Geometry of the generalized geodesic flow and inverse by Vesselin M. Petkov, Luchezar N. Stoyanov

By Vesselin M. Petkov, Luchezar N. Stoyanov

This booklet is a brand new variation of a title originally released in1992. No different publication has been released that treats inverse spectral and inverse scattering effects through the use of the so referred to as Poisson summation formulation and the comparable examine of singularities. This booklet provides these in a closed and accomplished shape, and the exposition is predicated on a mix of alternative instruments and effects from dynamical platforms, microlocal research, spectral and scattering theory.

The content material of the first edition is nonetheless proper, but the re-creation will contain a number of new effects tested after 1992; new textual content will comprise a few 3rd of the content material of the recent version. the most chapters within the first version together with the hot chapters will supply a greater and extra complete presentation of value for the purposes inverse effects. those effects are received via sleek mathematical thoughts which will be provided jointly that allows you to supply the readers the chance to fully comprehend them. additionally, a few easy familiar homes verified via the authors after the ebook of the 1st version setting up the big variety of applicability of the Poison relation might be offered for first time within the new version of the book.

Show description

Read or Download Geometry of the generalized geodesic flow and inverse spectral problems PDF

Best geometry books

Geometry for the Classroom

Meant to be used in university classes for potential or in-service secondary university academics of geometry. Designed to provide lecturers huge instruction within the content material of easy geometry in addition to heavily similar subject matters of a touch extra complex nature. The presentation and the modular structure are designed to include a versatile technique for the instructing of geometry, person who will be tailored to diversified lecture room settings.

Basic noncommutative geometry

"Basic Noncommutative Geometry offers an creation to noncommutative geometry and a few of its purposes. The e-book can be utilized both as a textbook for a graduate direction at the topic or for self-study. it is going to be beneficial for graduate scholars and researchers in arithmetic and theoretical physics and all people who find themselves attracted to gaining an figuring out of the topic.

Advances in Architectural Geometry 2014

This publication includes 24 technical papers awarded on the fourth version of the Advances in Architectural Geometry convention, AAG 2014, held in London, England, September 2014. It deals engineers, mathematicians, designers, and contractors perception into the effective layout, research, and manufacture of complicated shapes, with a view to aid open up new horizons for structure.

Additional info for Geometry of the generalized geodesic flow and inverse spectral problems

Example text

J ) ∈ Rn−1 and n−1 (t) zj = ξj t=1 ∂ϕj (t) ∂uj (0). ≥ 0. REFLECTING RAYS 37 Notice that for νj = ν(qj ), there exists λj > 0 such that vjj−1 + vjj+1 = −λj vj . Since the hypersurface Uj = ϕj (Rn−1 ) ⊂ Γ is convex at qj , the choice of the normal field ν shows that the second fundamental form Bj of Uj at qj is a non-positive definite. That is, n−1 Bj (ξj , ξj ) = νj , t,m=1 ∂ 2 ϕj (0) (t) (m) ∂uj ∂uj (t) (m) ξj ξj ≤0 for all ξj ∈ Rn−1 . Using the expressions for the second derivatives of G at 0 in the three possible cases, we get k n−1 ∂2G σ= (t) (t) (m) (m) j=1 t,m=1 ∂uj ∂uj k + ⎛ n−1 (0) ξj ξj ∂2G (t) (m) (0) ξj ξi (t) (m) j=1 i∈Ij t,m=1 ∂uj ∂ui k = ⎝− n−1 λj k n−1 + ∂ϕj aji (t) ∂uj j=1 i∈Ij t,m=1 k n−1 − ∂ϕj aji (t) ∂uj j=1 i∈Ij t,m=1 + ⎝− (t) (m) ∂uj ∂ui t,m=1 j=1 ⎛ ∂ 2 ϕj νj , k n−1 (t) ∂uj j=1 i∈Ij t,m=1 n−1 + aji j=1 i∈Ij t,m=1 ∂ϕj (t) ∂uj k =− ∂ϕj (m) ∂uj (0), vji ∂ϕj aji k (0), (0) (0), (0), vji (t) (m) ξj ξi (t) (m) (0) ξj ξj ⎞ ∂ϕj (m) (0), vji (0) ξj ξi ∂uj ∂ϕi (m) ∂ui (t) (m) ⎠ ξj ξi aji zj , vji j=1 i∈Ij k aji zj , zi + j=1 i∈Ij (0), vji k j=1 i∈Ij k − (m) ∂ui aji zj , zj − λj Bj (ξj , ξj ) + j=1 (t) (m) ⎞ ∂ϕi k (t) (m) ⎠ ξj ξj aji zj , vji zi , vji .

I and i+1 satisfy the law of reflection The points x1 , . . , xs will be called reflection points of γ, while k γ xi − xi+1 = i=1 will be called the length of γ. 1). If γ contains a segment orthogonal to X at some of its end points, then γ will be called symmetric, otherwise it will be called non-symmetric. 2). If γ has no such segments, we will say that γ is ordinary. 2 are not considered as reflection points. In general a periodic reflecting ray can pass two or more times through some of its reflection points, and two different periodic reflecting rays could have some common reflection points.

Choose an arbitrary > 0 such that ψ˜i ≥ I for every i = 1, . . , m and denote by Mk ( ) the subspace of Mk consisting of all M ∈ Mk such that M ≥ I. Notice that Ak (Mk−1 ( )) ⊂ Mk ( ) and for any A, B ∈ Mk−1 ( ) we have Ak (A) − Ak (B) = σk ((I + λk A)−1 (A − B)(I + λk B)−1 )σk . Therefore, Ak (A) − Ak (B) ≤ (1 + λk )−2 A − B ≤ A−B , (1 + λ)2 where λ = min λk . This shows that for any k the map Ak is a contraction from Mk−1 ( ) to Mk ( ). Then the map A = Am ◦ Am−1 ◦ · · · ◦ A1 is a contraction from M0 ( ) into M0 ( ).

Download PDF sample

Rated 4.71 of 5 – based on 26 votes