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Triafulvalene or cyclopropenylidenecyclopropene is a fulvalene hydrocarbon with chemical formula C 6 H 4 , composed of two linked cyclopropene rings. Triafulvalene has never been isolated, [ 1 ] since it can decompose via an isodesmic reaction . [ 2 ] However, its structure, [ 3 ] stability, and spectral properties are well-studied theoretically; [ citation needed ] the molecule is believed planar. [ 3 ]
This article about a hydrocarbon is a stub . You can help Wikipedia by expanding it .
This article about theoretical chemistry is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Triafulvalene |
The triakis truncated tetrahedral honeycomb is a space-filling tessellation (or honeycomb ) in Euclidean 3-space made up of triakis truncated tetrahedra . It was discovered in 1914. [ 1 ] [ 2 ]
It is the Voronoi tessellation of the carbon atoms in diamond , [ 3 ] [ 4 ] which lie in the diamond cubic crystal structure.
Being composed entirely of triakis truncated tetrahedra , it is cell-transitive .
It can be seen as the uniform quarter cubic honeycomb where its tetrahedral cells are subdivided by the center point into 4 shorter tetrahedra, and each adjoined to the adjacent truncated tetrahedral cells.
This 4-polytope article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Triakis_truncated_tetrahedral_honeycomb |
Trial and error is a fundamental method of problem-solving [ 1 ] characterized by repeated, varied attempts which are continued until success, [ 2 ] or until the practicer stops trying.
According to W.H. Thorpe , the term was devised by C. Lloyd Morgan (1852–1936) after trying out similar phrases "trial and failure" and "trial and practice". [ 3 ] Under Morgan's Canon , animal behaviour should be explained in the simplest possible way. Where behavior seems to imply higher mental processes, it might be explained by trial-and-error learning. An example is a skillful way in which his terrier Tony opened the garden gate, easily misunderstood as an insightful act by someone seeing the final behavior. Lloyd Morgan, however, had watched and recorded the series of approximations by which the dog had gradually learned the response, and could demonstrate that no insight was required to explain it.
Edward Lee Thorndike was the initiator of the theory of trial and error learning based on the findings he showed how to manage a trial-and-error experiment in the laboratory. In his famous experiment, a cat was placed in a series of puzzle boxes in order to study the law of effect in learning. [ 4 ] He plotted to learn curves which recorded the timing for each trial. Thorndike's key observation was that learning was promoted by positive results, which was later refined and extended by B. F. Skinner 's operant conditioning . [ citation needed ]
Trial and error is also a method of problem solving, repair , tuning, or obtaining knowledge . In the field of computer science , the method is called generate and test ( brute force ) . In elementary algebra , when solving equations, it is called guess and check . [ citation needed ]
This approach can be seen as one of the two basic approaches to problem-solving, contrasted with an approach using insight and theory . However, there are intermediate methods that, for example, use theory to guide the method, an approach known as guided empiricism . [ citation needed ]
This way of thinking has become a mainstay of Karl Popper 's critical rationalism . [ citation needed ]
The trial and error approach is used most successfully with simple problems and in games, and it is often the last resort when no apparent rule applies. This does not mean that the approach is inherently careless, for an individual can be methodical in manipulating the variables in an attempt to sort through possibilities that could result in success. Nevertheless, this method is often used by people who have little knowledge in the problem area. The trial-and-error approach has been studied from its natural computational point of view [ 5 ]
Ashby (1960, section 11/5) offers three simple strategies for dealing with the same basic exercise-problem, which have very different efficiencies. Suppose a collection of 1000 on/off switches have to be set to a particular combination by random-based testing, where each test is expected to take one second. [This is also discussed in Traill (1978–2006, section C1.2]. The strategies are:
Note the tacit assumption here that no intelligence or insight is brought to bear on the problem. However, the existence of different available strategies allows us to consider a separate ("superior") domain of processing — a "meta-level" above the mechanics of switch handling — where the various available strategies can be randomly chosen. Once again this is "trial and error", but of a different type.
Ashby's book develops this "meta-level" idea, and extends it into a whole recursive sequence of levels, successively above each other in a systematic hierarchy. On this basis, he argues that human intelligence emerges from such organization: relying heavily on trial-and-error (at least initially at each new stage), but emerging with what we would call "intelligence" at the end of it all. Thus presumably the topmost level of the hierarchy (at any stage) will still depend on simple trial-and-error.
Traill (1978–2006) suggests that this Ashby-hierarchy probably coincides with Piaget 's well-known theory of developmental stages. [This work also discusses Ashby's 1000-switch example; see §C1.2]. After all, it is part of Piagetian doctrine that children learn first by actively doing in a more-or-less random way, and then hopefully learn from the consequences — which all has a certain resemblance to Ashby's random "trial-and-error".
Traill (2008, espec. Table "S" on p.31 ) follows Jerne and Popper in seeing this strategy as probably underlying all knowledge-gathering systems — at least in their initial phase .
Four such systems are identified:
Trial and error has a number of features:
It is possible to use trial and error to find all solutions or the best solution, when a testably finite number of possible solutions exist. To find all solutions, one simply makes a note and continues, rather than ending the process, when a solution is found, until all solutions have been tried. To find the best solution, one finds all solutions by the method just described and then comparatively evaluates them based upon some predefined set of criteria, the existence of which is a condition for the possibility of finding a best solution. (Also, when only one solution can exist, as in assembling a jigsaw puzzle, then any solution found is the only solution and so is necessarily the best.)
Trial and error has traditionally been the main method of finding new drugs, such as antibiotics . Chemists simply try chemicals at random until they find one with the desired effect. In a more sophisticated version, chemists select a narrow range of chemicals it is thought may have some effect using a technique called structure–activity relationship . (The latter case can be alternatively considered as a changing of the problem rather than of the solution strategy: instead of "What chemical will work well as an antibiotic?" the problem in the sophisticated approach is "Which, if any, of the chemicals in this narrow range will work well as an antibiotic?") The method is used widely in many disciplines, such as polymer technology to find new polymer types or families.
Trial and error is also commonly seen in player responses to video games - when faced with an obstacle or boss , players often form a number of strategies to surpass the obstacle or defeat the boss, with each strategy being carried out before the player either succeeds or quits the game.
Sports teams also make use of trial and error to qualify for and/or progress through the playoffs and win the championship , attempting different strategies, plays, lineups and formations in hopes of defeating each and every opponent along the way to victory. This is especially crucial in playoff series in which multiple wins are required to advance , where a team that loses a game will have the opportunity to try new tactics to find a way to win, if they are not eliminated yet.
The scientific method can be regarded as containing an element of trial and error in its formulation and testing of hypotheses. Also compare genetic algorithms , simulated annealing and reinforcement learning – all varieties for search which apply the basic idea of trial and error.
Biological evolution can be considered as a form of trial and error. [ 6 ] Random mutations and sexual genetic variations can be viewed as trials and poor reproductive fitness, or lack of improved fitness, as the error. Thus after a long time 'knowledge' of well-adapted genomes accumulates simply by virtue of them being able to reproduce.
Bogosort , a conceptual sorting algorithm (that is extremely inefficient and impractical), can be viewed as a trial and error approach to sorting a list. However, typical simple examples of bogosort do not track which orders of the list have been tried and may try the same order any number of times, which violates one of the basic principles of trial and error. Trial and error is actually more efficient and practical than bogosort; unlike bogosort, it is guaranteed to halt in finite time on a finite list, and might even be a reasonable way to sort extremely short lists under some conditions.
Jumping spiders of the genus Portia use trial and error to find new tactics against unfamiliar prey or in unusual situations, and remember the new tactics. [ 7 ] Tests show that Portia fimbriata and Portia labiata can use trial and error in an artificial environment, where the spider's objective is to cross a miniature lagoon that is too wide for a simple jump, and must either jump then swim or only swim. [ 8 ] [ 9 ] | https://en.wikipedia.org/wiki/Trial_and_error |
In mathematics , a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle . The triangle can be an ordinary Euclidean triangle, a triangle on the sphere , or a hyperbolic triangle . Each triangle group is the symmetry group of a tiling of the Euclidean plane , the sphere , or the hyperbolic plane by congruent triangles called Möbius triangles , each one a fundamental domain for the action.
Let l , m , n be integers greater than or equal to 2. A triangle group Δ( l , m , n ) is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/ l , π/ m and π/ n (measured in radians ). The product of the reflections in two adjacent sides is a rotation by the angle which is twice the angle between those sides, 2π/ l , 2π/ m and 2π/ n . Therefore, if the generating reflections are labeled a , b , c and the angles between them in the cyclic order are as given above, then the following relations hold:
It is a theorem that all other relations between a, b, c are consequences of these relations and that Δ( l,m,n ) is a discrete group of motions of the corresponding space. Thus a triangle group is a reflection group that admits a group presentation
An abstract group with this presentation is a Coxeter group with three generators.
Given any natural numbers l , m , n > 1 exactly one of the classical two-dimensional geometries (Euclidean, spherical, or hyperbolic) admits a triangle with the angles (π/l, π/m, π/n), and the space is tiled by reflections of the triangle. The sum of the angles of the triangle determines the type of the geometry by the Gauss–Bonnet theorem : it is Euclidean if the angle sum is exactly π, spherical if it exceeds π and hyperbolic if it is strictly smaller than π. Moreover, any two triangles with the given angles are congruent. Each triangle group determines a tiling, which is conventionally colored in two colors, so that any two adjacent tiles have opposite colors.
In terms of the numbers l , m , n > 1 there are the following possibilities.
1 l + 1 m + 1 n = 1. {\displaystyle {\frac {1}{l}}+{\frac {1}{m}}+{\frac {1}{n}}=1.}
The triangle group is the infinite symmetry group of a certain tessellation (or tiling) of the Euclidean plane by triangles whose angles add up to π (or 180°). Up to permutations, the triple ( l , m , n ) is one of the triples (2,3,6), (2,4,4), (3,3,3). The corresponding triangle groups are instances of wallpaper groups .
The triangle group is the finite symmetry group of a tiling of a unit sphere by spherical triangles, or Möbius triangles , whose angles add up to a number greater than π. Up to permutations, the triple ( l , m , n ) has the form (2,3,3), (2,3,4), (2,3,5), or (2,2, n ), n > 1. Spherical triangle groups can be identified with the symmetry groups of regular polyhedra in the three-dimensional Euclidean space: Δ(2,3,3) corresponds to the tetrahedron , Δ(2,3,4) to both the cube and the octahedron (which have the same symmetry group), Δ(2,3,5) to both the dodecahedron and the icosahedron . The groups Δ(2,2, n ), n > 1 of dihedral symmetry can be interpreted as the symmetry groups of the family of dihedra , which are degenerate solids formed by two identical regular n -gons joined together, or dually hosohedra , which are formed by joining n digons together at two vertices.
The spherical tiling corresponding to a regular polyhedron is obtained by forming the barycentric subdivision of the polyhedron and projecting the resulting points and lines onto the circumscribed sphere. In the case of the tetrahedron, there are four faces and each face is an equilateral triangle that is subdivided into 6 smaller pieces by the medians intersecting in the center. The resulting tesselation has 4 × 6=24 spherical triangles (it is the spherical disdyakis cube ).
These groups are finite, which corresponds to the compactness of the sphere – areas of discs in the sphere initially grow in terms of radius, but eventually cover the entire sphere.
The triangular tilings are depicted below:
Spherical tilings corresponding to the octahedron and the icosahedron and dihedral spherical tilings with even n are centrally symmetric . Hence each of them determines a tiling of the real projective plane, an elliptic tiling . Its symmetry group is the quotient of the spherical triangle group by the reflection through the origin (- I ), which is a central element of order 2. Since the projective plane is a model of elliptic geometry , such groups are called elliptic triangle groups. [ 1 ]
The triangle group is the infinite symmetry group of a tiling of the hyperbolic plane by hyperbolic triangles whose angles add up to a number less than π. All triples not already listed represent tilings of the hyperbolic plane. For example, the triple (2,3,7) produces the (2,3,7) triangle group . There are infinitely many such groups; the tilings associated with some small values:
Hyperbolic triangle groups are examples of non-Euclidean crystallographic group and have been generalized in the theory of Gromov hyperbolic groups .
Denote by D ( l , m , n ) the subgroup of index 2 in Δ(l,m,n) generated by words of even length in the generators. Such subgroups are sometimes referred to as "ordinary" triangle groups [ 2 ] or von Dyck groups , after Walther von Dyck . For spherical, Euclidean, and hyperbolic triangles, these correspond to the elements of the group that preserve the orientation of the triangle – the group of rotations. For projective (elliptic) triangles, they cannot be so interpreted, as the projective plane is non-orientable, so there is no notion of "orientation-preserving". The reflections are however locally orientation-reversing (and every manifold is locally orientable, because locally Euclidean): they fix a line and at each point in the line are a reflection across the line. [ 3 ]
The group D ( l , m , n ) is defined by the following presentation:
In terms of the generators above, these are x = ab, y = ca, yx = cb . Geometrically, the three elements x , y , xy correspond to rotations by 2π/ l , 2π/ m and 2π/ n about the three vertices of the triangle.
Note that D ( l , m , n ) ≅ D ( m , l , n ) ≅ D ( n , m , l ), so D ( l , m , n ) is independent of the order of the l , m , n .
A hyperbolic von Dyck group is a Fuchsian group , a discrete group consisting of orientation-preserving isometries of the hyperbolic plane.
Triangle groups preserve a tiling by triangles, namely a fundamental domain for the action (the triangle defined by the lines of reflection), called a Möbius triangle , and are given by a triple of integers, ( l , m , n ), – integers correspond to (2 l ,2 m ,2 n ) triangles coming together at a vertex. There are also tilings by overlapping triangles, which correspond to Schwarz triangles with rational numbers ( l / a , m / b , n / c ), where the denominators are coprime to the numerators. This corresponds to edges meeting at angles of a π/ l (resp.), which corresponds to a rotation of 2 a π/ l (resp.), which has order l and is thus identical as an abstract group element, but distinct when represented by a reflection.
For example, the Schwarz triangle (2 3 3) yields a density 1 tiling of the sphere, while the triangle (2 3/2 3) yields a density 3 tiling of the sphere, but with the same abstract group. These symmetries of overlapping tilings are not considered triangle groups.
Triangle groups date at least to the presentation of the icosahedral group as the (rotational) (2,3,5) triangle group by William Rowan Hamilton in 1856, in his paper on icosian calculus . [ 4 ]
Triangle groups arise in arithmetic geometry . The modular group is generated by two elements, S and T , subject to the relations S ² = ( ST )³ = 1 (no relation on T ), is the rotational triangle group (2,3,∞) and maps onto all triangle groups (2,3, n ) by adding the relation T n = 1. More generally, the Hecke group H q is generated by two elements, S and T , subject to the relations S 2 = ( ST ) q = 1 (no relation on T ), is the rotational triangle group (2, q ,∞), and maps onto all triangle groups (2, q , n ) by adding the relation T n = 1 the modular group is the Hecke group H 3 . In Grothendieck 's theory of dessins d'enfants , a Belyi function gives rise to a tessellation of a Riemann surface by reflection domains of a triangle group.
All 26 sporadic groups are quotients of triangle groups, [ 6 ] of which 12 are Hurwitz groups (quotients of the (2,3,7) group).
This article incorporates material from Triangle groups on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . | https://en.wikipedia.org/wiki/Triangle_group |
In mathematics , the triangle inequality states that for any triangle , the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. [ 1 ] [ 2 ] This statement permits the inclusion of degenerate triangles , but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. [ 3 ] If a , b , and c are the lengths of the sides of a triangle then the triangle inequality states that
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths ( norms ):
where the length of the third side has been replaced by the length of the vector sum u + v . When u and v are real numbers , they can be viewed as vectors in R 1 {\displaystyle \mathbb {R} ^{1}} , and the triangle inequality expresses a relationship between absolute values .
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem , and for general triangles, a consequence of the law of cosines , although it may be proved without these theorems. The inequality can be viewed intuitively in either R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} . The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear , as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
In spherical geometry , the shortest distance between two points is an arc of a great circle , but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π ] ) with those endpoints. [ 4 ] [ 5 ]
The triangle inequality is a defining property of norms and measures of distance . This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers , Euclidean spaces , the L p spaces ( p ≥ 1 ), and inner product spaces .
The triangle inequality theorem is stated in Euclid's Elements , Book 1, Proposition 20:
[…] in the triangle ABC the sum of any two sides is greater than the remaining one, that is, the sum of BA and AC is greater than BC, the sum of AB and BC is greater than AC, and the sum of BC and CA is greater than AB. [ 6 ]
Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. [ 7 ] Beginning with triangle ABC , an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB . It then is argued that angle β has larger measure than angle α , so side AD is longer than side AC . However:
so the sum of the lengths of sides AB and BC is larger than the length of AC . This proof appears in Euclid's Elements , Book 1, Proposition 20. [ 8 ]
For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths a , b , c that are all positive and excludes the degenerate case of zero area):
A more succinct form of this inequality system can be shown to be
Another way to state it is
implying
and thus that the longest side length is less than the semiperimeter .
A mathematically equivalent formulation is that the area of a triangle with sides a , b , c must be a real number greater than zero. Heron's formula for the area is
In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).
The triangle inequality provides two more interesting constraints for triangles whose sides are a , b , c , where a ≥ b ≥ c and ϕ {\displaystyle \phi } is the golden ratio , as
In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum. [ 10 ]
The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle ADC . An isosceles triangle ABC is constructed with equal sides AB = AC . From the triangle postulate , the angles in the right triangle ADC satisfy:
Likewise, in the isosceles triangle ABC , the angles satisfy:
Therefore,
and so, in particular,
That means side AD , which is opposite to angle α , is shorter than side AB , which is opposite to the larger angle β . But AB = AC . Hence:
A similar construction shows AC > DC , establishing the theorem.
An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point B : [ 11 ] (i) as depicted (which is to be proved), or (ii) B coincident with D (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle γ , which would violate the triangle postulate ), or lastly, (iii) B interior to the right triangle between points A and D (in which case angle ABC is an exterior angle of a right triangle BDC and therefore larger than π /2 , meaning the other base angle of the isosceles triangle also is greater than π /2 and their sum exceeds π in violation of the triangle postulate).
This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.
Consider a triangle whose sides are in an arithmetic progression and let the sides be a , a + d , a + 2 d . Then the triangle inequality requires that
To satisfy all these inequalities requires
When d is chosen such that d = a /3 , it generates a right triangle that is always similar to the Pythagorean triple with sides 3 , 4 , 5 .
Now consider a triangle whose sides are in a geometric progression and let the sides be a , ar , ar 2 . Then the triangle inequality requires that
The first inequality requires a > 0 ; consequently it can be divided through and eliminated. With a > 0 , the middle inequality only requires r > 0 . This now leaves the first and third inequalities needing to satisfy
The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r 2 + r − 1 = 0 , i.e. r > φ − 1 where φ is the golden ratio . The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r 2 − r − 1 = 0 , i.e. 0 < r < φ . The combined requirements result in r being confined to the range
When r the common ratio is chosen such that r = √ φ it generates a right triangle that is always similar to the Kepler triangle .
The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths.
Consider a quadrilateral whose sides are in a geometric progression and let the sides be a , ar , ar 2 , ar 3 . Then the generalized polygon inequality requires that
These inequalities for a > 0 reduce to the following
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, r is limited to the range 1/ t < r < t where t is the tribonacci constant.
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.
No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path. [ 15 ]
The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.
In either case, if the side lengths are a , b , c we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number h consistent with the values a , b , and c , in which case this triangle exists.
By the Pythagorean theorem we have b 2 = h 2 + d 2 and a 2 = h 2 + ( c − d ) 2 according to the figure at the right. Subtracting these yields a 2 − b 2 = c 2 − 2 cd . This equation allows us to express d in terms of the sides of the triangle:
For the height of the triangle we have that h 2 = b 2 − d 2 . By replacing d with the formula given above, we have
For a real number h to satisfy this, h 2 must be non-negative:
which holds if the triangle inequality is satisfied for all sides. Therefore, there does exist a real number h {\displaystyle h} consistent with the sides a , b , c {\displaystyle a,b,c} , and the triangle exists. If each triangle inequality holds strictly , h > 0 {\displaystyle h>0} and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so h = 0 {\displaystyle h=0} , the triangle is degenerate.
The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an ( n − 1) - facet of an n - simplex is less than or equal to the sum of the hypervolumes of the other n facets.
Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.
In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points A , B , C , and Z in Euclidean space such that distances
and
However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle ABC is 169 3 {\textstyle 169{\sqrt {3}}} , which is larger than three times 39 3 {\textstyle 39{\sqrt {3}}} , the area of a 26–14–14 isosceles triangle (all by Heron's formula ), and so the arrangement is forbidden by the tetrahedral inequality.
In a normed vector space V , one of the defining properties of the norm is the triangle inequality:
That is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity . For any proposed function to behave as a norm, it must satisfy this requirement. [ 16 ]
If the normed space is Euclidean , or, more generally, strictly convex , then ‖ u + v ‖ = ‖ u ‖ + ‖ v ‖ {\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|\mathbf {u} \|+\|\mathbf {v} \|} if and only if the triangle formed by u , v , and u + v , is degenerate, that is, u and v are on the same ray, i.e., u = 0 or v = 0 , or u = α v for some α > 0 . This property characterizes strictly convex normed spaces such as the ℓ p spaces with 1 < p < ∞ . However, there are normed spaces in which this is not true. For instance, consider the plane with the ℓ 1 norm (the Manhattan distance ) and denote u = (1, 0) and v = (0, 1) . Then the triangle formed by u , v , and u + v , is non-degenerate but
The absolute value is a norm for the real line ; as required, the absolute value satisfies the triangle inequality for any real numbers u and v . If u and v have the same sign or either of them is zero, then | u + v | = | u | + | v | . {\displaystyle |u+v|=|u|+|v|.} If u and v have opposite signs, then without loss of generality assume | u | > | v | . {\displaystyle |u|>|v|.} Then | u + v | = | u | − | v | < | u | + | v | . {\displaystyle |u+v|=|u|-|v|<|u|+|v|.} Combining these cases: [ 17 ]
| u + v | ≤ | u | + | v | . {\displaystyle |u+v|\leq |u|+|v|.}
The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers. There is also a lower estimate, which can be found using the reverse triangle inequality which states that for any real numbers u and v , | u − v | ≥ | | u | − | v | | . {\displaystyle |u-v|\geq {\bigl |}|u|-|v|{\bigr |}.}
The taxicab norm or 1-norm is one generalization absolute value to higher dimensions. To find the norm of a vector v = ( v 1 , v 2 , … v n ) , {\displaystyle v=(v_{1},v_{2},\ldots v_{n}),} just add the absolute value of each component separately, ‖ v ‖ 1 = | v 1 | + | v 2 | + ⋯ + | v n | . {\displaystyle \|v\|_{1}=|v_{1}|+|v_{2}|+\dotsb +|v_{n}|.}
The Euclidean norm or 2-norm defines the length of translation vectors in an n -dimensional Euclidean space in terms of a Cartesian coordinate system . For a vector v = ( v 1 , v 2 , … v n ) , {\displaystyle v=(v_{1},v_{2},\ldots v_{n}),} its length is defined using the n -dimensional Pythagorean theorem : ‖ v ‖ 2 = | v 1 | 2 + | v 2 | 2 + ⋯ + | v n | 2 . {\displaystyle \|v\|_{2}={\sqrt {|v_{1}|^{2}+|v_{2}|^{2}+\dotsb +|v_{n}|^{2}}}.}
The inner product is norm in any inner product space , a generalization of Euclidean vector spaces including infinite-dimensional examples. The triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors u {\displaystyle u} and v {\displaystyle v} , and denoting the inner product as ⟨ u , v ⟩ {\displaystyle \langle u,v\rangle } : [ 18 ]
The Cauchy–Schwarz inequality turns into an equality if and only if u and v are linearly dependent. The inequality ⟨ u , v ⟩ + ⟨ v , u ⟩ ≤ 2 | ⟨ u , v ⟩ | {\displaystyle \langle u,v\rangle +\langle v,u\rangle \leq 2\left|\left\langle u,v\right\rangle \right|} turns into an equality for linearly dependent u {\displaystyle u} and v {\displaystyle v} if and only if one of the vectors u or v is a nonnegative scalar of the other. Taking the square root of the final result gives the triangle inequality.
The p -norm is a generalization of taxicab and Euclidean norms, using an arbitrary positive integer exponent, ‖ v ‖ p = ( | v 1 | p + | v 2 | p + ⋯ + | v n | p ) 1 / p , {\displaystyle \|v\|_{p}={\bigl (}|v_{1}|^{p}+|v_{2}|^{p}+\dotsb +|v_{n}|^{p}{\bigr )}^{1/p},} where the v i are the components of vector v .
Except for the case p = 2 , the p -norm is not an inner product norm, because it does not satisfy the parallelogram law . The triangle inequality for general values of p is called Minkowski's inequality . [ 19 ] It takes the form: ‖ u + v ‖ p ≤ ‖ u ‖ p + ‖ v ‖ p . {\displaystyle \|u+v\|_{p}\leq \|u\|_{p}+\|v\|_{p}\ .}
In a metric space M with metric d , the triangle inequality is a requirement upon distance :
for all points A , B , and C in M . That is, the distance from A to C is at most as large as the sum of the distance from A to B and the distance from B to C .
The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any x n and x m such that d ( x n , x ) < ε /2 and d ( x m , x ) < ε /2 , where ε > 0 is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, d ( x n , x m ) ≤ d ( x n , x ) + d ( x m , x ) < ε /2 + ε /2 = ε , so that the sequence { x n } is a Cauchy sequence, by definition.
This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via d ( u , v ) ≔ ‖ u − v ‖ , with u − v being the vector pointing from point v to u .
The reverse triangle inequality is an equivalent alternative formulation of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is: [ 20 ]
In the case of a normed vector space, the statement is:
or for metric spaces, | d ( A , C ) − d ( B , C ) | ≤ d ( A , B ) {\displaystyle |d(A,C)-d(B,C)|\leq d(A,B)} .
This implies that the norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} as well as the distance-from- z {\displaystyle z} function d ( z , ⋅ ) {\displaystyle d(z,\cdot )} are Lipschitz continuous with Lipschitz constant 1 , and therefore are in particular uniformly continuous .
The proof of the reverse triangle inequality from the usual one uses ‖ v − u ‖ = ‖ − 1 ( u − v ) ‖ = | − 1 | ⋅ ‖ u − v ‖ = ‖ u − v ‖ {\displaystyle \|v-u\|=\|{-}1(u-v)\|=|{-}1|\cdot \|u-v\|=\|u-v\|} to find:
Combining these two statements gives:
In the converse, the proof of the triangle inequality from the reverse triangle inequality works in two cases:
If ‖ u + v ‖ − ‖ u ‖ ≥ 0 , {\displaystyle \|u+v\|-\|u\|\geq 0,} then by the reverse triangle inequality, ‖ u + v ‖ − ‖ u ‖ = | ‖ u + v ‖ − ‖ u ‖ | ≤ ‖ ( u + v ) − u ‖ = ‖ v ‖ ⇒ ‖ u + v ‖ ≤ ‖ u ‖ + ‖ v ‖ {\displaystyle \|u+v\|-\|u\|={\big |}\|u+v\|-\|u\|{\big |}\leq \|(u+v)-u\|=\|v\|\Rightarrow \|u+v\|\leq \|u\|+\|v\|} ,
and if ‖ u + v ‖ − ‖ u ‖ < 0 , {\displaystyle \|u+v\|-\|u\|<0,} then trivially ‖ u ‖ + ‖ v ‖ ≥ ‖ u ‖ > ‖ u + v ‖ {\displaystyle \|u\|+\|v\|\geq \|u\|>\|u+v\|} by the nonnegativity of the norm.
Thus, in both cases, we find that ‖ u ‖ + ‖ v ‖ ≥ ‖ u + v ‖ {\displaystyle \|u\|+\|v\|\geq \|u+v\|} .
For metric spaces, the proof of the reverse triangle inequality is found similarly by:
d ( A , B ) + d ( B , C ) ≥ d ( A , C ) ⇒ d ( A , B ) ≥ d ( A , C ) − d ( B , C ) {\displaystyle d(A,B)+d(B,C)\geq d(A,C)\Rightarrow d(A,B)\geq d(A,C)-d(B,C)}
d ( C , A ) + d ( A , B ) ≥ d ( C , B ) ⇒ d ( A , B ) ≥ d ( B , C ) − d ( A , C ) {\displaystyle d(C,A)+d(A,B)\geq d(C,B)\Rightarrow d(A,B)\geq d(B,C)-d(A,C)}
Putting these equations together we find:
d ( A , B ) ≥ | d ( A , C ) − d ( B , C ) | {\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|}
And in the converse, beginning from the reverse triangle inequality, we can again use two cases:
If d ( A , C ) − d ( B , C ) ≥ 0 {\displaystyle d(A,C)-d(B,C)\geq 0} , then d ( A , B ) ≥ | d ( A , C ) − d ( B , C ) | = d ( A , C ) − d ( B , C ) ⇒ d ( A , B ) + d ( B , C ) ≥ d ( A , C ) {\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|=d(A,C)-d(B,C)\Rightarrow d(A,B)+d(B,C)\geq d(A,C)} ,
and if d ( A , C ) − d ( B , C ) < 0 , {\displaystyle d(A,C)-d(B,C)<0,} then d ( A , B ) + d ( B , C ) ≥ d ( B , C ) > d ( A , C ) {\displaystyle d(A,B)+d(B,C)\geq d(B,C)>d(A,C)} again by the nonnegativity of the metric.
Thus, in both cases, we find that d ( A , B ) + d ( B , C ) ≥ d ( A , C ) {\displaystyle d(A,B)+d(B,C)\geq d(A,C)} .
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that [ 21 ]
sim ( u , w ) ≥ sim ( u , v ) ⋅ sim ( v , w ) − ( 1 − sim ( u , v ) 2 ) ⋅ ( 1 − sim ( v , w ) 2 ) {\displaystyle \operatorname {sim} (u,w)\geq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)-{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}}
and
sim ( u , w ) ≤ sim ( u , v ) ⋅ sim ( v , w ) + ( 1 − sim ( u , v ) 2 ) ⋅ ( 1 − sim ( v , w ) 2 ) . {\displaystyle \operatorname {sim} (u,w)\leq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)+{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}\,.}
With these formulas, one needs to compute a square root for each triple of vectors { u , v , w } that is examined rather than arccos(sim( u , v )) for each pair of vectors { u , v } examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.
The Minkowski space metric η μ ν {\displaystyle \eta _{\mu \nu }} is not positive-definite, which means that ‖ u ‖ 2 = η μ ν u μ u ν {\displaystyle \|u\|^{2}=\eta _{\mu \nu }u^{\mu }u^{\nu }} can have either sign or vanish, even if the vector u is non-zero. Moreover, if u and v are both timelike vectors lying in the future light cone, the triangle inequality is reversed:
A physical example of this inequality is the twin paradox in special relativity . The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in n + 1 {\displaystyle n+1} dimensions for any n ≥ 1 {\displaystyle n\geq 1} . If the plane defined by u {\displaystyle u} and v {\displaystyle v} is space-like (and therefore a Euclidean subspace) then the usual triangle inequality holds. | https://en.wikipedia.org/wiki/Triangle_inequality |
The triangle of U ( / uː / OO ) is a theory about the evolution and relationships among the six most commonly known members of the plant genus Brassica . The theory states that the genomes of three ancestral diploid species of Brassica combined to create three common tetraploid vegetables and oilseed crop species. [ 1 ] It has since been confirmed by studies of DNA and proteins. [ 2 ]
The theory is summarized by a triangular diagram that shows the three ancestral genomes, denoted by AA, BB, and CC, at the corners of the triangle, and the three derived ones, denoted by AABB, AACC, and BBCC, along its sides.
The theory was first published in 1935 by Woo Jang-choon , [ 3 ] a Korean - Japanese botanist (writing under the Japanized name "U Nagaharu"). [ 4 ] Woo made synthetic hybrids between the diploid and tetraploid species and examined how the chromosomes paired in the resulting triploids.
The six species are
The code in the "Chr.count" column specifies the total number of chromosomes in each somatic cell, and how it relates to the number n of chromosomes in each full genome set (which is also the number found in the pollen or ovule ), and the number x of chromosomes in each component genome. For example, each somatic cell of the tetraploid species Brassica napus , with letter tags AACC and count "2 n =4 x =38", contains two copies of the A genome, each with 10 chromosomes, and two copies of the C genome, each with 9 chromosomes, which is 38 chromosomes in total. That is two full genome sets (one A and one C), hence "2 n =38" which means " n =19" (the number of chromosomes in each gamete ). It is also four component genomes (two A and two C), hence "4 x =38". [ 2 ]
The three diploid species exist in nature, but can easily interbreed because they are closely related. This interspecific breeding allowed for the creation of three new species of tetraploid Brassica . [ 3 ] (Critics, however, consider the geological separation too large.) These are said to be allotetraploid (containing four genomes from two or more different species); more specifically, amphidiploid (with two genomes each from two diploid species). [ 2 ]
The framework proposed by Woo, although backed by modern studies, leaves open questions about the time and place of hybridization and which species is the maternal or paternal parent. B. napus (AACC) is dated to have originated about 8,000 [ 5 ] or 38,000–51,000 [ 6 ] years ago. The homologous part of its constituent chromosomes has crossed over in many cultivars. [ 5 ] B. juncea (AABB) is estimated to have originated 39,000–55,000 years ago. [ 6 ] As of 2020, research on organellar genomes shows that B. nigra (BB) is likely the "mother" of B. carinata (BBCC) and that B. rapa (AA) likely mothered B. juncea . The situation with B. napus (AACC) is more complex: some specimens have a rapa -like organellar genome, while the rest indicate an ancient, unidentified maternal plant. [ 2 ]
Data from molecular studies indicate the three diploid species are themselves paleohexaploids . [ 7 ] [ 8 ]
In 2011 and 2018, novel allohexaploids (AABBCC) located at the "center" of the triangle of U were created by different means, [ 9 ] [ 10 ] [ 11 ] for example by crossing B. rapa (AA) with B. carinata (BBCC), or B. nigra (BB) with B. napus (AACC), or B. oleracea (CC) with B. juncea (AABB), followed by chromosome duplication of the triploid (ABC) offspring to generate doubled haploid (AABBCC) offspring. [ 11 ]
In addition, two stable allohexaploid (AABBSS) intergeneric hybrids between Indian mustard ( B. juncea , AABB) and white mustard ( Sinapis alba , SS) were created in 2020 by protoplast fusion . [ 12 ] | https://en.wikipedia.org/wiki/Triangle_of_U |
The triangle of auscultation is a relative thinning of the musculature of the back , situated along the medial border of the scapula which allows for improved listening to the lungs .
It has the following boundaries:
The superficial floor of the triangle is formed by the lateral portion of the erector spinae muscles. Deep to these muscles are the osseous portions of the 6th and 7th ribs and the internal and external intercostal muscles .
The triangle of auscultation is useful for assessment using a pulmonary auscultation and thoracic procedures. [ 1 ] Due to the relative thinning of the musculature of the back in the triangle, the posterior thoracic wall is closer to the skin surface, making respiratory sounds audible more clearly with a stethoscope . On the left side, the cardiac orifice of the stomach lies deep to the triangle. In days before X-rays were discovered, the sound of swallowed liquids were auscultated over this triangle to confirm an oesophageal tumour .
To better expose the floor of the triangle up of the posterior thoracic wall in the 6th and 7th intercostal space, a patient is asked to fold their arms across their chest , laterally rotating the scapulae, while bending forward at the trunk, somewhat resembling the fetal position .
The triangle of auscultation can be used as a surgical approach path. [ 2 ] [ 3 ] It can also be used for applying a nerve block known as the rhomboid intercostal block , which can be used to relieve pain after rib fractures , and a thoracotomy . This nerve block is usually achieved by injection of the local anesthetic agent into the fascial plane between the rhomboid upper intercostal muscle and the rhombic muscles. [ 4 ] [ 5 ] [ 6 ]
This article incorporates text in the public domain from page 434 of the 20th edition of Gray's Anatomy (1918) | https://en.wikipedia.org/wiki/Triangle_of_auscultation |
In the system of Aristotelian logic , the triangle of opposition is a diagram [ which? ] representing the different ways in which each of the three propositions of the system is logically related ('opposed') to each of the others. The system is also useful in the analysis of syllogistic logic, serving to identify the allowed logical conversions from one type to another.
In the 19th and 20th centuries, other triangles were proposed, including Nicolai A. Vasiliev 's triangle, [ 1 ] the Jespersenian Triangle, Ginzberg’s triangle of contraries and Sir William Hamilton’s subcontraries. [ 2 ]
Philosophy Documentation Center website A Triangle of Opposites for Types of Propositions in Aristotelian Logic , by Paul Jacoby, published in The New Scholasticism (Volume 24, Issue 1, January 1950, pages 32-56)
This logic -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Triangle_of_opposition |
The term triangular coordinates may refer to any of at least three related systems of coordinates in the Euclidean plane : | https://en.wikipedia.org/wiki/Triangular_coordinates |
In computer algebra , a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S 1 , ..., S e such that a point is a solution of S if and only if it is a solution of one of the systems S 1 , ..., S e .
When the purpose is to describe the solution set of S in the algebraic closure of its coefficient field , those simpler systems are regular chains . If the coefficients of the polynomial systems S 1 , ..., S e are real numbers, then the real solutions of S can be obtained by a triangular decomposition into regular semi-algebraic systems . In both cases, each of these simpler systems has a triangular shape and remarkable properties, which justifies the terminology.
The Characteristic Set Method is the first factorization-free algorithm, which was proposed for decomposing an algebraic variety into equidimensional components. Moreover, the Author, Wen-Tsun Wu , realized an implementation of this method and reported experimental data in his 1987 pioneer article titled "A zero structure theorem for polynomial equations solving". [ 1 ] To put this work into context, let us recall what was the common idea of an algebraic set decomposition at the time this article was written.
Let K be an algebraically closed field and k be a subfield of K . A subset V ⊂ K n is an (affine) algebraic variety over k if there exists a polynomial set F ⊂ k [ x 1 , ..., x n ] such that the zero set V ( F ) ⊂ K n of F equals V .
Recall that V is said irreducible if for all algebraic varieties V 1 , V 2 ⊂ K n the relation V = V 1 ∪ V 2 implies either V = V 1 or V = V 2 . A first algebraic variety decomposition result is the famous Lasker–Noether theorem which implies the following.
The varieties V 1 , ..., V e in the above Theorem are called the irreducible components of V and can be regarded as a natural output for a decomposition algorithm, or, in other words, for an algorithm solving a system of equations in k [ x 1 , ..., x n ] .
In order to lead to a computer program, this algorithm specification should prescribe how irreducible components are represented. Such an encoding is introduced by Joseph Ritt [ 2 ] through the following result.
We call the set C in Ritt's Theorem a Ritt characteristic set of the ideal ⟨ F ⟩ {\displaystyle \langle F\rangle } . Please refer to regular chain for the notion of a triangular set.
Joseph Ritt described a method for solving polynomial systems based on polynomial factorization over field extensions and computation of characteristic sets of prime ideals.
Deriving a practical implementation of this method, however, was and remains a difficult problem. In the 1980s, when the Characteristic set Method was introduced, polynomial factorization was an active research area and certain fundamental questions on this subject were solved recently [ 3 ]
Nowadays, decomposing an algebraic variety into irreducible components is not essential to process most application problems, since weaker notions of decompositions, less costly to compute, are sufficient.
The Characteristic Set Method relies on the following variant of Ritt's Theorem.
Different concepts and algorithms extended the work of Wen-Tsun Wu . In the early 1990s, the notion of a regular chain , introduced independently by Michael Kalkbrener in 1991 in his PhD Thesis and, by Lu Yang and Jingzhong Zhang [ 4 ] led to important algorithmic discoveries.
In Kalkbrener's vision, [ 5 ] regular chains are used to represent the generic zeros of the irreducible components of an algebraic variety. In the original work of Yang and Zhang, they are used to decide whether a hypersurface intersects a quasi-variety (given by a regular chain). Regular chains have, in fact, several interesting properties and are the key notion in many algorithms for decomposing systems of algebraic or differential equations.
Regular chains have been investigated in many papers. [ 6 ] [ 7 ] [ 8 ]
The abundant literature on the subject can be explained by the many equivalent definitions of a regular chain. Actually, the original formulation of Kalkbrener is quite different from that of Yang and Zhang. A bridge between these two notions, the point of view of Kalkbrener and that of Yang and Zhang, appears in Dongming Wang's paper. [ 9 ]
There are various algorithms available for obtaining triangular decomposition of V ( F ) both in the sense of Kalkbrener and in the sense of Lazard and Wen-Tsun Wu . The Lextriangular Algorithm by Daniel Lazard [ 10 ] and the Triade Algorithm by Marc Moreno Maza [ 11 ] together with the Characteristic Set Method are available in various computer algebra systems, including Axiom and Maple .
Let k be a field and x 1 < ... < x n be ordered variables. We denote by R = k [ x 1 , ..., x n ] the corresponding polynomial ring . For F ⊂ R , regarded as a system of polynomial equations, there are two notions of a triangular decomposition over the algebraic closure of k . The first one is to decompose lazily, by representing only the generic points of the algebraic set V ( F ) in the so-called sense of Kalkbrener.
The second is to describe explicitly all the points of V ( F ) in the so-called sense of in Lazard and Wen-Tsun Wu .
In both cases T 1 , ..., T e are finitely many regular chains of R and s a t ( T i ) {\displaystyle {\sqrt {\mathrm {sat} (T_{i})}}} denotes the radical of the saturated ideal of T i while W ( T i ) denotes the quasi-component of T i . Please refer to regular chain for definitions of these notions.
Assume from now on that k is a real closed field . Consider S a semi-algebraic system with polynomials in R . There exist [ 12 ] finitely many regular semi-algebraic systems S 1 , ..., S e such that we have
where Z k ( S ) denotes the points of k n which solve S . The regular semi-algebraic systems S 1 , ..., S e form a triangular decomposition of the semi-algebraic system S .
Denote Q the rational number field. In Q [ x , y , z ] {\displaystyle Q[x,y,z]} with variable ordering x > y > z {\displaystyle x>y>z} , consider the following polynomial system:
According to the Maple code:
One possible triangular decompositions of the solution set of S with using RegularChains library is: | https://en.wikipedia.org/wiki/Triangular_decomposition |
The triangular fibrocartilage complex (TFCC) is formed by the triangular fibrocartilage discus (TFC), the radioulnar ligaments (RULs) and the ulnocarpal ligaments (UCLs).
The triangular fibrocartilage disc (TFC) is an articular discus that lies on the pole of the distal ulna . It has a triangular shape and a biconcave body; the periphery is thicker than its center. The central portion of the TFC is thin and consists of chondroid fibrocartilage; this type of tissue is often seen in structures that can bear compressive loads. This central area is often so thin that it is translucent and in some cases it is even absent. [ 1 ] The peripheral portion of the TFC is well vascularized, while the central portion has no blood supply.
This discus is attached by thick tissue to the base of the ulnar styloid and by thinner tissue to the edge of the radius just proximal to the radiocarpal articular surface. [ 1 ]
The radioulnar ligaments (RULs) are the principal stabilizers of the distal radioulnar joint (DRUJ). There are two RULs: the palmar and dorsal radioulnar ligaments . [ citation needed ]
These ligaments arise from the distal radius medial border and insert on the ulna at two separate and distinct sites: the ulna styloid and the fovea (a groove that separates the ulnar styloid from the ulnar head). Each ligament consists of a superficial component and a deep component. The superficial components insert directly onto the ulna styloid. The deep components insert more anterior, into the fovea adjacent to the articular surface of the dome of the distal ulna. [ citation needed ]
The ligaments are composed of longitudinally oriented lamellar collagen to resist tensile loads and have a rich vascular supply to allow healing. [ 2 ] [ 3 ] [ 4 ]
The ulnocarpal ligaments (UCLs) consist of the ulnolunate and the ulnotriquetral ligaments. They originate from the ulnar styloid and insert into the carpal bones of the wrist: the ulnolunate ligament inserts into the lunate bone and the ulnotriquetral ligament into the triquetrum bone. These ligaments prevent dorsal migration of the distal ulna. They are more taut during supination , because in supination ulnar styloid moves away from the carpal bones volar side. [ 1 ] [ 5 ] [ 6 ]
The primary functions of the TFCC:
The TFCC is important in load transmission across the ulnar aspect of the wrist. The TFC transmits and absorbs compressive forces.
The ulnar variance influences the amount of load that is transmitted through the distal ulna. The load transmission is directly proportional to this ulnar variance. In neutral ulnar variance, approximately 20 percent of the load is transmitted. With negative ulnar variance, the load across the TFC is decreased. This occurs during supination, because the radius moves distally on the ulna and creates a negative ulnar variance. With positive ulnar variance it is reversed. The load that is transmitted across the TFC is then increased. This positive ulnar variance occurs during pronation. [ 7 ]
The TFCC is a major stabilizer of the DRUJ. To control the forearm rotation the DRUJ acts in concert with the proximal radioulnar joint. The connection between the distal radius and the distal ulna, maintain the congruency of the DRUJ. This attachment is mainly created by the RULs of the TFCC. These ligaments support the joint through its arc of rotation. [ 7 ]
The role of the TFCC in supination and in pronation is a matter of dispute. Some authors (Schuind et al. ) concluded that the dorsal fibers of the TFCC tighten in pronation, and the palmar fibers in supination. These conclusions are opposite of those published by Af Ekenstam and Hagert. [ 2 ]
Both parties are in fact right, as the RULs consists of two ligaments each made of another two components: the superficial and the deep ligaments. During supination, the superficial palmar and the deep dorsal ligaments are tightened, [ citation needed ] preventing palmar translation of the ulna. In pronation, this is reversed: the superficial dorsal and the deep palmar ligaments are tightened and prevent dorsal translation of the ulna. [ citation needed ]
The TFCC has a substantial risk for injury and degeneration because of its anatomic complexity and multiple functions.
Application of an extension-pronation force to an axial-load wrist, such as in a fall on an outstretched hand, causes most of the traumatic injuries of the TFCC.
Dorsal rotation injury, such as when a drill binds and rotates the wrist instead of the bit, can also cause traumatic injuries.
Injury may also occur from a distraction force applied to the volar forearm or wrist. Finally, tears of the TFCC are frequently found by patients with distal radius fractures. [ 7 ]
Perforations and defects in the TFCC are not all traumatic. There is an age related correlation with lesions in the TFCC, but many of these defects are asymptomatic. These lesions common occur by patients with positive ulnar variance. [ 7 ]
Chronic and excessive loading through the ulnocarpal joint, causes degenerative TFCC tears. These tears are a component of ulnar impaction syndrome.
Even though natural degeneration of the ulnocarpal joint is very common, it is important to recognize. In cadavaric examinations, 30% to 70% of the cases had TFCC perforations and chondromalacia of the ulnar head, lunate, and triquetrum. Cases with ulnar-negative variance had fewer degenerative changes. [ 1 ]
The Palmer classification is the most recognized scheme; it divides TFCC lesions into these two categories: traumatic and degenerative. This classification provides an anatomic description of tears, it does not guide treatment or indicate prognosis. [ 1 ]
Patients with a TFCC injury usually experience pain or discomfort located at the ulnar side of the wrist, often just above the ulnar styloid. However, there are also some patients who report diffuse pain throughout the entire wrist.
Rest can reduce pain and activity can make it worse, especially with rotating movements (supination and pronation) of the wrist or movements of the hand sideways in ulnar direction.
Other symptoms patients with a TFCC injury frequently mention are: swelling, loss of grip strength, instability, and grinding or clicking sounds ( crepitus ) that can occur during activity of the wrist. [ 8 ]
Injuries to the TFCC may be preceded by a fall on a pronated outstretched arm; a rotational injury to the forearm; an axial load trauma to the wrist; or a distraction injury of the wrist in ulnar direction. [ 8 ] However, not all patients can recall that a preceding trauma occurred.
Note: Imaging techniques can only be relevant together with the clinical findings of a carefully performed physical examination.
Other than a TFCC injury, there are many possible causes for ulnar-sided wrist pain.
The initial treatment for both traumatic and degenerative TFCC lesions, with a stable DRUJ, is conservative (nonsurgical) therapy. Patients may be advised to wear a temporary splint or cast to immobilize the wrist and forearm for four to six weeks. [ 10 ] The immobilization allows scar tissue to develop which can help heal the TFCC. In addition, oral NSAIDs and corticosteroid joint injections can be prescribed for pain relief. Physiotherapy and occupational therapy can help patients recover after immobilization or surgery. Wrist support straps used in sports can also be used in mild cases to compress and minimize movement of the area. [ 11 ]
Indications for acute TFCC surgery are: a clearly unstable DRUJ, or the existence of additional unstable or displaced fractures. TFCC surgery is also indicated when conservative treatment proves insufficient in about 8–12 weeks.
Fractures of the radius bone are often associated by TFCC damage. If the fracture is treated surgically it is recommended to evaluate and if necessary repair the TFCC as well. [ 12 ] Closed fractures (where the skin is still intact) of the radius bone are treated non-surgically with cast; the immobilization can also help heal the TFCC.
The central part of the TFC has no blood supply and therefore has no healing capacity. When a tear occurs in this area of the TFC, it typically creates an unstable flap of tissue that is likely to catch on other joint surfaces. Removing the damaged tissue (debridement) is then indicated. Arthroscopic debridement as a treatment for degenerative TFC tears associated with positive ulnar variance, unfortunately, show poor results. [ 10 ] [ 9 ]
Suturing TFCC ligaments can sometimes be performed arthroscopically. But only if there is no serious damage to the ligaments or other surrounding structures. Even after a short period of time torn ligaments tend to retract and therefore lose length. Retracted ligament ends are impossible to suture together again and a reconstruction may be necessary. [ citation needed ]
Open surgery is usually required for degenerative or more complex TFCC injuries, or if additional damage to the wrist or forearm caused instability or displacement. It is a more invasive surgical technique compared to arthroscopic treatment, but the surgeon has better visibility and access to the TFCC. [ citation needed ]
This article incorporates text in the public domain from page 325 of the 20th edition of Gray's Anatomy (1918) | https://en.wikipedia.org/wiki/Triangular_fibrocartilage |
The triangular interval (also known as the lateral triangular space , [ 1 ] lower triangular space , [ 2 ] and triceps hiatus ) is a space found in the axilla . It is one of the three intermuscular spaces found in the axillary space . The other two spaces are: quadrangular space and triangular space . [ 3 ]
Two of its borders are as follows:
Some sources state the lateral border is the humerus , [ 4 ] [ 5 ] while others define it as the lateral head of the triceps. [ 2 ] (The effective difference is relatively minor, though.)
The contents of its borders are as follows:
The radial nerve is visible through the triangular interval, on its way to the posterior compartment of the arm . Profunda brachii also passes through the triangular interval from anterior to posterior.
Triangular Interval Syndrome (TIS) was described as a differential diagnosis for radicular pain in the upper extremity. [ 6 ] It is a condition where the radial nerve is entrapped in the triangular interval resulting in upper extremity radicular pain. The radial nerve and profunda brachii pass through the triangular interval and are hence vulnerable. The triangular interval has a potential for compromise secondary alterations in thickness of the teres major and triceps. [ 7 ] It is described based on cadaveric studies that fibrous bands were commonly present between the teres major and triceps. When these bands were present, rotation of the shoulder caused a reduction in cross sectional area of the space. Normal resting postures of humeral adduction and internal rotation with scapular protraction may be speculated as a precedent for teres major contractures owing to the shortened position of this muscle in this position. In addition, hypertrophy of this muscle can occur secondary to weight training and potentially compromise the triangular interval with resultant entrapment of the radial nerve. [ 8 ] Shoulder dysfunctions have a potential for shortening and hypertrophy of the teres major. Shoulders that exhibit stiffness, secondary to capsular tightness, contribute to contracture and hypertrophy of the teres major. [ 9 ] Hence, restricted external rotation can encourage adaptive shortening and thickening of the internal rotators of the shoulder principally the teres major and subscapularis. One may speculate that the lateral arm pain presented in shoulder dysfunctions may be of a nerve origin secondary to adverse neural tension of the radial nerve. The triceps brachii has a potential to entrap the radial nerve in the triangular interval secondary to hypertrophy. The presence of a fibrous arch in the long head and lateral head further complicates the situation. Repeated forceful extension seen in weight training and sport involving punching may be a precedent to this scenario. The radial nerve is vulnerable as it passes through this space, for all of the reasons mentioned above. | https://en.wikipedia.org/wiki/Triangular_interval |
In coding theory , triangular network coding ( TNC ) is a non-linear network coding based packet coding scheme introduced by Qureshi, Foh & Cai (2012) . [ 1 ] Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large finite field is that it resulted in high encoding and decoding computational complexity . While linear encoding and decoding over GF(2) alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance.
The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of O ( n 3 ) {\displaystyle O(n^{3})} to O ( n 2 ) {\displaystyle O(n^{2})} (where n is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with code rate comparable to that of optimal coding schemes.
Triangular code has also been proposed as Fountain code [ 2 ] to achieve near-optimal performance with encoding and decoding computational complexity of O ( n log n ) {\displaystyle O(n\log n)} . It has been further shown that triangular based fountain code can even outperform optimized Luby transform code . [ 2 ]
In TNC, coding is performed in two stages. First redundant "0" bits are added at the head and tail of each packet such that all packets are of uniform bit length. Then the packets are XOR coded , bit-by-bit. The "0" bits are added in such a way that these redundant "0" bits added to each packet generate a triangular pattern .
In essence, the TNC decoding process, like the LNC decoding process involves Gaussian elimination . However, since the packets in TNC have been coded in such a manner that the resulting coded packets are in triangular pattern, the computational process of triangularization, [ 3 ] with complexity of O ( n 3 ) {\displaystyle O(n^{3})} , where n {\displaystyle n} is the number of packets, can be bypassed. The receiver now only needs to perform back-substitution, [ 3 ] with worst-case complexity given as O ( n 2 ) {\displaystyle O(n^{2})} for each bit location.
This article related to telecommunications is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Triangular_network_coding |
A triangular number or triangle number counts objects arranged in an equilateral triangle . Triangular numbers are a type of figurate number , other examples being square numbers and cube numbers . The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n . The first 100 terms sequence of triangular numbers, starting with the 0th triangular number , are
(sequence A000217 in the OEIS )
The triangular numbers are given by the following explicit formulas:
where ( n + 1 2 ) {\displaystyle \textstyle {n+1 \choose 2}} is notation for a binomial coefficient . It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as " n plus one choose two".
The fact that the n {\displaystyle n} th triangular number equals n ( n + 1 ) / 2 {\displaystyle n(n+1)/2} can be illustrated using a visual proof . [ 1 ] For every triangular number T n {\displaystyle T_{n}} , imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions n × ( n + 1 ) {\displaystyle n\times (n+1)} , which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or: T n = n ( n + 1 ) 2 {\displaystyle T_{n}={\frac {n(n+1)}{2}}} . The example T 4 {\displaystyle T_{4}} follows:
This formula can be proven formally using mathematical induction . [ 2 ] It is clearly true for 1 {\displaystyle 1} :
T 1 = ∑ k = 1 1 k = 1 ( 1 + 1 ) 2 = 2 2 = 1. {\displaystyle T_{1}=\sum _{k=1}^{1}k={\frac {1(1+1)}{2}}={\frac {2}{2}}=1.}
Now assume that, for some natural number m {\displaystyle m} , T m = ∑ k = 1 m k = m ( m + 1 ) 2 {\displaystyle T_{m}=\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}} . We can then verify it for m + 1 {\displaystyle m+1} : ∑ k = 1 m + 1 k = ∑ k = 1 m k + ( m + 1 ) = m ( m + 1 ) 2 + m + 1 = m 2 + m 2 + 2 m + 2 2 = m 2 + 3 m + 2 2 = ( m + 1 ) ( m + 2 ) 2 , {\displaystyle {\begin{aligned}\sum _{k=1}^{m+1}k&=\sum _{k=1}^{m}k+(m+1)\\&={\frac {m(m+1)}{2}}+m+1\\&={\frac {m^{2}+m}{2}}+{\frac {2m+2}{2}}\\&={\frac {m^{2}+3m+2}{2}}\\&={\frac {(m+1)(m+2)}{2}},\end{aligned}}}
so if the formula is true for m {\displaystyle m} , it is true for m + 1 {\displaystyle m+1} . Since it is clearly true for 1 {\displaystyle 1} , it is therefore true for 2 {\displaystyle 2} , 3 {\displaystyle 3} , and ultimately all natural numbers n {\displaystyle n} by induction.
The German mathematician and scientist, Carl Friedrich Gauss , is said to have found this relationship in his early youth, by multiplying n / 2 pairs of numbers in the sum by the values of each pair n + 1 . [ 3 ] However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans in the 5th century BC. [ 4 ] The two formulas were described by the Irish monk Dicuil in about 816 in his Computus . [ 5 ] An English translation of Dicuil's account is available. [ 6 ]
Occasionally it is necessary to compute large triangular numbers where the standard formula t = n*(n+1)/2 would suffer integer overflow before the final division by 2. For example, T 20 = 210 < 256, so will fit into an 8-bit byte , but not the intermediate product 420. This can be solved by dividing either n or n+1 by 2 before the multiplication, whichever is even. This does not require a conditional branch if implemented as t = (n|1) * ((n+1)/2) . If n is odd, the binary OR operation n|1 has no effect, so this is equivalent to t = n * ((n+1)/2) and thus correct. If n is even, setting the low bit with n|1 is the same as adding 1, while the 1 added before the division is truncated away , so this is equivalent to t = (n+1) * (n/2) and also correct.
Triangular numbers have a wide variety of relations to other figurate numbers.
Most simply, the sum of two consecutive triangular numbers is a square number, since: [ 7 ] [ 8 ]
with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum): T n + T n − 1 = ( n 2 2 + n 2 ) + ( ( n − 1 ) 2 2 + n − 1 ( n − 1 ) 2 2 ) = ( n 2 2 + n 2 ) + ( n 2 2 − n 2 ) = n 2 = ( T n − T n − 1 ) 2 . {\displaystyle T_{n}+T_{n-1}=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {\left(n-1\right)^{2}}{2}}+{\frac {n-1{\vphantom {\left(n-1\right)^{2}}}}{2}}\right)=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {n^{2}}{2}}-{\frac {n}{2}}\right)=n^{2}=(T_{n}-T_{n-1})^{2}.}
This property, colloquially known as the theorem of Theon of Smyrna , [ 9 ] is visually demonstrated in the following sum, which represents T 4 + T 5 = 5 2 {\displaystyle T_{4}+T_{5}=5^{2}} as digit sums :
4 3 2 1 + 1 2 3 4 5 5 5 5 5 5 {\displaystyle {\begin{array}{ccccccc}&4&3&2&1&\\+&1&2&3&4&5\\\hline &5&5&5&5&5\end{array}}}
This fact can also be demonstrated graphically by positioning the triangles in opposite directions to create a square:
The double of a triangular number, as in the visual proof from the above section § Formula , is called a pronic number .
There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36, 1225. Some of them can be generated by a simple recursive formula: S n + 1 = 4 S n ( 8 S n + 1 ) {\displaystyle S_{n+1}=4S_{n}\left(8S_{n}+1\right)} with S 1 = 1. {\displaystyle S_{1}=1.}
All square triangular numbers are found from the recursion S n = 34 S n − 1 − S n − 2 + 2 {\displaystyle S_{n}=34S_{n-1}-S_{n-2}+2} with S 0 = 0 {\displaystyle S_{0}=0} and S 1 = 1. {\displaystyle S_{1}=1.}
Also, the square of the n th triangular number is the same as the sum of the cubes of the integers 1 to n . This can also be expressed as ∑ k = 1 n k 3 = ( ∑ k = 1 n k ) 2 . {\displaystyle \sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.}
The sum of the first n triangular numbers is the n th tetrahedral number : ∑ k = 1 n T k = ∑ k = 1 n k ( k + 1 ) 2 = n ( n + 1 ) ( n + 2 ) 6 . {\displaystyle \sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}={\frac {n(n+1)(n+2)}{6}}.}
More generally, the difference between the n th m -gonal number and the n th ( m + 1) -gonal number is the ( n − 1) th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number ; the n th centered k -gonal number is obtained by the formula C k n = k T n − 1 + 1 {\displaystyle Ck_{n}=kT_{n-1}+1}
where T is a triangular number.
The positive difference of two triangular numbers is a trapezoidal number .
The pattern found for triangular numbers ∑ n 1 = 1 n 2 n 1 = ( n 2 + 1 2 ) {\displaystyle \sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}} and for tetrahedral numbers ∑ n 2 = 1 n 3 ∑ n 1 = 1 n 2 n 1 = ( n 3 + 2 3 ) , {\displaystyle \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{3}+2}{3}},} which uses binomial coefficients , can be generalized. This leads to the formula: [ 10 ] ∑ n k − 1 = 1 n k ∑ n k − 2 = 1 n k − 1 … ∑ n 2 = 1 n 3 ∑ n 1 = 1 n 2 n 1 = ( n k + k − 1 k ) {\displaystyle \sum _{n_{k-1}=1}^{n_{k}}\sum _{n_{k-2}=1}^{n_{k-1}}\dots \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{k}+k-1}{k}}}
Triangular numbers correspond to the first-degree case of Faulhaber's formula .
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Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers.
Every even perfect number is triangular (as well as hexagonal), given by the formula M p 2 p − 1 = M p ( M p + 1 ) 2 = T M p {\displaystyle M_{p}2^{p-1}={\frac {M_{p}(M_{p}+1)}{2}}=T_{M_{p}}} where M p is a Mersenne prime . No odd perfect numbers are known; hence, all known perfect numbers are triangular.
For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.
In base 10 , the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9:
The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".
The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.
If x is a triangular number, a is an odd square, and b = a − 1 / 8 , then ax + b is also a triangular number. Note that b will always be a triangular number, because 8 T n + 1 = (2 n + 1) 2 , which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for b given a is an odd square is the inverse of this operation.
The first several pairs of this form (not counting 1 x + 0 ) are: 9 x + 1 , 25 x + 3 , 49 x + 6 , 81 x + 10 , 121 x + 15 , 169 x + 21 , ... etc. Given x is equal to T n , these formulas yield T 3 n + 1 , T 5 n + 2 , T 7 n + 3 , T 9 n + 4 , and so on.
The sum of the reciprocals of all the nonzero triangular numbers is ∑ n = 1 ∞ 1 n 2 + n 2 = 2 ∑ n = 1 ∞ 1 n 2 + n = 2. {\displaystyle \sum _{n=1}^{\infty }{1 \over {{n^{2}+n} \over 2}}=2\sum _{n=1}^{\infty }{1 \over {n^{2}+n}}=2.}
This can be shown by using the basic sum of a telescoping series : ∑ n = 1 ∞ 1 n ( n + 1 ) = 1. {\displaystyle \sum _{n=1}^{\infty }{1 \over {n(n+1)}}=1.}
In addition, the n th partial sum of this series can be written as 2n / n + 1
Two other formulas regarding triangular numbers are T a + b = T a + T b + a b {\displaystyle T_{a+b}=T_{a}+T_{b}+ab} and T a b = T a T b + T a − 1 T b − 1 , {\displaystyle T_{ab}=T_{a}T_{b}+T_{a-1}T_{b-1},} both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra. The first formula are relevant to multiplication algorithm#Quarter square multiplication .
In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers, writing in his diary his famous words, " ΕΥΡΗΚΑ! num = Δ + Δ + Δ ". The three triangular numbers are not necessarily distinct, or nonzero; for example 20 = 10 + 10 + 0. This is a special case of the Fermat polygonal number theorem .
The largest triangular number of the form 2 k − 1 is 4095 (see Ramanujan–Nagell equation ).
Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in geometric progression . It was conjectured by Polish mathematician Kazimierz Szymiczek to be impossible and was later proven by Fang and Chen in 2007. [ 11 ] [ 12 ]
Formulas involving expressing an integer as the sum of triangular numbers are connected to theta functions , in particular the Ramanujan theta function . [ 13 ] [ 14 ]
The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation : L n = 3 T n − 1 = 3 ( n 2 ) ; L n = L n − 1 + 3 ( n − 1 ) , L 1 = 0. {\displaystyle L_{n}=3T_{n-1}=3{n \choose 2};\qquad L_{n}=L_{n-1}+3(n-1),~L_{1}=0.}
In the limit , the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n = 1 3 . {\displaystyle \lim _{n\to \infty }{\frac {T_{n}}{L_{n}}}={\frac {1}{3}}.}
The triangular number T n solves the handshake problem of counting the number of handshakes if each person in a room with n + 1 people shakes hands once with each person. In other words, the solution to the handshake problem of n people is T n −1 . [ 15 ]
Equivalently, a fully connected network of n computing devices requires the presence of T n − 1 cables or other connections.
In a tournament format that uses a round-robin group stage , the number of matches that need to be played between n teams is equal to the triangular number T n − 1 . For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.
One way of calculating the depreciation of an asset is the sum-of-years' digits method , which involves finding T n , where n is the length in years of the asset's useful life. Each year, the item loses ( b − s ) × n − y / T n , where b is the item's beginning value (in units of currency), s is its final salvage value, n is the total number of years the item is usable, and y the current year in the depreciation schedule. Under this method, an item with a usable life of n = 4 years would lose 4 / 10 of its "losable" value in the first year, 3 / 10 in the second, 2 / 10 in the third, and 1 / 10 in the fourth, accumulating a total depreciation of 10 / 10 (the whole) of the losable value.
Board game designers Geoffrey Engelstein and Isaac Shalev describe triangular numbers as having achieved "nearly the status of a mantra or koan among game designers ", describing them as "deeply intuitive" and "featured in an enormous number of games, [proving] incredibly versatile at providing escalating rewards for larger sets without overly incentivizing specialization to the exclusion of all other strategies". [ 16 ]
By analogy with the square root of x , one can define the (positive) triangular root of x as the number n such that T n = x : [ 17 ] n = 8 x + 1 − 1 2 {\displaystyle n={\frac {{\sqrt {8x+1}}-1}{2}}}
which follows immediately from the quadratic formula . So an integer x is triangular if and only if 8 x + 1 is a square. Equivalently, if the positive triangular root n of x is an integer, then x is the n th triangular number. [ 17 ]
By analogy with the factorial function, a product whose factors are the integers from 1 to n , Donald Knuth proposed the name Termial function , [ 18 ] with the notation n ? for the sum whose terms are the integers from 1 to n (the n th triangular number). Although some other sources use this name and notation, [ 19 ] they are not in wide use. | https://en.wikipedia.org/wiki/Triangular_number |
In trigonometry and geometry , triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Specifically in surveying , triangulation involves only angle measurements at known points, rather than measuring distances to the point directly as in trilateration ; the use of both angles and distance measurements is referred to as triangulateration .
Computer stereo vision and optical 3D measuring systems use this principle to determine the spatial dimensions and the geometry of an item. [ 2 ] Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object's surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the base b and must be known. By determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3D coordinate, is calculated from the triangular relations.
Triangulation today is used for many purposes, including surveying , navigation , metrology , astrometry , binocular vision , model rocketry and, in the military, the gun direction, the trajectory and distribution of fire power of weapons .
The use of triangles to estimate distances dates to antiquity. In the 6th century BC, about 250 years prior to the establishment of the Ptolemaic dynasty , the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids of ancient Egypt . He measured the length of the pyramids' shadows and that of his own at the same moment, and compared the ratios to his height ( intercept theorem ). [ 3 ] Thales also estimated the distances to ships at sea as seen from a clifftop by measuring the horizontal distance traversed by the line-of-sight for a known fall, and scaling up to the height of the whole cliff. [ 4 ] Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus , a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope , i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra , the forerunner of the Arabic alidade . A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria ( c. 10 –70 AD), which survived in Arabic translation; but the knowledge became lost in Europe until in 1615 Snellius , after the work of Eratosthenes , reworked the technique for an attempt to measure the circumference of the earth. In China, Pei Xiu (224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establish distances, [ 5 ] while Liu Hui ( c. 263 ) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places. [ 6 ] [ 7 ] | https://en.wikipedia.org/wiki/Triangulation |
In surveying , triangulation is the process of determining the location of a point by measuring only angles to it from known points at either end of a fixed baseline by using trigonometry , rather than measuring distances to the point directly as in trilateration . The point can then be fixed as the third point of a triangle with one known side and two known angles.
Triangulation can also refer to the accurate surveying of systems of very large triangles, called triangulation networks . This followed from the work of Willebrord Snell in 1615–17, who showed how a point could be located from the angles subtended from three known points, but measured at the new unknown point rather than the previously fixed points, a problem called resectioning . Surveying error is minimized if a mesh of triangles at the largest appropriate scale is established first. Points inside the triangles can all then be accurately located with reference to it. Such triangulation methods were used for accurate large-scale land surveying until the rise of global navigation satellite systems in the 1980s.
Triangulation may be used to find the position of the ship when the positions of A and B are known. An observer at A measures the angle α , while the observer at B measures β .
The position of any vertex of a triangle can be calculated if the position of one side, and two angles, are known. The following formulae are strictly correct only for a flat surface. If the curvature of the Earth must be allowed for, then spherical trigonometry must be used.
With ℓ {\displaystyle \ell } being the distance between A and B gives:
Using the trigonometric identities tan α = sin α / cos α and sin(α + β) = sin α cos β + cos α sin β, this is equivalent to:
therefore:
From this, it is easy to determine the distance of the unknown point from either observation point, its north/south and east/west offsets from the observation point, and finally its full coordinates.
Triangulation today is used for many purposes, including surveying , navigation , metrology , astrometry , binocular vision , model rocketry and gun direction of weapons .
In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the agrimensores ; but were introduced into medieval Spain through Arabic treatises on the astrolabe , such as that by Ibn al-Saffar (d. 1035). [ 1 ] Abu Rayhan Biruni (d. 1048) also introduced triangulation techniques to measure the size of the Earth and the distances between various places. [ 2 ] Simplified Roman techniques then seem to have co-existed with more sophisticated techniques used by professional surveyors. But it was rare for such methods to be translated into Latin (a manual on geometry, the eleventh century Geomatria incerti auctoris is a rare exception), and such techniques appear to have percolated only slowly into the rest of Europe. [ 1 ] Increased awareness and use of such techniques in Spain may be attested by the medieval Jacob's staff , used specifically for measuring angles, which dates from about 1300; and the appearance of accurately surveyed coastlines in the Portolan charts , the earliest of which that survives is dated 1296.
On land, the cartographer Gemma Frisius proposed using triangulation to accurately position far-away places for map-making in his 1533 pamphlet Libellus de Locorum describendorum ratione ( Booklet concerning a way of describing places ), which he bound in as an appendix in a new edition of Peter Apian 's best-selling 1524 Cosmographica . This became very influential, and the technique spread across Germany, Austria and the Netherlands. The astronomer Tycho Brahe applied the method in Scandinavia, completing a detailed triangulation in 1579 of the island of Hven , where his observatory was based, with reference to key landmarks on both sides of the Øresund , producing an estate plan of the island in 1584. [ 3 ] In England Frisius's method was included in the growing number of books on surveying which appeared from the middle of the century onwards, including William Cuningham 's Cosmographical Glasse (1559), Valentine Leigh's Treatise of Measuring All Kinds of Lands (1562), William Bourne 's Rules of Navigation (1571), Thomas Digges 's Geometrical Practise named Pantometria (1571), and John Norden 's Surveyor's Dialogue (1607). It has been suggested that Christopher Saxton may have used rough-and-ready triangulation to place features in his county maps of the 1570s; but others suppose that, having obtained rough bearings to features from key vantage points, he may have estimated the distances to them simply by guesswork. [ 4 ]
The modern systematic use of triangulation networks stems from the work of the Dutch mathematician Willebrord Snell , who in 1615 surveyed the distance from Alkmaar to Breda , approximately 72 miles (116 kilometres), using a chain of quadrangles containing 33 triangles in all. Snell underestimated the distance by 3.5%. The two towns were separated by one degree on the meridian , so from his measurement he was able to calculate a value for the circumference of the earth – a feat celebrated in the title of his book Eratosthenes Batavus ( The Dutch Eratosthenes ), published in 1617. Snell calculated how the planar formulae could be corrected to allow for the curvature of the earth. He also showed how to resection , or calculate, the position of a point inside a triangle using the angles cast between the vertices at the unknown point. These could be measured much more accurately than bearings of the vertices, which depended on a compass. This established the key idea of surveying a large-scale primary network of control points first, and then locating secondary subsidiary points later, within that primary network.
Snell's methods were taken up by Jean Picard who in 1669–70 surveyed one degree of latitude along the Paris Meridian using a chain of thirteen triangles stretching north from Paris to the clocktower of Sourdon , near Amiens . Thanks to improvements in instruments and accuracy, Picard's is rated as the first reasonably accurate measurement of the radius of the earth. Over the next century this work was extended most notably by the Cassini family: between 1683 and 1718 Jean-Dominique Cassini and his son Jacques Cassini surveyed the whole of the Paris meridian from Dunkirk to Perpignan ; and between 1733 and 1740 Jacques and his son César Cassini undertook the first triangulation of the whole country, including a re-surveying of the meridian arc , leading to the publication in 1745 of the first map of France constructed on rigorous principles.
Triangulation methods were by now well established for local mapmaking, but it was only towards the end of the 18th century that other countries began to establish detailed triangulation network surveys to map whole countries. The Principal Triangulation of Great Britain was begun by the Ordnance Survey in 1783, though not completed until 1853; and the Great Trigonometric Survey of India, which ultimately named and mapped Mount Everest and the other Himalayan peaks, was begun in 1801. For the Napoleonic French state, the French triangulation was extended by Jean-Joseph Tranchot into the German Rhineland from 1801, subsequently completed after 1815 by the Prussian general Karl von Müffling . Meanwhile, the mathematician Carl Friedrich Gauss was entrusted from 1821 to 1825 with the triangulation of the kingdom of Hanover ( Gaussian land survey [ de ] ), on which he applied the method of least squares to find the best fit solution for problems of large systems of simultaneous equations given more real-world measurements than unknowns.
Today, large-scale triangulation networks for positioning have largely been superseded by the global navigation satellite systems established since the 1980s, but many of the control points for the earlier surveys still survive as valued historical features in the landscape, such as the concrete triangulation pillars set up for retriangulation of Great Britain (1936–1962), or the triangulation points set up for the Struve Geodetic Arc (1816–1855), now scheduled as a UNESCO World Heritage Site . | https://en.wikipedia.org/wiki/Triangulation_(surveying) |
In mathematics, triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism . A space that admits such a homeomorphism is called a triangulable space . Triangulations can also be used to define a piecewise linear structure for a space, if one exists. Triangulation has various applications both in and outside of mathematics, for instance in algebraic topology, in complex analysis, and in modeling.
On the one hand, it is sometimes useful to forget about superfluous information of topological spaces: The replacement of the original spaces with simplicial complexes may help to recognize crucial properties and to gain a better understanding of the considered object.
On the other hand, simplicial complexes are objects of combinatorial character and therefore one can assign them quantities arising from their combinatorial pattern, for instance, the Euler characteristic . Triangulation allows now to assign such quantities to topological spaces.
Investigations concerning the existence and uniqueness of triangulations established a new branch in topology, namely piecewise linear topology (or PL topology). Its main purpose is to study the topological properties of simplicial complexes and their generalizations, cell-complexes .
An abstract simplicial complex above a set V {\displaystyle V} is a system T ⊂ P ( V ) {\displaystyle {\mathcal {T}}\subset {\mathcal {P}}(V)} of non-empty subsets such that:
The elements of T {\displaystyle {\mathcal {T}}} are called simplices, the elements of V {\displaystyle V} are called vertices. A simplex with n + 1 {\displaystyle n+1} vertices has dimension n {\displaystyle n} by definition. The dimension of an abstract simplicial complex is defined as dim ( T ) = sup { dim ( F ) : F ∈ T } ∈ N ∪ ∞ {\displaystyle {\text{dim}}({\mathcal {T}})={\text{sup}}\;\{{\text{dim}}(F):F\in {\mathcal {T}}\}\in \mathbb {N} \cup \infty } . [ 1 ]
Abstract simplicial complexes can be realized as geometrical objects by associating each abstract simplex with a geometric simplex, defined below.
Let p 0 , . . . p n {\displaystyle p_{0},...p_{n}} be n + 1 {\displaystyle n+1} affinely independent points in R n {\displaystyle \mathbb {R} ^{n}} ; i.e. the vectors ( p 1 − p 0 ) , ( p 2 − p 0 ) , … ( p n − p 0 ) {\displaystyle (p_{1}-p_{0}),(p_{2}-p_{0}),\dots (p_{n}-p_{0})} are linearly independent . The set Δ = { ∑ i = 0 n t i p i | each t i ∈ [ 0 , 1 ] and ∑ i = 0 n t i = 1 } {\textstyle \Delta =\{\sum _{i=0}^{n}t_{i}p_{i}\,|\,{\text{each}}\,t_{i}\in [0,1]\,{\text{and}}\,\sum _{i=0}^{n}t_{i}=1\}} is said to be the simplex spanned by p 0 , . . . p n {\displaystyle p_{0},...p_{n}} . It has dimension n {\displaystyle n} by definition. The points p 0 , . . . p n {\displaystyle p_{0},...p_{n}} are called the vertices of Δ {\displaystyle \Delta } , the simplices spanned by n {\displaystyle n} of the n + 1 {\displaystyle n+1} vertices are called faces, and the boundary ∂ Δ {\displaystyle \partial \Delta } is defined to be the union of the faces.
The n {\displaystyle n} -dimensional standard-simplex is the simplex spanned by the unit vectors e 0 , . . . e n {\displaystyle e_{0},...e_{n}} [ 2 ]
A geometric simplicial complex S ⊆ P ( R n ) {\displaystyle {\mathcal {S}}\subseteq {\mathcal {P}}(\mathbb {R} ^{n})} is a collection of geometric simplices such that
The union of all the simplices in S {\displaystyle {\mathcal {S}}} gives the set of points of S {\displaystyle {\mathcal {S}}} , denoted | S | = ⋃ S ∈ S S . {\textstyle |{\mathcal {S}}|=\bigcup _{S\in {\mathcal {S}}}S.} This set | S | {\displaystyle |{\mathcal {S}}|} is endowed with a topology by choosing the closed sets to be { A ⊆ | S | ∣ A ∩ Δ {\displaystyle \{A\subseteq |{\mathcal {S}}|\;\mid \;A\cap \Delta } is closed for all Δ ∈ S } {\displaystyle \Delta \in {\mathcal {S}}\}} . Note that, in general, this topology is not the same as the subspace topology that | S | {\displaystyle |{\mathcal {S}}|} inherits from R n {\displaystyle \mathbb {R} ^{n}} . The topologies do coincide in the case that each point in the complex lies only in finitely many simplices. [ 2 ]
Each geometric complex can be associated with an abstract complex by choosing as a ground set V {\displaystyle V} the set of vertices that appear in any simplex of S {\displaystyle {\mathcal {S}}} and as system of subsets the subsets of V {\displaystyle V} which correspond to vertex sets of simplices in S {\displaystyle {\mathcal {S}}} .
A natural question is if vice versa, any abstract simplicial complex corresponds to a geometric complex. In general, the geometric construction as mentioned here is not flexible enough: consider for instance an abstract simplicial complex of infinite dimension. However, the following more abstract construction provides a topological space for any kind of abstract simplicial complex:
Let T {\displaystyle {\mathcal {T}}} be an abstract simplicial complex above a set V {\displaystyle V} . Choose a union of simplices ( Δ F ) F ∈ T {\displaystyle (\Delta _{F})_{F\in {\mathcal {T}}}} , but each in R N {\displaystyle \mathbb {R} ^{N}} of dimension sufficiently large, such that the geometric simplex Δ F {\displaystyle \Delta _{F}} is of dimension n {\displaystyle n} if the abstract geometric simplex F {\displaystyle F} has dimension n {\displaystyle n} . If E ⊂ F {\displaystyle E\subset F} , Δ E ⊂ R N {\displaystyle \Delta _{E}\subset \mathbb {R} ^{N}} can be identified with a face of Δ F ⊂ R M {\displaystyle \Delta _{F}\subset \mathbb {R} ^{M}} and the resulting topological space is the gluing Δ E ∪ i Δ F . {\displaystyle \Delta _{E}\cup _{i}\Delta _{F}.} Effectuating the gluing for each inclusion, one ends up with the desired topological space. This space is actually unique up to homeomorphism for each choice of T , {\displaystyle {\mathcal {T}},} so it makes sense to talk about the geometric realization | T | {\displaystyle |{\mathcal {T}}|} of T . {\displaystyle {\mathcal {T}}.}
As in the previous construction, by the topology induced by gluing, the closed sets in this space are the subsets that are closed in the subspace topology of every simplex Δ F {\displaystyle \Delta _{F}} in the complex.
The simplicial complex T n {\displaystyle {\mathcal {T_{n}}}} which consists of all simplices T {\displaystyle {\mathcal {T}}} of dimension ≤ n {\displaystyle \leq n} is called the n {\displaystyle n} -th skeleton of T {\displaystyle {\mathcal {T}}} .
A natural neighbourhood of a vertex v ∈ V {\displaystyle v\in V} in a simplicial complex S {\displaystyle {\mathcal {S}}} is considered to be given by the star star ( v ) = { L ∈ S ∣ v ∈ L } {\displaystyle \operatorname {star} (v)=\{L\in {\mathcal {S}}\;\mid \;v\in L\}} of a simplex, whose boundary is the link link ( v ) {\displaystyle \operatorname {link} (v)} .
The maps considered in this category are simplicial maps: Let K {\displaystyle {\mathcal {K}}} , L {\displaystyle {\mathcal {L}}} be abstract simplicial complexes above sets V K {\displaystyle V_{K}} , V L {\displaystyle V_{L}} . A simplicial map is a function f : V K → V L {\displaystyle f:V_{K}\rightarrow V_{L}} which maps each simplex in K {\displaystyle {\mathcal {K}}} onto a simplex in L {\displaystyle {\mathcal {L}}} . By affine-linear extension on the simplices, f {\displaystyle f} induces a map between the geometric realizations of the complexes. [ 2 ]
A triangulation of a topological space X {\displaystyle X} is a homeomorphism t : | T | → X {\displaystyle t:|{\mathcal {T}}|\rightarrow X} where T {\displaystyle {\mathcal {T}}} is a simplicial complex. Topological spaces do not necessarily admit a triangulation and if they do, it is not necessarily unique.
Triangulations of spaces allow assigning combinatorial invariants rising from their dedicated simplicial complexes to spaces. These are characteristics that equal for complexes that are isomorphic via a simplicial map and thus have the same combinatorial pattern.
This data might be useful to classify topological spaces up to homeomorphism but only given that the characteristics are also topological invariants, meaning, they do not depend on the chosen triangulation. For the data listed here, this is the case. [ 4 ] For details and the link to singular homology , see topological invariance.
Via triangulation, one can assign a chain complex to topological spaces that arise from its simplicial complex and compute its simplicial homology . Compact spaces always admit finite triangulations and therefore their homology groups are finitely generated and only finitely many of them do not vanish. Other data as Betti-numbers or Euler characteristic can be derived from homology.
Let | S | {\displaystyle |{\mathcal {S}}|} be a finite simplicial complex. The n {\displaystyle n} -th Betti-number b n ( S ) {\displaystyle b_{n}({\mathcal {S}})} is defined to be the rank of the n {\displaystyle n} -th simplicial homology group of the spaces. These numbers encode geometric properties of the spaces: The Betti-number b 0 ( S ) {\displaystyle b_{0}({\mathcal {S}})} for instance represents the number of connected components. For a triangulated, closed orientable surfaces F {\displaystyle F} , b 1 ( F ) = 2 g {\displaystyle b_{1}(F)=2g} holds where g {\displaystyle g} denotes the genus of the surface: Therefore its first Betti-number represents the doubled number of handles of the surface. [ 5 ]
With the comments above, for compact spaces all Betti-numbers are finite and almost all are zero. Therefore, one can form their alternating sum
which is called the Euler characteristic of the complex, a catchy topological invariant.
To use these invariants for the classification of topological spaces up to homeomorphism one needs invariance of the characteristics regarding homeomorphism.
A famous approach to the question was at the beginning of the 20th century the attempt to show that any two triangulations of the same topological space admit a common subdivision . This assumption is known as Hauptvermutung ( German: Main assumption). Let | L | ⊂ R N {\displaystyle |{\mathcal {L}}|\subset \mathbb {R} ^{N}} be a simplicial complex. A complex | L ′ | ⊂ R N {\displaystyle |{\mathcal {L'}}|\subset \mathbb {R} ^{N}} is said to be a subdivision of L {\displaystyle {\mathcal {L}}} iff:
Those conditions ensure that subdivisions does not change the simplicial complex as a set or as a topological space. A map f : K → L {\displaystyle f:{\mathcal {K}}\rightarrow {\mathcal {L}}} between simplicial complexes is said to be piecewise linear if there is a refinement K ′ {\displaystyle {\mathcal {K'}}} of K {\displaystyle {\mathcal {K}}} such that f {\displaystyle f} is piecewise linear on each simplex of K {\displaystyle {\mathcal {K}}} . Two complexes that correspond to another via piecewise linear bijection are said to be combinatorial isomorphic. In particular, two complexes that have a common refinement are combinatorially equivalent. Homology groups are invariant to combinatorial equivalence and therefore the Hauptvermutung would give the topological invariance of simplicial homology groups. In 1918, Alexander introduced the concept of singular homology. Henceforth, most of the invariants arising from triangulation were replaced by invariants arising from singular homology. For those new invariants, it can be shown that they were invariant regarding homeomorphism and even regarding homotopy equivalence . [ 6 ] Furthermore it was shown that singular and simplicial homology groups coincide. [ 6 ] This workaround has shown the invariance of the data to homeomorphism. Hauptvermutung lost in importance but it was initial for a new branch in topology: The piecewise linear topology (short PL-topology). [ 7 ]
The Hauptvermutung ( German for main conjecture ) states that two triangulations always admit a common subdivision. Originally, its purpose was to prove invariance of combinatorial invariants regarding homeomorphisms. The assumption that such subdivisions exist in general is intuitive, as subdivision are easy to construct for simple spaces, for instance for low dimensional manifolds. Indeed the assumption was proven for manifolds of dimension ≤ 3 {\displaystyle \leq 3} and for differentiable manifolds but it was disproved in general: [ 8 ] An important tool to show that triangulations do not admit a common subdivision, that is, their underlying complexes are not combinatorially isomorphic is the combinatorial invariant of Reidemeister torsion.
To disprove the Hauptvermutung it is helpful to use combinatorial invariants which are not topological invariants. A famous example is Reidemeister torsion. It can be assigned to a tuple ( K , L ) {\displaystyle (K,L)} of CW-complexes: If L = ∅ {\displaystyle L=\emptyset } this characteristic will be a topological invariant but if L ≠ ∅ {\displaystyle L\neq \emptyset } in general not. An approach to Hauptvermutung was to find homeomorphic spaces with different values of Reidemeister torsion. This invariant was used initially to classify lens-spaces and first counterexamples to the Hauptvermutung were built based on lens-spaces: [ 8 ]
In its original formulation, lens spaces are 3-manifolds, constructed as quotient spaces of the 3-sphere: Let p , q {\displaystyle p,q} be natural numbers, such that p , q {\displaystyle p,q} are coprime. The lens space L ( p , q ) {\displaystyle L(p,q)} is defined to be the orbit space of the free group action
For different tuples ( p , q ) {\displaystyle (p,q)} , lens spaces will be homotopy equivalent but not homeomorphic. Therefore they can't be distinguished with the help of classical invariants as the fundamental group but by the use of Reidemeister torsion.
Two lens spaces L ( p , q 1 ) , L ( p , q 2 ) {\displaystyle L(p,q_{1}),L(p,q_{2})} are homeomorphic, if and only if q 1 ≡ ± q 2 ± 1 ( mod p ) {\displaystyle q_{1}\equiv \pm q_{2}^{\pm 1}{\pmod {p}}} . [ 9 ] This is the case if and only if two lens spaces are simple homotopy equivalent . The fact can be used to construct counterexamples for the Hauptvermutung as follows. Suppose there are spaces L 1 ′ , L 2 ′ {\displaystyle L'_{1},L'_{2}} derived from non-homeomorphic lens spaces L ( p , q 1 ) , L ( p , q 2 ) {\displaystyle L(p,q_{1}),L(p,q_{2})} having different Reidemeister torsion. Suppose further that the modification into L 1 ′ , L 2 ′ {\displaystyle L'_{1},L'_{2}} does not affect Reidemeister torsion but such that after modification L 1 ′ {\displaystyle L'_{1}} and L 2 ′ {\displaystyle L'_{2}} are homeomorphic. The resulting spaces will disprove the Hauptvermutung.
Besides the question of concrete triangulations for computational issues, there are statements about spaces that are easier to prove given that they are simplicial complexes. Especially manifolds are of interest. Topological manifolds of dimension ≤ 3 {\displaystyle \leq 3} are always triangulable [ 10 ] [ 11 ] [ 1 ] but there are non-triangulable manifolds for dimension n {\displaystyle n} , for n {\displaystyle n} arbitrary but greater than three. [ 12 ] [ 13 ] Further, differentiable manifolds always admit triangulations. [ 3 ]
Manifolds are an important class of spaces. It is natural to require them not only to be triangulable but moreover to admit a piecewise linear atlas, a PL-structure:
Let | X | {\displaystyle |X|} be a simplicial complex such that every point admits an open neighborhood U {\displaystyle U} such that there is a triangulation of U {\displaystyle U} and a piecewise linear homeomorphism f : U → R n {\displaystyle f:U\rightarrow \mathbb {R} ^{n}} . Then | X | {\displaystyle |X|} is said to be a piecewise linear (PL) manifold of dimension n {\displaystyle n} and the triangulation together with the PL-atlas is said to be a PL-structure on | X | {\displaystyle |X|} .
An important lemma is the following:
Let X {\displaystyle X} be a topological space. Then the following statements are equivalent:
The equivalence of the second and the third statement is because that the link of a vertex is independent of the chosen triangulation up to combinatorial isomorphism. [ 14 ] One can show that differentiable manifolds admit a PL-structure as well as manifolds of dimension ≤ 3 {\displaystyle \leq 3} . [ 15 ] Counterexamples for the triangulation conjecture are counterexamples for the conjecture of the existence of PL-structure of course.
Moreover, there are examples for triangulated spaces which do not admit a PL-structure. Consider an n − 2 {\displaystyle n-2} -dimensional PL-homology-sphere X {\displaystyle X} . The double suspension S 2 X {\displaystyle S^{2}X} is a topological n {\displaystyle n} -sphere. Choosing a triangulation t : | S | → S 2 X {\displaystyle t:|{\mathcal {S}}|\rightarrow S^{2}X} obtained via the suspension operation on triangulations the resulting simplicial complex is not a PL-manifold, because there is a vertex v {\displaystyle v} such that l i n k ( v ) {\displaystyle link(v)} is not a n − 1 {\displaystyle n-1} sphere. [ 16 ]
A question arising with the definition is if PL-structures are always unique: Given two PL-structures for the same space Y {\displaystyle Y} , is there a there a homeomorphism F : Y → Y {\displaystyle F:Y\rightarrow Y} which is piecewise linear with respect to both PL-structures? The assumption is similar to the Hauptvermutung and indeed there are spaces which have different PL-structures which are not equivalent. Triangulation of PL-equivalent spaces can be transformed into one another via Pachner moves:
Pachner moves are a way to manipulate triangulations: Let S {\displaystyle {\mathcal {S}}} be a simplicial complex. For two simplices K , L , {\displaystyle K,L,} the Join K ∗ L = { ( 1 − t ) k + t l | k ∈ K , l ∈ L , t ∈ [ 0 , 1 ] } {\textstyle K*L=\{(1-t)k+tl\;|\;k\in K,l\in L,t\in [0,1]\}} is the set of points that lie on straights between points in K {\displaystyle K} and in L {\displaystyle L} . Choose S ∈ S {\displaystyle S\in {\mathcal {S}}} such that l k ( S ) = ∂ K {\displaystyle lk(S)=\partial K} for any K {\displaystyle K} lying not in S {\displaystyle {\mathcal {S}}} . A new complex S ′ {\displaystyle {\mathcal {S'}}} , can be obtained by replacing S ∗ ∂ K {\displaystyle S*\partial K} by ∂ S ∗ K {\displaystyle \partial S*K} . This replacement is called a Pachner move. The theorem of Pachner states that whenever two triangulated manifolds are PL-equivalent, there is a series of Pachner moves transforming both into another. [ 17 ]
A similar but more flexible construction than simplicial complexes is the one of cellular complexes (or CW-complexes). Its construction is as follows:
An n {\displaystyle n} -cell is the closed n {\displaystyle n} -dimensional unit-ball B n = [ 0 , 1 ] n {\displaystyle B_{n}=[0,1]^{n}} , an open n {\displaystyle n} -cell is its inner B n = [ 0 , 1 ] n ∖ S n − 1 {\displaystyle B_{n}=[0,1]^{n}\setminus \mathbb {S} ^{n-1}} . Let X {\displaystyle X} be a topological space, let f : S n − 1 → X {\displaystyle f:\mathbb {S} ^{n-1}\rightarrow X} be a continuous map. The gluing X ∪ f B n {\displaystyle X\cup _{f}B_{n}} is said to be obtained by gluing on an n {\displaystyle n} -cell.
A cell complex is a union X = ∪ n ≥ 0 X n {\displaystyle X=\cup _{n\geq 0}X_{n}} of topological spaces such that
Each simplicial complex is a CW-complex, the inverse is not true. The construction of CW-complexes can be used to define cellular homology and one can show that cellular homology and simplicial homology coincide. [ 18 ] For computational issues, it is sometimes easier to assume spaces to be CW-complexes and determine their homology via cellular decomposition, an example is the projective plane P 2 {\displaystyle \mathbb {P} ^{2}} : Its construction as a CW-complex needs three cells, whereas its simplicial complex consists of 54 simplices.
By triangulating 1-dimensional manifolds, one can show that they are always homeomorphic to disjoint copies of the real line and the unit sphere S 1 {\displaystyle \mathbb {S} ^{1}} . The classification of closed surfaces , i.e. compact 2-manifolds, can also be proven by using triangulations. This is done by showing any such surface can be triangulated and then using the triangulation to construct a fundamental polygon for the surface. [ 19 ]
Giving spaces simplicial structures can help to understand continuous maps defined on the spaces. The maps can often be assumed to be simplicial maps via the simplicial approximation theorem:
Let K {\displaystyle {\mathcal {K}}} , L {\displaystyle {\mathcal {L}}} be abstract simplicial complexes above sets V K {\displaystyle V_{K}} , V L {\displaystyle V_{L}} . A simplicial map is a function f : V K → V L {\displaystyle f:V_{K}\rightarrow V_{L}} which maps each simplex in K {\displaystyle {\mathcal {K}}} onto a simplex in L {\displaystyle {\mathcal {L}}} . By affin-linear extension on the simplices, f {\displaystyle f} induces a map between the geometric realizations of the complexes. Each point in a geometric complex lies in the inner of exactly one simplex, its support. Consider now a continuous map f : K → L {\displaystyle f:{\mathcal {K}}\rightarrow {\mathcal {L}}} . A simplicial map g : K → L {\displaystyle g:{\mathcal {K}}\rightarrow {\mathcal {L}}} is said to be a simplicial approximation of f {\displaystyle f} if and only if each x ∈ K {\displaystyle x\in {\mathcal {K}}} is mapped by g {\displaystyle g} onto the support of f ( x ) {\displaystyle f(x)} in L {\displaystyle {\mathcal {L}}} . If such an approximation exists, one can construct a homotopy H {\displaystyle H} transforming f {\displaystyle f} into g {\displaystyle g} by defining it on each simplex; there it always exists, because simplices are contractible.
The simplicial approximation theorem guarantees for every continuous function f : V K → V L {\displaystyle f:V_{K}\rightarrow V_{L}} the existence of a simplicial approximation at least after refinement of K {\displaystyle {\mathcal {K}}} , for instance by replacing K {\displaystyle {\mathcal {K}}} by its iterated barycentric subdivision. [ 2 ] The theorem plays an important role for certain statements in algebraic topology in order to reduce the behavior of continuous maps on those of simplicial maps, for instance in Lefschetz's fixed-point theorem.
The Lefschetz number is a useful tool to find out whether a continuous function admits fixed-points. This data is computed as follows: Suppose that X {\displaystyle X} and Y {\displaystyle Y} are topological spaces that admit finite triangulations. A continuous map f : X → Y {\displaystyle f:X\rightarrow Y} induces homomorphisms f i : H i ( X , K ) → H i ( Y , K ) {\displaystyle f_{i}:H_{i}(X,K)\rightarrow H_{i}(Y,K)} between its simplicial homology groups with coefficients in a field K {\displaystyle K} . These are linear maps between K {\displaystyle K} -vector spaces, so their trace tr i {\displaystyle \operatorname {tr} _{i}} can be determined and their alternating sum
L K ( f ) = ∑ i ( − 1 ) i tr i ( f ) ∈ K {\displaystyle L_{K}(f)=\sum _{i}(-1)^{i}\operatorname {tr} _{i}(f)\in K}
is called the Lefschetz number of f {\displaystyle f} . If f = i d {\displaystyle f={\rm {id}}} , this number is the Euler characteristic of K {\displaystyle K} . The fixpoint theorem states that whenever L K ( f ) ≠ 0 {\displaystyle L_{K}(f)\neq 0} , f {\displaystyle f} has a fixed-point. In the proof this is first shown only for simplicial maps and then generalized for any continuous functions via the approximation theorem. Brouwer's fixpoint theorem treats the case where f : D n → D n {\displaystyle f:\mathbb {D} ^{n}\rightarrow \mathbb {D} ^{n}} is an endomorphism of the unit-ball. For k ≥ 1 {\displaystyle k\geq 1} all its homology groups H k ( D n ) {\displaystyle H_{k}(\mathbb {D} ^{n})} vanishes, and f 0 {\displaystyle f_{0}} is always the identity, so L K ( f ) = tr 0 ( f ) = 1 ≠ 0 {\displaystyle L_{K}(f)=\operatorname {tr} _{0}(f)=1\neq 0} , so f {\displaystyle f} has a fixpoint. [ 20 ]
The Riemann-Hurwitz formula allows to determine the genus of a compact, connected Riemann surface X {\displaystyle X} without using explicit triangulation. The proof needs the existence of triangulations for surfaces in an abstract sense: Let F : X → Y {\displaystyle F:X\rightarrow Y} be a non-constant holomorphic function on a surface with known genus. The relation between the genus g {\displaystyle g} of the surfaces X {\displaystyle X} and Y {\displaystyle Y} is
2 g ( X ) − 2 = deg ( F ) ( 2 g ( Y ) − 2 ) + ∑ x ∈ X ( ord ( F ) − 1 ) {\displaystyle 2g(X)-2=\deg(F)(2g(Y)-2)+\sum _{x\in X}(\operatorname {ord} (F)-1)}
where deg ( F ) {\displaystyle \deg(F)} denotes the degree of the map. The sum is well defined as it counts only the ramifying points of the function.
The background of this formula is that holomorphic functions on Riemann surfaces are ramified coverings. The formula can be found by examining the image of the simplicial structure near to ramifiying points. [ 21 ] | https://en.wikipedia.org/wiki/Triangulation_(topology) |
Triangulene (also known as Clar's hydrocarbon) is the smallest triplet-ground-state polybenzenoid . [ 1 ] It exists as a biradical with the chemical formula C 22 H 12 . [ 2 ] It was first hypothesized by Czech chemist Erich Clar in 1953. [ 3 ] Its first confirmed synthesis was published in a February 2017 issue of Nature Nanotechnology , in a project led by researchers David Fox and Anish Mistry at the University of Warwick in collaboration with IBM . [ 4 ] Other attempts by Japanese researchers have been successful only in making substituted triangulene derivatives. [ 5 ]
A six-step synthesis yielded two isomers of dihydrotriangulene which were then deposited on xenon or copper base. The researchers used a combined scanning tunneling and atomic force microscope (STM/AFM) to remove individual hydrogen atoms. The synthesized molecule of triangulene remained stable at high- vacuum low- temperature conditions for four days, giving the scientists plenty of time to characterize it (also using STM/AFM). [ 4 ]
Triangulene, as defined here, is a member of a wider class of [ n ]triangulenes, where n is the number of hexagons along an edge of the molecule. Thus, triangulene may also be referred to as [3]triangulene.
A tight-binding description of the molecular orbitals of [ n ]triangulenes predicts [ 6 ] that [ n ]triangulenes have ( n − 1) unpaired electrons, which are associated to ( n − 1) non-bonding states. When electron–electron interactions are included, theory predicts [ 6 ] [ 7 ] [ 8 ] that the ground state total spin quantum number S of [ n ]triangulenes is S = n − 1 / 2 . Thus, [3]triangulenes are predicted to have an S = 1 ground state. The intramolecular exchange interaction in triangulene, which determines the energy difference between the S = 1 ground state and the S = 0 excited state, is predicted to be the largest [ 9 ] among all polycyclic aromatic hydrocarbon (PAH) diradicals, due to maximum overlap of the wave function of the unpaired electrons.
The ground state spin of [ n ]triangulenes can be rationalized [ 6 ] in terms of a theorem [ 10 ] by Elliot H. Lieb , which relates, for a bipartite lattice, the ground state spin of the Hubbard model at half filling to the sublattice imbalance.
So far, the ultra-high vacuum on-surface syntheses of [ n ]triangulenes with n = 3, [ 4 ] 4, [ 11 ] 5 [ 12 ] and 7 [ 13 ] (the hitherto largest triangulene homologue) have been reported. In addition, the on-surface synthesis of [3]triangulene dimers [ 14 ] has also been reported, where inelastic electron tunneling spectroscopy provides a direct evidence of a strong antiferromagnetic coupling between the triangulenes. In 2021, an international team of researchers reported the fabrication of [3]triangulene-based quantum spin chains on a gold surface, [ 15 ] where signatures of both spin fractionalization and Haldane gap were observed. | https://en.wikipedia.org/wiki/Triangulene |
Triangulum Minus ( Latin for the Smaller Triangle ) was a constellation created by Johannes Hevelius . Its name is sometimes wrongly written as Triangulum Min or . It was formed from the southern parts of his Triangula (plural form of Triangulum ), alongside Triangulum Majus , but is no longer in use. The triangle was defined by the fifth-magnitude stars ι Trianguli (6 Tri), 10 Trianguli , and 12 Trianguli . [ 1 ]
Also known as TZ Trianguli, 6 Trianguli is a multiple star system with a combined magnitude of 4.7, whose main component is a yellow giant of spectral type G5III. [ 1 ]
This constellation -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Triangulum_Minus |
Triatomic hydrogen or H 3 is an unstable triatomic molecule containing only hydrogen . Since this molecule contains only three atoms of hydrogen it is the simplest triatomic molecule [ 1 ] and it is relatively simple to numerically solve the quantum mechanics description of the particles. Being unstable the molecule breaks up in under a millionth of a second. Its fleeting lifetime makes it rare, but it is quite commonly formed and destroyed in the universe thanks to the commonness of the trihydrogen cation . The infrared spectrum of H 3 due to vibration and rotation is very similar to that of the ion, H + 3 . In the early universe this ability to emit infrared light allowed the primordial hydrogen and helium gas to cool down so as to form stars.
The neutral molecule can be formed in a low pressure gas discharge tube . [ 2 ]
A neutral beam of H 3 can be formed from a beam of H + 3 ions passing through gaseous potassium , which donates an electron to the ion, forming K + . [ 3 ] Other gaseous alkali metals, such as caesium , can also be used to donate electrons. [ 4 ] H + 3 ions can be made in a duoplasmatron where an electric discharge passed through low pressure molecular hydrogen. This causes some H 2 to become H + 2 . Then H 2 + H + 2 → H + 3 + H. The reaction is exothermic with an energy of 1.7 eV, so the ions produced are hot with much vibrational energy. These can cool down via collisions with cooler gas if the pressure is high enough. This is significant because strongly vibrating ions produce strongly vibrating neutral molecules when neutralised according to the Franck–Condon principle . [ 3 ]
H 3 can break up in the following ways:
The molecule can only exist in an excited state. The different excited electronic states are represented by symbols for the outer electron nLΓ with n the principal quantum number, L is the electronic angular momentum, and Γ is the electronic symmetry selected from the D 3h group. Extra bracketed symbols can be attached showing vibration in the core: {s,d l } with s representing symmetrical stretch, d degenerate mode, and l vibrational angular momentum. Yet another term can be inserted to indicate molecular rotation: (N,G) with N angular momentum apart from electrons as projected on the molecular axis, and G the Hougen's convenient quantum number determined by G=l+λ-K. This is often (1,0), as the rotational states are restricted by the constituent particles all being fermions . Examples of these states are: [ 5 ] 2sA 1 ' 3sA 1 ' 2pA 2 " 3dE' 3DE" 3dA 1 ' 3pE' 3pA 2 ". The 2p 2 A 2 " state has a lifetime of 700 ns. If the molecule attempts to lose energy and go to the repulsive ground state , it spontaneously breaks up. The lowest energy metastable state, 2sA 1 ' has an energy -3.777 eV below the H + 3 and e − state but decays in around 1 ps . [ 5 ] The unstable ground state designated 2p 2 E' spontaneously breaks up into a H 2 molecule and an H atom. [ 1 ] Rotationless states have a longer life time than rotating molecules. [ 1 ]
The electronic state for a trihydrogen cation with an electron delocalized around it is a Rydberg state . [ 6 ]
The outer electron can be boosted to high Rydberg state, and can ionise if the energy gets to 29562.6 cm −1 above the 2pA 2 " state, in which case H + 3 forms. [ 7 ]
The shape of the molecule is predicted to be an equilateral triangle . [ 1 ] Vibrations can occur in the molecule in two ways, firstly the molecule can expand and contract retaining the equilateral triangle shape (breathing), or one atom can move relative to the others distorting the triangle (bending). The bending vibration has a dipole moment and thus couples to infrared radiation. [ 1 ]
Gerhard Herzberg was the first to find spectroscopic lines of neutral H 3 when he was 75 years old in 1979. Later he announced that this observation was one of his favourite discoveries. [ 8 ] The lines came about from a cathode discharge tube. [ 8 ] The reason that earlier observers could not see any H 3 spectral lines, was due to them being swamped by the spectrum of the much more abundant H 2 . The important advance was to separate out H 3 so it could be observed alone. Separation uses mass spectroscopy separation of the positive ions, so that H 3 with mass 3 can be separated from H 2 with mass 2. However there is still some contamination from HD , which also has mass 3. [ 3 ] The spectrum of H 3 is mainly due to transitions to the longer lived state of 2p 2 A 2 ". The spectrum can be measured via a two step photo-ionization method. [ 1 ]
Transitions dropping to the lower 2s 2 A 1 ' state are affected by its very short lifetime in what is called predissociation . The spectral lines involved are broadened. [ 3 ] In the spectrum there are bands due to rotation with P Q and R branches. The R branch is very weak in H 3 isotopomer but strong with D 3 (trideuterium). [ 3 ]
The symmetric stretch vibration mode has a wave number of 3213.1 cm −1 for the 3s 2 A 1 ' level and 3168 cm −1 for 3d 2 E" and 3254 cm −1 for 2p 2 A 2 ". [ 1 ] The bending vibrational frequencies are also quite similar to those for H + 3 . [ 1 ]
The related H + 3 ion is the most prevalent molecular ion in interstellar space. It is believed to have played a crucial role in the cooling of early stars in the history of the Universe through its ability readily to absorb and emit photons. [ 9 ] One of the most important chemical reactions in interstellar space is H + 3 + e − → H 3 and then → H 2 + H. [ 6 ]
Since the molecule is relatively simple, researchers have attempted to calculate the properties of the molecule ab-initio from quantum theory. The Hartree–Fock equations have been used. [ 10 ]
Triatomic hydrogen will be formed during the neutralization of H + 3 . This ion will be neutralised in the presence of gasses other than He or H 2 , as it can abstract an electron. Thus H 3 is formed in the aurora in the ionosphere of Jupiter and Saturn. [ 11 ]
J. J. Thomson observed H + 3 while experimenting with positive rays . He believed that it was an ionised form of H 3 from about 1911. He believed that H 3 was a stable molecule and wrote and lectured about it. He stated that the easiest way to make it was to target potassium hydroxide with cathode rays. [ 8 ] In 1913 Johannes Stark proposed that three hydrogen nuclei and electrons could form a stable ring shape. In 1919 Niels Bohr proposed a structure with three nuclei in a straight line, with three electrons orbiting in a circle around the central nucleus. He believed that H + 3 would be unstable, but that reacting H − 2 with H + could yield neutral H 3 . Stanley Allen 's structure was in the shape of a hexagon with alternating electrons and nuclei. [ 8 ]
In 1916 Arthur Dempster showed that H 3 gas was unstable, but at the same time also confirmed that the cation existed. In 1917 Gerald Wendt and William Duane discovered that hydrogen gas subjected to alpha particles shrank in volume and thought that diatomic hydrogen was converted to triatomic. [ 8 ] After this researchers thought that active hydrogen could be the triatomic form. [ 8 ] Joseph Lévine went so far as to postulate that low pressure systems on the Earth happened due to triatomic hydrogen high in the atmosphere. [ 8 ] In 1920 Wendt and Landauer named the substance "Hyzone" in analogy to ozone and its extra reactivity over normal hydrogen. [ 12 ] Earlier Gottfried Wilhelm Osann believed he had discovered a form of hydrogen analogous to ozone which he called "Ozonwasserstoff". It was made by electrolysis of dilute sulfuric acid. In those days no one knew that ozone was triatomic so he did not announce triatomic hydrogen. [ 13 ] This was later shown to be a mixture with sulfur dioxide, and not a new form of hydrogen. [ 12 ]
In the 1930s active hydrogen was found to be hydrogen with hydrogen sulfide contamination, and scientists stopped believing in triatomic hydrogen. [ 8 ] Quantum mechanical calculations showed that neutral H 3 was unstable but that ionised H + 3 could exist. [ 8 ] When the concept of isotopes came along, people such as Bohr then thought there may be an eka-hydrogen with atomic weight 3. This idea was later proven with the existence of tritium , but that was not the explanation of why molecular weight 3 was observed in mass spectrometers. [ 8 ] J. J. Thomson later believed that the molecular weight 3 molecule he observed was Hydrogen deuteride . [ 13 ] In the Orion nebula lines were observed that were attributed to nebulium which could have been the new element eka-hydrogen, especially when its atomic weight was calculated as near 3. Later this was shown to be ionised nitrogen and oxygen. [ 8 ]
Gerhard Herzberg was the first to actually observe the spectrum of neutral H 3 , and this triatomic molecule was the first to have a Rydberg spectrum measured where its own ground state was unstable. [ 1 ] | https://en.wikipedia.org/wiki/Triatomic_hydrogen |
Triatomic molecules are molecules composed of three atoms , of either the same or different chemical elements . Examples include H 2 O , CO 2 (pictured), HCN , O 3 ( ozone ) and NO 2 .
The vibrational modes of a triatomic molecule can be determined in specific cases.
A symmetric linear molecule ABA can perform:
In the previous formulas, M is the total mass of the molecule, m A and m B are the masses of the elements A and B, k 1 and k 2 are the spring constants of the molecule along its axis and perpendicular to it.
Homonuclear triatomic molecules contain three of the same kind of atom. That molecule will be an allotrope of that element .
Ozone , O 3 is an example of a triatomic molecule with all atoms the same. Triatomic hydrogen , H 3 , is unstable and breaks up spontaneously. H 3 + , the trihydrogen cation is stable by itself and is symmetric. 4 He 3 , the helium trimer is only weakly bound by van der Waals force and is in an Efimov state . [ 1 ] Trisulfur (S 3 ) is analogous to ozone.
All triatomic molecules may be classified as possessing either a linear , bent , or cyclic geometry. [ further explanation needed ]
Linear triatomic molecules owe their geometry to their sp or sp 3 d hybridised central atoms. Well-known linear triatomic molecules include carbon dioxide (CO 2 ) and hydrogen cyanide (HCN).
Xenon difluoride (XeF 2 ) is one of the rare examples of a linear triatomic molecule possessing non-bonded pairs of electrons on the central atom. | https://en.wikipedia.org/wiki/Triatomic_molecule |
In materials science , a triaxial shear test is a common method to measure the mechanical properties of many deformable solids, especially soil (e.g., sand , clay ) and rock , and other granular materials or powders . There are several variations on the test. [ 1 ] [ 2 ] [ 3 ] [ 4 ] In a triaxial shear test, stress is applied to a sample of the material being tested in a way which results in stresses along one axis being different from the stresses in perpendicular directions. This is typically achieved by placing the sample between two parallel platens which apply stress in one (usually vertical) direction, and applying fluid pressure to the specimen to apply stress in the perpendicular directions. (Testing apparatus which allows application of different levels of stress in each of three orthogonal directions are discussed below .)
The application of different compressive stresses in the test apparatus causes shear stress to develop in the sample; the loads can be increased and deflections monitored until failure of the sample. During the test, the surrounding fluid is pressurized, and the stress on the platens is increased until the material in the cylinder fails and forms sliding regions within itself, known as shear bands . The geometry of the shearing in a triaxial test typically causes the sample to become shorter while bulging out along the sides. The stress on the platen is then reduced and the water pressure pushes the sides back in, causing the sample to grow taller again. This cycle is usually repeated several times while collecting stress and strain data about the sample. During the test the pore pressures of fluids (e.g., water, oil) or gasses in the sample may be measured using Bishop's pore pressure apparatus.
From the triaxial test data, it is possible to extract fundamental material parameters about the sample, including its angle of shearing resistance, apparent cohesion, and dilatancy angle. These parameters are then used in computer models to predict how the material will behave in a larger-scale engineering application. An example would be to predict the stability of the soil on a slope, whether the slope will collapse or whether the soil will support the shear stresses of the slope and remain in place. Triaxial tests are used along with other tests to make such engineering predictions.
During the shearing, a granular material will typically have a net gain or loss of volume. If it had originally been in a dense state, then it typically gains volume, a characteristic known as Reynolds' dilatancy . If it had originally been in a very loose state, then contraction may occur before the shearing begins or in conjunction with the shearing.
Sometimes, testing of cohesive samples is done with no confining pressure, in an unconfined compression test. This requires much simpler and less expensive apparatus and sample preparation, though the applicability is limited to samples that the sides won't crumble when exposed, and the confining stress being lower than the in-situ stress gives results which may be overly conservative. The compression test performed for concrete strength testing is essentially the same test, on apparatus designed for the larger samples and higher loads typical of concrete testing.
For soil samples, the specimen is contained in a cylindrical latex sleeve with a flat, circular metal plate or platen closing off the top and bottom ends. This cylinder is placed into a bath of a hydraulic fluid to provide pressure along the sides of the cylinder. The top platen can then be mechanically driven up or down along the axis of the cylinder to squeeze the material. The distance that the upper platen travels is measured as a function of the force required to move it, as the pressure of the surrounding water is carefully controlled. The net change in volume of the material can also be measured by how much water moves in or out of the surrounding bath, but is typically measured - when the sample is saturated with water - by measuring the amount of water that flows into or out of the sample's pores.
For testing of high-strength rock, the sleeve may be a thin metal sheeting rather than latex. Triaxial testing on strong rock is fairly seldom done because the high forces and pressures required to break a rock sample require costly and cumbersome testing equipment.
The effective stress on the sample can be measured by using a porous surface on one platen, and measuring the pressure of the fluid (usually water) during the test, then calculating the effective stress from the total stress and pore pressure .
The triaxial test can be used to determine the shear strength of a discontinuity . A homogeneous and isotropic sample fails due to shear stresses in the sample. If a sample with a discontinuity is orientated such that the discontinuity is about parallel to the plane in which maximum shear stress will be developed during the test, the sample will fail due to shear displacement along the discontinuity, and hence, the shear strength of a discontinuity can be calculated. [ 5 ]
There are several variations of the triaxial test:
In a ' consolidated drained' test, the sample is consolidated and sheared in compression slowly to allow pore pressures built up by the shearing to dissipate. The rate of axial deformation is kept constant, i.e., strain is controlled. The test allows the sample and the pore pressures to fully consolidate (i.e., adjust ) to the surrounding stresses. The test may take a long time to allow the sample to adjust, in particular low permeability samples need a long time to drain and adjust strain to stress levels.
In a 'consolidated undrained' test, the sample is not allowed to drain. The shear characteristics are measured under undrained conditions, and the sample is assumed to be fully saturated. Measuring the pore pressures in the sample (sometimes called CUpp) allows for approximating the consolidated-drained strength. Shear speed is often calculated based on the rate of consolidation under a specific confining pressure (while saturated). Confining pressures can vary anywhere from 1 psi to 100 psi or greater, sometimes requiring special load cells capable of handling higher pressures.
In an ' unconsolidated undrained' test, the loads are applied quickly, and the sample is not allowed to consolidate during the test. The sample is compressed at a constant rate ( strain-controlled ).
Triaxial testing systems have been developed to allow independent stress control in three perpendicular directions. This enables the investigation of stress paths not capable of being generated in axisymmetric triaxial test machines, which can be useful in studies of cemented sands and anisotropic soils. The test cell is cubical, and there are six separate plates applying pressure to the specimen, with LVDTs reading the movement of each plate. [ 6 ] Pressure in the third direction can be applied using hydrostatic pressure in the test chamber, requiring only four stress application assemblies. The apparatus is significantly more complex than for axisymmetric triaxial tests and is, therefore, less commonly used.
Triaxial tests of classical construction had been criticized for their nonuniform stress and strain field imposed within the specimen during larger deformation amplitudes. [ 7 ] The highly localized discontinuity within a shear zone is caused by the combination of rough end plates and specimen height.
To test specimens during larger deformation amplitude, "new" [ 8 ] and "improved" [ 9 ] version of the triaxial apparatus were made. The "new" and the "improved" triaxial follow the same principle - sample height is reduced to one diameter height, and friction with the end plates is canceled.
The classical apparatus uses rough end plates - the whole surface of the piston head is made up of rough, porous filter. In upgraded apparatuses the tough end plates are replaced with smooth, polished glass, with a small filter at the center. This configuration allows a specimen to slide / expand horizontally while sliding along the polished glass. Thus, the contact zone between sample and the end plates does not buildup unnecessary shear friction, and a linear / isotropic stress field within the specimen is sustained.
Due to extremely uniform, near isotropic stress field - isotropic yielding takes place. During isotropic yielding volumetric (dilatational) strain is isotopically distributed within the specimen, this improves measurement of volumetric response during CD tests and pore water pressure during CU loading. Also, isotropic yielding makes the specimen expand radially in uniform manner, as it is compressed axially. The walls of a cylindrical specimen remain straight and vertical even during large strain amplitudes (50% strain amplitude was documented by Vardoulakis (1980), using "improved" triaxial, on non saturated sand). This is in contrast with classical setup, where the specimen forms a bugle in the center, while keeping a constant radius at the contact with the end plates.
The "new" apparatus has been upgraded to "the Danish triaxial" by L.B.Ibsen. [ 10 ] The Danish triaxial can be used for testing all soil types. It provides improved measurements of volumetric response - as during isotropic yielding, volumetric strain is distributed isotopically within the specimen. Isotropic volume change is especially important for CU testing, as cavitation of pore water sets the limit of undrained sand strength. [ 11 ] Measurement precision is improved by taking measurements near the specimen. The load cell is submerged and in direct contact with the upped pressure head of the specimen. Deformation transducers are attached directly to the piston heads as well. Control of the apparatus is highly automated, thus cyclic loading can be applied with great efficiency and precision.
The combination of high automation, improved sample durability and large deformation compatibility expands the scope of triaxial testing. The Danish triaxial can yield CD and CU sand specimens into plasticity without forming a shear rupture or bulging. A sample can be tested for yielding multiple times in a single, continuous loading sequence. Samples can even be liquefied to a large strain amplitude, then crushed to CU failure. CU tests can be allowed to transition into CD state, and cyclic tested in CD mode to observe post liquefaction recovery of stiffness and strength. [ 12 ] This allows to control the specimens to a very high degree, and observe sand response patterns which are not accessible using classical triaxial testing methods.
The list is not complete; only the main standards are included. For a more extensive listing, please refer to the websites of ASTM International (USA), British Standards (UK), International Organization for Standardization ( ISO ), or local organisations for standards. | https://en.wikipedia.org/wiki/Triaxial_shear_test |
Triazabicyclodecene ( 1,5,7-triazabicyclo[4.4.0]dec-5-ene or TBD ) is an organic compound consisting of a bicyclic guanidine . For a charge-neutral compound, it is a relatively strong base that is effective for a variety of organic transformations. TBD is colorless solid that is soluble in a variety of solvents. [ 4 ]
As a strong base, TBD fully deprotonates most phenols , carboxylic acids , and some carbon acids . [ 7 ] It catalyzes a variety of reactions including Michael reactions , Henry reactions , transesterification reactions, and Knoevenagel condensations . [ 8 ]
Deprotonation at the 7-position gives a particularly electron-rich ligand as manifested in the redox properties of ditungsten tetra(hpp) .
The conjugate acid of TBD is the preferred cation among the guanidinium hypoiodites , which are specialized oxidizing agents for various types of organic compounds . [ 9 ] | https://en.wikipedia.org/wiki/Triazabicyclodecene |
Triazenes are organic compounds that contain the functional group R 1 −N=N−NR 2 R 3 , where the R are each any of various types of substituent groups. [ 2 ] Some medications and dyes are triazenes. [ 3 ] Formally, the triazenes are related to the unstable chemical triazene , H 2 N−N=NH.
Triazenes are prepared from the N -coupling reaction between diazonium salts and primary or secondary amines . [ 4 ] The coupling reactions are typically mild, using a base such as sodium acetate , [ 5 ] sodium carbonate , [ 5 ] or sodium bicarbonate . [ 4 ]
The diazonium reagents are themselves available starting from amines. For symmetrical triazenes derived from primary amines, partial diazotization gives a mixture of the original amine and its diazo derivative that then couple with each other. For example, 1,3-diphenyltriazene (PhN=N−NHPh) can be made from aniline in a one-pot reaction . [ 5 ] [ 6 ] For asymmetrical triazenes, for example (phenyldiazenyl)pyrrolidine (PhN=N−NC 4 H 8 ), the phenyldiazonium salt must be pre-made.
Analogues of Tröger's base containing a symmetric pair of asymmetric triazene side-chains have been obtained similarly. [ 7 ]
Triazenes derived from primary amines engage in tautomerism . In the case of symmetric triazenes, the tautomers are identical.
Triazenes can be converted to diazonium salts . [ 8 ]
Triazenes decompose in the presence of protonating or alkylating agents into quaternary amines and diazonium salts; as such triazenes have been used as an in situ source of diazonium that reacted with sodium sulfide to give the corresponding thiophenols . [ 5 ] A strategy for the protection and deprotection of sensitive secondary amines is based on this principle. [ 9 ]
Polymeric triazenes are applied as conductive and absorbent materials. [ 10 ] Triazenes have been used in the synthesis of cinnoline , functionalized lactams , and coumarins . [ 9 ] [ 8 ] | https://en.wikipedia.org/wiki/Triazenes |
Tribasic may refer to: | https://en.wikipedia.org/wiki/Tribasic |
Tribimaximal mixing [ 1 ] is a specific postulated form for the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) lepton mixing matrix U . Tribimaximal mixing is defined by a particular choice of the matrix of moduli-squared of the elements of the PMNS matrix as follows:
This mixing is historically interesting as it is quite close to reality when compared to other simple hypotheses where the squares of matrix elements take exact ratios, and also compared to the naive supposition that the matrix would be approximately diagonal like the CKM matrix . However the precision of modern experiments mean that such a simple form is excluded by experiment at a level of over 5σ, mainly due to the fact the tribimaximal scheme has a zero in the U e 3 {\displaystyle U_{e3}} element, but also (to a much lesser extent) because it predicts no violation of CP symmetry .
The tribimaximal mixing form was compatible with pre-2011 neutrino oscillation experiments [ 2 ] and may be used as a zeroth-order approximation to more general forms for the PMNS matrix, [ 3 ] [ 4 ] including some that are consistent with the data. In the PDG [ 2 ] convention for the PMNS matrix, tribimaximal mixing may be specified in terms of lepton mixing angles as follows:
The above prediction has been falsified experimentally, because θ 13 was found to be nontrivial, θ 13 =8.5°. [ 5 ]
A non-negligible value of θ 13 has been foreseen in certain theoretical schemes that were put forward before tribimaximal mixing and that
supported a large solar mixing, before it was confirmed experimentally [ 6 ] [ 7 ] (these theoretical schemes do not have a special name, but for the reasons explained above, they could be called pre-tribimaximal or also non-tribimaximal). This situation is not new: also in the 1990s, the solar mixing angle was supposed to be small by most theorists, until KamLAND proved the contrary to be true.
The name tribimaximal reflects the commonality of the tribimaximal mixing matrix with two previously proposed specific forms for the PMNS matrix, the trimaximal [ 8 ] and bimaximal [ 9 ] [ 10 ] mixing schemes, both now ruled out by data. In tribimaximal mixing, [ 1 ] the ν 2 {\displaystyle \nu _{2}} neutrino mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of ν e {\displaystyle \nu _{e}} , ν μ {\displaystyle \nu _{\mu }} and ν τ {\displaystyle \nu _{\tau }} flavour eigenstates, i.e. maximal mixing among all three flavour states. The ν 3 {\displaystyle \nu _{3}} neutrino mass eigenstate, on the other hand, is "bimaximally mixed" in that it comprises a uniform admixture of only two flavour components, i.e. ν μ {\displaystyle \nu _{\mu }} and ν τ {\displaystyle \nu _{\tau }} maximal mixing, with effective decoupling of the ν e {\displaystyle \nu _{e}} from the ν 3 {\displaystyle \nu _{3}} , just as in the original bimaximal scheme. [ 10 ] [ 11 ]
By virtue of the zero ( | U e 3 | 2 = 0 {\displaystyle \ |U_{e3}|^{2}=0\ } ) in the tribimaximal mixing matrix, exact tribimaximal mixing would predict zero for all CP-violating asymmetries in the case of Dirac neutrinos (in the case of Majorana neutrinos , Majorana phases are still permitted, and could still lead to CP-violating effects).
For solar neutrinos the large angle MSW effect in tribimaximal mixing accounts for the experimental data , predicting average suppressions ⟨ P e e ⟩ ≃ 1 3 {\displaystyle \ \langle P_{ee}\rangle \simeq {\tfrac {1}{3}}\ } in the Sudbury Neutrino Observatory (SNO) and ⟨ P e e ⟩ ≃ 5 9 {\displaystyle \ \langle P_{ee}\rangle \simeq {\tfrac {5}{9}}\ } in lower energy solar neutrino experiments (and in long baseline reactor neutrino experiments). The bimaximally mixed ν 3 {\displaystyle \ \nu _{3}\ } in tribimaximal mixing accounts for the factor of two suppression ⟨ P μ μ ⟩ ≃ 1 2 {\displaystyle \ \langle P_{\mu \mu }\rangle \simeq {\tfrac {1}{2}}\ } observed for atmospheric muon-neutrinos (and confirmed in long-baseline accelerator experiments). Near-zero ν e {\displaystyle \ \nu _{e}\ } appearance in a ν μ {\displaystyle \ \nu _{\mu }\ } beam is predicted in exact tribimaximal mixing ( | U e 3 | 2 = 0 {\displaystyle \ |U_{e3}|^{2}=0\ } ), and this has been strongly ruled out by modern reactor neutrino experiments . Further characteristic predictions [ 1 ] of tribimaximal mixing – e.g. for very long baseline ν μ {\displaystyle \ \nu _{\mu }\ } and ν τ {\displaystyle \ \nu _{\tau }\ } vacuum survival probabilities P μ μ = P τ τ ≃ 7 18 {\displaystyle \ P_{\mu \mu }=P_{\tau \tau }\simeq {\tfrac {7}{18}}\ } – will be extremely hard to test experimentally.
The L/E flatness of the electron-like event ratio at Super-Kamiokande severely restricts the neutrino mixing matrices to the form given by Stancu & Ahluwalia (1999): [ 12 ]
Additional experimental data fixes sin θ = 1 3 . {\displaystyle \ \sin \theta ={\tfrac {1}{\sqrt {3\ }}}~.} The extension of this result to the CP-violating case is found in Ahluwalia, Liu, & Stancu (2002). [ 13 ]
The name tribimaximal first appeared in the literature in 2002 [ 1 ] although this specific scheme had been previously published in 1999 [ 14 ] as a viable alternative to the trimaximal [ 8 ] scheme. Tribimaximal mixing is sometimes confused with other mixing schemes, e.g. [ 15 ] which differ from tribimaximal mixing by row- and/or column-wise permutations of the mixing-matrix elements. Such permuted forms are experimentally distinct however, and are now ruled out by data. [ 2 ]
That the L/E flatness of the electron-like event ratio at Superkamiokande severely restricts the neutrino mixing matrices was first presented by D. V. Ahluwalia in a Nuclear and Particle Physics Seminar of the Los Alamos National Laboratory on June 5, 1998. It was just a few hours after the Super-Kamiokande press conference that announced the results on atmospheric neutrinos.
This particle physics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tribimaximal_mixing |
Tribochromism refers to a change in colour of a material caused by mechanical friction , similar to piezochromism , the change in colour of a material caused by pressure . It is a property of some materials and is often associated with thermochromism . [ 1 ] Tribochromism and piezochromism are often grouped together under the term mechanochromism . [ 2 ] [ 3 ]
Tribochromatic materials may be used in sensors when friction has to be detected. These materials generally change colour under mechanical stress conditions; the colour gradually fades once the stress is removed. [ 4 ]
This technology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tribochromism |
Tribocorrosion is a material degradation process due to the combined effect of corrosion and wear . [ 1 ] The name tribocorrosion expresses the underlying disciplines of tribology and corrosion . Tribology is concerned with the study of friction, lubrication and wear (its name comes from the Greek "tribo" meaning to rub) and corrosion is concerned with the chemical and electrochemical interactions between a material, normally a metal, and its environment. As a field of research tribocorrosion is relatively new, but tribocorrosion phenomena have been around ever since machines and installations are being used.
Wear is a mechanical material degradation process occurring on rubbing or impacting surfaces, while corrosion involves chemical or electrochemical reactions of the material. Corrosion may accelerate wear and wear may accelerate corrosion. [ 2 ] One then speaks of corrosion accelerated wear or wear accelerated corrosion. Both these phenomena, as well as fretting corrosion (which results from small amplitude oscillations between contacting surfaces) fall into the broader category of tribocorrosion. Erosion-corrosion is another tribocorrosion phenomenon involving mechanical and chemical effects: impacting particles or fluids erode a solid surface by abrasion, chipping or fatigue while simultaneously the surface corrodes. [ 3 ]
Tribocorrosion occurs in many engineering fields. It reduces the life-time of pipes, valves and pumps, of waste incinerators , of mining equipment or of medical implants , and it can affect the safety of nuclear reactors or of transport systems. On the other hand, tribocorrosion phenomena can also be applied to good use, for example in the chemical-mechanical planarization of wafers in the electronics industry [ 4 ] or in metal grinding and cutting in presence of aqueous emulsions. Keeping this in mind, we may define tribocorrosion in a more general way independently of the notion of usefulness or damage or of the particular type of mechanical interaction: Tribocorrosion concerns the irreversible transformation of materials or of their function as a result of simultaneous mechanical and chemical/electrochemical interactions between surfaces in relative motion.
Biotribocorrosion covers the science of surface transformations resulting from the interactions of mechanical loading and chemical/electrochemical reactions that occur between elements of a tribological system exposed to biological environments. [ 5 ] It has been studied for artificial joint prostheses. It is important to understand material degradation processes for joint implants to achieve longer service life and better safety issues for such devices.
While tribocorrosion phenomena may affect many materials, they are most critical for metals, especially the normally corrosion resistant so-called passive metals . The vast majority of corrosion resistant metals and alloys used in engineering (stainless steels, titanium , aluminium etc.) fall into this category. These metals are thermodynamically unstable in the presence of oxygen or water, and they derive their corrosion resistance from the presence at the surface of a thin oxide film, called the passive film , which acts as a protective barrier between the metal and its environment. [ 6 ] Passive films are usually just a few atomic layers thick. Nevertheless, they can provide excellent corrosion protection because if damaged accidentally they spontaneously self-heal by metal oxidation.
However, when a metal surface is subjected to severe rubbing or to a stream of impacting particles the passive film damage can become continuous and extensive. The self-healing process may no longer be effective and in addition it requires a high rate of metal oxidation. In other words, the underlying metal will strongly corrode before the protective passive film is reformed, if at all. In such a case, the total material loss due to tribocorrosion will be much higher than the sum of wear and corrosion one would measure in experiments with the same metal where only wear or only corrosion takes place.
The example illustrates the fact that the rate of tribocorrosion is not simply the addition of the rate of wear and the rate of corrosion but it is strongly affected by synergistic and antagonistic effects between mechanical and chemical mechanisms. To study such effects in the laboratory, one most often uses mechanical wear testing rigs which are equipped with an electrochemical cell. [ 7 ] This permits one to control independently the mechanical and chemical parameters. For example, by imposing a given potential to the rubbing metal one can simulate the oxidation potential of the environment and in addition, under certain conditions, the current flow is a measure of the instantaneous corrosion rate. Volume loss due to electrochemical dissolution can be measured by Faraday's laws of electrolysis and subtracted from total volume loss in tribocorrosion so the sum of mechanical wear loss and the synergies can be calculated. [ 8 ] For a deeper understanding tribocorrosion experiments are supplemented by detailed microscopic and analytical studies of the contacting surfaces.
At high temperatures, the more rapid generation of oxide due to a combination of temperature and tribological action during sliding wear can generate potentially wear resistant oxide layers known as ' glazes '. Under such circumstances, tribocorrosion can be used potentially in a beneficial way.
Erosion corrosion is a degradation of material surface due to mechanical action, often by impinging liquid, abrasion by a slurry, particles suspended in fast flowing liquid or gas, bubbles or droplets, cavitation , etc. [ 9 ] The mechanism can be described as follows:
The mechanism of erosion corrosion, the materials affected by it, and the conditions when it occurs are generally different from that of flow-accelerated corrosion , although the latter is sometimes classified as a sub-type of erosion corrosion. | https://en.wikipedia.org/wiki/Tribocorrosion |
Tribofilms ( boundary lubricant films , [ 1 ] boundary lubricating films , [ 2 ] tribo-boundary films [ 3 ] or boundary films [ 4 ] ) are films that form on tribologically stressed surfaces. Tribofilms are mostly solid surface films that result from a chemical reaction of lubricant components or tribological surfaces.
Tribofilms play an important role in reducing friction and wear in lubricated systems. They form as a result of complex mechanochemical interactions between surface materials and lubricants, [ 5 ] and the study of tribofilm formation processes is a major field of tribology .
Generally, a tribofilm is any film that forms in a tribosystem "as a result of interaction between chemical components of the [lubricant] with the lubricated surface". [ 6 ] The term is mostly used to describe strongly bound films that are formed on tribologically stressed surfaces, such as tribochemical reaction films [ 1 ] (for example produced by ZDDP -containing lubricants [ 7 ] [ 8 ] ) or polymeric and non-sacrificial reaction films (for example formed by complex esters). [ 1 ] | https://en.wikipedia.org/wiki/Tribofilm |
Tribology is the science and engineering of understanding friction , lubrication and wear phenomena for interacting surfaces in relative motion . It is highly interdisciplinary, drawing on many academic fields, including physics , chemistry , materials science , mathematics , biology and engineering . [ 1 ] The fundamental objects of study in tribology are tribosystems , which are physical systems of contacting surfaces. Subfields of tribology include biotribology , nanotribology and space tribology . It is also related to other areas such as the coupling of corrosion and tribology in tribocorrosion and the contact mechanics of how surfaces in contact deform.
Approximately 20% of the total energy expenditure of the world is due to the impact of friction and wear in the transportation, manufacturing, power generation, and residential sectors. [ 2 ]
The word tribology derives from the Greek root τριβ- of the verb τρίβω , tribo , "I rub" in classic Greek, and the suffix -logy from -λογία , -logia "study of", "knowledge of". Peter Jost coined the word in 1966, [ 1 ] in the eponymous report which highlighted the cost of friction , wear and corrosion to the UK economy. [ 2 ]
Despite the relatively recent naming of the field of tribology, quantitative studies of friction can be traced as far back as 1493, when Leonardo da Vinci first noted the two fundamental 'laws' of friction . [ 3 ] According to Leonardo, frictional resistance was the same for two different objects of the same weight but making contact over different widths and lengths. He also observed that the force needed to overcome friction doubles as weight doubles. However, Leonardo's findings remained unpublished in his notebooks. [ 3 ]
The two fundamental 'laws' of friction were first published (in 1699) by Guillaume Amontons , with whose name they are now usually associated. They state that: [ 3 ]
Although not universally applicable, these simple statements hold for a surprisingly wide range of systems. [ 4 ] These laws were further developed by Charles-Augustin de Coulomb (in 1785), who noticed that static friction force may depend on the contact time and sliding (kinetic) friction may depend on sliding velocity, normal force and contact area. [ 5 ] [ 6 ]
In 1798, Charles Hatchett and Henry Cavendish carried out the first reliable test on frictional wear . In a study commissioned by the Privy Council of the UK , they used a simple reciprocating machine to evaluate the wear rate of gold coins . They found that coins with grit between them wore at a faster rate compared to self-mated coins. [ 7 ] In 1860, Theodor Reye [ a ] proposed Reye's hypothesis [ it ] . [ 9 ] In 1953, John Frederick Archard developed the Archard equation which describes sliding wear and is based on the theory of asperity contact. [ 10 ]
Other pioneers of tribology research are Australian physicist Frank Philip Bowden [ 11 ] and British physicist David Tabor , [ 12 ] both of the Cavendish Laboratory at Cambridge University. Together they wrote the seminal textbook The Friction and Lubrication of Solids [ 13 ] (Part I originally published in 1950 and Part II in 1964). Michael J. Neale was another leader in the field during the mid-to-late 1900s. He specialized in solving problems in machine design by applying his knowledge of tribology. Neale was respected as an educator with a gift for integrating theoretical work with his own practical experience to produce easy-to-understand design guides. The Tribology Handbook , [ 14 ] which he first edited in 1973 and updated in 1995, is still used around the world and forms the basis of numerous training courses for engineering designers.
Duncan Dowson surveyed the history of tribology in his 1997 book History of Tribology (2nd edition). [ 5 ] This covers developments from prehistory, through early civilizations ( Mesopotamia , ancient Egypt ) and highlights the key developments up to the end of the twentieth century.
The term tribology became widely used following The Jost Report published in 1966. [ 1 ] The report highlighted the huge cost of friction, wear and corrosion to the UK economy (1.1–1.4% of GDP ). [ 1 ] As a result, the UK government established several national centres to address tribological problems. Since then the term has diffused into the international community, with many specialists now identifying as "tribologists".
Despite considerable research since the Jost Report, the global impact of friction and wear on energy consumption , economic expenditure, and carbon dioxide emissions are still considerable. In 2017, Kenneth Holmberg and Ali Erdemir attempted to quantify their impact worldwide. [ 15 ] They considered the four main energy consuming sectors: transport , manufacturing , power generation , and residential . The following were concluded: [ 15 ]
Classical tribology covering such applications as ball bearings, gear drives, clutches, brakes, etc. was developed in the context of mechanical engineering. But in the last decades tribology expanded to qualitatively new fields of applications, in particular micro- and nanotechnology as well as biology and medicine. [ 16 ]
The concept of tribosystems is used to provide a detailed assessment of relevant inputs, outputs and losses to tribological systems. Knowledge of these parameters allows tribologists to devise test procedures for tribological systems.
Tribofilms are thin films that form on tribologically stressed surfaces. They play an important role in reducing friction and wear in tribological systems.
The Stribeck curve shows how friction in fluid-lubricated contacts is a non-linear function of lubricant viscosity , entrainment velocity and contact load.
The word friction comes from the Latin "frictionem", which means rubbing. This term is used to describe all those dissipative phenomena, capable of producing heat and of opposing the relative motion between two surfaces. There are two main types of friction:
The study of friction phenomena is a predominantly empirical study and does not allow to reach precise results, but only to useful approximate conclusions. This inability to obtain a definite result is due to the extreme complexity of the phenomenon. If it is studied more closely it presents new elements, which, in turn, make the global description even more complex. [ 17 ]
All the theories and studies on friction can be simplified into three main laws, which are valid in most cases:
Coulomb later found deviations from Amontons' laws in some cases. [ 6 ] In systems with significant nonuniform stress fields, Amontons' laws are not satisfied macroscopically because local slip occurs before the entire system slides. [ 18 ]
Consider a block of a certain mass m, placed in a quiet position on a horizontal plane. If you want to move the block, an external force F → o u t {\displaystyle {\vec {F}}_{out}} must be applied, in this way we observe a certain resistance to the motion given by a force equal to and opposite to the applied force, which is precisely the static frictional force F → s . f . {\displaystyle {\vec {F}}_{s.f.}} . [ 19 ]
By continuously increasing the applied force, we obtain a value such that the block starts instantly to move. At this point, also taking into account the first two friction laws stated above, it is possible to define the static friction force as a force equal in modulus to the minimum force required to cause the motion of the block, and the coefficient of static friction μ {\displaystyle \mu } as the ratio of the static friction force F → s . f . {\displaystyle {\vec {F}}_{s.f.}} . and the normal force at block N → {\displaystyle {\vec {N}}} , obtaining | F → s . f . | ≤ μ | N → | {\displaystyle {\left|{\vec {F}}_{s.f.}\right|}\leq \mu {\left|{\vec {N}}\right|}}
Once the block has been put into motion, the block experiences a friction force with a lesser intensity than the static friction force F → s . f . {\displaystyle {\vec {F}}_{s.f.}} . The friction force during relative motion is known as the dynamic friction force F → d . f . {\displaystyle {\vec {F}}_{d.f.}} . In this case it is necessary to take into account not only the first two laws of Amontons, but also of the law of Coulomb, so as to be able to affirm that the relationship between dynamic friction force F → d . f . {\displaystyle {\vec {F}}_{d.f.}} , coefficient of dynamic friction k and normal force N is the following: | F → d . f . | = k | N → | {\displaystyle \left|{\vec {F}}_{d.f.}\right|=k{\left|{\vec {N}}\right|}}
At this point it is possible to summarize the main properties of the static friction coefficients μ {\displaystyle \mu } and the dynamic one k {\displaystyle k} .
These coefficients are dimensionless quantities, given by the ratio between the intensity of the friction force F → f {\displaystyle {\vec {F}}_{f}} and the intensity of the applied load W → {\displaystyle {\vec {W}}} , depending on the type of surfaces that are involved in a mutual contact, and in any case, the condition is always valid such that: μ > k {\displaystyle \mu >k} .
Usually, the value of both coefficients does not exceed the unit and can be considered constant only within certain ranges of forces and velocities, outside of which there are extreme conditions that modify these coefficients and variables.
In systems with significant nonuniform stress fields, the macroscopic static friction coefficient depends on the external pressure, system size, or shape because local slip occurs before the system slides. [ 18 ]
The following table shows the values of the static and dynamic friction coefficients for common materials:
In the case of bodies capable of rolling, there is a particular type of friction, in which the sliding phenomenon, typical of dynamic friction, does not occur, but there is also a force that opposes the motion, which also excludes the case of static friction. This type of friction is called rolling friction. Now we want to observe in detail what happens to a wheel that rolls on a horizontal plane. Initially the wheel is immobile and the forces acting on it are the weight force m g → {\displaystyle m{\vec {g}}} and the normal force N → {\displaystyle {\vec {N}}} given by the response to the weight of the floor.
At this point the wheel is set in motion, causing a displacement at the point of application of the normal force which is now applied in front of the center of the wheel, at a distance b , which is equal to the value of the rolling friction coefficient. The opposition to the motion is caused by the separation of the normal force and the weight force at the exact moment in which the rolling starts, so the value of the torque given by the rolling friction force is M → r . f . = b → × m g → {\displaystyle {{\vec {M}}_{r.f.}}={\vec {b}}\times m{\vec {g}}} What happens in detail at the microscopic level between the wheel and the supporting surface is described in Figure, where it is possible to observe what is the behavior of the reaction forces of the deformed plane acting on an immobile wheel.
Rolling the wheel continuously causes imperceptible deformations of the plane and, once passed to a subsequent point, the plane returns to its initial state. In the compression phase the plane opposes the motion of the wheel, while in the decompression phase it provides a positive contribution to the motion.
The force of rolling friction depends, therefore, on the small deformations suffered by the supporting surface and by the wheel itself, and can be expressed as | F → r | = b | N → | {\displaystyle |{\vec {F}}_{r}|=b|{\vec {N}}|} , where it is possible to express b in relation to the sliding friction coefficient μ {\displaystyle \mu } as b = μ v r {\textstyle b={\mu v \over r}} , with r being the wheel radius.
Going even deeper, it is possible to study not only the most external surface of the metal, but also the immediately more internal states, linked to the history of the metal, its composition and the manufacturing processes undergone by the latter.
it is possible to divide the metal into four different layers:
The layer of oxides and impurities (third body) has a fundamental tribological importance, in fact it usually contributes to reducing friction. Another fact of fundamental importance regarding oxides is that if you could clean and smooth the surface in order to obtain a pure "metal surface", what we would observe is the union of the two surfaces in contact. In fact, in the absence of thin layers of contaminants, the atoms of the metal in question, are not able to distinguish one body from another, thus going to form a single body if put in contact.
Contact between surfaces is made up of a large number of microscopic regions, in the literature called asperities or junctions of contact, where atom-to-atom contact takes place. The phenomenon of friction, and therefore of the dissipation of energy, is due precisely to the deformations that such regions undergo due to the load and relative movement. Plastic, elastic, or rupture deformations can be observed:
The energy that is dissipated during the phenomenon is transformed into heat, thus increasing the temperature of the surfaces in contact. The increase in temperature also depends on the relative speed and the roughness of the material, it can be so high as to even lead to the fusion of the materials involved.
In friction phenomena, temperature is fundamental in many areas of application. For example, a rise in temperature may result in a sharp reduction of the friction coefficient, and consequently, the effectiveness of the brakes.
The adhesion theory states that in the case of spherical asperities in contact with each other, subjected to a W → {\displaystyle {\vec {W}}} load, a deformation is observed, which, as the load increases, passes from an elastic to a plastic deformation. This phenomenon involves an enlargement of the real contact area A r {\displaystyle A_{r}} , which for this reason can be expressed as: A r = W → D {\displaystyle A_{r}={{\vec {W}} \over D}} where D is the hardness of the material definable as the applied load divided by the area of the contact surface.
If at this point the two surfaces are sliding between them, a resistance to shear stress t is observed, given by the presence of adhesive bonds, which were created precisely because of the plastic deformations, and therefore the frictional force will be given by F → a = A r t → {\displaystyle {\vec {F}}_{a}=A_{r}{\vec {t}}} At this point, since the coefficient of friction is the ratio between the intensity of the frictional force and that of the applied load, it is possible to state that μ = t D {\displaystyle \mu ={t \over D}} thus relating to the two material properties: shear strength t and hardness. To obtain low value friction coefficients μ {\displaystyle \mu } it is possible to resort to materials which require less shear stress, but which are also very hard. In the case of lubricants, in fact, we use a substrate of material with low cutting stress t , placed on a very hard material.
The force acting between two solids in contact will not only have normal components, as implied so far, but will also have tangential components. This further complicates the description of the interactions between roughness, because due to this tangential component plastic deformation comes with a lower load than when ignoring this component. A more realistic description then of the area of each single junction that is created is given by A i 2 = ( W → i D ) 2 + α ( F → i D ) 2 {\displaystyle {A_{i}}^{2}=\left({\frac {{\vec {W}}_{i}}{D}}\right)^{2}+\alpha \left({\frac {{\vec {F}}_{i}}{D}}\right)^{2}} with α {\displaystyle \alpha } constant and a "tangent" force F → i {\displaystyle {\vec {F}}_{i}} applied to the joint.
To obtain even more realistic considerations, the phenomenon of the third body should also be considered, i.e., the presence of foreign materials, such as moisture, oxides or lubricants, between the two solids in contact. A coefficient c is then introduced which is able to correlate the shear strength t of the pure "material" and that of the third body t t . b . {\displaystyle t_{t.b.}} t = c ⋅ t t . b . {\displaystyle t=c\cdot t_{t.b.}} with 0 < c < 1.
By studying the behavior at the limits it will be that for c = 0, t = 0 and for c = 1 it returns to the condition in which the surfaces are directly in contact and there is no presence of a third body. Keeping in mind what has just been said, it is possible to correct the friction coefficient formula as follows: μ = c [ α ( 1 − c 2 ) ] 1 / 2 {\displaystyle \mu ={\frac {c}{[\alpha (1-c^{2})]^{1/2}}}} In conclusion, the case of elastic bodies in interaction with each other is considered.
Similarly to what we have just seen, it is possible to define an equation of the type A = K W → {\displaystyle A=K{\vec {W}}} where, in this case, K depends on the elastic properties of the materials. Also for the elastic bodies the tangential force depends on the coefficient c seen above, and it will be F → T = c A s {\displaystyle {\vec {F}}_{T}=cAs} and therefore a fairly exhaustive description of the friction coefficient can be obtained μ = c K s {\displaystyle \mu =cKs}
The simplest and most immediate method for evaluating the friction coefficient of two surfaces is the use of an inclined plane on which a block of material is made to slide. As can be seen in the figure, the normal force of the plane is given by m g cos θ {\displaystyle mg\cos \theta } , while the frictional force is equal to m g sin θ {\displaystyle mg\sin \theta } . This allows us to state that the coefficient of friction can be calculated very easily, by means of the tangent of the angle in which the block begins to slip. In fact we have μ = F a N = m g sin θ m g cos θ = sin θ cos θ = tan θ {\displaystyle \mu ={F_{a} \over N}={mg\sin \theta \over mg\cos \theta }={\sin \theta \over \cos \theta }=\tan \theta } Then from the inclined plane we moved on to more sophisticated systems, which allow us to consider all the possible environmental conditions in which the measurement is made, such as the cross-roller machine or the pin and disk machine. Today there are digital machines such as the "Friction Tester" which allows, by means of a software support, to insert all the desired variables. Another widely used process is the ring compression test. A flat ring of the material to be studied is plastically deformed by means of a press, if the deformation is an expansion in both the inner and the outer circle, then there will be low or zero friction coefficients. Otherwise for a deformation that expands only in the inner circle there will be increasing friction coefficients.
To reduce friction between surfaces and keep wear under control, materials called lubricants are used. [ 20 ] Unlike what you might think, these are not just oils or fats, but any fluid material that is characterized by viscosity, such as air and water. Of course, some lubricants are more suitable than others, depending on the type of use they are intended for: air and water, for example, are readily available, but the former can only be used under limited load and speed conditions, while the second can contribute to the wear of materials.
What we try to achieve by means of these materials is a perfect fluid lubrication, or a lubrication such that it is possible to avoid direct contact between the surfaces in question, inserting a lubricant film between them. To do this there are two possibilities, depending on the type of application, the costs to address and the level of "perfection" of the lubrication desired to be achieved, there is a choice between:
The viscosity is the equivalent of friction in fluids, it describes, in fact, the ability of fluids to resist the forces that cause a change in shape.
Thanks to Newton's studies, a deeper understanding of the phenomenon has been achieved. He, in fact, introduced the concept of laminar flow : "a flow in which the velocity changes from layer to layer". It is possible to ideally divide a fluid between two surfaces ( S 1 {\displaystyle S_{1}} , S 2 {\displaystyle S_{2}} ) of area A, in various layers.
The layer in contact with the surface S 2 {\displaystyle S_{2}} , which moves with a velocity v due to an applied force F , will have the same velocity as v of the slab, while each immediately following layer will vary this velocity of a quantity dv , up to the layer in contact with the immobile surface S 1 {\displaystyle S_{1}} , which will have zero speed.
From what has been said, it is possible to state that the force F , necessary to cause a rolling motion in a fluid contained between two plates, is proportional to the area of the two surfaces and to the speed gradient: F ∝ A d v d y {\displaystyle F\propto A{dv \over dy}} At this point we can introduce a proportional constant μ {\displaystyle \mu } , which corresponds to the dynamic viscosity coefficient of the fluid, to obtain the following equation, known as Newton's law F = μ A d v d y {\displaystyle F=\mu A{dv \over dy}} The speed varies by the same amount dv of layer in layer and then the condition occurs so that dv / dy = v / L , where L is the distance between the surfaces S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} , and then we can simplify the equation by writing F = μ A v L {\displaystyle F=\mu A{v \over L}} The viscosity μ {\displaystyle \mu } is high in fluids that strongly oppose the motion, while it is contained for fluids that flow easily.
To determine what kind of flow is in the study, we observe its Reynolds number R e = ρ L v μ {\displaystyle Re={{\rho Lv} \over \mu }} This is a constant that depends on the fluid mass ρ {\displaystyle \rho } of the fluid, on its viscosity μ {\displaystyle \mu } and on the diameter L of the tube in which the fluid flows. If the Reynolds number is relatively low then there is a laminar flow, whereas for R e ≃ 2000 {\displaystyle Re\simeq 2000} the flow becomes turbulent.
To conclude we want to underline that it is possible to divide the fluids into two types according to their viscosity:
Temperature and pressure are two fundamental factors to evaluate when choosing a lubricant instead of another. Consider the effects of temperature initially.
There are three main causes of temperature variation that can affect the behavior of the lubricant:
In order to classify the various lubricants according to their viscosity behavior as a function of temperature, in 1929 the viscosity index (V.I.) was introduced by Dean and Davis. These assigned the best lubricant then available, namely the oil of Pennsylvania, the viscosity index 100, and at the worst, the American oil of the Gulf Coast, the value 0. To determine the value of the intermediate oil index, the following procedure is used: two reference oils are chosen so that the oil in question has the same viscosity at 100 °C, and the following equation is used to determine the viscosity index V . I . = L − O Test L − H × 100 {\displaystyle V.I.={{L-O_{\text{Test}}} \over {L-H}}\times 100} This process has some disadvantages:
In the case of oils with V.I. above 100 you can use a different relationship that allows you to get exact results V . I . = 10 N − 1 0.00715 + 100 {\displaystyle V.I.={{10^{N-1}} \over {0.00715}}+100} N = Log ( H ) − Log ( O Test ) Log ( v ) {\displaystyle N={{\operatorname {Log} (H)-\operatorname {Log} (O_{\text{Test}})} \over {\operatorname {Log} (v)}}} where, in this case, H is the viscosity at 100 °F (38 °C) of the oil with V.I. = 100 and v is the kinematic viscosity of the study oil at 210 °F (99 °C).
We can therefore say, in conclusion, that an increase in temperature leads to a decrease in the viscosity of the oil. It is also useful to keep in mind that, in the same way, an increase in pressure implies an increase in viscosity. To evaluate the effects of pressure on viscosity, the following equation is used μ = μ 0 exp ( α p ) {\displaystyle \mu =\mu _{0}\exp {(\alpha p)}} where μ {\displaystyle \mu } is the pressure viscosity coefficient p, μ 0 {\displaystyle \mu _{0}} is the viscosity coefficient at atmospheric pressure and α {\displaystyle \alpha } is a constant that describes the relationship between viscosity and pressure.
To determine the viscosity of a fluid, viscosimeters are used which can be divided into 3 main categories:
The first two types of viscometers are mainly used for Newtonian fluids, while the third is very versatile.
The wear is the progressive involuntary removal of material from a surface in relative motion with another or with a fluid. We can distinguish two different types of wear: moderate wear and severe wear. The first case concerns low loads and smooth surfaces, while the second concerns significantly higher loads and compatible and rough surfaces, in which the wear processes are much more violent. Wear plays a fundamental role in tribological studies, since it causes changes in the shape of the components used in the construction of machinery (for example). These worn parts must be replaced and this entails both a problem of an economic nature, due to the cost of replacement, and a functional problem, since if these components are not replaced in time, more serious damage could occur to the machine in its complex. This phenomenon, however, has not only negative sides, indeed, it is often used to reduce the roughness of some materials, eliminating the asperities. Erroneously we tend to imagine wear in a direct correlation with friction, in reality these two phenomena can not be easily connected. There may be conditions such that low friction can result in significant wear and vice versa. In order for this phenomenon to occur, certain implementation times are required, which may change depending on some variables, such as load, speed, lubrication and environmental conditions, and there are different wear mechanisms, which may occur simultaneously or even combined with each other:
As known, the contact between two surfaces occurs through the interaction between asperities. [ 21 ] If a shearing force is applied in the contact area, it may be possible to detach a small part of the weaker material, due to its adhesion to the harder surface. What is described is precisely the mechanism of the adhesive wear represented in the figure. This type of wear is very problematic, since it involves high wear speeds, but at the same time it is possible to reduce adhesion by increasing surface roughness and hardness of the surfaces involved, or by inserting layers of contaminants such as oxygen, oxides, water, or oils. In conclusion, the behavior of the adhesive wear volume can be described by means of three main laws
An important aspect of wear is emission of wear particles into the environment which increasingly threatens human health and ecology. The first researcher who investigated this topic was Ernest Rabinowicz . [ 22 ]
The abrasive wear consists of the cutting effort of hard surfaces that act on softer surfaces and can be caused either by the roughness that as tips cut off the material against which they rub (two-body abrasive wear), or from particles of hard material that interpose between two surfaces in relative motion (three-body abrasive wear). At application levels, the two-body wear is easily eliminated by means of an adequate surface finish, while the three-body wear can bring serious problems and must therefore be removed as much as possible by means of suitable filters, even before of a weighted machine design.
The fatigue wear is a type of wear that is caused by alternative loads, which cause local contact forces repeated over time, which in turn lead to deterioration of the materials involved. The most immediate example of this type of wear is that of a comb. If you slide a finger over the teeth of the comb over and over again, it is observed that at some point one or more teeth of the comb come off. This phenomenon can lead to the breaking of the surfaces due to mechanical or thermal causes. The first case is that described above in which a repeated load causes high contact stresses. The second case, however, is caused by the thermal expansion of the materials involved in the process. To reduce this type of wear, therefore, it is good to try to decrease both the contact forces and the thermal cycling, that is the frequency with which different temperatures intervene. For optimal results it is also good to eliminate, as much as possible, impurities between surfaces, local defects and inclusions of foreign materials in the bodies involved.
The corrosive wear occurs in the presence of metals that oxidize or corrode. When the pure metal surfaces come into contact with the surrounding environment, oxide films are created on their surfaces because of the contaminants present in the environment itself, such as water, oxygen or acids. These films are continually removed from the abrasive and adhesive wear mechanisms, continually recreated by pure-contaminating metal interactions. Clearly this type of wear can be reduced by trying to create an 'ad hoc' environment, free of pollutants and sensible to minimal thermal changes. Corrosive wear can also be positive in some applications. In fact, the oxides that are created, contribute to decrease the coefficient of friction between the surfaces, or, being in many cases harder than the metal to which they belong, can be used as excellent abrasives.
The rubbing wear occurs in systems subject to more or less intense vibrations, which cause relative movements between the surfaces in contact within the order of nanometers. These microscopic relative movements cause both adhesive wear, caused by the displacement itself, and abrasive wear, caused by the particles produced in the adhesive phase, which remain trapped between the surfaces. This type of wear can be accelerated by the presence of corrosive substances and the increase in temperature. [ 23 ]
The erosion wear occurs when free particles, which can be either solid or liquid, hit a surface, causing abrasion. The mechanisms involved are of various kinds and depend on certain parameters, such as the impact angle, the particle size, the impact velocity and the material of which the particles are made up.
Among the main factors influencing wear we find
It has been verified that the harder a material is, the more it decreases. In the same way, the less two materials are mutually soluble, the more the wear tends to decrease. Finally, as regards the crystalline structure, it is possible to state that some structures are more suitable to resist the wear of others, such as a hexagonal structure with a compact distribution, which can only deform by slipping along the base planes.
To provide an assessment of the damage caused by wear, we use a dimensionless coefficient called wear rate, given by the ratio between the height change of the body Δ h {\displaystyle \Delta h} and the length of the relative sliding Δ l {\displaystyle \Delta l} . T wear = δ h Δ l {\displaystyle T_{\text{wear}}={\delta h \over \Delta l}} This coefficient makes it possible to subdivide, depending on its size, the damage suffered by various materials in different situations, passing from a modest degree of wear, through a medium, to a degree of severe wear.
Instead, to express the volume of wear V it is possible to use the Holm equation
where W / H represents the real contact area, l the length of the distance traveled and k and k a {\displaystyle k_{a}} are experimental dimensional factors.
In experimental measurements of material wear, it is often necessary to recreate fairly small wear rates and to accelerate times. The phenomena, which in reality develop after years, in the laboratory must occur after a few days. A first evaluation of the wear processes is a visual inspection of the superficial profile of the body in the study, including a comparison before and after the occurrence of the wear phenomenon. In this first analysis the possible variations of the hardness and of the superficial geometry of the material are observed. Another method of investigation is that of the radioactive tracer, used to evaluate wear at macroscopic levels. One of the two materials in contact, involved in a wear process, is marked with a radioactive tracer. In this way, the particles of this material, which will be removed, will be easily visible and accessible. Finally, to accelerate wear times, one of the best-known techniques used is that of the high pressure contact tests. In this case, to obtain the desired results it is sufficient to apply the load on a very reduced contact area.
Historically, tribology research concentrated on the design and effective lubrication of machine components, particularly for bearings . However, the study of tribology extends into most aspects of modern technology and any system where one material slides over another can be affected by complex tribological interactions. [ 24 ]
Traditionally, tribology research in the transport industry focused on reliability, ensuring the safe, continuous operation of machine components. Nowadays, due to an increased focus on energy consumption , efficiency has become increasingly important and thus lubricants have become progressively more complex and sophisticated in order to achieve this. [ 24 ] Tribology also plays an important role in manufacturing . For example, in metal-forming operations, friction increases tool wear and the power required to work a piece. This results in increased costs due to more frequent tool replacement, loss of tolerance as tool dimensions shift, and greater forces required to shape a piece.
One of the more common materials in aerospace specifically is copper matrix self-lubricating composites, which use solid lubricants. These lubricants are mixed in with the copper and other materials, which is effectively done by resonant acoustic mixing. [ 25 ]
The use of lubricants which minimize direct surface contact reduces tool wear and power requirements. [ 26 ] It is also necessary to know the effects of manufacturing, all manufacturing methods leave a unique system fingerprint (i.e. surface topography ) which will influence the tribocontact (e.g. lubricant film formation).
Tribology research ranges from macro to nano scales, in areas as diverse as the movement of continental plates and glaciers to the locomotion of animals and insects. [ 24 ] Tribology research is traditionally concentrated on transport and manufacturing sectors, but this has considerably diversified. Tribology research can be loosely divided into the following fields (with some overlap):
Recently, intensive studies of superlubricity (phenomenon of vanishing friction) have sparked due to increasing demand for energy savings. [ 30 ] Furthermore, the development of new materials, such as graphene and ionic liquids , allows for fundamentally new approaches to solve tribological problems . [ 31 ]
There are now numerous national and international societies, including: the Society of Tribologists and Lubrication Engineers (STLE) in the US, the Institution of Mechanical Engineers and Institute of Physics (IMechE Tribology Group, IOP Tribology Group) in the UK, the German Society for Tribology (Gesellschaft für Tribologie), the Korean Tribology Society (KTS), the Malaysian Tribology Society (MYTRIBOS), the Japanese Society of Tribologists (JAST), the Tribology Society of India (TSI), the Chinese Mechanical Engineering Society (Chinese Tribology Institute) and the International Tribology Council.
Tribology research is mostly empirical, which can be explained by the vast number of parameters that influence friction and wear in tribological contacts. Thus, most research fields rely heavily on the use of standardized tribometers and test procedures as well component-level test rigs. | https://en.wikipedia.org/wiki/Tribology |
Triboluminescence is a phenomenon in which light is generated when a material is mechanically pulled apart, ripped, scratched, crushed, or rubbed (see tribology ). The phenomenon is not fully understood but appears in most cases to be caused by the separation and reunification of static electric charges , see also triboelectric effect . The term comes from the Greek τρίβειν ("to rub"; see tribology ) and the Latin lumen (light). Triboluminescence can be observed when breaking sugar crystals and peeling adhesive tapes.
Triboluminescence is often a synonym for fractoluminescence (a term mainly used when referring only to light emitted from fractured crystals). Triboluminescence differs from piezoluminescence in that a piezoluminescent material emits light when deformed, as opposed to broken. These are examples of mechanoluminescence , which is luminescence resulting from any mechanical action on a solid .
The Uncompahgre Ute indigenous people from Central Colorado are one of the first documented groups of people in the world credited with the application of mechanoluminescence involving the use of quartz crystals to generate light. [ 1 ] [ 2 ] The Ute constructed unique ceremonial rattles made from buffalo rawhide which they filled with clear quartz crystals collected from the mountains of Colorado and Utah. When the rattles were shaken at night during ceremonies, the friction and mechanical stress of the quartz crystals impacting together produced flashes of light visible through the translucent buffalo hide.
The first recorded observation is attributed to English scholar Francis Bacon when he recorded in his 1620 Novum Organum that "It is well known that all sugar , whether candied or plain, if it be hard, will sparkle when broken or scraped in the dark." [ 3 ] The scientist Robert Boyle also reported on some of his work on triboluminescence in 1663. [ 4 ] In 1675. Astronomer Jean-Felix Picard observed that his barometer was glowing in the dark as he carried it. His barometer consisted of a glass tube that was partially filled with mercury. The empty space above the mercury would glow whenever the mercury slid down the glass tube. [ 5 ]
In the late 1790s, sugar production began to produce more refined sugar crystals. These crystals were formed into a large solid cone for transport and sale. This solid sugar cone had to be broken into usable chunks using a sugar nips device. People began to notice that tiny bursts of light were visible as sugar was "nipped" in low light, an established example of triboluminescence. [ 6 ]
There remain a few ambiguities about the effect. The current theory of triboluminescence—based upon crystallographic, spectroscopic, and other experimental evidence—is that upon fracture of asymmetrical materials, charge is separated. When the charges recombine, the electrical discharge ionizes the surrounding air, causing a flash of light. Research further suggests that crystals that display triboluminescence often lack symmetry and are poor conductors. [ 7 ] However, there are substances which break this rule, and which do not possess asymmetry, yet display triboluminescence, such as hexakis(antipyrine)terbium iodide. [ 8 ] It is thought that these materials contain impurities, which make the substance locally asymmetric. Further information on some of the possible processes involved can be found in the page on the triboelectric effect.
The biological phenomenon of triboluminescence is thought to be controlled by recombination of free radicals during mechanical activation. [ 9 ]
Certain household materials and substances can be seen to exhibit the property:
A diamond may begin to glow while being rubbed; this occasionally happens to diamonds while a facet is being ground or the diamond is being sawn during the cutting process . Diamonds may fluoresce blue or red. Some other minerals, such as quartz , are triboluminescent, emitting light when rubbed together. [ 19 ]
Triboluminescence as a biological phenomenon is observed in mechanical deformation and contact electrification of epidermal surface of osseous and soft tissues, during chewing food, at friction in joints of vertebrae, during sexual intercourse, and during blood circulation . [ 20 ] [ 21 ]
Water jet abrasive cutting of ceramics (e.g., tiles ) creates a yellow/orange glow at the point of impact of very high-speed flow.
Fractoluminescence is often used as a synonym for triboluminescence. [ 27 ] It is the emission of light from the fracture (rather than rubbing) of a crystal , but fracturing often occurs with rubbing. Depending upon the atomic and molecular composition of the crystal, when the crystal fractures, a charge separation can occur, making one side of the fractured crystal positively charged and the other side negatively charged. Like in triboluminescence, if the charge separation results in a large enough electric potential , a discharge across the gap and through the bath gas between the interfaces can occur. The potential at which this occurs depends upon the dielectric properties of the bath gas. [ 28 ]
The emission of electromagnetic radiation (EMR) during plastic deformation and crack propagation in metals and rocks has been studied. The EMR emissions from metals and alloys have also been explored and confirmed. Molotskii presented a dislocation mechanism for this type of EMR emission. [ 29 ] In 2005, Srilakshmi and Misra reported an additional phenomenon of secondary EMR during plastic deformation and crack propagation in uncoated and metal-coated metals and alloys. [ 30 ]
EMR during the micro-plastic deformation and crack propagation from several metals and alloys and transient magnetic field generation during necking in ferromagnetic metals were reported by Misra (1973–75), which have been confirmed and explored by several researchers. [ 31 ] Tudik and Valuev (1980) were able to measure the EMR frequency during tensile fracture of iron and aluminum in the region 100 THz by using photomultipliers . Srilakshmi and Misra (2005a) also reported an additional phenomenon of secondary electromagnetic radiation in uncoated and metal-coated metals and alloys. If a solid material is subjected to stresses of large amplitudes, which can cause plastic deformation and fracture, emissions such as thermal, acoustic, ions, and exo-emissions occur.
The study of deformation is essential for the development of new materials. Deformation in metals depends on temperature, type of stress applied, strain rate, oxidation, and corrosion. Deformation-induced EMR can be divided into three categories: effects in ionic crystal materials, effects in rocks and granites, and effects in metals and alloys. EMR emission depends on the orientation of the grains in individual crystals since material properties are different in differing directions. [ 32 ] Amplitude of the EMR pulse increases as long as the crack grows as new atomic bonds are broken, leading to EMR. The pulse starts to decay as the cracking halts. [ 33 ] Observations from experiments showed that emitted EMR signals contain mixed frequency components.
The most widely used tensile test method is used to characterize the mechanical properties of materials. From any complete tensile test record, one can obtain important information about the material's elastic properties, the character and extent of plastic deformation, yield, and tensile strengths and toughness. The information obtained from one test justifies the extensive use of tensile tests in engineering materials research. Therefore, investigations of EMR emissions are mainly based on the tensile test of the specimens. From experiments, it can be shown that tensile crack formation excites more intensive EMR than shear cracking, increasing the elasticity, strength, and loading rate during uniaxial loading increases amplitude. Poisson's ratio is a key parameter for EMR characterization during triaxial compression. [ 34 ] If the Poisson's ratio is lower, it is harder for the material to strain transversally and hence there is a higher probability of new fractures. | https://en.wikipedia.org/wiki/Triboluminescence |
A tribometer is an instrument that measures tribological quantities, such as coefficient of friction , friction force, and wear volume, between two surfaces in contact. It was invented by the 18th century Dutch scientist Musschenbroek [ 1 ] [ 2 ]
A tribotester is the general name given to a machine or device used to perform tests and simulations of wear, friction and lubrication which are the subject of the study of tribology. [ citation needed ] Often tribotesters are extremely specific in their function and are fabricated by manufacturers who desire to test and analyze the long-term performance of their products. An example is that of orthopedic implant manufacturers who have spent considerable sums of money to develop tribotesters that accurately reproduce the motions and forces that occur in human hip joints so that they can perform accelerated wear tests of their products.
A simple tribometer is described by a hanging mass and a mass resting on a horizontal surface, connected to each other via a string and pulley. The coefficient of friction , μ , when the system is stationary, is determined by increasing the hanging mass until the moment that the resting mass begins to slide. Then using the general equation for friction force:
Where N , the normal force, is equal to the weight (mass x gravity) of the sitting mass ( m T ) and F , the loading force, is equal to the weight (mass x gravity) of the hanging mass ( m H ).
To determine the kinetic coefficient of friction the hanging mass is increased or decreased until the mass system moves at a constant speed.
In both cases, the coefficient of friction is simplified to the ratio of the two masses:
In most test applications using tribometers, wear is measured by comparing the mass or surfaces of test specimens before and after testing. Equipment and methods used to examine the worn surfaces include optical microscopes , scanning electron microscopes , optical interferometry and mechanical roughness testers.
Tribometers are often referred to by the specific contact arrangement they simulate or by the original equipment developer. Several arrangements are:
A bouncing ball tribometer consists of a ball which is impacted at an angle against a surface. During a typical test, a ball is slid on an angle along a track until it impacts a surface and then bounces off of the surface. The friction produced in the contact between the ball and the surface results in a horizontal force on the surface and a rotational force on the ball. Frictional force is determined by finding the rotational speed of the ball using high speed photography or by measuring the force on the horizontal surface. Pressure in the contact is very high due to the large instantaneous force caused by the impact with the ball.
Bouncing ball tribometers have been used to determine the shear characteristics of lubricants under high pressures such as is found in ball bearings or gears .
A pin on disc tribometer consists of a stationary pin that is normally loaded against a rotating disc. The pin can have any shape to simulate a specific contact, but cylindrical tips are often used to simplify the contact geometry. The coefficient of friction is determined by the ratio of the frictional force to the loading force on the pin.
The pin on disc test has proved useful in providing a simple wear and friction test for low friction coatings such as diamond-like carbon coatings on valve train components in internal combustion engines . | https://en.wikipedia.org/wiki/Tribometer |
The triboracyclopropenyl fragment is a cyclic structural motif in boron chemistry, named for its geometric similarity to cyclopropene . In contrast to nonplanar borane clusters that exhibit higher coordination numbers at boron (e.g., through 3-center 2-electron bonds to bridging hydrides or cations ), triboracyclopropenyl-type structures are rings of three boron atoms where substituents at each boron are also coplanar to the ring. Triboracyclopropenyl-containing compounds are extreme cases of inorganic aromaticity . They are the lightest and smallest cyclic structures known to display the bonding and magnetic properties that originate from fully delocalized electrons in orbitals of σ and π symmetry. Although three-membered rings of boron are frequently so highly strained as to be experimentally inaccessible, academic interest in their distinctive aromaticity and possible role as intermediates of borane pyrolysis motivated extensive computational studies by theoretical chemists . [ 1 ] [ 2 ] [ 3 ] [ 4 ] Beginning in the late 1980s with mass spectrometry work by Anderson et al . on all-boron clusters, experimental studies of triboracyclopropenyls were for decades exclusively limited to gas-phase investigations of the simplest rings (ions of B 3 ). [ 4 ] [ 5 ] [ 6 ] However, more recent work has stabilized the triboracyclopropenyl moiety via coordination to donor ligands or transition metals , dramatically expanding the scope of its chemistry. [ 7 ] [ 8 ] [ 9 ] [ 10 ]
For gas-phase spectroscopic studies, triboracyclopropenyl-containing compounds are obtained via laser ablation of boron targets and collimation of the resulting plasma cloud in a flow of inert carrier gas such as helium. The charged molecules of interest are then mass-selected by time-of-flight mass spectrometry. Addition of gases such as N 2 or CO to the gas stream affords the corresponding adducts , while addition of metals such as iridium and vanadium to the B target yields the corresponding metal-doped clusters. [ 11 ]
The sole isolable example of a triboracyclopropenyl anion that persists in solution and in the solid state was identified by Braunschweig and coworkers, who synthesized it by reducing the aminoborane Cl 2 B=NCy 2 (Cy = cyclohexyl ) with finely dispersed sodium metal in dimethoxyethane (DME). [ 7 ] Cooling of the resulting orange-red solution of the dimeric species Na 4 [B 3 (NCy 2 ) 3 ] 2 • 2 DME resulted in crystals suitable for X-ray diffraction , by which the structure was determined. Although the detailed reduction mechanism is unknown, it has been suggested that subvalent "R 2 N−B" intermediates are involved in the formation of such boron clusters. [ 12 ]
Due to their special status as the simplest aromatic cycles, the electronic structure of triboracyclopropenyl derivatives has been analyzed with a variety of techniques in computational chemistry . These have ranged from canonical molecular orbital theory to alternative formulations of bonding such as adaptive natural density partitioning theory, [ 13 ] the quantum theory of atoms in molecules , natural bond orbital theory , natural orbitals for chemical valence [ 14 ] and electron localization function analysis. NICS and ring current calculations have also been used to characterize the aromaticity in such systems by using magnetic criteria. [ 15 ] In general, the extremely small size of these cycles implies that their bonding electrons experience substantial Coulomb repulsion, resulting in abnormally high ring strain . [ 16 ] This effect is partially compensated for by the stabilization offered by aromatic delocalization.
B 3 + displays π aromaticity associated with its a 2 '' -symmetric HOMO . In its singlet electronic ground state, it is a Hückel 2π electron system analogous to the cyclopropenium cation , but it is too reactive to be isolated. It is triangular, with D 3h symmetry - all of its B atoms and B-B bond distances are chemically equivalent. The gas-phase adducts B 3 (N 2 ) 3 + and B 3 (CO) 3 + have been computationally studied through ETS-NOCV (extended transition state - natural orbitals for chemical valence) theory, which dissects the changes in energy and electron density that result as a molecule is prepared from a reference state of noninteracting fragments. [ 17 ] ETS-NOCV energy decomposition analysis suggests that the N 2 and CO adducts are primarily stabilized (by -83.6 and -112.3 kcal/mol respectively) through σ donation of the exocyclic ligands into the highly electron-deficient boron ring. As a result, each was interpreted as a B 3 + moiety supported by dative bonding from N 2 or CO. The electron deformation density constructed from the NOCVs of this system, together with charges derived from natural bond orbital populations, indicate electron flow from the exocyclic ligand into the ring, which induces all the equivalent bonds of the B 3 + core to shorten by approximately 4 pm. π-symmetry interactions are observed with both the weak σ donor N 2 and the strong π acceptor ligand CO. However, the out-of-plane π backdonation (from the π system of the B 3 ring to the π acceptor orbitals of each ligand) is less stabilizing than the in-plane π backdonation, with strengths of -26.7 and -19.6 kcal/mol for the [B 3 (CO) 2 + + CO] system. This suggests that the minimum-energy configuration of the molecule is one which preserves maximal π aromaticity in the B 3 + core.
Just as aromatic species like the cyclopentadienyl anion and the cyclopropenium cation can coordinate to transition metals, it was recently demonstrated that the B 3 + ring can bind to metal centers. [ 18 ] Laser ablation of a mixed B/Ir target produces two isomers of IrB 3 − , a B 3 + ring coordinated to a formal Ir 2- anion. These are a pseudo-planar η 2 adduct and a tetrahedral η 3 adduct, the latter of which contains an aromatic triboracyclopropenyl fragment. Both are nearly identical in energy and coexist in the generated cluster beam.
Computations suggest that B 3 + may even bind inert noble-gas atoms to form an unusual family of compounds B 3 (Rg) 3 + (Rg = rare/noble gas), with nonnegligible bond strengths (from 15 to 30 kcal/mol) that originate from Rg p-orbital σ donicity and a significant degree of charge transfer from Rg to B 3 + . The possibility of new noble-gas compounds that form exothermally and spontaneously is an opportunity for experimental work. [ 19 ] [ 21 ]
B 3 possesses a singly occupied a 1 ' HOMO (a SOMO) that consists of σ-symmetric orbitals oriented toward the core of the ring, associated with σ delocalization and slightly shorter B-B bond lengths as compared to B 3 + . It is paramagnetic with a doublet ground state. [ 22 ] It is nonpolar, flat and triangular, having D 3h symmetry.
B 3 − , with a filled a 1 ' HOMO in D 3h symmetry, is considered to be "doubly" aromatic and relatively stable - it simultaneously possesses highly delocalized σ and π electrons in its HOMO and HOMO-1 respectively. [ 23 ] [ 24 ]
B 3 R 3 2- , formulated with electron-sharing B−R bonds rather than dative arrows, is isoelectronic to B 3 + . 8 electrons are assigned to the triboracyclopenyl core, 6 in σ bonding orbitals and 2 in the π system, resulting in Hückel aromaticity. The only experimentally characterized compound of this class is Na 4 [B 3 (NCy 2 ) 3 ] 2 • 2 DME, a dimer of stacked B 3 R 3 2- units which are themselves aromatic. [ 7 ] Natural bond orbital analysis indicates that this compound is highly stabilized (by roughly 45 kcal/mol) by a donor-acceptor interaction of localized B−B bond orbitals with corresponding B−N antibonding orbital across the ring, in addition to being bound together by electrostatic attraction to bridging Na + cations identified in the crystal structure. DFT calculations show that the HOMO and HOMO-1 are antisymmetric and symmetric combinations of the π HOMO of an individual ring, respectively - a feature shared with metallocenes . As expected for a species with B−B bonds that have a formal MO bond order of 4 / 3 {\displaystyle 4/3} , the average B-B bond length of 1.62 Å is closer to those of diborene (R-B=B-R) radical cations than B−B single bonds of roughly 1.75 Å.
Triboracyclopropenyl-derived compounds were first identified by their mass-to-charge ratio, as transient species in the mass spectrometry of complex mixtures of cationic boron clusters. [ 5 ] Reactive scattering studies with O 2 soon followed, revealing the relatively strong bonding within light boron clusters. [ 6 ] Subsequently, B 3 was isolated in matrices of frozen noble gases and electron paramagnetic resonance spectra were recorded which confirmed its D 3h geometry. [ 22 ] Hyperfine coupling of the unpaired electron to the 11 B nucleus provided an estimate of 15% s-orbital character for the a 1 ' HOMO. The small and nonpolar B 3 rings were able to tumble and rotate freely even when confined in the matrix.
In general, triboracyclopropenyl-containing species have been too short-lived and produced in insufficient quantity for transmission-mode infrared spectroscopy . However, dissociating B 3 (N 2 ) 3 + with infrared light and observing the decay of the corresponding mass-to-charge signal via mass spectrometry allowed an effective infrared spectrum of B 3 (N 2 ) 3 + to be recorded. [ 17 ] [ 25 ] This vibrational photodissociation spectrum contained only a single detectable vibration with a redshift of 98 cm −1 relative to gaseous N 2 , suggesting a highly symmetric B 3 (N 2 ) 3 + adduct with slightly weakened N≡N bonding.
Negatively charged ions containing triboracyclopropenyl have proven amenable to study by photoelectron spectroscopy . By Koopman's theorem , neglecting the effects of strong electron correlation , the kinetic energies of electrons detached by X-rays can be mapped onto binding energies of individual orbitals and reveal the molecular electronic structure. [ 26 ] [ 27 ] Splitting of the resulting spectral peaks from " vibrational progression " (according to the Franck-Condon principle) indicates how ionization at different energies changes specific vibrational frequencies of the molecule, and such effects on bonding are interpreted in terms of changes to the electron configuration. In B 3 − , an unusually high-intensity and high energy band corresponding to a multielectron or "shake-up" transition (coupled electron detachment and electronic excitation) was observed, suggesting the strong electron correlation present in the triboracyclopropenyl fragment. [ 4 ] For IrB 3 − , vibrational progression from the stretching and breathing vibrations of IrB 3 could be assigned in the overlaid spectra of both isomers present in the cluster beam. By comparison to computations, the minimum energy structure of IrB 3 could then be formulated as a tetrahedron with an intact, aromatic B 3 + moiety. [ 8 ]
The reactivity of triboracyclopropenyl-containing compounds is relatively under-explored, as only one example has been prepared in the solution phase. The compound reported by Braunschweig, Na 4 [B 3 (NCy 2 ) 3 ] 2 • 2 DME, is an extremely potent reductant with an oxidation potential of -2.42 V vs. the ferrocene/ferrocenium couple. As a result, it is capable of reducing chloroboranes to afford tetrahedral B clusters, along with reducing PbCl 2 directly to metallic Pb. In addition, it will undergo a ring-opening reaction at the B 3 moiety by abstracting chlorine atoms from hexachloroethane . This level of reducing power is roughly comparable to an alkali metal, and has not been previously observed for any molecule based on an organic framework. [ 7 ]
Although most examples of transition metal-doped trinuclear boron clusters do not contain an aromatic triboracyclopropenyl fragment, the reactivity of such species with small molecules is likely to attract increasing scientific interest. It has been demonstrated under the conditions of mass spectrometry that VB 3 + dehydrogenates methane to afford the products VB 3 CH 2 + and H 2 . A minor side reaction that produces VH + and eliminates B 3 CH 3 is also operative. [ 10 ] | https://en.wikipedia.org/wiki/Triboracyclopropenyl |
A tribosystem is a tribological system that consists of at least two contacting bodies and any environmental factor that affects their interaction. [ 1 ] Tribologists study tribological systems in detail, and devise tribological test procedures.
According to ASTM G40-17 , a tribosystem is "any system that contains one or more triboelements, including all mechanical, chemical, and environmental factors relevant to tribological behavior." [ 2 ] Here, triboelement refers to "one of two or more solid bodies that comprise a sliding , rolling , or abrasive contact, or a body subjected to impingement or cavitation ." [ 2 ]
More simply speaking, a tribosystem is a tribological system that consists of at least two contacting bodies, including the environment in which the interaction takes place. [ 1 ] The complete description of a tribosystem includes all relevant factors that govern the tribological behavior of the system. Thus, depending on the aim of the tribological analysis, the tribosystem boundary is flexible and can be drawn more or less widely.
The description of tribosystems is based on a detailed assessment of relevant system inputs, outputs and losses, as well as an overall description of the system structure. [ 3 ] The following table gives an overview.
The complete description of a tribosystem is the first step when devising a tribological test procedure. Since tribological tests are often carried out on simplified model systems using standardized tribometers , a complete description of the tribosystem allows for tribological testing across different scales.
For example, if the tribological analysis aims to investigate a specific gear contact in a complex gearbox , exact knowledge of tribological inputs allows tribologists to devise a simplified test setup involving two gears only. Conversely, if the analysis aims to develop a new lubricant formulation for gearbox applications, a rough description of the gearbox-tribosystem allows to reduce testing to relevant system inputs. Thus, detailed knowledge of the tribosystem can significantly reduce the development effort for machines and lubricants . | https://en.wikipedia.org/wiki/Tribosystem |
Tribotronics is about the research on interaction between triboelectricity and semiconductor , which is using triboelectric potential controlling electrical transport and transformation in semiconductors for information sensing and active control (info-tribotronics), and using semiconductors managing triboelectric power transfer and conversion in circuits for power management and efficient utilization (power-tribotronics). [ 1 ]
The tribotronics can be divided into info-tribotronics and power-tribotronics. The tribotronic devices, such as tribotronic transistor, [ 2 ] contact-gated OLED , [ 3 ] touch memory , [ 4 ] wind-enhanced photocell, [ 5 ] sliding tunable diode, [ 6 ] tactile sensing array, [ 7 ] [ 8 ] stretchable transistor [ 9 ] and nanoscale transistor [ 10 ] have all demonstrated controlled electronics by triboelectric potential for information sensing and active control, which are belonging to info-tribotronics. On the other hand, the power-tribotronics can demonstrate manageable triboelectric power by electronics for power management and efficient utilization, such as the tribotronic energy extractor, [ 11 ] the power management module, [ 12 ] and so on.
As a fundamental info-tribotronic unit, contact electrification field-effect transistor (CE-FET) composed of a metal–oxide–semiconductor field-effect transistor (MOSFET) without top-gate electrode and a mobile layer is analyzed. [ 13 ] Different from the conventional MOSFET, the externally applied gate voltage source is replaced by the mobile layer, which can vertically contact to and separate from the insulator layer by the external force. When the fluorinated ethylene propylene (FEP) film contacts with the insulator layer, the SiO 2 has positive charges while the FEP has negative charges. When the mobile layer gradually separated, a positive inner gate voltage for the MOSFET is generated. Therefore, a depletion zone will be formed, which will decrease the channel width and thus the drain current. The CE-FET can be considered as the coupling of the MOSFET and TENG, in which the inner gate voltage can be generated and the carrier transport between drain and source can be tuned/controlled by the external contact instead of the conventional gate voltage.
To understand the potential maximal energy of TENG and develop the power management strategy, the cycles for maximized energy output of TENG (CMEO) are first elaborated. The output energy of TENG in one cycle E can be expressed in U-Q plot and calculated as the encircled area of the closed loop, where U is the built-up voltage and Q is the transferred charge. Meanwhile, the encircled area can be enlarged for CMEO by using a sequential switch. Although the energy could be maximally released to the resistor, the voltage is still a pulse high voltage that is not enough for directly powering the electronics. Therefore, the pulse high voltage should be converted to a steady low DC voltage, in which a classical DC–DC buck convertor is integrated to form an AC–DC buck conversion circuit. The DC–DC buck convertor is composed of a parallel freewheeling diode, a serial inductor, and a parallel capacitor that are connected in sequence between the switch and the resistor. | https://en.wikipedia.org/wiki/Tribotronics |
The tribovoltaic effect is a type of triboelectric current where a direct-current (DC) current is generated by sliding a P-type semiconductor on top of a N-type semiconductor or a metal surface without the illumination of photons, which was firstly proposed by Wang et al . [ 1 ] in 2019 and later observed experimentally in 2020. When a P-type semiconductor slides over a N-type semiconductor , electron-hole pairs can be produced at the interface, which separate in the built-in electric field ( contact potential difference ) at the semiconductor interface, generating a DC current. Research has shown that the tribovoltaic effect can occur at various interfaces, such as metal-semiconductor interface, [ 2 ] P-N semiconductors interface, [ 3 ] metal-insulator-semiconductor interface, [ 4 ] metal-insulator-metal interface, [ 5 ] and liquid-semiconductor interface. [ 6 ] [ 7 ] The tribovoltaic effect may find applications in the fields of energy harvesting and smart sensing. [ 3 ]
It has been suggested that the generation of tribo-current at the sliding PN junction or Schottky junction is analogous to the generation of photo-current in the photovoltaic effect , and the only difference is that the energy for exciting the electron-hole pairs is different, so it was named “tribovoltaic effect” by Wang et al . [ 1 ]
The tribovoltaic effect was observed at both macro- and nano-scale. It was found that a direct current can be generated by sliding the N-type diamond coated tip over the P-type Si samples, and the direction of the tribo-current depends on the direction of the built-in electric field at the PN and Schottky junctions .
Metal-semiconductor interface. When a Pt-coated silicon atomic force microscopy (AFM) tip rubs on molybdenum disulfide (MoS 2 ) surface, a DC current with a maximum density of 10 6 A/m 2 is generated. [ 2 ] Similarly using a pure Pt tip to rub both p-type and N-type silicon samples, the current follows the contact potential. [ 3 ]
P-N semiconductors interface. When using a N-type silicon to rub with a P-type Si , a DC current from the P-type Si to the N-type silicon is produced, with the same direction as the built-in electric field at the PN junction . [ 8 ] Furthermore, when a N-type diamond-coated silicon tip is used to rub with the surfaces of N-type silicon and P-type Si , tribocurrent can be generated at the interfaces of N-type tip and P-type Si . [ 3 ]
Metal-insulator-semiconductor interface. When a conducting tip rubs with a silicon, the tribovoltaic effect can induce water molecules to form an oxide layer on the silicon surface, and the tribo-current decreases gradually with increasing the thickness of oxide layer. [ 4 ]
Metal-insulator-metal interface. The studies of DC output characteristics of Al-TiO 2 -Ti heterojunctions show that the open-circuit voltage increases with increasing the thickness of TiO 2 , while the short-circuit current first increases and then decreases. The experiments have revealed that the tribo-current is contributed by quantum tunneling, thermionic emission and trap-assisted transport. [ 5 ]
Liquid-semiconductor interface. The tribovoltaic effect can also occur at aqueous solution and solid semiconductor interface, in which the aqueous solution is considered as a liquid semiconductor. [ 9 ] [ 10 ] [ 11 ] [ 12 ] The tribovoltaic effect at liquid-solid interface was also observed by Wang et al. [ 7 ] [ 13 ] | https://en.wikipedia.org/wiki/Tribovoltaic_effect |
Tribromine octoxide is a binary inorganic compound of bromine and oxygen with the chemical formula Br 3 O 8 . [ 1 ] [ 2 ] This is a free radical and one of the most complex bromine oxides .
A reaction of Br 2 with O 3 at 273 K and low pressure. [ 3 ] [ 4 ] [ 5 ]
The compound forms a white solid. [ 6 ] It exists in two forms which are both soluble in water. It is unstable above −73 °C. [ 3 ] [ 7 ] | https://en.wikipedia.org/wiki/Tribromine_octoxide |
Tribromoisocyanuric acid (C 3 Br 3 N 3 O 3 ) is a chemical compound used as a reagent for bromination in organic synthesis . [ 1 ] It is a white crystalline powder with a strong bromine odour. It is similar to trichloroisocyanuric acid .
Tribromoisocyanuric acid is used for the bromination of aromatics and alkenes. | https://en.wikipedia.org/wiki/Tribromoisocyanuric_acid |
Tributyltin ( TBT ) is an umbrella term for a class of organotin compounds which contain the (C 4 H 9 ) 3 Sn group , with a prominent example being tributyltin oxide . [ 1 ] For 40 years TBT was used as a biocide in anti-fouling paint , commonly known as bottom paint , applied to the hulls of oceangoing vessels. [ 2 ] Bottom paint improves ship performance and durability as it reduces the rate of biofouling , the growth of organisms on the ship's hull. The TBT slowly leaches out into the marine environment where it is highly toxic toward nontarget organisms. TBT toxicity can lead to biomagnification or bioaccumulation within such nontarget organisms like invertebrates, vertebrates, and a variety of mammals. TBT is also an obesogen . [ 3 ] After it led to collapse of local populations of organisms, TBT was banned. [ 4 ]
TBT, or tributyltin, tributylstannyl or tributyl stannic hydride compounds are organotin compounds . They have three butyl groups covalently bonded to a tin(IV) atom. [ 5 ] A general formula for these compounds is (CH 3 CH 2 CH 2 CH 2 ) 3 Sn−X . The −X is typically a chloride −Cl , hydroxide −OH , or a carboxylate RCO 2 − , where R is an organyl group . [ 6 ] TBT is also known to be an endocrine disrupting compound, which influences biological activities such as growth, reproduction and other physiological processes. [ 7 ]
TBT compounds have a low water solubility, a property that is ideal for antifouling agents. The toxicity of TBT prevents the growth of algae, barnacles, molluscs and other organisms on ships hulls. [ 8 ] When introduced into a marine or aquatic environment, TBT adheres to bed sediments. TBT has a low Log K ow of 3.19 – 3.84 in distilled water and 3.54 for sea water, this makes TBT moderately hydrophobic. TBT compounds have a high fat solubility and tend to absorb more readily to organic matter in soils or sediment. The bioaccumulation of TBT in organisms such as molluscs, oysters and dolphins, have extreme effects on their reproductive systems, central nervous systems and endocrine systems. [ 9 ] However, the adsorption of TBT to sediments is reversible and depends on pH level in the body of water.
TBT has a half-life of one or two weeks in marine water. [ 5 ] When it accumulates in sediments its half life is about 2 years. TBT often bonds to suspended material and sediments, where it can remain and be released for up to 30 years. [ 10 ] Studies have shown that 95% of TBT can be released from the sediments back into the aquatic environment. This absorption process can complicate quantification of TBT in an environment, since its concentration in the water is not representative of its availability. [ 1 ]
Tributyltin (TBT) compounds are biocides . TBT's antifouling properties were discovered in the 1950s in the Netherlands by van der Kerk and coworkers. It prevents microorganisms from settling on the hull of a ship and poisons the organisms that end up settling. By the mid-1960s, it had become the most popular anti-fouling paint around the globe. [ 4 ] TBT was mixed into paints to extend the life of antifouling coatings and ships were able to continue operations for a longer time frame. The paints ensured fuel efficiency and delayed costly ship repairs. It is also relatively inexpensive.
TBT is also an ingredient in some disinfectants , for example in combination with quaternary ammonium compounds. Additionally, TBT has been used in the fertilizer, textile, and wood industries. It has antifungal properties that make it useful for both the production of textiles and wood preservation, and in the creation of biocides for paired use with fertilizers. [ 11 ] Another use of TBT is that they were used as stabilizers in compounds like polyvinyl chlorides. [ 12 ] Due to this usage of TBT, there are a variety of consumer products where traces of TBT can be found, like in textile fabrics, plastic polymers, silicon, and many more.
The effects of antifouling paint go beyond the organisms that it is intended to kill. By poisoning barnacles, algae, and other organisms at the bottom of the food chain, the bioaccumulation of TBT increases over time affecting more and more of the bottom feeders of the aquatic food web environment, which are mainly invertebrates and are affected by TBT. There is a slight biomagnification of TBT that has been demonstrated in the lower part of the marine food chain (i.e., planktonic organisms, invertebrates, and fishes). However, the biomagnification of TBT into larger marine animals such as marine mammals is debatable. [ 13 ] Toxic effects in some species occur at 1 nanogram per liter of water. Air pollution from TBT has not been noticed or considered significant enough to effect the environment. In the water, photodegradation and microorganisms can break down TBT and leach into the soil sediments. [ 12 ]
As TBT is most often used as a biofouling agent, it bioaccumulates in marine wildlife such as molluscs, with levels being higher in organisms and sediments in and around areas of high maritime activity, such as ports and harbours. [ 12 ] The bioaccumulation increases over time, leading to a biomagnification in organisms higher up the food chain, although the biomagnification is not that considerable in size. [ 14 ] As TBT can remain in the environment for up to 30 years due to often bonding to suspended material and sediments, it can remain in an ecosystem for a very long time. This means that bioaccumulation readily occurs in marine environments, which can lead to very high amounts of TBT being accumulated, especially in smaller organisms at the bottom of the food chain, which in turn has various health effects.
Exposure to organotin compounds causes the development of male accessory sex organs in female prosobranch gastropods. This phenomenon has been termed imposex. TBT has been shown to affect invertebrate development. Marine snails, such as the dog whelk ( Nucella lapillus ), has often been used as an indicator species . [ 15 ] In gastropods, the normal process of accessory sex organ development is retinoid dependent, as has been proven by the effect 9cisRA has on male penises. TBTs mimic the endogenous ligand of Retinoid X Receptor (9cisRA), and thus activates the signalling cascades that are retinoid acid dependent, promoting female penis growth. [ 16 ] [ 17 ] [ 18 ] [ 19 ]
There have been many theories as to why molluscs are affected by TBT. For example, previous literature has stated that TBT would cause the inhibition of aromatase which would lead to an increase in testosterone and therefore, causing imposex. [ 20 ] It was theorized that TBT disrupts endocrine system by inhibiting cytochrome P450 molecule. Among its myriad functions, P450 converts androgen, which has male-hormone properties, into oestrogen, which has female hormone properties. It was theorized that the high concentration of androgen lead to the masculinization of females. [ 21 ] Another indicator species is Chironomus riparius , a species of non-biting midge , which has been used to test the effects of TBT on development and reproduction at sublethal concentrations found in marine environments. Higher concentrations of TBT were found to increase the female population and the results are interesting because unlike the masculinization of the stengoglassan gastropods, feminization was present. [ 6 ]
Vertebrates become affected by the waters contaminated with TBT, as well as by consuming organisms that have already been poisoned. Oryzias latipes , commonly called Japanese rice fish , has been used as a model vertebrate organism to test for effects of TBT at developmental stages of the embryo. It was observed that developmental rate was slowed by TBT in a concentration-related manner and that tail abnormalities occurred. Illustrating the infiltration of TBT in the food chain, one study showed that most samples of skipjack tuna tested positive for presence of TBT. Tuna from waters around developing Asian nations had particularly high levels of TBT. Regulation of TBT is not enforced in Asia as rigorously as in Europe or US. [ 22 ] Studies have shown that TBT is detrimental to the immune system. Research shows that TBT reduces resistance to infection in fish which live on the seabed and are exposed to high levels of TBT. These areas tend to have silty sediment like harbours and estuaries. [ 8 ] TBT compounds have been described to interfere with glucocorticoid metabolism in the liver by inhibiting the activity of the enzyme 11beta-hydroxysteroiddehydrogenase type 2, which converts cortisol to cortisone. [ 6 ]
TBT can enter the diet of humans and other mammals such as whales, dolphins, dugongs, and sea otters. As of 2008 high levels of tributyltin have been detected in the livers of sea otters ( Enhydra lutris ) and stranded bottlenose dolphins . [ 23 ] [ 24 ] Otters dying of infectious causes tended to have higher levels of tissue butyltins than those dying of trauma or other causes. It was also reported by scientists that sea otters typically stay near boats and closed off marinas, which may have led to these organisms experiencing higher levels of butyltins. [ 25 ] TBT has been shown to lead to immunosuppression in sea-otters and dolphins. TBT has also been linked to hearing loss in mammalian top predators such as toothed whales . [ 26 ] In rats, the hypothalamus-pituitary-adrenal (HPA) axis can be affected by TBT. In the pituitary and adrenal glands, there have been findings of morphophysiological changes within rats affected by TBT. [ 27 ] TBT can also affect humans as well. Humans can be exposed to these compounds and potentially experience headaches, fatigue, respiratory issues, and more. Long-term exposure can also lead to damage of some internal organs such as the kidneys and liver. [ 28 ]
Bans on TBT on boats less than 25 metres long first started in the 1980s. In 1990, the Marine Environment Protection Committee adopted Resolution MEPC 46(30), which recommended that the Government eliminate the use of TBT-containing antifouling paints on smaller vessels. This resolution was intended to be a temporary restriction until the International Maritime Organization could implement a ban of TBT anti-fouling agents for ships. Several countries followed and in 1997, Japan banned the production of TBT-based anti-fouling paints. The IMO began to use an Assembly resolution in 1999 that essentially wanted the MPEC to fix the severe environmental effects of the anti-fouling systems. This led to a worldwide ban on organotin compound applications on ships starting in 2003. [ 8 ] In 2008, organotin compounds acting as biocide like TBT compounds were banned entirely in anti-fouling paint and included in the Rotterdam Convention and have been banned by the International Convention on the Control of Harmful Anti-fouling Systems on Ships of the International Maritime Organization. [ 8 ] [ 29 ] It states that ships cannot bear organotin compounds on their hulls or external parts or surfaces, unless there is a coating that forms a barrier so that organotin compounds cannot leach out to reduce exposure by allowing recovery to occur. [ 1 ]
Although the ban on TBT use was proved to be effective on reducing the negative effects on the environment, some people that supplied them were still producing and selling them to other countries for a profit. Even though banned by some international agencies like the International Maritime Organization, TBT anti-fouling paints are still used in countries with poor regulation enforcement to this day, with the Caribbean being a prime example. [ 7 ]
In November 2018, the US Department of Justice announced that three people they had charged and arrested in New Jersey for manufacturing and selling tributyltin based marine paint had pleaded guilty. The sentencing of these people was scheduled for February 2019. [ 30 ] | https://en.wikipedia.org/wiki/Tributyltin |
Tributyltin azide is an organotin compound with the formula (C 4 H 9 ) 3 SnN 3 . It is a colorless solid although samples can appear as yellow oils. The compound is used as a reagent in organic synthesis .
Tributyltin azide is synthesized by the salt metathesis reaction of tributyltin chloride and sodium azide .
It is a reagent used in the synthesis of tetrazoles , which in turn are used to generate angiotensin II receptor antagonists . In some applications, tributyltin azide has been replaced by the less toxic trioctyltin azide and organoaluminium azides. [ 1 ]
Tributyl tin compounds are highly toxic and have penetrating odors. Tributyltin azide causes skin rashes, itching or blisters. [ 2 ] [ 3 ] | https://en.wikipedia.org/wiki/Tributyltin_azide |
In chemistry , the tricapped trigonal prismatic molecular geometry describes the shape of compounds where nine atoms, groups of atoms, or ligands are arranged around a central atom, defining the vertices of a triaugmented triangular prism (a trigonal prism with an extra atom attached to each of its three rectangular faces). [ 1 ]
It is very similar to the capped square antiprismatic molecular geometry , and there is some dispute over the specific geometry exhibited by certain molecules. [ citation needed ]
This geometry-related article is a stub . You can help Wikipedia by expanding it .
This stereochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tricapped_trigonal_prismatic_molecular_geometry |
Tricarbon (systematically named 1λ 2 ,3λ 2 -propadiene and catena -tricarbon ) is an inorganic compound with the chemical formula C 2 (μ-C) (also written [C(μ-C)C] or C 3 ). It is a colourless gas that only persists in dilution or solution as an adduct . It is one of the simplest unsaturated carbenes . Tricarbon can be found in interstellar space and can be produced in the laboratory by a process called laser ablation .
Tricarbon is a small carbon cluster first spectroscopically observed in the early 20th century in the tail of a comet by William Huggins and subsequently identified in stellar atmospheres . Small carbon clusters like tricarbon and dicarbon are regarded as soot precursors and are implicated in the formation of certain industrial diamonds and in the formation of fullerenes .
C 3 has also been identified as a transient species in various combustion reactions. [ 1 ] : 218–220, plate 20
The chemical properties of C 3 was investigated in the 1960s by Professor Emeritus Philip S. Skell of Pennsylvania State University , who showed that certain reactions of carbon vapor indicated its generation, such as the reaction with isobutylene to produce 1,1,1',1'-tetramethyl-bis-ethanoallene. [ 2 ]
The ground state molecular geometry of tricarbon has been identified as linear via its characteristic symmetric and antisymmetric stretching and bending vibrational modes and bears bond lengths of 129 to 130 picometer corresponding to those of alkenes . The ionization potential is determined experimentally at 11 to 13.5 electronvolts . [ 3 ] In contrast to the linear tricarbon molecule, the C + 3 cation is bent.
The systematic names 1λ 2 ,3λ 2 -propadiene , and μ-carbidodicarbon , valid IUPAC names, are constructed according to the substitutive and additive nomenclatures, respectively.
In appropriate contexts, tricarbon can be viewed as propadiene with four hydrogen atoms removed, or as propane with eight hydrogen atoms removed; and as such, propadienediylidene or propanetetraylidene , respectively, may be used as a context-specific systematic names, according to substitutive nomenclature. By default, these names pay no regard to the radicality of the tricarbon molecule. In even more specific context, these can also name the non-radical singlet state, whereas the diradical state is named propadienediylylidene , or propanediyldiylidene , and the tetraradical state is named propedienetetrayl or propanetetraylylidene . [ 2 ] | https://en.wikipedia.org/wiki/Tricarbon |
Trichloroacetonitrile is an organic compound with the formula CCl 3 CN. It is a colourless liquid, although commercial samples often are brownish. It is used commercially as a precursor to the fungicide etridiazole . It is prepared by dehydration of trichloroacetamide . [ 1 ] As a bifunctional compound , trichloroacetonitrile can react at both the trichloromethyl and the nitrile group. The electron-withdrawing effect of the trichloromethyl group activates the nitrile group for nucleophilic additions. The high reactivity makes trichloroacetonitrile a versatile reagent , but also causes its susceptibility towards hydrolysis .
The production of trichloroacetonitrile by dehydration of trichloroacetamide was first described in 1873 by L. Bisschopinck at the Katholieke Universiteit Leuven . [ 2 ]
Trichloroacetonitrile can be obtained by chlorination of acetonitrile on a zinc , copper and alkaline earth metal halide -impregnated activated carbon catalyst at 200–400 °C with a 54% yield. [ 3 ]
The high temperatures required by this process favours the formation of byproducts, such as tetrachloromethane . In contrast, the chlorination of acetonitrile saturated with hydrogen chloride leads to pure trichloroacetonitrile even at 50–80 °C in good yields. [ 4 ]
Like other halogenated acetonitriles, trichloroacetonitrile is produced from organic substances such as algae , humic acids and proteinaceous material in the disinfecting chlorination of water from natural sources. [ 5 ] [ 6 ]
Freshly-distilled trichloroacetonitrile is a colorless liquid with a pungent odor that discolours rapidly yellowish to light brown. It is sensitive towards water, acids and bases.
The bond lengths are 146.0 pm (C–C), 116.5 pm (C≡N) and 176.3 pm (C–Cl). The bond angle is 110.0° (Cl–C–Cl). [ 7 ]
The substitution of all electronegative substituents in trichloroacetonitrile by nucleophilic attack of alkoxide anions produces orthocarbonic acid esters in high yield.
Due to the high reactivity of the chlorine atoms, trichloroacetonitrile can be used (especially in combination with triphenylphosphine ) to convert allylic alcohols into the corresponding allylic chlorides. [ 8 ]
With carboxylic acids, acyl chlorides are obtained. [ 9 ]
Due to the mild reaction conditions, the Cl 3 CCN/PPh 3 system is also suitable for the activation of carboxylic acids and their linkage with supported amino compounds to amides ( peptides ) in solid-phase syntheses. [ 10 ] From sulfonic acids , the corresponding sulfochlorides are formed analogously. [ 11 ] In an analogous manner, the activation of diphenylphosphoric acid with Cl 3 CCN/PPh 3 and reaction with alcohols or amines proceeds to the corresponding phosphoric acid esters or amides in a gentle and efficient one-pot reaction. [ 12 ]
Also, phenolic hydroxy groups in nitrogen-containing aromatics can be converted into the chlorine compounds. [ 13 ]
In a Hoesch reaction , aromatic hydroxyketones are formed in the reaction of substituted phenols with trichloroacetonitrile, for example from 2-methyl phenol the 2-trichloroacyl derivative in 70% yield. [ 14 ]
The electron-withdrawing effect of the trichloromethyl group activates the nitrile group of trichloroacetonitrile for the attack of nucleophilic oxygen, nitrogen and sulfur compounds. For example, alcohols give O -alkyltrichloroacetimidates under basic catalysis in a direct and reversible addition, [ 15 ] which can be isolated as stable and less hydrolysis-sensitive adducts .
With primary and secondary amines, N -substituted trichloroacetamidines are formed in a smooth reaction with good yields, which can be purified by vacuum distillation and are obtained as colorless, malodorous liquids. [ 16 ] Reaction with ammonia and then with anhydrous hydrogen chloride gives the solid trichloroacetamidine hydrochloride, the starting compound for the fungicide etridiazole .
In academic research, trichloroacetonitrile is used as a reagent in the Overman rearrangement , converting allylic alcohols into allylic amines . [ 17 ] [ 18 ] [ 19 ] The reaction is based on a [3,3]- sigmatropic and diastereoselective rearrangement.
Benzyl trichloroacetimidate is easily accessible from benzyl alcohol and trichloroacetonitrile. [ 20 ] Benzyl trichloroacetimidate is useful as a benzylating reagent for sensitive alcohols under mild conditions and to preserve chirality. [ 21 ]
R. R. Schmidt and co-workers [ 22 ] have described the selective anomeric activation of O -protected hexopyranoses ( glucose , galactose , mannose , glucosamine , galactosamine ), hexofuranoses and pentopyranoses with trichloroacetonitrile in the presence of a base, as well as glycosylations under acid catalysis. [ 23 ] [ 24 ] [ 25 ]
Under kinetic control [ 26 ] with potassium carbonate as the base, β-trichloroacetimidates are formed selectively, whereas with sodium hydride , caesium carbonate or potassium hydroxide [ 27 ] and in the presence of phase-transfer catalysts [ 28 ] only α-trichloroacetimidates are obtained (thermodynamically controlled).
The trichloroacetimidates are reacted between −40 °C and room temperature with boron trifluoride etherate in dichloromethane with O -protected sugars. This method usually gives better results than the Koenigs–Knorr method using silver salts or the Helferich method which uses problematic mercury salts . Since an inversion occurs at the anomeric center, the reaction leads to β- O -glycosides (when using α-trichloroacetimidates). The trichloroacetimidate method often produces sterically uniform glycosides under mild reaction conditions in very good yields.
Thioacetic acid reacts with acetyl-protected α-galactosyl trichloroacetimidate even without additional acid catalysis to thioglycoside, from which (after cleavage of the protective groups) 1-thio-β- D -galactose is easily accessible, which is useful for the separation of racemates of amino acids . [ 29 ]
Trichloroacetonitrile was an important fumigant in the first half of the 20th century, but today it has become obsolete for this application. [ 30 ] | https://en.wikipedia.org/wiki/Trichloroacetonitrile |
Trichloroethylene ( TCE ) is a halocarbon with the formula C 2 HCl 3 , commonly used as an industrial metal-degreasing solvent . It is a clear, colourless, non-flammable, volatile liquid with a chloroform -like pleasant mild smell [ 3 ] and sweet taste. [ 9 ] Its IUPAC name is trichloroethene . Trichloroethylene has been sold under a variety of trade names. Industrial abbreviations include TCE , trichlor , Trike , Tricky and tri . Under the trade names Trimar and Trilene , it was used as a volatile anesthetic and as an inhaled obstetrical analgesic . It should not be confused with the similar 1,1,1-trichloroethane , which was commonly known as chlorothene .
The earliest record of trichloroethylene synthesis dates back to 1836. It was obtained from the action of potassium hydroxide on 1,1,2,2-tetrachloroethane and 1,1,1,2-tetrachloroethane by Auguste Laurent and notated as C 4 HCl 3 (then the atomic weight of carbon was thought to be the half of it really was). Laurent named it chlorétherise but did not investigate the compound further. [ 10 ] [ 11 ]
Trichloroethylene's discovery is widely attributed to E. Fischer who made it in 1864 via the reduction of hexachloroethane with hydrogen. Fischer investigated trichloroethylene and noted its boiling point as between 87 and 90 degrees Celsius. [ 12 ] [ 13 ] [ 14 ] Commercial production began in Germany, in 1920 and in the US in 1925. [ 15 ]
The use of trichloroethylene in the food and pharmaceutical industries has been banned in some parts of the world since the 1970s [ 16 ] due to concerns about its toxicity. Legislation has forced the replacement of trichloroethylene in many processes in Europe as the chemical was classified as a carcinogen carrying an R45 risk phrase , May cause cancer . Many degreasing chemical alternatives are being promoted such as Ensolv and Leksol; however, each of these is based on n -propyl bromide which carries an R60 risk phrase of May impair fertility , and would not be a legally acceptable substitute.
Trichloroethylene is a good analgesic at 0.35 to 0.5% concentrations. [ 17 ] Trichloroethylene was used in the treatment of trigeminal neuralgia beginning in 1916. [ 18 ]
Pioneered by Imperial Chemical Industries in Britain, under the trade name "Trilene" (from tri chloroethy lene ), its development was hailed as an anesthetic revolution. It was mostly known as "Trimar" in the United States. The –mar suffix indicates study and development at the University of Maryland , e.g., "Fluoromar" for fluroxene and "Vinamar" for ethyl vinyl ether ". [ 19 ] From the 1940s through the 1980s, both in Europe and North America, trichloroethylene was used as a volatile anesthetic almost invariably administered with nitrous oxide . Marketed in the UK by Imperial Chemical Industries under the trade name Trilene it was coloured blue (with a dye called waxoline blue in 1:200,000 concentration) [ 20 ] to avoid confusion with the similar-smelling chloroform. Trilene was stabilised with 0.01% thymol . [ 20 ]
Originally thought to possess less hepatotoxicity than chloroform , and without the unpleasant pungency and flammability of ether , TCE replaced earlier anesthetics chloroform and ether in the 1940s. TCE use was nonetheless soon found to have several pitfalls. These included promotion of cardiac arrhythmias , low volatility and high solubility preventing quick anesthetic induction, reactions with soda lime used in carbon dioxide absorbing systems, prolonged neurologic dysfunction when used with soda lime, and evidence of hepatotoxicity as had been found with chloroform. Alkali components of carbon dioxide absorbers reacted with trichloroethylene and released dichloroacetylene , a neurotoxin.
The introduction of halothane in 1956 greatly diminished the use of TCE as a general anesthetic in the 1960s, as halothane allowed much faster induction and recovery times and was considerably easier to administer. Trichloroethylene has been used in the production of halothane. [ 21 ]
Trilene was also used as an inhaled analgesic, mainly during childbirth, often self-applied by the patient. Trichloroethylene was introduced for obstetrical anaesthesia in 1943, and used until the 1980s. [ 17 ] Its anaesthetic use was banned in the United States in 1977 but the anaesthetic use in the United Kingdom remained until the late 1980s (especially for childbirth). [ 18 ] Fetal toxicity and concerns about the carcinogenic potential of TCE led to its abandonment in developed countries by the 1980s. TCE was used with halothane in the tri-service field anaesthetic apparatus used by the UK armed forces under field conditions. As of 2000 [update] , TCE was still in use as an anesthetic in Africa. [ 22 ] [ needs update ]
Today, most trichloroethylene is produced from ethylene . First, ethylene is chlorinated over a ferric chloride catalyst to produce 1,2-dichloroethane :
When heated to around 400 °C with additional chlorine, 1,2-dichloroethane is converted to trichloroethylene:
This reaction can be catalyzed by a variety of substances. The most commonly used catalyst is a mixture of potassium chloride and aluminum chloride . However, various forms of porous carbon can also be used. This reaction produces tetrachloroethylene as a byproduct and depending on the amount of chlorine fed to the reaction, tetrachloroethylene can even be the major product. Typically, trichloroethylene and tetrachloroethylene are collected together and then separated by distillation .
Prior to the early 1970s, however, most trichloroethylene was produced in a two-step process from acetylene . First, acetylene was treated with chlorine using a ferric chloride catalyst at 90 °C to produce 1,1,2,2-tetrachloroethane according to the chemical equation :
The 1,1,2,2-tetrachloroethane is then dehydrochlorinated to give trichloroethylene. This can be accomplished either with an aqueous solution of calcium hydroxide :
or in the vapor phase by heating it to 300–500 °C on a barium chloride or calcium chloride catalyst:
Common impurities in reagent and technical grade TCE are methyl chloroform , carbon tetrachloride , ethylene dichloride , tetrachloroethanes , benzene and phenol . However, these compounds are present in very small amounts and do not possess any risk. [ 18 ]
Trichloroethylene is an effective solvent for a variety of organic materials. It is mainly used for cleaning. Trichloroethylene is an active ingredient (solvent) in various printing ink , varnish and industrial paint formulations. [ 23 ] [ 18 ] Other uses include dyeing and finishing operations, adhesive formulations, rubber processing, adhesives, lacquers , and paint strippers . It is applied before plating , anodizing , and painting. [ 24 ]
When trichloroethylene was first widely produced in the 1920s, its major use was to extract vegetable oils from plant materials such as soy , coconut , and palm . Other uses in the food industry included coffee decaffeination (removal of caffeine ) and the preparation of flavoring extracts from hops and spices . [ 18 ] TCE was used a freezing point depressant in carbon tetrachloride fire extinguishers. [ 18 ]
Trichloroethylene is also a chain terminator for polyvinyl chloride . [ 18 ] Chlorination gives pentachloroethane .
TCE has also been used as a dry cleaning solvent, although mostly replaced by tetrachloroethylene , except for spot cleaning where it is still used under the trade name Picrin. [ citation needed ]
Perhaps the greatest use of TCE is as a degreaser for metal parts. It has been widely used in degreasing and cleaning since the 1920s because of its low cost, low flammability, low toxicity and high effectivity as a solvent. The demand for TCE as a degreaser began to decline in the 1950s in favor of the less toxic 1,1,1-trichloroethane . However, 1,1,1-trichloroethane production has been phased out in most of the world under the terms of the Montreal Protocol due to its contribution to the ozone depletion . As a result, trichloroethylene has experienced some resurgence in use as a degreaser. [ 18 ]
Trichloroethylene is used to remove grease and lanolin from wool before weaving . [ 18 ]
TCE has also been used in the United States to clean kerosene-fueled rocket engines (TCE was not used to clean hydrogen-fueled engines such as the Space Shuttle Main Engine ). During static firing, the RP-1 fuel would leave hydrocarbon deposits and vapors in the engine. These deposits had to be flushed from the engine to avoid the possibility of explosion during engine handling and future firing. TCE was used to flush the engine's fuel system immediately before and after each test firing. The flushing procedure involved pumping TCE through the engine's fuel system and letting the solvent overflow for a period ranging from several seconds to 30–35 minutes, depending upon the engine. For some engines, the engine's gas generator and liquid oxygen (LOX) dome were also flushed with TCE before test firing. [ 25 ] [ 26 ] The F-1 rocket engine had its LOX dome, gas generator, and thrust chamber fuel jacket flushed with TCE during launch preparations. [ 26 ]
TCE is also used in the manufacture of a range of fluorocarbon refrigerants [ 27 ] such as 1,1,1,2-tetrafluoroethane more commonly known as HFC 134a. TCE was also used in industrial refrigeration applications due to its high heat transfer capabilities and its low-temperature specification.
Despite its widespread use as a metal degreaser , trichloroethylene itself is unstable in the presence of metal over prolonged exposure. As early as 1961 this phenomenon was recognized by the manufacturing industry when stabilizing additives were added to the commercial formulation. Since the reactive instability is accentuated by higher temperatures, the search for stabilizing additives was conducted by heating trichloroethylene to its boiling point under a reflux condenser and observing decomposition. Definitive documentation of 1,4-dioxane as a stabilizing agent for TCE is scant due to the lack of specificity in early patent literature describing TCE formulations. [ 28 ] [ 29 ] Epichlorohydrin , butylene oxide , N -methyl pyrrole and ethyl acetate are common stabilisers for TCE, with epichlorohydrin being the most persistent and effective. [ 30 ] Other chemical stabilizers include ketones such as methyl ethyl ketone .
When inhaled, trichloroethylene produces central nervous system depression resulting in general anesthesia . These effects may be mediated by trichloroethylene acting as a positive allosteric modulator of inhibitory GABA A and glycine receptors . [ 31 ] [ 32 ] Its high blood solubility results in a less desirable slower induction of anesthesia. At low concentrations, it is relatively non-irritating to the respiratory tract. Higher concentrations result in tachypnea . Many types of cardiac arrhythmias can occur and are exacerbated by epinephrine (adrenaline). It was noted in the 1940s that TCE reacted with carbon dioxide (CO 2 ) absorbing systems ( soda lime ) to produce dichloroacetylene by dehydrochlorination and phosgene . [ 33 ] Cranial nerve dysfunction (especially the fifth cranial nerve) was common when TCE anesthesia was given using CO 2 absorbing systems. Muscle relaxation with TCE anesthesia sufficient for surgery was poor. For these reasons as well as problems with hepatotoxicity , TCE lost popularity in North America and Europe to more potent anesthetics such as halothane by the 1960s. [ 34 ]
The symptoms of acute non-medical exposure are similar to those of alcohol intoxication , beginning with headache, dizziness, and confusion and progressing with increasing exposure to unconsciousness. [ 35 ] Much of what is known about the chronic human health effects of trichloroethylene is based on occupational exposures. Besides the effects to the central nervous system, workplace exposure to trichloroethylene has been associated with toxic effects in the liver and kidney. [ 35 ] A history of long-term exposure to high concentrations of trichloroethylene is a suspected environmental risk of Parkinson's disease . [ 36 ]
Trichloroethylene is metabolised to trichloroepoxyethane (TCE oxide) which rapidly isomerises to trichloroacetaldehyde (chloral). [ 37 ] Chloral hydrates to chloral hydrate in the body. Chloral hydrate is either reduced to 2,2,2-trichloroethanol or oxidised to trichloroacetic acid . Monochloroacetic acid , [ 38 ] dichloroacetic acid [ 39 ] and trichloromethane [ 38 ] [ 40 ] [ 41 ] were also detected as minor metabolites of TCE.
With a specific gravity greater than 1 (denser than water), trichloroethylene can be present as a dense non-aqueous phase liquid (DNAPL) if sufficient quantities are spilt in the environment.
The first known report of TCE in groundwater was given in 1949 by two English public chemists who described two separate instances of well contamination by industrial releases of TCE. [ 42 ] Based on available federal and state surveys, between 9% and 34% of the drinking water supply sources tested in the US may have some TCE contamination, though EPA has reported that most water supplies comply with the maximum contaminant level (MCL) of 5 ppb. [ 43 ]
Generally, atmospheric levels of TCE are highest in areas of concentrated industry and population. Atmospheric levels tend to be lowest in rural and remote regions. Average TCE concentrations measured in air across the United States are generally between 0.01 ppb and 0.3 ppb, although mean levels as high as 3.4 ppb have been reported. [ 44 ] TCE levels in the low parts per billion range have been measured in food; however, levels as high as 140 ppb were measured in a few samples of food. [ 44 ] TCE levels above background [ how? ] have been found in homes undergoing renovation . [ 45 ]
State, federal, and international agencies classify trichloroethylene as a known or probable carcinogen for humans. In 2014, the International Agency for Research on Cancer updated its classification of trichloroethylene to Group 1 , indicating that sufficient evidence exists that it can cause cancer of the kidney in humans as well as some evidence of cancer of the liver and non-Hodgkin's lymphoma . [ 46 ]
In the European Union , the Scientific Committee on Occupational Exposure Limit Values (SCOEL) recommends an exposure limit for workers exposed to trichloroethylene of 10 ppm (54.7 mg/m 3 ) for 8-hour TWA and of 30 ppm (164.1 mg/m 3 ) for STEL (15 minutes). [ 47 ]
Existing EU legislation aimed at protection of workers against risks to their health (including Chemical Agents Directive 98/24/EC [ 48 ] and Carcinogens Directive 2004/37/EC [ 49 ] ) currently do not impose binding minimum requirements for controlling risks to workers' health during the use phase or throughout the life cycle of trichloroethylene.
In 2023, the United States United States Environmental Protection Agency (EPA) determined that trichloroethylene presents a risk of injury to human health in various uses, including during manufacturing, processing, mixing, recycling , vapor degreasing, as a lubricant, adhesive, sealant , cleaning product, and spray. It is dangerous from both inhalation and dermal exposure and was most strongly associated with immunosuppressive effects for acute exposure, as well as autoimmune effects for chronic exposures. [ 50 ] Chronic exposure to trichloroethylene has also been linked to an increased risk of Parkinson's disease . [ 51 ] [ 52 ] As of June 1, 2023, two U.S. states ( Minnesota and New York ) have acted on the EPA's findings and banned trichloroethylene in all cases but research and development. [ 53 ] [ 54 ] According to the US EPA, in October 2023 it "proposed to ban the manufacture (including import), processing, and distribution in commerce of TCE for all uses, with longer compliance time frames and workplace controls (including an exposure limit) for some processing and industrial and commercial uses until the prohibitions come into effect" to "protect everyone including bystanders from the harmful health effects of TCE". [ 55 ] Following the EPA's recommendation the Biden Administration announced a proposal to ban trichloroethylene later that month. [ 56 ] In December 2024 the EPA issued a final ruling on the regulation of trichloroethylene, with the rule taking effect on January 16, 2025. [ 57 ] The rule bans the manufacture (including import), processing, and distribution in commerce of trichloroethylene for all uses, with longer compliance timeframes and stringent worker protections for some processing and industrial and commercial uses until the prohibitions come into effect. [ 58 ] The EPA is prohibiting most uses of trichloroethylene within one year of the rule taking effect including manufacture and processing for most commercial and all consumer products, with only a limited number of commercial uses being allowed after January 16, 2026. [ 59 ] These uses will eventually be phased out as well, though an exact timeframe hasn't been determined yet, but until they have been phased out more stringent worker protections will be required with a lower inhalation exposure limit for airborne trichloroethylene being put in place. [ 58 ] Many of the trichloroethylene uses that are continuing for longer than one year occur in highly industrialized settings with critical uses such as the cleaning of parts used in medical devices , aircraft & other transportation, security and defense systems and the manufacture of battery separators and refrigerants . [ 58 ] These uses will ultimately be prohibited as well but are temporarily being allowed to continue in order to avoid negative impacts to national security or critical infrastructure , and to allow time to transition to alternative chemicals and methods. [ 58 ]
Recent research has focused on the in-place remediation of trichloroethylene in soil and groundwater using potassium permanganate instead of removal for off-site treatment and disposal. Naturally occurring bacteria have been identified with the ability to degrade TCE. Dehalococcoides sp. degrade trichloroethylene by reductive dechlorination under anaerobic conditions. Under aerobic conditions, Pseudomonas fluorescens can co-metabolize TCE. Soil and groundwater contamination by TCE has also been successfully remediated by chemical treatment and extraction. The bacteria Nitrosomonas europaea can degrade a variety of halogenated compounds including trichloroethylene. [ 60 ] Toluene dioxygenase has been reported to be involved in TCE degradation by Pseudomonas putida . [ 61 ] In some cases, Xanthobacter autotrophicus can convert up to 51% of TCE to CO and CO 2 . [ 61 ]
Groundwater and drinking water contamination from industrial discharge including trichloroethylene is a major concern for human health and has precipitated numerous incidents and lawsuits in the United States.
The 1995 non-fiction book A Civil Action was written about a lawsuit ( Anderson v. Cryovac ) against following the increase in cancer cases after trichloroethylene pollution incidents and it was adapted to cinema in 1998 .
TCE has been used as a recreational drug . [ 62 ] Common methods of taking trichloroethylene recreationally include inhalation from a rag and drinking. [ 63 ] Most TCE abusers were young people and workers who use the chemical in their workplace. The main reason for abuse is trichloroethylene's euphoriant and slight hallucinogenic effect. [ 63 ] Some workers had become addicted to TCE. [ 64 ] | https://en.wikipedia.org/wiki/Trichloroethylene |
This page provides supplementary chemical data on trichloroethylene .
The handling of this chemical may incur notable safety precautions. It is highly recommend that you seek the Material Safety Datasheet ( MSDS ) for this chemical from a reliable source such as eChemPortal , and follow its directions.
Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. | https://en.wikipedia.org/wiki/Trichloroethylene_(data_page) |
Trichlorofluoromethane, also called freon-11 , CFC-11 , or R-11 , is a chlorofluorocarbon (CFC). It is a colorless, faintly ethereal, and sweetish-smelling liquid that boils around room temperature. [ 5 ] CFC-11 is a Class 1 ozone -depleting substance which damages Earth's protective stratospheric ozone layer . [ 6 ] R-11 is not flammable at ambient temperature and pressure but it can become very combustible if heated and ignited by a strong ignition source.
Trichlorofluoromethane was first widely used as a refrigerant . Because of its high boiling point compared to most refrigerants, it can be used in systems with a low operating pressure, making the mechanical design of such systems less demanding than that of higher-pressure refrigerants R-12 or R-22 .
Trichlorofluoromethane is used as a reference compound for fluorine-19 NMR studies.
Trichlorofluoromethane was formerly used in the drinking bird novelty, largely because it has a boiling point of 23.77 °C (74.79 °F). The replacement, dichloromethane , boiling point 39.6 °C (103.3 °F), requires a higher ambient temperature to work.
Prior to the knowledge of the ozone depletion potential of chlorine in refrigerants and other possible harmful effects on the environment, trichlorofluoromethane was sometimes used as a cleaning/rinsing agent for low-pressure systems. [ 7 ]
Trichlorofluoromethane can be obtained by reacting carbon tetrachloride with hydrogen fluoride at 435 °C and 70 atm, producing a mixture of trichlorofluoromethane, tetrafluoromethane and dichlorodifluoromethane in a ratio of 77:18:5. The reaction can also be carried out in the presence of antimony(III) chloride or antimony(V) chloride : [ 8 ]
Trichlorofluoromethane is also formed as one of the byproducts when graphite reacts with chlorine and hydrogen fluoride at 500 °C. [ 8 ]
Sodium hexafluorosilicate under pressure at 270 °C, titanium(IV) fluoride , chlorine trifluoride , cobalt(III) fluoride , iodine pentafluoride , and bromine trifluoride are also suitable fluorinating agents for carbon tetrachloride. [ 8 ] [ 9 ]
Trichlorofluoromethane was included in the production moratorium in the Montreal Protocol of 1987. It is assigned an ozone depletion potential of 1.0, and U.S. production was ended on January 1, 1996. [ 6 ]
In 2018, the atmospheric concentration of CFC-11 was noted by researchers to be declining more slowly than expected, [ 10 ] [ 11 ] and it subsequently emerged that it remains in widespread use as a blowing agent for polyurethane foam insulation in the construction industry of China . [ 12 ] In 2021, researchers announced that emissions declined by 20,000 U.S. tons from 2018 to 2019, which mostly reversed the previous spike in emissions. [ 13 ] In 2022, the European Commission announced an updated regulation that mandates the recovery and prevention of emissions of CFC-11 blowing agents from foam insulation in demolition waste, which is still emitted at significant scale. [ 14 ]
R11, like most chlorofluorocarbons , forms phosgene gas when exposed to a naked flame. [ 15 ]
Because trichlorofluoromethane is one of the easiest to detect chlorofluorocarbons produced by anthropogenic activity, it has been used in attempting to detect industrial pollution in the atmospheres of earth-like exoplanets. [ 16 ] | https://en.wikipedia.org/wiki/Trichlorofluoromethane |
This is a supplementary data page for trichlorofluoromethane . | https://en.wikipedia.org/wiki/Trichlorofluoromethane_(data_page) |
The trichloromethyl group is a functional group that has the formula −CCl 3 . The naming of is group is derived from the methyl group (which has the formula −CH 3 ), by replacing each hydrogen atom by a chlorine atom. Compounds with this group are a subclass of the organochlorines . Some notable examples of compounds with this group are trichloromethane H−CCl 3 , 1,1,1-trichloroethane H 3 C−CCl 3 , and chloral HOC−CCl 3 .
The trichloromethyl group has a significant electronegativity . [ citation needed ] For this reason, trichloromethyl-substituted acids, such as trichloromethanesulfonic acid , are often stronger than the original. For example, the acidity constant (pK a ) of trichloroacetic acid HOOC – CCl 3 is 0.77, whereas that of acetic acid is 4.76. [ 1 ] [ 2 ]
By the same principle, the trichloromethyl group generally lowers the basicity of organic compounds, e.g. trichloroethanol .
This article about an organic halide is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Trichloromethyl_group |
Trichoderma asperellum Samuels, Lieckf & Nirenberg [ 1 ] is a species of fungus in the family Hypocreaceae . [ 2 ] It can be distinguished from T. viride by molecular and phenotypic characteristics. The most important molecular characteristics are divergent ITS-1 and 28S sequences and RFLP's of the endochitinase gene. Main phenotypic characters are conidial ornamentation and arrangement and branching of the conidiophores.
This species has been used commercially and experimentally as a biopesticide for plant disease control: some commercial isolates were previously placed in T. harzianum . | https://en.wikipedia.org/wiki/Trichoderma_asperellum |
Phymatotrichum hamatum (Bonord.) Oudem. 1903 [ 1 ] Pachybasium hamatum (Bonord.) Sacc. 1885 [ 2 ] Verticillium hamatum Bonord. 1851 [ 3 ]
Trichoderma hamatum [ 4 ] is a species of fungus in the family Hypocreaceae . [ 5 ] It has been used a biological control of certain plant diseases, including Sclerotinia lettuce drop caused by Sclerotinia minor . [ 6 ] | https://en.wikipedia.org/wiki/Trichoderma_hamatum |
Trichoderma stromaticum [ 1 ] is a species of fungus in the family Hypocreaceae . [ 2 ] It is a parasite of the cacao witches broom pathogen and has been used in its biological control . [ 3 ] | https://en.wikipedia.org/wiki/Trichoderma_stromaticum |
Trichophagia is a form of disordered eating in which persons with the disorder suck on, chew, swallow, or otherwise eat hair. [ 1 ] The term is derived from ancient Greek θρίξ, thrix ("hair") and φαγεῖν, phagein ("to eat"). [ 2 ] Tricho- phagy refers only to the chewing of hair, whereas tricho- phagia is ingestion of hair, but many texts refer to both habits as just trichophagia. [ 3 ] It is considered a chronic psychiatric disorder of impulse control. [ 4 ] Trichophagia belongs to a subset of pica disorders and is often associated with trichotillomania , the compulsive pulling out of ones own hair. [ 1 ] People with trichotillomania often also have trichophagia, with estimates ranging from 48-58% having an oral habit such as biting or chewing (i.e. trichophagy), and 4-20% actually swallowing and ingesting their hair (true trichophagia). Extreme cases have been reported in which patients consume hair found in the surrounding environment, including other people's and animals' hair. [ 5 ] In an even smaller subset of people with trichotillomania, their trichophagia can become so severe that they develop a hairball. [ 6 ] Termed a trichobezoar , these masses can be benign, or cause significant health concerns and require emergency surgery to remove them. Rapunzel syndrome is a further complication whereby the hairball extends past the stomach and can cause blockages of gastrointestinal system . [ 7 ]
Trichophagia occurs instinctively in many animal species and is not always a sign of a psychological disorder. Cats practice trichophagia as a form of regular grooming. [ 8 ]
Signs and symptoms of trichophagia are variable depending on the individual's behavior patterns. Trichophagia's loosest definition is the putting of hair in one's mouth, whether that be to chew it or suck on it, with the strictest definition being that the hair is swallowed and ingested. Trichophagia is most closely associated with trichotillomania , the pulling out of one's own hair, and thus any symptoms of trichotillomania could be predictive of trichophagia and must be ruled out. Rarely, persons with trichophagia do not exclusively have trichotillomania and instead will eat the hair of others. [ 9 ] [ 5 ]
Trichotillomania can be categorized as either "automatic", where the hair pulling is so habitual it is almost unconscious, or "focused", where the pulling is more deliberate, with the focused behavior thought to be more common among those with trichophagia. [ 10 ] Once the hair has been pulled out, persons with trichophagia might rub the hair against their lips, roll the hairs around and inspect them, bite off and swallow the bulb of the hair, or ingest the entire hair shaft as well. [ 9 ] Typically, ingested hair remains asymptomatic and is not harmful. However, if trichophagia is severe or chronic, a large mass of undigested hair can accumulate in the stomach, resulting in a trichobezoar . This can be symptomatic, including nausea, vomiting, and abdominal pain. [ 11 ] Once the trichobezoar grows large enough, it can extend beyond the stomach and lead to bowel obstructions , ulcers , perforations , acute pancreatitis and appendicitis (this is called Rapunzel syndrome ). [ 9 ] [ 7 ]
Along with the physical harm caused by the pulling out and ingestion of hair, the unpleasant symptoms and social stigma surrounding trichophagia negatively affect the quality of life of sufferers, leading to shame, guilt, and impairment of social functioning. [ 12 ] In one study, it was found that a significant percentage of patients with trichotillomania used drugs and alcohol to cope with negative feelings relating to pulling behaviors, with most sufferers reporting symptoms of anxiety and depression. [ 13 ] It is important for physicians to recognize and treat these secondary symptoms in order to relieve hair-pulling and eating behaviors. [ 14 ]
Trichophagia is estimated to have a prevalence of 0.6% in the general population with the most restrictive definition of hair ingestion, but looser definitions which are inclusive of sucking and chewing without swallowing, can be as high as 3.2%. [ 5 ] Its prevalence among patients with trichotillomania is estimated to be around 37.5%, with 33% developing trichobezoars. [ 15 ] Trichophagia can present at any age, with childhood cases typically being more common and of a more habitual nature, while in adulthood it is associated with underlying psychopathologies and more severe symptoms. [ 5 ] Among childhood cases the distribution between males and females is equal. However, in adolescents and adults, trichophagia is increasingly common among females, with a distribution of cases of 15:1, female-to-male. [ 16 ] Highest prevalence is in young adults. [ 9 ]
Many of the prevalence rates are thought to be underestimates due to stigma and inconsistent definitions of trichophagia. [ 1 ] Moreover, the discrepancy between rates in women and men could be explained by underreporting in men, either due to additional shame for men or the ease of shaving and hiding their underlying trichotillomania. [ 5 ] Trichophagia in men, while more rarely reported, is often more severe. [ 1 ]
Several etiological causes for trichotillomania and trichophagia have been hypothesized, suggesting that symptoms may be caused by disordered emotional regulation, autostimulation mechanisms, a response to stressors, behavioral conditioning, or addiction. [ 5 ] Research has also shown that there is a genetic component to the disorders; trichotillomania patients are more likely to have relatives who suffer from obsessive-compulsive disorder, excoriation disorder, and major depressive disorder. Decreased distress tolerance and increased impulsivity were also found in trichotillomania patients with a family history of obsessive-compulsive disorder. [ 17 ]
When assessing patients with trichophagia, common comorbid psychopathologies are anxiety disorders, eating disorders, depressive disorders, and addiction. [ 5 ] More comorbid psychopathologies are associated with more severe symptoms of trichotillomania and trichophagia. [ 17 ] The association between trichotillomania, obsessive-compulsive disorder, and related body-focused repetitive behaviors has been of particular interest to researchers, with studies finding that those with both trichotillomania and obsessive-compulsive disorder have higher levels of anxiety and depression as opposed to those who only suffer from trichotillomania. [ 18 ] The commonality of comorbid psychopathologies in individuals with trichotillomania and trichophagia could be indirectly caused by the social rejection sufferers face due to their symptoms. Researchers suggest that bringing awareness of the disorders to the general population could help relieve the stigmas faced by patients. [ 19 ]
Diagnosis of trichophagia can be difficult, as the behavior is easy to hide, and because of shame, individuals rarely admit they have trichophagia, even if they have stopped engaging in its related behaviors. [ 9 ] Often, individuals only seek medical help after they have developed gastrointestinal problems caused by a trichobezoar. [ 5 ] Any patient who has confirmed trichotillomania should be screened for trichophagia. [ 9 ] Some symptoms, such as hair loss, can be caused by other somatic conditions, the presence of which needs to be excluded before a diagnosis of trichotillomania or trichophagia. [ 5 ]
Psychotherapy has been widely used in the treatment of trichotillomania and trichophagia, with evidence supporting cognitive behavioral therapy , habit-reversal training , and mindfulness-based cognitive therapy as being effective treatments. [ 11 ] [ 20 ] [ 12 ] The most commonly used clinical treatment is the prescription of selective serotonin uptake inhibitors (SSRIs), such as fluoxetine and clomipramine , but the effectiveness of this treatment has not been supported empirically. [ 11 ] Evidence for efficacy has been found in studies using non-SSRIS, including N- acetylcysteine and olanzapine . [ 11 ]
Rapunzel syndrome , an extreme form of trichobezoar in which the "tail" of the hair ball extends into the intestines, and can be fatal if misdiagnosed. [ 6 ] [ 21 ] [ 22 ] [ 23 ] In some cases, surgery may be required to remove the mass. [ 24 ] In one case, a trichobezoar weighing 4.5 kilograms (9.9 lb) was removed from the stomach of an 18-year-old woman with trichophagia. [ 25 ]
Trichophagia and trichobezoars have been documented by physicians for centuries, even long before a medical definition was established for trichotillomania. [ 26 ] For example, in the 18th century, French doctor M. Baudamant described the condition in a 16-year-old boy. Trichophagia is most often covered in the medical literature only "as a rare symptom of trichotillomania." [ 1 ]
Despite its appearance in medical literature over the centuries, little research was conducted on trichophagia until the past decade. [ 12 ] Even now, most research focuses on Western cultures and European hair textures, with other groups with distinct hair textures, styles, and cultures, such as African Americans, often overlooked. [ 27 ]
Trichophagia is mentioned in the 1000 Ways to Die episode " Stupid Is As Stupid Dies " featuring a young woman who died from it. It is also mentioned in Grey's Anatomy season 9 episode 11 "The End Is the Beginning Is the End". As well as Season 3 episode 16 of The Resident , “Reverse Cinderella.” | https://en.wikipedia.org/wiki/Trichophagia |
Trichosclereids are hard needlelike branched cells found in some species of plants that serve the purpose of protecting the plant from herbivores . They are usually approximately 6 mm (1/4 inch) long, but in some species they grow to as long as 1 cm (0.4 inches). Trichosclereids are a type of sclereids that can be found in olive leaves and the aerial roots of the Swiss cheese plant ( Monstera deliciosa ). [ 1 ]
This cell biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Trichosclereid |
In group theory , the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Aschbacher ( 1981 , 1983 ) for rank 3 and by Gorenstein & Lyons (1983) for rank at least 4. The three classes are groups of GF(2) type (classified by Timmesfeld and others), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem and work by several others), and groups of uniqueness type , where Aschbacher proved that there are no simple groups .
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Trichotomy_theorem |
A trickle-bed reactor ( TBR ) is a chemical reactor that uses the downward movement of a liquid and the downward (co-current) or upward (counter-current) movement of gas over a packed bed of ( catalyst ) particles. It is considered to be the simplest reactor type for performing catalytic reactions where a gas and liquid (normally both reagents) are present in the reactor and accordingly it is extensively used in processing plants. Typical examples are liquid-phase hydrogenation , hydrodesulfurization , and hydrodenitrogenation in refineries (three phase hydrotreater ) and oxidation of harmful chemical compounds in wastewater streams or of cumene in the cumene process . [ 1 ] [ 2 ] [ 3 ] Also in the treatment of waste water trickle bed reactors are used where the required biomass resides on the packed bed surface.
Although the physical reactor is relatively simple, the hydrodynamics in the reactor are extremely complex. It is for this reason that TBRs have been extensively studied over the past five decades and currently [ when? ] the amount of open literature publications on TBRs is increasing, hinting that the understanding of the hydrodynamics is still limited.
A good introduction to the hydrodynamics of TBR can be found in the classic article by Satterfield. [ 2 ] Rate of reaction and mass transfer equations are derived by Fogler. [ 4 ] | https://en.wikipedia.org/wiki/Trickle-bed_reactor |
A trickling filter is a type of wastewater treatment system. It consists of a fixed bed of some material, such as rocks , coke , gravel , slag , polyurethane foam , sphagnum peat moss , ceramic , or plastic media, over which sewage or other wastewater flows downward and causes a layer of microbial slime ( biofilm ) to grow, covering the bed of media. Aerobic conditions are maintained by splashing, diffusion, and either by forced-air flowing through the bed or natural convection of air if the filter medium is porous. The treatment of sewage or other wastewater with trickling filters is among the oldest and most well characterized treatment technologies.
The fundamental components of a complete trickling filter system are:
The terms trickle filter, trickling biofilter, biofilter , biological filter and biological trickling filter are often used to refer to a trickling filter. These systems have also been described as roughing filters, intermittent filters, packed media bed filters, alternative septic systems, percolating filters, attached growth processes, and fixed film processes.
Typically, settled sewage flow enters at a high level and flows through the primary settlement tank. The supernatant from the tank flows into a dosing device, often a tipping bucket which delivers flow to the arms of the filter. The flush of water flows through the arms and exits through a series of holes pointing at an angle downwards. This propels the arms around distributing the liquid evenly over the surface of the filter media.
Most are uncovered (unlike the accompanying diagram) and are freely ventilated to the atmosphere.
The removal of pollutants from the waste water stream involves both absorption and adsorption of organic compounds and some inorganic species (such as nitrite and nitrate ions) by the layer of microbial biofilm . The filter media is typically chosen to provide a very high surface-to-volume ratio. Typical materials are often porous and have considerable internal surface area, in addition to the external surface of the medium. Passage of the wastewater over the media provides dissolved oxygen , which the biofilm layer requires for the biochemical oxidation of the organic compounds and releases carbon dioxide gas, water and other oxidized end products. As the biofilm layer thickens, it eventually sloughs off into the liquid flow and subsequently forms part of the secondary sludge. Typically, a trickling filter is followed by a clarifier or sedimentation tank for the separation and removal of the sloughed film. Filters utilizing higher-density media, such as sand, foam and peat moss do not produce a sludge that must be removed, but may require forced air blowers, backwashing, and/or an enclosed anaerobic environment. [ citation needed ]
The biofilm that develops in a trickling filter may become several millimetres thick and is typically a gelatinous matrix that may contain many species of bacteria , ciliates and amoeboid protozoa, annelids , round worms , insect larvae, other microfauna. (If annelids are abundant, the filter may be considered a vermifilter .) This is very different from many other biofilms, which may be less than 1 mm thick. Within the biofilm, both aerobic and anaerobic zones can exist supporting both oxidative and reductive biological processes. At certain times of year, especially in the spring, rapid growth of organisms in the film may cause the film to be too thick and it may slough off in patches leading to the "spring slough". [ 1 ]
A typical trickling filter is circular and between 10 metres and 20 metres across and between 2 metres to 3 metres deep. A circular wall, often of brick, contains a bed of filter media which in turn rests on a base of under-drains. These under-drains function both to remove liquid passing through the filter media but also to allow the free passage of air up through the filter media. Mounted in the center over the top of the filter media is a spindle supporting two or more horizontal perforated pipes which extend to the edge of the media. The perforations on the pipes are designed to allow an even flow of liquid over the whole area of the media and are also angled so that when liquid flows from the pipes the whole assembly rotates around the central spindle. [ 1 ] Settled sewage is delivered to a reservoir at the centre of the spindle via some form of dosing mechanism, often a tipping bucket device on small filters.
Larger filters may be rectangular and the distribution arms may be driven by hydraulic or electrical systems. [ 1 ]
Single trickling filters may be used for the treatment of small residential septic tank discharges and very small rural sewage treatment systems. Larger centralized sewage treatment plants typically use many trickling filters in parallel.
Systems can be configured for single-pass use where the treated water is applied to the trickling filter once before being disposed of, or for multi-pass use where a portion of the treated water is cycled back and re-treated via a closed loop . Multi-pass systems result in higher treatment quality and assist in removing Total Nitrogen (TN) levels by promoting nitrification in the aerobic media bed and denitrification in the anaerobic septic tank. Some systems use the filters in two banks operated in series so that the wastewater has two passes through a filter with a sedimentation stage between the two passes. Every few days the filters are switched round to balance the load. This method of treatment can improve nitrification and de-nitrification since much of the carbonaceous oxidative material is removed on the first pass through the filters.
Trickling may have a variety of types of filter media used to support the biofilm. Types of media most commonly used include coke , pumice , plastic matrix material, open-cell polyurethane foam , clinker, gravel, sand and geotextiles . Ideal filter medium optimizes surface area for microbial attachment, wastewater retention time, allows air flow, resists plugging, is mechanically robust in all weathers allowing walking access across the filter, and does not degrade. Some residential systems require forced aeration units which will increase maintenance and operational costs.
Synthetic filter media may pose a significant risk of flammability as demonstrated in Christchurch , New Zealand in May 2022 when two large trickling filters filled with plastic filter bales caught fire. The resultant smell had a significant impact on many city residents and this event put out of action a significant proportion of the sewage treatment capacity. [ 2 ]
The treatment of industrial wastewater may involve specialized trickling filters which use plastic media and high flow rates. Wastewaters from a variety of industrial processes have been treated in trickling filters. Such industrial wastewater trickling filters consist of two types:
The availability of inexpensive plastic tower packings has led to their use as trickling filter beds in tall towers, some as high as 20 meters. [ 6 ] As early as the 1960s, such towers were in use at: the Great Northern Oil's Pine Bend Refinery in Minnesota ; the Cities Service Oil Company Trafalgar Refinery in Oakville, Ontario and at a kraft paper mill. [ 7 ]
The treated water effluent from industrial wastewater trickling filters is typically processed in a clarifier to remove the sludge that sloughs off the microbial slime layer attached to the trickling filter media as for other trickling filter applications.
Some of the latest trickle filter technology involves aerated biofilters of plastic media in vessels using blowers to inject air at the bottom of the vessels, with either downflow or upflow of the wastewater. [ 8 ] | https://en.wikipedia.org/wiki/Trickling_filter |
Triclocarban (sometimes abbreviated as TCC ) is an antibacterial chemical once common in, but now phased out of, personal care products like soaps and lotions. It was originally developed for the medical field. [ 2 ] Although the mode of action is unknown, TCC can be effective in fighting infections by targeting the growth of bacteria such as Staphylococcus aureus . [ 3 ] Additional research seeks to understand its potential for causing antibacterial resistance and its effects on organismal and environmental health. [ 4 ]
Triclocarban has been used as an antimicrobial and antifungal compound since the 1960s. [ 5 ] It was commonly found in personal care products as an antimicrobial in soaps, lotions, deodorants, toothpaste, and plastic. [ 6 ] As of 2005 [update] about 80% of all antimicrobial bar soap sold in the United States contained triclocarban. [ 5 ] In 2011 United States consumers were spending nearly 1 billion dollars annually on products containing triclocarban and triclosan. [ 7 ]
In December 2013, the Food and Drug Administration (FDA) required all companies to prove within the next year, that triclocarban is not harmful to consumers. Companies like Johnson & Johnson , Procter & Gamble , Colgate-Palmolive , and Avon began phasing out antibacterial ingredients due to health concerns. [ 8 ]
By 2016 usage of triclocarban in soaps had declined to 40%, and that September the FDA banned triclocarban, triclosan and 17 other common antibacterial chemicals by September 2017, for their failure to be proven safe, or more effective than plain soap and water. [ 9 ]
Triclocarban, 3-(4-chlorophenyl)-1-(3,4-dichlorophenyl)urea, is a white powder that is insoluble in water. While triclocarban has two chlorinated phenyl rings, it is structurally similar to carbanilide compounds often found in pesticides (such as diuron ) and some drugs. Chlorination of ring structures is often associated with hydrophobicity, persistence in the environment, and bioaccumulation in fatty tissues of living organisms. For this reason, chlorine is also a common component of persistent organic pollutants . [ 10 ] Triclocarban is incompatible with strong oxidizing reagents and strong bases, reaction with which could result in safety concerns such as explosion, toxicity, gas, and heat.
There are two commercial routes used for the production of triclocarban, using the reaction of isocyanates with nucleophiles such as amines to form ureas : [ 11 ]
The purity specification in the draft USP monograph for triclocarban is: not less than 97.0% w/w. The purity of commercial production is greater, 98% w/w. [ 12 ]
Triclocarban is predominantly active against gram positive bacteria (bacteria with a thick peptidoglycan wall ). The precise mechanism of action of triclocarban is unknown, but it is shown to be bacteriostatic, which prevents bacterial proliferation. [ 13 ]
The specific mechanism of action for triclocarban's health effects on humans, like in bacteria, is unclear. Generally, in vitro , triclocarban enhances the gene expression of other steroid hormones, including androgens, estrogens, and cortisol. It is hypothesized that the compound acts similar to cofactors or coactivators that modulate the activity of estrogen receptors and androgen receptors . [ 14 ] [ 15 ] Experiments show that triclocarban activates constitutive androstane receptor and estrogen receptor alpha both in vivo and in vitro and might have the potential to alter normal physiological homeostasis. Activation of these receptors amplifies gene expression and, in doing so, may be the mechanistic base of triclocarban's health impact on humans. However, further investigation is needed to determine whether triclocarban increases the activity of sex steroid hormones by binding to the receptors or by binding to and sensitizing the receptor coactivators. [ 16 ] [ 17 ]
Triclocarban acts to treat both initial bacterial skin and mucosal infections as well as those infections at risk for superinfection . In vitro , triclocarban has been found to be effective against various strains of staphylococcus , streptococcus , and enterococcus bacteria. It has been shown to be effective as an antibacterial even at very low levels. Triclocarban's minimum inhibitory concentration has been found to range from 0.5 to 8 mg/L for these various strains. [ 18 ] Triclocarban is unquestionably bacteriostatic only for gram-positive bacteria such as Staphylococcus aureus , which suggests that the mechanism of triclocarban's antibacterial activity is through its destabilization of bacterial cell walls. [ 5 ]
Exposure of organisms like fish, algae, and humans to low levels of triclocarban and other antibacterial chemicals kills weak microbes and allows the stronger, resistant strains to proliferate. As microbes share genes , an increase in resistant strains increases the probability that weak microbes acquire these resistance genes. The consequence is a new colony of drug resistant microbes. [ 19 ]
When resistant microbes are exposed to antimicrobials, they increase their expression of genes that confer this resistance. The risk of bacterial antibiotic resistance has been studied by quantitatively monitoring the abundance of the tetQ gene in wastewater microcosms. As tetQ is the most common resistance gene in the environment and encodes for ribosomal protection proteins, the amount that it expresses correlates with the amount of resistance in a microbial population. The addition of triclocarban was shown to increase the expression of this tetQ gene. [ 19 ]
TetQ gene expression in bacteria was also found to be significantly increased when multiple antimicrobials such as tetracycline, triclosan, and triclocarban were added to an experimental system at the same time. Combining these compounds affects resistance by creating a situation where co-selection (or natural selection by more than one reagent) for resistance genes occurs. The complex nature of microbial communities and the multitude of antibiotics present in aquatic environments often leads to this sort of dynamic selection event and the multiple resistance patterns seen in naturally occurring bacteria. [ 19 ]
When triclocarban is manufactured, 139 toxic, carcinogenic byproducts, such as 4-chloroaniline and 3,4-dichloroaniline, are released. More of these carcinogens can be released upon chemical, physical and biological attack of triclocarban. [ 20 ] The duration of triclocarban chemical in personal product use is relatively short. Upon disposal, the triclocarban is washed down the drain to municipal wastewater treatment plants , where about 97-98% of triclocarban is removed from the water.
Discharge of effluent from these treatment plants and disposal of sludge on land is the primary route of environmental exposure to triclocarban. Research shows that triclocarban and triclosan have been detected in sewage effluents and sludge ( biosolids ) due to their incomplete removal during wastewater treatment. [ 21 ] Due to their hydrophobic nature, significant amounts of them in wastewater streams partition into sludge, with concentrations at mg/kg levels. The volume of triclocarban reentering the environment in sewage sludge after initial successful capture from wastewater is s 127,000 ± 194,000 kg/yr. This is equivalent to a 4.8 – 48.2% of its total U.S. consumption volume. Crops shown to take up antimicrobials from soil include barley, meadow fescue, carrots and pinto beans. [ 20 ] Studies show that substantial quantities of triclocarban (227,000 – 454,000 kg/y) can break through wastewater treatment plants and damage algae on surface waters. [ 20 ]
High concentrations of triclocarban may be found in wastewater . As of 2011 it was among the top ten most commonly detected organic wastewater compounds in terms of frequency and concentration. Triclocarban has been found in increasing concentrations over the past five years and is now more frequently detected than triclosan . [ 6 ]
Triclocarban has a hazard quotient rating of greater than one, which indicates the potential for adverse effects on organisms due to toxicity. [ 6 ] As triclocarban is found in high concentrations in aquatic environments, there are concerns regarding its toxicity to aquatic species. Specifically, triclocarban has been shown to be toxic to amphibians, fish, invertebrates, and aquatic plants, and traces of the compound have been found in Atlantic dolphins. [ 6 ] [ 22 ] Triclocarban may disrupt hormones critical to the developmental and endocrine processes in exposed animal wildlife. The neurological and reproductive systems are particularly affected through contact with this compound. Triclocarban may also affect animal wildlife behavior. [ 22 ] For example, triclosan and triclocarban are 100–1,000 times more effective in inhibiting and killing algae, crustaceans, and fish than they are in killing microbes. Triclocarban and triclosan have been observed in multiple organisms, including algae, aquatic blackworms, fish, and dolphins. [ 20 ]
Triclocarban bioaccumulation is possible in a number of organisms. Earthworms are known to store this chemical in their bodies and, because of their ecological role as a food source, they have the potential to move triclocarban up the food chain. [ 23 ] Microbial species found in soils also bioaccumulate triclocarban. However, the health of these microbes has not been found to be affected by the presence of the chemical. [ 24 ] Triclocarban is rapidly accumulated in both algae and adult caged snails. [ 25 ] Moreover, triclocarban is more likely than triclosan to bioaccumulate in aquatic organisms. [ 26 ]
Bioaccumulation occurs in plants treated with water containing triclocarban. However, it is estimated that less than 0.5% of the acceptable daily intake of triclocarban for humans is represented by vegetable consumption. Thus, the concentration of triclocarban in edible portions of plants is a negligible exposure pathway for humans. [ citation needed ]
The potential for triclocarban to bioaccumulate in plants has been exploited in the construction of wetlands meant to help remove triclocarban from wastewater. These constructed wetlands are considered a cost-effective treatment option for the removal of PPCPs , including triclocarban and triclosan, from domestic water effluent. Such compounds tend to concentrate in the roots of wetland plants. Potential ecological risks associated with this method are the decrease of root systems in wetland plants, reduced nutrient uptake, decreased competitive ability, and increased potential for uprooting. Due to these risks, the long term exposure of wetland ecosystems to wastewater containing triclocarban as a major solution to wastewater pollution is still under discussion. [ 27 ]
One study has investigated how triclocarban remains in the human system after using a bar of soap with traces of triclocarban. Analysis of urine samples from human test subjects shows that, after triclocarban has undergone glucuronidation , its oxidative metabolites are less readily excreted than triclocarban itself. This same study performed topical treatments of triclocarban on rats and, by analyzing urine and plasma levels, demonstrated that triclocarban does remain in the organism's system. [ 28 ]
Triclocarban induces weak responses mediated by aryl hydrocarbon , estrogen , and androgen receptors in vitro . This has yet to be confirmed in vivo . [ 29 ] In vitro , the dihydrotestosterone-dependent activation of androgen receptor-responsive gene expression is enhanced by triclocarban by up to 130%. [ 30 ] Triclocarban is also a potent inhibitor of the enzyme soluble epoxide hydrolase (sEH) in vitro . [ 28 ] Additionally, triclocarban amplifies the bioactivity of testosterone and other androgens. This increased activity may have adverse implications for reproductive health. [ 7 ] [ 23 ] Triclocarban studies on rats exhibited increased size of the specimens' prostate glands. [ 31 ] The amplification of sex hormones could promote the growth of breast and prostate cancer. [ citation needed ]
The chemical toxicity of triclocarban with respect to lethality is low ( LD 50 >5000 mg/kg). Its rate of skin absorption is also low. [ 32 ] Repeated low-dose exposure, however, can cause endocrine disruption over time. [ citation needed ]
Spillage may increase the risk of human, ecological, and environmental exposure to triclocarban. Immediate removal and restraint of the spill, including triclocarban as dust, is urged. [ 31 ] Although triclocarban has few to no direct detrimental effects on health aside from allergic reactions, preventing exposure to triclocarban is recommended. Since triclocarban enters the body through pores, wearing gloves, properly washing hands, and overall proper hygiene reduces the risk of skin exposure and irritation.
High concentrations of triclocarban dust may remain in the lungs and inhibit lung and respiratory function. For individuals with prior respiratory conditions, triclocarban exacerbates the severity of respiratory diseases, and proper protection is recommended as a precaution.
In case of exposure to triclocarban, the individual is suggested to wash the area with water or to clear the respiratory pathways. [ 31 ] In addition to its adverse effects on humans and the environment, solid triclocarban is a fire hazard. It is particularly combustible as dust. Contamination with other oxidizing agents may also result in combustion. [ 31 ]
The Food and Drug Administration began to review the safety of triclocarban and triclosan in the 1970s, but due to the difficulties of finding antimicrobial alternatives, no final policy, or "drug monograph," was established. [ 20 ] Legal action by the Natural Resources Defense Council in 2010 forced the FDA to review triclocarban and triclosan . [ 20 ] The United States Environmental Protection Agency maintains regulatory control over triclocarban and triclosan . [ 20 ]
On September 2, 2016, the Food and Drug Administration announced that triclosan and triclocarban must be removed from all antibacterial soap products by late 2017. [ 9 ] [ 33 ] Triclocarban is similar in its use and adverse health impacts as triclosan , and hexachlorophene which was already prohibited by the FDA. [ 20 ]
Scientists are searching for more sustainable antimicrobials that maintain their effectiveness while being minimally toxic to the environment, humans, and wildlife. This entails low degrees of bioaccumulation and rapid, clean biodegradation in existing wastewater treatment facilities. A lowered potential or no potential for resistance is also preferable. [ 20 ] These next generation chemicals should aim to act on a broad spectrum of microbes and pathogens while also being minimally toxic and bioaccumulating in non-target species. [ citation needed ]
Synthesis of these compounds could be improved upon by finding renewable sources for their production that lacks occupational hazards. [ 20 ] Research into sustainable chemical production is helping to formulate green pharmaceuticals. These same principles may be applied to the development of improved antimicrobials. [ 20 ] Developments in this area would benefit both people and the environment. [ 20 ] | https://en.wikipedia.org/wiki/Triclocarban |
Triclosan (sometimes abbreviated as TCS ) is an antibacterial and antifungal agent present in some consumer products, including toothpaste , soaps , detergents , toys, and surgical cleaning treatments. It is similar in its uses and mechanism of action to triclocarban . Its efficacy as an antimicrobial agent , the risk of antimicrobial resistance , and its possible role in disrupted hormonal development remains controversial. Additional research seeks to understand its potential effects on organisms and environmental health.
Triclosan was developed in 1966. [ 1 ] A 2006 study recommended showering with 2% triclosan as a regimen in surgical units to rid patients' skin of methicillin-resistant Staphylococcus aureus (MRSA). [ 2 ]
Triclosan was used as a hospital scrub in the 1970s. Prior to its change in regulatory status in the EU and US, it had expanded commercially and was a common ingredient in soaps (0.10–1.00%), shampoos , deodorants , toothpastes , mouthwashes , cleaning supplies, and pesticides . [ 3 ] It also was part of consumer products, including kitchen utensils, toys, bedding, socks, and trash bags. [ 3 ] [ 4 ]
Triclosan was registered as a pesticide in 1969. [ 5 ] U.S. EPA registration numbers are required for all EPA-registered pesticides. [ 6 ] As of 2017, there were five registrations for triclosan with the EPA. [ 7 ] Currently, there are 20 antimicrobial registrations with the EPA under the regulations of the Federal Insecticide, Fungicide, and Rodenticide Act (FIFRA) . [ 5 ] The antimicrobial active ingredient is added to a variety of products where it acts to slow or stop the growth of bacteria, fungi, and mildew. In commercial, institutional, and industrial equipment uses, triclosan is incorporated in conveyor belts, fire hoses, dye bath vats, or ice-making equipment as an antimicrobial. Triclosan may be directly applied to commercial HVAC coils, where it prevents microbial growth that contributes to product degradation. [ 5 ]
In the United States, by 2000, triclosan and triclocarban (TCC) could be found in 75% of liquid soaps and 29% of bar soaps, [ 8 ] and as of 2014 [update] triclosan was used in more than 2,000 consumer products. [ 9 ]
In healthcare, triclosan is used in surgical scrubs and hand washes. [ 10 ] Use in surgical units is effective with a minimum contact time of approximately two minutes. [ 11 ] [ 12 ] More recently, showering with 2% triclosan has become a recommended regimen in surgical units for the decolonization of patients whose skin carries methicillin-resistant Staphylococcus aureus (MRSA). [ 2 ] Two small uncontrolled case studies reported the use of triclosan correlated with reduction in MRSA infections. [ 11 ] [ 12 ]
Triclosan is also used in the coatings for some surgical sutures . [ 13 ] [ 14 ] There is good evidence these triclosan-coated sutures reduce the risk of surgical site infection. [ 15 ] [ 16 ] The World Health Organization, the American College of Surgeons and the Surgical Infection Society point out the benefit of triclosan-coated sutures in reducing the risk for surgical site infection. [ 17 ] [ 18 ]
Triclosan has been employed as a selective agent in molecular cloning. A bacterial host transformed by a plasmid harboring a triclosan-resistant mutant FabI gene (mFabI) as a selectable marker can grow in presence of high dose of triclosan in growth media. [ 19 ]
In surgery, triclosan coated sutures reduce the risk of surgical site infection. [ 15 ] [ 16 ] Some studies suggest that antimicrobial hand soaps containing triclosan provide a slightly greater bacterial reduction on the hands compared to plain soap. [ 20 ] As of 2013 [update] , the US FDA had found clear benefit to health for some consumer products containing triclosan, but not in others; for example the FDA had no evidence that triclosan in antibacterial soaps and body washes provides any benefit over washing with regular soap and water. [ 21 ]
A Cochrane review of 30 studies concluded that triclosan/copolymer-containing toothpastes produced a 22% reduction in both dental plaque and gingival inflammation when compared with fluoride toothpastes without triclosan/copolymer. [ 22 ] There was weak evidence of a reduction in tooth cavities , and no evidence of reduction in periodontitis . [ 22 ]
A study of triclosan toothpastes did not find any evidence that it causes an increase in serious adverse cardiac events such as heart attacks. [ 23 ]
A study by Colgate-Palmolive found a significant reduction in gingivitis, bleeding, and plaque with the use of triclosan-containing toothpaste. [ 24 ] An independent review by the Cochrane group suggests that the reduction in gingivitis, bleeding, and plaque is statistically significant (unlikely to occur by chance) but not clinically significant (unlikely to provide noticeable effects). [ 25 ]
Triclosan is used in food storage containers [ 26 ] : 417–423 [ 27 ] although this use is banned in the European Union since 2010. [ 27 ]
Veterinary use as a biocidal product in the EU is governed by the Biocidal Products Directive . [ 28 ]
This organic compound is a white powdered solid with a slight aromatic, phenolic odor. Categorized as a polychloro phenoxy phenol , triclosan is a chlorinated aromatic compound that has functional groups representative of both ethers and phenols . Phenols often demonstrate antibacterial properties. Triclosan is soluble in ethanol , methanol , diethyl ether , and strongly basic solutions such as a 1 M sodium hydroxide solution, but only slightly soluble in water. [ 29 ] Triclosan can be synthesized from 2,4-dichlorophenol .
Under a reflux process, 2,4,4'-trichloro-2'-methoxydiphenyl ether is treated with aluminium chloride . [ 30 ]
The United States Pharmacopeia formulary has published a monograph for triclosan that sets purity standards. [ 31 ]
At high concentrations, triclosan acts as a biocide with multiple cytoplasmic and membrane targets. [ 32 ] However, at the lower concentrations seen in commercial products, triclosan appears bacteriostatic , and it targets bacteria primarily by inhibiting fatty acid synthesis . [ citation needed ]
Triclosan binds to bacterial enoyl-acyl carrier protein reductase (ENR) enzyme, [ 33 ] which is encoded by the gene fabI . This binding increases the enzyme's affinity for nicotinamide adenine dinucleotide (NAD + ). This results in the formation of a stable, ternary complex of ENR-NAD + -triclosan, which is unable to participate in fatty acid synthesis. Fatty acids are necessary for building and reproducing cell membranes. Vertebrates do not have an ENR enzyme and thus are not affected by this mode of action. [ citation needed ]
Triclosan has been found to be a weak endocrine disruptor , though the relevance of this to humans is uncertain. [ 34 ] [ 35 ] The compound has been found to bind with low affinity to both the androgen receptor and the estrogen receptor , where both agonistic and antagonistic responses have been observed. [ 34 ]
Triclosan may upregulate or induce efflux pumps in bacteria causing them to become resistant against variety of other antibiotics. [ 36 ] [ 37 ]
Humans are exposed to triclosan through skin absorption when washing hands or in the shower, brushing teeth, using mouthwash or doing dishes, and through ingestion when swallowed. When triclosan is released into the environment, additional exposure to the chemical is possible through ingesting plants grown in soil treated with sewage sludge , or eating fish exposed to it. [ 9 ]
An article from the American Society of Agronomy refers to a study done by Monica Mendez et al. , in which the researchers irrigated plants with water containing triclosan and months later found it in all edible parts of tomato and onion plants. [ 38 ] Triclosan is found to kill a wide spectrum of bacteria, and the researchers are also concerned about the effect it has on the beneficial bacteria in soil. [ 39 ]
Once absorbed, triclosan is metabolized by humans primarily through conjugation reactions into glucuronide and sulfate conjugates that are excreted in feces and urine. Pharmacokinetic studies demonstrate that triclosan sulfate and glucuronide may be formed in the liver at approximately equal rates at the environmentally relevant concentration of 1 to 5 microMolar. When concentrations of triclosan are below 1 microMolar, sulfonation is expected to be the major metabolic pathway for elimination. [ 40 ]
Because of potential health concerns, due to the possibility of antimicrobial resistance, endocrine disruption and other issues as listed below, triclosan has been designated as a "contaminant of emerging concern (CEC)" by the United States Geological Survey , meaning it is under investigation for public health risk. "Emerging contaminants" can be broadly defined as any synthetic or naturally occurring chemical or any microorganism that is not commonly monitored in the environment but has the potential to enter the environment and cause known or suspected adverse ecological or human health effects. [ 41 ] Triclosan is thought to accumulate in wastewater and return to drinking water, thus propagating a buildup that could cause increasing effects with ongoing use. [ 42 ]
In the United States, after a decades-long review of the potential health issues from this contaminant of emerging concern, the FDA ruled on September 6, 2016, that 19 active ingredients including triclosan are not generally recognized as safe and effective (GRAS/GRAE). [ 43 ] (See policy section below).
Triclosan has been associated with a higher risk of food allergy . [ 44 ] This may be because exposure to bacteria reduces allergies , as predicted by the hygiene hypothesis , and not caused by toxicology of triclosan itself. This effect may also occur with chlorhexidine gluconate and chloroxylenol (PCMX), among other antibacterial agents. [ 45 ] [ 46 ] Other studies have linked triclosan to allergic contact dermatitis in some individuals. [ 47 ] [ 48 ] Additionally, triclosan concentrations have been associated with allergic sensitization, especially inhalant and seasonal allergens, rather than food allergens. [ 49 ]
Triclosan can react with the free chlorine in chlorinated tap water to produce lesser amounts of other compounds, such as 2,4-dichlorophenol . Some of these intermediates convert into dioxins upon exposure to UV radiation (from the sun or other sources). [ 50 ] The dioxins that can form from triclosan are not thought to be congeners of toxicologic concern for mammals, birds and fish. [ 51 ] [ 52 ]
Concerns on the health effects of triclosan have been raised after it was detected in human breast milk, blood, and urine samples. Studies on rats have shown that triclosan exposure modulates estrogen-dependent responses. [ 53 ] There have been many studies performed over the years both in vivo and in vitro, in male and female fish and rats and they support the conclusion that triclosan possesses (anti)estrogenic and (anti)androgenic properties depending on species, tissues, and cell types. [ 40 ]
Human studies on triclosan and hormone effects are fewer in number than those on animals, but are being conducted. In a 2017 study on 537 pregnant women in China, prenatal triclosan exposure was associated with increased cord testosterone levels in the infants. [ 54 ]
Triclosan (TCS) was patented in 1964 [ 9 ] by Swiss company Ciba-Geigy . The earliest known safety testing began in 1968. [ 55 ] It was introduced the next year, mainly for use in hospitals, and was in worldwide production and use by the early 1970s.
In 1997 Ciba-Geigy merged with another Swiss company, Sandoz , to form Novartis . [ 56 ] During the merger, Ciba-Geigy's chemical business was spun off to become Ciba Specialty Chemicals , which was acquired in 2008 by chemical giant BASF . BASF currently manufactures TCS under the brand name Irgasan DP300. [ 57 ]
Exposure to triclosan in personal product use is relatively short. Upon disposal, triclosan is sent to municipal sewage treatment plants, where, in the United States, about 97–98% of triclosan is removed. [ 9 ] Studies show that substantial quantities of triclosan (170,000–970,000 kg/yr) can escape from wastewater treatment plants and damage algae on surface waters. [ 9 ] In a study on effluent from wastewater treatment facilities, approximately 75% of triclocarban was present in sewage sludge . [ 58 ] This poses a potential environmental and ecological hazard, particularly for aquatic systems. The volume of triclosan, in the United States, re-entering the environment in sewage sludge after initial successful capture from wastewater is 44,000 ± 60,000 kg/yr. [ 9 ] Triclosan can attach to other substances suspended in aquatic environments, which potentially endangers marine organisms and may lead to further bioaccumulation. [ 59 ] Ozone is considered to be an effective tool for removing triclosan during sewage treatment. As little triclosan is released through plastic and textile household consumer products, these are not considered to be major sources of triclosan contamination. [ 59 ]
During wastewater treatment, a portion of triclosan is degraded, while the remainder adsorbs to sewage sludge or exits the plant as effluent. [ 60 ] [ 61 ] A mass balance in Athens (Greece) Sewage Treatment Plant (2013) showed that 43% of triclosan is accumulated to the primary and secondary sludge, 45% is lost due to degradation while the rest 12% is discharged to the environment via the secondary treated wastewater. [ 62 ] In the environment, triclosan may be degraded by microorganisms or react with sunlight, forming other compounds, which include chlorophenols and dioxins . [ citation needed ]
During 1999 to 2000, US Geological Survey detected TCS in 57.6% of streams and rivers sampled. [ 63 ]
While studies using semi-permeable membrane devices have found that triclosan does not strongly bioaccumulate , methyl-triclosan is comparatively more stable and lipophilic and thus poses a higher risk of bioaccumulation. The ability of triclosan to bioaccumulate is affected by its ionization state in different environmental conditions. [ 64 ]
Global warming may increase uptake and effects of triclosan in aquatic organisms. [ 65 ]
Triclosan is toxic to aquatic bacteria at levels found in the environment. It is highly toxic to various types of algae and has the potential to affect the structure of algal communities, particularly immediately downstream of effluents from wastewater treatment facilities that treat household wastewaters. [ 66 ] Triclosan has been observed in multiple organisms, including algae, aquatic blackworms , fish, and dolphins . It has also been found in land animals including earthworms and species higher up the food chain . [ 9 ] In toxicity experiments with Vibrio fischeri marine bacterium, an EC50 value of TCS equal to 0.22 mg/L has been determined. [ 67 ] Few data are available for the long-term toxicity of TCS to algae, daphnids and fish, while enough data are available for its acute toxicity on these groups of organisms. [ citation needed ]
A 2017 study that used risk quotient (RQ) methodology and evaluated the ecological threat due to the discharge of wastewater containing TCS in European rivers, reported that the probability that RQ values exceeds 1 ranged from 0.2% (for rivers with dilution factor of 1000) to 45% (for rivers with dilution factor 2). [ 68 ]
Triclosan favors anaerobic conditions which is typical in soil and sediment. The antimicrobial properties of Triclosan are resistant to anaerobic degradation which is the main contributor to its persistence in the environment. [ 69 ]
Concern pertains to the potential for cross-resistance (or co-resistance) to other antimicrobials. Numerous studies have been performed and there have been results indicating that the use of biocidal agents, such as triclosan, can cause cross-resistance.
A study done in a wide range of bacteria and different classes of antibiotics showed that Pseudomonas aeruginosa and Stenotrophomonas maltophilia , already resistant to triclosan, had increased resistance against antibiotics tetracycline and norfloxacin when exposed to triclosan. [ 70 ] Results from a study published in The American Journal of Infection Control showed that exposure to triclosan was associated with a high risk of developing resistance and cross-resistance in Staphylococcus aureus and Escherichia coli . This was not observed with exposure to chlorhexidine or a hydrogen peroxide -based agent (during the conditions in said study). [ 71 ]
A comprehensive meta-analysis published in 2007 indicated that, in community settings, plain soap was no less effective than soaps containing triclosan for "preventing infectious illness symptoms and reducing bacterial levels on the hands.". [ 72 ]
Nonorganic antibiotics and organic biocides are effective alternatives to triclosan, such as silver and copper ions and nanoparticles . [ 73 ]
In the US, triclosan is regulated as a pesticide by the EPA and as a drug by the FDA. The EPA generally regulates uses on solid surfaces, and FDA regulations cover uses in personal care products. [ citation needed ]
In 1974, the US FDA began the drug review monograph process for "over-the-counter (OTC) topical antimicrobial products", including triclosan and triclocarban. The advisory panel first met on June 29, 1972, and the agency published its proposed rule on Sept 13, 1974. The initial rule applied to, "OTC products containing antimicrobial ingredients for topical human use, which includes soaps, surgical scrubs, skin washes, skin cleansers, first aid preparations and additional products defined by the panel." [ 74 ] The proposed rule lists dozens of products that were already on the market at the time and the firms that produced them. [ citation needed ]
In 1978 the FDA published a tentative final monograph (TFM) for topical antimicrobial products. The record was re-opened in March 1979 to take into account six comments the agency received during the period for submitting objections to the TFM, including new data submitted by Procter & Gamble on the safety and effectiveness of triclocarban and by Ciba-Geigy on the proliferation of use of triclosan. The document states that, "significant amounts of new and previously unconsidered data were submitted with each of the above petitions." [ 75 ] It was re-opened again in October of that year to permit interested persons to submit further data establishing conditions for the safety, effectiveness and labeling of over-the-counter topical antimicrobial products for human use. [ citation needed ]
The next document issued was a proposed rule dated June 17, 1994, which states, the "FDA is issuing a notice of proposed rulemaking in the form of an amended tentative final monograph that would establish conditions under which OTC topical health-care antiseptic drug products are generally recognized as safe and effective and not misbranded. The FDA is issuing this notice of proposed rulemaking on topical antimicrobial drug products after considering the public comments on that notice and other information in the administrative record for this rulemaking. The FDA is also requesting data and information concerning the safety and effectiveness of topical antimicrobials for use as hand sanitizers or dips." [ 76 ] In the 1994 update to the rule, TCS was effectively removed from the drug category which made it available for use in consumer products. [ 9 ]
In 2010, the Natural Resources Defense Council forced the FDA to review triclosan after suing the agency for its inaction. Because the FDA prohibited hexachlorophene , a compound similar to triclosan, Halden and others argued that the FDA should also ban triclosan. [ 9 ] On December 17, 2013, the FDA issued a draft rule revoking the generally recognized as safe status of triclosan as an ingredient in hand wash products, citing the need for additional studies of its potential endrocrine and developmental effects; impact on bacterial resistance; and carcinogenic potential. [ 77 ]
On September 6, 2016, 44 years after its initial proposed rule, the FDA issued a final rule establishing that 19 active ingredients, including triclosan and triclocarban, used in over-the-counter (OTC) consumer antiseptic products intended for use with water (aka consumer antiseptic washes) are not generally recognized as safe and effective (GRAS/GRAE) and are misbranded, and are new drugs for which approved applications under section 505 of the FD&C Act are required for marketing. [ 78 ] Companies have one year to reformulate products without these ingredients, take them off the market or submit a New Drug Application (NDA) for the products. The 19 ingredients are:
In 2015 and 2016 FDA also issued proposed rules to amend the 1994 TFM regarding the safety and effectiveness of OTC health care antiseptics and OTC consumer antiseptic rubs.
The state of Minnesota took action against triclosan in advance of a federal rule. In May 2014, the governor signed a bill banning triclosan-containing products in the state. A CNN article quotes the new law, "In order to prevent the spread of infectious disease and avoidable infections and to promote best practices in sanitation, no person shall offer for retail sale in Minnesota any cleaning product that contains triclosan and is used by consumers for sanitizing or hand and body cleansing." [ 80 ] The law goes into effect on January 1, 2017. The exceptions to this rule are individual products that have received approval from the US Food and Drug Administration for consumer use.
In light of mounting evidence on the human health and ecotoxic effects of triclosan, some companies reformulated to remove it in advance of regulation: Colgate-Palmolive removed it from Palmolive Dish Soap and Softsoap in 2011 (but it remained in Colgate Total toothpaste until late 2018 or early 2019 [ 81 ] ); Johnson & Johnson removed it from baby products in 2012 and all products in 2015; Procter & Gamble from all products in 2014; in 2014 it was removed from Clearasil and Avon began phasing it out; and Unilever removed it from skin care and cleansing products in 2015, and says oral care by 2017. [ 82 ]
In Canada, triclosan is allowed in cosmetics, though FDA's recent announcement has prompted Health Canada spokeswoman Maryse Durette to state in an e-mail to Toronto newspaper The Globe and Mail that, "the government will publish a final assessment of the safety of triclosan 'in the near future' and take further action 'if warranted. ' " [ 83 ] Health Canada maintains a Cosmetic Ingredient Hotlist, including hundreds of chemicals that are not allowed or whose use is restricted in cosmetics. The list states that triclosan is currently allowed in cosmetics up to 0.3%, and 0.03% in mouthwashes and other oral products with required warnings to avoid swallowing and not for use in children under the age of 12. [ 84 ]
Triclosan was not approved by the European Commission as an active substance for use in biocidal products for product-type 1 in January 2016. [ 85 ] In the United States, manufacturers of products containing triclosan must indicate its presence on the label. In Europe, triclosan is regulated as a cosmetic preservative and must be listed on the label. [ 86 ] Use of triclosan in cosmetic products was restricted by the EU commission in 2014. [ 87 ] | https://en.wikipedia.org/wiki/Triclosan |
Triconex is a Schneider Electric brand that supplies products, systems, and services for safety, critical control, and turbo-machinery applications. Triconex also use its name for its hardware devices that use its TriStation application software. Triconex products are based on patented Triple modular redundancy (TMR) industrial safety-shutdown technology. Today, Triconex TMR products operate globally in more than 11,500 installations.
The history of Triconex was published in the book The History of a Safer World by Gary L. Wilkinson . The company was founded in September 1983 by Jon Wimer in Santa Ana, California and began operations in March 1984. The company was founded as a venture-capital funded private company. The business plan was written by Wimer and Peter Pitsker , an automation industry veteran and Stanford graduate. They presented the plan for a TMR (triple modular redundant) system named "Tricon" that would improve the safety and reliability of industrial applications. Among the customers they targeted were the petro-chemical giants, such as Exxon , Shell , Chevron , and BP .
Pitsker and Wimer presented the business plan to Los Angeles-based investor Chuck Cole, who was also a professor at USC. Cole was interested, so he contacted his personal attorney, future two-time Los Angeles Mayor Richard Riordan . Riordan agreed to invest $50,000 and Cole's venture capital team matched it, providing the seed money for Triconex. Wimer hired computer architect Ken Brody out of another computer manufacturer as Vice President of Research and Development and the number 2 employee. Ken Brody hired Wing N. Toy from Bell Labs . After two years, however, the company nearly failed due to the expense and complications of testing a new safety system. In February 1986, founder Wimer left the company and the board asked a seasoned executive, William K. Barkovitz, to become CEO; Barkovitz ended up leading the company for 9 years. At the end of his term, Triconex became the leading safety system in a market it largely created, made acquisitions, and completed an initial public offering . In January 1994, Triconex was acquired by British-based SIEBE for 90 million dollars.
The hardware architect of the company was Gary Hufton , and the software development manager was Glen Alleman . Along with Wing N. Toy (the lead engineer of the fault-tolerant ESS telephone switch ), they led a small successful engineering team that built the first Tricon system, sold in June 1986. Soon after, Exxon became a customer and Honeywell agreed to distribute the Tricon. Among the software engineers who worked for Triconex were Phil Huber and Dennis Morin , who later left the company to found Wonderware .
The Triconex system is based on the TMR patented technology that supports up to Safety Integrity Level 3 ( SIL 3 ) and is usually used as a safety rather than a control system. [ 1 ]
Fault tolerance in the Tricon is achieved by means of a Triple-Modular Redundant (TMR) architecture. The Tricon provides error-free, uninterrupted control in the presence of either hard failures of components, or transient faults from internal or external sources. The Tricon is designed with a fully triplicated architecture throughout, from the input modules through the Main Processors to the output modules. Every I/O module houses the circuitry for three independent legs. Each leg on the input modules reads the process data and passes that information to its respective Main Processor. The three Main Processors communicate with each other using a proprietary high-speed bus system called the TriBus. Once per scan, the three Main Processors synchronize and communicate with their two neighbors over the TriBus. The Tricon votes digital input data, compares output data, and sends copies of analog input data to each Main Processor. The Main Processors execute the user written application and send outputs generated by the application to the output modules. In addition to voting the input data, the TriBus votes the output data. This is done on the output modules as close to the field as possible to detect and compensate for any errors between the Tricon voting and the final output driven to the field.
The Triconex system usually consists of the following typical modules: [ 2 ]
The Triconex main processors can communicate with the so-called TriStation 1131 application software to download, update and/or monitor programs. [ 3 ] These programs are either written in:
(Function Block Diagram, Ladder diagram and Structured Text are defined in IEC1131-3)
Besides, a Sequence of Events (SOE) recorder software and Diagnostic monitor software are implemented.
In December 2017, it was reported that the safety systems of an unidentified power station, believed to be in Saudi Arabia were compromised when the Triconex industrial safety technology made by Schneider Electric SE was targeted in what is believed to have been a state sponsored attack. The computer security company Symantec claimed that the malware , known as "Triton", exploited a vulnerability in computers running the Microsoft Windows operating system. [ 4 ] | https://en.wikipedia.org/wiki/Triconex |
In systems engineering , the tricotyledon theory of system design ( T3SD ) is a mathematical theory of system design developed by A. Wayne Wymore . [ 1 ] T3SD consists of a language for describing systems and requirements , which is based on set theory , a mathematical systems model based on port automata, and a precise definition of the different types of system requirements and relationships between requirements.
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tricotyledon_theory_of_system_design |
A tricritical point is a point where a second order phase transition curve meets a first order phase transition curve. The notion was first introduced by Lev Landau in 1937, who referred to the tricritical point as the critical point of the continuous transition. [ 1 ] [ 2 ] The first example of a tricritical point was shown by Robert B. Griffiths in a helium-3 helium-4 mixture. [ 3 ] In condensed matter physics , dealing with the macroscopic physical properties of matter, a tricritical point is a point in the phase diagram of a system at which three-phase coexistence terminates. [ 4 ] This definition is clearly parallel to the definition of an ordinary critical point as the point at which two-phase coexistence terminates.
A point of three-phase coexistence is termed a triple point for a one-component system, since, from Gibbs' phase rule , this condition is only achieved for a single point in the phase diagram ( F = 2-3+1 = 0 ). For tricritical points to be observed, one needs a mixture with more components. It can be shown [ 5 ] that three is the minimum number of components for which these points can appear. In this case, one may have a two-dimensional region of three-phase coexistence ( F = 2-3+3 = 2 ) (thus, each point in this region corresponds to a triple point). This region will terminate in two critical lines of two-phase coexistence; these two critical lines may then terminate at a single tricritical point. This point is therefore "twice critical", since it belongs to two critical branches. Indeed, its critical behavior is different from that of a conventional critical point: the upper critical dimension is lowered from d=4 to d=3 so the classical exponents turn out to apply for real systems in three dimensions (but not for systems whose spatial dimension is 2 or lower).
It seems more convenient experimentally [ 6 ] to consider mixtures with four components for which one thermodynamic variable (usually the pressure or the volume) is kept fixed. The situation then reduces to the one described for mixtures of three components.
Historically, it was for a long time unclear whether a superconductor undergoes a first- or a second-order phase transition. The question was finally settled in 1982. [ 7 ] If the Ginzburg–Landau parameter κ {\displaystyle \kappa } that distinguishes type-I and type-II superconductors (see also here ) is large enough, vortex fluctuations become important which drive the transition to second order. [ 8 ] The tricritical point lies at roughly κ = 0.76 / 2 {\displaystyle \kappa =0.76/{\sqrt {2}}} , slightly below the value κ = 1 / 2 {\displaystyle \kappa =1/{\sqrt {2}}} where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo computer simulations . [ 9 ]
This condensed matter physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tricritical_point |
A tridentate ligand (or terdentate ligand ) is a ligand that has three atoms that can function as donor atoms in a coordination complex . [ 1 ]
Well-known tridentate ligands include diethylenetriamine with three nitrogen donor atoms, and the iminodiacetate anion which consists of one deprotonated amine nitrogen and a pair of carboxylate groups . [ 1 ]
An octahedrally coordinated atom has six positions around it. Two tridentate ligands may form a complex with such an atom. There are two possible arrangements for such a complex: fac where the coordination is in a triangle on one face of the octahedron, and mer where the coordinating atoms are in an arc around the central atom, with two atoms of the ligand opposite each other. Fac tridentate ligands are termed scorpionate ligands , especially in reference to polypyrazolylborates. [ 2 ]
If the tridentate ligand is not symmetrical, then in the fac complexes in octahedral coordination there are three possible isomers. In the mer complexes there are two enantiomers , mirror images of each other. | https://en.wikipedia.org/wiki/Tridentate_ligand |
Triethyl orthoacetate is the organic compound with the formula CH 3 C(OC 2 H 5 ) 3 . It is the ethyl orthoester of acetic acid . It is a colorless oily liquid.
Triethyl orthoacetate is used in organic synthesis for acetylation . [ 1 ]
It is also used in the Johnson-Claisen rearrangement . [ 2 ] | https://en.wikipedia.org/wiki/Triethyl_orthoacetate |
Triethyl phosphonoacetate is a reagent for organic synthesis used in the Horner-Wadsworth-Emmons reaction (HWE) or the Horner-Emmons modification.
Triethyl phosphonoacetate can be added dropwise to sodium methoxide solution to prepare a phosphonate anion. It has an acidic proton that can easily be abstracted by a weak base. When used in an HWE reaction with a carbonyl the resulting alkene formed is usually the E alkene, and is generated with excellent regioselectivity. [ 1 ]
This article about an ester is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Triethyl_phosphonoacetate |
Triethylene glycol dinitrate ( TEGDN ) is an, ether , nitrated alcohol ester of triethylene glycol . It is used as an energetic plasticizer in explosives and propellants . It is a pale yellow oily liquid. [ 1 ] It is somewhat similar to nitroglycerin .
TEGDN is often used together with trimethylolethane trinitrate (TMETN).
Triethylene glycol dinitrate, diethylene glycol dinitrate , and trimethylolethane trinitrate are being considered as replacements for nitroglycerin in propellants. [ 2 ] | https://en.wikipedia.org/wiki/Triethylene_glycol_dinitrate |
Triethylgallium is the organogallium compound with the formul Ga(C 2 H 5 ) 3 . Also called TEGa , it is a metalorganic source of gallium for metalorganic vapour phase epitaxy (MOVPE) of compound semiconductors . It is a colorless pyrophoric liquid, [ 2 ] typically handled with air-free techniques . It was discovered by Cornell University chemists L. M. Dennis and Winton Patnode in 1931. [ 3 ]
The main routes involve alkylation of gallium trichloride . When this alkylation is effected with ethyl Grignard reagent in ether, the product is the diethyl ether adduct of triethylgallium. The ether is not easily removed. Thus an alternative route involves transmetalation with triethylaluminium according to this simplified equation: [ 4 ]
Triethylgallium readily converts to the air-stable, colorless alkoxide by two routes, oxidation and alcoholysis : [ 4 ]
The sweet odor associated with triethylgallium is due to the alkoxide.
Redistribution reactions occur with gallium trichloride: [ 4 ]
TEGa can be a useful alternative to trimethylgallium in the metalorganic vapour phase epitaxy of compound semiconductors because films grown using TEGa have been shown to have a lower carbon impurity concentration. [ 5 ] | https://en.wikipedia.org/wiki/Triethylgallium |
Triethyloxonium tetrafluoroborate is the organic oxonium compound with the formula [(CH 3 CH 2 ) 3 O] + [BF 4 ] − . It is often called Meerwein's reagent or Meerwein's salt after its discoverer Hans Meerwein . [ 2 ] [ 3 ] Also well known and commercially available is the related trimethyloxonium tetrafluoroborate . The compounds are white solids that dissolve in polar organic solvents. They are strong alkylating agents . Aside from the BF − 4 salt, many related derivatives are available. [ 4 ]
Triethyloxonium tetrafluoroborate is prepared from boron trifluoride , diethyl ether , and epichlorohydrin : [ 5 ]
where the Et stands for ethyl . The trimethyloxonium salt is available from dimethyl ether via an analogous route. [ 6 ] These salts do not have long shelf-lives at room temperature. They degrade by hydrolysis:
The propensity of trialkyloxonium salts for alkyl-exchange can be advantageous. For example, trimethyloxonium tetrafluoroborate, which reacts sluggishly due to its low solubility in most compatible solvents, may be converted in situ to higher alkyl/more soluble oxoniums, thereby accelerating alkylation reactions. [ 7 ]
This reagent is useful for esterification of carboxylic acids under conditions where acid-catalyzed reactions are infeasible: [ 8 ]
The structure of triethyloxonium tetrafluoroborate has not been characterized by X-ray crystallography , but the structure of triethyloxonium hexafluorophosphate has been examined. The measurements confirm that the cation is pyramidal with C–O–C angles in the range 109.4°–115.5°. The average C–O distance is 1.49 Å. [ 9 ]
Triethyloxonium tetrafluoroborate is a very strong alkylating agent, although the hazards are diminished because it is non-volatile. It releases strong acid upon contact with water. The properties of the methyl derivative are similar. | https://en.wikipedia.org/wiki/Triethyloxonium_tetrafluoroborate |
Triethylsilane is the organosilicon compound with the formula (C 2 H 5 ) 3 SiH. It is a trialkylsilane. The Si-H bond is reactive.
It was first discovered by Albert Ladenburg in 1872 among the products of reduction of tetraethyl orthosilicate with sodium and diethylzinc . [ 2 ] He also prepared it by a stepwise reduction via ethoxytriethylsilane and named it silicoheptyl hydride , reflecting the idea of a silicon compound analogous to a seven-carbon hydrocarbon.
This colorless liquid is used in organic synthesis as a reducing agent and as a precursor to silyl ethers . [ 3 ] As one of the simplest trialkylsilanes that is a liquid at room temperature, triethylsilane is often used in studies of hydrosilylation catalysis. [ 4 ] | https://en.wikipedia.org/wiki/Triethylsilane |
Triflic acid , the short name for trifluoromethanesulfonic acid , TFMS , TFSA , HOTf or TfOH , is a sulfonic acid with the chemical formula CF 3 SO 3 H. It is one of the strongest known acids . Triflic acid is mainly used in research as a catalyst for esterification . [ 2 ] [ 3 ] It is a hygroscopic , colorless, slightly viscous liquid and is soluble in polar solvents .
Trifluoromethanesulfonic acid is produced industrially by electrochemical fluorination (ECF) of methanesulfonic acid
The resulting CF 3 SO 2 F is hydrolyzed , and the resulting triflate salt is reprotonated. Alternatively, trifluoromethanesulfonic acid arises by oxidation of trifluoromethylsulfenyl chloride: [ 4 ]
CF 3 SCl + 2 Cl 2 + 3 H 2 O ⟶ CF 3 SO 3 H + 5 HCl {\displaystyle {\ce {CF3SCl + 2 Cl2 + 3 H2O -> CF3SO3H + 5 HCl}}}
Triflic acid is purified by distillation from triflic anhydride . [ 3 ]
Trifluoromethanesulfonic acid was first synthesized in 1954 by Robert Haszeldine and Kidd by the following reaction: [ 5 ]
In the laboratory, triflic acid is useful in protonations because the conjugate base of triflic acid is nonnucleophilic. It is also used as an acidic titrant in nonaqueous acid-base titration because it behaves as a strong acid in many solvents ( acetonitrile , acetic acid , etc.) where common mineral acids (such as HCl or H 2 SO 4 ) are only moderately strong.
With a K a = 5 × 10 14 , p K a = −14.7 ± 2.0 , [ 1 ] triflic acid qualifies as a superacid . It owes many of its useful properties to its great thermal and chemical stability. Both the acid and its conjugate base CF 3 SO − 3 , known as triflate , resist oxidation/reduction reactions, whereas many strong acids are oxidizing, such as perchloric or nitric acid . Further recommending its use, triflic acid does not sulfonate substrates, which can be a problem with sulfuric acid , fluorosulfuric acid , and chlorosulfonic acid . Below is a prototypical sulfonation, which triflic acid does not undergo:
Triflic acid fumes in moist air and forms a stable solid monohydrate, CF 3 SO 3 H·H 2 O, melting point 34 °C.
The triflate ligand is labile, reflecting its low basicity. Trifluoromethanesulfonic acid reacts exothermically with metal carbonates , hydroxides , and oxides. Illustrative is the synthesis of Cu(OTf) 2 . [ 6 ]
Chloride ligands can be converted to the corresponding triflates:
This conversion is conducted in neat HOTf at 100 °C, followed by precipitation of the salt upon the addition of ether.
Triflic acid reacts with acyl halides to give mixed triflate anhydrides, which are strong acylating agents, e.g. in Friedel–Crafts reactions .
Triflic acid catalyzes the reaction of aromatic compounds with sulfonyl chlorides, probably also through the intermediacy of a mixed anhydride of the sulfonic acid.
Triflic acid promotes other Friedel–Crafts-like reactions including the cracking of alkanes and alkylation of alkenes, which are very important to the petroleum industry. These triflic acid derivative catalysts are very effective in isomerizing straight chain or slightly branched hydrocarbons that can increase the octane rating of a particular petroleum-based fuel.
Triflic acid reacts exothermically with alcohols to produce ethers and olefins.
Dehydration gives the acid anhydride , trifluoromethanesulfonic anhydride , (CF 3 SO 2 ) 2 O.
Triflic acid is one of the strongest acids. Contact with skin causes severe burns with delayed tissue destruction. On inhalation it causes fatal spasms, inflammation and edema . [ 7 ]
Like sulfuric acid, triflic acid must be slowly added to polar solvents to prevent thermal runaway . | https://en.wikipedia.org/wiki/Triflic_acid |
Trifluoroacetic acid ( TFA ) is a synthetic organofluorine compound with the chemical formula CF 3 CO 2 H. It is a haloacetic acid , with all three of the acetyl group's hydrogen atoms replaced by fluorine atoms. It is a colorless liquid with a vinegar -like odor. TFA is a stronger acid than acetic acid , having an acid ionisation constant , K a , that is approximately 34,000 times higher, [ 3 ] as the highly electronegative fluorine atoms and consequent electron-withdrawing nature of the trifluoromethyl group weakens the oxygen-hydrogen bond (allowing for greater acidity) and stabilises the anionic conjugate base . TFA is commonly used in organic chemistry for various purposes.
TFA is prepared industrially by the electrofluorination of acetyl chloride or acetic anhydride , followed by hydrolysis of the resulting trifluoroacetyl fluoride: [ 4 ]
Where desired, this compound may be dried by addition of trifluoroacetic anhydride . [ 5 ]
An older route to TFA proceeds via the oxidation of 1,1,1-trifluoro-2,3,3-trichloropropene with potassium permanganate . The trifluorotrichloropropene can be prepared by Swarts fluorination of hexachloropropene . [ 6 ]
TFA is the precursor to many other fluorinated compounds such as trifluoroacetic anhydride , trifluoroperacetic acid , and 2,2,2-trifluoroethanol . [ 4 ] It is a reagent used in organic synthesis because of a combination of convenient properties: volatility, solubility in organic solvents , and its strength as an acid. [ 7 ] TFA is also less oxidizing than sulfuric acid but more readily available in anhydrous form than many other acids. One complication to its use is that TFA forms an azeotrope with water (b. p. 105 °C).
TFA is used as a strong acid to remove protecting groups such as Boc used in organic chemistry and peptide synthesis . [ 8 ] [ 9 ]
At a low concentration, TFA is used as an ion pairing agent in liquid chromatography ( HPLC ) of organic compounds, particularly peptides and small proteins . TFA is a versatile solvent for NMR spectroscopy (for materials stable in acid). It is also used as a calibrant in mass spectrometry . [ 10 ]
TFA is used to produce trifluoroacetate salts. [ 11 ]
Trifluoroacetic acid is a strong acid. [ 12 ] TFA is harmful when inhaled, causes severe skin burns and is toxic for aquatic organisms even at low concentrations.
Skin burns are severe, heal poorly and can be necrotic . Vapour fumes have an LC 50 of 10.01 mg/L, tested on rats over 4 hours. Inhalation symptoms include mucus irritation, coughing, shortness of breath and possible formation of oedemas in the respiratory tract. Exposure damages the kidneys. [ 13 ]
Although trifluoroacetic acid is not produced biologically or abiotically, [ 14 ] it is a metabolic breakdown product of the volatile anesthetic agent halothane . It is also thought to be responsible for halothane-induced hepatitis . [ 15 ] It also may be formed by photooxidation of the commonly used refrigerant 1,1,1,2-tetrafluoroethane (R-134a). [ 16 ] [ 17 ] Moreover, it is formed as an atmospheric degradation product of almost all fourth-generation synthetic refrigerants, also called hydrofluoroolefins (HFO), such as 2,3,3,3-tetrafluoropropene . [ 18 ] [ 19 ]
Trifluoroacetic acid is also formed by the degradation of pesticides that contain a CF 3 group, like Flufenacet . [ 20 ] The German Umweltbundesamt has identified pesticides as the main source of TFA in water in agricultural areas. In November 2024, the Swiss authorities presented an overview of widespread groundwater contamination with TFA. [ 21 ]
Trifluoroacetic acid degrades very slowly in the environment and has been found in increasing amounts as a contaminant in water, soil, food, and the human body. [ 22 ] Median concentrations of a few micrograms per liter have been found in beer and tea. [ 23 ] Seawater can contain about 200 ng of TFA per liter. [ 24 ] [ 25 ] [ 26 ] Biotransformation by decarboxylation to fluoroform has been discussed. [ 27 ] In October 2024, a publication proposed classifying TFA as a planetary boundary threat, similar to how CFCs are treated. [ 28 ]
Trifluoroacetic acid is mildly phytotoxic . [ 29 ] In July 2024, the German Chemical Agency submitted a proposal to the European Chemicals Agency (ECHA) to link trifluoroacetic acid and its salts to reproductive toxicity and as suspected of damaging fertility. [ 30 ] | https://en.wikipedia.org/wiki/Trifluoroacetic_acid |
This page provides supplementary chemical data on trifluoroiodomethane .
The handling of this chemical may incur notable safety precautions. It is highly recommend that you seek the Material Safety Datasheet ( MSDS ) for this chemical from a reliable source such as SIRI , and follow its directions. | https://en.wikipedia.org/wiki/Trifluoroiodomethane_(data_page) |
Trifluoromethanesulfonic anhydride , also known as triflic anhydride, is the chemical compound with the formula (CF 3 SO 2 ) 2 O. It is the acid anhydride derived from triflic acid . This compound is a strong electrophile , useful for introducing the triflyl group , CF 3 SO 2 . Abbreviated Tf 2 O , triflic anhydride is the acid anhydride of the superacid triflic acid , CF 3 SO 2 OH. [ 2 ] [ 3 ]
Triflic anhydride is prepared by dehydration of triflic acid using P 4 O 10 . [ 2 ]
Triflic anhydride is useful for converting ketones into enol triflates. [ 4 ]
In a representative application, is used to convert an imine into a NTf group. [ 5 ] It will convert phenols into a triflic ester, which enables cleavage of the C-O bond. [ 6 ] [ 7 ]
The typical impurity in triflic anhydride is triflic acid, which is also a colorless liquid. Samples of triflic anhydride can be assayed by 19 F NMR spectroscopy: −72.6 ppm [ 8 ] vs. −77.3 for TfOH (std CFCl 3 ).
It is an aggressive electrophile and readily hydrolyzes to the strong acid triflic acid. It is very harmful to skin and eyes. [ 9 ] | https://en.wikipedia.org/wiki/Trifluoromethanesulfonic_anhydride |
Trifluoromethanesulfonyl azide or triflyl azide CF 3 SO 2 N 3 is an organic azide used as a reagent in organic synthesis . [ 2 ]
Trifluoromethanesulfonyl azide is prepared by treating trifluoromethanesulfonic anhydride with sodium azide , traditionally in dichloromethane . [ 1 ] However, the use of dichloromethane is avoided since it can generate highly explosive azido-chloromethane and diazidomethane. The reaction may also instead be conducted in toluene , [ 3 ] acetonitrile , or pyridine . [ 4 ]
An alternative route starts from imidazole-1-sulfonyl azide . [ 5 ]
Trifluoromethanesulfonyl azide generally converts amines to azides. | https://en.wikipedia.org/wiki/Trifluoromethanesulfonyl_azide |
The trifluoromethyl group is a functional group that has the formula −CF 3 . The naming of is group is derived from the methyl group (which has the formula −CH 3 ), by replacing each hydrogen atom by a fluorine atom. Some common examples are trifluoromethane H−CF 3 , 1,1,1-trifluoroethane H 3 C−CF 3 , and hexafluoroacetone F 3 C−CO−CF 3 . Compounds with this group are a subclass of the organofluorines .
The trifluoromethyl group has a significant electronegativity that is often described as being intermediate between the electronegativities of fluorine and chlorine. [ 1 ] For this reason, trifluoromethyl-substituted compounds are often strong acids, such as trifluoromethanesulfonic acid and trifluoroacetic acid . Conversely, the trifluoromethyl group lowers the basicity of compounds like trifluoroethanol .
The trifluoromethyl group occurs in certain pharmaceuticals, drugs, and abiotically synthesized natural fluorocarbon based compounds. The medicinal use of the trifloromethyl group dates from 1928, although research became more intense in the mid-1940s. [ 2 ] The trifluoromethyl group is often used as a bioisostere to create derivatives by replacing a chloride or a methyl group. This can be used to adjust the steric and electronic properties of a lead compound , or to protect a reactive methyl group from metabolic oxidation. Some notable drugs containing trifluoromethyl groups include efavirenz (Sustiva), an HIV reverse transcriptase inhibitor; fluoxetine (Prozac), an antidepressant; and celecoxib (Celebrex), a nonsteroidal anti-inflammatory drug .
Sulfoxaflor is used as a systemic insecticide. Trifluralin , as with several dinitritroaniline herbicides, is a trifluoromethyl herbicide. Fluazifop is another, a phenoxy herbicide .
The trifluoromethyl group can also be added to change the solubility of molecules containing other groups of interest.
Various methods exist to introduce this functionality. Carboxylic acids can be converted to trifluoromethyl groups by treatment with sulfur tetrafluoride and trihalomethyl compounds, particularly trifluoromethyl ethers and trifluoromethyl aromatics, are converted into trifluoromethyl compounds by treatment with antimony trifluoride / antimony pentachloride (the Swarts reaction ). Another route to trifluoromethyl aromatics is the reaction of aryl iodides with trifluoromethyl copper . Finally, trifluoromethyl carbonyls can be prepared by reaction of aldehydes and esters with Ruppert's reagent . [ 3 ] | https://en.wikipedia.org/wiki/Trifluoromethyl_group |
Trifluoromethylation in organic chemistry describes any organic reaction that introduces a trifluoromethyl group in an organic compound. [ 1 ] [ 2 ] [ 3 ] [ 4 ] Trifluoromethylated compounds are of some importance in pharmaceutical industry and agrochemicals . Several notable pharmaceutical compounds have a trifluoromethyl group incorporated: fluoxetine , mefloquine , leflunomide , nulitamide , dutasteride , bicalutamide , aprepitant , celecoxib , fipronil , fluazinam , penthiopyrad , picoxystrobin , fluridone , norflurazon , sorafenib , and triflurazin . A relevant agrochemical is trifluralin . The development of synthetic methods for adding trifluoromethyl groups to chemical compounds is actively pursued in academic research.
The first to investigate trifluoromethyl groups in relationship to biological activity was F. Lehmann in 1927. [ 5 ] An early review appeared in 1958. [ 6 ] An early synthetic method was developed by Frédéric Swarts in 1892, [ 7 ] based on antimony fluoride . In this reaction benzotrichloride was reacted with SbF 3 to form PhCF 2 Cl and
PhCF 3 . In the 1930s Kinetic Chemicals and IG Farben replaced SbF 3 with HF .
The McLoughlin-Thrower reaction (1968) is an early coupling reaction using iodofluoroalkanes, iodoaromatic compounds and copper . [ 8 ] In 1969 Kobayashi & Kumadaki adapted their protocol for trifluoromethylations. [ 9 ] [ 10 ]
Preparation of the trifluoromethyltrimethylsilane was reported by Ingo Ruppert in 1984. [ 11 ] In 1989, Prakash and Olah first reported activation of TMSCF 3 by fluoride to perform nucleophilic trifluoromethylation of carbonyl compounds. [ 12 ] In the same year, Stahly described similar reactions for the synthesis of trifluoromethylated phenols and anilines. [ 13 ] Since then TMSCF 3 has been widely used as a nucleophilic trifluoromethylating agent. [ 14 ] [ 15 ]
An example is the trifluoromethylation of cyclohexanone in THF using tetrabutylammonium fluoride . [ 16 ]
trifluoromethyltrimethylsilane [ 16 ]
The substrates can be aryl halides. [ 17 ] [ 18 ] Potassium (trifluoromethyl)trimethoxyborate for this purpose has been synthesised from B(OMe) 3 , CF 3 SiMe 3 and KF. [ 19 ] Aryl functionalization by C-H activation has also been reported. [ 20 ] [ 21 ]
Sodium trifluoroacetate as a reagent for trifluoromethylations was introduced by Matsui in 1981. In the original scope the substrate was an aromatic halide and the metal salt copper(I)iodide. [ 22 ] [ 23 ]
Fluoroform (CF 3 H) has been employed as a trifluoromethylation reagent for aldehydes in combination with a strong base. [ 24 ]
Trifluoroiodomethane is a reagent in aromatic coupling reactions. It has also been used with enones , for example with chalcone , a reaction catalysed by diethyl zinc and Wilkinson's catalyst : [ 25 ]
Trifluoromethyl sulfone (PhSO 2 CF 3 ) and trifluoromethyl sulfoxide (PhSOCF 3 ) can be used for trifluoromethylations of electrophiles [ 26 ]
Trifluoromethanesulfonyl chloride (or triflyl chloride, CF3SO2Cl) can be used in a highly efficient method to introduce a trifluoromethyl group to aromatic and heteroaromatic systems, including known pharmaceuticals such as Lipitor . The chemistry is general and mild, and uses a photoredox catalyst and a light source at room temperature. [ 27 ]
Sodium trifluoromethanesulfinate (CF 3 SO 2 Na) as a trifluoromethylation reagent was introduced by Langlois in 1991. [ 28 ] The reaction requires t-butyl hydroperoxide and generally a metal and proceeds through a radical mechanism. The reagent has been applied with heterocyclic substrates [ 29 ]
Umemoto reagents are (trifluoromethyl)dibenzoheterocyclic salts, such as 5-(trifluoromethyl)dibenzothiophenium triflate and 5-(trifluoromethyl)dibenzothiophenium tetrafluoroborate. [ 30 ] [ 31 ]
Many CF 3 -containing metal complexes have been prepared, and some are useful for trifluoromethylation. The most obvious reagent is CF 3 Li, which can be generated by lithium-iodide exchange. This compound is however unstable even at low temperatures. It degrades to lithium fluoride and difluorocarbene . Trifluoromethyl copper(I) reagents are more useful. These reagents are generated in situ by reaction of CF 3 I with copper powder in polar solvents. [ 32 ] Hg(CF 3 ) 2 , prepared by decarboxylation of the trifluoroacetate, has proven useful for the trifluoromethylation of other metals, [ 33 ] although for low-temperature reactions it may prove useful to transmetallate to bis(trifluoromethyl)cadmium. [ 34 ]
In coupling reactions between aromatic compounds and metal-trifluoromethyl complexes the metal is usually copper, Pd and Ni are less prominent. [ 1 ] The reactions are stoichiometric or catalytic. In the McLoughlin-Thrower reaction (1962) iodobenzene reacts with trifluoroiodomethane (CF 3 I) and copper powder in dimethylformamide at 150 °C to trifluorotoluene . The intermediate in this reaction type is a perfluoromethyl-metal complex.
A palladium acetate catalysed reaction described in 1982 used zinc powder with the main intermediate believed to be CF 3 ZnI with Pd(0) is the active catalyst. [ 35 ] [ 36 ] The first copper catalysed coupling was reported in 2009 and based on an iodoarene , a trifluoromethylsilane , copper iodide and 1,10-phenanthroline . [ 37 ] Variations include another CF 3 donor potassium (trifluoromethyl)trimethoxyborate , [ 38 ] the use of aryl boronic acids [ 39 ] [ 40 ] or the use of a trifluoromethyl sulfonium salt [ 41 ] or the use of a trifluoromethylcopper(I) phenanthroline complex. [ 42 ] A catalytic palladium catalysed reaction was reported in 2010 using aryl halides , (trifluoromethyl)triethylsilane and allylpalladium chloride dimer [ 43 ]
trifluoromethylation Oishi 2009 [ 37 ]
In radical trifluoromethylation the active species is the trifluoromethyl free radical . [ 44 ] Reagents such as bromotrifluoromethane and haloform have been used for this purpose [ 45 ] [ 46 ] [ 47 ] but in response to the Montreal Protocol alternatives such as trifluoroiodomethane have been developed as replacement. [ 48 ] [ 49 ] One particular combination is CF 3 I / triethylborane [ 50 ] [ 51 ] Other reagents that generate the CF 3 radical are sodium trifluoromethanesulfinate and bis(trifluoroacetyl) peroxide .
The base is 2,6-lutidine [ 50 ]
In the CF 3 radical the fluorine atom is an electron-withdrawing group via the inductive effect but also a weak pi donor through interaction of the fluorine lone pair with the radical center's SOMO . Compared to the methyl radical the CF 3 radical is pyramidal (angle 107.8 °C ) with a large inversion barrier , electrophilic and also more reactive. In reaction with styrene it is 440 times more reactive. [ 52 ] An early report (1949) describes the photochemical reaction of iodotrifluoromethane with ethylene to 3-iodo-1,1,1-trifluoropropane. [ 53 ] Reagents that have been reported for the direct trifluoromethylation of arenes are CF 3 I, CF 3 Br (thermal or photochemical), silver trifluoroacetate /TiO 2 (photochemical) and sodium trifluoromethanesulfinate /Cu(OSO 2 CF 3 ) 2 /tBuOOH.
In nucleophilic trifluoromethylation the active species is the CF 3 − anion. [ 54 ] It was, however, widely believed that the trifluoromethyl anion is a transient species and thus cannot be isolated or observed in the condensed phase. Contrary to the popular belief, the CF 3 anion, with [K(18-crown-6)] + as a countercation, was produced and characterized by Prakash and coworkers. [ 55 ] The challenges associated with observation of CF 3 anion are alluded to its strong basic nature and its tendency to form pentacoordinated silicon species, such as [Me 3 Si(CF 3 ) 2 ] − or [Me 3 Si(F)(CF 3 )] − .
The reactivity of fluoroform in combination with a strong base such as t-BuOK with carbonyl compounds in DMF is an example. [ 54 ] Here CF 3 − and DMF form an hemiaminolate adduct ([Me 2 NCH(O)CF 3 ]K). [ 24 ] [ 56 ] [ 57 ] [ 58 ]
The intermediate is CF 3 Cu [ 59 ]
In electrophilic trifluoromethylation the active trifluoromethyl donor group carries a positive charge. [ 60 ] [ 61 ] Production of an CF 3 + cation has been described as "extremely hard" [ 62 ] The first relevant reagent, a diaryl(trifluoromethyl) sulfonium salt (Ar 2 S + CF 3 SbF 6 − ) was developed in 1984 by reaction of an aryltrifluoromethyl sulfoxide 1 with SF 3 + SbF 6 − followed by reaction with an electron-rich arene. [ 63 ] The reagent was used in trifluoromethylation of a thiophenolate. S-(trifluoromethyl)dibenzothiophenium tetrafluoroborate is a commercially available and known trifluoromethylation reagent based on the same principle first documented in 1990. [ 64 ] [ 65 ] In this type of compound sulfur has been replaced by oxygen , selenium and tellurium . Examples of substrates that have been investigated are pyridine , aniline , triphenylphosphine and the lithium salt of phenylacetylene .
Another group of trifluoromethyl donors are hypervalent iodine(III) [ 66 ] –CF 3 reagents for example 3,3-dimethyl-1-(trifluoromethyl)-1,2-benziodoxole . [ 67 ] [ 68 ] [ 69 ] [ 70 ] Some of these are known as Togni reagents, such as Togni reagent II . Substrates are thiols, alcohols, phosphines, (hetero) arenes, [ 71 ] unactivated olefins [ 72 ] and unsaturated carboxylic acids. [ 73 ]
The reaction mechanism of electrophilic trifluoromethylations has been described as controversial with polar substitution or single electron transfer as likely candidates. [ 62 ]
In asymmetric trifluoromethylation the trifluoromethyl group is added to the substrate in an enantioselective way. [ 74 ] [ 75 ] Ruppert's reagent has been used for this purpose in an asymmetric induction approach to functionalise chiral amino acid derivates, [ 76 ] saccharides , [ 77 ] and steroids .
Because Ruppert's reagent requires a tetraalkylammonium fluoride, chiral ammonium fluorides have been employed in asymmetric catalysis . [ 78 ] [ 79 ] In the field of electrophilic trifluoromethylation an early contribution involved reaction of a metal enolate with a trifluoromethyl chalcogen salt in presence of a chiral boron catalyst. [ 80 ]
More recent examples of highly enantioselective methods for the α-trifluoromethylation of carbonyls are available through enamine catalysis of aldehydes ( photoredox [ 81 ] or iodonium [ 82 ] ), copper catalysis of β-ketoesters, [ 83 ] and radical addition to zirconium enolates. [ 84 ] | https://en.wikipedia.org/wiki/Trifluoromethylation |
Trifluoromethylsulfur pentafluoride , CF 3 SF 5 , is a rarely used industrial greenhouse gas . [ 1 ] It was first identified in the atmosphere in 2000. [ 2 ] Trifluoromethylsulfur pentafluoride is considered to be one of the several "super-greenhouse gases".
The chemistry of this compound is similar to that of sulfur hexafluoride (SF 6 ). [ 3 ]
On a per molecule basis, it is considered to be the most potent greenhouse gas present in Earth's atmosphere, having a global warming potential of about 18,000 times that of carbon dioxide . [ 4 ] The chemical is predicted to have a lifetime of 800 years in the atmosphere. [ 5 ] However, the current concentration of trifluoromethylsulfur pentafluoride remains at a level that is unlikely to measurably contribute to global warming . [ 4 ] The presence of the gas in the atmosphere is attributed to anthropogenic sources, possibly a by-product of the manufacture of fluorochemicals , originating from reactions of SF 6 with fluoropolymers used in electronic devices and microchips, or the formation can be associated with high voltage equipment created from SF 6 (a breakdown product of high voltage equipment) reacting with CF 3 to form the CF 3 SF 5 molecule. [ 4 ] | https://en.wikipedia.org/wiki/Trifluoromethylsulfur_pentafluoride |
Trifluoromethyltrimethylsilane (known as Ruppert-Prakash reagent , TMSCF 3 ) is an organosilicon compound with the formula CF 3 Si(CH 3 ) 3 . It is a colorless liquid. The compound is a reagent used in organic chemistry for the introduction of the trifluoromethyl group . The compound was first prepared in 1984 by Ingo Ruppert [ 1 ] and further developed as a reagent by G. K. Surya Prakash . [ 2 ] [ 3 ]
The reagent is prepared from trimethylsilyl chloride and bromotrifluoromethane in the presence of a phosphorus(III) reagent that serves as a halogen acceptor. [ 4 ]
In the presence of a metal salt (M + X − ), the reagent reacts with aldehydes and ketones to give a trimethylsilyl ether, the net product of insertion of the carbonyl into the Si-CF 3 bond. Hydrolysis gives trifluoromethyl methanols . The reagent also converts esters to trifluoromethyl ketones . A typical initiator is a soluble fluoride-containing species such as tetrabutylammonium fluoride ; however, simple alkoxides such as KOtBu are also effective. [ 5 ] The mechanism begins by generation of Si(CH 3 ) 3 X and a highly reactive [CF 3 ] − (trifluoromethide) intermediate. The [CF 3 ] − attacks the carbonyl to generate an alkoxide anion. The alkoxide is silylated by the reagent to give the overall addition product, plus [CF 3 ] − , thus propagating an anionic chain reaction. The reagent competes with the carbonyl for the reactive intermediate, rapidly sequestering [CF 3 ] − in a reversibly-generated -ate complex [(CF 3 ) 2 Si(CH 3 ) 3 ] − . This -ate complex is unable to react directly with the carbonyl, resulting in powerful inhibition of the chain reaction by the reagent. This inhibitory process is common to all anion-initiated reactions of the reagent, with the identity of the counter-cation (M + ) playing a major role in controlling the overall rate. [ 6 ]
The reagent has largely supplanted trifluoromethyllithium, which is not isolable and rapidly decomposes to yield lithium fluoride and difluorocarbene . | https://en.wikipedia.org/wiki/Trifluoromethyltrimethylsilane |
The trifluorooxonium cation is a hypothetical positively charged polyatomic ion with chemical formula O F + 3 . It is structurally equivalent to the hydronium ion where the hydrogen atoms surrounding the central oxygen atom have been replaced by fluorine , and is isoelectronic with nitrogen trifluoride . This cation would be an example of oxygen in the unprecedented +4 oxidation state .
The OF + 3 cation was shown to be vibrationally stable at all levels of theory applied (HF, MP2, CCSD(T)). OF + 3 was proposed to possess a pyramidal structure with an O–F bond length of 1.395 Å and F–O–F bond angles of 104.2° (CCSD(T) level of theory). The F + detachment energy of the OF + 3 cation was calculated to be +110.1 kcal mol −1 . However, the low-temperature reaction of F 2 , OF 2 and AsF 5 under UV irradiation, besides unreacted starting materials only yielded the dioxygenyl salt [O 2 ] + [AsF 6 ] − . The oxidation of OF 2 with KrF + salts also failed to produce evidence for the title cation. [ 1 ]
The formation of the hypothetical salt [OF 3 ] + [AsF 6 ] − was calculated to be about thermoneutral, but slightly unfavorable with OF 2 (g) + F 2 (g) + AsF 5 (g) → [OF 3 ] + [AsF 6 ] − (s) = +10.5 kcal mol −1 .
This article about theoretical chemistry is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Trifluorooxonium |
Trifluralin is a common pre-emergent selective herbicide , a dinitroaniline . With about 14 million pounds (6,400 t) used in the United States in 2001, [ 3 ] and 3–7 million pounds (1,400–3,200 t) in 2012, [ 4 ] it is one of the most widely used herbicides. Trifluralin is also used in Australia, [ 5 ] New Zealand, [ 6 ] Brazil and previously in the EU. Introduced in 1964, Trifluralin was the first organofluorine compound used as an agrochemical. [ 7 ]
Trifluralin is generally applied to the soil to control annual grass and broadleaf weed species. It inhibits root development by interrupting mitosis and controls weeds as they germinate. [ 5 ] [ 8 ] Trifluralin moves very little inside the plant, remaining in the roots. [ 9 ]
Selective herbicides were unavailable in the 1950s to protect soybean and cotton ( 2,4-DNP could have been used but had to be exactingly applied lest it destroy the crops), so Lilly Research Laboratories screened ~2000 compounds from 1958 to 1980 blindly looking for a result. Trifluralin was initially thought a failure, yet the plots stayed free of weeds weeks later. [ 7 ] Application by incorporation into the top soil instead was eight times more potent. [ 10 ] Pre-plant soil incorporation was a new technique at the time. It is unclear why trifluralin's exotic 4- trifluoromethyl was tested so early (1960), before more common candidates such as fluoro , bromo , or iodo . [ 7 ]
By 1968, trifluralin was internationally available, including Australia and New Zealand, [ 6 ] and trifluralin was the 5th most used herbicide in the US, at 22,960,000 pounds (10,410 t) by 1974. [ 11 ] The efficient post-emergent acetolactate synthase and ACCase inhibitors developed in 1980s significantly replaced trifluralin, though the trifluralin market has resurged, with weeds developing resistance to the post-emergents, and with rising no-till or low-till farming techniques. [ 12 ]
Trifluralin was introduced to Latin America. It is used on sugarcane and soybean in Brazil, where 4.16 million pounds (1,890 t) was used in 2019. [ 12 ]
Related compounds show similar herbicidal properties. In a study of 16, trifluoromethyl (as trifluralin is) compounds proved more active pre-emergence, and methyl compounds more active post-emergence. Replacing trifluralin's two propyl groups (with ethyl, allyl or butyl) yielded lower pre-emergent activity in all cases; post-emergence activity was highest in ethyl, allyl combination analogs. [ 13 ]
Nitralin replaces the trifluroromethyl group with a methylsulfonyl . Benfluralin replaces the propyl -propyl groups with ethyl - butyl . Profluralin replaces one propyl group with cyclopropylmethyl. Profluralin and nitralin are mostly obsolete, but benfluralin is commercially used, though less so than trifluralin.
Trifluralin, or other dinitroanilines, inhibits microtubule formation, [ 5 ] by binding to tubulin proteins. Tubulin polymerises into microtubules, which make up the cellular cytoskeleton . Trifluralin binds to tubulin, and this misshapen 'herbicide complex' is incorporated into the growing microtubule, blocking further tubulin binding, and halting cell-division. It also depolymerises (splits) microtubules. [ 12 ]
Dinitroanilines hit microtubules in plants and protists, but not animals, nor fungi, nor carrots, whose microtubules, even in purified form in laboratory work, are unaffected. [ 12 ]
Due to the low solubility, high soil-adhesion and high volatility, dinitroaniline herbicides are absorbed into plants primarily via gaseous vapour. [ 12 ]
Resistance, where evolved, can be through mutated α- or β-tubulin, particularly common in protists . This resistance is especially hard to evolve for weeds to tubulin disrupting herbicides because both α-tubulin and β-tubulin must mutate, as imbalance between their expressions is potentially lethal. Non-target-site resistance is usually though increased metabolism of trifluralin. Mobility-related mechanisms are not effective as minimal movement in the weed is needed to prevent germination. [ 12 ]
Resistance has been shown to devolve under repeated application of prosulfocarb on lolium rigidum (ryegrass). Supposedly, the mechanism of prosulfocarb-resistance is inverse to trifluralin resistance, requiring lower metabolism of herbicide, rather than greater. Therefore, when growing resistance for with one mechanism, the weeds undo their resistance to the other. [ 14 ] Some resistance mechanisms impose severe fitness cost on weeds, such as much reduced growth rate. [ 12 ] Resistance has also been reduced experimentally by applying the organo-phosphate insecticide phorate , which reduces plants' production of the enzyme P450s , thus reducing metabolism of trifluralin in-plant. [ 15 ]
Trifluralin is a Group D resistance class , (Aus), [ 16 ] K1 or 3. (global or numeric) [ 17 ] [ 18 ] Other Group D herbicides will experience resistance near identically.
Wheat and triticale, if affected by trifluralin, display reduced root extension, increased number of seminal roots, increased root diameter and decreased root dry weight. [ 19 ]
Trifluralin has been banned in the European Union since 20 March 2008, primarily due to high toxicity to aquatic life. [ 20 ] Specifically, due to aquatic risk, the toxicity of trifluralin's metabolites to sediment-organisms, and potential consumer exposure for non-cereal crops. They also had concerns over the aquatic toxicity, "high" potential for bioaccumulation, "high persistence in soil" and a potential for long range movement in the wind. [ 21 ]
The United Kingdom banned it under the same legislation. [ 22 ] With IPU banned at the same time, few options were left for farmers to control black-grass . [ 23 ]
Trifluralin is on the United States Environmental Protection Agency list of Hazardous Air Pollutants as a regulated substance under the Clean Air Act . [ 24 ]
Trifluralin is typically sold as emulsifiable concentrate [ 16 ] or granules. [ 2 ] Application rates vary, such as 0.8-3.0 L of 480 g/L formulation per hectare, typically diluted with water, [ 5 ] and other compatible herbicides, e.g. isoproturon , [ 2 ] to be sprayed in one go.
Trifluralin must be incorporated into soil within 24 hours of sowing, or in some cases sooner. Various methods achieve this; most involve machinery set to 5-13 cm deep. [ 5 ] This is to minimise volatilisation losses from trifluralin's relatively high vapour pressure. [ 25 ] Selectivity is possible even on susceptible crops, by sowing below the herbicide band, and shallower germinating weeds will be controlled. [ 5 ] Stubble-cover reduces effectiveness, but a greater water rate offsets the effect. A test saw 53% control of ryegrass at 30 L/Ha become 78% control at 150 L/Ha. Droplet size did not affect the results, and a similar effect was seen with pyroxasulfone , despite large differences in adsorption and solubility. [ 26 ]
Trifluralin breaks down into many products as it degrades, ultimately being incorporated into soil-bound residues or converted to carbon dioxide (mineralized). Among the more unusual behaviors of trifluralin is inactivation in wet soils. This has been linked to transformation of the herbicide by reduced soil minerals, which in turn had been previously reduced by soil microorganisms using them as electron acceptors in the absence of oxygen . This environmental degradation process has been reported for many structurally related herbicides ( dinitroanilines ) as well as a variety of explosives such as TNT and picric acid . [ 27 ]
Trifluralin has a long half-life in soil of ~180 days, but it is accepted at high application rates because of its low soil mobility and high volatility. [ 7 ] It is extremely resistant to leaching, and shows little lateral soil movement. [ 2 ] Repeated annual application shows steady and continuous decline in soil and does not accumulate, even applied well in excess of recommended rates. [ 28 ]
Ultraviolet light can cause degradation. Trifluralin is stable to hydrolysis. [ 2 ]
Trifluralin is safe for mammals and chickens, even in large amounts. [ 29 ] Mammals eliminate 85% after oral consumption within 72 hours. It is toxic to fish though: LC 50 for rainbow trout is 10-40 μg/L. [ 2 ] Metabolism involves the thyroid ; heavy and continuous exposure in rats can stress it via overstimulation. [ 30 ]
There is discussion of trifluralin being carcinogenic . Some studies have shown links, such as a 1986 study of three non-hodgkin lymphoma cases. A later, larger study found no significant relation. A review study examined trifluralin against kinds of cancer , finding no link except to colon cancer, which was found in only one studied cohort. Research on humans remains unconvincing, but EPA animal toxicity data "supports the possible carcinogenicity" of trifluralin. [ 31 ] No association exists with lung cancer . [ 32 ] Trifluralin exposure can reduce cell apoptosis . [ 33 ]
Trifluralin on mammalian ovaries (tested in mice, at 150 mg/kg/day) showed no effect on oocyte quality, but may induce a stress response in ovarian somatic cells . Fertility was unaffected. Levels of pRb stayed unchanged, though trifluralin raised levels of p53 , a tumor suppressing gene, by 2.5 times. The additional p53 appeared not to increase rates of apoptosis. [ 34 ]
Due to trifluralin's high vapour pressure, food residue is reduced in processing, especially in high temperature processes, such as in the mashing of beer . [ 35 ]
Trifluralin can be used as ointment to treat Leishmaniasis welts on the skin. It, and other dinitroanilines , are tubulin -binding agents with selective antileishmanial properties, leishmania being the parasite causing the disease, which killed 60,000 people in 2001. Research into expanding's trifluralin's medical use is stymied by its low water solubility and easy sublimation. Trifluralin analogues have been tried, and some show greater efficacy than miltefosine ; all trifluralin analogues have the benefits of being non- hemolytic and lower cell toxicity. [ 36 ]
Trifluralin also has anti- malarial properties and accumulates in parasite-infected erythrocytes , though low solubility makes effective administration of trifluralin difficult. [ 37 ] Treatment of Toxoplasma gondii and cryptosporidiosis is effective but limited due to solubility. [ 38 ] [ 39 ] Liposome -administered trifluralin has been used to treat leishmania in dogs successfully. [ 40 ]
Commercial formulations have included trifluralin mixtures with: linuron , napropamide , metribuzin , clomazone , tebutam , bromoxynil and ioxynil , isoproturon , terbutryn , trietazine , neburon and isoxaben . [ 2 ]
Crops trifluralin is used in include: Wheat, barley, cotton, triticale, rye, sunflowers, sugar cane, peas, canola, safflower, peanuts, tobacco, pigeon peas, lupins, lucerne, linseed, legume seed, strawberry, lentils, faba beans, chickpeas, cowpeas, lablab, mung beans, borlotti beans, red beans, adzuki beans, [ 5 ] citrus fruit, lettuce, capsicums, tomatoes, artichokes, onions, garlic, brassicas, sugar beet, parsnips, carrots or soya. [ 2 ] Trifluralin has had a huge market to control black-grass , but the weed was not listed on the label as being susceptible. [ 41 ] | https://en.wikipedia.org/wiki/Trifluralin |
In organic chemistry , the triflyl group ( systematic name : trifluoromethanesulfonyl group ) is a functional group with the formula R−SO 2 CF 3 and structure R−S(=O) 2 −CF 3 . The triflyl group is often represented by –Tf .
The related triflate group (trifluoromethanesulfonate) has the formula R−OSO 2 CF 3 , and is represented by –OTf. [ 1 ]
This organic chemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Triflyl_group |
In computer vision , the trifocal tensor (also tritensor ) is a 3×3×3 array of numbers (i.e., a tensor ) that incorporates all projective geometric relationships among three views. It relates the coordinates of corresponding points or lines in three views, being independent of the scene structure and depending only on the relative motion (i.e., pose ) among the three views and their intrinsic calibration parameters. Hence, the trifocal tensor can be considered as the generalization of the fundamental matrix in three views. It is noted that despite the tensor being made up of 27 elements, only 18 of them are actually independent.
There is also a so-called calibrated trifocal tensor , which relates the coordinates of points and lines in three views given their intrinsic parameters and encodes the relative pose of the cameras up to global scale, totalling 11 independent elements or degrees of freedom. The reduced degrees of freedom allow for fewer correspondences to fit the model, at the cost of increased nonlinearity. [ 1 ]
The tensor can also be seen as a collection of three rank-two 3 x 3 matrices T 1 , T 2 , T 3 {\displaystyle {\mathbf {T} }_{1},\;{\mathbf {T} }_{2},\;{\mathbf {T} }_{3}} known as its correlation slices . Assuming that the projection matrices of three views are P = [ I | 0 ] {\displaystyle {\mathbf {P} }=[{\mathbf {I} }\;|\;{\mathbf {0} }]} , P ′ = [ A | a 4 ] {\displaystyle {\mathbf {P} }'=[{\mathbf {A} }\;|\;{\mathbf {a} }_{4}]} and P ″ = [ B | b 4 ] {\displaystyle {\mathbf {P} ''}=[{\mathbf {B} }\;|\;{\mathbf {b} }_{4}]} , the correlation slices of the corresponding tensor can be expressed in closed form as T i = a i b 4 t − a 4 b i t , i = 1 … 3 {\displaystyle {\mathbf {T} }_{i}={\mathbf {a} }_{i}{\mathbf {b} }_{4}^{t}-{\mathbf {a} }_{4}{\mathbf {b} }_{i}^{t},\;i=1\ldots 3} , where a i , b i {\displaystyle {\mathbf {a} }_{i},\;{\mathbf {b} }_{i}} are respectively the i th columns of the camera matrices. In practice, however, the tensor is estimated from point and line matches across the three views.
One of the most important properties of the trifocal tensor is that it gives rise to linear relationships between lines and points in three images. More specifically, for triplets of corresponding points x ↔ x ′ ↔ x ″ {\displaystyle {\mathbf {x} }\;\leftrightarrow \;{\mathbf {x} }'\;\leftrightarrow \;{\mathbf {x} }''} and any corresponding lines l ↔ l ′ ↔ l ″ {\displaystyle {\mathbf {l} }\;\leftrightarrow \;{\mathbf {l} }'\;\leftrightarrow \;{\mathbf {l} }''} through them, the following trilinear constraints hold:
where [ ⋅ ] × {\displaystyle [\cdot ]_{\times }} denotes the skew-symmetric cross product matrix .
Given the trifocal tensor of three views and a pair of matched points in two views, it is possible to determine the location of the point in the third view without any further information. This is known as point transfer and a similar result holds for lines and conics. For general curves, the transfer can be realized through a local differential curve model of osculating circles (i.e., curvature), which can then be transferred as conics. [ 2 ] The transfer of third-order models reflecting space torsion using calibrated trifocal tensors have been studied, [ 3 ] but remains an open problem for uncalibrated trifocal tensors.
The classical case is 6 point correspondences [ 4 ] [ 5 ] giving 3 solutions.
The case estimating the trifocal tensor from 9 line correspondences has only recently been solved. [ 6 ]
Estimating the calibrated trifocal tensor has been cited as notoriously difficult, and requires 4 point correspondences. [ 7 ]
The case of using only three point correspondences has recently been solved, where the points are attributed with tangent directions or incident lines; with only two of the points having incident lines, this is a minimal problem of degree 312 (so there can be at most 312 solutions) and is relevant for the case of general curves (whose points have tangents), or feature points with attributed directions (such as SIFT directions). [ 8 ] The same technique solved the mixed case of three point correspondences and one line correspondence, which has also been shown to be minimal with degree 216. | https://en.wikipedia.org/wiki/Trifocal_tensor |
A trifunctional antibody is a monoclonal antibody with binding sites for two different antigens, typically CD3 and a tumor antigen , making it a type of bispecific monoclonal antibody . In addition, its intact Fc-part can bind to an Fc receptor on accessory cells like conventional monospecific antibodies. The net effect is that this type of drug links T cells (via CD3) and monocytes / macrophages , natural killer cells , dendritic cells or other Fc receptor expressing cells to the tumor cells, leading to their destruction. [ 1 ]
At an equivalent dose a trifunctional antibody is more potent (more than 1,000-fold) in eliminating tumor cells than conventional antibodies. [ 2 ] These drugs evoke the removal of tumor cells by means of (i) antibody-dependent cell-mediated cytoxicity , a process also described for conventional antibodies and more importantly by (ii) polyclonal cytotoxic T cell responses with emphasis on CD8 T cells. These trifunctional antibodies also elicit individual anti-tumor immune responses in cancer patients treated with e.g. catumaxomab; i.e. autologous antibodies as well as CD4 and CD8 T cells directed against the tumor were detected. [ 3 ] [ 4 ] Furthermore, putative cancer stem cells from malignant ascites fluid were eliminated due to catumaxomab treatment. [ 5 ]
Catumaxomab, was the first to be approved for clinical use (in 2009 for the treatment of malignant ascites in cancer patients).
Examples include catumaxomab ( EpCAM / CD3), [ 6 ] [ 7 ] ertumaxomab ( HER2/neu / CD3), [ 8 ] FBTA05 ( CD20 / CD3, proposed trade name Lymphomun) [ 9 ] [ 10 ] and TRBS07 ( GD2 / CD3, proposed trade name Ektomab), [ 11 ] drugs against various types of cancer.
Trifunctional antibodies were the first type of bispecific monoclonal antibodies to be produced. The first concepts date back to the mid-1980s. [ 12 ] [ 13 ] For over twenty years, no such antibody was approved for clinical use, mainly because of manufacturing difficulties. Immunogenicity results from the fact that appropriate parental antibodies are obtained from rat and mice. After application, the patient's immune system usually produces anti-drug antibodies, which represent early indicators for a beneficial clinical outcome. [ 14 ] Furthermore, despite the development of anti-drug antibody responses after the first catumaxomab application cycle a repeated cycle of catumaxomab also leads to a treatment success in recurrent malignant ascites. [ 15 ] Cross-linking leads to the release of cytokines , resulting in manageable adverse effects like fever, nausea and vomiting, that were generally reversible and mainly related to the immunological mode of action (e.g. catumaxomab). [ 16 ] Catumaxomab, which was approved in 2009 for the treatment of malignant ascites in cancer patients, satisfies these conditions. It was the first, and as of May 2011 the only approved one of these antibodies in clinical use.
Another way of immunotherapeutic intervention strategies is the exploration of bispecific antibodies with different structures, of which bi-specific T-cell engagers (BiTEs) have been produced since the mid-2000s. [ 17 ]
At first, mouse hybridoma cells whose monoclonal antibodies target one of the desired antigens are produced. Independently, rat hybridoma cells targeting the other antigen are produced. These two cell types are hybridised, yielding hybrid-hybridomas or quadromas , which produce hybrid (trifunctional) antibody as well as pure mouse and pure rat antibody. The trifunctional antibody is extracted chromatographically with protein A .
Using two different species (mouse and rat) has the advantage that less mismatched antibodies are produced because rat light chains preferably pair with rat heavy chains , and mouse light chains with mouse heavy chains. Single species (mouse/mouse or rat/rat) quadromas, by contrast, produce up to ten different kinds of antibody, most of which have mismatched heavy or light chains, or both. [ 18 ] | https://en.wikipedia.org/wiki/Trifunctional_antibody |
In neuroscience and neurology , a trigger zone is an area in the body, or of a cell, in which a specific type of stimulation triggers a specific type of response.
The term was first used in this context around 1914 by Hugh T. Patrick, who was writing about trigeminal neuralgia , a condition in which pain fibers in the trigeminal nerve become hypersensitive. [ 1 ] In people with trigeminal neuralgia, even a light touch to some part of the body—often a tooth or a part of the face—can give rise to an extended period of excruciating pain. Patrick referred to the sensitive part of the body as the "dolorogenic zone", and used the term "trigger zone" as a simpler equivalent. [ 2 ] Through the 1920s and 1930s the term came into steadily wider use, but almost always in the context of neuralgia . [ 3 ]
Starting in the late 1930s, other types of stimulation and other types of responses were characterized as having the properties of a trigger zone. In 1940, for example, Morison and Dempsey observed that a small area of the cerebral cortex could be triggered when electrical stimulation would evoke widespread activity in other parts of the cerebral cortex. [ 4 ] In 1944 Paul Wilcox described triggering of epileptic seizure by electrical stimulation of another area of the cerebral cortex. [ 5 ]
The chemoreceptor trigger zone is within the area postrema of the medulla oblongata in which many types of chemical stimulation can provoke nausea and vomiting . This area was first identified and named in 1951 by Herbert L. Borison and Kenneth R. Brizzee. [ 6 ]
Parts of cells, rather than parts of the body, can also behave as trigger zones. The axon hillock of a neuron possesses the highest density of voltage-gated Na + channels , and is therefore the region where it is easiest for the action potential threshold to be reached. [ 7 ] | https://en.wikipedia.org/wiki/Trigger_zone |
In chemistry , a trigonal bipyramid formation is a molecular geometry with one atom at the center and 5 more atoms at the corners of a triangular bipyramid . [ 1 ] This is one geometry for which the bond angles surrounding the central atom are not identical (see also pentagonal bipyramid ), because there is no geometrical arrangement with five terminal atoms in equivalent positions. Examples of this molecular geometry are phosphorus pentafluoride ( PF 5 ), and phosphorus pentachloride ( PCl 5 ) in the gas phase. [ 2 ]
The five atoms bonded to the central atom are not all equivalent, and two different types of position are defined. For phosphorus pentachloride as an example, the phosphorus atom shares a plane with three chlorine atoms at 120° angles to each other in equatorial positions, and two more chlorine atoms above and below the plane ( axial or apical positions).
According to the VSEPR theory of molecular geometry, an axial position is more crowded because an axial atom has three neighboring equatorial atoms (on the same central atom) at a 90° bond angle, whereas an equatorial atom has only two neighboring axial atoms at a 90° bond angle. For molecules with five identical ligands, the axial bond lengths tend to be longer because the ligand atom cannot approach the central atom as closely. As examples, in PF 5 the axial P−F bond length is 158 pm and the equatorial is 152 pm, and in PCl 5 the axial and equatorial are 214 and 202 pm respectively. [ 2 ]
In the mixed halide PF 3 Cl 2 the chlorines occupy two of the equatorial positions, [ 2 ] indicating that fluorine has a greater apicophilicity or tendency to occupy an axial position. In general ligand apicophilicity increases with electronegativity and also with pi-electron withdrawing ability, as in the sequence Cl < F < CN. [ 3 ] Both factors decrease electron density in the bonding region near the central atom so that crowding in the axial position is less important.
The VSEPR theory also predicts that substitution of a ligand at a central atom by a lone pair of valence electrons leaves the general form of the electron arrangement unchanged with the lone pair now occupying one position. For molecules with five pairs of valence electrons including both bonding pairs and lone pairs, the electron pairs are still arranged in a trigonal bipyramid but one or more equatorial positions is not attached to a ligand atom so that the molecular geometry (for the nuclei only) is different.
The seesaw molecular geometry is found in sulfur tetrafluoride (SF 4 ) with a central sulfur atom surrounded by four fluorine atoms occupying two axial and two equatorial positions, as well as one equatorial lone pair, corresponding to an AX 4 E molecule in the AXE notation . A T-shaped molecular geometry is found in chlorine trifluoride (ClF 3 ), an AX 3 E 2 molecule with fluorine atoms in two axial and one equatorial position, as well as two equatorial lone pairs. Finally, the triiodide ion ( I − 3 ) is also based upon a trigonal bipyramid, but the actual molecular geometry is linear with terminal iodine atoms in the two axial positions only and the three equatorial positions occupied by lone pairs of electrons (AX 2 E 3 ); another example of this geometry is provided by xenon difluoride , XeF 2 .
Isomers with a trigonal bipyramidal geometry are able to interconvert through a process known as Berry pseudorotation . Pseudorotation is similar in concept to the movement of a conformational diastereomer, though no full revolutions are completed. In the process of pseudorotation, two equatorial ligands (both of which have a shorter bond length than the third) "shift" toward the molecule's axis, while the axial ligands simultaneously "shift" toward the equator, creating a constant cyclical movement. Pseudorotation is particularly notable in simple molecules such as phosphorus pentafluoride (PF 5 ). | https://en.wikipedia.org/wiki/Trigonal_bipyramidal_molecular_geometry |
In chemistry , trigonal planar is a molecular geometry model with one atom at the center and three atoms at the corners of an equilateral triangle , called peripheral atoms, all in one plane. [ 1 ] In an ideal trigonal planar species, all three ligands are identical and all bond angles are 120°. Such species belong to the point group D 3h . Molecules where the three ligands are not identical, such as H 2 CO , deviate from this idealized geometry. Examples of molecules with trigonal planar geometry include boron trifluoride (BF 3 ), formaldehyde (H 2 CO), phosgene (COCl 2 ), and sulfur trioxide (SO 3 ). Some ions with trigonal planar geometry include nitrate ( NO − 3 ), carbonate ( CO 2− 3 ), and guanidinium ( C(NH 2 ) + 3 ). In organic chemistry, planar, three-connected carbon centers that are trigonal planar are often described as having sp 2 hybridization . [ 2 ] [ 3 ]
Nitrogen inversion is the distortion of pyramidal amines through a transition state that is trigonal planar.
Pyramidalization is a distortion of this molecular shape towards a tetrahedral molecular geometry . One way to observe this distortion is in pyramidal alkenes . [ 1 ] | https://en.wikipedia.org/wiki/Trigonal_planar_molecular_geometry |
In chemistry , the trigonal prismatic molecular geometry describes the shape of compounds where six atoms, groups of atoms, or ligands are arranged around a central atom, defining the vertices of a triangular prism . The structure commonly occurs for d 0 , d 1 and d 2 transition metal complexes with covalently-bound ligands and small charge separation. In d 0 complexes it may be ascribed to sd 5 hybridization, but in d 1 and d 2 complexes the d z 2 orbital is occupied by nonbonding electron (pair). Furthermore, when unoccupied, said orbital participates in bonding and causes C 3v distortion, like in W(CH 3 ) 6 .
Hexamethyltungsten (W(CH 3 ) 6 ) was the first example of a molecular trigonal prismatic complex. [ 2 ] The figure shows the six carbon atoms arranged at the vertices of a triangular prism with the tungsten at the centre. The hydrogen atoms are not shown.
Some other transition metals have trigonal prismatic hexamethyl complexes, including both neutral molecules such as Mo(CH 3 ) 6 and Re(CH 3 ) 6 and ions such as Ta(CH 3 ) − 6 and Zr(CH 3 ) 2− 6 . [ 3 ]
The complex Mo(S−CH=CH−S) 3 is also trigonal prismatic, with each S−CH=CH−S group acting as a bidentate ligand with two sulfur atoms binding the metal atom. [ 3 ] Here the coordination geometry of the six sulfur atoms around the molybdenum is similar to that in the extended structure of molybdenum disulfide (MoS 2 ).
This stereochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Trigonal_prismatic_molecular_geometry |
In chemistry , a trigonal pyramid is a molecular geometry with one atom at the apex and three atoms at the corners of a trigonal base, resembling a tetrahedron (not to be confused with the tetrahedral geometry ). When all three atoms at the corners are identical, the molecule belongs to point group C 3v . Some molecules and ions with trigonal pyramidal geometry are the pnictogen hydrides (XH 3 ), xenon trioxide (XeO 3 ) , the chlorate ion, ClO − 3 , and the sulfite ion, SO 2− 3 . In organic chemistry , molecules which have a trigonal pyramidal geometry are sometimes described as sp 3 hybridized . The AXE method for VSEPR theory states that the classification is AX 3 E 1 .
The nitrogen in ammonia has 5 valence electrons and bonds with three hydrogen atoms to complete the octet . This would result in the geometry of a regular tetrahedron with each bond angle equal to arccos (− 1 / 3 ) ≈ 109.5°. However, the three hydrogen atoms are repelled by the electron lone pair in a way that the geometry is distorted to a trigonal pyramid (regular 3-sided pyramid) with bond angles of 107°. In contrast, boron trifluoride is flat, adopting a trigonal planar geometry because the boron does not have a lone pair of electrons. In ammonia the trigonal pyramid undergoes rapid nitrogen inversion . [ 1 ] | https://en.wikipedia.org/wiki/Trigonal_pyramidal_molecular_geometry |
Trigonelline is an alkaloid with chemical formula C 7 H 7 N O 2 . It is a zwitterion formed by the methylation of the nitrogen atom of niacin (vitamin B 3 ). Trigonelline is a product of niacin metabolism that is excreted in the urine of mammals. [ 1 ]
Trigonelline occurs in many plants. It has been isolated from the Japanese radish [ 2 ] ( Raphanus sativus cv. Sakurajima Daikon), fenugreek seeds ( Trigonella foenum-graecum , hence the name), [ 3 ] garden peas, hemp seed, oats, [ 4 ] potatoes, Stachys species, dahlia , [ 5 ] Strophanthus species, [ 6 ] and Dichapetalum cymosum . [ 7 ] Trigonelline is also found in coffee . [ 8 ] Higher levels of trigonelline are found in arabica coffee .
Holtz, Kutscher, and Theilmann have recorded its presence in a number of animals. [ 9 ]
Trigonelline crystallizes as a monohydrate from alcohol in hygroscopic prisms (m.p. 130 °C or 218 °C [ dry, dec. ]). It is readily soluble in water or warm alcohol, less so in cold alcohol, and slightly so in chloroform or ether . The salts crystallize well, the monohydrochloride, in leaflets, sparingly soluble in dry alcohol. The picrate forms shining prisms (m.p. 198−200 °C) soluble in water but sparingly soluble in dry alcohol or ether. The alkaloid forms several aurichlorides: the normal salt, B•HCl•AuCl 3 , is precipitated when excess of gold chloride is added to the hydrochloride, and, after crystallization from dilute hydrochloric acid containing some gold chloride, has m.p. 198 °C. Crystallized from water or very dilute hydrochloric acid, slender needles of B 4 •3 HAuCl 4 (m.p. 186 °C) are obtained.
When trigonelline is heated in closed tubes with barium hydroxide at 120 °C, it gives rise to methylamine , and, if treated similarly with hydrochloric acid at 260 °C creates chloromethane and nicotinic acid (a form of vitamin B 3 ). Trigonelline is a methyl betaine of nicotinic acid. [ 10 ] | https://en.wikipedia.org/wiki/Trigonelline |
In mathematics , trigonometric interpolation is interpolation with trigonometric polynomials . Interpolation is the process of finding a function which goes through some given data points . For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods. This form is especially suited for interpolation of periodic functions .
An important special case is when the given data points are equally spaced, in which case the solution is given by the discrete Fourier transform .
A trigonometric polynomial of degree K has the form
This expression contains 2 K + 1 coefficients, a 0 , a 1 , … a K , b 1 , …, b K , and we wish to compute those coefficients so that the function passes through N points:
Since the trigonometric polynomial is periodic with period 2π, the N points can be distributed and ordered in one period as
(Note that we do not in general require these points to be equally spaced.) The interpolation problem is now to find coefficients such that the trigonometric polynomial p satisfies the interpolation conditions.
The problem becomes more natural if we formulate it in the complex plane . We can rewrite the formula for a trigonometric polynomial as p ( x ) = ∑ k = − K K c k e i k x , {\displaystyle p(x)=\sum _{k=-K}^{K}c_{k}e^{ikx},\,} where i is the imaginary unit . If we set z = e ix , then this becomes
with
This reduces the problem of trigonometric interpolation to that of polynomial interpolation on the unit circle . Existence and uniqueness for trigonometric interpolation now follows immediately from the corresponding results for polynomial interpolation.
For more information on formulation of trigonometric interpolating polynomials in the complex plane, see p. 156 of Interpolation using Fourier Polynomials .
Under the above conditions, there exists a solution to the problem for any given set of data points { x k , y k } as long as N , the number of data points, is not larger than the number of coefficients in the polynomial, i.e., N ≤ 2 K +1 (a solution may or may not exist if N >2 K +1 depending upon the particular set of data points). Moreover, the interpolating polynomial is unique if and only if the number of adjustable coefficients is equal to the number of data points, i.e., N = 2 K + 1. In the remainder of this article, we will assume this condition to hold true.
If the number of points N is odd, say N=2K+1 , applying the Lagrange formula for polynomial interpolation to the polynomial formulation in the complex plane yields that the solution can be written in the form
where
The factor e − i K x + i K x k {\displaystyle e^{-iKx+iKx_{k}}} in this formula compensates for the fact that the complex plane formulation contains also negative powers of e i x {\displaystyle e^{ix}} and is therefore not a polynomial expression in e i x {\displaystyle e^{ix}} . The correctness of this expression can easily be verified by observing that t k ( x k ) = 1 {\displaystyle t_{k}(x_{k})=1} and that t k ( x ) {\displaystyle t_{k}(x)} is a linear combination of the right powers of e i x {\displaystyle e^{ix}} .
Upon using the identity
the coefficient t k ( x ) {\displaystyle t_{k}(x)} can be written in the form
If the number of points N is even, say N=2K , applying the Lagrange formula for polynomial interpolation to the polynomial formulation in the complex plane yields that the solution can be written in the form
where
Here, the constants α k {\displaystyle \alpha _{k}} can be chosen freely. This is caused by the fact that the interpolating function ( 1 ) contains an odd number of unknown constants. A common choice is to require that the highest frequency is of the form a constant times cos ( K x ) {\displaystyle \cos(Kx)} , i.e. the sin ( K x ) {\displaystyle \sin(Kx)} term vanishes, but in general the phase of the highest frequency can be chosen to be φ K {\displaystyle \varphi _{K}} . To get an expression for α k {\displaystyle \alpha _{k}} , we obtain by using ( 2 ) that ( 3 ) can be written on the form
This yields
and
Note that care must be taken in order to avoid infinities caused by zeros in the denominators.
Further simplification of the problem is possible if nodes x m {\displaystyle x_{m}} are equidistant, i.e.
see Zygmund for more details.
Further simplification by using ( 4 ) would be an obvious approach, but is obviously involved. A much simpler approach is to consider the Dirichlet kernel
where N > 0 {\displaystyle N>0} is odd. It can easily be seen that D ( x , N ) {\displaystyle D(x,N)} is a linear combination of the right powers of e i x {\displaystyle e^{ix}} and satisfies
Since these two properties uniquely define the coefficients t k ( x ) {\displaystyle t_{k}(x)} in ( 5 ), it follows that
Here, the sinc -function prevents any singularities and is defined by
For N {\displaystyle N} even, we define the Dirichlet kernel as
Again, it can easily be seen that D ( x , N ) {\displaystyle D(x,N)} is a linear combination of the right powers of e i x {\displaystyle e^{ix}} , does not contain the term sin 1 2 N x {\displaystyle \sin {\tfrac {1}{2}}Nx} and satisfies
Using these properties, it follows that the coefficients t k ( x ) {\displaystyle t_{k}(x)} in ( 6 ) are given by
Note that t k ( x ) {\displaystyle t_{k}(x)} does not contain the sin 1 2 N x {\displaystyle \sin {\tfrac {1}{2}}Nx} as well. Finally, note that the function sin 1 2 N x {\displaystyle \sin {\tfrac {1}{2}}Nx} vanishes at all the points x m {\displaystyle x_{m}} . Multiples of this term can, therefore, always be added, but it is commonly left out.
A MATLAB implementation of the above can be found here and is given by:
The special case in which the points x n are equally spaced is especially important. In this case, we have
The transformation that maps the data points y n to the coefficients a k , b k is obtained from the discrete Fourier transform (DFT) of order N.
(Because of the way the problem was formulated above, we have restricted ourselves to odd numbers of points. This is not strictly necessary; for even numbers of points, one includes another cosine term corresponding to the Nyquist frequency .)
The case of the cosine-only interpolation for equally spaced points, corresponding to a trigonometric interpolation when the points have even symmetry , was treated by Alexis Clairaut in 1754. In this case the solution is equivalent to a discrete cosine transform . The sine-only expansion for equally spaced points, corresponding to odd symmetry, was solved by Joseph Louis Lagrange in 1762, for which the solution is a discrete sine transform . The full cosine and sine interpolating polynomial, which gives rise to the DFT, was solved by Carl Friedrich Gauss in unpublished work around 1805, at which point he also derived a fast Fourier transform algorithm to evaluate it rapidly. Clairaut, Lagrange, and Gauss were all concerned with studying the problem of inferring the orbit of planets , asteroids , etc., from a finite set of observation points; since the orbits are periodic, a trigonometric interpolation was a natural choice. See also Heideman et al. (1984).
Chebfun , a fully integrated software system written in MATLAB for computing with functions, uses trigonometric interpolation and Fourier expansions for computing with periodic functions. Many algorithms related to trigonometric interpolation are readily available in Chebfun ; several examples are available here . | https://en.wikipedia.org/wiki/Trigonometric_interpolation |
In the mathematical subfields of numerical analysis and mathematical analysis , a trigonometric polynomial is a finite linear combination of functions sin( nx ) and cos( nx ) with n taking on the values of one or more natural numbers . The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients , there is no difference between such a function and a finite Fourier series .
Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions . They are used also in the discrete Fourier transform .
The term trigonometric polynomial for the real-valued case can be seen as using the analogy : the functions sin( nx ) and cos( nx ) are similar to the monomial basis for polynomials . In the complex case the trigonometric polynomials are spanned by the positive and negative powers of e i x {\displaystyle e^{ix}} , i.e., Laurent polynomials in z {\displaystyle z} under the change of variables x ↦ z := e i x {\displaystyle x\mapsto z:=e^{ix}} .
Any function T of the form
T ( x ) = a 0 + ∑ n = 1 N a n cos ( n x ) + ∑ n = 1 N b n sin ( n x ) ( x ∈ R ) {\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx)\qquad (x\in \mathbb {R} )}
with coefficients a n , b n ∈ C {\displaystyle a_{n},b_{n}\in \mathbb {C} } and at least one of the highest-degree coefficients a N {\displaystyle a_{N}} and b N {\displaystyle b_{N}} non-zero, is called a complex trigonometric polynomial of degree N . [ 1 ] Using Euler's formula the polynomial can be rewritten as
T ( x ) = ∑ n = − N N c n e i n x ( x ∈ R ) . {\displaystyle T(x)=\sum _{n=-N}^{N}c_{n}e^{inx}\qquad (x\in \mathbb {R} ).} with c n ∈ C {\displaystyle c_{n}\in \mathbb {C} } .
Analogously, letting coefficients a n , b n ∈ R {\displaystyle a_{n},b_{n}\in \mathbb {R} } , and at least one of a N {\displaystyle a_{N}} and b N {\displaystyle b_{N}} non-zero or, equivalently, c n ∈ R {\displaystyle c_{n}\in \mathbb {R} } and c n = c ¯ − n {\displaystyle c_{n}={\bar {c}}_{-n}} for all n ∈ [ − N , N ] {\displaystyle n\in [-N,N]} , then
t ( x ) = a 0 + ∑ n = 1 N a n cos ( n x ) + ∑ n = 1 N b n sin ( n x ) ( x ∈ R ) {\displaystyle t(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx)\qquad (x\in \mathbb {R} )}
is called a real trigonometric polynomial of degree N . [ 2 ] [ 3 ]
A trigonometric polynomial can be considered a periodic function on the real line , with period some divisor of 2 π {\displaystyle 2\pi } , or as a function on the unit circle .
Trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm ; [ 4 ] this is a special case of the Stone–Weierstrass theorem . More concretely, for every continuous function f {\displaystyle f} and every ϵ > 0 {\displaystyle \epsilon >0} there exists a trigonometric polynomial T {\displaystyle T} such that | f ( z ) − T ( z ) | < ϵ {\displaystyle |f(z)-T(z)|<\epsilon } for all z {\displaystyle z} . Fejér's theorem states that the arithmetic means of the partial sums of the Fourier series of f {\displaystyle f} converge uniformly to f {\displaystyle f} provided f {\displaystyle f} is continuous on the circle; these partial sums can be used to approximate f {\displaystyle f} .
A trigonometric polynomial of degree N {\displaystyle N} has a maximum of 2 N {\displaystyle 2N} roots in a real interval [ a , a + 2 π ) {\displaystyle [a,a+2\pi )} unless it is the zero function. [ 5 ]
The Fejér-Riesz theorem states that every positive real trigonometric polynomial t ( x ) = ∑ n = − N N c n e i n x , {\displaystyle t(x)=\sum _{n=-N}^{N}c_{n}e^{inx},} satisfying t ( x ) > 0 {\displaystyle t(x)>0} for all x ∈ R {\displaystyle x\in \mathbb {R} } ,
can be represented as the square of the modulus of another (usually complex ) trigonometric polynomial q ( x ) {\displaystyle q(x)} such that: [ 6 ] t ( x ) = | q ( x ) | 2 = q ( x ) q ¯ ( x ) . {\displaystyle t(x)=|q(x)|^{2}=q(x){\bar {q}}(x).} Or, equivalently, every Laurent polynomial w ( z ) = ∑ n = − N N w n z n , {\displaystyle w(z)=\sum _{n=-N}^{N}w_{n}z^{n},} with w n ∈ C {\displaystyle w_{n}\in \mathbb {C} } that satisfies w ( ζ ) ≥ 0 {\displaystyle w(\zeta )\geq 0} for all ζ ∈ T {\displaystyle \zeta \in \mathbb {T} } can be written as: w ( ζ ) = | p ( ζ ) | 2 = p ( ζ ) p ¯ ( ζ ¯ ) , {\displaystyle w(\zeta )=|p(\zeta )|^{2}=p(\zeta ){\bar {p}}({\bar {\zeta }}),} for some polynomial p ( z ) {\displaystyle p(z)} . [ 7 ] | https://en.wikipedia.org/wiki/Trigonometric_polynomial |
In mathematics , a trigonometric substitution replaces a trigonometric function for another expression. In calculus , trigonometric substitutions are a technique for evaluating integrals. In this case, an expression involving a radical function is replaced with a trigonometric one. Trigonometric identities may help simplify the answer. [ 1 ] [ 2 ] Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.
Let x = a sin θ , {\displaystyle x=a\sin \theta ,} and use the identity 1 − sin 2 θ = cos 2 θ . {\displaystyle 1-\sin ^{2}\theta =\cos ^{2}\theta .}
In the integral
∫ d x a 2 − x 2 , {\displaystyle \int {\frac {dx}{\sqrt {a^{2}-x^{2}}}},}
we may use
x = a sin θ , d x = a cos θ d θ , θ = arcsin x a . {\displaystyle x=a\sin \theta ,\quad dx=a\cos \theta \,d\theta ,\quad \theta =\arcsin {\frac {x}{a}}.}
Then, ∫ d x a 2 − x 2 = ∫ a cos θ d θ a 2 − a 2 sin 2 θ = ∫ a cos θ d θ a 2 ( 1 − sin 2 θ ) = ∫ a cos θ d θ a 2 cos 2 θ = ∫ d θ = θ + C = arcsin x a + C . {\displaystyle {\begin{aligned}\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}&=\int {\frac {a\cos \theta \,d\theta }{\sqrt {a^{2}-a^{2}\sin ^{2}\theta }}}\\[6pt]&=\int {\frac {a\cos \theta \,d\theta }{\sqrt {a^{2}(1-\sin ^{2}\theta )}}}\\[6pt]&=\int {\frac {a\cos \theta \,d\theta }{\sqrt {a^{2}\cos ^{2}\theta }}}\\[6pt]&=\int d\theta \\[6pt]&=\theta +C\\[6pt]&=\arcsin {\frac {x}{a}}+C.\end{aligned}}}
The above step requires that a > 0 {\displaystyle a>0} and cos θ > 0. {\displaystyle \cos \theta >0.} We can choose a {\displaystyle a} to be the principal root of a 2 , {\displaystyle a^{2},} and impose the restriction − π / 2 < θ < π / 2 {\displaystyle -\pi /2<\theta <\pi /2} by using the inverse sine function.
For a definite integral, one must figure out how the bounds of integration change. For example, as x {\displaystyle x} goes from 0 {\displaystyle 0} to a / 2 , {\displaystyle a/2,} then sin θ {\displaystyle \sin \theta } goes from 0 {\displaystyle 0} to 1 / 2 , {\displaystyle 1/2,} so θ {\displaystyle \theta } goes from 0 {\displaystyle 0} to π / 6. {\displaystyle \pi /6.} Then,
∫ 0 a / 2 d x a 2 − x 2 = ∫ 0 π / 6 d θ = π 6 . {\displaystyle \int _{0}^{a/2}{\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\int _{0}^{\pi /6}d\theta ={\frac {\pi }{6}}.}
Some care is needed when picking the bounds. Because integration above requires that − π / 2 < θ < π / 2 {\displaystyle -\pi /2<\theta <\pi /2} , θ {\displaystyle \theta } can only go from 0 {\displaystyle 0} to π / 6. {\displaystyle \pi /6.} Neglecting this restriction, one might have picked θ {\displaystyle \theta } to go from π {\displaystyle \pi } to 5 π / 6 , {\displaystyle 5\pi /6,} which would have resulted in the negative of the actual value.
Alternatively, fully evaluate the indefinite integrals before applying the boundary conditions. In that case, the antiderivative gives
∫ 0 a / 2 d x a 2 − x 2 = arcsin ( x a ) | 0 a / 2 = arcsin ( 1 2 ) − arcsin ( 0 ) = π 6 {\displaystyle \int _{0}^{a/2}{\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin \left({\frac {x}{a}}\right){\Biggl |}_{0}^{a/2}=\arcsin \left({\frac {1}{2}}\right)-\arcsin(0)={\frac {\pi }{6}}} as before.
The integral
∫ a 2 − x 2 d x , {\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,dx,}
may be evaluated by letting x = a sin θ , d x = a cos θ d θ , θ = arcsin x a , {\textstyle x=a\sin \theta ,\,dx=a\cos \theta \,d\theta ,\,\theta =\arcsin {\dfrac {x}{a}},} where a > 0 {\displaystyle a>0} so that a 2 = a , {\textstyle {\sqrt {a^{2}}}=a,} and − π / 2 ≤ θ ≤ π / 2 {\textstyle -\pi /2\leq \theta \leq \pi /2} by the range of arcsine, so that cos θ ≥ 0 {\displaystyle \cos \theta \geq 0} and cos 2 θ = cos θ . {\textstyle {\sqrt {\cos ^{2}\theta }}=\cos \theta .}
Then, ∫ a 2 − x 2 d x = ∫ a 2 − a 2 sin 2 θ ( a cos θ ) d θ = ∫ a 2 ( 1 − sin 2 θ ) ( a cos θ ) d θ = ∫ a 2 ( cos 2 θ ) ( a cos θ ) d θ = ∫ ( a cos θ ) ( a cos θ ) d θ = a 2 ∫ cos 2 θ d θ = a 2 ∫ ( 1 + cos 2 θ 2 ) d θ = a 2 2 ( θ + 1 2 sin 2 θ ) + C = a 2 2 ( θ + sin θ cos θ ) + C = a 2 2 ( arcsin x a + x a 1 − x 2 a 2 ) + C = a 2 2 arcsin x a + x 2 a 2 − x 2 + C . {\displaystyle {\begin{aligned}\int {\sqrt {a^{2}-x^{2}}}\,dx&=\int {\sqrt {a^{2}-a^{2}\sin ^{2}\theta }}\,(a\cos \theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(1-\sin ^{2}\theta )}}\,(a\cos \theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(\cos ^{2}\theta )}}\,(a\cos \theta )\,d\theta \\[6pt]&=\int (a\cos \theta )(a\cos \theta )\,d\theta \\[6pt]&=a^{2}\int \cos ^{2}\theta \,d\theta \\[6pt]&=a^{2}\int \left({\frac {1+\cos 2\theta }{2}}\right)\,d\theta \\[6pt]&={\frac {a^{2}}{2}}\left(\theta +{\frac {1}{2}}\sin 2\theta \right)+C\\[6pt]&={\frac {a^{2}}{2}}(\theta +\sin \theta \cos \theta )+C\\[6pt]&={\frac {a^{2}}{2}}\left(\arcsin {\frac {x}{a}}+{\frac {x}{a}}{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}\right)+C\\[6pt]&={\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+C.\end{aligned}}}
For a definite integral, the bounds change once the substitution is performed and are determined using the equation θ = arcsin x a , {\textstyle \theta =\arcsin {\dfrac {x}{a}},} with values in the range − π / 2 ≤ θ ≤ π / 2. {\textstyle -\pi /2\leq \theta \leq \pi /2.} Alternatively, apply the boundary terms directly to the formula for the antiderivative.
For example, the definite integral
∫ − 1 1 4 − x 2 d x , {\displaystyle \int _{-1}^{1}{\sqrt {4-x^{2}}}\,dx,}
may be evaluated by substituting x = 2 sin θ , d x = 2 cos θ d θ , {\displaystyle x=2\sin \theta ,\,dx=2\cos \theta \,d\theta ,} with the bounds determined using θ = arcsin x 2 . {\textstyle \theta =\arcsin {\dfrac {x}{2}}.}
Because arcsin ( 1 / 2 ) = π / 6 {\displaystyle \arcsin(1/{2})=\pi /6} and arcsin ( − 1 / 2 ) = − π / 6 , {\displaystyle \arcsin(-1/2)=-\pi /6,} ∫ − 1 1 4 − x 2 d x = ∫ − π / 6 π / 6 4 − 4 sin 2 θ ( 2 cos θ ) d θ = ∫ − π / 6 π / 6 4 ( 1 − sin 2 θ ) ( 2 cos θ ) d θ = ∫ − π / 6 π / 6 4 ( cos 2 θ ) ( 2 cos θ ) d θ = ∫ − π / 6 π / 6 ( 2 cos θ ) ( 2 cos θ ) d θ = 4 ∫ − π / 6 π / 6 cos 2 θ d θ = 4 ∫ − π / 6 π / 6 ( 1 + cos 2 θ 2 ) d θ = 2 [ θ + 1 2 sin 2 θ ] − π / 6 π / 6 = [ 2 θ + sin 2 θ ] | − π / 6 π / 6 = ( π 3 + sin π 3 ) − ( − π 3 + sin ( − π 3 ) ) = 2 π 3 + 3 . {\displaystyle {\begin{aligned}\int _{-1}^{1}{\sqrt {4-x^{2}}}\,dx&=\int _{-\pi /6}^{\pi /6}{\sqrt {4-4\sin ^{2}\theta }}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}{\sqrt {4(1-\sin ^{2}\theta )}}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}{\sqrt {4(\cos ^{2}\theta )}}\,(2\cos \theta )\,d\theta \\[6pt]&=\int _{-\pi /6}^{\pi /6}(2\cos \theta )(2\cos \theta )\,d\theta \\[6pt]&=4\int _{-\pi /6}^{\pi /6}\cos ^{2}\theta \,d\theta \\[6pt]&=4\int _{-\pi /6}^{\pi /6}\left({\frac {1+\cos 2\theta }{2}}\right)\,d\theta \\[6pt]&=2\left[\theta +{\frac {1}{2}}\sin 2\theta \right]_{-\pi /6}^{\pi /6}=[2\theta +\sin 2\theta ]{\Biggl |}_{-\pi /6}^{\pi /6}\\[6pt]&=\left({\frac {\pi }{3}}+\sin {\frac {\pi }{3}}\right)-\left(-{\frac {\pi }{3}}+\sin \left(-{\frac {\pi }{3}}\right)\right)={\frac {2\pi }{3}}+{\sqrt {3}}.\end{aligned}}}
On the other hand, direct application of the boundary terms to the previously obtained formula for the antiderivative yields ∫ − 1 1 4 − x 2 d x = [ 2 2 2 arcsin x 2 + x 2 2 2 − x 2 ] − 1 1 = ( 2 arcsin 1 2 + 1 2 4 − 1 ) − ( 2 arcsin ( − 1 2 ) + − 1 2 4 − 1 ) = ( 2 ⋅ π 6 + 3 2 ) − ( 2 ⋅ ( − π 6 ) − 3 2 ) = 2 π 3 + 3 {\displaystyle {\begin{aligned}\int _{-1}^{1}{\sqrt {4-x^{2}}}\,dx&=\left[{\frac {2^{2}}{2}}\arcsin {\frac {x}{2}}+{\frac {x}{2}}{\sqrt {2^{2}-x^{2}}}\right]_{-1}^{1}\\[6pt]&=\left(2\arcsin {\frac {1}{2}}+{\frac {1}{2}}{\sqrt {4-1}}\right)-\left(2\arcsin \left(-{\frac {1}{2}}\right)+{\frac {-1}{2}}{\sqrt {4-1}}\right)\\[6pt]&=\left(2\cdot {\frac {\pi }{6}}+{\frac {\sqrt {3}}{2}}\right)-\left(2\cdot \left(-{\frac {\pi }{6}}\right)-{\frac {\sqrt {3}}{2}}\right)\\[6pt]&={\frac {2\pi }{3}}+{\sqrt {3}}\end{aligned}}} as before.
Let x = a tan θ , {\displaystyle x=a\tan \theta ,} and use the identity 1 + tan 2 θ = sec 2 θ . {\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta .}
In the integral
∫ d x a 2 + x 2 {\displaystyle \int {\frac {dx}{a^{2}+x^{2}}}}
we may write
x = a tan θ , d x = a sec 2 θ d θ , θ = arctan x a , {\displaystyle x=a\tan \theta ,\quad dx=a\sec ^{2}\theta \,d\theta ,\quad \theta =\arctan {\frac {x}{a}},}
so that the integral becomes
∫ d x a 2 + x 2 = ∫ a sec 2 θ d θ a 2 + a 2 tan 2 θ = ∫ a sec 2 θ d θ a 2 ( 1 + tan 2 θ ) = ∫ a sec 2 θ d θ a 2 sec 2 θ = ∫ d θ a = θ a + C = 1 a arctan x a + C , {\displaystyle {\begin{aligned}\int {\frac {dx}{a^{2}+x^{2}}}&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}+a^{2}\tan ^{2}\theta }}\\[6pt]&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}(1+\tan ^{2}\theta )}}\\[6pt]&=\int {\frac {a\sec ^{2}\theta \,d\theta }{a^{2}\sec ^{2}\theta }}\\[6pt]&=\int {\frac {d\theta }{a}}\\[6pt]&={\frac {\theta }{a}}+C\\[6pt]&={\frac {1}{a}}\arctan {\frac {x}{a}}+C,\end{aligned}}}
provided a ≠ 0. {\displaystyle a\neq 0.}
For a definite integral, the bounds change once the substitution is performed and are determined using the equation θ = arctan x a , {\displaystyle \theta =\arctan {\frac {x}{a}},} with values in the range − π 2 < θ < π 2 . {\displaystyle -{\frac {\pi }{2}}<\theta <{\frac {\pi }{2}}.} Alternatively, apply the boundary terms directly to the formula for the antiderivative.
For example, the definite integral
∫ 0 1 4 d x 1 + x 2 {\displaystyle \int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}\,}
may be evaluated by substituting x = tan θ , d x = sec 2 θ d θ , {\displaystyle x=\tan \theta ,\,dx=\sec ^{2}\theta \,d\theta ,} with the bounds determined using θ = arctan x . {\displaystyle \theta =\arctan x.}
Since arctan 0 = 0 {\displaystyle \arctan 0=0} and arctan 1 = π / 4 , {\displaystyle \arctan 1=\pi /4,} ∫ 0 1 4 d x 1 + x 2 = 4 ∫ 0 1 d x 1 + x 2 = 4 ∫ 0 π / 4 sec 2 θ d θ 1 + tan 2 θ = 4 ∫ 0 π / 4 sec 2 θ d θ sec 2 θ = 4 ∫ 0 π / 4 d θ = ( 4 θ ) | 0 π / 4 = 4 ( π 4 − 0 ) = π . {\displaystyle {\begin{aligned}\int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}&=4\int _{0}^{1}{\frac {dx}{1+x^{2}}}\\[6pt]&=4\int _{0}^{\pi /4}{\frac {\sec ^{2}\theta \,d\theta }{1+\tan ^{2}\theta }}\\[6pt]&=4\int _{0}^{\pi /4}{\frac {\sec ^{2}\theta \,d\theta }{\sec ^{2}\theta }}\\[6pt]&=4\int _{0}^{\pi /4}d\theta \\[6pt]&=(4\theta ){\Bigg |}_{0}^{\pi /4}=4\left({\frac {\pi }{4}}-0\right)=\pi .\end{aligned}}}
Meanwhile, direct application of the boundary terms to the formula for the antiderivative yields ∫ 0 1 4 d x 1 + x 2 = 4 ∫ 0 1 d x 1 + x 2 = 4 [ 1 1 arctan x 1 ] 0 1 = 4 ( arctan x ) | 0 1 = 4 ( arctan 1 − arctan 0 ) = 4 ( π 4 − 0 ) = π , {\displaystyle {\begin{aligned}\int _{0}^{1}{\frac {4\,dx}{1+x^{2}}}\,&=4\int _{0}^{1}{\frac {dx}{1+x^{2}}}\\[6pt]&=4\left[{\frac {1}{1}}\arctan {\frac {x}{1}}\right]_{0}^{1}\\[6pt]&=4(\arctan x){\Bigg |}_{0}^{1}\\[6pt]&=4(\arctan 1-\arctan 0)\\[6pt]&=4\left({\frac {\pi }{4}}-0\right)=\pi ,\end{aligned}}} same as before.
The integral
∫ a 2 + x 2 d x {\displaystyle \int {\sqrt {a^{2}+x^{2}}}\,{dx}}
may be evaluated by letting x = a tan θ , d x = a sec 2 θ d θ , θ = arctan x a , {\displaystyle x=a\tan \theta ,\,dx=a\sec ^{2}\theta \,d\theta ,\,\theta =\arctan {\frac {x}{a}},}
where a > 0 {\displaystyle a>0} so that a 2 = a , {\displaystyle {\sqrt {a^{2}}}=a,} and − π 2 < θ < π 2 {\displaystyle -{\frac {\pi }{2}}<\theta <{\frac {\pi }{2}}} by the range of arctangent, so that sec θ > 0 {\displaystyle \sec \theta >0} and sec 2 θ = sec θ . {\displaystyle {\sqrt {\sec ^{2}\theta }}=\sec \theta .}
Then, ∫ a 2 + x 2 d x = ∫ a 2 + a 2 tan 2 θ ( a sec 2 θ ) d θ = ∫ a 2 ( 1 + tan 2 θ ) ( a sec 2 θ ) d θ = ∫ a 2 sec 2 θ ( a sec 2 θ ) d θ = ∫ ( a sec θ ) ( a sec 2 θ ) d θ = a 2 ∫ sec 3 θ d θ . {\displaystyle {\begin{aligned}\int {\sqrt {a^{2}+x^{2}}}\,dx&=\int {\sqrt {a^{2}+a^{2}\tan ^{2}\theta }}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}(1+\tan ^{2}\theta )}}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int {\sqrt {a^{2}\sec ^{2}\theta }}\,(a\sec ^{2}\theta )\,d\theta \\[6pt]&=\int (a\sec \theta )(a\sec ^{2}\theta )\,d\theta \\[6pt]&=a^{2}\int \sec ^{3}\theta \,d\theta .\\[6pt]\end{aligned}}} The integral of secant cubed may be evaluated using integration by parts . As a result, ∫ a 2 + x 2 d x = a 2 2 ( sec θ tan θ + ln | sec θ + tan θ | ) + C = a 2 2 ( 1 + x 2 a 2 ⋅ x a + ln | 1 + x 2 a 2 + x a | ) + C = 1 2 ( x a 2 + x 2 + a 2 ln | x + a 2 + x 2 a | ) + C . {\displaystyle {\begin{aligned}\int {\sqrt {a^{2}+x^{2}}}\,dx&={\frac {a^{2}}{2}}(\sec \theta \tan \theta +\ln |\sec \theta +\tan \theta |)+C\\[6pt]&={\frac {a^{2}}{2}}\left({\sqrt {1+{\frac {x^{2}}{a^{2}}}}}\cdot {\frac {x}{a}}+\ln \left|{\sqrt {1+{\frac {x^{2}}{a^{2}}}}}+{\frac {x}{a}}\right|\right)+C\\[6pt]&={\frac {1}{2}}\left(x{\sqrt {a^{2}+x^{2}}}+a^{2}\ln \left|{\frac {x+{\sqrt {a^{2}+x^{2}}}}{a}}\right|\right)+C.\end{aligned}}}
Let x = a sec θ , {\displaystyle x=a\sec \theta ,} and use the identity sec 2 θ − 1 = tan 2 θ . {\displaystyle \sec ^{2}\theta -1=\tan ^{2}\theta .}
Integrals such as
∫ d x x 2 − a 2 {\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}}
can also be evaluated by partial fractions rather than trigonometric substitutions. However, the integral
∫ x 2 − a 2 d x {\displaystyle \int {\sqrt {x^{2}-a^{2}}}\,dx}
cannot. In this case, an appropriate substitution is: x = a sec θ , d x = a sec θ tan θ d θ , θ = arcsec x a , {\displaystyle x=a\sec \theta ,\,dx=a\sec \theta \tan \theta \,d\theta ,\,\theta =\operatorname {arcsec} {\frac {x}{a}},}
where a > 0 {\displaystyle a>0} so that a 2 = a , {\displaystyle {\sqrt {a^{2}}}=a,} and 0 ≤ θ < π 2 {\displaystyle 0\leq \theta <{\frac {\pi }{2}}} by assuming x > 0 , {\displaystyle x>0,} so that tan θ ≥ 0 {\displaystyle \tan \theta \geq 0} and tan 2 θ = tan θ . {\displaystyle {\sqrt {\tan ^{2}\theta }}=\tan \theta .}
Then, ∫ x 2 − a 2 d x = ∫ a 2 sec 2 θ − a 2 ⋅ a sec θ tan θ d θ = ∫ a 2 ( sec 2 θ − 1 ) ⋅ a sec θ tan θ d θ = ∫ a 2 tan 2 θ ⋅ a sec θ tan θ d θ = ∫ a 2 sec θ tan 2 θ d θ = a 2 ∫ ( sec θ ) ( sec 2 θ − 1 ) d θ = a 2 ∫ ( sec 3 θ − sec θ ) d θ . {\displaystyle {\begin{aligned}\int {\sqrt {x^{2}-a^{2}}}\,dx&=\int {\sqrt {a^{2}\sec ^{2}\theta -a^{2}}}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int {\sqrt {a^{2}(\sec ^{2}\theta -1)}}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int {\sqrt {a^{2}\tan ^{2}\theta }}\cdot a\sec \theta \tan \theta \,d\theta \\&=\int a^{2}\sec \theta \tan ^{2}\theta \,d\theta \\&=a^{2}\int (\sec \theta )(\sec ^{2}\theta -1)\,d\theta \\&=a^{2}\int (\sec ^{3}\theta -\sec \theta )\,d\theta .\end{aligned}}}
One may evaluate the integral of the secant function by multiplying the numerator and denominator by ( sec θ + tan θ ) {\displaystyle (\sec \theta +\tan \theta )} and the integral of secant cubed by parts. [ 3 ] As a result, ∫ x 2 − a 2 d x = a 2 2 ( sec θ tan θ + ln | sec θ + tan θ | ) − a 2 ln | sec θ + tan θ | + C = a 2 2 ( sec θ tan θ − ln | sec θ + tan θ | ) + C = a 2 2 ( x a ⋅ x 2 a 2 − 1 − ln | x a + x 2 a 2 − 1 | ) + C = 1 2 ( x x 2 − a 2 − a 2 ln | x + x 2 − a 2 a | ) + C . {\displaystyle {\begin{aligned}\int {\sqrt {x^{2}-a^{2}}}\,dx&={\frac {a^{2}}{2}}(\sec \theta \tan \theta +\ln |\sec \theta +\tan \theta |)-a^{2}\ln |\sec \theta +\tan \theta |+C\\[6pt]&={\frac {a^{2}}{2}}(\sec \theta \tan \theta -\ln |\sec \theta +\tan \theta |)+C\\[6pt]&={\frac {a^{2}}{2}}\left({\frac {x}{a}}\cdot {\sqrt {{\frac {x^{2}}{a^{2}}}-1}}-\ln \left|{\frac {x}{a}}+{\sqrt {{\frac {x^{2}}{a^{2}}}-1}}\right|\right)+C\\[6pt]&={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}-a^{2}\ln \left|{\frac {x+{\sqrt {x^{2}-a^{2}}}}{a}}\right|\right)+C.\end{aligned}}}
When π 2 < θ ≤ π , {\displaystyle {\frac {\pi }{2}}<\theta \leq \pi ,} which happens when x < 0 {\displaystyle x<0} given the range of arcsecant, tan θ ≤ 0 , {\displaystyle \tan \theta \leq 0,} meaning tan 2 θ = − tan θ {\displaystyle {\sqrt {\tan ^{2}\theta }}=-\tan \theta } instead in that case.
Substitution can be used to remove trigonometric functions.
For instance,
∫ f ( sin ( x ) , cos ( x ) ) d x = ∫ 1 ± 1 − u 2 f ( u , ± 1 − u 2 ) d u u = sin ( x ) ∫ f ( sin ( x ) , cos ( x ) ) d x = ∫ 1 ∓ 1 − u 2 f ( ± 1 − u 2 , u ) d u u = cos ( x ) ∫ f ( sin ( x ) , cos ( x ) ) d x = ∫ 2 1 + u 2 f ( 2 u 1 + u 2 , 1 − u 2 1 + u 2 ) d u u = tan ( x 2 ) {\displaystyle {\begin{aligned}\int f(\sin(x),\cos(x))\,dx&=\int {\frac {1}{\pm {\sqrt {1-u^{2}}}}}f\left(u,\pm {\sqrt {1-u^{2}}}\right)\,du&&u=\sin(x)\\[6pt]\int f(\sin(x),\cos(x))\,dx&=\int {\frac {1}{\mp {\sqrt {1-u^{2}}}}}f\left(\pm {\sqrt {1-u^{2}}},u\right)\,du&&u=\cos(x)\\[6pt]\int f(\sin(x),\cos(x))\,dx&=\int {\frac {2}{1+u^{2}}}f\left({\frac {2u}{1+u^{2}}},{\frac {1-u^{2}}{1+u^{2}}}\right)\,du&&u=\tan \left({\frac {x}{2}}\right)\\[6pt]\end{aligned}}}
The last substitution is known as the Weierstrass substitution , which makes use of tangent half-angle formulas .
For example,
∫ 4 cos x ( 1 + cos x ) 3 d x = ∫ 2 1 + u 2 4 ( 1 − u 2 1 + u 2 ) ( 1 + 1 − u 2 1 + u 2 ) 3 d u = ∫ ( 1 − u 2 ) ( 1 + u 2 ) d u = ∫ ( 1 − u 4 ) d u = u − u 5 5 + C = tan x 2 − 1 5 tan 5 x 2 + C . {\displaystyle {\begin{aligned}\int {\frac {4\cos x}{(1+\cos x)^{3}}}\,dx&=\int {\frac {2}{1+u^{2}}}{\frac {4\left({\frac {1-u^{2}}{1+u^{2}}}\right)}{\left(1+{\frac {1-u^{2}}{1+u^{2}}}\right)^{3}}}\,du=\int (1-u^{2})(1+u^{2})\,du\\&=\int (1-u^{4})\,du=u-{\frac {u^{5}}{5}}+C=\tan {\frac {x}{2}}-{\frac {1}{5}}\tan ^{5}{\frac {x}{2}}+C.\end{aligned}}}
Substitutions of hyperbolic functions can also be used to simplify integrals. [ 4 ]
For example, to integrate 1 / a 2 + x 2 {\displaystyle 1/{\sqrt {a^{2}+x^{2}}}} , introduce the substitution x = a sinh u {\displaystyle x=a\sinh {u}} (and hence d x = a cosh u d u {\displaystyle dx=a\cosh u\,du} ), then use the identity cosh 2 ( x ) − sinh 2 ( x ) = 1 {\displaystyle \cosh ^{2}(x)-\sinh ^{2}(x)=1} to find:
∫ d x a 2 + x 2 = ∫ a cosh u d u a 2 + a 2 sinh 2 u = ∫ cosh u d u 1 + sinh 2 u = ∫ cosh u cosh u d u = u + C = sinh − 1 x a + C . {\displaystyle {\begin{aligned}\int {\frac {dx}{\sqrt {a^{2}+x^{2}}}}&=\int {\frac {a\cosh u\,du}{\sqrt {a^{2}+a^{2}\sinh ^{2}u}}}\\[6pt]&=\int {\frac {\cosh {u}\,du}{\sqrt {1+\sinh ^{2}{u}}}}\\[6pt]&=\int {\frac {\cosh {u}}{\cosh u}}\,du\\[6pt]&=u+C\\[6pt]&=\sinh ^{-1}{\frac {x}{a}}+C.\end{aligned}}}
If desired, this result may be further transformed using other identities, such as using the relation sinh − 1 z = arsinh z = ln ( z + z 2 + 1 ) {\displaystyle \sinh ^{-1}{z}=\operatorname {arsinh} {z}=\ln(z+{\sqrt {z^{2}+1}})} : sinh − 1 x a + C = ln ( x a + x 2 a 2 + 1 ) + C = ln ( x + x 2 + a 2 a ) + C . {\displaystyle {\begin{aligned}\sinh ^{-1}{\frac {x}{a}}+C&=\ln \left({\frac {x}{a}}+{\sqrt {{\frac {x^{2}}{a^{2}}}+1}}\,\right)+C\\[6pt]&=\ln \left({\frac {x+{\sqrt {x^{2}+a^{2}}}}{a}}\,\right)+C.\end{aligned}}} | https://en.wikipedia.org/wiki/Trigonometric_substitution |
In mathematics , tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators , trigonometric tables were essential for navigation , science and engineering . The calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices .
Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of trigonometric functions is still used in computer graphics , where only modest accuracy may be required and speed is often paramount.
Another important application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called twiddle factors ) must be evaluated many times in a given transform, especially in the common case where many transforms of the same size are computed. In this case, calling generic library routines every time is unacceptably slow. One option is to call the library routines once, to build up a table of those trigonometric values that will be needed, but this requires significant memory to store the table. The other possibility, since a regular sequence of values is required, is to use a recurrence formula to compute the trigonometric values on the fly. Significant research has been devoted to finding accurate, stable recurrence schemes in order to preserve the accuracy of the FFT (which is very sensitive to trigonometric errors).
A trigonometry table is essentially a reference chart that presents the values of sine, cosine, tangent, and other trigonometric functions for various angles. These angles are usually arranged across the top row of the table, while the different trigonometric functions are labeled in the first column on the left. To locate the value of a specific trigonometric function at a certain angle, you would find the row for the function and follow it across to the column under the desired angle. [ 1 ]
Modern computers and calculators use a variety of techniques to provide trigonometric function values on demand for arbitrary angles (Kantabutra, 1996). One common method, especially on higher-end processors with floating-point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation , best uniform approximation, Padé approximation , and typically for higher or variable precisions, Taylor and Laurent series ) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction. Maintaining precision while performing such interpolation is nontrivial, but methods like Gal's accurate tables , Cody and Waite range reduction, and Payne and Hanek radian reduction algorithms can be used for this purpose. On simpler devices that lack a hardware multiplier , there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons.
The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a minimax approximation algorithm .
For very high precision calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean , which itself approximates the trigonometric function by the ( complex ) elliptic integral (Brent, 1976).
Trigonometric functions of angles that are rational multiples of 2π are algebraic numbers . The values for a/b·2π can be found by applying de Moivre's identity for n = a to a b th root of unity , which is also a root of the polynomial x b - 1 in the complex plane . For example, the cosine and sine of 2π ⋅ 5/37 are the real and imaginary parts , respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i, which is a root of the degree -37 polynomial x 37 − 1. For this case, a root-finding algorithm such as Newton's method is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for transcendental trigonometric constants, however.
Historically, the earliest method by which trigonometric tables were computed, and probably the most common until the advent of computers, was to repeatedly apply the half-angle and angle-addition trigonometric identities starting from a known value (such as sin(π/2) = 1, cos(π/2) = 0). This method was used by the ancient astronomer Ptolemy , who derived them in the Almagest , a treatise on astronomy . In modern form, the identities he derived are stated as follows (with signs determined by the quadrant in which x lies):
These were used to construct Ptolemy's table of chords , which was applied to astronomical problems.
Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and versine .
A quick, but inaccurate, algorithm for calculating a table of N approximations s n for sin (2 π n / N ) and c n for cos (2π n / N ) is:
for n = 0,..., N − 1, where d = 2π/ N .
This is simply the Euler method for integrating the differential equation :
with initial conditions s (0) = 0 and c (0) = 1, whose analytical solution is s = sin( t ) and c = cos( t ).
Unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/ N .
For example, for N = 256 the maximum error in the sine values is ~0.061 ( s 202 = −1.0368 instead of −0.9757). For N = 1024, the maximum error in the sine values is ~0.015 ( s 803 = −0.99321 instead of −0.97832), about 4 times smaller. If the sine and cosine values obtained were to be plotted, this algorithm would draw a logarithmic spiral rather than a circle.
A simple recurrence formula to generate trigonometric tables is based on Euler's formula and the relation:
This leads to the following recurrence to compute trigonometric values s n and c n as above:
for n = 0, ..., N − 1, where w r = cos(2π/ N ) and w i = sin(2π/ N ). These two starting trigonometric values are usually computed using existing library functions (but could also be found e.g. by employing Newton's method in the complex plane to solve for the primitive root of z N − 1).
This method would produce an exact table in exact arithmetic, but has errors in finite-precision floating-point arithmetic. In fact, the errors grow as O(ε N ) (in both the worst and average cases), where ε is the floating-point precision.
A significant improvement is to use the following modification to the above, a trick (due to Singleton [ 2 ] ) often used to generate trigonometric values for FFT implementations:
where α = 2 sin 2 (π/ N ) and β = sin(2π/ N ). The errors of this method are much smaller, O(ε √ N ) on average and O(ε N ) in the worst case, but this is still large enough to substantially degrade the accuracy of FFTs of large sizes. | https://en.wikipedia.org/wiki/Trigonometric_tables |
The trigonometry of a tetrahedron [ 1 ] explains the relationships between the lengths and various types of angles of a general tetrahedron .
The following are trigonometric quantities generally associated to a general tetrahedron:
Let X = P 1 P 2 P 3 P 4 ¯ {\displaystyle X={\overline {P_{1}P_{2}P_{3}P_{4}}}} be a general tetrahedron, where P 1 , P 2 , P 3 , P 4 {\displaystyle P_{1},P_{2},P_{3},P_{4}} are arbitrary points in three-dimensional space .
Furthermore, let e i j {\displaystyle e_{ij}} be the edge that joins P i {\displaystyle P_{i}} and P j {\displaystyle P_{j}} and let F i {\displaystyle F_{i}} be the face of the tetrahedron opposite the point P i {\displaystyle P_{i}} ; in other words:
where i , j , k , l ∈ { 1 , 2 , 3 , 4 } {\displaystyle i,j,k,l\in \{1,2,3,4\}} and i ≠ j ≠ k ≠ l {\displaystyle i\neq j\neq k\neq l} .
Define the following quantities:
Let Δ i {\displaystyle \Delta _{i}} be the area of the face F i {\displaystyle F_{i}} . Such area may be calculated by Heron's formula (if all three edge lengths are known):
or by the following formula (if an angle and two corresponding edges are known):
Let h i {\displaystyle h_{i}} be the altitude from the point P i {\displaystyle P_{i}} to the face F i {\displaystyle F_{i}} . The volume V {\displaystyle V} of the tetrahedron X {\displaystyle X} is given by the following formula: V = 1 3 Δ i h i {\displaystyle V={\frac {1}{3}}\Delta _{i}h_{i}} It satisfies the following relation: [ 2 ]
where Q i j = d i j 2 {\displaystyle Q_{ij}=d_{ij}^{2}} are the quadrances (length squared) of the edges.
Take the face F i {\displaystyle F_{i}} ; the edges will have lengths d j k , d j l , d k l {\displaystyle d_{jk},d_{jl},d_{kl}} and the respective opposite angles are given by α l , i , α k , i , α j , i {\displaystyle \alpha _{l,i},\alpha _{k,i},\alpha _{j,i}} .
The usual laws for planar trigonometry of a triangle hold for this triangle.
Consider the projective (spherical) triangle at the point P i {\displaystyle P_{i}} ; the vertices of this projective triangle are the three lines that join P i {\displaystyle P_{i}} with the other three vertices of the tetrahedron. The edges will have spherical lengths α i , j , α i , k , α i , l {\displaystyle \alpha _{i,j},\alpha _{i,k},\alpha _{i,l}} and the respective opposite spherical angles are given by θ i j , θ i k , θ i l {\displaystyle \theta _{ij},\theta _{ik},\theta _{il}} .
The usual laws for spherical trigonometry hold for this projective triangle.
Take the tetrahedron X {\displaystyle X} , and consider the point P i {\displaystyle P_{i}} as an apex. The Alternating sines theorem is given by the following identity: sin ( α j , l ) sin ( α k , j ) sin ( α l , k ) = sin ( α j , k ) sin ( α k , l ) sin ( α l , j ) {\displaystyle \sin(\alpha _{j,l})\sin(\alpha _{k,j})\sin(\alpha _{l,k})=\sin(\alpha _{j,k})\sin(\alpha _{k,l})\sin(\alpha _{l,j})} One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.
Putting any of the four vertices in the role of O yields four such identities, but at most three of them are independent; if the "clockwise" sides of three of the four identities are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity.
Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by the sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional. [ 3 ]
See: Law of sines
The law of cosines for the tetrahedron [ 4 ] relates the areas of each face of the tetrahedron and the dihedral angles about a point. It is given by the following identity:
Take the general tetrahedron X {\displaystyle X} and project the faces F i , F j , F k {\displaystyle F_{i},F_{j},F_{k}} onto the plane with the face F l {\displaystyle F_{l}} . Let c i j = cos θ i j {\displaystyle c_{ij}=\cos \theta _{ij}} .
Then the area of the face F l {\displaystyle F_{l}} is given by the sum of the projected areas, as follows: Δ l = Δ i c j k + Δ j c i k + Δ k c i j {\displaystyle \Delta _{l}=\Delta _{i}c_{jk}+\Delta _{j}c_{ik}+\Delta _{k}c_{ij}} By substitution of i , j , k , l ∈ { 1 , 2 , 3 , 4 } {\displaystyle i,j,k,l\in \{1,2,3,4\}} with each of the four faces of the tetrahedron, one obtains the following homogeneous system of linear equations: { − Δ 1 + Δ 2 c 34 + Δ 3 c 24 + Δ 4 c 23 = 0 Δ 1 c 34 − Δ 2 + Δ 3 c 14 + Δ 4 c 13 = 0 Δ 1 c 24 + Δ 2 c 14 − Δ 3 + Δ 4 c 12 = 0 Δ 1 c 23 + Δ 2 c 13 + Δ 3 c 12 − Δ 4 = 0 {\displaystyle {\begin{cases}-\Delta _{1}+\Delta _{2}c_{34}+\Delta _{3}c_{24}+\Delta _{4}c_{23}=0\\\Delta _{1}c_{34}-\Delta _{2}+\Delta _{3}c_{14}+\Delta _{4}c_{13}=0\\\Delta _{1}c_{24}+\Delta _{2}c_{14}-\Delta _{3}+\Delta _{4}c_{12}=0\\\Delta _{1}c_{23}+\Delta _{2}c_{13}+\Delta _{3}c_{12}-\Delta _{4}=0\end{cases}}} This homogeneous system will have solutions precisely when: | − 1 c 34 c 24 c 23 c 34 − 1 c 14 c 13 c 24 c 14 − 1 c 12 c 23 c 13 c 12 − 1 | = 0 {\displaystyle {\begin{vmatrix}-1&c_{34}&c_{24}&c_{23}\\c_{34}&-1&c_{14}&c_{13}\\c_{24}&c_{14}&-1&c_{12}\\c_{23}&c_{13}&c_{12}&-1\end{vmatrix}}=0} By expanding this determinant, one obtains the relationship between the dihedral angles of the tetrahedron, [ 1 ] as follows: 1 − ∑ 1 ≤ i < j ≤ 4 c i j 2 + ∑ j = 2 k ≠ l ≠ j 4 c 1 j 2 c k l 2 = 2 ( ∑ i = 1 j ≠ k ≠ l ≠ i 4 c i j c i k c i l + ∑ 2 ≤ j < k ≤ 4 l ≠ j , k c 1 j c 1 k c j l c k l ) {\displaystyle 1-\sum _{1\leq i<j\leq 4}c_{ij}^{2}+\sum _{j=2 \atop k\neq l\neq j}^{4}c_{1j}^{2}c_{kl}^{2}=2\left(\sum _{i=1 \atop j\neq k\neq l\neq i}^{4}c_{ij}c_{ik}c_{il}+\sum _{2\leq j<k\leq 4 \atop l\neq j,k}c_{1j}c_{1k}c_{jl}c_{kl}\right)}
Take the general tetrahedron X {\displaystyle X} and let P i j {\displaystyle P_{ij}} be the point on the edge e i j {\displaystyle e_{ij}} and P k l {\displaystyle P_{kl}} be the point on the edge e k l {\displaystyle e_{kl}} such that the line segment P i j P k l ¯ {\displaystyle {\overline {P_{ij}P_{kl}}}} is perpendicular to both e i j {\displaystyle e_{ij}} & e k l {\displaystyle e_{kl}} . Let R i j {\displaystyle R_{ij}} be the length of the line segment P i j P k l ¯ {\displaystyle {\overline {P_{ij}P_{kl}}}} .
To find R i j {\displaystyle R_{ij}} : [ 1 ]
First, construct a line through P k {\displaystyle P_{k}} parallel to e i l {\displaystyle e_{il}} and another line through P i {\displaystyle P_{i}} parallel to e k l {\displaystyle e_{kl}} . Let O {\displaystyle O} be the intersection of these two lines. Join the points O {\displaystyle O} and P j {\displaystyle P_{j}} . By construction, O P i P l P k ¯ {\displaystyle {\overline {OP_{i}P_{l}P_{k}}}} is a parallelogram and thus O P k P i ¯ {\displaystyle {\overline {OP_{k}P_{i}}}} and O P l P i ¯ {\displaystyle {\overline {OP_{l}P_{i}}}} are congruent triangles. Thus, the tetrahedron X {\displaystyle X} and Y = O P i P j P k ¯ {\displaystyle Y={\overline {OP_{i}P_{j}P_{k}}}} are equal in volume.
As a consequence, the quantity R i j {\displaystyle R_{ij}} is equal to the altitude from the point P k {\displaystyle P_{k}} to the face O P i P j ¯ {\displaystyle {\overline {OP_{i}P_{j}}}} of the tetrahedron Y {\displaystyle Y} ; this is shown by translation of the line segment P i j P k l ¯ {\displaystyle {\overline {P_{ij}P_{kl}}}} .
By the volume formula, the tetrahedron Y {\displaystyle Y} satisfies the following relation: 3 V = R i j × Δ ( O P i P j ¯ ) {\displaystyle 3V=R_{ij}\times \Delta ({\overline {OP_{i}P_{j}}})} where Δ ( O P i P j ¯ ) {\displaystyle \Delta ({\overline {OP_{i}P_{j}}})} is the area of the triangle O P i P j ¯ {\displaystyle {\overline {OP_{i}P_{j}}}} . Since the length of the line segment O P i ¯ {\displaystyle {\overline {OP_{i}}}} is equal to d k l {\displaystyle d_{kl}} (as O P i P l P k ¯ {\displaystyle {\overline {OP_{i}P_{l}P_{k}}}} is a parallelogram): Δ ( O P i P j ¯ ) = 1 2 d i j d k l sin λ {\displaystyle \Delta ({\overline {OP_{i}P_{j}}})={\frac {1}{2}}d_{ij}d_{kl}\sin \lambda } where λ = ∠ O P i P j {\displaystyle \lambda =\angle OP_{i}P_{j}} . Thus, the previous relation becomes: 6 V = R i j d i j d k l sin λ {\displaystyle 6V=R_{ij}d_{ij}d_{kl}\sin \lambda } To obtain sin λ {\displaystyle \sin \lambda } , consider two spherical triangles:
Combining the two equations gives the following result: cos α i , k sin α k , j + cos λ sin α i , j = cos α i , l ( cos α i , j sin α k , j + sin α i , j cos α k , j ) = cos α i , l sin α l , j {\displaystyle \cos \alpha _{i,k}\sin \alpha _{k,j}+\cos \lambda \sin \alpha _{i,j}=\cos \alpha _{i,l}\left(\cos \alpha _{i,j}\sin \alpha _{k,j}+\sin \alpha _{i,j}\cos \alpha _{k,j}\right)=\cos \alpha _{i,l}\sin \alpha _{l,j}}
Making cos λ {\displaystyle \cos \lambda } the subject: cos λ = cos α i , l sin α l , j sin α i , j − cos α i , k sin α k , j sin α i , j {\displaystyle \cos \lambda =\cos \alpha _{i,l}{\frac {\sin \alpha _{l,j}}{\sin \alpha _{i,j}}}-\cos \alpha _{i,k}{\frac {\sin \alpha _{k,j}}{\sin \alpha _{i,j}}}} Thus, using the cosine law and some basic trigonometry: cos λ = d i j 2 + d i k 2 − d j k 2 2 d i j d i k d i k d k l − d i j 2 + d i l 2 − d j l 2 2 d i j d i l d i l d k l = d i k 2 + d j l 2 − d i l 2 − d j k 2 2 d i j d k l {\displaystyle \cos \lambda ={\frac {d_{ij}^{2}+d_{ik}^{2}-d_{jk}^{2}}{2d_{ij}d_{ik}}}{\frac {d_{ik}}{d_{kl}}}-{\frac {d_{ij}^{2}+d_{il}^{2}-d_{jl}^{2}}{2d_{ij}d_{il}}}{\frac {d_{il}}{d_{kl}}}={\frac {d_{ik}^{2}+d_{jl}^{2}-d_{il}^{2}-d_{jk}^{2}}{2d_{ij}d_{kl}}}} Thus: sin λ = 1 − ( d i k 2 + d j l 2 − d i l 2 − d j k 2 2 d i j d k l ) 2 = 4 d i j 2 d k l 2 − ( d i k 2 + d j l 2 − d i l 2 − d j k 2 ) 2 2 d i j d k l {\displaystyle \sin \lambda ={\sqrt {1-\left({\frac {d_{ik}^{2}+d_{jl}^{2}-d_{il}^{2}-d_{jk}^{2}}{2d_{ij}d_{kl}}}\right)^{2}}}={\frac {\sqrt {4d_{ij}^{2}d_{kl}^{2}-(d_{ik}^{2}+d_{jl}^{2}-d_{il}^{2}-d_{jk}^{2})^{2}}}{2d_{ij}d_{kl}}}} So: R i j = 12 V 4 d i j 2 d k l 2 − ( d i k 2 + d j l 2 − d i l 2 − d j k 2 ) 2 {\displaystyle R_{ij}={\frac {12V}{\sqrt {4d_{ij}^{2}d_{kl}^{2}-(d_{ik}^{2}+d_{jl}^{2}-d_{il}^{2}-d_{jk}^{2})^{2}}}}} R i k {\displaystyle R_{ik}} and R i l {\displaystyle R_{il}} are obtained by permutation of the edge lengths.
Note that the denominator is a re-formulation of the Bretschneider-von Staudt formula , which evaluates the area of a general convex quadrilateral. | https://en.wikipedia.org/wiki/Trigonometry_of_a_tetrahedron |
In geometry , the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons . [ 1 ] It consists of equilateral triangles and regular hexagons , arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling . Two hexagons and two triangles alternate around each vertex , and its edges form an infinite arrangement of lines . Its dual is the rhombille tiling . [ 2 ]
This pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi . [ 3 ] The pattern has long been used in Japanese basketry , where it is called kagome . The Japanese term for this pattern has been taken up in physics, where it is called a kagome lattice . It occurs also in the crystal structures of certain minerals. Conway calls it a hexadeltille , combining alternate elements from a hexagonal tiling (hextille) and triangular tiling (deltille). [ 4 ]
Kagome ( Japanese : 籠目 ) is a traditional Japanese woven bamboo pattern; its name is composed from the words kago , meaning "basket", and me , meaning "eye(s)", referring to the pattern of holes in a woven basket.
The kagome pattern is common in bamboo weaving in East Asia. In 2022, archaeologists found bamboo weaving remains at the Dongsunba ruins in Chongqing, China, 200 BC. After 2200 years, the kagome pattern is still clear. [ 5 ] [ 6 ]
It is a woven arrangement of laths composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The woven process gives the Kagome a chiral wallpaper group symmetry, p6 (632).
The term kagome lattice was coined by Japanese physicist Kôdi Husimi , and first appeared in a 1951 paper by his assistant Ichirō Shōji. [ 7 ] The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling.
Despite the name, these crossing points do not form a mathematical lattice .
A related three dimensional structure formed by the vertices and edges of the quarter cubic honeycomb , filling space by regular tetrahedra and truncated tetrahedra , has been called a hyper-kagome lattice . [ 8 ] It is represented by the vertices and edges of the quarter cubic honeycomb , filling space by regular tetrahedra and truncated tetrahedra . It contains four sets of parallel planes of points and lines, each plane being a two dimensional kagome lattice. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an orthorhombic-kagome lattice . [ 8 ] The trihexagonal prismatic honeycomb represents its edges and vertices.
Some minerals , namely jarosites and herbertsmithite , contain two-dimensional layers or three-dimensional kagome lattice arrangement of atoms in their crystal structure . These minerals display novel physical properties connected with geometrically frustrated magnetism . For instance, the spin arrangement of the magnetic ions in Co 3 V 2 O 8 rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures. [ 9 ] Quantum magnets realized on Kagome metals have been discovered to exhibit many unexpected electronic and magnetic phenomena. [ 10 ] [ 11 ] [ 12 ] [ 13 ] It is also proposed that SYK behavior can be observed in two dimensional kagome lattice with impurities. [ 14 ]
The term is much in use nowadays in the scientific literature, especially by theorists studying the magnetic properties of a theoretical kagome lattice.
See also: Kagome crests .
The trihexagonal tiling has Schläfli symbol of r{6,3}, or Coxeter diagram , , symbolizing the fact that it is a rectified hexagonal tiling , {6,3}. Its symmetries can be described by the wallpaper group p6mm, (*632), [ 15 ] and the tiling can be derived as a Wythoff construction within the reflectional fundamental domains of this group . The trihexagonal tiling is a quasiregular tiling , alternating two types of polygons, with vertex configuration (3.6) 2 . It is also a uniform tiling , one of eight derived from the regular hexagonal tiling.
There are two distinct uniform colorings of a trihexagonal tiling. Naming the colors by indices on the 4 faces around a vertex (3.6.3.6): 1212, 1232. [ 1 ] The second is called a cantic hexagonal tiling , h 2 {6,3}, with two colors of triangles, existing in p3m1 (*333) symmetry.
The trihexagonal tiling can be used as a circle packing , placing equal diameter circles at the center of every point. [ 16 ] Every circle is in contact with 4 other circles in the packing ( kissing number ).
The trihexagonal tiling can be geometrically distorted into topologically equivalent tilings of lower symmetry. [ 1 ] In these variants of the tiling, the edges do not necessarily line up to form straight lines.
The trihexagonal tiling exists in a sequence of symmetries of quasiregular tilings with vertex configurations (3. n ) 2 , progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of * n 32 all of these tilings are wythoff construction within a fundamental domain of symmetry, with generator points at the right angle corner of the domain. [ 17 ] [ 18 ]
There are 2 regular complex apeirogons , sharing the vertices of the trihexagonal tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p { q } r are constrained by: 1/ p + 2/ q + 1/ r = 1. Edges have p vertices arranged like a regular polygon , and vertex figures are r -gonal. [ 19 ]
The first is made of triangular edges, two around every vertex, second has hexagonal edges, two around every vertex. | https://en.wikipedia.org/wiki/Trihexagonal_tiling |
The trihydrogen cation or protonated molecular hydrogen ( IUPAC name : hydrogen onium ion ) is a cation (positive ion ) with formula H + 3 , consisting of three hydrogen nuclei ( protons ) sharing two electrons .
The trihydrogen cation is one of the most abundant ions in the universe. It is stable in the interstellar medium (ISM) due to the low temperature and low density of interstellar space. The role that H + 3 plays in the gas-phase chemistry of the ISM is unparalleled by any other polyatomic ion .
The trihydrogen cation is the simplest triatomic molecule , because its two electrons are the only valence electrons in the system. It is also the simplest example of a three-center two-electron bond system.
H + 3 was first discovered by J. J. Thomson in 1911. [ 1 ] While using an early form of mass spectrometry to study the resultant species of plasma discharges, he discovered a large abundance of a polyatomic ion with a mass-to-charge ratio of 3. He stated that the only two possibilities were C 4+ or H + 3 . Since the signal grew stronger in pure hydrogen gas, he correctly assigned the species as H + 3 .
The formation pathway was discovered by Hogness & Lunn in 1925. [ 2 ] They also used an early form of mass spectrometry to study hydrogen discharges. They found that as the pressure of hydrogen increased, the amount of H + 3 increased linearly and the amount of H + 2 decreased linearly. In addition, there was little H + at any pressure. These data suggested the proton exchange formation pathway discussed below.
In 1961, Martin et al. first suggested that H + 3 may be present in interstellar space given the large amount of hydrogen in interstellar space and its reaction pathway was exothermic (~1.5 eV ). [ 3 ] This led to the suggestion of Watson and Herbst & Klemperer in 1973 that H + 3 is responsible for the formation of many observed molecular ions. [ 4 ] [ 5 ]
It was not until 1980 that the first spectrum of H + 3 was discovered by Takeshi Oka, [ 6 ] which was of the ν 2 fundamental band (see #Spectroscopy ) using a technique called frequency modulation detection. This started the search for extraterrestrial H + 3 . Emission lines were detected in the late 1980s and early 1990s in the ionospheres of Jupiter , Saturn , and Uranus . [ 7 ] [ 8 ] [ 9 ] In the textbook by Bunker and Jensen [ 10 ] Figure 1.1 reproduces part of the ν 2 emission band from a region of auroral activity in the upper atmosphere of Jupiter, [ 11 ] and its Table 12.3 lists the transition wavenumbers of
the lines in the band observed by Oka [ 6 ] with their assignments.
In 1996, H + 3 was finally detected in the interstellar medium (ISM) by Geballe & Oka in two molecular interstellar clouds in the sightlines GL2136 and W33A. [ 12 ] In 1998, H + 3 was unexpectedly detected by McCall et al. in a diffuse interstellar cloud in the sightline Cygnus OB2#12 . [ 13 ] In 2006 Oka announced that H + 3 was ubiquitous in interstellar medium, and that the Central Molecular Zone contained a million times the concentration of ISM generally. [ 14 ]
The three hydrogen atoms in the molecule form an equilateral triangle , with a bond length of 0.90 Å on each side. The bonding among the atoms is a three-center two-electron bond , a delocalized resonance hybrid type of structure. The strength of the bond has been calculated to be around 4.5 eV (104 kcal/mol). [ 15 ]
In theory, the cation has 10 isotopologues , resulting from the replacement of one or more protons by nuclei of the other hydrogen isotopes ; namely, deuterium nuclei ( deuterons , 2 H + ) or tritium nuclei ( tritons , 3 H + ). Some of them have been detected in interstellar clouds. [ 16 ] They differ in the atomic mass number A and the number of neutrons N :
The deuterium isotopologues have been implicated in the fractionation of deuterium in dense interstellar cloud cores. [ 17 ]
The main pathway for the production of H + 3 is by the reaction of H + 2 and H 2 . [ 18 ]
The concentration of H + 2 is what limits the rate of this reaction in nature - the only known natural source of it is via ionization of H 2 by a cosmic ray in interstellar space:
The cosmic ray has so much energy, it is almost unaffected by the relatively small energy transferred to the hydrogen when ionizing an H 2 molecule. In interstellar clouds, cosmic rays leave behind a trail of H + 2 , and therefore H + 3 . In laboratories, H + 3 is produced by the same mechanism in plasma discharge cells, with the discharge potential providing the energy to ionize the H 2 .
There are many destruction reactions for H + 3 . The dominant destruction pathway in dense interstellar clouds is by proton transfer with a neutral collision partner. The most likely candidate for a destructive collision partner is the second most abundant molecule in space, CO .
The significant product of this reaction is HCO + , an important molecule for interstellar chemistry. Its strong dipole and high abundance make it easily detectable by radioastronomy . H + 3 can also react with atomic oxygen to form OH + and H 2 .
OH + then usually reacts with more H 2 to create further hydrogenated molecules.
At this point, the reaction between OH + 3 and H 2 is no longer exothermic in interstellar clouds. The most common destruction pathway for OH + 3 is dissociative recombination , yielding four possible sets of products: H 2 O + H, OH + H 2 , OH + 2H, and O + H 2 + H . While water is a possible product of this reaction, it is not a very efficient product. Different experiments have suggested that water is created anywhere from 5–33% of the time. Water formation on grains is still considered the primary source of water in the interstellar medium.
The most common destruction pathway of H + 3 in diffuse interstellar clouds is dissociative recombination. This reaction has multiple products. The major product is dissociation into three hydrogen atoms, which occurs roughly 75% of the time. The minor product is H 2 and H, which occurs roughly 25% of the time. [ 18 ]
The protons of [ 1 H 3 ] + can be in two different spin configurations , called ortho and para . Ortho - H + 3 has all three proton spins parallel, yielding a total nuclear spin of 3/2. Para - H + 3 has two proton spins parallel while the other is anti-parallel, yielding a total nuclear spin of 1/2.
The most abundant molecule in dense interstellar clouds is 1 H 2 which also has ortho and para states , with total nuclear spins 1 and 0, respectively. When a H + 3 molecule collides with a H 2 molecule, a proton transfer can take place. The transfer still yields a H + 3 molecule and a H 2 molecule, but can potentially change the total nuclear spin of the two molecules depending on the nuclear spins of the protons. When an ortho - H + 3 and a para - H 2 collide, the result may be a para - H + 3 and an ortho - H 2 . [ 18 ]
The spectroscopy of H + 3 is challenging. The pure rotational spectrum is exceedingly weak. [ 19 ] Ultraviolet light is too energetic and would dissociate the molecule. Rovibronic (infrared) spectroscopy provides the ability to observe H + 3 . Rovibronic spectroscopy is possible with H + 3 because one of the vibrational modes of H + 3 , the ν 2 asymmetric bend mode (see example of ν 2 ) has a weak transition dipole moment. Since Oka's initial spectrum, [ 6 ] over 900 absorption lines have been detected in the infrared region. H + 3 emission lines have also been found by observing the atmospheres of the Jovian planets. H + 3 emission lines are found by observing molecular hydrogen and finding a line that cannot be attributed to molecular hydrogen.
H + 3 has been detected in two types of the universe environments: jovian planets and interstellar clouds . In jovian planets, it has been detected in the planets' ionospheres , the region where the Sun 's high energy radiation ionizes the particles in the planets' atmospheres . Since there is a high level of H 2 in these atmospheres, this radiation can produce a significant amount of H + 3 . Also, with a broadband source like the Sun, there is plenty of radiation to pump the H + 3 to higher energy states from which it can relax by spontaneous emission .
The detection of the first H + 3 emission lines was reported in 1989 by Drossart et al. , [ 7 ] found in the ionosphere of Jupiter. Drossart found a total of 23 H + 3 lines with a column density of 1.39 × 10 9 /cm 2 . Using these lines, they were able to assign a temperature to the H + 3 of around 1,100 K (830 °C), which is comparable to temperatures determined from emission lines of other species like H 2 . In 1993, H + 3 was found in Saturn by Geballe et al. [ 8 ] and in Uranus by Trafton et al. [ 9 ]
H + 3 was not detected in the interstellar medium until 1996, when Geballe & Oka reported the detection of H + 3 in two molecular cloud sightlines, GL 2136 and W33A . [ 12 ] Both sources had temperatures of H + 3 of about 35 K (−238 °C) and column densities of about 10 14 /cm 2 . Since then, H + 3 has been detected in numerous other molecular cloud sightlines, such as AFGL 2136 , [ 20 ] Mon R2 IRS 3 , [ 20 ] GCS 3–2 , [ 21 ] GC IRS 3 , [ 21 ] and LkHα 101 . [ 22 ]
Unexpectedly, three H + 3 lines were detected in 1998 by McCall et al. in the diffuse interstellar cloud sightline of Cyg OB2 No. 12 . [ 13 ] Before 1998, the density of H 2 was thought to be too low to produce a detectable amount of H + 3 . McCall detected a temperature of ~27 K (−246 °C) and a column density of ~10 14 /cm 2 , the same column density as Geballe & Oka . Since then, H + 3 has been detected in many other diffuse cloud sightlines, such as GCS 3–2, [ 21 ] GC IRS 3, [ 21 ] and ζ Persei . [ 23 ]
To approximate the path length of H + 3 in these clouds, Oka [ 24 ] used the steady-state model to determine the predicted number densities in diffuse and dense clouds. As explained above, both diffuse and dense clouds have the same formation mechanism for H + 3 , but different dominating destruction mechanisms. In dense clouds, proton transfer with CO is the dominating destruction mechanism. This corresponds to a predicted number density of 10 −4 cm −3 in dense clouds.
In diffuse clouds, the dominating destruction mechanism is dissociative recombination. This corresponds to a predicted number density of 10 −6 /cm 3 in diffuse clouds. Therefore, since column densities for diffuse and dense clouds are roughly the same order of magnitude, diffuse clouds must have a path length 100 times greater than that for dense clouds. Therefore, by using H + 3 as a probe of these clouds, their relative sizes can be determined. | https://en.wikipedia.org/wiki/Trihydrogen_cation |
Trihydrogen oxide is a predicted inorganic compound of hydrogen and oxygen with the chemical formula H 3 O . [ 2 ] [ 3 ] This is still a hypothetical compound , one of the unstable hydrogen polyoxides . It is hypothesized that the compound could constitute a thin layer of metallic liquid around the cores of Uranus and Neptune , and that this could be the source of their magnetic fields. [ 4 ] Calculations indicate the stability of H 3 O in solid, superionic, and fluid metallic states at the deep interior conditions of these planets.
Trihydrogen oxide has not been observed experimentally as of 2023, but its existence is predicted by calculation using the CALYPSO method. [ 5 ] The compound should be stable in the pressure range 450–600 GPa and could be produced by the reaction:
The compound is considered not a true molecular trihydrogen oxide compound. Instead, each oxygen atom is linked by a strong (covalent) bond to only two hydrogen atoms, as a water molecule, and there are molecules of dihydrogen inserted in the voids of the water molecules network. [ 6 ] Structurally, it is thus a 2(H 2 O)·H 2 stoichiometric combination.
At 600 GPa and 7000 K, the compound density is calculated to be 4.3 g/cm 3 . Molecular dynamics simulations were carried out at constant density for different temperatures: [ 6 ]
The magnetic fields of both Uranus and Neptune are special—non-dipolar and non-axisymmetric. This fact can be explained if the magnetic fields are produced by dynamo effect within a sufficiently thin conductive layer. However, the origin of the fields is still problematic because the cores of these planets are probably solid (thus too rigid), and the thick mantles of ice are too poorly conductive to create the effect. [ 7 ] [ 8 ] | https://en.wikipedia.org/wiki/Trihydrogen_oxide |
In chemistry, triiodide usually refers to the triiodide ion, I − 3 . This anion, one of the polyhalogen ions , is composed of three iodine atoms. It is formed by combining aqueous solutions of iodide salts and iodine . Some salts of the anion have been isolated, including thallium(I) triiodide (Tl + [I 3 ] − ) and ammonium triiodide ([NH 4 ] + [I 3 ] − ). Triiodide is observed to be a red colour in solution. [ 1 ]
Other chemical compounds with "triiodide" in their name may contain three iodide centers that are not bonded to each other as the triiodide ion, but exist instead as separate iodine atoms or iodide ions. Examples include nitrogen triiodide (NI 3 ) and phosphorus triiodide (PI 3 ), where individual iodine atoms are covalently bonded to a central atom. As some cations have the theoretical possibility to form compounds with both triiodide and iodide ions, such as ammonium , compounds containing iodide anions in a 3:1 stoichiometric ratio should only be referred to as triiodides in cases where the triiodide anion is present. It may also be helpful to indicate the oxidation number of a metal cation, where appropriate. For example, the covalent molecule gallium triiodide (Ga 2 I 6 ) is better referred to as gallium(III) iodide to emphasise that it is iodide anions that are present, and not triiodide.
The following exergonic equilibrium gives rise to the triiodide ion :
In this reaction, iodide is viewed as a Lewis base , and the iodine is a Lewis acid . The process is analogous to the reaction of S 8 with sodium sulfide (which forms polysulfides ) except that the higher polyiodides have branched structures. [ 2 ]
The ion is linear and symmetrical. According to valence shell electron pair repulsion theory , the central iodine atom has three equatorial lone pairs, and the terminal iodine atoms are bonded axially in a linear fashion, due to the three lone pairs bonding to the central iodine-atom. In the molecular orbital model , a common explanation for the hypervalent bonding on the central iodine involves a three-center four-electron bond . The I−I bond is longer than in diatomic iodine, I 2 .
In ionic compounds , the bond lengths and angles of triiodide vary depending on the nature of the cation . The triiodide anion is easily polarised and in many salts, one I−I bond becomes shorter than the other. Only in combination with large cations, e.g. a quaternary ammonium such as [N(CH 3 ) 4 ] + , may the triiodide remain roughly symmetrical. [ 3 ]
In solution phase, the bond lengths and angles of triiodide vary depending on the nature of solvent . The protic solvents tend to localize the triiodide anion's excess charge, resulting in the triiodide anion's asymmetric structure. [ 4 ] [ 5 ] For example, the triiodide anion in methanol has an asymmetric bent structure with a charge localized on the longer end of the anion. [ 6 ]
The dimensions of the triiodide [I a −I b −I c ] − bonds in a few sample compounds are shown below:
The triiodide ion is the simplest polyiodide ; several higher polyiodides exist. In solution, it appears yellow in low concentrations, and brown at higher concentrations. The triiodide ion is responsible for the well-known blue-black color which arises when iodine solutions interact with starch . Iodide does not react with starch; nor do solutions of iodine in nonpolar solvents .
Lugol's iodine contains potassium iodide and a stoichiometric amount of elemental iodine, so that significant amounts of triiodide ion exist in this solution. Tincture of iodine , although nominally a solution of elemental iodine in ethanol, also contains significant amounts of triiodide, due to its content of both iodide and water.
Triiodide is a model system in photochemistry . Its reaction mechanism has been studied in gas phase, solution and the solid state. In gas phase, the reaction proceeds in multiple pathways that include iodine molecule, metastable ions and iodine radicals as photoproducts, which are formed by two-body and three-body dissociation . [ 7 ] [ 8 ] In condensed phases, due to confinement, geminate recombination is more common. In solution, only two-body dissociation of triiodide has been observed. [ 9 ] [ 10 ] In the protic solvents, an iodine atom at the shorter end of the triiodide anion dissociates upon photoexcitation showing two-body dissociation. [ 6 ] In the solid state, the triiodide photochemistry has been studied in compounds involving quaternary ammonium cations , such as tetrabutylammonium triiodide . [ 11 ] It has been shown that the solid state photoreaction mechanism depends on the light wavelength, yielding fast recovery in a few picoseconds [ 12 ] or going through a two-stage process that involves the formation and break-up of a tetraiodide intermediate on longer timescales. [ 13 ] Besides, triiodide photochemistry is an important contributor in the environmental cycle of iodine . [ 14 ] Because of the presence of heavy iodine atoms and the well-calibrated chemical pathways, triiodide has also become a computational benchmark system for relativistic quantum chemistry . [ 15 ]
The redox reactions of triiodide and iodide has been proposed as critical steps in dye-sensitized solar cells . [ 16 ] and rechargeable batteries . [ 17 ] | https://en.wikipedia.org/wiki/Triiodide |
Triiodothyronine , also known as T 3 , is a thyroid hormone . It affects almost every physiological process in the body, including growth and development , metabolism , body temperature , and heart rate . [ 1 ]
Production of T 3 and its prohormone thyroxine (T 4 ) is activated by thyroid-stimulating hormone (TSH), which is released from the anterior pituitary gland. This pathway is part of a closed-loop feedback process: Elevated concentrations of T 3 , and T 4 in the blood plasma inhibit the production of TSH in the anterior pituitary gland. As concentrations of these hormones decrease, the anterior pituitary gland increases production of TSH, and by these processes, a feedback control system stabilizes the level of thyroid hormones in the bloodstream .
At the cellular level, T 3 is the body's more active and potent thyroid hormone. [ 2 ] T 3 helps deliver oxygen and energy to all of the body's cells, its effects on target tissues being roughly four times more potent than those of T 4 . [ 2 ] Of the thyroid hormone that is produced, just about 20% is T 3 , whereas 80% is produced as T 4 . Roughly 85% of the circulating T 3 is later formed in the liver and anterior pituitary by removal of the iodine atom from the carbon atom number five of the outer ring of T 4 . In any case, the concentration of T 3 in the human blood plasma is about one-fortieth that of T 4 . The half-life of T 3 is about 2.5 days. [ 3 ] The half-life of T 4 is about 6.5 days. [ 4 ] T 3 levels start to rise 45 minutes after administration and peak at about 2.5 hours. Although manufacturer of Cytomel states half-life to be 2.5 days the half-life variability is great and can vary depending on the thyroid status of the patient. Newer studies have found the pharmakokinetics of T 3 to be complex and the half-life to vary between 10 – 22 hours. [ 5 ]
T 3 is the more metabolically active hormone produced from T 4 . T 4 is deiodinated by three deiodinase enzymes to produce the more-active triiodothyronine:
T 4 is synthesised in the thyroid follicular cells as follows.
The thyroid gland also produces small amounts of T 3 directly. In the follicular lumen , tyrosine residues become iodinated. This reaction requires hydrogen peroxide . Iodine bonds carbon 3 or carbon 5 of tyrosine residues of thyroglobulin in a process called organification of iodine. The iodination of specific tyrosines yields monoiodotyrosine (MIT) and diiodotyrosine (DIT). One MIT and one DIT are enzymatically coupled to form T 3 . The enzyme is thyroid peroxidase .
The small amount of T 3 could be important because different tissues have different sensitivities to T 4 due to differences in deiodinase ubiquitination in different tissues. [ 7 ] This once again raises the question if T 3 should be included in thyroid hormone replacement therapy (THRT).
T 3 and T 4 bind to nuclear receptors ( thyroid hormone receptors ). [ 8 ] T 3 and T 4 , although being lipophilic, are not able to passively diffuse through the phospholipid bilayers of target cells, [ 9 ] instead relying on transmembrane iodothyronine transporters . The lipophilicity of T 3 and T 4 requires their binding to the protein carrier thyroid-binding protein (TBG) ( thyroxine-binding globulins , thyroxine binding prealbumins , and albumins ) for transport in the blood. The thyroid receptors bind to response elements in gene promoters, thus enabling them to activate or inhibit transcription. The sensitivity of a tissue to T 3 is modulated through the thyroid receptors.
T 3 and T 4 are carried in the blood, bound to plasma proteins. This has the effect of increasing the half-life of the hormone and decreasing the rate at which it is taken up by peripheral tissues. There are three main proteins that the two hormones are bound to. Thyroxine-binding globulin (TBG) is a glycoprotein that has a higher affinity for T 4 than for T 3 . Transthyretin is also a glycoprotein, but only carries T 4 , with hardly any affinity at all for T 3 . Finally, both hormones bind with a low affinity to serum albumin , but, due to the large availability of albumin, it has a high capacity.
The saturation of binding spots on thyronine-binding globulin (TBG) by endogenous T 3 can be estimated by the triiodothyronine resin uptake test . The test is performed by taking a blood sample , to which an excess of radioactive exogenous T 3 is added, followed by a resin that also binds T 3 . A fraction of the radioactive T 3 binds to sites on TBG not already occupied by endogenous thyroid hormone, and the remainder binds to the resin. The amount of labeled hormones bound to the resin is then subtracted from the total that was added, with the remainder thus being the amount that was bound to the unoccupied binding sites on TBG. [ 11 ]
T 3 increases the basal metabolic rate and, thus, increases the body's oxygen and energy consumption. The basal metabolic rate is the minimal caloric requirement needed to sustain life in a resting individual. T 3 acts on the majority of tissues within the body, with a few exceptions including the spleen. It increases the synthesis and activity of the Na + /K + -ATPase (which normally constitutes a substantial fraction of total cellular ATP expenditure) without disrupting transmembrane ion balance. [ 12 ] In general, it increases the turnover of different endogenous macromolecules by increasing their synthesis and degradation.
Thyroid hormones are essential for normal growth and skeletal maturation. [ 13 ] They potentiate the effect of growth hormone and somatomedins to promote bone growth , epiphysial closure and bone maturation . [ 12 ] [ 13 ]
T 3 stimulates the production of RNA polymerase I and II and, therefore, increases the rate of protein synthesis. It also increases the rate of protein degradation, and, in excess, the rate of protein degradation exceeds the rate of protein synthesis. In such situations, the body may go into negative ion balance. [ further explanation needed ]
T 3 stimulates the breakdown of cholesterol and increases the number of LDL receptors, thereby increasing the rate of lipolysis .
T 3 increases the heart rate and force of contraction, thus increasing cardiac output , by increasing β-adrenergic receptor levels in myocardium. [ 14 ] This results in increased systolic blood pressure and decreased diastolic blood pressure . The latter two effects act to produce the typical bounding pulse seen in hyperthyroidism . [ citation needed ] It also upregulates the thick filament protein myosin, which helps to increase contractility. A helpful clinical measure to assess contractility is the time between the QRS complex and the second heart sound. This is often decreased in hyperthyroidism .
T 3 has profound effect upon the developing embryo and infants. It affects the lungs and influences the postnatal growth of the central nervous system. It stimulates the production of myelin , the production of neurotransmitters , and the growth of axons. It is also important in the linear growth of bones.
T 3 may increase serotonin in the brain, in particular in the cerebral cortex, and down-regulate 5HT-2 receptors, based on studies in which T 3 reversed learned helplessness in rats and physiological studies of the rat brain. [ 15 ]
Thyroid hormones act to increase protein turnover. This might serve an adaptive function in regard to long-term calorie restriction with adequate protein. [ 16 ] [ 17 ] When calories are in short supply, reduction in protein turnover may ameliorate the effects of the shortage.
Triiodothyronine can be measured as free triiodothyronine , which is an indicator of triiodothyronine activity in the body. It can also be measured as total triiodothyronine , which also depends on the triiodothyronine that is bound to thyroxine-binding globulin . [ 18 ]
The addition of triiodothyronine to existing treatments such as SSRIs is one of the most widely studied augmentation strategies for refractory depression , [ 19 ] however success may depend on the dosage of T 3 . A long-term case series study by Kelly and Lieberman of 17 patients with major refractory unipolar depression found that 14 patients showed sustained improvement of symptoms over an average timespan of two years, in some cases with higher doses of T 3 than the traditional 50 μg required to achieve therapeutic effect, with an average of 80 μg and a dosage span of 24 months; dose range: 25–150 μg. [ 19 ] The same authors published a retrospective study of 125 patients with the two most common categories of bipolar disorders II and NOS whose treatment had previously been resistant to an average of 14 other medications. They found that 84% experienced improvement and 33% experienced full remission over a period of an average of 20.3 months (standard deviation of 9.7). None of the patients experienced hypomania while on T 3 . [ 20 ]
3,5-Diiodo- L -thyronine and 3,3′-diiodo- L -thyronine are used as ingredients in certain over-the-counter fat-loss supplements, designed for bodybuilding . Several studies have shown that these compounds increase the metabolization of fatty acids and the burning of adipose fat tissue in rats. [ 21 ] [ 22 ]
Triiodothyronine has been used to treat Wilson's syndrome , an alternative medical diagnosis not recognized as a medical condition by mainstream medicine . This diagnosis involves various non-specific symptoms that are attributed to the thyroid, despite normal thyroid function tests . The American Thyroid Association has raised concern that the prescribed treatment with triiodothyronine is potentially harmful. [ 23 ]
In 1950 Dr Jack Gross , a Canadian endocrinologist, came to the British National Institute for Medical Research to work with Rosalind Pitt-Rivers as a postdoctoral fellow. Gross had previous experience working at McGill University under Professor Charles Leblond , where they used radioactive iodine to study the physiology of thyroid hormone and applied chromatography to analyze radioiodinated proteins in human blood after radioiodine therapy. Gross and Leblond found an unknown radioactive compound in the blood of rats given radioactive iodine. The compound migrated close to thyroxine in chromatography and they initially named it 'unknown 1' . Around that time a group led by Jean Roche in Paris described a deiodinating activity in the sheep thyroid gland, raising the possibility that 'unknown 1' is the less iodinated analogue of T4, triiodothyronine. [ 24 ] In march of 1952 Gross & Pitt-Rivers published a paper in The Lancet titled "The identification of 3: 5: 3'-L-triiodothyronine in human plasma". [ 25 ]
While Gross & Pitt-Rivers are normally credited with discovering T 3 , this compound was actually first isolated by the biochemists Hird & Trikojus at the University of Melbourne in 1948. [ 26 ] It has been suggested that their published paper was little-known and therefore easily ignored. [ 27 ] It has also been stated that Pitt-Rivers had read this paper but failed to mention it. [ 28 ]
Between 2020 and 2024, in numerous studies, an association was observed between serum free triiodothyronine (fT3) concentrations and the prognosis of severe COVID-19 in patients with SARS-CoV-2 infection. Serum fT3 concentrations are significantly lower in patients with severe COVID-19 compared to those who are not severely ill, and they predict all-cause mortality in patients with severe COVID-19. [ 29 ] [ 30 ] [ 31 ] [ 32 ] [ 33 ] | https://en.wikipedia.org/wiki/Triiodothyronine |
The compound with empirical formula Fe 3 Sn 2 is the first known kagome magnet . It is an intermetallic compound composed of iron (Fe) and tin (Sn) , with alternating planes of Fe 3 Sn and Sn. [ 1 ]
The iron-tin intermetallic forms at around 750 °C (1,380 °F) and naturally assumes a kagome structure. [ 2 ] Quenching in an ice bath then cools the material to room temperature without disrupting the atomic structure. [ 3 ]
The compound's band structure exhibits a double Dirac cone , enabling Dirac fermions . A 30 meV gap separates the cones, which indicates the quantum Hall effect and massive Dirac fermions. [ 4 ] Close measurement of the Fermi surface via the de Haas-van Alphen effect suggests that the massive fermions also exhibit Kane-Mele-type spin-orbit coupling . [ 5 ]
Fe 3 Sn 2 can also host magnetic skyrmions , but these typically require high magnetic fields to nucleate . For samples with a small (but nonzero) thickness gradient, only a small- amplitude (5-10 mT), direction-variant magnetic field suffices to nucleate the quasiparticles . [ 6 ] | https://en.wikipedia.org/wiki/Triiron_ditin_intermetallic |
Triisopropyl silane ( TIPS ) is an organosilicon compound with the formula ( i -Pr) 3 SiH ( i -Pr = isopropyl). This colorless liquid is used as a scavenger in peptide synthesis . It can also act as a mild reducing agent.
In peptide synthesis, TIPS is used as a scavenger for peptide groups being removed from the peptide sequence at the global deprotection. TIPS is able to scavenge carbocations formed in the deprotection of a peptide as it can act as a hydride donor in acidic conditions. [ 1 ] Silanes may be preferred as scavengers in place of sulfur -based scavengers. | https://en.wikipedia.org/wiki/Triisopropylsilane |
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