International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
ON OPTIMALITY OF THE INDEX OF SUM, PRODUCT, MAXIMUM, AND MINIMUM OF FINITE BA...UniversitasGadjahMada
Chaatit, Mascioni, and Rosenthal de ned nite Baire index for a bounded real-valued function f on a separable metric space, denoted by i(f), and proved that for any bounded functions f and g of nite Baire index, i(h) i(f) + i(g), where h is any of the functions f + g, fg, f ˅g, f ^ g. In this paper, we prove that the result is optimal in the following sense : for each n; k < ω, there exist functions f; g such that i(f) = n, i(g) = k, and i(h) = i(f) + i(g).
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
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This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
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In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
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II Subalternation and Theology
III Theology and Dogmatic Declarations
IV The Mixed Principles of Theology
V Virtual Revelation: The Unity of Theology
VI Theology as a Natural Science
VII Theology’s Certitude
VIII Conclusion
Notes
Bibliography
All the contents are fully attributable to the author, Doctor Victor Salas. Should you wish to get this text republished, get in touch with the author or the editorial committee of the Studia Poinsotiana. Insofar as possible, we will be happy to broker your contact.
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Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...
A generalisation of the ratio-of-uniform algorithm
1. On a generalisation of the
ratio-of-uniform algorithm
Christian P. Robert
Universit´e Paris-Dauphine, CEREMADE, and CREST, Paris
1. CORE
One specific approach to random number generation (Devroye, 1986) that pro-
ceeds from the fundamental lemma of simulation Robert and Casella (2004) is the
ratio-of-uniform method (Kinderman et al., 1977). It is commonly used for the
most standard distributions as it can be calibrated to produce high acceptance
probability with a minimal number of operations. The said method is based on
the result that the uniform distribution on the set
A = (u; v) ∈ R+
× X; 0 ≤ u2
≤ f(v/u)
induces a marginal distribution with density proportional to f for the random
variable V = U. This lemma explains for the name of the method, despite neither
U nor V being marginally uniform variates. The proof of this result is a straight-
forward application of the transform method, since, if (U, V ) ∼ U(A), then the
density of (W, X) = (U2, V/U) is
˜f(w, x) ∝ IA(w
1/2
, w
1/2
x) × w
1/2
×
1
2w1/2
∝ I0≤w≤f(x)
As mentioned already, this is therefore a consequence of the fundamental lemma
of simulation, since we recover the uniform distribution on the set
B = (u; v) ∈ R+
× X; 0 ≤ u ≤ f(v)
which clearly induces the marginal distribution with density proportional to f
on V . While there is thus no mathematical issue with the marginalisation result
behind the ratio-of-uniform method, it is much less straightforward to picture the
construction of efficient random number generators based on this principle, when
compared with the fundamental lemma.
The first difficulty is to determine the shape of the set A, for which there is
little intuition if any. Figure 1 displays the resulting sets A for the Normal N(0, 1)
and the Gamma Ga(1/2, 1) distributions. The later is unbounded in u, due to the
asymptote of the density f at x = 0. In the general case, the boundary of the set
A is given by the parametric curve (Devroye, 1986)
u(x) = f(x), v(x) = x f(x), x ∈ X .
∗
Christian P. Robert, CEREMADE, Universit´e Paris-Dauphine, 75775 Paris cedex 16, France
xian@ceremade.dauphine.fr. Research partly supported by a Institut Universitaire de France 2016–
2021 senior chair. C.P. Robert is also affiliated as a part-time professor in the Department of
Statistics of the University of Warwick, Coventry, UK.
1
imsart-sts ver. 2009/02/27 file: ratiU.tex date: November 18, 2016
2. 2
Fig 1. Representation of the ratio-of-uniform set A for the Normal N(0, 1) and the Gamma
Ga(1/2, 1) distributions.
In the event where X ⊂ R and both f(x) and x2f(x) are bounded over X, it
is formally possible to create a cube in R+ × X around A and to deduce an
accept-reject strategy, based on uniform simulations in that cube. Using a two
component normal mixture as a benchmark, the following pictures (Figures 2
and 3) show that the set A is then bounded, albeit quite sparse in the cube that
contains it. This implies that the ratio-of-uniform method would not be efficient
in that situation.
As pointed out in Devroye (1986), since a simulation from f can also be derived
by uniform simulations on sets like
C = (u; v) ∈ R+
× X; 0 ≤ u ≤ f(u + v)
with a marginalisation in U + V , or
D = (u; v) ∈ R+
× X; 0 ≤ u2
≤ f(v/
√
u)3
with a marginalisation in V/
√
U. As we will now demonstrate, there exists a
generic construction of such sets. However, it seems impossible to rank those sets
in terms of efficiency in the general case. (This comparison only stands under the
assumption that all relevant functions are properly bounded to allow for bound-
ing boxes.) Even without seeking a bounding box and the associated uniform
distribution on that set, it seems delicate to compare slice samplers on the three
sets A, C, D, and their generalisations.
2. EXTENSION
There exists a generic family of transforms that generalises the original ratio-
of- uniform method. Namely, considering a differentiable monotone function h
over the positive half-line, R+, the uniform distribution over the set
H = (u; v) ∈ R+
× X; 0 ≤ u ≤ h ◦ f(v/g(u))
induces the right marginal f on the ratio V/g(U) if the primitive G of g is the
inverse of h, i.e., G ◦ h(x) = x. The proof is rather straightforward. Considering
imsart-sts ver. 2009/02/27 file: ratiU.tex date: November 18, 2016
3. 3
Fig 2. Set A for a two component Normal mixture.
Fig 3. Set A for a four component Normal mixture.
imsart-sts ver. 2009/02/27 file: ratiU.tex date: November 18, 2016
4. 4
the change of variable from (u, v) to (u, x) = (u, v/g(u)) produces a Jacobian of
g(u) and a joint density
U, X ∼ I0≤u≤h◦f(x)(u, x) g(u)
which integrates into
X ∼
h◦f(x)
0
g(u) du = G ◦ h ◦ f(x) = f(x)
when G ◦ f = id. The corresponding boundary of H is then provided by the
parameterised curve
u(x) = h ◦ f(x), v(x) = x(g ◦ h ◦ f)(x), x ∈ X .
For instance, when h(x) = xa, a power transform, the boundary can be written
as
u(x) = f(x)a
, v(x) = xf(x)1−a
, x ∈ X .
Similarly, when h(x) = exp(x), the ratio-of-uniform set is defined as
H = (u; v) ∈ R+
× X; 1 ≤ u ≤ exp{f(vu)}
with the curve
u(x) = exp{f(x)}, v(x) = x exp{−f(x)}, x ∈ X .
Note the change of range for u in that case.
One appeal of this generalisation is the formal possibility to include unbounded
densities and still produce compact boxes, as this seems essential for accept-reject
simulation if not for slice sampling. One possible choice for h is the generalised
logistic transform
h(ω) =
ωa
1 + ωa
which ensures that the [ratio-of-almost-uniform] set H is bounded in u. Since the
transform g is the derivative of the inverse of h,
g(y) =
a−1y(1−a)/a
(1 − y)(1+a)/a
.
the parametrisation of the boundary of H is
u(x) =
f(x)a
(1 + f(x))a
, v(x) = a−1
xf(x)a−1
(1 + f(x))−2
[(1 + f(x)a
− f(x)a
]−1+a/a
which means H remains bounded if (a) a ≤ 1 [to ensure boundedness at infinity]
and (b) the limit of v(x) at zero [where the asymptote of f must stand] is bounded.
This is satisfied if
lim
x→0
xf(x)
a+1/a
< +∞ .
For instance, this constraint holds for Gamma distributions with shape parameter
larger than 1/2.
imsart-sts ver. 2009/02/27 file: ratiU.tex date: November 18, 2016
5. 5
However, resorting to an arbitrary cdf Φ instead of the generalised logistic cdf
solves the difficulty for most distributions, including all Gamma distributions.
(Note that the power a is superfluous since Φa is also a cdf. Nonetheless, using this
representation brings an easier calibration of the proposal.) Indeed, the relevant
set is now
H = (u; v) ∈ R+
× X; 0 ≤ u ≤ Φ(f(v/g(u)))a
while the boundary of H is
u(x) = Φ(f(x))a
, v(x) = a−1
xf(x)
1−a/a
/ϕ ◦ f(x), x ∈ X ,
when ϕ is the derivative of Φ. This can be seen from G(u) = Φ−1(u1/a
) and
g(u) = a−1
u
1−a/a
ϕ(Φ−1
(u
1/a
))
which implies that
g(u(x)) = a−1
f(x)
1−a/a
ϕ(Φ−1
(Φ(f(x)))) .
This result means that the set remains bounded if ϕ has heavy enough tails, like
x−2, to handle the explosion at x = 0. Obviously, the density ϕ must further
enjoy an asymptote at zero to handle the limit at infinity when f(x) goes to zero.
For instance, consider the case when a = 1, meaning the boundary of H is
u(x) = Φ(f(x)), v(x) = 1/ϕ ◦ f(x),
and assume f(x) ≡ x− at x = 0 and ϕ(y) ≡ y−α−1 at y = +∞. At x = 0 we then
have v(x) ≡ x1− (α+1), which is a positive power for α small enough. Furthermore,
if f(x) ≡ x−δ−1 at x = ∞ and ϕ(y) ≡ y−β at y = 0, then at x = +∞, we have
v(x) ≡ x1−β(δ+1), which is a negative power for β close enough to 1.
However, this type of tail behaviour in Φ is not sufficient to handle a Gamma
distribution Ga(1 − , 1) (0 < < 1) since the exponential term is dominant at
∞. Take thus a transform such that φ(x) is equivalent to log{1/x}b (b > 0) near
zero. Then, at ∞
x/ϕ◦f(x) ≡ x {(−(1 − ) log(x) + x}−b
≡ x1−b
which remains bounded when b > 1, as for instance b = 2. The behaviour of
x/ϕ◦f(x) is easier to manage since f is then equivalent to x1− . If the tail behaviour
of ϕ at infinity is polynomial, i.e., ϕ(y) ≡ y−1−ν, we get
x/ϕ◦f(x) ≡ x x−1+ −(1+ν)
≡ x1−(1− )(1+ν)
at infinity, which remains bounded when (1 − )(1 + ν) < 1, i.e., ν < /1− . For
practical purposes, we consider the mixture density 0 < γ < 1)
ϕ(x) = γ
2 log(x)2
1 + (1 + log(2))2
I(0,1/2)(x) + (1 − γ)
ν{3/2}ν
(1 + x)ν+1
I(0,1/2)c (x)
which is normalised, continuous at x = 1/2 when
γ =
1
1 + 3ν log(2)2
/1+(1+log(2))2
imsart-sts ver. 2009/02/27 file: ratiU.tex date: November 18, 2016
6. 6
Fig 4. Set H for (a) Ga(0.1, 1) target; (b) Ga(0.5, 1) target; (c) Ga(0.9, 1) target, when ν =
.9 /(1 − ).
imsart-sts ver. 2009/02/27 file: ratiU.tex date: November 18, 2016
7. 7
and associated with the cdf
Φ(x) = γ
2x[(1 − log(x))2 + 1]
1 + (1 + log(2))2
I(0,1/2)(x)+ γ + (1 − γ) 1 −
(3/2)ν
(1 + x)ν
I(0,1/2)c (x)
Figure 4 produces three different bounded sets for Ga(α, 1) distributions based
on the above mixture transform Φ. Obviously, given the shape of those sets,
using a rectangular box does not lead to an efficient simulation algorithm. (This
goes without mentioning that a Ga(1 − , 1) distribution can be derived from a
Ga(2 − , 1) by multiplying a simulation from the former by U−1/1−
.)
REFERENCES
Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer-Verlag, New York.
URL http://cgm.cs.mcgill.ca/~luc/rnbookindex.html.
Kinderman, A., Monahan, J. and Ramage, J. (1977). Computer methods for sampling from
Student’s t-distribution. Math. Comput., 31 1009–1018.
Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. 2nd ed. Springer-Verlag,
New York.
imsart-sts ver. 2009/02/27 file: ratiU.tex date: November 18, 2016