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This Presentation will Use to develop your knowledge and doubts in Knapsack problem. This Slide also include Memory function part. Use this Slides to Develop your knowledge on Knapsack and Memory function
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For any intuitionistic multi-fuzzy set A = { < x , µA(x) , νA(x) > : x∈X} of an universe set X, we study the set [A](α, β) called the (α, β)–lower cut of A. It is the crisp multi-set { x∈X : µi(x) ≤ αi , νi(x) ≥ βi , ∀i } of X. In this paper, an attempt has been made to study some algebraic structure of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (α, β)–lower cut sets
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In this paper, we introduce the concept of pseudo bipolar fuzzy cosets, pseudo bipolar fuzzy double cosets of a bipolar fuzzy and bipolar anti-fuzzy subgroups. We also establish these concepts to bipolar fuzzy and bipolar anti-fuzzy HX subgroups of a HX group with suitable examples. Also we discuss some of their relative properties.
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notion of cubic BF- Algebra i.e., an interval-valued BF-Algebra and an anti fuzzy BF-Algebra. Intersection of two cubic
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Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...
Simultaneous Triple Series Equations Involving Konhauser Biorthogonal Polynomials
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 42-45
www.iosrjournals.org
www.iosrjournals.org 42 | Page
Simultaneous Triple Series Equations Involving Konhauser
Biorthogonal Polynomials
P.K. Mathur,
Department of Mathematics, Saifia College, Bhopal, (M.P.)
Abstract: Biorthogonal polynomials are of great interest for Physicists.Spencer and Fano [9]
used the
biorthogonal polynomials (for the case k = 2) in carrying out calculations involving penetration of gamma rays
through matter.In the present paper an exact solution of simultaneous triple series equations involving
Konhauser-biorthogonal polynomials of first kind of different indices is obtained by multiplying factor technique
due to Noble.[4]
This technique has been modified by Thakare [10, 11]
to solve dual series equations involving
orthogonal polynomials which led to disprove a possible conjecture of Askey [1]
that a dual series equation
involving Jacobi polynomials of different indices can not be solved. In this paper the solution of simultaneous
triple series equations involving generalized Laguerre polynomials also have been discussed as a charmfull
particular case.
Key Words: 45F 10 Simultaneous Triple Series Equations, 33C45 Konhasure bi-orthogonal polynomials,
Laguerre polynomials,42C O5 Orthogonal Functions and polynomials, General Theory, 33D45 Basic
Orthogonal polynomials, 26A33 Fractional derivatives and integrals.
I. Introduction
Konhauser [3]
introduced a pair of sets of bi-orthogonal polynomials Z
α
n
x ; k and Y
α
n
x ; k with
respect to the weight function x
α
exp −x over the interval 0 , ∞ based on the study of
Preiser [7]
. In fact Konhauser defined
Z
α
n
x; k =
Γ kn+α+1
n!
−1
jn
j=0
n
j
xkj
Γ kj+α+1
and
Y
α
n
x ; k =
1
n!
x
p
n!
n
p=0 −1
qp
q=0
p
q
q+α+1
k n
Z
α
n
x; k is called Konhauser-biorthogonal set of the first kind and Y
α
n
x ; k the Konhauser-biorthogonal set of
the second kind. For k = 1, both the polynomials reduce to the generalized Laguerre polynomials L
α
n
x .
In the present paper we shall solve simultaneous triple series equations of the form
aij
s
j=1
∞
n=0
AnjZ
α
ni+p(x;k)
Γ α+kni +p+1
= fi x , 0 ≤ x < a [1.1]
bij
s
j=1
∞
n=0
Anj Z
β+δ−1
ni+p
x;k
Γ δ+β+kni +p
= gi x , a < 𝑥 < 𝑏 [1.2]
cij
s
j=1
∞
n=0
AnjZ
σ
ni+p(x;k)
Γ δ+β+kni +p
= hi x , b < 𝑥 < ∞ [1.3]
Where Z
α
n
x; k is the Konhauser-biorthogonal polynomial, fi x , gi x , and
hi x are prescribed functions, aij, bij and cij are known constants for
i = 1,2,……….,s and j = 1,2,…………,s. δ + β + m > 𝛼 + 1 > 0 and
2. Simultaneous Triple Series Equations Involving Konhauser Biorthogonal Polynomials
www.iosrjournals.org 43 | Page
σ + 1 > 𝛿 + 𝛽 > 0, m being some positive integer and p is a non-negative integer. Anj′s are the unknown
constants for j= 1,2,…………..,s. in the series equations which are to be determine.
II. Results Used In The Sequel
During the course of analysis the following results shall be needed:
(i) Biorthogonal relation was given by Konhauser [3]
as follows:
e
−x∞
0
x
α
Z
α
n
x; k Y
α
m
x; k dx =
Γ kn+α+1
n!
δ
n
m
[2.1]
where δ
n
m
is Kranecker’s delta.
(ii) Prabhakar [6]
introduced the following mth
differential formula in the form
d
m
dx
m x
α + m
Z
α + m
n
x;k =
Γ kn +α+m+1 xα
Z
α
n
(x;k)
Γ kn +α+1
[2.2]
With α > −1.
(iii) Prabhakar [6]
introduced the following fractional integrals, the first being the Riemann- Liouville
fractional integral:
ξ − x
β − 1ξ
0
x
α
Z
α
n
(x; k)dx
=
Γ kn+α+1 Γ β
Γ kn+α+β+1
ξ
α + β
Z
α + β
n
(ξ; k) [2.3]
when β > 0, 𝛼 + 𝛽 + 1 > 0
and the second, the Weyl fractional integral
e
−x∞
ξ
x − ξ
β − 1
Z
α
n
(x; k)dx = Γ β e
−ξ
Z
α − β
n
(ξ; k) [2.4]
where α + 1 > 𝛽 > 0
III. The Solution Of Triple Series Equations :
Multiplying both the sides of the equation [1.1] by x
α
ξ − x
−α + β + δ + m − 2
and
Integrating with respect to x over (0, ξ) and further using first fractional integral formula [2.3]
we get aij
s
j=1
∞
n=0
Anj Γ −α+β+δ+m−1
Γ kni +p+β+δ+m
ξ
β + δ + m − 1
Z
β + δ + m − 1
ni + p
(ξ ; k)
= x
αξ
0
ξ − x
−α + β + δ + m − 2
fi(x) dx [3.1]
From the relation [3.1], we have
aij
s
j=1
∞
n=0
AnjZ
β+δ+m−1
ni+p
(ξ;k)
Γ kni +p+β+δ+m
=
ξ−β−δ−m+1
Γ −α+β+δ+m−1
x
αξ
0
ξ − x
−α + β + δ + m − 2
fi(x) dx [3.2]
with 0 < x < ξ and β + δ + m > 𝛼 + 1 > 0
Now multiplying both sides of equation [3.2] by ξ
β + δ + m − 1
and differentiating both sides ‘m’ times with
respect to ξ and using the derivative formula [2.2] we get
bij
s
j=1
∞
n=0
AnjZ
β+δ−1
ni+p
(ξ;k)
Γ kni +p+β+δ
= dij
s
j=1
ξ−β−δ+1
Γ −α+β+δ+m−1
Fi ξ , 0 < ξ < 𝑎 [3.3]
where Fi ξ =
d
m
dx
m x
αξ
0
ξ − x
−α + β + δ + m − 2
fi(x) dx ,
i = 1,2,………………….,s. [3.4]
and dij are the elements of the matrix [bij][aij]
−1
.
Next we multiply both the sides of the equation [1.3] by e
−x
x − ξ
σ − δ − β
and integrate with respect to x
over (ξ , ∞) and using second fractional integral formula [2.4], we get
cij
s
j=1
∞
n=0
AnjΓ σ−δ−β+1
Γ kni +p+δ+β
e
−ξ
Z
β + δ − 1
ni + p
ξ ; k = e
−x∞
ξ
x − ξ
σ − δ − β
hi(x) dx
[3.5]
which can be written as
3. Simultaneous Triple Series Equations Involving Konhauser Biorthogonal Polynomials
www.iosrjournals.org 44 | Page
cij
s
j=1
∞
n=0
Anj Z
β+δ−1
ni+p
(ξ ;k)
Γ kni +p+δ+β
=
eξ
Γ σ−δ−β+1
e
−x∞
ξ
x − ξ
σ − δ − β
hi(x) dx [3.6]
where ξ < 𝑥 < ∞ , σ + 1 > 𝛿 + 𝛽 > 0.
From equation [3.6], we get
bij
s
j=1
∞
n=0
Anj Z
β+δ−1
ni+p
(ξ ;k)
Γ kni +p+δ+β
= eij
s
j=1
eξ
Γ σ−δ−β+1
Hi ξ , b < ξ < ∞. [3.7]
where Hi(ξ) = e
−x∞
ξ
x − ξ
σ − δ − β
hi(x) dx , i = 1,2,……………………..,s. [3.8]
and eij are the elements of the matrix [bij][cij]
−1
.
Now left hand sides of the equations [3.3], [1.2] and [3.7] are identical. Applying the biorthogonal relation [2.1]
of Konhauser polynomials , we get the solution of the simultaneous triple series equations in the form
Anj = qij
s
j=1 ni + p ! X
dij
s
j=1
1
Γ −α+β+δ+m−1
e
−ξ
Y
β + δ − 1
ni + p
ξ ;k Fi ξ dξa
0
+ e
−ξb
a
ξ
β + δ − 1
Y
β + δ − 1
ni + p
ξ ; k gi ξ dξ
+ eij
s
j=1
1
Γ σ−δ−β+1
∞
b
ξ
β + δ − 1
Y
β + δ − 1
ni + p
ξ; k Hi ξ dξ
where qij are the elements of the matrix [bij]
−1
and n = 0,1,2,……and j = 1,2,…….,s and Fi ξ , Hi ξ are
defined by [3.4] and [3.8] respectively.
IV. Particular Case
It is very interesting for us if we put k =1 in equations [1.1], [1.2] and [1.3] then Konhauser polynomials
involved in these equations are reduced to generalized Laguerre polynomials and we receive the following
Simultaneous triple series equations involving generalized Laguerre polynomials
aij
s
j=1
∞
n=0
Anj L
α
ni+p
x
Γ α+kni +p+1
= fi x , 0 ≤ x < a [4.1]
bij
s
j=1
∞
n=0
Anj L
β+δ−1
ni+p
x
Γ δ+β+kni +p
= gi x , a < 𝑥 < 𝑏 [4.2]
cij
s
j=1
∞
n=0
Anj L
σ
ni+p
x
Γ δ+β+kni +p
= hi x , b < 𝑥 < ∞ [4.3]
with the solution in the form
Anj = qij
s
j=1 ni + p ! X
dij
s
j=1
1
Γ −α+β+δ+m−1
e
−ξ
L
β + δ − 1
ni + p
x Fi ξ dξa
0
+ e
−ξb
a
ξ
β + δ − 1
L
β + δ − 1
ni + p
x gi ξ dξ
+ eij
s
j=1
1
Γ σ−δ−β+1
∞
b
ξ
β + δ − 1
L
β + δ − 1
ni + p
x Hi ξ dξ
where qij are the elements of the matrix [bij]
−1
and n = 0,1,2,……and j = 1,2,…….,s and Fi ξ , Hi ξ are
defined by [3.4] and [3.8] respectively when
k = 1.
REFERENCES
[1]. Askey, Richard: Dual equations and classical orthogonal polynomials, J. Math. Anal. Applic., 24, pp. 677- 685, (1968).
[2]. Konhasure, J.D.E. : Some properties of biorthogonal polynomials, Ibid, 11, pp.242- 260, (1965).
[3]. Konhasure, J.D.E. : Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. , 21, pp. 303- 314, (1967).
[4]. Noble, B.: Some dual series equations involving Jacobi polynomials,Proc. Camb.
[5]. phil. Soc. , 59, pp. 363- 372, (1963).
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[6]. Patil, K.R. and Thakare, N.K. : Certain dual series equations involving the Konhasure orthogonal polynomials, J. Math. Phys. , 18,
pp. 1724- 1726, (1977).
[7]. Prabhakar, T.R. : On a set of polynomials suggested by Laguerre polynomials, Pacific J. Math. , 35, pp. 213- 219, (1970).
[8]. Preiser, S. : An investigation of biorthogonal polynomials derivable from ordinary differential equation of the third order, J. Math.
Anal. Applic., 4, pp. 38- 64, (1962).
[9]. Rainville, E.D. : Special Function, Macmillan and Co., New York, (1967).
[10]. Spencer, L. and Fano, U. : Penetration and diffusion of X- rays, Calculation of spatial distributions by polynomial expansion, J. Res.
natn. Bur. Std. , 46, pp.446- 461, (1951).
[11]. Thakare, N.K. : Remarks on dual series equations involving orthogonal polynomials, J. Shivaji Univ. , 5, pp. 67- 72, (1972).
[12]. Thakare, N.K. : Dual series equations involving Jacobi polynomials, Zeitschrift fur Aug. Math. Mech. 54, pp. 283- 284, (1974).