In quantum mechanics any such multiplicity in the eigenvalues of the Hamiltonian operator gives rise to degenerate energy levels.

Similarly, the multiplicity of calls gives the impression that the tit ’ s territory is more densely occupied than it actually is.

In addition to probability estimates, Algorithmic Probability " has for AI another important value: its multiplicity of models gives us many different ways to understand our data ;

This gives a multiplicity ( 2S + 1 ) of 1 or 3, respectively

The number of ordinary values of the function f{t} at T will be called its multiplicity at T.

Thus the multiplicity of Af for a given T must be an even number.

Because approach number four is often based on hardware mechanisms and avoids abstractions and a multiplicity of degrees of freedom, it is more practical.

** In statistical mechanics Ω refers to the multiplicity ( number of microstates ) in a system.

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

Yet, one must consider Boltzmann to be the " father " of statistical thermodynamics with his 1875 derivation of the relationship between entropy S and multiplicity Ω, the number of microscopic arrangements ( microstates ) producing the same macroscopic state ( macrostate ) for a particular system.

Zeno propounded a number of celebrated paradoxes, much debated by later philosophers, which try to show that supposing that there is any change or multiplicity leads to contradictions.

The multiplicity of vulcanization effects demanded cannot be achieved with one universal substance ; a large number of diverse additives, comprising the " cure package ," are necessary.

where Ω ( n ) is the number of prime factors of n, counted with multiplicity.

" in order to illustrate how in this first principle are combined unity and multiplicity, perfect simplicity with the most contradictory attributes, we are referred to the Greek letter ι, the single stroke of which represents units ; and which also represents the number ten.

* Ω ( n ), the big Omega function, is the number of prime factors of n counted with multiplicity ( so it is the sum of all prime factor multiplicities ).

* A powerful number ( also called squareful ) has multiplicity above 1 for all prime factors.

A number where some but not all prime factors have multiplicity above 1 is neither square-free nor squareful.

* A k-powersmooth number has all p < sup > m </ sup > ≤ k where p is a prime factor with multiplicity m.

Most multiple star systems known are triple ; for higher multiplicities, the number of known systems with a given multiplicity decreases exponentially with multiplicity.

Singularities may be classified by the triple δ, r, where m is the multiplicity, δ is the delta-invariant, and r is the branching number.

The number of times an element belongs to the multiset is the multiplicity of that member.

A number of coinciding knots is sometimes referred to as a knot with a certain multiplicity.

The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.

* Given an eigenvalue λ < sub > i </ sub >, its geometric multiplicity is the dimension of Ker ( A − λ < sub > i </ sub > I ), and it is the number of Jordan blocks corresponding to λ < sub > i </ sub >.

Generalized eigenvectors are needed to form a complete basis of a defective matrix, which is a matrix in which there are fewer linearly independent eigenvectors than eigenvalues ( counting multiplicity ).

The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i. e. number of linearly independent eigenvectors with that eigenvalue.

# From multiplicity to unity: (' Classic art achieves its unity by making the parts independent as free members, and the baroque abolishes the uniform independence of the parts in favour of a more unified total motive.

At the time, in addition to records released by larger record companies, there was a huge number of soul releases by a wide variety of artists on a multiplicity of obscure, independent labels.

In particular, suppose that an eigenvalue λ of a matrix A has an algebraic multiplicity m but fewer corresponding eigenvectors.

If the geometric multiplicity ( dimension of the eigenspace ) of λ is p, one can choose the first p vectors to be eigenvectors, but the remaining m − p vectors are only generalized eigenvectors.

Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in D ; if the coefficient in D at z is negative, then we require that h has a zero of at least that multiplicity at z – if the coefficient in D is positive, h can have a pole of at most that order.

* The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λ < sub > i </ sub > is its algebraic multiplicity.

Formally, let G be a Coxeter group with reduced root system R and k < sub > v </ sub > a multiplicity function on R ( so k < sub > u </ sub > = k < sub > v </ sub > whenever the reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to the roots u and v are conjugate in G ).

A system of imprimitivity is homogeneous of multiplicity n, where 1 ≤ n ≤ ω if and only if the corresponding projection-valued measure π on X is homogeneous of multiplicity n. In fact, X breaks up into a countable disjoint family

They are called multiplicities: the algebraic multiplicity of an eigenvalue is defined as the multiplicity of the corresponding root of the characteristic polynomial.

Sometimes this theorem is stated as follows: every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity.

Individual human consciousness, in the process of reaching for autonomy and freedom, has no choice but to deal with an obvious reality: the collective identities ( such as the multiplicity of world views, ethnic, cultural and national identities ) that divide the human race and set ( and always have set ) different groups in violent conflict with each other.

This statement must be qualified in several important ways, by considering points at infinity, allowing complex coordinates ( or more generally, coordinates from the algebraic closure of the ground field ), assigning an appropriate multiplicity to each intersection point, and excluding a degenerate case when X and Y have a common component.

Yet at the same time, Laing insisted that such a socially real grouping “ can be nothing else than the multiplicity of the points of view and actions of its members ... even where, through the interiorization of this multiplicity as synthesized by each, this synthesized multiplicity becomes ubiquitous in space and enduring in time ”.

Another effect was the multiplicity of models: Sevens, Eights, Nines, Tens, Elevens, Twelves, Fourteens, Sixteens etc each to fit with a taxation class.

The glycome exceeds the complexity of the proteome as a result of the even greater diversity of the glycome's constituent carbohydrates and is further complicated by the sheer multiplicity of possibilities in the combination and interaction of the carbohydrates with each other and with proteins.

* The multiplicity of a prime factor p of n is the largest exponent m for which p < sup > m </ sup > divides n. The tables show the multiplicity for each prime factor.

# W so that W: F < big >→</ big > Z is an arc multiset, so that the count for each arc is a measure arc multiplicity.

This notion of one ( multiplicity ) culture to an organization leads to the classification of culture along its path of emergence into nascent, adolescent, and mature types, each of which is distinct in terms of the pattern of the three cognitive components and behavior.

This is because " for each participant the task has total significance and dynamic closure " as well as the requirement to deploy a multiplicity of skills and to have the responsible autonomy in order to select when and how to do so.

The necessary and sufficient condition is for each eigenvalue to have geometric multiplicity equal to its algebraic multiplicity.

In particular, L < sup > 2 </ sup >( G ) decomposes into an orthogonal direct sum of all the irreducible unitary representations, in which the multiplicity of each irreducible representation is equal to its degree ( that is, the dimension of the underlying space of the representation ).

Although Vasubandhu had postulated numerous ālaya-vijñāna-s, a separate one for each individual person in the para-kalpita, this multiplicity was later eliminated in the Fa Hsiang and Hua Yen metaphysics.

There is a unique function assigning to each triplet ( P, Q, p ) consisting of a pair of polynomials, P and Q, in K and a point p in K < sup > 2 </ sup > a number I < sub > p </ sub >( P, Q ) called the intersection multiplicity of P and Q at p that satisfies the following properties:

A striking trait of higher education in France, compared to other countries such as the United States, is the small size and multiplicity of establishments, each specialized in a more or less broad spectrum of disciplines.

This line of thinking provides a much more flexible theological account of work, both better suited for the dynamic contemporary societies in which people engage in multiple and changing kinds of work over the course of a life-time and better coordinated with the multiplicity of ministries that each person can have in the church.

It is however often complicated by the multiplicity of events that have taken place throughout the history of individual lineages, leaving only distorted and superimposed traces in the genome of each living organism.

' The multiple details of the former are each uniquely contemplated: the multiplicity of the latter serves to diminish the dominance of line, and to enhance the unification of the multifarious whole.

The multiplicity of a projective indecomposable module as a summand of the group algebra ( viewed as the regular module ) is the dimension of its socle ( for large enough fields of characteristic zero, this recovers the fact that each simple module occurs with multiplicity equal to its dimension as a direct summand of the regular module ).