Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les Cordeliers
Slides of Richard Everitt's presentation
After we applied the stochastic Galerkin method to solve stochastic PDE, and solve large linear system, we obtain stochastic solution (random field), which is represented in Karhunen Loeve and PCE basis. No sampling error is involved, only algebraic truncation error. Now we would like to escape classical MCMC path to compute the posterior. We develop an Bayesian* update formula for KLE-PCE coefficients.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les Cordeliers
Slides of Richard Everitt's presentation
After we applied the stochastic Galerkin method to solve stochastic PDE, and solve large linear system, we obtain stochastic solution (random field), which is represented in Karhunen Loeve and PCE basis. No sampling error is involved, only algebraic truncation error. Now we would like to escape classical MCMC path to compute the posterior. We develop an Bayesian* update formula for KLE-PCE coefficients.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.
Mobile computing application risks in ZimbabweIOSR Journals
Abstract: Mobile technology has now become the order of the day. Everyone seems to own one or more mobile
devices. Everyone is so excited because this has made life easier for a lot of people. Though mobile technology
has brought many benefits to people’s lives, its application has some risks that come with it. This paper
therefore looked at mobile computing application areas in Zimbabwe, the risks brought about by mobile
computing application in Zimbabwe and how mobile computing application risks are mitigated in a developing
country like Zimbabwe where the technology level seems to be behind. The study findings showed that mobile
application areas in Zimbabwe include paying bills, social networking and playing games. Most Zimbabweans
revealed that they do not know how to mitigate mobile computing application risks. The study finally
recommended that mobile computing application risks should be taught in schools as well as in televisions and
radios so that everyone is aware of them.
Keywords: application risks, mobile computing, mobile device,mobile technology
Implementation of Vertical Handoff Algorithm between IEEE 802.11 WLAN & CDMA ...IOSR Journals
Wireless communications is the fastest growing segment of the communications industry. Everyone
wants the quality of services anytime & anywhere. Wireless networks can integrate various heterogeneous radio
access technologies as GSM, WLAN, Wimax etc. WiMAX is an IP based, wireless broadband access technology
that provides performance similar to 802.11/Wi-Fi networks with the coverage and QOS (quality of service) of
cellular networks. WiMAX is also an acronym meaning "Worldwide Interoperability for Microwave Access
(WiMAX). The main promise of interconnecting these heterogeneous networks is to provide high performance in
achieving a high data rate and support real time applications. These services require various networks (such as
CDMA2000 and Wireless LAN) to be integrated into IP-based networks, which further require a seamless
vertical handoff to 4th generation wireless networks. When a mobile host (MH) changes its point of attachment,
its IP address gets changed. MH should be able to maintain all the existing connections using the new IP
address. This process of changing a connection from one IP address to another one in IP network is called
handoff. Vertical handoff is switching from one network to another while maintaining the session. Vertical
Handoff (VHO) is a major concern for different heterogeneous networks. VHO can be user requested or based
on some criteria already designed by the researcher of that particular algorithm. The main objective is to
implement efficient & effective handoff scheme between two heterogeneous network ie. 802.11 WLAN &
CDMA
Fluorescence technique involves the optical detection and spectral analysis of light emitted by a substance undergoing a transition from an excited electronic state to a lower electronic state. The aim of this study is to assess the -amino levulinic acid (-ALA) uptake. Based on image processing technique, Matlab was used to analyze the fluorescence images resulted from activation of (-ALA) and follow its uptake along one week. Analyzing the RGB colours pixel profile from obtained results showed different profiles for malignant tissues, normal tissues, treated just after PDT and finally at one week post PDT. The treated tissues fluorescence profile showed changes from closer to malignant tissue profile till been closed to normal one.
Mathematical Modelling: A Comparatively Mathematical Study Model Base between...IOSR Journals
In this paper, we have studied on the topic of „Corruption‟. Also, I will try to find or study the effect of corruption on the Development of the country or any country of the world. Therefore, how find the solution of the problem of corruption will be destroyed completely from the society. We have observed that the Development of the country depends upon Corruption. That is, when the Corruption increases, Development decreases automatically of any country of the world. Therefore, I will try to find the formula on the problem of „Relation between the Corruption and Development of any field or any country of the world‟. Also, I have to highlight the concept of „Application of Mathematical modeling in the interesting problem “corruption” in every field of our country or world .Also, Applied Mathematics focuses on the formulation and study of Mathematical Models .Thus the activity of Applied Mathematics is vitally connected with Research in Pure Mathematics. So I will try to study on it and find, what is corruption and quantity of corruption and also find the growth of corruption and how it will decay? Now we convert this areal world problem to mathematics problem and find some formulae on it such as Mathematical Corruption Growth formula, Mathematical Constant corruption level formula and Mathematical decay of corruption formula.
Perishable Inventory Model Having Weibull Lifetime and Time Dependent DemandIOSR Journals
In this paper we develop and analyse an inventory model for deteriorating items with Weibull rate of decay and time dependent demand. Using the differential equations, the instantaneous state of inventory at time‘t’, the amount of deterioration etc. are derived. With suitable cost considerations the total cost function and profit rate function are also obtained by maximizing the profit rate function, the optimal ordering and pricing policies of the model are derived. The sensitivity of the model with respect to the parameters is discussed through numerical illustration. It is observed that the deteriorating parameters have a tremendous influence on the optimal selling price and ordering quantity.
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.
Shape restrictions such as monotonicity often naturally arise. In this talk, we consider a Bayesian approach to monotone nonparametric regression with a normal error. We assign a prior through piecewise constant functions and impose a conjugate normal prior on the coefficient. Since the resulting functions need not be monotone, we project samples from the posterior on the allowed parameter space to construct a “projection posterior”. We obtain the limit posterior distribution of a suitably centered and scaled posterior distribution for the function value at a point. The limit distribution has some interesting similarity and difference with the corresponding limit distribution for the maximum likelihood estimator. By comparing the quantiles of these two distributions, we observe an interesting new phenomenon that coverage of a credible interval can be more than the credibility level, the exact opposite of a phenomenon observed by Cox for smooth regression. We describe a recalibration strategy to modify the credible interval to meet the correct level of coverage.
This talk is based on joint work with Moumita Chakraborty, a doctoral student at North Carolina State University.
Code of the Multidimensional Fractional Quasi-Newton Method using Recursive P...mathsjournal
The following paper presents one way to define and classify the fractional quasi-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems.
Best Approximation in Real Linear 2-Normed SpacesIOSR Journals
This pape r d e l i n e a t e s existence, characterizations and st rong unicity of best uniform
approximations in real linear 2-normed spaces.
AMS Su ject Classification: 41A50, 41A52, 41A99, 41A28.
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.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
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.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 33-36
www.iosrjournals.org
DOI: 10.9790/5728-11133336 www.iosrjournals.org 33 |Page
A Generalization of QN-Maps
S. C. Arora1
, Vagisha Sharma2
1
Former Professor& Head, Department of Mathematics, University of Delhi, Delhi, INDIA
2
Department of Mathematics, IP College for Women, (University of Delhi), Delhi, INDIA
Abstract: The notion of GQN-Maps is introduced and some results regarding these maps are obtained.
Keywords: Quasi-nonexpansive maps, GQN-maps, convex set, fixed point set, continuous maps, retract,
retraction mapping, locally weakly compact, conditional fixed point property.
AMS subject classification codes: 47H10, 54H25
I. Introduction
A self mapping T of a subset C of a normed linear space X is said to be nonexpansive if Tx – Ty ≤
x – y for allx , y in C [3]. It is quasi- nonexpansiveif T has at least one fixed point p of T in C and Tx – p
≤ x – p for allx in C and for each fixed point p of T in C [5,6].Many results have been proved for
nonexpansive and quasi-nonexpansive mappings. One may referBrowder and Petryshyn [1], Bruck [4],
Chidume [5], Das and Debata [6], Dotson [7],Petryshyn and Williamson [8], Rhoades [9], Singh and Nelson
[11], Senter and Dotson [10]and many more.
The purpose of the present paper is to introduce the notion of generalized quasi-nonexpansive
mappings (GQN-maps).
Throughout the paper, unless stated otherwise, X denotes a Banach space, , the field of real numbers,
A, the closure of A and F(T), the fixed point set of a mapping T. A subset C of X is locally compact if each point
of C has a compact neighbourhood in C [12]. The mapping r from a set C onto A, A being a subset of C, is a
retraction mapping if ra = afor alla in A[2].
II. Definition
2.1:A selfmapping T of a subset C of X is said to be generalised quasi-nonexpansive mapping (GQN-map)
provided T has at least one fixed point and corresponding to each fixed point T, there exists a constant M
depending on the fixed point p (referred as M(p)) in such that for each x belonging to C,
Tx – p ≤ M(p) x – p
Clearly, every quasi-nonexpansive map is a GQN map. However, the converse may not be true.
Example 1.2 establishes the same. It is well known that for a linear map, the fixed point setF(T) is convex and
for a continuous map, the fixed point set is closed. But there are non-linear discontinuous GQN-maps whose
fixed point sets are closed and convex.
Example 2.2:
(i) Define T:[0,
π
2
]→ [0,
π
2
] by
Tx = x + ( x −
π
4
)( cosx + 1)
Then F(T) = {
π
4
}
(ii) Define T: [0,1]→ [0,1] by
Tx= n + 1 x − 1 ,
1
n+1
< x ≤
1
n
, n = 1,2, . . ..
T(0) = 1
Then F(T) = {1,
1
2
,
1
3
,
1
4
, ........}
(iii) Define T: +
→ +
by
Tx =
1−x
n
,
1
n+1
< x ≤
1
n
, n = 0,1,2,3, . . . . ..
Then F(T) = {
1
n+1
: n = 0,1,2,3,.....}
(iv) Consider the Banach space n
= {(x1, x2, x3 … . xn): xifor all i = 1,2,3,. . . . . . n}.
Set C = {(x1, x2, x3 … . xn):xn = 0for all n > 2,x2 ≠ 0, x2 ≠1}.
2. A Generalization of QN-Maps
DOI: 10.9790/5728-11133336 www.iosrjournals.org 34 |Page
Define T : C → C by
T(x1, x2, 0, 0 … .0) = (2x2 − x2 − 1 x1, x2, 0, 0, … .0)
Then F(T) = {(2, x2, 0,0, . . . ,0): x2 ~{0,1}}.
The above examples show that F(T) may or may not be closed and convex for a GQN-map. Note that except in
example (i), the GQN-maps are discontinuous also. The exact set of conditions under which the fixed point set
of a GQN-map is closed and convex, areyet to obtained, but the conditions for F(T) to be a GQN-retract are
obtained in the next section.
III. Main results
IV.
In this section, C always denotes a closed, bounded and convex subset of the space X.
Definition3.1:A subset A of C is said to be a GQN-retract of C if there exists a retraction mapping r from C
onto A which is a GQN-map.
To find the set of conditions for any nonempty subset of a locally weakly compact set to be aGQN-retract, we
prove the following two lemmas:
Lemma3.2: Suppose A is a nonempty subset of a locally weakly compact set C and let
G(A) = {f: C → C is a GQN-map and f(x) = x for each x in A}. Then G(A) is compact in the topology of
weak-pointwise convergence.
Proof: Fix x0A. For each f G(A), there exists a real number Mf(x0) such that
f(x) – f (x0) ≤ Mf(x0)x – x0for allxC.
Let M x0 = Mff∈G A
Sup
(x0).
Case (i): Let M x0 be finite. For each xC, define Ax= {yC: y – x0≤ M(x0)x – x0}. Then Ax contains
f(x) for each x in C and f in G(A) which gives that G(A) is a subset of the Cartesian product P =x∈C Ax. Now
Ax is convex and weakly compact. So if Axis given the weak topology and P is given the product topology, by
Tychonoff’s theorem for the product of compact sets, P is compact.
Case (ii): Let M x0 be infinite. Then P = C and hence P is compact.
Now to show that G(A) is closed in P, let f be a limit point of G(A) in Pand <f>, a net in G(A) such that f →f.
Then, using lower semi-continuity of the norm function and the fact that f is in G(A), we get that G(A) is a
closed subset of the compact set P and hence is compact as desired.
Lemma3.3: Suppose A is nonempty subset of C and C is locally weakly compact. Then there exists an r in G(A)
such that for each f G(A) we have rx – ry ≤ f x – f(y) for all x, yin C.
Proof: Define an order < on G(A) by setting f < 𝑔 if f(x) – f(y) ≤g(x) – g(y) for each x, y in C with
inequality holding for at least one pair of x and y. Also f ≤ g means either f < 𝑔or f = g. Clearly ≤ is a partial
order on G(A).For each f in G(A), we define the initial segmentIs(f) = { gG(A): g ≤ f}. Then, as shown in
lemma 2.2, Is(f) is closed and compact in G(A). Now consider a chain in G(A). Then T = {Is(f): f } is a
chain of compact sets under set- inclusion as a partial order relation.By the finite intersection property for
compact sets, T is bounded below, say, by Is(h). Thenf ≤ h ∀ f G(A).
Now we prove the desired result in the following form:
Theorem3.4: Suppose C is locally weakly compact and A is a nonempty subset of C. Suppose further that for
each z in C, there exists an h ∈ G(A) such that h(z) A. Then A is a GQN-retract of C.
Proof: By lemma 2.3, there exists an r G(A) such that for each x, y in C and f G(A)
r(x) – r(y) ≤f(x) – f(y) ...................................................................... (2.1)
Also, it can be easily verified that for each f G(A), the composite map f ∘ r G(A).
Since r G(A), it is sufficient to show that for each x C, r(x) A. For this, let x C and put z = r(x).
Then as z C, the hypothesis assures the existence of an h G(A) such that h(r(x)) A. Now, let h(r(x)) =
y then as h ∘ r G(A), the inequality 2.1 implies
3. A Generalization of QN-Maps
DOI: 10.9790/5728-11133336 www.iosrjournals.org 35 |Page
r(x) – r(y) ≤ h ∘ r (x) – h ∘ r (y) ....................................................................... (2.2)
Since y = h(r(x)) A and r G(A), therefore, r(y) = y which further implies h(r(y)) = h(y) = y =
h(r(x)). So we get , in view of 2.2, that r(x) A.
Since for a GQN-map T, the fixed point set F(T) is always nonempty, so we have the following :
Corollary 3.5: Let C be a locally weakly compact set and T: C → C is a GQN-map. Suppose that for each z C
there exists an h in G(F(T)) such that h(z) F(T). Then F T is a GQN-retract of C.
Theorem 3.6: Under the conditions of Theorem 2.4, the class of GQN-retracts is closed under arbitrary
intersection.
Proof:By theorem 2.4, the collection { Is(f): f}, where is a chain in G(A), has a minimal element f in G(A)
which is a GQN-retract of C. Let = {Af C: f G(A) and Af is the corresponding GQN-retract of C}. Clearly
≠ φas A . Order by AfAgiff ≤ g f and g in G(A). By Zorn’s lemma, has a minimal element, say,
Ag. It can be seen that g is minimal in G(A).
Put F = Af.f∈G(A) AsA F(f) for every f, therefore, F is nonempty. Also minimality of g in G(A) implies that
Agis contained in each GQN-retract of C and hence in F. Then F= Ag. Thus F is a GQN-retract of C.
We now establish that the set of common fixed points of an increasing sequence of GQN-maps is a GQN-retract
of C .
Theorem 3.7: Let C be a locally weakly compact subset of X. If <rn> is a sequence of GQN-maps in G(A) such
that the corresponding GQN-retracts F(rn) form an increasing sequence with F(n rn ) ≠ φ then there exists a
GQN-map r from C to C such that F(r) = F(n rn ).
Proof: Consider = {F(rn): rn is a GQN-retraction of C onto F(rn)}.Order as A ≤ B if A B. By Zorn’s
lemma, there exists a minimal element, say, F.ThenF= F(n rn). Thus F(n rn ) is a GQN-retract of C.
By hypothesis, F(n rn) ≠ φ. So let F(n rn). Then pF(rn) for each n. Choose a sequence <n > of
positive numbers such that nn = 1 and let r = nn rn.For each p F(n rn) and x C,
r(x) – r(p) ≤ ( nn rn ) (x)( nn rn)(p)
≤ nn rn x − rn p
≤ M(p)x p
as nn = 1 and M(p) = maxn{Mrn
(p):Mrn
(p) is a constant corresponding to the GQN-map rn}. Thus r is a
GQN-map. Further, using nn = 1, it can be shown that F(r) = F(n rn) which proves the result.
Definition3.8: [3]:A mapping T: C →X is said to satisfy the conditional fixed point property (CFPP) if either T
has no fixed point or T has a fixed point in each nonempty bounded closed set it leaves invariant.
Definition 3.9: A nonempty subset C is said to have the hereditary fixed point property (HFPP) for GQN maps
if every nonempty bounded closed convex subset of C has a fixed point for GQN- mappings.
Following Bruck [3], we prove the following:
Theorem 3.10: If C is locally weakly compact and T: C →C is a GQN-map which satisfies CFPP then F(T) is a
GQN retract of C.
Proof: By definition of T, F(T) is nonempty. For a fixed z in C, define K = {f(z) ∶ f G(F(T))}. In view of the
compactness of G(F(T)), following [3], K is weakly compact and hence bounded. Also, K . For f and g in
G(F(T) and 0 ≤ ≤ 1, consider f + (1 − )g. If y0 F(T) then F(y0) = y0= g(y0) so that for all x, y in C,
(f + (1 )g)(x)y0 ≤ (Mf (y0)+ (1 )Mg (y0)xx0
where Mf (y0 ) andMg (y0 ) are real numbers corresponding to the fixed point y0 and for mappings f and g
respectively. Let us putM(Mf + (1)Mg)
(y0) = Mf (y0) + (1 )Mg (y0) then f + (1 )gis a GQN-map.
Also every fixed point x of T is a fixed point of f + (1 )g and hence K is convex. Also for fG(F(T)),
T ∘ f G(F(T)) i.e. T(K) K. Therefore, by hypothesis T has a fixed point in K i.e. ∃f G(F(T)) such that
f(z) F(T) for each z C. Thus, by theorem 2.4, F(T) is a GQN-retract of C.
Corollary3.11: Suppose T: C →C is a GQN-map satisfying CFPP and the convex closure conv(T C ) of the
range of T is locally weakly compact then F(T) is a GQN-retract of C.
4. A Generalization of QN-Maps
DOI: 10.9790/5728-11133336 www.iosrjournals.org 36 |Page
The following result can be proved following the arguments of Bruck [3].
Theorem3.12: Let C be locally weakly compact and {Fα: } be a family of weakly closed GQN retracts of C.
Then
(a) If this family is directed by, then Fαα is a generalised quasi-nonexpansive retract of C.
(b) If each Fα is convex and the family is directed by then ( Fαα ), the closure of( Fαα ), is a generalised
quasi-nonexpansive retract of C.
Lemma3.13: Let C be weakly compact and satisfies HFPP for GQN-maps. Let F be nonempty GQN- retract of
C and T: C → C is a GQN-map which leaves F invariant. Then F(T) ∩F is a nonempty GQN-retract of C.
Theorem3.14: Suppose C is weakly compact and has HFPP for GQN-maps. If {Tj : 1 ≤ j ≤ n} is a finite family
of commuting GQN-mapsTj: C → C then F(Tj)n
j=1 is a nonempty GQN-retract of C.
Theorem 3.15: Let {Tα: } is a family of GQN-maps of C, where, is some index set. If exactly one map,
sayTα, of the family is linear and continuous and commutes with each of the remaining then F( Tα ) ∩
( conv. F(Tβ)β≠α ) is nonempty.
Proof: Without loss of generality, we may assume that T1 is linear and continuous such that T1Tα= TαT1 for
allα. Clearly conv (F T1 ) = F(T1). Also for each α, conv (F Tα ) is a nonempty compact convex subset
of C. Linearity and continuity of T1 implies T1(conv (F Tα )conv (F Tα ). So, by Tychonoff’s theorems for
fixed points, T1 has fixed points in conv (F Tα ) and hence the result.
Remark3.16: In the proof of the above result, the condition of the self mapping being GQN-map is required to
assume that F(Tα)’s are nonempty. So if the hypothesis of the theorem contains the fact that F(Tα) ≠ for
allα, the result remains true for an ordinary family of mappings with exactly one map of the family being
linear and continuous.
The result of theorem 2.15 can be extended to a countable intersection of convex closures of F(Ti)’s but the least
conditions required are yet to be traced though the result is trivially true for the family of linear and continuous
maps.
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