text:
Singapore Hokkien Huay Kuan Cultural Academy © 2019 | Privacy Policy
The Singapore Hokkien Huay Kuan Cultural Academy (SHHKCA) is established by the Singapore Hokkien Huay Kuan. It is founded based on 170 years of excellent foundation in education, especially in the preservation and promotion of Chinese Language and culture. Supported by the Singapore Government, Prime Minister Lee Hsien Loong first announced the plans at the 2012 National Day Rally. As Singapore's first Chinese Cultural Academy, it will focus on promoting Chinese Language and proactively grow and preserve the traditional arts.
The Academy offers Chinese language, cultural and humanities programmes for toddlers, pre-schoolers, children, youths and adults.
About Singapore Hokkien Huay Kuan
Singapore Hokkien Huay Kuan (SHHK) is a not-for-profit organisation established in 1840. Her main objectives are to preserve and promote Chinese language and culture; advance the cause of education; and contribute to social service. It manages 6 affiliated schools: Tao Nan School, Ai Tong School, Chongfu School, Nan Chiau Primary School, Nan Chiau High School and Kong Hwa School. It also operates and manages the Arts and Cultural Troupe and formerly the Singapore Hokkien Huay Kuan Dance Theatre (now known as the Singapore Chinese Dance Theatre). The opening of the Cultural Academy marks a new milestone for SHHK.
The SHHK Cultural Academy aims to:
Promote, teach and preserve Chinese Language as a living language that can be used in our daily lives;
Promote a holistic education, including an in-depth understanding of the Chinese culture;
Grow the Chinese traditional fine arts and performance arts and develop creative talents including production and arts management;
Instil keen interest and motivation for the learning of Chinese Language and Chinese culture amongst the young; and
Develop pedagogy for Chinese language and Chinese culture to complement mainstream schools.
Mr Tan Cheng Gay
Mr Perng Peck Seng
Mr Chan Hock Keng
Mr Chua Kee Seng Thomas
Mr Tan Aik Hock
Mr Lee Swee Keng
Mr Pok Cheng Chong
Mr Ong Chong Hua
Mr Kuah Geok Lin James
Mr Loo Lip Giam
Mr Teo Cher Cheong James
Mr Tan Thiam Soon
Children Programmes
Hokkien Class


When was your favorite star born?
Other Birthday Lists: All My Children, Another World, As the World Turns, Bold & Beautiful, Days of Our Lives, General Hospital, GH: Night Shift, Guiding Light, One Life to Live, Passions, Port Charles, Sunset Beach, Young & the Restless, 24, 30 Rock, Alias, Almost Human, American Gothic, Angel, Arrow, Babylon 5, Battlestar Galactica, Beauty & The Beast, Being Human, Big Bang Theory, Blackadder, Bones, Boston Legal, Buffy the Vampire Slayer, Castle, Charmed, Covert Affairs, CSI, Curb Your Enthusiasm, Dawson's Creek, Drew Carey Show, Earth 2, Eerie Indiana, Fawlty Towers, Ghost Whisperer, Gotham, Grey's Anatomy, Grosse Pointe, Haven, Herman's Head, Heroes, Hi Honey I'm Home, Hollywood Heights, Howard Stern Show, House, Jericho, Larry Sanders Show, Law & Order, Lois and Clark, Lost, Mad About You, Mork & Mindy, The O.C., Pacific Palisades, Police Squad!, Red Dwarf, Remington Steele, Savannah, Sledgehammer, Smallville, Space: 1999, Star Trek, True Blood, TV Nation and Vampire Diaries.
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LONDON, UK / ACCESSWIRE / July 27, 2016 / Active Wall St. announces the list of stocks for today's coverage. Pre-market the Active Wall St. team provides the technical notes impacting selected stocks trading on the Toronto Exchange and belonging under the Basic Materials sector. Companies recently under review include Canadian Natural Resources Limited, Seven Generations Energy, AltaGas Ltd., and Inter Pipeline Ltd. Get all of our research notes free by signing up at: http://www.activewallst.com/register/.
On Tuesday, July 26, 2016, the TSX Composite Index edged 0.36% higher, to finish at 14,550.00. Active Wall St. has initiated coverage on the following equities: Canadian Natural Resources Ltd. (TSX: CNQ), Seven Generations Energy Ltd. (TSX: VII), AltaGas Ltd. (TSX: ALA), and Inter Pipeline Ltd. (TSX: IPL).
Register with us now for your free membership and more at: http://www.activewallst.com/register/.
Calgary, Canada headquartered Canadian Natural Resources Ltd.'s stock finished Tuesday's session 0.08% higher at $39.77 with a total volume of 2.58 million shares traded. Over the last one month and the previous three months, Canadian Natural Resources Ltd.'s shares have advanced 5.60% and 2.76%, respectively. Furthermore, the Company's stock has gained 30.95% in the past one year. Shares of the Company, which acquires, explores for, develops, produces, markets, and sells crude oil, natural gas, and natural gas liquids, are trading above its 50-day and 200-day moving averages. Canadian Natural Resources' 50-day moving average of $39.44 is above its 200-day moving average of $34.75. See our notes on CNQ.TO at: http://www.activewallst.com/registration-3/?symbol=CNQ.
Calgary, Canada headquartered independent petroleum company, Seven Generations Energy Ltd.'s stock advanced 1.54%, to close the day at $26.98. The stock recorded a trading volume of 2.15 million shares, which was above its three months average volume of 728,157 shares. Seven Generations Energy's shares have surged 11.81% in the last one month, 22.64% in the past three months and 92.17% in the previous one year. The company's shares are trading above their 50-day and 200-day moving averages. Moreover, the stock's 50-day moving average of $25.96 is greater than its 200-day moving average of $20.89. Shares of the Company, which focuses on the acquisition, exploration, development, and production of oil and natural gas properties in western Canada, traded at a PE ratio of 202.86. The complimentary notes on VII.TO at: http://www.activewallst.com/registration-3/?symbol=VII.
On Tuesday, shares in AltaGas Ltd. Calgary, Canada headquartered ended the session 0.24% lower at $32.75 with a total volume of 360,623 shares traded. AltaGas' shares have advanced 7.98% in the last one month and 7.80% in the previous three months. The stock is trading above its 50-day and 200-day moving averages. The company's 200-day moving average of $31.27 is greater than its 50-day moving average of $31.17. Shares of AltaGas, which operates as a diversified energy infrastructure company in North America, traded at a PE ratio of 129.96. Register for free and access the latest notes on ALA.TO at: http://www.activewallst.com/registration-3/?symbol=ALA.
On Tuesday, shares in Calgary, Canada headquartered Inter Pipeline Ltd. recorded a trading volume of 327,192 shares. The stock ended the day 0.29% higher at $27.30. Inter Pipeline's stock has advanced 4.68% in the last one month, 1.45% in the previous three months and 1.83% in the past one year. The Company is trading above its 200-day moving average. The stock's 50-day moving average of $27.33 is above its 200-day moving average of $25.31. Shares of the Company, which engages in the petroleum transportation, natural gas liquids extraction, and bulk liquid storage businesses in Canada and Europe, traded at a PE ratio of 22.40. Get free access to your notes on IPL.TO at: http://www.activewallst.com/registration-3/?symbol=IPL.
The non-sponsored content contained herein has been prepared by a writer (the "Author") and is fact checked and reviewed by a third party research service company (the "Reviewer") represented by a credentialed financial analyst [for further information on analyst credentials, please email [email protected]. Rohit Tuli, a CFA® charterholder (the "Sponsor"), provides necessary guidance in preparing the document templates. The Reviewer has reviewed and revised the content, as necessary, based on publicly available information which is believed to be reliable. Content is researched, written and reviewed on a reasonable-effort basis. The Reviewer has not performed any independent investigations or forensic audits to validate the information herein. The Reviewer has only independently reviewed the information provided by the Author according to the procedures outlined by AWS. AWS is not entitled to veto or interfere in the application of such procedures by the third-party research service company to the articles, documents or reports, as the case may be. Unless otherwise noted, any content outside of this document has no association with the Author or the Reviewer in any way.


text:
Gheorghe Man ( - d. februarie 1972) a fost un scrimer român. A concurat în proba de sabie pe echipe la Jocurile Olimpice de vară din 1936.

Note

Nașteri în 1914
Scrimeri olimpici români
Scrimeri la Jocurile Olimpice de vară din 1936

redpajama_set_name: RedPajamaWikipedia


The New York Public Library's exhibit "The ABC of It: Why Children's Books Matter" curated by Leonard S. Marcus has been extended to September 7, ….
The theme for the ….
The National Children's Museum, for children ages 8 and under, includes exhibits on health, global citizenship, the environment, civic engagement and the ….
The Torpedo Factory Art Center on the waterfront in Old Alexandria, VA, was once a building where torpedoes and ….
Pennsylvania has a hamburger festival, a chili pepper festival, an apple festival, and now a pretzel ….
Joanna's Furnace, owned by the Hay Creek Historical Association , was a charcoal iron furnace from ….
The Giant Heart: A Healthy Interactive Experience is an icon at ….
The United States Botanic Garden was the vision of George Washington, Thomas Jefferson and James Madison. It was established on the National Mall in Washington, D.C. in 1820 and opened to the public in ….
The Pennsylvania Horticultural Society's Philadelphia International Flower Show began in 1829 and ….


text:
package org.kordamp.ezmorph.object;

import org.apache.commons.lang3.builder.EqualsBuilder;
import org.apache.commons.lang3.builder.HashCodeBuilder;
import org.kordamp.ezmorph.MorphException;

import java.util.Calendar;
import java.util.Date;
import java.util.Map;

/**
 * Morphs a Map into a Date.<br>
 * The Map should have at least one of the following keys
 * [yer,month,day,hour,minutes,seconds,milliseconds] and the values should be
 * instances of Number. Any key that is not defined will have zero (0) assigned
 * as its value.
 *
 * @author Andres Almiray
 */
public class MapToDateMorpher extends AbstractObjectMorpher {
    private Date defaultValue;

    public MapToDateMorpher() {
        super();
    }

    public MapToDateMorpher(Date defaultValue) {
        super(true);
        this.defaultValue = defaultValue;
    }

    public boolean equals(Object obj) {
        if (this == obj) {
            return true;
        }
        if (obj == null) {
            return false;
        }

        if (!(obj instanceof MapToDateMorpher)) {
            return false;
        }

        MapToDateMorpher other = (MapToDateMorpher) obj;
        EqualsBuilder builder = new EqualsBuilder();
        if (isUseDefault() && other.isUseDefault()) {
            builder.append(getDefaultValue(), other.getDefaultValue());
            return builder.isEquals();
        }
        return !isUseDefault() && !other.isUseDefault() && builder.isEquals();
    }

    /**
     * Returns the default value for this Morpher.
     */
    public Date getDefaultValue() {
        return (Date) defaultValue.clone();
    }

    public int hashCode() {
        HashCodeBuilder builder = new HashCodeBuilder();
        if (isUseDefault()) {
            builder.append(getDefaultValue());
        }
        return builder.toHashCode();
    }

    public Object morph(Object value) {
        if (value == null) {
            return null;
        }

        if (Date.class.isAssignableFrom(value.getClass())) {
            return (Date) value;
        }

        if (!supports(value.getClass())) {
            throw new MorphException(value.getClass() + " is not supported");
        }

        Map map = (Map) value;
        if (map.isEmpty()) {
            if (isUseDefault()) {
                return defaultValue;
            } else {
                throw new MorphException("Unable to parse the date " + value);
            }
        }

        Calendar c = Calendar.getInstance();
        c.set(Calendar.YEAR, getValue(map, "year"));
        c.set(Calendar.MONTH, getValue(map, "month"));
        c.set(Calendar.DATE, getValue(map, "day"));
        c.set(Calendar.HOUR_OF_DAY, getValue(map, "hour"));
        c.set(Calendar.MINUTE, getValue(map, "minutes"));
        c.set(Calendar.SECOND, getValue(map, "seconds"));
        c.set(Calendar.MILLISECOND, getValue(map, "milliseconds"));
        return c.getTime();
    }

    public Class<?> morphsTo() {
        return Date.class;
    }

    /**
     * Sets the defaultValue to use if the value to be morphed is null.
     *
     * @param defaultValue return value if the value to be morphed is null
     */
    public void setDefaultValue(Date defaultValue) {
        this.defaultValue = (Date) defaultValue.clone();
    }

    public boolean supports(Class<?> clazz) {
        return clazz != null && Map.class.isAssignableFrom(clazz);
    }

    private int getValue(Map map, String key) {
        Object value = map.get(key);
        if (value == null || !(value instanceof Number)) {
            return 0;
        }

        Number n = (Number) value;
        return n.intValue();
    }
}

redpajama_set_name: RedPajamaGithub


Foodpyramid.com is ranked #4,409,528 among the most popular web sites, the lower the rank indicator the more popular is the web site and the more visitors it has. Foodpyramid.com is registered in 1998-09-14 owned by , website's age 20 years, 7 months, 5 days years. The number of unique users visiting this website every day is 123. Each individual visits 1.5 unique pages per day. Approximate time spent on a web site 01:15. It has an average of 303 pages indexed in major search engines like Google™. There are an average of 102 links pointing back to foodpyramid.com from other websites. Foodpyramid.com daily advertisement revenue is $2 USD. According to our analysis the approximate value of the website is $886 US Dollars. Most popular search engine request "foodpyramid.com". Website's IP address is 64.6.97.196 which is hosted by A1COLO.COM, site server is located in San Diego CA in United States. Google PageRank™ of the website is 0 of 10. Foodpyramid.com reputation rank is of 100. The web site is not listed in DMOZ open directory project.
Food Pyramid - info about the USDA Food Pyramid, food groups, and dietary guidelines. Your free guide to healthy eating, nutrition, diets and weight loss.
Foodpyramid.com is visited by around 123 unique visitors per day. Every visitor of this website viewing approximately 1.5 pages per day.
We found 102 backlinks to Foodpyramid.com from other web sites. The largest number of backward links are 0, the smallest number of links are 0, the average number of links are 0 refer to this web site.
Using information from users, and other reliable sources, Foodpyramid.com has the reputation 100 out of 100.
The top queries driving traffic to Foodpyramid.com from search engines.
The IP address of Foodpyramid.com is 64.6.97.196.
From the HTTP headers of Foodpyramid.com, you will know that HTTP status code is HTTP/1.1 200 OK, server name is Apache/2.
The chart shows how Foodpyramid.com indexed in major search engines.
Google indexed 303 pages. The maximum value is 0, the minimum value is 0, while the average value is 0.
Bing indexed 337 pages. The maximum value is 0, the minimum value is 0, while the average value is 0.


Keinton Butler is senior curator of design and architecture at Museum of Applied Arts and Sciences (MAAS) in Sydney. Keinton graduated from Furniture Technology at RMIT University in Melbourne, after studying key developments in furniture production. She recently returned to Sydney after ten years in London, where she worked as an independent design curator, collaborating with leading cultural organisations, including the British Council. She was a curator for Damien Hirst's contemporary art publishing project Other Criteria, where she developed a unique multidisciplinary curatorial program for two new gallery spaces in the UK. Keinton holds a Master of Arts in Curating Contemporary Design from the Design Museum in London and Kingston University. Keinton joined MAAS most recently from Precinct, a project space she established in London for cultural exchange between artists, designers and local communities.


text:
Ashcroft v. Abbasi (formerly Turkman v. Ashcroft) – U.S. Supreme Court
Ashcroft v. Abbasi (formerly Turkman v. Ashcroft) – U.S. Supreme Court
Preserving Federal Court Review in Bivens/FTCA cases
The Council, along with the National Immigration Project of the National Lawyers Guild (NIPNLG), is seeking to preserve federal court review of damages actions brought by noncitizens for abuse of authority by immigration agents. In actions brought under Bivens v. Six Unknown Named Agents of the Federal Bureau of Narcotics, 403 U.S. 388 (1971), the government routinely moves to dismiss these cases on a variety of jurisdictional grounds, including by arguing that INA § 242(g) bars the court's review of damages claims in any case involving removal procedures, and that a remedy under Bivens is not available in immigration-related actions. In essence, the government is attempting to deprive those who have been harmed by immigration agents of any remedy in federal court.
We have filed amicus briefs which counter these arguments in various federal district and circuit courts. One such case is Ashcroft v. Abbasi, No. 15-1359 (S.Ct. amicus brief filed Dec. 27, 2016) (Turkman v. Ashcroft when it was pending at the Second Circuit below). The Council and NIPNLG filed an amicus brief while this case was pending at the Second Circuit, in support of eight noncitizens of Arab and South Asian descent who sued individual federal officials for physical and emotional abuse that they suffered when detained on immigration charges in the aftermath of September 11. These plaintiffs were held in particularly egregious conditions while the FBI decided whether they were terrorists, or had any value to the 9/11 investigation. In the decision below, the Second Circuit accepted amici's argument that the Supreme Court has provided a remedy for the type of constitutional misconduct by federal agents suffered by the plaintiffs in this case. Turkman v. Ashcroft, 789 F.3d 218 (2d Cir. 2015). Former Attorney General Ashcroft and the other defendants filed a petition for certiorari to the Supreme Court, which granted review. The Council, joined by NIPNLG, ACLU Immigrant Rights Project, Northwest Immigrant Rights Project, and the National Police Accountability Project, filed an amicus brief arguing, among other things, that the immigration context of the case is not a basis for denying a damages remedy for the plaintiffs. The Court held oral argument on January 18, 2017.
For more information about these amicus briefs, please contact the Council's legal department.

redpajama_set_name: RedPajamaCommonCrawl


text:
Provinsen El Bayadh () er en af Algeriets 48 provinser. Administrationscenteret er El Bayadh.

Eksterne henvisninger

Provinser i Algeriet


text:
Updated news stories weekly. Published online only, every Thursday.
Issue:- 21 March 2013
On screen and online
A BRAND new website showcasing Liverpool as one of the UK's top filming locations has launched. For the first time, the new site - liverpoolfilmoffice.tv will have details the hundreds of movies, television programmes, adverts and music videos, which have been filmed in Liverpool. This dates back to as far as 1896, when a film called Liverpool Scenes was shot in the City by Alexandre Promio for the Lumiere Brothers. It's the earliest recorded footage of Liverpool's Lime Street Station and the electric Overhead Railway.
The list is bang up to date and includes details of the movie Noble which is currently being filmed in the City and tells the true story of a woman who overcame her horrific childhood in the slums to help underprivileged children in Vietnam and Mongolia.
A dedicated You Tube channel which shows examples of different filming has also been set up, ranging from a McCain's pizza advert from 1994, a music video for The Coral's Dreaming of You right through to BBC drama Good Cop and Hollywood blockbuster Captain America.
Liverpool City Council's cabinet member for culture and tourism, Councillor Wendy Simon, said:- "Year on year, we've seen an increase in the number of productions coming to the City. The feedback we get from producers is that they want to return to the City again and again; not just because of the amazing variety of locations but because of the advice and support they receive from the Film Office team. This new website allows industry experts, as well as the general public, a real insight into the work of this City Council team in attracting people to film in the City, and also the extent of the portfolio over the years which is huge. Already in 2013, a movie is being filmed in the City along with a national advert to be aired on Channel 4, so it looks set to be another bumper year making Liverpool one of the most popular filming locations in the UK."
The website is also a one stop shop for producers looking to film in the City; giving details of the variety of locations available, advice on crew and facility options, information about local casting agencies and recommendations for where units can be based.
There is also an option for businesses, community groups, schools or even residents to register their location or home as a potential filming venue.
In 2012, filming levels increased by 22% on the previous year, with 894 filming days taking place in the City. In total, 227 films, programmes and adverts were filmed; bringing in �19million to the local economy.
Find out the latest by �liking' the Liverpool Film Office page on Facebook or follow FilmLiverpool on Twitter.
A SWEET WAY TO SHOW YOU CARE - NATIONAL DOUGHNUT WEEK IS BACK!
THE popular fundraising event, National Doughnut Week returns in 2013 from Saturday, 11 May to Saturday, 18 May 2013, to raise some dough for The Children's Trust. Now in its 22nd year, and having raised over �755,000 for children's charities, through the sale of 14 million doughnuts, it is hoped that more independent craft bakers will take part in the event and that food retailers and coffee shops across the UK will join in as well to help to make this year the most successful ever. The annual event, sponsored by CSM United Kingdom Limited, is a fun week of enjoying delicious doughnut treats. A percentage from each doughnut sale through participating bakers and businesses will be donated to The Children's Trust, a national charity that provides specialist care to some of the UK's most severely disabled children, and rehabilitation to children with an acquired brain injury. Phil Tufnell, Vice-President of The Children's Trust, said:- "I do hope that bakers, food retailers, coffee shops and the public will get behind this year's fundraising event and help to raise as much money as possible for a very special charity." There are lots of fun ways for the general public to raise money: taking on a doughnut-eating challenge with family, friends or work colleagues, or by holding a doughnut-themed event.
For more details on fundraising ideas and doughnut facts, includingwhat the UK's most popular doughnut is, visit:- thechildrenstrust.org.uk/doughnuts.
HMRC to replace Enquiry Centres with home visits for customers who need help
A new, flexible support service for 1.5 million customers who need extra help with their tax affairs will be piloted by HM Revenue and Customs (HMRC) from June 2013. Yet many are asking if this is wise as Enquiry Centres offer more security for some people and phone services can be very unhelpful. But the HMRC say that this new service will provide mobile, one-to-one support in a range of convenient locations, including a person's own home or business. It will also provide expert advisors on the phone, trained to deal with people who need extra support, to help a caller until the issue is resolved. HMRC will also make more funding and support available for voluntary sector organisations to help them to deal with customers who turn to them for help.
Lin Homer, HMRC's Chief Executive, said:- "HMRC is dedicated to providing help to customers who need it. This new service will enable us to tailor that help in a way that works better for customers and is more flexible and affordable than the service we currently provide. We will give a more specialised phone service for customers whose affairs can be resolved over the telephone, and face-to-face help to those who need it, visiting them at a place convenient to them, saving them both travel and time. HMRC will provide a more modern and accessible service that will target the right support to customers who need it, where and when they want it."
HMRC has calculated that the new service will:- "save customers almost �12 million a year in lost time and travel costs, and will be more than �13 million a year cheaper to run than the current service, as a result of the closure of the Enquiry Centre network in 2014. Face-to-face support will be provided to suit the customers who need it; no longer constrained by the fixed location and limited opening times of Enquiry Centres. HMRC's 281 Enquiry Centres have seen customer demand halved, from five million visitors in 2005/06 to fewer than 2.5 million in 2011/12. Some centres are now open just a day a week as a result of the sharp drop in demand. Of the customers using an Enquiry Centre in 2012, just 16% needed a face-to-face appointment. The cost of Enquiry Centre appointments is also high; with the average cost was �152 per appointment last year, and in 1 centre the cost reached �500 per appointment. In contrast, the average cost of serving a customer by phone is �3 per call, and an online transaction costs just 9p. The vast majority of customers never use an Enquiry Centre - HMRC handles more than 60 million calls a year and millions of online transactions, such as for filing self assessment and VAT returns. Recent investment and improvements in our contact centres mean that we are now answering 90 per cent of all calls, and moving our numbers from:- 0845 to cheaper:- 03 alternatives makes calling us more affordable. For those who cannot afford a call, HMRC can also arrange for a call back."
HMRC is discussing the impacts of these changes with staff in Enquiry Centres and its unions, and will do everything possible to redeploy Enquiry Centre staff within HMRC, or help them to find another role in the Civil Service. Almost 1,000 of the 1,300 staff employed in our 281 Enquiry Centres are already based in areas where HMRC has other offices. At this point the HMRC has said that a 5 month pilot is now underway to test these new services. The test is currently running in the North East of England until 31 October 2013, and HMRC has also launched a nationwide consultation on the new service.
So what do our readers think? Are you in HMRC and affected by this? Do you think it is a good move or will it result in added stress and problems for all sides? Email us your views to:- news24@southportreporter.com and let us know what you really think!
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text:
Foundations and NPOs
Intellectual and industrial property
Commercial and civil procedural law
Cross-border conflicts
You are here: Home1 / SERVICES2 / Legal
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We effectively meet the needs of our clients, addressing all legal issues that may affect them in a rigorous, swift and precise manner with technical accuracy.
The Legal Department of AddVANTE offers in-depth knowledge of each field of law linked to the world of companies and business, and specialises in commercial and corporate law and M&A specifically.
It benefits from highly qualified lawyers integrated within a cohesive multidisciplinary team allowing us to work in a coordinated manner with professionals from other disciplines according to the needs of each client in various sectors, industries and fields of activity.
Furthermore, it is necessary to highlight the extensive experience possessed by the Procedural Department in solving and mediating conflicts before the law courts and the courts of arbitration, and the expertise of the Labour Department which is on hand to help in all aspects of the labour world in the broadest sense.
Key issues we address
Representation and counselling for private clients, public and private companies, national groups and multinationals with all legal areas relating to their business activity.
We provide a global perspective from a legal standpoint with regard to the strategy for the start-up or expansion of your business on all levels.
Expert advice is given to businesses and private parties for all manner of national and international corporate operations and business, financial and commercial transactions.
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We have the best lawyers and professionals for you at state level in Commercial, Procedural and Labour Law, among other disciplines.
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Our professionals have developed tailor-made solutions for each case that will help you make the best decisions and obtain excellent results. Our proposals will enable you to obtain competitive advantages.
Intervention in the complex world of works of art and cultural property calls for expert counselling.
Moreover, art financing and the granting of guarantees by way of repayment of loans also calls for specialist advice.
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Initial Coin Offerings (ICOs) and Security Token Offerings (STOs)
Our aim is to provide first class services in the field of Blockchain, Smart Contracts and Project Tokenisation. We focus on the needs of each client. We seek to connect organizations and individuals with blockchain technology and its regulatory and technical requirements to achieve successful projects with greater added value.
Crime prevention management system
The implementation of a suitable control and supervision model to prevent crime can serve as grounds for exemption in a criminal liability lawsuit.
AddVANTE, thanks to its significant experience implementing ISO management systems, in addition to its in-depth knowledge of the regulations, has developed a specific methodology to assist organisations in designing and implementing a crime prevention programme.
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For the processing of personal data the RGPD requires free, informed, specific, unequivocal and express consent. In order for consent to be considered express, there must be an express declaration by the data subject or a positive action stating that he/she consents to the processing of his/her personal data. Therefore, tacit consent does not legitimise the processing of personal data.
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Nowadays, minimising the legal risks concerning the "Law and Regulation on Money Laundering Prevention"and investing in prevention and anticipation is key for the responsible governance of your organisation.
Furthermore, every day more companies are "required to certify the transparency of the origin and traceability of funds", whether national or international, used in financial transactions and corporate finance and M&A operations.
We have a team of lawyers who specialise in all legal matters relating to sport and entertainment.
Our extensive experience is based on advice for national and international athletes, clubs, federations, public limited sports companies and investors over a wide range of sports. We also assist sponsors in the negotiating, renewal and termination of sponsorship agreements.
Below you will find opinion and analysis articles written by our professionals related to the Legal practice.
Article 21 June, 2021
What is decentralised finance (DeFi)?
DeFi-nending DeFi. What is decentralised finance (DeFi)? In the past decade we have seen how digital innovation was transformative of business models, but failed to play a disruptive role. The ecosystem merely copied…
https://www.addvante.com/wp-content/uploads/2021/06/Defi-web_376062848.jpg 380 380 Xavier Gasia https://www.addvante.com/wp-content/uploads/2021/10/ADDVANTE-30-ok.svg Xavier Gasia2021-06-21 12:05:522021-06-25 09:12:20What is decentralised finance (DeFi)?
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text:
Q: How to read the contents of a json post with asp before processing the request? What I want to do is extremely simple in php. I just want to read the contents of the post. It also extremely simple on sailsjs / node ... I just return the result from within the async function.
In c# asp the answer is eluding me. I want the function to read the contents of the post before it attempts to process the post.
Sometimes the following code works. Sometimes the reading of the json from the post happens too slowly and jsonText is read as "" so nothing is processed.
In all of the test runs the json is being sent in the body of the post.
What is the best way to return a httpResponse after making sure the contents of the post is read first?
public HttpResponseMessage Post()
    {
        string content;
        try
        {
            string result = String.Empty;
            Newtonsoft.Json.Linq.JObject jObject = null;

            string jsonText = String.Empty;
            var syncTask = new Task<string>( () =>  {
                  return Request.Content.ReadAsStringAsync().Result;
            });
            /* I'm expecting that this will finish */
            syncTask.RunSynchronously();
            jsonText = syncTask.Result;

            /* before this line of code executes */
            System.Net.Http.HttpResponseMessage response = new HttpResponseMessage();

            if (jsonText == "")
            {
                result = "{\"error\":\"body is empty\"}";
                response.StatusCode = System.Net.HttpStatusCode.InternalServerError;
            }
            else
            {
                jObject = (Newtonsoft.Json.Linq.JObject)Newtonsoft.Json.Linq.JRaw.Parse(jsonText);

                string ipAddress = System.Web.HttpContext.Current.Request.ServerVariables["REMOTE_ADDR"];
                jObject["ipAddress"] = ipAddress;

                Models.JsonXML jsonXml = new JsonXML(jObject.ToString(Newtonsoft.Json.Formatting.None));
                System.Xml.XmlDocument document = new System.Xml.XmlDocument();
                document.LoadXml(jsonXml.xml);
                result = ReferralsManager.ProcessReferral(document);
                if (result == "")
                {
                    result = "{}";
                }

                response.StatusCode = System.Net.HttpStatusCode.OK;
            }
            response.Content = new StringContent(result);
            response.Content.Headers.ContentType = new System.Net.Http.Headers.MediaTypeHeaderValue("application/json");

            return response;
        }
        catch (Exception ex)
        {
            content = ErrorMessage.ServerException(Converter, ex);
            return Request.ToResponseMessage(content);
        }
        finally
        {
            LogManager.GetCurrentClassLogger().Info(InfoMessage.FUNC_ENDS, "Process Referral");
        }
    }

The working modified code after the answer from @Mekap is
    public class ProcessReferralAddressModel {

        public ProcessReferralAddressModel() { }

        public string address { get; set; }
        public string name { get; set; }

    }

    public class ProcessReferralModel
    {
        public ProcessReferralModel()
        {

        }
        public string uuid { get; set; }
        public DateTime date { get; set; }
        public ProcessReferralAddressModel from { get; set; }
        public ProcessReferralAddressModel[] to { get; set; }
        public string subject { get; set; }
        public string text { get; set; }
        public string html { get; set; }

    }

    /// <summary>
    /// Process a referral.
    /// </summary>
    /// <param name="userid">The userid.</param>
    /// <returns></returns>
    public HttpResponseMessage Post([FromBody] ProcessReferralModel processReferralModel)
    {
        string content;
        string jsonText = Newtonsoft.Json.JsonConvert.SerializeObject(processReferralModel) ;

        try
        {
            string result = String.Empty;
            Newtonsoft.Json.Linq.JObject jObject = null;

            System.Net.Http.HttpResponseMessage response = new HttpResponseMessage();

            if (jsonText == "" || jsonText == null )
            {
                result = "{\"error\":\"body is empty\"}";
                response.StatusCode = System.Net.HttpStatusCode.InternalServerError;
            }
            else
            {
                jObject = (Newtonsoft.Json.Linq.JObject)Newtonsoft.Json.Linq.JRaw.Parse(jsonText);

                string ipAddress = System.Web.HttpContext.Current.Request.ServerVariables["REMOTE_ADDR"];
                jObject["ipAddress"] = ipAddress;

                Models.JsonXML jsonXml = new JsonXML(jObject.ToString(Newtonsoft.Json.Formatting.None));
                System.Xml.XmlDocument document = new System.Xml.XmlDocument();
                document.LoadXml(jsonXml.xml);
                result = ReferralsManager.ProcessReferral(document);
                if (result == "")
                {
                    result = "{}";
                }

                response.StatusCode = System.Net.HttpStatusCode.OK;
            }
            response.Content = new StringContent(result);
            response.Content.Headers.ContentType = new System.Net.Http.Headers.MediaTypeHeaderValue("application/json");

            return response;
        }
        catch (Exception ex)
        {
            content = ErrorMessage.ServerException(Converter, ex);
            return Request.ToResponseMessage(content);
        }
        finally
        {
            LogManager.GetCurrentClassLogger().Info(InfoMessage.FUNC_ENDS, "Process Referral");
        }
    }

A: The json you're fetching, for our example will look something like
{ "ID" : 3,
 "StringCmd" : "ls -l"
}

For starters, we are going to write a small class who's representing our data in your web api
public class StringCmdModel
{
    public StringCmdModel()
    {
    }
    public int ID { get; set; }
    public string StringCmd { get; set; }
}

Now, we just have to write our Entry point in our WebAPI :
 [HttpPost]
 public HttpResponseMessage PostFonction([FromBody] StringCmdModel NewEntry)

You don't have to check for the existence of the data inside the function. But you should still do proper checks on theirs values, in case you get bad formated json or malicious calls.
But, if you get a call  with json that is not matching the StringCmdModel you gave in parameter from the body, this function will not be executed, and the server will throw on its own a 500 error.

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}
\label{introduction}

An infiltration process takes place when the external surface of an initially dry porous
medium is in contact with water source at a fixed pressure head.
In this circumstance, assuming that the Darcy law holds, the gradient of the pressure head will move
moisture into and through the pore spaces, as schematically shown in Fig. \ref{infiltratedmedia}(a).
If the domain geometry constrains the moisture to a one-dimensional horizontal flow, we can invoke a
Green-Ampt \cite{GreenAmpt} approximation and assume a well defined moving front $S\left( t\right)$
between filled and empty pore spaces.
Under this scenario, it would be expected that the filled volume $F\left( t\right)$ will increase in
time as
\begin{equation}
F \sim t^n ,
\label{defn}
\end{equation}
with the infiltration exponent $n=1/2$ \cite{campbellnorman, Peng, Chen}.
However, in many systems a different exponent is measured
\cite{logsdon,lockington,Kuntz,wilson,elabd2004,gerasimov,Voller_wrr2,elabd2015,berli2017},
which characterizes some type of anomalous transport (subdiffusive if $n<1/2$,
superdiffusive if $n>1/2$).

A related process occurs when the porous medium is filled with a static fluid and the external
surface is in contact with a reservoir of the same fluid but with a different concentration
of a solute.
This solute then infiltrates into or out of that medium, depending on the concentration difference,
with the solute molecules moving diffusively; see Fig. \ref{infiltratedmedia}(b).
A model for this process in a two-dimensional lattice was proposed by Sapoval, Rosso, and Gouyet
\cite{sapovalJPL1985}, and it was recently extended to fractal lattices \cite{Reis2016}:
the solute is represented by particles that execute random walks in the lattice and that have
a constant concentration at an external boundary.
Here, this is termed random walk infiltration (RWI).
Ref. \protect\cite{Reis2016} used a scaling approach and numerical simulations to show the
relation between the exponent $n$, the fractal dimensions of the infiltrated medium ($D_F$) and
of the infiltration boundary ($D_B$), and the random walk dimension in that medium ($D_W$).

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.45\textwidth,angle=0]{infiltratedmedia-eps-converted-to.pdf}
\caption{
(a) Schematic of fluid (blue) infiltration from a high pressure reservoir at the left into
a porous media (solid in tan color, non-filled pores in white).
(b) Infiltration of a solute (red) from a reservoir at the left into a porous media initially
without that solute.
The characteristic lengths $s_{RMS}$ and $S$ are indicated for one-dimensional infiltration.
}
\label{infiltratedmedia}
\end{figure}

In this work, we study RWI in homogeneous and fractal media with a small source at an external
boundary and discuss the close relation between this problem and fluid infiltration in the same media.
For two- and three-dimensional media, the time evolutions of the infiltrated area and volume
are obtained from the solution of the diffusion equation and confirmed by numerical simulations.
A scaling approach is then used to show that the infiltration has the same time evolution of the
number of distinct sites visited by a single random walker in the infiltrated lattice.
This leads to estimates of the infiltration exponent in fractal media, including the prediction
of an apparently normal infiltration in fractals where $D_F>D_W$, as illustrated for a Menger sponge.
In fluid infiltration, where the sharp fluid front moves with a velocity proportional
to the pressure gradient, the analytical form of the time evolution of the infiltrated area
or volume in two- and three-dimensional media exactly matches that seen in  geometrically
equivalent RWI processes.
Additional  numerical simulations also reveal similarities for fluid infiltration and RWI
in Sierpinski obstacle carpets. Together these results confirm the
scaling equivalence of fluid infiltration with RWI, and establish a connection between
different infiltration models with localized sources and the recurrence properties of random walks.

Despite the simple features of the fractal media studied here, we recall that they were used in
recent laboratory studies of anomalous transport.
For instance, infiltration of a fluid in horizontal Hele-Shaw cells with obstacles distributed
as in the Sierpinski carpets was studied in Ref. \protect\cite{Voller_wrr2} and Darcy-like flow
in media formed by sequences of Menger sponges was studied in Ref. \protect\cite{balankin2016}.
In both cases, good agreement with simulation results or with scaling approaches were obtained.
Hence it is expected that the scope of the work herein will benefit our understanding of
transport in porous materials of current geological and technological interest.

This work is organized as follows.
In Sec. \ref{model}, the RWI model and the basic quantities to be measured are presented,
and the example of infiltration in two dimensions with a linear boundary is discussed.
In Sec. \ref{localized}, we study RWI with localized sources in two- and three-dimensional
homogeneous media and in fractal lattices embedded in two and three dimensions, and establish
the connection with recurrence properties of random walks.
In Sec. \ref{fluid}, we study fluid infiltration in homogneous media and in a fractal.
In Sec. \ref{conclusion}, we summarize our results and present our conclusions.

\section{Random walk infiltration}
\label{model}

\subsection{Model definition}
\label{definition}

RWI is defined as a process in which particles are released from a static source
with constant concentration and in which those particles execute random walks in a lattice with
excluded volume interaction.
We denote as $a$ the edge of a lattice site, which can be occupied by at most one particle at a time.
In the time unit $\tau$, each particle attempts to hop to a randomly chosen nearest neighbor (NN) site;
the hop is allowed only if that site is empty, otherwise the particle does not move.
The diffusion coefficient of a free particle in a free lattice of dimension $d$ is
\begin{equation}
D=\frac{a^2}{2d\tau} .
\label{defD}
\end{equation}

Figs. \ref{diffmodel1d}(a),(b) illustrate the RWI in one dimension with a point
source at $x=0$ and all sites with $x>0$ empty at $t=0$.
As the particle at the source moves to $x=a$, another particle immediately refills the source
[Fig. \ref{diffmodel1d}(a)]; in this situation, only the particle at $x=a$ can move and the only
hop allowed for this particle is to the right.
Fig. \ref{diffmodel1d}(b) shows a configuration after several particles have entered the lattice,
in which only the particles at large distance from the origin can move.
The particle at the source is also unable to move in this situation; it will move only when the
site at $x=a$ is vacant, which requires diffusion of several intermediate particles.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.45\textwidth,angle=0]{diffmodel1d-eps-converted-to.pdf}
\caption{
(a) Initial configurations of RWI in a line with a particle at the source $x=0$.
A new particle is inserted at the origin as soon as the initial particle moves
to the neighboring site.
(b) Configuration developed after some time, in which the particle at the source is blocked
and the closest vacancy is at $x=5a$.
(c) Configuration after a few steps of RWI in a square lattice with a line source at $x=0$.
The arrows indicate the possible hops of the infiltrated and of the source particles.
}
\label{diffmodel1d}
\end{figure}

An equivalent model is RWI in the region $x>0$ of a square lattice with a line source at $x=0$,
as shown in Fig. \ref{diffmodel1d}(c).
Again, when one particle leaves the line $x=0$, another particle is immediately added to that point.

For the simulation of RWI, we adopt an algorithm in which the particles that attempt to hop are
sequentially chosen at random from a list that includes all infiltrated particles and all particles
at the source.
Thus, there is no preferential choice of particles at the front or at the source for the first
hop attempts.
This brings to the model an additional randomization factor to that of the choice of the hop directions.

\subsection{Continuous limit}
\label{continuous}

Let $\rho\left( \vec{r},t\right)$ be the particle density at position $\vec{r}$ at time $t$,
given in number of particles per lattice site.
Here we derive an equation for its evolution in the continuous limit.

For simplicity, consider the one-dimensional case shown in Figs. \ref{diffmodel1d}(a),(b).
The density at position $x$ changes from time $t-\tau$ to time $t$ as the master equation
\begin{eqnarray}
\rho\left( x,t\right) - \rho\left( x,t-\tau\right) &=&
\frac{1}{2}\rho\left( x-a,t-\tau\right) \left[ 1 - \rho\left( x,t-\tau\right) \right] \nonumber \\
&& +\frac{1}{2}\rho\left( x+a,t-\tau\right) \left[ 1 - \rho\left( x,t-\tau\right) \right] \nonumber \\
&& -\frac{1}{2}\rho\left( x,t-\tau\right) \left[ 1 - \rho\left( x-a,t-\tau\right) \right] \nonumber \\
&& -\frac{1}{2}\rho\left( x,t-\tau\right) \left[ 1 - \rho\left( x+a,t-\tau\right) \right] .\nonumber \\
&&
\label{master}
\end{eqnarray}
In the right rand size of Eq. (\ref{master}), the first and second terms account for the
hops from neighboring sites to the position $x$, which are possible only if this site is not occupied;
the third and fourth terms account for the hops from site $x$ to neighboring sites, which are
possible only if those sites are not occupied.
The joint probabilities in Eq. (\ref{master}) were factorized, which considers that
the densities at neighboring positions are uncorrelated;
this assumption is similar to the independent cluster approximation which is also applicable for
problems of diffusion, aggregation, and fragmentation, with or without particle injection
\cite{majumdarPRL1998,reisstinchcombe2005a}.
This assumption cancels the crossed terms in the master equation.
In the continuous limit, we consider
$\tau \frac{\partial \rho}{\partial t}\approx \rho\left( x,t\right) - \rho\left( x,t-\tau\right)$
and $\frac{a^2}{2} \frac{\partial^2 \rho}{\partial x^2}\approx
\left[ \rho\left( x-a,t-\tau\right)-2\rho\left( x,t-\tau\right) + \rho\left( x+a,t-\tau\right)\right]$,
which give
\begin{equation}
\frac{\partial \rho}{\partial t} = D \frac{\partial^2 \rho}{\partial x^2} ,
\label{diffusioneq1d}
\end{equation}
where $D$ is given in Eq. (\ref{defD}).
This is the one-dimensional diffusion equation.
Generalization of the above arguments to higher dimensions is straightforward, which leads to
\begin{equation}
\frac{\partial \rho}{\partial t} = D\nabla^2 \rho .
\label{diffusioneq}
\end{equation}
The boundary conditions are $\rho =1$ at the source at all times
and $\rho\left( \vec{r},t=0\right) = 0$ at all points but the source.

Within the approximations in Eq. (\ref{master}), the excluded volume condition of the RWI model does
not affect the collective diffusion coefficient $D$ because the frustrated hop of a particle in
one direction is compensated by the frustration of the hop of another particle in the opposite direction.
However, if the tracer diffusion coefficient is measured (i. e. the coefficient characterizing the
motion of a particular particle), it is expected to be smaller than $D$ due to the frustration of
many hop attempts.

This continuous limit is expected to be valid only for $t\gg\tau$, where average particle
displacements are much larger than the lattice constant $a$.
In this case, the density $\rho\left( \vec{r},t\right)$ slowly varies in time
and we can compare RWI data with the solution of the diffusion equation, if available.

\subsection{Basic quantities}
\label{basic}

In RWI, we denote as $I\left( t\right)$ the number of infiltrated sites of the lattice at time $t$;
this quantity is hereafter termed the infiltration.
In the continuous description, $I$ is obtained by integration of the density $\rho$ through
the available space.
In one, two, and three-dimensions, $I$ respectively represent the total length, area, and volume
infiltrated by the random walkers.
The infiltration is expected to scale as a power law in time with exponent $n$,
in a similar fashion the fluid infiltration in Eq. (\ref{defn})
(a discussion on the relation between RWI and fluid infiltration is postponed to Sec. \ref{fluid}).

We also calculate the root mean square (rms) distance of the infiltrated particles from the source,
$s_{RMS}\left( t\right)$; see illustration in Fig. \ref{infiltratedmedia}(a).
In RWI, a large fraction of the infiltrated particles move in regions with low density,
so that $s_{RMS}$ is expected to follow the same scaling law of free random walks in that medium:
\begin{equation}
s_{RMS} \sim t^{1/D_W} ,
\label{defDW}
\end{equation}
where $D_W$ is the random walk dimension.
In Euclidean lattices, we have normal diffusion with $D_W=2$, while in fractal lattices we
expect subdiffusion with $D_W>2$.

If the medium has a fractal dimension $D_F$ ($D_F=d$ in the cases of Euclidean lattices
in $d$ dimensions), then we define a characteristic infiltrated size $S$ by the equation
\begin{equation}
I = S^{D_F} .
\label{defS}
\end{equation}
$S$ may be interpreted as a characteristic radius of a region in which all infiltrated particles
are symmetrically located around the source with no empty site between them.
In fluid infiltration, the radius of the infiltrated region has the same scaling as $S$
(Sec. \ref{fluid}).
Figs. \ref{infiltratedmedia}(a),(b) illustrate this definition in effectively one-dimensional
infiltration problems.

The quantities calculated here are presented in terms of the dimensionless time
\begin{equation}
T\equiv \frac{t}{\tau} .
\label{defT}
\end{equation}
This is the average number of hop attempts of a particle that was located at the infiltration
boundary at $t=0$.

\subsection{Example: RWI in a square lattice with a line source}
\label{numerical1d}

Figs. \ref{infsquare}(a) and \ref{infsquare}(b) show configurations of an infiltrated square
lattice at $T=1000$ and $T=4000$, respectively, considering a line source at $x=0$.
Figs. \ref{infsquare}(c) and \ref{infsquare}(d) show the corresponding density distributions.
In each time, the numerical data are averages over $20$ configurations and source length $1000a$.
Figs. \ref{infsquare}(c) and \ref{infsquare}(d) also show the exact solution of
Eq. (\ref{diffusioneq1d}), $\rho\left( x,t\right) = 1-erf\left[ x/\left( 2\sqrt{Dt}\right)\right]$,
where $erf\left( u\right)$ denotes the error function of $u$.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=0.45\textwidth,angle=0]{infsquare-eps-converted-to.pdf}
\caption{
(a), (b) Sections of size $100a$ of an infiltrated square lattice with a line source at the left
in the two indicated times.
(c), (d) The corresponding concentration distributions calculated numerically (green squares) and
analytically (blue curves).
}
\label{infsquare}
\end{figure}

There is a very good agreement between RWI data and the solution of the diffusion equation in this case.
This was formerly noted in Ref. \protect\cite{sapovalJPL1985}.
Discrepancies between the RWI and the continuous model are observed only at positions $x$ larger than
$\sqrt{Dt}$ by a factor of order $10$ [not shown in Figs. \ref{infsquare}(c),(d)],
since the diffusion equation fails to represent the RWI in these regions (e. g. the infinitely long
empty region in RWI due to the finite particle velocity).

The infiltration in this geometry scales as $I\sim T^{1/2}$ and Eq. (\ref{defS}) implies that the
characteristic infiltration length also scales as $S\sim T^{1/2}$.
This is the same scaling observed for the rms distance $s_{RMS}\left( t\right)$.

\subsection{Interpretation and relation with other models}
\label{relation}

RWI may represent the diffusive motion of a chemical species in a fluid that fills
the pores of a medium in contact with a reservoir (the source) of that species.
This interpretation suggests the application of the model to infiltration of solutes in rocks
or soils in cases where advective transport is negligible.
A particle in this model is not necessarily equivalent to a solute molecule, but may
represent a large number of molecules that entered the medium; in this case, the presence
or absence of a particle represents a local solute concentration above or below a given threshold,
respectively.
Consider, for instance, the application of this molecular interpretation to the configuration
of Fig. \ref{diffmodel1d}(b), in which particle hops from the source are not allowed due to
excluded volume effect.
It represents a case in which the region of the medium near the reservoir has a density
of molecules approximately equal to that of the reservoir.
In lengthscales smaller than the size of a lattice site, molecular diffusion has not ceased
inside and outside the medium, but the net flux from the source to the medium is zero.

The infiltrated region has many empty sites (vacancies) in the neighborhood of the infiltrated
particles, as illustrated in Figs. \ref{infsquare}(a) and \ref{infsquare}(b) for the case of a
line source in two dimensions.
The random walks of particles are equivalent to random walks of these vacancies, with excluded volume
conditions and with the same diffusion coefficient $D$.
When a vacancy reaches the source, it is immediately annihilated; this corresponds to the instantaneous
refilling of the source in RWI.
Thus, from the point of view of vacancy diffusion, we have a classical trapping problem
\cite{Havlin,rednerbook}.
Letting $\sigma\equiv 1-\rho$ be the density of vacancies, it also obeys the diffusion equation
(\ref{diffusioneq}); however, the boundary condition is $\sigma =0$ at the source.

The one-dimensional version of RWI [Fig. \ref{diffmodel1d}(a)] is equivalent to the trapping
of the vacancies by a static trap at the origin.
This model was originally proposed by Weiss, Kopelman, and Havlin \cite{weiss1989}.
The increase of the infiltration as $t^{1/2}$, which is observed in RWI,
is equivalent to the trapping rate decay as $t^{-1/2}$.
Despite the apparent simplicity, that trapping model has some nontrivial features; for instance,
the distance from the origin of the nearest vacancy increases anomalously as $t^{1/4}$
\cite{weiss1989}.
Similar anomalies were recently observed in other trapping models \cite{flekkoy2017}.

The square lattice version of RWI [Fig. \ref{diffmodel1d}(c)] was first proposed
by Sapoval, Rosso, and Gouyet \cite{sapovalJPL1985} for investigating the properties of the
interface of a conducting cluster formed by the particles connected to the source.
Subsequently, it motivated the proposal of the gradient percolation problem \cite{bunde1985,rosso1986}.
In a recent work, RWI in deterministic fractals was studied, also with the sources at flat boundaries
\cite{Reis2016}, which helped to understand scaling properties of infiltration models.

\section{RWI with localized sources}
\label{localized}

Here we consider the RWI models defined in Sec. \ref{definition}
with a point source localized in the neighborhood of an impenetrable boundary,
as illustrated in Figs. \ref{diffmodel2d3d}(a) ($d=2$) and \ref{diffmodel2d3d}(b) ($d=3$).
This geometry is chosen to represent a large porous medium with a narrow hole at a
surface which is in contact with a reservoir of the solute.
The same geometry is also considered in simulations of RWI in determininistic fractals
embedded in two- and three-dimensional lattices.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.45\textwidth,angle=0]{diffmodel2d3d-eps-converted-to.pdf}
\caption{
Initial condition of RWI with a point source (red circle and sphere) and an impenetrable
boundary at $x=0$ (blue) in (a) two and (b) three dimensions.
}
\label{diffmodel2d3d}
\end{figure}

For simplicity, solutions of the diffusion equation (\ref{diffusioneq}) will be obtained
with the source in media that are infinitely large in all directions (i.e., without the
impenetrable boundary).
This is expected to provide the same time scaling of the relevant quantities.

\subsection{Infiltration in homogeneous media}
\label{homogeneous}

\subsubsection{Two dimensions}
\label{twodimensions}

In this case, an equivalent diffusion model considers an infinite line source of radius $a$
and constant density $\rho(r \le a, t\geq 0)=1$ placed in an infinite three-dimensional domain
initially with $\rho(r>a,t=0)=0$.
The subsequent change in the density in a plain of fixed $z$ is governed by the diffusion equation
\begin{equation}
\label{eqdiff2d}
\frac{\partial \rho}{\partial t}=\frac{D}{r} \frac{\partial}{\partial r}\left(r\frac{\partial \rho}
{\partial r} \right), \ \ \ r>a, \ \rho\left(r \rightarrow \infty, t\right)=0 .
\end{equation}

Ref. \protect\cite{crank} (pp. 87-88) provides a closed solution for this problem,
from which we can calculate the amount fluxed into the region $r>a$ per unit area and time,
$F\equiv -D\left(\frac{\partial \rho}{\partial r} \right)_{r=a}$.
Assuming for convenience that $D=1/4$ and $a=1$ [which leads to $T=t$ in Eq.(\ref{defT})],
the solution for large times is
\begin{equation}
\label{flux2d}
F=\frac{1}{2} \left[ \left(\frac{1}{\ln{T}-2\gamma}  -\frac{1}{(\ln{T}-2\gamma)^2} \right)
+\frac{1-\gamma} {(\ln{T}-2\gamma)^2}  \ldots \right] ,
\end{equation}
where $\gamma=0.57722$ is Euler's constant and $T$ is defined in Eq. (\ref{defT}).
From this we can calculate the infiltration through a simple integration in time;
this is denoted as $I_{DE}$ to stress that it results from the solution of a diffusion equation.
Integrating the first bracketed term on the right hand side of Eq. (\ref{flux2d}),
the infiltration in the half plane $x\geq 0$ [as in the RWI geometry of Fig. \ref{diffmodel2d3d}(a)]
is, to leading order,
\begin{equation}
\label{I2d}
I_{DE}\left( t\right) = \pi \int_0^t F dT = \frac{\pi}{2} \frac{T}{\log(T)-2\gamma} .
\end{equation}

We performed simulations of RWI in square lattices of lateral size $L=2000$ with a point source in
the middle of a border up to $T=2\times {10}^4$, which led to the injection of more than
$3\times {10}^3$ particles.
The number of configurations used to calculate $I$ and $s_{RMS}$ was ${10}^3$ (i. e. this is the number of
times in which particles were injected in an empty lattice).

Fig. \ref{infilt2d}(a) shows $I$ (the number of infiltrated lattice sites) as a function of $T/\ln{T}$.
The inset of Fig. \ref{infilt2d}(a) shows that the data obtained in two different configurations are
very close to each other.
This means that the accuracy of the averaged data is very high, so that the standard deviations in
the main plot are much smaller than the size of the data points.
The excellent linear fit in that plot confirms that $I$ has the same
logarithmic corrected time scaling predicted by the continuous model [Eq. (\ref{I2d})].
Fig. \ref{infilt2d}(b) shows a bilogarithmic plot of $s_{RMS}$ as a function of $T$, which
confirms the diffusive increase of that average displacement; the accuracy was similar to that of $I$.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=0.45\textwidth,angle=0]{infilt2d-eps-converted-to.pdf}
\caption{
RWI from a point source in two dimensions:
(a) infiltration versus $T/\ln{T}$ (red squares) and a linear fit of the data (dashed blue line);
(b) rms distance from the source versus $T^{1/2}$ (blue triangles) and a linear fit of the data
(dashed green line) whose slope is $0.499$.
The inset in (a) compares results in two configurations (blue line and orange squares).
}
\label{infilt2d}
\end{figure}

The characteristic length of the infiltration [Eq. (\ref{defS})] scales as
$S\sim {\left( T/\ln{T}\right)}^{1/2}$.
Thus, it is smaller than the rms distance $s_{RMS}$ due to the logarithmic correction.

\subsubsection{Three dimensions}
\label{threedimensions}

In this case, the equivalent diffusion model has a spherical point source of radius $a$ and
constant density $\rho\left(r \le a, t>0\right)=1$, placed in a infinite domain initially
with $\rho\left( r>a,t=0\right) =0$.
The subsequent change in the potential in $r>a$ is governed by the diffusion equation
\begin{equation}
\label{diffeq3d}
\frac{\partial \rho}{\partial t}=\frac{D}{r^2} \frac{\partial}{\partial r}
\left(r^2\frac{\partial \rho}{\partial r} \right) , \ \ \ r>a, \
u\left( r \rightarrow \infty, t\right) =0
\end{equation}
The analytical solution (Ref. \protect\cite{crank}, pp. 102-103) is
\begin{equation}
\label{rho3d}
\rho=\frac{a}{r} \text{erfc}\left(\frac{r-a}{2\sqrt{Dt}}\right)
\end{equation}
The amount fluxed into the region $r>a$ per unit area and time is given as
\begin{equation}\label{F3d}
F=-D\left(\frac{\partial u}{\partial r} \right)_{r=a}=\frac{D}{a} + \sqrt{\frac{D}{\pi t}}
\end{equation}
From this we can calculate the infiltration $I_{DE}$ through a simple integration in time.
Considering $D=1/6$ and $a=1$ [which leads to $T=t$ in Eq.(\ref{defT})] and
integrating only in the region $x\geq 0$ [as in Fig. \ref{diffmodel2d3d}(b)], we obtain
\begin{equation}\label{I3d}
I_{DE} = 2 \pi \int_0^t F dt = \frac{\pi}{3} T+\frac{4 \sqrt{\pi}}{ \sqrt{T}}  .
\end{equation}

Our simulations of RWI were run up to $T=2\times {10}^4$ in a lattice with lateral size $L=700$;
$I$ and $s_{RMS}$ were averaged over ${10}^3$ configurations, which also provides
these quantities with high accuracy.
Approximately ${10}^4$ particles were injected in each configuration.

Fig. \ref{infilt3d}(a) shows $I$ as a function of $T$; the linear fit confirms that it has
the same scaling as $I_{DE}$ in Eq. (\ref{I3d}).
The characteristic length of the infiltration [Eq. (\ref{defS})] consequently scales as
$S\sim T^{1/3}$.
Fig. \ref{infilt3d}(b) shows a bilogarithmic plot of $s_{RMS}$ as a function of $T$, which
confirms the diffusive increase of that average distance.
In this case, we also observe that the characteristic length $S$ defined from the infiltrated
volume does not have the diffusive scaling of the rms distance from the source.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=0.45\textwidth,angle=0]{infilt3d-eps-converted-to.pdf}
\caption{
Results for infiltration from a point source in three dimensions:
(a) infiltration versus $T$ (orange squares) and a linear fit of the data (dashed green line);
(b) rms distance from the source versus $T$ (green triangles) and a linear fit of the data
(dashed blue line) whose slope is $0.500$.
}
\label{infilt3d}
\end{figure}

\subsection{Scaling approach for the infiltration}
\label{scaling}

Here we explore the observation of Sec. \ref{relation} that RWI is equivalent to the problem
of trapping of vacancies at the source.
It implies that the infiltration rate $\frac{\text{d}I}{\text{d}t}$ is equal to the probability
$P_{trapping}\left( t\right)$ of a vacancy to be trapped at the source at time $t$.
This is the probability that the first passage of the vacancy at the source occurs at time $t$.
For long times, it is independent of the source position,
but it is strongly dependent on the spatial dimension \cite{montroll,rednerbook}.

$I$ is obtained by the integration of the first passage probability.
Since this probability can be obtained as a property of a single random walk, $I$ is equal to
the sum of this probability over the hops of a single walker up to time $t$.
When the walker hops to a site for the first time, the number of distinct visited sites increases
by one unit; thus, $I\left( t\right)$ equals the number of distinct sites visited by the walker
in the medium up to time $t$, which is denoted as $N_D\left( t\right)$ \cite{montroll,rednerbook}.
The infiltration exponent $n$ in Eq. (\ref{defn}) is the same exponent of the time scaling
of the number of distinct sites visited by a random walker; an amplitude relating $I$ and
$t^n$, which is omitted in Eq. (\ref{defn}), may depend on the boundary conditions.

In two dimensions, the dominant term in the scaling of the number of distinct sites visited
by a random walker is $N_D\sim t/\ln{t}$ \cite{montroll}; this agrees with the simulation
result for the RWI in Sec. \ref{twodimensions}.
In three dimensions, the dominant term is $N_D\sim t$ \cite{montroll}, which also agrees with
our simulation results for RWI (Sec. \ref{threedimensions}).

These results establish a relation between RWI and recurrence properties of lattice random walks.
Moreover, since the time evolution of the infiltration in RWI may also be predicted by the
diffusion equation with constant concentration at the source, our approach also connects
the solution of this equation with the recurrence properties of random walks.

\subsection{Infiltration in Sierpinski carpets}
\label{carpets}

Here we study RWI infiltration with localized sources in fractals embedded in two
dimensions combining the previous scaling approach, numerical results of RWI, and results
of numerical integration of the diffusion equation.

The iterative construction of a Sierpinski carpet is illustrated in Fig. \ref{diffmodelcarpet}(a).
We denote as $b$ the scaling factor of the carpet and $m$ the number of
subsquares removed in the first iteration of the construction.
At each stage of construction, each subsquare is replaced by the generator (stage $n=1$).
The removed subsquares represent the impenetrable (solid) part of the medium and the empty
subsquares form the porous medium.
The length of an outer border of the carpet after $n$ iterations is $L=b^n$.
The fractal is obtained after an infinite number of iterations, with fractal dimension
$D_F=\ln{\left( b^2-m\right)}/\ln{b}$.
The initial condition for RWI in a carpet with a point source is shown in
Fig. \ref{diffmodelcarpet}(b).

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.45\textwidth,angle=0]{diffmodelcarpet-eps-converted-to.pdf}
\caption{
(a) First two stages of construction of the carpet with $b=3$ and $m=1$, with the impenetrable
solid in blue.
(b) Initial condition for RWI in a carpet with a point source (red) at an outer boundary.
(c) Initial condition for RWI in a sponge with a point source (red) at an outer boundary.
}
\label{diffmodelcarpet}
\end{figure}

The number of distinct sites visited by a random walker in this fractal is expected to scale as
\begin{equation}
N_D \sim t^{D_S/2} \qquad , \qquad D_S = 2 \frac{D_F}{D_W} ,
\label{AOrelation}
\end{equation}
where $D_S$ is the spectral (or fracton) dimension.
Eq. (\ref{AOrelation}) is known as Alexander-Orbach (AO) relation \cite{AO,Havlin}; it is based on
the assumption of homogeneous distribution of the random walker position within the accessible
fractal volume at time $t$.
However, the AO relation has been criticized in the past due to deviations observed in some
numerical results; for a discussion, see Ref. \protect\cite{Havlin}, Ch. 7,
and Ref. \protect\cite{sokolov}.

For RWI, Eqs. (\ref{defn}) and (\ref{AOrelation}) give the infiltration exponent $n=D_F/D_W$.
In the carpet of Fig. \ref{diffmodelcarpet}(a), $D_F=\ln{8}/\ln{3}=1.8928\dots$.
Ref. \protect\cite{balankin2017} conjectured that the exact value of the random walk dimension
in this carpet is $D_W=2+\ln{\left( 10/9\right)}/\ln{3} =2.0959\dots$; this value agrees with the
numerical estimates $2.101\left( 22\right)$ \cite{reisJPA1995} and
$2.106\left( 16\right)$ \cite{kimJPA1993}.
Using the exact $D_F$ and the conjectured $D_W$, we obtain $n=0.903\dots$.

We performed simulations of RWI in the $7^{\text{th}}$ stage of construction of this carpet
up to $T={10}^5$.
Averages were also taken over ${10}^3$ configurations, with more than ${10}^4$
injected particles in each one; thus, the accuracy of the data is similar to that in
Fig. \ref{infilt2d}.

In Fig. \ref{infpont31}(a), we show a bilogarithmic plot of $I$ versus $T$ in this fractal.
The slope at long times is still slightly smaller than the proposed value of $n$.
This is consistent with previous calculations of $N_D$ in the carpets, which suggested corrections
to the AO relation \cite{reisPLA1996,dasgupta1999}.
In Fig. \ref{infpont31}(b), we show the ratio $I/T^{0.903}$  as a function of $1/T^{0.27}$;
the variable in the abscissa is the one that provides the best linear fit of the data in the
range $5\times {10}^2\leq T\leq {10}^5$ (among other variables in the form $T^{-\lambda}$ with
positive $\lambda$).
The convergence to a finite value at long times confirms that the infiltration scales
asymptotically with the proposed value of $n$, and the small exponent in the abscissa of
Fig. \ref{infpont31}(b) indicates the presence of large subdominant corrections to the AO relation.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.45\textwidth,angle=0]{infpont31-eps-converted-to.pdf}
\caption{
(a) Time evolution of the infiltration from a point source in the SC with $b=3$ and $m=1$.
The dashed line has the theoretically predicted slope $0.903$.
(b) Scaled infiltration as a function of $1/t^{0.27}$ for RWI in the Sierpinski carpet.
The solid line is a linear fit of the data for $5\times {10}^2\leq T\leq {10}^5$.
}
\label{infpont31}
\end{figure}

For comparing different methods of solution for the RWI problem,
we also solved Eq. (\ref{diffusioneq}) numerically.
We use a time explicit finite-difference control volume solution for the
solute infiltration from a point source ($\rho =1$) into the stage $n=3$ of the same
Sierpinski carpet [Fig. \ref{diffmodelcarpet}(a)], with diffsuivity $D=1$ at the empty subsquares
and with the solid squares representing areas of very low diffusivity $D<<1$.
The numerical solutions, similar to that outlined in  \cite{Voller_wrr}, involves a $3^6 \times 3^6$
mesh of square node-centered control volumes of side length $\Delta =1/3^6$ and, to retain stability
of the explicit scheme, a  time step $\Delta t = 0.125\times\Delta^2$.

Fig. \ref{VVadd1REV}(a) shows some concentration contours after 120,000 time steps.
The measure over time of this simulation is the solute content $I_{DE}$, see Eq. (\ref{I2d}),
from which we obtain the local time exponent $n= d{\text ln}{I_{DE}}/d{\text ln}t$.
The key observation in Fig. \ref{VVadd1REV}(b) is that this exponent approaches,
in long times, a consatnt  value that is within fractions of a percent of the expected value
of $\sim$0.903.
Consequently, this solution significantly improves that of direct simulation of RWI
(Fig. \ref{infpont31}).
More importantly, it confirms with very good accuracy
that the AO relation is valid in a Sierpinski carpet,
advancing over the current and previous random walk based methods.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.5\textwidth,angle=0]{VVadd1REV.pdf}
\caption{
Solute diffusion into a 3$^{\mbox{rd}}$ stage Sierpinski carpet with $b=3, m=1$.
(a) The concentration contours after 120,000 time steps: from right to left,
$\rho=0.0.005$, $0.01$, $0.05$, and from $0.1$ to $0.9$ in intervals of $0.1$.
Solid squares and pores are not distinguished by different colors here.
(b) The local time exponent $n= d{\text ln}I_{DE}/d{\text ln}t$; the dashed line has the theoretically
predicted value $0.903$.}
\label{VVadd1REV}
\end{figure}

The oscillations observed in Fig. \ref{VVadd1REV}(b) at long times resemble the
log-periodic oscillations shown in Refs. \protect\cite{babJCP2008,akkermans} in models
of random walks in regular fractals.
Some methods were developed to treat data with this type of oscillation \cite{babJCP2008,haber2014},
which may be helpful whenever accurate exponent estimates are not available.

In Ref. \protect\cite{Reis2016}, a scaling approach was used to relate the infiltration exponent
$n$ [Eq. (\ref{defn})] and the random walk dimension $D_W$ in cases where the source was
one of the outer boundaries of the fractal.
For a border with dimension $D_B$, it was shown that
\begin{equation}
n = \frac{D_F -D_B}{D_W} .
\label{scalingnnu}
\end{equation}
This result may be extended to infiltration from a point source in a carpet because, in this case,
$D_B=0$, which gives $n=D_F/D_W$.

\subsection{RWI in Menger sponges}
\label{sponge}

A Menger sponge is iteratively constructed here by extending to three dimensions the
procedure used to construct the SCs:
first, a cube is divided in $b^3$ subcubes and $m$ symmetrically distributed subcubes are removed,
which forms the generator; each one of the subcubes is then divided following the same rule,
and this process is repeated indefinitely.
The fractal dimension of the sponge is $D_F=\ln{\left( b^3-m\right)}/\ln{b}$.

Here we study RWI in the sponge with $b=5$ and $m=27$.
The initial condition for RWI with a point source in this sponge
is illustrated in Fig. \ref{diffmodelcarpet}(c).
The fractal dimension is $D_F=\ln{98}/\ln{5}\approx 2.84880$ and the
estimate of the random walk dimension is $D_W = 2.09(6)$ \cite{Reis2016}.

These values imply $D_F > D_W$.
The AO relation [Eq. (\ref{AOrelation})] is expected to be valid only when random walks are
recurrent in a given fractal, but this is not the case here.
Instead, here the accessible region for the infiltrated particles has a radius $\sim s_{RMS}$ and
a volume of order $s_{RMS}^{D_F}\sim t^{D_F/D_W}\gg t$, i.e. this region is much larger than the
number of sites that the walker can visit within a time $t$.
This means that only a small fraction of the accessible sites is visited, and this fraction tends
to zero asymptotically.
In such cases, we expect that $N_D$ scales linearly in time, and so $I$ is also linear in time.

We simulated RWI in the $4^{\text{th}}$ stage of construction of the sponge up to $T=2\times {10}^4$.
Averages were taken over ${10}^3$ configurations and more than $1.5\times {10}^4$ particles were
injected in each configuration.
Fig. \ref{infpontm53} shows that the infiltrated volume $I$ increases linearly in time in all
the simulated time interval, which confirms the above prediction.
Note that the same scaling was obtained in a homogeneous three-dimensional medium with a point source,
as shown in Sec. \ref{threedimensions}.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.35\textwidth,angle=0]{infpontm53-eps-converted-to.pdf}
\caption{Infiltrated volume as a function of time for RWI in the Menger sponge.
The solid line is a linear fit of the data.
}
\label{infpontm53}
\end{figure}

It is interesting to observe that the infiltration in this fractal is normal,
although the diffusion of a single particle is clearly anomalous ($D_W >2$).
The same observation is applicable to other fractals where random walks are not recurrent,
i.e. $D_W<D_F$.
Consequently, in an infiltration experiment on such a fractal with a localized source, the
measurements of infiltrated mass or volume will not show any anomaly, but the same linear increase
observed in a homogeneous medium.
In order to observe the anomaly, it is necessary to measure the diffusion lengths
of the infiltrated molecules (or of other tracer particles).

RWI in the sponge also shows a failure of the scaling relation (\ref{scalingnnu}), since that relation
implies the AO relation for $D_B=0$.
Indeed, Eq. (\ref{scalingnnu}) was obtained in Ref. \protect\cite{Reis2016} under the assumption
that a finite fraction of a region of lateral size $s_{RMS}$ was filled with infiltrated particles.
This is actually the case in two-dimensional systems with linear sources or three-dimensional
systems with planar sources.
However, in a fractal with large $D_F$, a point source is too small to fill a finite fraction
of a region with that size.

\section{Relation with fluid infiltration}
\label{fluid}

Here we consider infiltration of a fluid with a constant pressure head $h=1$ at a fixed boundary,
which is the source of the fluid molecules, and assume that the head vanishes at the infiltration
front.
In the region limited by these boundaries, the slowly varying pressure head $h$ obeys the
Laplace equation
\begin{equation}
\nabla^2 h = 0 ,
\label{laplacianh}
\end{equation}
and the velocity of the infiltration front obeys the Darcy law
\begin{equation}
\vec{v} = -K \vec{\nabla} h ,
\label{darcy}
\end{equation}
where $K$ is the hydraulic conductivity.
For simplicity, we consider $K=1$ in a homogeneous medium.

This model was numerically studied in several carpets with different boundaries in Refs.
\protect\cite{Voller_wrr} and \protect\cite{reisbolstervoller}.
The estimates of the infiltration exponent $n$ were in excellent agreement with estimates from
the RWI model and with the scaling relation (\ref{scalingnnu}) \cite{Reis2016}.
Experiments with infiltration in finite stages of construction of some carpets
also showed good agreement with those estimates \cite{Voller_wrr2}.

Here we extend this comparison to infiltration with localized sources.

\subsection{Two dimensions and localized source}
\label{fluid2d}

We consider the problem in a two-dimensional cylindrical geometry, where a small source is
located at $r=a$.
Eq. (\ref{laplacianh}) becomes
\begin{equation}
\frac{\partial}{\partial r}\left(r \frac{\partial h}{\partial r}\right)=0, \  \  \ h(a)=1, \ h(R)=0 ,
\label{eqh2dpoint}
\end{equation}
which gives the head profile
\begin{equation}
h=\frac{\ln{r}- \ln{R}}{\ln{a} -\ln{R}}, \  \  \ a\le r \le R .
\label{h2dpoint}
\end{equation}
Eq. (\ref{darcy}) for the moving front becomes
\begin{equation}
- \left(\frac{\partial h}{\partial r} \right)_{r=R}=\frac{dR}{dt}
\label{front2dpoint}
\end{equation}
and substitution of Eq. (\ref{h2dpoint}) gives
\begin{equation}
\frac{dR}{dt}=-\frac{1}{R\left(\ln{a} - \ln{R}\right) }, \  \  \ R\left( 0\right) =a
\label{eqR2dpoint}
\end{equation}
The solution for $R(t)$ is
\begin{equation}
\frac{R^2}{2} \left(\ln{R}- \log a \right)+\frac{a^2-R^2}{4}=t
\label{R2dpoint}
\end{equation}
In limit of $R>>a$, i.e. when the front displacement is much larger than the radius of the source,
we have
\begin{equation}
R\approx 2{\left( \frac{t}{\ln{t}} \right)}^{\frac{1}{2}}  .
\label{R2dpointapprox}
\end{equation}
The infiltrated area in this limit is
\begin{equation}
A\approx 4\pi\left( \frac{t}{\ln{t}} \right)  .
\label{A2dpointapprox}
\end{equation}

This is the same linear increase depleted by a logarithmic correction that we observed in RWI
(Sec. \ref{twodimensions}).
The front radius $R$ is equivalent to the length $S$ defined for RWI [Eq. (\ref{defS})],
and $R$ may be directly measured in an experiment.

\subsection{Three dimensions and localized source}
\label{fluid3d}

The model can be extended to three dimensions, where Eq. (\ref{laplacianh}) for the head leads to
\begin{equation}
\frac{\partial}{\partial r}\left(r^2 \frac{\partial h}{\partial r}\right)=0, \  \  \ h(a)=1, \ h(R)=0
\label{eqh3dpoint}
\end{equation}
with the same moving front condition as in the cylindrical case
\begin{equation}
- \left(\frac{\partial h}{\partial r} \right)_{r=R}=\frac{dR}{dt} .
\label{front3dpoint}
\end{equation}
The solution of Eq. (\ref{eqh3dpoint}) in $ a\le r \le R$ is
\begin{equation}
h=\frac{1}{r}\frac{aR}{R-a}-\frac{a}{R-a} .
\label{h3dpoint}
\end{equation}
Substituting Eq. (\ref{h3dpoint}) in Eq. (\ref{front3dpoint}), we obtain the equation for the
movement of the front as
\begin{equation}
\frac{dR}{dt}=\frac{a}{R^2-aR}, \  \  \ R(0)=a .
\label{eqR3dpointapprox}
\end{equation}
The solution for $R(t)$ is
\begin{equation}
\frac{R^3}{3a}-\frac{R^2}{2}+\frac{a^2}{6}=t .
\label{R3dpoint}
\end{equation}
In limit where the front displacement is much larger than the source radius, $R>>a$, we have
\begin{equation}
R\approx {\left( 3at\right)}^{1/3}
\label{R3dpointapprox}
\end{equation}
and the infiltrated volume is
\begin{equation}
V\approx 4\pi at
\label{V3dpointapprox}
\end{equation}

Here, the fluid infiltration shows the same linear time increase of the RWI in three dimensional
homogeneous media.
Note that the radius $R$, which may be experimentally measured, has a subdiffusive scaling,
which contrasts the rms displacement of RWI; again, $R$ is analogous to the length $S$ defined for RWI.

\subsection{Extension to fractal media}
\label{universality}

We simulated the fluid infiltration model in the $3^{\text{rd}}$ stage of construction of
the carpet with $b=3$ and $m=1$ and with a point source in an external boundary using
the same control volume finite difference methods of Ref. \protect\cite{Voller_wrr}.
Essentially, at each time step, this involves the numerical solution of
$\nabla (K \nabla h)=0$, with $K=1$ in the spaces and $K<<1$ in the obstacles.
In the current simulations, we use a  $3^5 \times 3^5$ mesh of square node centered control
volumes of size $\Delta =1/3^5$ and a time step of $\Delta t= 0.005$;
as fully explained in Ref. \protect\cite{Chen}, the accuracy of the calculation is independent
of the time step size.

In Fig. \ref{VVadd2}a, we show the infiltration front at the time in which the fluid reaches
the horizontal domain boundaries.
In Fig. \ref{VVadd2}b, we show the time evolution of the infiltrated area.
This area scales with an exponent close to the theoretical value $0.903$, which is also in
agreement with that of RWI in the same carpet (Sec. \ref{carpets}).

In this simulation, the dimension of the gate $a$ for the infiltration is 1/9 the size of the
smallest obstacle dimension in the carpet; we would expect values closer to the theory as the
relative size of the gate is reduced.
In fact, simulations on the geometry in Fig. \ref{VVadd2} with coarse grids and thus wider gates
show a quadratic dependence between the exponent $n$ and gate size;
in the continuous limit ($a\rightarrow 0$), this quadratic fit suggests an exponent $n=0.8994$, which differs less
than $0.5\%$ from the theoretical value.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.5\textwidth,angle=0]{VVFig11redo.pdf}
\caption{Area filled with fluid as a function of time in the carpet with $b=3$ and $m=1$.
(a) The infiltration front at the point the fluid reaches the horizontal boundary.
(b) The log-log plot of the infiltration versus time.
}
\label{VVadd2}
\end{figure}

\section{Conclusion}
\label{conclusion}

We studied a random walk infiltration model with small sources at the boundaries of homogeneous
and fractal media.
The model represents the penetration of a solute from a reservoir in contact with a porous medium by
a narrow hole, viz. the source.
A scaling approach proposes that the number of infiltrated particles increases in time
as the number of distinct sites visited by a single random walker in the same medium,
$N_D\left( t\right)$.
This is confirmed by numerical results obtained by direct simulations of the lattice model and
by integration of the diffusion equation that represents the model in the continuous limit.
In a Sierpinski carpet, the numerical integration of the diffusion equation provides an exponent
for the time evolution of the infiltration which is very close to the value of the
Alexander-Orbach conjecture for $N_D$, improving previous results obtained from random walk models.
In a Menger sponge where the fractal dimension is larger than the random walk dimension, the
infiltrated volume increases linearly in time, similarly to a homogeneous medium.

We also studied a fluid infiltration model in the same geometry, with a constant pressure head
at the source and with the front displacement driven by the local gradient of that head.
Exact and numerical solutions in two and three dimensions and in Sierpinski carpets
show that the fluid infiltration model is in the same universality class of the random walk
infiltration.
The main difference between these models is the sharp front of fluid infiltration, while in RWI
the front is diffuse; in Fig. \ref{VVadd3REV}, these differences are highlighted, as well as the
differences between the random walk infiltration in a lattice and the solution of the
corresponding diffusion equation.
However, in the two models, the evolution of the fronts are driven by the gradient
of two analogous fields, which are the fluid head and the particle concentration.
Note that the equivalence between the two problems is observed in cases where random walks are
recurrent (carpets or two-dimensional media) and not recurrent (three-dimensional homogeneous media).
The scaling approach then sets a relation between the different infiltration
models from localized sources and the recurrence properties of random walks.

\begin{figure}[!ht]
\center
\includegraphics[clip,width=.35\textwidth,angle=0]{VVadd3REV.pdf}
\caption{Despite the different natures of the conservation of the RWI, diffusion equation, and
fluid infiltration processes, they all have identical scaling,
$I_{RWI}\sim I_{DE}\sim I_{fluid} \sim t^n$.
}
\label{VVadd3REV}
\end{figure}

In fractal media where random walks are not recurrent, we show the failure of a previous exponent
relation for infiltration from flat boundaries \cite{Reis2016}.
This is the case of the Menger sponge studied here and of other fractals embedded in three dimensions
with the fractal dimension exceeding the random walk dimension (typically fractals with large
dimensions and weak subdiffusion).
In these cases, the normal infiltration scaling of three dimensional systems, $I\sim t$, is obtained,
which hides the fractality of the infiltrated medium.

As a final note, we recall that normal diffusion was already observed in several fractals,
i.e. $D_W=2$ \cite{burioni1994,forte2014,balankin2018}.
However, infiltration may be anomalous in those systems because the exponent
$n$ also depends on the dimension of the medium and on the dimension of the source,
as shown here for the case of small localized sources and in previous works for extended sources.

\begin{acknowledgments}

FDAAR acknowledges support by the Brazilian agencies CNPq (304766/2014-3) and FAPERJ
(E-26/202941/2015) and thanks the hospitality of the Department of Civil, Environmental,
and Geo-Engineering of University of Minnesota, where part of this work was done.
VRV acknowledges support from the James L. Record Professorship.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv