Reorthogonalization for the Golub–Kahan–Lanczos bidiagonal reduction

The Golub–Kahan–Lanczos (GKL) bidiagonal reduction generates, by recurrence, the matrix factorization of $X \in \mathbb{R }^{m \times n}, m \ge n$ , given by $$\begin{aligned} X = UBV^T \end{aligned}$$ where $U \in \mathbb{R }^{m \times n}$ is left orthogonal, $V \in \mathbb{R }^{n \times n}$ is orthogonal, and $B \in \mathbb{R }^{n \times n}$ is bidiagonal. When the GKL recurrence is implemented in finite precision arithmetic, the columns of $U$ and $V$ tend to lose orthogonality, making a reorthogonalization strategy necessary to preserve convergence of the singular values. The use of an approach started by Simon and Zha (SIAM J Sci Stat Comput, 21:2257–2274, 2000) that reorthogonalizes only one of the two left orthogonal matrices $U$ and $V$ is shown to be very effective by the results presented here. Supposing that $V$ is the matrix reorthogonalized, the reorthogonalized GKL algorithm proposed here is modeled as the Householder Q–R factorization of $\left( \begin{array}{c} 0_{n \times k} \\ X V_k \end{array}\right) $ where $V_k = V(:,1:k)$ . That model is used to show that if $\varepsilon _M $ is the machine unit and $$\begin{aligned} \bar{\eta }= \Vert \mathbf{tril }(I-V^T\!~V)\Vert _F, \end{aligned}$$ where $\mathbf{tril }(\cdot )$ is the strictly lower triangular part of the contents, then: (1) the GKL recurrence produces Krylov spaces generated by a nearby matrix $X + \delta X$ , $\Vert \delta X\Vert _F = \mathcal O (\varepsilon _M + \bar{\eta }) \Vert X\Vert _F$ ; (2) singular values converge in the Lanczos process at the rate expected from the GKL algorithm in exact arithmetic on a nearby matrix; (3) a new proposed algorithm for recovering leading left singular vectors produces better bounds on loss of orthogonality and residual errors.     

Multiple fire interactions: A further investigation by burning rate data of square fire arrays

This paper presents the first effort to explore the spatial distributions of the burning rates in group fires consisting of a large number of fire points, by analyzing burn-out time data from experimental square fire arrays ranging from 3 × 3 to 15 × 15. A new concept termed fire layer is introduced and defined to characterize the spatial locations of fire points by which the complex spatial variations of burning rates, under different conditions, are analyzed and physically interpreted. Analysis shows that the fire layer burning rates vary from outer to inner in definite nonlinear modes. This indicates that the two fire interaction effects, heat feedback enhancement and air supply restriction, involve distinct spatial fluctuations in fire arrays. The spatial fluctuations of the two interaction effects are significantly affected by the two major parameters, fire spacing and fire array size. Definite spatial regions and parameter ranges for the spatial fluctuations and high competitions of the two interaction effects are clearly distinguished. It is demonstrated that the average burning rates of all fire layers involve consistent variations versus fire spacing or fire array size, especially with high comparability to the entire fire array. It is found that by varying fire spacing, the average burning rates for all fire layers vary linearly versus the fire area ratio, within the same ranges as the entire fire array, while there exist different fluctuation modes of fire layer burning rates with respect to fire array size. Furthermore, analysis shows that the burning rates of all fire layers will be significantly affected by fire merging when it occurs. Finally, a new approach is presented to simulate fire propagation among discrete fuel sources, by which the positive effect of the surrounding new fire points on the burning rates of the original ones is definitely indicated.     

Experimental research on flame revolution and precession of fire whirls

This paper presents the first experimental effort to explore the large scale 3-D flame instabilities of fire whirls, including the inclined flame revolution during the transition from a general pool fire to fire whirl, and the swirling flame precession in a quasi-steady fire whirl. The experimental medium-scale fire whirls were produced by a fixed-frame facility. Experimental observations indicate that flame revolution is an important flame instability during the formation of fire whirl, showing that the entire flame is inclined and revolves around the geometrical axis of symmetry with increasing angular velocity until the critical point, without the self-rotation of the flame. It is found that the inlet velocity fluctuates synchronously with the flame revolution. As soon as the fire whirl forms, the erect swirling flame starts to precess around the geometrical axis of symmetry. Analysis indicates that during flame precession the periodic fluctuations of inlet velocity disappear and a local annular external recirculation zone (ERZ) is produced outside the flame (vortex core), while the flow is upward inside. It is found that the inlet velocities are nearly constant within the continuous flame in order to maintain a stable generating eddy. A good linear correlation exists between the average inlet velocities and average ambient circulations for all fuel pan sizes. The precession frequency is relatively stable during one test. The frequencies of flame revolution and precession are both proportional to the average inlet velocity, and the corresponding Strouhal numbers are constants of 0.42 and 0.80, respectively. The flame revolves and precesses in the same direction as the self-rotation of the fire whirl flame in all tests. The flame revolution is related to the periodical fluctuations of inlet flow, while the flame precession is considered to be linked to the occurrence of ERZ in fire whirls.     

Heterotrimetallic Mixed-Valent Molecular Precursors Containing Periodic Table Neighbors: Assignment of Metal Positions and Oxidation States

A known trinuclear structure was used to design the heterobimetallic mixed-valent, mixed-ligand molecule [Co^II(hfac)₃−Na−Co^III(acac)₃] ( 1 ). This was used as a template structure to develop heterotrimetallic molecules [Co^II(hfac)₃−Na−Fe^III(acac)₃] ( 2 ) and [Ni^II(hfac)₃−Na−Co^III(acac)₃] ( 3 ) via isovalent site-specific substitution at either of the cobalt positions. Diffraction methods, synchrotron resonant diffraction, and multiple-wavelength anomalous diffraction were applied beyond simple structural investigation to provide an unambiguous assignment of the positions and oxidation states for the periodic table neighbors in the heterometallic assemblies. Molecules of 2 and 3 are true heterotrimetallic rather than a statistical mixture of two heterobimetallic counterparts. Trinuclear platform 1 exhibits flexibility in accommodating a variety of di- and trivalent metals, which can be further utilized in the design of molecular precursors for the NaMM′O₄ functional oxide materials.     

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in Davis et al. (1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length 4, sharpening the lower bound of $\frac{1}{3}$ from LeSaulnier and Vijay (2011).     

Tetrazole Azasydnone (C2N7O2H) And Its Salts: High-Performing Zwitterionic Energetic Materials Containing A Unique Explosophore

This work reports the first compound containing both a tetrazole and an azasydnone ring, a unique energetic material. Several energetic salts of the tetrazole azasydnone were synthesized and characterized, leading to the creation of new secondary and primary explosives. Molecular structures are confirmed by ¹H and ¹³C NMR, IR spectroscopy, and X-ray crystallographic analysis. The high heats of formation, fast detonation velocities, and straight-forward synthesis of energetic azasydnones should capture the attention of future energetics research.     

Pressure‐Tuneable Visible‐Range Band Gap in the Ionic Spinel Tin Nitride

The application of pressure allows systematic tuning of the charge density of a material cleanly, that is, without changes to the chemical composition via dopants, and exploratory high‐pressure experiments can inform the design of bulk syntheses of materials that benefit from their properties under compression. The electronic and structural response of semiconducting tin nitride Sn₃N₄ under compression is now reported. A continuous opening of the optical band gap was observed from 1.3 eV to 3.0 eV over a range of 100 GPa, a 540 nm blue‐shift spanning the entire visible spectrum. The pressure‐mediated band gap opening is general to this material across numerous high‐density polymorphs, implicating the predominant ionic bonding in the material as the cause. The rate of decompression to ambient conditions permits access to recoverable metastable states with varying band gaps energies, opening the possibility of pressure‐tuneable electronic properties for future applications.