


<p>For the Jacobian resulting from the previously considered problem of the path
integral reduction in Wiener path integrals for a mechanical system with
symmetry describing the motion of two interacting scalar particles on a
manifold that is the product of a smooth compact finitedimensional Riemannian
manifold and a finitedimensional vector space, a geometric representation is
obtained. This representation follows from the formula for the scalar curvature
of the original manifold endowed by definition with a free isometric smooth
action of a compact semisimple Lie group. The derivation of this formula is
performed using adapted coordinates, which can be determined in the principal
fiber bundle associated with the problem under the study. These coordinates are
similar to those used in the standard approach to quantization of YangMills
fields interacting with scalar fields.
</p>




<p>We study several families of vertex operator superalgebras from a jet
(super)scheme point of view. We provide new examples of vertex algebras which
are "chiralizations" of their Zhu's Poisson algebras $R_V$. Our examples come
from affine $C_\ell^{(1)}$series vertex algebras ($\ell \geq 1$), certain
$N=1$ superconformal vertex algebras, FeiginStoyanovsky principal subspaces,
FeiginStoyanovsky type subspaces, graph vertex algebras $W_{\Gamma}$, and
extended Virasoro vertex algebra. We also give a counterexample to the
chiralization property for the $N=2$ superconformal vertex algebra of central
charge $1$.
</p>




<p>We construct an integrable colored sixvertex model whose partition function
is a double Grothendieck polynomial. This gives an integrable systems
interpretation of bumpless pipe dreams and recent results of Weigandt
[<a href="/abs/2003.07342">arXiv:2003.07342</a>]. We then construct a new model that we call the semidual
version model for vexillary permutations. We use our semidual model and the
fivevertex model of Motegi and Sakai to given a new proof that double
Grothendieck polynomials for vexillary permutations are equal to flagged
factorial Grothendieck polynomials. Taking the stable limit, we obtain a new
proof that the stable limit is a factorial Grothendieck polynomial as defined
by McNamara. The states of our semidual model naturally correspond to families
of nonintersecting lattice paths, where we can then use the
Lindstr\"omGesselViennot lemma to give a determinant formula for double
Schubert polynomials corresponding to vexillary permutations.
</p>




<p>For arbitrarily small values of $\varepsilon>0,$ we formulate and analyse the
Maxwell system of equations of electromagnetism on $\varepsilon$periodic sets
$S^\varepsilon\subset{\mathbb R}^3.$ Assuming that a family of Borel measures
$\mu^\varepsilon,$ such that ${\rm supp}(\mu^\varepsilon)=S^\varepsilon,$ is
obtained by $\varepsilon$contraction of a fixed 1periodic measure $\mu,$ and
for righthand sides $f^\varepsilon\in L^2({\mathbb R}^3, d\mu^\varepsilon),$
we prove ordersharp normresolvent convergence estimates for the solutions of
the system. In the resent work we address the case of nonzero current density
in the Maxwell system and complete the analysis of the general setup including
nonconstant permittivity and permeability coefficients.
</p>




<p>We study the algebra of invariant representative functions over the Nfold
Cartesian product of copies of a compact Lie group G modulo the action of
conjugation by the diagonal subgroup. We construct a basis of invariant
representative functions referred to as quasicharacters. The form of the
quasicharacters depends on the choice of a reduction scheme. We determine the
multiplication law of quasicharacters and express their structure constants in
terms of recoupling coefficients. Via this link, the choice of the reduction
scheme acquires an interpretation in terms of binary trees. We show explicitly
that the structure constants decompose into products over primitive elements of
9j symbol type. For SU(2), everything boils down to the combinatorics of
angular momentum theory. Finally, we apply this theory to the construction of
the Hilbert space costratification of (finite) lattice quantum gauge theory.
The methods developed in this paper may be useful in the study of virtually all
quantum models with polynomial constraints related to some symmetry.
</p>




<p>A fluid flow is described by fictitious particles hopping on homogeneously
distributed nodes with a given finite set of discrete velocities. We emphasize
that the existence of a fictitious particle having a discrete velocity among
the set in a node is given by a probability. We describe a compressible thermal
flow of the level of accuracy of the NavierStokes equation by 25 or 33
discrete velocities for twodimensional space and perform simulations for
investigating internal structural evolution of a shock wave.
</p>




<p>For arbitrarily small values of $\varepsilon>0,$ we formulate and analyse the
Maxwell system of equations of electromagnetism on $\varepsilon$periodic sets
$S^\varepsilon\subset{\mathbb R}^3.$ Assuming that a family of Borel measures
$\mu^\varepsilon,$ such that ${\rm supp}(\mu^\varepsilon)=S^\varepsilon,$ is
obtained by $\varepsilon$contraction of a fixed 1periodic measure $\mu,$ and
for righthand sides $f^\varepsilon\in L^2({\mathbb R}^3, d\mu^\varepsilon),$
we prove ordersharp normresolvent convergence estimates for the solutions of
the system. Our analysis includes the case of periodic "singular structures",
when $\mu$ is supported by lowerdimensional manifolds.
</p>




<p>We develop a novel method to analyze the dynamics of stochastic rewriting
systems evolving over finitary adhesive, extensive categories. Our formalism is
based on the socalled rule algebra framework and exhibits an intimate
relationship between the combinatorics of the rewriting rules (as encoded in
the rule algebra) and the dynamics which these rules generate on observables
(as encoded in the stochastic mechanics formalism). We introduce the concept of
combinatorial conversion, whereby under certain technical conditions the
evolution equation for (the exponential generating function of) the statistical
moments of observables can be expressed as the action of certain differential
operators on formal power series. This permits us to formulate the novel
concept of momentbisimulation, whereby two dynamical systems are compared in
terms of their evolution of sets of observables that are in bijection. In
particular, we exhibit nontrivial examples of graphical rewriting systems that
are momentbisimilar to certain discrete rewriting systems (such as branching
processes or the larger class of stochastic chemical reaction systems). Our
results point towards applications of a vast number of existing
wellestablished exact and approximate analysis techniques developed for
chemical reaction systems to the far richer class of general stochastic
rewriting systems.
</p>




<p>The magnetic Laplacian (also called the line bundle Laplacian) on a connected
weighted graph is a selfadjoint operator wherein the realvalued adjacency
weights are replaced by unit complexvalued weights $\{\omega_{xy}\}_{xy\in
E}$, satisfying the condition that $\omega_{xy}=\overline{\omega_{yx}}$ for
every directed edge $xy$. When properly interpreted, these complex weights give
rise to magnetic fluxes through cycles in the graph.
</p>
<p>In this paper we establish the spectrum of the magnetic Laplacian, as a set
of real numbers with multiplicities, on the Sierpinski gasket graph ($SG$)
where the magnetic fluxes equal $\alpha$ through the upright triangles, and
$\beta$ through the downright triangles. This is achieved upon showing the
spectral selfsimilarity of the magnetic Laplacian via a 3parameter map
$\mathcal{U}$ involving nonrational functions, which takes into account
$\alpha$, $\beta$, and the spectral parameter $\lambda$. In doing so we provide
a quantitative answer to a question of Bellissard [Renormalization Group
Analysis and Quasicrystals (1992)] on the relationship between the dynamical
spectrum and the actual magnetic spectrum.
</p>
<p>Our main theorems lead to two applications. In the case $\alpha=\beta$, we
demonstrate the approximation of the magnetic spectrum by the filled Julia set
of $\mathcal{U}$, the Sierpinski gasket counterpart to Hofstadter's butterfly.
Meanwhile, in the case $\alpha,\beta\in \{0,\frac{1}{2}\}$, we can compute the
determinant of the magnetic Laplacian determinant and the corresponding
asymptotic complexity.
</p>




<p>We solve rigorously the time dependent Schr\"odinger equation describing
electron emission from a metal surface by a laser field perpendicular to the
surface. We consider the system to be onedimensional, with the halfline $x<0$
corresponding to the bulk of the metal and $x>0$ to the vacuum. The laser field
is modeled as a classical electric field oscillating with frequency $\omega$,
acting only at $x>0$. We consider an initial condition which is a stationary
state of the system without a field, and, at time $t=0$, the field is switched
on. We prove the existence of a solution $\psi(x,t)$ of the Schr\"odinger
equation for $t>0$, and compute the surface current. The current exhibits a
complex oscillatory behavior, which is not captured by the "simple" three step
scenario. As $t\to\infty$, $\psi(x,t)$ converges with a rate $t^{\frac32}$ to
a time periodic function with period $\frac{2\pi}{\omega}$ which coincides with
that found by Faisal, Kami\'nski and Saczuk (Phys Rev A 72, 023412, 2015).
However, for realistic values of the parameters, we have found that it can take
quite a long time (over 50 laser periods) for the system to converge to its
asymptote. Of particular physical importance is the current averaged over a
laser period $\frac{2\pi}\omega$, which exhibits a dramatic increase when
$\hbar\omega$ becomes larger than the work function of the metal, which is
consistent with the original photoelectric effect.
</p>




<p>In this note we address the question whether one can recover from the vertex
operator algebra associated with a fourdimensional N=2 superconformal field
theory the deformation quantization of the Higgs branch of vacua that appears
as a protected subsector in the threedimensional circlereduced theory. We
answer this question positively if the UV Rsymmetries do not mix with
accidental (topological) symmetries along the renormalization group flow from
the fourdimensional theory on a circle to the threedimensional theory. If
they do mix, we still find a deformation quantization but at different values
of its period.
</p>




<p>A twodimensional superintegrable system of singular oscillators with
internal degrees of freedom is identified and exactly solved. Its symmetry
algebra is seen to be the dual $1$ Hahn algebra which describes the bispectral
properties of the polynomials with the same name that are essentially the
ClebschGordan coefficients of the superconformal algebra
$\mathfrak{osp}(12)$. It is also shown how this superintegrable model is
obtained under dimensional reduction from a set of uncoupled harmonic
oscillators in four dimensions.
</p>




<p>We show that the Killing spinor equations of all supergravity theories which
may include higher order corrections on a (r,s)signature spacetime are
associated with twisted covariant form hierarchies. These hierarchies are
characterized by a connection on the space of forms which may not be degree
preserving. As a consequence we demonstrate that the form Killing spinor
bilinears of all supersymmetric backgrounds satisfy a suitable generalization
of conformal KillingYano equation with respect to this connection. To
illustrate the general proof the twisted covariant form hierarchies of some
supergravity theories in 4, 5, 6, 10 and 11 dimensions are also presented.
</p>




<p>We estimate the mixing time of the a nonreversible finite Markov chain called
Repeated BallsintoBins (RBB) process. This process is a discrete time
conservative interacting particle system with parallel updates. Place initially
in $L$ bins $rL$ balls, where $r$ is a fixed positive constant. At each time
step a ball is taken from each nonempty bin. Then \emph{all the balls} are
uniformly reassigned into bins. We prove that the mixing time of the RBB
process depends linearly on the maximum occupation number of balls of the
initial state. Thus if the initial configuration is such that the maximum
occupation number of balls is of order $L$ then the mixing time is of the same
correct order. While if the initial configuration is more diluted then the
equilibrium is reached in a time of order $(\log L)^c$.
</p>




<p>We give a systematic local description of invariant metrics and other
invariant fields on a spacetime under the action of a (nonabelian) group. This
includes the invariant fields in a neighbourhood of a principal and a special
orbit. The construction is illustrated with examples. We also apply the
formalism to give the Rsymmetry invariant metrics of some AdS backgrounds and
comment on applications to KaluzaKlein theory.
</p>




<p>We identify the local scaling limit of multiple boundarytoboundary branches
in a uniform spanning tree (UST) as a local multiple SLE(2), i.e., an SLE(2)
process weighted by a suitable partition function. By recent results, this also
characterizes the "global" scaling limit of the full collection of full curves.
The identification is based on a martingale observable in the UST with $N$
branches, obtained by weighting the wellknown martingale in the UST with one
branch by the discrete partition functions of the models. The obtained
weighting transforms of the discrete martingales and the limiting SLE
processes, respectively, only rely on a discrete domain Markov property and
(essentially) the convergence of partition functions. We illustrate their
generalizability by sketching an analogous convergence proof for a
boundaryvisiting UST branch and a boundaryvisiting SLE(2).
</p>




<p>We present a novel 8parameter integrable map in $\mathbb{R}^4$. The map is
measurepreserving and possesses two functionally independent 2integrals, as
well as a measurepreserving 2symmetry.
</p>




<p>It is shown that the carrier of a bounded localized free Dirac wavefunction
shrinks from infinity and subsequently expands to infinity again. The motion
occurs isotropicly at the speed of light. In between there is the phase of
rebound, which is limited in time and space in the order of the diameter of the
carrier at its minimal extension. This motion proceeds anisotropicly and
abruptly as for every direction in space there is a specific time, at which the
change from shrinking to expanding happens instantaneously. Asymptotically,
regarding the past and the future as well, the probability of position
concentrates up to 1 within any spherical shell whose outer radius increases at
light speed.
</p>




<p>The structure function of a random matrix ensemble can be specified as the
covariance of the linear statistics $\sum_{j=1}^N e^{i k_1 \lambda_j}$,
$\sum_{j=1}^N e^{i k_2 \lambda_j}$ for Hermitian matrices, and the same with
the eigenvalues $\lambda_j$ replaced by the eigenangles $\theta_j$ for unitary
matrices. As such it can be written in terms of the Fourier transform of the
densitydensity correlation $\rho_{(2)}$. For the circular $\beta$ensemble of
unitary matrices, and with $\beta$ even, we characterise the bulk scaling limit
of $\rho_{(2)}$ as the solution of a linear differential equation of order
$\beta + 1$  a duality relates $\rho_{(2)}$ with $\beta$ replaced by
$4/\beta$ to the same equation. Asymptotics obtained in the case $\beta = 6$
from this characterisation are combined with previously established results to
determine the explicit form of the degree 10 palindromic polynomial in
$\beta/2$ which determines the coefficient of $k^{11}$ in the small $k$
expansion of the structure function for general $\beta > 0$. For the Gaussian
unitary ensemble we give a reworking of a recent derivation and generalisation,
due to Okuyama, of an identity relating the structure function to simpler
quantities in the Laguerre unitary ensemble first derived in random matrix
theory by Br\'ezin and Hikami. This is used to determine various scaling
limits, many of which relate to the diprampplateau effect emphasised in
recent studies of many body quantum chaos, and allows too for rates of
convergence to be established.
</p>




<p>This paper studies how the static nonlinear electromagneticvacuum spacetime
of a point nucleus with negative bare mass affects the selfadjointness of the
generalrelativistic Dirac Hamiltonian for a test electron, without and with an
anomalous magnetic moment.
</p>
<p>The study interpolates between the previously studied extreme cases of a test
electron in (a) the ReissnerWeylNordstr\"om spacetime (Maxwell's
electromagnetic vacuum), which supports a very strong curvature singularity
with negative infinite bare mass, and (b) the Hoffmann spacetime (Born or
BornInfeld's electromagnetic vacuum) with vanishing bare mass, which features
the mildest possible curvature singularity.
</p>
<p>The main conclusion reached is: {on electrostatic spacetimes of a point
nucleus with a strictly negative bare mass} (which may be $\infty$) essential
selfadjointness fails unless the radial electric field diverges sufficiently
fast at the nucleus and the anomalous magnetic moment of the electron is taken
into account.
</p>
<p>Thus on the Hoffmann spacetime with (strictly) negative bare mass the Dirac
Hamiltonian of a test electron, with or without anomalous magnetic moment, is
not essentially selfadjoint.
</p>
<p>All these operators have selfadjoint extensions, though, with the usual
essential spectrum $(\infty,\mEL c^2]\cup[\mEL c^2,\infty)$ and an infinite
discrete spectrum located in the gap $(\mEL c^2,\mEL c^2)$
</p>
