f(u) \\ 0 \\ -g(\gamma) \end{pmatrix} \quad \text{for} \quad U\in\mathbb{V}^1.  \notag
\end{align}
Due to the two embeddings, $H^1(\Omega)\hookrightarrow L^{s_1}(\Omega)$, $s_1\in[1,6]$, and $H^1(\Gamma)\hookrightarrow L^{s_2}(\Omega)$, $s_2\in[1,\infty)$, one can show that when $r_1\in[1,3]$ in \eqref{growth}, then $\mathcal{F}:\mathbb{V}^1\times\mathbb{X}^2\rightarrow\mathbb{V}^1\times\mathbb{X}^2$ is locally Lipschitz (indeed, cf. e.g. \cite[Lemma 2.6]{Graber-Shomberg-16}).
With $r_1\ge1$ arbitrary, this motivates us to set
\[
\widetilde{\mathbb{V}}^{s,r_1} = \left\{U=(u,\gamma)^{tr}\in \left[ H^s(\Omega)\cap L^{r_1}(\Omega)\right] \times H^s(\Gamma) : tr_D(u)=\gamma\right\}
\]
with the canonical norm whose square is given by
\[
\|U\|^2_{\widetilde{\mathbb{V}}^{s,r_1}}:=\|u\|^2_{H^s(\Omega)}+\|u\|^{r_1}_{L^{r_1}(\Omega)}+\|\gamma\|^2_{H^s(\Gamma)},
\]
and also set $\mathcal{H}_0:=\widetilde{\mathbb{V}}^{1,r_1}\times\mathbb{X}^2$.
The space $\mathcal{H}_0$ is Hilbert with the norm whose square is given by, for $\mathcal{X} = (U,V)\in\mathcal{H}_0$,
\begin{align}
\|\mathcal{X}\|^2_{\mathcal{H}_0} &:= \|U\|^2_{\widetilde{\mathbb{V}}^{1,r_1}}+\|V\|^2_{\mathbb{X}^2} \notag \\
& = \|u\|^2_{H^1(\Omega)}+\|u\|^{r_1}_{L^{r_1}(\Omega)} + \|v\|^2_{L^2(\Omega)} + \|\gamma\|^2_{H^1(\Gamma)} + \|\delta\|^2_{L^2(\Gamma)}   \notag \\
& = \left( \|\nabla u\|^2_{L^2(\Omega)} + \|u\|^2_{L^2(\Omega)} \right)+\|u\|^{r_1}_{L^{r_1}(\Omega)} + \|v\|^2_{L^2(\Omega)} + \left( \|\nabla_\Gamma\gamma\|^2_{L^2(\Gamma)} + \|\gamma\|^2_{L^2(\Gamma)} \right) + \|\delta\|^2_{L^2(\Gamma)}.   \notag
\end{align}
The space $\mathcal{H}_0$ is our weak energy phase space.
Moreover, given $\mathcal{X}_0=(U_0,U_1)\in\mathcal{H}_0=\widetilde{\mathbb{V}}^{1,r_1}\times\mathbb{X}^2,$ the abstract formulation of Problem {\textbf P} takes the form
\[
\left\{ \begin{array}{ll}
\displaystyle\frac{d}{dt}\mathcal{X}(t) = \mathcal{A}_{\theta,\alpha,\omega}\mathcal{X}(t)+\mathcal{F}(\mathcal{X}(t)) & t>0, \\ ~ \\
\mathcal{X}(0) = \mathcal{X}_0. \end{array} \right.
\]

We can now introduce the variational formulation of Problem {\textbf P}.

\begin{definition}  \label{d:ws}
Let $\theta\in[\frac{1}{2},1),$ $\alpha\in(0,1]$ and $\omega\in(0,1].$
Let $T>0$ and $\mathcal{X}_0=(U_0,U_1)\in\mathcal{H}_0.$
A function $\mathcal{X}(t)=(U(t),\partial_tU(t))=(u(t),u_{\mid\Gamma}(t),\partial_t u(t),\partial_t u_{\mid\Gamma}(t))$ satisfying
\begin{align}
U & \in L^\infty(0,T;\mathbb{V}^1),  \label{wk-reg-1} \\
\partial_tU & \in L^\infty(0,T;\mathbb{X}^2),  \label{wk-reg-2} \\
\sqrt{\omega}\partial_tu & \in L^2(0,T;H^\theta(\Omega)),  \label{wk-reg-3} \\
\partial^2_tU & \in L^\infty(0,T;(\mathbb{V}^1)^*),  \label{wk-reg-4}
\end{align}
for almost all $t\in(0,T]$ is called a {\sc{weak solution}} to Problem {\textbf P} with initial data $\mathcal{X}_0$ if the following identities hold almost everywhere on $[0,T]$, and for all $\Xi=(\Xi_1,\Xi_2)\in \mathbb{V}^1\times\mathbb{V}^1$:
\begin{align}
\frac{d}{dt}\left\langle \mathcal{X}(t),\Xi \right\rangle_{\mathcal{V}^{-1}\times\mathcal{V}^1} = \left\langle \mathcal{A}_{\theta,\alpha,\omega} \mathcal{X}(t),\Xi \right\rangle_{\mathcal{H}_0} + \left\langle \mathcal{F}(\mathcal{X}(t)),\Xi \right\rangle_{\mathcal{H}_0}.  \label{var1}
\end{align}
Also, the initial conditions \eqref{ic1}-\eqref{ic2} hold in the $L^2$-sense; i.e.,
\begin{align}
\left\langle \mathcal{X}(0),\Xi \right\rangle_{\mathcal{H}_0}=\left\langle \mathcal{X}_0,\Xi \right\rangle_{\mathcal{H}_0}, \quad \text{for every}\quad \Xi\in \mathbb{V}^1\times\mathbb{V}^1.  \label{var-ic}
\end{align}
We say $\mathcal{X}(t)=(U(t),\partial_tU(t))$ is a {\sc{global weak solution}} of Problem {\textbf P} if it is a weak solution on $[0,T]$, for any $T>0.$
\end{definition}

\begin{remark}
Observe that we are solving a more general problem because $\gamma_0$ and $\gamma_1$, from $U_0$ and $U_1$ respectively, may be taken to be initial data {\em independent} of $u$ and $\partial_tu$.
However, if $\partial_tu(t)\in H^{s}(\Omega)$, for all $t>0$ and for some $s>1/2$, then $\gamma_t(t)=\partial_tu_{\mid\Gamma}(t)$.
\end{remark}

\section{Global existence}  \label{s:wp}

\begin{theorem}
Let $\mathcal{X}_0=(U_0,U_1)\in\mathcal{H}_0$ satisfy $\|\mathcal{X}_0\|_{\mathcal{H}_0}\le R$ for some $R>0$.
Then there exists a global weak solution to Problem {\textbf P} satisfying the additional regularity,
\begin{align}
\sqrt{\alpha\omega}\partial_tu & \in L^2(0,T;H^1(\Gamma)), \label{adreg1}
\end{align}
\end{theorem}

\begin{proof}
{\underline{Step 1.}} (An {\em a priori} estimate.)
In \eqref{var1} take $\Xi=(\partial_tU,\partial_t U)$ to find the differential identity
\begin{align}
\frac{1}{2}\frac{d}{dt} & \left\{ \|\partial_tU\|^2_{\mathbb{X}^2} + \|U\|^2_{\mathbb{V}^1} + 2(F(u),1)_{L^2(\Omega)} + 2(G(u),1)_{L^2(\Gamma)} \right\}  \notag \\
& + \omega\|\nabla^\theta \partial_tu\|^2_{L^2(\Omega)} + \|\partial_tu\|^2_{L^2(\Omega)} + \alpha\omega\|\nabla \partial_tu\|^2_{L^2(\Gamma)} + \|\partial_tu\|^2_{L^2(\Gamma)} = 0.  \label{po1}
\end{align}
Using \eqref{growth} and setting $\tilde F'=f$ and $\tilde G'=g$, a simple integration by parts on \eqref{sign} shows, for all $u\in H^1(\Omega)$, and $\gamma\in H^1(\Gamma),$
\begin{align}
(\tilde F(u),1)_{L^2(\Omega)} & \ge (f(u),u)_{L^2(\Omega)}+\frac{M_1}{2}\|u\|^2_{L^2(\Omega)} \label{cons1}
\end{align}
and
\begin{align}
(\tilde G(\gamma),1)_{L^2(\Gamma)} & \ge (g(\gamma),\gamma)_{L^2(\Gamma)}+\frac{M_2}{2}\|\gamma\|^2_{L^2(\Gamma)}. \label{cons2}
\end{align}
To bound the products on the right-hand sides of \eqref{cons1} and \eqref{cons2} from below, we utilize \eqref{balance}.
Following \cite[(2.22)]{Gal12-2}, \cite[(3.34)]{Gal-Shomberg15-2} and \cite[(3.11)]{RBT01}, we estimate the products as
\begin{align}
& (f(u),u)_{L^2(\Omega)}+(g(u),u)_{L^2(\Gamma)} \notag \\
& = \int_\Omega \left( f(u)u+\frac{|\Gamma|}{|\Omega|}g(u)u \right) dx - \frac{|\Gamma|}{|\Omega|}\int_\Omega \left( g(u)u-\frac{1}{|\Gamma|}\int_\Gamma g(u)u\mathrm{d}\sigma \right) dx,  \label{po3}
\end{align}
whereby we exploit the Poincar\'{e} inequality (\ref{eq:Poincare}) and Young's
inequality to see that, for all $\varepsilon>0$,
\begin{align}
\frac{|\Gamma|}{|\Omega|}\int_\Omega\left( g(u)u-\frac{1}{
|\Gamma|}\int_\Gamma g(u)u d\sigma \right) dx & \le C_\Omega
\frac{|\Gamma|}{|\Omega|}\int_\Omega|\nabla (g(u)u)|dx \notag \\
& =C_\Omega\frac{|\Gamma|}{|\Omega|}\int_\Omega|\nabla u(g'(u)u+g(u))|dx  \notag \\
& \le \varepsilon \|\nabla u\|_{L^2(\Omega)}^2+\frac{C_\Omega^2|\Gamma|^2}{4\varepsilon |\Omega|^2}\int_\Omega|g'(u)u+g(u)|^2dx. \label{po4}
\end{align}
Then combining \eqref{po3} and \eqref{po4}, and applying assumption \eqref{balance} yields
\begin{equation}
(f(u),u)_{L^2(\Omega)}+(g(u),u)_{L^2(\Gamma)} \ge \|u\|_{L^{r_1}(\Omega)}^{r_1}-\varepsilon \|\nabla u\|_{L^2(\Omega)}^2-C_\delta,  \label{po5}
\end{equation}
for some positive constants $\delta$ and $C_\delta$ that are independent
of $t$ and $\varepsilon$.
Hence, together \eqref{cons1} and \eqref{cons2} become
\begin{align}
(F(u),1)_{L^2(\Omega)}+(G(u),1)_{L^2(\Gamma)} & \ge \|u\|_{L^{r_1}(\Omega)}^{r_1}+\frac{M_1}{2}\|u\|^2_{L^2(\Omega)}+\frac{M_2}{2}\|u\|^2_{L^2(\Gamma)}-\varepsilon \|\nabla u\|_{L^2(\Omega)}^2-C_\delta. \label{cons3}
\end{align}
Moreover, \eqref{cons3} provides a lower-bound to the functional
\begin{align}
E(t) := \|\partial_tU(t)\|^2_{\mathbb{X}^2} + \|U(t)\|^2_{\mathbb{V}^1} + 2(F(u(t)),1)_{L^2(\Omega)} + 2(G(u(t)),1)_{L^2(\Gamma)}.  \notag
\end{align}
Integrating the identity \eqref{po1} over $(0,t)$, yields
\begin{align}
E(t) + 2\int_0^t \left( \omega\|\nabla^\theta \partial_tu(\tau)\|^2_{L^2(\Omega)} + \alpha\omega\|\nabla \partial_tu(\tau)\|^2_{L^2(\Gamma)} + \|\partial_tU(\tau)\|^2_{\mathbb{X}^2} \right)d\tau = E(0).  \label{po1.5}
\end{align}
We can find an upper-bound on $E(0)$ with \eqref{growth}.
Evidently
\begin{align}
& 2(F(u(0)),1)_{L^2(\Omega)} + 2(G(u(0)),1)_{L^2(\Gamma)} \notag \\
& \le \ell_1(\|u(0)\|_{L^1(\Omega)}+\|u(0)\|^{r_1}_{L^{r_1}(\Omega)})+\ell_2(\|u(0)\|_{L^1(\Gamma)}+\|u(0)\|^{r_2}_{L^{r_2}(\Gamma)}).  \label{po1.6}
\end{align}
Hence, \eqref{po1.6} and the embedding $\mathbb{V}^1\hookrightarrow\mathbb{X}^2$ show
\begin{align}
E(0) & \le \|\partial_tu(0)\|^2_{L^2(\Omega)} + \|\nabla u(0)\|^2_{L^2(\Omega)} + \|u(0)\|^2_{L^2(\Omega)} + \|\partial_tu(0)\|^2_{L^2(\Gamma)} + \|\nabla u(0)\|^2_{L^2(\Gamma)} + \|u(0)\|^2_{L^2(\Gamma)}  \notag \\
& + \ell_1(\|u(0)\|_{L^1(\Omega)}+\|u(0)\|^{r_1}_{L^{r_1}(\Omega)})+\ell_2(\|u(0)\|_{L^1(\Gamma)}+\|u(0)\|^{r_2}_{L^{r_2}(\Gamma)}) \notag \\
& \le \|\partial_tU(0)\|^2_{\mathbb{X}^2} + \|U(0)\|^2_{\mathbb{V}^1} + C\left( \|U(0)\|_{\mathbb{V}^1} + \|u(0)\|^{r_1}_{L^{r_1}(\Omega)} + \|u(0)\|^{r_2}_{L^{r_2}(\Gamma)} \right).  \label{po1.7}
\end{align}
Thus \eqref{po1.5} and \eqref{po1.7} yield, for all $t\ge0,$
\begin{align}
& \|\partial_tU(t)\|^2_{\mathbb{X}^2} + \|U(t)\|^2_{\mathbb{V}^1} + 2(F(u(t)),1)_{L^2(\Omega)} + 2(G(u(t)),1)_{L^2(\Gamma)} \notag \\
& + 2\int_0^t \left( \omega\|\nabla^\theta \partial_tu(\tau)\|^2_{L^2(\Omega)} + \alpha\omega\|\nabla \partial_tu(\tau)\|^2_{L^2(\Gamma)} + \|\partial_tU(\tau)\|^2_{\mathbb{X}^2} \right)d\tau \notag \\
& \le \|\partial_tU(0)\|^2_{\mathbb{X}^2} + \|U(0)\|^2_{\mathbb{V}^1} + C\left( \|U(0)\|_{\mathbb{V}^1} + \|u(0)\|^{r_1}_{L^{r_1}(\Omega)} + \|u(0)\|^{r_2}_{L^{r_2}(\Gamma)} \right) \notag \\
& \le \|\partial_tU(0)\|^2_{\mathbb{X}^2} + \|U(0)\|^2_{\mathbb{V}^1} + C\left( \|U(0)\|_{\mathbb{V}^1} + \|u(0)\|^{r_1}_{L^{r_1}(\Omega)} + 1 \right), \label{po1.8}
\end{align}
where the last inequality follows from Lemma \ref{t:pre-bnd-reg}.

Now we see that, for any $T>0$, there hold
\begin{align}
U & \in L^\infty(0,T;\mathbb{V}^1),  \label{po6} \\
\partial_tU & \in L^\infty(0,T;\mathbb{X}^2),  \label{po7} \\
\sqrt{\omega} \partial_tu & \in L^2(0,T;H^\theta(\Omega)), \label{po7.1} \\
\sqrt{\alpha\omega} \partial_tu & \in L^2(0,T;H^1(\Gamma)), \label{po7.3} \\
F(u) & \in L^\infty(0,T;L^1(\Omega)), \label{po8} \\
G(u) & \in L^\infty(0,T;L^1(\Gamma)). \label{po9}
\end{align}
We have found $\mathcal{X}\in L^\infty(0,T;\mathcal{H}_0)$.
Moreover, since $U\in L^\infty(0,T;\mathbb{V}^1)$, we have $\Delta_WU \in L^\infty(0,T;(\mathbb{V}^1)^*)$ and as $\sqrt{\alpha\omega}\partial_tU\in L^2(0,T;\mathbb{V}^1)$, we also have $\Delta_W^{\theta,\alpha,\omega}\partial_tU \in L^2(0,T;(\mathbb{V}^1)^*)$.
Therefore, after comparing terms in the first equation of \eqref{po1}, we see that
\begin{equation}
\partial_t^2U\in L^2(0,T;(\mathbb{V}^1)^*). \label{po10}
\end{equation}
Hence, this justifies our choice of test function in \eqref{po1}.
With \eqref{po7.3}, we also find \eqref{adreg1} as claimed.
This concludes Step 1.

{\underline{Step 2.}} (A Galerkin basis.)
According to Section \ref{s:prelim}, for each $\theta\in[\frac{1}{2},1),$ the operator $\mathcal{A}_{\theta,\alpha,\omega}$ admits a system of eigenfunctions $\Psi_i^{\theta,\alpha,\omega}=(\psi^{\theta,\alpha,\omega},\phi^{\theta,\alpha,\omega},\psi_{\mid\Gamma}^{\theta,\alpha,\omega},\phi_{\mid\Gamma}^{\theta,\alpha,\omega})$ satisfying $\{\Psi_i^{\theta,\alpha,\omega}\}_{i=1}^\infty \subset D(\mathcal{A}_{\theta,\alpha,\omega})\cap (C^2({\overline{\Omega}}) \times C^2(\Gamma) \times C^2({\overline{\Omega}}) \times C^2(\Gamma))$ and
\[
\mathcal{A}_{\theta,\alpha,\omega} \Psi_i^{\theta,\alpha,\omega}=\Lambda_i\Psi_i^{\theta,\alpha,\omega}, \quad i=1,2,\dots,
\]
where the eigenvalues $\Lambda_i=\Lambda_i^{\theta,\alpha,\omega}\in(0,+\infty)$ may be put into increasing order and counted according to their multiplicity to form a diverging sequence.
This means the pair $(\Lambda_i,\Psi_i)$, $\Psi_i=\Psi_i^{\theta,\alpha,\omega}$ is a classical solution of the elliptic problem
\[
\left\{ \begin{array}{ll}
-\Delta\psi_i + \psi_i + \omega(-\Delta)^\theta \phi_i + \phi_i  = \Lambda_i\psi_i & \text{in}\ \Omega \\
-\alpha\omega\Delta_\Gamma \phi_{i\mid\Gamma} + \phi_{i\mid\Gamma} - \Delta_\Gamma\psi_{i\mid\Gamma} + \psi_{i\mid\Gamma} = \Lambda_i\psi_{i\mid\Gamma} & \text{on}\ \Gamma. \end{array} \right.
\]
Also due to standard spectral theory, these eigenfunctions form an orthogonal basis in $\mathcal{H}_0$ that is orthonormal in $L^2(\Omega)\times L^2(\Omega)\times L^2(\Gamma)\times L^2(\Gamma)$.

Let $T>0$ be fixed.
For $n\in \mathbb{N}$, set the spaces
\begin{equation*}
\mathbb{H}_n := \mathrm{span}\left\{ \Psi_1^{\theta,\alpha,\omega},\dots,\Psi_n^{\theta,\alpha,\omega}\right\} \subset \mathcal{H}_0 \quad \text{and} \quad \mathbb{H}_\infty := \bigcup_{n=1}^\infty \mathbb{H}_n.
\end{equation*}
Obviously, $\mathbb{H}_\infty$ is a dense subspace of $\mathcal{H}_0$.
For each $n\in \mathbb{N}$, let $\mathbb{P}_n:\mathcal{H}_0\rightarrow \mathbb{H}_n$ denote the orthogonal projection of $\mathcal{H}_0$ onto $\mathbb{H}_n$.
Thus, we seek functions of the form
\begin{equation}
\mathcal{X}^{(n)}(t)=\sum_{i=1}^n A_i(t)\Psi_i^{\theta,\alpha,\omega}  \label{po11}
\end{equation}
that will satisfy the associated discretized Problem \textbf{P}$_n$
described below.
The functions $A_i$ are assumed to be (at least) $C^2((0,T))$ for $i=1,\dots,n$.
Precisely,
\begin{equation}
u^{(n)}(t)=\sum_{i=1}^n A_i(t)\psi_i^{\theta,\alpha,\omega}, \quad \partial_t u^{(n)}(t)=\sum_{i=1}^n A_i'(t)\psi_i^{\theta,\alpha,\omega}, \label{po12-1}
\end{equation}
and
\begin{equation}
u^{(n)}_{\mid\Gamma}(t)=\sum_{i=1}^n A_i(t)\phi_{i\mid\Gamma}^{\theta,\alpha,\omega}, \quad \partial_t u^{(n)}_{\mid\Gamma}(t)=\sum_{i=1}^n A_i'(t)\phi_{i\mid\Gamma}^{\theta,\alpha,\omega}. \label{po12-2}
\end{equation}
Using semigroup properties of $\mathcal{A}_{\theta,\alpha,\omega}$, the domain $D(\mathcal{A}^{\theta,\alpha,\omega})$ is dense in $\mathcal{H}_0$.
So to approximate the given initial data $\mathcal{X}_0\in \mathcal{H}_0$, we may take $\mathcal{X}_0^{(n)}\in D(\mathcal{A}^{\theta,\alpha,\omega})$ such that $\mathcal{X}_0^{(n)}\rightarrow \mathcal{X}_0$ in $\mathcal{H}_0$.

For $T>0$ and for each integer $n\ge1$, the weak formulation of the
approximate Problem \textbf{P}$_n$ is: to find $\mathcal{X}^{(n)}$ given by \eqref{po11} such that, for all ${\overline{\mathcal{X}}}=(\overline{U},\overline{V})\in \mathbb{H}_n$, the equation
\begin{equation}
\left\langle \partial_t\mathcal{X}^{(n)},{\overline{\mathcal{X}}}\right\rangle_{\mathcal{H}_0} + \left\langle \mathcal{A}_{\theta,\alpha,\omega}\mathcal{X}^{(n)},{\overline{\mathcal{X}}} \right\rangle_{\mathcal{H}_0}+\left\langle \mathbb{P}_n\mathcal{F}\left( \mathcal{X}^{(n)}\right),{\overline{\mathcal{X}}}\right\rangle_{\mathcal{H}_0}=0 \label{po13}
\end{equation}
holds for almost all $t\in(0,T)$, subject to the initial conditions
\begin{equation}
\left\langle \mathcal{X}^{(n)}(0),{\overline{\mathcal{X}}} \right\rangle _{\mathcal{H}_0}=\left\langle \mathcal{X}_0^{(n)},{\overline{\mathcal{X}}}\right\rangle_{\mathcal{H}_0}.  \label{po14}
\end{equation}

To show the existence of at least one solution to \eqref{po13}-\eqref{po14}, we now suppose that $n$ is fixed and we take ${\overline{\mathcal{X}}}=\mathcal{X}^{(k)}$ for some $1\le k\le n$.
Then substituting the discretized functions \eqref{po12-1}-\eqref{po12-2} into \eqref{po13}-\eqref{po14}, we find a system of ordinary differential equations in the unknowns $A_k=A_k(t)$ on $\mathcal{X}^{(n)}$.
Also, we recall that
\begin{equation*}
\left\langle \mathbb{P}_n\mathcal{F}\left( \mathcal{X}^{(n)}\right),\mathcal{X}^{(k)} \right\rangle_{\mathcal{H}_0}=\left\langle \mathcal{F}\left( \mathcal{X}^{(n)}\right),\mathbb{P}_n\mathcal{X}^{(k)} \right\rangle_{\mathcal{H}_0}=\left\langle \mathcal{F}\left( \mathcal{X}^{(n)}\right),\mathcal{X}^{(k)} \right\rangle_{\mathcal{H}_0}.
\end{equation*}
Since $f\in C(\mathbb{R})$ and $g\in C^1(\mathbb{R})$, we may apply Cauchy's theorem for ODEs to find that there is $T_n\in (0,T)$ such that $A_k\in C^2((0,T_n))$, for $1\le k\le n$, and \eqref{po13} holds in the classical sense for all $t\in [0,T_n]$.
This shows the existence of at least one local solution to the approximate Problem \textbf{P}$_n$ and ends Step 2.

\underline{Step 3}. (Boundedness and continuation of approximate maximal
solutions.)
We begin by noticing that the {\em a priori} estimate \eqref{po1.8} holds for any approximate solution $\mathcal{X}^{(n)}$ of Problem \textbf{P}$_n$ on the interval $[0,T_n)$, where $T_n<T$.
Thanks to the boundedness of the projector $\mathbb{P}_n$, we infer
\begin{align}
& \|\partial_tU^{(n)}(t)\|^2_{\mathbb{X}^2} + \|U^{(n)}(t)\|^2_{\mathbb{V}^1} + 2(F(u^{(n)}(t)),1)_{L^2(\Omega)} + 2(G(u^{(n)}(t)),1)_{L^2(\Gamma)} \notag \\
& + 2\int_0^t \left( \omega\|\nabla^\theta \partial_tu^{(n)}(\tau)\|^2_{L^2(\Omega)} + \alpha\omega\|\nabla \partial_tu^{(n)}(\tau)\|^2_{L^2(\Gamma)} + \|\partial_tU^{(n)}(\tau)\|^2_{\mathbb{X}^2} \right)d\tau \notag \\
& \le \|\partial_tU(0)\|^2_{\mathbb{X}^2} + \|U(0)\|^2_{\mathbb{V}^1} + C\left( \|U(0)\|_{\mathbb{V}^1} + \|u(0)\|^{r_1}_{L^{r_1}(\Omega)} + \|u(0)\|^{r_2}_{L^{r_2}(\Gamma)} \right).  \label{po15}
\end{align}
Since the right-hand side of \eqref{po15} is independent of $n$ and $t$, every
approximate solution may be extended to the whole interval $[0,T]$, and
because $T>0$ is arbitrary, any approximate solution is a global one.
From above in Step 1, we also obtain the uniform bounds \eqref{po6}-\eqref{po10} for each approximate solution $\mathcal{X}^{(n)}$.
Thus,
\begin{align}
U^{(n)} & \in L^\infty(0,T;\mathbb{V}^1),  \label{po16} \\
\partial_tU^{(n)} & \in L^\infty(0,T;\mathbb{X}^2),  \label{po17} \\
\sqrt{\omega} \partial_tu^{(n)} & \in L^2(0,T;H^\theta(\Omega)), \label{po17.1} \\
\sqrt{\alpha\omega} \partial_tu^{(n)} & \in L^2(0,T;H^1(\Gamma)), \label{po17.3} \\
F(u^{(n)}) & \in L^\infty(0,T;L^1(\Omega)), \label{po18} \\
G(u^{(n)}) & \in L^\infty(0,T;L^1(\Gamma)). \label{po19}
\end{align}
This concludes Step 3.

\underline{Step 4}. (Convergence of approximate solutions.)
We begin this step by applying Alaoglu's theorem (cf. e.g. \cite[Theorem 6.64]{Renardy&Rogers04}) to the uniform bounds \eqref{po16}-\eqref{po19} to find that there is a subsequence of $\mathcal{X}^{(n)}$, generally not relabelled, and a function $\mathcal{X}=(u,\partial_tu,u_{\mid\Gamma},\partial_tu_{\mid\Gamma})$, obeying \eqref{po6}-\eqref{po10}, such that as $n\rightarrow \infty$,
\begin{align}
U^{(n)}\rightharpoonup U & \quad\text{weakly$-*$ in}\quad L^\infty(0,T;\mathbb{V}^1), \label{po20} \\
\partial_tU^{(n)}\rightharpoonup \partial_tU & \quad\text{weakly$-*$ in}\quad L^\infty(0,T;\mathbb{X}^2), \label{po21} \\
\sqrt{\omega}\partial_tu^{(n)}\rightharpoonup u & \quad\text{weakly in}\quad L^2(0,T;H^\theta(\Omega)), \label{po22} \\
\sqrt{\alpha\omega}\partial_tu^{(n)}\rightharpoonup u & \quad\text{weakly in}\quad L^2(0,T;H^1(\Gamma)), \label{po23} \\
\partial_tU^{(n)}\rightharpoonup \partial_tU & \quad\text{weakly in}\quad L^2(0,T;(\mathbb{V}^1)^*). \label{po24}
\end{align}
Using the above convergences \eqref{po20} and \eqref{po21}, as well as the fact that the injection $\mathbb{V}^1\hookrightarrow\mathbb{X}^2$ is compact, we draw upon the conclusion of the Aubin-Lions Lemma (cf. Lemma \ref{t:Lions}) to deduce the following embedding is compact
\begin{align}
\mathcal{W}:=\{ U\in L^2(0,T;\mathbb{V}^1) : \partial_tU\in L^2(0,T;\mathbb{X}^2)\} \hookrightarrow L^2(0,T;\mathbb{X}^2) \label{compact-u}
\end{align}
(see, e.g., \cite{Temam88}).
Thus,
\begin{eqnarray}
U^{(n)} \rightarrow U & \text{strongly in} & L^2(0,T;\mathbb{X}^2), \label{st-con-u}
\end{eqnarray}
and deduce that $U^{(n)}$ converges to $U$, almost everywhere in $\Omega\times (0,T)$.
The last strong convergence property is enough to pass to the limit in the
nonlinear terms since $f,g\in C^1(\mathbb{R})$ (see, e.g., \cite{Gal12-2,Gal&Warma10}).
Indeed, on account of standard arguments (cf. also \cite{CGGM10}) we have
\begin{eqnarray}
\mathbb{P}_n\mathcal{F}(\mathcal{X}^{(n)}) \rightharpoonup \mathcal{F}(\mathcal{X}) & \text{weakly in} & L^2(0,T;\mathcal{H}_0).  \label{po25}
\end{eqnarray}
At this point the convergence properties \eqref{po20}-\eqref{po25} are sufficient to pass to the limit as $n\rightarrow\infty$ in equation \eqref{po13}.
Additionally, we recover \eqref{var1} using standard density arguments.
The proof of the theorem is finished.
\end{proof}

\noindent {\bf{Concerning uniqueness.}}
A proof of the following conjecture is needed to show that the weak solutions to Problem {\textbf{P}} constructed above depend continuously on initial data, and hence, are unique.

\begin{conjecture}
Let $T>0$, $R>0$ and $\mathcal{X}_{01}=(U_{01},U_{11}),\mathcal{X}_{02}=(U_{02},U_{12})\in\mathcal{H}_0$ be such that $\|\mathcal{X}_{01}\|_{\mathcal{H}_0}\le R$ and $\|\mathcal{X}_{02}\|_{\mathcal{H}_0}\le R$.
Any two weak solutions, $\mathcal{X}^1(t)$ and $\mathcal{X}^2(t)$, to Problem {\textbf{P}} on $[0,T]$ corresponding to the initial data $\mathcal{X}_{01}$ and $\mathcal{X}_{02}$, respectively, satisfy for all $t\in[0,T]$,
\begin{equation}  \label{contdep}
\left\|\mathcal{X}^1(t)-\mathcal{X}^2(t)\right\|_{\mathcal{H}_0} \le e^{Q(R)t} \left\|\mathcal{X}_{01} - \mathcal{X}_{02}\right\|_{\mathcal{H}_0}.
\end{equation}
\end{conjecture}

In order to prove the conjecture, typically one needs to control products of the form
\[
(f(u^1)-f(u^2),\partial_t\bar u)_{L^2(\Omega)} \quad \text{and} \quad (g(u^1)-g(u^2),\partial_t\bar u)_{L^2(\Gamma)}
\]
where $u^1$ and $u^2$ are two weak solutions corresponding to (possibly the same) data $\mathcal{X}_{01}=(U_{01},U_{11})=(u_{01},\gamma_{01},u_{11},\gamma_{11})$ and $\mathcal{X}_{02}=(U_{02},U_{12})=(u_{02},\gamma_{02},u_{12},\gamma_{12})$.
A suitable control on $\|f(u^1)-f(u^2)\|_{L^q(\Omega)}$, for example, is readily available when we assume \eqref{growth} with $r_1\in[1,3]$ (cf. \cite[Lemma 2.6]{Graber-Shomberg-16})), but this is no longer valid when we assume $r_1\ge1$ is arbitrary.
In the later case it would be interesting to investigate whether a {\em generalized semiflow} in the sense of \cite{Ball00,Ball04} exists.
Under certain conditions, such generalized semiflows admit global attractors which have similar properties to their well-posed counterparts (cf. \cite{Hale88}).

redpajama_set_name: RedPajamaArXiv


text:
Q: floating layouts spilling over Yet another web guy trying to learn floating layouts after using HTML tables for years ...
I am dynamically creating floating divs using a jQuery call.
Each div is constrained by width, height, min-width, min-height, max-width, and max-height. For the most part they line up OK, but sometimes one of the divs will be too short or too long.
If I finagle the page (pop up the ctrl+shft+j in chrome or resize the window) the offending section zips back into place.
Is this a browser shortcoming? Am I dancing on the broken glass of undefined behavior?
function createChartsHTML( size ) {
    var htmlString = "";
    for( var i = 0; i < size; i++ ) {
            htmlString += '
';           // hammer this one into size!
            else
                htmlString += '" style="width: 45%; min-width: 45%; max-width: 45%; height: 300px; margin: 0 100; float: right">';      // bam !
            htmlString += '
';
            if ( (i+1) % 2 == 0 )
                htmlString += '
';
    }
    $('#charts_area').html( htmlString );
}

A: The max-width and min-width directives are not understood by every browser. As you want the elements to float to a fixes width anyways just define width.
The percentage you give is always to the parent element. So if you did not define the width of the parent (either the page or another div) the browser figures out the width by itself. Some browsers will do that based on the content.
You should also absolutely define the overflow-behavior as some browsers extend the divs if you did not tell them not to.
Last: don't mix up the float-direction as that complicates everything. Just line up elements with the same float-direction one after another (in your case 'right' to add each following element right to the preceding one) and they will be ordered that way.
Your style could looke like this:
style="width: 45%; min-width: 45%; max-width: 45%; height: 300px; margin: 0 100; float: right;


text:
Charles Louis Théodore van Malcote de Kessel (Ledeberg, 2 september 1834 - Ukkel, 7 juni 1907) was een Belgisch edelman en burgemeester.

Levensloop
Charles van Malcote was een zoon van Julien van Malcote en van Virginie Lateur. Hij werd burgemeester van Zulte en schreef verschillende historische werken.

In 1861 kreeg hij erkenning en, voor zoveel als nodig, verheffing in de Belgische adel. In 1891 kreeg hij vergunning om aan de familienaam de Kessel toe te voegen.

Hij trouwde in 1855 in Zaffelare met Prudence du Pont (1834-1893) en ze kregen twee zoons.

Georges de Malcote de Kessel
Georges de Malcote (1856-1905) trouwde in Pétange in 1888 met Madeleine Nothomb (1865-1935), dochter van Alphonse Nothomb en Wilhelmine Pescatore. Ze kregen twee zoons en een dochter, maar zonder verdere afstammelingen.

Albert de Malcote de Kessel
 Albert de Malcote (1858-1905) trouwde in Elsene in 1890 met zijn schoonzus, Lucie Nothomb (1870-1921).
Antoine de Malcote de Kessel (1902-1981) trouwde in 1923 in Zedelgem met Alida Timmerman (1900-1967). Met een beperkte afstamming tot heden.

Literatuur
 Généalogie Malcote, in: Annuaire de la noblesse de Belgique, Brussel, 1873.
 P. DU CHASTEL, Généalogie de la familie van Malcote de Kessel, 1877.
 Oscar COOMANS DE BRACHÈNE, État présent de la noblesse belge, Annuaire 1993, Brussel, 1993.

Burgemeester van Zulte
malcote, charles van

redpajama_set_name: RedPajamaWikipedia


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