Thus, the binding energy to contain or trap a particle in a smaller region of space, increases without bound, as the region of space grows smaller.

Thus, one nucleus offers a more tightly bound position to an electron than does another nucleus, with the result that one atom may transfer an electron to the other.

Thus the entire society from top to bottom is bound by feudalism.

Thus they were not gravitationally bound to the Milky Way, and were unlikely to be a part of the galaxy.

Thus, because of the O ( n log n ) upper bound on heapsort's running time and constant upper bound on its auxiliary storage, embedded systems with real-time constraints or systems concerned with security often use heapsort.

Thus in 1992, the Catechism of the Catholic Church, while affirming that " the Church does not know of any means other than Baptism that assures entry into eternal beatitude ", but also stating that " God has bound salvation to the sacrament of Baptism, but he himself is not bound by his sacraments ", stated: " As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them.

Thus, a federal district court that falls within the geographic boundaries of the Third Circuit Court of Appeals is bound by rulings of the Third Circuit Court, but not by rulings in the Ninth Circuit, since the Circuit Courts of Appeals have jurisdiction defined by geography.

Thus the inferior court decision may remain in effect even though it does not obey the superior court decision, as the only way a decision can enter the appeal process is by application of one of the parties bound by it.

Thus it provides an upper bound on what can be achieved with rational ( self-interested ) coordination or knowledge of others ' preferences.

Thus the courts had no power to compel the sovereign to be bound by the courts, as they were created by the sovereign for the protection of his or her subjects.

Thus range encoding can achieve greater compression ratios than the one-bit-per-symbol lower bound on Huffman encoding and it does not suffer the inefficiencies that Huffman does when dealing with probabilities that are not exact powers of two.

Thus, the plate will contain enzyme in proportion to the amount of secondary antibody bound to the plate.

Thus, if energy is removed ( or emitted ) from the system at the time it is bound, the loss of energy from the system will also result in the loss of the mass of the energy, from the system.

Thus, it can be seen that reflexives and reciprocals are bound variables ( known technically as anaphors ) while true pronouns can be free variables in some grammatical structures or variables that cannot be bound in other grammatical structures.

Thus in Acts 23: 21, over forty men, enemies of Paul, bound themselves, under a curse, neither to eat nor to drink till they had slain him.

Thus the tyranny of prescience will end with Siona — humanity can never again be bound by a powerful prescient like Muad ' dib or the God Emperor himself.

Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number.

) Thus an Archimedean field is one whose natural numbers grow without bound.

Thus, siderophile elements are bound through metallic bonds with iron in the dense layer of the Earth's core where pressures may be high enough to keep the iron solid.

Thus, the maximum margin of error represents an upper bound to the uncertainty ; one is at least 95 % certain that the " true " percentage is within the maximum margin of error of a reported percentage for any reported percentage.

Thus causality is intimately bound up with time's arrow.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Thus, the vector space B ( X, Y ) can be given the operator norm

Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, i. e. acting on every point in it.

Thus, the divergence of the vector field F can be expressed as:

Thus, the Jones vector represents ( relative ) amplitude and ( relative ) phase of electric field in x and y directions.

This identifies the tangent space T < sub > e </ sub > at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by = < sub > e </ sub >.

Thus the area of S < sub > D </ sub > is obtained by integrating the length of the normal vector to the surface over the appropriate region D in the parametric uv plane.

Thus force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action.

Thus the spinor rotation γ ( ψ ) = γψ ( ordinary quaternionic multiplication ) will rotate the spinor ψ through an angle one-half the measure of the angle of the corresponding vector rotation.

Thus, it is easy to convert from a vector file to a range of bitmap / raster file formats but it is much more difficult to go in the opposite direction, especially if subsequent editing of the vector picture is required.

Thus two arrows and in space represent the same free vector if they have the same magnitude and direction: that is, they are equivalent if the quadrilateral ABB ′ A ′ is a parallelogram.

Thus, this equation is sometimes known as the vector wave equation.

Thus for example atomic processes that are observed as rates, will necessarily be adjusted in a way consistent with relativity, while those involving the measurement of energy and momentum, which themselves form a relativistic vector, will undergo parallel adjustment which preserves the relativistic covariance of the observed values.

Thus if γ ( s ) is the arclength parametrization of C then the unit tangent vector T ( s ) is given by

Thus, suppose that ( x < sub > 1 </ sub >,..., x < sub > n </ sub >) is a choice of Cartesian coordinates, in terms of which the coordinates of the vector V are

Thus the act of measurement induces a fundamental irreversible change on the density matrix ; this is analogous to the " collapse " of the state vector, or wavefunction collapse.

Thus, acceleration is the second derivative of the position vector that defines the trajectory of a particle.

Thus, the magnitude of the angular velocity vector at a given time t is consistent with the two dimensions case.

Thus Tychonoff's theorem joins several other basic theorems ( e. g. that every nonzero vector space has a basis ) in being equivalent to AC.

Thus, G < sub > t </ sub >( V ) = exp ( tV ) is the exponential map of the vector tV.

Thus, for every vector

Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the z-axis.