text:
Plant Components, AerialPlant LeavesFruitSemenSpermatozoaThyroid Gland
WithaniaPhysalisDaturaDioscoreaceaeVanillaPeasLycium
Diseases 3
Thyroid NeoplasmsThyroid DiseasesFood Hypersensitivity
WithanolidesErgosterolPlant ExtractsPowdersSuperoxide DismutaseAntioxidantsAmino Acids, EssentialAmino AcidsFlavoring AgentsAmino Acids, Branched-ChainThyroid Hormones
Medicine, AyurvedicPhytotherapySperm CountSperm MotilityDrug Labeling
Psychiatry and Psychology 2
LibidoPeer Review, Research
Phenomena and Processes 6
Molecular StructureSperm CountEjaculationSperm MotilityEnergy IntakeEnergy Metabolism
BotanyChemistry, OrganicPeer Review, ResearchBacteriology
Anthropology, Education, Sociology and Social Phenomena 4
Medicine, AyurvedicPatents as TopicPeer Review, ResearchAccess to Information
Technology, Industry, Agriculture 6
PeasFruitFlavoring AgentsCommerceProduct LabelingDrug Labeling
Periodicals as TopicJournal Impact FactorPublishingBibliometricsPeer Review, ResearchAuthorshipAccess to InformationInformation ServicesInternet
Patents as TopicPeer Review, ResearchSafety
Geographicals 2
OmanIndia
Withanolide derivatives from the roots of Withania somnifera and their neurite outgrowth activities. (1/77)
Five new withanolide derivatives (1, 9-12) were isolated from the roots of Withania somnifera together with fourteen known compounds (2-8, 13-19). On the basis of spectroscopic and physiochemical evidence, compounds 1 and 9-12 were determined to be (20S,22R)-3 alpha,6 alpha-epoxy-4 beta,5 beta,27-trihydroxy-1-oxowitha-24-enolide (1), 27-O-beta-D-glucopyranosylpubesenolide 3-O-beta-D-glucopyranosyl (1-->6)-beta-D-glucopyranoside (withanoside VIII, 9), 27-O-beta-D-glucopyranosyl (1-->6)-beta-D-glucopyranosylpubesenolide 3-O-beta-D-glucopyranosyl (1-->6)-beta-D-glucopyranoside (withanoside IX, 10), 27-O-beta-D-glucopyranosylpubesenolide 3-O-beta-D-glucopyranoside (withanoside X, 11), and (20R,22R)-1 alpha,3 beta,20,27-tetrahydroxywitha-5,24-dienolide 3-O-beta-D-glucopyranoside (withanoside XI, 12). Of the isolated compounds, 1, withanolide A (2), (20S,22R)-4 beta,5 beta,6 alpha,27-tetrahydroxy-1-oxowitha-2,24-dienolide (6), withanoside IV (14), withanoside VI (15) and coagulin Q (16) showed significant neurite outgrowth activity at a concentration of 1 microM on a human neuroblastoma SH-SY5Y cell line. (+info)
Radiosensitizing effect of withaferin A combined with hyperthermia on mouse fibrosarcoma and melanoma. (2/77)
The effect of withaferin A, a plant withanolide, alone or in combination with acute and fractionated radiotherapy and/or hyperthermia, was tested on two mouse tumors, B16F1 melanoma and fibrosarcoma, grown in C57BL and Swiss albino mice, respectively. Tumors were exposed locally to 30 or 50 Gy gamma radiation as acute dose, or 5 fractions of 10 Gy. Withaferin A, 40 mg/kg, was injected intraperitoneally, 1h before acute irradiation, or 30 mg/kg before every 10 Gy fraction. Local hyperthermia, 43 degrees C for 30 min, followed acute RT or first fraction of 10 Gy. Withaferin A, radiation and hyperthermia, individually and in bimodality treatments, produced no complete response (CR) in melanoma. Some CR were seen in fibrosarcoma, which increased after bimodality treatments. Trimodality treatment synergistically increased CR to 37% in melanoma and to 64% in fibrosarcoma. Fractionated radiotherapy (10 Gy x 5) was more effective (25% CR) than acute dose of 50 Gy (0% CR) on melanoma, while there was no difference between the response of fibrosarcoma to the two regimens. Withaferin A with fractionated radiotherapy synergistically increased the CR of both tumors; hyperthermia further enhanced this effect. Utility of withaferin A in increasing the clinical response of radioresistant tumors to fractionated radiotherapy has to be explored. (+info)
Development of withaferin A analogs as probes of angiogenesis. (3/77)
The natural product withaferin A (WFA) is a potent angiogenesis inhibitor and it targets the ubiquitin-proteasome pathway in vascular endothelial cells. We generated a biotinylated affinity analog WFA-LC(2)B for use as a probe to study angiogenesis. WFA-LC(2)B inhibits angiogenic sprouting in vitro and it causes levels of ubiquitinated proteins to increase in tumor necrosis factor-alpha-treated human umbilical vein endothelial cells, confirming the retention of WFA's biological activity. We show that WFA-LC(2)B forms protein adducts in endothelial cells which are competed by free WFA in vivo. This WFA-LC(2)B analog will be useful to isolate the biological target of WFA. (+info)
Apoptosis is induced in leishmanial cells by a novel protein kinase inhibitor withaferin A and is facilitated by apoptotic topoisomerase I-DNA complex. (4/77)
Protein kinase C (PKC) is an important constituent of the signaling pathways involved in apoptosis. We report here that like staurosporine, withaferin A is a potent inhibitor of PKC. In Leishmania donovani, the inhibition of PKC by withaferin A causes depolarization of DeltaPsim and generates ROS inside cells. Loss of DeltaPsim leads to the release of cytochrome c into the cytosol and subsequently activates caspase-like proteases and oligonucleosomal DNA cleavage. Moreover, in treated cells, oxidative DNA lesions facilitate the stabilization of topoisomerase I-mediated cleavable complexes, which also contribute to DNA fragmentation. However, withaferin A and staurosporine cannot induce cleavable complex formation in vitro with recombinant topoisomerase I nor with nuclear extracts from control cells. Taken together, our results indicate that inhibition of PKC by withaferin A is a central event for the induction of apoptosis and that the stabilization of topoisomerase I-DNA complex is necessary to amplify apoptotic process. (+info)
Small molecule anti-angiogenic probes of the ubiquitin proteasome pathway: potential application to choroidal neovascularization. (5/77)
PURPOSE: To characterize the angiogenic and inflammatory responses of human choroidal endothelial cells (HCECs) to stimulators and inhibitors of the ubiquitin proteasome pathway (UPP). METHODS: The regulation of the UPP by the inhibitor withaferin A and its congener, withanolide D, two natural products derived from the medicinal plant Withania somnifera was assessed in the three-dimensional endothelial cell sprouting assay (3D-ECSA), by using HCEC- and human umbilical vein endothelial cell (HUVEC)-derived spheroids embedded in a collagen I matrix. Western blot analysis was used to investigate the effect of withanolides on IkappaB-alpha, polyubiquitination, and heme oxygenase (HO)-1 regulation in HCEC and HUVEC cultures. RESULTS: HCECs, like HUVECs, responded to fibroblast growth factor-2, vascular endothelial growth factor, and tumor necrosis factor (TNF)-alpha stimulation and sprouted vessel-like structures in collagen I matrix. However, HCECs were slower to generate these sprouting vessels, when compared with HUVECs. The extent of inhibition of endothelial cell sprouting in 3D matrix, the blockade of TNF-alpha-induced IkappaB-alpha degradation, levels of global polyubiquitinated proteins, and induced production of HO-1 in response to treatment by the withanolides in cultured endothelial cells was similarly regulated between HCECs and HUVECs. CONCLUSIONS: HCECs share with HUVECs a similar response to UPP inhibitors, suggesting that this well-conserved pathway that regulates angioinflammatory mechanisms could be exploited for drug-targeting in the development of novel agents for CNV treatment. (+info)
The tumor proteasome is a primary target for the natural anticancer compound Withaferin A isolated from "Indian winter cherry". (6/77)
Withaferin A (WA) is a steroidal lactone purified from medicinal plant "Indian Winter Cherry" that is widely researched for its variety of properties, including antitumor effects. However, the primary molecular target of WA is unknown. By chemical structure analysis, we hypothesized that Withaferin A might be a natural proteasome inhibitor. Computational modeling studies consistently predict that C1 and C24 of WA are highly susceptible toward a nucleophilic attack by the hydroxyl group of N-terminal threonine of the proteasomal chymotrypsin subunit beta5. Furthermore, WA potently inhibits the chymotrypsin-like activity of a purified rabbit 20S proteasome (IC50=4.5 microM) and 26S proteasome in human prostate cancer cultures (at 5-10 microM) and xenografts (4-8 mg/kg/day). Inhibition of prostate tumor cellular proteasome activity in cultures and in vivo by WA results in accumulation of ubiquitinated proteins and three proteasome target proteins (Bax, p27, and IkappaB-alpha) accompanied by androgen receptor protein suppression (in androgen-dependent LNCaP cells) and apoptosis induction. Treatment of WA under conditions of the aromatic ketone reduction, or reduced form of Celastrol, had significantly decreased the proteasome-inhibitory and apoptosis-inducing activities. Treatment of human prostate PC-3 xenografts with WA for 24 days resulted in 70% inhibition of tumor growth in nude mice, associated with 56% inhibition of the tumor tissue proteasomal chymotrypsinlike activity. Our results demonstrate that the tumor proteasome beta5 subunit is the primary target of WA, and inhibition of the proteasomal chymotrypsin-like activity by WA in vivo is responsible for, or contributes to, the antitumor effect of this ancient medicinal compound. (+info)
Withaferin a strongly elicits IkappaB kinase beta hyperphosphorylation concomitant with potent inhibition of its kinase activity. (7/77)
The transcription factor NFkappaB plays a critical role in normal and pathophysiological immune responses. Therefore, NFkappaB and the signaling pathways that regulate its activation have become a major focus of drug development programs. Withania somnifera (WS) is a medicinal plant that is widely used in Palestine for the treatment of various inflammatory disorders. In this study we show that the leave extract of WS, as well as its major constituent withaferin A (WA), potently inhibits NFkappaB activation by preventing the tumor necrosis factor-induced activation of IkappaB kinase beta via a thioalkylation-sensitive redox mechanism, whereas other WS-derived steroidal lactones, such as withanolide A and 12-deoxywithastramonolide, are far less effective. To our knowledge, this is the first communication of IkappaB kinase beta inhibition by a plant-derived inhibitor, coinciding with MEK1/ERK-dependent Ser-181 hyperphosphorylation. This prevents IkappaB phosphorylation and degradation, which subsequently blocks NFkappaB translocation, NFkappaB/DNA binding, and gene transcription. Taken together, our results indicate that pure WA or WA-enriched WS extracts can be considered as a novel class of NFkappaB inhibitors, which hold promise as novel anti-inflammatory agents for treatment of various inflammatory disorders and/or cancer. (+info)
Par-4-dependent apoptosis by the dietary compound withaferin A in prostate cancer cells. (8/77)
Deletion or mutation of the androgen receptor (AR) renders prostate tumors refractory to apoptosis by androgen ablation, the mainstay of prostate cancer therapy. To identify novel therapeutics that can induce apoptosis regardless of the AR status of prostate cancer cells, we screened dietary herbal compounds using a reporter assay for the prostate apoptosis response-4 (Par-4) gene, which induces p53- and PTEN-independent and cancer-selective apoptosis. One of the compounds, withaferin A (WA), a major constituent of the dietary compound Withania somnifera, induced Par-4-dependent apoptosis in androgen-refractory prostate cancer cells and regression of PC-3 xenografts in nude mice. Interestingly, restoration of wild-type AR in PC-3 (AR negative) cells abrogated both Par-4 induction and apoptosis by WA. Individually, WA and anti-androgens induced neither Par-4 nor apoptosis in androgen-responsive prostate cancer cells, yet in combination, WA and anti-androgen synergistically induced Par-4 and apoptosis in androgen-responsive prostate cancer cells. Thus, when judiciously combined with anti-androgens, WA inhibits survival of both androgen-responsive and androgen-refractory prostate cancer cells by a Par-4-dependent mechanism. As Par-4 up-regulation induces apoptosis in most tumor cells, our findings can be extended to high-throughput screens to identify synergistic combinations for both therapy-sensitive and therapy-resistant cancers. (+info)


In Commissioner of Taxation v Bamford, the High Court considered a number of long-debated issues in relation to the taxation of trusts. In this case, a taxpayer sought to reclassify the capital gain from the sale of a capital asset as income of the trust rather than capital pursuant to the terms of the Trust Deed. This was objected to by the Commissioner of Taxation on the grounds that the amount of capital gain did not fall within the definition of income according to ordinary concepts under the Income Tax Assessment Act 1936. Despite the Commissioner's objection, the Court allowed the reclassification of the amount as income confirming in its decision that the income of a trust is to be determined in accordance with the terms of the trust deed. The Court in its decision also considered the amount that a taxpayer receiving distributions from a trust should include in their assessable income. Here, the Court decided that if a beneficiary of a trust is presently entitled to a percentage of the income of that trust, then that beneficiary must include that percentage of the trust's taxable income in that beneficiary's own assessable income.
In light of the Court's decision , it is important that Trust Deeds and Testamentary Trust Wills are reviewed to ensure that the definition of income in the documents are appropriate given the objectives of the trust. The decision also means that it may be beneficial to ensure that the terms of the trust deed/testamentary trust Will gives trustees the discretion to alter the definition of income of the trust fund as well as the discretion to decide if amounts should be recognised as income or capital in order to provide maximum flexibility in calculating trust income so that tax savings may be achieved.
If you have a Trust Deed or Testamentary Trust Will which needs to be amended or reviewed, please contact Owen Hodge Lawyers.


text:
* write comments for minimum deletion and the helper function reorder
* track size, so we can not delete from an empty heap
* implement different strategies for arbitrary deletion and filling
* play around with strictness
* write Soft Heap selection algorithm
* write tests
* write examples
* make meld nicer using linear types

redpajama_set_name: RedPajamaGithub


text:
Q: Are recently accessed Realm objects kept in memory? Suppose I fetch a list of Realm objects and access all the data from the results (causing the data to be loaded into memory). Next, with the reference to the list of objects still around, I fetch one of those objects directly by its primary key. Am I correct to assume that, in this case, the object will be returned without having to hit the disk? What if I no longer had a reference to that original list of results? Might it still be in memory?
Just curious to understand how objects are cached internally by Realm, so I don't unnecessarily try to optimize things in my code (e.g. holding onto objects that I think I'll use again soon) if it's not needed.

A: Realm memory-maps the file, and only accesses it when you actually read the properties on objects. This means that recent-accessed data will still be in memory until the OS has to discard it to free up that RAM for something else, and there is no difference in terms of disk i/o from obtaining a new object from Realm and holding on to an existing object.

redpajama_set_name: RedPajamaStackExchange


text:
Q: php - prepared statement not executing So I have this php that I am using to validate a form that would be submitted. it takes a Name and ID that the User generates, and checks it against a database.
error_reporting(E_ALL);
require 'connection.php';
$conn = mysqli_connect($servername, $username, $password) or die("Error connecting to database: ".mysqli_error());
mysqli_select_db($conn,$database) or die(mysqli_error($conn));

$PName=$_POST['Name'];
$PID=$_POST['ID'];
$query = $PID;

$min_length = 0;
if(strlen($query) >= $min_length){
    $query = htmlspecialchars($query);
    $query = mysqli_real_escape_string($conn,$query);
    $raw_results = mysqli_query($conn,"SELECT DISTINCT `Person_ID` FROM `Person_DT` WHERE `Person_ID` LIKE '%".$query."%'")" or die(mysqli_error($conn));

    if(mysqli_num_rows($raw_results) > 0){
    // this section was omitted to make the post shorter
    // i.e. this reissues the previous form, but has all the previous information the user submitted
    }
else{// this is the bit where I am having trouble
$addstmt=mysqli_stmt_prepare("INSERT INTO Person_dt (PName,PID) VALUES (?,?");
mysqli_stmt_bind_param('ss',$PName,$PID);
mysqli_stmt_execute($addstmt);
printf("%d Row Inserted.\n",$addstmt->affected_rows);
}
    }
else{ // if query length is less than minimum
        echo "Minimum length is ".$min_length;
}

So on running this code, I return 0 rows inserted. I also return no errors; Is there something wrong with my syntax? should I change to another method?

A: I've not demoed the code, but looking at your example, I see a few typo-class issues that may be effecting your output. I'm noticing, at least in your posted code, that you have an extra quotation mark in line 14 after your mysqli_query closing parenthesis. Similarly, you don't close the parenthesis inside your prepare statement on line 21.
So, try again in line 21 with $addstmt=mysqli_stmt_prepare("INSERT INTO Person_dt (PName,PID) VALUES (?,?)"); -- since the closing parenthesis here is part of the statement.
If you're looking for alternative methods, the object-oriented interface for mysqli() looks a little cleaner (see The PHP mysqli docs). Personally, I prefer using PDO (docs here).
Good luck! I hope the solution is as simple as quote control.


The Virtual Assistants Association of South Africa gives you the opportunity to become part of a virtual team, and the opportunity to be recognised as a professional in your field by becoming an accredited member of the Association and adhering to our CODE OF CONDUCT.
Because VAASA is the only organisation in existence for VAs in South Africa, we receive many RFPs from entrepreneurs and business professionals looking for help from people like YOU! These requests are delivered to members via the email discussion list.
Sure there are many VA organisations you can join, and even though VAASA is the only one you really need to belong to, we don't discourage you from joining any of the others out there, but please remember that this one specifically relates to the South African market.
A free private Facebook page where you can feel comfortable reaching out to fellow South Africans for advice, sharing of ideas, etc.
When someone in need of a VA comes to the VAASA website, the first thing they do is browse our member directory. Businesses are searchable by region, and you're given lots of room to list detailed information about the services you provide.
Regularly, we host educational webinars on cutting edge technologies, business practices and industry resources. VAASA members are invited to attend these events for FREE!!
Attend information webinars / zoom sessions hosted by guest speakers in the industry and other business experts.
Access to our Success Circles – monthly mind-storm sessions, attended by VAs who are interested in taking their business to the next level.
Access and discounted rates towards the Success Partner Programs – Mentoring, Coaching, Consulting all intended to support you with your business in a one-on-one environment with successful, established VAs.
Everything you just finished reading about – is going to set you back R500.00 per year.
That's right – only R500.00 per year! Not per month, or bi-weekly. VAASA membership is yours for R500.00 per year.
Exclusive access to our RFP system ALONE is worth more than a yearly payment of R500.00 – retain one client and your membership has paid for itself!
This is one of the most cost effective ways you can possibly market your VA business. For that R500.00, your business is listed in our directory, you will be networking with VAs all over South Africa and timely notification of new RFPs are provided on the VAASA Facebook page. You really can't go wrong with this tiny monetary investment.
If you're still exploring the idea of becoming a VA we have a special 3 month membership that allows you to see what VAASA has to offer. There are many benefits of full membership that won't be available to you, like access to the latest RFPs, but you will get an opportunity to appreciate how important it is to become fully involved with VAASA.
We would love to have you as VAASA's newest member.
All you have to do to gain access to all member-only areas of VAASA is Join Us.
As a Virtual Assistant, this will potentially be the best and most important R500.00 investment you'll ever make. (That works out to just over R40.00 a month!).
A virtual assistant, or VA, is a small business owner who offers many different services in the administrative, technical, or creative areas. A VA has spent a lot of time developing expertise in their chosen area of specialization. Services are provided remotely and communication is done via email, phone, mail, or instant messaging. Work product is delivered electronically or via the various delivery methods available in South Africa.
A VA can help you with administrative tasks (e.g., proofreading a proposal, preparing documents, bulk mailings), technical tasks (e.g., website development), or creative tasks (e.g., designing marketing materials, desktop publishing, social media marketing).
VAs offer different services so it is important that you choose someone who has expertise in the type of work you need to have done.
VAs work with any business that wants to cut overhead costs, increase productivity, and gain more time in each day to focus on their core business objectives.
The important thing is good communication between a client and a VA.
A VA will start by discussing what work a client requires to have done and how the client prefers to communicate – email, phone, instant messaging, etc. A client will be given periodic updates to keep you informed about the status of the work the VA is are doing for the client.
From the clients end, a VA would ask that the client provide timely communication of any changes to the client's work needs, such as changed deadlines, tasks, etc.. Once the work is completed, the VA would send it to the client in whatever way has been chosen.
How does a client choose a Virtual Assistant?
The first step is to decide what work the client needs assistance with. From there, the client can determine what type of services they are looking for. The client should also determine their other requirements, such as an estimate of the number of hours of help that is needed per day, per week, or per month.
You can then view the list of VAs listed on our Member Directory, seeing who is able to provide those services.
Couldn't a client just use a temporary employee?
You could but you would likely not get the same level of professionalism, collaboration, and dedication that you would with a VA. A temporary employee may be available one day or one week but the next time you need them, they may no longer be available, causing you to have to start all over again with someone new. A VA is there when you need them and has an interest in making sure you succeed.


text:
COVID-19 Griffin-Spaulding County Schools Policy Teaching in Atlanta
Griffin-Spalding Schools: masks optional, continuing pandemic measures 2021-22 school year
Jessica Orozco July 19, 2021 July 20, 2021 0 Comments Griffin-Spalding County Schools, Masks
Griffin-Spalding County Schools will not require masks for the 2021-22 school year, the district announced Monday.
The district will continue COVID-19 mitigation efforts including three-foot physical distancing and sanitization of large areas, according to the announcement. Many schools have been outfitted with filtration systems in HVAC units and touchless water fountains with water bottle filling stations to mitigate the spread of viruses. These installations are ongoing.
Masks will not be required in any capacity, though students can choose to wear them, Laura Beaber, communications specialist for the district, said.
The decision comes after an administrative retreat with 95 Griffin-Spalding administrators gathering at Griffin High School, the announcement states.
The school year begins Aug. 4, according to the district's website, less than a week after a district-wide open house.
"We look forward to reconnecting with our students and families," Superintendent Keith Simmons stated. "I encourage families to participate in the open house activities as we will be providing resources and information to ensure a smooth start to the school year."
Griffin-Spalding County Schools welcoming new administrators
Henry County elementary schools welcomes new principals
Jessica Orozco
Jessica Orozco is an intern reporter for Education in Atlanta.
← Henry County Schools to hold drive-thru backpack giveaway
Georgia Department of Education Releases Reward Schools →
Gwinnett and Cobb County Schools Transition to Four Day Work Weeks
Staff Writer April 10, 2020 April 11, 2020 0
Griffin-Spalding Schools to consult community on spending American Rescue Plan funds
Jessica Orozco July 6, 2021 July 9, 2021 0

redpajama_set_name: RedPajamaCommonCrawl


Showing 1 - 15 of 15 articles tagged "Drew Barrymore"
Drew Barrymore and Timothy Olyphant announced the renewal in a video posted to the show's Twitter account.
"I will never do my own stunts again," says the actress.
Drew Barrymore: Santa Clarita Diet brought me back to acting when I was in a "dark place"
Drew Barrymore recalls flashing David Letterman: "It's like a distant memory that doesn't seem like me -- but it is me"
The Drew Barrymore-Timothy Olyphant horror-comedy series returns on March 23.
Drew Barrymore will serve as producer of Black Rose Anthology.


text:
Conte invites board to make judgement on his Chelsea reign
Chelsea have put their 2018 struggles behind them with a fine run of form but uncertainty over the future of boss Antonio Conte remains.
Chelsea head coach Antonio Conte - Getty Images
Antonio Conte insists he remains committed to his contract at Chelsea but feels the final say on his Stamford Bridge tenure must come from those above him.
Despite leading the Blues to a stunning Premier League title triumph in his first season at the helm last year, widespread reports of Conte's unhappiness over Chelsea's transfer policy suggested he could be the latest coach to head through the club's revolving door.
The Blues opened with a shock 3-2 defeat at home to Burnley and their title defence crumbled entirely in the new year, all the while with Conte making thinly veiled swipes at the pre-season recruitment drive.
Nevertheless, Chelsea entertain Huddersfield Town on Wednesday retaining a chance of sneaking into the top four and are on a run of five consecutive wins in all competitions, with the FA Cup final against Manchester United still to come.
It is a streak that casts a new light on talk linking Conte's countrymen Maurizio Sarri and Massimiliano Allegri with his post and the 48-year-old pounded a familiar drum on Tuesday.
"I think and I'm sure that I am Chelsea's coach and my focus is only on this," he told reporters.
"As I said before it is very difficult to comment on any speculations because there have been speculations around me from the start of the season, after the first game against Burnley and I think you understood that this speculation is not important for me.
"My focus is only to do my work in the best possible way with my players. There are only two weeks and this season will finish. Then you will know if there is a different situation or if you see me again next season and we start again."
Conte signed a fresh contract last July that did not extend the length of his initial terms, running until the end of 2018-19.
Asked whether he envisaged a long-term stay at Stamford Bridge, the former Italy and Juventus boss stated any such preference on his part would be meaningless without the club being in agreement.
"I think that in this decision there are always two parties to take the decision, not only one side and I don't know," he said.
"I must be honest to tell you that this is not my task to tell if I did a good job or if I have a good team for next season or if we are building something important.
"There is the club and people able to make judgements about this season and also the situation because we have worked together on this project and I think I am the last person to make a judgement on this season."
Conte added: "For sure the reality in this moment is I have a contract I signed with Chelsea to be committed for another year.
"Now my present is with Chelsea. My focus is only to Chelsea. The future you never know what will happen because our job is very difficult. You have to be ready for everything."


This New Zealand Firm Trialed A Four-Day Work Week: Did It Work?
03 Dec This New Zealand Firm Trialed A Four-Day Work Week: Did It Work?
There are many modern-day business and workplace strategies, with one of the most commonly talked about this year being shorter work weeks. However, it does not just refer losing working hours; it refers to the strategic approach of having less work days in the week, with no change in salary or wages and producing the same, if not more, amount of productivity than within in a normal five-day working week.
According to The Guardian, a New Zealand based firm has trialed and tested the workplace strategy of changing their employees work week from the traditional five days, to a shorter four days, whilst their pay and daily eight hours, remains the same.
"Academics who studied the trial found staff at the firm's offices around the country reported lower stress levels, higher levels of job satisfaction and an improved sense of work-life balance," (Ainge Roy, 2018).
The flexible four-day work week strategy offered employees to opt-in or opt-out, depending on work preferences, starting times, and childcare commitments. Not only this, the well-thought-out strategy was cleared by, and abided to, the New Zealand employment law and conditions requirements.
The trial itself came about after the firm's founder observed how much pressure some staff were under, and their unfocused working patterns, as they tried to manage their personal and working life. From there, the founder of the firm developed the trial to provide the staff with the option of having an extra day within the week for their personal lives, in which it was found to be a great success as it boosted staff morale, productivity and efficiency.
Perhaps this is a lesson to be learnt by many more organisations whom are noticing a decrease in staff attention and an increase in presenteeism..


Bionorm natural products provides a wide range of pharmaceutical products which includes hawthorn extract. It is a traditionally used plant and it contains active substances acting as free radical scavengers. Contact us for more information.
Your message will be sent directly to Bionorm Natural Products and will be answered at their discretion.


text:
Q: How to make DNS record differ inside and oustide the network I am trying to allow access to our NAS from inside and outside our SOHO network. I have setup a public IP with DDNS, setup the NAT so it let through :80 and :443, and I can access my NAS from outside the network.
However, my router (a Huawei H122-373) doesn't support NAT hairpinning, so all requests made from within the network are dropped. I wanted to try this answer by setting up a DNS in my LAN. However, it won't work with non-recursive DNS clients as they will first ask the "reals" DNS providers.
Here is my setup :
My DNS provider:
my-office.company.com IN A 88.99.11.22 # DDNS to my public IP

My LAN DNS
my-office.company.com IN A 192.168.1.2 # My NAS

I tried setting up NS records from my DNS provide toward my LAN DNS and redirect :53 to my LAN DNS, but again, the clients within the network can't contact the LAN DNS.

A:
However, it won't work with non-recursive DNS clients as they will first ask the "reals" DNS providers.

The missing part is that your DNS server is supposed to be the "real" DNS provider for LAN devices.
It is already common for LANs to have an internal DNS resolver (e.g. embedded in the router), with its address being announced to devices via DHCP instead of directly announcing an external resolver – your router might be doing that already. (Some apps will nowadays bypass it and deliberately contact an external DNS resolver, often via TLS or DoH, but in most cases the resolver hosted within the LAN is as "real" as any other.)
So in your case, you're supposed to change the router's DHCP server configuration to start advertising the custom DNS server's IP address instead of whatever it currently has.


The azaleas are having their moment. Don't miss this year's blooms.
Spend over $49, receive free 8" shears. Great for pruning, trimming and shaping.
Enjoy the seasonal rhythms of a maple. Foliage without a fuss.
Tools from our own collection, now available for your use at home.
Serious specimens for serious bonsai enthusiasts.
Brussel's Bonsai has been growing premium quality bonsai since 1975 and is now the largest grower of classic and specimen bonsai trees in America. Here's what that means for you.
Choice: find the best selection of tree species, indoor/outdoor types and prices right here.
Smooth shipping: our custom designed shipping cartons ensure healthy tree delivery anywhere in the continental U.S.


text:
\section{Introduction}

Common wisdom in emulsion science tells that, in order to kinetically stabilize an emulsion of water and oil,
say, surfactants are needed in order to decrease the interfacial tension thereby decreasing the thermodynamic
force causing droplet coalescence \cite{Safran2003}.
This picture has been upset by Leunissen et al.\ who showed experimentally that in certain additive-free
water-oil mixtures micron-sized water droplets in oil may be stabilized electrostatically by absorbing ions
present in the system \cite{Leunissen2007a,Leunissen2007b}.
Several aspects of these experiments such as the proposed charging of the water droplets due to an unequal
partitioning \cite{Zwanikken2007,Bier2008,Zwanikken2008} and the formation of a colloidal crystal of water droplets
\cite{deGraaf2008} can be understood theoretically within a simple Poisson-Boltzmann model.
However, the rather unimodal size distribution of the water droplets in the above-mentioned experiments has not
been explained so far.
A similar observation has been made by Sacanna et al.\ who found experimental indications of the existence of
thermodynamically favored droplet radii in certain emulsions stabilized by nano-sized colloids \cite{Sacanna2007}.
A thermodynamically favored droplet radius requires a radius dependent water-oil interfacial tension because
otherwise the global minimum of the free energy would be attained for one single macroscopic drop.
One is thereby led to the problem of analyzing the liquid-liquid interfacial tension as a function of the droplet
radius.

The study of the curvature dependence of liquid-vapor surface tensions has been pioneered
by Gibbs \cite{Gibbs1961}, Tolman \cite{Tolman1949}, and Kirkwood and Buff \cite{Kirkwood1949}.
Tolman introduced a low-curvature expansion of the form
$\gamma(a)/\gamma(\infty)\simeq 1/(1+2\delta/a)\simeq 1-2\delta/a$ where $a$ denotes the radius of curvature,
$\gamma(a)$ is the surface tension of the curved surface, and $\gamma(\infty)$ is its planar value.
The parameter $\delta$, which has the dimension of length, is called the \emph{Tolman length} and it can be
identified with the spatial distance between the Gibbs dividing surface and the surface of tension.
In the last decades the concept of a curvature dependent liquid-vapor surface tension has been taken up within
various studies on critical phenomena \cite{Fisher1984}, interface elasticity \cite{Blokhuis1992}, and nucleation
\cite{Talanquer1995,Barrett1999}.

However, whereas in all these investigations the droplet and the surrounding bulk were composed of the same
substance, albeit in different phases, here a mixture of two different liquids and ions is studied.
Moreover, only the excess interfacial tension due to the electrolyte is of interest here while the two liquids
forming droplet and bulk merely act as external fields onto the ions.

The present investigation is carried out within the spherical version of the model studied in Ref.~\cite{Bier2008}
(Sec.~\ref{sec:model}).
As in Ref.~\cite{Bier2008} linearization of the Poisson-Boltzmann equation offers the possibility of closed
analytical expressions for the interfacial tension (Sec.~\ref{sec:linear}).
In Sec.~\ref{sec:discuss} the approximative analytical expressions for the interfacial tension will be shown
to at least qualitatively, in many realistic cases even quantitatively, agree with the numerical results
obtained within the full, non-linear theory.
The main conclusion will be that it is precisely the electrostatic contribution to the interfacial tension that
brings about a curvature dependence, which, however, is usually insignificant to serve as an explanation for
unimodal radius distributions in the emulsions by Leunissen et al.\ mentioned above (Sec.~\ref{sec:conclusions}).
On the other hand, the curvature-dependent electrostatic contribution to the interfacial tension can be
expected to increase considerably in magnitude if highly charged colloids instead of monovalent ions are present.
Under these conditions, however, the approximations made in the present work are not a priori justified, and it
is left for future studies to investigate the influence of valency on the qualitative picture to be drawn
here, which corresponds to the low-valency limit.

\section{\label{sec:model}Model and Formalism}

In the following dimensionless quantities are expessed in units of the thermal energy $k_\m{B}T$, the elementary
charge $e$, and the vacuum Bjerrum length $\displaystyle\ell=\frac{e^2}{4\pi\varepsilon_\m{vac}k_\m{B}T}$ with the
permeability of the vacuum $\varepsilon_\m{vac}$.
Dimensionful quantities are denoted by the same symbol as the corresponding dimensionless quantities.

Consider a liquid spherical droplet of radius $a$ and relative dielectric constant $\varepsilon_\m{d}$ surrounded
by bulk liquid of relative dielectric constant $\varepsilon_\m{b}$.
Due to the spherical symmetry of the setting the only relevant positional variable is the distance
$r\in[0,\infty)$ from the droplet center.
Monovalent cations ($+$ ions) and anions ($-$ ions) are distributed in both liquids.
The difference in solvation free energy of a $\pm$ ion in the droplet with respect to the
bulk liquid is denoted by $f_\pm$, which, within the Born approximation \cite{Born1920}, can be estimated by
$\displaystyle f_\pm=\frac{1}{2a_\pm}\Big(\frac{1}{\varepsilon_\m{d}}-\frac{1}{\varepsilon_\m{b}}\Big)$ with the ion radius
$a_\pm$.
As in Ref.~\cite{Bier2008} all interfacial effects due to, e.g., smooth interfaces, finite ion size, van der
Waals forces, and image charges, which are short ranged as compared to the electrostatic potential, are
accounted for by introducing solvent-induced ion potentials $V_\pm(r)=f_\pm\Theta(a+s-r)$ with $\Theta$ the
Heaviside function.
Note that the parameter $s$, which describes the radial offset of the solvent induced ion potentials $V_\pm$
with respect to the dielectric interface at $r=a$, can be positive and negative, depending on the net effect
of the above-mentioned interfacial effects.
More detailed representations of the interfacial effects are possible at the expense of more phenomenological
parameters \cite{Onuki2006,Onuki2008a,Onuki2008b}, but for the sake of convenience and because handy analytical
expressions are desired the present most simple choice is made here.

A convenient approach to calculate the interfacial tension of the system under consideration is to first
determine the equilibrium ion number density profiles $\varrho_\pm$ by means of density functional theory
\cite{Evans1979,Evans1989,Evans1991} and then to infer the interfacial tension from inserting these equilibrium
profiles into the grand potential density functional.
Poisson-Boltzmann theory corresponds to the mean-field grand potential density functional
\begin{eqnarray}
   \Omega[\varrho_\pm]
   & = &
   4\pi\sum_{\alpha=\pm}\Int{0}{\infty}{r} r^2\varrho_\alpha(r)\Big(\ln(\varrho_\alpha(r))-1-\mu_\alpha
   \nonumber\\
   &&
   \phantom{4\pi\sum_{\alpha=\pm}\Int{0}{\infty}{r}}
   +V_\alpha(r)+\frac{\alpha}{2}\phi(r,[\varrho_\pm])\Big)
   \label{eq:df}
\end{eqnarray}
with $\mu_\alpha$ the chemical potential of $\alpha$ ions and $\phi(r,[\varrho_\pm])$ the electrostatic potential
functional at radius $r$, which fulfills the Poisson equation
\begin{equation}
   \frac{1}{r^2}\Big(\varepsilon(r)r^2\phi'(r,[\varrho_\pm])\Big)'
   =
   -4\pi\sum_{\alpha=\pm}\alpha\varrho_\alpha(r)
   \label{eq:pe}
\end{equation}
subject to the boundary conditions $\phi'(r=0)=0$ and $\phi(r=\infty)=0$, where a prime denotes a derivative with
respect to $r$ and $\varepsilon(r):=\varepsilon_\m{d}\Theta(a-r)+\varepsilon_\m{b}\Theta(r-a)$.
The electrostatic potential is a continuous function of $r$, and at the dielectric interface ($r=a$) the radial
component of the dielectric displacement is continuous:
$\varepsilon_\m{d}\phi'(r\nearrow a)=\varepsilon_\m{b}\phi'(r\searrow a)$.

Minimizing the density functional in Eq.~\Eq{df} gives rise to the Euler-Lagrange equations
\begin{equation}
   \varrho_\alpha(r) = \exp(\mu_\alpha - V_\alpha(r) - \alpha\phi(r,[\varrho_\pm])).
   \label{eq:ele1}
\end{equation}
Due to the local charge neutrality in the bulk liquid far away from the droplet ($\varrho_+(r=\infty)=\varrho_-(r=\infty)$)
one infers $\mu_+=\mu_-=:\mu$.
Upon introducing the reference densities $\varrho^\m{ref}_\m{b}:=\exp(\mu)$ and
$\varrho^\m{ref}_\m{d}:=\varrho^\m{ref}_\m{b}\exp(-(f_++f_-)/2)$, the sharp-kink reference density profile
$\varrho^\m{ref}(r,x):=\varrho^\m{ref}_\m{b}\Theta(r-x) + \varrho^\m{ref}_\m{d}\Theta(x-r)$ with the discontinuity located
at radius $x$, and the shifted electrostatic potential $\psi(r):=\phi(r)-\phi_D\Theta(a+s-r)$ with the Donnan
potential $\phi_D:=(f_--f_+)/2$, the Euler-Lagrange equation~\Eq{ele1} can be rewritten as
\begin{equation}
   \varrho_\alpha(r) = \varrho^\m{ref}(r,a+s)\exp(-\alpha\psi(r)).
   \label{eq:ele2}
\end{equation}

Inserting Eq.~\Eq{ele2} into the Poisson equation~\Eq{pe} leads to the Poisson-Boltzmann equation
\begin{equation}
   \psi''(r) = \kappa(r)^2\sinh(\psi(r)), \qquad r\not=a,a+s
   \label{eq:pbe}
\end{equation}
with $\kappa(r):=\sqrt{8\pi\varrho^\m{ref}(r,a+s)/\varepsilon(r)}$ the Debye screening factor.
Given a solution $\psi$, the interfacial tension with respect to the dielectric interface at $r=a$ in excess to
the pure, salt-free liquid-liquid interfacial tension between the droplet and the bulk liquid is, after
inserting Eq.~\Eq{ele2} into Eq.~\Eq{df}, determined by
\begin{eqnarray}
   \gamma^\m{ex}
   &\!\! = \!\!&
   \frac{\Omega[\varrho_\pm] - \Omega[\varrho^\m{ref}(\cdot,a)]}{4\pi a^2}
   \nonumber\\
   &\!\! = \!\!&
   -\frac{1}{a^2}\sum_{\alpha=\pm}\Int{0}{\infty}{r}
   r^2\Big(\varrho_\alpha(r) - \varrho^\m{ref}(r,a)
   \nonumber\\
   & &
   \phantom{-\frac{1}{a^2}\sum_{\alpha=\pm}\Int{0}{\infty}{r}}
   +\frac{\alpha}{2}\varrho_\alpha(r)\phi(r,[\varrho_\pm])\Big).
   \label{eq:gamma1}
\end{eqnarray}
As solutions of the non-linear Poisson-Boltzmann equation~\Eq{pbe} in the spherical geometry can be obtained only
numerically, the same holds for the excess interfacial tension $\gamma^\m{ex}$ in Eq.~\Eq{gamma1}.
However, upon linearizing the Euler-Lagrange equation~\Eq{ele2} and the Poisson-Boltzmann equation~\Eq{pbe} one
obtains analytical expressions for the excess interfacial tension $\gamma^\m{ex}$, which will be derived in the
next section.

\section{\label{sec:linear}Linearized theory}

For a sufficiently small Donnan potential $|\phi_D| < 1$ the Euler-Lagrange equations~\Eq{ele2} and the
Poisson-Boltzmann equation~\Eq{pbe} can be linearized leading to
\begin{equation}
   \varrho_\alpha(r) = \varrho^\m{ref}(r,a+s)(1-\alpha\psi(r))
\end{equation}
and
\begin{equation}
   \psi''(r) = \kappa(r)^2\psi(r), \qquad r\not=a,a+s,
   \label{eq:lpbe}
\end{equation}
respectively.
Inserting both expressions into Eq.~\Eq{gamma1} one obtains
\begin{eqnarray}
   \gamma^\m{ex}
   & = &
   2s(\varrho^\m{ref}_b - \varrho^\m{ref}_d)\Big(1+\frac{s}{a}+\frac{1}{3}\Big(\frac{s}{a}\Big)^2\Big)
   \nonumber\\
   & &
   - \frac{\phi_D}{2}\Big(1+\frac{s}{a}\Big)^2\sigma(a+s)
   \label{eq:gamma2}
\end{eqnarray}
with
\begin{equation}
   \sigma(r):=-\frac{\varepsilon(r)\psi'(r)}{4\pi}
   \label{eq:sigma1}
\end{equation}
the charge enclosed by a sphere of radius $r$ around the origin per sphere surface area.

The linear Poisson-Boltzmann equation~\Eq{lpbe} is analytically soluble which gives rise to an expression of
the form
\begin{equation}
   \sigma(a+s) = \frac{\phi_D}{\displaystyle\Big(1+\frac{s}{a}\Big)^2}\sqrt{\frac{\varepsilon_\m{b}\varrho^\m{ref}_\m{b}}{2\pi}}
                 F(s/a,\kappa_\m{b}a,n,p),
   \label{eq:sigma2}
\end{equation}
where $\kappa_\m{b}:=\sqrt{8\pi\varrho^\m{ref}_\m{b}/\varepsilon_\m{b}}$, $n:=\sqrt{\varepsilon_\m{d}/\varepsilon_\m{b}}$,
and $p:=\sqrt{\varrho^\m{ref}_\m{d}/\varrho^\m{ref}_\m{b}}$.
The full scaling function $F$, which is recorded in the appendix, appears somewhat lengthy but is straightforward
to obtain in principle.

However, since the effective interfacial width parameter $s$ is usually very much smaller than the droplet radius
and the local Debye lengths, $|s|\ll a,\kappa(r)^{-1}$, the first argument of the scaling function $F$ can, within
an excellent approximation, be set to zero.
Inserting $x=0$ into Eqs.~\Eq{F} and \Eq{T} leads to
\begin{widetext}
\begin{equation}
   F(0,y,n,p) =
   \frac{\displaystyle np + \frac{n(p-n)}{y} - \Big(\frac{n}{y}\Big)^2 +
              \exp\Big(-2y\frac{p}{n}\Big)\Big(np + \frac{n(p+n)}{y} + \Big(\frac{n}{y}\Big)^2\Big)}
        {\displaystyle 1 + np + \frac{1-n^2}{y} +
              \exp\Big(-2y\frac{p}{n}\Big)\Big(-1 + np - \frac{1-n^2}{y}\Big)}.
   \label{eq:F0}
\end{equation}
\end{widetext}
At this level of approximation Eqs.~\Eq{gamma2} and \Eq{sigma2} reduce to
\begin{equation}
   \gamma^\m{ex} = 2s\varrho^\m{ref}_\m{b}(1-p^2) - \frac{\phi_D}{2}\sigma(a)
   \label{eq:gamma3}
\end{equation}
and
\begin{equation}
   \sigma(a) = \phi_D\sqrt{\frac{\varepsilon_b\varrho^\m{ref}_\m{b}}{2\pi}}F(0,\kappa_\m{b}a,n,p),
   \label{eq:sigma3}
\end{equation}
respectively.
According to Eq.~\Eq{sigma3}, the droplet charge per droplet surface area $\sigma(a)$ is (almost) independent
of the interfacial width $s$.
On the other hand, the excess interfacial tension $\gamma^\m{ex}$ in Eq.~\Eq{gamma3} comprises a contribution
describing the ion exclusion due to the short-ranged \emph{interfacial effects}
\begin{equation}
   \gamma^\m{ex}_\m{ie} := 2s\varrho^\m{ref}_\m{b}(1-p^2),
   \label{eq:gammaie}
\end{equation}
which is (essentially) linear in $s$ and (almost) independent of the droplet radius $a$, as well as an
\emph{electrostatic} contribution
\begin{equation}
   \gamma^\m{ex}_\m{es} := - \frac{\phi_D}{2}\sigma(a),
   \label{eq:gammaes}
\end{equation}
which is (almost) independent of the effective interfacial width $s$.

In order to understand the involved dependence of the scaling function $F(0,y,n,p)$ on $y$ it is useful to
investigate the asymptotic behavior for large and small values of $y$.
If $y \gg y^\times_1 := n/p$, the terms in Eq.~\Eq{F0} proportional to the exponentials may be neglected such that
\begin{equation}
   F(0,y\gg y^\times_1,n,p) \simeq \frac{np}{1+np}G(y,n,p)
   \label{eq:F0lowcurv1}
\end{equation}
with
\begin{equation}
   G(y,n,p) := \frac{\displaystyle \Big(1 -\frac{n}{py}\Big)\Big(1 + \frac{1}{y}\Big)}{\displaystyle 1 + \frac{1-n^2}{(1+np)y}}.
   \label{eq:G1}
\end{equation}
Defining $\displaystyle y^\times_2 := \frac{|1-n^2|}{1+np}$ and $y^\times_3 := 1$ and noting that
$\displaystyle\frac{1-n^2}{1+np}\in[-y^\times_1,y^\times_3]$ one infers from Eq.~\Eq{F0lowcurv1} the leading order
asymptotic behavior
\begin{equation}
   F(0,y \gg y^\times_1,n,p)\simeq
   \left\{\begin{array}{ll}
      \displaystyle \frac{np}{1+np}       & \m{(I)}\,   y \gg y^\times_3                \\[8pt]
      \displaystyle \frac{np}{1+np}y^{-1} & \m{(II)}\,  y^\times_2 \ll y \ll y^\times_3 \\[8pt]
      \displaystyle \frac{np}{1-n^2}      & \m{(III)}\, y \ll y^\times_2.
   \end{array}\right.
   \label{eq:F0lowcurv2}
\end{equation}
The three cases considered in Eq.~\Eq{F0lowcurv2} are exhaustive and mutually exclusive for $y \gg y^\times_1$
because $y^\times_2 \leq \max(y^\times_1,y^\times_3)$.
If $y \ll y^\times_1$, Eq.~\Eq{F0} leads to
\begin{equation}
   F(0,y \ll y^\times_1,n,p) \simeq \frac{p^2}{3}y.
   \label{eq:F0highcurv}
\end{equation}

\begin{figure}[!t]
   \includegraphics[width=8.5cm]{Fig1.ps}
   \caption{\label{fig:1}Scaling function $F(0,y,n,p)$ as a function of $y$ for the relation
           $y^\times_1 \ll y^\times_2 \ll y^\times_3$ of the crossover positions (see main text) in a log-log plot.
           The asymptotic regimes I--IV corresponding to Eqs.~\Eq{F0lowcurv2} and \Eq{F0highcurv} are apparent.
           For $y^\times_1 \gg y^\times_2$ regime III is absent, and for $y^\times_1 \gg y^\times_3$ also regime II.}
\end{figure}
Figure~\ref{fig:1} displays $F(0,y,n,p)$ for the case $y^\times_1 \ll y^\times_2 \ll y^\times_3$, where all four
asymptotic regimes I--IV of Eqs.~\Eq{F0lowcurv2} and \Eq{F0highcurv} are apparent.
If $y^\times_2 \ll y^\times_1 \ll y^\times_3$, however, regime III in Fig.~\ref{fig:1} is absent, and a
crossover between regimes II and IV takes place at $y=y^\times_1$.
Moreover, if $y^\times_1 \gg y^\times_3$ regime II is also absent, and $F(0,y,n,p)$ exhibits a single
crossover at $y=y^\times_1$ between regimes I and IV.

It will turn out in the next section that the analytical expressions based on the linearized theory derived in the
present section agree qualitatively, in typical cases even quantitatively, with numerically calculated
interfacial tensions within the non-linear theory.

\section{\label{sec:discuss}Discussion}

Here the closed analytical expressions obtained within the linearized Poisson-Boltzmann theory of the previous
section are discussed and compared with numerical results obtained within the non-linear theory based on
Eqs.~\Eq{ele2}--\Eq{gamma1}.
Some of the numerical data presented here have already been considered in Ref.~\cite{deGraaf2008}.
Throughout this section one of the liquids is water with dielectric constant $\varepsilon_\m{w}=80$.
Moreover, the largely arbitrary but representative choice of ion radii $a_+=0.36\,\m{nm}$ and
$a_-=0.30\,\m{nm}$ is made throughout.
Given the dielectric constant of the second liquid, called ``oil'', the parameters $n$, $p$, and $\phi_D$ are
known within the Born approximation (see Sec.~\ref{sec:model}).
The cases of an oil droplet in water (O/W) and of a water droplet in oil (W/O) will be distinguished.

\begin{figure}[!t]
   \includegraphics[width=8.5cm]{Fig2.ps}
   \caption{\label{fig:2}Electrostatic contribution to the excess interfacial tension $\gamma^\m{ex}_\m{es}$
           in mixtures of oil ($\varepsilon_\m{o}\in\{5,7.5,10\}$) and water ($\varepsilon_\m{w}=80$) as a function
           of the radius $a$ of an oil droplet in water (O/W, ascending curves) and a water droplet in oil (W/O,
           descending curves) with ion radii $a_+=0.36\,\m{nm}$ and $a_-=0.30\,\m{nm}$ as well as an ionic strength
           in water $I_\m{w}=1\,\m{mM}$.
           The thin solid curves are calculated by means of the analytical expressions within the linear theory
           of Sec.~\ref{sec:linear} whereas the thick dotted curves are obtained by numerically solving the
           non-linear Poisson-Boltzmann equation~\Eq{pbe}.
           Upon swapping oil and water (O/W $\leftrightarrow$ W/O) the slope of the curves at $a^{-1}=0$ (planar
           system) changes its sign.}
\end{figure}
\begin{table}[!t]
   \begin{minipage}{8.5cm}
   (a) O/W\\[5pt]
   \begin{tabular}{c||c|c|c|c|c}
      $\varepsilon_\m{o}$ & $n$     & $p$        & $\phi_D$ & $y^\times_1$ & $y^\times_2$ \\
      \hline
      $5$              & $0.25$  & $0.000285$ & $1.48$   & $876$        & $0.937$      \\
      $7.5$            & $0.306$ & $0.00520$  & $0.956$  & $58.9$       & $0.905$      \\
      $10$             & $0.354$ & $0.0222$   & $0.692$  & $15.9$       & $0.868$      \\
   \end{tabular}

   \vspace{10pt}

   (b) W/O\\[5pt]
   \begin{tabular}{c||c|c|c|c|c}
      $\varepsilon_\m{o}$ & $n$    & $p$    & $\phi_D$ & $y^\times_1$ & $y^\times_2$ \\
      \hline
      $5$              & $4$    & $3500$ & $-1.48$  & $0.00114$    & $0.00107$    \\
      $7.5$            & $3.27$ & $192$  & $-0.956$ & $0.0170$     & $0.0154$     \\
      $10$             & $2.83$ & $45.1$ & $-0.692$ & $0.0627$     & $0.0545$
   \end{tabular}
   \end{minipage}
   \caption{\label{tab:1}Quantities $n$, $p$, and $\phi_D$ as well as the crossover values $y^\times_1$ and
           $y^\times_2$ (see Secs.~\ref{sec:model} and \ref{sec:linear}) within the Born approximaion for (a)
           O/W and (b) W/O systems with the oil dielectric constant $\varepsilon_\m{o}\in\{5,7.5,10\}$ and
           ion radii $a_+=0.36\,\m{nm}$ and $a_-=0.30\,\m{nm}$.}
\end{table}
Figure~\ref{fig:2} displays the electrostatic contribution to the excess interfacial tension $\gamma^\m{ex}_\m{es}$
(see Eq.~\Eq{gammaes}) of oil droplets in water (O/W, ascending curves) and water droplets in oil (W/O, descending
curves) for an ionic strength in water $I_\m{w}=1\,\m{mM}$, where $I_\m{w}:=\varrho^\m{ref}_\m{b}$ for O/W and
$I_\m{w}:=\varrho^\m{ref}_\m{d}$ for W/O, as a function of the droplet radius $a$.
The analytical expression Eq.~\Eq{gammaes} within linearized Poisson-Boltzmann theory (thin solid curves) is
compared with numerical results of the non-linear Poisson-Boltzmann theory (thick dotted curves).
The slight quantitative differences are due to the linearization approximation and they are already present in the
planar system ($a^{-1}=0$).
The quantities $n$, $p$, and $\phi_D$ as well as the crossover values $y^\times_1$ and $y^\times_2$ correponding
to the curves in Fig.~\ref{fig:2} are displayed in Tab.~\ref{tab:1}.
According to Sec.~\ref{sec:linear}, regime III is expected to be absent for the W/O systems because
$y^\times_3 > y^\times_1 > y^\times_2$, whereas regimes II and III are absent for the O/W systems because
$y^\times_1 > y^\times_3 > y^\times_2$ (see also the discussion of Fig.~\ref{fig:5} at the end of this section).

Due to Eq.~\Eq{gammaes} the relative change of the electrostatic excess
interfacial tension $\gamma^\m{ex}_\m{es}(a)$ and the droplet charge per droplet surface area $\sigma(a)$ with
respect to their planar values $\gamma^\m{ex}_\m{es}(\infty)$ and $\sigma(\infty)$, respectively, are equal, and
they exhibit the low-curvature asymptotic behavior (see Eqs.~\Eq{sigma3} and \Eq{F0lowcurv1})
\begin{equation}
   \frac{\gamma^\m{ex}_\m{es}(a)}{\gamma^\m{ex}_\m{es}(\infty)} =
   \frac{\sigma(a)}{\sigma(\infty)} \simeq
   G(\kappa_\m{b}a,n,p)
   \qquad , \kappa_\m{b}a \gg y^\times_1,
   \label{eq:gammasigmalowcurv}
\end{equation}
where $G$ was defined in Eq.~\Eq{G1}.
Upon rewriting Eq.~\Eq{G1} one recognizes the asymptotic behavior
\begin{eqnarray}
   & & G(y \gg y^\times_2,n,p)
   \nonumber\\
   & = &
   \displaystyle 1 - \frac{n(1-p^2)}{p(1+np)}y^{-1}
        - \frac{\displaystyle \frac{n(p+n)^2}{p(1+np)^2}y^{-2}}{\displaystyle 1 + \mathop{\mathrm{sign}}(1-n)\frac{y^\times_2}{y}}
   \nonumber\\
   & \simeq &
   1 - \frac{n(1-p^2)}{p(1+np)}y^{-1} - \frac{n(p+n)^2}{p(1+np)^2}y^{-2},
   \label{eq:G2}
\end{eqnarray}
which equals the expansion in $y^{-1}$ up to second order.
Hence, the low-curvature expansion up to second order in $a^{-1}$ obtained by combining Eqs.~\Eq{gammasigmalowcurv}
and \Eq{G2} is expected to be acurate if $\kappa_\m{b}a \gg y^\times_1, y^\times_2$.
Traditionally, empirically motivated expansions in $a^{-1}$ have been used to represent the curvature dependence of
the interfacial tension without knowing their applicability a priori.
However, it has been argued by K\"{o}nig, Roth, and Mecke on the basis of a morphometrical approach that the
deviation of intensive thermodynamic quantities from their planar values are linear combinations of the
mean and the Gaussian curvature \emph{provided} the geometrical length scales are much larger than any
correlation length \cite{Koenig2004}, i.e., $a \gg \kappa_\m{d}^{-1},\kappa_\m{b}^{-1}$ with the inverse Debye
length in the droplet $\kappa_\m{d} := \kappa_\m{b}p/n$, or equivalently $\kappa_\m{b}a \gg y^\times_1,y^\times_3$.
This condition is only sufficient but not necessary for the validity of the above low-curvature expansion because
it already implies $\kappa_\m{b}a \gg y^\times_2$ due to $y^\times_2 \leq \max(y^\times_1,y^\times_3)$.
For $n,p \gg 1$, the low-curvature expansion is valid if $\kappa_\m{d}a \gg 1$, \emph{independent} of the bulk
Debye length $\kappa_\m{b}^{-1}$, because in this case $y^\times_2 \approx y^\times_1$.
This is the case, e.g., for the W/O systems considered in Tab.~\ref{tab:1}.

From Eq.~\Eq{G1} one straightforwardly recognizes the symmetry $G(py/n,1/n,1/p)=G(-y,n,p)$ which means that
swapping droplet and bulk liquid, i.e., $p \mapsto 1/p$, $n \mapsto 1/n$, $\kappa_\m{b} \mapsto \kappa_\m{d}$, while
keeping the droplet radius $a$ fixed has numerically the same effect on function $G$ as inverting the sign of the
droplet radius.
Due to this symmetry one concludes for the coefficients of an expansion in inverse powers of $a$ as in Eq.~\Eq{G2}
for $y=\kappa_\m{b}a$ that upon swapping droplet and bulk liquid the odd-order coefficients merely invert their
sign, whereas the even-order coefficients do not change.
This phenomenon can be observed in Fig.~\ref{fig:2}, where the slope close to the planar limit ($a^{-1}=0$), which is
proportional to the excess Tolman length due to the presence of ions, simply changes its sign upon swapping oil
and water (O/W $\leftrightarrow$ W/O).

\begin{figure}[!t]
   \includegraphics[width=8.5cm]{Fig3.ps}
   \caption{\label{fig:3}Electrostatic contribution to the excess interfacial tension in mixtures of oil
           and water ($\varepsilon_\m{w}=80$) as a function of the dielectric constant $\varepsilon_\m{o}$ of the oil
           for droplet radii $a\in\{50\,\m{nm},100\,\m{nm},250\,\m{nm},500\,\m{nm},1000\,\m{nm},\infty\}$
           of an oil droplet in water (O/W) and a water droplet in oil (W/O)
           with ion radii $a_+=0.36\,\m{nm}$ and $a_-=0.30\,\m{nm}$ as well as the ionic strength in water
           $I_\m{w}=1\,\m{mM}$.
           The thin solid curves are calculated by means of the analytical expressions within the linear theory
           of Sec.~\ref{sec:linear} whereas the thick dashed curves are obtained by numerically solving the
           non-linear Poisson-Boltzmann equation~\Eq{pbe}.
           There is quantitative agreement for all $a$ and $\varepsilon_\m{o}$ considered here.}
\end{figure}
Figure~\ref{fig:3} exhibits the electrostatic contribution to the excess interfacial tension $\gamma^\m{ex}_\m{es}$
as a function of the dielectric constant $\varepsilon_\m{o}$ of the oil for the ionic strength in water
$I_\m{w}=1\,\m{mM}$ and for various droplet radii
$a\in\{50\,\m{nm},100\,\m{nm},250\,\m{nm},500\,\m{nm},1000\,\m{nm},\infty\}$.
As in Fig.~\ref{fig:2}, the thin solid curves correspond to the analytic linear theory of Sec.~\ref{sec:linear}
whereas the thick dotted curves are the numerical results of the non-linear scheme.
Quantitative agreement is observed, even in the low-$\varepsilon_\m{o}$ range where the Donnan potential $\phi_D$ is
\emph{not} small and the linearization approximation is \emph{not a priori} justified.
From the linearized theory of Sec.~\ref{sec:linear} one can derive the asymptotic behavior
$\gamma^\m{ex}_\m{es}=\mathcal{O}(-(\varepsilon_\m{o}-\varepsilon_\m{w})^2)$ for $\varepsilon_\m{o}\rightarrow\varepsilon_\m{w}$
as well as $\gamma^\m{ex}_\m{es}=\mathcal{O}(-\exp(-\m{const}/\varepsilon_\m{o}))$ for an O/W system and
$\gamma^\m{ex}_\m{es}=\mathcal{O}(-1/\varepsilon_\m{o})$ for a W/O system as $\varepsilon_\m{o} \rightarrow 0$.
This behavior is apparent in Fig.~\ref{fig:3}, too.

The total excess interfacial tension $\gamma^\m{ex}$ comprises not only the electrostatic part
$\gamma^\m{ex}_\m{es}$ but also the contribution $\gamma^\m{ex}_\m{ie}$ due to the interfacial effects (see
Eq.~\Eq{gammaie}).
It is readily seen that $\gamma^\m{ex}_\m{ie}=\pm\mathcal{O}(s(\varepsilon_\m{w}-\varepsilon_\m{o}))$ for
$\varepsilon_\m{o}\rightarrow\varepsilon_\m{w}$ and $\gamma^\m{ex}_\m{ie}=\pm\mathcal{O}(s)$ for
$\varepsilon_\m{o} \rightarrow 0$ where the upper ($+$) and the lower ($-$) sign correspond to an O/W and a W/O
system, respectively.
Hence, if $s \not= 0$, the interfacial effects will dominate over the electrostatic effects in the limits
$\varepsilon_\m{o} \rightarrow 0$ for O/W systems and $\varepsilon_\m{o}\rightarrow\varepsilon_\m{w}$ for arbitrary systems.
\begin{figure}[!t]
   \includegraphics[width=8.5cm]{Fig4.ps}
   \caption{\label{fig:4}Total excess interfacial tension in mixtures of oil
           and water ($\varepsilon_\m{w}=80$) as a function of the dielectric constant $\varepsilon_\m{o}$ of the oil
           for droplet radii $a\in\{50\,\m{nm},100\,\m{nm},250\,\m{nm},500\,\m{nm},1000\,\m{nm},\infty\}$
           of an oil droplet in water (O/W) and a water droplet in oil (W/O)
           with ion radii $a_+=0.36\,\m{nm}$ and $a_-=0.30\,\m{nm}$, interfacial width parameter $|s|=0.33\,\m{nm}$,
           as well as the ionic strength in water $I_\m{w}=1\,\m{mM}$.}
\end{figure}
Figure~\ref{fig:4} displays the total excess interfacial tension corresponding to the parameters used in
Fig.~\ref{fig:3} and an interfacial width parameter $s$ with $|s|=0.33\,\m{nm}$ on the water side of the interface,
i.e., $s>0$ for O/W and $s<0$ for W/O.

According to the results of Sec.~\ref{sec:linear} the electrostatic excess interfacial tension $\gamma^\m{ex}_\m{es}$
as a function of the bulk ionic strength $I_\m{b}:=\varrho^\m{ref}_\m{b}$ can be asymptotically described by
\begin{equation}
   \gamma^\m{ex}_\m{es} \simeq
   \left\{\begin{array}{ll}
      \displaystyle -\phi_D^2\sqrt{\frac{\varepsilon_\m{b}}{8\pi}}\frac{np}{1+np}I_\m{b}^{1/2}
      &
      \m{(I)}\,   I_\m{b} \gg I^\times_{\m{b}1},I^\times_{\m{b}3}
      \\[10pt]
      \displaystyle -\phi_D^2\frac{\varepsilon_\m{b}}{8\pi}\frac{np}{(1+np)a}
      &
      \m{(II)}\,  I^\times_{\m{b}1},I^\times_{\m{b}2} \ll I_\m{b} \ll I^\times_{\m{b}3}
       \\[10pt]
      \displaystyle -\phi_D^2\sqrt{\frac{\varepsilon_\m{b}}{8\pi}}\frac{np}{1-n^2}I_\m{b}^{1/2}
      &
      \m{(III)}\, I^\times_{\m{b}1} \ll I_\m{b} \ll I^\times_{\m{b}2}
      \\[10pt]
      \displaystyle -\phi_D^2\frac{p^2}{3}aI_\m{b}
      &
      \m{(IV)}\,  I_\m{b} \ll I^\times_{\m{b}1}
   \end{array}\right.
   \label{eq:gammaesasymp}
\end{equation}
with the crossover bulk ionic strengths $\displaystyle I^\times_{\m{b}k} := \frac{\varepsilon_\m{b}(y^\times_k)^2}{8\pi a^2},
k\in\{1,2,3\}$, where the $y^\times_k, k\in\{1,2,3\}$ are defined in Sec.~\ref{sec:linear}.
For a planar system ($a=\infty$) the crossovers are at zero ionic strength, hence only the high-ionic strength
regime I in Fig.~\ref{fig:1} ($I_\m{b} \gg I^\times_{\m{b}1},I^\times_{\m{b}3}$) is present, which coincides
exactly with the electrostatic contribution to the excess interfacial tension in Ref.~\cite{Bier2008}.

For an oil dielectric constant $\varepsilon_\m{o}=5$ and a droplet radius $a=1\,\m{\mu m}$ the crossover bulk ionic
strengths are $I^\times_{\m{b}1} \approx 71\,\m{mM}, I^\times_{\m{b}2} \approx 82\,\m{nM}, I^\times_{\m{b}3}
\approx 93\,\m{nM}$ for an O/W system, where $I_\m{b}=I_\m{w}$ is the ionic strength in water, and
$I^\times_{\m{b}1} \approx 7.6\,\m{fM}, I^\times_{\m{b}2} \approx 6.6\,\m{fM}, I^\times_{\m{b}3} \approx
5.8\,\m{nM}$ for a W/O system, where $I_\m{b}=I_\m{o}$ is the ionic strength in oil (see Tab.~\ref{tab:1}).
Here, ionic strengths in oil, $I_\m{o}$, and in water, $I_\m{w}$, are related to each other by $I_\m{o}/I_\m{w}
\approx 8.1\cdot 10^{-8}$.
\begin{figure}[!t]
   \includegraphics[width=8.5cm]{Fig5.ps}
   \caption{\label{fig:5}Electrostatic excess interfacial tension in mixtures of oil ($\varepsilon_\m{o}=5$)
           and water ($\varepsilon_\m{w}=80$) as a function of the ionic strength in oil ($I_\m{o}$) or water
           ($I_\m{w}$) for droplet radii $a\in\{0.1\,\m{\mu m},1\,\m{\mu m}\}$
           of an oil droplet in water (O/W) and a water droplet in oil (W/O)
           with ion radii $a_+=0.36\,\m{nm}$ and $a_-=0.30\,\m{nm}$.
           The O/W system exhibits only the regimes I and IV (see main text and Fig.~\ref{fig:1}), whereas
           for the W/O system the regimes I, II, and IV are present.
           Upon changing the droplet size $a$ the crossover ionic strengths shift by a factor $a^{-2}$.}
\end{figure}
Figure~\ref{fig:5} displays $\gamma^\m{ex}_\m{es}$ as a function of the ionic strength in the physical
range $I_\m{w}\in[10^{-7}\,\m{M},10\,\m{M}]$ for the droplet radii $a=1\,\m{\mu m}$ and $a=0.1\,\m{\mu m}$.
The crossover ionic strengths of the latter droplet size are larger by a factor $100$ as compared to the former
because $I^\times_{\m{b}k} \sim a^{-2}$.
By inspection of the values of the crossover bulk ionic strengths one expects only the regimes I and IV of
Fig.~\ref{fig:1} to be present for the O/W system whereas the regimes I, II, and IV are expected for the W/O
system.
The occurrence of the regimes I and IV for the O/W system and I, II, and IV for the W/O system can be
inferred from Fig.~\ref{fig:5} in conjunction with Eq.~\Eq{gammaesasymp}.

\section{\label{sec:conclusions}Conclusions and Summary}

It turned out in the previous section that the analytical theory of Sec.~\ref{sec:linear} based on a linearized
Poisson-Boltzmann theory is in good (at least) qualitative agreement with the results from the full non-linear
theory.
It can therefore be expected that the general conclusions drawn from that linear theory apply to more
elaborate models \cite{Onuki2006,Onuki2008a,Onuki2008b}, too.

According to Eqs~\Eq{gamma3}, \Eq{gammaie}, and \Eq{gammaes} the excess liquid-liquid interfacial tension is
$\gamma^\m{ex}=\gamma^\m{ex}_\m{ie}+\gamma^\m{ex}_\m{es}$ where the curvature dependence is
essentially only due to the electrostatic part $\gamma^\m{ex}_\m{es}$ and not due to the contribution of the
short-ranged interfacial effects $\gamma^\m{ex}_\m{ie}$.
While $\gamma^\m{ex}$ can indeed be negative, thereby decreasing the total interfacial tension, the largest
magnitude $|\gamma^\m{ex}|$ is attained at high ionic strengths where $\gamma^\m{ex}\approx\gamma^\m{ex}_\m{ie}$,
i.e., where $\gamma^\m{ex}$ is essentially curvature-independent.
One has to conclude that the unimodal droplet size distribution of W/O emulsions observed by Leunissen
et al.\ \cite{Leunissen2007a,Leunissen2007b} \emph{cannot} be explained by the curvature dependence of the
interfacial tension due to electrostatic effects alone.
However, this conclusion does not apply to the experiments by Sacanna et al.\ \cite{Sacanna2007}, where
highly charged colloids instead of monovalent ions are present, as the linearized theory of
Sec.~\ref{sec:linear} is not a priori justified for multivalent ions or highly charged colloids.
Instead it is an interesting open question to be addressed in future studies as to what extent the qualitative
low-valency picture drawn here is valid for the presence of high-valency particles.

In summary, the curvature dependence of the electrolytic liquid-liquid interfacial tension within a simple linear
Poisson-Boltzmann model in the spherical geometry has been calculated analytically.
This linear theory turned out to be at least qualitatively reliable as has been checked by numerically solving
the corresponding non-linear Poisson-Boltzmann model.
Novel low ionic strength regimes, which are not present for a planar liquid-liquid interface, have been identified.
Low and high curvature asymptotics of the interfacial tension have been discussed.
In particular, it has been found that in systems where the ionic strength and the dielectric constant in the droplet
are much larger than in the bulk the range of validity of low-curvature expansions up to second order in the
inverse radius of curvature is independent of the bulk Debye length.

\begin{acknowledgments}
   This work is part of the research program of the ``Stichting voor
   Fundamenteel Onderzoek der Materie (FOM)'', which is financially supported by
   the ``Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)''.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Designed with relaxation in mind, the Big Agnes Big Six has a tall back, wide seat and sits an accessible 18" off the ground. Set-up is quick and easy with a shock-cord pole set.
Packed size: 17" x 5".
Made with durable, lightweight nylon 100-denier Robic with a 300-denier polyester rip-stop.
Lightweight shock-corded, aircraft aluminum pole system.
Packable enough for stashing in your vehicle or travel bag so you can take it anywhere you go.
Extra height and width for outstanding comfort.
Lightweight and compact size makes it perfect for base camp.


text:
Savoir Flair Honors M·A·C Cosmetics for Championing Diversity Through 'Best of Beauty' Awards
Submitted by M·A·C Cosmetics
Savoir Flair recently released its annual "Best of Beauty" awards with M·A·C Cosmetics taking the top spot for "The Diversity Champion" category.
As a brand with the founding credo of "All Ages, All Races, All Genders," M·A·C has long been a champion for inclusivity and diversity.
As Savoir Flair notes: "There's no denying that the beauty world is more diverse than ever, with brands realizing that 40 shades of beige no longer cut it. One brand that's been putting diversity on the map since it burst onto the scene way back in 1984 is MAC, whose inclusive culture and "All Ages. All Races. All Genders" ethos is inspiring and pioneering. All about using style and experimentation as a form of self-expression, its campaigns feature gender-fluid people of all colors, shapes, sizes and ages. To this day, it celebrates beauty without gender boundaries while working to support numerous initiatives around the world fighting for the rights of minority groups. As one of the most inclusive beauty brands in the world, it celebrates "Every Shade of You" with extensive color ranges that work with all skin tones and under tones, with its best-selling 'Studio Fix' boasting 64 shades!"
Continue reading here
M·A·C Cosmetics
About M·A·C Cosmetics
M·A·C (Make-up Art Cosmetics), a leading brand of professional cosmetics, is part of The Estée Lauder Companies Inc. Since its creation in Toronto, Canada, over 30 years ago, the brand's popularity has grown through a tradition of word-of-mouth endorsement from makeup artists, models, photographers and journalists around the world. M·A·C is now sold in over 130 countries/territories worldwide. Follow M·A·C on Twitter, Snapchat and Pinterest at @MACcosmetics, become a M·A·C fan on Facebook, follow M·A·C on Instagram, watch M·A·C videos on YouTube. For a M·A·C location, visit maccosmetics.com.
Learn more about M·A·C's commitment to doing good for our global community and the planet at M·A·C Cares.
More from M·A·C Cosmetics


text:
Q: Least squares approximation of trigonometric DFT polynomial So my task is to interpolate the following 4 points: $(0,0) (0.25,1) (0.5,0) (0.75,-1)$. So I begin my using the DFT on $[0,1,0,-1]$ which results in $[0,-i,0,i]$. This gives the interpolation: $P_4(t)=sin2\pi t$ (which I have verified is correct).
Now my task is to find the best order 2 least squares approximation to this data, using the basis functions $1$ and $cos2\pi t$ but I am unsure on how to proceed. Usually using least squares approximation would just mean chopping off from the trigonometric polynomial, but now I am supposed to use basis functions.
Does this mean that I simply only want to keep the two first parts of the trigonometric polynomial (since I am using the first two basis functions), however since $P_4(t)=sin2\pi t$ only has the third part, the answer is $0$?

A: I'm not entirely sure, but if I understand correctly, aren't you simply supposed to minimize
$$f(a,b)=\int_0^1\Big(\big(a+b\cos(2\pi t)\big)-\sin(2\pi t)\Big)^2\,dt$$
over $(a,b)\in\mathbb{R}^2$?