A. Savchuk and A. Shkalikov [17] gave thorough spectral analysis of such operators. In particular, they consider a broad class of boundary conditions (bc) – see (1.6), Theorem 1.5 there – in terms of a function y and its quasi–derivative u = y′ −Qy. The natural form of periodic or antiperiodic (Per±) bc is the following one: (1.3) Per± : y(π) = ±y(0), u(π) = ±u(0) If the potential v happens to ...

If l 6⊂ M , let π2 : M2−→M1 be the blow-up at y and let E2 be its exceptional divisor. Then LM2 −π ∗ 2E1−E2 is nef as in (0.5), and we get a contradiction by using vanishing theorem. If l ⊂ M , there is a smooth member D of |L| such that l ⊂ D, provided n ≥ 3. Let D1 be the proper transform of D on M1. Then y ∈ D1 and [π (K + tL)−E1]D1 = [π ∗(KD + (t− 1)LD)− E1]D1 is spanned at y by the inducti...

A spatial coherenceies of SR beam at visible light region were measured both σ and π-polarized components at Photon Factory. A wave-front division type polarized interferometer was designed and constructed for this experiment. Interferograms were observed clearly in the vertical direction. π phase difference of the interference fringe was observed between interferograms corresponding to σ and π...

http://www.csc.kth.se/~lauria/sos14/ Let G and H be two graphs, we say that two graphs are isomorphic, denoted as G ≅ H, if there is a bijection π ∶ V(G) → V(H), such that {u, v} ∈ E(G) if and only if {π(u), π(v)} ∈ E(H). In this lecture, we discuss a recent paper by O’Donnell, Wright, Wu and Zhou 1 which proved the 1 R. O’Donnell, J. Wright, C. Wu, and Y. Zhou. Hardness of Robust Graph Isomorp...

This result could have been proved by the Greeks more than two thousand years ago, so we certainly did not expect it to be a new theorem. However, since we found it (1996) we have asked a large number of experts, we have searched the literature, and we have posted queries on the internet, all without finding anyone who has heard of the theorem. We thus feel comfortable in concluding that the th...

The prime number theorem is one of the most fundamental theorems of analytic number theory, stating that the prime counting function, π(x), is asymptotic to x/ log x. However, it says little about the parity of π(n) as an arithmetic function. Using Selberg’s sieve, we prove a positive lower bound for the proportion of positive integers n such that π(n) is r mod t for any fixed integers r and t....

This article is concerned with Euler’s theorem and small Fermat’s theorem that play important roles in public-key cryptograms. In the first section, we present some selected theorems on integers. In the following section, we remake definitions about the finite sequence of natural, the function of natural times finite sequence of natural and π of the finite sequence of natural. We also prove som...

The Grätzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. A lattice is algebraic if it is complete and generated by its compact elements. We show that the set of indices of computable lattices that are complete is Π 1 -complete; the set of indices of computable lattices that are algebraic is Π 1 -complete; and that th...

webπ is a recent process calculus introduced to formally specify Web Services composition. It extends the π-calculus with timed workunits, namely an asynchronous and temporized mechanism for events raising and catching. In this paper we encode Berger-Honda Timed-π in webπ timed workunits and we prove a simulation theorem. The overall perspective of this work is to make webπ comparable with both...