On elementary proof of AGT relations from six dimensions

The actual definition of the Nekrasov functions participating in the AGT relations implies a peculiar choice of contours in the LMNS and Dotsenko-Fateev integrals. Once made explicit and applied to the original triply-deformed (6-dimensional) version of these integrals, this approach reduces the AGT relations to symmetry in q_{1,2,3}, which is just an elementary identity for an appropriate choice of the integration contour (which is, however, a little non-traditional). We illustrate this idea with the simplest example of N=(1,1) U(1) SYM in six dimensions, however, all other cases can be evidently considered in a completely similar way.Comment: 5 page

Ding-Iohara-Miki symmetry of network matrix models

Ward identities in the most general "network matrix model" can be described in terms of the Ding-Iohara-Miki algebras (DIM). This confirms an expectation that such algebras and their various limits/reductions are the relevant substitutes/deformations of the Virasoro/W-algebra for (q, t) and (q_1, q_2, q_3) deformed network matrix models. Exhaustive for these purposes should be the Pagoda triple-affine elliptic DIM, which corresponds to networks associated with 6d gauge theories with adjoint matter (double elliptic systems). We provide some details on elliptic qq-characters.Comment: 20 pages, 2 figure

Molecular Motors Interacting with Their Own Tracks

Dynamics of molecular motors that move along linear lattices and interact with them via reversible destruction of specific lattice bonds is investigated theoretically by analyzing exactly solvable discrete-state ``burnt-bridge'' models. Molecular motors are viewed as diffusing particles that can asymmetrically break or rebuild periodically distributed weak links when passing over them. Our explicit calculations of dynamic properties show that coupling the transport of the unbiased molecular motor with the bridge-burning mechanism leads to a directed motion that lowers fluctuations and produces a dynamic transition in the limit of low concentration of weak links. Interaction between the backward biased molecular motor and the bridge-burning mechanism yields a complex dynamic behavior. For the reversible dissociation the backward motion of the molecular motor is slowed down. There is a change in the direction of the molecular motor's motion for some range of parameters. The molecular motor also experiences non-monotonic fluctuations due to the action of two opposing mechanisms: the reduced activity after the burned sites and locking of large fluctuations. Large spatial fluctuations are observed when two mechanisms are comparable. The properties of the molecular motor are different for the irreversible burning of bridges where the velocity and fluctuations are suppressed for some concentration range, and the dynamic transition is also observed. Dynamics of the system is discussed in terms of the effective driving forces and transitions between different diffusional regimes

Analitic Investigation of the Regularities of Changing Dust Concentration During the Abrasive Decrease of Stone Structures

In the process of repair or restoration of building structures, it is often necessary to strengthen building structures from limestone-shell rock, concrete, reinforced concrete, hard materials-granite, basalt, etc. by cutting or making cuts of the required size with detachable circles of synthetic diamond and cubic boron nitride (CA and CBN)The cutting process is accompanied by considerable dust formation, which can be both harmful and dangerous factor in the work.The aim of the work is studying the process of dust sedimentation and the regularity of the change in dust concentration during the abrasive cutting of concrete and stone materials.Mathematical models have been developed – dust emission from under the wheel, speed of sedimentation of dust particles depending on their material, size and shape, and also depending on temperature, pressure and humidity, the concentration of dust in the working space and the concentration change during the cutting cycle are calculated.It is shown that the velocity of the sedimentation of particles depends significantly on the shape. The higher the sphericity, the higher the sedimentation rate. The ambient temperature has little effect on the sedimentation rate, in the temperature range (-20 → + 40 °C) at which the operation takes place.The sedimentation rate of dust particles generated by cutting the most common building stone materials also differs slightly. Almost the same sedimentation rate has dust particles obtained by cutting basalt and concrete. A bit higher is the sedimentation rate of particles from granite.The sedimentation rate of particles of generated dust is about 600-700 cm/h or 10-11 cm/min for particles measuring 6 μm. This means that at a production height of about 2 m (200 cm) during the operating cycle (about 3 min), the dust will remain at an altitude of about 1.5 m, i.е. practically remains in the working area. This gives grounds to assert about a high concentration of dust during the cutting cycle (about 4.8 108/m3)

The MacMahon R-matrix

We introduce an $R$-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$. This $R$-matrix acts on pairs of $3d$ Young diagrams and retains the nice symmetry of the DIM algebra under the permutation of three deformation parameters $q$, $t^{-1}$ and $\frac{t}{q}$. We construct the intertwining operator for a tensor product of the horizontal Fock representation and the vertical MacMahon representation and show that the intertwiners are permuted using the MacMahon $R$-matrix.Comment: 39 page

Duality in elliptic Ruijsenaars system and elliptic symmetric functions

We demonstrate that the symmetric elliptic polynomials $E_\lambda(x)$ originally discovered in the study of generalized Noumi-Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians that act on the mother function variable $y_i$ (substitute of the Young-diagram variable $\lambda$). This means they are eigenfunctions of the dual eRS system. At the same time, their orthogonal complements in the Schur scalar product, $P_R(x)$ are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov (KS) Hamiltonians. This means that these latter are related to the dual eRS Hamiltonians by a somewhat mysterious orthogonality transformation, which is well defined only on the full space of time variables, while the coordinates $x_i$ appear only after the Miwa transform. This observation explains the difficulties with getting the apparently self-dual Hamiltonians from the double elliptic version of the KS Hamiltonians.Comment: 15 page

Controlled Formation Of Emissive Silver Nanoclusters Using Rationally Designed Metal-binding Proteins

The metal-binding properties of rationally designed, synthetic proteins were used to prepare a series of emissive silver nanoclusters having predictable sizes and emission energies. Metal-binding a-helical coiled coils were designed to exist as peptide trimers, tetramers, and hexamers and found to uniquely bind 6, 8, and 12 Ag+ ions, respectively. Subsequent treatment with a chemical reducing agent produced a series of peptide-bound Ag-0 nanoclusters that display a strong visible fluorescence whose emission energies depend on the number of bound metal ions in excellent agreement with theory