To see the other types of publications on this topic, follow the link: Linear equations in primes.
Dissertations / Theses on the topic 'Linear equations in primes'
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the top 50 dissertations / theses for your research on the topic 'Linear equations in primes.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
Browse dissertations / theses on a wide variety of disciplines and organise your bibliography correctly.
1
Kong, Yafang, and 孔亚方. "On linear equations in primes and powers of two." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2013. http://hub.hku.hk/bib/B50533769.
Full textAbstract:
It is known that the binary Goldbach problem is one of the open problems on linear equations in primes, and it has the Goldbach-Linnik problem, that is, representation of an even integer in the form of two odd primes and powers of two, as its approximate problem. The theme of my research is on linear equations in primes and powers of two. Precisely, there are two cases: one pair of linear equations in primes and powers of two, and one class of pairs of linear equations in primes and powers of two, in this thesis.
In 2002, D.R. Heath-Brown and P.C. Puchta obtained that every sufficiently large even integer is the sum of two odd primes and k powers of two. Here k = 13, or = 7 under the generalized Riemann hypothesis. In 2010, B. Green and T. Tao obtained that every pair of linear equations in four prime variables with coefficients matrix A = (a_ij)s×t with s ≤ t, satisfying nondegenerate condition, that is, A has full rank and the only elements of the row-space of A over Q with two or fewer nonzero entries is the zero vector, is solvable. The restriction on the coefficient matrix means that they excluded the case of the binary Goldbach problem. Motivated by the above results, it is obtained that for every pair of sufficiently large positive even integers B1, B2, the simultaneous equation
{█({B1 = p1 + p2 + 2v1 + 2v2 + · · · + 2vk ,@B2 = p3 + p4 + 2v1 + 2v2 + · · · + 2vk ,)┤ (1)
is solvable, where p1, · · · , p4 are odd primes, each vi is a positive integer, and the positive integer k ≥ 63 or ≥ 31 under the generalized Riemann
hypothesis. Note that, in 1989, M.C. Liu and K.M. Tsang have obtained that subject to some natural conditions on the coefficients, every pair of linear equations in five prime variables is solvable. Therefore one class of pairs of linear equations in four prime variables with special coefficient matrix and powers of two is considered. Indeed, it is deduced that every pair of integers B1 and B2 satisfying B1 ≡ 0 (mod 2), 3BB1 > e^(eB^48 ), B2 ≡ ∑_1^4▒= 1^(a_i ) (mod 2) and |B2| < BB1, where B = max1≤j≤4(2, |aj|), can be represented as
{█(B1 = 〖p1〗_1 + p2 + 2^(v_1 ) + 2^(v_2 )+ · · · + 2^(v_k )@B2 = a1p1 + a2p2 + a3p3 + a4p4 + 2^(v_1 )+ 2^(v_2 )+ · · · + 2^(v_k ) )┤ (2)
with k being a positive integer. Here p1, · · · p4 are odd primes, each 〖v 〗_iis a positive integer and the integral coefficients ai (i = 1, 2, 3, 4) satisfy
{█((〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) = 1,@〖a 〗_1 〖a 〗_2< 0, 〖a 〗_3 〖a 〗_4<0,)┤
Moreover it is calculated that the positive integer k ≥ g(〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) where g(〖a 〗_21- 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = [(log〖G(〖a 〗_21, …, 〖a 〗_24 〗)-log〖F (〖a 〗_21, …, 〖a 〗_24)〗)/log0.975805-84.0285], (3)
G(〖a 〗_21, 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = (min(1/(|a_24 |), 1/(|a_23 |)) - (〖|a〗_(21 )- a_22 |)/(|〖a_23 a〗_24 |) 〖(3B)〗^(-1) ×〖(3B)〗^(-1) (1-0.000001)- 〖(3B)〗^(-1-4), with B = max1≤j≤4(2, |a2j|), and
F(a_21, …, a_24) = √(f(a_21)f〖(a〗_22 ))
with
f(a_2i) = {█(4414.15h (a_21-1)+5.088331 if a_21≠1@59.8411 if a_21=1,)┤
for i = 1, 2, and h(n) =∏_(p|n,p>2)▒(p-1)/(p-2). This result, if without the powers of
two, can make up some of the cases excluded in Green and Tao’s paper.published_or_final_version
Mathematics
Doctoral
Doctor of Philosophy
2
Fan, Ka-wing. "Prime solutions in arithmetic progressions of some quadratic equations and linear equations /." Hong Kong : University of HOng Kong, 2000. http://sunzi.lib.hku.hk/hkuto/record.jsp?B23540308.
Full text
3
張勁光 and King-kwong Cheung. "Prime solutions in arithmetic progressions of some linear ternary equations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B42575874.
Full text
4
Cheung, King-kwong. "Prime solutions in arithmetic progressions of some linear ternary equations." Click to view the E-thesis via HKUTO, 2000. http://sunzi.lib.hku.hk/hkuto/record/B42575874.
Full text
5
蔡國光 and Kwok-kwong Stephen Choi. "Some explicit estimates on linear diophantine equations in three primevariables." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1990. http://hub.hku.hk/bib/B3120966X.
Full text
6
Choi, Kwok-kwong Stephen. "Some explicit estimates on linear diophantine equations in three prime variables /." [Hong Kong] : University of Hong Kong, 1990. http://sunzi.lib.hku.hk/hkuto/record.jsp?B12907236.
Full text
7
樊家榮 and Ka-wing Fan. "Prime solutions in arithmetic progressions of some quadratic equationsand linear equations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B31225962.
Full text
8
趙善衡 and Shin-hang Chiu. "The solubility and the insolubility of systems of linear equations in prime variables." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1991. http://hub.hku.hk/bib/B31209646.
Full text
9
Chiu, Shin-hang. "The solubility and the insolubility of systems of linear equations in prime variables /." [Hong Kong : University of Hong Kong], 1991. http://sunzi.lib.hku.hk/hkuto/record.jsp?B12996622.
Full text
10
Gärtner, Christian [Verfasser], and Christian [Akademischer Betreuer] Bender. "Primal-dual methods for dynamic programming equations arising in non-linear option pricing / Christian Gärtner ; Betreuer: Christian Bender." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2017. http://d-nb.info/1154438392/34.
Full text
11
Cook, Brian Michael. "Equations in the primes." Thesis, University of British Columbia, 2012. http://hdl.handle.net/2429/42217.
Full textAbstract:
We provide results related to the study of prime points on level
sets of homogeneous integral forms which are linear or quadratic.
In the linear case we present an extension of the Green-Tao Theorem,
which finds affine copies of finite intervals in relatively dense
subsets of the primes, to a higher dimensional setting in which one
finds affine copies of suitably generic point configurations in
relatively dense subsets of a Cartesian product of the primes.
For general integral quadratic forms we present a result which is a
Birch-Goldbach type theorem for a single quadratic form with
sufficient rank. This guarantees solubility among the primes on the
level set of a quadratic form subject to local conditions. This is
an extension of a well known result of Hua.
12
Yesilyurt, Deniz. "Solving Linear Diophantine Equations And Linear Congruential Equations." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-19247.
Full textAbstract:
This report represents GCD, euclidean algorithm, linear diophantine equation and linear congruential equation. It investigates the methods for solving linear diophantine equations and linear congruential equations in several variables. There are many examples which illustrate the methods for solving equations.
13
Bienvenu, Pierre. "Linear, bilinear and polynomial structures in function fields and the primes." Thesis, University of Bristol, 2018. http://hdl.handle.net/1983/65dafc60-53df-4151-89b2-5efc0eff2b5c.
Full textAbstract:
This thesis is a contribution to arithmetic combinatorics. We present the Green-Tao method and the Green-Tao-Ziegler theorem concerning asymptotics for linear configura- tions of primes. Then we show extensions of our own to this theorem: first to some family of quadratic configurations, and secondly to configurations with unbounded coefficients. As a result, we are able to provide an asymptotic for configurations of primes inside the set of shifted squarefree numbers. We then leave integers and move to vector spaces over finite fields. In this context, we prove a bidirectional additive smoothing result for sets of pairs P ⊂ Fnp × Fnp . This is a bilinear version of Bogolyubov’s theorem. We then equip these vector spaces with a multiplicative structure, that is, we consider polynomial rings over finite fields. Using the Croot-Lev-Pach method, we show that sets of polynomials of degree less than n that contain no nontrivial solution to a given polynomial equation (of some specific type) is exponentially small. Finally, we seek to apply the Green-Tao method on polynomial rings, with the intention of deriving asymptotics for configurations of irreducible polynomials. To this aim, we bound correlations of the M ̈obius function with linear and quadratic phases.
14
Chen, Huyuan. "Fully linear elliptic equations and semilinear fractionnal elliptic equations." Thesis, Tours, 2014. http://www.theses.fr/2014TOUR4001/document.
Full textAbstract:
Cette thèse est divisée en six parties. La première partie est consacrée à l'étude de propriétés de Hadamard et à l'obtention de théorèmes de Liouville pour des solutions de viscosité d'équations aux dérivées partielles elliptiques complètement non-linéaires avec des termes de gradient,This thesis is divided into six parts. The first part is devoted to prove Hadamard properties and Liouville type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations with gradient term
15
Goedhart, Eva Govinda. "Explicit bounds for linear difference equations /." Electronic thesis, 2005. http://etd.wfu.edu/theses/available/etd-05102005-222845/.
Full text
16
Jonklass, Raymond. "Learners' strategies for solving linear equations." Thesis, Stellenbosch : Stellenbosch University, 2002. http://hdl.handle.net/10019.1/52915.
Full textAbstract:
Thesis (MEd)--University of Stellenbosch, 2002.ENGLISH ABSTRACT: Algebra deals amongst others with the relationship between variables. It differs from Arithmetic amongst others as there is not always a numerical solution to the problem. An algebraic expression can even be the solution to the problem in Algebra. The variables found in Algebra are often represented by letters such as X, y, etc. Equations are an integral part of Algebra. To solve an equation, the value of an unknown must be determined so that the left hand side of the equation is equal to the right hand side. There are various ways in which the solving of equations can be taught. The purpose of this study is to determine the existence of a cognitive gap as described by Herseovies & Linchevski (1994) in relation to solving linear equations. When solving linear equations, an arithmetical approach is not always effective. A new way of structural thinking is needed when solving linear equations in their different forms. In this study, learners' intuitive, informal ways of solving linear equations were examined prior to any formal instruction and before the introduction of algebraic symbols and notation. This information could help educators to identify the difficulties learners have when moving from solving arithmetical equations to algebraic equations. The learners' errors could help educators plan effective ways of teaching strategies when solving linear equations. The research strategy for this study was both quantitative and qualitative. Forty-two Grade 8 learners were chosen to individually do assignments involving different types of linear equations. Their responses were recorded, coded and summarised. Thereafter the learners' responses were interpreted, evaluated and analysed. Then a representative sample of fourteen learners was chosen randomly from the same class and semi-structured interviews were conducted with them From these interviews the learners' ways of thinking when solving linear equations, were probed. This study concludes that a cognitive gap does exist in the context of the investigation. Moving from arithmetical thinking to algebraic thinking requires a paradigm shift. To make adequate provision for this change in thinking, careful curriculum planning is required.
AFRIKAANSE OPSOMMING: Algebra behels onder andere die verwantskap tussen veranderlikes. Algebra verskil van Rekenkunde onder andere omdat daar in Algebra nie altyd 'n numeriese oplossing vir die probleem is nie. InAlgebra kan 'n algebraïese uitdrukking somtyds die oplossing van 'n probleem wees. Die veranderlikes in Algebra word dikwels deur letters soos x, y, ens. voorgestel. Vergelykings is 'n integrale deel van Algebra. Om vergelykings op te los, moet 'n onbekende se waarde bepaal word, om die linkerkant van die vergelyking gelyk te maak aan die regterkant. Daar is verskillende maniere om die oplossing van algebraïese vergelykings te onderrig. Die doel van hierdie studie is om die bestaan van 'n sogenaamde "kognitiewe gaping" soos beskryf deur Herseovies & Linchevski (1994), met die klem op lineêre vergelykings, te ondersoek. Wanneer die oplossing van 'n linêere vergelyking bepaal word, is 'n rekenkundige benadering nie altyd effektiefnie. 'n Heel nuwe, strukturele manier van denke word benodig wanneer verskillende tipes linêere vergelykings opgelos word. In hierdie studie word leerders se intuitiewe, informele metodes ondersoek wanneer hulle lineêre vergelykings oplos, voordat hulle enige formele metodes onderrig is en voordat hulle kennis gemaak het met algebraïese simbole en notasie. Hierdie inligting kan opvoeders help om leerders se kognitiewe probleme in verband met die verskil tussen rekenkundige en algebraïese metodes te identifiseer.Die foute wat leerders maak, kan opvoeders ook help om effektiewe onderrigmetodes te beplan, wanneer hulle lineêre vergelykings onderrig. As leerders eers die skuif van rekenkundige metodes na algebrarese metodes gemaak het, kan hulle besef dat hul primitiewe metodes nie altyd effektief is nie. Die navorsingstrategie wat in hierdie studie aangewend is, is kwalitatief en kwantitatief Twee-en-veertig Graad 8 leerders is gekies om verskillende tipes lineêre vergelykings individueel op te los. Hul antwoorde is daarna geïnterpreteer, geëvalueer en geanaliseer. Daarna is veertien leerders uit hierdie groep gekies en semigestruktureerde onderhoude is met hulle gevoer. Vanuit die onderhoude kon 'n dieper studie van die leerders se informele metodes van oplossing gemaak word. Die gevolgtrekking wat in hierdie studie gemaak word, is dat daar wel 'n kognitiewe gaping bestaan in die konteks van die studie. Leerders moet 'n paradigmaskuif maak wanneer hulle van rekenkundige metodes na algebraïese metodes beweeg. Hierdie klemverskuiwing vereis deeglike kurrikulumbeplanning.
17
Altassan, Alaa Abdullah. "Linear equations over free Lie algebras." Thesis, University of Manchester, 2013. https://www.research.manchester.ac.uk/portal/en/theses/linear-equations-over-free-liealgebras(6e29b286-1869-4207-b054-8baab98e70df).html.
Full textAbstract:
In this thesis, we study equations of the form $[x_1,u_1]+[x_2, u_2]+\ldots+[x_k,u_k]=0$ over a free Lie algebra $L$, where $k>1$ and the coefficients $u_1, u_2, \ldots,u_k$ belong to $L$. The starting point of this research is a paper [22], in which the authors embarked on a systematic study of very concrete linear equations over free Lie algebras. They focused on the given equations in the case where $k=2$. We generalise and develop a number of the results on equations with two variables to equations with an arbitrary number of indeterminates. Most of the results refer to the case where the coefficients coincide with the free generators of $L$. Throughout our research, we study some features of the solution space of these equations such as the homogenous structure and the fine homogenous structure. The main achievement in this work is that we give a detailed description of the solution space. Then we obtain explicit bases for some specific fine homogeneous components of the solution space, in particular, we give a basis for the "multilinear'' fine homogenous component. Moreover, we generalise earlier results on commutator calculus using the "language'' of free Lie algebras and apply them to determine the radical and the coordinate algebra of the solution space of the given equations.
18
Chen, Hua, Wei-Xi Li, and Chao-Jiang Xu. "Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations." Universität Potsdam, 2007. http://opus.kobv.de/ubp/volltexte/2009/3028/.
Full text
19
Hafez, Salah Taha. "Continued fractions and solutions of linear and non-linear lattice equations." Thesis, University of Kent, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236725.
Full text
20
Jimenez, Amelia. "Hands-on equations program: An approach to teaching linear equations using manipulatives." Scholarly Commons, 2011. https://scholarlycommons.pacific.edu/uop_etds/94.
Full textAbstract:
Recently there has been a keen interest in the area of mathematics and finding the best methods of instruction. For instance, the No Child Left Behind Act (NCLB) has placed new levels of accountability on educators for the success of students with their education, especially in mathematics. Certain areas of mathematics, such as Algebra, have been known to challenge students to think abstractly. This has become a difficult task for educators to accomplish. The challenge of teaching algebra becomes apparent when students do not comprehend the abstract reasoning of algebra. Many students need help with the transition from numerical calculation to the abstract reasoning required for algebra. This dissertation focuses on the best approaches to helping students with this transition. This dissertation investigates a mathematics program called Hands-On Equations (HOE), which is designed to help students learn abstract concepts taught in algebra with less difficulty. The program concentrates on the transition from numerical calculation to abstract reasoning by utilizing manipulatives. The objective of the study is to investigate the effectiveness of HOE in 9 th and 10 th grade. The research uses three pretests, three posttests, a three week retention test, a six week retention test, and benchmark tests to evaluate the academic growth of students in two set groups. The collected data is then quantitatively analyzed by applying simple t-tests and an ANOVA. Analysis of the data endorses HOE as being effective with solving linear equations at Level 1, Level 2, Level 3, and with the three-week retention tests, which indicates HOE may be a positive factor in achieving success with linear equations. However, analysis of the data revealed HOE is not as effective with the six-week retention test or the benchmark test which indicates after six weeks and beyond, students do not retain the information to be successful on end of the year exams such as benchmarks. The findings from this study may be useful to educators who are contemplating implementing HOE or other such programs at their schools.
21
Torshage, Axel. "Linear Functional Equations and Convergence of Iterates." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-56450.
Full textAbstract:
The subject of this work is functional equations with direction towards linear functional equations. The .rst part describes function sets where iterates of the functions converge to a .xed point. In the second part the convergence property is used to provide solutions to linear functional equations by de.ning solutions as in.nite sums. Furthermore, this work contains some transforms to linear form, examples of functions that belong to di¤erent classes and corresponding linear functional equations. We use Mathematica to generate solutions and solve itera- tively equations.
22
Grey, David John. "Parallel solution of power system linear equations." Thesis, Durham University, 1995. http://etheses.dur.ac.uk/5429/.
Full textAbstract:
At the heart of many power system computations lies the solution of a large sparse set of linear equations. These equations arise from the modelling of the network and are the cause of a computational bottleneck in power system analysis applications. Efficient sequential techniques have been developed to solve these equations but the solution is still too slow for applications such as real-time dynamic simulation and on-line security analysis. Parallel computing techniques have been explored in the attempt to find faster solutions but the methods developed to date have not efficiently exploited the full power of parallel processing. This thesis considers the solution of the linear network equations encountered in power system computations. Based on the insight provided by the elimination tree, it is proposed that a novel matrix structure is adopted to allow the exploitation of parallelism which exists within the cutset of a typical parallel solution. Using this matrix structure it is possible to reduce the size of the sequential part of the problem and to increase the speed and efficiency of typical LU-based parallel solution. A method for transforming the admittance matrix into the required form is presented along with network partitioning and load balancing techniques. Sequential solution techniques are considered and existing parallel methods are surveyed to determine their strengths and weaknesses. Combining the benefits of existing solutions with the new matrix structure allows an improved LU-based parallel solution to be derived. A simulation of the improved LU solution is used to show the improvements in performance over a standard LU-based solution that result from the adoption of the new techniques. The results of a multiprocessor implementation of the method are presented and the new method is shown to have a better performance than existing methods for distributed memory multiprocessors.
23
PEI, HUILING. "EXPLORING BOOTSTRAP APPLICATIONS TO LINEAR STRUCTURAL EQUATIONS." University of Cincinnati / OhioLINK, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1021928281.
Full text
24
Serna, Rodrigo. "Solving Linear Systems of Equations in Hardware." Thesis, KTH, Skolan för elektro- och systemteknik (EES), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-200610.
Full text
25
Dimakos, Michail. "Linear, linearisable and integrable nonlinear PDEs." Thesis, University of Cambridge, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.607875.
Full text
26
Porter, Annabelle Louise. "The evolution of equation-solving: Linear, quadratic, and cubic." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3069.
Full textAbstract:
This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work. It uses a historical perspective to highlight many of the concepts underlying modern equation solving.
27
Johnson, Solomon Nathan. "Best simultaneous approximation in normed linear spaces." Thesis, Rhodes University, 2018. http://hdl.handle.net/10962/58985.
Full textAbstract:
In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.
28
Le, Gia Quoc Thong. "Approximation of linear partial differential equations on spheres." Texas A&M University, 2003. http://hdl.handle.net/1969.1/22.
Full textAbstract:
The theory of interpolation and approximation of solutions to
differential and integral equations on spheres has attracted
considerable interest in recent years; it has also been applied
fruitfully in fields such as physical geodesy, potential theory,
oceanography, and meteorology.
In this dissertation we study the approximation of linear
partial differential equations on spheres, namely a class of
elliptic partial differential equations
and the heat equation on the unit sphere.
The shifts of a spherical basis
function are used to construct the approximate solution. In the
elliptic case, both the finite element method and the collocation method
are discussed. In the heat equation, only the collocation method is
considered. Error estimates in the supremum norms and the Sobolev norms
are obtained when certain regularity conditions are imposed on
the spherical basis functions.
29
Becker, Dulcenéia. "Parallel unstructured solvers for linear partial differential equations." Thesis, Cranfield University, 2006. http://hdl.handle.net/1826/4140.
Full textAbstract:
This thesis presents the development of a parallel algorithm to solve symmetric
systems of linear equations and the computational implementation of a parallel
partial differential equations solver for unstructured meshes. The proposed
method, called distributive conjugate gradient - DCG, is based on a single-level
domain decomposition method and the conjugate gradient method to obtain a
highly scalable parallel algorithm.
An overview on methods for the discretization of domains and partial differential
equations is given. The partition and refinement of meshes is discussed and
the formulation of the weighted residual method for two- and three-dimensions
presented. Some of the methods to solve systems of linear equations are introduced,
highlighting the conjugate gradient method and domain decomposition
methods. A parallel unstructured PDE solver is proposed and its actual implementation
presented. Emphasis is given to the data partition adopted and the
scheme used for communication among adjacent subdomains is explained. A series
of experiments in processor scalability is also reported.
The derivation and parallelization of DCG are presented and the method validated
throughout numerical experiments. The method capabilities and limitations
were investigated by the solution of the Poisson equation with various source
terms. The experimental results obtained using the parallel solver developed as
part of this work show that the algorithm presented is accurate and highly scalable,
achieving roughly linear parallel speed-up in many of the cases tested.
30
Hendriks, Peter Anne. "Algebraic aspects of linear differential and difference equations." [S.l. : [Groningen] : s.n.] ; [University Library Groningen] [Host], 1996. http://irs.ub.rug.nl/ppn/153769580.
Full text
31
Qi, Yuan-Wei. "The blow-up of quasi-linear parabolic equations." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.253381.
Full text
32
Davidson, Bryan Duncan. "Recursive projection for semi-linear partial differential equations." Thesis, University of Bristol, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.294932.
Full text
33
Caramanis, Constantine (Constantine Michael) 1977. "Solving linear partial differential equations via semidefinite optimization." Thesis, Massachusetts Institute of Technology, 2001. http://hdl.handle.net/1721.1/8949.
Full textAbstract:
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 49-51).
Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this thesis a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on both linear and nonlinear functionals defined on solutions of linear partial differential equations. We apply the proposed methods to examples of PDEs in one and two dimensions with very encouraging results. We also provide computational evidence that the semidefinite constraints are critically important in improving the quality of the bounds, that is without them the bounds are weak.
by Constantine Caramanis.
S.M.
34
van, Heerden Francois A. "Nonlinear Schrödinger Type Equations with Asymptotically Linear Terms." DigitalCommons@USU, 2002. https://digitalcommons.usu.edu/etd/7089.
Full textAbstract:
We study the nonlinear Schrödinger type equation
- Δu + (λg(x) + l)u = f(u)
on the whole space R^N. The nonlinearity f is assumed to be asymptotically linear and g(x) ≥ 0 has a potential well. We do not assume a limit for g(x) as lxl →∞ . Using variational techniques, we prove the existence of a positive solution for λ large. In the case where f is odd we obtain multiple pairs of solutions. The limiting behavior of solutions as λ →∞ is also considered.
35
Benner, Peter, Enrique Quintana-Ortí, and Gregorio Quintana-Ortí. "Solving Linear Matrix Equations via Rational Iterative Schemes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601460.
Full textAbstract:
We investigate the numerical solution of stable Sylvester equations via iterative schemes proposed for computing the sign function of a matrix. In particular, we discuss how the rational iterations for the matrix sign function can efficiently be adapted to the special structure implied by the Sylvester equation. For Sylvester equations with factored constant term as those arising in model reduction or image restoration, we derive an algorithm that computes the solution in factored form directly. We also suggest convergence criteria for the resulting iterations and compare the accuracy and performance of the resulting methods with existing Sylvester solvers. The algorithms proposed here are easy to parallelize. We report on the parallelization of those algorithms and demonstrate their high efficiency and scalability using experimental results obtained on a cluster of Intel Pentium Xeon processors.
36
Henriot, Kevin. "Structures linéaires dans les ensembles à faible densité." Thèse, Paris 7, 2014. http://hdl.handle.net/1866/11116.
Full textAbstract:
Réalisé en cotutelle avec l'Université Paris-Diderot.Nous présentons trois résultats en combinatoire additive, un domaine récent à la croisée de la combinatoire, l'analyse harmonique et la théorie analytique des nombres. Le thème unificateur de notre thèse est la détection de structures additives dans les ensembles arithmétiques à faible densité, avec un intérêt particulier pour les aspects quantitatifs. Notre première contribution est une estimation de densité améliorée pour le problème, initié entre autres par Bourgain, de trouver une longue progression arithmétique dans un ensemble somme triple. Notre deuxième résultat consiste en une généralisation des bornes de Sanders pour le théorème de Roth, du cas d'un ensemble dense dans les entiers à celui d'un ensemble à faible croissance additive dans un groupe abélien arbitraire. Finalement, nous étendons les meilleures bornes quantitatives connues pour le théorème de Roth dans les premiers, à tous les systèmes d'équations linéaires invariants par translation et de complexité un.
We present three results in additive combinatorics, a recent field at the interface of combinatorics, harmonic analysis and analytic number theory. The unifying theme in our thesis is the detection of additive structure in arithmetic sets of low density, with an emphasis on quantitative aspects. Our first contribution is an improved density estimate for the problem, initiated by Bourgain and others, of finding a long arithmetic progression in a triple sumset. Our second result is a generalization of Sanders' bounds for Roth's theorem from the dense setting, to the setting of small doubling in an arbitrary abelian group. Finally, we extend the best known quantitative results for Roth's theorem in the primes, to all translation-invariant systems of equations of complexity one.
37
Saravi, Masoud. "Numerical solution of linear ordinary differential equations and differential-algebraic equations by spectral methods." Thesis, Open University, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.446280.
Full textAbstract:
This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudospectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriatec hoice of Gauss-Chebyshev-Radapuo ints, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear.
38
Mathia, Karl. "Solutions of linear equations and a class of nonlinear equations using recurrent neural networks." PDXScholar, 1996. https://pdxscholar.library.pdx.edu/open_access_etds/1355.
Full textAbstract:
Artificial neural networks are computational paradigms which are inspired by biological neural networks (the human brain). Recurrent neural networks (RNNs) are characterized by neuron connections which include feedback paths. This dissertation uses the dynamics of RNN architectures for solving linear and certain nonlinear equations. Neural network with linear dynamics (variants of the well-known Hopfield network) are used to solve systems of linear equations, where the network structure is adapted to match properties of the linear system in question. Nonlinear equations inturn are solved using the dynamics of nonlinear RNNs, which are based on feedforward multilayer perceptrons. Neural networks are well-suited for implementation on special parallel hardware, due to their intrinsic parallelism. The RNNs developed here are implemented on a neural network processor (NNP) designed specifically for fast neural type processing, and are applied to the inverse kinematics problem in robotics, demonstrating their superior performance over alternative approaches.
39
Spence, Euan Alastair. "Boundary value problems for linear elliptic PDEs." Thesis, University of Cambridge, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.609476.
Full text
40
Tsang, Siu Chung. "Preconditioners for linear parabolic optimal control problems." HKBU Institutional Repository, 2017. https://repository.hkbu.edu.hk/etd_oa/464.
Full textAbstract:
In this thesis, we consider the computational methods for linear parabolic optimal control problems. We wish to minimize the cost functional while fulfilling the parabolic partial differential equations (PDE) constraint. This type of problems arises in many fields of science and engineering. Since solving such parabolic PDE optimal control problems often lead to a demanding computational cost and time, an effective algorithm is desired. In this research, we focus on the distributed control problems. Three types of cost functional are considered: Target States problems, Tracking problems, and All-time problems. Our major contribution in this research is that we developed a preconditioner for each kind of problems, so our iterative method is accelerated. In chapter 1, we gave a brief introduction to our problems with a literature review. In chapter 2, we demonstrated how to derive the first-order optimality conditions from the parabolic optimal control problems. Afterwards, we showed how to use the shooting method along with the flexible generalized minimal residual to find the solution. In chapter 3, we offered three preconditioners to enhance our shooting method for the problems with symmetric differential operator. Next, in chapter 4, we proposed another three preconditioners to speed up our scheme for the problems with non-symmetric differential operator. Lastly, we have the conclusion and the future development in chapter 5.
41
Zhang, Junchi. "GPU computing of Heat Equations." Digital WPI, 2015. https://digitalcommons.wpi.edu/etd-theses/515.
Full textAbstract:
There is an increasing amount of evidence in scientific research and industrial engineering indicating that the graphic processing unit (GPU) has a higher efficiency and a stronger ability over CPUs to process certain computations. The heat equation is one of the most well-known partial differential equations with well-developed theories, and application in engineering. Thus, we chose in this report to use the heat equation to numerically solve for the heat distributions at different time points using both GPU and CPU programs. The heat equation with three different boundary conditions (Dirichlet, Neumann and Periodic) were calculated on the given domain and discretized by finite difference approximations. The programs solving the linear system from the heat equation with different boundary conditions were implemented on GPU and CPU. A convergence analysis and stability analysis for the finite difference method was performed to guarantee the success of the program. Iterative methods and direct methods to solve the linear system are also discussed for the GPU. The results show that the GPU has a huge advantage in terms of time spent compared with CPU in large size problems.
42
Duelli, Markus Michael. "Functional calculus for bisectorial operators and applications to linear and non-linear evolution equations /." Berlin : Logos-Verl, 2005. http://deposit.ddb.de/cgi-bin/dokserv?id=2619697&prov=M&dok_var=1&dok_ext=htm.
Full text
43
Roidot, Kristelle. "Numerical study of non-linear dispersive partial differential equations." Phd thesis, Université de Bourgogne, 2011. http://tel.archives-ouvertes.fr/tel-00692445.
Full textAbstract:
Numerical analysis becomes a powerful resource in the study of partial differential equations (PDEs), allowing to illustrate existing theorems and find conjectures. By using sophisticated methods, questions which seem inaccessible before, like rapid oscillations or blow-up of solutions can be addressed in an approached way. Rapid oscillations in solutions are observed in dispersive PDEs without dissipation where solutions of the corresponding PDEs without dispersion present shocks. To solve numerically these oscillations, the use of efficient methods without using artificial numerical dissipation is necessary, in particular in the study of PDEs in some dimensions, done in this work. As studied PDEs in this context are typically stiff, efficient integration in time is the main problem. An analysis of exponential and symplectic integrators allowed to select and find the more efficient method for each PDE studied. The use of parallel computing permitted to address numerically questions of stability and blow-up in the Davey-Stewartson equation, in both stiff and non-stiff regimes.
44
Knight, Christopher J. K. "Freonts in non-linear wave equations with spatial inhomogeneity." Thesis, University of Surrey, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.549460.
Full text
45
Smith, James A. "“Looking for nothing" : Bayes linear methods for solving equations." Thesis, Durham University, 1993. http://etheses.dur.ac.uk/2207/.
Full textAbstract:
Here I will describe and implement Bayes linear methods for finding zeros of deterministic functions. We assume that the zero is known to be unique. Initially, the value of the function is modelled simply as the product of two independent factors, the position of the point from the zero and a "slope" which is assumed to vary "smoothly” with position. Additional prior information specifies first and second order properties of the slopes and the position of the zero: in particular, smoothness is specified by modelling the slope process to be stationary with a decreasing correlation function. This research is motivated by problems arising in large scale computer simulation of mathematical models of complex physical phenomena, where a single run of the code can be expensive and the output difficult to assimilate. Scientists are often confident about the structure of their model as a description of a physical process but may be uncertain about the values of certain model "parameters". Such parameters usually refer directly to physical attributes, and so collateral information about their values is usually available. In some applications, the physical process itself has been observed, and several runs of the code are made at different parameter settings in an attempt to match the realisation of the code with the actual realisation. The eventual aim is to aid scientists to search through the "parameter space” efficiently and systematically, using their knowledge of the process. Obviously, there are several respects in which this formulation does not tackle the real problem, as we mainly consider a single-valued function of a real variable. As well as considering this problem I will review the current state of play in the more general field of statistical numerical analysis and its relationship to deterministic computer experiments; and partial belief specification or Bayes linear methods
46
Sproule, Olive Elizabeth. "The development of concepts of linear and quadratic equations." Thesis, Queen's University Belfast, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.314224.
Full text
47
Lee, Sang-Gu. "Linear Operators Strongly Preserving Polynomial Equations Over Antinegative Semirings." DigitalCommons@USU, 1991. https://digitalcommons.usu.edu/etd/6984.
Full textAbstract:
We characterized the group of linear operators that strongly preserve r-potent matrices over the binary Boolean semiring, nonbinary Boolean semirings, and zero-divisor free antinegative semirings. We extended these results to show that linear operators that strongly preserve r-potent matrices are equivalent to those linear operators that strongly preserve the matrix polynomial equation p(X) = X. where p(X) = Xr1 + Xr2 + ... + Xrt and r1>r2>...>rt≥2.
In addition, we characterized the group of linear operators that strongly preserve r-cyclic matrices over the same semirings. We also extended these results to linear operators that strongly preserve the matrix polynomial equation p(X) = I where p(X) is as above.
Chapters I and II of this thesis contain background material and summaries of the work done by other researchers on the linear preserver problem. Characterizations of linear operators in chapters III, IV, V, and VI of this thesis are new.
48
Kisela, Tomáš. "Basics of Qualitative Theory of Linear Fractional Difference Equations." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2012. http://www.nusl.cz/ntk/nusl-234025.
Full textAbstract:
Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.
49
Torres, Ledesma César Enrique. "Non linear ellipter equations with non-local regional operators." Tesis, Universidad de Chile, 2013. http://www.repositorio.uchile.cl/handle/2250/115927.
Full textAbstract:
Doctor en Ciencias de la Ingeniería, Mención Modelación MatemáticaEsta tesis consiste de cinco partes. En la primera parte se considera el problema de Dirichlet lineal y no lineal con una difusi\'on no local regional definido implicitamente por \!\!donde $0< \alpha < 1$, $\rho \in C(\overline)$ y $\lambda dist(x,\partial \Omega) \leq \rho (x) \leq dist(x, \partial \Omega)$ con $\lambda \in (0,1]$, $x\in \Omega$. Haciendo uso del teorema de Lax-Milgran y el Teorema del paso de la monta\~na se demuestra la existencia de soluciones d\'ebiles. En la segunda parte, se considera la ecuaci\'on de Schr\"odinger no lineal con difusi\'on no local regional {\small \begin{eqnarray}\label{Aeq04-} \epsilon^{2\alpha} (-\Delta)_{\rho}^{\alpha}u + u = f(u) \quad \mbox{in}\quad \mathbb{R}^{n},\quad u \in H^{\alpha}(\mathbb{R}^{n}), \end{eqnarray}} \!\!donde $0< \alpha <1$, $\epsilon>0$, $n\geq 2$ y $f:\mathbb{R} \to \mathbb{R}$ es super-lineal y tiene un crecimiento sub-critico. El operador $(-\Delta)_{\rho}^{\alpha}$ es el laplaciano no local regional, con rango de alcance determinado por una funci\'on positiva $\rho \in C(\mathbb{R}^{n}, \mathbb{R}^{+})$ y definido por {\small \begin{eqnarray}\label{Aeq05-} \int_{\mathbb{R}^{n}} \!\!\!\!(-\Delta)_{\rho}^{\alpha} uvdx = \int_{\mathbb{R}^{n}}\!\!\int_{B(0,\rho (x))} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\frac{[u(x+z) - u(x)][v(x+z) - v(x)]}{|z|^{n+2\alpha}}dzdx. \end{eqnarray}} \!\!Se prueba la existencia de soluci\'on d\'ebil para (\ref{Aeq04-}) aplicando el Teorema del paso de la monta\~na al funcional $I_{\rho}$ definido en $H_{\rho}^{\alpha}(\mathbb{R}^{n})$, combinado con un argumento de comparaci\'on creado por Rabinowitz. El objetivo principal de la tercera parte es estudiar el comportamiento de concentraci\'on de la soluci\'on d\'ebil de la ecuaci\'on (\ref{Aeq04-}) con $f(s) = s^{p}$, cuando $\epsilon \to 0$. En la cuarta parte se estudia el resultado de simetr\'ia para las soluciones ground state de (\ref{Aeq04-}). Para tal prop\'osito, se combina los rearreglos de funciones con los m\'etodos variacionales. Finalmente, se considera un sistema Hamiltoniano fraccionario {\small \begin{eqnarray}\label{Aeq08-} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla W(t,u(t)) \end{eqnarray}} \!\!donde $\alpha \in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n\times n})$ es una matriz sim\'etrica positiva definida para todo $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R} \times \mathbb{R}^{n}, \mathbb{R})$ y $\nabla W (t,u)$ es el gradiente de $W$ en $u$. Se demuestra que (\ref{Aeq08-}) posee al menos una soluci\'on no trivial via el Teorema del paso de la monta\~na.
50
Nabti, Abderrazak. "Non linear, non-local evolution equations : theory and application." Thesis, La Rochelle, 2015. http://www.theses.fr/2015LAROS032.
Full textAbstract:
Cette thèse concerne l’étude qualitative (existence locale, existence globale, explosion en temps fini) de quelques équations de Schrödinger non-linéaires non-locales. Dans le cas où les solutions explosent en temps fini, l’estimation du temps maximal d’existence des solutions sera présentée. Le chapitre 1 concerne l’étude d’une équation de Schrödinger non-linéaire sur RN. On s’intéresse à l’existence locale d’une solution pour toute condition initiale donnée dans L2(RN). De plus, on montre que la norme-L2 de la solution explose en temps fini T < 1. Les démonstrations reposent essentiellement sur le théorème de point fixe de Banach et les estimations de Strichartz, et aussi sur le choix convenable de la fonction test dans la formulation faible du problème. Dans le chapitre 2, on considère une équation de Schrödinger non-linéaire non-locale en temps, et on démontre que les solutions de notre problème explosent en temps fini ; ensuite on obtient des conditions nécessaires d’existence globale. Finalement, on obtient une borne inférieure du temps maximal d’existence de la solution. Le chapitre 3 porte sur la non-existence de solutions d’une équation de Schrödinger non-linéaire posée dans RN. Dans un premier temps, sous certaines conditions sur la donnée initiale, on montre qu’il n’existe pas de solution faible globale ; puis on donne une estimation du temps maximal d’existence de la solution. Enfin, on établit des conditions d’existence locale, ou globale de l’équation considérée. En plus, on généralise les résultats précédents au cas d’un système 2 _ 2. Le dernier chapitre traite une équation de Schrödinger non-linéaire non-locale en temps sur le groupe de Heisenberg H. En utilisant la méthode de la fonction test, on démontre que l’équation n’admet pas de solution faible globale. De plus, on obtient, sous certaines conditions sur les données initiales, une estimation inférieure du temps maximal d’existence de la solutionOur objective in this thesis is to study the existence of local solutions, existence global and blow up of solutions at a finite time to some nonlinear nonlocal Schrödinger equations. In the case when a solution blows-up at a finite time T < 1, we obtain an upper estimate of the life span of solutions. In the first chapter, we consider a nonlinear Schrödinger equation on RN. We first prove local existence of solution for any initial condition in L2 space. Then we prove nonexistence of a nontrivial global weak solution. Furthermore, we prove that the L2-norm of the local intime L2-solution blows up at a finite time. The second chapter is dedicated to study an initial value problem for the nonlocal intime nonlinear Schrödinger equation. Using the test function method, we derive a blow-up result. Then based on integral inequalities, we estimate the life span of blowing-up solutions. In the chapter 3, we prove nonexistence result of a space higher-order nonlinear Schrödinger equation. Then, we obtain an upper bound of the life span of solutions. Furthermore, the necessary conditions for the existence of local or global solutions are provided. Next, we extend our results to the 2 _ 2-system. Our method of proof rests on a judicious choice of the test function in the weak formulation of the equation. Finally, we consider a nonlinear nonlocal in time Schrödinger equation on the Heisenberg group. We prove nonexistence of non-trivial global weak solution of our problem. Furthermore, we give an upper bound of the life span of blowing up solutions