Inverse Spectral Problem for Analytic Domains I: Balian-Bloch Trace Formula

(II) fedN(X) = ^ e ± i ^ i J k is a tempered distributional function of t. The classical trace formula says that the singular support of (II) is contained in the length spectrum of X. Moreover, under suitable hypotheses on geodesic flow, the trace formula gives considerable information about the singularities in (II). (See [DG and C].) There is a fairly straightforward (and not terribly interesting) generalization of the trace formula to vector bundles. (See, for instance, the introduction to [DG].) We will be concerned in this article with a much more subtle generalization inspired by recent articles of Hogreve, Potthoff, and Schrader [HPS], and Schrader and Taylor [ST] in Communications in Mathematical Physics. Let G be a compact Lie group and ir: P —> X a principle G-bundle with connection. Given a finite-dimensional unitary representation, p, of G we will denote by Ep the vector bundle over X associated with p and by Dp the associated connection. Now consider a ladder {pe, e = 1,2,...} of irreducible representations of G. (This means that the maximal weight of pe is e times the maximal weight of p\.) For given e let Afc,e? # = 1, 2, O, . . . be the spectrum of the Laplace operator on C(Epe): Ae = D*peDpe + e .

Parents of children with genetically determined leukoencephalopathies play a major role in their children's health care. Because of the COVID-19 pandemic, many health care services were suspended, delayed or delivered remotely with telemedicine. We sought to explore the experience of parents of children with genetically determined leukoencephalopathies during the pandemic given the adapted health care services. We conducted semistructured interviews with 13 parents of 13 affected children. Three main themes were identified using thematic analysis: perceived impact of COVID-19 on health care services, benefits and challenges of telemedicine, and expectations of health care after the pandemic. Parents perceived a loss/delay in health care services while having a positive response to telemedicine. Parents wished telemedicine would remain in their care after the pandemic. This is the first study assessing the impact of COVID-19 on health care services in this population. Our results suggest that parents experience a higher level of stress owing to the shortage of services and the children's vulnerability.

In this paper we study the inverse spectral problem for singular Sturm–Liouville operator with sign valued weight. We define the spectral data of the problem, we construct the main integral equation. Solve the inverse spectral problem and use the inverse problem by two spectrum.

We show that trapezoids with identical Neumann spectra are congruent up to rigid motions of the plane. The proof is based on heat trace invariants and some new wave trace invariants associated to certain diffractive billiard trajectories. The reason we can only treat the Neumann case is that the wave trace is "more singular" for the Neumann case compared to the Dirichlet case. This is a new observation which is interesting on its own.

Nonlinear Cauchy/Robin inverse problems solved by an optimal splitting-linearizing method

Editorial

Our aim in this section is to establish an explicit formula allowing one to calculate the Euler characteristic of the metric graph directly from the spectrum of the standard Laplace operator.

A linear inverse model for the temperature-dependent thermal conductivity determination in one-dimensional problems

In this paper, a system of nonlinear Kaup equations with a loaded additional term in the class of periodic functions with respect to the spatial variable is considered. The invariance of the spectrum is proved and an analog of the Dubrovin system is derived for the evolution of the spectral parameters of a quadratic pencil of Sturm-Liouville operators on the entire line, the periodic coefficients of which are the solution of the Cauchy problem posed for the loaded system of nonlinear Kaup equations. Using trace formulas and an analog of the Dubrovin system, it is shown that the loaded system of nonlinear Kaup equations can be integrated by the inverse spectral problem method. An algorithm for solving the Cauchy problem for the loaded nonlinear Kaup’s system in the class of periodic functions with respect to the spatial variable is obtained. It is shown that if the initial functions are real analytic functions, then the solution will also be an analytic function with respect to the spatial variable. The 𝜋/2 -periodicity of the solution with respect to the spatial variable is revealed for the 𝜋/2 -periodicity of the initial functions.

We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains.

This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.

The inverse spectral problem is concerned with the question to what extent the spectrum of a domain determines its geometry. We find that, associated to a convex domain Ω in ℝ2, there is a convexfunction which is a length spectrum invariant under continuous deformations. It includes several geometric quantities, such as the lengths and Lazutkin parameters of caustics, as well as the asymptotic invariants discovered by Marvizi and Melrose. Via a Poisson relation, we also find invariants determined by the Laplace spectrum of Ω.

Inverse spectral problems are studied for the non-self-adjoint matrix Sturm–Liouville differential equation on a finite interval. We give formulations of the inverse problems, prove the corresponding uniqueness theorems, and provide a constructive procedure for the solution of the inverse problems by the method of spectral mappings. The obtained results are a natural generalization of the classical results in inverse problem theory for scalar Sturm–Liouville operators.

In this chapter we state and solve some Cauchy type problems in the complex domain formulated in terms of associated integro-differential operators \(\mathbb{L}_{{s + 1/2}}^{*}\left( {s = 1,2,...} \right) \) and \( {\mathbb{L}^ * }_{s + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) (s= 1, 2,…) of fractional order. We represent explicitly the solutions of these problems by the Mittag-Leffler type function \( {E_{s + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\) (z;µ), and we prove an analog of the classical Lagrange formula for these solutions. Then we state some special boundary value problems in the complex domain by means of the mentioned operators. Namely, we assume that the solutions of the mentioned Cauchy type problems satisfy some boundary conditions at the ends of the sum of odd segments $$\gamma 2s + \left( \sigma \right) = \mathop{{\mathop{ \cup }\limits_{{h \to - s}} }}\limits^{s} \left\{ {z = r \exp \left( {i\pi \left( {h + 1/2} \right)} \right),0 \leqslant r \leqslant \sigma } \right\} $$ situated in the Riemann surface G∞ of Ln z.KeywordsEntire FunctionVector FunctionComplex DomainRiesz BaseCommon EndpointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Physics-informed neural networks for high-speed flows