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Bankruptcy fraud should be distinguished from "strategic bankruptcy", which is not a criminal act since it creates a real (not a fake) bankruptcy state. However, it may still work against the filer.
All assets must be disclosed in bankruptcy schedules whether or not the debtor believes the asset has a net value. This is because once a bankruptcy petition is filed, it is for the creditors, not the debtor, to decide whether a particular asset has value. The future ramifications of omitting assets from schedules can be quite serious for the offending debtor. In the United States, a closed bankruptcy may be reopened by motion of a creditor or the U.S. trustee if a debtor attempts to later assert ownership of such an "unscheduled asset" after being discharged of all debt in the bankruptcy. The trustee may then seize the asset and liquidate it to benefit the (formerly discharged) creditors. Whether or not a concealment of such an asset should also be considered for prosecution as fraud or perjury would then be at the discretion of the judge or U.S. Trustee.
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By country.
In some countries, such as the United Kingdom, bankruptcy is limited to individuals; other forms of insolvency proceedings (such as liquidation and administration) are applied to companies. In the United States, "bankruptcy" is applied more broadly to formal insolvency proceedings. In some countries, such as in Finland, bankruptcy is limited only to companies and individuals who are insolvent are condemned to de facto indentured servitude or minimum social benefits until their debts are paid in full, with accrued interest except when the court decides to show rare clemency by accepting a debtors application for debt restructuring, in which case an individual may have the amount of remaining debt reduced or be released from the debt.
Argentina.
In Argentina, the national Act "24.522 de Concursos y Quiebras" regulates Bankruptcy and Reorganization of individuals and companies; public entities are not included.
Armenia.
A person may be declared bankrupt with an application submitted to the court by the creditor or with an application to recognize his own bankruptcy. Legal and natural persons, including individual entrepreneurs, who have an indisputable payment obligation exceeding 60 days and amounting to more than one million AMD can be declared bankrupt. All creditors, including the state and municipalities, to whom the person has an obligation that meets the above-mentioned minimum criteria can submit an application to declare a person bankrupt by compulsory procedure. Basically, these obligations are derived from the legal acts of the court, transactions, the obligation of the debtor to pay taxes, duties, and other fees defined by law.
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At the same time, when being declared bankrupt with a voluntary bankruptcy application, the applicant bears the obligation to prove the fact that the value of his assets is less than his assets by one million AMD or more.
Australia.
In Australia, bankruptcy is a status which applies to individuals and is governed by the federal "Bankruptcy Act 1966". Companies do not go bankrupt but rather go into liquidation or administration, which is governed by the federal "Corporations Act 2001".
If a person commits an act of bankruptcy, then a creditor can apply to the Federal Circuit Court or the Federal Court for a sequestration order. Acts of bankruptcy are defined in the legislation, and include the failure to comply with a bankruptcy notice. A bankruptcy notice can be issued where, among other cases, a person fails to pay a judgment debt of at least $5,000. A person can also seek to have themselves declared bankrupt for any amount of debt by lodging a debtor's petition with the "Official Receiver", which is the Australian Financial Security Authority (AFSA).
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All bankrupts must lodge a Statement of Affairs document, also known as a Bankruptcy Form, with AFSA, which includes important information about their assets and liabilities. A bankruptcy cannot be discharged until this document has been lodged.
Ordinarily, a bankruptcy lasts three years from the filing of the Statement of Affairs with AFSA.
A Bankruptcy Trustee (in most cases, the Official Trustee at AFSA) is appointed to deal with all matters regarding the administration of the bankrupt estate. The Trustee's job includes notifying creditors of the estate and dealing with creditor inquiries; ensuring that the bankrupt complies with their obligations under the Bankruptcy Act; investigating the bankrupt's financial affairs; realising funds to which the estate is entitled under the Bankruptcy Act and distributing dividends to creditors if sufficient funds become available.
For the duration of their bankruptcy, all bankrupts have certain restrictions placed upon them. For example, a bankrupt must obtain the permission of their trustee to travel overseas. Failure to do so may result in the bankrupt being stopped at the airport by the Australian Federal Police. Additionally, a bankrupt is required to provide their trustee with details of income and assets. If the bankrupt does not comply with the Trustee's request to provide details of income, the trustee may have grounds to lodge an Objection to Discharge, which has the effect of extending the bankruptcy for a further three or five years depending on the type of Objection.
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The realisation of funds usually comes from two main sources: the bankrupt's assets and the bankrupt's wages. There are certain assets that are protected, referred to as "protected assets". These include household furniture and appliances, tools of the trade and vehicles up to a certain value. All other assets of value can be sold. If a house, including the main residence, or car is above a certain value, a third party can buy the interest from the estate in order for the bankrupt to utilise the asset. If this is not done, the interest vests in the estate and the trustee is able to take possession of the asset and sell it.
The bankrupt must pay income contributions if their income is above a certain threshold. If the bankrupt fails to pay, the trustee can ask the Official Receiver to issue a notice to garnishee the bankrupt's wages. If that is not possible, the Trustee may seek to extend the bankruptcy for a further three or five years.
Bankruptcies can be annulled, and the bankrupt released from bankruptcy, prior to the expiration of the normal three-year period if all debts are paid out in full. Sometimes a bankrupt may be able to raise enough funds to make an Offer of Composition to creditors, which would have the effect of paying the creditors some of the money they are owed. If the creditors accept the offer, the bankruptcy can be annulled after the funds are received.
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After the bankruptcy is annulled or the bankrupt has been automatically discharged, the bankrupt's credit report status is shown as "discharged bankrupt" for some years. The maximum number of years this information can be held is subject to the retention limits under the Privacy Act. How long such information is on a credit report may be shorter, depending on the issuing company, but the report must cease to record that information based on the criteria in the Privacy Act.
Brazil.
In Brazil, the Bankruptcy Law (11.101/05) governs court-ordered or out-of-court receivership and bankruptcy and only applies to public companies (publicly traded companies) with the exception of financial institutions, credit cooperatives, consortia, supplementary scheme entities, companies administering health care plans, equity companies and a few other legal entities. It does not apply to state-run companies.
Current law covers three legal proceedings. The first one is bankruptcy itself ("Falência"). Bankruptcy is a court-ordered liquidation procedure for an insolvent business. The final goal of bankruptcy is to liquidate company assets and pay its creditors.
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The second one is Court-ordered Restructuring ("Recuperação Judicial"). The goal is to overcome the business crisis situation of the debtor in order to allow the continuation of the producer, the employment of workers and the interests of creditors, leading, thus, to preserving company, its corporate function and develop economic activity. It is a court procedure required by the debtor which has been in business for more than two years and requires approval by a judge.
The Extrajudicial Restructuring ("Recuperação Extrajudicial") is a private negotiation that involves creditors and debtors and, as with court-ordered restructuring, also must be approved by courts.
Canada.
Bankruptcy, also referred to as insolvency in Canada, is governed by the Bankruptcy and Insolvency Act and is applicable to businesses and individuals. For example, Target Canada, the Canadian subsidiary of the Target Corporation, the second-largest discount retailer in the United States filed for bankruptcy on 15 January 2015, and closed all of its stores by 12 April. The office of the Superintendent of Bankruptcy, a federal agency, is responsible for ensuring that bankruptcies are administered in a fair and orderly manner by all licensed Trustees in Canada.
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Trustees in bankruptcy, 1041 individuals licensed to administer insolvencies, bankruptcy and proposal estates are governed by the Bankruptcy and Insolvency Act of Canada.
Bankruptcy is filed when a person or a company becomes insolvent and cannot pay their debts as they become due and if they have at least $1,000 in debt.
In 2011, the Superintendent of Bankruptcy reported that trustees in Canada filed 127,774 insolvent estates. Consumer estates were the vast majority, with 122,999 estates. The consumer portion of the 2011 volume is divided into 77,993 bankruptcies and 45,006 consumer proposals. This represented a reduction of 8.9% from 2010. Commercial estates filed by Canadian trustees in 2011 4,775 estates, 3,643 bankruptcies and 1,132 Division 1 proposals. This represents a reduction of 8.6% over 2010.
Duties of trustees.
Some of the duties of the trustee in bankruptcy are to:
Creditors' meetings.
Creditors become involved by attending creditors' meetings. The trustee calls the first meeting of creditors for the following purposes:
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Consumer proposals.
In Canada, a person can file a consumer proposal as an alternative to bankruptcy. A consumer proposal is a negotiated settlement between a debtor and their creditors.
A typical proposal would involve a debtor making monthly payments for a maximum of five years, with the funds distributed to their creditors. Even though most proposals call for payments of less than the full amount of the debt owing, in most cases, the creditors accept the deal—because if they do not, the next alternative may be personal bankruptcy, in which the creditors get even less money. The creditors have 45 days to accept or reject the consumer proposal. Once the proposal is accepted by both the creditors and the Court, the debtor makes the payments to the Proposal Administrator each month (or as otherwise stipulated in their proposal), and the general creditors are prevented from taking any further legal or collection action. If the proposal is rejected, the debtor is returned to his prior insolvent state and may have no alternative but to declare personal bankruptcy.
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A consumer proposal can only be made by a debtor with debts to a maximum of $250,000 (not including the mortgage on their principal residence). If debts are greater than $250,000, the proposal must be filed under Division 1 of Part III of the Bankruptcy and Insolvency Act. An Administrator is required in the Consumer Proposal, and a Trustee in the Division I Proposal (these are virtually the same although the terms are not interchangeable). A Proposal Administrator is almost always a licensed trustee in bankruptcy, although the Superintendent of Bankruptcy may appoint other people to serve as administrators.
In 2006, there were 98,450 personal insolvency filings in Canada: 79,218 bankruptcies and 19,232 consumer proposals.
Commercial restructuring.
In Canada, bankruptcy always means liquidation. There is no way for a company to emerge from bankruptcy after restructuring, as is the case in the United States with a Chapter 11 bankruptcy filing. Canada does, however, have laws that allow for businesses to restructure and emerge later with a smaller debt load and a more positive financial future. While not technically a form of bankruptcy, businesses with $5M or more in debt may make use of the Companies' Creditors Arrangement Act to halt all debt recovery efforts against the company while they formulate a plan to restructure.
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China.
The People's Republic of China legalized bankruptcy in 1986, and a revised law that was more expansive and complete was enacted in 2007.
Ireland.
Bankruptcy in Ireland applies only to natural persons. Other insolvency processes including liquidation and examinership are used to deal with corporate insolvency.
Irish bankruptcy law has been the subject of significant comment, from both government sources and the media, as being in need of reform. Part 7 of the Civil Law (Miscellaneous Provisions) Act 2011 has started this process and the government has committed to further reform.
Israel.
Bankruptcy in Israel is governed by the Insolvency and Rehabilitation Law, 2018. Insolvency proceedings below ₪150,000 will be administered entirely by the Enforcement and Collection Authority. Insolvency proceedings above ₪150,000 individual debtors file the documents will be conducted before the official receiver (the Insolvency Commissioner) and, if a creditor want to file against a debtor, he needs to open process, before the magistrate's court that hears in the district. Company bankruptcy will be conducted before District Court. Simultaneously, with the issue of the order for the commencement of insolvency proceedings, the Insolvency Commissioner shall appoint a trustee for the debtor and an audit will be carried out, in which the debtor's economic capability and his conduct will be examined (lasting approximately 12 months). At the end of this audit a payment plan is established, at the end of which the debtor will receive a discharge. The default scenario is a payment period of three years; however, the court reserves the right to increase or decrease the period depending upon the circumstances of the case. If the debtor has no proven financial ability to pay the creditors, he may be granted an immediate discharge. Since 1996, Israeli personal bankruptcy law has shifted to a relatively debtor-friendly regime, not unlike the American model.
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India.
In May 2016, the Parliament of India passed the Insolvency and Bankruptcy Code (IBC), updating outdated corporate insolvency laws. The IBC streamlined the process, reducing delays from a decade to 180 days, and replaced the Board for Industrial and Financial Reconstruction (BIFR) with a market-driven approach.
The Netherlands.
Dutch bankruptcy law is governed by the Dutch Bankruptcy Code (). The code covers three separate legal proceedings:
Russia.
Federal Law No. 127-FZ "On Insolvency (Bankruptcy)" dated 26 October 2002 (as amended) (the "Bankruptcy Act"), replacing the previous law in 1998, to better address the above problems and a broader failure of the action. Russian insolvency law is intended for a wide range of borrowers: individuals and companies of all sizes, with the exception of state-owned enterprises, government agencies, political parties and religious organizations. There are also special rules for insurance companies, professional participants of the securities market, agricultural organizations and other special laws for financial institutions and companies in the natural monopolies in the energy industry. Federal Law No. 40-FZ "On Insolvency (Bankruptcy)" dated 25 February 1999 (as amended) (the "Insolvency Law of Credit Institutions") contains special provisions in relation to the opening of insolvency proceedings in relation to the credit company. Insolvency Provisions Act, credit organizations used in conjunction with the provisions of the Bankruptcy Act.
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Bankruptcy law provides for the following stages of insolvency proceedings:
The main face of the bankruptcy process is the insolvency officer (trustee in bankruptcy, bankruptcy manager). At various stages of bankruptcy, he must be determined: the temporary officer in monitoring procedure, external manager in external control, the receiver or administrative officer in the economic recovery, the liquidator. During the bankruptcy trustee in bankruptcy (insolvency officer) has a decisive influence on the movement of assets (property) of the debtor – the debtor and has a key influence on the economic and legal aspects of its operations.
Spain.
In Spain, people who cannot repay their home mortgages may declare bankruptcy. Bankruptcy and foreclosure discharges the obligation to pay mortgage interest, but not mortgage principal. If mortgage principal is not paid, the debtor is placed on a list of untrustworthy people.
Switzerland.
Under Swiss law, bankruptcy can be a consequence of insolvency. It is a court-ordered form of debt enforcement proceedings that applies, in general, to registered commercial entities only. In a bankruptcy, all assets of the debtor are liquidated under the administration of the creditors, although the law provides for debt restructuring options similar to those under Chapter 11 of the U.S. Bankruptcy code.
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Sweden.
In Sweden, bankruptcy () is a formal process that may involve a company or individual. A creditor or the company itself can apply for bankruptcy. An external bankruptcy manager takes over the company or the assets of the person, and tries to sell as much as possible. A person or a company in bankruptcy cannot access its assets (with some exceptions).
The formal bankruptcy process is rarely carried out for individuals. Creditors can claim money through the Enforcement Administration anyway, and creditors do not usually benefit from the bankruptcy of individuals because there are costs of a bankruptcy manager which has priority. Unpaid debts remain after bankruptcy for individuals. People who are deeply in debt can obtain a debt arrangement procedure (). On application, they obtain a payment plan under which they pay as much as they can for five years, and then all remaining debts are cancelled. Debts that derive from a ban on business operations (issued by court, commonly for tax fraud or fraudulent business practices) or owed to a crime victim as compensation for damages, are exempted from this—and, as before this process was introduced in 2006, remain lifelong. Debts that have not been claimed during a 3–10 year period are cancelled. Often crime victims stop their claims after a few years since criminals often do not have job incomes and might be hard to locate, while banks make sure their claims are not cancelled. The most common reasons for personal insolvency in Sweden are illness, unemployment, divorce or company bankruptcy.
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For companies, formal bankruptcy is a normal effect of insolvency, even if there is a reconstruction mechanism where the company can be given time to solve its situation, e.g. by finding an investor. The government can pay salaries to employees in insolvent companies which do not pay them, but only if the company is declared bankrupt. Therefore, it is normal that trade union do the application for bankruptcy if a supplier has not already done so.
The formal bankruptcy involves contracting a bankruptcy manager, who makes certain that assets are sold and money divided by the priority the law claims, and no other way. Banks have such a priority. After a finished bankruptcy for a company, it is terminated. The activities might continue in a new company which has bought important assets from the bankrupted company.
United Arab Emirates.
The United Arab Emirates Bankruptcy Law came into force on 29 December 2016, and created a single law governing bankruptcy procedures, which had previously been spread across multiple sources. There are two court procedures: first, a procedure for a company that is not yet insolvent, known as a protective composition, and second, a formal bankruptcy that is split into a rescue process (similar to protective composition) or liquidation.
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Directors of a company can be held personally liable for its debts.
The Bankruptcy Law does not apply to government bodies, or to companies trading in free zones such as the Dubai International Financial Centre or the Abu Dhabi Global Market, which have their own insolvency laws.
United Kingdom.
Bankruptcy in the United Kingdom (in a strict legal sense) relates only to individuals (including sole proprietors) and partnerships. Companies and other corporations enter into differently named legal insolvency procedures: liquidation and administration (administration order and administrative receivership). However, the term 'bankruptcy' is often used when referring to companies in the media and in general conversation. Bankruptcy in Scotland is referred to as sequestration. To apply for bankruptcy in Scotland, an individual must have more than £1,500 of debt.
A trustee in bankruptcy must be either an Official Receiver (a civil servant) or a licensed insolvency practitioner. Current law in England and Wales derives in large part from the Insolvency Act 1986. Following the introduction of the Enterprise Act 2002, bankruptcy in England and Wales now normally lasts no longer than 12 months, and may be less if the Official Receiver files in court a certificate that investigations are complete. It was expected that the UK Government's liberalisation of the bankruptcy regime would increase the number of bankruptcy cases; initially, cases increased, as the Insolvency Service statistics appear to bear out. Since 2009, the introduction of the Debt Relief Order has resulted in a dramatic fall in bankruptcies, the latest estimates for year 2014/15 being significantly less than 30,000 cases.
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Pensions.
The UK bankruptcy law was changed in May 2000, effective from 29 May 2000. Debtors may now retain occupational pensions while in bankruptcy, except in rare cases.
United States.
Bankruptcy in the United States is a matter placed under federal jurisdiction by the United States Constitution (in Article 1, Section 8, Clause 4), which empowers Congress to enact "uniform Laws on the subject of Bankruptcies throughout the United States". Congress has enacted statutes governing bankruptcy, primarily in the form of the Bankruptcy Code, located at Title 11 of the United States Code.
A debtor declares bankruptcy to obtain relief from debt, and this is normally accomplished either through a discharge of the debt or through a restructuring of the debt. When a debtor files a voluntary petition, their bankruptcy case commences.
Debts and exemptions.
While bankruptcy cases are always filed in United States Bankruptcy Court (an adjunct to the U.S. District Courts), bankruptcy cases, particularly with respect to the validity of claims and exemptions, are often dependent upon State law. A Bankruptcy Exemption defines the property a debtor may retain and preserve through bankruptcy. Certain real and personal property can be exempted on "Schedule C" of a debtor's bankruptcy forms, and effectively be taken outside the debtor's bankruptcy estate. Bankruptcy exemptions are available only to individuals filing bankruptcy.
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There are two alternative systems that can be used to exempt property from a bankruptcy estate, federal exemptions (available in some states but not all), and state exemptions (which vary widely between states). For example, Maryland and Virginia, which are adjoining states, have different personal exemption amounts that cannot be seized for payment of debts. This amount is the first $6,000 in property or cash in Maryland, but normally only the first $5,000 in Virginia. State law therefore plays a major role in many bankruptcy cases, such that there may be significant differences in the outcome of a bankruptcy case depending upon the state in which it is filed.
After a bankruptcy petition is filed, the court schedules a hearing called a "341 meeting" or "meeting of creditors", at which the bankruptcy trustee and creditors review the petitioner's petition and supporting schedules, question the petitioner, and can challenge exemptions they believe are improper.
Chapters.
There are six types of bankruptcy under the Bankruptcy Code, located at Title 11 of the United States Code:
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An important feature applicable to all types of bankruptcy filings is the automatic stay. The automatic stay means that the mere request for bankruptcy protection automatically halts most lawsuits, repossessions, foreclosures, evictions, garnishments, attachments, utility shut-offs, and debt collection activity.
The most common types of personal bankruptcy for individuals are Chapter 7 and Chapter 13. Chapter 7, known as a "straight bankruptcy", involves the discharge of certain debts without repayment. Chapter 13 involves a plan of repayment of debts over a period of years. Whether a person qualifies for Chapter 7 or Chapter 13 is in part determined by income. As many as 65% of all US consumer bankruptcy filings are Chapter 7 cases.
Before a consumer may obtain bankruptcy relief under either Chapter 7 or Chapter 13, the debtor is to undertake credit counseling with approved counseling agencies prior to filing a bankruptcy petition and to undertake education in personal financial management from approved agencies prior to being granted a discharge of debts under either Chapter 7 or Chapter 13. Some studies of the operation of the credit counseling requirement suggest that it provides little benefit to debtors who receive the counseling because the only realistic option for many is to seek relief under the Bankruptcy Code.
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Corporations and other business forms normally file under Chapters 7 or 11.
Chapter 7.
Often called "straight bankruptcy" or "simple bankruptcy", a Chapter 7 bankruptcy potentially allows debtors to eliminate most or all of their debts over a period of as little as three or four months. In a typical consumer bankruptcy, the only debts that survive a Chapter 7 are student loans, child support obligations, some tax bills, and criminal fines. Credit cards, pay day loans, personal loans, medical bills, and just about all other bills are discharged.
In Chapter 7, a debtor surrenders non-exempt property to a bankruptcy trustee, who then liquidates the property and distributes the proceeds to the debtor's unsecured creditors. In exchange, the debtor is entitled to a discharge of some debt. However, the debtor is not granted a discharge if guilty of certain types of inappropriate behavior (e.g., concealing records relating to financial condition) and certain debts (e.g., spousal and child support and most student loans). Some taxes are not discharged even though the debtor is generally discharged from debt. Many individuals in financial distress own only exempt property (e.g., clothes, household goods, an older car, or the tools of their trade or profession) and do not have to surrender any property to the trustee. The amount of property that a debtor may exempt varies from state to state (as noted above, Virginia and Maryland have a $1,000 difference). Chapter 7 relief is available only once in any eight-year period. Generally, the rights of secured creditors to their collateral continues, even though their debt is discharged. For example, absent some arrangement by a debtor to surrender a car or "reaffirm" a debt, the creditor with a security interest in the debtor's car may repossess the car even if the debt to the creditor is discharged.
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Ninety-one percent of US individuals who petition for relief under Chapter 7 hire an attorney to file their petitions. The typical cost of an attorney is $1,170.00. Alternatives to filing with an attorney are: filing pro se, hiring a non-lawyer petition preparer, or using online software to generate the petition.
To be eligible to file a consumer bankruptcy under Chapter 7, a debtor must qualify under a statutory means test. The means test was intended to make it more difficult for a significant number of financially distressed individual debtors whose debts are primarily consumer debts to qualify for relief under Chapter 7 of the Bankruptcy Code. The "means test" is employed in cases where an individual with primarily consumer debts has more than the average annual income for a household of equivalent size, computed over a 180-day period prior to filing. If the individual must take the means test, their average monthly income over this 180-day period is reduced by a series of allowances for living expenses and secured debt payments in a very complex calculation that may or may not accurately reflect that individual's actual monthly budget. If the results of the means test show no disposable income (or in some cases a very small amount) then the individual qualifies for Chapter 7 relief. An individual who fails the means test will have their Chapter 7 case dismissed, or may have to convert the case to a Chapter 13 bankruptcy.
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If a debtor does not qualify for relief under Chapter 7 of the Bankruptcy Code, either because of the Means Test or because Chapter 7 does not provide a permanent solution to delinquent payments for secured debts, such as mortgages or vehicle loans, the debtor may still seek relief under Chapter 13 of the Code.
Generally, a trustee sells most of the debtor's assets to pay off creditors. However, certain debtor assets will be protected to some extent by bankruptcy exemptions. These include Social Security payments, unemployment compensation, limited equity in a home, car, or truck, household goods and appliances, trade tools, and books. However, these exemptions vary from state to state.
Chapter 11.
In Chapter 11 bankruptcy, the debtor retains ownership and control of assets and is re-termed a debtor in possession (DIP). The debtor in possession runs the day-to-day operations of the business while creditors and the debtor work with the Bankruptcy Court in order to negotiate and complete a plan. Upon meeting certain requirements (e.g., fairness among creditors, priority of certain creditors) creditors are permitted to vote on the proposed plan. If a plan is confirmed, the debtor continues to operate and pay debts under the terms of the confirmed plan. If a specified majority of creditors do not vote to confirm a plan, additional requirements may be imposed by the court in order to confirm the plan. Debtors filing for Chapter 11 protection a second time are known informally as "Chapter 22" filers.
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In a corporate or business bankruptcy, an indebted company is typically recapitalized so that it emerges from bankruptcy with more equity and less debt, with potential for dispute over the valuation of the reorganized business.
Chapter 13.
In Chapter 13, debtors retain ownership and possession of all their assets but must devote some portion of future income to repaying creditors, generally over three to five years. The amount of payment and period of the repayment plan depend upon a variety of factors, including the value of the debtor's property and the amount of a debtor's income and expenses. Under this chapter, the debtor can propose a repayment plan in which to pay creditors over three to five years. If the monthly income is less than the state's median income, the plan is for three years, unless the court finds just cause to extend the plan for a longer period. If the debtor's monthly income is greater than the median income for individuals in the debtor's state, the plan must generally be for five years. A plan cannot exceed the five-year limit.
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Relief under Chapter 13 is available only to individuals with regular income whose debts do not exceed prescribed limits. If the debtor is an individual or a sole proprietor, the debtor is allowed to file for a Chapter 13 bankruptcy to repay all or part of the debts. Secured creditors may be entitled to greater payment than unsecured creditors.
In contrast to Chapter 7, the debtor in Chapter 13 may keep all property, whether or not exempt. If the plan appears feasible and if the debtor complies with all the other requirements, the bankruptcy court typically confirms the plan and the debtor and creditors are bound by its terms. Creditors have no say in the formulation of the plan, other than to object to it, if appropriate, on the grounds that it does not comply with one of the Code's statutory requirements. Generally, the debtor makes payments to a trustee who disburses the funds in accordance with the terms of the confirmed plan.
When the debtor completes payments pursuant to the terms of the plan, the court formally grant the debtor a discharge of the debts provided for in the plan. However, if the debtor fails to make the agreed upon payments or fails to seek or gain court approval of a modified plan, a bankruptcy court will normally dismiss the case on the motion of the trustee. After a dismissal, creditors may resume pursuit of state law remedies to recover the unpaid debt.
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European Union.
In 2004, the number of insolvencies reached record highs in many European countries. In France, company insolvencies rose by more than 4%, in Austria by more than 10%, and in Greece by more than 20%. The increase in the number of insolvencies, however, does not indicate the total financial impact of insolvencies in each country because there is no indication of the size of each case. An increase in the number of bankruptcy cases does not necessarily entail an increase in bad debt write-off rates for the economy as a whole.
Bankruptcy statistics are also a trailing indicator. There is a time delay between financial difficulties and bankruptcy. In most cases, several months or even years pass between the financial problems and the start of bankruptcy proceedings. Legal, tax, and cultural issues may further distort bankruptcy figures, especially when comparing on an international basis. Two examples:
The insolvency numbers for private individuals also do not show the whole picture. Only a fraction of heavily indebted households file for insolvency. Two of the main reasons for this are the stigma of declaring themselves insolvent and the potential business disadvantage.
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Following the soar in insolvencies in the previous decade, a number of European countries, such as France, Germany, Spain and Italy, began to revamp their bankruptcy laws in 2013. They modelled these new laws on Chapter 11 of the U.S. Bankruptcy Code. Currently, the majority of insolvency cases have ended in liquidation in Europe rather than the businesses surviving the crisis. These new law models are meant to change this; lawmakers are hoping to turn bankruptcy into a chance for restructuring rather than a death sentence for the companies.
EU policy aims to ensure that "honest entrepreneurs" are afforded a second chance at business development. A faster start-up programme for people affected by bankruptcy operating in Denmark and a scheme to support Belgian business owners and self-employed persons were highlighted in a 2008 European Commission "Communication" as good practice examples in this field.
Effective sovereign bankruptcy.
Technically, states do not collapse directly due to a sovereign default event itself. However, the tumultuous events that follow may bring down the state, so in common language, states would be described as being bankrupted.
An example of this is when the Goguryeo–Sui War in 614 A.D. ended in the disintegration of Sui dynasty China within 4 years, although their enemy Goguryeo (occupying modern Korea) also seemingly entered into decline and fell some 56 years later. |
Blissymbols
Blissymbols or Blissymbolics is a constructed language conceived as an ideographic writing system called Semantography consisting of several hundred basic symbols, each representing a concept, which can be composed together to generate new symbols that represent new concepts. Blissymbols differ from most of the world's major writing systems in that the characters do not correspond at all to the sounds of any spoken language.
"Semantography" was published by Charles K. Bliss in 1949 and found use in the education of people with communication difficulties.
History.
Semantography was invented by Charles K. Bliss (1897–1985), born Karl Kasiel Blitz to a Jewish family in Chernivtsi (then Czernowitz, Austria-Hungary), which had a mixture of different nationalities that "hated each other, mainly because they spoke and thought in different languages."
Bliss graduated as a chemical engineer at the Vienna University of Technology, and joined an electronics company. After the Nazi annexation of Austria in 1938, Bliss was sent to concentration camps but his German wife Claire managed to get him released, and they finally became exiles in Shanghai, where Bliss had a cousin.
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Bliss devised the symbols while a refugee at the Shanghai Ghetto and Sydney, from 1942 to 1949. He wanted to create an easy-to-learn international auxiliary language to allow communication between different linguistic communities. He was inspired by Chinese characters, with which he became familiar at Shanghai.
Bliss published his system in "Semantography" (1949, exp. 2nd ed. 1965, 3rd ed. 1978.)
It had several names:
As the "tourist explosion" took place in the 1960s, a number of researchers were looking for new standard symbols to be used at roads, stations, airports, etc. Bliss then adopted the name "Blissymbolics" in order that no researcher could plagiarize his system of symbols.
Since the 1960s/1970s, Blissymbols have become popular as a method to teach disabled people to communicate.
In 1971, Shirley McNaughton started a pioneer program at the Ontario Crippled Children's Centre (OCCC), aimed at children with cerebral palsy, from the approach of augmentative and alternative communication (AAC). According to Arika Okrent, Bliss used to complain about the way the teachers at the OCCC were using the symbols, in relation with the proportions of the symbols and other questions: for example, they used "fancy" terms like "nouns" and "verbs", to describe what Bliss called "things" and "actions". (2009, p. 173-4).
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The ultimate objective of the OCCC program was to use Blissymbols as a practical way to teach the children to express themselves in their mother tongue, since the Blissymbols provided visual keys to understand the meaning of the English words, especially the abstract words.
In "Semantography," Bliss had not provided a systematic set of definitions for his symbols (there was a provisional vocabulary index instead (1965, pp. 827–67)), so McNaughton's team might often interpret a certain symbol in a way that Bliss would later criticize as a "misinterpretation". For example, they might interpret a tomato as a vegetable —according to the English definition of tomato— even though the ideal Blissymbol of vegetable was restricted by Bliss to just vegetables growing underground. Eventually the OCCC staff modified and adapted Bliss's system in order to make it serve as a bridge to English. (2009, p. 189) Bliss' complaints about his symbols "being abused" by the OCCC became so intense that the director of the OCCC told Bliss, on his 1974 visit, never to come back. In spite of this, in 1975, Bliss granted an exclusive world license, for use with disabled children, to the new Blissymbolics Communication Foundation directed by Shirley McNaughton (later called Blissymbolics Communication International, BCI). Nevertheless, in 1977, Bliss claimed that this agreement was violated so that he was deprived of effective control of his symbol system.
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According to Okrent (2009, p. 190), there was a final period of conflict, as Bliss would make continuous criticisms to McNaughton often followed by apologies. Bliss finally brought his lawyers back to the OCCC, reaching a settlement:
Blissymbolic Communication International now claims an exclusive license from Bliss, for the use and publication of Blissymbols for persons with communication, language, and learning difficulties.
The Blissymbol method has been used in Canada, Sweden, and a few other countries. Practitioners of Blissymbolics (that is, speech and language therapists and users) maintain that some users who have learned to communicate with Blissymbolics find it easier to learn to read and write traditional orthography in the local spoken language than do users who did not know Blissymbolics.
The speech question.
Unlike similar constructed languages like aUI, Blissymbolics was conceived as a written language with no phonology, on the premise that "interlinguistic communication is mainly carried on by reading and writing". Nevertheless, Bliss suggested that a set of international words could be adopted, so that "a kind of spoken language could be established – as a travelling aid only". (1965, p. 89–90).
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Whether Blissymbolics constitutes an unspoken language is a controversial question, whatever its practical utility may be. Some linguists, such as John DeFrancis and J. Marshall Unger have argued that genuine ideographic writing systems with the same capacities as natural languages do not exist.
Semantics.
Bliss' concern about semantics finds an early referent in John Locke, whose "Essay Concerning Human Understanding" prevented people from those "vague and insignificant forms of speech" that may give the impression of being deep learning.
Another vital referent is Gottfried Wilhelm Leibniz's project of an ideographic language "characteristica universalis", based on the principles of Chinese characters. It would contain small figures representing "visible things by their lines, and the invisible, by the visible which accompany them", adding "certain additional marks, suitable to make understood the flexions and the particles." Bliss stated that his own work was an attempt to take up the thread of Leibniz's project.
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Finally there is a strong influence by "The Meaning of Meaning" (1923) by C. K. Ogden and I. A. Richards, which was considered a standard work on semantics. Bliss found especially useful their "triangle of reference": the physical thing or "referent" that we perceive would be represented at the right vertex; the meaning that we know by experience (our implicit definition of the thing), at the top vertex; and the physical word that we speak or symbol we write, at the left vertex. The reversed process would happen when we read or listen to words: from the words, we recall meanings, related to referents which may be real things or unreal "fictions". Bliss was particularly concerned with political propaganda, whose discourses would tend to contain words that correspond to unreal or ambiguous referents.
Grammar.
The grammar of Blissymbols is based on a certain interpretation of nature, dividing it into matter (material things), energy (actions), and human values (mental evaluations). In a natural language, these would give place respectively to nouns, verbs, and adjectives. In Blissymbols, they are marked respectively by a small square symbol, a small cone symbol, and a small V or inverted cone. These symbols may be placed above any other symbol, turning it respectively into a "thing", an "action", and an "evaluation":
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When a symbol is not marked by any of the three grammar symbols (square, cone, inverted cone), it may refer to a non-material thing, a grammatical particle, etc.
Examples.
The symbol represents the expression "world language", which was a first tentative name for Blissymbols. It combines the symbol for "writing tool" or "pen" (a line inclined, as a pen being used) with the symbol for "world", which in its turn combines "ground" or "earth" (a horizontal line below) and its counterpart derivate "sky" (a horizontal line above). Thus the world would be seen as "what is among the ground and the sky", and "Blissymbols" would be seen as "the writing tool to express the world". This is clearly distinct from the symbol of "language", which is a combination of "mouth" and "ear". Thus natural languages are mainly oral, while Blissymbols is just a writing system dealing with semantics, not phonetics.
The 900 individual symbols of the system are called "Bliss-characters"; these may be "ideographic" – representing abstract concepts, "pictographic" – a direct representation of objects, or "composite" – in which two or more existing Bliss-characters are superimposed to represent a new meaning. Size, orientation and relation to the "skyline" and "earthline" affects the meaning of each symbol. A single concept is called a "Bliss-word", which can consist of one or more Bliss-characters. In multiple-character Bliss-words, the main character is called the "classifier" which "indicates the semantic or grammatical category to which the Bliss-word belongs". To this can be added Bliss-characters as prefixes or suffixes called "modifiers" which amend the meaning of the first symbol. A further symbol called an "indicator" can be added above one of the characters in the Bliss-word (typically the classifier); these are used as "grammatical and/or semantic markers."
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Sentence on the right means "I want to go to the cinema.", showing several features of Blissymbolics:
Towards the international standardization of the script.
Blissymbolics was used in 1971 to help children at the Ontario Crippled Children's Centre (OCCC, now the Holland Bloorview Kids Rehabilitation Hospital) in Toronto, Ontario, Canada. Since it was important that the children see consistent pictures, OCCC had a draftsman named Jim Grice draw the symbols. Both Charles K. Bliss and Margrit Beesley at the OCCC worked with Grice to ensure consistency. In 1975, a new organization named Blissymbolics Communication Foundation directed by Shirley McNaughton led this effort. Over the years, this organization changed its name to Blissymbolics Communication Institute, Easter Seal Communication Institute, and ultimately to Blissymbolics Communication International (BCI).
BCI is an international group of people who act as an authority regarding the standardization of the Blissymbolics language. It has taken responsibility for any extensions of the Blissymbolics language as well as any maintenance needed for the language. BCI has coordinated usage of the language since 1971 for augmentative and alternative communication. BCI received a licence and copyright through legal agreements with Charles K. Bliss in 1975 and 1982. Limiting the count of Bliss-characters (there are currently about 900) is very useful in order to help the user community. It also helps when implementing Blissymbolics using technology such as computers.
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In 1991, BCI published a reference guide containing 2300 vocabulary items and detailed rules for the graphic design of additional characters, so they settled a first set of approved "Bliss-words" for general use.
The Standards Council of Canada then sponsored, on January 21, 1993, the registration of an encoded character set for use in ISO/IEC 2022, in the ISO-IR international registry of coded character sets.
After many years of requests, the Blissymbolic language was finally approved as an encoded language, with code , into the ISO 639-2 and ISO 639-3 standards.
A proposal was posted by Michael Everson for the Blissymbolics script to be included in the Universal Character Set (UCS) and encoded for use with the ISO/IEC 10646 and Unicode standards. BCI would cooperate with the Unicode Technical Committee (UTC) and the ISO Working Group.
The proposed encoding does not use the lexical encoding model used in the existing ISO-IR/169 registered character set, but instead applies the Unicode and ISO character-glyph model to the "Bliss-character" model already adopted by BCI, since this would significantly reduce the number of needed characters. Bliss-characters can now be used in a creative way to create many new arbitrary concepts, by surrounding the invented words with special Bliss indicators (similar to punctuation), something which was not possible in the ISO-IR/169 encoding.
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However, by the end of 2009, the Blissymbolic script was not encoded in the UCS. Some questions are still unanswered, such as the inclusion in the BCI repertoire of some characters (currently about 24) that are already encoded in the UCS (like digits, punctuation signs, spaces and some markers), but whose unification may cause problems due to the very strict graphical layouts required by the published Bliss reference guides. In addition, the character metrics use a specific layout where the usual baseline is not used, and the ideographic em-square is not relevant for Bliss character designs that use additional "earth line" and "sky line" to define the composition square.
Some fonts supporting the BCI repertoire are available and usable with texts encoded with private-use assignments (PUA) within the UCS. But only the private BCI encoding based on ISO-IR/169 registration is available for text interchange. |
Bessel function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
formula_1
for an arbitrary complex number formula_2, which represents the "order" of the Bessel function. Although formula_2 and formula_4 produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of formula_2.
The most important cases are when formula_2 is an integer or half-integer. Bessel functions for integer formula_2 are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer formula_2 are obtained when solving the Helmholtz equation in spherical coordinates.
Applications.
Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (); in spherical problems, one obtains half-integer orders (). For example:
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Bessel functions also appear in other problems, such as signal processing (e.g., see FM audio synthesis, Kaiser window, or Bessel filter).
Definitions.
Because this is a linear differential equation, solutions can be scaled to any amplitude. The amplitudes chosen for the functions originate from the early work in which the functions appeared as solutions to definite integrals rather than solutions to differential equations. Because the differential equation is second-order, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.
Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by and , respectively, rather than and .
Bessel functions of the first kind:.
Bessel functions of the first kind, denoted as , are solutions of Bessel's differential equation. For integer or positive , Bessel functions of the first kind are finite at the origin (); while for negative non-integer , Bessel functions of the first kind diverge as approaches zero. It is possible to define the function by formula_9 times a Maclaurin series (note that need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying the Frobenius method to Bessel's equation:
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formula_10
where is the gamma function, a shifted generalization of the factorial function to non-integer values. Some earlier authors define the Bessel function of the first kind differently, essentially without the division by formula_11 in formula_12; this definition is not used in this article. The Bessel function of the first kind is an entire function if is an integer, otherwise it is a multivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to formula_13 (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large . (The series indicates that is the derivative of , much like is the derivative of ; more generally, the derivative of can be expressed in terms of by the identities below.)
For non-integer , the functions and are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order , the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):
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formula_14
This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.
Bessel's integrals.
Another definition of the Bessel function, for integer values of , is possible using an integral representation:
formula_15
which is also called Hansen-Bessel formula.
This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for :
formula_16
Relation to hypergeometric series.
The Bessel functions can be expressed in terms of the generalized hypergeometric series as
formula_17
This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.
Relation to Laguerre polynomials.
In terms of the Laguerre polynomials and arbitrarily chosen parameter , the Bessel function can be expressed as
formula_18
Bessel functions of the second kind:.
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The Bessel functions of the second kind, denoted by , occasionally denoted instead by , are solutions of the Bessel differential equation that have a singularity at the origin () and are multivalued. These are sometimes called Weber functions, as they were introduced by , and also Neumann functions after Carl Neumann.
For non-integer , is related to by
formula_19
In the case of integer order , the function is defined by taking the limit as a non-integer tends to :
formula_20
If is a nonnegative integer, we have the series
formula_21
where formula_22 is the digamma function, the logarithmic derivative of the gamma function.
There is also a corresponding integral formula (for ):
formula_23
In the case where : (with formula_24 being Euler's constant)formula_25
is necessary as the second linearly independent solution of the Bessel's equation when is an integer. But has more meaning than that. It can be considered as a "natural" partner of . See also the subsection on Hankel functions below.
When is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:
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formula_26
Both and are holomorphic functions of on the complex plane cut along the negative real axis. When is an integer, the Bessel functions are entire functions of . If is held fixed at a non-zero value, then the Bessel functions are entire functions of .
The Bessel functions of the second kind when is an integer is an example of the second kind of solution in Fuchs's theorem.
Hankel functions: ,.
Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, and , defined as
formula_27
where is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. They are named after Hermann Hankel.
These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form . For real formula_28 where formula_29, formula_30 are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of Euler's formula, substituting , for formula_31 and formula_29, formula_30 for formula_34, formula_35, as explicitly shown in the asymptotic expansion.
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The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the sign convention for the frequency).
Using the previous relationships, they can be expressed as
formula_36
If is an integer, the limit has to be calculated. The following relationships are valid, whether is an integer or not:
formula_37
In particular, if with a nonnegative integer, the above relations imply directly that
formula_38
These are useful in developing the spherical Bessel functions (see below).
The Hankel functions admit the following integral representations for :
formula_39
where the integration limits indicate integration along a contour that can be chosen as follows: from to 0 along the negative real axis, from 0 to along the imaginary axis, and from to along a contour parallel to the real axis.
Modified Bessel functions: ,.
The Bessel functions are valid even for complex arguments , and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as
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formula_40
when is not an integer; when is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments . The series expansion for is thus similar to that for , but without the alternating factor.
formula_41 can be expressed in terms of Hankel functions:
formula_42
Using these two formulae the result to formula_43+formula_44, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following
formula_45
given that the condition is met. It can also be shown that
formula_46
only when || < and but not when .
We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ):
formula_47
and are the two linearly independent solutions to the modified Bessel's equation:
formula_48
Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, and are exponentially growing and decaying functions respectively. Like the ordinary Bessel function , the function goes to zero at for and is finite at for . Analogously, diverges at with the singularity being of logarithmic type for , and otherwise.
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Two integral formulas for the modified Bessel functions are (for ):
formula_49
Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for ):
formula_50
It can be proven by showing equality to the above integral definition for . This is done by integrating a closed curve in the first quadrant of the complex plane.
Modified Bessel functions of the second kind may be represented with Bassett's integral
formula_51
Modified Bessel functions and can be represented in terms of rapidly convergent integrals
formula_52
The modified Bessel function formula_53 is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions.
The modified Bessel function of the second kind has also been called by the following names (now rare):
Spherical Bessel functions: ,.
When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form
formula_54
The two linearly independent solutions to this equation are called the spherical Bessel functions and , and are related to the ordinary Bessel functions and by
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formula_55
is also denoted or ; some authors call these functions the spherical Neumann functions.
From the relations to the ordinary Bessel functions it is directly seen that:
formula_56
The spherical Bessel functions can also be written as ()
formula_57
The zeroth spherical Bessel function is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:
formula_58
and
formula_59
The first few non-zero roots of the first few spherical Bessel functions are:
Generating function.
The spherical Bessel functions have the generating functions
formula_60
Finite series expansions.
In contrast to the whole integer Bessel functions , the spherical Bessel functions have a finite series expression:
formula_61
Differential relations.
In the following, is any of , , , for
formula_62
Spherical Hankel functions: ,.
There are also spherical analogues of the Hankel functions:
formula_63
In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers :
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formula_64
and is the complex-conjugate of this (for real ). It follows, for example, that and , and so on.
The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.
Riccati–Bessel functions: , , ,.
Riccati–Bessel functions only slightly differ from spherical Bessel functions:
formula_65
They satisfy the differential equation
formula_66
For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger's equation with hypothetical cylindrical infinite potential barrier. This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g., Du (2004) for recent developments and references.
Following Debye (1909), the notation , is sometimes used instead of , .
Asymptotic forms.
The Bessel functions have the following asymptotic forms. For small arguments formula_67, one obtains, when formula_2 is not a negative integer:
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formula_69
When is a negative integer, we have
formula_70
For the Bessel function of the second kind we have three cases:
formula_71
where is the Euler–Mascheroni constant (0.5772...).
For large real arguments , one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless is half-integer) because they have zeros all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of one can write an equation containing a term of order :
formula_72
The asymptotic forms for the Hankel functions are:
formula_73
These can be extended to other values of using equations relating and to and .
It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, is not asymptotic to the average of these two asymptotic forms when is negative (because one or the other will not be correct there, depending on the used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for "complex" (non-real) so long as goes to infinity at a constant phase angle (using the square root having positive real part):
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formula_74
For the modified Bessel functions, Hankel developed asymptotic expansions as well:
formula_75
There is also the asymptotic form (for large real formula_76)
formula_77
When , all the terms except the first vanish, and we have
formula_78
For small arguments formula_79, we have
formula_80
Properties.
For integer order , is often defined via a Laurent series for a generating function:
formula_81
an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.)
Infinite series of Bessel functions in the form formula_82 where formula_83 arise in many physical systems and are defined in closed form by the Sung series. For example, when N = 3: formula_84. More generally, the Sung series and the alternating Sung series are written as:
formula_85
formula_86
A series expansion using Bessel functions (Kapteyn series) is
Another important relation for integer orders is the "Jacobi–Anger expansion":
formula_88
and
formula_89
which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal.
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More generally, a series
formula_90
is called Neumann expansion of . The coefficients for have the explicit form
formula_91
where is Neumann's polynomial.
Selected functions admit the special representation
formula_92
with
formula_93
due to the orthogonality relation
formula_94
More generally, if has a branch-point near the origin of such a nature that
formula_95
then
formula_96
or
formula_97
where formula_98 is the Laplace transform of .
Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:
formula_99
where and .
This formula is useful especially when working with Fourier transforms.
Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by , the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:
formula_100
where , is the Kronecker delta, and is the th zero of . This orthogonality relation can then be used to extract the coefficients in the Fourier–Bessel series, where a function is expanded in the basis of the functions for fixed and varying .
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An analogous relationship for the spherical Bessel functions follows immediately:
formula_101
If one defines a boxcar function of that depends on a small parameter as:
formula_102
(where is the rectangle function) then the Hankel transform of it (of any given order ), , approaches as approaches zero, for any given . Conversely, the Hankel transform (of the same order) of is :
formula_103
which is zero everywhere except near 1. As approaches zero, the right-hand side approaches , where is the Dirac delta function. This admits the limit (in the distributional sense):
formula_104
A change of variables then yields the "closure equation":
formula_105
for . The Hankel transform can express a fairly arbitrary function as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is:
formula_106
for .
Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:
formula_107
where and are any two solutions of Bessel's equation, and is a constant independent of (which depends on α and on the particular Bessel functions considered). In particular,
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formula_108
and
formula_109
for .
For , the even entire function of genus 1, , has only real zeros. Let
formula_110
be all its positive zeros, then
formula_111
Recurrence relations.
The functions , , , and all satisfy the recurrence relations
formula_112
and
formula_113
where denotes , , , or . These two identities are often combined, e.g. added or subtracted, to yield various other relations. In this way, for example, one can compute Bessel functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives). In particular, it follows that
formula_114
"Modified" Bessel functions follow similar relations:
formula_115
and
formula_116
and
formula_117
The recurrence relation reads
formula_118
where denotes or . These recurrence relations are useful for discrete diffusion problems.
Transcendence.
In 1929, Carl Ludwig Siegel proved that , , and the logarithmic derivative are transcendental numbers when "ν" is rational and "x" is algebraic and nonzero. The same proof also implies that formula_119 is transcendental under the same assumptions.
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Sums with Bessel functions.
The product of two Bessel functions admits the following sum:
formula_120
formula_121
From these equalities it follows that
formula_122
and as a consequence
formula_123
These sums can be extended for a polynomial prefactor. For example,
formula_124
formula_125
formula_126
formula_127
Multiplication theorem.
The Bessel functions obey a multiplication theorem
formula_128
where and may be taken as arbitrary complex numbers. For , the above expression also holds if is replaced by . The analogous identities for modified Bessel functions and are
formula_129
and
formula_130
Zeros of the Bessel function.
Bourget's hypothesis.
Bessel himself originally proved that for nonnegative integers , the equation has an infinite number of solutions in . When the functions are plotted on the same graph, though, none of the zeros seem to coincide for different values of except for the zero at . This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integers and , the functions and have no common zeros other than the one at . The hypothesis was proved by Carl Ludwig Siegel in 1929.
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Transcendence.
Siegel proved in 1929 that when "ν" is rational, all nonzero roots of and are transcendental, as are all the roots of . It is also known that all roots of the higher derivatives formula_131 for are transcendental, except for the special values formula_132 and formula_133.
Numerical approaches.
For numerical studies about the zeros of the Bessel function, see , and .
Numerical values.
The first zeros in J0 (i.e., j0,1, j0,2 and j0,3) occur at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.
History.
Waves and elasticity problems.
The first appearance of a Bessel function appears in the work of Daniel Bernoulli in 1732, while working on the analysis of a vibrating string, a problem that was tackled before by his father Johann Bernoulli. Daniel considered a flexible chain suspended from a fixed point above and free at its lower end. The solution of the differential equation led to the introduction of a function that is now considered formula_134. Bernoulli also developed a method to find the zeros of the function.
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Leonhard Euler in 1736, found a link between other functions (now known as Laguerre polynomials) and Bernoulli's solution. Euler also introduced a non-uniform chain that lead to the introduction of functions now related to modified Bessel functions formula_135.
In the middle of the eighteen century, Jean le Rond d'Alembert had found a formula to solve the wave equation. By 1771 there was dispute between Bernoulli, Euler, d'Alembert and Joseph-Louis Lagrange on the nature of the solutions vibrating strings.
Euler worked in 1778 on buckling, introducing the concept of Euler's critical load. To solve the problem he introduced the series for formula_136. Euler also worked out the solutions of vibrating 2D membranes in cylindrical coordinates in 1780. In order to solve his differential equation he introduced a power series associated to formula_137, for integer "n".
During the end of the 19th century Lagrange, Pierre-Simon Laplace and Marc-Antoine Parseval also found equivalents to the Bessel functions. Parseval for example found an integral represantion of formula_134 using cosine.
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At the beginning of the 1800s, Joseph Fourier used formula_134 to solve the heat equation in a problem with cylindrical symmetry. Fourier won a prize of the French Academy of Sciences for this work in 1811. But most of the details of his work, including the use of a Fourier series, remained unpublished until 1822. Poisson in rivalry with Fourier, extended Fourier's work in 1823, introducing new properties of Bessel functions including Bessel functions of half-integer order (now known as spherical Bessel functions).
Astronomical problems.
In 1770, Lagrangre introduced the series expansion of Bessel functions to solve Kepler's equation, a trascendental equation in astronomy. Friedrich Wilhelm Bessel had seen Lagrange solution but found it difficult to handle. In 1813 in a letter to Carl Friedrich Gauss, Bessel simplified the calculation using trigonometric functions. Bessel published his work in 1819, independently introducing the method of Fourier series unaware of the work of Fourier who was published later.
In 1824, Bessel carried out a systematic investigation of the Bessel functions which earned the function his name. In older literature the functions were called cylindrical functions or even Bessel–Fourier functions. |
Brahui language
Brahui ( ; ; also romanised as Brahvi or Brohi) is a Dravidian language, spoken by the Brahui primarily in central areas (Brahuistan) of the Pakistani province of Balochistan; with smaller communities of speakers scattered in parts of Iranian Baluchestan, Afghanistan, and Turkmenistan (around Merv). It is also spoken by expatriate Brahui communities in Iraq, Qatar, and the United Arab Emirates. It is isolated from the nearest Dravidian-speaking neighbouring population of South India by a distance of more than . The Kalat, Khuzdar, Mastung, Quetta, Bolan, Nasirabad, Nushki, and Kharan districts of Balochistan Province are predominantly Brahui-speaking.
Brahui is the only Dravidian language that is primarily written in the Perso-Arabic script. It is also written in the Latin script.
Distribution.
Brahui is spoken in the central part of Pakistani Balochistan, mainly in the Kalat, Khuzdar and Mastung districts, but also in smaller numbers in neighboring districts, as well as in Afghanistan which borders Pakistani Balochistan; however, many members of the ethnic group no longer speak Brahui. There are also an unknown (but very small) number of expatriate Brahuis in the Arab States of the Persian Gulf, and Turkmenistan.
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History.
There is no consensus as to whether Brahui is a relatively recent language introduced into Balochistan or the remnant of a formerly more widespread Dravidian language family. According to Josef Elfenbein (1989), the most common theory is that the Brahui were part of a Dravidian migration into north-western parts of the Indian subcontinent in the 3rd millennium BC, but unlike other Dravidians who migrated to the south, they remained in Sarawan and Jahlawan since before 2000 BC. However, some other scholars see it as a recent migrant language to its present region. They postulate that Brahui could only have migrated to Balochistan from central India after 1000 AD. This is contradicted by genetic evidence that shows the Brahui population to be indistinguishable from neighbouring Balochi speakers, and genetically distant from central Dravidian speakers. The main Iranian contributor to Brahui vocabulary, Balochi, is a Northwestern Iranian language, and moved to the area from the west only around 1000 AD. One scholar places the migration as late as the 13th or 14th century. The Brahui lexicon is believed to be of: 35% Perso-Arabic origin, 20% Balochi origin, 20% Indo-Aryan origin, 15% Dravidian origin, and 10% unknown origin.
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Franklin Southworth proposed that Brahui is not a Dravidian language, but can be linked with the remaining Dravidian languages and Elamite to form the "Zagrosian family," which originated in Southwest Asia (southern Iran) and was widely distributed in South Asia and parts of eastern West Asia before the Indo-Aryan migration.
Dialects.
There are no important dialectal differences. Jhalawani (southern, centered on Khuzdar) and Sarawani (northern, centered on Kalat) dialects are distinguished by the pronunciation of *h, which is retained only in the north (Elfenbein 1997).
Brahui has been influenced by the Iranian languages spoken in the area, including Persian, Balochi and Pashto.
Phonology.
Brahui vowels show a partial length distinction between long and diphthongs and short . Brahui does not have short /e, o/ due to influence from neighbouring Indo-Aryan and Iranic languages, the PD short *e was replaced by a, ē and i, and ∗o by ō, u and a in root syllables.
Brahui consonants show patterns of retroflexion but lack the aspiration distinctions found in surrounding languages and include several fricatives such as the voiceless lateral fricative , a sound not otherwise found in the region.
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Consonants are also very similar to those of Balochi, but Brahui has more fricatives and nasals (Elfenbein 1993).
Stress.
Stress in Brahui follows a quantity-based pattern, occurring either on the first long vowel or diphthong, or on the first syllable if all vowels are short.
Orthography.
Perso-Arabic script.
Brahui is the only Dravidian language which is not known to have been written in a Brahmi-based script; instead, it has been written in the Arabic script since the second half of the 20th century. Other Dravidian languages have also been historically written in the Arabic script by the Muslim minority speakers of each respective language, namely Arabi-Tamil and Arabi-Malayalam.
In Pakistan, an Urdu based Nastaʿlīq script is used in writing. Brahui orthography is unique in having the letter representing the sound . Table below presents the letters adopted for Brahui orthography:
Latin script.
More recently, a Roman-based orthography named Brolikva (an abbreviation of "Brahui Roman Likvar") was developed by the Brahui Language Board of the University of Balochistan in Quetta and adopted by the newspaper Talár.
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Below is the new promoted Bráhuí Báşágal Brolikva orthography:
The letters with diacritics are the long vowels, post-alveolar and retroflex consonants, the voiced velar fricative and the voiceless lateral fricative.
Sample text.
English.
All human beings are born free and equal in dignity and rights. They are endowed with reason and conscience and should act towards one another in a spirit of brotherhood.
Latin script.
Muccá insáńk ájo o izzat ná rid aŧ barebar vadí massuno. Ofte puhí o dalíl raseńgáne. andáde ofte asi elo ton ílumí e vaddifoí e.
Endangerment.
According to a 2009 UNESCO report, Brahui is one of the 27 languages of Pakistan that are facing the danger of extinction. It was classified as "unsafe", the least endangered level out of the five levels of concern (Unsafe, Definitely Endangered, Severely Endangered, Critically Endangered and Extinct). This status has since been renamed to "vulnerable".
Publications.
Talár is the first daily newspaper in the Brahui language. It uses the new Roman orthography and is "an attempt to standardize and develop [the] Brahui language to meet the requirements of modern political, social and scientific discourse." |
Berkeley DB
Berkeley DB (BDB) is an embedded database software library for key/value data, historically significant in open-source software. Berkeley DB is written in C with API bindings for many other programming languages. BDB stores arbitrary key/data pairs as byte arrays and supports multiple data items for a single key. Berkeley DB is not a relational database, although it has database features including database transactions, multiversion concurrency control and write-ahead logging. BDB runs on a wide variety of operating systems, including most Unix-like and Windows systems, and real-time operating systems.
BDB was commercially supported and developed by Sleepycat Software from 1996 to 2006. Sleepycat Software was acquired by Oracle Corporation in February 2006, who continued to develop and sell the C Berkeley DB library. In 2013 Oracle re-licensed BDB under the AGPL license and released new versions until May 2020. Bloomberg L.P. continues to develop a fork of the 2013 version of BDB within their Comdb2 database, under the original Sleepycat permissive license.
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Origin.
Berkeley DB originated at the University of California, Berkeley as part of BSD, Berkeley's version of the Unix operating system. After 4.3BSD (1986), the BSD developers attempted to remove or replace all code originating in the original AT&T Unix from which BSD was derived. In doing so, they needed to rewrite the Unix database package. Seltzer and Yigit created a new database, unencumbered by any AT&T patents: an on-disk hash table that outperformed the existing dbm libraries. Berkeley DB itself was first released in 1991 and later included with 4.4BSD. In 1996 Netscape requested that the authors of Berkeley DB improve and extend the library, then at version 1.86, to suit Netscape's requirements for an LDAP server and for use in the Netscape browser. That request led to the creation of Sleepycat Software. This company was acquired by Oracle Corporation in February 2006.
Berkeley DB 1.x releases focused on managing key/value data storage and are referred to as "Data Store" (DS). The 2.x releases added a locking system enabling concurrent access to data. This is what is known as "Concurrent Data Store" (CDS). The 3.x releases added a logging system for transactions and recovery, called "Transactional Data Store" (TDS). The 4.x releases added the ability to replicate log records and create a distributed highly available single-master multi-replica database. This is called the "High Availability" (HA) feature set. Berkeley DB's evolution has sometimes led to minor API changes or log format changes, but very rarely have database formats changed. Berkeley DB HA supports online upgrades from one version to the next by maintaining the ability to read and apply the prior release's log records.
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Starting with the 6.0.21 (Oracle 12c) release, all Berkeley DB products are licensed under the GNU AGPL. Previously, Berkeley DB was redistributed under the 4-clause BSD license (before version 2.0), and the Sleepycat Public License, which is an OSI-approved open-source license as well as an FSF-approved free software license. The product ships with complete source code, build script, test suite, and documentation. The comprehensive feature along with the licensing terms have led to its use in a multitude of free and open-source software. Those who do not wish to abide by the terms of the GNU AGPL, or use an older version with the Sleepycat Public License, have the option of purchasing another proprietary license for redistribution from Oracle Corporation. This technique is called dual licensing.
Berkeley DB includes compatibility interfaces for some historic Unix database libraries: dbm, ndbm and hsearch (a System V and POSIX library for creating in-memory hash tables).
Architecture.
Berkeley DB has an architecture notably simpler than relational database management systems. Like SQLite and LMDB, it is not based on a server/client model, and does not provide support for network access programs access the database using in-process API calls. Oracle added support for SQL in 11g R2 release based on the popular SQLite API by including a version of SQLite in Berkeley DB (it uses Berkeley DB for storage).
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A program accessing the database is free to decide how the data is to be stored in a record. Berkeley DB puts no constraints on the record's data. The record and its key can both be up to four gigabytes long.
Berkeley DB supports database features such as ACID transactions, fine-grained locking, hot backups and replication.
Oracle Corporation use of name "Berkeley DB".
The name "Berkeley DB" is used by Oracle Corporation for three different products, only one of which is BDB:
Open-source programs still using Berkeley DB.
BDB was once very widespread, but usage dropped steeply from 2013 (see licensing section). Notable software that still uses Berkeley DB for data storage include:
Open-source operating systems and languages such as Perl and Python still support old BerkelyDB interfaces. The FreeBSD and OpenBSD operating systems ship Berkeley DB 1.8x to support the codice_1 operating system call used by password programs such as codice_2. Linux operating systems, including those based on Debian, and Fedora ship Berkeley DB 5.3 libraries.
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Licensing.
Berkeley DB V2.0 and higher is available under a dual license:
Switching the open source license in 2013 from the Sleepycat license to the AGPL had a major effect on open source software. Since BDB is a library, any application linking to it must be under an AGPL-compatible license. Many open source applications and all closed source applications would need to be relicensed to become AGPL-compatible, which was not acceptable to many developers and open source operating systems. By 2013 there were many alternatives to BDB, and Debian Linux was typical in their decision to completely phase out Berkeley DB, with a preference for the Lightning Memory-Mapped Database (LMDB). |
Boolean satisfiability problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) asks whether there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the formula's variables can be consistently replaced by the values TRUE or FALSE to make the formula evaluate to TRUE. If this is the case, the formula is called "satisfiable", else "unsatisfiable". For example, the formula ""a" AND NOT "b" is satisfiable because one can find the values "a" = TRUE and "b" = FALSE, which make ("a" AND NOT "b") = TRUE. In contrast, "a" AND NOT "a"" is unsatisfiable.
SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem. This means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization problems, are at most as difficult to solve as SAT. There is no known algorithm that efficiently solves each SAT problem, and it is generally believed that no such algorithm exists, but this belief has not been proven mathematically, and resolving the question of whether SAT has a polynomial-time algorithm is equivalent to the P versus NP problem, which is a famous open problem in the theory of computing.
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Nevertheless, as of 2007, heuristic SAT-algorithms are able to solve problem instances involving tens of thousands of variables and formulas consisting of millions of symbols, which is sufficient for many practical SAT problems from, e.g., artificial intelligence, circuit design, and automatic theorem proving.
Definitions.
A "propositional logic formula", also called "Boolean expression", is built from variables, operators AND (conjunction, also denoted by ∧), OR (disjunction, ∨), NOT (negation, ¬), and parentheses. A formula is said to be "satisfiable" if it can be made TRUE by assigning appropriate logical values (i.e. TRUE, FALSE) to its variables. The "Boolean satisfiability problem" (SAT) is, given a formula, to check whether it is satisfiable. This decision problem is of central importance in many areas of computer science, including theoretical computer science, complexity theory, algorithmics, cryptography and artificial intelligence.
Conjunctive normal form.
A "literal" is either a variable (in which case it is called a "positive literal") or the negation of a variable (called a "negative literal"). A "clause" is a disjunction of literals (or a single literal). A clause is called a "Horn clause" if it contains at most one positive literal. A formula is in "conjunctive normal form" (CNF) if it is a conjunction of clauses (or a single clause).
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For example, is a positive literal, is a negative literal, and is a clause. The formula is in conjunctive normal form; its first and third clauses are Horn clauses, but its second clause is not. The formula is satisfiable, by choosing "x"1 = FALSE, "x"2 = FALSE, and "x"3 arbitrarily, since (FALSE ∨ ¬FALSE) ∧ (¬FALSE ∨ FALSE ∨ "x"3) ∧ ¬FALSE evaluates to (FALSE ∨ TRUE) ∧ (TRUE ∨ FALSE ∨ "x"3) ∧ TRUE, and in turn to TRUE ∧ TRUE ∧ TRUE (i.e. to TRUE). In contrast, the CNF formula "a" ∧ ¬"a", consisting of two clauses of one literal, is unsatisfiable, since for "a"=TRUE or "a"=FALSE it evaluates to TRUE ∧ ¬TRUE (i.e., FALSE) or FALSE ∧ ¬FALSE (i.e., again FALSE), respectively.
For some versions of the SAT problem, it is useful to define the notion of a "generalized conjunctive normal form" formula, viz. as a conjunction of arbitrarily many "generalized clauses", the latter being of the form for some Boolean function "R" and (ordinary) literals . Different sets of allowed Boolean functions lead to different problem versions. As an example, "R"(¬"x","a","b") is a generalized clause, and "R"(¬"x","a","b") ∧ "R"("b","y","c") ∧ "R"("c","d",¬"z") is a generalized conjunctive normal form. This formula is used below, with "R" being the ternary operator that is TRUE just when exactly one of its arguments is.
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Using the laws of Boolean algebra, every propositional logic formula can be transformed into an equivalent conjunctive normal form, which may, however, be exponentially longer. For example, transforming the formula ("x"1∧"y"1) ∨ ("x"2∧"y"2) ∨ ... ∨ ("x""n"∧"y""n") into conjunctive normal form yields
while the former is a disjunction of "n" conjunctions of 2 variables, the latter consists of 2"n" clauses of "n" variables.
However, with use of the Tseytin transformation, we may find an equisatisfiable conjunctive normal form formula with length linear in the size of the original propositional logic formula.
Complexity.
SAT was the first problem known to be NP-complete, as proved by Stephen Cook at the University of Toronto in 1971 and independently by Leonid Levin at the Russian Academy of Sciences in 1973. Until that time, the concept of an NP-complete problem did not even exist. The proof shows how every decision problem in the complexity class NP can be reduced to the SAT problem for CNF formulas, sometimes called CNFSAT. A useful property of Cook's reduction is that it preserves the number of accepting answers. For example, deciding whether a given graph has a 3-coloring is another problem in NP; if a graph has 17 valid 3-colorings, then the SAT formula produced by the Cook–Levin reduction will have 17 satisfying assignments.
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NP-completeness only refers to the run-time of the worst case instances. Many of the instances that occur in practical applications can be solved much more quickly. See §Algorithms for solving SAT below.
3-satisfiability.
Like the satisfiability problem for arbitrary formulas, determining the satisfiability of a formula in conjunctive normal form where each clause is limited to at most three literals is NP-complete also; this problem is called 3-SAT, 3CNFSAT, or 3-satisfiability. To reduce the unrestricted SAT problem to 3-SAT, transform each clause to a conjunction of clauses
where are fresh variables not occurring elsewhere. Although the two formulas are not logically equivalent, they are equisatisfiable. The formula resulting from transforming all clauses is at most 3 times as long as its original; that is, the length growth is polynomial.
3-SAT is one of Karp's 21 NP-complete problems, and it is used as a starting point for proving that other problems are also NP-hard. This is done by polynomial-time reduction from 3-SAT to the other problem. An example of a problem where this method has been used is the clique problem: given a CNF formula consisting of "c" clauses, the corresponding graph consists of a vertex for each literal, and an edge between each two non-contradicting literals from different clauses; see the picture. The graph has a "c"-clique if and only if the formula is satisfiable.
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There is a simple randomized algorithm due to Schöning (1999) that runs in time (4/3)"n" where "n" is the number of variables in the 3-SAT proposition, and succeeds with high probability to correctly decide 3-SAT.
The exponential time hypothesis asserts that no algorithm can solve 3-SAT (or indeed "k"-SAT for any ) in time (that is, fundamentally faster than exponential in "n").
Selman, Mitchell, and Levesque (1996) give empirical data on the difficulty of randomly generated 3-SAT formulas, depending on their size parameters. Difficulty is measured in number recursive calls made by a DPLL algorithm. They identified a phase transition region from almost-certainly-satisfiable to almost-certainly-unsatisfiable formulas at the clauses-to-variables ratio at about 4.26.
3-satisfiability can be generalized to k-satisfiability (k-SAT, also k-CNF-SAT), when formulas in CNF are considered with each clause containing up to "k" literals. However, since for any "k" ≥ 3, this problem can neither be easier than 3-SAT nor harder than SAT, and the latter two are NP-complete, so must be k-SAT.
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Some authors restrict k-SAT to CNF formulas with exactly k literals. This does not lead to a different complexity class either, as each clause with "j" < "k" literals can be padded with fixed dummy variables to . After padding all clauses, 2"k"–1 extra clauses must be appended to ensure that only can lead to a satisfying assignment. Since "k" does not depend on the formula length, the extra clauses lead to a constant increase in length. For the same reason, it does not matter whether duplicate literals are allowed in clauses, as in .
Special cases of SAT.
Conjunctive normal form.
Conjunctive normal form (in particular with 3 literals per clause) is often considered the canonical representation for SAT formulas. As shown above, the general SAT problem reduces to 3-SAT, the problem of determining satisfiability for formulas in this form.
Disjunctive normal form.
SAT is trivial if the formulas are restricted to those in disjunctive normal form, that is, they are a disjunction of conjunctions of literals. Such a formula is indeed satisfiable if and only if at least one of its conjunctions is satisfiable, and a conjunction is satisfiable if and only if it does not contain both "x" and NOT "x" for some variable "x". This can be checked in linear time. Furthermore, if they are restricted to being in full disjunctive normal form, in which every variable appears exactly once in every conjunction, they can be checked in constant time (each conjunction represents one satisfying assignment). But it can take exponential time and space to convert a general SAT problem to disjunctive normal form; to obtain an example, exchange "∧" and "∨" in the above exponential blow-up example for conjunctive normal forms.
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Exactly-1 3-satisfiability.
A variant of the 3-satisfiability problem is the one-in-three 3-SAT (also known variously as 1-in-3-SAT and exactly-1 3-SAT). Given a conjunctive normal form with three literals per clause, the problem is to determine whether there exists a truth assignment to the variables so that each clause has "exactly" one TRUE literal (and thus exactly two FALSE literals). In contrast, ordinary 3-SAT requires that every clause has "at least" one TRUE literal. Formally, a one-in-three 3-SAT problem is given as a generalized conjunctive normal form with all generalized clauses using a ternary operator "R" that is TRUE just if exactly one of its arguments is. When all literals of a one-in-three 3-SAT formula are positive, the satisfiability problem is called one-in-three positive 3-SAT.
One-in-three 3-SAT, together with its positive case, is listed as NP-complete problem "LO4" in the standard reference "Computers and Intractability: A Guide to the Theory of NP-Completeness" by Michael R. Garey and David S. Johnson. One-in-three 3-SAT was proved to be NP-complete by Thomas Jerome Schaefer as a special case of Schaefer's dichotomy theorem, which asserts that any problem generalizing Boolean satisfiability in a certain way is either in the class P or is NP-complete.
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Schaefer gives a construction allowing an easy polynomial-time reduction from 3-SAT to one-in-three 3-SAT. Let "("x" or "y" or "z")" be a clause in a 3CNF formula. Add six fresh Boolean variables "a", "b", "c", "d", "e", and "f", to be used to simulate this clause and no other. Then the formula "R"("x","a","d") ∧ "R"("y","b","d") ∧ "R"("a","b","e") ∧ "R"("c","d","f") ∧ "R"("z","c",FALSE) is satisfiable by some setting of the fresh variables if and only if at least one of "x", "y", or "z" is TRUE, see picture (left). Thus any 3-SAT instance with "m" clauses and "n" variables may be converted into an equisatisfiable one-in-three 3-SAT instance with 5"m" clauses and "n" + 6"m" variables. Another reduction involves only four fresh variables and three clauses: "R"(¬"x","a","b") ∧ "R"("b","y","c") ∧ R("c","d",¬"z"), see picture (right).
Not-all-equal 3-satisfiability.
Another variant is the not-all-equal 3-satisfiability problem (also called NAE3SAT). Given a conjunctive normal form with three literals per clause, the problem is to determine if an assignment to the variables exists such that in no clause all three literals have the same truth value. This problem is NP-complete, too, even if no negation symbols are admitted, by Schaefer's dichotomy theorem.
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Linear SAT.
A 3-SAT formula is "Linear SAT" ("LSAT") if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. An LSAT formula can be depicted as a set of disjoint semi-closed intervals on a line. Deciding whether an LSAT formula is satisfiable is NP-complete.
2-satisfiability.
SAT is easier if the number of literals in a clause is limited to at most 2, in which case the problem is called 2-SAT. This problem can be solved in polynomial time, and in fact is complete for the complexity class NL. If additionally all OR operations in literals are changed to XOR operations, then the result is called exclusive-or 2-satisfiability, which is a problem complete for the complexity class SL = L.
Horn-satisfiability.
The problem of deciding the satisfiability of a given conjunction of Horn clauses is called Horn-satisfiability, or HORN-SAT. It can be solved in polynomial time by a single step of the unit propagation algorithm, which produces the single minimal model of the set of Horn clauses (w.r.t. the set of literals assigned to TRUE). Horn-satisfiability is P-complete. It can be seen as P's version of the Boolean satisfiability problem. Also, deciding the truth of quantified Horn formulas can be done in polynomial time.
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Horn clauses are of interest because they are able to express implication of one variable from a set of other variables. Indeed, one such clause ¬"x"1 ∨ ... ∨ ¬"x""n" ∨ "y" can be rewritten as "x"1 ∧ ... ∧ "x""n" → "y"; that is, if "x"1...,"x""n" are all TRUE, then "y" must be TRUE as well.
A generalization of the class of Horn formulas is that of renameable-Horn formulae, which is the set of formulas that can be placed in Horn form by replacing some variables with their respective negation. For example, ("x"1 ∨ ¬"x"2) ∧ (¬"x"1 ∨ "x"2 ∨ "x"3) ∧ ¬"x"1 is not a Horn formula, but can be renamed to the Horn formula ("x"1 ∨ ¬"x"2) ∧ (¬"x"1 ∨ "x"2 ∨ ¬"y"3) ∧ ¬"x"1 by introducing "y"3 as negation of "x"3. In contrast, no renaming of ("x"1 ∨ ¬"x"2 ∨ ¬"x"3) ∧ (¬"x"1 ∨ "x"2 ∨ "x"3) ∧ ¬"x"1 leads to a Horn formula. Checking the existence of such a replacement can be done in linear time; therefore, the satisfiability of such formulae is in P as it can be solved by first performing this replacement and then checking the satisfiability of the resulting Horn formula.
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XOR-satisfiability.
Another special case is the class of problems where each clause contains XOR (i.e. exclusive or) rather than (plain) OR operators. This is in P, since an XOR-SAT formula can also be viewed as a system of linear equations mod 2, and can be solved in cubic time by Gaussian elimination; see the box for an example. This recast is based on the kinship between Boolean algebras and Boolean rings, and the fact that arithmetic modulo two forms a finite field. Since "a" XOR "b" XOR "c" evaluates to TRUE if and only if exactly 1 or 3 members of {"a","b","c"} are TRUE, each solution of the 1-in-3-SAT problem for a given CNF formula is also a solution of the XOR-3-SAT problem, and in turn each solution of XOR-3-SAT is a solution of 3-SAT; see the picture. As a consequence, for each CNF formula, it is possible to solve the XOR-3-SAT problem defined by the formula, and based on the result infer either that the 3-SAT problem is solvable or that the 1-in-3-SAT problem is unsolvable.
Provided that the complexity classes P and NP are not equal, neither 2-, nor Horn-, nor XOR-satisfiability is NP-complete, unlike SAT.
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Schaefer's dichotomy theorem.
The restrictions above (CNF, 2CNF, 3CNF, Horn, XOR-SAT) bound the considered formulae to be conjunctions of subformulas; each restriction states a specific form for all subformulas: for example, only binary clauses can be subformulas in 2CNF.
Schaefer's dichotomy theorem states that, for any restriction to Boolean functions that can be used to form these subformulas, the corresponding satisfiability problem is in P or NP-complete. The membership in P of the satisfiability of 2CNF, Horn, and XOR-SAT formulae are special cases of this theorem.
The following table summarizes some common variants of SAT.
Extensions of SAT.
An extension that has gained significant popularity since 2003 is satisfiability modulo theories (SMT) that can enrich CNF formulas with linear constraints, arrays, all-different constraints, uninterpreted functions, etc. Such extensions typically remain NP-complete, but very efficient solvers are now available that can handle many such kinds of constraints.
The satisfiability problem becomes more difficult if both "for all" (∀) and "there exists" (∃) quantifiers are allowed to bind the Boolean variables. An example of such an expression would be ; it is valid, since for all values of "x" and "y", an appropriate value of "z" can be found, viz. "z"=TRUE if both "x" and "y" are FALSE, and "z"=FALSE else. SAT itself (tacitly) uses only ∃ quantifiers. If only ∀ quantifiers are allowed instead, the so-called tautology problem is obtained, which is co-NP-complete. If any number of both quantifiers are allowed, the problem is called the quantified Boolean formula problem (QBF), which can be shown to be PSPACE-complete. It is widely believed that PSPACE-complete problems are strictly harder than any problem in NP, although this has not yet been proved. Using highly parallel "P systems", QBF-SAT problems can be solved in linear time.
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Ordinary SAT asks if there is at least one variable assignment that makes the formula true. A variety of variants deal with the number of such assignments:
Other generalizations include satisfiability for first- and second-order logic, constraint satisfaction problems, 0-1 integer programming.
Finding a satisfying assignment.
While SAT is a decision problem, the search problem of finding a satisfying assignment reduces to SAT. That is, each algorithm which correctly answers whether an instance of SAT is solvable can be used to find a satisfying assignment. First, the question is asked on the given formula Φ. If the answer is "no", the formula is unsatisfiable. Otherwise, the question is asked on the partly instantiated formula Φ{"x"1=TRUE}, that is, Φ with the first variable "x"1 replaced by TRUE, and simplified accordingly. If the answer is "yes", then "x"1=TRUE, otherwise "x"1=FALSE. Values of other variables can be found subsequently in the same way. In total, "n"+1 runs of the algorithm are required, where "n" is the number of distinct variables in Φ.
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This property is used in several theorems in complexity theory:
Algorithms for solving SAT.
Since the SAT problem is NP-complete, only algorithms with exponential worst-case complexity are known for it. In spite of this, efficient and scalable algorithms for SAT were developed during the 2000s and have contributed to dramatic advances in the ability to automatically solve problem instances involving tens of thousands of variables and millions of constraints (i.e. clauses). Examples of such problems in electronic design automation (EDA) include formal equivalence checking, model checking, formal verification of pipelined microprocessors, automatic test pattern generation, routing of FPGAs, planning, and scheduling problems, and so on. A SAT-solving engine is also considered to be an essential component in the electronic design automation toolbox.
Major techniques used by modern SAT solvers include the Davis–Putnam–Logemann–Loveland algorithm (or DPLL), conflict-driven clause learning (CDCL), and stochastic local search algorithms such as WalkSAT. Almost all SAT solvers include time-outs, so they will terminate in reasonable time even if they cannot find a solution. Different SAT solvers will find different instances easy or hard, and some excel at proving unsatisfiability, and others at finding solutions. Recent attempts have been made to learn an instance's satisfiability using deep learning techniques.
SAT solvers are developed and compared in SAT-solving contests. Modern SAT solvers are also having significant impact on the fields of software verification, constraint solving in artificial intelligence, and operations research, among others. |
Bob Jones University
Bob Jones University (BJU) is a private university in Greenville, South Carolina, United States. It is known for its conservative and evangelical cultural and religious positions. The university, with approximately 3,000 students, is accredited by the Southern Association of Colleges and Schools Commission on Colleges (SACSCOC) and the Transnational Association of Christian Colleges and Schools. In 2017, the university estimated the number of its graduates at 40,184.
History.
During the Fundamentalist-Modernist controversy of the 1920s, Christian evangelist Bob Jones Sr. grew increasingly concerned about what he perceived to be the secularization of higher education and the influence of religious liberalism in denominational colleges. Jones recalled that in 1924, his friend William Jennings Bryan leaned over to him at a Bible conference service in Winona Lake, Indiana, and said, "If schools and colleges do not quit teaching evolution as a fact, we are going to become a nation of atheists." Though Jones was not a college graduate, he was determined to found a college. On September 12, 1927, Jones opened Bob Jones College in Panama City, Florida, with 88 students. Jones said that although he had been averse to naming the school after himself, his friends overcame his reluctance "with the argument that the school would be called by that name because of my connection with it, and to attempt to give it any other name would confuse the people".
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Bob Jones took no salary from the college. He supported the school with personal savings and income from his evangelistic campaigns. The Florida land boom had peaked in 1925, and a hurricane in September 1926 further reduced land values. Bob Jones College barely survived bankruptcy and its move to Cleveland, Tennessee, in 1933. In the same year, the college also ended participation in intercollegiate sports. Bankrupt at the nadir of the Depression, without a home and with barely enough money to move its library and office furniture, the college became the largest liberal arts college in Tennessee thirteen years later. With the enactment of the GI Bill at the end of World War II, the need for campus expansion to accommodate increased enrollment led to a relocation to South Carolina.
Though Jones had served as acting president as early as 1934, his son, Bob Jones Jr. became the school's second president in 1947 before the college moved to Greenville, South Carolina, and became Bob Jones University. In Greenville, the university more than doubled in size within two years and started an AM radio station in 1949 (1260 WMUU with 94.5 WMUU-FM signing on in 1960), film department, and art gallery—the latter of which eventually became one of the largest collections of religious art in the Western Hemisphere.
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During the late 1950s, BJU and alumnus Billy Graham, who had attended Bob Jones College for one semester in 1936 and received an honorary degree from the university in 1948, had a dispute over the propriety of theological conservatives cooperating with theological liberals to support evangelistic campaigns, a controversy that widened an already growing rift between separatist fundamentalists and other evangelicals. Negative publicity caused by the dispute precipitated a decline in BJU enrollment of about 10% in the years 1956–59, and seven members of the university board (of about a hundred) also resigned in support of Graham, including Graham himself and two of his staff members. When, in 1966, Graham held his only American campaign in Greenville, the university forbade BJU dormitory students to attend under penalty of expulsion. Enrollment quickly rebounded, and by 1970, there were 3,300 students, approximately 60% more than in 1958.
In 1971, Bob Jones III became president at age 32, though his father, with the title of Chancellor, continued to exercise considerable administrative authority into the late 1990s. At the 2005 commencement, Stephen Jones was installed as the fourth president, and Bob Jones III assumed the title of chancellor. Stephen Jones resigned in 2014 for health reasons, and evangelist Steve Pettit was named president, the first president unrelated to the Jones family.
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In 2011, the university became a member of the Transnational Association of Christian Colleges and Schools (TRACS) and reinstated intercollegiate athletics. In March 2017, the university regained its federal tax exemption after a complicated restructuring divided the organization into for-profit and non-profit entities, and in June 2017, it was granted accreditation by the Southern Association of Colleges and Schools.
In March 2023, Pettit resigned, effective May 5, citing his inability to work with the chairman of the university's board of trustees. Shortly thereafter, the president of the board also resigned. Vice President Alan Benson became the interim president for the 2023–24 school year. In May of 2024, Baptist pastor and BJU alumnus Joshua Crockett was elected the university's sixth president.
Academics.
The university comprises seven colleges and schools offering more than 60 undergraduate majors, including fourteen associate degree programs. Many of the university employees consider their positions as much ministries as jobs. It is common for retiring professors to have served the university for more than forty years, a circumstance that has contributed to the stability and conservatism of an institution that has virtually no endowment and at which faculty salaries are "sacrificial".
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Religious education.
School of Religion.
The School of Religion includes majors for both men and women, although only men train as ministerial students. Many of these students go on to a seminary after completing their undergraduate education. Others take ministry positions straight from college. In 1995, 1,290 BJU graduates were serving as senior or associate pastors in churches across the United States. In 2017 more than 100 pastors in the Upstate (South Carolina) alone were BJU graduates.
Fine arts.
The Division of Fine Arts has the largest faculty of the university's six undergraduate schools. Each year, the university presents an opera in the spring semester and Shakespearean plays in both the fall and spring semesters. The Division of Fine Arts includes an RTV department with a campus radio and television station, WBJU. More than a hundred concerts, recitals, and laboratory theater productions are also presented annually.
Each fall, as a recruiting tool, the university sponsors a "High School Festival" in which students compete in music, art, and speech (including preaching) contests with their peers from around the country. In the spring, a similar competition sponsored by the American Association of Christian Schools, and hosted by BJU since 1977, brings thousands of national finalists to the university from around the country. In 2005, 120 of the finalists from previous years returned to BJU as freshmen.
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Science.
Bob Jones University supports young-earth creationism, all their biology faculty are young Earth creationists and the university rejects evolution, calling it "at best an unsupportable and unworkable hypothesis".
According to the BJU website, "More than 80% of our premed graduates are accepted to medical or dental school within a year of graduation." The Department of Biology hosts two research programs on campus, one in cancer research, the other in animal behavior. Although ten of the sixteen members of the science faculty have bachelor's degrees from BJU, all earned their doctorates from accredited, non-religious institutions.
The university's nursing major is approved by the South Carolina State Board of Nursing, and a BJU graduate with a BSN is eligible to take the National Council Licensure Examination to become a registered nurse. The BJU engineering program is accredited by the Accreditation Board for Engineering and Technology (ABET).
Accreditation and rankings.
Bob Jones Sr. was leery of academic accreditation almost from the founding of the college, and by the early 1930s, he had publicly stated his opposition to holding regional accreditation. Jones and the college were criticized for this stance, and academic recognition, as well as student and faculty recruitment, were hindered.
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In 1944, Jones wrote to John Walvoord of Dallas Theological Seminary that while the university had "no objection to educational work highly standardized…. We, however, cannot conscientiously let some group of educational experts or some committee of experts who may have a behavioristic or atheistic slant on education control or even influence the administrative policies of our college." Five years later, Jones reflected that "it cost us something to stay out of an association, but we stayed out. We have lived up to our convictions."
Because graduates did not benefit from accredited degrees, the faculty felt an increased responsibility to prepare their students. Early in the history of the college, there had been some hesitancy on the part of other institutions to accept BJU credits at face value, but by the 1960s, BJU alumni were being accepted by most of the major graduate and professional schools in the United States. Undoubtedly helpful was that some of the university's strongest programs were in the areas of music, speech, and art, disciplines in which ability could be measured by audition or portfolio rather than through paper qualifications.
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Nevertheless, by the early 2000s, the university quietly reexamined its position on accreditation as degree mills proliferated, and some government agencies, such as local police departments, began excluding BJU graduates because the university did not appear on appropriate federal lists. In 2004, the university began the process of joining the Transnational Association of Christian Colleges and Schools. Candidate status—effectively, accreditation—was obtained in April 2005, and full membership in the Association was conferred in November 2006. In December 2011, BJU announced its intention to apply for regional accreditation with the Southern Association of Colleges and Schools (SACSCOC), and it received that accreditation in 2017.
In 2017, "US News" ranked BJU as #61 (tie) in Regional Universities South and #7 in Best Value Schools.
Political involvement.
As a twelve-year-old, Bob Jones Sr. made a twenty-minute speech in defense of the Populist Party. Jones was a friend and admirer of William Jennings Bryan but also campaigned throughout the South for Herbert Hoover (and against Al Smith) during the 1928 presidential election. The authorized history of BJU notes that both Bob Jones Sr. and Bob Jones Jr. "played political hardball" when dealing with the three municipalities in which the school was successively located. For instance, in 1962, Bob Jones Sr. warned the Greenville City Council that he had "four hundred votes in his pocket and in any election he would have control over who would be elected."
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Bob Jones Sr.'s April 17, 1960, Easter Sunday sermon, broadcast on the radio, entitled "Is Segregation Scriptural?" served as the university position paper on race in the 60s, 70s, and 80s. The transcript was sent in pamphlet form in fund-raising letters and sold in the university bookstore. In the sermon, Jones states, "If you are against segregation and against racial separation, then you are against God Almighty." The school began a long history of supporting politicians who were considered aligned with racial segregation.
Republican Party ties.
From nearly the inception of Bob Jones College, a majority of students and faculty were from the northern United States, where there was a larger ratio of Republicans to Democrats than in the South (which was solidly Democratic). Therefore, almost from its founding year, BJU had a larger portion of Republicans than the surrounding community. After South Carolina Senator Strom Thurmond switched his allegiance to the Republican Party in 1964, BJU faculty members became increasingly influential in the new state Republican party. BJU alumni were elected to local political and party offices. In 1976, candidates supported by BJU faculty and alumni captured the local Republican party with unfortunate short-term political consequences, but by 1980 the religious right and the "country club" Republicans had joined forces. From then on, most Republican candidates for local and statewide offices sought the endorsement of Bob Jones III and greeted faculty/staff voters at the University Dining Common.
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National Republicans soon followed. Ronald Reagan spoke at the school in 1980, although the Joneses supported his opponent, John Connally, in the South Carolina primary. Later, Bob Jones III denounced Reagan as "a traitor to God's people" for choosing George H. W. Bush—whom Jones called a "devil"—as his vice president. Even later, Jones III shook Bush's hand and thanked him for being a good president. In the 1990s, other Republicans such as Dan Quayle, Pat Buchanan, Phil Gramm, Bob Dole, and Alan Keyes also spoke at BJU. Democrats were rarely invited to speak at the university, in part because they took political and social positions (especially support for abortion rights) opposed by the Religious Right.
2000 election.
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Withdrawal from politics.
Although the March 2007 issue of "Foreign Policy" listed BJU as one of "The World's Most Controversial Religious Sites" because of its past influence on American politics, BJU has seen little political controversy since Stephen Jones became president. When asked by a "Newsweek" reporter if he wished to play a political role, Stephen Jones replied, "It would not be my choice." Further, when asked if he felt ideologically closer to his father's engagement with politics or to other evangelicals who have tried to avoid civic involvement, Jones answered, "The gospel is for individuals. The main message we have is to individuals. We're not here to save the culture." In a 2005 "Washington Post" interview, Jones dodged political questions and even admitted that he was embarrassed by "some of the more vitriolic comments" made by his predecessors. "I don't want to get specific," Jones said, "But there were things said back then that I wouldn't say today." In October 2007, when Bob Jones III, as "a private citizen," endorsed Mitt Romney for the Republican nomination for president, Stephen Jones made it clear that he wished "to stay out of politics" and that neither he nor the university had endorsed anyone. Despite a hotly contested South Carolina primary, none of the candidates appeared on the platform of BJU's Founders' Memorial Amphitorium during the 2008 election cycle. In April 2008, Stephen Jones told a reporter, "I don't think I have a political bone in my body."
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Renewed political engagement.
In 2015 BJU reemerged as a campaign stop for conservative Republicans. Ben Carson and Ted Cruz held large on-campus rallies on two successive days in November. BJU president Steve Pettit met with Marco Rubio, Rick Perry, Mike Huckabee, and Scott Walker. Jeb Bush, Carson, Cruz, and Rubio also appeared at a 2016 Republican presidential forum at BJU. Chip Felkel, a Greenville Republican consultant, noted that some candidates closely identified "with the folks at Bob Jones. So it makes sense for them to want to be there." Nevertheless, unlike BJU's earlier periods of political involvement, Pettit did not endorse a candidate.
According to Furman University political science professor Jim Guth, because Greenville has grown so much recently, it is unlikely BJU will ever again have the same political influence it had between the 1960s and the 1980s. Nevertheless, about a quarter of all BJU graduates continue to live in the Upstate, and as long-time mayor Knox White has said, "The alumni have had a big impact on every profession and walk of life in Greenville."
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Campus.
The university occupies 205 acres at the eastern city limit of Greenville. The institution moved into its initial 25 buildings during the 1947–48 school year, and later buildings were also faced with the light yellow brick chosen for the originals.
Museum and gallery.
Bob Jones Jr. was a connoisseur of European art from his teen years and began collecting after World War II on about $30,000 a year authorized by the University Board of Directors. Jones first concentrated on the Italian Baroque, a style then out of favor and relatively inexpensive in the years immediately following the war. The museum's collection currently includes more than 400 European paintings from the 14th through the 19th centuries, period furniture, and a notable collection of Russian icons. The museum also includes a variety of Holy Land antiquities. The gallery is strong in Baroque paintings and includes notable works by Rubens, Tintoretto, Veronese, Cranach, Gerard David, Murillo, Mattia Preti, Ribera, van Dyck, and Gustave Doré. Included in the Museum & Gallery collection are seven large canvases, part of a series by Benjamin West painted for George III, called "The Progress of Revealed Religion", which are displayed in the War Memorial Chapel. The museum also includes a variety of Holy Land antiquities collected in the early 20th century by missionaries Frank and Barbara Bowen.
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Every Easter, the university and the Museum & Gallery present the "Living Gallery", a series of tableaux vivants recreating noted works of religious art using live models disguised as part of two-dimensional paintings.
BJU has been criticized by some fundamentalists for promoting "false Catholic doctrine" through its art gallery because much of Baroque art was created for the Counter-Reformation.
A painting by Lucas van Leyden that had been displayed in the gallery's collection for more than ten years and had been consigned to Sotheby's for sale was recognized by Interpol as art that had been stolen by the Nazis from the Mittelrhein-Museum in Koblenz. The painting was eventually returned to Germany after months of negotiations between the Mittelrhein-Museum and Julius H. Weitzner, a dealer in Old Master paintings.
After the death of Bob Jones Jr., Erin Jones, the wife of BJU president Stephen Jones, became director. According to David Steel, curator of European art at the North Carolina Museum of Art, Erin Jones "brought that museum into the modern era", employing "a top-notch curator, John Nolan", and following "best practices in conservation and restoration". The museum cooperates with other institutions, lending works for outside shows such as a Rembrandt exhibit in 2011.
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In 2008, the BJU Museum & Gallery opened a satellite location, the Museum & Gallery at Heritage Green near downtown Greenville, which featured rotating exhibitions from the main museum and interactive children's activities. In February 2017, the Museum & Gallery closed both locations permanently. In 2018, the museum announced that a new home would be built at a yet undetermined located off the BJU campus. In 2021, Erin Jones said the museum was exploring a permanent home near the proposed downtown conference center.
Library.
The Mack Library (named for John Sephus Mack) holds a collection of more than 300,000 books and includes seating for 1,200 as well as a computer lab and a computer classroom. (Its ancillary, a music library, is included in the Gustafson Fine Arts Center.) Mack Library's Special Collections includes an American Hymnody Collection of about 700 titles. The "Jerusalem Chamber" is a replica of the room in Westminster Abbey in which work on the King James Version of the Bible was conducted, and it displays a collection of rare Bibles. An adjoining Memorabilia Room commemorates the life of Bob Jones Sr. and the history of the university.
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The library's Fundamentalism File collects periodical articles and ephemera about social and religious matters of interest to evangelicals and fundamentalists. The University Archives holds copies of all university publications, oral histories of faculty and staff members, surviving remnants of university correspondence, and pictures and artifacts related to the Jones family and the history of the university.
Ancillary ministries.
"Unusual Films".
Both Bob Jones Sr. and Bob Jones Jr. believed that film could be an excellent medium for mass evangelism, and in 1950, the university established "Unusual Films" within the School of Fine Arts. (The studio name derives from a former BJU promotional slogan, "The World's Most Unusual University".) Bob Jones Jr. selected a speech teacher, Katherine Stenholm, as the first director. Although she had no experience in cinema, she took summer courses at the University of Southern California and received personal instruction from Hollywood specialists, such as Rudolph Sternad.
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Unusual Films has produced seven feature-length films, each with an evangelistic emphasis: "Wine of Morning", "Red Runs the River", "Flame in the Wind", "Sheffey", "Beyond the Night", "The Printing", and "Milltown Pride". "Wine of Morning" (1955), based on a novel by Bob Jones Jr., represented the United States at the Cannes Film Festival. The first four films are historical dramas set, respectively, in the time of Christ, the U.S. Civil War, 16th-century Spain, and the late 19th-century South—the latter a fictionalized treatment of the life of Methodist evangelist, Robert Sayers Sheffey. "Beyond the Night" closely follows an actual 20th-century missionary saga in Central Africa, and "The Printing" uses composite characters to portray the persecution of believers in the former Soviet Union. According to The Dove Foundation, "The Printing" "no doubt will urge Christian believers everywhere to appreciate the freedoms they enjoy. It is inspiring!" In 1999, Unusual Films began producing feature films for children, including "The Treasure Map", "Project Dinosaur", and "Appalachian Trial".
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BJU Press.
BJU Press originated from the need for textbooks for the burgeoning Christian school movement. The press publishes a full range of K–12 textbooks.
BJU Press also offers distance learning courses online, via DVD and hard drive. Another ancillary, the Academy of Home Education, is a "service organization for homeschooling families" that maintains student records, administers achievement testing, and issues high school diplomas. The press sold its music division, SoundForth, to Lorenz Publishing on October 1, 2012.
Pre-college programs.
The university operates Bob Jones Academy, which enrolls students from preschool through 12th grade. With about 1100 students, the school's demographic makeup leans heavily white (90.3%), with non-Black minorities making up the bulk of other ethnicities. Black students make up 0.5% of enrollment.
Controversies.
Sexual abuse reports.
In December 2011, in response to accusations of mishandling of student reports of sexual abuse (most of which had occurred in their home churches when the students were minors) and a concurrent reporting issue at a church pastored by a university board member, the BJU board of trustees hired an independent ombudsman, GRACE (Godly Response to Abuse in the Christian Environment), to investigate. Released in December 2014, the GRACE report suggested that BJU had discouraged students from reporting past sexual abuse, and though the university declined to implement many of the report's recommendations, President Steve Pettit formally apologized "to those who felt they did not receive from us genuine love, compassion, understanding, and support after suffering sexual abuse or assault". The university's mishandling of sexual abuse in the past came into light again in August 2020 when a student filed a lawsuit against Bob Jones University and Furman University alleging both administrations ignored the sexual assault report and expelled the student for consuming alcohol, which is against the Student Code of Conduct handbook.
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Racial policies and ban on interracial dating.
Although BJU had admitted Asian students and other ethnic groups from its inception, it did not enroll Black students until 1971. From 1971 to 1975, BJU admitted only married Black people. However, the Internal Revenue Service (IRS) had already determined in 1970 that "private schools with racially discriminatory admissions policies" were not entitled to federal tax exemption. In 1975, the University Board of Trustees authorized a policy change to admit Black students, a move that occurred shortly before the announcement of the Supreme Court decision in "Runyon v. McCrary" (427 U.S. 160 [1976]), which prohibited racial exclusion in private schools. In May 1975, BJU expanded rules against interracial dating and marriage.
In 1976, the Internal Revenue Service revoked the university's tax exemption retroactively to December 1, 1970, because it practiced racial discrimination. The case was heard by the U.S. Supreme Court in 1982. After BJU lost the decision in "Bob Jones University v. United States" (461 U.S. 574)[1983], the university chose to maintain its interracial dating policy and pay a million dollars in back taxes. The year following the Court decision, contributions to the university declined by 13 percent. In 2000, following a media uproar prompted by the visit of presidential candidate George W. Bush to the university, Bob Jones III dropped the university's interracial dating rule, announcing the change on CNN's "Larry King Live". In the same year, Bob Jones III drew criticism after reposting a letter on the university's web page referring to Mormons and Catholics as being members of "cults which call themselves Christian".
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