Modeling transport and mean age of dense core vesicles in large axonal arbors

A model simulating transport of dense core vesicles (DCVs) in type II axonal terminals of Drosophila motoneurons has been developed. The morphology of type II terminals is characterized by the large number of en passant boutons. The lack of both scaled up DCV transport and scaled down DCV capture in boutons results in a less efficient supply of DCVs to distal boutons. Furthermore, the large number of boutons that DCVs pass as they move anterogradely, until they reach the most distal bouton, may lead to the capture of a majority of DCVs before they turn around in the most distal bouton to move in the retrograde direction. This may lead to a reduced retrograde flux of DCVs and a lack of DCV circulation in type II terminals. The developed model simulates DCV concentrations in boutons, DCV fluxes between the boutons, age density distributions of DCVs, and the mean age of DCVs in various boutons. Unlike published experimental observations, our model predicts DCV circulation in type II terminals after these terminals are filled to saturation. This disagreement is likely because experimentally observed terminals were not at steady-state, but rather were accumulating DCVs for later release. Our estimates show that the number of DCVs in the transiting state is much smaller than that in the resident state. DCVs traveling in the axon, rather than DCVs transiting in the terminal, may provide a reserve of DCVs for replenishing boutons after a release. The techniques for modeling transport of DCVs developed in our paper can be used to model the transport of other organelles in axons.Comment: Added an acknowledgement to Holger Metzle

On free regular and Bondesson convolution semigroups

Free regular convolution semigroups describe the distribution of free subortinators, while Bondesson class convolution semigroups correspond to classical subordinators with completely monotone Levy density. We show that these two classes of convolution semigroups are in bijection with the class of complete Bernstein functions and we establish an integral identity linking the two semigroups. We provide several explicit examples that illustrate this result.Comment: 11 page

Computing the Mertens function on a GPU

A GPU implementation of an algorithm to compute the Mertens function in O(x2/3+{\ko}) time is discussed. Results for x up to $10^{22}$, and a new extreme value for $M(x)/x^{1/2}$, -0.585768 ($M(x) \approx -1.996 \ast 10^9$ at $x \approx 1.161 \ast 10^{19}$), are reported.An approximate algorithm is used to examine values of M(x) for x up to $\exp{(10^{15})}$

Height of exceptional collections and Hochschild cohomology of quasiphantom categories

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define its new invariant (the height) and show that the orthogonal to an exceptional collection of height $h$ in the derived category of a smooth projective variety $X$ has the same Hochschild cohomology as $X$ in degrees up to $h - 2$. We use this to describe the second Hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type. We also give necessary and sufficient conditions of fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.Comment: 23 pages, the construction of the \v{C}ech enhancement is correcte

Mean value properties of harmonic functions and related topics (a survey)

Results involving various mean value properties are reviewed for harmonic, biharmonic and metaharmonic functions. It is also considered how the standard mean value property can be weakened to imply harmonicity and belonging to other classes of functions.Comment: 21 page

A simple counterexample to the Jordan-H\"older property for derived categories

A counterexample to the Jordan-H\"older property for semiorthogonal decompositions of derived categories of smooth projective varieties was constructed by B\"ohning, Graf von Bothmer and Sosna. In this short note we present a simpler example by realizing Bondal's quiver in the derived category of a blowup of the projective space

Semiorthogonal decompositions in algebraic geometry

In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related issues such as categorical resolutions of singularities.Comment: Contribution to the ICM 2014; v2: acknowledgements updated; v3: a reference adde

Computational lower limits on small Ramsey numbers

Computer-based attempts to construct lower bounds for small Ramsey numbers are discussed. A systematic review of cyclic Ramsey graphs is attempted. Many known lower bounds are reproduced. Several new bounds are reported

Two-dimensional water waves in the presence of a freely floating body: conditions for the absence of trapped modes

The coupled motion is investigated for a mechanical system consisting of water and a body freely floating in it. Water occupies either a half-space or a layer of constant depth into which an infinitely long surface-piercing cylinder is immersed, thus allowing us to study two-dimensional modes. Under the assumption that the motion is of small amplitude near equilibrium, a linear setting is applicable and for the time-harmonic oscillations it reduces to a spectral problem with the frequency of oscillations as the spectral parameter. It is essential that one of the problem's relations is linear with respect to the parameter, whereas two others are quadratic with respect to it. Within this framework, it is shown that the total energy of the water motion is finite and the equipartition of energy holds for the whole system. On this basis, it is proved that no wave modes can be trapped provided their frequencies exceed a bound depending on cylinder's properties, whereas its geometry is subject to some restrictions and, in some cases, certain restrictions are imposed on the type of mode.Comment: 11 pages, 1 figur

Lefschetz decompositions and Categorical resolutions of singularities

Let $Y$ be a singular algebraic variety and let \TY be a resolution of singularities of $Y$. Assume that the exceptional locus of \TY over $Y$ is an irreducible divisor \TZ in \TY. For every Lefschetz decomposition of \TZ we construct a triangulated subcategory \TD \subset \D^b(\TY) which gives a desingularization of \D^b(Y). If the Lefschetz decomposition is generated by a vector bundle tilting over $Y$ then \TD is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then \TD is a crepant resolution.Comment: 24 pages; the proof of the main theorem rewritten, a section on functoriality is adde