This paper proposes a novel consensus-based distributed filter over directed graphs under the collectively observability condition. The distributed filter is designed using an augmented leader-following information fusion strategy, and the gain parameter is determined exclusively using local information. Additionally, the lower bound of the fusion step number is derived to ensure that the estimation error covariance remains uniformly upper-bounded. Furthermore, the lower bounds for the convergence rates of the steady-state performance gap between the proposed filter and the centralized filter are provided as the fusion step number approaches infinity. The analysis demonstrates that the convergence rate is at least as fast as exponential convergence, provided the communication topology satisfies the spectral norm condition. Finally, the theoretical results are validated through two simulation examples.

In this paper, we explore linear representations of skew left braces, which are known to provide bijective non-degenerate set-theoretical solutions to the Yang--Baxter equation that are not necessarily involutive. A skew left brace $(A, \cdot, \circ)$ induces an action $λ^{\op}: (A, \circ) \to \Aut (A, \cdot)$, which gives rise to the group $Λ_{A^{\op}} = (A, \cdot) \rtimes_{λ^{\op}} (A, \circ)$. We prove that if $A$ and $B$ are isoclinic skew left braces, then $Λ_{A^{\op}}$ and $Λ_{B^{\op}}$ are also isoclinic under some mild restrictions on the centers of the respective groups. Our key observation is that there is a one-to-one correspondence between the set of equivalence classes of irreducible representations of $(A, \cdot, \circ)$ and that of the group $Λ_{A^{\op}}$. We obtain a decomposition of the induced representation of the additive group $(A, \cdot)$ and of the multiplicative group $(A, \circ)$ corresponding to the regular representation of the group $Λ_{A^{\op}}$. As examples, we compute the dimensions of the irreducible representations for several skew left braces with prime power orders.

Repeatedly adding or subtracting the digital reversal to or from an integer, depending on which one is larger, can be treated as a dynamical system. On one hand, a three-digit version of this map running only two steps is the 1089 mathematical trick problem; on the other hand, this mapping can be compared to John Conway's reverse-add-then-sort (RATS) iteration, as well as the 3x+1 problem, also known as Collatz's map. We numerically run this map and find interesting dynamics, including limiting cycles with unusual periodicity and length-8 diverging trajectories.

Developing and testing modern RDF-based applications often requires access to RDF datasets with certain characteristics. Unfortunately, it is very difficult to publicly find domain-specific knowledge graphs that conform to a particular set of characteristics. Hence, in this paper we propose RDFGraphGen, an open-source RDF graph generator that uses characteristics provided in the form of SHACL (Shapes Constraint Language) shapes to generate synthetic RDF graphs. RDFGraphGen is domain-agnostic, with configurable graph structure, value constraints, and distributions. It also comes with a number of predefined values for popular schema.org classes and properties, for more realistic graphs. Our results show that RDFGraphGen is scalable and can generate small, medium, and large RDF graphs in any domain.

We use the correlation matrix of the generating distribution to determine the mixing time for random walks on the torus $(\mathbb{Z}/q\mathbb{Z})^n$. We present our method in the context of the Diaconis-Gangolli random walk on both the $1 \times n$ and $m \times n$ contingency tables over $\mathbb{Z}/q\mathbb{Z}$. In the $1 \times n$ case, we prove that the random walk exhibits cutoff at time $\dfrac{n q^2 \log(n)}{8 π^2}$ when $q \gg n$; in the $m \times n$ case, where $m, n$ are of the same order, we establish cutoff for the random walk at time $\dfrac{mn q^2 \log(mn)}{16 π^2}$ when $q \gg n^2$. Our method reveals that a general class of random walks on the torus $(\mathbb{Z}/q\mathbb{Z})^n$ has cutoff. If each coordinate of the lifted random walk onto $\mathbb{Z}^n$ has variance $σ^2/n$ in each jump, and the between-coordinate correlations are sufficiently low, then cutoff occurs at time $\dfrac{nq^2 \log(n)}{4π^2 σ^2}$.

We consider a system of particles performing a discrete-time binary branching random walk with independent standard normal increments subject to a penalty $\b$ for every pair of particles that get within distance $\e$ of each other at every time. We study the optimal configurations that minimise the sum of the spread out cost and the repulsion cost up to a given time horizon $N$. We show that at time $N$ particles are spread out over a distance $\asymp (\b\e)^{1/3} 2^{2N/3}$. We also show that the total cost of the optimal configurations up to time $N$ is $\asymp (\b\e)^{2/3} 2^{4N/3}$.

Eylee Jung \textit{et.al}\cite{jung2008} had conjectured that $P_{max}=\frac{1}{2}$ is a necessary and sufficient condition for the perfect two-party teleportation and consequently the Groverian measure of entanglement for the entanglement resource must be $\frac{1}{\sqrt{2}}$. It is also known that prototype $W$ state is not useful for standard teleportation. Agrawal and Pati\cite{pati2006} have successfully executed perfect (standard) teleportation with non-prototype $W$ state. Aligned with Pati's protocol\cite{pati2006} we have considered here $Star$ type tripartite states and have shown that perfect teleportation is suitable with such states. Moreover, we have taken the linear superposition of non-prototype $W$ state and its spin-flipped version and shown that it belongs to $Star$ class. Also, standard teleportation is possible with these states. It is observed that genuine tripartite entanglement is not necessary requirement for a state to be used as a channel for successful standard teleportation. We have also shown that these $Star$ class states are $P_{max}=\frac{1}{4}$ states and their Groverian entanglement is $\frac{\sqrt{3}}{2}$, thus concluding that Jung conjecture is not a necessary condition.

Higher dimensions provide fertile ground for diverse topological phases and their associated localization phenomena, thanks to the rich geometric features of boundaries and defects. In this paper, we investigate non-Hermitian lattices with defects and establish a correspondence between spectral winding topology, fractal structures, and defect-localized states in arbitrary dimensions. Through analytical derivation and numerical simulations, we demonstrate that defect states emerge only when the spectral winding number exceeds a threshold determined by the defect size, which is linked to their fractal characteristics. By utilizing the Green's function, we identify amplified responses at defects under external driving fields, strengthening the physical correspondence between these topological and fractal features. Our findings offer a universal framework for understanding defect-localized states in higher-dimensional non-Hermitian systems.

Networks with two nodes are previously grouped into either two classes (mutually interactive, master-slave) or five classes (mutualism, competition, predator-prey, commensalism, amensalism). By allowing self-loops, the number of signed regulatory graphs increases to 39. We provide a complete summary of dynamical behaviors of the 39 two-node McCulloch-Pitts models when the link weights are constrained to three values [$-1$,0,$+1$] and Boolean node variables. Depending on whether the Boolean values are [$-1,1$] (bipolar) or [0,1] (binary), we show that the dynamics could also be different with the same signed regulatory graphs. We demonstrate that slight variations in the McCulloch-Pitts model (called variants) may lead to fundamentally different dynamics. We study the full model space and three kinds of robustness or stability: a) of a rule against parameter change on its overall dynamics, b) for a given state against parameter change on its final state, and c) against an initial state change on its final state. All these stability properties are loosely related to a model's limiting dynamics, with the fixed-point rules to be more stable in the first two types of robustness, but less stable in the third robustness type. These analyses pave the way towards a better understanding of a minimum complex system.

We propose a modeling framework to develop a continuum description of opinion dynamics on networks as an alternative to other models, like the ones based on graphons. In a nutshell, the continuum model that we propose aims at approximating the distribution of opinions as well as the probability that two given opinions are connected. To illustrate our framework, we focus on a simple model of consensus dynamics on a network and derive a continuum description using techniques inspired by mean-field limits. We also discuss the limitations of this approach and suggest extensions to account for dynamic networks with evolving connections, stochastic effects, and directional interactions.

In this paper, we introduce matrix weighted Bourgain-Morrey spaces and obtain two sufficient conditions for precompact sets in matrix weighted Bourgain-Morrey spaces. We prove that the dyadic average operator is bounded on some matrix weighted Bourgain-Morrey spaces. With this result, we obtain the necessity for precompact sets in some matrix weighted Bourgain-Morrey spaces. The results are new even for the unweighted Bourgain-Morrey spaces.

Cellulose ethers represent a class of water-soluble polymers widely utilized across diverse sectors, spanning from healthcare to the construction industry. This experimental study specifically delves into aqueous suspensions of carboxymethylcellulose (CMC), a polymer that undergoes gel formation in acidic environments due to attractive interactions between hydrophobic patches along its molecular chain. We use rheometry to determine the linear viscoelastic properties of both CMC suspensions and acid-induced gels at various temperatures. Then, applying the time-temperature superposition principle, we construct master curves for the viscoelastic spectra, effectively described by fractional models. The horizontal shift factors exhibit an Arrhenius-like temperature dependence, allowing us to extract activation energies compatible with hydrophobic interactions. Furthermore, we show that acid-induced CMC gels are physical gels that display a reversible yielding transition under external shear. In particular, we discuss the influence of pH on the non-linear viscoelastic response under large-amplitude oscillatory shear. Overall, our results offer a comprehensive description of the linear and non-linear rheological properties of a compelling case of physical hydrogel involving hydrophobic interactions.

This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda. The addenda include proofs, in the style of the book, showing that (A1) Albert algebras are exceptional and in particular that a central simple Jordan algebra over a field is exceptional if and only if it is an Albert algebra; (A2) A regular lattice in a real Albert algebra is also an Albert algebra; (A3) a Freudenthal algebra over a field is split by an extension of degree dividing 6; and (A4) a Freudenthal subalgebra of rank 9 in an Albert algebra can be used to describe the Albert algebra as a Tits construction.

This paper proposes a data-driven framework to learn a finite-dimensional approximation of a Koopman operator for approximating the state evolution of a dynamical system under noisy observations. To this end, our proposed solution has two main advantages. First, the proposed method only requires the measurement noise to be bounded. Second, the proposed method modifies the existing deep Koopman operator formulations by characterizing the effect of the measurement noise on the Koopman operator learning and then mitigating it by updating the tunable parameter of the observable functions of the Koopman operator, making it easy to implement. The performance of the proposed method is demonstrated on several standard benchmarks. We then compare the presented method with similar methods proposed in the latest literature on Koopman learning.

Let $u$ be either a second eigenfunction of the fractional $p$-Laplacian or a least energy nodal solution of the equation $(-Δ)^s_p \, u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $Ω$ and under the nonlocal zero Dirichlet conditions. Assuming only that $Ω$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\partialΩ$. As a consequence, the nodal set of $u$ has the same property whenever $Ω$ is connected. The proof is based on the analysis of equality cases in certain polarization inequalities involving positive and negative parts of $u$, and on alternative characterizations of second eigenfunctions and least energy nodal solutions.

The shock initiation of energetic materials is mediated by the localization of mechanical energy into hotspots. These originate through the interaction of the shock and material microstructure; the most potent hotspots are formed by the collapse of porosity. Recent work using molecular dynamics (MD) has shed light on the molecular mechanisms responsible for the shock-to-deflagration transition following pore collapse in pure energetic materials. However, explosive formulations are composites of energetic crystals and a polymer binder, which differs from the prior focus on pure materials. The role of polymer phases on hotspot formation and its criticality is not well-understood. We use reactive MD simulations to investigate the role of polystyrene and polyvinyl nitrate films around pores in the shock-induced pore collapse of RDX. The polymer affects the hotspots' temperature and their criticality. While the presence of inert polymer often delays or hinders chemical reactions of the energetic material, certain geometries accelerate chemistry. The simulations provide a mechanistic understanding of these phenomena.

We prove generic differentiability in $P$-minimal theories, strengthening an earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and Pillay's $P$-minimal analogue of Pillay's conjectures on o-minimal groups. Specifically, let $G$ be an $n$-dimensional definable group in a highly saturated model $M$ of a $P$-minimal theory. Then there is an open definable subgroup $H \subseteq G$ such that $H$ is compactly dominated by $H/H^{00}$, and $H/H^{00}$ is a $p$-adic Lie group of the expected dimension. Additionally, the generic differentiability theorem immediately implies a classification of interpretable fields in $P$-minimal theories, by work of Halevi, Hasson, and Peterzil.

The existence of a radio synchrotron background (RSB) excess is implied by a number of measurements, including excess emission seen by the ARCADE~2 and LWA experiments. Highly sensitive wideband radio arrays, of the kind used to measure the cosmic 21cm signal, provide a promising way to further constrain the RSB excess through its anisotropy, providing additional insight into its origin. We present a framework for evaluating the potential of 21cm arrays to disentangle different components of the diffuse radio sky based on the combination of their frequency spectrum and angular power spectrum (APS). The formalism is designed to calculate uncertainties due to the intrinsic cosmic variance alone or together with instrumental noise. In particular, we predict the potential for measuring the anisotropy of a broad generalised class of excess radio background models using the low-frequency HERA array as an example. We find that a HERA-like array can distinguish an RSB excess from other sky components based on its angular clustering and spectral dependence, even if these are quite similar to one or more of the other components -- but only in the case that the RSB excess is relatively bright.

We introduce the trace operator for quasi-plurisubharmonic functions on compact Kähler manifolds, allowing to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish. Using this construction we obtain novel $L^2$ extension theorems and give applications to restricted volumes of big line bundles.

We study the local-triviality dimensions of actions on $C^*$-algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fibers. However, in certain circumstances upper semicontinuity of the weak local-triviality dimension is guaranteed. We examine these results and counterexamples with a focus on noncommutative tori and noncommutative spheres, both in terms of computation and theory.

Device-independent certification of quantum states enables the characterization of states within a device under minimal physical assumptions. A major problem in this regard is to certify quantum states using minimal resources. Aiming to address this problem, we consider a multipartite quantum steering scenario involving an arbitrary number of parties, of which only one is trusted, meaning that the measurements performed by this party are known. Consequently, the self-testing scheme is almost device-independent. Importantly, all the parties can only perform two measurements each, which is the minimal number of measurements required to observe any form of quantum nonlocality. Then, we propose steering inequalities that are maximally violated by three major classes of genuinely multipartite entangled (GME) states: graph states of arbitrary local dimension, Schmidt states of arbitrary local dimension, and $N$-qubit generalized W states. Using the proposed inequalities, we then provide an almost device-independent certification of the above GME states. Restricting to qubits, we also lift our almost device-independent scheme to device-independent self-testing.

The Gamow shell model provides the open quantum system formulation of nuclear shell model. In the coupled-channel representation, Gamow shell model provides the unified theory of nuclear structure and reactions which is well suited for the study of resonances and clusterization. In this work, we apply this approach to study the continuum-coupling correlation energy for selected near-threshold states of $^7$Li, $^7$Be, using a translationally invariant Hamiltonian with an effective finite-range two-body interaction.

Let $R$ be a ring with unity and $\mathcal{X}$ a semibrick in the module category $\mathrm{Mod}\,R$, that is, a class of pairwise orthogonal finitely presented modules whose endomorphism rings are division rings. We study the full subcategory $\mathrm{Filt}(\mathcal{X})$ consisting of all modules admitting a filtration with factors in $\mathcal{X}$. We show that $\mathrm{Filt}(\mathcal{X})$ is a wide subcategory of $\mathrm{Mod}\,R$. For the Ext-orthogonal class \[ \mathcal{X}^{\perp} = \{M \in \mathrm{Mod}\,R \mid \operatorname{Ext}^1_R(X,M)=0 \text{ for all } X \in \mathcal{X}\} \] we construct, for every module $Y$, an $\mathcal{X}^{\perp}$-envelope $Y_{\mathcal{X}}(\infty)$ as a direct limit of iterated universal short exact sequences. Assume that every $X \in \mathcal{X}$ has projective dimension at most one and that $\operatorname{Hom}_R(X,R)=0$ for all $X \in \mathcal{X}$. Then the envelope $R_{\mathcal{X}}(\infty)$ of the regular module is isomorphic to the universal localization $R_{\mathcal{X}}$ of $R$ at $\mathcal{X}$ in the sense of Schofield. The $\mathcal{X}^{\perp}$-envelopes of modules in $\mathcal{X}$ are called Prüfer modules since they share many properties with classical Prüfer groups and with Prüfer modules over tame hereditary algebras. We prove that every injective object in $\mathrm{Filt}(\mathcal{X})$ is a direct sum of such Prüfer modules.

To apply the Euler system method to a $p$-adic Galois representation $T$, one needs the existence of a $σ\in G_{\mathbb{Q}(μ_{p^{\infty}})}$ such that $V/(σ-1)V$ is free of rank one over the coefficient ring: we say that such a $σ$ is an Euler-suitable element for $V$. Given a non-CM classical newform $f$ of weight $k \geq 2$ and character $χ$, a classical newform $g$ of weight $1$ and character $ψ$, and a prime ideal $\mathfrak{p}$ of residue characteristic $p$ of a sufficiently large number field, we consider the situation where $V=V_{f,g,\mathfrak{p}}$ is the tensor product of the $\mathfrak{p}$-adic representations attached to $f$ and $g$. D. Loeffler asked the following question: is is true that if $χψ\neq 1$, then there is an Euler-suitable element for $V_{f,g,\mathfrak{p}}$ for all but finitely many $\mathfrak{p}$? He gave a positive answer when $f,g$ had coprime conductors. We give several weaker sufficient conditions to answer this question in the affirmative. As an application, we remove some of the technical assumptions in the version of the Bloch-Kato Conjecture proved in arXiv:1503.02888. We also show that the general answer to the question is negative, by constructing a family of counter-examples, and giving additional counter-examples that do not fit in this family.

Due to the sheer size of software logs, developers rely on automated log analysis. Log parsing, which parses semi-structured logs into a structured format, is a prerequisite of automated log analysis. However, existing log parsers are unsatisfactory when applied in practice because they 1) ignore categories of variables, and 2) need labor-intensive model tuning. To address these limitations, we propose LogPTR, a variable-aware log parser that can extract the static and dynamic parts in logs, and further identify categories of variables. The key of LogPTR is formulating log parsing as a text summarization problem and using a pointer mechanism to copy words from the log message and label tokens indicating categories of variables. The experimental results on widely-used benchmark datasets show that LogPTR outperforms state-of-the-art log parsers on both general log parsing that extracts log templates and variable-aware log parsing that further identifies categories of variables.

For dynamical systems with infinite topological entropy, the classical entropy fails to quantify their complexity effectively, while the metric mean dimension provides a natural extension in this context. In this paper, we study the complexity of saturated sets from the perspective of Bowen upper and lower metric mean dimensions. We show that if a dynamical system $(X,f)$ satisfies the $g$-almost product property, then for any compact connected non-empty subset $K$ of a set of the convex combination of finitely many invariant measures, the saturated set $G_K$ satisfies $$ \overline{\operatorname{mdim}}^B_{M}\left(G_K, f, d\right)=\limsup_{\varepsilon \rightarrow 0} \frac{1}{|\ln \varepsilon|} \inf_{μ\in K} \inf _{\operatorname{diam}(ξ)<\varepsilon} h_μ(f, ξ), $$ $$ \underline{\operatorname{mdim}}^B_{M}\left(G_K , f, d\right)=\liminf_{\varepsilon \rightarrow 0} \frac{1}{|\ln \varepsilon|} \inf_{μ\in K} \inf_{\operatorname{diam}(ξ)<\varepsilon} h_μ(f, ξ), $$ where $\overline{\operatorname{mdim}}^B_{M}$ and $\underline{\operatorname{mdim}}^B_{M}$ denote Bowen upper and lower metric mean dimensions of $f$ on $G_K$, respectively, and $h_μ(f,ξ)$ is the measure-theoretic entropy of the measure $μ$ with respect to the partition $ξ$. As an application, we give an abstract framework for multifractal analysis of general continuous functions, which extends the prior work of Backes (2023, Trans. Inform. Theory, 69, 5485-5496) and Liu (2024, J. Math. Anal. Appl., 534, 128043).

Inference of the reproduction number through time is of vital importance during an epidemic outbreak. Typically, epidemiologists tackle this using observed prevalence or incidence data. However, prevalence and incidence data alone is often noisy or partial. Models can also have identifiability issues with determining whether a large amount of a small epidemic or a small amount of a large epidemic has been observed. Sequencing data however is becoming more abundant, so approaches which can incorporate genetic data are an active area of research. We propose using particle MCMC methods to infer the time-varying reproduction number from a combination of prevalence data reported at a set of discrete times and a dated phylogeny reconstructed from sequences. We validate our approach on simulated epidemics with a variety of scenarios. We then apply the method to real data sets of HIV-1 in North Carolina, USA and tuberculosis in Buenos Aires, Argentina. The models and algorithms are implemented in an open source R package called EpiSky which is available at https://github.com/alicia-gill/EpiSky.

Let $τ_k(Ω)$ be the $k$-th eigenvalue of the Laplace operator in a bounded domain $Ω$ of the form $Ω_{\text{out}} \setminus \overline{B_α}$ under the Neumann boundary condition on $\partial Ω_{\text{out}}$ and the Robin boundary condition with parameter $h \in (-\infty,+\infty]$ on the sphere $\partial B_α$ of radius $α>0$ centered at the origin, the limiting case $h=+\infty$ being understood as the Dirichlet boundary condition on $\partial B_α$. In the case $h>0$, it is known that the first eigenvalue $τ_1(Ω)$ does not exceed $τ_1(B_β\setminus \overline{B_α})$, where $β>0$ is chosen such that $|Ω| = |B_β\setminus \overline{B_α}|$, which can be regarded as a reverse Faber-Krahn type inequality. We establish this result for any $h \in (-\infty,+\infty]$. Moreover, we provide related estimates for higher eigenvalues under additional geometric assumptions on $Ω$, which can be seen as Szegő-Weinberger type inequalities. A few counterexamples to the obtained inequalities for domains violating imposed geometric assumptions are given. As auxiliary information, we investigate shapes of eigenfunctions associated with several eigenvalues $τ_{i}(B_β\setminus \overline{B_α})$ and show that they are nonradial at least for all positive and all sufficiently negative $h$ when $i \in \{2,\ldots,N+2\}$. At the same time, we give numerical evidence that, in the planar case $N=2$, already second eigenfunctions can be radial for some $h<0$. The latter fact provides a simple counterexample to the Payne nodal line conjecture in the case of the mixed boundary conditions.

In the presence of topologically nontrivial bosonic field configurations, the fermion number operator may take on fractional eigenvalues, because of the existence of zero-energy fermion modes. The simplest examples of this occur in 1+1 dimensions, with zero modes attached to kink-type solitons. In the presence of a kink-antikink pair, the two associated zero modes bifurcate into positive and negative energy levels with energies $\pm ge^{-gΔ}$, in terms of the Yukawa coupling $g\ll 1$ and the distance $Δ$ between the kink and antikink centers. When the kink and antikink are moving, it seems that there could be Landau-Zener-like transitions between these two fermionic modes, which would be interpretable as the creation or annihilation of fermion-antifermion pairs; however, with only two solitons in relative motion, this does not occur. If a third solitary wave is introduced farther away to perturb the kink-antikink system, a movement of the faraway kink can induce transitions between the discrete fermion modes bound to the solitons. These state changes can be interpreted globally as creation or destruction of a novel type of pair: a half-fermion and a half-antifermion. The production of the half-integral pairs will dominate over other particle production channels as long as the solitary waves remain well separated, so that there is a manifold of discrete fermion states whose energies are either zero or exponentially close to zero.

We investigate the existence and multiplicity of abstract weak solutions of the equation $-Δ_p u -Δ_q u=α|u|^{p-2}u + β|u|^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and $α,β\in \mathbb{R}$. We determine three generally different ranges of parameters $α$ and $β$ for which the problem possesses a given number of distinct pairs of solutions with a prescribed sign of energy. As auxiliary results, which are also of independent interest, we provide alternative characterizations of variational eigenvalues of the $q$-Laplacian using narrower and larger constraint sets than in the standard minimax definition.