Algebraic Montgomery�CYang problem: the non-cyclic case

Montgomery?CYang problem predicts that every pseudofree circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery?CYang problem: every projective surface S with quotient singularities such that the second Betti number b ₂(S) = 1 has at most 3 singular points if its smooth locus S ⁰ is simply connected. We prove the conjecture under the assumption that S has at least one non-cyclic singularity. In the course of the proof, we classify projective surfaces S with quotient singularities such that (i) b ₂(S) = 1, (ii) ${H_1(S^0, \mathbb{Z}) = 0}$ , and (iii) S has 4 or more singular points, not all cyclic, and prove that all such surfaces have ${\pi_1(S^0)\cong \mathfrak{A}_5}$ , the icosahedral group.     

Synchronization of inferior olive neurons via {\mathcal {L}}_1 adaptive feedback

This paper considers the synchronization of inferior olive neurons based on the \({\mathcal {L}}_1\) adaptive control theory. The ION model treated here is the cascade connection of two nonlinear subsystems, termed ZW and UV subsystems. It is assumed that the structure of the nonlinear functions and certain parameters of the IONs are not known, and disturbance inputs are present in the system. First, an \({\mathcal {L}}_1\) adaptive control system is designed to achieve global synchrony of the ZW subsystems using a single control input. This controller can accomplish local synchrony of the UV subsystems if the linearized UV subsystem is exponentially stable. For global synchrony of the UV subsystems, an \({\mathcal {L}}_1\) adaptive control law is designed. Each of these controllers includes a state predictor, an update law, and a control law. In the closed-loop system, global synchrony of the complete models of the IONs (the interconnected ZW and UV subsystems) is accomplished using these two adaptive controllers. Simulations results show that in the closed-loop system, the IONs are synchronized, despite unmodeled nonlinearities, disturbance inputs, and parameter uncertainties in the system.     

An efficient synthesis of 2,5-diketopiperazine derivatives by the Ugi four-center three-component reaction

A facile synthetic approach to 2,5-diketopiperazines 4 by the Ugi four-center three-component reaction using commercially available dipeptides as a bifunctional component, aldehydes, and isocyanides was described.     

The Alternating Group of Degree 6 in the Geometry of the Leech Lattice and K3 Surfaces

The alternating group of degree 6 is located at the junctionof three series of simple non-commutative groups: simple sporadicgroups, alternating groups and simple groups of Lie type. Itplays a very special role in the theory of finite groups. Weshall study its new roles both in a finite geometry of a certainpentagon in the Leech lattice and also in the complex algebraicgeometry of K3 surfaces. 2000 Mathematics Subject Classification14J28, 11H06, 20D06, 20D08.     

Colorimetric determination of Hg2+ via thiosemicarbazide-to-oxadiazole transformation of a coumarin-benzopyrylium dye

A dual signaling reaction-based probe derived from a hybrid coumarin-benzopyrylium dye was developed for Hg²⁺ signaling. The probe exhibited selective colorimetric Hg²⁺ signaling via the Hg²⁺-induced thiosemicarbazide-oxadiazole transformation. Interference from Cu²⁺ ions was successfully circumvented using a citrate buffer as a masking agent. The detection limit for Hg²⁺ ions was found to be 1.88?×?10^–7 M. Practical application to Hg²⁺ detection in simulated wastewater was realized using a typical smartphone as a stand-alone data capture and data processing device.     

In/InCl(3)-Mediated cross-coupling reactions of methyl vinyl ketone with benzaldehyde in aqueous media

Various beta,gamma-unsaturated ketones were successfully prepared by cross-coupling reactions of methyl vinyl ketone (MVK) with benzaldehydes mediated by In/InCl(3) in aqueous media.     

Automorphisms of a Generic Jacobian Kummer Surface

For a generic quartic surface F in ³ with sixteen nodes, we shall prove that every birational automorphism of F is induced by a Cremona transformation of ³.     

1H and 13C NMR assignments for new heterocyclic TAM leuco dyes, (2Z,2'E)-2,2'-(2-phenyl propane-1,3-diylidene) bis(1,3,3-trimethylindoline) derivatives. Part II

The (1)H and (13)C NMR spectra of the novel heterocyclic Leuco-TAM dyes, (2Z, 2'E)-2,2'-(2-phenyl propane-1,3-diylidene) bis(1,3,3-trimethylindoline) derivatives 1-4 as precursors of triarylmethane (TAM)(+) (Malachite Green FB-analog) dyes were completely assigned by 1D and 2D NMR experiments, including DEPT, COSY, HSQC, HMBC, and NOESY. Especially, the diastereotopic gem-dimethyl protons at the C3 and C3' positions of the FB rings were definitively assigned. The (Z,E) isomers adopt the energetically favored three-bladed propeller conformation in solution.     

The automorphism groups of Kummer surfaces associated with the product of two elliptic curves

We calculate the automorphism groups of several Kummer surfaces associated with the product of two elliptic curves. We give their generators explicitly.