title: Theorem

abstract:
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the most important results, and use the terms lemma, proposition and corollary for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language. A theory consists of some basis statements called axioms, and some deducing rules. The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to proof theory, which allows proving general theorems about theorems and proofs. In particular, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory. As the axioms are often abstractions of properties of the physical world, theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law, which is experimental, the justification of the truth of a theorem is purely deductive.
A conjecture is a tentative proposition that may evolve to become a theorem if proven true.

text:
# Theorem

In mathematics and formal logic, a **theorem** is a statement that has been proven, or can be proven. The *proof* of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.

In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as *theorems* only the most important results, and use the terms *lemma*, *proposition* and *corollary* for less important theorems.

In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language. A theory consists of some basis statements called *axioms*, and some *deducing rules* (sometimes included in the axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to proof theory, which allows proving general theorems about theorems and proofs. In particular, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory (that is they cannot be proved inside the theory).

As the axioms are often abstractions of properties of the physical world, theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law, which is *experimental*, the justification of the truth of a theorem is purely deductive. A *conjecture* is a tentative proposition that may evolve to become a theorem if proven true.

## Theoremhood and truth

Until the end of the 19th century and the foundational crisis of mathematics, all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every natural number has a successor, and that there is exactly one line that passes through two given distinct points. These basic properties that were considered as absolutely evident were called postulates or axioms; for example Euclid's postulates. All theorems were proved by using implicitly or explicitly these basic properties, and, because of the evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the interior angles of a triangle equals 180°, and this was considered as an undoubtable fact.

One aspect of the foundational crisis of mathematics was the discovery of non-Euclidean geometries that do not lead to any contradiction, although, in such geometries, the sum of the angles of a triangle is different from 180°. So, the property *"the sum of the angles of a triangle equals 180°"* is either true or false, depending whether Euclid's fifth postulate is assumed or denied. Similarly, the use of "evident" basic properties of sets leads to the contradiction of Russell's paradox. This has been resolved by elaborating the rules that are allowed for manipulating sets.

This crisis has been resolved by revisiting the foundations of mathematics to make them more rigorous. In these new foundations, a theorem is a well-formed formula of a mathematical theory that can be proved from the axioms and inference rules of the theory. So, the above theorem on the sum of the angles of a triangle becomes: *Under the axioms and inference rules of Euclidean geometry, the sum of the interior angles of a triangle equals 180°*. Similarly, Russell's paradox disappears because, in an axiomatized set theory, the *set of all sets* cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is inconsistent, and every well-formed assertion, as well as its negation, is a theorem.

In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even the significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but only that the validity of a theorem is independent from the significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas.

An important consequence of this way of thinking about mathematics is that it allows defining mathematical theories and theorems as mathematical objects, and to prove theorems about them. Examples are Gödel's incompleteness theorems. In particular, there are well-formed assertions than can be proved to not be a theorem of the ambient theory, although they can be proved in a wider theory. An example is Goodstein's theorem, which can be stated in Peano arithmetic, but is proved to be not provable in Peano arithmetic. However, it is provable in some more general theories, such as Zermelo–Fraenkel set theory.

## Epistemological considerations

Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as **hypotheses** or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic).

Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.

In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way *why* it is obviously true. In some cases, one might even be able to substantiate a theorem by using a picture as its proof.

Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.

## Informal account of theorems

Logically, many theorems are of the form of an indicative conditional: *If A, then B*. Such a theorem does not assert *B* — only that *B* is a necessary consequence of *A*. In this case, *A* is called the *hypothesis* of the theorem ("hypothesis" here means something very different from a conjecture), and *B* the *conclusion* of the theorem. The two together (without the proof) are called the *proposition* or *statement* of the theorem (e.g. "*If A, then B*" is the *proposition*). Alternatively, *A* and *B* can be also termed the *antecedent* and the *consequent*, respectively. The theorem "If *n* is an even natural number, then *n*/2 is a natural number" is a typical example in which the hypothesis is "*n* is an even natural number", and the conclusion is "*n*/2 is also a natural number".

In order for a theorem to be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in a completely symbolic form—with the presumption that a formal statement can be derived from the informal one.

It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms and the structure of proofs.

Some theorems are "trivial", in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas.

Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search that is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities.

## Relation with scientific theories

Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.

Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find a single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs.

For example, both the Collatz conjecture and the Riemann hypothesis are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The Collatz conjecture has been verified for start values up to about 2.88 × 1018. The Riemann hypothesis has been verified to hold for the first 10 trillion non-trivial zeroes of the zeta function. Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved.

Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number *n* for which the Mertens function *M*(*n*) equals or exceeds the square root of *n*) is known: all numbers less than 1014 have the Mertens property, and the smallest number that does not have this property is only known to be less than the exponential of 1.59 × 1040, which is approximately 10 to the power 4.3 × 1039. Since the number of particles in the universe is generally considered less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search.

The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory (see mathematical theory). There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.

## Terminology

A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time.

* An *axiom* or *postulate* is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a *definition*, which gives the meaning of a word or a phrase in terms of known concepts. Classical geometry discerns between axioms, which are general statements; and postulates, which are statements about geometrical objects. Historically, axioms were regarded as "self-evident"; today they are merely *assumed* to be true.
* A *conjecture* is an unproved statement that is believed to be true. Conjectures are usually made in public, and named after their maker (for example, Goldbach's conjecture and Collatz conjecture). The term *hypothesis* is also used in this sense (for example, Riemann hypothesis), which should not be confused with "hypothesis" as the premise of a proof. Other terms are also used on occasion, for example *problem* when people are not sure whether the statement should be believed to be true. Fermat's Last Theorem was historically called a theorem, although, for centuries, it was only a conjecture.
* A *theorem* is a statement that has been proven to be true based on axioms and other theorems.
* A *proposition* is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in propositional logic. In classical geometry the term "proposition" was used differently: in Euclid's *Elements* (c.300 BCE), all theorems and geometric constructions were called "propositions" regardless of their importance.
* A *lemma* is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a *theorem*, though the term "lemma" is usually kept as part of its name (e.g. Gauss's lemma, Zorn's lemma, and the fundamental lemma).
* A *corollary* is a proposition that follows immediately from another theorem or axiom, with little or no required proof. A corollary may also be a restatement of a theorem in a simpler form, or for a special case: for example, the theorem "all internal angles in a rectangle are right angles" has a corollary that "all internal angles in a *square* are right angles" - a square being a special case of a rectangle.
* A *generalization* of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a special case (a *corollary*).

Other terms may also be used for historical or customary reasons, for example:

A few well-known theorems have even more idiosyncratic names, for example, the division algorithm, Euler's formula, and the Banach–Tarski paradox.

## Layout

A theorem and its proof are typically laid out as follows:

*Theorem* (name of the person who proved it, along with year of discovery or publication of the proof) *Statement of theorem (sometimes called the* proposition*)* *Proof* *Description of proof* *End*

The end of the proof may be signaled by the letters Q.E.D. (*quod erat demonstrandum*) or by one of the tombstone marks, such as "□" or "∎", meaning "end of proof", introduced by Paul Halmos following their use in magazines to mark the end of an article.

The exact style depends on the author or publication. Many publications provide instructions or macros for typesetting in the house style.

It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem.

Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem.

## Lore

It has been estimated that over a quarter of a million theorems are proved every year.

The well-known aphorism, "A mathematician is a device for turning coffee into theorems", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.

The classification of finite simple groups is regarded by some to be the longest proof of a theorem. It comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give a complete proof, and several ongoing projects hope to shorten and simplify this proof. Another theorem of this type is the four color theorem whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by a layman.

## Theorems in logic

In mathematical logic, a formal theory is a set of sentences within a formal language. A sentence is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of logical consequence. Some accounts define a theory to be closed under the semantic consequence relation ( $\models $), while others define it to be closed under the syntactic consequence, or derivability relation ( $\vdash $).

For a theory to be closed under a derivability relation, it must be associated with a deductive system that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from the context. The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system.

In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be unsound relative to a given semantics, or relative to the standard interpretation of the underlying language. A theory that is inconsistent has all sentences as theorems.

The definition of theorems as sentences of a formal language is useful within proof theory, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory, which is concerned with the relationship between formal theories and structures that are able to provide a semantics for them through interpretation.

Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express. What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement *about* a formal system (as opposed to *within* a formal system) is called a *metatheorem*.

Some important theorems in mathematical logic are:

### Syntax and semantics

The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a *true proposition,* which introduces semantics. Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e. belief, justification or other modalities). The soundness of a formal system depends on whether or not all of its theorems are also validities. A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are tautologies). A formal system is considered semantically complete when all of its theorems are also tautologies.

### Interpretation of a formal theorem

### Theorems and theories

1. Elisha Scott Loomis. "The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sources for data of the four kinds of proofs" (PDF). *Education Resources Information Center*. Institute of Education Sciences (IES) of the U.S. Department of Education. Retrieved 2010-09-26. Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics.
2. "Definition of THEOREM". *Merriam-Webster*. Retrieved 2019-11-02.
3. "Theorem | Definition of Theorem by Lexico". *Lexico Dictionaries | English*. Archived from the original on November 2, 2019. Retrieved 2019-11-02.
4. McLarty, Colin (2010). "What does it take to prove Fermat's last theorem? Grothendieck and the logic of number theory". *The Review of Symbolic Logic*. **13** (3). Cambridge University Press: 359–377. doi:10.2178/bsl/1286284558. S2CID 13475845.
5. McLarty, Colin (2020). "The large structures of Grothendieck founded on finite order arithmetic". *Bulletin of Symbolic Logic*. **16** (2). Cambridge University Press: 296–325. arXiv:1102.1773. doi:10.1017/S1755020319000340. S2CID 118395028.
6. Markie, Peter (2017), "Rationalism vs. Empiricism", in Zalta, Edward N. (ed.), *The Stanford Encyclopedia of Philosophy* (Fall 2017 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-11-02
7. However, both theorems and scientific law are the result of investigations. See Heath 1897 Introduction, The terminology of Archimedes, p. clxxxii:"theorem (θεὼρνμα) from θεωρεἳν to investigate"
8. Weisstein, Eric W. "Theorem". *mathworld.wolfram.com*. Retrieved 2019-11-02.
9. Darmon, Henri; Diamond, Fred; Taylor, Richard (2007-09-09). "Fermat's Last Theorem" (PDF). *McGill University – Department of Mathematics and Statistics*. Retrieved 2019-11-01.
10. "Implication". *intrologic.stanford.edu*. Retrieved 2019-11-02.
11. Weisstein, Eric W. "Deep Theorem". *MathWorld*.
12. Doron Zeilberger. "Opinion 51".
13. Such as the derivation of the formula for $\tan(\alpha +\beta )$ from the addition formulas of sine and cosine.
14. Petkovsek et al. 1996.
15. Wentworth, G.; Smith, D.E. (1913). *Plane Geometry*. Ginn & Co. Articles 46, 47.
16. Wentworth & Smith, article 51
17. "Earliest Uses of Symbols of Set Theory and Logic". *jeff560.tripod.com*. Retrieved 2 November 2019.
18. Hoffman 1998, p. 204.
19. Hoffman 1998, p. 7.
20. An enormous theorem: the classification of finite simple groups, Richard Elwes, Plus Magazine, Issue 41 December 2006.
21. Boolos, et al 2007, p. 191.
22. Chiswell and Hodges, p. 172.
23. Enderton, p. 148
24. Hedman, p. 89.
25. Hinman, p. 139.
26. Hodges, p. 33.
27. Johnstone, p. 21.
28. Monk, p. 208.
29. Rautenberg, p. 81.
30. van Dalen, p. 104.

## References

* Boolos, George; Burgess, John; Jeffrey, Richard (2007). *Computability and Logic* (5th ed.). Cambridge University Press.
* Chiswell, Ian; Hodges, Wilfred (2007). *Mathematical Logic*. Oxford University Press.
* Enderton, Herbert (2001). *A Mathematical Introduction to Logic* (2nd ed.). Harcourt Academic Press.
* Heath, Sir Thomas Little (1897). *The works of Archimedes*. Dover. Retrieved 2009-11-15.
* Hedman, Shawn (2004). *A First Course in Logic*. Oxford University Press.
* Hinman, Peter (2005). *Fundamentals of Mathematical Logic*. Wellesley, MA: A K Peters.
* Hoffman, P. (1998). *The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth*. Hyperion, New York. ISBN 1-85702-829-5.
* Hodges, Wilfrid (1993). *Model Theory*. Cambridge University Press.
* Hunter, Geoffrey (1996) \[1973\]. *Metalogic: An Introduction to the Metatheory of Standard First Order Logic*. University of California Press. ISBN 0-520-02356-0.
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* Mates, Benson (1972). *Elementary Logic*. Oxford University Press. ISBN 0-19-501491-X.
* Monk, J. Donald (1976). *Mathematical Logic*. Springer-Verlag.
* Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). *A = B*. A.K. Peters, Wellesley, Massachusetts. ISBN 1-56881-063-6.
* Rautenberg, Wolfgang (2010). *A Concise Introduction to Mathematical Logic* (3rd ed.). Springer.
* van Dalen, Dirk (1994). *Logic and Structure* (3rd ed.). Springer-Verlag.


title: Dunguib

abstract: Dunguib is an Irish Thoroughbred racehorse who became the second horse to complete the Cheltenham and Punchestown double in 2009 for trainer Philip Fenton and owners Daniel Harnett and Lily Lawlor. However he was later disqualified from his Punchestown Champion Bumper victory

text:
# Dunguib

**Dunguib** (foaled 8 April 2003) is an Irish Thoroughbred racehorse who became the second horse to complete the Cheltenham and Punchestown double in 2009 for trainer Philip Fenton and owners Daniel Harnett and Lily Lawlor. However he was later disqualified from his Punchestown Champion Bumper victory

## Background

Dunguib was sold by breeder Liam Meade as a foal at the Tattersalls Ireland horse sales in November 2003 to Daniel Harnett and Lily Lawlor, who intended to sell him on as a three-year-old like they had many horses before, including Irish Grand National winner Niche Market.

As they lived near to the town of Killenaule in County Tipperary, both Harnett and Lawlor knew Philip Fenton well as he rode for local trainer Edward O'Grady as an amateur for many years, securing a Cheltenham Festival success for the trainer on Loving Around in the 1996 National Hunt Chase. Both told Fenton that if their son of Presenting was not sold on then they would put him in training at his Carrick On Suir base in the neighbouring county of Waterford.

The name Dunguib comes from a small townland outside the village Killenaule of the same name.

## Racing career

### 2008/9: National Hunt Flat races

Dunguib had his first racing start at the 2008 Punchestown Festival where he was beaten by On Raglan Road in a National Hunt flat race. He went on to win his next two starts in National Hunt flat races, including the Grade 2 Future Champ INH Flat Race. After that convincing victory, Dunguib was moved to the highest level to contest the Champion Bumper at the 2009 Cheltenham Festival, a race Harnett and Lawlor had dreamed of winning for some time.

Under amateur rider Brian O'Connell, his only rider to date, Dunguib stayed on strongly up the Cheltenham hill to beat Some Present and Rite Of Passage with ease. Having landed his first Grade 1 connections were quick to win another naming the Punchestown Champion Bumper as his next target and gave him a short break before producing him in top condition that April. Dunguib finished first, but was later found to have been running with a banned substance in his system and was disqualified, awarding the race to the second home Sweeps Hill.

### 2009/10: Novice Hurdles

#### Before Cheltenham

Fenton had schooled Dunguib over hurdles before his Punchestown run and gave him his first start over those obstacles in October 2009 at Galway. Conditions were very testing, but the competition was not strong and the six-year-old won by four lengths, once again in the hands of O'Connell. Connections had wanted to go straight to the Royal Bond Novice Hurdle after this race using a racecourse gallop at Thurles as a preparation. This plan was changed however, when Thurles became waterlogged by excessive levels of rain and Dunguib was taken to Punchestown for another hurdle race. In similar style to his first run of the season, Dunguib won by several lengths with virtually no urgings from O'Connell.

Dunguib's next test was to be his most challenging so far. A return to Grade 1 action for the Royal Bond Novice Hurdle at Fairyhouse was planned and preparations were going well, but a freak rain storm waterlogged Fairyhouse on the Sunday of the big race, forcing its abandonment. Horse Racing Ireland worked quickly to ensure the day, which was also set to feature the Hatton's Grace Hurdle and Drinmore Novice Chase, was saved, moving the entire card to the following Wednesday and the course's original fixture set for that date to the next Thursday (10 December). Still, the opposition was no match for Dunguib and he went seven lengths clear of Some Present to record another win. Afterwards, Fenton was considering a run at the Future Champions Novice Hurdle at Leopardstown because his star had won so easily again, but all concerned decided to stick with their original plan and Dunguib was rested until the Deloitte Novice Hurdle in February.

He won this without being pressured, but made several jumping errors. These did not harm him though and he was described by Fenton as "bright as a button" the following day.

That contest was his final preparation for the Supreme Novices' Hurdle at the Cheltenham Festival in March 2010.

#### Cheltenham and After

Despite widespread anticipation that Dunguib would win the opening race of the 2010 Cheltenham Festival the horse failed to live up to expectations on the day. Sent to post an odd-on favourite at a price of 4/5, Dunguib was ridden wide throughout the 2 mile, Grade 1 race by jockey Brian O'Connell. This may have been a deliberate tactic to give the horse a clear view of the hurdles. His prospects were not helped by the slow pace of the race which ensured that almost a dozen horses were in contention turning into the straight. This forced the horse even wider on the track and serious ground was lost. Despite quickening up the Cheltenham hill, O'Connell and Dunguib were unable to catch the Philip Hobbs trained Menorah, ridden by Richard Johnson, with Get Me Out Of Here a close second.

Immediately after the race, some commentators were quick to criticise jockey Brian O'Connell, claiming he was too inexperienced to ride the horse in such a high-profile event. Channel 4 commentator John McCririck called for O’Connell to be sacked after riding the horse. However, such a comment was not taken well by other jockeys and McCririck came under immediate attack from many others ridding at the Festival who jumped to the defense of O’Connell with Davy Russell demanding an apology saying "John McCririck doesn’t know one end of the horse from the other. I would have been insulted by the comments said on the television. He wasn’t the best horse on the day and if you lost your money you should take it on the chin. I would like a public apology from him (McCririck)."

The official distances were a head and one and three quarter lengths. The form of the race has looked better in retrospect, with both Menorah and the fourth placed Oscar Whisky going on to win Grade I races.

Dunguib suffered the second defeat of his career at the Punchestown Festival in April 2010. Again sent off favourite, at a price of 5/2, the Philip Fenton novice was only able to finish eighth behind a top class field in the Grade 1 event. Hurricane Fly, the future Champion Hurdler went on to win the race at a price of 3–1.

### 2010/11 season

Dunguib's 2010–2011 campaign was seriously disrupted due to a combination of illness and inclement weather. A harsh Irish winter ensured an almost blanket cancellation of all racing during November and December due to heavy rain, frozen ground and snowfalls. All three ensured, that despite being entered in various races, Philip Fenton's horse would not run again in 2010. Dunguib's long-awaited re-appearance did not occur until February 2011 when he was sent off an 8-11 favourite in the three-horse-race, Red Mills Trial Hurdle at Gowran. Dunguib travelled well throughout the race and came through to challenge leader Luska Lad for the lead two hurdles from home. Once taking the lead, the race was never in doubt, with the official winning distance being three and a half lengths with Gimli's Rock seven lengths back in third.

At the 2011 Cheltenham Festival, Dunguib failed to produce his best form on good ground in the Champion Hurdle, finishing eighth of the eleven runners behind Hurricane Fly.

### 2013/14 season

Dunguib missed the next two seasons with leg injuries. In January 2014, Dunguib began his comeback and finished third to Rule the World in the Grade III Limestone Lad Hurdle at Naas. On 16 February, the gelding recorded his first win in three years when he easily defeated the 5/4 favourite Zaidpour by six lengths in the Grade II Boyne Hurdle at Navan. When asked if the gelding was better that ever, Fenton replied "He could be, even though he is not getting any younger". On 12 March, Dunguib carried top weight of 166 pounds in the Coral Cup over two miles five furlongs at the Cheltenham Festival. He was never travelling well and was pulled up by O'Connell before the third last. He reportedly finished lame.

## Racing results

16 March 2010, 23 April 2010, 19 February 2011, 15 March 2011, 18 January 2014, 16 February 2014, 12 March 2014

## InfoBox

| Dunguib | |
| --- | --- |
| Sire | Presenting |
| Dam | Edermine Berry |
| Damsire | Durgam |
| Sex | Gelding |
| Foaled | 8 April 2003 |
| Country | Ireland |
| Colour | Bay |
| Breeder | Liam Meade |
| Owner | Daniel Harnett and Lily Lawlor |
| Trainer | Philip Fenton |
| Record | 16: 9-1-2 |
| Earnings | £234,306 |
| Major wins | |
| Future Champions INH Flat Race (2008)
Cheltenham Champion Bumper (2009)
Punchestown Champion Bumper (disqualified)
Royal Bond Novice Hurdle (2009)
Deloitte Novice Hurdle (2010)
Boyne Hurdle (2014) | |
| Last updated on 20 February 2010 | |


title: There You Are (Martina McBride song)

abstract: "There You Are" is a song written by Ed Hill, Mark D. Sanders and Bob DiPiero, and recorded by American country music artist Martina McBride. It was released in May 2000 as the third single from her album Emotion. The song was McBride's twelfth Top 10 hit on the U.S. Billboard Hot Country Singles & Tracks, and was her first Top 20 hit on the U.S. Adult Contemporary charts. In 2001, it was included as a track on McBride's Greatest Hits album.

text:
# There You Are (Martina McBride song)

"**There You Are**" is a song written by Ed Hill, Mark D. Sanders and Bob DiPiero, and recorded by American country music artist Martina McBride. It was released in May 2000 as the third single from her album *Emotion*. The song was McBride's twelfth Top 10 hit on the U.S. *Billboard* Hot Country Singles & Tracks, and was her first Top 20 hit on the U.S. Adult Contemporary charts.

In 2001, it was included as a track on McBride's *Greatest Hits* album.

## Charts

"There You Are" re-entered the chart as a single at number 67 on the U.S. *Billboard* Hot Country Singles & Tracks for the week of May 20, 2000.

| Chart (2000) | Peak
position |
| --- | --- |
| Canada Country Tracks (*RPM*) | 19 |
| US Hot Country Songs (*Billboard*) | 10 |
| US *Billboard* Hot 100 | 60 |
| US Adult Contemporary (*Billboard*) | 15 |
|

### Year-end charts

| Chart (2000) | Position |
| --- | --- |
| US Country Songs (*Billboard*) | 39 |
|

## Release history

Release dates and format(s) for "There You Are"
| Region | Date | Format(s) | Label(s) | Ref. |
| --- | --- | --- | --- | --- |
| United States | May 30, 2000 | Country radio | RCA Nashville |  |
| | July 24, 2000 | Adult contemporary radio | |  |

1. "Top RPM Country Tracks: Issue 7268." *RPM*. Library and Archives Canada. Retrieved January 17, 2022.
2. "Martina McBride Chart History (Hot Country Songs)". *Billboard*.
3. "Martina McBride Chart History (Hot 100)". *Billboard*.
4. "Martina McBride Chart History (Adult Contemporary)". *Billboard*.
5. "Best of 2000: Country Songs". *Billboard*. Prometheus Global Media. 2000. Retrieved August 15, 2012.
6. "Going for Adds | Country" (PDF). *Radio & Records*. No. 1352. May 26, 2000. p. 93. Retrieved April 22, 2024.
7. "AC/Hot AC Impact" (PDF). *Gavin Report*. No. 2313. July 14, 2000. p. 23. Retrieved June 1, 2024.

## InfoBox

| "There You Are" | |
| --- | --- |
|  | |
| Single by Martina McBride | |
| from the album  *Emotion* | |
| Released | May 30, 2000 |
| Genre | Country |
| Length | 3:26 |
| Label | RCA Nashville |
| Songwriter(s) |  |
| Producer(s) |  |
| Martina McBride singles chronology | |
| "Love's the Only House"
(1999) "**There You Are**"
(2000) "It's My Time"
(2000) | |


title: Dual loop

abstract: Dual-loop is a method of electrical circuit termination used in electronic security applications, particularly modern intruder alarms. It is called 'dual-loop' because two circuits are combined into one using resistors. Its use became widespread in the early 21st century, replacing the basic closed-circuit system, mainly because of changes in international standards and practices. Dual-loop allows the burglar alarm control panel to read the values of end-of-line resistors for the purpose of telling a zone's status. For example: if an alarm system's software uses 2K ohms as its non-alarm value, an inactive detector will give a reading of 2K ohms as the circuit is passing through just one resistor. When the detector goes into an active state, the circuit path has been altered and it must now pass through a second resistor wired in series with the first. This gives a reading of 4K ohms and will trigger an intruder alarm. If a resistance reading is not recognised by the system either due to short-circuit or open-circuit, an anti-tamper alarm will trigger. Dual loop is more commonly known as Balanced EOL resistors. It is more secure than the former Double Pole loop, but nevertheless can be bypassed by someone with sufficient knowledge of security alarm systems.

text:
# Dual loop

**Dual-loop** is a method of electrical circuit termination used in electronic security applications, particularly modern intruder alarms. It is called 'dual-loop' because two circuits (alarm and anti-tamper) are combined into one using resistors. Its use became widespread in the early 21st century, replacing the basic closed-circuit system, mainly because of changes in international standards and practices.

Dual-loop allows the burglar alarm control panel to read the values of end-of-line resistors for the purpose of telling a zone's status. For example: if an alarm system's software uses 2K ohms as its non-alarm value, an inactive detector will give a reading of 2K ohms as the circuit is passing through just one resistor. When the detector goes into an active state (i.e. a door contact being opened), the circuit path has been altered and it must now pass through a second resistor wired in series with the first. This gives a reading of 4K ohms and will trigger an intruder alarm. If a resistance reading is not recognised by the system either due to short-circuit or open-circuit, an anti-tamper alarm will trigger.

Dual loop is more commonly known as Balanced EOL (End of Line) resistors. It is more secure than the former Double Pole loop, but nevertheless can be bypassed by someone with sufficient knowledge of security alarm systems.

1. Mandatory UK standard Archived November 21, 2008, at the Wayback Machine


title: Dual linear program

abstract:
The dual of a given linear program (LP) is another LP that is derived from the original LP in the following schematic way:
  - Each variable in the primal LP becomes a constraint in the dual LP;
  - Each constraint in the primal LP becomes a variable in the dual LP;
  - The objective direction is inversed – maximum in the primal becomes minimum in the dual and vice versa. The weak duality theorem states that the objective value of the dual LP at any feasible solution is always a bound on the objective of the primal LP at any feasible solution. In fact, this bounding property holds for the optimal values of the dual and primal LPs. The strong duality theorem states that, moreover, if the primal has an optimal solution then the dual has an optimal solution too, and the two optima are equal. These theorems belong to a larger class of duality theorems in optimization. The strong duality theorem is one of the cases in which the duality gap is 0.

text:
# Dual linear program

The **dual** of a given linear program (LP) is another LP that is derived from the original (the **primal**) LP in the following schematic way:

* Each variable in the primal LP becomes a constraint in the dual LP;
* Each constraint in the primal LP becomes a variable in the dual LP;
* The objective direction is inversed – maximum in the primal becomes minimum in the dual and vice versa.

The **weak duality theorem** states that the objective value of the dual LP at any feasible solution is always a bound on the objective of the primal LP at any feasible solution (upper or lower bound, depending on whether it is a maximization or minimization problem). In fact, this bounding property holds for the optimal values of the dual and primal LPs.

The **strong duality theorem** states that, moreover, if the primal has an optimal solution then the dual has an optimal solution too, *and the two optima are equal*.

These theorems belong to a larger class of duality theorems in optimization. The strong duality theorem is one of the cases in which the duality gap (the gap between the optimum of the primal and the optimum of the dual) is 0.

## Form of the dual LP

Suppose we have the linear program:

> Maximize **c**T**x** subject to *A***x****b**, **x** ≥ 0.

We would like to construct an upper bound on the solution. So we create a linear combination of the constraints, with positive coefficients, such that the coefficients of **x** in the constraints are at least **c**T. This linear combination gives us an upper bound on the objective. The variables **y** of the dual LP are the coefficients of this linear combination. The dual LP tries to find such coefficients that *minimize* the resulting upper bound. This gives the following LP:

> Minimize **b**T**y** subject to *A*T**y****c**, **y** ≥ 0

This LP is called the *dual of* the original LP.

### Interpretation

The duality theorem has an economic interpretation. If we interpret the primal LP as a classical "resource allocation" problem, its dual LP can be interpreted as a "resource valuation" problem.

Consider a factory that is planning its production of goods. Let $x$ be its production schedule (make $x\_{i}$ amount of good $i$), let $c\geq 0$ be the list of market prices (a unit of good $i$ can sell for $c\_{i}$). The constraints it has are $x\geq 0$ (it cannot produce negative goods) and raw-material constraints. Let $b$ be the raw material it has available, and let $A\geq 0$ be the matrix of material costs (producing one unit of good $i$ requires $A\_{ji}$ units of raw material $j$).

Then, the constrained revenue maximization is the primal LP:

> Maximize **c**T**x** subject to *A***x****b**, **x** ≥ 0.

Now consider another factory that has no raw material, and wishes to purchase the entire stock of raw material from the previous factory. It offers a price vector of $y$ (a unit of raw material $i$ for $y\_{i}$). For the offer to be accepted, it should be the case that $A^{T}y\geq c$, since otherwise the factory could earn more cash by producing a certain product than selling off the raw material used to produce the good. It also should be $y\geq 0$, since the factory would not sell any raw material with negative price. Then, the second factory's optimization problem is the dual LP:

> Minimize **b**T**y** subject to *A*T**y****c**, **y** ≥ 0

The duality theorem states that the duality gap between the two LP problems is at least zero. Economically, it means that if the first factory is given an offer to buy its entire stock of raw material, at a per-item price of **y**, such that *A*T**y****c**, **y** ≥ 0, then it should take the offer. It will make at least as much revenue as it could producing finished goods.

The strong duality theorem further states that the duality gap is zero. With strong duality, the dual solution $y^{\*}$ is, economically speaking, the "equilibrium price" (see shadow price) for the raw material that a factory with production matrix $A$ and raw material stock $b$ would accept for raw material, given the market price for finished goods $c$. (Note that $y^{\*}$ may not be unique, so the equilibrium price may not be fully determined by $A$, $b$, and $c$.)

To see why, consider if the raw material prices $y\geq 0$ are such that $(A^{T}y)\_{i}\<c\_{i}$ for some $i$, then the factory would purchase more raw material to produce more of good $i$, since the prices are "too low". Conversely, if the raw material prices satisfy $A^{T}y\geq c,y\geq 0$, but does not minimize $b^{T}y$, then the factory would make more money by selling its raw material than producing goods, since the prices are "too high". At equilibrium price $y^{\*}$, the factory cannot increase its profit by purchasing or selling off raw material.

The duality theorem has a physical interpretation too.

## Constructing the dual LP

In general, given a primal LP, the following algorithm can be used to construct its dual LP. The primal LP is defined by:

* A set of *n* variables: $x\_{1},\ldots ,x\_{n}$.
* For each variable $x\_{i}$, a *sign constraint* – it should be either non-negative ( $x\_{i}\geq 0$), or non-positive ( $x\_{i}\leq 0$), or unconstrained ( $x\_{i}\in \mathbb {R} $).
* An objective function: $\text{maximize}~~~c\_{1}x\_{1}+\cdots +c\_{n}x\_{n}$
* A list of *m* constraints. Each constraint *j* is: $a\_{j1}x\_{1}+\cdots +a\_{jn}x\_{n}\lesseqqgtr b\_{j}$where the symbol before the $b\_{j}$ can be one of $\geq $ or $\leq $ or $=$.

The dual LP is constructed as follows.

* Each primal constraint becomes a dual variable. So there are *m* variables: $y\_{1},\ldots ,y\_{m}$.
* The sign constraint of each dual variable is "opposite" to the sign of its primal constraint. So " $\geq b\_{j}$" becomes $y\_{j}\leq 0$ and " $\leq b\_{j}$" becomes $y\_{j}\geq 0$ and " $=b\_{j}$" becomes $y\_{j}\in \mathbb {R} $.
* The dual objective function is $\text{minimize }~~~b\_{1}y\_{1}+\cdots +b\_{m}y\_{m}$
* Each primal variable becomes a dual constraint. So there are *n* constraints. The coefficient of a dual variable in the dual constraint is the coefficient of its primal variable in its primal constraint. So each constraint *i* is: $a\_{1i}y\_{1}+\cdots +a\_{mi}y\_{m}\lesseqqgtr c\_{i}$, where the symbol before the $c\_{i}$ is similar to the sign constraint on variable *i* in the primal LP. So $x\_{i}\leq 0$ becomes " $\leq c\_{i}$" and $x\_{i}\geq 0$ becomes " $\geq c\_{i}$" and $x\_{i}\in \mathbb {R} $ becomes " $=c\_{i}$".

From this algorithm, it is easy to see that the dual of the dual is the primal.

## Vector formulations

If all constraints have the same sign, it is possible to present the above recipe in a shorter way using matrices and vectors. The following table shows the relation between various kinds of primals and duals.

| Primal | Dual | Note |
| --- | --- | --- |
| Maximize **c**T**x** subject to *A***x****b**, **x** ≥ 0 | Minimize **b**T**y** subject to *A*T**y****c**, **y** ≥ 0 | This is called a "symmetric" dual problem |
| Maximize **c**T**x** subject to *A***x****b** | Minimize **b**T**y** subject to *A*T**y** = **c**, **y** ≥ 0 | This is called an "asymmetric" dual problem |
| Maximize **c**T**x** subject to *A***x** = **b**, **x** ≥ 0 | Minimize **b**T**y** subject to *A*T**y****c** |  |

## The duality theorems

Below, suppose the primal LP is "maximize **c**T**x** subject to \[constraints\]" and the dual LP is "minimize **b**T**y** subject to \[constraints\]".

### Weak duality

The **weak duality theorem** says that, for each feasible solution **x** of the primal and each feasible solution **y** of the dual: **c**T**x****b**T**y**. In other words, the objective value in each feasible solution of the dual is an upper-bound on the objective value of the primal, and objective value in each feasible solution of the primal is a lower-bound on the objective value of the dual. Here is a proof for the primal LP "Maximize **c**T**x** subject to *A***x****b**, **x** ≥ 0":

* **c**T**x**
* = **xTc** \[since this just a scalar product of the two vectors\]
* **xT**(***A*****T****y**) \[since *A*T**y****c** by the dual constraints, and **x** ≥ 0\]
* = (**xT*A*T**)**y** \[by associativity\]
* = (**Ax**)**T****y** \[by properties of transpose\]
* **bTy** \[since *A***x****b** by the primal constraints\]

Weak duality implies:

> max**x** **c**T**x** ≤ min**y** **b**T**y**

In particular, if the primal is unbounded (from above) then the dual has no feasible solution, and if the dual is unbounded (from below) then the primal has no feasible solution.

### Strong duality

The **strong duality theorem** says that if one of the two problems has an optimal solution, so does the other one and that the bounds given by the weak duality theorem are tight, i.e.:

> max**x** **c**T**x** = min**y** **b**T**y**

The strong duality theorem is harder to prove; the proofs usually use the weak duality theorem as a sub-routine.

One proof uses the simplex algorithm and relies on the proof that, with the suitable pivot rule, it provides a correct solution. The proof establishes that, once the simplex algorithm finishes with a solution to the primal LP, it is possible to read from the final tableau, a solution to the dual LP. So, by running the simplex algorithm, we obtain solutions to both the primal and the dual simultaneously.

Another proof uses the Farkas lemma.

### Theoretical implications

1. The weak duality theorem implies that finding a *single* feasible solution is as hard as finding an *optimal* feasible solution. Suppose we have an oracle that, given an LP, finds an arbitrary feasible solution (if one exists). Given the LP "Maximize **c**T**x** subject to *A***x****b**, **x** ≥ 0", we can construct another LP by combining this LP with its dual. The combined LP has both **x** and **y** as variables:

> Maximize **1**

> subject to *A***x****b**, *A*T**y****c**, **c**T**x****b**T**y**, **x** ≥ 0, **y** ≥ 0

If the combined LP has a feasible solution (**x**,**y**), then by weak duality, **c**T**x** = **b**T**y**. So **x** must be a maximal solution of the primal LP and **y** must be a minimal solution of the dual LP. If the combined LP has no feasible solution, then the primal LP has no feasible solution either.

2. The strong duality theorem provides a "good characterization" of the optimal value of an LP in that it allows us to easily prove that some value *t* is the optimum of some LP. The proof proceeds in two steps:

* Show a feasible solution to the primal LP with value *t*; this proves that the optimum is at least *t*.
* Show a feasible solution to the dual LP with value *t*; this proves that the optimum is at most *t*.

## Examples

### Tiny example

Consider the primal LP, with two variables and one constraint:

$$\begin{aligned}{\text{maximize }&3x\_{1}+4x\_{2}\\\{\text{subject to }&5x\_{1}+6x\_{2}=7\\\&x\_{1}\geq 0,x\_{2}\geq 0\end{aligned}}}$$

Applying the recipe above gives the following dual LP, with one variable and two constraints:

$$\begin{aligned}{\text{minimize }&7y\_{1}\\\{\text{subject to }&5y\_{1}\geq 3\\\&6y\_{1}\geq 4\\\&y\_{1}\in \mathbb {R} \end{aligned}}}$$

It is easy to see that the maximum of the primal LP is attained when *x*1 is minimized to its lower bound (0) and *x*2 is maximized to its upper bound under the constraint (7/6). The maximum is 4  7/6 = 14/3.

Similarly, the minimum of the dual LP is attained when *y*1 is minimized to its lower bound under the constraints: the first constraint gives a lower bound of 3/5 while the second constraint gives a stricter lower bound of 4/6, so the actual lower bound is 4/6 and the minimum is 7  4/6 = 14/3.

In accordance with the strong duality theorem, the maximum of the primal equals the minimum of the dual.

We use this example to illustrate the proof of the weak duality theorem. Suppose that, in the primal LP, we want to get an upper bound on the objective $3x\_{1}+4x\_{2}$. We can use the constraint multiplied by some coefficient, say $y\_{1}$. For any $y\_{1}$ we get: $y\_{1}\cdot (5x\_{1}+6x\_{2})=7y\_{1}$. Now, if $y\_{1}\cdot 5x\_{1}\geq 3x\_{1}$and $y\_{1}\cdot 6x\_{2}\geq 4x\_{2}$, then $y\_{1}\cdot (5x\_{1}+6x\_{2})\geq 3x\_{1}+4x\_{2}$, so $7y\_{1}\geq 3x\_{1}+4x\_{2}$. Hence, the objective of the dual LP is an upper bound on the objective of the primal LP.

### Farmer example

Consider a farmer who may grow wheat and barley with the set provision of some *L* land, *F* fertilizer and *P* pesticide. To grow one unit of wheat, one unit of land, $F\_{1}$ units of fertilizer and $P\_{1}$ units of pesticide must be used. Similarly, to grow one unit of barley, one unit of land, $F\_{2}$ units of fertilizer and $P\_{2}$ units of pesticide must be used.

The primal problem would be the farmer deciding how much wheat ( $x\_{1}$) and barley ( $x\_{2}$) to grow if their sell prices are $S\_{1}$ and $S\_{2}$ per unit.

| Maximize: $S\_{1}\cdot x\_{1}+S\_{2}\cdot x\_{2}$ | | (maximize the revenue from producing wheat and barley) |
| --- | --- | --- |
| subject to: | $x\_{1}+x\_{2}\leq L$ | (cannot use more land than available) |
|  | $F\_{1}\cdot x\_{1}+F\_{2}\cdot x\_{2}\leq F$ | (cannot use more fertilizer than available) |
|  | $P\_{1}\cdot x\_{1}+P\_{2}\cdot x\_{2}\leq P$ | (cannot use more pesticide than available) |
|  | $x\_{1},x\_{2}\geq 0$ | (cannot produce negative quantities of wheat or barley). |

In matrix form this becomes:

Maximize: $\begin{bmatrix}S\_{1}&S\_{2}\end{bmatrix}{\begin{bmatrix}x\_{1}\\x\_{2}\end{bmatrix}}$ subject to: $\begin{bmatrix}1&1\\F\_{1}&F\_{2}\\P\_{1}&P\_{2}\end{bmatrix}{\begin{bmatrix}x\_{1}\\x\_{2}\end{bmatrix}\leq {\begin{bmatrix}L\\F\\P\end{bmatrix},\\,{\begin{bmatrix}x\_{1}\\x\_{2}\end{bmatrix}\geq 0.}}}$

For the dual problem assume that *y* unit prices for each of these means of production (inputs) are set by a planning board. The planning board's job is to minimize the total cost of procuring the set amounts of inputs while providing the farmer with a floor on the unit price of each of his crops (outputs), *S*1 for wheat and *S*2 for barley. This corresponds to the following LP:

| Minimize: $L\cdot y\_{L}+F\cdot y\_{F}+P\cdot y\_{P}$ | | (minimize the total cost of the means of production as the "objective function") |
| --- | --- | --- |
| subject to: | $y\_{L}+F\_{1}\cdot y\_{F}+P\_{1}\cdot y\_{P}\geq S\_{1}$ | (the farmer must receive no less than *S*1 for his wheat) |
|  | $y\_{L}+F\_{2}\cdot y\_{F}+P\_{2}\cdot y\_{P}\geq S\_{2}$ | (the farmer must receive no less than *S*2 for his barley) |
|  | $y\_{L},y\_{F},y\_{P}\geq 0$ | (prices cannot be negative). |

In matrix form this becomes:

Minimize: $\begin{bmatrix}L&F&P\end{bmatrix}{\begin{bmatrix}y\_{L}\\y\_{F}\\y\_{P}\end{bmatrix}}$ subject to: $\begin{bmatrix}1&F\_{1}&P\_{1}\\1&F\_{2}&P\_{2}\end{bmatrix}{\begin{bmatrix}y\_{L}\\y\_{F}\\y\_{P}\end{bmatrix}\geq {\begin{bmatrix}S\_{1}\\S\_{2}\end{bmatrix},\\,{\begin{bmatrix}y\_{L}\\y\_{F}\\y\_{P}\end{bmatrix}\geq 0.}}}$

The primal problem deals with physical quantities. With all inputs available in limited quantities, and assuming the unit prices of all outputs is known, what quantities of outputs to produce so as to maximize total revenue? The dual problem deals with economic values. With floor guarantees on all output unit prices, and assuming the available quantity of all inputs is known, what input unit pricing scheme to set so as to minimize total expenditure?

To each variable in the primal space corresponds an inequality to satisfy in the dual space, both indexed by output type. To each inequality to satisfy in the primal space corresponds a variable in the dual space, both indexed by input type.

The coefficients that bound the inequalities in the primal space are used to compute the objective in the dual space, input quantities in this example. The coefficients used to compute the objective in the primal space bound the inequalities in the dual space, output unit prices in this example.

Both the primal and the dual problems make use of the same matrix. In the primal space, this matrix expresses the consumption of physical quantities of inputs necessary to produce set quantities of outputs. In the dual space, it expresses the creation of the economic values associated with the outputs from set input unit prices.

Since each inequality can be replaced by an equality and a slack variable, this means each primal variable corresponds to a dual slack variable, and each dual variable corresponds to a primal slack variable. This relation allows us to speak about complementary slackness.

### Infeasible program

A LP can also be unbounded or infeasible. Duality theory tells us that:

* If the primal is unbounded, then the dual is infeasible;
* If the dual is unbounded, then the primal is infeasible.

However, it is possible for both the dual and the primal to be infeasible. Here is an example:

| Maximize: $2x\_{1}-x\_{2}$ | |
| --- | --- |
| Subject to: | $x\_{1}-x\_{2}\leq 1$ |
|  | $-x\_{1}+x\_{2}\leq -2$ |
|  | $x\_{1},x\_{2}\geq 0.$ |

## Viewing the solution to a linear programming problem as a (generalized) eigenvector

There is a close connection between linear programming problems, eigenequations, and von Neumann's general equilibrium model. The solution to a linear programming problem can be regarded as a generalized eigenvector.

The eigenequations of a square matrix are as follows:

$\begin{matrix}\mathbf {p} ^{T}\mathbf {A} =\rho \mathbf {p} ^{T}\\\mathbf {A} \mathbf {z} =\rho {\mathbf {z} }\\\end{matrix}$

where $\mathbf {p} ^{T}$ and $\mathbf {z} $ are the left and right eigenvectors of the square matrix $\mathbf {A} $, respectively, and $\rho $ is the eigenvalue.

The above eigenequations for the square matrix can be extended to von Neumann's general equilibrium model:

$\begin{matrix}\mathbf {p} ^{T}\mathbf {A} \geq \rho \mathbf {p} ^{T}\mathbf {B} \\\mathbf {A} \mathbf {z} \leq \rho \mathbf {B} {\mathbf {z} }\\\end{matrix}$

where the economic meanings of $\mathbf {p} $ and $\mathbf {z} $ are the equilibrium prices of various goods and the equilibrium activity levels of various economic agents, respectively.

The von Neumann's equilibrium model can be further extended to the following structural equilibrium model with $\mathbf {A} $ and $\mathbf {B} $ as matrix-valued functions:

$\begin{matrix}\mathbf {p} ^{T}\mathbf {A} (\mathbf {p} ,\mathbf {u} ,\mathbf {z} )\geq \rho \mathbf {p} ^{T}\mathbf {B} (\mathbf {p} ,\mathbf {u} ,\mathbf {z} )\\\mathbf {A} (\mathbf {p} ,\mathbf {u} ,\mathbf {z} )\mathbf {z} \leq \rho \mathbf {B} (\mathbf {p} ,\mathbf {u} ,\mathbf {z} ){\mathbf {z} }\\\end{matrix}$

where the economic meaning of $\mathbf {u} $ is the utility levels of various consumers. A special case of the above model is

$\begin{matrix}\mathbf {p} ^{T}\mathbf {A} (u)\geq \mathbf {p} ^{T}\mathbf {B} \\\mathbf {A} (u)\mathbf {z} \leq \mathbf {B} {\mathbf {z} }\end{matrix}$

This form of the structural equilibrium model and linear programming problems can often be converted to each other, that is, the solutions to these two types of problems are often consistent.

If we define $\mathbf {A} (u)={\begin{bmatrix}\mathbf {0} &u\\\mathbf {A} &\mathbf {0} \\\end{bmatrix}}$, $\mathbf {B} ={\begin{bmatrix}\mathbf {c} ^{T}&0\\\mathbf {0} &\mathbf {b} \\\end{bmatrix}}$, $\mathbf {p} ={\begin{bmatrix}1\\\mathbf {y} \\\end{bmatrix}}$, $\mathbf {z} ={\begin{bmatrix}\mathbf {x} \\1\\\end{bmatrix}}$, then the structural equilibrium model can be written as

$\begin{bmatrix}\mathbf {y} ^{T}\mathbf {A} &u\\\end{bmatrix}\geq {\begin{bmatrix}\mathbf {c} ^{T}&\mathbf {y} ^{T}\mathbf {b} \\\end{bmatrix}}$

$\begin{bmatrix}u\\\mathbf {A} \mathbf {x} \\\end{bmatrix}\leq {\begin{bmatrix}\mathbf {c} ^{T}\mathbf {x} \\\mathbf {b} \\\end{bmatrix}}$

Let us illustrate the structural equilibrium model with the previously discussed tiny example. In this example, we have $\mathbf {A} ={\begin{bmatrix}5&6\end{bmatrix}}$, $\mathbf {A} (u)={\begin{bmatrix}0&0&u\\5&6&0\\\end{bmatrix}}$ and $\mathbf {B} ={\begin{bmatrix}3&4&0\\0&0&7\\\end{bmatrix}}$.

To solve the structural equilibrium model, we obtain

$\mathbf {p} ^{\*}=(1,2/3)^{T},\quad \mathbf {z} ^{\*}=(0,7/6,1)^{T},\quad u^{\*}=14/3$

These are consistent with the solutions to the linear programming problems.

We substitute the above calculation results into the structural equilibrium model, obtaining $\begin{matrix}\mathbf {p} ^{T}\mathbf {A} (u)=(10/3,4,14/3)\geq (3,4,14/3)=\mathbf {p} ^{T}\mathbf {B} \\\mathbf {A} (u)\mathbf {z} =(14/3,7)^{T}\leq (14/3,7)^{T}=\mathbf {B} {\mathbf {z} }\end{matrix}$

## Applications

The max-flow min-cut theorem is a special case of the strong duality theorem: flow-maximization is the primal LP, and cut-minimization is the dual LP. See Max-flow min-cut theorem#Linear program formulation.

Other graph-related theorems can be proved using the strong duality theorem, in particular, Konig's theorem.

The Minimax theorem for zero-sum games can be proved using the strong-duality theorem.

## Alternative algorithm

Sometimes, one may find it more intuitive to obtain the dual program without looking at the program matrix. Consider the following linear program:

| Minimize | $\sum \_{i=1}^{m}c\_{i}x\_{i}+\sum \_{j=1}^{n}d\_{j}t\_{j}$ | |
| --- | --- | --- |
| subject to | $\sum \_{i=1}^{m}a\_{ij}x\_{i}+e\_{j}t\_{j}\geq g\_{j},$ |  | $1\leq j\leq n$ |
|  | $f\_{i}x\_{i}+\sum \_{j=1}^{n}b\_{ij}t\_{j}\geq h\_{i},$ |  | $1\leq i\leq m$ |
|  | $x\_{i}\geq 0,\\,t\_{j}\geq 0,$ |  | $1\leq i\leq m,1\leq j\leq n$ |

We have *m* + *n* conditions and all variables are non-negative. We shall define *m* + *n* dual variables: **y**j and **s***i*. We get:

| Minimize | $\sum \_{i=1}^{m}c\_{i}x\_{i}+\sum \_{j=1}^{n}d\_{j}t\_{j}$ | |
| --- | --- | --- |
| subject to | $\sum \_{i=1}^{m}a\_{ij}x\_{i}\cdot y\_{j}+e\_{j}t\_{j}\cdot y\_{j}\geq g\_{j}\cdot y\_{j},$ |  | $1\leq j\leq n$ |
|  | $f\_{i}x\_{i}\cdot s\_{i}+\sum \_{j=1}^{n}b\_{ij}t\_{j}\cdot s\_{i}\geq h\_{i}\cdot s\_{i},$ |  | $1\leq i\leq m$ |
|  | $x\_{i}\geq 0,\\,t\_{j}\geq 0,$ |  | $1\leq i\leq m,1\leq j\leq n$ |
|  | $y\_{j}\geq 0,\\,s\_{i}\geq 0,$ |  | $1\leq j\leq n,1\leq i\leq m$ |

Since this is a minimization problem, we would like to obtain a dual program that is a lower bound of the primal. In other words, we would like the sum of all right hand side of the constraints to be the maximal under the condition that for each primal variable the sum of its coefficients do not exceed its coefficient in the linear function. For example, **x**1 appears in *n* + 1 constraints. If we sum its constraints' coefficients we get *a*1,1**y**1 + *a*1,2**y**2 + ... + *a*1,;;n;;**y***n* + *f*1**s**1. This sum must be at most **c**1. As a result, we get:

| Maximize | $\sum \_{j=1}^{n}g\_{j}y\_{j}+\sum \_{i=1}^{m}h\_{i}s\_{i}$ | |
| --- | --- | --- |
| subject to | $\sum \_{j=1}^{n}a\_{ij}y\_{j}+f\_{i}s\_{i}\leq c\_{i},$ |  | $1\leq i\leq m$ |
|  | $e\_{j}y\_{j}+\sum \_{i=1}^{m}b\_{ij}s\_{i}\leq d\_{j},$ |  | $1\leq j\leq n$ |
|  | $y\_{j}\geq 0,\\,s\_{i}\geq 0,$ |  | $1\leq j\leq n,1\leq i\leq m$ |

Note that we assume in our calculations steps that the program is in standard form. However, any linear program may be transformed to standard form and it is therefore not a limiting factor.


title: Dual loyalty

abstract: In politics, dual loyalty is loyalty to two separate interests that potentially conflict with each other, leading to a conflict of interest.

text:
# Dual loyalty

In politics, **dual loyalty** is loyalty to two separate interests that potentially conflict with each other, leading to a conflict of interest.

## Historical examples

Other historical examples of actual or perceived "dual loyalty" include the following:

## Transnationalist interpretations

Some scholars refer to a growing trend of transnationalism and suggest that as societies become more heterogeneous and multicultural, the term "dual loyalty" had increasingly become a meaningless bromide. According to the theory of transnationalism, migration and other factors, including improved global communication, produce new forms of identity that transcend traditional notions of physical and cultural space. Nina Glick Schiller, Linda Basch, and Cristina Blanc-Szanton define a process by which immigrants "link together" their country of origin and their country of settlement.

The transnationalist view is that "dual loyalty" is a potentially-positive expression of multi-culturalism and can contribute to the diversity and strength of civil society. That view is popular in many academic circles, but others are skeptical of the idea. As one paper describes it,

> On occasion, these imagined communities conform to the root meaning of transnational, extending beyond loyalties that connect to any specific place of origin or ethnic or national group. Yet what immigration scholars describe as transnationalism is usually its opposite... highly particularistic attachments antithetical to those by-products of globalization denoted by the concept of "transnational civil society" and its related manifestations.

Beyond its usage in particular instances, the terms "dual loyalty" and "transnationalism" continue to be the subject of much debate. As one academic wrote:

> Although the events of September 11th may have shaken some assumptions – at least in the United States – about the nature of transnational networks and their capacity to facilitate flows of people, goods, and ideas across borders, the terms "globalization" and "transnationalism" remain relatively stable, albeit frustratingly imprecise additions to the language of social sciences, including anthropology.


title: Dunn Field (Elmira)

abstract: Dunn Field is a stadium in Elmira, New York primarily used for baseball. Located on the banks of the Chemung River at the end of Luce St, it has been the home of various incarnations of the Elmira Pioneers since its opening in 1939.

text:
# Dunn Field (Elmira)

**Dunn Field** is a stadium in Elmira, New York primarily used for baseball. Located on the banks of the Chemung River at the end of Luce St, it has been the home of various incarnations of the Elmira Pioneers since its opening in 1939.

## History

The first stadium at the site was known as the **Maple Avenue Driving Park**. On Friday, November 21, 1902, the stadium was the site of the first ever professional football night game. The Philadelphia Athletics football team, of the first National Football League, defeated the Kanaweola Athletic Club, 39–0. The stadium also served as an alternate home for baseball's National League Buffalo Bisons in the late 19th century.

The Maple Avenue Driving Park was replaced by Recreation Park, which burned down in 1938. A new stadium built to replace it was named after a local businessman, Edward Joseph Dunn, who donated some land for the stadium to the city.

Besides being the home field of the Elmira Pioneers baseball team, a variety of events have taken place in the stadium over the years, such as wrestling and boxing matches, high school sports, and music concerts (The Beach Boys played there in 1984).

Dunn Field served as the home field for both Elmira Free Academy and Southside High School football teams for many years until each team built their own stadiums. The Erie Bell game was often a special occasion at Dunn Field, since both Elmira Free Academy and Southside High School played their home games at Dunn Field.

Many great baseball players played their home games on this field during part of their minor league years, including Wade Boggs, Curt Schilling, and Jim Palmer. Former player and MLB manager Don Zimmer was even married on Dunn Field in 1951 (pictured in Sports Illustrated magazine, and his autobiography).

## Stadium Operations Staff

| Stadium Staff | | | | | | |
| --- | --- | --- | --- | --- | --- | --- |
| Title | **STAFF MEMBER** |
| **Owner** | City Of Elmira |
| **President of Stadium Operations** | Robbie Nichols |
| **Field Crew** | City of Elmira and Kane. |

## Renovations 2012

What was supposed to be $12,000 quickly climbed to almost $40,000 in renovations. The work is being paid for by the City of Elmira, owner of the 73 year old stadium, and will go from the entrance way all the way to some of the seating being done and repainted. Bathrooms without tile will be a thing of the past. Most notable is a section of stadium seating ripped out. Robbie Nichols is the new co-owner of the Elmira Pioneers, along with his wife Nellie Franco-Nichols. Workers are now working to fill in that hole with new steel. Other sections of the stadium will also need the same repair.

While fans also enjoy a new beer garden just behind 3rd base, with live entertainment during games, the players won't be forgotten. "There were posters hanging in the locker room. You take down a poster and there was either a bat hole or ball hole. We put 10 new sheets of sheet rock up,″ Lewis said. The new locker room and other events are in hopes Dunn Field will return to its once famed reputation for Elmira.

"We want to open it the right way. Have a great impact to the city,” Lewis said.

## The Numbers

Dunn Field has a capacity of 4,020 people, including 312 box seats. Its field dimensions are:

* Left - 325'
* Left Center - 358'
* Center - 386'
* Right Center - 358'
* Right - 325'
* Fences are 12' high (20' in LF).

## InfoBox

Dunn Field
| The Dunn | |
| --- | --- |
|  | |
| Location | 546 Luce Street Elmira, NY 14904 |
| Coordinates | 42°04′49″N 76°46′50″W / 42.080296°N 76.780674°W / 42.080296; -76.780674 |
| Owner | City of Elmira, NY |
| Operator | Elmira Baseball Amateur LLC. |
| Field size | Left - 325' Left Center - 358'
Center - 386' Right Center - 358'

Right - 325' Fences are 12' high (20' in LF) |
| Surface | Grass |
| Construction | |
| Broke ground | 1938 |
| Opened | 1939 |
| Renovated | 2012 |
| Expanded | 1993 |
| Tenants | |
| Elmira Pioneers (PGCBL) (1939–present)
Elmira College (2015–present)
Elmira Southside High School Baseball
Elmira Southside High School Football | |


title: Duncan Tanner

abstract: Duncan Tanner was a political historian and academic. His best-known work covered the British Labour Party and voting in the early 20th century. He held the post of director of the Welsh Institute for Social and Cultural Affairs at Bangor University.

text:
# Duncan Tanner

**Duncan Tanner** (19 February 1958 – 11 February 2010) was a political historian and academic. His best-known work covered the British Labour Party and voting in the early 20th century. He held the post of director of the Welsh Institute for Social and Cultural Affairs at Bangor University.

## Selected bibliography

* *Political Change and the Labour Party 1900-1918* (1990)
* *Labour's First Century* (2000)
* *Debating nationhood and governance in Britain, 1885-1945* (2006)
* *The Strange Survival of Liberal England* (2007)


title: Durga Mohan Bhattacharyya

abstract: Durga Mohan Bhattacharyya was an Indian scholar of Sanskrit. He had served as a professor of Sanskrit at the Scottish Church College in Calcutta. He was a key figure in reviving many manuscripts of the Paippalāda-Saṃhitā and its ancillary literature like the Āṅgirasakalpa after painstaking search over years in Orissa and south-west Bengal. Durgamohan Bhattacharyya’s discovery of a living tradition of the Paippalāda-Saṃhitā, unknown until then, was hailed in the Indological world as epoch making. Ludwig Alsdorf went so far as to say that it was the greatest event in Indology. Bhattacharyya died in 1965 leaving his edition of the text incomplete. This task was completed by his son Dipak, whose critical edition of the first 18 kāṇḍas was published by the Asiatic Society, Calcutta came out in three volumes in 1997, 2008 and 2011.

text:
# Durga Mohan Bhattacharyya

**Durga Mohan Bhattacharyya** was an Indian scholar of Sanskrit. He had served as a professor of Sanskrit at the Scottish Church College in Calcutta.

He was a key figure in reviving many manuscripts of the Paippalāda-Saṃhitā and its ancillary literature like the Āṅgirasakalpa after painstaking search over years in Orissa and south-west Bengal. Durgamohan Bhattacharyya’s discovery of a living tradition of the Paippalāda-Saṃhitā, unknown until then, was hailed in the Indological world as epoch making. Ludwig Alsdorf went so far as to say that it was the greatest event in Indology. Bhattacharyya died in 1965 leaving his edition of the text incomplete. This task was completed by his son Dipak, whose critical edition of the first 18 kāṇḍas was published by the Asiatic Society, Calcutta came out in three volumes in 1997, 2008 and 2011.

## Early life

In the early 1900 he with other members of his family, migrated to Sahanagar, Lalbag in the district of Murshidabad. The family was poor and could not send its young children to an English medium school. His early education was derived from tols and chatuspathis, where the main subjects taught were Bengali and Sanskrit, the medium of education primarily being Bengali. Durgamohan was an exceptionally brilliant student and by the year 1915 he had appeared at several Sanskrit Upadhi examinations and topped the list of candidates for the several examinations on Sanskrit conducted by the Government of Bengal. He acquired the highest degrees in Kavya, Sankhya and Purana and got the title of Bhagavataratna.

Durgamohan with his widowed mother (Sarada) and only younger brother moved to his maternal uncle's house in Calcutta. Coming to know about the keen desire of Durgamohan to study English, his senior maternal uncle took him to Suresh Chandra Kundu, then the headmaster of Town School, Calcutta, an institution of great reputation. It was an immense task for Durgamohan to achieve as he had already reached the age of 16 and he was required to complete the normal course of ten years in a single year. He successfully completed the task and in 1917 he sat for the Entrance Examination of the University of Calcutta and was declared successful, obtaining a place in the First Division of successful candidates.

The Intermediate Examination (F.A.) was achieved in 1919 at the Vidyasagar College, the B.A. Examination with a First Class honours Degree in Sanskrit from the Scottish Church College was gained in 1921 and the master's degree in Sanskrit was obtained in 1923 from the University of Calcutta.

## Career

### Early career

After completing his studies in the University, Durgamohan decided to take up the educational line as his field of activities. Having served as a Professor of Sanskrit in the Narasinha Dutt College of Howrah for some time, he joined the Scottish Church College as a professor of Sanskrit and Bengali and eventually became the head of the department of Sanskrit in the early thirties. In 1952 he was inducted in the West Bengal Senior Educational Service as Professor of Vedic Language, Literature and Culture in the Postgraduate Training and Research Department of the Sanskrit College, which position he occupied till the date of his death.

He used to be invited by learned societies like the Asiatic Society of Bengal, the Asiatic Society of Bombay, the Bhandarkar Oriental Research Institute, and others to deliver talks on specific topics particularly the Vedas. He was awarded gold medals by the Asiatic Societies for his services in the field of Sanskrit.

### Work on Paippalāda-Saṃhitā

He had come to infer from many sources that of the four Vedas, the Atharva Veda and its practice had not become extinct in India as many scholars of repute used to hold and propagate. To prove his conviction in this regard he visited a large number of places all over India, and, ultimately a few years before his death, he was able to locate a place in Orissa, Guhiapal to be precise where he found the Atharva Veda to be actively practiced and there he discovered several Oriya manuscripts in which the Paippalāda-Saṃhitā, one of the nine versions of the Atharva Veda was faithfully reproduced. The discovery was made known to the world and the belief about the extinction of the practice of Atharva Veda was proved incorrect. He was hailed for his painstaking effort and perseverance in the unearthing of the Paippalāda-Saṃhitā as an epoch making discovery.

He started serious work on the Paippalāda-Saṃhitā, and publications also started which received acclamations from scholars all over the world. But unfortunately Durgamohan fell ill with cancer and died on 12 November 1965. His task was completed by his son Dipak Bhattacharya whose critical edition of the first 18 kāṇḍas published by the Asiatic Society, Calcutta came out in three volumes in 1997, 2008 and 2011.

(Input source Mrs Lila Chakraborty, Mr Joy Bhattacharya)

## Personal life

Durgamohan had five sons and four daughters. His wife Bhavani Bhattacharya died in 1992.

## InfoBox

| Durga Mohan Bhattacharyya | |
| --- | --- |
|  | |
| Born | 13 October 1899
Dhaka, British India |
| Died | 12 November 1965(1965-11-12) (aged 66)
Kolkata West Bengal, India |
| Nationality | Indian |
| Occupation(s) | Writer, Reformer, Professor |
| Known for | Paippalāda-Saṃhitā |
| Spouse | Bhavani Bhattacharya |
| Children | Sukumar Bhattacharya
Lila Chakraborty
Amalendu Bhattacharya
Chhaya Chakraborty
Bani Chakraborty
Dipak Bhatacharya
Dilip Bhattacharya |


title: ThermoSpas

abstract: ThermoSpas is an American brand of hot tubs, headquartered in Wallingford, Connecticut. Founded in 1983, it was purchased by a subsidiary of Jacuzzi Group Worldwide in 2012 and now operates under the name ThermoSpas Hot Tub Products, Inc.

text:
# ThermoSpas

**ThermoSpas** is an American brand of hot tubs, headquartered in Wallingford, Connecticut. Founded in 1983, it was purchased by a subsidiary of Jacuzzi Group Worldwide in 2012 and now operates under the name ThermoSpas Hot Tub Products, Inc.

## History

ThermoSpas was founded in 1983 and began in a 30,000 square foot facility in Shelton, Connecticut. It continued to expand and relocated to a 140,000 square foot manufacturing plant a few years later in Wallingford.

ThermoSpas originally sold through hot tub dealers in the mid-1980s. In 1995, the company cut ties with most dealers in favor of a new approach to hot tub sales. The company's business model became unique in the industry. While major hot tub companies sold through independent dealers, ThermoSpas used a direct-to-consumer model and performed everything from the initial site inspection through installation of its tubs. Its best lead system was from infomercials which generated hundreds of thousands of leads for the company.

ThermoSpas ran into financial difficulty in 2008-2009 during the United States recession. It became unable to manage every aspect of its business, reporting $2.6 million in assets with almost $18 million in debt, eventually filing bankruptcy. A subsidiary of Jacuzzi Group Worldwide purchased some of the assets of ThermoSpas, giving ThermoSpas the resources and improved infrastructure to continue it use of the direct-to-consumer business model. Although considered a subsidiary of Jacuzzi Group Worldwide, ThermoSpas operates as an independent company officially named ThermoSpas Hot Tub Products, Inc.

Since its inception, the ThermoSpas has been known for development and use of technology such as a patented control therapy system that allows each person in a hot tub to have a personalized therapeutic experience in addition to various filtration systems that help purify water used in a hot tub.

## Products

ThermoSpas sells pre-designed hot tubs as well as personally designed hot tubs for personal use. In addition to personal hot tubs, ThermoSpas is known for healing spas and swim spa trainers. It is also attributed as the first to develop a healing spa specifically for arthritis sufferers, a design that they partnered with the Arthritis Foundation to complete.

## InfoBox

ThermoSpas Hot Tub Products, Inc.
|
|  |
| Founded | 1983 |
| Headquarters | Wallingford, Connecticut, U.S. |
| Website | www.thermospas.com |


title: Third Kohl cabinet

abstract: The Third Kohl cabinet led by Helmut Kohl was sworn in on March 12, 1987. The cabinet was formed after the 1987 elections. This cabinet oversaw the German Reunification. It laid down its function on January 18, 1991, after the formation of the Fourth Kohl cabinet, which was formed following the 1990 elections.

text:
# Third Kohl cabinet

The **Third Kohl cabinet** led by **Helmut Kohl** was sworn in on March 12, 1987. The cabinet was formed after the 1987 elections. This cabinet oversaw the German Reunification. It laid down its function on January 18, 1991, after the formation of the Fourth Kohl cabinet, which was formed following the 1990 elections.

## Composition

| Portfolio | Minister | Tookoffice | Leftoffice | Party | |
| --- | --- | --- | --- | --- | --- |
| Chancellor | Helmut Kohl | 12 March 1987 | 18 January 1991 |  | CDU |
| Vice Chancellor &
Federal Minister of Foreign Affairs | Hans-Dietrich Genscher | 12 March 1987 | 18 January 1991 |  | FDP |
| Federal Minister of Defense | Manfred Wörner | 12 March 1987 | 18 May 1988 |  | CDU |
| | Rupert Scholz | 18 May 1988 | 21 April 1989 |  | CDU |
| | Gerhard Stoltenberg | 21 April 1989 | 18 January 1991 |  | CDU |
| Federal Minister of the Interior | Friedrich Zimmermann | 12 March 1987 | 21 April 1989 |  | CSU |
| | Wolfgang Schäuble | 21 April 1989 | 18 January 1991 |  | CDU |
| Federal Minister of Finance | Gerhard Stoltenberg | 12 March 1987 | 21 April 1989 |  | CDU |
| | Theo Waigel | 21 April 1989 | 18 January 1991 |  | CSU |
| Federal Minister of Justice | Hans A. Engelhard | 12 March 1987 | 18 January 1991 |  | FDP |
| Federal Minister of Economics | Martin Bangemann | 12 March 1987 | 9 December 1988 |  | FDP |
| | Helmut Haussmann | 9 December 1988 | 18 January 1991 |  | FDP |
| Federal Minister of Labour and Social Affairs | Norbert Blüm | 12 March 1987 | 18 January 1991 |  | CDU |
| Federal Minister of Food, Agriculture, and Forestry | Ignaz Kiechle | 12 March 1987 | 18 January 1991 |  | CSU |
| Federal Minister of Transport | Jürgen Warnke | 12 March 1987 | 21 April 1989 |  | CSU |
| | Friedrich Zimmermann | 21 April 1989 | 18 January 1991 |  | CSU |
| Federal Minister of Construction | Oscar Schneider | 12 March 1987 | 21 April 1989 |  | CSU |
| | Gerda Hasselfeldt | 21 April 1989 | 18 January 1991 |  | CSU |
| Federal Minister of Youth, Family, and Health | Rita Süssmuth | 12 March 1987 | 9 December 1988 |  | CDU |
| | Ursula Lehr | 9 December 1988 | 18 January 1991 |  | CDU |
| Federal Minister of Research and Technology | Heinz Riesenhuber | 12 March 1987 | 18 January 1991 |  | CDU |
| Federal Minister of Education and Science | Jürgen Möllemann | 12 March 1987 | 18 January 1991 |  | FDP |
| Federal Minister of Economic Cooperation | Hans Klein | 12 March 1987 | 21 April 1989 |  | CSU |
| | Jürgen Warnke | 21 April 1989 | 18 January 1991 |  | CSU |
| Federal Minister of Environment,
Nature Conservation, and Reactor Security | Walter Wallmann | 12 March 1987 | 22 April 1987 |  | CDU |
| | Klaus Töpfer | 22 April 1987 | 18 January 1991 |  | CDU |
| Federal Minister of Posts and Communications | Christian Schwarz-Schilling | 12 March 1987 | 18 January 1991 |  | CDU |
| Federal Minister of Intra-German Relations | Dorothee Wilms | 12 March 1987 | 18 January 1991 |  | CDU |
| Federal Minister of Special Affairs &
Head of the Chancellery | Wolfgang Schäuble | 12 March 1987 | 21 April 1989 |  | CDU |
| | Rudolf Seiters | 21 April 1989 | 18 January 1991 |  | CDU |
| Federal Minister of Special Affairs &
Government Spokesman-in-chief | Hans Klein | 21 April 1989 | 20. Dezember 1990 |  | CSU |
| Federal Minister of Special Affairs | Lothar de Maizière | 3 October 1990 | 19 December 1990 |  | CDU |
| | Sabine Bergmann-Pohl | 3 October 1990 | 18 January 1991 |  | CDU |
| | Günther Krause | 3 October 1990 | 18 January 1991 |  | CDU |
| | Rainer Ortleb | 3 October 1990 | 18 January 1991 |  | FDP |
| | Hansjoachim Walther | 3 October 1990 | 18 January 1991 |  | DSU |

## InfoBox

| Third Cabinet of Helmut Kohl
Cabinet Kohl III | |
| --- | --- |
|
15th Cabinet of the Federal Republic of Germany | |
| 12 March 1987 – 20 December 1990
*(until 18 January 1991 as caretaker government)* | |
| Helmut Kohl | |
| Date formed | 12 March 1987 |
| Date dissolved | 18 January 1991
(3 years, 10 months and 5 days) |
| People and organisations | |
| President | Richard von Weizsäcker |
| Chancellor | Helmut Kohl |
| Vice-Chancellor | Hans-Dietrich Genscher |
| Member party | Christian Democratic Union
Christian Social Union
Free Democratic Party
German Social Union
(October 1990 – January 1991) |
| Status in legislature | Coalition government led by CDU/CSU
269/497 (54%)

 |
| Opposition party | Social Democratic Party
The Greens |
| Opposition leader |  |
| History | |
| Election | 1987 federal election |
| Legislature terms | 11th Bundestag |
| Predecessor | Kohl II |
| Successor | Kohl IV |


title: Dunkirk (town), New York

abstract: Dunkirk is a town in Chautauqua County, New York, United States. The population was 1,257 at the 2020 census. The town of Dunkirk is split into two sections, east and west, by the city of Dunkirk, which is a separate entity.

text:
# Dunkirk (town), New York

**Dunkirk** is a town in Chautauqua County, New York, United States. The population was 1,257 at the 2020 census.

The town of Dunkirk is split into two sections, east and west, by the city of Dunkirk, which is a separate entity.

## History

The area was first settled *circa* 1808. Development was slow until the New York and Erie Railroad established a station in Dunkirk village in 1851. The town of Dunkirk was established in 1859 from the northern portion of the town of Pomfret. The village of Dunkirk was incorporated in 1837, becoming a city in 1888.

The first naval skirmish in the War of 1812 began in Lake Erie near the mouth of Canadaway Creek, which is now within the boundaries of the western part of the town. A small marker commemorating the event sits near the intersection of Route 5 and Temple Street.

## Geography

According to the United States Census Bureau, the town has a total area of 6.3 square miles (16.3 km2), of which 6.2 square miles (16.1 km2) is land and 0.04 square miles (0.1 km2), or 0.85%, is water.

The New York State Thruway (Interstate 90) and NY Route 5 pass through the town. The town of Dunkirk is served by the Buffalo-Chicago main lines of both CSX Transportation and the Norfolk Southern Railway.

The town of Dunkirk is split into an east section and a west section by the city of Dunkirk, which is located in the middle.

### Adjacent towns and areas

(Clockwise)

## Demographics

Historical population
| Census | Pop. | Note |  |
| --- | --- | --- | --- |
| 1860 | 5,616 |  |  |
| 1900 | 454 |  |  |
| 1910 | 429 |  | −5.5% |
| 1920 | 512 |  | 19.3% |
| 1930 | 726 |  | 41.8% |
| 1940 | 672 |  | −7.4% |
| 1950 | 887 |  | 32.0% |
| 1960 | 1,541 |  | 73.7% |
| 1970 | 1,646 |  | 6.8% |
| 1980 | 1,584 |  | −3.8% |
| 1990 | 1,482 |  | −6.4% |
| 2000 | 1,387 |  | −6.4% |
| 2010 | 1,318 |  | −5.0% |
| 2020 | 1,257 |  | −4.6% |
| 2021 (est.) | 1,248 |  | −0.7% |
| U.S. Decennial Census | | | |

As of the census of 2000, there were 1,387 people, 519 households, and 333 families residing in the town. The population density was 222.1 inhabitants per square mile (85.8/km2). There were 602 housing units at an average density of 96.4 per square mile (37.2/km2). The racial makeup of the town was 94.16% White, 2.24% Black or African American, 0.58% Native American, 0.07% Asian, 1.66% from other races, and 1.30% from two or more races. Hispanic or Latino of any race were 4.33% of the population.

There were 519 households, out of which 22.5% had children under the age of 18 living with them, 49.1% were married couples living together, 10.4% had a female householder with no husband present, and 35.8% were non-families. 30.3% of all households were made up of individuals, and 13.5% had someone living alone who was 65 years of age or older. The average household size was 2.27 and the average family size was 2.78.

In the town, the population was spread out, with 15.6% under the age of 18, 5.8% from 18 to 24, 20.3% from 25 to 44, 27.2% from 45 to 64, and 31.1% who were 65 years of age or older. The median age was 50 years. For every 100 females, there were 85.9 males. For every 100 females age 18 and over, there were 81.4 males.

The median income for a household in the town was $33,553, and the median income for a family was $40,000. Males had a median income of $32,692 versus $23,194 for females. The per capita income for the town was $19,124. About 7.1% of families and 11.7% of the population were below the poverty line, including 19.5% of those under age 18 and 3.8% of those age 65 or over.

## Communities and locations in Dunkirk

* **Canadaway Creek**  The War of 1812 began near the mouth of Canadaway Creek.

## InfoBox

| Town of Dunkirk, New York | |
| --- | --- |
| Town | |
| Location within Chautauqua County and New York | |
| Dunkirk (town)Location within the state of New York | |
| Coordinates: 42°27′51″N 79°20′33″W / 42.46417°N 79.34250°W / 42.46417; -79.34250 | |
| Country | United States |
| State | New York |
| County | Chautauqua |
| Government | |
| Type | Town Council |
| Town Supervisor | Richard A. Purol () |
| Town Council | Members' List |
| Area | |
| Total | 6.28 sq mi (16.27 km2) |
| Land | 6.23 sq mi (16.13 km2) |
| Water | 0.05 sq mi (0.14 km2) |
| Elevation | 673 ft (205 m) |
| Population (2020) | |
| Total | 1,257 |
| Estimate (2021) | 1,248 |
| Density | 170.36/sq mi (65.78/km2) |
| Time zone | UTC-5 (Eastern (EST)) |
| Summer (DST) | UTC-4 (EDT) |
| ZIP codes | 14048, 14166 |
| Area code | 716 |
| FIPS code | 36-21116 |
| GNIS feature ID | 0978912 |
| Website | dunkirkny.org |


title: Dubočani, Konjic

abstract: Dubočani is a village in the municipality of Konjic, Bosnia and Herzegovina.

text:
# Dubočani, Konjic

**Dubočani** is a village in the municipality of Konjic, Bosnia and Herzegovina.

## Demographics

According to the 2013 census, its population was 54, all Bosniaks.

## InfoBox

| Dubočani | |
| --- | --- |
| Village | |
|  | |
| Dubočani | |
| Coordinates: 43°35′06″N 18°04′28″E / 43.58500°N 18.07444°E / 43.58500; 18.07444 | |
| Country | Bosnia and Herzegovina |
| Entity | Federation of Bosnia and Herzegovina |
| Canton | Herzegovina-Neretva |
| Municipality | Konjic |
| Area | |
| Total | 5.40 sq mi (13.99 km2) |
| Population (2013) | |
| Total | 54 |
| Density | 10.0/sq mi (3.9/km2) |
| Time zone | UTC+1 (CET) |
| Summer (DST) | UTC+2 (CEST) |


title: Dubočane

abstract: Dubočane is a village located in the city of Zaječar, Serbia. According to the 2011 census, the village has a population of 361 inhabitants.

text:
# Dubočane

**Dubočane** is a village located in the city of Zaječar, Serbia. According to the 2011 census, the village has a population of 361 inhabitants.

## InfoBox

| Dubočane Дубочане | |
| --- | --- |
| Village | |
| Dubočane | |
| Coordinates: 44°06′23″N 22°16′13″E / 44.10639°N 22.27028°E / 44.10639; 22.27028 | |
| Country | Serbia |
| District | Zaječar District |
| Municipality | Zaječar |
| Area | |
| Total | 32.31 km2 (12.47 sq mi) |
| Elevation | 383 m (1,257 ft) |
| Population (2011) | |
| Total | 361 |
| Density | 11/km2 (29/sq mi) |
| Time zone | UTC+1 (CET) |
| Summer (DST) | UTC+2 (CEST) |