There are infinitely many limit points of the fractional parts of powers

Suppose that α > 1 is an algebraic number and ξ > 0 is a real number. We prove that the sequence of fractional partsξα ⁿ , n = 1, 2, 3, …, has infinitely many limit points except when α is a PV-number and ξ ∈ ℚ(α). For ξ = 1 and α being a rational non-integer number, this result was proved by Vijayaraghavan.     

Seizure warning algorithm based on optimization and nonlinear dynamics

There is growing evidence that temporal lobe seizures are preceded by a preictal transition, characterized by a gradual dynamical change from asymptomatic interictal state to seizure. We herein report the first prospective analysis of the online automated algorithm for detecting the preictal transition in ongoing EEG signals. Such, the algorithm constitutes a seizure warning system. The algorithm estimates STL_max, a measure of the order or disorder of the signal, of EEG signals recorded from individual electrode sites. The optimization techniques were employed to select critical brain electrode sites that exhibit the preictal transition for the warning of epileptic seizures. Specifically, a quadratically constrained quadratic 0-1 programming problem is formulated to identify critical electrode sites. The automated seizure warning algorithm was tested in continuous, long-term EEG recordings obtained from 5 patients with temporal lobe epilepsy. For individual patient, we use the first half of seizures to train the parameter settings, which is evaluated by ROC (Receiver Operating Characteristic) curve analysis. With the best parameter setting, the algorithm applied to all cases predicted an average of 91.7% of seizures with an average false prediction rate of 0.196 per hour. These results indicate that it may be possible to develop automated seizure warning devices for diagnostic and therapeutic purposes.Mathematics Subject Classification (1991):20E28, 20G40, 20C20     

On the distribution of the number of customers in the symmetric M/G/1 queue

We consider an M/G/1 queue with symmetric service discipline. The class of symmetric service disciplines contains, in particular, the preemptive last-come-first-served discipline and the processor-sharing discipline. It has been conjectured in Kella et al. [1] that the marginal distribution of the queue length at any time is identical for all symmetric disciplines if the queue starts empty. In this paper we show that this conjecture is true if service requirements have an Erlang distribution. We also show by a counterexample, involving the hyperexponential distribution, that the conjecture is generally not true. AMS Subject Classifications Primary—60K25; Secondary—90B22     

Electroencephalogram (EEG) time series classification: Applications in epilepsy

Epilepsy is among the most common brain disorders. Approximately 25–30% of epilepsy patients remain unresponsive to anti-epileptic drug treatment, which is the standard therapy for epilepsy. In this study, we apply optimization-based data mining techniques to classify the brain's normal and epilepsy activity using intracranial electroencephalogram (EEG), which is a tool for evaluating the physiological state of the brain. A statistical cross validation and support vector machines were implemented to classify the brain's normal and abnormal activities. The results of this study indicate that it may be possible to design and develop efficient seizure warning algorithms for diagnostic and therapeutic purposes. Research was partially supported by the Rutgers Research Council grant-202018, the NSF grants DBI-980821, CCF-0546574, IIS-0611998, and NIH grant R01-NS-39687-01A1.     

Phase Transfer Catalysis Preparation of Acyl Isothiocyanates

The use of phase transfer catalysis, PTC, for nucleophillic substitution reactions is well documented.^1,2 Included among these reactions is the preparation of both alkyl³ and aryl⁴ thiocyanates. However, PTC reactions at an acyl carbon are much less common. Brándstróm⁵ has reported the preparation of acyl azides and Weber⁶ has used PTC to prepare benzoyl cyanide. Recently Illi⁷ has used PTC to acylate sterically crowded phenols.     

Phase Transfer Catalysis Preparation of Arvl,Thiocyanates

Phase transfer catalvsis, PTC, has become a common preparative method in organic chemistry. Numerous aliphatic substitution reactions have been performed yielding cvanides,¹ azides,² thiocyanates,³ ethers,⁴ and many other types of substitution products.¹     

Removal of the PsaF Polypeptide Biases Electron Transfer in Favor of the PsaB Branch of Cofactors in Triton X‐100 Photosystem I Complexes from Synechococcus sp. PCC 7002†

Continuous wave (CW) and transient electron paramagnetic resonance studies have implied that when PsaF is removed genetically, the double reduction of A_1A is facile, the lifetime of A_1A^? is shorter and the ratio of fast to slow kinetic phases increases in PS I complexes isolated with Triton X‐100 (Van der Est, A., A. I. Valieva, Y. E. Kandrashkin, G. Shen, D. A. Bryant and J. H. Golbeck [2004] Biochemistry 43 , 1264–1275). Changes in the lifetimes of A_1A^? and A_1B^? are characteristic of mutants involving the quinone binding sites, but changes in the relative amplitudes of A_1A^? and A_1B^? are characteristic of mutants involving the primary electron acceptors, A_0A and A_0B. Here, we measured the fast and slow phases of electron transfer from A_1B^? and A_1A^? to F_X in psaF and psaE psaF null mutants using time‐resolved CW and pump‐probe optical absorption spectroscopy. The lifetime of the fast kinetic phase was found to be unaltered, but the lifetime of the slow kinetic phase was shorter in the psaF null mutant and even more so in the psaE psaF null mutant. Concomitantly, the amplitude of the fast kinetic phase increased by a factor of 1.8 and 2.0 in the psaF and psaE psaF null mutants, respectively, at the expense of the slow kinetic phase. The change in ratio of the fast to slow kinetic phases is explained as either a redirection of electron transfer through A_1B at the expense of A_1A, or a shortening of the lifetime of A_1A^? to become identical to that of A_1B^?. The constant lifetime and the characteristics of the near‐UV spectrum of the fast kinetic phase favor the former explanation. A unified hypothesis is presented of a displacement of the A‐jk(1) α‐helix and switchback loop, which would weaken the H‐bond from Leu₇₂₂ to A_1A, accounting for the acceleration of the slow kinetic phase, as well as weaken the H‐bond from Tyr₆₉₆ to A_0A, accounting for the bias of electron transfer in favor of the PsaB branch of cofactors.     

Iterated homology and decompositions of simplicial complexes

Kalai has conjectured that a simplicial complex can be partitioned into Boolean algebras at least as roughly, as a shifting-preserving collapse sequence of its algebraically shifted complex. In particular, then, a simplicial complex could (conjecturally) be partitioned into Boolean intervals whose sizes are indexed by its iterated Betti numbers, a generalization of ordinary homology Betti numbers. This would imply a long-standing conjecture made (separately) by Garsia and Stanley concerning partitions of Cohen-Macaulay complexes into Boolean intervals. We prove a relaxation of Kalai’s conjecture, showing that a simplicial complex can be partitioned into recursively defined spanning trees of Boolean intervals indexed by its iterated Betti numbers.