TRANSMISSION EIGENVALUES FOR MULTIPOINT SCATTERERS

We study the transmission eigenvalues for the multipoint scatterers of the Bethe- Peierls-Fermi-Zeldovich-Beresin-Faddeev type in dimensions d = 2 and d = 3. We show that for these scatterers: 1) each positive energy E is a transmission eigenvalue (in the strong sense) of infinite multiplicity; 2) each complex E is an interior transmission eigenvalue of infinite multiplicity. The case of dimension d = 1 is also discussed.

RECONSTRUCTION OF GAUSSIAN RANDOM FUNCTIONS FROM SPECTRUM DATA

It is known that a stationary random process is represented as a superposition of harmonic oscillations with real frequencies and uncorrelated amplitudes. In the study of nonstationary processes, it is natural to have increasing or declining oscillationсs. This raises the problem of constructing algorithms that would allow constructing broad classes of nonstationary processes from elementary nonstationary random processes. A natural generalization of the concept of the spectrum of a nonstationary random process is the transition from the real spectrum in the case of stationary to a complex or infinite multiple spectrum in the nonstationary case. There is also the problem of describing within the correlation theory of random processes in which the spectrum has no analogues in the case of stationary random processes, namely, the spectrum point is real, but it has infinite multiplicity for the operator image of the corresponding operator, and when the spectrum itself is complex. Reconstruction of the complex spectrum of a nonstationary random function is a very important problem in both theoretical and applied aspects. In the paper the procedure of reconstruction of random process, sequence, field from a spectrum for Gaussian random functions is developed. Compared to the stationary case, there are wider possibilities, for example, the construction of a nonstationary random process with a real spectrum, which has infinite multiplicity and which can be distributed over the entire finite segment of the real axis. The presence of such a spectrum leads, in contrast to the case of a stationary random process, to the appearance of new components in the spectral decomposition of random functions that correspond to the internal states of «strings», i.e. generated by solutions of systems of equations in partial derivatives of hyperbolic type. The paper deals with various cases of the spectrum of a non-self-adjoint operator , namely, the case of a discrete spectrum and the case of a continuous spectrum, which is located on a finite segment of the real axis, which is the range of values of the real non-decreasing function a(x). The cases a(x)=0, a(x)=a0, a(x)=x and a(x) is a piecewise constant function are studied. The authors consider the recovery of nonstationary sequences for different cases of the spectrum of a non-self-adjoint operator promising since spectral decompositions are a superposition of discrete or continuous internal states of oscillators with complex frequencies and uncorrelated amplitudes and therefore have deep physical meaning.

Membranes with thin and heavy inclusions: Asymptotics of spectra

We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator is non-self-adjoint and possesses the Jordan chains of length 2. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity. The complete asymptotic analysis of eigenvalues has been carried out.

A Note on the Three Dimensional Dirac Operator with Zigzag Type Boundary Conditions

In this note the three dimensional Dirac operator $$A_m$$ A m with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $$A_m$$ A m is self-adjoint in $$L^2(\Omega ;{\mathbb {C}}^4)$$ L 2 ( Ω ; C 4 ) for any open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in $$\Omega $$ Ω . In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of $$A_m$$ A m consists of discrete eigenvalues that accumulate at $$\pm \infty $$ ± ∞ and one additional eigenvalue of infinite multiplicity.

Interview with Patricia Hill Collins on Critical Thinking, Intersectionality and Educational: key objectives for critical articulation on Inclusive Education

The interview with Patricia Hill Collins, a prominent social theorist whose research and studies have examined issues of race, gender, social class, sexuality and / or nation, make her a significant reference in the field of Education and Intersectionality. The content of this interview can be described in Deleuzian and Guattarian terms as crucial elements in the configuration of a Pedagogy of the minor, that is, centered on the multiplicity of differences that inhabit the school space, of which semiological, citizen and political force demand the reconfiguration of the school space. In such a case, Inclusive Education involves a complex change in the way of thinking and practicing a variety of problems and issues that relate to the totality of students known as multiple singularities. Hill Collins thinks that intersectionality is seen as a form of research and critical practice that academics and activists have used to develop a more complex understanding of social inequality and social injustice, emerging in many places by people who dealt with the common social problem to respond to social injustice. The construction of a public education through the lens of inclusion demands the building – positive labor– of an educational architecture capable of critically examining issues related to race, schools, the common benefit and democratic possibilities, recognizing the endemic nature of the violence. All this suggests a change of paradigm in the way in which intersection power systems inform the structures and organizational practices of schools. A public education system adheres to broader ethical principles of equity, equality, justice and inclusion. Thereby assuming, as a complex, relational, structural and multidimensional term. Countering the effects of differential inclusion, that is, through inequalities, proposes the challenge to educational systems to address the production of socially unfair results through education as a mechanism to reproduce inequality. American social theory points out that it is vital to offer an understanding of social justice in a more complex way to address educational inequalities. The interview addresses topics related to social and educational justice, the contribution of feminism as critical elements in the construction of the epistemology of Inclusive Education, the contributions of the intersectional current as a heuristic and methodological device key in the examination of law in the education of the infinite multiplicity of differences

Conclusions

Reviewing the ways this book examines anthropology’s fraught and contradictory relationship with theology and the potential the latter offers for revitalizing the former, this chapter extends this book’s exercise of critique. Examining anthropology’s fealty to secular sovereign reason, and by extension the sovereign state, it questions the discipline’s fear of revelation. Should anthropology reintegrate revelation, not only within its catalogue of topics for examination, but with its very own reason, it could attune to reason’s fragility, acquire an alertness to integrative capacities disavowed by the modern university, and more fully consider infinite multiplicity. In reconstructing paths back to “Athens” and “Jerusalem,” as well as beyond, we could wonder not only, “what is different?” but also “what is?”

Infinite multiplicity of stable entire solutions for a semilinear elliptic equation with exponential nonlinearity

We consider the infinite multiplicity of entire solutions for the elliptic equation Δu + K(x)eu + μf(x) = 0 in ℝn, n ⩾ 3. Under suitable conditions on K and f, the equation with small μ ⩾ 0 possesses a continuum of entire solutions with a specific asymptotic behaviour. Typically, K behaves like |x|ℓ at ∞ for some ℓ > −2 and the entire solutions behave asymptotically like − (2 + ℓ)log |x| near ∞. Main tools of the analysis are comparison principle for separation structure, asymptotic expansion of solutions near ∞, barrier method and strong maximum principle. The linearized operator for the equation has two characteristic behaviours related with the stability and the weak asymptotic stability of the solutions as steady states for the corresponding parabolic equation.

Respectable uncertainty and pathetic truth in Amazonian Quichua-speaking culture

It is argued in this chapter, on the basis of evidence from grammar, discourse, and verbal art, that for Amazonian Quichua speakers, there is a cultural preference for expressing uncertainty, which is linked with animistic perspectivism. Animistic perspectivism endows nonhumans with subjectivity and implies that there is an infinite multiplicity of perspectives, thereby making a single, totalizing truth impossible. Respectable uncertainty is also apparent in the system of evidentiality, in speech reports, echo questions, and verbal art, all of which emphasize perspective over certainty. A type of certainty that Runa do value, however, and which would not be valid within a rational framework of inquiry, is that of emotional truth, involving feelings of empathy for others, including nonhumans. Emotional truth, then, provides an exception to the preference for uncertainty, and may lead people to confidently reason about ethical matters.​