text:
A boat is a watercraft
A watercraft is a vessel or craft designed to move across or through water. The name is derived from the term "craft" which was used to describe all types of water going vessels...
of any size designed to float or plane
Planing (sailing)
Planing is the mode of operation for a waterborne craft in which its weight is predominantly supported by hydrodynamic lift, rather than hydrostatic lift .-History:...
, to provide passage across water. Usually this water will be inland (lake
A lake is a body of relatively still fresh or salt water of considerable size, localized in a basin, that is surrounded by land. Lakes are inland and not part of the ocean and therefore are distinct from lagoons, and are larger and deeper than ponds. Lakes can be contrasted with rivers or streams,...
s) or in protected coastal areas. However, boats such as the whaleboat
Whaleboat
A whaleboat is a type of open boat that is relatively narrow and pointed at both ends, enabling it to move either forwards or backwards equally well. It was originally developed for whaling, and later became popular for work along beaches, since it does not need to be turned around for beaching or...
were designed to be operated from a ship
Since the end of the age of sail a ship has been any large buoyant marine vessel. Ships are generally distinguished from boats based on size and cargo or passenger capacity. Ships are used on lakes, seas, and rivers for a variety of activities, such as the transport of people or goods, fishing,...
in an offshore environment. In naval
A navy is the branch of a nation's armed forces principally designated for naval and amphibious warfare; namely, lake- or ocean-borne combat operations and related functions...
terms, a boat is a vessel small enough to be carried aboard another vessel (a ship). Another less restrictive definition is a vessel that can be lifted out of the water. Strictly speaking and uniquely a submarine
A submarine is a watercraft capable of independent operation below the surface of the water. It differs from a submersible, which has more limited underwater capability...
is a boat as defined by the Royal Navy
The Royal Navy is the naval warfare service branch of the British Armed Forces. Founded in the 16th century, it is the oldest service branch and is known as the Senior Service...
. Some boats too large for the naval definition include the Great Lakes freighter
Lake freighter
Lake freighters, or Lakers, are bulk carrier vessels that ply the Great Lakes. The best known was the , the most recent and largest major vessel to be wrecked on the Lakes. These vessels are traditionally called boats, although classified as ships. In the mid-20th century, 300 lakers worked the...
, riverboat
A riverboat is a ship built boat designed for inland navigation on lakes, rivers, and artificial waterways. They are generally equipped and outfitted as work boats in one of the carrying trades, for freight or people transport, including luxury units constructed for entertainment enterprises, such...
, narrowboat
A narrowboat or narrow boat is a boat of a distinctive design, made to fit the narrow canals of Great Britain.In the context of British Inland Waterways, "narrow boat" refers to the original working boats built in the 18th, 19th, and 20th centuries for carrying goods on the narrow canals...
and ferry
A ferry is a form of transportation, usually a boat, but sometimes a ship, used to carry primarily passengers, and sometimes vehicles and cargo as well, across a body of water. Most ferries operate on regular, frequent, return services...
boat. The term armed boat, used primarily by English speaking naval forces, referred to any boat carrying either a cannon
A cannon is any piece of artillery that uses gunpowder or other usually explosive-based propellents to launch a projectile. Cannon vary in caliber, range, mobility, rate of fire, angle of fire, and firepower; different forms of cannon combine and balance these attributes in varying degrees,...
or armed occupants, such as marines.
Boats have served as short-distance transportation since early times. Circumstantial evidence, such as the early settlement of Australia over 40,000 years ago, and findings in Crete dated 130,000 years ago, suggests that boats have been used since ancient times. The earliest boats have been predicted to be logboats
Dugout (boat)
A dugout or dugout canoe is a boat made from a hollowed tree trunk. Other names for this type of boat are logboat and monoxylon. Monoxylon is Greek -- mono- + ξύλον xylon -- and is mostly used in classic Greek texts. In Germany they are called einbaum )...
. The oldest boats to be found by archaeological excavation are logboats from around 7,000–10,000 years ago. The oldest recovered boat in the world is the canoe of Pesse; it is a dugout
or hollowed tree trunk from a Pinus sylvestris. It was constructed somewhere between 8200 and 7600 B.C. This canoe is exhibited in the Drents Museum in Assen, Netherlands; other very old dugout boats have been recovered.
A 7,000 year-old seagoing boat made from reeds
Reed boat
Reed boats and rafts, along with dugout canoes and other rafts, are among the oldest known types of boats. Often used as traditional fishing boats, they are still used in a few places around the world, though they have generally been replaced with planked boats. Reed boats can be distinguished from...
and tar has been found in Kuwait
The State of Kuwait is a sovereign Arab state situated in the north-east of the Arabian Peninsula in Western Asia. It is bordered by Saudi Arabia to the south at Khafji, and Iraq to the north at Basra. It lies on the north-western shore of the Persian Gulf. The name Kuwait is derived from the...
Boats were used between 4000 BCE and 3000 BCE in Sumer
Sumer was a civilization and historical region in southern Mesopotamia, modern Iraq during the Chalcolithic and Early Bronze Age....
, ancient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
and in the Indian Ocean
The Indian Ocean is the third largest of the world's oceanic divisions, covering approximately 20% of the water on the Earth's surface. It is bounded on the north by the Indian Subcontinent and Arabian Peninsula ; on the west by eastern Africa; on the east by Indochina, the Sunda Islands, and...
Boats played a very important part in the commerce between the Indus Valley Civilization
The Indus Valley Civilization was a Bronze Age civilization that was located in the northwestern region of the Indian subcontinent, consisting of what is now mainly modern-day Pakistan and northwest India...
and Mesopotamia
Mesopotamia is a toponym for the area of the Tigris–Euphrates river system, largely corresponding to modern-day Iraq, northeastern Syria, southeastern Turkey and southwestern Iran.Widely considered to be the cradle of civilization, Bronze Age Mesopotamia included Sumer and the...
. Evidence of varying models of boats has also been discovered in various Indus Valley sites.
The Uru wooden big boat made in Beypore
Beypore
Beypore or Beypur is a ancient port town and part of Calicut municipal corporation in Kozhikode district in the state of Kerala, India. The place was formerly known as Vaypura / Vadaparappanad. Tippu Sultan named the town "Sultan Pattanam".This place also has a marina and a beautiful beach...
a village in south Calicut, Kerala
or Keralam is an Indian state located on the Malabar coast of south-west India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
, in southwestern India
. These have been used by the Arabs and Greeks since ancient times as trading vessels. This mammoth wooden ship was constructed using teak, without any iron or blueprints which has transportation capacity of 400 tonnes.
The accounts of historians Herodotus
Herodotus
Herodotus was an ancient Greek historian who was born in Halicarnassus, Caria and lived in the 5th century BC . He has been called the "Father of History", and was the first historian known to collect his materials systematically, test their accuracy to a certain extent and arrange them in a...
, Pliny the Elder
Pliny the Elder
Gaius Plinius Secundus , better known as Pliny the Elder, was a Roman author, naturalist, and natural philosopher, as well as naval and army commander of the early Roman Empire, and personal friend of the emperor Vespasian...
, and Strabo
Strabo
Strabo, also written Strabon was a Greek historian, geographer and philosopher.-Life:Strabo was born to an affluent family from Amaseia in Pontus , a city which he said was situated the approximate equivalent of 75 km from the Black Sea...
suggest that boats were being used for commerce and traveling.
Boats can be categorized into three types:
Unpowered or human-powered boats. (Unpowered boats include raft
A raft is any flat structure for support or transportation over water. It is the most basic of boat design, characterized by the absence of a hull...
s and floats meant for one-way downstream travel. Human-powered boats include canoe
A canoe or Canadian canoe is a small narrow boat, typically human-powered, though it may also be powered by sails or small electric or gas motors. Canoes are usually pointed at both bow and stern and are normally open on top, but can be decked over A canoe (North American English) or Canadian...
s, kayak
A kayak is a small, relatively narrow, human-powered boat primarily designed to be manually propelled by means of a double blade paddle.The traditional kayak has a covered deck and one or more cockpits, each seating one paddler...
s, gondola
The gondola is a traditional, flat-bottomed Venetian rowing boat, well suited to the conditions of the Venetian Lagoon. For centuries gondolas were the chief means of transportation and most common watercraft within Venice. In modern times the iconic boats still have a role in public transport in...
s and boats propelled by poles like a punt
Punt (boat)
A punt is a flat-bottomed boat with a square-cut bow, designed for use in small rivers or other shallow water. Punting refers to boating in a punt. The punter generally propels the punt by pushing against the river bed with a pole...
The term sailing ship is now used to refer to any large wind-powered vessel. In technical terms, a ship was a sailing vessel with a specific rig of at least three masts, square rigged on all of them, making the sailing adjective redundant. In popular usage "ship" became associated with all large...
, which are boats propelled solely by means of sail
A sail is any type of surface intended to move a vessel, vehicle or rotor by being placed in a wind—in essence a propulsion wing. Sails are used in sailing.-History of sails:...
A motorboat is a boat which is powered by an engine. Some motorboats are fitted with inboard engines, others have an outboard motor installed on the rear, containing the internal combustion engine, the gearbox and the propeller in one portable unit.An inboard/outboard contains a hybrid of a...
s, which are boats propelled by mechanical means, such as engines.
Parts and terminology
Several key components make up the main structure of most boats. The hull
Hull (watercraft)
A hull is the watertight body of a ship or boat. Above the hull is the superstructure and/or deckhouse, where present. The line where the hull meets the water surface is called the waterline.The structure of the hull varies depending on the vessel type...
is the main structural component of the boat which actually provides buoyancy for the boat. The roughly horizontal, but chambered structures spanning the hull of the boat are referred to as the deck. In a ship there are often several decks, but a boat is unlikely to have more than one, if any at all. Above the deck are the superstructures. The underside of a deck is the deck head.
An enclosed space on a boat is referred to as a cabin. Several structures make up a cabin: the similar but usually lighter structure which spans a raised cabin is a coach-roof. The "floor" of a cabin is properly known as the sole, but is more likely to be called the floor (a floor is properly, a structural member which ties a frame to the keelson and keel). The vertical surfaces dividing the internal space are bulkheads.
The keel
In boats and ships, keel can refer to either of two parts: a structural element, or a hydrodynamic element. These parts overlap. As the laying down of the keel is the initial step in construction of a ship, in British and American shipbuilding traditions the construction is dated from this event...
is a lengthwise structural member to which the frames are fixed (sometimes referred to as a backbone).
The front (or forward end) of a boat is called the bow. Boats of earlier times often featured a figurehead protruding from the front of the bows. The rear (or aft end) of the boat is called the stern. The right side (facing forward) is starboard and the left side is port.
Until the mid 19th century most boats were of all natural materials; primarily wood although reed
Reed (plant)
Reed is a generic polyphyletic botanical term used to describe numerous tall, grass-like plants of wet places, which are the namesake vegetation of reed beds...
, bark and animal skins were also used. Early boats include the bound-reed style of boat seen in Ancient Egypt, the birch bark
Birch bark or birchbark is the bark of several Eurasian and North American birch trees of the genus Betula.The strong and water-resistant cardboard-like bark can be easily cut, bent, and sewn, which made it a valuable building, crafting, and writing material, since pre-historic times...
, the animal hide-covered kayak
and coracle
The coracle is a small, lightweight boat of the sort traditionally used in Wales but also in parts of Western and South Western England, Ireland , and Scotland ; the word is also used of similar boats found in India, Vietnam, Iraq and Tibet...
and the dugout canoe made from a single log. By the mid 19th century, many boats had been built with iron or steel frames but still planked in wood. In 1855 ferro-cement boat construction was patented by the French. They called it Ferciment. This is a system by which a steel or iron wire framework is built in the shape of a boat's hull and covered (trowell
Trowell
Trowell is a village and civil parish in Nottinghamshire, England. It lies a few miles west of Nottingham, in the borough of Broxtowe on the border with Derbyshire. According to the 2001 census it had a population of 2,568....
ed) over with cement. Reinforced with bulkheads and other internal structure, it is strong but heavy, easily repaired, and, if sealed properly, will not leak or corrode. These materials and methods were copied all over the world, and have faded in and out of popularity to the present. As the forests of Britain and Europe continued to be over-harvested to supply the keels of larger wooden boats, and the Bessemer process
Bessemer process
The Bessemer process was the first inexpensive industrial process for the mass-production of steel from molten pig iron. The process is named after its inventor, Henry Bessemer, who took out a patent on the process in 1855. The process was independently discovered in 1851 by William Kelly...
(patent
A patent is a form of intellectual property. It consists of a set of exclusive rights granted by a sovereign state to an inventor or their assignee for a limited period of time in exchange for the public disclosure of an invention....
ed in 1855) cheapened the cost of steel, steel ships and boats began to be more common. By the 1930s boats built of all steel from frames to plating were seen replacing wooden boats in many industrial uses, even the fishing fleets. Private recreational boats in steel are uncommon. In the mid 20th century aluminium
Aluminium or aluminum is a silvery white member of the boron group of chemical elements. It has the symbol Al, and its atomic number is 13. It is not soluble in water under normal circumstances....
gained popularity. Though much more expensive than steel, there are now aluminium alloys available that will not corrode in salt water, and an aluminium boat built to similar load carrying standards could be built lighter than steel.
Around the mid 1960s, boats made of glass-reinforced plastic
Glass-reinforced plastic
Fiberglass , is a fiber reinforced polymer made of a plastic matrix reinforced by fine fibers of glass. It is also known as GFK ....
, more commonly known as fibreglass, became popular, especially for recreational boats. The United States Coast Guard
United States Coast Guard
The United States Coast Guard is a branch of the United States Armed Forces and one of the seven U.S. uniformed services. The Coast Guard is a maritime, military, multi-mission service unique among the military branches for having a maritime law enforcement mission and a federal regulatory agency...
refers to such boats as 'FRP' (for Fibre Reinforced Plastic) boats.
Fibreglass boats are extremely strong, and do not rust (iron oxide), corrode, or rot. They are, however susceptible to structural degradation from sunlight and extremes in temperature over their lifespan. Fibreglass provides structural strength, especially when long woven strands are laid, sometimes from bow to stern, and then soaked in epoxy or polyester resin to form the hull of the boat. Whether hand laid or built in a mould, FRP boats usually have an outer coating of gelcoat which is a thin solid colored layer of polyester resin that adds no structural strength, but does create a smooth surface which can be buffed to a high shine and also acts as a protective layer against sunlight. FRP structures can be made stiffer with sandwich panels, where the FRP encloses a lightweight core such as balsa or foam. Cored FRP is most often found in decking which helps keep down weight that will be carried above the waterline. The addition of wood makes the cored structure of the boat susceptible to rotting which puts a greater emphasis on not allowing damaged sandwich structures to go unrepaired. Plastic based foam cores are less vulnerable. The phrase 'advanced composites' in FRP construction may indicate the addition of carbon fibre, kevlar(tm) or other similar materials, but it may also indicate other methods designed to introduce less expensive and, by at least one yacht surveyor's eyewitness accounts, less structurally sound materials.
Cold molding is similar to FRP in as much as it involves the use of epoxy or polyester resins, but the structural component is wood instead of fibreglass. In cold moulding very thin strips of wood are laid over a form or mould in layers. This layer is then coated with resin and another directionally alternating layer is laid on top. In some processes the subsequent layers are stapled or otherwise mechanically fastened to the previous layers, but in other processes the layers are weighted or even vacuum bagged to hold layers together while the resin sets. Layers are built up thus to create the required thickness of hull.
People have even made their own boats or watercraft out of materials such as foam or plastic, but most homebuilts today are built of plywood and either painted or covered in a layer of fibreglass and resin.
The most common means are:
human power (rowing
Watercraft rowing
Watercraft rowing is the act of propelling a boat using the motion of oars in the water. The difference between paddling and rowing is that with rowing the oars have a mechanical connection with the boat whereas with paddling the paddles are hand-held with no mechanical connection.This article...
, paddling
Watercraft paddling
Paddling with regard to watercraft is the act of manually propelling or navigating a small boat using a blade that is joined to a shaft, known as paddle, in the water. The paddle is also used to steer the vessel and may either be a single bladed or double bladed...
, setting pole
Setting pole
A setting pole is a pole, handled by a single individual, made to move watercraft by pushing the craft in the desired direction. Because it is a pushing tool, it is generally used from the stern of the craft....
and so on.)
wind power (sailing)
Motor powered screws
A propeller is a type of fan that transmits power by converting rotational motion into thrust. A pressure difference is produced between the forward and rear surfaces of the airfoil-shaped blade, and a fluid is accelerated behind the blade. Propeller dynamics can be modeled by both Bernoulli's...
Inboard motor
An inboard motor is a marine propulsion system for boats. As opposed to an outboard motor where an engine is mounted outside of the hull of the craft, an inboard motor is an engine enclosed within the hull of the boat, usually connected to a propulsion screw by adriveshaft.-History:The first...
internal combustion (gasoline, diesel, heavy fuel oil)
steam (coal, fuel oil
Fuel oil is a fraction obtained from petroleum distillation, either as a distillate or a residue. Broadly speaking, fuel oil is any liquid petroleum product that is burned in a furnace or boiler for the generation of heat or used in an engine for the generation of power, except oils having a flash...
nuclear (for submarines and large naval ships)
Inboard/Outboard (sterndrive)
A diesel engine is an internal combustion engine that uses the heat of compression to initiate ignition to burn the fuel, which is injected into the combustion chamber...
An outboard motor is a propulsion system for boats, consisting of a self-contained unit that includes engine, gearbox and propeller or jet drive, designed to be affixed to the outside of the transom and are the most common motorized method of propelling small watercraft...
paddle wheel
Paddle steamer
A paddle steamer is a steamship or riverboat, powered by a steam engine, using paddle wheels to propel it through the water. In antiquity, Paddle wheelers followed the development of poles, oars and sails, where the first uses were wheelers driven by animals or humans...
Water Jet (Personal water craft
Personal water craft
A personal water craft , also called water scooter, is a recreational watercraft that the rider rides or stands on, rather than inside of, as in a boat....
, Jetboat
Jetboat
A jetboat is a boat propelled by a jet of water ejected from the back of the craft. Unlike a powerboat or motorboat that uses a propeller in the water below or behind the boat, a jetboat draws the water from under the boat into a pump inside the boat, then expels it through a nozzle at the...
Air Fans (Hovercraft
A hovercraft is a craft capable of traveling over surfaces while supported by a cushion of slow moving, high-pressure air which is ejected against the surface below and contained within a "skirt." Although supported by air, a hovercraft is not considered an aircraft.Hovercraft are used throughout...
, Air boat
Air boat
An airboat, also known as a fanboat, is a flat-bottomed vessel propelled in a forward direction by an aircraft-type propeller and powered by either an aircraft or automotive engine. Airboats are a very popular means of transportation in the Florida Everglades, parts of the Indian River Lagoon, the...
Track-driven propulsion
An early uncommon means of boat propulsion was referred to as the water caterpillar which is similar in construction to paddles on a conveyor belt and preceded the development of tracked vehicles such as military tanks and earth moving equipment. A series of paddles on chains moved along the bottom of the boat to propel it over the water.
The first water caterpillar was developed by Desblancs in 1782 and propelled by a steam engine. In the United States the first water caterpillar was patented in 1839 by William Leavenworth of New York.
A floating boat displaces
Displacement (fluid)
In fluid mechanics, displacement occurs when an object is immersed in a fluid, pushing it out of the way and taking its place. The volume of the fluid displaced can then be measured, as in the illustration, and from this the volume of the immersed object can be deduced .An object that sinks...
its weight in water. The material of the boat hull may be denser than water, but if this is the case then it forms only the outer layer. If the boat floats, the mass of the boat (plus contents) as a whole divided by the volume below the waterline is equal to the density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of water (1 kg/l). If weight is added to the boat, the volume below the waterline will increase to keep the weight balance equal, and so the boat sinks a little to compensate.
Abora
Abora is the name of an ancestal solar deity of La Palma and a traditional god of the Azore Islands, and of two reed boats.-Abora, reed boat:...
Boat building, one of the oldest branches of engineering, is concerned with constructing the hulls of boats and, for sailboats, the masts, spars and rigging.-Parts:* Bow - the front and generally sharp end of the hull...
Cabin cruiser
A cabin cruiser is a type of power boat that provides accommodation for its crew and passengers inside the structure of the craft.A cabin cruiser usually ranges in size from in length, with larger pleasure craft usually considered yachts. Many cabin cruisers can be recovered and towed with a...
The dory is a small, shallow-draft boat, about long. It is a lightweight and versatile boat with high sides, a flat bottom and sharp bows. They are easy to build because of their simple lines. For centuries, dories have been used as traditional fishing boats, both in coastal waters and in the...
Flat-bottomed boat
A flat-bottomed boat is a boat with a flat bottomed, two-chined hull, which allows it be used in shallow bodies of water, such as rivers, because it is less likely to ground....
Glossary of nautical terms
This is a glossary of nautical terms; some remain current, many date from the 17th-19th century. See also Wiktionary's nautical terms, :Category:Nautical terms, and Nautical metaphors in English.- A :...
Halkett boat
A Halkett boat is a type of lightweight inflatable boat designed by during the 1840s. Halkett had long been interested in the difficulties of travelling in the Canadian Arctic, and the problems involved in designing boats light enough to be carried over arduous terrain, but robust enough to be...
An inflatable boat is a lightweight boat constructed with its sides and bow made of flexible tubes containing pressurised gas. For smaller boats, the floor and hull beneath it is often flexible. On boats longer than , the floor often consists of three to five rigid plywood or aluminium sheets fixed...
Launch (boat)
A launch in contemporary usage refers to a large motorboat. The name originally referred to the largest boat carried by a warship. The etymology of the word is given as Portuguese lancha "barge", from Malay lancha, lancharan, "boat," from lanchar "velocity without effort," "action of gliding...
Lifeboat (rescue)
A rescue lifeboat is a boat rescue craft which is used to attend a vessel in distress, or its survivors, to rescue crewmen and passengers. It can be hand pulled, sail powered or powered by an engine...
Lifeboat (shipboard)
A lifeboat is a small, rigid or inflatable watercraft carried for emergency evacuation in the event of a disaster aboard ship. In the military, a lifeboat may be referred to as a whaleboat, dinghy, or gig. The ship's tenders of cruise ships often double as lifeboats. Recreational sailors sometimes...
Log canoe
The log canoe is a type of sailboat developed in the Chesapeake Bay region. Based on the dugout, it was the principal traditional fishing boat of the bay until superseded by the bugeye and the skipjack. However, it is most famous as a racing sailboat, and races continue to be held.The history of...
Motor Launch
A Motor Launch is a small military vessel in British navy service. It was designed for harbour defence and submarine chasing or for armed high speed air-sea rescue....
Naval architecture is an engineering discipline dealing with the design, construction, maintenance and operation of marine vessels and structures. Naval architecture involves basic and applied research, design, development, design evaluation and calculations during all stages of the life of a...
Panga (boat)
The Panga is a type of modest-sized, open, outboard-powered, fishing boat common throughout much of the developing world, including Central America, the Caribbean, parts of Africa, the Middle East, and much of Asia. The original panga design was developed by Yamaha as part of a World Bank project...
Pirogue
A pirogue is a small, flat-bottomed boat of a design associated particularly with the Cajuns of the Louisiana marsh. In West Africa they were used as traditional fishing boats. These boats are not usually intended for overnight travel but are light and small enough to be easily taken onto land...
Rescue craft
A rescue craft is a boat, ship or aircraft used in rescuing.The most common are Lifeboat for Inshore and closer to shore rescues. For operations further out from shore helicopters & ships are mainly used....
Row boat
A sailboat or sailing boat is a boat propelled partly or entirely by sails. The term covers a variety of boats, larger than small vessels such as sailboards and smaller than sailing ships, but distinctions in the size are not strictly defined and what constitutes a sailing ship, sailboat, or a...
Sampan
A sampan is a relatively flat bottomed Chinese wooden boat from long. Some sampans include a small shelter on board, and may be used as a permanent habitation on inland waters. Sampans are generally used for transportation in coastal areas or rivers, and are often used as traditional fishing boats...
The term skiff is used for a number of essentially unrelated styles of small boat. The word is related to ship and has a complicated etymology: "skiff" comes from the Middle English skif, which derives from the Old French esquif, which in turn derives from the Old Italian schifo, which is itself of...
Traditional fishing boats
Traditionally, many different kinds of boats have been used as fishing boats to catch fish in the sea, or on a lake or river. Even today, many traditional fishing boats are still in use. According to the United Nations Food and Agriculture Organization , at the end of 2004, the world fishing fleet...
A yacht is a recreational boat or ship. The term originated from the Dutch Jacht meaning "hunt". It was originally defined as a light fast sailing vessel used by the Dutch navy to pursue pirates and other transgressors around and into the shallow waters of the Low Countries...
A barge is a flat-bottomed boat, built mainly for river and canal transport of heavy goods. Some barges are not self-propelled and need to be towed by tugboats or pushed by towboats...
PDRacer - The Easiest Sailboat In the World To Build
Sailboats database: sailboat data sheets all over the world
University of Washington Libraries Digital Collections -- http://content.lib.washington.edu/cgi-bin/queryresults.exe?CISOOP=adv&CISORESTMP=%2Fsite-templates%2Fsearch_results-sub.html&CISOVIEWTMP=%2Fsite-templates%2Fitem_viewer.html&CISOMODE=thumb&CISOGRID=thumbnail%2CA%2C1%3Btitle%2CA%2C1%3Bsubjec%2CA%2C0%3Bdescri%2C200%2C0%3B0%2CA%2C0%3B10&CISOBIB=title%2CA%2C1%2CN%3Bsubjec%2CA%2C0%2CN%3Bdescri%2CK%2C0%2CN%3B0%2CA%2C0%2CN%3B0%2CA%2C0%2CN%3B10&CISOTHUMB=3%2C5&CISOTITLE=10&CISOPARM=%2Ffishimages%3Asubjec%3Avessels&x=51&y=5Freshwater and Marine Image Bank -- Vessels] - images of boats and vessels


At the end of last week, UAE Team Emirates rider Dan Martin won Stage 6 of the Tour de France in a new jersey from Champion Systems that looked slightly off. What was missing was the full length front zipper we've all come to expect from top-level road jerseys. The as yet unreleased Champ Sys Apex Lite jersey apparently performs so well that several riders on the UAE team are switching for its aero benefits.
Pro riders have been doing time trials in skinsuits that zip up the back for added aero benefits for years, but it looks like this new Champ Sys racing jersey is the first to be raced in a World Tour road race.
The Apex Lite Zipperless jersey was actually developed together with the UAE team to provide an aerodynamic race jersey that was both comfortable and breathable. Pretty much everyone looking for a pro-level jersey has demanded a full-length zipper for years, mostly so you can unzip for more ventilation. But as we've seen more clothing makers stitch together more aero jerseys and everyday racing skinsuits, maybe it's no surprise that the zipper could get ditched as well.
After racing in the standard Apex Lite jersey last summer, UAE realized that the four-way stretch Agile mesh body fabric was vented enough that most riders didn't actually need to unzip to cool down. So why not drop the zip altogether and save some Watts.
Champion Systems says the stretchy jersey is easy to pull over your head, and gets a tight cut that fits close to the body. The UAE team riders are racing the Tour de France in a standard off the rack sizing (it will come in XS-XL, but the team uses only XS-M.) "The feedback from the riders has been excellent," tells Champ Sys operations director Chris Reynolds. "Dan Martin is very particular with what products he'll use in races, as he fully understands the demands and the benefits of certain products." They know that not every rider or even racer will want to ditch the zip, but always want to offer consumer the same race tech their pros use.
The Apex Lite Zipperless jerseys look like they will get an official introduction later this summer, with stock & custom availability soon after. Until then, it looks like Champion Systems is offering a 15% discount on all custom clothing ordered in July with coupon code: "july15off".


I think whoever rides this bike must be a very interesting person.
Yesterday – playing around in a beautiful meadow in Boulder.
Today – I finally paid attention to how much time it takes to develop, scan, and edit three rolls of film. Ten hours.


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Global burden foodborne diseases - from data to action
The World Health Organization (WHO) launched an initiative to generate the first ever estimates of the global burden of foodborne disease and established an external advisory group—the Foodborne Disease Burden Epidemiology Reference Group (FERG)—to take this forward. The first ever WHO estimates of the global burden of foodborne diseases, developed by FERG, were published in December 2015. They provide estimates by age, sex, and region for foodborne diseases caused by a defined list of microbes, parasites and chemicals.
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The FERG symposium 'The Global burden of foodborne diseases' was held in Amsterdam, The Netherlands, on 15-16 December 2015. This FERG symposium was organized by the World Health Organization and the Local Organizing Committee at RIVM, supported by the Scientific Committee.
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Lawrence Man Engineers a Future
Submitted by review on May 23, 2012 – 4:43 pm
Herinell Linares
Herinell Linares has an impressive aptitude for math that he never knew he had until he enrolled at Northern Essex Community College.
In May, the 23-year-old Lawrence resident graduated from NECC with an associate degree in Engineering Science. In the fall he begins classes toward his Bachelor of Science in Electrical Engineering at UMass Lowell.
Born in the Dominican Republic, Herinell immigrated to the United States when he was 17. He arrived with a high school diploma and no grasp of the English language. Eager to learn English, he followed in his mother's footsteps and enrolled in English as a Second Language classes at NECC in the summer of 2007.
By the time he was seated in a college-level math course, the professor noticed he was getting great grades with very little effort. Recognizing his potential, the professor encouraged him to think about pursuing an engineering degree.
"I signed up for a digital logic class," he says. "After I took that class I knew I wanted to go into engineering."
By the spring of 2010 Herinell was taking classes toward his associate degree in Engineering Science; all the while working nearly full time as a security guard at an apartment complex in Lawrence and occasionally as a banquet server for an area hotel. Herinell relied on the NECC shuttle to travel between Lawrence and Haverhill each day.
After earning his bachelor's degree, Herinell hopes to find an engineering position with a large firm.
He says NECC launched his education and it can help others as well. "Northern Essex is a great place to start," he said. "The classes are small and the professors get to know their students. You are not just an ID number here. "
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text:
\section{Introduction}
Knowledge of  $D_{s}^{\pm}$  decay properties   are vital input for studies of
the $B^0_s$  hadron, whose decay channels are dominated by the final states
involving $D_{s}^{\pm}$ mesons~\cite{PDG}. Furthermore, hadronic $D_{s}^{\pm}$
decays probe the interplay of short-distance weak-decay matrix elements and
long-distance QCD interactions, and the measured branching fractions (BFs)
provide valuable information concerning the amplitudes and phases that the
strong force induces in the decay process~\cite{PRD79-034016, PRD81-074021, PRD84-074019}.
The singly Cabibbo-suppressed (SCS) decay $D_{s}^{+} \to K^{0}\pi^{+}\pi^{0}$
has a large BF of the order of $10^{-2}$~\cite{PDG}. This decay, therefore, is
often used as a reference channel for the other decays of $D_{s}^{\pm}$ mesons.
Accurate knowledge of its substructure is essential to reduce the systematic
uncertainties in those analyses using this channel. To date, there have been
few measurements of charge-parity asymmetries $A_{\rm CP}$ in SCS $D_{s}^{\pm}$
decay modes in general and none for the mode discussed here.

An amplitude analysis of the $D^+_s$ decay to a three-body pseudoscalar meson
final state is a powerful tool for studying the vector-pseudoscalar channels of
the SCS $D^+_s$ decay. Table~\ref{tab:BF_prediction} shows the current measured
values and theoretical predictions, in various models, for the BFs of
$D^+_s\to K_{S}^{0}\rho^+$, $K^{*}(892)^{0}\pi^+$, and $K^{*}(892)^{+}\pi^0$
($\rho^+$ denotes $\rho(770)^+$ throughout this paper).
References~\cite{YLWu} and \cite{HYCheng} took into account quark flavor SU(3)
symmetry and its breaking effects. Reference~\cite{PRD84-074019} used a
generalized factorization method considering the resonance effects in the pole
model for the annihilation contributions and introducing large strong phases
between different topological diagrams. More precise  experimental results are
required to validate or falsify  these theoretical predictions.

The CLEO collaboration has reported a measurement of
$\mathcal{B}(D_{s}^{+} \to K^{0}\pi^{+}\pi^{0}) = (1.00\pm 0.18)\%$~\cite{CLEO-BF},
using 600~pb$^{-1}$ of $e^+e^-$ collisions recorded at a
center-of-mass energy~($\sqrt{s}$) of 4.17~GeV. In this paper, by using
6.32~$\rm fb^{-1}$ of  data  collected with the BESIII detector  at
$\sqrt{s}=4.178$-$4.226$~GeV, we perform the first amplitude analysis of
$D^+_s\to K^0_S\pi^+\pi^0$ and improve the measurement of its absolute BF.
\begin{table*}[htp]
 \renewcommand\arraystretch{1.25}
\begin{center}
  \caption{Summary of $D_s^+$ decays to a vector and pseudoscalar meson,
    showing the measured BFs and theoretical predictions from various models ($\times 10^{-3}$).}
\begin{tabular}{lcccc}
\hline
Channel                               & PDG~\cite{PDG}     & Y.L.~Wu {\it et~al.}~\cite{YLWu} & H.Y.~Cheng {\it et~al.}~\cite{HYCheng}& F.S.~Yu {\it et~al.}~\cite{PRD84-074019}\\
\hline
$K^{0}\rho^{+}$         & --                 & $9.1 \pm 7.7$                     & $11.47 \pm 0.48$                       & $7.5 \pm 2.1$                \\
$K^{*}(892)^{0}\pi^{+}$ & $2.13 \pm 0.36$    & $3.3 \pm 3.5$                     & $\phantom{0}3.65 \pm 0.24$             & $1.5 \pm 0.7$                \\
$K^{*}(892)^{+}\pi^{0}$ & --                 & $1.3 \pm 1.3$                     & $\phantom{0}1.02 \pm 0.07$             & $0.1 \pm 0.1$                \\
\hline
\end{tabular}
\label{tab:BF_prediction}
\end{center}
\end{table*}

\section{Detector and data sets}
\label{sec:detector_dataset}
The BESIII detector is a magnetic spectrometer~\cite{Ablikim:2009aa, Ablikim:2019hff}
located at the Beijing Electron Positron Collider~(BEPCII)~\cite{Yu:IPAC2016-TUYA01}.
The cylindrical core of the BESIII detector consists of a helium-based
multilayer drift chamber~(MDC), a plastic scintillator time-of-flight
system~(TOF), and a CsI(Tl) electromagnetic calorimeter~(EMC), which are all
enclosed in a superconducting solenoidal magnet providing a 1.0~T magnetic
field. The solenoid is supported by an octagonal flux-return yoke with
resistive plate counter muon identifier modules interleaved with steel. The
acceptance of charged particles and photons is 93\% over a $4\pi$ solid angle.
The charged-particle momenta resolution at 1.0~GeV/$c$ is $0.5\%$, and the
specific energy loss~(d$E$/d$x$) resolution is $6\%$ for the electrons from
Bhabha scattering. The EMC measures photon energies with a resolution of
$2.5\%$~($5\%$) at $1$~GeV in the barrel (end-cap) region. The time resolution
of the TOF barrel part is 68~ps, while that of the end-cap part is 110~ps. The
end-cap TOF was  upgraded in 2015 with multi-gap resistive plate chamber
technology, providing a time resolution of 60~ps

The data samples used in this analysis are listed in Table~\ref{energe}.
For some aspects of the analysis, these samples are organized into three
sample groups, 4.178~GeV, 4.189-4.219~GeV, and 4.226~GeV, that were acquired
during the same year under consistent running conditions. Since the cross
section of $D_{s}^{*\pm}D_{s}^{\mp}$ production in $e^{+}e^{-}$ annihilation is
about a factor of twenty larger than that of $D_{s}^{+}D_{s}^{-}$~\cite{DsStrDs},
and the $D_{s}^{*\pm}$ meson decays to $\gamma D_{s}^{\pm}$ with a dominant BF
of $(93.5\pm0.7)$\%~\cite{PDG}, the signal events discussed in this paper are
selected from the process $e^+e^-\to D_{s}^{*\pm}D_{s}^{\mp}\to \gamma D_{s}^{+}D_{s}^{-}$.

 \begin{table}[htb]
 \renewcommand\arraystretch{1.25}
 \centering
 \caption{The integrated luminosities ($\mathcal{L}_{\rm int}$) and the
   requirements on $M_{\rm rec}$ for various collision energies. The definition
   of $M_{\rm rec}$ is given in Eq.~(\ref{eq:mrec}). The first and second
   uncertainties are statistical and systematic, respectively.}
 \begin{tabular}{ccc}
 \hline
 $\sqrt{s}$ (GeV) & $\mathcal{L}_{\rm int}$ (pb$^{-1}$) & $M_{\rm rec}$ (GeV/$c^2$)\\
 \hline
  4.178 &3189.0$\pm$0.2$\pm$31.9&[2.050, 2.180] \\
  4.189 &526.7$\pm$0.1$\pm$2.2&[2.048, 2.190] \\
  4.199 &526.0$\pm$0.1$\pm$2.1&[2.046, 2.200] \\
  4.209 &517.1$\pm$0.1$\pm$1.8&[2.044, 2.210] \\
  4.219 &514.6$\pm$0.1$\pm$1.8&[2.042, 2.220] \\
  4.226 &1047.3$\pm$0.1$\pm10.2$&[2.040, 2.220] \\
  \hline
 \end{tabular}
 \label{energe}
\end{table}

Simulated samples are produced with the {\sc geant4}-based~\cite{GEANT4}
Monte Carlo~(MC) package, which includes the geometric description of the BESIII
detector and the detector response. These samples are used to determine the detection
efficiency and to estimate the background. The simulation includes the beam-energy
spread and initial-state radiation~(ISR) in  $e^+e^-$ annihilations
modeled with the generator {\sc kkmc}~\cite{KKMC}. The generic MC samples
consist of the production of $D\bar{D}$ pairs with consideration of quantum
coherence for all neutral $D$ modes, the non-$D\bar{D}$ decays of the $\psi(3770)$,
the ISR production of the $J/\psi$ and $\psi(3686)$ states, and the continuum
processes. The known decay modes are modeled with {\sc evtgen}~\cite{EVTGEN1, EVTGEN2}
using the BFs taken from the Particle Data Group~(PDG)~\cite{PDG}, and the
remaining unknown decays from the charmonium states with
{\sc lundcharm}~\cite{LUNDCHARM1, LUNDCHARM2}. Final-state radiation from charged particles is
incorporated with the {\sc photos}~\cite{PHOTOS} package.

\section{Event selection}
\label{ST-selection}
The data samples were collected just above the $D_s^{*\pm}D_s^{\mp}$ threshold.
The tag method allows clean signal samples to be selected, which provide an
opportunity to perform amplitude analyses and to measure the absolute BFs of
the hadronic $D^+_s$ meson decays. In the tag method, a single-tag~(ST)
candidate requires only one of the $D_{s}^{\pm}$ mesons to be reconstructed via
a hadronic decay; a double-tag~(DT) candidate has both $D_s^+D_s^-$ mesons
reconstructed via hadronic decays. The DT candidates are required to have the
$D_{s}^{+}$ meson decaying to the signal mode
$D_{s}^{+} \to K^0_{S}\pi^{+}\pi^{0}$ and the $D_{s}^{-}$ meson decaying to a tag
mode. (Charge conjugation is implied throughout this paper.) Nine tag modes are
reconstructed and the corresponding mass windows on the tagging $D_{s}^{-}$
mass~($M_{\rm tag}$) are listed in Table~\ref{tab:tag-cut}. The $D_{s}^{\pm}$
candidates are constructed from individual $\pi^\pm$, $K^\pm$, $\eta$,
$\eta^{\prime}$, $K_{S}^{0}$ and $\pi^{0}$ particles.

\begin{table}[htbp]
 \renewcommand\arraystretch{1.25}
 \centering
\caption{Requirements on $M_{\rm tag}$ for various tag modes, where the
  $\eta$ and $\eta^\prime$ subscripts denote the decay modes used to reconstruct
  these particles.}\label{tab:tag-cut}
     \begin{tabular}{lc}
        \hline
        Tag mode                                     & Mass window (GeV/$c^{2}$) \\
        \hline
        $D_{s}^{-} \to K_{S}^{0}K^{-}$               & [1.948, 1.991]            \\
        $D_{s}^{-} \to K^{+}K^{-}\pi^{-}$            & [1.950, 1.986]            \\
        $D_{s}^{-} \to K_{S}^{0}K^{+}\pi^{0}$        & [1.946, 1.987]            \\
        $D_{s}^{-} \to K^{+}K^{-}\pi^{-}\pi^{0}$     & [1.947, 1.982]            \\
        $D_{s}^{-} \to K_{S}^{0}K^{+}\pi^{-}\pi^{-}$ & [1.953, 1.983]            \\
        $D_{s}^{-} \to \pi^{-}\pi^{-}\pi^{+}$        & [1.952, 1.982]            \\
        $D_{s}^{-} \to \pi^{-}\eta_{\gamma\gamma}$   & [1.930, 2.000]            \\
        $D_{s}^{-} \to \pi^{-}\pi^{0}\eta_{\gamma\gamma}$   & [1.920, 2.000]     \\
        $D_{s}^{-} \to \pi^{-}\eta_{\pi^{+}\pi^{-}\eta_{\gamma\gamma}}^{\prime}$
                                                     & [1.940, 1.996]            \\
        \hline
      \end{tabular}
\end{table}

Charged track candidates from the MDC must satisfy $|\!\cos\theta|<0.93$,
where $\theta$ is the polar angle with respect to the direction of the positron
beam. The closest approach to the interaction point is required to be less
than 10~cm along the beam direction and less than 1 cm in the plane perpendicular
to the beam. Particle identification (PID) of charged particles is implemented
by combining the d$E$/d$x$ information in the MDC and the time-of-flight information
from the TOF system. For charged kaon (pion) candidates, the probability for the
kaon (pion) hypothesis is required to be larger than that for the pion (kaon)
hypothesis.

The $K_{S}^{0}$ mesons are reconstructed with pairs of two oppositely
charged tracks, which satisfy $|\!\cos\theta|< 0.93$ and the distances of
closest approach along the beam direction must be less than 20~cm.
The decay length of the reconstructed $K_{S}^{0}$ in the signal side decay is
required to be more than twice that of the vertex resolution away from the
interaction point. The invariant masses of these charged track pairs are
required to be in the range $[0.487, 0.511]$~GeV/$c^{2}$.

Photons are reconstructed from the clusters of deposited energy in the EMC. The shower time
is required to be within $[0, 700]$~ns of the event start time in order to
suppress electronics noise or $e^+e^-$ beam background. Photon candidates within
$|\!\cos\theta| < 0.80$ (barrel) are required to have an energy deposition
larger than $25$~MeV and those with $0.86<|\!\cos\theta|<0.92$ (end-cap) must
have an energy deposition larger than $50$~MeV. To suppress the noise from
hadronic shower splitoffs, the calorimeter positions of photon candidates must
lie outside a cone of $10\degree$ from all charged tracks.
The $\pi^0$ $(\eta)$ candidates are reconstructed through
$\pi^0\to \gamma\gamma$ ($\eta \to \gamma\gamma$) decays,
with at least one barrel photon. The invariant mass of the photon pair for
$\pi^{0}$ and $\eta$ candidates must be in the ranges $[0.115, 0.150]$~GeV/$c^{2}$
and $[0.490, 0.580]$~GeV/$c^{2}$, respectively.
A kinematic fit that constrains the $\gamma\gamma$ invariant mass to the $\pi^{0}$
or $\eta$ nominal mass~\cite{PDG} is performed to improve the mass resolution.
The $\chi^2$ of the kinematic fit is required to be less than 30.
The $\eta^{\prime}$ candidates are formed from the $\pi^{+}\pi^{-}\eta$
combinations with an invariant mass within a range of
$[0.946, 0.970]$~GeV/$c^{2}$.

$D_{s}^{\pm}$ candidates with $M_{\rm rec}$ lying with the mass windows listed in Table~\ref{energe} are retained for further
study. The quantity $M_{\rm rec}$ is defined as
\begin{eqnarray}
\begin{aligned}
    M_{\rm rec} = \sqrt{(E_{\rm cm} - \sqrt{|\vec{p}_{D_{s}}|^{2}+m_{D_{s}}^{2}})^{2} - |\vec{p}_{D_{s}} | ^{2}} \; , \label{eq:mrec}
\end{aligned}\end{eqnarray}
where $E_{\rm cm}$ is the initial energy of the $e^+e^-$ center-of-mass system,
$\vec{p}_{D_{s}}$ is the three-momentum of the $D_{s}^{\pm}$ candidate in the
$e^+e^-$ center-of-mass frame, and $m_{D_{s}}$ is the $D_{s}^{\pm}$ nominal mass~\cite{PDG}.

A ``$K_{S}^{0}K$'' veto and a ``$D^{0}$'' veto are applied on the signal
$D_{s}^{+}$ candidates. The Cabibbo-favored $D_{s}^{+} \to K_{S}^{0}K^{+}$
decay contributes background when the $K^{+}$ is misreconstructed as a $\pi^{+}$.
This background is reduced by a veto on the signal $D_{s}^{+}$, with
$M_{K_{S}^{0}K^{+}}-m_{D_{s}}< -20$~MeV/$c^{2}$, where $M_{K_{S}^{0}K^{+}}$ is the
invariant mass of the $K_{S}^{0}$ and reconstructed $\pi^{+}$ track, when it is assumed to
be a kaon.
There is also track-swap background where $D^{0} \to K_{S}^{0}K^{-}\pi^{+}$
versus~$\bar{D}^{0} \to K^{+}\pi^{-}\pi^{0}$  fake
$D_{s}^{+} \to K_{S}^{0}\pi^+\pi^0$ versus~$D_{s}^{-} \to K^{+}K^{-}\pi^-$ events through the
exchange of $K_{S}^{0}$ and $K^{+}$, or $\pi^{0}$ and $K^{-}$ tracks. Events which
simultaneously satisfy $|M_{K_{S}^{0}K^{-}\pi^{+}} -m_{D^{0}}|<30$~MeV/$c^{2}$ and
$|M_{K^{+}\pi^{-}\pi^{0}}-m_{D^{0}}|<30$~MeV/$c^{2}$ are rejected, where
$M_{K_{S}^{0}K^{-}\pi^{+}}$ ($M_{K^{+}\pi^{-}\pi^{0}}$) is the invariant mass
of the $K_{S}^{0}K^{-}\pi^{+}$ ($K^{+}\pi^{-}\pi^{0}$) combination and $m_{D^{0}}$ is the $D^{0}$
nominal mass~\cite{PDG}.

\section{Amplitude analysis of $D_{s}^{+} \to K^0_{S}\pi^{+}\pi^{0}$}
\label{Amplitude-Analysis}
\subsection{Event selection}
\label{AASelection}
An eight-constraint kinematic fit is performed assuming the process
$e^+e^-\to D_{s}^{*\pm}D_{s}^{\mp}\to \gamma D_{s}^{+}D_{s}^{-}$,
with $D_{s}^{-}$ decaying to one of the tag modes and $D_{s}^{+}$ decaying to the
signal mode. The combination with the  minimum $\chi^2$ is chosen, assuming that a $D_s^{*+}$
meson decays to $D_{s}^{+}\gamma$ or a $D_s^{*-}$ meson decays to
$D_{s}^{-}\gamma$. In addition to the constraints of four-momentum
conservation in the $e^+e^-$ center-of-mass system, the invariant masses of $(\gamma\gamma)_{\pi^{0}}$,
$(\pi^{+}\pi^{-})_{K_S^0}$, tag $D_{s}^{-}$, and $D_{s}^{*\pm}$ candidates are
constrained to the corresponding nominal masses~\cite{PDG}.
In order to ensure that all candidates fall within the phase-space boundary,
the constraint of the signal $D_s^{+}$ mass is added to the kinematic fit
and the updated four-momenta are used for the amplitude analysis.

Furthermore, it is required that the energy of the transition photon from
$D_s^{*\pm}\to \gamma D_s^{\pm}$ is smaller than 0.18~GeV and the
mass recoiling against this photon and the signal $D_s^{+}$ candidate lies
within the range $[1.952, 1.995]$~GeV/$c^2$. Finally, a mass window,
$[1.930, 1.990]$~GeV/$c^2$, is applied on the signal $D_s^{+}$ candidates.
Fig.~\ref{fig:fit_Ds} shows the fits to the invariant-mass distributions of
the accepted signal $D_s^{+}$ candidates, $M_{\rm sig}$, for various data
samples. In the fits, the signal is described by an MC-simulated shape
convolved with a Gaussian resolution function, and the background is described by a
second-order Chebyshev function.
There are $352$, $193$, and $64$ events retained for
the amplitude analysis with purities, $w_{\rm sig}$, of $88.9\pm6.8\%$, $84.6\pm8.3\%$, and
$75.9\pm14.3\%$ for the data samples at $\sqrt{s}=4.178$~GeV, 4.189-4.219~GeV, and 4.226~GeV,
respectively.

\begin{figure*}[!htbp]
  \centering
  \includegraphics[width=0.32\textwidth]{f1_a.eps}
  \includegraphics[width=0.32\textwidth]{f1_b.eps}
  \includegraphics[width=0.32\textwidth]{f1_c.eps}
  \caption{
    Fits to the $M_{\rm sig}$ distributions of the data samples at $\sqrt{s}=$ (a) 4.178~GeV,
    (b) 4.189-4.219~GeV, and (c) 4.226~GeV. The black points with error
    bars are data. The blue solid lines are the total fits. The red dotted and the black
    dashed lines are the fitted signal and background, respectively. The pairs of red
    arrows indicate the signal regions.
  } \label{fig:fit_Ds}
\end{figure*}

\subsection{Fit method}
The intermediate-resonance composition in the decay $D_{s}^{+}\to K_{S}^{0}\pi^+\pi^0$
is determined by an unbinned maximum-likelihood fit to data. The likelihood
function is constructed with a probability density function (PDF), which depends
on the momenta of the three daughter particles. The amplitude of the $n$th
intermediate state~($A_{n}$) is
\begin{eqnarray}
\begin{aligned}
  A_{n} = P_{n}S_{n}F_{n}^{r}F_{n}^{D}\,, \label{base-amplitude}
\end{aligned}\end{eqnarray}
where $S_{n}$ and $F_{n}^{r(D)}$ are the spin factor and the Blatt-Weisskopf
barriers of the intermediate state (the $D_{s}^{\pm}$ meson), respectively, and
$P_{n}$ is the propagator of the intermediate resonance.

The total amplitude $M$ is then the coherent sum of the amplitudes of
intermediate processes, $M=\begin{matrix}\sum c_{n}A_{n}\end{matrix}$,
where $c_{n}=\rho_{n}e^{i\phi_{n}}$ is the corresponding complex coefficient.
The magnitude $\rho_{n}$ and phase $\phi_{n}$ are free parameters in the fit,
and are defined relative to those of a reference mode, for which they are fixed.
The signal PDF $f_{S}(p_{j})$ is given by
\begin{eqnarray}\begin{aligned}
  f_{S}(p_{j}) = \frac{\epsilon(p_{j})\left|M(p_{j})\right|^{2}R_{3}(p_{j})}{\int \epsilon(p_{j})\left|M(p_{j})\right|^{2}R_{3}(p_{j})\,dp_{j}}\,, \label{signal-PDF}
\end{aligned}\end{eqnarray}
where $\epsilon(p_{j})$ is the detection efficiency parameterized in terms of
the final four-momenta $p_{j}$. The index $j$ refers to the different particles
in the final states, and $R_{3}(p_{j})$ is the standard element of three-body
phase space. The normalization integral is determined by an MC integration,
\begin{eqnarray}\begin{aligned}
  \int \epsilon(p_{j})\left|M(p_{j})\right|^{2}R_{3}(p_{j})\,dp_{j} \approx
\frac{1}{N_{M}}\sum_{k}^{N_{M}} \frac{\left|M(p_{j}^{k})\right|^{2}}{\left|M^{g}(p_{j}^{k})\right|^{2}}\,, \label{MC-intergral}
\end{aligned}\end{eqnarray}
where $k$ is the index of the $k$th event and $N_{M}$ is the number of the
selected MC events. Here $M^{g}(p_{j})$ is the PDF used to generate the MC
samples in MC integration. To account for any bias caused by differences in
PID and tracking  efficiency  between data and MC simulation, each signal MC event is
weighted with a ratio, $\gamma_{\epsilon}(p)$, of the efficiency of data to
that of MC simulation and the MC integration then becomes
\begin{eqnarray}\begin{aligned}
    &\int \epsilon(p_{j})\left|M(p_{j})\right|^{2}R_{3}(p_{j})\,dp_{j} \approx
&\frac{1}{N_{M}} \sum_{k}^{N_{M}} \frac{\left|M(p_{j}^{k})\right|^{2}\gamma_{\epsilon}(p_{j}^{k})}{\left|M^{g}(p_{j}^{k})\right|^{2}}\,.
\label{MC-intergral-corrected}
\end{aligned}\end{eqnarray}
Finally, the log-likelihood is written as
\begin{eqnarray}\begin{aligned}
  \ln{\mathcal{L}} = \sum_{k}^{N_{D}} \ln f_{S}(p_{j}^{k})\,,  \label{likelihood}
\end{aligned}\end{eqnarray}
where $N_{D}$ is the number of candidate events in data.

Equation~(\ref{likelihood}), however,  is only appropriate for a data sample with
negligible background. A signal-background combined PDF is introduced to
account for the approximate 15\% of background in this analysis. The background PDF
is given by
\begin{eqnarray}\begin{aligned}
  f_{B}(p_{j}) = \frac{B(p_{j})R_{3}(p_{j})}{\int B(p_{j})R_{3}(p_{j})\,dp_{j}}\,.\label{bkg-PDF}
\end{aligned}\end{eqnarray}
The background shape $B(p_{j})$ is derived from about 3500 background
events selected from generic MC samples using RooNDKeysPdf~\cite{Verkerke} with the Dalitz
distribution of $M^2_{K_S^0\pi^+}$ versus~$M^2_{K_S^0\pi^0}$ as input.
This background PDF is added to the signal PDF incoherently. Then, the combined
PDF is written as
\begin{eqnarray}
\begin{aligned}
  w_{\rm sig}f_{S}(p_{j})&+(1-w_{\rm sig})f_{B}(p_{j})\\
  &=w_{\rm sig}\frac{\epsilon(p_{j})\left|M(p_{j})\right|^{2}R_{3}(p_{j})}{\int \epsilon(p_{j})\left|M(p_{j})\right|^{2}R_{3}(p_{j})\,dp_{j}}\\
  &+(1-w_{\rm sig})\frac{B(p_{j})R_{3}(p_{j})}{\int B(p_{j})R_{3}(p_{j})\,dp_{j}}\,. \label{combined-PDF}
\end{aligned}
\end{eqnarray}
Thus, $\epsilon(p_{j})$ is factorized from the background PDF to make it a
comparable equation with the signal PDF. In this way, the $\epsilon(p_{j})$ term,
which is independent of the fitted variables, is regarded as a constant and can be
dropped during the log-likelihood fit. The background shape is extracted from the
selected events, hence one has to manually divide the background PDF by the
efficiency, $B_{\epsilon}\equiv B/\epsilon$. As a consequence, the combined PDF
becomes
\begin{eqnarray}
\begin{aligned}
  w_{\rm sig}f_{S}(p_{j})+(1-w_{\rm sig})&f_{B}(p_{j})\\
  =\epsilon(p_{j})R_{3}(p_{j})&\left[\frac{w_{\rm sig}\left|M(p_{j})\right|^{2}}{\int \epsilon(p_{j})\left|M(p_{j})\right|^{2}R_{3}(p_{j})\,dp_{j}}\right.\\
  &+\left.\frac{(1-w_{\rm sig})B_{\epsilon}(p_{j})}{\int \epsilon(p_{j})B_{\epsilon}(p_{j})R_{3}(p_{j})\,dp_{j}}
\right]\,. \label{combined-PDF-2}
\end{aligned}
\end{eqnarray}
Next, the integration in the denominator of the background term can also be
handled by the MC integration method in the same way as for the signal only sample:
\begin{eqnarray}\begin{aligned}
  \int \epsilon(p_{j})B_{\epsilon}(p_{j})R_{3}(p_{j})\,dp_{j} \approx
\frac{1}{N_{M}}\sum_{k}^{N_{M}} \frac{B_{\epsilon}(p_{j}^{k})}{\left|M^{g}(p_{j}^{k})\right|^{2}}\,.
\end{aligned}\end{eqnarray}
Eventually, for the case with a  non-negligible background, Eq.~(\ref{likelihood}) is extended
to become
\begin{eqnarray}\begin{aligned}
  \ln{\mathcal{L}} = \sum_{i=1}^{3}\sum_{k}^{N_{D, i}} \ln\left[w_{\rm sig}^{i}f_{S}(p_{j}^{k})+(1-w_{\rm sig}^{i})f_{B}(p_{j})\right]\,,  \label{likelihood3}
\end{aligned}\end{eqnarray}
where $i$ indicate the data sample and $w_{\rm sig}^{i}$ are fixed during
the log-likelihood fit.

\subsubsection{Blatt-Weisskopf barrier factors}
For the process $a \to bc$, the Blatt-Weisskopf barrier $F_L(p_j)$ is
parameterized as a function of the angular momenta $L$ and the momenta $q$ of
the daughter $b$ or $c$ in the rest system of $a$,
\begin{eqnarray}
\begin{aligned}
 F_{L=0}(q)&=1,\\
 F_{L=1}(q)&=\sqrt{\frac{z_0^2+1}{z^2+1}},\\
 F_{L=2}(q)&=\sqrt{\frac{z_0^4+3z_0^2+9}{z^4+3z^2+9}}\,,
\end{aligned}
\end{eqnarray}
where $z=qR$ and $z_0=q_0R$. The effective radius of the barrier $R$ is fixed to
3.0~GeV$^{-1}$ for the intermediate resonances and 5.0~GeV$^{-1}$ for the $D_s^+$
meson.

\subsubsection{Propagator}
The intermediate resonances $K^{*}(892)^{0, +}$ and $K^{*}(1410)^{0}$
are parameterized as relativistic Breit-Wigner
functions,
\begin{eqnarray}\begin{aligned}
  \begin{array}{lr}
    P = \frac{1}{(m_{0}^{2} - s_{a} ) - im_{0}\Gamma(m)}\,, &\\
    \Gamma(m) = \Gamma_{0}\left(\frac{q}{q_{0}}\right)^{2L+1}\left(\frac{m_{0}}{m}\right)\left(\frac{F_{L}(q)}{F_{L}(q_{0})}\right)^{2}\,, &
  \end{array}\label{RBW}
\end{aligned}\end{eqnarray}
where $s_{a}$ denotes the invariant-mass squared of the parent particle;
$m_{0}$ and $\Gamma_{0}$ are the rest masses and the widths of the intermediate
resonances, respectively, and are fixed to the PDG values~\cite{PDG}.

We parameterize the $\rho^{+}$ and $\rho(1450)^+$ resonances by the Gounaris-Sakurai
lineshape~\cite{PhysRevLett.21.244}, which is given by
\begin{eqnarray}\begin{aligned}
P_{\rm GS}(m)=\frac{1+d\frac{\Gamma_0}{m_0}}{(m_0^2-m^2)+f(m)-im_0\Gamma(m)}\,.
\end{aligned}\end{eqnarray}
The function $f(m)$ is given by
\begin{eqnarray}
\begin{aligned}
f(m)&=\Gamma_0\frac{m_0^2}{q_0^3}\left[q^2(h(m)-h(m_0))\phantom{\left.\frac{d h}{d(m^2)}\right|_{m_0^2}}\right.\\
    &+\left.(m_0^2-m^2)q_0^2\left.\frac{dh}{d(m^2)}\right|_{m_0^2}\right]\,,
\end{aligned}
\end{eqnarray}
where
\begin{eqnarray}\begin{aligned}
h(m)=\frac{2q}{\pi m}\ln\left(\frac{m+2q}{2m_{\pi}}\right)\,,
\end{aligned}\end{eqnarray}
and
\begin{eqnarray}\begin{aligned}
&\left.\frac{dh}{d(m^2)}\right|_{m_0^2}=
&h(m_0)\left[(8q_0^2)^{-1}-(2m_0^2)^{-1}\right]+(2\pi m_0^2)^{-1}\,.
\end{aligned}\end{eqnarray}
The normalization condition at $P_{\rm GS}(0)$ fixes the parameter
$d=f(0)/(\Gamma_0 m_0)$. It is found to be
\begin{eqnarray}\begin{aligned}
d=\frac{3m^2_\pi}{\pi q_0^2}\ln\left(\frac{m_0+2q_0}{2m_\pi}\right)+\frac{m_0}{2\pi q_0}-\frac{m^2_\pi m_0}{\pi q^3_0}\,.
\end{aligned}\end{eqnarray}

\subsubsection{Spin factors}
The spin-projection operators are defined as~\cite{covariant-tensors}
\begin{eqnarray}
\begin{aligned}
  P^{(1)}_{\mu\mu^{\prime}}(a) &= -g_{\mu\mu^{\prime}}+\frac{p_{a,\mu}p_{a,\mu^{\prime}}}{p_{a}^{2}}\,,\\
  P^{(2)}_{\mu\nu\mu^{\prime}\nu^{\prime}}(a) &= \frac{1}{2}(P^{(1)}_{\mu\mu^{\prime}}(a)P^{(1)}_{\nu\nu^{\prime}}(a)+P^{(1)}_{\mu\nu^{\prime}}(a)P^{(1)}_{\nu\mu^{\prime}}(a))\\
  &-\frac{1}{3}P^{(1)}_{\mu\nu}(a)P^{(1)}_{\mu^{\prime}\nu^{\prime}}(a)\,.
 \label{spin-projection-operators}
\end{aligned}
\end{eqnarray}
The quantities $p_a$, $p_b$, and $p_c$ are the momenta of particles $a$,
$b$, and $c$, respectively, and $r_a = p_b-p_c$.
The covariant tensors are given by
\begin{eqnarray}
\begin{aligned}
    \tilde{t}^{(1)}_{\mu}(a) &= -P^{(1)}_{\mu\mu^{\prime}}(a)r^{\mu^{\prime}}_{a}\,,\\
    \tilde{t}^{(2)}_{\mu\nu}(a) &= P^{(2)}_{\mu\nu\mu^{\prime}\nu^{\prime}}(a)r^{\mu{\prime}}_{a}r^{\nu^{\prime}}_{a}\,.\\
\label{covariant-tensors}
\end{aligned}
\end{eqnarray}
The spin factors for $S$, $P$, and $D$ wave decays are
\begin{eqnarray}
\begin{aligned}
    S &= 1\,, &(S\ \text{wave}), &\\
    S &= \tilde{T}^{(1)\mu}(D_{s}^{\pm})\tilde{t}^{(1)}_{\mu}(a)\,,         &(P\ \text{wave}),\\
    S &= \tilde{T}^{(2)\mu\nu}(D_{s}^{\pm})\tilde{t}^{(2)}_{\mu\nu}(a)\,,         &(D\ \text{wave}),
\label{spin-factor}
\end{aligned}
\end{eqnarray}
where the $\tilde{T}^{(l)}$ factors have the same definition as $\tilde{t}^{(l)}$. The
tensor describing the $D_{s}^{+}$ decay is denoted by $\tilde{T}$ and
that of the $a$ decay is denoted by $\tilde{t}$.

\subsection{Fit results}
The Dalitz plot of $M^{2}_{K_S^0\pi^0}$ versus~$M^{2}_{K_S^0\pi^{+}}$ summed over all the
data samples is shown in Fig.~\ref{dalitz}. We can see an anti-diagonal band
corresponding to $K_{S}^{0}\rho^+$. In the fit, the magnitude and phase of the
reference amplitude $D_{s}^{+} \to K_S^0\rho^+$ are fixed to 1.0 and 0.0,
respectively, and the masses and widths of all resonances are fixed to the
corresponding PDG averages~\cite{PDG}. In addition to the dominating amplitude
$D_{s}^{+} \to K_S^0\rho^+$, we have tested for the contribution of all possible intermediate
resonances including $K^{*}(892)^{0}$, $K^{*}(892)^{+}$, $K^*(1410)$,
$K^*_0(1430)$, $K^*_2(1430)$, $\rho(1450)$, $K^*(1680)$, $\rho(1700)$, etc.
We find that $D^+_s\to K_{S}^{0}\rho(1450)^{+}$,
$D^+_s\to K^{*+}(892)\pi^0$, and $K^{*}(1410)^0\pi^+$ have a statistical
significances greater than three standard deviations and retain these amplitudes in the final model.

\begin{figure}[htbp]
  \centering
  \mbox{
    \begin{overpic}[width=0.5\textwidth]{plots_dalitz.eps}
    \end{overpic}
  }
  \caption{The Dalitz plot of the DT candidates from the data samples at
    $\sqrt{s}= 4.178$-$4.226$~GeV.}
  \label{dalitz}
\end{figure}

The calculation of the fit fractions (FFs) for individual amplitudes, involves the
phase-space MC truth information without  detector acceptance or resolution effects.
The FF for the $n$th amplitude is defined as
\begin{eqnarray}\begin{aligned}
  {\rm FF}_{n} = \frac{\sum^{N_{\rm gen}} \left|c_{n}A_{n}\right|^{2}}{\sum^{N_{\rm gen}} \left|M\right|^{2}}\,, \label{Fit-Fraction-Definition}
\end{aligned}\end{eqnarray}
where $N_{\rm gen}$ is the number of phase-space MC events at generator
level. These FFs will not sum to unity if there is net constructive or
destructive interference. Interference  IN between the $n$th and the
$n^{\prime}$th amplitudes is defined as (for $n<n^{\prime}$ only)
\begin{eqnarray}\begin{aligned}
  {\rm IN}_{nn^{\prime}} = \frac{\sum^{N_{\rm gen}} 2Re[c_{n}c^{*}_{n^{\prime}}A_{n}A^{*}_{n^{\prime}}]}{\sum^{N_{\rm gen}} \left|M\right|^{2}}\,. \label{interferenceFF-Definition}
\end{aligned}\end{eqnarray}
In order to determine the statistical uncertainties of FFs the amplitude
coefficients are
randomly sampled by a Gaussian-distributed amount set by the fit
uncertainty and the covariance matrix. Then the distribution of each FF is
fitted with a Gaussian function and the width of the Gaussian function is
defined as the uncertainty of the FF.

The magnitudes, phases, FFs, and significances for the amplitudes are listed in
Table~\ref{fit-result}. The interference between amplitudes is listed in
Table~\ref{fit-interference}. The Dalitz plot projections are shown in
Fig.~\ref{dalitz-projection}. The assignment of  systematic uncertainties is
discussed in next section.

\begin{table*}[htbp]
    \caption{Magnitudes, phases, FFs, and significances for the amplitudes.
      The uncertainties in the magnitudes are statistical only. The first and the
      second uncertainties in the phases and FFs are statistical and systematic,
      respectively. The total FF is 86.9$\%$.}
    \label{fit-result}
    \begin{center}
    \begin{tabular}{lcccc}
      \hline
      Amplitude                               & Magnitude ($\rho_n$) & Phase ($\phi_n$)         & FF~(\%)                           &Significance~($\sigma$)\\
      \hline
      $D_{s}^{+} \to K_{S}^{0}\rho^{+}$       & 1.0(fixed)           & 0.0(fixed)               & $50.2 \pm 7.2 \pm 3.9$            &$>$10 \\
      $D_{s}^{+} \to K_{S}^{0}\rho(1450)^{+}$ & $2.7 \pm 0.5$        & $ 2.2 \pm 0.2 \pm 0.1$   & $20.4 \pm 4.3 \pm 4.4$            &$>$10\\
      $D_{s}^{+} \to K^{*}(892)^{0}\pi^{+}$   & $0.4 \pm 0.1$        & $ 3.2 \pm 0.2 \pm 0.1$   & $\phantom{0}8.4  \pm 2.2 \pm 0.9$ &5.0\\
      $D_{s}^{+} \to K^{*}(892)^{+}\pi^{0}$   & $0.3 \pm 0.1$        & $ 0.2 \pm 0.2 \pm 0.2$   & $\phantom{0}4.6  \pm 1.4 \pm 0.4$ &4.0\\
      $D_{s}^{+} \to K^{*}(1410)^{0}\pi^{+}$  & $0.8 \pm 0.2$        & $ 0.2 \pm 0.3 \pm 0.1$   & $\phantom{0}3.3  \pm 1.6 \pm 0.5$ &3.7\\
      \hline
    \end{tabular}
    \end{center}
\end{table*}
\begin{table*}[htbp]
    \caption{Interference between amplitudes, in  \% of the total amplitude.
      A denotes $D_{s}^{+} \to K_{S}^{0}\rho^{+}$,
      B $D_{s}^{+} \to K_{S}^{0}\rho(1450)^{+}$,
      C $D_{s}^{+} \to K^{*}(892)^{0}\pi^{+}$,
      D $D_{s}^{+} \to K^{*}(892)^{+}\pi^{0}$, and
      E $D_{s}^{+} \to K^{*}(1410)^{0}\pi^{+}$.
      The uncertainties are statistical only.}
    \label{fit-interference}
    \begin{center}
    \begin{tabular}{ccccc}
      \hline
                    & B              & C                         & D                         & E\\
    A                & 20.3 $\pm$ 5.3 & -4.1 $\pm$ 1.0            & -2.6 $\pm$ 0.9            & 5.1 $\pm$ 1.6\\
    B                &                & -4.5 $\pm$ 0.9            & -3.2 $\pm$ 0.7            & 0.8 $\pm$ 1.7\\
    C                &                &                           & -0.5 $\pm$ 0.1            & 0.4 $\pm$ 1.0\\
    D                &                &                           &                           & 0.5 $\pm$ 0.4 \\
    \hline
    \end{tabular}
    \end{center}
\end{table*}

\begin{figure*}[!htbp]
 \centering
 \includegraphics[width=0.32\textwidth]{mKsPi0.eps}
 \includegraphics[width=0.32\textwidth]{mKsPi.eps}
 \includegraphics[width=0.32\textwidth]{mPiPi0.eps}
 \caption{
   The projections of (a) $M_{K_S^0\pi^0}$, (b) $K_{S}^0\pi^+$, and (c)
   $M_{\pi^+\pi^0}$ from the nominal fit. The data samples at
   $\sqrt{s}= 4.178$-$4.226$~GeV are represented by points with error
   bars, the fit results by the solid blue lines, and the background
   estimated from generic MC samples by the black dashed lines.}
 \label{dalitz-projection}
\end{figure*}

\subsection{Systematic uncertainties for amplitude analysis}
\label{sec:PWA-Sys}
The systematic uncertainties for the amplitude analysis are summarized
in Table~\ref{systematic-uncertainties}, with their assignment described below.
\begin{itemize}
\item[\lowercase\expandafter{\romannumeral1}]
Resonance parameters. The masses and the widths of $\rho^{+}$, $\rho(1450)^{+}$,
$K^*(892)^{0(+)}$, and $K^*(1410)^{0}$ are shifted by their
corresponding uncertainties~\cite{PDG}.

\item[\lowercase\expandafter{\romannumeral2}]
R values. The radii of the nonresonant state and $D_s^{\pm}$ mesons
are varied within the range $[2.0, 4.0]$~GeV$^{-1}$ for intermediate
resonances and $[3.0, 7.0]$~GeV$^{-1}$ for $D_s^{\pm}$ mesons.

\item[\lowercase\expandafter{\romannumeral3}]
Background estimation.
The uncertainties associated with background are studied by varying the
fractions of signal (equivalent to the fractions of background),
i.e.~$w_{\rm sig}$ in Eq.~(\ref{likelihood3}). The fractions of signal
for the three sample groups are varied by one corresponding
statistical uncertainty. The largest differences from the nominal results
are assigned as the uncertainties.

The other source of potential bias arised from  the knowledge  of the background distributions.
We follow an alternative procedure by determining the background
shape with another two variables, $M^2_{K_{S}^{0}\pi^{+}}$ versus
$M^2_{\pi^{+}\pi^{0}}$, and change the smooth parameters in RooNDKeysPdf~\cite{Verkerke}.
This resulting change in results is small enough to be ignored and so we assign no uncertainty from this source.

\item[\lowercase\expandafter{\romannumeral4}]
Experimental effects.
To estimate the systematic uncertainty related to the difference in acceptance
between MC and data associated with the PID and tracking efficiencies, that is
$\gamma_{\epsilon}$ in Eq.~(\ref{MC-intergral-corrected}), the amplitude
fit is performed varying the PID and tracking efficiencies according to their
uncertainties.

\item[\lowercase\expandafter{\romannumeral5}]
Fit bias.
The amplitude analysis is performed on three-hundred data-sized signal MC samples and the
pulls, which are the normalized-residual distributions of the fit, are inspected to look for biases or
significant excursions from a normal distribution.
These studies indicate that the FFs of $D_{s}^{+} \to K^{*}(892)^{0}\pi^{+}$ and $D_{s}^{+} \to K^{*}(1410)^{0}\pi^{+}$ are slightly biased. In addition, the statistical uncertainties of the FF and the phase of $D_{s}^{+} \to K^{*}(892)^{0}\pi^{+}$ are underestimated. Therefore, we correct the biased FFs by the mean values of the pull distributions and scale the underestimated uncertainties by the widths of the pulls.
The systematic
uncertainty due to the correction is assigned as the uncertainty of the mean
value. An additional systematic uncertainty due to the normalization is taken
into account  by $\sqrt{2f\Delta f}$, where $f$ is the fitted width and
$\Delta f$ is its uncertainty~\cite{err_on_err1, err_on_err2}.
\end{itemize}

\begin{table*}[tp]
  \renewcommand\arraystretch{1.25}
  \centering
  \caption{Systematic uncertainties on the $\phi$ and FFs for each
    amplitude in units of the corresponding statistical uncertainties.  The sources are:
    (\lowercase\expandafter{\romannumeral1}) Fixed parameters in the amplitudes,
    (\lowercase\expandafter{\romannumeral2}) The R values,
    (\lowercase\expandafter{\romannumeral3}) Background,
    (\lowercase\expandafter{\romannumeral4}) Experimental effects,
    (\lowercase\expandafter{\romannumeral5}) Fit bias.
}
  \label{systematic-uncertainties}
  \begin{tabular}{lccccccc}
    \hline
    \multirow{2}{*}{Amplitude}&\multicolumn{7}{c}{Source}\cr
    & & \lowercase\expandafter{\romannumeral1} &\lowercase\expandafter{\romannumeral2} &\lowercase\expandafter{\romannumeral3} &\lowercase\expandafter{\romannumeral4} &\lowercase\expandafter{\romannumeral5}& Total   \\
    \hline
    $D_{s}^{+} \to K_{S}^{0}\rho^{+}$                        & FF     & 0.03 & 0.49 & 0.02 & 0.23 & 0.03 & 0.54  \\
    \hline
    \multirow{2}{*}{$D_{s}^{+} \to K_{S}^{0}\rho(1450)^{+}$} & $\phi$ & 0.34 & 0.38 & 0.15 & 0.16 & 0.06 & 0.56  \\
                                                             & FF     & 0.34 & 0.95 & 0.05 & 0.19 & 0.03 & 1.03  \\
    \hline
    \multirow{2}{*}{ $D_{s}^{+} \to K^{*}(892)^{0}\pi^{+}$}  & $\phi$ & 0.03 & 0.16 & 0.26 & 0.27 & 0.23 & 0.47  \\
                                                             & FF     & 0.06 & 0.29 & 0.10 & 0.21 & 0.18 & 0.42  \\
    \hline
    \multirow{2}{*}{$D_{s}^{+} \to K^{*}(892)^{+}\pi^{0}$ }  & $\phi$ & 0.03 & 0.33 & 0.70 & 0.21 & 0.06 & 0.80  \\
                                                             & FF     & 0.10 & 0.03 & 0.15 & 0.25 & 0.03 & 0.31  \\
    \hline
    \multirow{2}{*}{$D_{s}^{+} \to K^{*}(1410)^{0}\pi^{+}$}  & $\phi$ & 0.15 & 0.02 & 0.25 & 0.14 & 0.06 & 0.33  \\
                                                             & FF     & 0.14 & 0.22 & 0.13 & 0.09 & 0.03 & 0.31  \\
    \hline
  \end{tabular}
\end{table*}

\section{Branching fraction measurement of $D_{s}^{+} \to K^0_{S}\pi^{+}\pi^{0}$}
\label{BFSelection}
With the selection criteria described in Sec.~\ref{ST-selection}, the best tag
candidate with $M_{\rm rec}$ closest to the $D_{s}^{\pm}$ nominal
mass~\cite{PDG} is chosen if there are multiple ST candidates. The yields for
various tag modes are listed in Table~\ref{ST-eff} and obtained by fitting the
corresponding $M_{\rm tag}$ distributions. As an example, the fits to the data
sample at $\sqrt s=4.178$~GeV are shown in Fig.~\ref{fit:Mass-data-Ds_4180}. In
the fits, the signal is modeled by an MC-simulated shape convolved with a
Gaussian function to take into account the data-MC resolution difference. The
background is described by a second-order Chebyshev function. MC studies show
that there is no significant peaking background in any tag mode, except for
$D^{-} \to K_{S}^{0} \pi^-$ and $D_{s}^{-} \to \eta\pi^+\pi^-\pi^-$ faking the
$D_{s}^{-} \to K_{S}^{0} K^-$ and $D_{s}^{-} \to \pi^-\eta^{\prime}$ tags,
respectively. Therefore, in the fits, the MC-simulated shapes of these two
peaking background sources are added to the background polynomial functions.
\begin{table*}[htbp]
  \caption{The ST yields for the samples collected at $\sqrt{s} =$ (I) 4.178, (II) 4.199-4.219,
    and (III) 4.23~GeV. The uncertainties are statistical.}\label{ST-eff}
    \begin{center}
      \begin{tabular}{lccc}
        \hline
        Tag mode                                     & (I) $N_{\rm ST}$                          & (II) $N_{\rm ST}$                       & (III) $N_{\rm ST}$    \\
        \hline
        $D_{s}^{-}\to K_{S}^{0}K^{-}$               & $\phantom{0}31668\pm315\phantom{0}$   & $18340\pm260\phantom{0}$            & $\phantom{0}6550\pm158\phantom{0}$  \\
        $D_{s}^{-}\to K^{+}K^{-}\pi^{-}$            & $135867\pm610\phantom{0}$             & $80417\pm507\phantom{0}$            & $28289\pm328\phantom{0}$ \\
        $D_{s}^{-}\to K_{S}^{0}K^{-}\pi^{0}$        & $\phantom{0}11284\pm512\phantom{0}$   & $\phantom{0}6729\pm462\phantom{0}$  & $\phantom{0}2144\pm218\phantom{0}$  \\
        $D_{s}^{-}\to K^{+}K^{-}\pi^{-}\pi^{0}$     & $\phantom{0}38421\pm767\phantom{0}$   & $22894\pm645\phantom{0}$            & $\phantom{0}7855\pm439\phantom{0}$  \\
        $D_{s}^{-}\to K_{S}^{0}K^{+}\pi^{-}\pi^{-}$ & $\phantom{0}15644\pm289\phantom{0}$   & $\phantom{0}8922\pm229\phantom{0}$  & $\phantom{0}3241\pm169\phantom{0}$  \\
        $D_{s}^{-}\to \pi^{-}\pi^{-}\pi^{+}$        & $\phantom{0}37702\pm853\phantom{0}$   & $21675\pm772\phantom{0}$            & $\phantom{0}7506\pm392\phantom{0}$  \\
        $D_{s}^{-}\to \pi^{-}\eta_{\gamma\gamma}$   & $\phantom{0}18070\pm560\phantom{0}$   & $10033\pm355\phantom{0}$            & $\phantom{0}3699\pm244\phantom{0}$  \\
        $D_{s}^{-}\to \pi^{-}\pi^{0}\eta_{\gamma\gamma}$
                                                    & $\phantom{0}40862\pm1313$             & $25877\pm1823$                      & $10659\pm1060$\\
        $D_{s}^{-}\to \pi^{-}\eta_{\pi^{+}\pi^{-}\eta_{\gamma\gamma}}^{'}$
                                                    & $\phantom{00}7773\pm143\phantom{0}$   & $\phantom{0}4464\pm111\phantom{0}$  & $\phantom{0}1676\pm74\phantom{00}$   \\
        \hline
      \end{tabular}
    \end{center}
\end{table*}

\begin{figure*}[htp]
\begin{center}
\includegraphics[width=0.90\textwidth]{bf_ST_zdraft_3v4.eps}
\caption{Fits to the $M_{\rm tag}$ distributions of the ST candidates
         from the data sample at $\sqrt{s}=4.178$~GeV. The points with
         error bars are data, the blue solid lines are the total fits, and the black
         dashed lines are background. The pairs of red arrows denote the
         signal regions.
         }
\label{fit:Mass-data-Ds_4180}
\end{center}
\end{figure*}

Once a tag mode is identified, we search for the signal decay
$D_{s}^{+} \to K_S^{0}\pi^{+}\pi^{0}$. In the case of multiple candidates, the
DT candidate with the average mass, $(M_{\rm sig}+M_{\rm tag})/2$,
closest to the $D_{s}^{\pm}$ nominal mass is retained.

To measure the BF, we start from the following equations for a single tag mode:
\begin{eqnarray}\begin{aligned}
  N_{\text{tag}}^{\text{ST}} = 2N_{D_{s}^{+}D_{s}^{-}}\mathcal{B}_{\text{tag}}\epsilon_{\text{tag}}^{\text{ST}}\,, \label{eq-ST}
\end{aligned}\end{eqnarray}
\begin{equation}
    N_{\text{tag,sig}}^{\text{DT}}=2N_{D_{s}^{+}D_{s}^{-}}\mathcal{B}_{\text{tag}}\mathcal{B}_{\text{sig}}\epsilon_{\text{tag,sig}}^{\text{DT}}\,,
  \label{eq-DT}
\end{equation}
where $N_{D_{s}^{+}D_{s}^{-}}$ is the total number of $D_{s}^{*\pm}D_{s}^{\mp}$
pairs produced from the $e^{+}e^{-}$ collisions; $N_{\text{tag}}^{\text{ST}}$ is
the ST yield for the tag mode; $N_{\text{tag,sig}}^{\text{DT}}$ is the DT yield;
$\mathcal{B}_{\text{tag}}$ and $\mathcal{B}_{\text{sig}}$ are the BFs of the
tag and signal modes, respectively; $\epsilon_{\text{tag}}^{\text{ST}}$ is the
ST efficiency to reconstruct the tag mode; and $\epsilon_{\text{tag,sig}}^{\text{DT}}$
is the DT efficiency to reconstruct both the tag and the signal decay modes. In
the case of more than one tag modes and sample groups,
\begin{eqnarray}
\begin{aligned}
  \begin{array}{lr}
    N_{\text{total}}^{\text{DT}}=\Sigma_{\alpha, i}N_{\alpha,\text{sig},i}^{\text{DT}}   = \mathcal{B}_{\text{sig}}
 \Sigma_{\alpha, i}2N_{D_{s}^{+}D_{s}^{-}}\mathcal{B}_{\alpha}\epsilon_{\alpha,\text{sig}, i}^{\text{DT}}\,,
  \end{array}
  \label{eq-DTtotal}
\end{aligned}
\end{eqnarray}
where $\alpha$ represents tag modes in the $i$th sample group.
By isolating $\mathcal{B}_{\text{sig}}$, we find
\begin{eqnarray}\begin{aligned}
  \mathcal{B}_{\text{sig}} =
  \frac{N_{\text{total}}^{\text{DT}}}{ \mathcal{B}_{K_S^0\to\pi^+\pi^-}\mathcal{B}_{\pi^0\to\gamma\gamma}\begin{matrix}\sum_{\alpha, i} N_{\alpha, i}^{\text{ST}}\epsilon^{\text{DT}}_{\alpha,\text{sig},i}/\epsilon_{\alpha,i}^{\text{ST}}\end{matrix}}\,.
\end{aligned}\end{eqnarray}
where $N_{\alpha,i}^{\text{ST}}$ and $\epsilon_{\alpha,i}^{\text{ST}}$ are
obtained from the data and generic MC samples, respectively, while
$\epsilon_{\alpha,\text{sig},i}^{\text{DT}}$ is determined with signal MC
samples, where $D_{s}^{+} \to K_S^{0}\pi^{+}\pi^{0}$ events are generated
according to the results of the  amplitude analysis.  The two branching ratios
$ \mathcal{B}_{K_S^0\to\pi^+\pi^-}$ and $\mathcal{B}_{\pi^0\to\gamma\gamma}$
have been introduced to account for the fact that the signal is reconstructed through
these decays.

The DT yield $N_{\text{total}}^{\text{DT}}$ is found to be $666\pm37$ from the
fit to the $M_{\rm sig}$ distribution of the selected $D^+_s\to K^0_S\pi^+\pi^0$
candidates. The fit result is shown in Fig.~\ref{DT-fit}. In the fit, the signal
shape is described by an MC-simulated shape convolved with a Gaussian function
to take into account the data-MC resolution difference. The background shape is
described by an MC-simulated shape, which includes the small peaking
background~($2.1\%$) that is mainly from
$D_{s}^{+} \to \pi^{+}\pi^{+}\pi^{-}\pi^{0}$ decays. The width of the Gaussian
function is fixed to be $1.9\pm 1.1$~MeV/$c^{2}$, which is extracted from the
control sample of  $D_{s}^{+} \to K_{S}^{0}K^{+}\pi^{0}$ decays. Note that
the DT yield is larger than the fit yields of Fig.~\ref{fig:fit_Ds} since
the selection for the BF measurement is looser than that for the amplitude analysis
and no kinematic fit is applied in the BF measurement.

\begin{figure}[!htbp]
  \centering
  \includegraphics[width=0.6\textwidth]{bf_zfit_ds.eps}
 \caption{Fit to the $M_{\rm sig}$ distribution of the DT candidates from the
   data samples  at $\sqrt{s}= 4.178$-$4.226$~GeV. The data are represented by
   points with error bars, the total fit by the blue solid line, and the fitted
   signal and the fitted background by the red dotted and the black dashed
   lines, respectively.
 }
  \label{DT-fit}
\end{figure}

We take the differences in pion tracking efficiency between data and MC
simulation into account, and apply a correction to the MC signal efficiency of $+0.3\%$.
The differences in PID efficiency are negligible.
The BF is determined to be
$\mathcal{B}(D_{s}^{+} \to K_{S}^0\pi^{+}\pi^{0})=(5.43\pm0.30_{\rm stat}\pm 0.15_{\rm syst})\times 10^{-3}$.

In order to test $CP$ conservation in the decay,
the BFs are measured separately for the charge-conjugated modes.
The BFs of
$D_{s}^{+} \to K_{S}^0\pi^{+}\pi^{0}$ and $D_{s}^{-} \to K_{S}^0\pi^{-}\pi^{0}$
are measured to be $(5.33\pm0.41_{\rm stat}\pm 0.15_{\rm syst})\times 10^{-3}$
and $(5.63\pm0.44_{\rm stat}\pm 0.16_{\rm syst})\times 10^{-3}$, respectively.
From these measurements the asymmetry
of the BFs is determined to be $(2.7\pm5.5_{\rm stat}\pm0.9_{\rm syst})\%$ by
$A_{CP}=\frac{B(D_{s}^{+})-B(D_{s}^{-})}{B(D_{s}^{+})+B(D_{s}^{-})}$, where
$B(D_{s}^{+(-)})$ is the BF of the decay $D_{s}^{+(-)} \to K_{S}^0\pi^{+(-)}\pi^{0}$.
Hence, no $CP$ violation is observed.
Note that the systematic uncertainties related to $K_{S}^0$ and $\pi^{0}$
reconstructions cancel in the $A_{CP}$ calculation.

The following sources of the systematic uncertainties are taken into account
for the BF measurement.
\begin{itemize}
\item Signal shape. The systematic uncertainty due to the signal shape is
  studied by repeating the fit with an alternative width of the convolved
  Gaussian. This width is varied according to the uncertainty of the control
  sample.

  \item Background shape. Since $q\bar{q}$ or non-$D_{s}^{*\pm}D_{s}^{\mp}$
  open charm are the major background sources, we alter the MC shapes by varying
  the relative fractions of the background from $q\bar{q}$ or
  non-$D_{s}^{*+}D_{s}^{-}$ open charm by $\pm 30\%$. The largest change is
  taken as the corresponding systematic uncertainty.

\item $\pi^{+}$ tracking/PID efficiency. The $\pi^{+}$ tracking and PID
  efficiencies are studied with $e^+e^-\to K^+K^-\pi^+\pi^-$ events. The
  data-MC efficiency ratios of the $\pi^{+}$ tracking and PID efficiencies are
  $1.003\pm0.002$ and $1.000\pm0.002$, respectively. After multiplying the
  signal efficiencies by the factor $1.003$, we assign $0.2\%$ and $0.2\%$ as
  the systematic uncertainties arising from $\pi^{+}$ PID and tracking,
  respectively.

\item $K_S^0$ reconstruction. The systematic uncertainty from the $K^0_S$ reconstruction
  efficiency is assigned to be $1.5\%$, determined  from studying a control
  sample of $\psi(3770)\to D\bar{D}$ events containing hadronic $D$ decays.

\item $\pi^{0}$ reconstruction. A control sample of the process
  $e^+e^-\to K^+K^-\pi^+\pi^-\pi^0$ is used to study the uncertainty
  due to $\pi^0$ reconstruction, which is assigned as 2.0\%.

\item MC statistics. The uncertainty due to the limited MC statistics is
  obtained by
  $\sqrt{\begin{matrix} \sum_{i} (f_{i}\frac{\delta_{\epsilon_{i}}}{\epsilon_{i}}\end{matrix}})^2$,
  where $f_{i}$ is the tag yield fraction, and $\epsilon_{i}$ and
  $\delta_{\epsilon_{i}}$ are the signal efficiency and the corresponding
  uncertainty of tag mode $i$, respectively.

\item Dalitz model. The uncertainty from the Dalitz model is estimated by
  varying the Dalitz model parameters based on their error matrix. The
  distribution of 600 efficiencies resulting from this variation  is fitted by a Gaussian function
  and the deviation from the nominal mean value is taken as an uncertainty.

\item Peaking background. The uncertainty of the peaking background is about
8\% and corresponds to only one event in the $D_s^+\to K_{S}^{0}\pi^+\pi^0$
decay. Therefore the associated uncertainty in the BF measurement is negligible.
\end{itemize}

All of the systematic uncertainties are summarized in Table~\ref{BF-Sys}.
Adding them in quadrature gives a total systematic uncertainty in the BF
measurement of 2.8\%.
\begin{table}[htbp]
  \caption{Systematic uncertainties in the BF measurement.}
  \label{BF-Sys}
  \begin{center}
    \begin{tabular}{lccc}
      \hline
      Source   & Sys. Uncertainty (\%)\\
      \hline
      Signal shape                        & 0.8 \\
      Background shape                    & 0.5 \\
      $\pi^{+}$ PID efficiency            & 0.2 \\
      $\pi^{+}$ tracking efficiency       & 0.2 \\
      $K_S^0$ reconstruction              & 1.5 \\
      $\pi^0$ reconstruction              & 2.0 \\
      MC statistics                       & 0.3 \\
      Dalitz model                        & 0.8 \\
      \hline
      Total                               & 2.8 \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\section{Summary}
An amplitude analysis has been performed for the decay
$D_{s}^{+} \to K_{S}^0\pi^{+}\pi^{0}$. The results for the  FFs and phases among the
different intermediate processes are listed in Table~\ref{fit-result}.
After calculating a detection efficiency that accounts for the variation of decays over phase space
found in the amplitude analysis, the
BF for the decay $D^+_s\to K^0_S\pi^+\pi^0$ is measured to be
$(5.43\,\pm\,0.30_{\rm stat}\,\pm\,0.15_{\rm syst})\times 10^{-3}$ with an
improved precision by about a factor of 3 compared to the PDG value~\cite{PDG}.
The BFs for the intermediate processes are calculated with
$\mathcal{B}_{i} = {\rm FF}_{i} \times \mathcal{B}(D_{s}^{+} \to K_{S}^0\pi^{+}\pi^{0})$
and listed in Table~\ref{inter-processes}. Assuming
$\mathcal{B}(K^0\to K^0_S)=0.5$, we determine
$\mathcal{B}(D_{s}^{+} \to K^{0}\rho^{+}) = (5.46\,\pm\,0.84_{\rm stat}\,\pm\,0.44_{\rm syst})\times 10^{-3}$,
$\mathcal{B}(D_{s}^{+} \to K^{*}(892)^{0}\pi^{+}) = (2.71\,\pm\,0.72_{\rm stat}\,\pm\,0.30_{\rm syst})\times 10^{-3}$, and
$\mathcal{B}(D_{s}^{+} \to K^{*}(892)^{+}\pi^{0}) = (0.75\,\pm\,0.24_{\rm stat}\,\pm\,0.06_{\rm syst})\times 10^{-3}$.
Our results are valuable for a deeper understanding of quark flavor SU(3) symmetry, SU(3)
breaking effects, and other related theoretical issues.

These results can be compared to the current theoretical predictions~\cite{YLWu, HYCheng, PRD84-074019}. The predictions in Ref~\cite{YLWu} are consistent with our results, but their large uncertainties make the comparisons less conclusive. The calculations in Ref~\cite{HYCheng} have small uncertainties, while the predicted  $\mathcal{B}(D_{s}^{+} \to K^{0}\rho^{+})$ is over five standard deviations off the measured one. The predictions in Ref~\cite{PRD84-074019} have moderate uncertainties and match our measurements in principle, but the predicted $\mathcal{B}(D_{s}^{+} \to K^{*}(892)^{+}\pi^{0})$ is only marginally consistent with our measurement. Based on the current experimental and theoretical precisions, it is difficult to draw a definite conclusion to discriminate between models yet.

The asymmetry for the BFs of the decays $D_{s}^{+} \to K_{S}^0\pi^{+}\pi^{0}$
and $D_{s}^{-} \to K_{S}^0\pi^{-}\pi^{0}$ is determined to be
$(2.7\pm5.5_{\rm stat}\pm0.9_{\rm syst})\%$. No evidence for $CP$ violation is
found.

\begin{table}[htbp]
  \caption{The BFs for various intermediate processes with the final state
    $K_{S}^0\pi^{+}\pi^{0}$. The first and second uncertainties are
    statistical and systematic, respectively.}\label{inter-processes}
  \begin{center}
    \begin{tabular}{lc}
      \hline
      Intermediate process & BF ($10^{-3}$)\\
      \hline
      $D_{s}^{+} \rightarrow K_{S}^{0}\rho^{+}$        & $ 2.73 \pm 0.42 \pm 0.22$  \\
      $D_{s}^{+} \rightarrow K_{S}^{0}\rho(1450)^{+}$  & $ 1.11 \pm 0.24 \pm 0.24$  \\
      $D_{s}^{+} \rightarrow K^{*}(892)^{0}\pi^{+}$    & $ 0.45 \pm 0.12 \pm 0.05$  \\
      $D_{s}^{+} \rightarrow K^{*}(892)^{+}\pi^{0}$    & $ 0.25 \pm 0.08 \pm 0.02$  \\
      $D_{s}^{+} \rightarrow K^{*}(1410)^{0}\pi^{+}$   & $ 0.18 \pm 0.09 \pm 0.03$  \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\acknowledgments
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Research and Development Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11625523, 11635010, 11735014, 11775027, 11822506, 11835012, 11875054, 11935015, 11935016, 11935018, 11961141012; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1732263, U1832207, U2032104; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462; European Union Horizon 2020 research and innovation programme under Contract No. Marie Sklodowska-Curie grant agreement No 894790; German Research Foundation DFG under Contracts Nos. 443159800, Collaborative Research Center CRC 1044, FOR 2359, FOR 2359, GRK 214; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; Olle Engkvist Foundation under Contract No. 200-0605; STFC (United Kingdom); The Knut and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The Royal Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0012069.

\bibliographystyle{JHEP}

redpajama_set_name: RedPajamaArXiv


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