Affine subgroups having the same Coxeter number with the affine Coxeter groups W(An), W(Dn), and W(En) are constructed by graph folding technique. The affine groups W(Cn) and W(Bn) are obtained from the Coxeter groups W(A2n-1) and W(D2n-1) respectively. The affine groups W(E6), W(D6) and W(E8) lead to the affine groups W(F4), W(H3), and W(H4) respectively by graph folding. The latter two are the non-crystallographic groups where W(H3) plays a special role in the quasicrystallographic structures with icosahedral symmetry. A general construction of the affine dihedral subgroups is introduced, some of which, describe the existing planar quasicrystallography. In the construction of the root systems, sets of orthonormal vectors are used but a special non-orthogonal set of vectors in the formulation of the root system of W(An) is also introduced which has practical applications in the construction of the lattices An and An* and their Delone and Voronoi cells.

We consider strong approximations of $1+1$-dimensional stochastic PDEs driven by additive space-time white noise. It has been long proposed (Davie-Gaines '01, Jentzen-Kloeden '08), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. The present paper proves this. For a large class of nonlinearities, with possibly superlinear growth, a temporal rate of (almost) $1$ is proven, a major improvement on the rate $1/4$ that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) $1/2$ as it is standard in the literature.

We investigate the effective coupling between heat and fluid dynamics within a thin fluid layer in contact with a solid structure via a rough surface. Moreover, the opposing vertical surfaces of the thin layer are in relative motion. This setup is particularly relevant to grinding processes, where cooling lubricants interact with the rough surface of a rotating grinding wheel. The resulting model is non-linearly coupled through(i) temperature-dependent viscosity and (ii) convective heat transport. The underlying geometry is highly heterogeneous due to the thin, rough surface characterized by a small parameter representing both the height of the layer and the periodicity of the roughness. We analyze this non-linear system for existence, uniqueness, and energy estimates and study the limit behavior within the framework of two-scale convergence in thin domains. In this limit, we derive an effective interface model in 3D (a line in 2D) for the heat and fluid interactions inside the fluid. We implement the system numerically and validate the limit problem through direct comparison with the micromodel. Additionally, we investigate the influence of the temperature-dependent viscosity and various geometrical configurations via simulation experiments. The corresponding numerical code is freely available on GitHub.

The complex relationship between Einstein's second postulate and the Maxwell electromagnetic theory is elucidated. A simple deduction of the main results of the Ignatowski approach to the theory of relativity is given. The peculiar status of the principle of relativity among the Maxwellians is illustrated.

We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2<p<\infty$. The main simplification is an alternative construction of a smooth and compactly supported unstable vortex, which is split into two steps: Firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This simpler structure of the unstable vortex yields a simplification of the other parts of Vishik's proof.

It is known that in Friedmann-Lemaitre-Robertson-Walker cosmology one has more number of dynamical equations, compared to the number of unknown variables. This fact makes some equations redundant. The situation becomes complicated because all the relevant differential equations in cosmology are not of the same order. In this article we study the fate of the redundant equations. We show that this redundancy is inevitable in general relativity. It is shown that this redundancy is primarily responsible for a special role of one of the Friedmann equations, which constrains the initial values of the problem. Our method of analyzing the dynamical structure of the theories relies on an operational approach and can be generalized further.

We study the design of an auction for an income-generating asset such as an intellectual property license. Each bidder has a signal about his future income from acquiring the asset. After the asset is allocated, the winner's income from the asset is realized privately. The principal can audit the winner, at a cost, and then charge a payment contingent on the winner's realized income. We solve for an auction that maximizes the principal's revenue, net of auditing costs. The winning bidder is charged linear royalties up to a cap, beyond which there is no auditing. A higher bidder pays more in cash upfront and faces a lower royalty cap.

Single molecule X-ray scattering experiments using free electron lasers hold the potential to resolve both single structures and structural ensembles of biomolecules. However, molecular electron density determination has so far not been achieved due to low photon counts, high noise levels and low hit rates. Most analysis approaches therefore focus on large specimen like entire viruses, which scatter substantially more photons per image, such that it becomes possible to determine the molecular orientation for each image. In contrast, for small specimen like proteins, the molecular orientation cannot be determined for each image, and must be considered random and unknown. Here we developed and tested a rigorous Bayesian approach to overcome these limitations, and also taking into account intensity fluctuations, beam polarization, irregular detector shapes, incoherent scattering and background scattering. We demonstrate using synthetic scattering images that it is possible to determine electron densities of small proteins in this extreme high noise Poisson regime. Tests on published experimental data from the coliphage PR772 achieved the detector-limited resolution of $9\,\mathrm{nm}$, using only $0.01\,\%$ of the available photons per image.

In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \\ As an application, the structure of the connected components of every quasi-compact quasi-separated (qcqs) scheme $X$ is fully characterized. They are exactly of the form $f^{-1}(C)$ where $f:X\rightarrow\Spec(R)$ is the canonical morphism, $C$ is a connected component of $\Spec(R)$ and $R=\mathscr{O}_{X}(X)$ is the ring of global sections of $X$. \\ Next, we make new advances in understanding the structure of the connected components of projective spaces. In general, for an $\mathbb{N}$-graded ring $R=\bigoplus\limits_{n\geqslant0}R_{n}$, the structure of the connected components of scheme $\Proj(R)$ is still unknown. However, we show that for any scheme $S$ the connected components of the projective space $\mathbb{P}^{n}_{S}= \mathbb{P}^{n}_{\mathbb{Z}}\times_{\Spec(\mathbb{Z})}S$ are exactly of the form $\mathbb{P}^{n}_{C}$ where $C$ is a connected component of $S$ which is equipped with a closed subscheme structure.

Nobody has discovered any perfect cuboid and there is no formula to deliver all possible Euler bricks. During investigations of famous open problems regarding the perfect cuboid and Euler brick; I have found new important conjectures on Pythagorean triples and biquadratic Diophantine equations [4] which are reduced $\&$ complete form for perfect cuboid and Euler brick problems. The details of the conjectures have been provided in Sections 2-3. If any perfect cuboid exists, it will be only among the solutions of six conjectures and all the Euler bricks are only among the solutions of next three conjectures [4]. For example, if any odd $n\in \mathbb{N}$ satisfy $n=e^2-f^2=g^2-h^2=k^2-l^2$ and $e^2f^2=g^2h^2+k^2l^2$; then we can discover a perfect cuboid of type 1 as $\{e^2-f^2,2gh,2kl,g^2+h^2,k^2+l^2,2ef,e^2+f^2\}$ having $(e^2-f^2,2gh,2kl)$ as its edges; $(g^2+h^2,k^2+l^2,2ef)$ as its face diagonals and $e^2+f^2$ as its body diagonal where $e,f,g,h,k,l~(>1)\in \mathbb{N}$. Equivalently, biquadratic Diophantine equation conjectures have been introduced for these perfect cuboid conjectures. For the benefit of readers, along with the original contribution for new important conjectures on perfect cuboid and Euler brick problems; brief review related to Pythagorean Triple, perfect cuboid and Euler brick problems as well as on Diophantine Equation and Biquadratic Diophantine Equation; studied in the past by previous researchers, have been discussed in the paper.

The main result of this article is a fantastic generalization of a classical result in graded ring theory. In fact, our result states that if $S$ is a multiplicative set of homogeneous elements of an $M$-graded commutative ring $R=\bigoplus\limits_{m\in M}R_{m}$ with $M$ a commutative monoid, then the localization ring $S^{-1}R=\bigoplus\limits_{x\in G}(S^{-1}R)_{x}$ is a $G$-graded ring where $G$ is the Grothendieck group of $M$ and each homogeneous component $(S^{-1}R)_{x}$ is the set of all fractions $f\in S^{-1}R$ such that $f=0$ or it is of the form $f=r/s$ where $r$ is a homogeneous element of $R$ and $x=[\dg(r),\dg(s)]$. As an application, ...

Let g be an affine Lie algebra with associated Yangian Y_hg. We prove the existence of two meromorphic R-matrices associated to any pair of representations of Y_hg in the category O. They are related by a unitary constraint and constructed as products of the form R(s)=R^+(s)R^0(s)R^-(s), where R^+(s) = R^-_{21}(-s)^{-1}. The factor R^0(s) is a meromorphic, abelian R-matrix, and R^-(s) is a rational twist. Our proof relies on two novel ingredients. The first is an irregular, abelian, additive difference equation whose difference operator is given in terms of the q-Cartan matrix of g. The regularization of this difference equation gives rise to R^0(s) as the exponentials of the two canonical fundamental solutions. The second key ingredient is a higher order analogue of the adjoint action of the affine Cartan subalgebra of g on Y_hg. This action has no classical counterpart, and produces a system of linear equations from which R^-(s) is recovered as the unique solution. Moreover, we show that both operators give rise to the same rational R-matrix on the tensor product of any two highest-weight representations.

Cloud droplets containing ice-nucleating particles (INPs) may freeze at temperatures above the homogeneous freezing threshold temperature. This process, referred to as immersion freezing, is one of the modulators of aerosol-cloud interactions in the Earth's atmosphere. In modeling studies, immersion freezing is often described using either so-called "singular" or "time-dependent" parameterizations. Here, we juxtapose both approaches and discuss them in the context of probabilistic particle-based cloud microphysics modeling. First, using a box model, we contrast how both parameterizations respond to different idealized ambient cooling rate profiles and quantify the impact of the polydispersity of the immersed surface spectrum on the frozen fraction evolution. Second, using a prescribed-flow two-dimensional cloud model, we illustrate the implications of applying the singular model in simulations with flow regimes relevant to ambient cloud conditions rather than to the cloud-chamber experiments on which these parameterizations are built upon. We discuss the critical role of the attribute-space sampling strategy for particle-based model simulations in modeling heterogeneous ice nucleation which is contingent on the presence of relatively sparse immersed INPs. The key takeaways include: (i) The singular approach, constituting a time-integrated form of a more general time-dependent approach, is only applicable under a limited range of ambient cooling rates. (ii) The time-dependent approach, especially when based on water-activity, is suitable for integration with particle-based model components of detailed aerosol composition and collisional growth/breakup. (iii) A flow-coupled aerosol-budget-resolving simulation shows the benefits and challenges of modeling cloud condensation nuclei activation and immersion freezing on insoluble ice nuclei with super-particle methods.

In the Metric Dimension problem, one asks for a minimum-size set $R$ of vertices such that for any pair of vertices of the graph, there is a vertex from $R$ whose two distances to the vertices of the pair are distinct. This problem has mainly been studied on undirected graphs and has gained a lot of attention in the recent years. We focus on directed graphs, and show how to solve the problem in linear time on digraphs whose underlying undirected graph (ignoring multiple edges) is a tree. This (non-trivially) extends a previous algorithm for oriented trees. We then extend the method to orientations of unicyclic graphs. We also give a fixed-parameter-tractable algorithm for digraphs when parameterized by the directed modular-width, extending a known result for undirected graphs. Finally, we show that Metric Dimension is NP-hard even on planar triangle-free acyclic digraphs of maximum degree 6.

The celebrated holographic duality posits a correspondence between a quantum gravity in a bulk spacetime and a quantum field theory (QFT) defined on its lower-dimensional boundary. This duality not only offers deep insights into the enigmatic nature of quantum gravity but also provides an efficient methodology for studying strongly correlated systems. However, despite its profound significance in modern physics, holographic duality remains a conjecture, and further experimental exploration is highly sought after. Here, we present the first experimental test of holographic duality between a three-dimensional bulk gravity and a two-dimensional boundary QFT using hyperbolic lattices. By experimentally measuring the classical scalar field propagator in hyperbolic circuits, we reproduce the equal-time two-point correlation function of the dual boundary conformal field theory (CFT), verifying its exponential dependence on the boundary separation and the conformal dimension-scalar mass relation. Furthermore, by leveraging the two-point correlation function, we reconstruct the entanglement entropy for a boundary CFT subsystem, confirming that it follows the Ryu-Takayanagi formula. These results constitute the first direct experimental evidence that quantum properties of the QFT can be holographically reproduced through its dual classical field in curved space. This heuristic experimental effort opens a new avenue for in-depth investigations on the holographic duality and extensive exploration of quantum-gravity-inspired phenomena in classical systems.

We prove a fundamental theorem for tropical partial differential equations, analogous to the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al., Falkensteiner et al. and from Fink and Toghani for the case of trivial valuation as introduced by Grigoriev to differential equations with power series coefficients over any valued field. Crucial ingredients are the framework for tropical partial differential equations introduced by Giansiracusa and Mereta and a result on infinite intersections of projections of fibers of tropicalizations, which we prove using Hrushovski and Loeser's model-theoretic interpretation of Berkovich analytification. As a corollary of the fundamental theorem, we show that the radius of convergence of solutions of an ordinary differential equation over a nontrivially valued field can be computed tropically.

An updating rule specifies how an agent reacts to information. An updating rule is Blackwell monotone if more information is always better for an agent in a decision problem and strictly Blackwell monotone if, in addition, there is always a decision problem in which more information is strictly better for an agent. Bayes' law is strictly Blackwell monotone, and I show that within a broad class of updating rules--those that distort the Bayesian posteriors in a signal-independent manner--it is the only strictly Blackwell-monotone updating rule. If an agent's decisions are evaluated non-paternalistically (according to her beliefs), the Blackwell-monotone updating rules are affine distortions of the Bayesian posteriors.

We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author. Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots, a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible. Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots a_n-1$. We prove that this cell complex gives cellular minimal free resolution of this of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range examples all polarizations of the Artin monomial ideal. We also show that the squeezed balls of G.Kalai \cite{Ka} derive from polarizations of Artin monomial ideals.

We study what changes to an agent's decision problem increase her value for information. We prove that information becomes more valuable if and only if the agent's reduced-form payoff in her belief becomes more convex. When the transformation corresponds to the addition of an action, the requisite increase in convexity occurs if and only if a simple geometric condition holds, which extends in a natural way to the addition of multiple actions. We apply these findings to two scenarios: a monopolistic screening problem in which the good is information and delegation with information acquisition.

As the first main result of this article, we prove that if $e$ and $e'$ are idempotents of a commutative ring $A$, then there is a canonical isomorphism of $A$-modules: $$Ae\oplus Ae'\simeq Ae/Ae(1-e')\oplus Ae'/Ae'(1-e)\oplus A(e+e'-2ee').$$ This result plays an important role in proving several results on the Grothendieck ring $K_{0}(A)$. Especially, we first show that for any ring $A$ there is a complex of Abelian groups which is exact at the beginning and end: $$\xymatrix{0\ar[r]&\Pic(A)\ar[r]&K_{0}(A)^{\ast} \ar[r]&\mathscr{B}(A)\ar[r]&0.}$$ Then we show that the above sequence is split exact for some certain rings $A$ (including Dedekind domains or more generally Noetherian one dimensional rings). The next main result asserts that for any ring $A$ we have the canonical isomorphisms of Abelian groups $\mathscr{B}(A)\simeq\mathscr{B}\big(K_{0}(A)\big)\simeq H_{0}(A)^{\ast}$. As an application, we show that a morphism of rings $A\rightarrow B$ lifts idempotents if and only if the induced ring map $K_{0}(A)\rightarrow K_{0}(B)$ lifts idempotents. If moreover, $B$ has finitely many maximal ideals then the map $K_{0}(A)\rightarrow K_{0}(B)$ is surjective. Finally, we show that the support of a finitely generated projective module is the whole prime spectrum if and only if its trace ideal is the whole unit ideal.

DNA-coated colloids can crystallize into a multitude of lattices, ranging from face-centered cubic to diamond and thereby contribute to our understanding of crystallization and open avenues to producing structures with useful photonic properties. Despite the broad potential design space of DNA-coated colloids, the design cycle for synthesizing DNA-coated particles is slow: preparing a particle with a new type of DNA sequence takes more than one day and requires custom-made and chemically modified DNA that typically takes the supplier over a month to synthesize. Here, we introduce a method to generate particles with custom sequences from a single feed stock in under an hour at ambient conditions. Our method appends new DNA domains onto the DNA grafted to colloidal particles based on a template that takes the supplier less than a week to produce. The resultant particles crystallize as readily and at the same temperature as those produced via direct chemical synthesis. Moreover, we show that particles coated with a single sequence can be converted into a variety of building blocks with differing specificities by appending different DNA sequences to them. This approach to DNA-coated particle preparation will make it practical to identify optimal and complex particle sequence designs and to expand the use of DNA-coated colloids to a much broader range of investigators and commercial entities.

A $k$-edge-weighting of $G$ is a mapping $ω:E(G)\longrightarrow \{1,\ldots,k\}$. The edge-weighting of $G$ naturally induces a vertex-colouring $σ_ω:V(G)\longrightarrow \mathbb{N}$ given by$σ_ω(v)=\sum_{u\in N_G(v)}ω(vu)$ for every $v\in V(G)$. The edge-weighting $ω$ is neighbour sum distinguishing if it yields a proper vertex-colouring $σ_ω$, \emph{i.e.}, $σ_ω(u)\neq σ_ω(v)$ for every edge $uv$ of $G$.We investigate a neighbour sum distinguishing edge-weighting with local constraints, namely, we assume that the set of edges incident to a vertex of large degree is not monochromatic. A graph is nice if it has no components isomorphic to $K_2$. We prove that every nice graph with maximum degree at most~5 admits a neighbour sum distinguishing $(Δ(G)+2)$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights. Furthermore, we prove that every nice graph admits a neighbour sum distinguishing $7$-edge-weighting such that all the vertices of degree at least~6 are incident with at least two edges of different weights. Finally, we show that nice bipartite graphs admit a neighbour sum distinguishing $6$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights.

For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. The proof of sufficiency is based on an intricate Peierls argument. We present a simplified version of this Peierls argument. Our main motivation is the open problem of determining stability of monotone cellular automata with intrinsic randomness, in which for the unperturbed evolution the local update rules at different space-time points are chosen in an i.i.d. fashion according to some fixed law. We apply Toom's Peierls argument to prove stability of a class of cellular automata with intrinsic randomness and also derive lower bounds on the critical parameter for some deterministic cellular automata.

Increases in national concentration have been a salient feature of industry dynamics in the U.S. and have contributed to concerns about increasing market power. Yet, local trends may be more informative about market power, particularly in the retail sector where consumers have traditionally shopped at nearby stores. We find that local concentration has increased almost in parallel with national concentration using novel Census data on product-level revenue for all U.S. retail stores between 1992 and 2012. The increases in concentration are broad based, affecting most markets, products, and retail industries. We show that the expansion of multi-market firms into new markets explains most of the increase in national retail concentration, with consolidation via increases in local market shares increasing in importance between 1997 and 2007, and single-market firms playing a negligible role. Finally, we find that increases in local concentration can explain one-quarter to one-third of the observed rise in retail gross margins.

A finite-time singularity of 2D harmonic map flow will be called "strictly type-II" if the outer energy scale satisfies $λ(t) = O(T - t)^{\frac{1 + α}{2}}.$ We prove that the body map at a strict type-II blowup is Hölder continuous.

Physical dynamic networks most commonly consist of interconnections of physical components that can be described by diffusive couplings. These diffusive couplings imply that the cause-effect relationships in the interconnections are symmetric and therefore physical dynamic networks can be represented by undirected graphs. This paper shows how prediction error identification methods developed for linear time-invariant systems in polynomial form can be configured to consistently identify the parameters and the interconnection structure of diffusively coupled networks. Further, a multi-step least squares convex optimization algorithm is developed to solve the nonconvex optimization problem that results from the identification method.

In this note, we present a new proof of the solvability of the electrostatic Born-Infeld equation with radial charge, based on the conformal method and the Spacetime Positive Energy Theorem. An advantage of this approach is that the resulting solutions are automatically classical and spacelike.

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges is in each color class. If there exists an integer $k$ such that, for $n$ sufficiently large, every 2-coloring of $K_n$ with more than $k$ edges in each color contains a balanced copy of $G$, then we say that $G$ is balanceable. The smallest integer $k$ such that this holds is called the balancing number of $G$. In this paper, we define a more general variant of the balancing number, the list balancing number, by considering 2-list edge colorings of $K_n$, where every edge $e$ has an associated list $L(e)$ which is a nonempty subset of the color set $\{r,b\}$. In this case, edges $e$ with $L(e) = \{r,b\}$ act as jokers in the sense that their color can be chosen $r$ or $b$ as needed. In contrast to the balancing number, every graph has a list balancing number. Moreover, if the balancing number exists, then it coincides with the list balancing number. We give the exact value of the list balancing number for all cycles except for $4k$-cycles for which we give tight bounds. In addition, we give general bounds for the list balancing number of non-balanceable graphs based on the extremal number of its subgraphs, and study the list balancing number of $K_5$, which turns out to be surprisingly large.

Let $X \subset\mathbb{P}^r$ be a projective $d$-variety with isolated determinantal singularities and $ω$ be a $1$-form on $X$ with a finite number of singularities (in the stratified sense). Under some technical conditions on $r$ we use two generalization of Poincaré-Hopf index with the goal of proving a Poincaré-Hopf Type Theorem for $X$.

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges are in each color class. If, for every sufficiently large $n$, there exists an integer $k$ such that every 2-coloring of $K_n$ with more than $k$ edges in each color class contains a balanced copy of $G$, then we say that $G$ is balanceable. Balanceability was introduced by Caro, Hansberg and Montejano, who also gave a structural characterization of balanceable graphs. In this paper, we extend the study of balanceability by finding new sufficient conditions for a graph to be balanceable or not. We use those conditions to fully characterize the balanceability of graph classes such as rectangular and triangular grids, as well as a special class of circulant graphs.