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"Fields (mathematics)"
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Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities
In this work we deal with linear second order partial differential operators of the following type:
eBook
Quadratic Vector Equations On Complex Upper Half-Plane
The authors consider the nonlinear equation -\\frac 1m=z+Sm with a parameter z in the complex upper half plane \\mathbb H , where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \\mathbb H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on \\mathbb R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z\\in \\mathbb H, including the vicinity of the singularities.
eBook
Fundamental solutions and local solvability for nonsmooth Hörmander’s operators
The authors consider operators of the form L=\\sum_{i=1}^{n}X_{i}^{2}+X_{0} in a bounded domain of \\mathbb{R}^{p} where X_{0},X_{1},\\ldots,X_{n} are nonsmooth Hörmander's vector fields of step r such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution \\gamma for L and provide growth estimates for \\gamma and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that \\gamma also possesses second derivatives, and they deduce the local solvability of L, constructing, by means of \\gamma, a solution to Lu=f with Hölder continuous f. The authors also prove C_{X,loc}^{2,\\alpha} estimates on this solution.
eBook
Wave Front Set of Solutions to Sums of Squares of Vector Fields
We study the (micro)hypoanalyticity and the Gevrey hypoellipticity of sums of squares of vector fields in terms of the Poisson–Treves
stratification. The FBI transform is used. We prove hypoanalyticity for several classes of sums of squares and show that our method,
though not general, includes almost every known hypoanalyticity result. Examples are discussed.
eBook
The Pinch Technique and its Applications to Non-Abelian Gauge Theories
Non-Abelian gauge theories, such as quantum chromodynamics (QCD) or electroweak theory, are best studied with the aid of Green's functions that are gauge-invariant off-shell, but unlike for the photon in quantum electrodynamics, conventional graphical constructions fail. The Pinch Technique provides a systematic framework for constructing such Green's functions, and has many useful applications. Beginning with elementary one-loop examples, this book goes on to extend the method to all orders, showing that the Pinch Technique is equivalent to calculations in the background field Feynman gauge. The Pinch Technique Schwinger-Dyson equations are derived, and used to show how a dynamical gluon mass arises in QCD. Applications are given to the center vortex picture of confinement, the gauge-invariant treatment of resonant amplitudes, the definition of non-Abelian effective charges, high-temperature effects, and even supersymmetry. This book is ideal for elementary particle theorists and graduate students.
eBook
Teaching knowledge and difficulties of In-field and Out-of-field Junior High School mathematics teachers in algebra
This paper sought to explore the Knowledge of Algebra for Teaching and the algebra difficulties of In-field and Out-of-field Junior High School mathematics teachers using the expanded KAT framework. The study employed the descriptive survey design and involved the participation of 374 mathematics teachers using an achievement test instrument. The study projected School Algebra Knowledge as the prevailing knowledge domain and revealed that the two categories of Junior High School mathematics teachers possess knowledge that falls below average for five (5) out of the seven (7) algebra knowledge domains and also for the overall Knowledge of Algebra for Teaching. However, In-field mathematics teachers showed higher knowledge as compared to Out-of-field mathematics teachers in six (6) out of the seven (7) algebra knowledge domains as well as the overall Knowledge of Algebra for Teaching. Also, the research revealed that mathematics teachers with 5 or more years of teaching experience have higher Knowledge of Algebra for Teaching as compared to those with below 5 years of teaching experience. The study finally identified eight major algebra difficulties among Junior High School mathematics teachers. The findings of the study have implications for teacher preparation, policy and practice.
Journal Article
On MDS and MDR Codes over the Ring 2,q ≔ qu, υ/⟨u2 – u, υ2 – υ
Linear block codes that achieve equality in the Singleton bound are called maximum distance separable (MDS) codes. In codes over finite fields, MDS codes are the class of linear codes whose distance is the largest possible among linear codes of the same length and dimension. In that sense, for codes over finite rings, MDS codes are generalized to maximum distance with respect to rank (MDR) codes. In this article, we discuss the MDS and MDR codes over a specific ring 2,q ≔ q[u, υ]/⟨u2 – u, υ2 – υ⟩, together with their properties related to the MDS codes over a finite field q and the MDS matrices.
Journal Article
Structure Preserving Schemes for Coupled Nonlinear Schrödinger Equation
The numerical solution of CNLS equations are studied for periodic wave solutions. We use the first order partitioned average vector field method, the second order partitioned average vector field composition method and plus method. The nonlinear implicit schemes preserve the energy and the momentum. The results show that the methods are successful to get approximation.
Journal Article