Improved quantum algorithms for the ordered search problem via semidefinite programming

One of the most basic computational problems is the task of finding a desired item in an ordered list of N items. While the best classical algorithm for this problem uses log_2 N queries to the list, a quantum computer can solve the problem using a constant factor fewer queries. However, the precise value of this constant is unknown. By characterizing a class of quantum query algorithms for ordered search in terms of a semidefinite program, we find new quantum algorithms for small instances of the ordered search problem. Extending these algorithms to arbitrarily large instances using recursion, we show that there is an exact quantum ordered search algorithm using 4 log_{605} N \approx 0.433 log_2 N queries, which improves upon the previously best known exact algorithm.Comment: 8 pages, 4 figure

On the order of summability of the Fourier inversion formula

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems

Comparative histology of pineal calcification

The pineal organ (pineal gland, epiphysis cerebri) contains several calcified concretions called "brain sand7' or acervuli (corpora arenacea). These concretions are conspicuous with imaging techniques and provide a useful landmark for orientation in the diagnosis of intracranial diseases. Predominantly composed of calcium and magnesium salts, corpora arenacea are numerous in old patients. In smaller number they can be present in children as well. The degree of calcification was associated to various diseases. However, the presence of calcified concretions seems not to reflect a specific pathological state. Corpora arenacea occur not only in the actual pineal tissue but also in the leptomeninges, in the habenular commissure and in the choroid plexus.Studies with the potassium pyroantimonate (PPA) method on the ultrastructural localization of free calcium ions in the human pineal, revealed the presence of calcium alongside the cell membanes, a finding that underlines the importance of membrane functions in the production of calcium deposits. Intrapineal corpora arenacea are characterized by a surface with globular structures. Meningeal acervuli that are present in the arachnoid cover of the organ, differ in structure from intrapineal ones and show a prominent concentric lamination of alternating dark and light lines. The electron-lucent lines contain more calcium than the dark ones. There is a correlation between the age of the subject and the number of layers in the largest acervuli. This suggests that the formation of these layers is connected to circannual changes in the calcium level of the organ. The histological organization of the human pineal is basically the same as that of mammalian experimental animals.Pineal concretions present in mammalian animal species are mainly of the meningeal type. Meningeal cells around acervuli contain active cytoplasmic organelles and exhibit alkaline phosphatase reaction in the rat and mink, an indication of a presumable osteoblast-like activity. Using Kossa's method for the staining of calcium deposits, a higher calcium concentration was detected in the rat pineal than in the surrounding brain tissue. Since in parathyroidectomised rats calcified deposits are larger and more numerous than in controls, the regulation of the production of acervuli by the parathyroid gland has also been postulated.In most of submammalian species, the pineal organs (pineal-, parapineal organ, frontal organ, parietal eye) are photoreceptive and organized similarly to the retina. Acervuli were found in the pineal of some birds. The pineal organs of lower vertebrates (fish, amphibians, reptiles) exhibit a high calcium content by ultrastructural calcium histochemistry (PPA-method). However, concrements are not formed. The accumulation of ca2+ seems to depend on the receptor function of the organ. Comparing pineal and retinal photoreceptors in the frog, the photoreceptor outer segments of pinealocytes as well as retinal cones and rods show a large amount of Capyroantimonate deposits. In dark adapted animals calcium ions are present in both sides of the photoreceptor membranes of the outer segment, whereas calcium is shifted extra-cellularly following light adaptation.Overviewing the data available about the pineal calcification, we can conclude that a multifactorial mechanism may be responsible for the calcification. The pineal of higher vertebrates is not just a simple endocrine gland, rather, its histological organization resembles a folded retina having both hormonal and neural efferentation. Mammalian pinealocytes preserve several characteristics of submammalian receptor cells and accumulate free ca2+ on their membranes (1). In the thin walled retina and in the similarly organized pineal of submammalian species, the diffusion of extracellular calcium is probably easy and there is a lesser tendency to form concrements. In the larger mammalian pineal the compaction of a high amount of pineal cells actively exchanging calcium ions is supposed to increase the local concentration of calcium (2). Further, neural elements of more developed species exhibit a higher rate of calcium exchange and consequently, a higher number of calcium deposits (3). Finally, the barrier effect of the multilayered pineal arachnoid and the tight-junctions among their cells in mammals presumably promote the intrapincal concentration of calcium ions. The formaction of acervuli may be regulated by calcitonin, and the periacervular arachnoid cells ("acervuloblasts") act like ostoblasts (4