Since there is a continual loss of micrometeoritic material in space because of the radiation effects, there must be a continual replenishment: otherwise, micrometeorites would have disappeared from interplanetary space.

Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

Since the earth is rotating and the unleveled gyro-stabilized platform is fixed with respect to a reference in space, an observer on the earth will see the platform rotating ( with respect to the earth ).

Since the block headers occupied space, the usable capacity of the drive was dependent on the block size.

Since there are infinitely many vectors in the basis, this is an infinite-dimensional Hilbert space.

Since commercial signage usually has limited physical space, signage is normally in French alone.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

Since every compact Hausdorff space is a Tychonoff space, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space.

Since the map d restricts to 0 on I < sub > x </ sub >< sup > 2 </ sup > ( the reader should verify this ), d descends to a map from I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup > to the dual of the tangent space, ( T < sub > x </ sub > M )< sup >*</ sup >.

Since the subarachnoid space around the brain and spinal cord can contain only 135 to 150 ml, large amounts are drained primarily into the blood through arachnoid granulations in the superior sagittal sinus.

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure.

Since the corona has been photographated at high resolution in the X-rays by the satellite Skylab in 1973, and then later by Yohkoh and the other following space instruments, it has been seen that the structure of the corona is very various and complex: different zones have been immediately classified on the coronal disc

Since traditional " Euclidean " space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required ' plane at infinity '.

Since work is defined as a force acting through a distance ( a length of space ), energy is always equivalent to the ability to exert pulls or pushes against the basic forces of nature, along a path of a certain length.

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.

Since this implies that one particle is communicating with the other instantaneously across space, i. e. faster than light, this is the " paradox ".

Since 1944 he has also conducted regularly at the San Francisco Opera, where he made his debut with a memorable performance of Verdi's Falstaff.

Since 1954 the Negro's desire for social justice has led to an ironically anarchical rebellion.

Since the great flood of these dystopias has appeared only in the last twelve years, it seems fairly reasonable to assume that the chief impetus was the 1949 publication of Nineteen Eighty-Four, an assumption which is supported by the frequent echoes of such details as Room 101, along with education by conditioning from Brave New World, a book to which science-fiction writers may well have returned with new interest after reading the more powerful Orwell dystopia.

Since 1953 California has led the nation in enacting guarantees that public business shall be publicly conducted, but not until this year did the lawmakers in Sacramento plug the remaining loopholes in the Brown Act.

Since little is known about autism, and almost nothing has been written for the layman, we'd like to share one experienced mother's comments.

Since it was issued in the spring of 1611, the King James Version has been most generally considered the most poetic and beautiful of all translations of the Bible.

Since the Connally amendment has the effect of giving the same right to the other party to a dispute with the United States, it also prevents us from using the court effectively.

Since he has just shown who is top dog, he may not be ready to receive this highest honor in the gift of the Soviet people.

Since then nothing has happened to save the life of Pope.

Since accurate base maps are necessary for any planning program, the first step taken by the planning division to implement the long-range state plan has been to prepare two series of base maps -- one at a scale of 1 inch to a mile, and the second a series of 26 sheets at a scale of 1 inch to 2000 feet, covering the entire state.

Since then, there has been a notable increase in the number of stations and also the accumulation of additional data and the development of new techniques for using it, leading to a better understanding of propagation phenomena.

Since the Industrial Revolution, when factories emerged, this classical pattern has been followed.

Since C is rational, this correspondence has K coincidences, each of which implies a line of the pencil which meets its image.

Since the course, one member has lost her husband.

Since her bereavement this individual has reported to the writer on numerous occasions about how helpful the class discussions were to her in this adjustment crisis.

Since on the one hand school desegregation has come in Virginia hand-in-glove with pupil assignment, shall we support the plan??

Since this book is concerned only incidentally with railroad rates, it will not attempt to analyze the methods by which the staff of the Interstate Commerce Commission has estimated out-of-pocket costs and apportioned residue costs.

Since that time the telegraph office has shifted in location from the railroad station at the Depot and shops at the Center back to the town clerk's office and drugstore at the Village.

Since 1948 the working headquarters has been the Department of Geology and Mineralogy.

Since the removal force is a function of coating thickness, a differential transformer pickup has been incorporated into the instrument to accurately measure film thickness.

Since the mid 1950s, when urethane foam first made its appearance in the American market, growth has been little short of fantastic.

Since interviewing is the basic therapeutic and diagnostic instrument of modern psychiatry, the recording of interviews for playbacks and study has been a boost of Redstone proportions in new research and training.

Since 1949, the St. Louis club has been below on May 1 just four times.

Since 1927, fewer than a dozen men have made serious runs at Babe Ruth's record and each, in turn, has been thwarted.

Since the column rank and the row rank are the same, we can also define the rank of A as the dimension of the row space of A, or the number of rows in a basis of the row space.

Since ω is a differential two-form the skew-symmetric condition implies that M has even dimension.

Since no " perfect " forms would work within the dimension of the plaza, his apparent oval in the paving is actually egg-shaped, narrower at one end than at the other.

Since Keeler also had sexual relations with Yevgeni Ivanov, the senior naval attaché at the Soviet Embassy, the Profumo Affair took on a national security dimension.

Since the inauguration of Israeli commercial television, a high-quality local drama developed — adding another dimension to Israel Television, which had previously relied on imported series from England and the United States.

Since cell widths must be matched with minimum dimension of containment, any wave overdriven by the initiator will be quenched.

Since the dimension of ker T is m and the dimension of V is m + n, it suffices to show that the dimension of image T is n.

Since the Lorentz group has no non-trivial unitary representation of finite dimension, it naively seems that one cannot construct a state with finite, non-zero spin and positive, Lorentz-invariant norm.

Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation.

Since the basic physical phenomenon is based on diffraction, the periodicity of the photonic crystal structure has to be of the same length-scale as half the wavelength of the EM waves i. e. ~ 200 nm ( blue ) to 350 nm ( red ) for photonic crystals operating in the visible part of the spectrum-the repeating regions of high and low dielectric constants have to be of this dimension.

Since February 2006, Lindesfarne has had her own blog cleverly titled “ Virtual Quill .” Originally, she was posting it as a sort of private journal, using the interdimensional portal to make sure that if it were ever seen by others, it would only be seen by humans from her native dimension ( namely, our world ), not the dimension she ’ s called home for the vast majority of her life.

Since the trace of a commutator is zero, and the trace of the identity is the dimension of the matrix, the representation must be zero dimensional.

Since is singular, the column space is of lesser dimension.

Since surgery theory requires working up to the middle dimension, once one is in dimension 5, the middle dimension has codimension greater than 2, and hence one avoids knots.

Since they have the same dimension, there is an isomorphism of vector spaces.

Since there is 1 superdiagonal entry, there will be one generalized eigenvector ( or you could note that the vector space is of dimension 2, so there can be only one generalized eigenvector ).

Since 1974 the IPO has organized a series of lectures and conferences on the cultural dimension of international affairs ( Amman 1974 ; Innsbruck 1974 ; Vienna 1979 ; Rome 1981 ; Nicosia 1984 ); these activities preceded the current global discourse on the dialogue of civilizations in which the I. P. O.

Eisenstein series form the most explicit examples of modular forms for the full modular group Since the space of modular forms of weight has dimension 1 for, different products of Eisenstein series having those weights have to be proportional.