An approach to generate three-term recurrence relations with arbitrary degree step is proposed. Specifically, given any class of orthogonal polynomials $\{Q_{p}(x)\}_{p=0}^{\infty}$ defined by Favard's theorem, we employ the adjacent members $Q_{p}(x)$ and $Q_{p-1}(x)$ to compute the one of high degree $Q_{p+s}(x)$ and that of low degree $Q_{p-t}(x)$, where $(s,t)\in \mathbb{N^{+}}$. Therefore, it is able to derive a three-term recurrence relation with respect to $Q_{p+s}(x)$, $Q_{p}(x)$ and $Q_{p-t}(x)$ by taking $Q_{p-1}(x)$ as the bridge. Moreover, as the extensions of standard recursive formula which is characterized with degree increase, the ones for degree decrease and end-to-middle directions are formulated as well. The explicit recurrence relations with two-degree step are offered for Hermite, Gegenbauer and Legendre polynomials. Precision comparison is also performed between the standard three-term recurrence relation and the proposed ones.

Theorems by Bray-Brendle-Neves and Zhu about positive scalar curvature metrics on products of a 2-sphere and an n-torus suggests that positive scalar curvature suggests and appropriate topological assumptions should lead to the existence of a topologically non trivial 2-spheres of small area, which can be stated as upper bound on the 2-systole of such manifolds. Recent progress have been made in this direction by Sha and Tsiamis under the additional K{ä}hler assumption while Checcini-Hirsh-Ziedler, Stryker and Tsiamis showed similar upper bounds on the stable 2-systole using index theoretic methods. We prove here similar inequalities for some K{ä}hler manifold which control the relative sizes of the representatives of a well chosen set of homology classes. For instance on $(\mathbb{CP}^1x\mathbb{CP}^1 , ω)$ with a positive scalar curvature K{ä}hler metric we quantitavely show that largeness of one factor imposes smallness of the other one.

In this paper, we establish a unified geometric description for the propagation behavior of multidimensional nonlocal diffusion equations. By extending the classical Freidlin-Gärtner framework to general asymmetric, nonlocal operators, our theory naturally captures biased propagation--a regime where the intrinsic spreading set may exclude the origin. A key consequence is the representation of the spreading set as a Minkowski sum, which holds for both bounded and unbounded initial supports. Within this framework, we derive uniform spreading estimates and prove the local Hausdorff convergence of level sets in arbitrary dimensions. Our work therefore not only recovers known isotropic results but also provides a complete characterization of the biased case. Moreover, the methods developed here are readily adaptable to a broader class of diffusion problems featuring general operators and nonlinearities.

We investigate three clean-like properties for arbitrary rings, for endomorphism rings of abelian groups and for matrix rings over finite fields. Specifically, we study and establish when a ring is weakly strongly $k$-nil-clean for some fixed natural number $k\geq 2$, when the matrix ring is either quasi $2$-nil-clean or quasi $3$-nil-clean, and when the endomorphism ring is weakly clean. Our theorems improve substantially on some results due to Goldsmith-Vámos in Rend. Sem. Mat. Univ. Padova (2007), Breaz {\it et al}. in Linear Algebra \& Appl. (2013), Koşan-Zhou in Front. Math. China (2016), Su {\it et al}. in J. Algebra \& Appl. (2027), and some other existing results in this current topic.

Providing reliable communication for unmanned aerial vehicles (UAVs) via existing cellular networks is crucial for enabling the rapid growth of the low-altitude economy. However, UAV jittering significantly degrades communication quality due to induced beam misalignment. Inspired by recent advances in integrated sensing and communication, we propose a novel two-stage active sensing-assisted communication framework tailored for ground-to-UAV links with jittering. Specifically, two schemes are conceived to leverage sensing for enhancing communication performance, namely the communication-oriented scheme and the sensing-oriented scheme. For the sensing-oriented scheme, deterministic signals are employed in the first stage to facilitate angle-of-arrival (AoA) acquisition at the UAV side, followed by pure communication service in the second stage by using the estimated AoA. In contrast, the communication-oriented scheme employs Gaussian information-bearing signals throughout both stages, with AoA estimation relying on Gaussian random signals. For both schemes, we provide maximum likelihood estimators for AoA, along with analytical results characterizing the Cramér-Rao bound. To capture the performance limit, closed-form expressions for the achievable rates of the two schemes are derived, unveiling a fundamental tradeoff between sensing and communication quality across the two stages by tuning the time allocated to the first stage. The optimal time allocation that maximizes the overall rate is obtained in semi-closed-form. Based on these results, we unveil a sufficient condition under which the communication-oriented scheme outperforms the sensing-oriented scheme, which admits an interesting threshold-based structure. Asymptotic analysis demonstrates that the performance loss of the proposed schemes relative to the jitter-free upper bound approaches zero in the high transmit power regime.

Community detection is a fundamental problem in network science that consists of identifying groups of vertices exhibiting stronger internal connectivity than external connectivity. From an Operations Research perspective, the problem can be interpreted as a family of combinatorial optimization and clustering models defined over network structures. This survey provides a unified framework for the study of community detection, with particular emphasis on modularity-based approaches. We first review existing surveys and analyze the taxonomic criteria used to classify the literature, highlighting the absence of a common conceptual framework. Based on this analysis, we propose a multidimensional taxonomy that organizes community detection methods according to network characteristics, community structure, objective functions, methodological paradigms, evaluation criteria, and application domains. We then introduce a general mathematical formalization of the Community Detection Problem that accommodates disjoint, overlapping, and fuzzy community structures within a unified assignment framework. Building on this formalization, we review representative modularity functions, discussing their underlying assumptions, null models, and known limitations. We also survey modularity-based community detection methods, distinguishing between algorithmic and mathematical programming approaches. Finally, we review commonly used benchmark datasets and discuss their role in evaluation and reproducibility. By integrating taxonomy, mathematical modeling, modularity analysis, and benchmarking practices, this survey provides a structured reference for researchers and practitioners working on community detection and related network optimization problems.

We consider the torsion function and the first Laplacian eigenfunction in a convex curvilinear sector in the plane, under homogeneous Neumann conditions on the two straight lateral sides and a homogeneous Dirichlet condition on the remaining part of the boundary. We prove that they are, respectively, strictly $(\frac 1 2)$-concave and strictly log-concave, provided the interior angles at the vertices are at most $\frac π2$. Under the same assumption, we further establish a billiard-type concavity, obtained through reflections across the Neumann sides. As a consequence, we deduce that there exists a unique hot spot, located at the Neumann--Neumann vertex, and that some monotonicity properties hold along suitable segments. Finally, we prove that the associated variational energies, namely the mixed torsional rigidity and the first mixed Laplacian eigenvalue, satisfy Brunn--Minkowski type inequalities in the class of convex curvilinear sectors with a fixed opening angle.

This technical report presents the design and analysis of filter bank multicarrier (FBMC)/QAM multi-user MISO systems. We describe the complete uplink and downlink signal processing chains and characterize the end-to-end effective channel, including inter-carrier interference, inter-symbol interference, intrinsic interference, and residual self-interference. We compare FBMC/QAM with CP-OFDM and FBMC/OQAM through the lens of the Balian-Low theorem, and analyze prototype filter choices (PHYDYAS, Type-I, and Type-II), including the interference power breakdown under MRT and ZF precoding. Furthermore, we present an online stochastic successive convex approximation framework for ergodic sum-rate maximization with closed-form power updates, and contrast it with offline Monte Carlo-based approaches. Simulation results demonstrate the BER and network spectral efficiency advantages of FBMC/QAM over CP-OFDM under residual carrier frequency offset.

Let $N=pq$ be a semiprime and let $\ell\nmid Na$ be an odd prime. The hyperbolic sieve set $H_a(N;\ell)=\{ax+Nx^{-1}:x\in\mathbb F_\ell^*\}$ contains the residue of the linear form $ap+q$ modulo $\ell$ and has exact cardinality $(\ell+χ(aN))/2$, where $χ$ is the Legendre symbol modulo $\ell$. We study simultaneous sieving for several linear forms and give a complete local analysis of the two-target primitive-root case proposed in connection with deterministic integer factorization. For two distinct coefficients $a,b$, with $A=4aN$ and $B=4bN$, we prove an exact formula for $|H_a(N;\ell)\cup H_b(N;\ell)|$ in terms of the degree-four character sum \[ K(A,B)=\sum_{z\in\mathbb F_\ell}χ((z^2-A)(z^2-B)).\] For a smooth projective genus-one curve $E/\mathbb{F}_\ell$, we write $t_E=\ell+1-\#E(\mathbb{F}_\ell)$ for its Frobenius trace. With this convention, $K(A,B)$ is the Frobenius trace, up to sign and an additive constant, of the genus-one curve $Y^2=(X^2-A)(X^2-B)$. Hence Hasse--Weil gives a uniform $O(\sqrt\ell)$ error from the main term $3\ell/4$, and negative traces explain the counterexamples to the pointwise bound $3\ell/4+1$. We also prove a multi-target estimate \[\left|\left|\bigcup_{j=1}^k H_{a_j}(N;\ell)\right|-\ell(1-2^{-k})\right|\le (k-1+2^{-k})\sqrt\ell+k\] for distinct coefficients $a_1,\ldots,a_k$, together with the corresponding CRT product bound. Finally, for special-shape inputs $N=u^rv$, we study the $r$-power-constrained image $H_{a,r}(N;\ell)$ and determine its exact size by an elementary involution argument. These results recast the proposed local sieve questions as explicit finite-field statements with verified local tests.

Guarding a target against a fast Attacker with a slower Guard is posed as a differential game. Both players follow nonlinear, thrust-free, aerodynamic motion in the plane. The capturability problem inherent in this scenario is addressed by formulating the game such that the Attacker is constrained to terminate at its Target, and the game value is the flyby distance. The latter is made accessible at termination by introducing an auxiliary state variable that records the players' minimum separation distance throughout their flight. This novel combination of termination condition and auxiliary state removes a structural limitation of current pursuit games and opens a new class of solvable problems. It also enables us to embed attacker mission constraints directly. Taking an indirect approach, we formulate the game's Two-Point Boundary Value Problem, solve it numerically, and identify several solution types. When the Attacker's only constraint is to reach the Target, he can do so without being captured. These types of solutions, however, are less practical due to long flight times and slow terminal speeds. Imposing an Attacker terminal speed constraint yields simpler solutions and reveals the trade-off between terminal speed and flyby distance. We demonstrate that in such a setting, capture is possible in a relevant region of the parameter space.

In this article, we employ a gluing construction and a perturbation argument to produce complete Calabi--Yau metrics on Lefschetz K3 fibrations $M$ over $\mathbb{C}$, whose sectional curvature is unbounded at infinity.

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Teich}(S_g)$ be the Teichmüller space of $S_g$. In this paper, we provide an algorithm for determining the isometries induced in $\mathrm{Teich}(S_g)$ by certain irreducible periodic mapping classes in Fenchel-Nielsen coordinates. As a demonstration of this method, we provide a description of the isometry induced by periodic mapping class of order $4g+2$ and also derive some of its geometric properties.

In lattice-based cryptographic schemes, both encoded messages and accumulated decryption noise are represented in a modulo $q$ space. Therefore, it is natural to study toroidal distances and maximum toroidal distance (MTD) codes. In this paper, we derive some upper bounds for minimum toroidal distance of a code, including a Plotkin-type bound, a local ball--Plotkin bound, and a Delsarte linear programming bound. We also exhibit examples showing that these bounds are sharp in some cases. Moreover, we present several code constructions with good minimum distance, some of which are MTD codes. For $\ell=2$, we obtain a family of four-point MTD codes in $\mathbb Z_q^2$. For $\ell=4$, we propose a general code construction and exhibit several explicit instances for specific values of $q$, some of which are proven to be MTD codes. For $\ell=8$, using the $E_8$ lattice, we construct codes $C=2mE_8\cap \mathbb Z_q^8$, where $q=4m$ and show that they are MTD codes. These results give explicit optimal constructions of MTD codes for $\ell=2,4,8$. In the case $\ell=16$, we construct a code with minimum toroidal distance $3$ for $q=4$, while the known upper bound in this case is $2\sqrt{3}$. Our main tools are geometric and linear programming methods.

We prove that all reasonable definitions of multiple-valued functions with continuous derivatives are equivalent, which eliminates inconsistencies in the literature. As an intermediate result we prove that affinely approximable functions defined on intervals admit a differentiable selection.

For an $\aleph_0$-saturated weakly o-minimal expansion of an ordered group $\mathcal{M}$, it is shown that $\mathcal{M}^{\text{eq}}$ has uniform finiteness if and only if the collection of definable convex subgroups of $\mathcal{M}$ has uniform finiteness. If $\mathcal{M}$ expands an ordered field, then considering definable convex valuation subrings is sufficient. The results use a criterion of Johnson for uniform finiteness in $\mathcal{M}^{\text{eq}}$, [8]. In addition, it is shown that uniform finiteness in $\mathcal{M}^{\text{eq}}$ may fail for weakly o-minimal expansions of fields.

For a tilting complex over a finite-dimensional symmetric algebra over an algebraically closed field, whose distinct graded components have no indecomposable projective direct summands in common, we explicitly (re)construct a two-sided tilting complex that corresponds to the tilting complex, starting from a bimodule inducing a stable equivalence of Morita type.

Time-harmonic elastic wave scattering involves both compressional (P-) and shear (S-) waves, which propagate with different wavenumbers and polarization characteristics. The naive construction of perfectly matched layer (PML)-type methods based on complex coordinate stretching may lack robustness, or even fail, particularly when the wavenumbers are highly contrasted. The recently developed real compressed layer (RCL) technique build upon real compression transformations and explicit extraction of resulting oscillatory patterns for time-harmonic Helmholtz problems may not work, since the oscillations cannot be explicitly extracted by a single change of variables. This paper intends to bridge this gap by developing a robust RCL method for two-dimensional time-harmonic elastic wave scattering in unbounded domains with compactly supported inhomogeneities. A key observation is that, through the Helmholtz decomposition, the displacement field in the exterior homogeneous region decoupled into P-wave and S-wave and each has a distinctive separation of its oscillatory pattern and decaying behaviours in polar coordinates. We then apply the real compression coordinate transformation in the radial direction to each component. We further propose a coupled displacement-potential RCL formulation that seamlessly integrates the Helmholtz-decomposed wave components with the interior displacement field. We show that, under this framework, the essential oscillations in the layer can be effectively removed. We prove the well-posedness of the resulting coupled problem and establish the exponential convergence of the RCL solution to the original scattering solution in the truncated domain of interest. We discretize the RCL-system using high-order spectral element method and demonstrate the effectiveness and robustness of the proposed method through ample numerical results.

In this paper, we establish quantitative estimates for the orbital stability of the first Laplacian eigenstates of the incompressible Euler equation on a two-dimensional flat torus. We focus mainly on the hexagonal torus, where the first Laplacian eigenspace has a more intricate structure and the Casimir functionals may exhibit strong degeneracy at special amplitude and phase configurations. The main novelty of the proof is to reduce the estimates for the amplitude parameters of the perturbed solution to a root-stability problem for a cubic polynomial under coefficient perturbations, thereby overcoming the strong degeneracy in an effective way. These estimates appear to indicate that stronger degeneracy in the amplitude-phase configuration leads to weaker stability.

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold with nonpositive sectional curvature. We prove sharp, logarithmically improved spectral cluster bounds for orthonormal systems in the supercritical range. More precisely, for spectral windows of size $(\log λ)^{-1}$, we obtain the orthonormal analogue of the logarithmically improved $L^q$ estimates of Hassell-Tacy. Our argument combines the universal orthonormal spectral cluster bounds of Frank-Sabin with Bérard-type kernel estimates and a generalization of the Bourgain-Shao-Sogge-Yao multiplier estimate to the orthonormal setting.

In this article, we study the reverse mathematics of variable word theorems used in the proof of the Dual Ramsey theorem. We prove that the Ordered Variable Word theorem does not imply Ramsey's theorem for pairs and that every computable instance admits a solution of low${}_2$ degree. This is used to prove the Carlson-Simpson Lemma and the Open Dual Ramsey theorem over~$\mathsf{ACA}_0$, thereby answering some 40-years old open questions. These results have consequences in the reverse mathematics of structural Ramsey theory.

We prove the following dichotomy result for $L^2(\mathbb R)$ solutions to the Benjamin--Ono equation: On windows traveling at any speed, the solution either converges to zero or to a soliton dictated by the spectral properties of the Lax operator associated to the initial data. As an application of this result, we prove asymptotic stability of Benjamin--Ono multisolitons in $L^2(\mathbb R)$. Specifically, we show that solutions to the Benjamin--Ono equation emanating from small $L^2(\mathbb R)$ perturbations of multisolitons evolve towards a series of separating one-solitons when viewed in windows traveling with these solitons.

An edge-cut of a graph is said to be essential if its removal results in a graph with at least two non-trivial components. The essential edge-connectivity of a graph $G$ is the minimum cardinality among all essential edge-cuts of $G$. The spectral gap of $G$ is the difference between its largest and second largest eigenvalues. In this paper, we prove that for any integers $t$ and $r$ with $6\leq r\leq t\leq 2r-3$, the maximum spectral gap among all connected $r$-regular graphs with essential edge-connectivity at most $t$ is equal to $\frac{1}{2}(r+7-\sqrt{(r+7)^2-8t-32})$ when $t-r$ is odd and $\frac{1}{2}(r+6-\sqrt{(r+6)^2-8t-32})$ when $t-r$ is even. We construct a family of connected $r$-regular graphs achieving these bounds.

Given an ordered tuple $\mathbf v=(v_0,\ldots,v_d)$ of vectors in $\mathbb{R}^d$, let $A_{\mathbf v}=[\,v_1-v_0\ \cdots\ v_d-v_0\,]$ be its edge matrix. We prove that, in every finite colouring of $\mathbb{R}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \[ (c_2(A_{\mathbf v}),\ldots,c_d(A_{\mathbf v})). \] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\mathbf v})=\det(A_{\mathbf v})$. We also prove a discrete analogue: if $E\subseteq\mathbb{Z}^d$ has positive upper Banach density, then, for some $q\geq 1$, the set of coefficient tuples realized by ordered tuples in $E$ contains \[ q^2\mathbb{Z}\times q^3\mathbb{Z}\times\cdots\times q^d\mathbb{Z}. \] Finally, we show that the ordinary trace $c_1(A_{\mathbf v})$ cannot be added to these conclusions. The proof combines a quantitative directional expansion result for ergodic actions of free abelian groups with a trace calculation for a family of model edge matrices.

Consider Schott's statistic (Schott, 2005) defined as the squared Frobenius norm of the sample correlation matrix for data from $α$-regularly varying populations. We investigate its asymptotic distribution in a general framework characterized by data dimension p, sample size n, and regularly varying coefficients $α$. In particular, we identify a phase transition phenomenon in the asymptotic behavior. For light-tailed populations ($α> 3$), we revisit the $α$-free asymptotic distribution but relax the constraint on the ratio of $p/n$. For heavy-tailed populations ($α< 3$), we derive a new asymptotic normal distribution whose variance explicitly depends on $α$. We also propose a consistent estimator for the asymptotic variance such that the standardized Schott's test statistic remains applicable for unknown location parameters and all $α> 0$.

We extend the basic theory of the groupoids introduced by Brink and Howlett in their study of normalizers of parabolic subgroups of Coxeter groups, by studying both their abstract root systems and root systems realized in real vector spaces. Such root systems have some properties formally analogous to those of root systems of Borcherds-Kac-Moody Lie algebras; in particular, some contain imaginary simple roots. Further, positive roots correspond to certain reflection subgroups. We also extend the most basic properties of the Tits cone and imaginary cone of Coxeter groups to corresponding cones defined for Brink-Howlett groupoids. The results linearize the study of certain classes of reflection subgroups of Coxeter groups in a similar way as root systems of Coxeter groups linearize the study of reflections.

Flag-rank-metric codes arise as a natural generalization of rank-metric codes in the context of network communication. While recent research has mainly focused on algebraic and structural properties of these codes, the combinatorial geometry underlying the flag-rank metric remains largely unexplored. In this paper, we initiate a detailed investigation of this geometry. We explicitly determine the size of spheres of small flag-rank radius in the space $\mathrm{U}(n,\mathbb{F}_q)$ of upper triangular matrices over the finite field $\mathbb{F}_q$, and consequently obtain formulas for the size of balls of radius at most $3$. Using these enumerative results, we derive a sphere-packing bound for flag-rank-metric codes and introduce the notion of perfect codes with respect to the flag-rank metric. We observe that no non-trivial perfect flag-rank-metric codes exist in $\mathrm{U}(n,\mathbb{F}_q)$ for $n\in\{2,3\}$. We then investigate the possible parameters of perfect codes in higher dimensions. For minimum distance $3$, we obtain a characterization in terms of the codimension of the code, and show that suitable maximum flag-rank distance codes with minimum distance $3$ yield non-trivial perfect codes. For minimum distances $5$ and $7$, we derive explicit quadratic and cubic conditions, respectively, that any perfect code must satisfy. Finally, using asymptotic estimates for balls of fixed radius, we prove that for fixed length $n$ and $δ\in\{3,5,7,9,11\}$, perfect linear flag-rank-metric codes with minimum distance $δ$ do not exist over $\mathbb{F}_q$ for all sufficiently large $q$.

Sogge studied Kakeya problems on two extreme types of three dimensional Riemannian manifolds: Manifolds with the most symmetries (manifolds of constant sectional curvature) and manifolds with the least symmetries, which he called manifolds with chaotic curvature and variably curved manifolds. In the same paper, Sogge proposed studying manifolds with intermediate symmetry, such as (locally) symmetric spaces. In the current paper, we propose a classification of curvature conditions in the spirit of Sogge's program. In particular, these curvature conditions give a complete geometric characterization of the contact order conditions (for Riemannian distance functions), introduced when people were studying Hörmander-type oscillatory integral operators. One of these conditions generalizes Sogge's chaotic curvature condition to all finite orders: The chaotic curvature condition of order $\le k$ for every $k\ge 1,$ with the case $k=1$ corresponding to Sogge's original condition for variably curved manifolds. As byproducts of our main results, we show that there are no manifolds satisfying the chaotic curvature condition of order $\le 1$. We also show that both the chaotic curvature condition of order $\le 2$ and its failure can occur robustly under small smooth perturbations, and for every $k\ge 3$, a ``generic" manifold satisfies the chaotic curvature condition of order $\le k$. It turns out that the chaotic curvature condition of order $\le k$ is precisely the same as the notion of non-$(k+2)$-exceptional, where $k$-exceptional is introduced by Lytchak and Petrunin \cite{LP22} when studying convex sets and the non-existence of totally geodesic sub-manifolds. Thus our results imply, in particular, that every manifold is $3$-exceptional.

Let $S_{\mathbf n}f$ denote the symmetric rectangular partial sums of the trigonometric Fourier series of a function on the $d$-dimensional torus. We prove a summable endpoint criterion at the Zhizhiashvili critical scale for all $d\ge2$ and $1\le p\le2$. The criterion allows a general summable secondary weight at the iterated-logarithmic level and contains, as special cases, a double-logarithmic endpoint criterion and an $L^p$ Oskolkov-type corollary. In particular, it answers the Zhizhiashvili--Marcinkiewicz problem for $1<p<2$ and sharpens Zhizhiashvili's classical sufficient conditions in the endpoint cases $p=1$ and $p=2$.

Balanced words on a finite alphabet are those words in which every letter of the alphabet occurs the same number of times. The notion of descents and major index extends in a natural way to words. It is known that the bivariate generating polynomials for descents and major index over balanced words on the alphabet $[k]$ with $n$ occurrences each is palindromic, but a bijective proof has been missing even for balanced binary words. We give an explicit bijection proving this result. For permutations (which are also balanced), our bijection is different from the complementation map. We also show that for balanced binary words, this bijection simultaneously flips the ascent and comajor index as well.

Let $D_{2}(Q)$ denote the sum of squared distances between consecutive Farey fractions in the full interval $(0, 1]$. Daniele Mundici conjectured that $C(Q):=D_{2}(Q)\cdot Q^2/\log Q$ is less than 3 for all $Q\geq 2$, which is confirmed true in \cite{DLN2026}. In this paper, we generalize this result to subintervals of $(0, 1]$ and to $h$-spacings. As applications, we obtain Mundici-type bounds in these two settings, extending the full-interval consecutive-spacing case of Mundici's conjecture.