These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

These individuals and the work of other biostatisticians, mathematical biologists, and statistically inclined geneticists helped bring together evolutionary biology and genetics into a consistent, coherent whole that could begin to be quantitatively modeled.

These estimates assume no major breakthroughs in solving the underlying mathematical problems that ECC is based on.

These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system.

These descriptions refer to the mathematical properties of the filter ( that is, the frequency and phase response ).

These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics.

These models are black box systems, using mathematical and statistical concepts to link a certain input ( for instance rainfall ) to the model output ( for instance runoff ).

: Indices: These are quite apart from the above, their use is for mathematical formalism, see Index notation.

These documents provide important information on ancient writings ; they give us the only extant copy of Menander, the Egyptian Book of the Dead, Egyptian treatises on medicine ( the Ebers Papyrus ) and on surgery ( the Edwin Smith papyrus ), Egyptian mathematical treatises ( the Rhind papyrus ), and Egyptian folk tales ( the Westcar papyrus ).

And the " Program Evaluation and Review Technique " or PERT, was developed by Booz Allen Hamilton as part of the United States Navy's ( in conjunction with the Lockheed Corporation ) Polaris missile submarine program ; These mathematical techniques quickly spread into many private enterprises.

These factors have changed dramatically in recent decades, both with the decreasing cost of computing power and with new mathematical discoveries.

These allow mathematical and logical operators to be written in their traditional form:

These restrictions correspond roughly to a particular mathematical model which differs from Euclidean space in its manifest symmetry.

These studies used mathematical models of several cities and their emission sources in order to compare the cost and effectiveness of various control strategies.

These matrices, and the form of the wave function, have a deep mathematical significance.

These are generally adverbs such as " very ", or " somewhat ", which modify the meaning of a set using a mathematical formula.

These recovered mathematical methods were later enhanced and developed by other Islamic scholars, notably by Persian scientists Al-Biruni and Abu Nasr Mansur.

These are objects defined in mathematical terms that do transform under the given rotation group ( see group theory ) but however their properties cannot be visualized with the idea of rotating a rigid object.

These meanings of open and closed, and the mathematical meanings, give rise to ambiguity because the terms can also refer to a closed manifold i. e. compact without boundary, not to be confused with a closed set.

These assumptions make up a mathematical model that is an idealized approximation, at best, to any realistic situation.

These units allowed direct hardware acceleration of very specific but complex mathematical problems, like the Fourier-transform or matrix operations.

These disruptions have made the earlier mathematical relationship inaccurate, but have in fact made Old Faithful more predictable.

These concepts do not generalize numbers but combine notions of functions and sets which are not yet formalized, breaking away from familiar mathematical objects.

These experimental methods also helped the mathematical modeling of polymers and even for a better understanding of the properties of polymers.

These terms also refer to clinical disciplines involving diagnosis and treatment of these diseases.

These diverse topics reflect multiple disciplines that oceanographers blend to further knowledge of the world ocean and understanding of processes within it.

These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic.

These disciplines study gender and sexuality in the fields of literature and language, history, political science, sociology, anthropology, cinema and media studies, human development, law, and medicine.

These disciplines are carried out as policies and procedures for establishing baselines and performing a standard change management process.

These diverse topics reflect multiple disciplines that oceanographers blend to further knowledge of the world ocean and understanding of processes within it: biology, chemistry, geology, meteorology, and physics as well as geography.

These terms are used in scientific disciplines from mathematics and logic to economics and psychology to denote similar investigative procedures.

These experiments grew into the heuristics and biases research program which spread beyond academic psychology into other disciplines including medicine and political science.

These cross-sectional images are used for diagnostic and therapeutic purposes in various medical disciplines.

These disciplines study gender and sexuality in the fields of literature and language, history, political science, sociology, anthropology, cinema and media studies, human development, law, and medicine.

These disciplines sometimes differ in their approaches to how and why they study gender.

These disciplines are less well understood than those of service management and therefore often some of their content is believed to be covered ' by implication ' in service management disciplines.

( These phenomena are of course central to many spiritual and religious disciplines, and are regularly employed by evangelists, cult leaders and holy men of all kinds ).

These two divergent disciplines stem from their unique cultural backgrounds and philosophies that went into their founding.

These five key points, when taken together, formulate a philosophy and world view which embraces bio-ethical abolitionism, life extension, singularitarianism, technogaianism, freedom of information and several other related disciplines and philosophies.

These disciplines included kyūdō ( Japanese archery ), calligraphy, ikebana ( flower arranging ), sado ( Japanese tea ceremony ), dance, theater, film, poetry, as well as health care and psychotherapy.

These are sub-qualifications denoting the air traffic control discipline or disciplines in which the person has been trained.

These horses are versatile and can be used in many disciplines.

These often involve fasting or conducting life according to certain rigid disciplines.

These phenomena include the causes, effects and correlates of transpersonal experiences and development, as well as the disciplines and practices inspired by them.

These authors define transpersonal psychology as being the branch of psychology that is concerned with transpersonal experiences and related phenomena, noting that " These phenomena include the causes, effects and correlates of transpersonal experiences, as well as the disciplines and practices inspired by them.

These disciplines include but are not limited to archaeology, Egyptology, textual criticism, linguistics, history, sociology and theology.