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In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of the center. Median income, for example, may be a better way to describe the center of the income distribution because increases in the largest incomes alone have no effect on the median. For this reason, the median is of central importance in robust statistics.
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.
In mathematical analysis, the Dirac delta function, also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Since there is no function having this property, to model the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.
In computer science, best, worst, and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively. Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource. Best case is the function which performs the minimum number of steps on input data of n elements. Worst case is the function which performs the maximum number of steps on input data of size n. Average case is the function which performs an average number of steps on input data of n elements.
The Heaviside step function, or the unit step function, usually denoted by H or θ, is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one.
In quantum physics, a wave function is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ. Wave functions are complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.
Rounding or rounding off means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $23.4476 with $23.45, the fraction 312/937 with 1/3, or the expression √2 with 1.414.
Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent. Exponential growth is the inverse of logarithmic growth.
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal x-axis, called the real axis, is formed by the real numbers, and the vertical y-axis, called the imaginary axis, is formed by the imaginary numbers.
A powiat is the second-level unit of local government and administration in Poland, equivalent to a county, district or prefecture in other countries. The term "powiat" is most often translated into English as "county" or "district". In historical contexts, this may be confusing because the Polish term hrabstwo (an administrative unit administered/owned by a hrabia is also literally translated as "county".
Paraplegia, or paraparesis, is an impairment in motor or sensory function of the lower extremities. The word comes from Ionic Greek (παραπληγίη) "half-stricken". It is usually caused by spinal cord injury or a congenital condition that affects the neural (brain) elements of the spinal canal. The area of the spinal canal that is affected in paraplegia is either the thoracic, lumbar, or sacral regions. If four limbs are affected by paralysis, tetraplegia or quadriplegia is the correct term. If only one limb is affected, the correct term is monoplegia. Spastic paraplegia is a form of paraplegia defined by spasticity of the affected muscles, rather than flaccid paralysis.
In cryptography, the avalanche effect is the desirable property of cryptographic algorithms, typically block ciphers and cryptographic hash functions, wherein if an input is changed slightly, the output changes significantly. In the case of high-quality block ciphers, such a small change in either the key or the plaintext should cause a drastic change in the ciphertext. The actual term was first used by Horst Feistel, although the concept dates back to at least Shannon's diffusion.
The New York Court of Appeals is the highest court in the Unified Court System of the State of New York. The Court of Appeals consists of seven judges: the Chief Judge and six associate judges who are appointed by the governor and confirmed by the state senate to 14-year terms. The Chief Judge of the Court of Appeals also heads administration of the state's court system, and thus is also known as the Chief Judge of the State of New York. Its courthouse is located in New York's capital, Albany. The court of appeals was found in 1847.
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.
The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .
In computational complexity theory, the average-case complexity of an algorithm is the amount of some computational resource used by the algorithm, averaged over all possible inputs. It is frequently contrasted with worst-case complexity which considers the maximal complexity of the algorithm over all possible inputs.
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is undecidable, meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs.
Diphallia, penile duplication (PD), diphallic terata, or diphallasparatus, is an extremely rare developmental abnormality in which a male is born with two penises. The first reported case was by Johannes Jacob Wecker in 1609. Its occurrence is 1 in 5.5 million boys in the United States.
References
- Greco F, Finocchiaro M, Pavone P, Trifiletti RR, Parano E. Hemihydranencephaly: case report and literature review. J Child Neurol. 2001;16 :218–221
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- ↑ "Hemihydranencephaly: living with half brain dysfunction" (PDF). Retrieved 23 February 2024.