An elementary approach to toy models for D. H. Lehmer's conjecture

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice \ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.Comment: 18 page

The isodiametric problem with lattice-point constraints

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space R^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices.Comment: 12 pages, 4 figures, (v2) referee comments and suggestions incorporated, accepted in Monatshefte fuer Mathemati

The trace of the heat kernel on a compact hyperbolic 3-orbifold

The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold H^3/\Ga are evaluated in the case in which the discrete group \Ga contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.Comment: 11 page

Conformal Field Theories, Representations and Lattice Constructions

An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFT's), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted and $Z_2$-twisted theories, $H(\Lambda)$ and $\tilde H(\Lambda)$ respectively, which may be constructed from a suitable even Euclidean lattice $\Lambda$. Similarly, one may construct lattices $\Lambda_C$ and $\tilde\Lambda_C$ by analogous constructions from a doubly-even binary code $C$. In the case when $C$ is self-dual, the corresponding lattices are also. Similarly, $H(\Lambda)$ and $\tilde H(\Lambda)$ are self-dual if and only if $\Lambda$ is. We show that $H(\Lambda_C)$ has a natural ``triality'' structure, which induces an isomorphism $H(\tilde\Lambda_C)\equiv\tilde H(\Lambda_C)$ and also a triality structure on $\tilde H(\tilde\Lambda_C)$. For $C$ the Golay code, $\tilde\Lambda_C$ is the Leech lattice, and the triality on $\tilde H(\tilde\Lambda_C)$ is the symmetry which extends the natural action of (an extension of) Conway's group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurman's construction of the natural Monster module in a more general context. The results also serve to shed some light on the classification of self-dual CFT's. We find that of the 48 theories $H(\Lambda)$ and $\tilde H(\Lambda)$ with central charge 24 that there are 39 distinct ones, and further that all 9 coincidences are accounted for by the isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.Comment: 65 page

Support varieties for selfinjective algebras

Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb's theorem is true

ИССЛЕДОВАНИЕ РАДОНОЗАЩИТНЫХ ХАРАКТЕРИСТИК ОБЛИЦОВОЧНЫХ ИЗДЕЛИЙ И МАТЕРИАЛОВ

An article considers the issues connected with radon-protective characteristics of facing products and materials. Experimental estimations are obtained of the radon diffusion factor for ceramic and stoneware facing products and for the number of rolled gas- and waterproof materials with the use of developed original methodic of radon diffusion factor estimation in the porous medium. It is shown that the use of these products and materials in the process of modern buildings construction could give considerable effect of the radon inflow reducing into indoor air from the block of enclosing constructions.В статье рассматриваются вопросы, связанные с радонозащитными характеристиками облицовочных изделий и материалов. С использованием разработанной оригинальной методики определения коэффициента диффузии радона в пористых средах получены экспериментальные оценки его величины для керамических и керамогранитных облицовочных изделий, а также для ряда рулонных газо- и гидроизоляционных материалов. Показано, что использование этих изделий и материалов в строительстве современных зданий может давать значительный эффект снижения поступления радона в воздух помещений из массива ограждающих конструкций

Kernel reconstruction for delayed neural field equations

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues. In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for the inverse problem. We employ spectral regularization techniques for its stable solution. A sensitivity analysis of the regularized kernel reconstruction with respect to the input signal u is carried out, investigating the Frechet differentiability of the kernel with respect to the signal. Finally, we use numerical examples to show the feasibility of the approach for kernel reconstruction, including numerical sensitivity tests, which show that the integral equation approach is a very stable and promising approach for practical computational neuroscience

Оценка доз внешнего облучения населения с учетом структуры аппаратурного обеспечения для контроля мощности дозы фотонного излучения в организациях Роспотребнадзора

This paper provides review of equipment and methodology for measurement of photon radiation dose; analysis of possible reasons for considerable deviation between the Russian Federation population annual effective external irradiation doses and the relevant average global value. Data on Rospotrebnadzor bodies dosimetry equipment used for measurement of gamma radiation dose are collected and systematized. Over 60 kinds of dosimeters are used for monitoring of population external irradiation doses. Most of dosimeters used in the country have gas-discharge detectors (Geiger-Mueller counters, minor biochemical annunciators, etc.) which have higher total values of own background level and of space radiation response than the modern dosimeters with scintillation detectors. This feature of dosimeters is apparently one of most plausible reasons of a bit overstating assessment of population external irradiation doses. The options for specification of population external irradiation doses assessment are: correction of gamma radiation dose measurement results with consideration of dosimeters own background level and space radiation response, introduction of more up-to-date dosimeters with scintillation detectors, etc. The most promising direction of research in verification of population external irradiation doses assessment is account of dosimetry equipment.Рассматриваются вопросы аппаратурно-методического обеспечения измерений мощности дозы фотонного излучения, анализируются возможные причины значительной разницы между годовыми эффективными дозами внешнего облучения населения Российской Федерации и среднемировым значением этих величин. Собраны и систематизированы сведения о парке дозиметрического оборудования, которое используется для измерений мощности дозы гамма-излучения в организациях Роспотребнадзора. Установлено, что для контроля доз внешнего облучения населения используются более 60 типов дозиметров. Большая часть применяемых в стране дозиметров имеют газоразрядные детекторы (счетчики Гейгера – Мюллера, СБМ и т.п.), которые характеризуются более высокими суммарными значениями уровня собственного фона и отклика на космическое излучение, чем современные дозиметры со сцинтилляционными детекторами. Показано, что эта характеристика дозиметров является, по-видимому, одной из наиболее вероятных причин того, что оценки доз внешнего облучения населения оказываются несколько завышенными. Рассматриваются возможные варианты уточнения оценки доз внешнего облучения населения: введение поправки в результаты измерений мощности дозы гамма-излучения с учетом уровня собственного фона дозиметров и отклика на космическое излучение, применение более современных дозиметров со сцинтилляционными детекторами и др. Показано, что учет структуры парка дозиметрического оборудования является наиболее перспективным направлением исследований по верификации оценки доз внешнего облучения населения

Effects of cognac on coronary flow reserve and plasma antioxidant status in healthy young men

<p>Abstract</p> <p>Background</p> <p>The cardioprotective effects of certain alcoholic beverages are partly related to their polyphenol content, which may improve the vasodilatory reactivity of arteries. Effect of cognac on coronary circulation, however, remains unknown. The purpose of this randomized controlled cross-over study was to determine whether moderate doses of cognac improve coronary reactivity as assessed with cold pressor testing (CPT) and coronary flow reserve (CFR) measument.</p> <p>Methods</p> <p>Study group consisted of 23 subjects. Coronary flow velocity and epicardial diameter was assessed using transthoracic echocardiography at rest, during CPT and adenosine infusion-derived CFR measurements before drinking, after a moderate (1.2 ± 0.1 dl) and an escalating high dose (total amount 2.4 ± 0.3 dl) of cognac. To explore the bioavailability of antioxidants, the antioxidant contents of cognac was measured and the absorption from the digestive tract was verified by plasma antioxidant capacity determination.</p> <p>Results</p> <p>Serum alcohol levels increased to 1.2 ± 0.2‰ and plasma antioxidant capacity from 301 ± 43.9 μmol/l to 320 ± 25.0 μmol/l by 7.6 ± 11.8%, (p = 0.01) after high doses of cognac. There was no significant change in flow velocity during CPT after cognac ingestion compared to control day. CFR was 4.4 ± 0.8, 4.1 ± 0.9 (p = NS), and 4.5 ± 1.2 (p = NS) before drinking and after moderate and high doses on cognac day, and 4.5 ± 1.4, and 4.0 ± 1.2 (p = NS) on control day.</p> <p>Conclusion</p> <p>Cognac increased plasma antioxidant capacity, but it had no effect on coronary circulation in healthy young men.</p> <p>Trial Registration</p> <p>NCT00330213</p