Generalized equivalence of pairs of matrices

Pairs (A ₁ B ₁) and (A ₂ B ₂) of matrices over a principal ideal domain R are called the generalized equivalent pairs if A ₂=UA ₁ V ₁ B ₂=UB ₁ V ₂ for some invertible matrices U V ₁ V ₂ over R. A special form is established to which a pair of matrices can be reduced by means of generalized equivalent transformations. Besides necessary and sufficient conditions are found, under which a pair of matrices is generalized equivalent to a pair of diagonal matrices. Applications are made to study the divisibility of matrices and multiplicative property of the Smith normal form.     

Monitoring the mean of multivariate financial time series

Timely detection of changes in the mean vector of multivariate financial time series is of great practical importance. In this paper, the covariance dynamics of the multivariate stochastic processes is assessed by either the RiskMetrics approach, the constant conditional correlation, or the dynamic conditional correlation models. For online monitoring of mean changes, we introduce several control schemes based on exponential smoothing and cumulative sums, which explicitly account for heteroscedasticity. The detecting ability of the introduced charts is compared for different processes in a Monte Carlo simulation study. The empirical study illustrates monitoring of changes in the mean vector of daily returns of exchange rates. Copyright © 2013 John Wiley & Sons, Ltd.     

On Families of Graphs of Large Cycle Indicator, Matrices of Large Order and Key Exchange Protocols With Nonlinear Polynomial Maps of Small Degree

This paper studies the group theoretical protocol of Diffie?CHellman key exchange in the case of symmetrical group ${S_{p^n}}$ and more general Cremona group ${C(\mathbb K^n)}$ of polynomial automorphisms of free module ${\mathbb K^n}$ over arbitrary commutative ring ${\mathbb K}$ . This algorithm depends very much on the choice of the base ${g_n \in C( \mathbb K^n)}$ . It is important to work with the base ${g_n \in C( \mathbb K^n)}$ , which is a polynomial map of a small degree and a large order such that the degrees of all powers ${g_n^k}$ are also bounded by a small constant. We suggest fast algorithms for generation of a map ${g_n={f_n} \xi_nf_n^{-1}}$ , where ?? _n is an affine transformation (degree is 1) of a large order and f _n is a fixed nonlinear polynomial map in n variables such that ${f_n^{-1}}$ is also a polynomial map and both maps f _n and ${f_n^{-1}}$ are of small degrees. The method is based on properties of infinite families of graphs with a large cycle indicator and families of graphs of a large girth in particular. It guaranties that the order of g _n is tending to infinity as the dimension n tends to infinity. We propose methods of fast generation of special families of cubical maps f _n such that ${f_n^{-1}}$ is also of degree 3 based on properties of families of graphs of a large girth and graphs with a large cycle indicator. At the end we discuss cryptographical applications of maps of the kind ?? f _n ??^?1 and some graph theoretical problems motivated by such applications.     

White‐Fluorescent Dual‐Emission Mechanosensitive Membrane Probes that Function by Bending Rather than Twisting

Bent N,N′‐diphenyl‐dihydrodibenzo[a,c]phenazine amphiphiles are introduced as mechanosensitive membrane probes that operate by an unprecedented mechanism, namely, unbending in the excited state as opposed to the previously reported untwisting in the ground and twisting in the excited state. Their dual emission from bent or “closed” and planarized or “open” excited states is shown to discriminate between micelles in water and monomers in solid‐ordered (S_o), liquid‐disordered (L_d) and bulk membranes. The dual‐emission spectra cover enough of the visible range to produce vesicles that emit white light with ratiometrically encoded information. Strategies to improve the bent mechanophores with expanded π systems and auxochromes are reported, and compatibility with imaging of membrane domains in giant unilamellar vesicles by two‐photon excitation fluorescence (TPEF) microscopy is demonstrated.     

Synthesis of enantiopure (R)-2-(4-methoxy-3-(3-methoxypropoxy)-benzyl)-3-methylbutanoic acid—a key intermediate for the preparation of Aliskiren

The enantioselective hydrogenation of (E)-2-(4-methoxy-3-(3-methoxypropoxy)-benzylidene)-3-methylbutanoic acid (1) to (R)-2-(4-methoxy-3-(3-methoxypropoxy)-benzyl)-3-methylbutanoic acid (2)—a key intermediate in the synthesis of the pharmacologically important renin inhibitor Aliskiren—is described. The stereochemistry of the catalytic transformation has been studied using a number of homogeneous chiral Rh(I) and Ru(II) complexes bearing ferrocene-based phosphine ligands. The highest enantioselectivity for the homogeneous hydrogenation of 1 (up to 95% ee) was achieved with a [Rh(NBD)₂]BF₄ pre-catalyst (substrate/catalyst ratio 100:1, 10 bar H₂, 40 °C, in MeOH). To bring the enantioselectivity to perfection an effective method for the isolation of the enantiopure carboxylic acid is suggested likewise.     

Reactions of trialkoxynitridomolybdenum with low-coordinate phosphorus compounds containing a P=N double bond

The reactions of N≡Mo(OR)(3) (R = (t)Bu, (i)Pr) with (Me(3)Si)(2)NPNSiMe(3) (1), (Me(3)Si)(2)NPN(t)Bu (2), (Me(3)Si)(2)NPS(N(t)Bu) (3) and (Me(3)Si)(2)NP(NSiMe(3))(2) (4) have been studied. Reported complexes were synthesized via 1,2-addition of an Mo-OR bond across the P=N bond, resulting in four-membered metallacycles of the corresponding σ(2)λ(3)-iminophosphine or σ(3)λ(5)-iminophosphorane with trialkoxynitridomolybdenum. The structure of all new compounds was elucidated by (1)H, (13)C and (31)P NMR spectroscopy. Compounds [(Me(3)Si)(2)N-P(NSiMe(3))(O-(t)Bu)]{((t)BuO)(2)Mo≡N} (5), [(Me(3)Si)(2)N-PS(N(t)Bu)(O-(t)Bu)]{((t)BuO)(2)Mo≡N} (7), [(Me(3)Si)(2)N-P(NSiMe(3))(2)(O-(t)Bu)]{((t)BuO)(2)Mo≡N} (8) and [(Me(3)Si)(2)N-P(NSiMe(3))(2)(O-(i)Pr)]{((i)PrO)(2)Mo≡N} (12) were also characterized by single X-ray analysis and shown to be metallacycles containing the Mo atom with an intact terminal nitrido ligand.     

Synthesis of calixarene-based ketocyanine fluorophores

Cup-shaped calix[4]arenes bearing one or two ketocyanine fluorophore fragments at the wide rim of the macrocycle are synthesized by condensation of formyl calix[4]arenes with arylmethylene(hetarylmethylene)cyclopentanones in the presence of the ionic liquid, dimethylammonium dimethylcarbamate (DIMCARB) and characterized using UV-vis and fluorescence spectroscopy. Strong positive solvatofluorochromism for the calixarene ketocyanines is observed.     

Oxa-spirocycles: synthesis,properties and applications

A general approach to a new generation of spirocyclic molecules – oxa-spirocycles – was developed. The key synthetic step was iodocyclization. More than 150 oxa-spirocyclic compounds were prepared. Incorporation of an oxygen atom into the spirocyclic unit dramatically improved water solubility (by up to 40 times) and lowered lipophilicity. More potent oxa-spirocyclic analogues of antihypertensive drug terazosin were synthesized and studied in vivo.

A general practical approach to a new generation of spirocyclic molecules – oxa-spirocycles – is developed.     

On minimal lightlike surfaces in Minkowski space–time

We distinguish two particular classes of lightlike surfaces in the Minkowski space Mⁿ, which may be viewed as natural analogues of space- and timelike minimal surfaces.