By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

Thus, we might as well treat that eigenspace as the actual Hilbert space of the system.

Thus one is led to consider the idea of a rigged Hilbert space.

Thus the dynamics of the system alone, treated in isolation, are non-unitary and, as such, are represented by irreversible transformations acting on the system's Hilbert space,.

Thus, unitary operators are just automorphisms of Hilbert spaces, i. e., they preserve the structure ( in this case, the linear space structure, the inner product, and hence the topology ) of the space on which they act.

Thus we define the representations of G on an Hilbert space H to be those group homomorphisms, ρ, which arise from continuous actions of G on H. We say that a representation ρ is unitary if ρ ( g ) is a unitary operator for all g ∈ G ; i. e., for all v, w ∈ H. ( I. e.

Thus the conjecture of Pólya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.

Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space.

* Any Hilbert space H is a Hilbert manifold with a single global chart given by the identity function on H. Moreover, since H is a vector space, the tangent space T < sub > p </ sub > H to H at any point p ∈ H is canonically isomorphic to H itself, and so has a natural inner product, the " same " as the one on H. Thus, H can be given the structure of a Riemannian manifold with metric

Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.

Thus jazz is transmuted into something holy, the sacred road to integration of being.

Thus the fictional detective is much more than a simple businessman.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus in both types attention is focused on the community itself, and its phenomenological life.

Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

Thus human perception and human volition is the immanent cause of all social change and this most truly when the change reaches the civilizational level.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.

Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.

Thus there is a clearer division of authority, administrative and legislative.

Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.

Thus, the need for the B-70 as a strategic weapon system is doubtful.

Thus technical efficiency is achieved at the expense of actual experience.

Thus, there is an added incentive to stay on the job.

Thus, the Span of its ossification was shortened and the center's ability to `` catch up '' in ossification is demonstrated.

Thus T is not diagonalizable.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Thus, the study of the solutions to the equation Af is reduced to the study of the space of solutions of a differential equation of the form Af.

Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.

Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.

Thus, it rejected " Palestinization " of the conflict with Israel, insisting on the necessary involvement of the greater Arab nation.

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 three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines ( e. g. Fuzzy Sets and Systems ), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which Intuitionism attempts to construct / refute / refound are taken as intuitively given.

Thus the actual mathematics of mathematical games may not be apparent to the average player.

Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same.

Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game.

Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.

Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed.

Thus Babylonian mathematics remained stale in character and content, with very little progress or innovation, for nearly two millennia.

Thus Grosseteste concluded, following very much in what Boethius had argued, that mathematics was the highest of all sciences, and the basis for all others, since every natural science ultimately depended on mathematics.

Thus, while Fessenden was only a teenager, he was teaching mathematics to the young children at the school while simultaneously studying with the older students at Bishop's University.

Thus, providing students of archaeology with a solid background in quantitative sciences such as mathematics, statistics and computer sciences seems today more important than ever.

Thus, inverse problems are one of the most important, and well-studied mathematical problems in science and mathematics.

Thus in 1900 he enrolled to École Normale Supérieure to study mathematics.

Thus the root of mathematics and scientific pursuits in Pythagoreanism is also based on a spiritual desire to free oneself from the cycle of birth and death.

Thus the order of arguments in some of the constructions below is exactly the opposite from those in many mathematics textbooks.

Thus humans do not invent mathematics, but rather discover and experience it, and any other intelligent beings in the universe would presumably do the same.

Thus the rationalists took mathematics as their model for knowledge, and the empiricists took the physical sciences.

Thus Kant arrives at the conclusion that all pure mathematics is synthetic though a priori ; the number 7 is seven and the number 5 is five and the number 12 is twelve and the same principle applies to other numerals ; in other words, they are universal and necessary.

Thus Bardas founded the Magnaura School with seats for philosophy, grammar, astronomy and mathematics, supported scholars like Leo the Mathematician and promoted the missionary activities of Cyril and Methodius to Greater Moravia.

In mathematics, a P-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their k-th power for 1 ≤ k ≤ P. Thus, a magic cube is bimagic when it is 2-multimagic, and trimagic when it is 3-multimagic, tetramagic when it is 4-multimagic.