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wiki_53_chunk_15 | Biochemistry | Sugar can be characterized by having reducing or non-reducing ends. A reducing end of a carbohydrate is a carbon atom that can be in equilibrium with the open-chain aldehyde (aldose) or keto form (ketose). If the joining of monomers takes place at such a carbon atom, the free hydroxy group of the pyranose or furanose f... | wikipedia |
wiki_53_chunk_16 | Biochemistry | Lipids | wikipedia |
wiki_53_chunk_17 | Biochemistry | Lipids comprise a diverse range of molecules and to some extent is a catchall for relatively water-insoluble or nonpolar compounds of biological origin, including waxes, fatty acids, fatty-acid derived phospholipids, sphingolipids, glycolipids, and terpenoids (e.g., retinoids and steroids). Some lipids are linear, open... | wikipedia |
wiki_53_chunk_18 | Biochemistry | Lipids are usually made from one molecule of glycerol combined with other molecules. In triglycerides, the main group of bulk lipids, there is one molecule of glycerol and three fatty acids. Fatty acids are considered the monomer in that case, and may be saturated (no double bonds in the carbon chain) or unsaturated (o... | wikipedia |
wiki_53_chunk_19 | Biochemistry | Most lipids have some polar character in addition to being largely nonpolar. In general, the bulk of their structure is nonpolar or hydrophobic ("water-fearing"), meaning that it does not interact well with polar solvents like water. Another part of their structure is polar or hydrophilic ("water-loving") and will tend... | wikipedia |
wiki_53_chunk_20 | Biochemistry | Lipids are an integral part of our daily diet. Most oils and milk products that we use for cooking and eating like butter, cheese, ghee etc., are composed of fats. Vegetable oils are rich in various polyunsaturated fatty acids (PUFA). Lipid-containing foods undergo digestion within the body and are broken into fatty ac... | wikipedia |
wiki_53_chunk_21 | Biochemistry | Proteins | wikipedia |
wiki_53_chunk_22 | Biochemistry | Proteins are very large molecules—macro-biopolymers—made from monomers called amino acids. An amino acid consists of an alpha carbon atom attached to an amino group, –NH2, a carboxylic acid group, –COOH (although these exist as –NH3+ and –COO− under physiologic conditions), a simple hydrogen atom, and a side chain comm... | wikipedia |
wiki_53_chunk_23 | Biochemistry | Proteins can have structural and/or functional roles. For instance, movements of the proteins actin and myosin ultimately are responsible for the contraction of skeletal muscle. One property many proteins have is that they specifically bind to a certain molecule or class of molecules—they may be extremely selective in ... | wikipedia |
wiki_53_chunk_24 | Biochemistry | The enzyme-linked immunosorbent assay (ELISA), which uses antibodies, is one of the most sensitive tests modern medicine uses to detect various biomolecules. Probably the most important proteins, however, are the enzymes. Virtually every reaction in a living cell requires an enzyme to lower the activation energy of the... | wikipedia |
wiki_53_chunk_25 | Biochemistry | The structure of proteins is traditionally described in a hierarchy of four levels. The primary structure of a protein consists of its linear sequence of amino acids; for instance, "alanine-glycine-tryptophan-serine-glutamate-asparagine-glycine-lysine-…". Secondary structure is concerned with local morphology (morpholo... | wikipedia |
wiki_53_chunk_26 | Biochemistry | Ingested proteins are usually broken up into single amino acids or dipeptides in the small intestine and then absorbed. They can then be joined to form new proteins. Intermediate products of glycolysis, the citric acid cycle, and the pentose phosphate pathway can be used to form all twenty amino acids, and most bacteri... | wikipedia |
wiki_53_chunk_27 | Biochemistry | If the amino group is removed from an amino acid, it leaves behind a carbon skeleton called an α-keto acid. Enzymes called transaminases can easily transfer the amino group from one amino acid (making it an α-keto acid) to another α-keto acid (making it an amino acid). This is important in the biosynthesis of amino aci... | wikipedia |
wiki_53_chunk_28 | Biochemistry | A similar process is used to break down proteins. It is first hydrolyzed into its component amino acids. Free ammonia (NH3), existing as the ammonium ion (NH4+) in blood, is toxic to life forms. A suitable method for excreting it must therefore exist. Different tactics have evolved in different animals, depending on th... | wikipedia |
wiki_53_chunk_29 | Biochemistry | In order to determine whether two proteins are related, or in other words to decide whether they are homologous or not, scientists use sequence-comparison methods. Methods like sequence alignments and structural alignments are powerful tools that help scientists identify homologies between related molecules. The releva... | wikipedia |
wiki_53_chunk_30 | Biochemistry | Nucleic acids, so-called because of their prevalence in cellular nuclei, is the generic name of the family of biopolymers. They are complex, high-molecular-weight biochemical macromolecules that can convey genetic information in all living cells and viruses. The monomers are called nucleotides, and each consists of thr... | wikipedia |
wiki_53_chunk_31 | Biochemistry | The most common nucleic acids are deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). The phosphate group and the sugar of each nucleotide bond with each other to form the backbone of the nucleic acid, while the sequence of nitrogenous bases stores the information. The most common nitrogenous bases are adenine, cyt... | wikipedia |
wiki_53_chunk_32 | Biochemistry | Aside from the genetic material of the cell, nucleic acids often play a role as second messengers, as well as forming the base molecule for adenosine triphosphate (ATP), the primary energy-carrier molecule found in all living organisms. Also, the nitrogenous bases possible in the two nucleic acids are different: adenin... | wikipedia |
wiki_53_chunk_33 | Biochemistry | Glucose is an energy source in most life forms. For instance, polysaccharides are broken down into their monomers by enzymes (glycogen phosphorylase removes glucose residues from glycogen, a polysaccharide). Disaccharides like lactose or sucrose are cleaved into their two component monosaccharides. Glycolysis (anaerobi... | wikipedia |
wiki_53_chunk_34 | Biochemistry | Glucose is mainly metabolized by a very important ten-step pathway called glycolysis, the net result of which is to break down one molecule of glucose into two molecules of pyruvate. This also produces a net two molecules of ATP, the energy currency of cells, along with two reducing equivalents of converting NAD+ (nico... | wikipedia |
wiki_53_chunk_35 | Biochemistry | Aerobic
In aerobic cells with sufficient oxygen, as in most human cells, the pyruvate is further metabolized. It is irreversibly converted to acetyl-CoA, giving off one carbon atom as the waste product carbon dioxide, generating another reducing equivalent as NADH. The two molecules acetyl-CoA (from one molecule of glu... | wikipedia |
wiki_53_chunk_36 | Biochemistry | Gluconeogenesis | wikipedia |
wiki_53_chunk_37 | Biochemistry | In vertebrates, vigorously contracting skeletal muscles (during weightlifting or sprinting, for example) do not receive enough oxygen to meet the energy demand, and so they shift to anaerobic metabolism, converting glucose to lactate.
The combination of glucose from noncarbohydrates origin, such as fat and proteins. T... | wikipedia |
wiki_53_chunk_38 | Biochemistry | Relationship to other "molecular-scale" biological sciences Researchers in biochemistry use specific techniques native to biochemistry, but increasingly combine these with techniques and ideas developed in the fields of genetics, molecular biology, and biophysics. There is not a defined line between these disciplines. ... | wikipedia |
wiki_53_chunk_39 | Biochemistry | Biochemistry is the study of the chemical substances and vital processes occurring in live organisms. Biochemists focus heavily on the role, function, and structure of biomolecules. The study of the chemistry behind biological processes and the synthesis of biologically active molecules are applications of biochemistry... | wikipedia |
wiki_53_chunk_40 | Biochemistry | See also Lists Important publications in biochemistry (chemistry)
List of biochemistry topics
List of biochemists
List of biomolecules See also Astrobiology
Biochemistry (journal)
Biological Chemistry (journal)
Biophysics
Chemical ecology
Computational biomodeling
Dedicated bio-based chemical
EC number
Hypot... | wikipedia |
wiki_53_chunk_41 | Biochemistry | a. Fructose is not the only sugar found in fruits. Glucose and sucrose are also found in varying quantities in various fruits, and sometimes exceed the fructose present. For example, 32% of the edible portion of a date is glucose, compared with 24% fructose and 8% sucrose. However, peaches contain more sucrose (6.66%)... | wikipedia |
wiki_53_chunk_42 | Biochemistry | Fruton, Joseph S. Proteins, Enzymes, Genes: The Interplay of Chemistry and Biology. Yale University Press: New Haven, 1999.
Keith Roberts, Martin Raff, Bruce Alberts, Peter Walter, Julian Lewis and Alexander Johnson, Molecular Biology of the Cell
4th Edition, Routledge, March, 2002, hardcover, 1616 pp.
3rd Edit... | wikipedia |
wiki_53_chunk_43 | Biochemistry | The Virtual Library of Biochemistry, Molecular Biology and Cell Biology
Biochemistry, 5th ed. Full text of Berg, Tymoczko, and Stryer, courtesy of NCBI.
SystemsX.ch – The Swiss Initiative in Systems Biology
Full text of Biochemistry by Kevin and Indira, an introductory biochemistry textbook. Biotechnology
Molecular ... | wikipedia |
wiki_54_chunk_0 | Boolean algebra (structure) | In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can b... | wikipedia |
wiki_54_chunk_1 | Boolean algebra (structure) | Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the... | wikipedia |
wiki_54_chunk_2 | Boolean algebra (structure) | The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and lat... | wikipedia |
wiki_54_chunk_3 | Boolean algebra (structure) | Definition A Boolean algebra is a six-tuple consisting of a set A, equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (called "complement" or "not") and two elements 0 and 1 in A (called "bottom" and "top", or "least" and "greatest" element, also denoted by th... | wikipedia |
wiki_54_chunk_4 | Boolean algebra (structure) | {| cellpadding=5
|a ∨ (b ∨ c) = (a ∨ b) ∨ c
|a ∧ (b ∧ c) = (a ∧ b) ∧ c
| associativity
|-
|a ∨ b = b ∨ a
|a ∧ b = b ∧ a
| commutativity
|-
|a ∨ (a ∧ b) = a
|a ∧ (a ∨ b) = a
| absorption
|-
|a ∨ 0 = a
|a ∧ 1 = a
| identity
|-
|a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
|a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
| distributivity
|-
|a ∨ ... | wikipedia |
wiki_54_chunk_5 | Boolean algebra (structure) | A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (In older works, some authors required 0 and 1 to be distinct elements in order to exclude this case.) | wikipedia |
wiki_54_chunk_6 | Boolean algebra (structure) | It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that
a = b ∧ a if and only if a ∨ b = b.
The relation ≤ defined by a ≤ b if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet a ... | wikipedia |
wiki_54_chunk_7 | Boolean algebra (structure) | It follows from the first five pairs of axioms that any complement is unique. The set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean al... | wikipedia |
wiki_54_chunk_8 | Boolean algebra (structure) | It has applications in logic, interpreting 0 as false, 1 as true, ∧ as and, ∨ as or, and ¬ as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically... | wikipedia |
wiki_54_chunk_9 | Boolean algebra (structure) | The two-element Boolean algebra is also used for circuit design in electrical engineering; here 0 and 1 represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the... | wikipedia |
wiki_54_chunk_10 | Boolean algebra (structure) | The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers o... | wikipedia |
wiki_54_chunk_11 | Boolean algebra (structure) | The power set (set of all subsets) of any given nonempty set S forms a Boolean algebra, an algebra of sets, with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the empty set and the largest element 1 is the set S itself. After the two-element Boolean algebra, the simplest Boolean... | wikipedia |
wiki_54_chunk_12 | Boolean algebra (structure) | The set of all subsets of that are either finite or cofinite is a Boolean algebra and an algebra of sets called the finite–cofinite algebra. If is infinite then the set of all cofinite subsets of which is called the Fréchet filter, is a free ultrafilter on However, the Fréchet filter is not an ultrafilter on the p... | wikipedia |
wiki_54_chunk_13 | Boolean algebra (structure) | For any natural number n, the set of all positive divisors of n, defining if a divides b, forms a distributive lattice. This lattice is a Boolean algebra if and only if n is square-free. The bottom and the top element of this Boolean algebra is the natural number 1 and n, respectively. The complement of a is given by ... | wikipedia |
wiki_54_chunk_14 | Boolean algebra (structure) | Homomorphisms and isomorphisms A homomorphism between two Boolean algebras A and B is a function f : A → B such that for all a, b in A: f(a ∨ b) = f(a) ∨ f(b),
f(a ∧ b) = f(a) ∧ f(b),
f(0) = 0,
f(1) = 1. | wikipedia |
wiki_54_chunk_15 | Boolean algebra (structure) | It then follows that f(¬a) = ¬f(a) for all a in A. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices. | wikipedia |
wiki_54_chunk_16 | Boolean algebra (structure) | An isomorphism between two Boolean algebras A and B is a homomorphism f : A → B with an inverse homomorphism, that is, a homomorphism g : B → A such that the composition g ∘ f: A → A is the identity function on A, and the composition f ∘ g: B → B is the identity function on B. A homomorphism of Boolean algebras is an ... | wikipedia |
wiki_54_chunk_17 | Boolean algebra (structure) | Every Boolean algebra (A, ∧, ∨) gives rise to a ring (A, +, ·) by defining a + b := (a ∧ ¬b) ∨ (b ∧ ¬a) = (a ∨ b) ∧ ¬(a ∧ b) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra; the... | wikipedia |
wiki_54_chunk_18 | Boolean algebra (structure) | Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + (x · y) and x ∧ y := x · y.
Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f : A → B is a homomorph... | wikipedia |
wiki_54_chunk_19 | Boolean algebra (structure) | Hsiang (1985) gave a rule-based algorithm to check whether two arbitrary expressions denote the same value in every Boolean ring. More generally, Boudet, Jouannaud, and Schmidt-Schauß (1989) gave an algorithm to solve equations between arbitrary Boolean-ring expressions.
Employing the similarity of Boolean rings and Bo... | wikipedia |
wiki_54_chunk_20 | Boolean algebra (structure) | An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I. This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. An ideal I of A is called prime if I ≠ A and if a ∧ b in I always implies a in I or b in I. Furthermor... | wikipedia |
wiki_54_chunk_21 | Boolean algebra (structure) | The dual of an ideal is a filter. A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x ∧ y in p and for all a in A we have a ∨ x in p. The dual of a maximal (or prime) ideal in a Boolean algebra is ultrafilter. Ultrafilters can alternatively be described as 2-valued morphisms from A to ... | wikipedia |
wiki_54_chunk_22 | Boolean algebra (structure) | Representations It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two. Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra A ... | wikipedia |
wiki_54_chunk_23 | Boolean algebra (structure) | The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematician Alfred North Whitehead in 1898.
It included the above axioms and additionally x∨1=1 and x∧0=0.
In 1904, the American mathematician Edward V. Huntington (1874–1952) gave probably the most parsimonious ... | wikipedia |
wiki_54_chunk_24 | Boolean algebra (structure) | Commutativity: x + y = y + x.
Associativity: (x + y) + z = x + (y + z).
Huntington equation: n(n(x) + y) + n(n(x) + n(y)) = x. Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit: 4. Robbins Equation: n(n(x + y) + n(x + n(y))) = x, | wikipedia |
wiki_54_chunk_25 | Boolean algebra (structure) | do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the Robbins conjecture) remained open for decades, and became a favorite question of Alfred Tarski and his s... | wikipedia |
wiki_54_chunk_26 | Boolean algebra (structure) | Further work has been done for reducing the number of axioms; see Minimal axioms for Boolean algebra. Generalizations | wikipedia |
wiki_54_chunk_27 | Boolean algebra (structure) | Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice B is a generalized Boolean lattice, if it has a smallest element 0 and for any elements a and b in B such that a ≤ b, there exists an element x such that a ∧ x = 0 a... | wikipedia |
wiki_54_chunk_28 | Boolean algebra (structure) | A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed subspaces for separable Hilbert spaces. See also Notes References Works cited . . General references .... | wikipedia |
wiki_54_chunk_29 | Boolean algebra (structure) | Stanford Encyclopedia of Philosophy: "The Mathematics of Boolean Algebra," by J. Donald Monk.
McCune W., 1997. Robbins Algebras Are Boolean JAR 19(3), 263—276
"Boolean Algebra" by Eric W. Weisstein, Wolfram Demonstrations Project, 2007.
Burris, Stanley N.; Sankappanavar, H. P., 1981. A Course in Universal Algebra. ... | wikipedia |
wiki_55_chunk_0 | Bandwidth (signal processing) | Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in hertz, and depending on context, may specifically refer to passband bandwidth or baseband bandwidth. Passband bandwidth is the difference between the upper and lower cutoff frequencies o... | wikipedia |
wiki_55_chunk_1 | Bandwidth (signal processing) | Bandwidth in hertz is a central concept in many fields, including electronics, information theory, digital communications, radio communications, signal processing, and spectroscopy and is one of the determinants of the capacity of a given communication channel. A key characteristic of bandwidth is that any band of a gi... | wikipedia |
wiki_55_chunk_2 | Bandwidth (signal processing) | Bandwidth is a key concept in many telecommunications applications. In radio communications, for example, bandwidth is the frequency range occupied by a modulated carrier signal. An FM radio receiver's tuner spans a limited range of frequencies. A government agency (such as the Federal Communications Commission in the... | wikipedia |
wiki_55_chunk_3 | Bandwidth (signal processing) | For other applications, there are other definitions. One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a specified level of performance. A less strict and more practically useful definition will refer to the frequencies beyond which performance is degraded. In ... | wikipedia |
wiki_55_chunk_4 | Bandwidth (signal processing) | In the context of, for example, the sampling theorem and Nyquist sampling rate, bandwidth typically refers to baseband bandwidth. In the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems it refers to passband bandwidth. The of a simple radar pulse is defined as the inverse of... | wikipedia |
wiki_55_chunk_5 | Bandwidth (signal processing) | In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal's spectral density (in W/Hz or V2/Hz) is nonzero or above a small threshold value. The threshold value is often defined relative to the maximum value, and is most commonly the , that is the point where the spectral densit... | wikipedia |
wiki_55_chunk_6 | Bandwidth (signal processing) | The bandwidth is also used to denote system bandwidth, for example in filter or communication channel systems. To say that a system has a certain bandwidth means that the system can process signals with that range of frequencies, or that the system reduces the bandwidth of a white noise input to that bandwidth. | wikipedia |
wiki_55_chunk_7 | Bandwidth (signal processing) | The 3 dB bandwidth of an electronic filter or communication channel is the part of the system's frequency response that lies within 3 dB of the response at its peak, which, in the passband filter case, is typically at or near its center frequency, and in the low-pass filter is at or near its cutoff frequency. If the ma... | wikipedia |
wiki_55_chunk_8 | Bandwidth (signal processing) | In electronic filter design, a filter specification may require that within the filter passband, the gain is nominally 0 dB with a small variation, for example within the ±1 dB interval. In the stopband(s), the required attenuation in decibels is above a certain level, for example >100 dB. In a transition band the gain... | wikipedia |
wiki_55_chunk_9 | Bandwidth (signal processing) | In signal processing and control theory the bandwidth is the frequency at which the closed-loop system gain drops 3 dB below peak. In communication systems, in calculations of the Shannon–Hartley channel capacity, bandwidth refers to the 3 dB-bandwidth. In calculations of the maximum symbol rate, the Nyquist sampling r... | wikipedia |
wiki_55_chunk_10 | Bandwidth (signal processing) | The fact that in equivalent baseband models of communication systems, the signal spectrum consists of both negative and positive frequencies, can lead to confusion about bandwidth since they are sometimes referred to only by the positive half, and one will occasionally see expressions such as , where is the total band... | wikipedia |
wiki_55_chunk_11 | Bandwidth (signal processing) | Relative bandwidth The absolute bandwidth is not always the most appropriate or useful measure of bandwidth. For instance, in the field of antennas the difficulty of constructing an antenna to meet a specified absolute bandwidth is easier at a higher frequency than at a lower frequency. For this reason, bandwidth is ... | wikipedia |
wiki_55_chunk_12 | Bandwidth (signal processing) | There are two different measures of relative bandwidth in common use: fractional bandwidth () and ratio bandwidth (). In the following, the absolute bandwidth is defined as follows,
where and are the upper and lower frequency limits respectively of the band in question. Fractional bandwidth
Fractional bandwidth i... | wikipedia |
wiki_55_chunk_13 | Bandwidth (signal processing) | While the geometric mean is more rarely used than the arithmetic mean (and the latter can be assumed if not stated explicitly) the former is considered more mathematically rigorous. It more properly reflects the logarithmic relationship of fractional bandwidth with increasing frequency. For narrowband applications, t... | wikipedia |
wiki_55_chunk_14 | Bandwidth (signal processing) | Ratio bandwidth is defined as the ratio of the upper and lower limits of the band, Ratio bandwidth may be notated as . The relationship between ratio bandwidth and fractional bandwidth is given by, and Percent bandwidth is a less meaningful measure in wideband applications. A percent bandwidth of 100% corresponds to ... | wikipedia |
wiki_55_chunk_15 | Bandwidth (signal processing) | In photonics, the term bandwidth carries a variety of meanings:
the bandwidth of the output of some light source, e.g., an ASE source or a laser; the bandwidth of ultrashort optical pulses can be particularly large
the width of the frequency range that can be transmitted by some element, e.g. an optical fiber
the gain ... | wikipedia |
wiki_55_chunk_16 | Bandwidth (signal processing) | A related concept is the spectral linewidth of the radiation emitted by excited atoms. See also Bandwidth extension
Broadband
Rise time
Spectral efficiency Notes References Signal processing
Telecommunication theory
Filter frequency response | wikipedia |
wiki_56_chunk_0 | BPP (complexity) | In computational complexity theory, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/3 for all instances.
BPP is one of the largest practical classes of problems, meaning most p... | wikipedia |
wiki_56_chunk_1 | BPP (complexity) | Informally, a problem is in BPP if there is an algorithm for it that has the following properties:
It is allowed to flip coins and make random decisions
It is guaranteed to run in polynomial time
On any given run of the algorithm, it has a probability of at most 1/3 of giving the wrong answer, whether the answer is YES... | wikipedia |
wiki_56_chunk_2 | BPP (complexity) | Definition
A language L is in BPP if and only if there exists a probabilistic Turing machine M, such that
M runs for polynomial time on all inputs
For all x in L, M outputs 1 with probability greater than or equal to 2/3
For all x not in L, M outputs 1 with probability less than or equal to 1/3
Unlike the complexit... | wikipedia |
wiki_56_chunk_3 | BPP (complexity) | Alternatively, BPP can be defined using only deterministic Turing machines. A language L is in BPP if and only if there exists a polynomial p and deterministic Turing machine M, such that
M runs for polynomial time on all inputs
For all x in L, the fraction of strings y of length p(|x|) which satisfy is greater than... | wikipedia |
wiki_56_chunk_4 | BPP (complexity) | In practice, an error probability of 1/3 might not be acceptable, however, the choice of 1/3 in the definition is arbitrary. Modifying the definition to use any constant between 0 and 1/2 (exclusive) in place of 1/3 would not change the resulting set BPP. For example, if one defined the class with the restriction that ... | wikipedia |
wiki_56_chunk_5 | BPP (complexity) | Problems All problems in P are obviously also in BPP. However, many problems have been known to be in BPP but not known to be in P. The number of such problems is decreasing, and it is conjectured that P = BPP. | wikipedia |
wiki_56_chunk_6 | BPP (complexity) | For a long time, one of the most famous problems known to be in BPP but not known to be in P was the problem of determining whether a given number is prime. However, in the 2002 paper PRIMES is in P, Manindra Agrawal and his students Neeraj Kayal and Nitin Saxena found a deterministic polynomial-time algorithm for this... | wikipedia |
wiki_56_chunk_7 | BPP (complexity) | An important example of a problem in BPP (in fact in co-RP) still not known to be in P is polynomial identity testing, the problem of determining whether a polynomial is identically equal to the zero polynomial, when you have access to the value of the polynomial for any given input, but not to the coefficients. In oth... | wikipedia |
wiki_56_chunk_8 | BPP (complexity) | Related classes
If the access to randomness is removed from the definition of BPP, we get the complexity class P. In the definition of the class, if we replace the ordinary Turing machine with a quantum computer, we get the class BQP. Adding postselection to BPP, or allowing computation paths to have different lengths... | wikipedia |
wiki_56_chunk_9 | BPP (complexity) | A Monte Carlo algorithm is a randomized algorithm which is likely to be correct. Problems in the class BPP have Monte Carlo algorithms with polynomial bounded running time. This is compared to a Las Vegas algorithm which is a randomized algorithm which either outputs the correct answer, or outputs "fail" with low proba... | wikipedia |
wiki_56_chunk_10 | BPP (complexity) | Complexity-theoretic properties It is known that BPP is closed under complement; that is, BPP = co-BPP. BPP is low for itself, meaning that a BPP machine with the power to solve BPP problems instantly (a BPP oracle machine) is not any more powerful than the machine without this extra power. In symbols, BPPBPP = BPP. | wikipedia |
wiki_56_chunk_11 | BPP (complexity) | The relationship between BPP and NP is unknown: it is not known whether BPP is a subset of NP, NP is a subset of BPP or neither. If NP is contained in BPP, which is considered unlikely since it would imply practical solutions for NP-complete problems, then NP = RP and PH ⊆ BPP. | wikipedia |
wiki_56_chunk_12 | BPP (complexity) | It is known that RP is a subset of BPP, and BPP is a subset of PP. It is not known whether those two are strict subsets, since we don't even know if P is a strict subset of PSPACE. BPP is contained in the second level of the polynomial hierarchy and therefore it is contained in PH. More precisely, the Sipser–Lautemann... | wikipedia |
wiki_56_chunk_13 | BPP (complexity) | Adleman's theorem states that membership in any language in BPP can be determined by a family of polynomial-size Boolean circuits, which means BPP is contained in P/poly. Indeed, as a consequence of the proof of this fact, every BPP algorithm operating on inputs of bounded length can be derandomized into a deterministi... | wikipedia |
wiki_56_chunk_14 | BPP (complexity) | Relativization
Relative to oracles, we know that there exist oracles A and B, such that PA = BPPA and PB ≠ BPPB. Moreover, relative to a random oracle with probability 1, P = BPP and BPP is strictly contained in NP and co-NP. | wikipedia |
wiki_56_chunk_15 | BPP (complexity) | There is even an oracle in which BPP=EXPNP (and hence P<NP<BPP=EXP=NEXP), which can be iteratively constructed as follows. For a fixed ENP (relativized) complete problem, the oracle will give correct answers with high probability if queried with the problem instance followed by a random string of length kn (n is insta... | wikipedia |
wiki_56_chunk_16 | BPP (complexity) | Lemma: Given a problem (specifically, an oracle machine code and time constraint) in relativized ENP, for every partially constructed oracle and input of length n, the output can be fixed by specifying 2O(n) oracle answers.
Proof: The machine is simulated, and the oracle answers (that are not already fixed) are fixed s... | wikipedia |
wiki_56_chunk_17 | BPP (complexity) | The lemma ensures that (for a large enough k), it is possible to do the construction while leaving enough strings for the relativized ENP answers. Also, we can ensure that for the relativized ENP, linear time suffices, even for function problems (if given a function oracle and linear output size) and with exponentiall... | wikipedia |
wiki_56_chunk_18 | BPP (complexity) | Derandomization
The existence of certain strong pseudorandom number generators is conjectured by most experts of the field. This conjecture implies that randomness does not give additional computational power to polynomial time computation, that is, P = RP = BPP. Note that ordinary generators are not sufficient to sho... | wikipedia |
wiki_56_chunk_19 | BPP (complexity) | László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson showed that unless EXPTIME collapses to MA, BPP is contained in
The class i.o.-SUBEXP, which stands for infinitely often SUBEXP, contains problems which have sub-exponential time algorithms for infinitely many input sizes. They also showed that P = BPP if th... | wikipedia |
wiki_56_chunk_20 | BPP (complexity) | Russell Impagliazzo and Avi Wigderson showed that if any problem in E, where
has circuit complexity 2Ω(n) then P = BPP. See also
RP
ZPP
BQP
List of complexity classes References | wikipedia |
wiki_56_chunk_21 | BPP (complexity) | Valentine Kabanets (2003). "CMPT 710 – Complexity Theory: Lecture 16". Simon Fraser University.
Pages 257–259 of section 11.3: Random Sources. Pages 269–271 of section 11.4: Circuit complexity.
Section 10.2.1: The class BPP, pp. 336–339.
Arora, Sanjeev; Boaz Barak (2009). "Computational Complexity: A Modern Ap... | wikipedia |
wiki_57_chunk_0 | Bioinformatics | Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combines biology, chemistry, physics, computer science, information engineering, ma... | wikipedia |
wiki_57_chunk_1 | Bioinformatics | Bioinformatics includes biological studies that use computer programming as part of their methodology, as well as specific analysis "pipelines" that are repeatedly used, particularly in the field of genomics. Common uses of bioinformatics include the identification of candidates genes and single nucleotide polymorphism... | wikipedia |
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