Measure (mathematics)In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
ManuscriptA manuscript (abbreviated MS for singular and MSS for plural) was, traditionally, any document written by hand or typewritten, as opposed to mechanically printed or reproduced in some indirect or automated way. More recently, the term has come to be understood to further include any written, typed, or word-processed copy of an author's work, as distinguished from the rendition as a printed version of the same. Before the arrival of prints, all documents and books were manuscripts.
Outer measureIn the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.
Biblical manuscriptA biblical manuscript is any handwritten copy of a portion of the text of the Bible. Biblical manuscripts vary in size from tiny scrolls containing individual verses of the Jewish scriptures (see Tefillin) to huge polyglot codices (multi-lingual books) containing both the Hebrew Bible (Tanakh) and the New Testament, as well as extracanonical works. The study of biblical manuscripts is important because handwritten copies of books can contain errors.
Manuscript cultureA manuscript culture is a culture that depends on hand-written manuscripts to store and disseminate information. It is a stage that most developed cultures went through in between oral culture and print culture. Europe entered the stage in classical antiquity. In early medieval manuscript culture, monks copied manuscripts by hand. They copied not just religious works, but a variety of texts including some on astronomy, herbals, and bestiaries.
Σ-finite measureIn mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.
Complete measureIn mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure on the plane as a product measure.
Borel measureIn mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Let be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of ; this is known as the σ-algebra of Borel sets. A Borel measure is any measure defined on the σ-algebra of Borel sets.
Convergence of measuresIn mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly.
Radon measureIn mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
Vector measureIn mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.
Palm-leaf manuscriptPalm-leaf manuscripts are manuscripts made out of dried palm leaves. Palm leaves were used as writing materials in the Indian subcontinent and in Southeast Asia reportedly dating back to the 5th century BCE. Their use began in South Asia and spread to other regions, as texts on dried and smoke-treated palm leaves of Palmyra palm or the talipot palm. Their use continued until the 19th century when printing presses replaced hand-written manuscripts.
Equivalence (measure theory)In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero. Let and be two measures on the measurable space and let and be the sets of -null sets and -null sets, respectively.
Illuminated manuscriptAn illuminated manuscript is a formally prepared document where the text is decorated with flourishes such as borders and miniature illustrations. Often used in the Roman Catholic Church for prayers, liturgical services and psalms, the practice continued into secular texts from the 13th century onward and typically include proclamations, enrolled bills, laws, charters, inventories and deeds. The earliest extant illuminated manuscripts come from the Kingdom of the Ostrogoths and the Eastern Roman Empire and date from between 400 and 600 CE.
ResearchResearch is "creative and systematic work undertaken to increase the stock of knowledge". It involves the collection, organization and analysis of evidence to increase understanding of a topic, characterized by a particular attentiveness to controlling sources of bias and error. These activities are characterized by accounting and controlling for biases. A research project may be an expansion on past work in the field. To test the validity of instruments, procedures, or experiments, research may replicate elements of prior projects or the project as a whole.
ProductivityProductivity is the efficiency of production of goods or services expressed by some measure. Measurements of productivity are often expressed as a ratio of an aggregate output to a single input or an aggregate input used in a production process, i.e. output per unit of input, typically over a specific period of time. The most common example is the (aggregate) labour productivity measure, one example of which is GDP per worker.
Review articleA review article is an article that summarizes the current state of understanding on a topic within a certain discipline. A review article is generally considered a secondary source since it may analyze and discuss the method and conclusions in previously published studies. It resembles a survey article or, in news publishing, overview article, which also surveys and summarizes previously published primary and secondary sources, instead of reporting new facts and results.
ScientistA scientist is a person who researches to advance knowledge in an area of the natural sciences. In classical antiquity, there was no real ancient analog of a modern scientist. Instead, philosophers engaged in the philosophical study of nature called natural philosophy, a precursor of natural science. Though Thales (circa 624–545 BC) was arguably the first scientist for describing how cosmic events may be seen as natural, not necessarily caused by gods, it was not until the 19th century that the term scientist came into regular use after it was coined by the theologian, philosopher, and historian of science William Whewell in 1833.
Medical researchMedical research (or biomedical research), also known as experimental medicine, encompasses a wide array of research, extending from "basic research" (also called bench science or bench research), – involving fundamental scientific principles that may apply to a preclinical understanding – to clinical research, which involves studies of people who may be subjects in clinical trials. Within this spectrum is applied research, or translational research, conducted to expand knowledge in the field of medicine.
