In the Low Countries, stadtholder was an office of steward, designated a medieval official and a national leader. The stadtholder was the replacement of the duke or earl of a province during the Burgundian and Habsburg period; the title was used for the official tasked with maintaining peace and provincial order in the early Dutch Republic and, at times, became de facto head of state of the Dutch Republic during the 16th to 18th centuries, an hereditary role. For the last half century of its existence, it became an hereditary role and thus a monarchy under Prince William IV of Orange, his son, Prince William V of Orange, was the last stadtholder of the republic, whose own son, King William I of the Netherlands, became the first king of the Netherlands. The current Dutch monarchy is only distantly related to the first stadtholder of the Dutch Republic, William I of Orange, the leader of the successful Dutch Revolt against the Spanish Empire, his line having died out with William III; the title stadtholder is comparable to England's historic title Lord Protector.

Stadtholder means "steward". Its component parts translate as "place holder," or as a direct cognate, "stead holder", it was a term for a "steward" or "lieutenant". Note, not the word for the military rank of lieutenant, luitenant in Dutch. Stadtholders in the Middle Ages were appointed by feudal lords to represent them in their absence. If a lord had several dominions, some of these could be ruled by a permanent stadtholder, to whom was delegated the full authority of the lord. A stadtholder was thus more powerful than a governor, who had only limited authority, but the stadtholder was not a vassal himself, having no title to the land; the local rulers of the independent provinces of the Low Countries made extensive use of stadtholders, e.g. the Duke of Guelders appointed a stadtholder to represent him in Groningen. In the 15th century the Dukes of Burgundy acquired most of the Low Countries, these Burgundian Netherlands each had their own stadtholder. In the 16th century, the Habsburg Holy Roman Emperor Charles V King of Spain, who had inherited the Burgundian Netherlands, completed this process by becoming the sole feudal overlord: Lord of the Netherlands.

Only the Prince-Bishopric of Liège and two smaller territories remained outside his domains. Stadtholders continued to be appointed to represent Charles and King Philip II, his son and successor in Spain and the Low Countries. Due to the centralist and absolutist policies of Philip, the actual power of the stadtholders diminished. When, in 1581, during the Dutch Revolt, most of the Dutch provinces declared their independence with the Act of Abjuration, the representative function of the stadtholder became obsolete in the rebellious northern Netherlands – the feudal lord himself having been abolished – but the office continued in these provinces who now united themselves into the Republic of the Seven United Netherlands; the United Provinces were struggling to adapt existing feudal concepts and institutions to the new situation and tended to be conservative in this matter, as they had after all rebelled against the king to defend their ancient rights. The stadtholder no longer represented the lord but became the highest executive official, appointed by the states of each province.

Although each province could assign its own stadtholder, most stadtholders held appointments from several provinces at the same time. The highest executive power was exerted by the sovereign states of each province, but the stadtholder had some prerogatives, like appointing lower officials and sometimes having the ancient right to affirm the appointment of the members of regent councils or choose burgomasters from a shortlist of candidates; as these councils themselves appointed most members of the states, the stadtholder could indirectly influence the general policy. In Zeeland the Princes of Orange, who after the Dutch Revolt most held the office of stadtholder there, held the dignity of First Noble, were as such a member of the states of that province, because they held the title of Marquis of Veere and Flushing as one of their patrimonial titles. On the Republic's central'confederal' level, the stadtholder of the provinces of Holland and Zealand was also appointed Captain-General of the confederate army and Admiral-General of the confederate fleet, though no stadtholder actually commanded a fleet in battle.

In the army, he could appoint officers by himself. Legal powers of the stadtholder were thus rather limited, by law he was a mere official, his real powers, were sometimes greater given the martial law atmosphere of the'permanent' Eighty Years War. Maurice of Orange after 1618 ruled as a military dictator, William II of Orange attempted the same; the leader of the Dutch Revolt was William the Silent. His personal influence and reputation was subsequently associated with the office and transferred to members of his house. After his assassination, there was a short-lived move to install Robert D

Travis Sauter is an American stock car racing driver. A regular competitor in several midwestern racing series, he is second on the all-time win list with four Oktoberfest wins at the LaCrosse Fairgrounds Speedway. Travis is the grandson of Jim Sauter, son of Tim, nephew of Johnny and Jay. Sauter began his racing career in 2002, finishing second in the track championship at Madison International Speedway in 2006 before moving full-time to touring series events in 2007, having started running the ASA Late Model Series in 2004. Competing in the ASA Midwest Tour, Sauter has won six times in 53 starts in the series, with a best points finish of seventh in 2008. Owning his own race team, Sauter competes in selected, high-profile events throughout the Midwest. In 2012, he won the inaugural Howie Lettow Memorial 150 at the Milwaukee Mile, topping a field of over 80 late models to win the event, he finished second to Kyle Busch in the Howie Lettow Memorial 150 in 2013, before making his debut in NASCAR competition at Iowa Speedway in August, driving for Joe Nemechek.

Sauter set the all-time track record at the 1/3 mile Dells Raceway Park on August 17, 2013 with a time of 13.108 seconds in his Super Late Model. Sauter started his 2015 season by winning the inaugural Icebreaker 100 super late model at Dells Raceway Park in a feature field of 24 drivers. * Season still in progress1 Ineligible for series points Official website Travis Sauter driver statistics at Racing-Reference

In graph theory, the Shannon capacity of a graph is a graph invariant defined from the number of independent sets of strong graph products. It measures the Shannon capacity of a communications channel defined from the graph, is upper bounded by the Lovász number, which can be computed in polynomial time. However, the computational complexity of the Shannon capacity itself remains unknown; the Shannon capacity models the amount of information that can be transmitted across a noisy communication channel in which certain signal values can be confused with each other. In this application, the confusion graph or confusability graph describes the pairs of values that can be confused. For instance, suppose that a communications channel has five discrete signal values, any one of which can be transmitted in a single time step; these values may be modeled mathematically as the five numbers 0, 1, 2, 3, or 4 in modular arithmetic modulo 5. However, suppose that when a value i is sent across the channel, the value, received is i + ξ where ξ represents the noise on the channel and may be any real number in the open interval −1 < ξ < 1.

Thus, if the recipient receives a value such as 3.6, it is impossible to determine whether it was transmitted as a 3 or as a 4. This situation can be modeled by a graph, a cycle C5 of length 5, in which the vertices correspond to the five values that can be transmitted and the edges of the graph represent values that can be confused with each other. For this example, it is possible to choose two values that can be transmitted in each time step without ambiguity, for instance, the values 1 and 3; these values are far enough apart that they can't be confused with each other: when the recipient receives a value x in the range 0 < x < 2, it can deduce that the value, sent must have been 1, when the recipient receives a value x in the range 2 < x < 4, it can deduce that the value, sent must have been 3. In this way, in n steps of communication, the sender can communicate up to 2n different messages. Two is the maximum number of values that the recipient can distinguish from each other: every subset of three or more of the values 0, 1, 2, 3, 4 includes at least one pair that can be confused with each other.

Though the channel has five values that can be sent per time step only two of them can be used with this coding scheme. However, more complicated coding schemes allow a greater amount of information to be sent across the same channel, by using codewords of length greater than one. For instance, suppose that in two consecutive steps the sender transmits one of the five code words "11", "23", "35", "54", or "42"; each pair of these code words includes at least one position where its values differ by two or more modulo 5. Therefore, a recipient of one of these code words will always be able to determine unambiguously which one was sent: no two of these code words can be confused with each other. By using this method, in n steps of communication, the sender can communicate up to 5n/2 messages more than the 2n that could be transmitted with the simpler one-digit code; the effective number of values that can be transmitted per unit time step is 1/n = √5. In graph-theoretic terms, this means that the Shannon capacity of the 5-cycle is at least √5.

As Lovász showed, this bound is tight: it is not possible to find a more complicated system of code words that allows more different messages to be sent in the same amount of time, so the Shannon capacity of the 5-cycle is √5. If a graph G represents a set of symbols and the pairs of symbols that can be confused with each other a subset S of symbols avoids all confusable pairs if and only if S is an independent set in the graph, a subset of vertices that does not include both endpoints of any edge; the maximum possible size of a subset of the symbols that can all be distinguished from each other is the independence number α of the graph, the size of its maximum independent set. For instance, α = 2: not larger. For codewords of longer lengths, one can use independent sets in larger graphs to describe the sets of codewords that can be transmitted without confusion. For instance, for the same example of five symbols whose confusion graph is C5, there are 25 strings of length two that can be used in a length-2 coding scheme.

These strings may be represented by the vertices of a graph with 25 vertices. In this graph, each vertex has the eight strings that it can be confused with. A subset of length-two strings forms a code with no possible confusion if and only if it corresponds to an independent set of this graph; the set of code words forms one of maximum size. If G is a graph representing the signals and confusable pairs of a channel the graph representing the length-two codewords and their confusable pairs is G ⊠ G, where the symbol "⊠" represents the strong product of graphs; this is a graph that has a vertex for each pair of a vertex in the first argument of the product and a vertex in the second argument of the product. Two distinct pairs and are adjacent in the strong product if and only if u1 and u2 are identical or adjacent, v1 and v2 are identical or adjacent. More the codewords of length k can be represented by the graph Gk, the k-fold strong product of G with itself, the maximum number of codewords of this length that can be transmitt