<p>The aim of the paper is to study the group scheme $G:=\operatorname{SL}(2,
A)$, it's comodules and a virtual Clebsch-Gordan formula. Here $A$ is a field
or a Dedekind domain. If $V:=A\{e_1,e_2\}$ is the free rank $2$ module on $A$
and if we give $V$ the "standard" structure as comodule on $G$, we may form the
symmetric powers $\operatorname{Sym}^n(V)$ for $n \geq 1$ an integer. If $A$ is
a field of characteristic zero, there is a direct sum decomposition of the
tensor product $\operatorname{Sym}^n(V) \otimes \operatorname{Sym}^m(V)$ into
irreducible $G$-comodules and the main aim of the paper is to investigate if
similar resu…
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<p>As quantum theory allows for information processing and computing tasks that
otherwise are not possible with classical systems, there is a need and use of
quantum Internet beyond existing network systems. At the same time, the
realization of a desirably functional quantum Internet is hindered by
fundamental and practical challenges such as high loss during transmission of
quantum systems, decoherence due to interaction with the environment, fragility
of quantum states, etc. We study the implications of these constraints by
analyzing the limitations on the scaling and robustness of quantum Internet.
Considering quantum networks, we present p…
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<p>Given a square pencil $A+ \lambda B$, where $A$ and $B$ are complex matrices,
we consider the problem of finding the singular pencil nearest to it in the
Frobenius distance. This problem is known to be very difficult, and the few
algorithms available in the literature can only deal efficiently with pencils
of very small size. We show that the problem is equivalent to minimizing a
certain objective function over the Riemannian manifold $SU(n) \times SU(n)$,
where $SU(n)$ denotes the special unitary group. With minor modifications, the
same approach extends to the case of finding a nearest singular pencil with a
specified minimal index. This …
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<p>In our study, we conduct magnetization and heat capacity measurements to
investigate field-induced magnetic phase transitions within the newly
synthesized compound K2Ni2(SeO3)3, a spin-1 dimer system arranged on a
triangular lattice. The Ni-Ni dimers exhibit a ferromagnetic intra-dimer
interaction, effectively behaving as an ensemble with a total spin of S=2. In
contrast, antiferromagnetic interactions manifest between these dimers on the
triangular lattice. The trigonal distortion of the NiO6 octahedra introduces
easy-axis magnetic anisotropy, accounting for the distinct magnetic phase
diagrams observed when applying c-axis directional and…
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<p>In this article, we present a novel data assimilation strategy in pore-scale
imaging and demonstrate that this makes it possible to robustly address
reactive inverse problems incorporating Uncertainty Quantification (UQ).
Pore-scale modeling of reactive flow offers a valuable opportunity to
investigate the evolution of macro-scale properties subject to dynamic
processes. Yet, they suffer from imaging limitations arising from the
associated X-ray microtomography (X-ray microCT) process, which induces
discrepancies in the properties estimates. Assessment of the kinetic parameters
also raises challenges, as reactive coefficients are critical p…
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<p>In this paper, we present a fast divergence-free spectral algorithm (FDSA)
for the curl-curl problem. Divergence-free bases in two and three dimensions
are constructed by using the generalized Jacobi polynomials. An accurate
spectral method with exact preservation of the divergence-free constraint
point-wisely is then proposed, and its corresponding error estimate is
established. We then present a highly efficient solution algorithm based on a
combination of matrix-free preconditioned Krylov subspace iterative method and
a fully diagonalizable auxiliary problem, which is derived from the spectral
discretisations of generalized eigenvalue pr…
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<p>The increasing trend in adopting electric vehicles (EVs) will significantly
impact the residential electricity demand, which results in an increased risk
of transformer overload in the distribution grid. To mitigate such risks, there
are urgent needs to develop effective EV charging controllers. Currently, the
majority of the EV charge controllers are based on a centralized approach for
managing individual EVs or a group of EVs. In this paper, we introduce a
decentralized Multi-agent Reinforcement Learning (MARL) charging framework that
prioritizes the preservation of privacy for EV owners. We employ the
Centralized Training Decentralized E…
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<p>Decision-making problems can be represented as mathematical optimization
models, finding wide applications in fields such as economics, engineering and
manufacturing, transportation, and health care. Optimization models are
mathematical abstractions of the problem of making the best decision while
satisfying a set of requirements or constraints. One of the primary barriers to
deploying these models in practice is the challenge of helping practitioners
understand and interpret such models, particularly when they are infeasible,
meaning no decision satisfies all the constraints. Existing methods for
diagnosing infeasible optimization models o…
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<p>We show that the range of a critical branching random walk conditioned to
survive forever and the Minkowski sum of two independent simple random walk
ranges are intersection-equivalent in any dimension $d\ge 5$, in the sense that
they hit any finite set with comparable probability, as their common starting
point is sufficiently far away from the set to be hit. Furthermore, we extend a
discrete version of Kesten, Spitzer and Whitman's result on the law of large
numbers for the volume of a Wiener sausage. Here, the sausage is made of the
Minkowski sum of $N$ independent simple random walk ranges in $\mathbb{Z}^d$,
with $d>2N$, and…
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<p>In the article, we investigate the average behaviour of normalised Hecke
eigenvalues over certain polynomials and establish an estimate for the power
moments of the normalised Hecke eigenvalues of a normalised Hecke eigenform of
weight $k \ge 2$ for the full modular group $SL_2(\mathbb{Z})$ over certain
polynomial, given by a sum of triangular numbers with certain positive
coefficients. More precisely, for each $r \in \mathbb{N}$, we obtain an
asymptotic for the following sum \begin{equation*} \begin{split}
\displaystyle{\sideset{}{^{\flat }}\sum_{ \alpha(\underline{x}))+1 \le X \atop
\underline{x} \in {\mathbb Z}^{4}} } \lambda_{f}^{r}(\al…
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<p>We develop a novel asymptotic theory for local polynomial (quasi-)
maximum-likelihood estimators of time-varying parameters in a broad class of
nonlinear time series models. Under weak regularity conditions, we show the
proposed estimators are consistent and follow normal distributions in large
samples. Our conditions impose weaker smoothness and moment conditions on the
data-generating process and its likelihood compared to existing theories.
Furthermore, the bias terms of the estimators take a simpler form. We
demonstrate the usefulness of our general results by applying our theory to
local (quasi-)maximum-likelihood estimators of a time-…
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<p>Consider the following prediction problem. Assume that there is a block box
that produces bits according to some unknown computable distribution on the
binary tree. We know first $n$ bits $x_1 x_2 \ldots x_n$. We want to know the
probability of the event that that the next bit is equal to $1$. Solomonoff
suggested to use universal semimeasure $m$ for solving this task. He proved
that for every computable distribution $P$ and for every $b \in \{0,1\}$ the
following holds: $$\sum_{n=1}^{\infty}\sum_{x: l(x)=n} P(x) (P(b | x) - m(b |
x))^2 < \infty\ .$$ However, Solomonoff's method has a negative aspect: Hutter
and Muchnik proved t…
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<p>Given a monoid $S$ with $E$ any non-empty subset of its idempotents, we
present a novel one-sided version of idempotent completion we call left
$E$-completion. In general, the construction yields a one-sided variant of a
small category called a constellation by Gould and Hollings. Under certain
conditions, this constellation is inductive, meaning that its partial
multiplication may be extended to give a left restriction semigroup, a type of
unary semigroup whose unary operation models domain. We study the properties of
those pairs $S,E$ for which this happens, and characterise those left
restriction semigroups that arise as such left $E$-co…
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<p>This paper shows that immersed totally geodesic $m$-dimensional suborbifolds
of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite
subgroups of the commensurator whenever $m \geqslant \lfloor \frac{n}{2}
\rfloor$. We call such totally geodesic suborbifolds finite centraliser
subspaces (or fc-subspaces) and use them to formulate an arithmeticity
criterion for hyperbolic lattices.
</p>
<p>We show that a hyperbolic orbifold $M$ is arithmetic if and only if it has
infinitely many fc-subspaces, exhibiting examples of non-arithmetic orbifolds
that contain non-fc subspaces of codimension one. We provid…
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<p>We revisit Vafa-Witten theory in the more general setting whereby the
underlying moduli space is not that of instantons, but of the full Vafa-Witten
equations. We physically derive (i) a novel Vafa-Witten four-manifold invariant
associated with this moduli space, (ii) their relation to Gromov-Witten
invariants, (iii) a novel Vafa-Witten Floer homology assigned to three-manifold
boundaries, (iv) a novel Vafa-Witten Atiyah-Floer correspondence, (v) a proof
and generalization of a conjecture by Abouzaid-Manolescu in [1] about the
hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands
duality of these invariants, Floer homolo…
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<p>In this paper, we study min-max optimization problems on Riemannian
manifolds. We introduce a Riemannian Hamiltonian function, minimization of
which serves as a proxy for solving the original min-max problems. Under the
Riemannian Polyak--{\L}ojasiewicz condition on the Hamiltonian function, its
minimizer corresponds to the desired min-max saddle point. We also provide
cases where this condition is satisfied. For geodesic-bilinear optimization in
particular, solving the proxy problem leads to the correct search direction
towards global optimality, which becomes challenging with the min-max
formulation. To minimize the Hamiltonian function, …
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<p>The classical Ham Sandwich theorem states that any $d$ point sets in
$\mathbb{R}^d$ can be simultaneously bisected by a single affine hyperplane. A
generalization of Dolnikov asserts that any $d$ families of pairwise
intersecting compact, convex sets in $\mathbb{R}^d$ admit a common hyperplane
transversal. We extend Dolnikov's theorem by showing that families of compact
convex sets satisfying more general non-disjointness conditions admit common
transversals by multiple hyperplanes. In particular, these generalize all known
optimal results to the long-standing Gr\"unbaum--Hadwiger--Ramos measure
equipartition problem in the case o…
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<p>We work over a perfect field. Recent work of the third-named author
established a Derived Auslander Correspondence that relates finite-dimensional
self-injective algebras that are twisted $3$-periodic to algebraic triangulated
categories of finite type. Moreover, the aforementioned work also shows that
the latter triangulated categories admit a unique differential graded
enhancement. In this article we prove a higher-dimensional version of this
result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic
algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster
tilting object. We also show that the latter triang…
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<p>The optimal stopping problem is one of the core problems in financial
markets, with broad applications such as pricing American and Bermudan options.
The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has
shown great power in solving high-dimensional forward-backward stochastic
differential equations (FBSDEs), and inspired many applications. However, the
method solves backward stochastic differential equations (BSDEs) in a forward
manner, which can not be used for optimal stopping problems that in general
require running BSDE backwardly. To overcome this difficulty, a recent paper
[Wang, Chen, Sudjianto, Liu and Shen, …
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<p>Geometric quantiles are location parameters which extend classical univariate
quantiles to normed spaces (possibly infinite-dimensional) and which include
the geometric median as a special case. The infinite-dimensional setting is
highly relevant in the modeling and analysis of functional data, as well as for
kernel methods.
</p>
<p>We begin by providing new results on the existence and uniqueness of
geometric quantiles. Estimation is then performed with an approximate
M-estimator and we investigate its large-sample properties in infinite
dimension.
</p>
<p>When the population quantile…
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<p>Copy-move forgery on speech (CMF), coupled with post-processing techniques,
presents a great challenge to the forensic detection and localization of
tampered areas. Most of the existing CMF detection approaches necessitate
pre-segmentation of speech to facilitate similarity calculations among these
segments. However, these approaches usually suffer from the problems of
uncontrollable computational complexity and sensitivity to the presence of a
word that is read multiple times within a speech recording. To address these
issues, we propose a local feature tensors-based CMF detection algorithm that
can transform duplicate detection and locali…
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<p>Recently, [Phys. Rev. Lett. 130, 221501 (2023)] Jacobson and Visser
calculated the quantum partition function of a fixed, finite volume of a region
with the topology of a ball in the saddle point approximation within the
context of Einstein's gravity with or without a cosmological constant. The
result can be interpreted as the dimension of Hilbert space of the theory. Here
we extend their computation to a theory defined in principle with infinitely
many powers of curvature in three dimensions. We confirm their result: The
partition function of a spatial region in the leading saddle point
approximation is given as the exponential of the…
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<p>The Hopf formula for Hamilton-Jacobi reachability (HJR) analysis has been
proposed to solve high-dimensional differential games, producing the set of
initial states and corresponding controller required to reach (or avoid) a
target despite bounded disturbances. As a space-parallelizable method, the Hopf
formula avoids the curse of dimensionality that afflicts standard
dynamic-programming HJR, but is restricted to linear time-varying systems. To
compute reachable sets for high-dimensional nonlinear systems, we pair the Hopf
solution with Koopman theory for global linearization. By first lifting a
nonlinear system to a linear space and then s…
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<p>In this paper, we consider stochastic monotone Nash games where each player's
strategy set is characterized by possibly a large number of explicit convex
constraint inequalities. Notably, the functional constraints of each player may
depend on the strategies of other players, allowing for capturing a subclass of
generalized Nash equilibrium problems (GNEP). While there is limited work that
provide guarantees for this class of stochastic GNEPs, even when the functional
constraints of the players are independent of each other, the majority of the
existing methods rely on employing projected stochastic approximation (SA)
methods. However,…
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<p>Recently, [Phys. Rev. Lett. 130, 221501 (2023)] Jacobson and Visser
calculated the quantum partition function of a fixed, finite volume of a region
with the topology of a ball in the saddle point approximation within the
context of Einstein's gravity with or without a cosmological constant. The
result can be interpreted as the dimension of Hilbert space of the theory. Here
we extend their computation to a theory defined in principle with infinitely
many powers of curvature in three dimensions. We confirm their result: The
partition function of a spatial region in the leading saddle point
approximation is given as the exponential of the…
Voters: 0
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Latest: Aug. 26, 2023, 7:32 a.m.
