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Basic method for pseudo-random number sampling
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov
Inverse_transform_sampling
Fourier analysis technique applied to sequences
{\displaystyle s_{_{N}}} sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic. The array of | S k
Discrete-time Fourier transform
Discrete-time_Fourier_transform
Computational statistics technique
zero as the number of iterations grows). Inverse transform sampling Ratio of uniforms Pseudo-random number sampling Ziggurat algorithm Casella, George; Robert
Rejection_sampling
Statistical transform
functions. The Box–Muller transform was developed as a more computationally efficient alternative to the inverse transform sampling method. The ziggurat algorithm
Box–Muller_transform
Function in discrete mathematics
halves of the result of the transform. Using the standard definition of the DFT omits the sampling interval (or sampling distance) Δ t {\displaystyle
Discrete_Fourier_transform
Probability theory operation
distribution to have a selected distribution: this is known as inverse transform sampling. Suppose that a random variable X {\displaystyle X} has a continuous
Probability integral transform
Probability_integral_transform
Integral transform useful in probability theory, physics, and engineering
X ( s ) {\displaystyle X(s)} . Once solved, the inverse Laplace transform can be used to transform it to the original domain. This is often aided by
Laplace_transform
Probability distribution
divergence for all sample sizes n > 0. A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random
Exponential_distribution
Branch of mathematics
harmonic analysis. Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis
Fourier_analysis
Topics referred to by the same term
Inverse method may refer to: The inverse transform sampling method The inverse method in automated reasoning This disambiguation page lists articles associated
Inverse_method
Uniform distribution on an interval
uniform distribution is useful for sampling from arbitrary distributions. A general method is the inverse transform sampling method, which uses the cumulative
Continuous uniform distribution
Continuous_uniform_distribution
Technique used in signal processing and data compression
cosine transform is the type-II DCT, which is often called simply the DCT. This was the original DCT as first proposed by Ahmed. Its inverse, the type-III
Discrete_cosine_transform
Discrete probability distribution
for large values of λ include rejection sampling and using Gaussian approximation. Inverse transform sampling is simple and efficient for small values
Poisson_distribution
Mathematical transform that expresses a function of time as a function of frequency
theorem, i.e., Inverse transform The functions f {\displaystyle f} and f ^ {\displaystyle {\widehat {f}}} are referred to as a Fourier transform pair. A common
Fourier_transform
Probability that random variable X is less than or equal to x
{\displaystyle F} . This is used in random number generation using the inverse transform sampling-method. If { X α } {\displaystyle \{X_{\alpha }\}} is a collection
Cumulative distribution function
Cumulative_distribution_function
Sufficiency theorem for reconstructing signals from samples
and the sampling theorem Balian–Low theorem, a similar theoretical lower bound on sampling rates, but which applies to time–frequency transforms Cheung–Marks
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Discrete Fourier transform algorithm
Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform converts
Fast_Fourier_transform
Statistical function that defines the quantiles of a probability distribution
suitable series for live Monte Carlo use. Inverse transform sampling Percentage point Probability integral transform Quantile Rank–size distribution For the
Quantile_function
Linear transform from the time domain to the frequency domain
the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or Fourier series, of the periodic values of the Z-transform around
Z-transform
Probability distribution
σ {\displaystyle \sigma } . This is obtained by applying the inverse transform sampling-method. R ∼ R a y l e i g h ( σ ) {\displaystyle R\sim \mathrm
Rayleigh_distribution
Topics referred to by the same term
tracking system, computer software that manages product issues Inverse transform sampling, a method for generating random numbers from various probability
Its
(number theory) Information bottleneck method Inverse chain rule method (calculus) Inverse transform sampling method (probability) Iterative method (numerical
List of mathematics-based methods
List_of_mathematics-based_methods
Generating pseudo-random numbers that follow a probability distribution
methods for generating independent samples: Rejection sampling for arbitrary density functions Inverse transform sampling for distributions whose CDF is known
Non-uniform random variate generation
Non-uniform_random_variate_generation
Fourier-related transform for signals that change over time
transform is called the "Gabor transform". It can also be explained with reference to the sampling and Nyquist frequency. Take a window of N samples from
Short-time_Fourier_transform
Transform in numerical harmonic analysis
discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage
Discrete_wavelet_transform
Discrete probability distribution
methods, but the most common way to sample from a categorical distribution uses a type of inverse transform sampling: Assume a distribution is expressed
Categorical_distribution
number sampling, that is, for drawing random samples from a statistical distribution. Like rejection sampling and inverse transform sampling, it is an
Ratio_of_uniforms
Probability distribution
distributed on (0, 1], then −ln U is distributed Gamma(1, 1) (i.e. inverse transform sampling). Now, using the "α-addition" property of gamma distribution,
Gamma_distribution
Mathematical algorithm
Z-transform can be computed in O(n log n) operations where n = max ( M , N ) n=\max(M,N) . An O(N log N) algorithm for the inverse chirp Z-transform (ICZT)
Chirp_Z-transform
Measurement of a signal at discrete time intervals
{\displaystyle T} seconds, which is called the sampling interval or sampling period. Then the sampled function is given by the sequence: s ( n T ) {\displaystyle
Sampling_(signal_processing)
Probability distribution
{1}{2\alpha -1}}} (see Aaberge 2005). Random samples can be generated using inverse transform sampling. Given a random variate U drawn from the uniform
Pareto_distribution
Special case of the short-time Fourier transform
The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency
Gabor_transform
Technique to analyze the infrared spectrum of matter
called the Fourier transform. The Fourier transform converts one domain (in this case displacement of the mirror in cm) into its inverse domain (wavenumbers
Fourier-transform infrared spectroscopy
Fourier-transform_infrared_spectroscopy
Probability distribution
(folded normal distribution). Random samples from the Lévy distribution can be generated using inverse transform sampling. Given a random variate U drawn from
Lévy_distribution
Mathematical operation
Wei-Qiang; Wang, Yue (2008). "Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain". IEEE Transactions
Fractional_Fourier_transform
Technique altering AI content for easier detection
distribution. Distortion-free schemes based on the Gumbel-max trick or inverse transform sampling (Aaronson 2022; Kuditipudi et al. 2023; Christ et al. 2024) preserve
AI_content_watermarking
Short-time Fourier transform with variable resolution
_{n=0}^{N-1}W[n-m]x[n]e^{-j2\pi kn/N}.} Given a data series at sampling frequency fs = 1/T, T being the sampling period of our data, for each frequency bin we can
Constant-Q_transform
Algorithm used in data compression
STX and ETX. def inverse_bwt(r: str, start=chr(STX), end=chr(ETX)) -> str: r""" Apply inverse Burrows–Wheeler transform. >>> inverse_bwt('\x03ANNB\x02AA')
Burrows–Wheeler_transform
Discipline for structuring uncertainties as coherent data models
regardless of platform. This is accomplished through inverse transform sampling, also known as the F-Inverse method, coupled to a portable pseudo random number
Probability_management
Mathematical analysis of frequency content of signals
multidimensional Fourier transform, m stands for multidimensional dimension. Define f as a multidimensional discrete-domain signal. The inverse multidimensional
Multidimensional_transform
Algorithm that generates an approximation of a random number sequence
b ) {\displaystyle f(b)} . This is based on the inverse transform sampling. For example, the inverse of cumulative Gaussian distribution erf − 1 ( x
Pseudorandom_number_generator
Probability distribution
will produce a different value. It is also possible to use the inverse transform sampling. A beta distribution B ( α , β ) {\displaystyle \mathrm {B} (\alpha
Beta_distribution
Integral transform and linear operator
inverse transform is − H {\displaystyle -\operatorname {H} } . This fact can most easily be seen by considering the effect of the Hilbert transform on
Hilbert_transform
Radar 2D mapping technique
pipes. Errors in the 2D planar Inverse ISAR transform include: Image blocking modeling errors: The Inverse ISAR transform currently assumes that scatterers
Inverse synthetic-aperture radar
Inverse_synthetic-aperture_radar
Partial order between random variables
{\displaystyle Z\sim \mathrm {Uniform} (0,1)} , then use the inverse transform sampling to get X = F X − 1 ( Z ) , Y = F Y − 1 ( Z ) {\displaystyle X=F_{X}^{-1}(Z)
Stochastic_dominance
{\displaystyle \rho } is achieved by using inverse transform sampling. U {\displaystyle U} denotes the uniform sampling of a value in the given interval. Because
Hyperbolic_geometric_graph
Probabilistically checkable proof Box–Muller transform Metropolis algorithm Gibbs sampling Inverse transform sampling method Walk-on-spheres method Risk Value
List_of_probability_topics
Probability distribution
b . {\displaystyle \ b~.} This is obtained by applying the inverse transform sampling-method. The special case b = 1 {\displaystyle \ b=1\ } yields
Type-2_Gumbel_distribution
First known wavelet basis
{2}}\end{bmatrix}}} The Haar transform can be thought of as a sampling process in which rows of the transformation matrix act as samples of finer and finer resolution
Haar_wavelet
In mathematics, a type of conformal map
named after Nikolai Zhukovsky, who published it in 1910. The transform and its right-inverse are z = ζ + 1 ζ , ζ = 1 2 z ± ( 1 2 z ) 2 − 1 = 1 1 2 z ∓ (
Joukowsky_transform
coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted
List of Fourier-related transforms
List_of_Fourier-related_transforms
Process of calculating the causal factors that produced a set of observations
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating
Inverse_problem
Change of basis applied in quantum computing
exponent vary; here the quantum Fourier transform has the same effect as the inverse discrete Fourier transform, and conversely.) Since ω N l {\displaystyle
Quantum_Fourier_transform
these more complex Poisson point processes in a similar manner to inverse transform sampling. Let X , Y {\displaystyle X,Y} be locally compact and polish and
Mapping theorem (point process)
Mapping_theorem_(point_process)
Concept in applied mathematics
uniform case, however, this substitution is unrelated to the inverse Fourier transform. The inversion of the NUDFT is a separate problem, discussed below
Non-uniform discrete Fourier transform
Non-uniform_discrete_Fourier_transform
FTIR spectroscopy applied on powder samples without prior preparation
Fourier transform spectroscopy (DRIFTS) is an infrared spectroscopy sampling technique used on powder samples without prior preparation. The sample is added
Diffuse reflectance infrared Fourier transform spectroscopy
Diffuse_reflectance_infrared_Fourier_transform_spectroscopy
Mathematical transform using in signal processing
{1}{2}}\right)\right].} Like for the DCT-IV, an orthogonal transform, the inverse has the same form as the forward transform. In the case of a windowed MDCT with the usual
Modified discrete cosine transform
Modified_discrete_cosine_transform
Transform in mathematics
mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real
Discrete_sine_transform
Time-frequency transform in geophysics
\Delta _{T}} is the sampling interval and Δ F {\displaystyle \Delta _{F}} is the sampling frequency. The Discrete time S-transform can then be expressed
S_transform
Generalization of exponential distribution
{\displaystyle X\sim Y.} Random deviates can be drawn using inverse transform sampling. Given a variable U that is uniformly distributed on the interval
Q-exponential_distribution
Signal processing operation
size of the trapezoidal rule used in the bilinear transform derivation; or, in other words, the sampling period. The above bilinear approximation can be
Bilinear_transform
Theorem in mathematics
sampling U {\displaystyle U} and V {\displaystyle V} at intervals of 1 / N {\displaystyle 1/N} and performing an inverse discrete Fourier transform (DFT)
Convolution_theorem
Derived representation of a digital image
Mathematica Morphological Inverse Distance Transform function in Mathematica A general algorithm for computing distance transforms in linear time [1]
Distance_transform
Type of signal filter
looped signal, the Fourier transform is taken, filtered in the frequency domain, followed by an inverse Fourier transform. Only O(n log(n)) operations
Low-pass_filter
Statistical transformation
z-transformation) of a Pearson correlation coefficient is its inverse hyperbolic tangent (artanh). When the sample correlation coefficient r is near 1 or -1, its distribution
Fisher_transformation
Property of many linear time-invariant (LTI) systems
filter is y(t), which is the inverse Laplace transform of Y(s). If sampled every T seconds, it is y(n), which is the inverse conversion of Y(z).These signals
Infinite_impulse_response
Integral transform
The Fourier transform is the fractional Fourier transform when θ = 90 ∘ . {\displaystyle \theta =90^{\circ }.} The inverse Fourier transform corresponds
Linear canonical transformation
Linear_canonical_transformation
Mathematical technique used in data compression and analysis
subsequent FFT of this function Multiplication with the transformed signal YFFT of the first step Inverse transformation of the product into the time domain
Wavelet_transform
probability Inverse probability weighting Inverse relationship Inverse-chi-squared distribution Inverse-gamma distribution Inverse transform sampling Inverse-variance
List_of_statistics_articles
Estimate of the importance of a word in a document
In information retrieval, tf–idf (term frequency–inverse document frequency, TF*IDF, TFIDF, TF–IDF, or Tf–idf) is a measure of importance of a word to
Tf–idf
Function for integral Fourier-like transform
forming a continuous wavelet transform (CWT) are subject to the uncertainty principle of Fourier analysis respective sampling theory: given a signal with
Wavelet
periodic sampling is by far the simplest scheme. Theoretically, sampling can be performed with respect to any set of points. But practically, sampling is carried
Hexagonal_sampling
Method of detecting shapes within images
{\displaystyle S} called the shape space, one can interpret the Hough transform as the inverse transform of a probability distribution on the image space to the shape
Hough_transform
Quantum algorithm for eigenvalue estimation
algorithm. The final part of the circuit involves applying the inverse quantum Fourier transform (QFT) Q F T {\displaystyle {\mathcal {QFT}}} on the first
Quantum phase estimation algorithm
Quantum_phase_estimation_algorithm
Probabilistic problem-solving algorithm
use adaptive routines such as stratified sampling, recursive stratified sampling, adaptive umbrella sampling or the VEGAS algorithm. A similar approach
Monte_Carlo_method
Statistical measure used in survey research
the sampling design is correlated with the outcome of interest. For example, a possible sampling design might be such that each element in the sample may
Design_effect
Unlike the discrete wavelet transform, SWT does not downsample the signal at each level. Instead, it maintains the original sampling rate throughout the decomposition
Stationary_wavelet_transform
Pseudo-random number sampling Inverse transform sampling — general and straightforward method but computationally expensive Rejection sampling — sample from a simpler
List of numerical analysis topics
List_of_numerical_analysis_topics
Changing the resolution of a digital image
two-dimensional example of sample-rate conversion, the conversion of a discrete signal from a sampling rate (in this case, the local sampling rate) to another.
Image_scaling
relationship Algorithmic Lovász local lemma Box–Muller transform Gibbs sampling Inverse transform sampling method Las Vegas algorithm Metropolis algorithm Monte
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Measure of linear correlation
of non-normal observed values if sample sizes are large enough. For determining the critical values for r the inverse function is needed: r = t n − 2 +
Pearson correlation coefficient
Pearson_correlation_coefficient
Lobe in a discrete aperture antenna
2{\frac {samples}{cycle}}\times {\frac {k{\frac {radians}{meter}}}{2\pi {\frac {radians}{cycle}}}}} . The sampling interval, which is the inverse of the
Grating_lobes
Mathematical algorithm
multiresolution analysis (MRA). In the terms given there, one selects a sampling scale J with sampling rate of 2J per unit interval, and projects the given signal
Fast_wavelet_transform
Spectroscopy based on time- or space-domain data
d{\tilde {\nu }}.\end{aligned}}} This is just a Fourier cosine transform. The inverse gives us our desired result in terms of the measured quantity I
Fourier-transform spectroscopy
Fourier-transform_spectroscopy
Statistical model used in machine learning
transformed from z 0 {\displaystyle z_{0}} . The functions f 1 , . . . , f K {\displaystyle f_{1},...,f_{K}} should be invertible, i.e. the inverse function
Flow-based_generative_model
Mathematical function for the probability a given outcome occurs in an experiment
a fixed number of total occurrences, sampling using a Pólya urn model (in some sense, the "opposite" of sampling without replacement) Categorical distribution
Probability_distribution
Audio effect
the sampling period in seconds }}[s].\end{cases}}} The discrete-time domain filter for integer delay M {\displaystyle M} as the inverse zeta transform of
Digital_delay_line
Probability distribution
arises as the sampling distribution of the t statistic. Below the one-sample t statistic is discussed, for the corresponding two-sample t statistic see
Student's_t-distribution
Frequency of a chirp pulse
To carry out the inverse process, i.e. to find the time domain function s(t) given frequency domain data, the inverse Fourier transform is derived. s (
Chirp_spectrum
Family of continuous probability distributions
{1-2t}})].} An inverse Gaussian distribution in double parameter form f ( x ; μ , λ ) {\displaystyle f(x;\mu ,\lambda )} can be transformed into a single
Inverse_Gaussian_distribution
Algorithmic determination of wave cycle parts
1-D inverse scattering problem, were proven by Klibanov and his collaborators (see References). Here we consider 1-D discrete Fourier transform (DFT)
Phase_retrieval
Integral expressing the amount of overlap of one function as it is shifted over another
f ∗ g ) ( t ) {\displaystyle (f*g)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle
Convolution
Mathematical algorithm
In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate
Inverse_iteration
Method of evaluating certain integrals along paths in the complex plane
the contour are determined by its values along the contour. The inverse Laplace transform is defined by a complex contour integral known as the Bromwich
Contour_integration
Inverse of the average of the inverses of a set of numbers
Pharm Sci 74(2) 229-231 Cox DR (1969) Some sampling problems in technology. In: New developments in survey sampling. U.L. Johnson, H Smith eds. New York: Wiley
Harmonic_mean
Statistical considerations on how many observations to make
complicated sampling techniques, such as stratified sampling, the sample can often be split up into sub-samples. Typically, if there are H such sub-samples (from
Sample_size_determination
Probability distribution
density as in inverse Mills ratio, so here we have σ 2 {\textstyle \sigma ^{2}} instead of σ {\displaystyle \sigma } . The Fourier transform of a normal
Normal_distribution
Generalization of the one-dimensional normal distribution to higher dimensions
{\displaystyle 1\leq i\leq k} and 1 ≤ j ≤ k {\displaystyle 1\leq j\leq k} . The inverse of the covariance matrix is called the precision matrix, denoted by Q =
Multivariate normal distribution
Multivariate_normal_distribution
Mathematical model which is both linear and time-invariant
possible in general to determine causality from the Z transform, because the inverse transform is not unique[dubious – discuss]. When a region of convergence
Linear_time-invariant_system
Standard linear arrays
occurs in time series analysis for an under-sampled signal. Per Shannon's sampling theorem, the sampling rate must be at least twice the highest frequency
Standard_linear_array
INVERSE TRANSFORM-SAMPLING
INVERSE TRANSFORM-SAMPLING
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Tamil
Universe
Girl/Female
Gujarati, Hindu, Indian, Sanskrit
Transformer
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Indian
Universe
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Indian
Universe
Boy/Male
Muslim
Universe
Girl/Female
Muslim
Universe
Girl/Female
Muslim
Universe
Girl/Female
Australian, Greek
Kind; Innocent
Surname or Lastname
Danish and Norwegian
Danish and Norwegian : patronymic from the personal name Ivar, from Old Norse Ãvarr, a compound of either Ãv ‘yew tree’, ‘bow’ or Ing (the name of a god) + ar ‘warrior’ or ‘spear’.North German (Frisian) : patronymic from a Germanic personal name composed of the elements Ä«wa ‘yew (tree)’ + hard ‘strong’, ‘firm’.English : variant spelling of Iverson.
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Universe
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English : habitational name from a place in Worcestershire, named Bransford, from Old English brægen ‘hill’ + ford ‘hill’.
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English : habitational name, probably from Ramsfold Farm in Lurgashall, Sussex. In a 14th-century record the name occurs as de Rammesford.
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English : from Middle English, Old French convers ‘convert’ (Latin conversus, past participle of convertere ‘to turn’), hence a nickname for a Jew converted to Christianity, or more often an occupational name for someone converted to the religious way of life, a lay member of a convent.
Girl/Female
Greek
Kind or innocent.
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Anglo, Australian, British, English
From the Raven Ford
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INVERSE TRANSFORM-SAMPLING
INVERSE TRANSFORM-SAMPLING
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Safety, Security, Welfare, Tranquility, Goddess Durga
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Molten.
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Thirumalai | திரூமாலாஈÂ
Abode of Lord venkateswara, Holy place
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Goddess Laxmi; Good Initiation
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Moderateness; Clemency
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Christian, German, Italian
Hard Working; Industrious; Rival; Work
Surname or Lastname
English and German
English and German : from a medieval personal name, a pet form of Martin or Marta.English and French : metonymic occupational name for a smith or a nickname for a forceful person, from Old French martel ‘hammer’ (Late Latin martellus). Charles Martel, the grandfather of Charlemagne, gained his byname from the force with which he struck down his enemies in battle.Spanish and Portuguese : from Portuguese martelo, Old Spanish martel ‘hammer’ (Late Latin martellus), or an Iberianized form of the Italian cognate Martello.
Boy/Male
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Pratyakash | பà¯à®°à®¤à¯à®¯à®•à¯à®·Â Â
In front
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English : habitational name from Pusey in Oxfordshire (formerly in Berkshire ), so called from Old English peose, piosu ‘pea(s)’ + ēg ‘island’, ‘low-lying land’, or from Pewsey in Wiltshire, recorded in Domesday Book as Pevesie, apparently from the genitive case of an Old English personal name Pefe, not independently attested + Old English ēg ‘island’.French : habitational name form Pusey in Haute-Saône, so named from a Gallo-Roman personal name, Pusius, + the locative suffix -acum.
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End
INVERSE TRANSFORM-SAMPLING
INVERSE TRANSFORM-SAMPLING
INVERSE TRANSFORM-SAMPLING
INVERSE TRANSFORM-SAMPLING
INVERSE TRANSFORM-SAMPLING
a.
Opposite in nature and effect; -- said with reference to any two operations, which, when both are performed in succession upon any quantity, reproduce that quantity; as, multiplication is the inverse operation to division. The symbol of an inverse operation is the symbol of the direct operation with -1 as an index. Thus sin-1 x means the arc whose sine is x.
n.
That which is inverse.
v. t.
To change; to transform; to invert.
a.
The back side; as, the reverse of a drum or trench; the reverse of a medal or coin, that is, the side opposite to the obverse. See Obverse.
v. t.
To transfer or transform the nature of.
v. t.
To carry or bear from one place to another; to remove; to convey; as, to transport goods; to transport troops.
a.
Inverted; having a position or mode of attachment the reverse of that which is usual.
v. t.
To reverse.
a.
Alt. of Renverse
a.
Subjected to the process of inversion; inverted; converted; as, invert sugar.
n.
To offer incense to. See Incense.
imp. & p. p.
of Transform
v. t.
To convey from one place or person another; to transport, remove, or cause to pass, to another place or person; as, to transfer the laws of one country to another; to transfer suspicion.
a.
Opposite in order, relation, or effect; reversed; inverted; reciprocal; -- opposed to direct.
imp. & p. p.
of Invert
v. t.
To transform anew or back.
v. t.
See Inhearse.
adv.
In an inverse order or manner; by inversion; -- opposed to directly.
v. t.
To change the form of; to change in shape or appearance; to metamorphose; as, a caterpillar is ultimately transformed into a butterfly.
v. t.
To change into another substance; to transmute; as, the alchemists sought to transform lead into gold.