Invariante Charakterisierung von Kurvennetzen

Ohne Zusammenfassung     

Transition‐Metal Complexes Based on the 1,3,5‐Triethynyl Benzene Linking Unit

The synthesis of a unique series of heteromultinuclear transition metal compounds is reported. Complexes 1‐I‐3‐Br‐5‐(FcC≡C)‐C₆H₃ ( 4 ), 1‐Br‐3‐(bpy‐C≡C)‐5‐(FcC≡C)‐C₆H₃ ( 6 ), 1,3‐(bpy‐C≡C)₂‐5‐(FcC≡C)‐C₆H₃ ( 7 ), 1‐(XC≡C)‐3‐(bpy‐C≡C)‐5‐(FcC≡C)‐C₆H₃ ( 8 , X = SiMe₃; 9 , X = H), 1‐(HC≡C)‐3‐[(CO)₃ClRe(bpy‐C≡C)]‐5‐(FcC≡C)‐C₆H₃ ( 11 ), 1‐[(Ph₃P)AuC≡C]‐3‐[(CO)₃ClRe(bpy‐C≡C)]‐5‐(FcC≡C)‐C₆H₃ ( 13 ), 1‐[(Ph₃P)AuC≡C]‐3‐(bpy‐C≡C)‐5‐(FcC≡C)‐C₆H₃ ( 14 ), [1‐[(Ph₃PAuC≡C]‐3‐[{[Ti](C≡CSiMe₃)₂}Cu(bpy‐C≡C)]‐5‐(FcC≡C)‐C₆H₃]PF₆ ( 16 ), and [1,3‐[(tBu₂bpy)₂Ru(bpy‐C≡C)]₂‐5‐(FcC≡C)‐C₆H₃](PF₆)₄ ( 18 ) (Fc = (η⁵‐C₅H₄)(η⁵‐C₅H₅)Fe, bpy = 2,2′‐bipyridiyl‐5‐yl, [Ti] = (η⁵‐C₅H₄SiMe₃)₂Ti) were prepared by using consecutive synthesis methodologies including metathesis, desilylation, dehydrohalogenation, and carbon–carbon cross‐coupling reactions. In these complexes the corresponding metal atoms are connected by carbon‐rich bridging units comprising 1,3‐diethynyl‐, 1,3,5‐triethynylbenzene and bipyridyl units. They were characterized by elemental analysis, IR and NMR spectroscopy, and partly by ESI‐TOF mass spectrometry., The structures of 4 and 11 in the solid state are reported. Both molecules are characterized by the central benzene core bridging the individual transition metal complex fragments. The corresponding acetylide entities are, as typical, found in a linear arrangement with representative M–C, C–C_C≡C and C≡C bond lengths.     

Solvent-dependent formation of two different Cd(II) complexes with p-BrC6H4C(S)NHP(O)(OiPr)2 (HL). Crystallographic modification of Cd(HL)2L2

Reaction of the potassium salt of N-(diisopropoxyphosphoryl)-p-bromothiobenzamide p-BrC₆H₄C(S)NHP(O)(OiPr)₂ (HL) with Cd(II) cations in freshly dried and distilled EtOH leads exclusively to the complex [Cd(p-BrC₆H₄C(S)NH₂-S)(L-O,S)₂] ([Cd(L^I)L₂]), while the same reaction in H₂O leads to the complex [Cd(HL-O)₂(L-O,S)₂] ([Cd(HL)₂L₂]). The corresponding reactions with Zn(II) always lead to the complex [Zn(L-O,S)₂] ([ZnL₂]) regardless of the solvent. The crystal structure of [Cd(HL)₂L₂]^.2/3H₂O reveals to be a polymorph to the previously reported anhydrous [Cd(HL)₂L₂].     

Realization of reflectionless potentials in photonic lattices

We realize experimentally a true reflectionless potential, which facilitates nonresonant unity transmission for all incident waves and at the same time supports localized modes. We utilize arrays of evanescently coupled optical waveguides, where a particular modulation of the transverse waveguide separations provides a physical realization of reflectionless Ablowitz-Ladik soliton potentials.     

Dispersion and localization of electronic states at a ferrocene/Cu(111) interface

Low-temperature scanning tunneling microscopy and spectroscopy combined with first-principles simulations reveal a nondissociative physisorption of ferrocene molecules on a Cu(111) surface, giving rise to ordered molecular layers. At the interface, a 2D-like electronic band is found, which shows an identical dispersion as the Cu(111) Shockley surface-state band. Subsequent deposition of Cu atoms forms charged organometallic compounds that localize interface-state electrons.     

Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices

Coherent states and their generalizations, displaced Fock states, are of fundamental importance to quantum optics. Here we present a direct observation of a classical analogue for the emergence of these states from the eigenstates of the harmonic oscillator. To this end, the light propagation in a Glauber-Fock waveguide lattice serves as equivalent for the displacement of Fock states in phase space. Theoretical calculations and analogue classical experiments show that the square-root distribution of the coupling parameter in such lattices supports a new family of intriguing quantum correlations not encountered in uniform arrays. Because of the broken shift invariance of the lattice, these correlations strongly depend on the transverse position. Consequently, quantum random walks with this extra degree of freedom may be realized in Glauber-Fock lattices.     

Central limit theorems for empirical product densities of stationary point processes

We prove the asymptotic normality of kernel estimators of second- and higher-order product densities (and of the pair correlation function) for spatially homogeneous (and isotropic) point processes observed on a sampling window \(W_n\) , which is assumed to expand unboundedly in all directions as \(n \rightarrow \infty \,\) . We first study the asymptotic behavior of the covariances of the empirical product densities under minimal moment and weak dependence assumptions. The proof of the main results is based on the Brillinger-mixing property of the underlying point process and certain smoothness conditions on the higher-order reduced cumulant measures. Finally, the obtained limit theorems enable us to construct \(\chi ^2\) -goodness-of-fit tests for hypothetical product densities.