The Ministry of Education, Culture, Sports, Science and Technology (文部科学省, Monbu-kagaku-shō, lit. Ministry of Letters and Science), also known as MEXT or Monka-shō, is one of the eleven Ministries of Japan that composes part of the executive branch of the Government of Japan. Its goal is to improve the development of Japan in relation with the international community. The ministry is responsible for funding research under its jurisdiction, some of which includes: children's health in relation to home environment, delta-sigma modulations utilizing graphs, gender equality in sciences, neutrino detection which contributes to the study of supernovas around the world, and other general research for the future.
For a positive elliptic operator A, the logarithmic zeta determinant ln_ζA=-ζ_A'(0) combines UV information encoded by local heat-kernel coefficients with finite contributions determined by the full spectrum. We introduce a finite-difference zeta readout based on ζ_A(0) and ζ_A(q-1), defining a one-parameter meromorphic family whose node-coalescence limit q→ 1 recovers the standard logarithmic zeta determinant. The parameter q fixes the Mellin evaluation point s=q-1, organising genuine poles, regular local special values, generic full-spectrum values, and the logarithmic determinant limit along a common coordinate, while simultaneously determining the spectral weight λ^-q in the q-dependent variational response. In relative spectral problems, this coordinate distinguishes systems retaining a leading local hierarchy from those in which the entire local power-law hierarchy cancels, illustrated respectively by a reflectionless soliton and a twisted circle. In four dimensions, the framework recovers the standard local scale response at q=1, governed by the heat-kernel coefficient a_4, whereas at generic regular values of q it retains finite mass-sensitive information beyond the local hierarchy. The construction thereby provides a unified analytic framework for comparing local UV structure, finite full-spectrum information, variational response, and relative spectral behaviour within fixed operator spectra.
We study compact orientable essential surfaces in knot exteriors in the 3-sphere. The genus g, the number of boundary components b, and the boundary slope p/q are fundamental invariants of an essential surface. The realization problem asks whether, for a given triple (g, b, q) with g ≥ 0, b ≥ 1, and q ≥ 1, there exists a knot K ⊂ S^3 whose exterior E(K) contains a compact orientable essential surface F of genus g with b boundary components and boundary slope p/q for some p. In general, not all combinations of (g, b, q) are realizable. First, we show that if b is odd, then q must be equal to 1. Our main theorem states that for any given even b ≥ 2 and q ≥ 1, there exist a genus g ≥ 0 and a knot K such that E(K) contains a compact orientable essential surface with these parameters.
